research paper

4 Mechanical Properties of Polymers

4.1 Fundamental Principles of Mechanical Behavior

The mechanical properties of polymers often play a key role for their application. The demands placed on test methodology are correspondingly high. They can be fulfilled only if the fundamentals of mechanical behavior are given due consideration from the perspective of both continuum mechanical and materials science when test strategies are being worked out.

Numerous extensive studies are available that describe the behavior of materials in general and polymers in particular [1.16, 4.1 – 4.3]. By ‘mechanical behavior’ we mean the reaction of any material under mechanical loading. When a force acts on a body, deformation is the result. Just how the body is deformed depends on its mechanical behavior and geometry, as well as on load value and loading direction. To describe material behavior under mechanical loading, it is useful to consider the influence of geometry by introducing loading parameters in the form of stress and strain.

4.1.1 Mechanical Loading Parameters

4.1.1.1 Stress

By stress we mean the force F per unit area acting on a plane in the material. Two principal cases can be distinguished depending on the loading direction. If the normal of the reference plane and loading direction lie parallel to each other, we refer to the resulting stress as normal stress . Normal stresses occur, for example, in the cross-sectional area of prismatic rods under uniaxial loading. For the simple example in Fig. 4.1a it holds that:

0A

F (4.1)

A0 represents the cross-sectional area of the undeformed specimen and is used as a reference quantity.

74 4 Mechanical Properties of Polymers

L

F

F

L 0

L

F

F

A0

�

a b

L 0

Fig. 4.1: Diagram of deformation (a) under normal stress loading and (b) under shear stress loading

If the loading direction and the normal of the reference plane are perpendicular to each other, as in Fig. 4.1b, the resulting stress is termed shear stress . By analogy to Eq. 4.1, then:

0A

F (4.2)

Generally speaking, when the stress vector (force vector per unit area) and the normal of the reference plane are oriented neither parallel nor perpendicular to one another, the rules of vector analysis can be used to break the stress down into a normal stress component zz and two perpendicular shear stress components xz and yz. This is shown in Fig. 4.2.

Under complex loading conditions, it is necessary to describe the spatial stress state independently of any concrete reference plane. To do so, nine stress components are required that act on the interfaces of an infinitesimally small cubic volume element, as shown in Fig. 4.3. Equal stresses act on the facing surfaces of the volume elements to uphold the balance of forces, but in opposing directions.

z

x

y

��zz

�yz

�xz

A

C

D

B Fig. 4.2: Breakdown of stress acting on reference plane ABCD into normal stress component zz

and shear stress components xz and yz

4.1 Fundamental Principles of Mechanical Behavior 75

�

�

�

�

� �

y

zz

xz yz

yy

zy

xy

xx

zx

yx

x

z

�

�

�

Fig. 4.3: Three-dimensional stress state

The stress components can be represented in matrix form as elements of a second order tensor:

zzzyzx

yzyyyx

xzxyxx

iij (4.3)

Due to the symmetry properties of the tensor ( ij = ji), the number of independent stress components reduces to six.

Using coordinate transformation, the size of the stress components can be calculated with reference to variously oriented coordinate systems x, y, z. The coordinate system in which stress tensor all shear stress components disappear ( ij = 0 for all i j) is of particular importance. The axes of this coordinate system are termed principal axes 1, 2, 3, the remaining normal stresses ( ij with i = j) being principal stresses 1, 2, 3. Based on the invariants I1, I2 and I3 of the stress tensor, the stress state can be described independently of the selected coordinate system:

2 xyzz

2 zxyy

2 yzxxzxyzxyzzyyxx3

2 zx

2 yz

2 xyxxzzzzyyyyxx2

zzyyxx1

2I

I

I

(4.4)

With regard to the effects of the stresses, we can distinguish between volume and shape changes. Correspondingly, the stress tensor can be divided into a hydrostatic (dilatational component) p

3

I )(

3

1 p 1zzyyxx (4.5)

76 4 Mechanical Properties of Polymers

and a deviatoric component (shape-change component) ij .

)p(

)p(

)p(

zzzyzx

yzyyyx

xzxyxx

ij (4.6)

4.1.1.2 Strain

Due to the effect of stresses, relative shape changes called strains (normal strains) and shear strains (normal shear strains), respectively, are induced in mechanically loaded bodies. For a simple case of uniaxial loading, as illustrated in Fig. 4.1a, the normal strain is a non-dimensional function of the length change L = L – L0 and initial length L0 of an unloaded body:

0

0

0 L

LL

L

L (4.7)

Alternatively, draw ratio and true strain (Hencky strain) w are often used as strain values to describe relatively large deformations:

1 L

L

0

(4.8)

1lnln L

L ln

L

dL

0

L

L w

0

(4.9)

Thus, in cases of simple shear loading (Fig. 4.1b), it holds for shear strain that:

tan L

L

0

. (4.10)

Under more complex loading conditions, the relative displacements of adjacent mass points must be precisely analyzed in order to describe the deformation state. As the result of such an analysis, the deformation state is described by a strain tensor ij whose components are arranged in the form of a symmetric matrix analogous to a stress tensor (Eq. 4.3):

zzzyzx

yzyyyx

xzxyxx

ij (4.11)

4.1 Fundamental Principles of Mechanical Behavior 77

Relative length changes of the system in terms of the x, y, z axes of the coordinate system are described by normal strains xx, yy and zz. By contrast, angle changes result in shear components xy, yz and zx.

The strain tensor exhibits properties formally similar to those of the stress tensor. Thus it is possible to assume a system of principal axes 1, 2, 3 relative to which shearing disappears and only the principal strains 1, 2 and 3 exist. Furthermore, it is possible to determine three invariants as well as to distinguish a hydrostatic volume change component from a deviatoric (shape change) component.

4.1.2 Material Behavior and Constitutive Equations

The relationship between the mechanical loading parameters stress and strain is determined by material behavior and described by constitutive equations. It occurs in an enormous variety of combinations depending on the structural state of the investigated material, as well as the loading conditions. In the area of polymers alone, its spectrum ranges from brittle glassy solidified amorphous polymers to ductile semicrystalline thermoplastics to soft elastomers all the way to fluid-like polymer melts. Due to the multiplicity of observable phenomena, a uniform description is scarcely possible. Instead, basic types of mechanical behavior are defined using simplified assumptions that allow us to approximate a description of the stress–strain relationship within a narrow range of validity.

4.1.2.1 Elastic Behavior

The mechanical behavior of a material is called elastic as long as there is a bijective relationship between its stress and deformation states, i.e., entirely reversible in the mechanical as well as thermodynamic sense. With respect to different thermodynamic causes, we distinguish between energy elasticity and entropy elasticity.

Energy Elasticity

The structural cause of energy-elastic behavior is a change in median interatomic distances and bond angles under the influence of mechanical loading. The required mechanical work is stored in the form of potential energy (increase in internal energy) and entirely regained when loading is removed (first law of thermodynamics). Due to its structural causes, energy-elastic behavior remains limited to relatively small deformations. Here we can observe a linear relationship

78 4 Mechanical Properties of Polymers

between stress and strain as described by Hooke’s law. In a simple case of uniaxial tensile loading (see Fig. 4.1a), it holds that:

E (4.12)

The proportionality constant between stress and strain is called the modulus of elasticity E. It is related to the bonding forces in the material. Alternatively, compliance C can also be determined:

C (4.13)

In addition to length change, a tensile loaded specimen simultaneously undergoes reduction in cross-section. The magnitude of this cross-sectional change is described by Poisson’s ratio . It expresses the relationship between strains in the latitudinal ( y,

z) and longitudinal ( x) directions. In cases of uniaxial loading, it holds that:

x

z

x

y (4.14)

For the general case of multiaxial loading, energy-elastic behavior is described by the generalized Hooke’s law. It is based on the assumption that each of the six components of the stress tensor ij is linear-dependent on the six components of the deformation tensor kl:

klij ijklC (4.15)

klij ijklD (4.16)

The proportionality constants between the components of the stress and deformation tensors form a forth order tensor, also called the elasticity tensor Cijkl or compliance tensor Dijkl. This tensor consists of 81 components of which, however, only 21 are independent of one another in static equilibrium. Symmetry properties of the material can lead to a further reduction in the number of independent components. Two components are required in order to completely describe the elasticity and/or compliance tensor of an isotropic material. The relationship between stress and deformation state of an isotropic material can be vectorially expressed as follows [4.4]:

4.1 Fundamental Principles of Mechanical Behavior 79

zx

yz

xy

zz

yy

xx

1211

1211

1211

111212

121112

121211

zx

yz

xy

zz

yy

xx

2

CC 00000

0 2

CC 0000

00 2

CC 000

000CCC

000CCC

000CCC

(4.17)

The elastic constants C11 and C12 stand in relation to the modulus of elasticity E and Poisson’s ratio of an isotropic material:

)21)(1(

)1(E C11 (4.18)

)21)(1(

E C12 (4.19)

From the modulus of elasticity E and Poisson’s ratio , further material parameters can be calculated such as shear modulus G and compression modulus K:

2

CC

)1(2

E G 1211 (4.20)

3

C2C

)21(3

E

VV

p K 1211

0

(4.21)

Energy elasticity dominates the behavior of polymer materials for relatively small deformations, especially at low temperatures and high loading rates. Here, energy elasticity theory contributes strongly to our understanding of the deformation behavior. Moreover, it provides workable approximating solutions for a quantitative description of the stress–strain relationship.

Entropy Elasticity

By entropy elasticity we mean the tendency of macromolecules to return to their entropically most advantageous, i.e., coiled, state subsequent to deformation. If a flexible-chained polymer material is subjected to mechanical loading, its macromolecules orientate in the stress field. The state of molecular order is accompanied by a reduction in system entropy. If irreversible chain slip can be prevented, for example by crosslinking, the molecules tend to maximize entropy

80 4 Mechanical Properties of Polymers

upon being released (second law of thermodynamics). They assume a permanent unordered state of equilibrium.

Entropy-elastic behavior up to strains of several hundred percent can be observed, whereby the relationship between stress and deformation is non-linear. Simple continuum mechanical considerations, as well as molecular statistical models [4.5] in the case of uniaxial load, lead to the following relation:

)( 3

E 2 (4.22)

The material’s parameter modulus of elasticity E is determined by the crosslink density N or average molecular weight between the crosslinking points of the polymer MC . Moreover, it is dependent on temperature T as well as on the Boltzmann number k or the universal gas constant R and density :

RT M

3 NkT3E

C

(4.23)

Using Eq. 4.22, essential phenomena of mechanical behavior of vulcanized rubber can be illustrated. Their quantitative validity often remains limited to strains of less than 100 %. For this reason, the simple rubber elasticity theory has undergone a series of further developments which are covered in [4.6], for example.

Entropy elasticity is not limited to covalent crosslinked polymers. It also plays an important role above the glass transition temperature in amorphous and semicrystalline thermoplastics of sufficiently high molecular weight. Here, molecular entanglements assume the role of temporary crosslinking points [4.7 – 4.9].

4.1.2.2 Viscous Behavior

In contrast to elastic behavior, viscous behavior is characterized by the total irreversibility of deformation processes. Therefore,

1. Once deformation has been effected, it remains in place even after unloading; the relationship between stress and strain is unambiguous only with respect to prehistory; however, it is no unique reversible relationship.

2. Work expended on deformation is entirely dissipated by the material.

Structurally speaking, viscous behavior is characterized by relative displacement among adjacent structure units (molecules and/or molecule sequences in polymer materials). Any frictional forces to be overcome are dependent on deformation velocity. When the relationship observed between stress and deformation velocity is

4.1 Fundamental Principles of Mechanical Behavior 81

linear, we speak of Newtonian material behavior. This is characterized by the material parameter viscosity . In cases of simple shear loading (shear flow) it holds that:

dt

d (4.24)

By analogy in cases of elongational flow under normal stress loading, it holds that:

TT

dt

d (4.25)

The viscosity T is called elongational viscosity or Trouton viscosity. At low shear rates, it is three times greater than shear viscosity (Trouton ratio T/ = 3) [4.10].

Newtonian behavior is found in polymer melts. Here, however, it is generally limited to low shear rates. At higher shear rates, shear softening, also called pseudoplasticity, often occurs . More rarely observed is shear hardening (dilatancy). As it deviates from Newtonian behavior, viscosity becomes a function of deformation rate. Various rheological methods are available to describe the occurring non-linearities [4.10].

A viscosity theory focusing on structural consideration has been developed by Eyring [4.11] (Rate Theory). It describes the irreversible deformation process resulting from local interchange of sites by stress-aided thermal activation. The relationship between shear rate and shear stress depends on the characteristic material parameters of

the energy barrier height to be overcome during site change (activation enthalpy H0), the activation volume v and a pre-exponential factor 0 , as well as on the

Boltzmann number k and temperature T. This relationship can be expressed as:

Tk

v sinh

Tk

H exp 00 (4.26)

To overcome potential barriers in polymer melts, the proportion of mechanical energy is generally small compared to that of thermal energy (v << kT). Thereby Eq. 4.26 becomes a borderline case of Newtonian behavior ( ~ ). By analogy with Eq.

4.24, the resulting viscosity is:

Tk

H exp 00 (4.27)

with

82 4 Mechanical Properties of Polymers

v

Tk

0

0 (4.28)

Equation 4.27 describes the temperature dependence of viscosity in the form of an Arrhenius relationship. A relationship of this type has been demonstrated in experiments with melts of semicrystalline thermoplastic (far from glass transition temperature). For amorphous polymer melts near glass transition temperature, it is often more advantageous to use the Vogel/Fulcher/Tammann equation (Eq. 4.29) which is related to the free volume theory with the constants A and B as well as

temperature 0T [1.16].

0TT

B expA (4.29)

4.1.2.3 Viscoelastic Behavior

Viscosity and elasticity are the characteristic properties of fluids and solid bodies in the area of low-molecular materials. For polymers they represent merely the limits of a broad spectrum of properties that are characterized by the simultaneous occurrence of viscous and elastic effects called viscoelasticity. The characteristic feature of viscoelastic behavior is the time dependence of material properties. This is expressed, for example, by relaxation and retardation phenomena under static loading. A detailed presentation and interpretation of viscoelastic properties of polymers can be found in the works of Ferry [4.12] or Aklonis and MacKnight [4.13].

Linear Viscoelasticity

When material properties depend only on time, but not on the level of mechanical loading, the material’s behavior is called linear-viscoelastic. Linear viscoelasticity is exactly defined only for the range of infinitesimally small loads. In practice, the validity for solid polymers is limited to strains less than 1 %, but for polymer melts it can reach 100 % [4.14].

Linear-viscoelastic behavior can be expressed by a combination of linear-elastic and linear-viscous processes (laws of Hooke and Newton). Mechanical models can be used for illustration, in which elastic behavior is symbolized by a spring and viscous behavior by a dashpot. In the simplest case, both basic elements are arranged either in series or parallel, as shown in Fig. 4.4.

4.1 Fundamental Principles of Mechanical Behavior 83

Maxwell Voigt Kelvin- Fig. 4.4: Analogy model for describing viscoelastic behavior

The spring and dashpot series is called the Maxwell model. It describes the phenomenon of stress relaxation (reaction to a sudden change of deformation). One characteristic of this model is the additivity of elastic and viscous deformation segments:

ve (4.30)

By substituting Eq. 4.12 and 4.25 in Eq. 4.30, we obtain differential equation Eq.4.31. Its solution in the case of stress relaxation ( 0 ) results in a temporally exponentially falling stress (t) (Eq. 4.32), or as a material function in a temporally exponentially falling relaxation modulus E(t) (Eq. 4.33).

1

E

1 (4.31)

t expt

E exp)t( 0T0 (4.32)

t expt

E exp

)t( )t(E

0

0

0

0

0

(4.33)

The quotient /E represents the model time constant, also called relaxation time . Once a relaxation time has to be considered, the Maxwell model is incapable of describing the complex relaxation behavior of real polymers. Correspondence between model and experiment can be achieved by introducing a discrete relaxation time spectrum. This can be illustrated in the analogy model by several Maxwell elements arranged in parallel, as is shown in Fig. 4.5.

The relaxation modulus E(t) of this generalized Maxwell model results from the sum of individual relaxation moduli Ei(t):

84 4 Mechanical Properties of Polymers

E 1

� 1

E 2

�

2

E 3

�

3

E∞

�1 �2 �3

E i

�

i

� i Fig. 4.5: Generalized Maxwell model

t expEE)t(E

n

1i i (4.34)

As n , transition takes place to a continuous relaxation time spectrum H( ):

)(lnd t

exp)(HE)t(E (4.35)

In contrast to the Maxwell model, the parallel arrangement of spring and dashpot known as the Voigt-Kelvin model characterizes retardation behavior (reaction to a sudden change of stress). Analogous to the procedure described above, compliance C(t) can be calculated as a characteristic value function.

t exp1

)t( )t(C

0

0

0

(4.36)

Introduction of the retardation time spectrum L( ) results in:

)(lnd t

exp1)(LJ)t(C (4.37)

Besides the Maxwell and Voigt-Kelvin models, rheology uses numerous other rheological models to describe linear-viscoelastic behavior. Regardless of the approach used, its mathematical description leads to a linear differential equation with the form

constb,a

...bbbb...aaaa

ii

32103210 (4.38)

4.1 Fundamental Principles of Mechanical Behavior 85

which forms the basis for linear viscoelasticity theory. It is the theoretical foundation for a series of rules whose applicability has contributed to the acceptance of the theory.

Boltzmann Superposition Principle

The Boltzmann superposition principle describes the influence of mechanical prehistory on material behavior. It states that the time-dependent effects of sequential changes in the loading state combine linearly to the overall effect. Figure 4.6 illustrates this with a creep recovery experiment.

0

0

� (t) 1

time tt t

� (t)

� (t) 2

� (t) 1

� (t) 2

� (t) = � (t) + � (t)1 2

21

s tr

a in

� s tr

e s s

�

Fig. 4.6: Linear overlapping of strains 1(t) and 2(t) occurring as a result of sudden stress changes 1

and 2 using a creep recovery experiment as an example

At time t1, stress change 1 is generated, effecting a time-dependent deformation change 1(t). Any further stress change 2 = – 1 at time t2 has the same effect. However, 2(t) lags behind and has an opposite direction. The total effect (t) of sequential stress changes results from the sum of individual effects 1(t) + 2(t). For n stress steps, it holds that:

n

1i ii

n

1i i )tt(C)tt()t( (4.39)

From this and by transition to differentially small load changes, the Boltzmann superposition integral results

d d

d )t(C)t(

t

(4.40)

86 4 Mechanical Properties of Polymers

or

d d

d )t(E)t(

t

(4.41)

which describes the behavior for any given loading history and which can be regarded as a constitutive equation for linear-viscoelastic materials.

Time–Temperature Superposition Principle

Viscoelastic material properties exhibit strong temperature dependence in addition to their pronounced time dependence, because molecular motion and transformation processes determine the relaxation and/or retardation spectrum of the material. These molecular processes are thermally activated and proceed more rapidly as temperature increases. During the process, the relaxation and retardation time spectrum shifts to shorter times. If, in dependence on temperature, only the speed, but not the type or number of molecular processes changes, the relaxation and/or retardation spectrum is maintained, and along with it, the shape of viscoelastic functions along the logarithmic time axis. Their temporal position, however, changes in accordance with the temperature. Such behavior is usually defined as thermorheologically simple. As a consequence of this behavior time-temperature equivalence, applied as a time-temperature superposition principle, has gained great practical significance for predicting long-term behavior. If the progression of a viscoelastic parameter, e.g., modulus E(log t), is known for a certain time interval at varying temperatures, then individual curves, such as in Fig. 4.7, can be horizontally shifted to coincide with curve E0(log t) acquired at reference temperature T0. The

T

lo g E

effective

master

log (t) log tlog (t )0

0

T 3

T 1

T 2

log aT

range

curve

Fig. 4.7: Master curve drawn using time-temperature superposition (diagram)

4.1 Fundamental Principles of Mechanical Behavior 87

result is a master curve illustrating material behavior over a wide time range. The shift function log aT = log t – log t0 is temperature-dependent. In many cases it can be described using an Arrhenius approach:

00

T T

1

T

1

k3.2

H

t

t logalog (4.42)

In the glass transition temperature range, however, it often follows the Williams/ Landel/Ferry (WLF) equation:

02

01

0

T TTC

)TT(C

t

t logalog (4.43)

with universal constants C1 and C2 [4.15].

Correspondence Principle

Practical work with linear-viscoelastic constitutive equations can be considerably simplified using Laplace’s transformation. That means that a function y(t) is transformed into a function y with the new variable s according to the following rule:

dt)st(expyy 0

(4.44)

If, for example, this procedure is applied to Boltzmann’s superposition integral (Eq. 4.41) it results in:

)s(Es (4.45)

Formally this corresponds to Hooke’s law (Eq. 4.12), by analogy with which Eq. 4.45 can be modified according to normal rules of algebra, thus leading to the results known from linear elasticity theory. After inverse transformation, it provides a solution to the loading problem for viscoelastic material behavior.

Non-Linear Viscoelasticity

Once the limit of validity of linear viscoelasticity is exceeded, the time and temperature-dependent viscoelastic properties are additionally influenced by load magnitude. Therefore, mechanical behavior can now no longer be described in the form of a linear differential equation. Because the solution of the resulting non-linear differential equations is mathematically extremely complicated and cannot be solved without simplifications, it did not become standard practice. For simple applications,

88 4 Mechanical Properties of Polymers

the Leadermann approach [4.16] has often proved satisfactory. It supplements the Boltzmann superposition integral (Eq. 4.41) with the load-dependent empirical function f( ).

d d

)(df )t(E)t(

t

(4.46)

For stress relaxation as an example, it results in:

)(f)t(E)t( (4.47)

In addition to the procedure described by Leadermann, the literature provides numerous additional mathematical approaches for treating non-linear viscoelastic problems [4.10]. Since the introduction of Fourier transform rheology, considerable progress has been made in describing non-linear viscoelastic behavior under oscillating loading [4.17].

4.1.2.4 Plastic Behavior

Similar to viscoelastic behavior, a combination of reversible and irreversible processes characterizes plastic behavior. However, in contrast to viscoelastic behavior, they do not occur simultaneously, but are separated from each other by a yield point F. Below the yield point, material behavior is elastic; above it irreversible flow processes take place (see Fig. 4.8a). Using the equations of Hooke (Eq. 4.12) and Newton (Eq. 4.25), the stress–strain relationship can be formulated as follows:

FF T

F

for

forE (4.48)

Plastic deformation behavior is detected in many amorphous and semicrystalline polymers. Under uniaxial tensile loading, as illustrated in Fig. 4.8b, it takes the form of yield stress s, characterized by a local maximum in the stress–strain curve and usually observed for elongations ranging from 5 to 25 %.

Instances of yield strain are accompanied by reduction in local cross-section of the specimen also known as necking. In the necking zone, irreversible deformations of several hundred percent occur. Due to this inhomogeneity, considerable discrepancies arise between nominal and true stress and/or strain. By establishing true stress–strain diagrams, it could be shown that the stress reduction after yield stress has been exceeded often is only an apparent geometry effect [4.18].

4.1 Fundamental Principles of Mechanical Behavior 89

� S

1

2

strain �

F

a b

strain �

s tr

e s s

�

�

s tr

e s s

�

Fig. 4.8: Relationship between stress and strain in plastic material behavior: Model (a) and polymer

material (b) (1: nominal (engineering) stress–strain curve; 2: (true stress–strain curve)

The level of yield stress required for plastic flow processes to start depends on the stress state as well as on temperature and loading rate. The influence of stress state can generally be described by the yield criteria of classical mechanics [4.19]. The temperature and velocity dependence of yield flow allows for the thermally active nature of the underlying deformation processes. For amorphous as well as semicrystalline polymers, it often conforms to the Eyring equation (Eq. 4.26). With respect to the occuring deformation mechanisms, however, amorphous and semicrystalline polymers exhibit considerable differences. In amorphous polymers, plastic deformation occurs in the glassy state where local molecular motion under the effect of stress forms plasticized microdomains whose growth and conjunction on the macroscopic scale lead to plastic deformation in the form of shear bands or crazes. [4.20, 4.21]. In semicrystalline polymers, plastic deformation generally occurs above the glass temperature of the amorphous phase. Here, crystallographic slip processes represent the decisive step in deformation [4.22 – 4.24] as a result of which the lamellar structure is transformed into a fibrillar structure [4.25, 4.26]. Observation of the deformation mechanisms makes clear that the microscopic processes leading to plastic deformation begin to occur far below the yield point. They can often be identified as early as during loading in the linear-viscoelastic range, so that relationships can be established between relaxation time spectrum and plastic behavior [4.27].

An orientation of the macromolecules takes place as a result of plastic deformation. The achievable changes in properties are the focus of numerous polymer processing technologies. Due to molecular orientation, entropy-elastic restoring forces are triggered that resist plastic deformation and cause strain hardening that are observed with large deformations. If loading is increased further, breaking occurs in overloaded polymer chains, preceding the macroscopic fracture of the material.

90 4 Mechanical Properties of Polymers

4.2 Mechanical Spectroscopy

One characteristic feature of polymer materials is the pronounced time dependence of their mechanical properties. This is caused by the different relaxation times of a wide spectrum of molecular relaxation processes. The relationship between relaxation time spectrum H( ) and modulus of elasticity E(t) can be established on the basis of linear viscoelasticity theory using Eq. 4.35. For the sake of simplicity, Alfery’s approximation solution [4.28] can often be used:

ttlnd

)t(dE )(H (4.49)

The determination and analysis of the relaxation time spectrum on the basis of mechanical investigations is the subject of mechanical spectroscopy. More specifically, a type of absorption spectroscopy is involved that determines the energy- absorption in the material due to internal friction as a function of the frequency and duration of mechanical loading. The position of an absorption process on the frequency or time axis, as well as its intensity, provides information on the type of underlaying molecular rearrangement as well as on the magnitude and number of structural elements involved. Thus, mechanical spectroscopy is an effective tool for characterizing the structure and explaining molecular relaxation processes.

4.2.1 Experimental Determination of Time Dependent Mechanical Properties

Mechanical spectroscopy is primarily oriented to investigations involving small loads where there is no irreversible structural change in the material, and linear viscoelasticity theory is valid. It is based on experimental data acquirable within a time span of approx. 10-8 s to 108 s. A combination of various testing methods is required to characterize mechanical behavior over such a wide time span. For long- term loading times lasting more than a minute, static test methods (stress relaxation, retardation) are used. The short-term, however, is dominated by dynamic test methods with oscillating loading that fall into the category of dynamic–mechanical analysis. Relationships between statically and dynamically determined values can be established on the basis of linear viscoelasticity theory [1.16, 4.12].

4.2 Mechanical Spectroscopy 91

4.2.1.1 Static Testing Methods

Static testing methods are based on the analysis of material behavior subsequent to a sudden change in mechanical loading. We can make a principal distinction between two cases as illustrated by the diagrams in Fig. 4.9.

In stress relaxation (Fig. 4.9a), the change of stress with time (t) at 0 = const. caused by a sudden change in deformation is measured. From this we obtain the material parameter called time-dependent modulus of elasticity E(t):

0

)t( )t(E (4.50)

By analogy in the retardation or creep test (Fig. 4.9b), the change of deformation (t) with time to the value 0 = const. as a result of a sudden change in stress is used to determine compliance value C(t):

0

)t( )t(C (4.51)

Static tests can extend over very long time periods. They therefore place strong demands on the constancy of test conditions, especially temperature and humidity. In addition, structural changes such as chemical reactions and physical ageing can influence material behavior. During relatively short loading times, experimental results are influenced by the loading velocity.

time ttime t

s tr

e s s

�

s tr

a in

�

0� = const.

time t

s tr

e s s

�

� (t)

� (t)

s tr

a in

� 0� = const.

time t

Fig. 4.9: Stress relaxation and retardation (creep) for characterizing viscoelastic behavior at long

loading times

92 4 Mechanical Properties of Polymers

4.2.1.2 Dynamic–Mechanical Analysis (DMA)

In dynamic–mechanical analysis, specimens are subjected to oscillating loading. The time dependence of material behavior can be characterized by varying the frequency. For the relation between loading time t and frequency f or angular frequency , the rule is:

1

f2

1 t (4.52)

DMA is especially useful since only relatively short test times are required to acquire viscoelastic values over a wide frequency range. Moreover, dynamic–mechanical– thermal analysis (DMTA) makes it relatively easy to investigate material behavior as a function of temperature.

A number of procedures are available for performing DMA that differ with respect to the achievable frequency range, type of mechanical loading and magnitude of measurable material properties. These procedures can be classified according to the type of vibrational excitation, such as forced vibrations, freely damped vibrations and resonant vibrations. In higher frequency ranges, the propagation of sonic and ultrasonic waves is also used to determine characteristic values. The various methods of DMA are standardized by ISO 6721.

Tests Using Forced Vibrations

When forced vibrations are used to characterize viscoelastic properties, specimens are subjected to sinusoidal alternating mechanical loading at constant frequency and constant amplitude. In cases of linear-viscoelastic material behavior, steady-state changes of stress and deformation with time exhibit the same frequency, but varying phase positions. For cases of normal stress loading, the rule is:

tsin)t( 0 (4.53)

)tsin()t( 0 (4.54)

This is illustrated in Fig. 4.10.

Due to the phase shift between stress and strain, its modulus has to be introduced as the complex modulus E* to describe the stress–strain relationship.

EiEE* (4.55)

4.2 Mechanical Spectroscopy 93

0

� (t)

s tr

a in

�

time ts tr

e s s

�

1/f ���/

�

� � (t)

0

Fig. 4.10: Change in stress and strain with time in dynamic–mechanical analysis using forced

vibrations

This complex modulus can be regarded as a vector in the complex plane (Fig. 4.11) whose direction is given by the phase angle and its amount by the ratio of stress and strain amplitudes:

0

0*E (4.56)

E’’

�

E’

i

j

E

Fig. 4.11: Diagram of modulus E* in the complex plane

Using simple trigonometric relations, a real part E and an imaginary part E can be distinguished:

coscosEE 0

0* and (4.57)

sinsinEE 0

0* (4.58)

The real part E is called the storage modulus. It is a measure of the energy storable during the oscillation period Wrev. By contrast, the imaginary part E of the complex modulus is related to the energy dissipated during the oscillation period Wirrev. For this reason, it is called the loss modulus.

2 0rev E

2

1 W (4.59)

94 4 Mechanical Properties of Polymers

2 0irrev EW (4.60)

From the ratio of loss and storage moduli, we obtain the loss factor tan that characterizes the damping behavior of the material:

rev

irrev

W

W

2

1

E

E tan (4.61)

The forced vibration procedure is limited to frequencies below specimen resonance frequency. Commercial devices have a range of approx. 10-2 Hz to 102 Hz. Measurement can be controlled by monitoring both strain and stress, thus enabling the determination of complex modulus E* and complex compliance C*. Using axial and torsional testing equipment and an appropriate specimen adapter, various loading types (tensile, compression, bending, shear, torsion) can be employed. This enables the acquisition of complex elasticity and shear moduli over a wide stiffness range from 10-3 MPa to 106 MPa. The most significant disadvantage of this procedure lies in its lack of sensitivity when measuring relatively low damping (tan < 0.01).

Thanks to their wide application range, test methods with forced vibrations play a dominant role in the dynamic–mechanical analysis of polymer materials.

Tests Using Free Damped Vibrations (Torsion Pendulum)

When a specimen is deflected from its state of equilibrium by pulsed deformation, it returns to its state of equilibrium in free damped vibrations. The natural frequency of vibration, as well as the decrease of vibration amplitudes with time, depends on the viscoelastic properties of the material.

a b

2

1

3

2

1

3

4

Fig. 4.12: Diagram of a torsion pendulum setup without counterweighting (a) and with

counterweighting (b)

4.2 Mechanical Spectroscopy 95

The principle of free damped vibrations has its practical application in torsion pendulum testing as standardized in ISO 6721-2. The basic setup of the torsion pendulum is illustrated in Fig. 4.12.

Usually a prismatic specimen (1) is firmly clamped at one end (2). At the other end it is connected to an oscillating weight (3) that influences the moment of inertia and thereby the eigenfrequency of the entire system. In order to eliminate longitudinal normal stresses in the specimen, a counterweight (4) can be applied. The specimen is excited to freely decreasing torsional oscillations by pulsed deformation of the oscillating weight as shown in Fig. 4.13.

time t

1/f

l0

0

A n

A n

+ 1

d e fle

ct io

n

l

Fig. 4.13: Freely decaying damped vibration

The storage modulus G can be determined from vibration eigenfrequency. According to ISO 6721-2:

g 2 0d

2 d FfFfI4G (4.62)

Here fd is the eigenfrequency of the pendulum with, and f0 the eigenfrequency of the pendulum without the specimen (if work without counterweighting, then f0 = 0). Additional influencing factors to be considered include the moment of inertia I of the oscillation weight with clamping, as well as the damping correction Fd and the geometry factor Fg. When prismatic specimens are used (clamping length L, width b, thickness h), with h/b 6 and a geometry correction factor Fc = 1 – 0.63 h/b, then:

c 32

d0 22

d 2 FbhLff21If12G (4.63)

The logarithmic decrement characterizes system damping. It is determined from the ratio of sequential vibration amplitudes:

1n

n

A

A ln (4.64)

96 4 Mechanical Properties of Polymers

Using logarithmic decrement, the loss modulus G can be calculated as

g0 2 d FIf4G (4.65)

For work without counterweighting, the logarithmic decrement of the pendulum without specimen 0 is 0. For specimens with square cross-sections and small h/b ratio and with counterweighting at low internal damping of the pendulum ( 0 << ), the loss modulus can be calculated as:

c 32

d FbhLIf12G (4.66)

With the storage and loss moduli, the complex modulus G* and the loss factor tan can be determined analogous to Eqs. 4.57, 4.58 and 4.61.

The torsion pendulum works at frequencies ranging from 0.1 to 10 Hz. It is preferred for investigating materials with low damping (tan 0.1). When investigating with temperature dependence, a change takes place in system eigenfrequency due to modulus changes (see, e.g., Eq. 4.62). Modulus temperature curves are therefore generally measured at sliding frequency. To be sure, the frequency changes can be compensated in principle via changes in the moment of inertia of the oscillating weight. Basic advantages of the torsion pendulum are the simplicity of its setup and measurement, as well as its high level of sensitivity.

Tests Using Forced Resonant Oscillation

When forced oscillations are generated at a frequency whose wave length approaches the dimensions of the specimen, resonant phenomena occur. If the specimen is excited in the range of resonance at constant force amplitude, the amplitude of deflection follows a peaked curve (Fig. 4.14). Resonant frequency fi and width at half maximum fi are related to the viscoelastic properties of the material being investigated.

frequeny f

f i

0

1

0.707

fi

a m

p lit

u d

e A

/A m

a x

Fig. 4.14: Resonance curve of a viscoelastic material

4.2 Mechanical Spectroscopy 97

from generator

to amplifier

specimen

clamp

method A method B

specimen

to a m

p li fi e

r

fr o

m g e n e ra

to r

textile filaments

Fig. 4.15: Test setup for acquiring viscoelastic properties with forced resonant oscillation

Forced resonant oscillations are applied to determine the complex modulus by the flexural vibration-resonance-curve method (ISO 6721-3). A prismatic rod is used as specimen and clamped either on one side (procedure A) or hung on textile fibers at the vibration nodes (procedure B). The diagram in Fig. 4.15 illustrates both arrangements. Non-contacting excitation and measurement are performed via electromagnetic transducers connected to the polymer material via thin metal tabs glued to the specimen surface. Using a frequency synthesizer, the excitation frequency can be continuously varied over a range from approx. 101 Hz to 103 Hz.

While scanning this frequency range, the detector registers several peaks in oscillation amplitude corresponding to different order resonances i (i = 1, 2, 3,…). The real part of the complex elasticity modulus E can be determined from resonance frequency fi at the i-th resonance point, density of the investigated material and specimen dimensions (free length L, thickness h):

2 i

i 22

k

f

h

L 34'E (4.67)

The numerical value ki 2 depends on the order number i of the resonance point and

clamping conditions (see Table 4.1).

Table 4.1: Numerical value ki 2 for determining storage modulus E with the flexural vibration test

Order number

i

ki 2

(Method A)

ki 2

(Method B)

1

2

> 2

3.52

22.0

(i –1/2)2 2

22.4

61.7

(i –1/2)2 2

98 4 Mechanical Properties of Polymers

The additional parameter loss factor tan can be calculated from the width at half- maximum of resonance curve fi and resonance frequency fi :

i

i

f

f tan (4.68)

For materials with low internal damping (tan < 0.01) and resonance frequency, the analysis of freely decreasing vibrations after the exciter has been shut off can be recommended as an alternative. Here, the falling amplitudes of sequential vibrations is observed (see Fig. 4.13), from which the loss factor can be determined using the logarithmic decrement :

1n

n

A

A ln

1 tan (4.69)

The flexural vibration test is especially suited for characterizing rigid materials whose loss factor does not significantly exceed the value tan = 0.1. A decisive disadvantage of this procedure lies in the fact that relatively few resonance points are available and that the position of resonance points can be influenced only by altering specimen dimensions. During temperature-dependent measurements, changes take place in resonance frequency so that it is impossible to acquire values at constant frequency. Based on the achievable frequency range, the characterization of structure-borne sound insulating materials is the preferred application area for flexural vibration tests.

Tests Based on Wave Propagation (Ultrasonic Technique)

Above resonance frequency, the wave length of oscillating mechanical loading is short compared to specimen dimensions. That makes it possible to use the wave propagation characteristic of a material to determine its viscoelastic properties. Such investigations are generally performed using an ultrasonic (f > 20 kHz) pulse-echo or pulse-transmission technique [4.29, 4.30]. Sound velocity v and sound absorption coefficient are obtained from acoustic path length l and the corresponding pulse transit time, as well as from pulse amplitudes I1 and I2 at various path lengths l1 and l2.

t

l v (4.70)

2

1

12 I

I ln

ll

1 (4.71)

4.2 Mechanical Spectroscopy 99

When the density of the material is known, longitudinal wave modulus L and shear modulus G can be determined using longitudinal waves (vl , l) and transverse waves (vt , t) respectively. At low damping ( /2 << 1), we can approximate that [4.31]:

3 ll2

l v2

''Landv'L (4.72)

3 tt2

t v2

''Gandv'G (4.73)

According to elasticity theory, the modulus of elasticity E can be calculated from longitudinal wave modulus and shear modulus. It depends on the ratio of propagation velocities of transverse and longitudinal waves:

2 lt

2 lt2

t vv1

vv75.0 v4

1'G'L

'G4'L3 'E (4.74)

Ultrasonic methods have gained great importance for determining the properties of oriented polymers and polymer composites. All components of the stiffness matrix can be acquired from a single specimen by varying the polarization direction and propagation direction of the ultrasonic waves.

Ultrasonic measurements are typically made at frequencies between 100 kHz and 100 MHz. The top end of the frequency range is defined by the strong increase in damping. Various vibration exciters have to be used when working in this wide frequency range. Relatively large frequency ranges can be covered following broadband excitation by applying Fourier analysis [4.32].

4.2.2 Time and Temperature Dependence of Viscoelastic Properties

The time and temperature dependence of viscoelastic properties is quite significant for polymer material engineering. That is why it provides the basis for classification into elastomers, thermoplastic elastomers, thermoplastics and thermosets. The diagram in Fig. 4.16 illustrates the dependence of the viscoelastic properties storage modulus E , loss modulus E and loss factor tan on loading time t for an amorphous thermoplastic.

Based on the discontinuous curve of the storage modulus E , viscoelastic behavior can be divided into four characteristic regions. Under short-term loading (t << ), the material exists in its glassy state. Here, at values between 109 Pa and 1010 Pa, the storage modulus is only slightly time-dependent. In the glass transition region

100 4 Mechanical Properties of Polymers

g la

s s

tr a n s it io

n10 10

108

106

104

102

log t

glassy state E

‘‘ (P

a )

� � �� � �

10-1

100

101

ru b b e r-

e la

s ti c

p la

te a u

fl o w

r e g io

n

ta n

�

E ‘ (P

a )

Fig. 4.16: Time dependence of viscoelastic properties of an amorphous thermoplastic

(t ), a drastic reduction of the storage modulus by 3 to 4 decades occurs over a relatively short time span. This is followed by a more or less pronounced rubber- elastic plateau where the material is soft and deformable like rubber. At very long loading times, viscous properties dominate mechanical behavior in the flow region.

The time dependence of the storage modulus is calculated according to Eq. 4.35 using a relaxation time spectrum H( ) whose structural cause is a spectrum of molecular relaxation processes. The discontinuous changes in curve shape are due to changes in the dominant relaxation mechanism. At the same time, the mechanical losses (E , tan ) pass through a local maximum due to molecular friction.

The most significant relaxation process in amorphous thermoplastics is the glass transition, also called primary relaxation or process. It is related to the activation of micro Browninan motion. By this we mean cooperative rearrangements of rather long sections of polymer chains (approx. 50 to 100 CH2 units). In the glassy state of amorphous polymers, additional secondary relaxation processes ( , process) can occur that are caused by molecular motion of substituents, side chains, or short mainchain segments. A simultaneous increase in relaxation time can generally be observed that is on the order of the structural units partaking in the relaxation process. Secondary relaxation processes have relatively little influence on the storage modulus E of the material; however, they can effect sometimes distinct partial change in toughness [4.33].

4.2 Mechanical Spectroscopy 101

1010

108

106

104

E ‘‘

(P a )

-1

0

1

ta n

�

E ‘ (P

a )

T (°C)

10

10

10

10 -2

-150 -100 -50 0 50 100 150

T �

Tg

Fig. 4.17: Modulus–temperature curves and mechanical loss factor dependent on temperature for polyvinylbutyrate (frequency f = 1 Hz)

After passing through the glass transition region, entropy elasticity (see Section 4.1.2.1) dominates mechanical behavior in the rubber-elastic plateau. In this region, entanglements act as temporary crosslinks in a flexible-chained polymer network. Disentanglement processes lead, after very long loading times, to a loosening of the network crosslinks. Thereby, irreversible flow processes are enabled.

The changes in viscoelastic properties presented in Fig. 4.16 take place in amorphous thermoplastics over a time span of 15 to 20 decades. Since this large area can only be partially covered experimentally, investigations to characterize viscoelastic behavior are often performed on the basis of temperature dependence. According to the time– temperature equivalence (see Section 4.1.2.3), it is possible to acquire the entire spectrum of viscoelastic properties at constant loading time or frequency in one temperature run. As a result of such measurements, Fig. 4.17 shows the temperature dependence of storage modulus E , loss modulus E , as well as loss factor tan for amorphous thermoplastic polyvinylbutyrate. The tests were performed at 1 Hz constant frequency under dynamic tensile loading in the range of – 120 °C T +120 °C. Glassy state, glass transition, rubber-elastic plateau and flow region can be clearly distinguished. The dynamic glass transition temperature Tg is an important engineering parameter and can be determined in practice by the maximum of the loss factor. However, the peak of the storage modulus shifted to lower temperatures is sometimes used as a reference point. In contrast to the results of static testing methods (calorimetry, dilatometry), glass temperature under dynamic loading

102 4 Mechanical Properties of Polymers

10

8

7

E ‘ (P

a )

1/T (1000/K)

10 10 10 10 10 10 10 -6 -4 -2 0 2 4 6

10

10

10

10

10

f (Hz)

effective

0.1 ... 50 Hz

T = 0 °C

T = 50 °C

master curve

T = 25 °C0

ln (

a )

T

9

6

3.1

-5

5

15 Arrhenius-plot

H = 430 kJ/mol

3.3 3.5 3.7 -15

range

Fig. 4.18: Master curve of the storage modulus of polyvinylbutyrate for reference temperature

T0 = 25 °C

is generally observed at higher temperatures due to the time dependence of viscoelastic properties.

If temperature-dependent measurements of viscoelastic properties are performed at various frequencies, a master curve can be constructed using the time-temperature superpositon principle (see Section 4.1.2.3) that provides for an estimation of visco- elastic behavior beyond the experimentally covered time period for a reference temperature T0. Such a master curve is illustrated in Fig. 4.18.

Based on experimental data acquired at temperatures from 0 °C to 50 °C over a frequency range of 0.1 Hz to 50 Hz (effective range) in the glass transition range, mechanical behavior can be estimated over a time span of more than 10 decades.

4.2.3 Structural Factors Influencing Viscoelastic Properties

Relationships between chemical structure, molecular relaxation processes and viscoelastic properties are of great practical interest for characterizing and developing materials. That is why they are the object of intensive theoretical [4.34, 4.35] and experimental studies [4.36 – 4.38].

From the time and temperature dependence of the viscoelastic properties of amorphous polymers, information can be acquired as to their chain stiffness (chemical structure of backbone chains and substituents), molecular interactions

4.2 Mechanical Spectroscopy 103

lo g

E

‘

T

increasing molecular weight

increasing crystallinity

increasing crosslink density

Fig. 4.19: Influence of molecular weight, crosslink density and degree of crystallinity on the

temperature dependence of storage modulus

(molecular coefficient of friction), molecular weight and molecular weight distribu- tion, crosslink density and molecular orientation, among others. For semicrystalline polymers, statements can be made as to the degree of crystallinity and crystallite morphology (lamellar thickness). Viscoelastic properties also have great significance for investigating the composition, phase morphology and interface effects of copolymers and polymer blends. Figure 4.19 illustrates the influence of molecular weight, crosslink density and degree of crystallinity on the temperature dependence of storage modulus.

Figure 4.20 illustrates the differences in temperature dependence of the viscoelastic properties of homogenous and heterogeneous polymer systems based on styrene– butadiene copolymers. The two-phase block copolymer (SBS) exhibits two separate glass transitions in the transition temperature regions of the base components polybutene (PB) and polystyrene (PS). However, only one glass transition is detected in the single phase statistical copolymer (SBR) whose temperature position, corresponding to the composition of the mixed phase, is shifted relative to the values of its base components.

104 4 Mechanical Properties of Polymers

10 E

‘ (P

a )

10

0

1

2

3

T (°C)

PB

PS

SBR

SBS

-150 -100 -50 0 50 100 150 200 250

ta n

�

9 10

8 10

7 10

6 10

5 10

Fig. 4.20: Comparison of temperature dependence of storage modulus and loss factor of homogenous (SBR) and heterogenous (SBS) styrene–butadiene copolymers with the corresponding homopolymers (PB and PS)

4.3 Quasi-Static Test Methods

4.3.1 Deformation Behavior of Polymers

By quasi-static test methods, we mean mechanical tests involving strain rates ranging from approx. 10-5 to 10-1 s-1 in which specimen fracture or a predetermined load limit is reached in an economically reasonable time span. It is thereby assumed that loading proceeds slowly, impact-free and increases continuously to fracture. Thus, universal test machines employed in such tests must ensure that the cross-head speed remains constant, regardless of load level and loading speed, when using the preferred conventional test methods. Under these formal conditions, quasi-static test methods such as tensile or flexural tests mainly serve to acquire material values or material characteristic functions. They are also applied in quality assurance, failure analysis and pre-selection of polymers for specified applications, as well as for solving simple design problems [1.38].

4.3 Quasi-Static Test Methods 105

The total deformation of mechanically loaded polymers has the following components, whereby the absolute amount of such components depends on effective loading time and acting temperature:

• elastic deformation, • linear-viscoelastic deformation, • non-linear viscoelastic deformation and • plastic deformation.

In unreinforced plastics, the region of elastic deformation, which corresponds to changes in atomic distance and valency angle in the macromolecule while simultaneously storing elastic potential energy, is typically very small . The molecular links generated in the production process are not released in cases of energy-elastic deformation. In such materials, this region corresponds to a reversible strain of < 0.1 % and, given a linear relation between stress and strain, is completely described by Hooke’s law. In thermosets with network structures or highly filled or reinforced thermoplastic materials, this behavior can be detected especially at short loading times and/or low temperatures up to 40 % of the corresponding stress at break [4.39]. Especially in unidirectional reinforced fiber composites, this deformation behavior is dominant up to break of the specimen used.

Other than in metallic materials, polymers exhibit a mechanically reversible, but time-dependent deformation behavior (viscoelasticity), even on small deformations and at temperatures dictated by application. Based on load level, a principle distinction is made between the linear-viscoelastic and the non-linear viscoelastic deformation component (see Section 4.1.2.3).

Linear-viscoelastic deformation is characterized by mechanically stimulated molecular rearranging processes in which the existing molecular links are not released. This deformation region in thermoplastic polymers lies empirically in an interval between 0.1 and approx. 0.5 % of the total applied strain and passes over into non-linear viscoelastic deformation. The strain occuring as a reaction to loading is reversible, but time- and temperature-dependent, as the diagrams in Fig. 4.21a, b show for static loading. During loading and unloading, it amounts to less than the applied stress. A hysteresis curve (Fig. 4.21c) is created whereby, in cases of quasi-static loading, special initial effects can result, such as zero drift or stress relaxation [4.40].

We define non-linear viscoelastic deformation behavior with the release of molecular entanglements when the polymer properties depend not only on time and temperature, but also on the level of mechanical load. In this deformation region, which is characterized by the beginning of microstructural material damage, molecular migrations take place leading to irreversible yield processes and thereby to

106 4 Mechanical Properties of Polymers

t

�

0

a

�

t

�

0

b

�

�

0

c

�

0 � �

Fig. 4.21: Linear-viscoelastic deformation behavior of polymers in a diagram illustration of stress-time function (a) and strain-time function (b) given static as well as stress–strain function c) under quasi-static loading

permanent deformation [4.41]. The plastic deformation of polymer products is often mentioned in connection with “cold” yielding and stretching and hardening processes. The polymers then often exhibit clearly definable yield stresses depending on the type of loading selected, as well as on testing speed and temperature. Depending on the mentioned factors and the type of polymer, the dominant deformation mechanisms are crazes and shear band formation, both demonstrable using microstructural investigation methods [4.42], but which are sometimes even macroscopically visible as well.

The great variety of polymers created by chemical modification, mixing, filling and reinforcing leads, due to interaction among the various organic and inorganic components, to new internal top or border surfaces from which numerous different damage mechanisms may result. These mechanisms, such as fiber breaking or debonding, as well as the formation of voids and microcracks, begin to act immediately in the transition range between linear- and non-linear viscoelastic deformation and affect the durability and reliability of such materials in use. Here the problem arises that such effects are not visible on the stress–strain diagram. Therefore, they can only be demonstrated indirectly via stiffness loss in the specimen employed, or using one the hybrid methods of polymer diagnostics (cf. Chapter 9).

Regardless of the deformation behavior described and the occurring damage mechanisms, the production related internal state of the specimen employed has a decisive effect on stress– deformation behavior. Due to their incomparable residual stresses and orientations, specimens produced by different manufacturing methods (e.g. compression and injection molding) have to be considered component parts. That means that molding material properties can only be determined under idealized conditions (cf. Chapter 2). Based on the test conditions in quasi-static test methods, varying residual stresses and orientations affect the E modulus, while varying orientation states especially influence strength and deformation behavior.

4.3 Quasi-Static Test Methods 107

Values acquired by quasi-static test methods are influenced by retardation and/or relaxation behavior overlaying the mechanical experiment, as well as by the particular specimen state, test conditions and any occurring damage mechanisms. When the stress–strain behavior of a hypothetical polymer without the influence of time is considered under this aspect in the tensile test, a stress–strain curve results such as the one presented Fig. 4.22a. If, in the experiment, influence from retardation mechanisms occurs, strain increases due to simultaneously occurring creep (Fig. 4.22b). If in addition, relaxation mechanisms are active, strain at break remains constant, but tensile strength is significantly reduced (Fig. 4.22c). In actual tensile tests, both components occur simultaneously, thus influencing strength, E modulus and deformation (Fig. 4.22d).

Any variation in test conditions regarding temperature and cross-head speed vT (1 to 500 mm min-1), as is sometimes permitted in the common test specifications, leads to a wide spectrum of material values. The occurring relaxation and creep processes over the range of practically relevant temperatures and application times cannot be ignored when dimensioning plastic components and testing polymers. In thermosets and thermoplastics used for design purposes in the glassy state, the real loads do not

without influence of timepure retardation pure relaxation

with influence of time

�

�

�

�

�

�

�

�

b a c

d

��

� �

Fig. 4.22: Diagram of the stress–strain behavior under quasi-static loading without time influence (a), with retardation influence (b), with stress relaxation influence (c) and with time influences (d)

usually lie in the immediate neighborhood of the transition regions. Thus, for these materials, defined E modulus values can be given that depend very little on loading time and temperature. However, for many thermoplastic polymers the temperature range of their engineering application is within their transition ranges, thus strongly

108 4 Mechanical Properties of Polymers

involving dependence of the material values acquired on loading time, ambient temperature and load level (Fig. 4.23). For selected thermoplastic polymers, Fig. 4.23 shows the relationship between the E modulus Et acquired by tensile test, loading time t and test temperature T. Here, an obvious impact on value levels is being measured dependent on the type of polymer and its corresponding transition ranges that can significantly limit the material’s area of application.

The behavior illustrated in Fig. 4.23 can be permanently influenced by adding fillers or reinforcers such as chalk, talc, carbon or glass-fibers [4.43], as well as nanoparticles [4.44].

a

b

t (s)

PS

E

( M

P a

)

10 2

10 3

10101010101010 1010 3210-1-2-3 4-4

PVC

PS-HI

PE-HD

PE-LD

10 2

10 3

T (°C)

40200-20-40 60

PS

PVC

PS-HI

PE-HD

PE-LD

t E

(

M P

a )

t

10 4

10 4

Fig. 4.23: E modulus acquired by tensile test for selected polymers as a function of time (a) and temperature (b)

As can be clearly seen from the functional relationships illustrated, the strength and stiffness behavior of polymers cannot be described by single-point data. These are acceptable only when the test conditions are univocal and reproducible. As a matter of principle, the mechanical values of polymers should be acquired on the basis of time and temperature dependence and be described by functional relationships. In summary, it can be said that every value measured for polymers is influenced by a

4.3 Quasi-Static Test Methods 109

series of factors determined by measurement and test techniques, as well as by the production of test specimens:

• State and properties (M) of the molding material to be tested, such as chemical structure, viscosity, molecular weight and its distribution, as well as fillers and reinforcers employed,

• The process used to produce the specimen and the resulting internal state (S) in the specimen such as morphology, residual stresses, orientation and degree of crystallinity,

• Specimen geometry (G), such as on dumbell specimens or flat specimens, notch stress, dynamic and static weld lines, as well as structural inhomogeneities, such as cavities or agglomerates, and

• Test strategy and test technique that can be subsumed in the concept of test conditions (T ), i.e., type of loading, testing temperature and testing velocity, as well as ambient influences (moisture, UV radiation, etc.).

Consequently, any property P measured on a specimen is a function of the para- meters listed above:

T)G, S, (M,fP (4.75)

Characteristic values can only be measured reproducibly, if these relationships are known and appropriately considered. That means that the prerequisites for reproducible measurements include comparable chemical and physical structure and morphology, as well as identical geometric conditions and identical test methodology.

General tendencies can be stated for the relationship between strength and deformation behavior and internal state:

• Polymer strength increases, for example, with increasing molecular weight, increasing degree of crosslinking, increased orientation or by filling or reinforcing the polymer whereby deformation behavior is generally reduced;

• Strength is reduced by increased moisture, increasing age, degradation or decomposition; by contrast, deformability can increase or decrease depending on various deformation processes.

However, these statements are specific to the material and not generally valid, as for example is shown by positive post-crystallization effects due to ageing.

Based on the relations illustrated between structure and properties, it is clear that values acquired on standardized specimens cannot be carried over directly to plastics components.

110 4 Mechanical Properties of Polymers

The most important – and in practice most significant – mechanical tests are the tensile test, bend test and compression test, as well as the tear test relevant for foils. Torsional testing, however, bears little relevance for polymers. In addition to these basic tests, there are various test methods based on comparable measurement technology. They serve especially to characterize joints (glued and welded lines), and to determine adhesive strength and interlaminar shear strength. They include the shear tension or shear bend and the peel test [4.46] (see Chapter 10).

4.3.2 Tensile Tests on Polymers

4.3.2.1 Theoretical Basis of the Tensile Test

Among static and quasi-static testing and measuring methods, the tensile test is regarded as the fundamental test in mechanical material testing. In spite of the fact that, in practice, pure tensile loading is rather the exception and in spite of its experimental and interpretive problems, this test ranks high in polymer testing as well. Because of the great variety of modifications available with polymers, various approaches to executing tensile tests are known requiring different specimens, loading conditions and/or clamping devices. The practical objectives are relatively simply measurable, informative material parameters that can be used to evaluate properties for quality assurance, material selection and simple dimensioning tasks. Consequently, the main areas of application in polymers testing include:

• acquisition of tensile properties of molding and extrusion plastics and thermosets, • characterization of tensile properties of polymer sheets and films, • determination of properties of isotropic and orthotropic fiber-reinforced plastics.

The conventional tensile test, i.e., the tensile test with constant cross-head speed, is a quasi-static test with fundamental assumptions regarding testing conditions and technique, as well as the specimens used. Loading must be applied without impact and increase slowly and steadily until fracture of the specimen occurs. A uniaxial loading and stress state should be generated in the specimen. This means that, at a sufficient distance from the top and bottom clamp, there exists a homogenous uniaxial stress state not influenced by Hertzian stress, whereby a homogenous normal stress and strain arises that is evenly distributed over the cross-section (Fig. 4.24). A homogenous, isotropic materials state is assumed with respect to the specimen.

There are no geometric imperfections (e.g., notches or lumps); the specimens are prismatic. Influences from the testing technique have to be eliminated, such as may be due to compliance by the universal testing machine, or setting motions that may affect loading, or strain effects if the adapter slips. Given these prerequisities, the total

4.3 Quasi-Static Test Methods 111

A 0

�(t)

t = 0 t

x

F

cross-head

x

z

y

cross-section

b L

L 1

L (t)

L

L

L

0 1

0 L

traverse path

L

0 2

v T

F

L

0 0

L

L 2

L 3

L 4

L(t)= L + L + L + L1 2 3 4

L (x)

Fig. 4.24: Temporal and local deformation behavior with tensile testing

increase in prismatic specimen length L is obtained at any point in time as the sum of the elongation of equidistant specimen sections Li(x) (Fig. 4.24) and is thus identical with the traverse path.

The reaction force arising in the specimen due to external load F is also longitudinally constant due to the uniform cross-section A0 and therefore only a function of time. If specimens with altered cross-section or length are used, the measured force F and elongation L have to be normalized in order to evaluate material properties. To do so, the acting force is related to the initial cross-section area, whereby stress is obtained as follows:

0A

F (4.76)

The elongation resulting from external load L0 is related to the defined initial gauge length L0 and termed normative strain this can be stated dimensionless or as a percentage:

%100 L

L

0

0 (4.77)

112 4 Mechanical Properties of Polymers

Thus it is clear that the recorded load–extension diagram is identical with the stress– strain diagram, since both quantities are related to constant initial values. Value Li thereby corresponds to the actual length of the specimen and is a function of time or the duration of the tensile test, respectively. In addition to normal stress, there is a maximum shear stress at less than 45° to the surface normal direction x when the prismatic specimen is loaded. Under certain conditions, this leads to visual deformations (shear bands) on the specimen surface, which are not evaluated. On narrow prismatic specimens in a plane stress state, a transversal contraction in the y and z directions occurs simultaneously with the arising elongation, thus effecting a reduction in the initial cross-section. As a result of the uniaxial load state, there is a tendency for a uniaxial stress state and a three-dimensional strain state to arise. From these statements and the fact that measured elongation should be related to current specimen length, we derive the term “apparent” or “engineering” stress and strain for Eqs. 4.76 and 4.77. Depending on whether the traverse path or the extensometer or a clip gauge is utilized to eliminate compliance effects from the testing machine (grips and load cell), either Eq. 4.77 or 4.78 has to be used to calculate strain:

%100 L

L t (4.78)

The strain given in Eq. 4.77 is called normative strain; the strain given in Eq. 4.78 with index t is called nominal strain. From the derivation of strain with time, one can obtain normative strain rate or nominal strain rate in the deformed volume d /dt:

dt

)L(d

L

1

dt

d 0

0

(4.79)

or

L

v

dt

)L(d

L

1

dt

d Tt t (4.80)

The value acquired from Eq. 4.79 corresponds to the strain rate between the knife edges of the strain measurement system used and can be utilized especially in strain- controlled tensile tests. Nominal strain rate (Eq. 4.80) states the relation between the required testing speed in % min-1 or s-1 and the testing speed of traverse vT to be set on the testing machine. Given the above mentioned requirements for the tensile test and the prismatic specimen corresponding to Fig. 4.24, nominal and normative strain rates are identical, i.e., these values are identical in every section of the specimen.

Prismatic specimens usually break directly at the top or bottom clamping grip. For this reason, so-called dumbbell specimens are used to ensure satisfactory clamping

4.3 Quasi-Static Test Methods 113

F

� (s ) -1

normative strain rate of dumb-

bell specimen

normative = nominal strain

rate of prismatic specimen

average nominal strain rate of dumbbell specimen

L

L L 0

Fig. 4.25: Comparison of strain rates on dumbbell and prismatic specimens

conditions. Due to their geometry, a smaller and non-constant normative strain rate occurs within the clamping length of dumbbell specimens (Fig. 4.25). The differences between the median strain rate of a dumbbell specimen and the nominal strain rate of a prismatic specimen depend on the actual specimen geometries. This circumstance illustrates the problematics for reproducibly determining material values compared to the requirements stated in theory (Fig. 4.24). In addition, we have to consider the intrinsic specimen state, which is especially influenced by processing conditions and generally not isotropic and homogenous, as well as the influence of temperature and testing speed, so that the actual measurement result represents more than just the property of the molding compound.

In order to interpret results of tensile tests as single-point data according to ISO 10350, it must be ensured that the acquired values represent a summary of the properties of the molding material, the processing conditions and the selected test conditions, and that they can therefore only be used for simple material preselection. For a detailed selection of materials, comparable multipoint data (ISO 11403) should be used that represent properties as functions of essential parameters such as temperature, time and environmental conditions.

4.3.2.2 Conventional Tensile Tests

The preferred standard for performing tensile tests on plastics is ISO 527, which includes the testing of molding compounds, films and sheets, as well as fiber composite materials. One essential prerequisite for performing such tests is that suitable polymer-compatible specimens are used, such as those illustrated in Fig. 4.26.

114 4 Mechanical Properties of Polymers

1A 1B 1BA 5A 2 5 4

1BB 5B

b

r

h

b1

l l 2 l 1 L 03 L

2 Fig. 4.26: Specimens for tensile tests on plastics according to ISO 527

Type 1A and 1B specimens are basic specimens corresponding to ISO 3167 and suitable for use in a variety of testing techniques (see Table 4.2). Specimen 1A, which is generally produced by direct shaping, i.e., injection molding, has the parallel length l1 = 80 mm and is known as a multipurpose specimen. Thanks to its parallel length, this popular specimen can also be used in other types of testing such as the bend test, compression test and impact test where it has the advantage of comparable internal state. Type 1B specimens are normally produced by indirect shaping, e.g., sawing and milling, from semi-finished products in the form of sheets. Versions 1BA and 1BB are

Table 4.2: Specimens type 1A and 1B

Specimen type 1A 1B

Dimensions in mm

L 3

Overall length 150

L 1

Length of narrow portion 80 2 60 0.5

R Radius 20 – 25 60

L 2

Distance between broad parallel portions 104 – 113 106 – 120

B 2

Width at ends 20.0 0.2

B 1

Width of narrow portion 10.0 0.2

H Thickness 4.0 0.2

L 0

Initial gauge length 75.0 0.7 or 50.0 0.5 50.0 0.5

L Initial clamping length (grip distance) 115 1 l 2

4.3 Quasi-Static Test Methods 115

proportional small versions of type 1B to a scale of 1 : 2 and 1 : 5, respectively. This also makes them suitable specimens for characterizing tensile properties when cut from component parts. Specimens 5A and 5B correspond to types 2 and 4, respectively, of ISO 37 and are recommended for tensile tests on rubber and other elastomeric materials. Specimens of type 2 and 4 are recommended for characterizing tensile properties of foils and sheets as per ISO 527-3. Type 5 specimens, also called spoon-like specimens, are preferred for testing of ductile materials with high tensile strain at break (e.g., PE-LD). For fiber-reinforced plastics materials, specimens of type 2 or the similar type 3, with optional center hole or cap strip (load application plate), are used to ensure sufficient clamping conditions according to ISO 527-4 (see also Chapter 10).

For characterizing tensile properties of plastics, we distinguish between two tests with different cross-head speeds: the test for acquiring elastic constants, especially the E modulus, and the tensile test for acquiring strength and deformation properties. For E modulus determination, a cross-head speed of vT = 1 mm min

-1 is specified, approximately corresponding to a 1 % min-1 strain rate at measurement intervals of L0 = 50 mm.

For the actual tensile test according to ISO 527, there exists a wide range of vT from 1 to 500 mm min-1 for testing many very different plastics, so that ISO 10350 requires more specifics. Brittle breaking plastics whose elongation at break is less than 10 % have to be tested at vT = 5 mm min

-1 while most other materials are tested at vT = 50 mm min

-1 [4.47]. According to ISO 527 it is allowed to determine the E modulus for one specimen at vT = 1 mm min

-1 in a first step up to = 0.25 % and then the tensile properties are measured by switching to vT = 50 mm min

-1. But the favoured procedure of the standard includes an unloading of specimen after E modulus determination followed by the tensile test up to the fracture of the sample.

Due to the narrow elastic deformation range of plastics, the E modulus is determined as a secant modulus (Fig. 4.27a). It includes the elastic and linear-viscoelastic deformation range of the stress–strain diagram. Evaluation is then limited to the deformation range between 0.05 % and 0.25 % of the normative strain between strain gauges. The required cross-head speed is related to initial clamping length and calculated according to Eq. 4.80. The E modulus is calculated according to Hooke’s law from force change F = F2 – F1 and strain change = 2 – 1 = 0.2 % = 0.002:

0

12

12

12

A002.0

FF E (4.81)

116 4 Mechanical Properties of Polymers

a b 0 b�

2

� (M

P a

)

F 2

F 1

Fv

L

� 0

b 1

� q

q2 b

2

1 2

� 1

� 0

F (N

)

F

� = f(�)

L - L0102

Lv 01 L02 L (mm)0

� 1

� 2

� (%)

� = f(�)

� 1

� 2

� (%)

L01 L02 L (mm)0

L - L0102

b

= b

- b

b (

m m

)

q1

�

� (%

) q

Fig. 4.27: Acquiring elastic constants in tensile tests: stress–strain diagram (a) and transverse strain–

strain diagram (b)

The product of E modulus and cross-sectional area is called tensile stiffness EA0. If pronounced transient behavior occurs during the test, a prestress 0 or a preload Fv can be applied, but should not result in elongation > 0.05 %. The evaluation software in a computer-aided evaluation provides in part for the subsequent correction of such transient effects. If a second strain gauge is used for acquiring change in thickness or, if preferred, width simultaneously with elongation, the Poisson ratio μb (width) or μh (thickness) can be measured (Fig. 4.27b). This elastic parameter should be measured in a linear range within the interval 0.3 % < y and can be calculated in cases of changes in width as follows:

00

0qb b

bL

Lb (4.82)

In order to obtain correct values for tensile loading, the measuring system has to be prestressed during setup at approx. 90 % of the expected nominal load and using a rigid setup specimen compared to the specimen to be measured. This serves to avoid possible setting motion in the load line.

To eliminate any influence from superimposed flexural stress, precise load line alignment has to be monitored at sufficient intervals. When measuring for elastic values, strain gauges with a minimum resolution 1 μm (better: 0.1 μm) should be used together with parallel tensile clamping devices. The use of traverse path to determine E modulus – often seen in practice – leads to decidedly lower values, since the internal deformations of the test system (slipping in clamps or load cell deformation in the measuring path) lead to faulty paths. Such values can only be used under identical

4.3 Quasi-Static Test Methods 117

conditions for quality assurance and are in no way comparable with values acquired using strain gauges.

Tensile tests for determining strength and deformation properties of plastics are usually performed at a testing speed of 50 mm min-1. Typical stress–strain diagrams for various plastics are shown in Fig. 4.28. The parameters derivable from the curves correspond to characteristic points in the diagrams. Diagram (a) can be assigned to brittle material behavior with relatively high tensile strength M, whereby the tensile strain at break B achieved can be as much as 10 %. Typical examples for such material behavior include PS and other brittle thermoplastics, thermosets, as well as filled and reinforced plastics. Stress–strain diagrams of types (b) through (d) represent ductile deformation behavior with strains at break of several hundred percent, but relatively low tensile strengths. Some typical examples for such material behavior include polyolefines and polyamides. Particularly thermoplastics with type (b) and (c) stress– strain behavior exhibit a yield stress y , at which local necking followed by a constant stress plateau occurs, also called cold yielding. The stress plateau is the result of stretching with yield zone formation, whereby the material is stretched and pulls itself simultaneously out of the unstretched part of the specimen. Depending on the testing speed set, the yield zone can extend over the entire prismatic section of the specimen. This process leads to orientation by aligning the molecules in the direction of loading, as can be demonstrated, e.g., by measuring density.

� (%)

� = �

� (M

P a

)

�

�

�

�

�

� = �

a

b

c

d

B

y

B

B

x

t

tB

tB tM

� = �B M

y

M

� = �B M

� = �y M

� = �B M

� (%)

e

�

� = �y M x� � = �B M

� tB

Fig. 4.28: Stress–strain diagrams and parameters of various plastics: brittle materials (a), tough materials with yield point (b and c), tough materials without yield point (d) and elastomeric materials (e)

118 4 Mechanical Properties of Polymers

In principle, any deformation of a material, besides being a change in its internal energy state, is connected with heat tone that can be shown by video thermography. Since in normal climate (23 °C, 50 % humidity) plastics are in a temperature range where even slight temperature changes can influence their behavior considerably, reactions to their deformation behavior occur due to such heat effects. The above mentioned, irreversible cold yielding thus corresponds to a hot stretching due to internal heating of the plastics material. At very low strain rates close to equilibrium, total orientation in the plane parallel region of the specimen is achieved. At this point load is finally no longer absorbed by intermolecular forces, but by primary valence bonds. Due to this fact, a type of stress known as work hardening increases in these plastics. The second peak in the stress–strain curve in this case is called tensile strength M (type b). If no work hardening occurs (type c), yield stress y is identical with tensile strength M.

Tensile stress at break B and tensile strain at break B rarely provide values that are physically meaningful and applicable in practice. Of these, tensile strain at break exhibits especially high statistical scatter and tensile stress at break is strongly dependent on the degree to which the test software is suited for evaluation. According to the standard B should be determined at the stress level equal to 10 % of M.

The type e stress–strain diagram in Fig. 4.28 corresponds to the typical S-shaped curve of rubbery materials with very low strength and E moduli, but very high tensile strains at break (as much as 1000 %). This materials group includes, for example, PVC-P as well as natural and synthetic rubber.

The region of final break or initial geometric instabilities, such as necking at yield point, is often called the macrodamage limit. Because of local deformation processes, significant changes in specimen cross-section and a markedly inhomogenous strain rate relative to specimen length, the equations listed below actually lose their validity for the parameters. The acquired values thus have only limited use for dimensioning purposes.

Depending on the deformation behavior of the investigated plastic, the parameters listed below can be determined according to ISO 527:

Yield stress y : The first stress value at which an increase in strain occurs without an increase in stress. This value is generally determined via the slope d /d of the stress– strain curve. This value can be identical to tensile strength M.

0

y y

A

F (4.83)

4.3 Quasi-Static Test Methods 119

Tensile strength M : the maximum tensile stress registered during the experiment. Depending on material behavior, this value can be identical to yield stress y or tensile stress at break B.

0

max M

A

F (4.84)

Stress x at x % of strain : If the stress–strain diagram exhibits no macroscopic visual yield stress (type d in Fig. 4.28), this parameter can be used for the comparison of materials. This value, sometimes called offset yield strength, is the stress in MPa at which strain x reaches the defined value x in %. The test software of state-of-the-art universal test machines often allows for the selection of several values for x.

0

x x

A

F (4.85)

Tensile stress at break B : the stress at which the specimen breaks is determined. This value, often called fracture strength in testing practice, depends on machine-related parameters such as break detector threshold as well as sampling rate and can thus be subject to considerable statistical variation. Tensile stress at break B can be identical to tensile strength M.

0

B B

A

F (4.86)

Strains assignable to stress parameters characterize the deformation behavior of a material. According to ISO 527, normative strain for plastics has to be recorded up to yield stress or for brittle materials up to tensile stress at break, or tensile strength has to be recorded using an extensometer or strain gauge (Fig. 4.28 Type (a) and (d) to break, Type (b) and (c) to yield stress). In this case, the measuring distance amounts to L0 = 75 or 50 mm (type 1A) or L0 = 50 mm (type 1B). Beyond yield stress, nominal strain t has to be used for ductile plastics (Fig. 4.29) and the total strain is calculated as sum of = y + t with L = 75 mm. This assumption is valid if no relevant amount of strain is measurend in the range of the shoulders. In case of L0 = 50 mm the total strain is determined in the same way but using the clamping length of L = 115 mm. If customers wish to compare older results of measurements it is also allowed to carry out the measurements without extensometer only using the cross head motion. These conditions make it necessary to use fracture resistant automatic strain sensors and to record cross-head and extensometer elongation simultaneously. According to the standard, tensile test results have to be rejected if the specimen breaks directly at or in the grip. In these cases, nominal strain t is to be preferred in order to ensure

120 4 Mechanical Properties of Polymers

� (M

P a

)

� = �y M

� B

���

�

� or � (%)

� = �y M

t � �t

� B

� y

� M

y

t Fig. 4.29: Use of normative and nominal strain according to ISO 527

comparability of the results. The following parameters of strain can be determined:

Yield strain y : the strain at yield stress stated dimensionless or as a percentage.

%100 L

L

0

0 y (4.87)

Strain at tensile strength: this is the strain corresponding to tensile strength M. For materials without deformations above yield strain, either normative strain , or nominal strain t is used:

%100 L

L

0

M0 M or (4.88)

%100 L

LM tM (4.89)

Tensile strain at break B : the strain when tensile stress at break is reached can be stated as a nominal or normative parameter, depending on deformation behavior:

%100 L

L

0

B0 B or (4.90)

%100 L

LB tB (4.91)

For the deformation behavior of plastics, the test conditions, e.g., testing speed, temperature or ambient conditions are extremely important, since they have a lasting influence on the relaxation and retardation mechanisms in progress. Figure 4.30

4.3 Quasi-Static Test Methods 121

� (%)

� (M

P a

)

b 125

100

75

50

25

0 0 50 100 150 200

� (M

P a

)

� increasing

a 125

100

75

50

25

0 0 50 100 150 200

� (%)

T decreasing

Fig. 4.30: Stress–strain diagrams of a thermoplastic material as a function of testing speed (a) and temperature (b)

shows diagrams of the influence of temperature and strain rate on the stress–strain behavior of a ductile thermoplastic. As speed increases or temperature decreases, tensile strength increases and tensile strain at break decreases, thus changing the progression of the stress–strain diagram.

Table 4.3 lists the relevant values for E modulus and tensile strength of selected plastics.

Table 4.3: Modulus of elasticity in tension and tensile strength of selected polymers [1.54]

Material E (MPa) M (MPa) Material E (MPa) M (MPa)

Thermoplastics unreinforced Thermoplastics reinforced

PE-HD 1040 28 PP + 30 wt.-% GF 6200 73

PE-LD 280 - PA 6 + 30 wt.-% GF 6500 110

PS 3200 51 PA + 30 wt.-% CF 18000 190

PA 6 1300 32 PP + talc 3000 32

PC 2300 71 PP + chalk 3000 32

PMMA 3300 74 PVC + chalk 3200 -

PVC-U 3200 50 Thermosets

PVC-P - 21 Phenole resin 8800 -

PP 1300 34 Urea resin 8750 44

ABS 2400 38 Melamine– formaldehyde resin

7000 -

POM 3000 68 Unsaturated polyester resin

3700 53

122 4 Mechanical Properties of Polymers

Table 4.3: Modulus of elasticity in tension and tensile strength of selected polymers [1.54]

Material E (MPa) M (MPa) Material E (MPa) M (MPa)

Thermoplastics unreinforced Thermosets

PET 2700 40 Epoxy resin 2840 62

PEEK 3900 - Silicone resin 10400 24

PUR 1900 - PUR 1230 26

SAN 3700 76

PBT 2600 53

4.3.2.3 Enhanced Information of Tensile Tests

Macrodamage limits, such as yield stress or tensile strength can be derived from the stress–strain diagram. However, the stress–strain diagram does not give sufficient information on the irreversible material damage, or microdamage, induced at low load levels. Due to structural changes in materials under mechanical loading as well as to machine compliance, the dominant normative strain rate varies constantly within the specimen. Also dependent on the internal state of the specimen, this strain rate differs from cross-head speed by amounts determined by the speed of the plastic deformation component and the stiffness ratio between the specimen and the universal testing machine. Since tensile test values for plastics are strongly influenced by strain rates, knowledge of these metrological and test-technological effects is essential for evaluating and interpreting the results.

Today’s material development requires material-descriptive, structurally or morphologically based parameters that provide information as to the dependence of load limits on loading conditions and that enable the suitable selection and dimensioning of polymers in conformance with the relevant material laws. Conventional tensile testing cannot fulfill these requirements, since in most cases the acquired values cannot be explained structurally or physically and are based on varying testing conditions.

Various methods of tensile testing are available for better adapting properties’ characterization to materials and loads:

• Event related evaluation and interpretation of the stress–strain diagram (Fig. 4.31),

• Qualification of the tensile test by improved measuring and evaluation techniques, such as video or laser extensometry [4.48, 4.49],

• Correction of geometric and technological influencing factors in conventional tensile tests and

4.3 Quasi-Static Test Methods 123

• Load or strain-controlled tensile tests with constant deformation conditions [4.49, 4.50].

The simultaneous coupling of tensile test with damage-sensitive NDT methods is an essential requirement for event-related interpretation of polymer deformation phases (Fig. 4.31a) and for increasing the informational value of conventional materials parameters. This hybrid approach is also called polymer diagnostics (see Chapter 9). Thanks to the coupling of tensile tests, e.g., with acoustic emission analysis [4.51, 4.52], or with modern methods of thermography [4.53], information can be obtained on early microdamage taking place in the plastic. This enables the formulation of damage functions or critical load limits. Taking a ductile, stretched plastic, as in the example in Fig. 4.31b, it can be clearly seen that specimen cross-section decreases with increasing plastic deformation, i.e., there is a discrepancy between nominal and normative strain rates. This reaction of the material to loading shows that polymers do not fulfill the condition of homogenous isotropic and velocity-independent material behavior.

Simultaneous application of volume dilatometry enables the use of defect density QD for quantifying microdamage. Irreversible microdamage occurs in the form of an increase in defect density QD (Fig. 4.31b) at the start of non-linear viscoelastic deformation regions. This decrease is due to stretching and the resulting increase in orientation once yield stress has been exceeded. By using event-related and supplementary morphological examination, material damage can be described as the precursor of ultimate failure, and it becomes possible to formulate material limit states or diagnostic functions [1.17, 4.54].

In conjunction with event-related interpretation of deformation behavior, strain measuring techniques with local resolution are especially required in order to qualify the informational value of conventional material values from tensile tests. These include laser interferometry (electronic speckle-pattern interferometry – ESPI or shearography) [4.55, 4.56] or laser extensometry [4.57] (see Chapter 9). When these contactless and inertia-free measurement methods are used, local deformation fields or strain distributions can be detected that enable statements on near-surface defects, orientation state, or heterogeneity of deformation. They also have special advantages with respect to notch-sensitive plastics.

Various procedures are available for correcting measurement and technological influencing factors arising from the stiffness ratio between the specimen and test machine, the fundamental problems of detecting apparent values in tensile tests, as well as geometric inadequacies. The difference between normative and nominal strain rates (Fig. 4.25), which depends on internal state and specimen geometry, can be reduced by experiments comparing the deformation behaviors of dumbbell and

124 4 Mechanical Properties of Polymers

71 2 3 4 5 6

elongation without

elongation with necking

� (M

P a

)

� (%)

linear-viscoelastic region linear-elastic region

non-linear viscoelastic region necking region steady-state plastic yielding strain-hardening region ultimative failure – fracture

1 2 3 4 5 6 7

� = f (�)

d e

fe c t

d e

n s it y

Q D

� (%

/m in

)

� = f (�)

�

Q = f (�)

t

D

� (%)

a

b

necking

Fig. 4.31: Event-related interpretation of deformation phases in tensile testing

prismatic specimens from identical materials. The median strain rate is determined on a flat multipurpose specimen d S /dt and a prismatic specimen d P /dt. To evaluate the influence of specimen geometry, the quotient of strain rates can be included as correction factor k:

S

Pk (4.92)

Taking Eq. 4.92 into consideration, Eq. 4.80 can be expanded:

k L

v T t (4.93)

DIN 53 455 provides an evaluation method that allows for the geometry effect which, however, has lost its validity since the introduction of ISO 527. There the difference of strain rate is corrected by determining reduced clamping length lred (Eqs. 4.94 – 4.97 and Fig. 4.32).

em1 l2l2lL (4.94)

4.3 Quasi-Static Test Methods 125

r2

b 1a 1 (4.95)

r

l arcsin

2

1

1a

r

l arcsin

2

1 tan1a

arctan 1a

a

l b

m

2

m

2

m m (4.96)

2

e

m

m

1

1 1red

b

l2

b

l2

b

l bl (4.97)

In place of Eq. 4.80, the following relation is to be used for calculating nominal strain rate:

red

T t

l

v (4.98)

In the region of elongation without necking (see regions 1 to 3 in Fig. 4.31), cross- section or transverse contraction decreases as deformation increases. The actual measurement length changes constantly, i.e., the apparent parameters of stress and strain turn out to be faulty. In order to acquire true stress, minimal cross-section has to be recorded at all times; this presents a technological problem, especially for prismatic specimens. The measuring equipment to be used must be capable of recording minimum cross-section A regardless of place and time of its occurrence. Given this requirement, true stress w is obtained:

A

F w (4.99)

b 1

b m

b 2

l e l 2

1

lred

L

r

lm

l

Fig. 4.32: Dimensions of the specimen for calculating reduced clamping length

126 4 Mechanical Properties of Polymers

For large elongations, integrating the infinitesimal strains as per Eq. 4.9 leads to true strain w. If nominal strain, i.e., the faulty traverse path, is used, this equation can be used to calculate true strain. Direct measurement of true strain on the specimen can only be performed by special laser optical measuring systems that trace the variation of surface roughness on two surface regions at a constant distance from each other [4.47, 4.58]. The true strain rate is obtained by the differentiation of Eq. 4.9:

L

L

dt

dL

L

1

L

L ln

dt

d

0

w (4.100)

For nominal strain, the rate dL/dt is identical with cross-head speed vT:

1L

v

LL

v TT w (4.101)

In order to estimate material behavior and machine influence on nominal test speed, Eqs. 4.80 and 4.101 can be expanded, in which case measurement length L0 is identical with clamping length L.

MKAL

v

00

T (4.102)

MKAL1

v

00

T w (4.103)

The factor K corresponds to the test machine compliance in mm kN-1 and is valid only for the particular test configuration chosen. If changes are made, e.g., in the selection of a load cell or an extension joint, compliance has to be redetermined. Value A0 is the initial cross-section area of the specimen and M is the rise in stress– strain diagram d /d . In the elastic deformation region, M corresponds to the modulus of elasticity E. For Eqs. 4.9 as well as 4.101 through 4.103, it must be kept in mind that these are valid only for homogenous deformations in the region of elongation without necking. At neck formation, observations have to be limited to restricted regions of deformation, or strain measurement methods with local resolution have to be used that enable neck fronts to be traced on the specimen surface with sufficient precision.

Regardless of the corrective measures undertaken, rate differences occur in the specimen volume investigated. Depending on the particular material behavior, the strain rate varies during test duration and thereby influences especially the relaxation and retardation behavior overlying the tensile test. Two different versions of the

4.3 Quasi-Static Test Methods 127

controlled tensile test are available for achieving constant load condition using either analogue or incremental closed-loop systems.

The load-controlled test often applied on metallic materials is equivalent to a ramp function with constant load or stress rate. The required control is ensured by variable cross-head speed. For plastics, this type of test method represents the worst possible case, since there is a constant increase in force per unit of time regardless of the dominant deformation mechanism. For this reason, significantly higher tensile strengths are recorded together with reduced strain at break.

The use of strain measurement devices is a prerequisite for the strain-controlled tensile test and determines strain directly on the specimen as the actual value. Analogous to the load-controlled tensile test, deviation from the selected reference value is compensated by changes in cross-head speed. By this type of test procedure, normative engineering strain rate in the specimen volume is controlled at a constant level between the knife edges of the strain sensor. Since the difference between apparent and true strain increases with increasing test duration, Eq. 4.104 can be applied to correct the calculations and alter cross-head speed. Thus, a constant true strain rate is obtained.

00wT LLv (4.104)

Regardless of the type of test procedure, meaningful and useful results can only be obtained in the region of elongation without necking. At very fast changes in strain rate or local formation of neck, instabilities can occur in the control loop requiring the test to be interrupted. The parameter of control loop (PID) and the E modulus of the investigated material are very important for such tests, since rigid plastics normally require proportionally less amplification P than do materials with low E moduli. This can be explained by the fact that the specimen itself, with its load- dependent stiffness changes d /d represents a segment in the control loop. Figure 4.33 provides a comparison between conventional and strain-controlled tensile tests on a PA6 with 20 wt.-% GF at a strain rate of 1 % min-1. The nominal strain rate calculated from traverse motion is constant. The normative strain rate measured by an extensometer directly on the specimen is subject to strong variations (Fig. 4.33a). In the strain-controlled tensile test, the integral normative strain rate does not vary between the measurement sensors of the extensometer (Fig. 4.33b). Due to these constant relaxation conditions, tensile strength in Fig. 4.33b is smaller and strain at break is markedly higher.

Although the loading conditions in the investigated specimen volume are clearly improved in a strain-controlled test method, it can be demonstrated, e.g., with laser extensometry (see Chapter 9), that there are local differences in the strain rate in spite

128 4 Mechanical Properties of Polymers

� (%) �

(M P

a )

b 100

80

60

40

20

0 0 3 6 9 10 12

� (%

/m in

)

�

a 1.5

1.2

0.9

0.6

0.3

0 0 2 4 6 8 10

� (%)

� (%

/m in

)

1.5

1.2

0.9

0.6

0.3

0

t

� = f (�)

� = f (�) � = f (�)

� = f (�)

Fig. 4.33: Stress–strain behavior and strain rate of PA6 with 20 wt.-% short-glass-fiber reinforcement in conventional (a) and strain-controlled (b) tensile tests at 1 % min-1 nominal strain rate

of constant integral strain rate. Advanced types of strain control are described in [4.59] that use strain measurement techniques with local resolution.

4.3.3 Tear Test

Technological problems, such as those caused by wrinkle formation at the clamping grips, are the reason why tensile tests fail to provide satisfactory results when used for testing elastomers, soft foam materials and polymer films.

With the goal of describing tear resistance in a manner adequate to the material, the experimental methods for determining tear resistance were introduced in polymer testing. These techniques determine the resistance that a specimen exerts against tear propagation under tensile loading. The test conditions are so complicated and disparate from those of tensile tests that this test has to be regarded as a test of plastic moldings or components, i.e., an engineering test.

Strip, angle and crescent specimens are used in the tear strength test (Fig. 4.34). A special technique for determining tear resistance in small elastomeric specimens (small (Delft) specimens) is covered by ISO 34-2. Cuts are made in the specimens whereby the effective notch stress triggers the tear strength process under tensile load. Depending on the technique, the maximum or median force in N from the load–time or load–elongation diagram is calculated relative to the specimen median thickness according to Eq. 4.105 as tear resistance TS.

B

F TS (4.105)

B specimen thickness in mm

4.3 Quasi-Static Test Methods 129

2525

2 5

120

5 0

15°

75°

notch

clamp mark

notch

100

90°

b

a

R 25

.4

R 12.7

28.4 27

Fig. 4.34: Angle specimen for the tear propagation test according to ISO 34-1 (a) and trapezodial

specimen according to DIN 53363 (b)

The value acquired by this test method provides only for a relative comparison of different materials, however. It is particularly dependent on:

• Material specifications and treating state, • The state of orientation induced by calendering, injection molding or blowing, • Duration of vulcanization in elastomers and • Test temperature and deformation speed.

Since the properties of films may vary greatly due to lateral and longitudinal orientation of the polymer chains relative to the processing direction, at least 5 specimens have to be tested in each direction. Figure 4.35 shows a clamped trapezoidal specimen (Fig. 4.35a) and typical load–deformation diagrams of 50 μm thick PE-LD films produced by film blowing (Fig. 4.35b). Owing to the special form of the trapezoidal specimen, the highest stress concentration appears directly at the notch tip from the beginning of the test. Therefore, ideally no further deformation of the specimen volume away from the notch/crack tip takes place and all deformation energy is expended only for the tear process. This is a precondition for a meaningful analysis of such diagrams.

130 4 Mechanical Properties of Polymers

a

v

F

F

T

10

12

14 b

Fmax

6

8

10

F (N

)

0

2

4 parallel

perdendicular

to the processing direction

0 10 20 30 40

0

Δl (mm)

Fig. 4.35: Clamped trapezoidal specimen for the tear test with films (a) and typical load–deformation diagrams from tear tests of 50 μm thick PE-LD films produced by film blowing (b)

4.3.4 Compression Test on Polymers

4.3.4.1 Theoretical Basis of the Compression Test

Compression testing is used to evaluate material behavior under uniaxial compression load. Specimens include rectangular prisms, cylinders or pipe sections. Although there is a number of different, mostly material-oriented, standards for testing mechanical properties under uniaxial loading, the compression test has not, with a few exceptions, achieved the same significance as the tensile or bend test or hardness measurement. This is due to the relative irrelevance of compression loading and to practical measurement problems, so that the application of compression testing has been limited to special application cases and/or selected materials. Among these are mainly building materials (concrete, polymer concrete, brick, tile, wood and foams), materials used in dampers, friction bearings or fluid seals (copper alloys, polyamides, polyethylenes or rubber) and packaging materials (cardboard and foams). For polymers, there are a number of different standards defining the conditions for testing elastomers, polymer concrete, foams and fiber-reinforced plastics. In testing practice, ISO 604, which is generally valid for polymers, is the preferred standard in use. This standard can be used for:

• Rigid and semi-rigid thermoplastic injection molding and extrusion molding compounds, including filled and reinforced molding compounds,

• Rigid and semi-rigid thermosetting molding compounds, including filled and reinforced molding compounds and

• Thermotropic liquid-crystalline polymers.

This standard is not appropriate for textile-fiber reinforced materials, rigid foams or layered composites with foam or honeycomb cores. Depending on the material,

4.3 Quasi-Static Test Methods 131

compression testing can be used for characterizing compression properties as well as in quality assurance.

Analogous to the tensile test, identical fundamental conditions are valid for the standardized compression with constant traverse speed. Loading must be impact free and increase slowly to break or to a defined load limit. The specimen should also be homogenous and isotropic, and there must be no influences exerted by the testing method used (see Section 4.3.2.1).

Under these conditions and under compression load, a homogenous uniaxial stress state arises in the specimen at sufficient distance from the top and bottom compression plates that corresponds to normal stress and strain distributed uniformly over the cross-section (Fig. 4.36).

x

z

y

A 0

� (t)

cross-section

b

L

L

0 1

L

0 2

F

upper pressure plate

traverse path L

F

A = b d 0

F

lower pressure plate

L 0

L= L − L02 01

Fig. 4.36: Stress state in the specimen under uniaxial compression load

According to the definition, a negative prefix is assigned to the arising compression stresses and compressions; however, in testing practice only absolute values are assigned. The resulting compression stress can be calculated by analogy to Eq. 4.76:

0A

F (4.106)

When mechanical or optical strain sensors are used, compression results either from path difference as a normative value (Eq. 4.107) or, when traverse path is used, as a nominal value (Eq. 4.108) (Fig. 4.36).

%100 L

L

0

0 (4.107)

%100 L

L c (4.108)

132 4 Mechanical Properties of Polymers

The presumed uniaxial compressive stress state is influenced by friction between the specimen and the compression plates. This expresses itself in hindered deformation in the y and z directions. Conically deformed elastic zones extend to the center of the specimen, beginning at its base areas. Thus, in ductile materials, the zones of plastic deformation are located mainly in the center of the specimen, and bulging followed by shear fracture occurs. The real stress field is strongly dependent on geometry. Acquired values are comparable only for identical dimensions, and practical applications of this test are limited. To minimize this influence, friction between specimen and compression plates can be reduced by lubricants or with fine sand- paper. The use of such aids has to be expressly stated in the test protocol.

Geometric dimensions are an additional influencing factor in compression test performance. In order to avoid a case of Euler stability, i.e., specimen buckling, there must be a sufficient ratio between specimen length and the dimension determining the axial moment of inertia Iy. Therefore, a slenderness ratio of 10, or at least 6, has to be maintained in order to eliminate buckling. The slenderness ratio is defined as the ratio of specimen length to the smallest radius of inertia i of its base area:

i

l (4.109)

The smallest radius of inertia of base area i of the specimen results from:

0

y

A

I i (4.110)

whereby Iy is the smallest axial inertia moment and A0 is the cross-section area of the specimen in the initial state. The required relations for possible types of specimens are illustrated in Fig. 4.37. Buckling problems can occur in thin-walled pipe-shaped specimens that are caused by stress distribution in the pipe wall.

If the required slenderness ratios are not maintained, specimens can buckle during compression testing with the accompanying danger of shrapnel flying off the specimen. Such effects can also occur if specimens are not positioned precisely in the load line of the compression plate, so that additional bending moments lead to premature break.

Technology related errors in value acquisition can, however, also arise due to excessive production related surface roughness on specimen surfaces or due to inexact plane parallel surfaces. Such errors can lead to strong initial effects or to erroneous E modulus values due to surface roughness or tilting of the specimen.

4.3 Quasi-Static Test Methods 133

x

y

prism cylinder tube

I =y b d

3

12

A = b d0

I =y �d

4

64

A =0 �d

2

4

l = � d

3.46 l =

4

l l

l

d d ib

z

da

� d

I =y �

64 (d − d )a

4 4

A =0 �

4

l = �

4

i

(d − d )a 2 2

i

(d − d )a 4 4

i

(d − d )a 2 2

i Fig. 4.37: Specimen geometry and slenderness ratio of specimens for compression tests

When fiber-reinforced plastics are compression tested, splicing can occur on the specimen, thus rendering value acquisition difficult or even impossible.

4.3.4.2 Performance and Evaluation of Compression Tests

ISO 604 is used for performing compression tests on polymers. In contrast to the statements above, this standard specifies the use of prismatic specimens produced either by direct shaping (injection molding) or indirectly by cutting from finished or semi-finished parts (laminates, extruded or calendered sheet). For the characterization of molding materials, for compression tests, as well as to obtain E modulus under compression load, specimens should be produced that comply with the multipurpose specimen defined in ISO 3167 (Fig. 4.38). This provides the advantage of a comparable internal state (orientation, residual stress) and comparability with other mechanical test methods. The dimensions of the specimen specified in the standard for determining E modulus are 50 10 4 mm3 (Fig. 4.38b). For the compression test for recording the compressive stress–compressive strain diagram, the dimensions are 10 10 4 mm3 (Fig. 4.38c). In contrast to the above rules for the slenderness ratio for specimens in compression tests, ISO 604 specifies Eq. 4.111 for defining an adequate ratio between specimen dimensions and the clamping length between compression plates.

134 4 Mechanical Properties of Polymers

2

c l

x 4.0 (4.111)

In this equation, c is the maximum dimensionless nominal compression strain occurring during the test, l is specimen length and x is the diameter of a cylinder or the thickness of a prismatic compression specimen.

This standard recommends a ratio of x/l 0.08 for determining compression modulus Ec . For the compression test, x/l 0.4 should be used, corresponding approximately to a compression or negative strain of 6 %. Since Eq. 4.111 was formulated under the assumption of linear-elastic behavior, when compression increases, or ductile materials are tested for c, values have to be used that are 2 to 3 times higher than maximum compression. In anisotropic materials, such as laminates, specimens should always be tested in their particular direction of main orientation, since the results of this test depend, in analogy to other mechanical test methods, very strongly on test direction. According to standard ISO 604 for determining Ec, test speed vT amounts to 1 mm min

-1. For determining compression properties with the specimen in Fig. 4.38c, it also amounts to 1 mm min-1 in most cases. For ductile plastics, the traverse speed should be 5 mm min-1, whereby practical testing experience shows that maximum compression should not exceed 50 % in the test. For different specimen dimensions or shapes, Eq. 4.111 is to be used for checking experimental conditions and, in compliance with the standard, test speed has to be adjusted in the range of 1 to 20 mm min-1.

1 0

10

5 0

50

10

80

10

b

a

c

Fig. 4.38: Specimen preparation for the compression test on a multipurpose specimen according to

ISO 3167 (a); a specimen for acquiring the E modulus (b); and a specimen for determining compressive stress–compressive strain behavior (c)

4.3 Quasi-Static Test Methods 135

�c (%)

� (M

P a

)

�M = �B

�x

�y

�M = �B �y

a

b

c

d

�x

�M = �B

�M = �B

�cM = �cB�cy

� (%)

�M

�B

�M

�cM

�B

�cB Fig. 4.39: Compressive stress–compressive strain behavior of brittle plastics (EP) (a); ductile plastics

with compressive yield stress (PS) (b); ductile plastics without compressive yield stress (PMMA) (c); and ductile plastics without break (PA) (d)

By analogy to the tensile test, Ec is acquired as in Fig. 4.27 and Eq. 4.81 as a secant modulus in the interval of 0.05 to 0.25 % compression. For this, a strain gauge with a resolution of 0.1 μm is specified. In case a preload Fv is used to minimize initial effects, it must not generate a compression greater than 0.05 %. If a specimen as in Fig. 4.38c is used, and taking specific material behavior into consideration, the following parameters can be determined from the compressive stress–compressive strain diagram (Fig. 4.39).

Compressive stress at yield y : the stress value at which a first increase in compression occurs without an increase in stress. This value is also called compressive yield stress or compressive yield point and is determined via the increase d /d from the compressive stress–compressive strain diagram. If microcracks occur when com- pressive yield stress is reached, the result of the experiment may be contaminated.

0

y y

A

F (4.112)

Compressive strength M : the maximum compressive stress recorded during the test. This value can be identical to compressive stress at break (see Fig. 4.39 curves (a) and (c).

0

max M

A

F (4.113)

136 4 Mechanical Properties of Polymers

Compressive stress at break B : this value results from the compressive stress at the time of break. In cases of material behavior corresponding to diagram (a) in Fig. 4.39, compressive strength and compressive stress at break have the same value. This value depends very strongly on the break detector threshold setting.

0

B B

A

F (4.114)

Compressive stress x at x % strain : this value is determined when the material exhibits no compressive yield stress or when no break occurs during the test (curve (d) Fig. 4.39). The value of x can be taken in compliance with mold material standards, or agreed upon with the customer, whereby x always has to be smaller than the compression corresponding to compressive strength. Common value used in testing practice is 50 %.

0

x x

A

F (4.115)

Compressions assignable to strength values can be determined with mechanical or optical extensometers, whereby normative compressions can be stated. Since specimens have a relatively short length of 10 mm, nominal compression c is mostly used in practice. It results from the change in distance of the compression plates and corresponds to the traverse path. Compressive values can be stated either dimensionless or in percentages.

Compressive yield strain y or cy : the compression value achieved at pressure yield strength y. This value is often called compressive strain at compressive stress at yield.

0

y0 y

L

L (4.116)

l

l y cy (4.117)

Here l is initial specimen length and l is compression in mm.

Nominal compressive strain at compressive strength M or cM : the compression at which compressive strength is reached. For the material behavior in curve (a) of Fig. 4.39 it can be identical with compressive strain at break

0

M0 M

L

L (4.118)

4.3 Quasi-Static Test Methods 137

l

lM cM (4.119)

Compressive strain at break B or cB : this value is determined when break is reached.

0

B0 B

L

L (4.120)

l

lB cB (4.121)

By analogy with tensile tests, the results of compression tests are dependent on loading conditions, especially temperature and test speed. For this reason, only when production and test conditions are identical, are test results comparable. They are not readily applicable to the behavior of components. Special problems can arise in the acquisition of compressive strength or yield stress at break, since specimen break is not always clearly classifiable. For this reason, the visual observation of deformation behavior is very important, while taking appropriate safety precautions. This test is mainly used in quality assurance or for material characterization. Due to the different deformation mechanisms of plastics under tensile and compression loading, clearly deviant – diagrams are obtained. Taking PS as an example (Fig. 4.40), it can be seen that, due to the dominant craze mechanism, no yield point occurs in the tensile test, and material behavior is brittle. Due to shear flow under compression load, strength in the compression test is at 100 MPa almost twice as high as in the tensile test (50 MPa), and recorded strain at break is significantly higher. Compressive strength values are listed in Table 4.4 for selected polymers.

� (M

P a )

�y

a

b

�M = �B

�M

shear bands

crazes

� (%)�y

Fig. 4.40: Deformation behavior of PS in tensile tests (a) and in compression tests (b)

138 4 Mechanical Properties of Polymers

Table 4.4: Compressive strength of selected polymers [1.48]

Material M (MPa) Material M (MPa)

Thermosets Thermoplastics unreinforced

Phenole resin 170 PMMA 110

Urea resin 200 PTFE 12

Melamine–formaldehyde resin 200 Thermoplastics reinforced

UP resin 150 PP + 30 wt.-% GF 60

EP resin 150 PA 6 + 30 wt.-% GF 160

PUR 110 PA 66 + 30 wt.-% GF 170

4.3.5 Bend Tests on Polymers

4.3.5.1 Theoretical Basis of the Bend Test

Flexural loading is one of the most common types of load encountered in practice. Thus it is highly significant for determining characteristic values of polymers and fiber composite materials. This type of load is used in the following test procedures:

• Bend test for characterizing thermoplastic and thermosetting molding compounds and filled as well as reinforced composite materials,

• Mechanical-thermal flexural loading for measuring heat-distortion resistance in the HDT test, as well as

• Mechanical-environmental flexural loading for measuring environmental stress cracking resistance.

The quasi-static bend test is used especially for testing brittle materials whose failure behavior causes technical problems with tensile tests. For polymers, this test is used on the following materials according to the specifications of test standards:

• Thermoplastic injection and extrusion molding compounds, including filled and reinforced molding compounds, as well as rigid thermoplastic sheets,

• Thermosetting molding compounds, including filled and reinforced composite materials,

• Thermosetting sheets, including laminates, • Fiber-reinforced thermosetting and thermoplastic composite materials containing

both unidirectional and non-unidirectional reinforcements, and • Thermotropic liquid-crystalline polymers.

However, this test method is not suited for rigid foams or sandwich structures with foam cores.

4.3 Quasi-Static Test Methods 139

Under flexural load, as under tensile or compression load, the various deformation components have to be considered that are dependent on time and load. Depending on the type of polymer, linear-elastic, linear-viscoelastic, non-linear viscoelastic and plastic deformation components also occur. The ratio of deformation components to total deformation depends on the particular polymer as well as on loading conditions (temperature and test speed). Therefore, the value acquired in the bend test is a function of deformation, strain rate, load or stress, temperature and the internal state of the specimen. In actual testing practice, three-point and four-point bend test equipment is available for performing bend tests (Fig. 4.41). In view of the occurring loads, the four-point bend test is the fundamentally more suitable method due to the constant bending moment and the freedom from transverse force.

With this arrangement, no corrective measures are required in the case of off-center fracture of the specimen. Its disadvantages are its technically more complicated construction and work-intensive handling; in addition, it requires an extremely precise deflection measuring device. For these reasons in particular, the three-point bend test is specified in ISO 178 as the test method for plastics, even though the four- point bend test produces more precise and reproducible results. Due to structural deficiencies in specimens, eccentric fracture can occur under three-point bend loading; as long as fracture occurs in the median third of the specimen that is acceptable for value acquisition.

The same fundamental requirements hold for performing bend tests as for tensile and compression tests (see also Section 4.3.2.1), i.e., load must be applied without impact and increase steadily. A uniaxial normal stress state should arise in the specimen, whereby influences from test equipment, shearing by transverse force, as well as compression at the supports, have to be negligible. The specimen must also be free of geometric imperfections, such as notches, and its cross-section must remain plane during the test, i.e., no warping is to occur.

With these requirements and the general differential equation of the elastic bending line (Eq. 4.122), the relation between deflection, E modulus and specimen geometry can be derived for the case of three-point bending (Fig. 4.42).

IE

)x(M

xf1

xf b 2/32

(4.122)

In this equation, Mb (x) is local variable bending moment, EI flexural stiffness and df/dx the slope of the bending line. Since this differential equation of the deformed cantilever beam can only be solved numerically, and is thus difficult to handle, the simplified Eq. 4.123 is used in actual practice, being valid for small deflections f and

140 4 Mechanical Properties of Polymers

specimen

traverse

bending jaw

ll l

F

a ab

Q = 0

support

specimen

F

variable radii

positioning slide

traverse

v

anvil

L

T

L/3

Mmax

v T

b

a

= F L 4

F l

2 a

F 2

Q = F

2 Q = +

bending moment M

transverse force QF 2

Q =

b

F

2 Q = +

support

variable radii

positioning slide

Mmax =

bending moment Mb

transverse force Q

Fig. 4.41: Construction diagram of three- (a) and four-point bend test equipment (b)

slope df/dx 0 [4.60].

IE

)x(M )x(f b (4.123)

The elastic bending line for three-point bending, taking boundary conditions into consideration, yields the following:

4.3 Quasi-Static Test Methods 141

32 x 12

F xL

16

F

IE

1 )x(f (4.124)

For technical measurement purposes, the middle deflection f at point x = L/2 is especially interesting:

IE48

LF f

3

(4.125)

Assuming linear-elastic material behavior, strain and stress are distributed symmet- rically over the cross-section (Fig. 4.42b), whereby an unstressed and unstrained fiber, also called a neutral fiber, appears in the center of the specimen. Since external stress is distributed linearly over the cross-section, the highest tensile or compression stress always occurs in the peripheral fibers of the bend specimen, whereby the sign is generally ignored in actual practice (Eq. 4.126).

2

h

I

Mb f (4.126)

Using the maximum bending moment in the specimen center Mb = FL/4 and the axial moment of inertia in prismatic specimen I (Eq. 4.127), we obtain the equation for calculating flexural stress f in the three-point bend test (Eq. 4.128).

x

z y

�

-�

+�

x

y

z

h

-�

+�

x

y

z

h

max

maxmax

max

h

b

z y

F EI

L/2

y

�max

b

a

c

f

Fig. 4.42: Three-point bend specimen (a), distribution of normal stress and strain (b), as well as shear

stress distribution (c) over specimen cross-section

142 4 Mechanical Properties of Polymers

12

hb I

3

(4.127)

2 f

hb2

LF3 (4.128)

This yields a load level varying over specimen height or thickness, whereby the individual deformation components can occur simultaneously in time and location.

When tensile and compression stress deformation behavior are identical, only singular, planar symmetrical layers in the specimen reach yield stress (tensile load) or compressive strain at yield (compression load). Thus the yield stress observed is never as pronounced as in the tensile test. When a yield point is reached, the effects on the actually developing stress and deformation region can be seen particularly in the peripheral fiber of the specimen, in contrast to theoretical compressive behavior. Due to these circumstances, a stress distribution over cross-section results that diverges from linearity, particularly when the yield point, i.e., beginning plastic deformation is reached. If there are significant differences of tensile and compressive behavior for the material investigated (Fig. 4.40), displacement of neutral fibers can occur, leading to asymmetrical stress distribution over the cross-section.

Using Hooke’s law and maximum flexural stress f in Eq. 4.128 we obtain the relationship between measured quantity f and dimensionless peripheral fiber strain f:

2 f

L

hf6 (4.129)

In order to measure traverse path, i.e., to determine the deflection between bend anvil and supports (Fig. 4.42a), the elasticity modulus Ef is obtained as follows:

3

3

f hbf4

LF E (4.130)

The relationship between cross-head speed vT and desired peripheral fiber strain rate d f /dt is important for setting the universal testing machine:

2

Tf f

L

hv6

dt

d (4.131)

Besides normal stress, additional shear stress (Fig. 4.42c) develops in the specimen that can influence the result of the bend test. In order to minimize this effect, the ratio of support length L to specimen height h must be considered when testing plastics:

4.3 Quasi-Static Test Methods 143

h116L (4.132)

For very thick specimens or materials containing coarse fillers, a relatively large L/h ratio may be required to avoid delaminations due to shearing. This is especially true for laminates or other layered plastics, except when testing for interlaminar shear strength (short-beam test). In this case, the ratio L/h should range from 20 to 25 to eliminate the shear stress portion. For very soft plastics, such as PE, the support span can be increased or the support radius altered in order to reduce the amount of indention caused by the supports on the specimen. If only very short specimens are available, deflection can also be determined by strain gauges instead of measuring traverse path (Fig. 4.43).

When measuring middle deflection using a deflection sensor (Fig. 4.43a), the measurement result does not include indenting into the specimen by the bend anvil, i.e., Eqs. 4.125 to 4.131 are still valid. If fork sensors such as in Fig. 4.43b are used, indenting by the supports is also eliminated; however, different equations have to be used to calculate deflection f, peripheral fiber strain and modulus Ef :

G

2 G LL3

IE96

LF f (4.133)

deflection sensor anvil

F

traverse

v T

a

F

traverse

v T

fork sensor

anvil

b

support

support

f f

Fig. 4.43: Use of different strain measurement systems for determining deflection with a deflection

sensor (a) and a fork sensor (b)

144 4 Mechanical Properties of Polymers

G 2 G

f LL3L

Lhf12 (4.134)

3

G 2 G

f hbf8

LL3LF E (4.135)

If the specimens are very thick or very short, support indentation can become a problem for the measurement technique. To eliminate the problem, larger support radii are used, which, if deflection is considerable, can cause specimen roll-off, resulting in a shortening of the support span and thus influencing the test result. In addition, normal stresses often are generated by friction, bringing about a displacement of neutral fiber and, thereby, stress distribution. More extensive information on numerical correction for such measurement effects is provided by [4.61] among others. When bend tests are being performed, care must be taken, as with other basic mechanical tests, to position the specimens as precisely in the load line as possible; otherwise bending can be skewed, or torsional moment can influence the measurement results. For this reason, no flexible, self-adjusting supports should be used. The forces occurring in actual testing practice are not sufficient to maintain plane position of the specimen. If the test equipment has centering devices, such as arresting stoppers on the supports, these should always be utilized.

By analogy with the tensile or compression test on plastics, pure bending is overlaid by relaxation and creep processes in the quasi-static bend test as well, as exhibited by the strong dependence of test results on test speed and temperature. Strict adherence to test conditions is indispensable to obtain comparable results.

4.3.5.2 The Standardized Bend Test

ISO 178 is used for performing bend tests on plastics. The specimen of preference exhibits dimensions of 80 10 4 mm3 and can be produced directly by injection molding or by removing the shoulders from type 1A multipurpose specimens (Fig. 4.44a, Table 4.2). The latter procedure has the advantage of comparable orientation and residual stress state for evaluating material properties.

When specimens with different dimensions are used, Eq. 4.136 has to be fulfilled:

h120l (4.136)

With thermoplastic or thermosetting sheets or with textile and long-glass-fiber reinforced plastics, the specimen can be up to 80 mm wide and up to 50 mm thick; however, a correspondingly long specimen and wide supports have to be used. When testing anisotropic materials, such as laminates or laminated plastics, specimen types

4.3 Quasi-Static Test Methods 145

(Fig. 4.44b) should be selected such that the main loading direction is used for acquiring values. If the differences in the various directions are considerable, tests are required on specimens with differing orientations, whereby collation of the measurement results with the orientation direction must be assured. For all types of specimens, it is essential that they not exhibit any kinking, edge rounding, scratches or fissures. Specimens with very sharp cooling contraction and shrink marks cannot be used.

Test speed in the bend test is generally set to correspond with the particular product standard; ISO 178 permits test speeds from 1 mm min-1 to 500 mm min-1.

If there are no such specifications, according to the standard, the traverse speed vT is selected so that it is close to the normal flexural strain rate of d f /dt = 1 % min

-1. For preferred specimens with dimensions of 80 10 4 mm3 and a support span L = 64 mm, this yields a cross-head speed of 2 mm min-1 (Eq. 4.131). With materials with pronounced initial behavior, a preload Fv can be used; however, the resulting strains must not exceed 0.05 % with reference to measurement of the E modulus. By analogy with the tensile and compression test, it is recommended that the test apparatus for bend tests be set up with a system prestress of 90 to 95 % nominal load and with rigid set-up specimens to avoid setting motion during the test. Elasticity modulus Ef under flexural loading is acquired under the same conditions as in tensile

a

b

1 0

80

B

D

A C

hb

h

b

h

h

b

width direction of the product

length direction

of the product

(processing direction)

Fig. 4.44: Specimens for the bend test according to ISO 178

146 4 Mechanical Properties of Polymers

and compression tests, i.e., as a secant modulus (Fig. 4.27a) in a range of 0.05 to 0.25 % peripheral fiber strain (Eq. 4.136). The same test speed is used as in the bend test itself.

002.0 E

1f2f

1f2f f (4.137)

Figure 4.45 shows typical flexural stress–peripheral fiber strain diagrams of various polymers. Diagram (a) in Fig. 4.45 shows the brittle fracture behavior of materials such as PS or PMMA.

a

b

c

s (mm)

� (M

P a )

�x

�fM = �fB

�fM

�f (%)�x �fB

�fB

�fC

f

�fM �fB

sB sM sB sC Fig. 4.45: Typical flexural stress–peripheral fiber strain diagrams of polymers in the bend test

When the material behavior is ductile (curve (b) in Fig. 4.45), the force reaches a peak and fracture occurs prior to reaching so-called conventional deflection sC. In this case, flexural strength can be measured at maximum load. If neither a peak in force, nor fracture occurs prior to conventional deflection, flexural stress at conventional deflection fC is determined at this position; this material behavior is still considered ductile. According to Fig. 4.45, the material parameters explained in the following can be determined for various plastic materials.

Flexural strength fM : the maximum flexural stress tolerated by the specimen during the experiment. In the case of material behavior as shown in diagram (a) in Fig. 4.45, this value is identical to fB.

2

max fM

hb2

LF3 (4.138)

4.3 Quasi-Static Test Methods 147

Flexural stress at break fB : this value is determined if specimen break occurs during the experiment. Of course, it depends to a high degree on the conditions set for the break detector threshold on the universal testing machine.

2

B fB

hb2

LF3 (4.139)

Flexural stress at conventional deflection fC : this parameter is determined if the specimen fails to break, or if no peak force occurs. In this case, the flexural stress is calculated with the recorded conventional deflection sC = 1.5 h. Given a support span of L = 16 h, this deflection value corresponds to a peripheral fiber strain f of 3.5 %.

2

C fC

hb2

LF3 (4.140)

Flexural strain at flexural strength fM : the peripheral fiber strain determined at the position of flexural strength. This value can be stated dimensionless or as a percentile. The value can be identical with flexural strain at break if the material behavior corresponds to curve (a) in Fig. 4.45.

2

M fM

L

hf6 (4.141)

Flexural strain at break fB : the normal flexural strain reached at specimen break.

2

B fB

L

hf6 (4.142)

ISO 178 does not comply with these rules. Here, deflection is not represented by f, but by s. As stated in Section 4.3.5.1, the equations presented here are valid only for deflections that are relatively small compared to geometrical dimensions.

Corresponding to the standard, however, values as high as 6 mm for maximum possible deflection are distinctly higher. Thus under relatively large normal flexural strains and in the presence of inhomogeneities, off-center breaks can occur that corrupt the bend test result.

Figure 4.46 shows flexural stress–peripheral fiber strain diagrams of PP as a function of GF content. Flexural strength increases with fiber content, while tensile strain at break decreases. At the same time, it becomes clear that flexural strength can only be determined for high fiber contents (40 and 50 wt.-% in Fig. 4.46). In PP materials with low fiber content, flexural stress at conventional deflection is reached at 3.5 % peripheral fiber strain or 6 mm deflection, without a peak maximum or specimen

148 4 Mechanical Properties of Polymers

� (M

P a

)

� f (%)

f

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0

20

40

60

80

100

120

140

160

180

200

0 wt.-%

10 wt.-%

20 wt.-%

30 wt.-%

50 wt.-%

PP/GF

-%40 wt.

Fig. 4.46: Flexural stress–peripheral fiber strain diagrams of PP/GF composites as functions of GF

content

break occurring. The flexural stress recorded for these materials at conventional deflection fC is not comparable with flexural strength fM. For this reason, the bend test mainly serves for quality control and the determination of material values for simple dimensioning tasks. The use of stress x at x % of strain, which is not explicitly

Table 4.5: Selected characteristic values obtained from bend tests

Material fM

(MPa) fC

(MPa) Material fM

(MPa) fC

(MPa)

Thermoplastics unreinforced Thermoplastics unreinforced

PE-HD 35 SAN 135

PE-LD 10 PBT 85 70

PS 100 Thermoplastics reinforced

PA 6 50 PP + 20 wt.-% GF 90

PC 70 PET + 30 wt.-% GF 220

PMMA 110 PBT + 30 wt.-% GF 210

PVC-U 100 Thermosets

PVC-P 65 Phenole resin 70

PP 35 Urea resin 70

ABS 55 UP resin 60

POM 120 PUR 110

4.4 Impact Loading 149

described in the standard, enables meaningful comparison of these materials. The expected tendency for strength to increase with increasing fiber content ( in Fig. 4.46) and the reduction in peripheral fiber strain ( in Fig. 4.46) is reflected correctly with these parameters. Table 4.5 lists values from bend tests for selected plastics.

4.4 Impact Loading

4.4.1 Introduction

When products made from polymers are utilized in industrial practice, impact loading often occurs in addition to static loading. Examples of this include

• Demolding, • Traffic accidents (crash), • Underground assembly and laying of pipes, • Hail impact on plastic rooves and window profiles, • Stone impact on frontal surfaces of automotives and railroad rolling stock, • Loading on plastic safety films and security glazing, and • Accidents involving two-wheel vehicles (bicycle and motorcycle helmets).

Impact loading results in increased strain rate, significantly altering the strength and break behavior of most plastics. Besides increased strain rate, factors contributing to brittle fracture include low temperatures and multiaxial stress states including residual stresses. Stress concentrations at notches contribute especially to the formation of brittle fracture, so that tests are often performed on notched specimens.

The tests mostly used for evaluating the toughness of plastics under impact loading are the Charpy or notched Charpy impact test, the uniaxial impact or notched impact test, and the biaxial free-falling dart test owing to their relatively simple applicability. On specimens with rectangular cross-sections, plastic sheets and films, conventional toughness values are acquired that generally decrease with increasing strain rate, i.e., the occurrence of macroscopic brittle fracture phenomena is promoted.

The Charpy impact test in various arrangements has gained the greatest importance in the quality control of plastics because of its methodological simplicity, short testing time and relatively low consumption of materials. Although its use in quality assurance is undisputed, its applicability is limited in the area of material development and optimization (cf. Chapter 5).

When thermoplastics are tested, injection molded specimens are preferred because of their simple production technology (cf. Chapter 2). At the same time, internal states

150 4 Mechanical Properties of Polymers

related to manufacturing process, i.e., varying orientation states of macromolecules, residual stresses and developing morphology have to be taken into consideration. Orientation influence can be reduced by using different manufacturing processes, such as sheet molding.

4.4.2 Charpy Impact Test and Charpy Notched Impact Test

Three different configuration can be distinguished for impact loading with pendulum impact testers. The specimen either lies with its notched side centrally between two supports (Charpy configuration) or it is firmly clamped on one side (Izod configuration) (Fig. 4.47).

CHARPY arrangement IZOD arrangement

support span

support

F

impact direction

F impact direction

specimen

specimen

support

Fig. 4.47: Impact loading in Charpy and Izod configurations

When testing relatively small specimens, the Dynstat configuration is preferred; here, an unnotched specimen is held on one side over its entire width between two supports.

The Charpy impact test is performed on notched and unnotched specimens with three-point support and serves to evaluate the toughness behavior of plastics under impact loading. It is standardized in ISO 179. Prismatic specimens have to be produced according to the corresponding molding material standard. The specimens can be produced directly by injection molding or by cutting from pressed or cast sheets (Table 4.6). The type 1 specimens, mainly used for thermoplastics, can be taken from type 1A multipurpose specimens according to ISO 3167. Type 2 and 3

Table 4.6: Specimen types, dimensions and support spans for impact testing according to ISO 179

Length l

(mm)

Width b

(mm)

Thickness h

(mm)

Support span L

(mm)

Type 1 80 ± 2 10.0 ± 0.2 4.0 ± 0.2 62

Type 2 25 h 10 or 15 3 20 h

Type 3 (11 or 13) h 10 or 5 3 (6 or 8) h

4.4 Impact Loading 151

specimens are used for composite materials exhibiting interlaminar shear fracture, e.g., long-fiber reinforced polymers.

In the notched Charpy impact test, a notch is cut into the specimen. By notching, a stress concentration as well as an increase in crack propagation rate is achieved at the front of the crack tip. In this way, a break can be achieved even on tough plastics that do not break when unnotched specimens are used. When the notch is cut, care must be taken that the margin area is cut on the tensile loaded side of the specimen.

The test standard distinguishes between a configuration in which the direction of impact is parallel to dimension b with impact on the narrow longitudinal surface h l of the specimen (edgewise) and one in which the direction of impact is parallel to dimension h with impact on the broad longitudinal surface b l (flatwise). For the most commonly used Type 1 specimen, the test is performed unnotched on the broad surface especially when surface effects are to be investigated. The test method of preference is designated as ISO 179/1eA. Here, notch shape A is used with notch base radius rN = 0.25 0.05 mm and remaining width bN at notch base of the specimen of 8.0 0.2 mm. There are additional notch types B with rN = 1.00 0.05 mm and C with rN = 0.10 0.02 mm.

For the test, pendulum hammers according to ISO 13802 are used with nominal impact energies of 0.5 J to 50 J and impact velocities of 2.9 m s-1 and 3.8 m s-1, respectively in Charpy configuration, and 3.5 m s-1 in Izod configuration.

When the test is performed, the energy W absorbed by the specimen is calculated from the difference between the pendulum hammer height over specimen before and after impact and the mass mp of the pendulum hammer.

)cos(coslgm)hh(gmWWW 2121 (4.143) W1 pendulum hammer energy before specimen breaking W2 pendulum hammer energy after specimen breaking h1 height of pendulum hammer before impact h2 height of pendulum hammer after impact l distance between the axis of rotation of the pendulum and the center of gravity of the pendulum g local acceleration due to gravity (g = 9.81 m s-2)

starting angle angle of rise

To determine the Charpy impact strength of an unnotched specimen acU, the energy Wc absorbed by breaking the specimen is related to the initial cross-section area of the specimen:

hb

W a ccU (4.144)

152 4 Mechanical Properties of Polymers

In order to determine Charpy impact strength, the surface quality of the specimens must fulfill special requirements.

Besides the Charpy impact strength of a polymer, its notched Charpy impact strength has special technical relevance, since notches often develop in component parts. These can be surface flaws and sharp-edged cross-sectional borders, such as fins, edges and cut-outs.

To determine notched Charpy impact strength, the notched specimen is positioned centrally on the supports and with the notch on its tension surface. Thus, impact occurs on the side opposite the notch. Notched Charpy impact strength acN ist calculated from the absorbed energy Wc, related to the smallest initial cross-section of the specimen at notch base:

hb

W a

N

c cN (4.145)

bN remaining specimen width at notch base

The difference between Charpy impact strength acU and notched Charpy impact strength acN indicates how sensitive a plastic is to external notches, i.e., takes the problematic of notch effect for the Charpy impact test into consideration and indicates how effective fillers are. Thus, notch sensitivity can be calculated from the quotients of acN and acU:

%100 a

a k

cU

cN z (4.146)

Notched Charpy impact strength is decisively influenced by the notch radius selected and the notching procedure (sawing, milling, planing). Besides such machining methods, the machining parameters, such as feed or cutting speed, affect the measurement of notched Charpy impact strength. The magnitude of difference depends on the particular material, although generally speaking, notched Charpy impact strength increases with increasing notch radius (Fig. 4.48). Differences between the notched Charpy impact strengths measured are most pronounced in the notch radius range dictated by the ISO test standard of 0.1 mm to 1 mm. Figure 4.48a shows that PVC, Nylon (PA) and POM exhibit a qualitatively similar behavior in their dependence on notch radius, whereby the notch radius influence is strongest for PVC. ABS and PMMA show less dependence at a different level of toughness. Within the notch radii range considered, ABS turns out to be the most notch-insensitive material. In Fig. 4.48 the investigated notch radius range was extended downward to notch radii of 2.5 μm by making razor blade notches. At notch radii < 0.1 mm, polyamide 66, impact modified polyamide 66 and PC exhibit brittle behavior, as

4.4 Impact Loading 153

characterized by small Izod notched impact strength values aiN that are independent of notch radius. For high-impact polyamide 66, by contrast, only small influence on Izod notched impact strength was observed for notch sharpnesses from 2.5 μm to 0.25 mm. Due to their different loadings, Charpy and Izod notched impact strengths are not quantitatively comparable. However, similar qualitative tendencies result from their dependence on notch radius. Of the materials mentioned here, ABS and high- impact polyamide 66 prove to be notch insensitive over the entire notch radius range. Loading temperature and velocity are additional parameters that have to be considered.

Measuring the temperature dependence of toughness with the goal of stating material specific brittle-to-tough transition temperatures is meaningful only where sharp notches are involved, as can be seen from Fig. 4.48b.

� (mm)

PVC Nylon

POM

ABS

PMMA

40

30

20

10

0 10 10

0

a

1

a

(

k J m

) c N

-2

A

D B

C

1.6

1.2

0.8

0.4

0

10 10 10 -2 -1

b

� (mm)

0

a

(k

J m

) iN

-2

Fig. 4.48: Dependence of notched Charpy impact strength acN on notch radius for selected plastics (a) and of notched Izod impact strength aiN on for polyamide (nylon) materials (A - PA 66 high impact, B - PA 66 impact, C - PA 66) and PC, curve D (b) [4.62, 4.63]

Analogous statements can be made regarding the dependence of crack toughness on the rate of change of load parameters (cross-head speed of testing machine, impact speed of pendulum or falling hammer or the impact speed of a projectile in an arrest experiment). Thus it is clear that critical notch radii can only be determined by investigatiing the dependence of fracture mechanic values on notch radius.

Table 4.7 lists typical values for Charpy impact strength and notched Charpy impact strength of selected polymers.

154 4 Mechanical Properties of Polymers

Table 4.7: Charpy impact and notched Charpy impact strengths according to ISO 179 (data taken from CAMPUS [1.51] and FORMAT [1.54]; N = without fracture)

Thermoplastics unreinforced

acU

(kJ m-2)

acN

(kJ m-2) Thermoplastics

reinforced acU

(kJ m-2)

acN

(kJ m-2)

PE-HD N 4.9 PP + GF 45 15

PE-LD N N PA 6 + 30 wt.-% GF 85 19

PP 100 10 PA + CF 70 15

PS 21.5 2.8 PP + 20 wt.-% talc 40 3.5

SAN 19 2.5 PP + 20 wt.-% chalk 40 3.5

ABS 120 20 PVC + chalk 9

PC N 18 Thermosets unreinforced

PMMA 25 2.9 PF resin 8.5 2.9

PVC-U 80 3.2 UF resin 6.3 1.3

PVC-P N 50 MF resin 4.3 1.8

PA N 50 UP resin 11 3

POM N 12 EP resin 22 1.5

PET N 3.9

PTFE N 16

The disadvantage of impact energy measured by the notched Charpy impact test lies in the fact that, due to the relation

cff

0f c dfFE (4.147)

f deflection fc deflection at break F load (force)

it consists of both a strength and a deformation component. Thus, the same impact energy is obtained from very different load and deflection values. For this reason, it is impossible to utilize impact energy as a dimensioning parameter for impact loaded components. The meaningfulness of the notched Charpy impact test can be significantly increased by the electronic recording of load–deflection diagrams or load–time diagrams in the instrumented notched Charpy impact test (cf. Section 5.4.2 and [1.38]). Such curves are presented in Fig. 4.49 to illustrate the fundamental problem using recorded F–f diagrams of two polyolefine materials with comparable notched Charpy impact strengths.

4.4 Impact Loading 155

F (N

)

500

400

300

200

100

0

0 0.5 1.0 1.5 2.0 2.5

acN

a cN

f (mm)

a a cN cN

~~

1

1

2

2

Fig. 4.49: Comparison of load–deflection diagrams for PP/GF and PB-1/GF composites with different material behaviors but comparable notched Charpy impact strengths

Although no conclusions can be drawn regarding the behavior of component parts under impact loading from the results of impact and notched Charpy impact tests, standardized notched Charpy impact tests continue to be routinely used on specially produced specimens in production monitoring and in developmental laboratories, also to evaluate temperature dependence. The relevance of Charpy impact and notched Charpy impact tests ends at the point where, despite notching, the specimens no longer break because of their insufficient stiffness. In such cases, the energy absorbed by the specimen for deformation is just a measure of its flexural stiffness. In order to obtain material values even from these materials, the tensile-impact test can be used.

4.4.3 Tensile-Impact and Notched Tensile-Impact Tests

The conventional tensile-impact test according to ISO 8256 is a uniaxial tensile test with a relatively high strain rate which is used for polymers that, according to ISO 179 or ISO 180, are too flexible or too thin for Charpy impact tests.

For this test, commercial pendulum impact testers with a multiple-member arm for holding the head are used (Fig. 4.50). The standard specifies two procedures for determining the energy required to break plastic specimens under uniaxial impact loading. In procedure A, the specimen lies horizontally and is held at one end in a

156 4 Mechanical Properties of Polymers

Fig. 4.50: Tensile-impact test device

specimen clamp and at the other end in a cross-head. The cross-head lies loosely on the specimen clamp support. The bifurcate arm impacts the cross-head at the lowest point of pendulum motion. In procedure B, the cross-head moves together with the striker.

For procedure A, specimen strips are used that are notched on both sides and have a length of 80 mm, a width of 10 mm and a thickness of up to 4 mm, or unnotched dumbbell specimens with the same length and a shoulder width of 15 mm and a parallel midsection 10 mm long and 10 mm wide. The notches are either molded in or formed mechanically. Radius at notch base has to be 1.0 0.02 mm, notch angle 45 1°; notch depth is always 2 mm. Narrow tolerances must be maintained for notch profile and radius for most polymers, since these factors decisively determine the stress concentration at the notch base (cf. Fig. 4.48). In addition, it has to be taken into consideration that specimens with molded-in notches tend to provide different results than specimens with mechanically made notches. Special unnotched dumbbell specimens are used in procedure B. The specimens are produced either by injection molding or mechanically from components and semi-finished products, such as extruded parts, foils and laminates, extruded and cast sheets. These procedures are suitable both for production monitoring as well as for quality assurance.

Tensile-impact strength E or notched tensile-impact strength En (Eq. 4.148) is measured. If the cross-head is thrown off (procedure A) or bounces off (procedure B), the test standard precisely specifies methodological corrections.

dx

E EoderE cn (4.148)

Ec corrected impact energy x width of the narrow parallel side-section of the specimen or distance between notches d total specimen thickness

4.4 Impact Loading 157

By analogy, permanent strain bl can be measured with the tensile test. To do so, the change in length lbl measured by fitting the pieces together after the test is related to the initial gauge length l0 :

%100 l

ll

0

0bl bl (4.149)

The test results obtained on specimens with different dimensions do not have to be identical. Just as in the notched Charpy impact test, mechanically cut and directly injection-molded specimens do not provide identical results due to their different internal material states. The experimental results from procedures A and B do not necessarily have to be comparable. Thus it is clear that tensile-impact strengths and notched tensile-impact strengths are not suitable data sources for calculating design components.

Figure 4.51 provides an example for the application of tensile-impact tests in the development of PE-HD/NBR blends [4.64]. Generally, compatibilizers are used in blends to improve phase distribution and/or interaction between the phases. This example illustrates the influence of the compatibilizers maleic acid hydride and carbolic acid, whereby only MAH leads to a significant improvement of tensile- impact strength in the PE-HD/NBR blend. This improvement in toughness is due to the fact that MAH positively influences the size, shape and particle distance of the NBR-phase, as well as the uniformity and especially the adhesion between the phases.

0 0.2 0.4 0.6 0.8

120

160

200

240

280

320

compatibilizer content (wt.-%)

E (

k J m

)

MAH

phenol

-2

Fig. 4.51: Tensile-impact strength of a PE-HD/NBR blend as a function of compatibilizer content for

the compatibilizers MAH (maleic acid hydride) and carbolic acid [4.64]

158 4 Mechanical Properties of Polymers

This shows the principle suitability of the tensile-impact test for material development, whereby the informational value of this test procedure, too, can be significantly increased by the electronic recording of load–time diagrams (cf. Section 5.5.2).

In addition to impact or notched impact testing with pendulum impact testers, further experimental methods for testing polymers include dynamic tensile tests with servo-hydraulic high-speed test machines, rotary impact testers and free-fall dart testers.

4.4.4 Free-falling Dart Test and Puncture Impact Test

Because of biaxial loading, conventional puncture impact test can be used to test materials under conditions similar to actual use, i.e., better simulating the often complex loading conditions of components than pure impact bending or tensile- impact tests. The procedure is standardized in ISO 6603-1. As test equipment, a device is used that provides for the impact of a guided striker perpendicular to the plane of the specimen (Fig. 4.52). The striker of preference has a polished hemispherical striking surface with a diameter of 20 0.2 mm and can be provided with additional weights. Round or square specimens are used with an edge length or diameter of 60 2 mm, a thickness of 2 0.1 mm using a clamping ring of 40 mm in diameter.

So-called 50 % impact-failure energy E50 is a material parameter measured using a multi-specimen technique. By varying the height of fall or striker mass, a step-by-step

guiding device for the drop weight

arrest and trigger device

drop weight

striker

support

frame

base plate

specimen

clamp

D2 D3 D4

R

H H

Fig. 4.52: Test arrangement for the puncture impact test

4.4 Impact Loading 159

Fig. 4.53: Views of specimen surfaces damaged by penetration (a), breaks (b) and brittle fracture (c)

variation of the energy exerted takes place in the following manner: if damage occurs at the exerted energy selected, the energy exerted on the next specimen is reduced by an amount to be determined in preliminary tests; if no damage occurs, the energy is increased by the corresponding amount.

Depending on the material and test procedure, characteristic damage features occuring are penetration, initial cracks, breaks and brittle fracture (Fig. 4.53). Based on the damage characteristic, behavior in actual use can be estimated for defined impact loads.

When testing sheets and particularly films , electronic recording of the relationships between force and time or deformation can significantly enhance the informational value.

Figure 4.54 illustrates this with a plotted graph obtained from an instrumented puncture impact test on a PE-LD film. Besides the total penetration energy measured by conventional puncture testing, which corresponds to the surface beneath the curve, additional parameters can be determined as they arise due to initial cracking, including peak force Fp or time tp at maximum load and energy to peak load (force).

The instrumented free-falling dart test can also be used as a technical test method to evaluate the behavior of welded pipes under impact loading (Fig. 4.55 left side). This type of investigation requires the use of a special, V-shaped support adjustable to the pipe diameter. With this type of test equipment, the weld seam can be tested under impact loading at various points.

160 4 Mechanical Properties of Polymers

t (ms)

3.6

3.8

4.0

4.2

F (N

)

H

test speed

energy

load deformation

100

80

60

40

20

0

30

25

20

15

10

5

0

1.0

0.8

0.6

0.4

0.2

0

0 1 2 3 4 5 6 7

4.4

E (

J )

s (

m m

)

v

( m

s )

-1

Fig. 4.54: Characteristic force (F)–path (s)–energy (E)–time (t) diagram of a PE-LD film in the instru- mented biaxial puncture impact test

For example, a compound pipe system consisting of an inner and an outer pipe extruded from PE or PE-X and an aluminum core with a wall thickness of approx. 1 mm, we can illustrate the influence of various welding methods (single V-groove weld using TIG and overlapping USW). The different load–deformation behavior for both types of welding can be seen from falling dart tests at position = 0°.

Upon impact by the falling dart on the pipe, force increases and the pipe becomes dented. As loading increases, a second increase in force (curve (a) in Fig. 4.55, right side) occurs due to the limiting effect of the support. Microscopic evaluation of the cut surfaces revealed clearly:

pipe specimen

support

strain

drop weight

F

� = 0°� = 45°� = 90°

test arrangement

0 10 20 30

0

200

400

600

800

F (N

)

t (ms)

welded joints

� = 0°

� = 0°

a

b

gauge

Fig. 4.55: Testing weld seams with the instrumented free-falling dart test

4.5 Fatigue Behavior 161

• for V-seams: mainly failure due to tangential fissures in the root area • for overlapped welds: radial fissures and delaminations

(Fig. 4.55 right side, curve (b)).

4.5 Fatigue Behavior

4.5.1 Fundamentals

In addition to static long-term loading, plastic components are often subject to dynamic loading in practice. Such dynamic loading can lead to component failure at markedly lower stress or deformations than under static load conditions.

When material-dependent limits are exceeded, damage phenomena arise in the region of linear-viscoelastic material behavior, leading to fatigue. This type of material failure can also be observed with fiber-reinforced polymers (FRP) and has to be taken into consideration for their application, since it is the inclusion of reinforcing fibers in composite materials that makes them suitable for load-bearing structural parts.

Fatigue behavior in polymeric materials is essentially determined by the specific behavior of the polymeric structure.

At the start of a periodically alternating, cyclic load sequence there is a deviation from linear-elastic behavior, and a hysteresis loop develops due to the phase shift between forced vibration and deformation. Acting forces and forced deformations occur in staggered sequence, whereby additional energy must be exerted for elastic recovery. As loading increases, the deformation energy absorbed by the material changes, the hysteresis area (loss energy) grows, and an increase in temperature takes place in the polymer material. This increase in temperature is caused in particular by the structurally determined low heat conductivity of plastics – two to three decimal powers lower than that of metals. This heating phenomenon depends on the frequency of cyclic loading, so that early failure can occur due to increased teperature and/or mechanical damage.

The continuous vibration test provides the basis for determining fatigue behavior of plastics [1.38]. For testing polymers, we assume, by and large, the concept definitions and specifications according to DIN 50100, the test standard for continuous vibration tests on metallic materials (Fig. 4.56).

162 4 Mechanical Properties of Polymers

�

t

� 0

1 stress cycle

� m

� a

� a

� u

Fig. 4.56: Stress–time diagram under cyclic loading taking the pulsating tensile area as an example ( o maximum stress, u minimum stress, m mean stress, a alternating stress amplitude)

In the continuous vibration test we distinguish between stress-controlled loading, in which a constant alternating stress amplitude a is overlaid by a constant mean stress

m, and strain-controlled loading, in which a constant strain amplitude a is overlaid by a constant mean strain m. Depending upon the load on the material to be tested, this test can be performed in three loading ranges in a total of seven cases of loading (Fig. 4.57). Either mean stress and stress amplitude, or the maximum and minimum stress are predefined as loading values, depending on the test procedure. The parameter stated in the stress-controlled continuous vibration test is the stress ratio R = u / o . Here we distinguish between the

• Range for pulsating compressive stresses: o and u are negative, m a ; 0 R < +1

• Range for alternating stresses: o and u have contrary signs and m < a; 0 > R -1

• Range for pulsating tensile stresses: o and u are positive. m a; 0 R < +1

If a constant mean stress is assumed, the object of the test is to determine endurance strength or fatigue strength D. Fatigue strength D characterizes the largest stress amplitude a that a specimen can sustain indefinitely without unacceptable deformations. Specimen break occurs at all stress amplitudes above D. For a practical determination of D, the Wöhler test can be performed, reflecting the dependence between the magnitude of loading and the established number of cycles until fracture. For plastics, the Wöhler test is performed at stress cycles reaching N 107.

4.5 Fatigue Behavior 163

� >

� m

a

+ te

n si

o n

range for pulsating range for pulsating

1 2 3 4 5 6 7

co m

p re

ss io

n

� =

� m

a

� <

� m

a

� <

� m

a

� =

� m

a

� >

� m

a

� =

0 m

range for pulsating stresses compressive stresses tensile stresses

Fig. 4.57: Loading states during the continuous vibration test

4.5.2 Experimental Determination of Fatigue Behavior

Test specifications and standards are required for determining service life curves for the fatigue behavior of polymers. An overview and evaluation of the current situation has been undertaken by Oberbach [4.65] with emphasis on thermoplastic materials and by Ehrenstein [4.66] for FRP systems. At this time, compulsory standards exist only for a a limited number of special cases.

Wöhler curves (S–N curves) are determined by single-stage flexural fatigue testing, i.e., using load cycles with constant amplitude a and constant mean stress values

m or constant stress ratio s. For several years, DIN 53442 has provided a test standard

f

measurement

rotation axis

specimen directing

drive motion link

zero position

eccentric hub

eccentric drive

supporting bracket of rotating axis

load cell

motion link

spring

Fig. 4.58: Working principle of the flexural fatigue test according to DIN 53442

164 4 Mechanical Properties of Polymers

for performing flexural fatigue tests on flat specimens under flexural fatigue load. The testing principle is illustrated in Fig. 4.58.

A flat specimen is fastened to both the drive and measurement arms. The specimen is made to bend by the eccentric crank mechanism, whereby the rotation center of the measurement arm is fixed by the specimen and two springs. The test stress is set and the stiffness decrease during duration of load is recorded via a deformation measure- ment unit on the measurement arm (in the simple case: dial gauges). Waisted flat specimens with a thickness of 2 to 8 mm are used for the test. The reduction in cross- section at the specimen center defines the region of expected failure or break.

While the test is being performed, specimen surface temperature is monitored and recorded to check its self-heating. The number of cycles recorded until fracture is represented with S–N curves as functions of their dependence on the gradated initial stresses (Fig. 4.59). Instead of fracture failure, stress drop (generally 20 %; for FRC also 10 %) can also be defined as a damage criterion. The advantage of this test procedure lies in its simplicity and the modest equipment and manpower require- ments. Disadvantages include the limited controllability while the test is being performed, as well as the lack of clarity for defining and checking loading conditions (stress state). S–N curves obtained by this method mainly provide general material information regarding the service life of components [4.67]. Nonetheless, such mechanical pulsators will continue to be used for testing polymers. In this respect, it should be mentioned that changes have been made in the test arrangement according to ASTM D 671 for bending vibration loading on flat specimens with a constant deformation presetting [4.65].

Moreover, thermoplastics are fatigue tested on mechanical pulsators designed as rotary bend test machines with eccentric drive (DIN 50113) [4.65]. The advantages of this method include constant bending moment, as well as good regulation of the loading frequency, providing defined specimen load. A disadvantage lies in the fact that the equipment requires round specimens, otherwise uncommon in polymer testing.

The state-of-the-art for determining fatigue behavior of polymers and FRP is represented by the application of electrical servo-hydraulic test equipment. Practical testing advantages are provided by the defined control engineering of the testing machine or testing system (force, strain, path), variability of types of load (alternating load, pulsating load in the tensile and compression area), presetting of vibration modes (sinus, triangular, trapeze, random, etc.), as well as presetting of test frequencies and defined stress ratios s.

4.5 Fatigue Behavior 165

failure by fracture

S–N curve

temperature

damage line

10 10 10 104 5 6 7

N

160

140

120

100

80

60

40

20

0 10 10 10 10

4 5 6 7

N

80

60

40

20

0

T (°

C )

f = 11.2 Hz �a T

ba T

(° C

)

� (M

P a

) a

� (M

P a

) a

1

11

Fig. 4.59: Initial stress amplitude a1 (N = 1) as a function of the number of stress cycles N (Wöhler curve, S–N curve): plotted according to DIN 53442 (a) using PA as an example (b)

The essential components of such an electro-servo hydraulic (ESH) test system consist of a column test bed with test cylinder, load cell, strain measurement system and digital control system. These are illustrated in Fig. 4.60 (a), which shows a measuring station for flexural fatigue testing as an example. Figure 4.60 (b) illustrates a test arrangement for pulsating tensile testing.

Use of these methods is not limited to standardized specimen shapes; strip specimens, various shaped dumbbell specimens and even compact specimens can also be used (Fig. 4.61).

One test specification requiring the use of ESH technology is DIN standard 65586 for FRP aeronautical applications. This standard is aimed at the fatigue testing of oriented laminates (UD layers, prepregs and woven fabric laminates). Very thin strip specimens are used for testing with a preferential stress ratio of s = -1 or s = 0.1.

a specimen

load cell

strain-controlled

test device

clamp b

controller

Fig. 4.60: Electro-servo hydraulic (ESH) measuring stations for flexural fatigue test (a) and pulsating tensile test (b)

166 4 Mechanical Properties of Polymers

Fig. 4.61: Specimen shapes for fatigue tests

To keep specimens from buckling under compression load, they are guided on special low-friction buckling columns.

Not only the number of cycles until fracture has to be recorded, but specimen temperature (limiting temperatures of 50 °C or 40 °C, depending on resin system) has to be monitored, and stiffness decrease determined (preferably 20 %). Stiffness decrease is monitored by recording the stress–strain hysteresis periodically. Damage

10 10 10 10 10 10 100 1 2 3 4 5 6

1000 900 800

700

600

500

400

300

200

average curve

P = 90 % - curve c

N

s = -1

� (M

P a )

a

Fig. 4.62: S–N curve according to DIN 65586

4.5 Fatigue Behavior 167

� (M

P a )

p u l

PA/GF

P c10

P c90

P c90

P c10

CFK

260

210

160

110

60

10 10 10 10 10 10 10

2 3 4 5 6 7

N

s = 0.1

Fig. 4.63: Pulsating fatigue strength sch as a function of the number of stress cycles N ( sch = 2 a for

m = a)

progress in fiber-reinforced plastics can be monitored simultaneously using non- destructive testing (ultrasound, X-ray, thermography).

Figure 4.62 shows the graphic illustration provided in DIN 65586 of a mid-range S–N curve, as well as the lower confidence limit (Pc = 90 % curve). The variation can be calculated with a dual-parametric Weibull distribution. The tensile strength values m obtained are included in evaluation as fatigue strength values at N = 1. Once a preset number of stress cycles, e.g., N = 2 106, has been reached, it is recommended that remaining strength be measured on undistroyed specimens (so-called through- runners) as a further indicator value for progressing materials fatigue. S–N curves plotted experimentally on the basis of DIN 65586 are shown in Fig. 4.63 using a PA glass-fiber composite (PA/GF) and a carbon-fiber reinforced polymer (CFRP) as examples. The pulsating fatigue strength plotted represents a special case of fatigue strength for stress oscillating between zero and a maximum value, i.e., u = 0 and

m = a (cf. load case 2 in Fig. 4.57).

4.5.3 Planning and Evaluating Fatigue Tests

Theoretically, a S–N curve can be expressed as two linear curve segments (Fig. 4.64):

Segment 1: linear regressing low-cycle fatigue strength in the preferred presentation in log–log scale log – log N and semi-logarithmic plot – log N

Segment 2: fatigue strength as a stress value sustained for any given number of stress cycles without failure (type I) or connected to a second, often

168 4 Mechanical Properties of Polymers

tapering S–N curve (type II); for segment 2, K = (type I) and K = Kx (type II).

In the literature [1.38, 4.65, 4.66], the circumstance is well-known that the endurance strength parameter D cannot generally be determined for polymers and FCP, and that, therefore, low-cycle fatigue strength i is stated. Thus the determination of the fatigue behavior is superficially limited to determining, as precisely as possible, the curve of low-cycle fatigue strength as a function of the number of stress cycles. This relationship is expressed as:

k

1

D

i Di NN (4.150)

Ni number of stress cycles ND number of stress cycles at the slope break low-cycle fatigue strength / fatigue strength

i low-cycle fatigue strength k rise of low-cycle fatigue strength-no. of stress cycles curve

lo g

�

alternating fatigue strength

fatigue strength

N K K log N D

I

II

� D

x Fig. 4.64: Linearized S–N curve represented in log-log scale [4.68]

There are basically two approaches to planning the test:

1. In a procedure known as the “pearl string” method (Fig. 4.65a) the Wöhler curve is determined with Pc = 50 %, i.e., a median Wöhler line with 50 % survival probability. Single tests have to be performed at as many test horizons as possible, i.e., variously alternating stress amplitudes a . As the sample size increases ( 6 to 20), the precision of the Wöhler line being drawn also increases. The testing points of the individual test horizons can be projected onto a median test horizon in order to estimate scatter. This is also the basis for Wöhler curve evaluation according to DIN 65586 (cf. Fig. 4.62), stating a Pc = 90 % Wöhler line.

4.5 Fatigue Behavior 169

arctan k

N K log ND

arctan k

� D

T �

P = 90 %c

50 % 10 %

� a

TN i

a lo

g �

b

� D

� a

� a

iN K log ND i

a lo

g �

i lo

g �

1

2

3

Fig. 4.65: Evaluation of S–N curves (Wöhler curves) from fatigue test using the “pearl string” procedure (a) and for statements at > 50 % survival probability (b) (TN, TG Weibull parameters) [4.68]

2. If specific statements of Pc > 50 % survival probability are to be secured, 6 to 10 single tests have to be performed on three or four stress horizons (Fig. 4.65b). Mean values and scatter can be determined for each stress horizon, enabling the statistical calculation of tapering lines thus secured.

To optimize time, effort and expense, it is advisable, as early as in the planning phase of the investigation, to clearly define goals regarding the desired reliability of the material information to be gathered on fatigue behavior for the particular application

4.5.4 Factors Influencing the Fatigue Behavior and Service-Life Prediction of Service Life for Polymers

Due to the multiplicity and complexity of actual use requirements for polymers, as well as continuing knowledge deficits regarding their fatigue behavior, Oberbach et.al. [4.65] try to formulate evaluation criteria for the shape and position of S–N curves for polymers. This approach has been expanded by [4.66] to include fiber composite materials with polymer matrix (Fig. 4.66).

The shape of the S–N curve of any polymer is influenced by material-related aspects and load criteria. Whereas the influence of processing technology on materials has to be emphasized, load criteria are defined both by testing and application technology. When plastic parts are produced by injection molding, the direction of flow has to be considered, for example. Depending on the direction in which the specimen is removed, significant differences in flexural fatigue behavior result across and parallel to the direction of flow for a composite material such as PA 66 GF 30 (30 wt.-% glass- fiber) (Fig. 4.67).

When transposing test results to service cases, it must be remembered that loading in the compression, tensile and compression/tensile interaction range leads to different fatigue strengths. Figure 4.68 provides an overview over the influence of load types

170 4 Mechanical Properties of Polymers

material:

fiber reinforcing: CF, GF, AF

matrix material : thermoplastic resin, thermoset

reinforcement : UD, fabric, mat

fiber orientation, positioning

fiber content, filler content

material treatment: post-curing, conditioning

loading:

tensile, compression, bending, load ratio

loading type: sine, rectangle, triangle

frequency

environment: temperature, humidity, medium

S–N curve

following fracture

or failure criteria

thermal failure

fatigue stress failure

stress cycle number N

lo a d l e v e l

Fig. 4.66: Factors influencing fatigue behavior

using PA 66 as an example. Given the complexity of acting influences, as well as their potential interactions, it is impossible to predict the service life of either polymers or fiber-reinforced polymers under dynamic loading. Experimental data for various material groups [1.18, 1.22], further types of loading [1.18, 1.47] and modern fiber- reinforced polymers 4.69] are available to the designer as points of reference for the

� (M

P a )

a

100

90

80

70

60

50

40

30 10 10 10 10

3 4 5 6

perpendicular to

flow direction

N

flow direction

Fig. 4.67: Fatigue limit of PA 66 - GF 30 in the pulsating tensile range as a function of the direction of

specimen removal [1.18]

4.6 Long-Term Static Behavior 171

� (M

P a )

a

60

40

20

0 10 10 10 10 10

3 4 5 6 7

N

compressive cyclic loading

tensile cyclic loading

tension - compression loading

Fig. 4.68: Influence of load type on the fatigue limit of PA 66 [1.18]

approximating calculation of components under cyclic loading. Arbitrary reduction factors for the fatigue strength of polymers in the form of a reduced strength level of 30 % – 50 % of initial strength do not represent a practical solution in the sense of a techno-ecological shortcut. Such an approach regularly leads to overdesigning and a loss of innovative approach.

To generalize the overall situation, the following actions are recommended for future cost- and work-intensive research on the fatigue behavior of polymers:

• as comprehensive a characterization of the materials structure and it changes over the entire test period as possible

• collection and documentation of test conditions • description and qualification of damage behavior • as comprehensive information as possible in documentation of test results

(comparability, reproducibility) • creation of knowledge-based material information systems and material data

banks [4.70, 4.71].

4.6 Long-Term Static Behavior

4.6.1 Fundamentals

In order to design long-term loaded component parts and products from polymers for reliability, information is required as to their material behavior under long acting static loading. Long-term experiments can be performed under tensile, compression,

172 4 Mechanical Properties of Polymers

and flexural loading stress as functions of loading temperature and with environ- mental exposure (see Chapter 7). These investigation methods are especially important for polymers, since these materials clearly exhibit non-linear viscoelastic behavior even at room temperature.

The sudden application of a static load causes plastics to alter their shape, depending on their particular stiffness, at first in a linear-elastic way. At a constant loading level and as the loading time increases, the linear-elastic deformation component becomes overlaid by a second, time-dependent deformation component, i.e., viscoelastic deformation (creep deformation; Fig. 4.9b). Creep behavior (cold flow) qualitatively describes total time and stress-dependent deformation; quantitative characteristic functions of materials are determined by means of creep curves.

By analogy, a gradual, time-determined drop in stress occurs at a given constant deformation to which a particular stress value, called stress relaxation (Fig. 4.9a), is assigned. Thus, the static long-term behavior of plastic components is characterized by retardation and stress relaxation determined by molecular structure. Due to structure differences, e.g,. for thermoplastics with amorphous or semicrystalline structure and three-dimensional crosslinking in thermosets, there are significant differences in static long-term behavior among the individual material groups [4.65].

Material failure in the form of fracture can occur as the result of time-dependent creep activity. Long before creep-rupture failure, the durability or service life of many plastics components is limited, however, by the occurrence of excessive time- dependent creep deformations that can lead to unacceptable deviations in shape and dimensions and thus to the loss of component functionality.

To characterize the creep behavior of plastics, time-dependent material parameters determined on the basis of standardized test specifications (see List of Standards) are used in various calculation guidelines for dimensioning mechanically loaded plastic components.

The goal of creep tests is to establish a multi-parametric relationship between stress, strain and time, that can be presented in the form of a three-dimensional illustration (Fig. 4.69). The relationship = f ( 0, t) described as the objective function of the creep experiments forms a spatial, three-dimensional plane in the deformation– stress–time diagram [4.72, 4.73], illustrating the complex interaction of parameters under loading and measured parameters.

4.6 Long-Term Static Behavior 173

0� = const.

t = const.

� = const.

� = f (� ,t) 0

log t

� 0

�

Fig. 4.69: Stress–strain–time behavior in a creep experiment [4.72]

4.6.2 Tensile Creep Test

The creep behavior of plastics is determined experimentally by the tensile creep test under static uniaxial tensile loading according to ISO 899-1, which is used most often to determine the long-term mechanical behavior of plastics.

Generally, specimens of the type used in the tensile test according to ISO 527 are used, whereby dumbbell types 1A and 1B are recommended. These specimens correspond to the multipurpose specimens according to ISO 3167 that are mainly used for testing amorphous or semicrystalline thermoplastics. Especially with FCP, strip specimens with cap strips in the clamp zone are preferred.

The main components of creep-test equipment are a base with clamping devices for specimens, loading system and strain measuring arrangement (Fig. 4.70). Further information on test technology is provided by [4.74] and [4.75].

When performing creep experiments, special care must be taken that force is introduced free of impact in the loading phase, that strain on the specimen is measured continuously and without contact, and that ambient conditions are kept constant during the entire test period. The equipment measures the increase in time- dependent elongation:

174 4 Mechanical Properties of Polymers

specimen

clamping device

base frame

loading device

optical deformation

measurement sensor

mass

Fig. 4.70: Structural drawing of a creep testing station

0LtLtL (4.151)

from which creep strain (t) is determined:

100 L

L)t(L .resp100

L

)t(L )t(

0

0

0

(%) (4.152)

The time-dependent strain values acquired by the creep test under constant load (stress) are called creep curves, from which the relationships illustrated in Fig. 4.71 can be derived:

• Creep curves (creep–time diagram) = f (t) with 0 = const. ( 1, 2 ,...) provide the basis for deriving creep diagrams and isochronous – diagrams (Fig. 4.71a). Creep–time curves are linear when loading is in the linear-viscoelastic range.

• Isochronous stress–strain diagrams result from creep–time curves by adding perpendicular cuts at specified times (Fig. 4.71b). Each of these – curves corresponds to a particular loading period, e.g., 1, 102, 104 h.

4.6 Long-Term Static Behavior 175

t t t t1 2 3 4t t t t1 2 3 4

log t

�1

�2

�3

�4

�1

�2

� 3

�4

� (%

)

�1

�2

�3

�4

log t

t1

t1 < t 2< t 3< t4

� (%)

� 1

�2

� 3

� 4

log t

a b

dc

� (M

P a )

� (M

P a )

E (M

P a )

c

�1 < �2 < �3 < �4

�1 < �2 < �3 < �4

t2

t3

t4

�1 < �2 < �3 < �4

t t t t1 2 3 4

Fig. 4.71: Diagram of the functional relationship in the tensile creep test: creep curves (creep–time

diagrams) (a), isochronous stress–strain diagrams (b), creep diagram (time–stress curves) (c) and creep modulus curves (d)

• Creep diagrams = f (t) for = const. ( 1, 2 ,...) result from the creep–time field by adding horizontal cuts at specified strains (Fig. 4.71c) In extreme cases, the time–stress curve is the creep–rupture curve.

Creep modulus Ec (t) is introduced to describe time-dependent material behavior of plastics (Fig. 4.71d). It is derived as the quotient of applied stress in initial state 0 and time-dependent deformation (t):

)t( )t(E 0c (4.153)

Creep rate d /dt is a further parameter used to describe static long-term behavior; it is calculated from the quotients of deformation increase and time difference:

12

tt

ttt

12 (4.154)

176 4 Mechanical Properties of Polymers

In order to reach the stated goal of determining material values for design purposes, test standard ISO 899-1 recommends testing over “a wide range of stresses, times and ambient conditions”. Fulfillment of this directive has to be considered during the conception of cost and time intensive creep experiments to be performed. This can succeed if a large number of test stresses with single specimens is given preference over the performance of parallel tests at a single testing stress.

Creep tests have to be conceived in such a way that specimens normally survive a test period of at least 103 h without fracture. 30 to 50 % short-term tensile strength is recommended as a reference value, whereby at least 4, preferably 6 stress steps are to be defined below this stress level [4.76]. Such a procedure increases the informational value of creep experiments, if rigorous test evaluation is done simultaneously on the basis of the functional relationships illustrated in Fig. 4.71.

Various physical-mathematical models have been developed to describe creep behavior over the measured region. They are compiled in, e.g., [4.77] and [4.78]. Most often, Findley’s power law is applied

n 0 tmt (4.155)

m, n materials constant

which is based on a description of the time dependence of experimentally determined measurement data.

A different approach using four parameters is described in [4.77]

G

n G sinh

a

t 1

E t (4.156)

E, G materials parameters for the elastic deformation component A, n materials parameter t test period

that presumes the approximation of stress dependence of the isochronous stress– strain curves obtained in the experiment and then determines the time dependence of the parameters. Creep behavior values obtained in this manner concur in respect to dependencies with the original, experimentally obtained measurement data, in which the creep modulus values Ec(t) are also included. Figure 4.72 illustrates this procedure using as an example the evaluation of systematic creep experiments with PP. The extrapolation of creep data over application-relevant time periods of 10 years requires a basis of experimentally secured creep curves for measurement times of

104 h. However, such tensile-creep experiments generally assume room temperature or standard atmosphere and apply to selected plastics. Under standard atmospheric conditions, extrapolation times of up to twenty years are considered quite realistic,

4.6 Long-Term Static Behavior 177

1.0

0.8

0.6

0.4

0.2

0 10 10 10 10 10 10-1 0 1 2 3 4

0� (MPa)

2 4 5

6 8 10

t (h)

2

10 1

8 6

4

2

10 0

2 4 6 810 2 4 6 8 10 0 1

100

10 1

10 2

10 3

104

10 -1

b 5

4

3

2

1

0

10 MPa

8 MPa

6 MPa

2 MPa

4 MPa

5 MPa

10

8

6

4

2

0

3.0 %

2.5 %

2.0 %

1.5 %

1.0 %

0.5 %

� (%

)

t (h)

� (%)

a

dc

� (M

P a )

E (M

P a )

c 0

� (M

P a )

0

t (h)

t (h)

10 10 10 10 10 10-1 0 1 2 3 4

10 10 10 10 10 10-1 0 1 2 3 4

Fig. 4.72: Tensile creep behavior of PP under various loadings: creep curves (a),

isochronous stress–strain diagrams (b), creep diagram (c) and creep modulus curves (d)

assuming of course that material behavior is not significantly affected by aging. A far more critical question is how important the limits of extrapolation are for long-term mechanical behavior under simultaneous temperature and/or environmental loading due to structural material changes in cases of diffusion, hydrolysis, etc. (cf. Chapter 7). In such cases, the expected loading on a plastic component has to be secured by creep experiments under conditions as realistic as possible; for extrapolation, creep experiments over test periods t 2 103 h are regarded as mandatory (Fig. 4.73).

In order to quantify the potential influence of the ambient conditions temperature and environment, it is advisable, parallel to the creep experiments, to store mechanically non-loaded specimens (immersion test) and to test them with regard to mechanical and/or changes in physical-thermal properties at suitably large time intervals.

In order to apply the material data gained from creep experiments, normally a deformation range up to 5 % is considered adequate, in special cases 10 %. For many polymers, especially for semicrystalline thermoplastics, these deformation values lie far below tensile strain at break and fracture strength, so that creep tests do

178 4 Mechanical Properties of Polymers

10 10 10 10 10 10-1 0 1 2 3 4

b 14

12

8

6

4

0

t (h)

a

t (h)

10 10 10 10 10 10-1 0 1 2 3 4

� (%

)

0 � (MPa)

water 20 °C

23 20 18

15

12

9 6 3

tensile creep strength

wash lye 20 °C

12

9 6 3

15

18

19

20 21�

(% )

10

2

14

12

8

6

4

0

10

2

0 � (MPa)

Fig. 4.73: Creep curves of PP under environmental load in tap water (a) and detergent solution (b)

not require that rupture failure be achieved. Creep-rupture tests represent a special type of creep test for determining tensile creep strength. They should be conceived separately from creep experiments and performed preferably on plastics with low tensile strain at break, e.g., thermosetting mold materials. Deformation measurement is not required; stress-rupture time is the value to be measured.

Since tensile creep strengths are quite prone to error, they have to be determined according to statistical methods of properly designed experiments. Beginning with the assumption of short-term tensile strength, fracture times should be acquired at selected, high loading stresses and at least 3 stress horizons using up to ten specimens, thus securing the shape of the creep curve statistically (Fig. 4.74). Proof of creep strength according to the test principle of the tensile creep test is also used in other test specifications. Here, the proof of stress-cracking resistance under the influence of

� (M

P a

)

10 10 10 10 10 10 1 2 3 4 5 6

t (h)B

� < � < �1 2 3

�

�

�

3

2

1

Fig. 4.74: Determination of the creep-rupture curve (diagram) taking measuring error into

consideration

4.6 Long-Term Static Behavior 179

clamping jig

frame

elongation

specimen

load cell

measurement device

Fig. 4.75: Test arrangement for determining the relaxation behavior of plastics according to DIN 53 441

complex loading conditions (stress level, test temperature, ambient conditions), e.g., according to ISO 6252 (cf. Chapter 7) and by various long-term service-life tests of plastics joints according to DVS guidelines 2203 and 2226, has gained special engineering importance.

Compared to the retardation test under static uniaxial tensile load, the stress relaxation test (cf. Fig. 4.9a) has only secondary significance for actual applications, although the corresponding material parameters are useful for designing bonded joints, for example. According to DIN 53441, a tensile specimen is loaded with a defined deformation which is kept constant over the test period (Fig. 4.75). A time- dependent stress drop occurs due to viscoelastic material behavior of the polymer.

The relaxation modulus is a useful, essential design value, where:

0

r )t(

)t(E (4.157)

To illustrate the analogy between relaxation and creep modulus, Fig. 4.76 shows the dependence of the relaxation modulus Er on loading time using the example of PE- HD.

Since the differences between creep and relaxation behavior are small, creep modulus can be used for approximating calculations [4.78].

180 4 Mechanical Properties of Polymers

10 10 10 10 10 -1 0 1 2 3

1000

� = 1 %

600

400

200

100

t (h)

� = 2 %

� = 3 %

E (

M P

a )

r

Fig. 4.76: Relaxation modulus of PE-HD as a function of loading duration at T = 23 °C [1.18]

4.6.3 Flexural Creep Test

Creep behavior determination under three-point bend loading is standardized in ISO 899-2. It specifies specimens of the same shape and dimensions as are specified for determining flexural properties according to ISO 178. Under three-point bend loading, the peak in bending moment occurs at the point of application that was described in Chapter 4.3. By contrast to the short-beam test, the change in deflection with time fb (t) is introduced to calculate peripheral fiber strain f (t):

100 L

)t(fh6 )t(

2

b f (4.158)

The statements made in Section 4.6.2 regarding the tensile creep test apply in principle to the performance and evaluation of creep experiments under bending load. Figure 4.77 presents evaluated test results for PVC.

A large amount of practical experience and knowledge in connection with flexural creep behavior is based on the meanwhile withdrawn DIN 54852, which involved three- and four-point flexural loading. Chapter 4.3 covers the advantages and disadvantages of various flexural loads.

Creep fracture tests using three-point bend loading are preferably performed for polymers with low flexural strain at break. Experiments for determining flexural creep strength should be performed analogous to the tensile creep test (cf. Fig. 4.74). However, flexural creep strength curves are not frequently used.

4.6 Long-Term Static Behavior 181

ba

d 4.5

4.0

3.5

3.0

2.5

2.0 10 10 10 10 10 10

-1

t (h)

10 MPa

20 MPa

30 MPa 40 MPa

50 MPa

60 MPa

0 1 2 3 4

t (h)

10 1

80 60

40

20

10 2

10 2 4 6 8 10

�r (%)

100

10 1 10

2

10 3

104

10 -1

2 4

8 6

2.5

2.0

1.5

1.0

0.5

0

60 MPa

50 MPa 40 MPa

10 MPa

20 MPa

30 MPa

10 10 10 10 10 10-1

t (h)

0 1 2 3 4

60

48

36

24

12

0

1.50 %

0.25 %

1.00 %

1.25 %

0.75 %

0.50 %

c

10 10 10 10 10 10 -1

t (h)

0 1 2 3 4

� (%

)

� (M

P a )

E (G

P a )

c 0

� (M

P a )

0

-1 0

Fig. 4.77 Flexural creep behavior of PVC under various loads: creep curves (a), isochronous flexural

stress–peripheral fiber strain diagrams (b), creep diagram (c) and flexural creep modulus curves (d) [4.73]

4.6.4 Creep Compression Test

The creep compression test is used to investigate polymers under long-term compression loading, such as bearing materials, seals, building materials and thermal insulation materials. This test is standardized neither for structurally used thermoplastic and thermosetting polymers, nor for FCM in general, although creep compression tests can be performed quite simply using the equipment for and the analogy of tensile and flexural creep tests.

The specimen shapes specified by DIN EN 826 for the compression test are suitable in principle for determining compression deformation as a function of loading time (creep compression curves). The results from creep compression tests are evaluated and presented analogous to the tensile creep test. Figure 4.78 shows exemplary creep compression curves (a), isochronous compressive stress–compressive strain diagrams (b), creep diagrams (c) and creep compression modulus curves (d) of a PTFE often used as seal and bearing material.

182 4 Mechanical Properties of Polymers

ba

d 1.0

0.8

0.6

0.4

0.2

0 10 10 10 10 10 10

-1

t (h)

0 1 2 3 4

2

0

10 8

4

2

1

6 8 10 2 4 6 8 10

�r (%)

2

10

12.5

10.0

7.5

5.0

2.5

0 10 10 10 10 10 10-1

t (h)

0 1 2 3 4

12.5

10.0

7.5

5.0

2.5

0

c

10 10 10 10 10 10 -1

t (h)

0 1 2 3 4

� (%

)

� (M

P a

) E

(G

P a

) c

0

� (M

P a

) 0

0 1

10 MPa

8 MPa 6 MPa

5 MPa 4 MPa

2 MPa

t (h)

10 0

10 1 10 2

10 3

10 -1

5.0 % 4.0 %

3.0 %

2.0 %

1.0 %

0.5 %

�0 (MPa)

2 4 5

6 8 10

6

Fig. 4.78: Creep compression behavior of PTFE [4.73]

Regardless of the current standards situation, there are product standards for individual material groups that specify test equipment and procedures for determining creep compression behavior, e.g., for insulation materials for the construction industry. Evaluation of long-term creep behavior under compression load of insulation materials for the construction industry is performed according to DIN EN 1606. The principle is based on measurement of specimen deformation (compression strain) in a special test arrangement under constant compression stress and defined conditions of temperature, moisture and time. The various loading steps in the creep test have to be determined either from compression strength m or from compression stress at 10 % compression strain. Creep behavior should be measured at equidistant intervals (e.g., log) over a period of at least 90 days. Essentially, this procedure corresponds to the specifications in ISO 899. Test period is defined by the corresponding product standards. When an appropriate mathematical extrapolation method is used, values can be obtained that are reliable in the long term for up to several times as the test period (e.g., 10 years). In addition to the uniaxial compression test, quality assurance and product certification require parameters for plastics components that take time-dependent behavior under multiaxial loading into consideration. These include, e.g., the vertex compression test on pipe sections to

4.7 Hardness Test Methods 183

establish a minimal creep modulus for pressure pipes and the internal pressure creep rupture test as proof of durability for plastic pipes (cf. Section 11.3, Component Testing).

4.7 Hardness Test Methods

4.7.1 Principles of Hardness Testing

Hardness testing on polymers is based on test methods originally developed for metallic materials, especially for steels, and the material values determined thereby. In 1908, Martens defined the material property technical hardness as resistance against indentation by a harder body; it is a parameter for describing a material and its physical state, respectively. This equally simple and descriptive definition has become standard in spite of a certain fuzziness for practical application [1.38]. In the standardized hardness tests used most often today, a hard indenter is pressed into the surface of the specimens under investigation. A triaxial stress state is thereby formed within the specimen.

The hardness test is among the most often applied methods in mechanical material testing. That is because it can be performed comparatively simply, quickly and, from an equipment point of view, efficiently. Since slight damage to a component surface in the form of one or more relatively small indentations usually has little effect on its function, the hardness test is mentioned among the nearly non-destructive test methods. That makes it possible to test very small components and thin layers that can hardly provide information on the profiles of other properties. This approach is supported by the fact that there are statistically secured correlations between hardness and other mechanical properties, such as yield point or abrasion, at least within one group of materials.

The test methods, individually standardized for particular material groups and areas of application, differ fundamentally with respect to the shape of the indenter (e.g., ball, cone, pyramid), material (stainless steel, hard metal, diamond), load level and loading time, as well as their mode of application (under total test load, after unloading). Hardness values dependent on test methods and test conditions cannot be extrapolated from one to the other, or only to a limited extent. In industrial testing practice, however, a trend can be seen to a few universal procedures.

Hardness testing on plastics is performed taking material-specific behavior into con- sideration. The type of deformation under load can be observed on the indentations, ranging from rubber-elastic (elastomers), viscoelastic-plastic (thermoplastics, e.g.,

184 4 Mechanical Properties of Polymers

PE-LD) or predominantly plastic (thermosets, even thermoplastics at low temper- atures and e.g. ABS) deformations (Fig. 4.79). Therefore, the following influencing factors have to be noted:

• Nominal test temperature, • Load rise time,

material

behavior

related to

deformation

and time

indentations

after

unloading

mostly

plastic

viscoelastic -

plastic

rubber-

elastic

timet t 1

d e

fo rm

a ti o

n

2 t t 1 2

t t 1 2

Fig. 4.79: Relationship between material behavior and indentation shape

Table 4.8: Overview of conventional hardness test methods for plastics and rubber (see also guideline VDI/VDE 2616 Part 2)

Measurement under load Measurement after unloading

IIndentation depth measurment Hardness value from the indentation surface

Ball indentation hardness

ISO 2039-1 Vickers hardness

IRHD-hardness ISO 48 Buchholz indentation resistance ISO 2815

Rockwell hardness ISO 2039-2 Hardness value from depth of indentation

Shore hardness DIN 53505 Rockwell hardness ASTM D 785

Barcol hardness DIN EN 59 Hardness value from the ratio of test force to projected area of indentation surface

Determination of indentation diagonal

Vickers hardness under load

Special testing methods

Ultrasonic contact impedance (UCI) process

Knoop hardness

4.7 Hardness Test Methods 185

• Full load duration and • Prehistory of the material (processing and storage).

Moreover, the test result is influenced by orientation, residual stresses and morphology (supermolecular structure, fillers and reinforcements).

In principle, it is possible to measure indentation magnitude after unloading or under load (Table 4.8) the latter method being preferable for plastics and, when testing elastomers, unavoidable due to their rubber-elastic redeformation.

4.7.2 Conventional Hardness Testing Methods

4.7.2.1 Test Methods for Determining Hardness Values after Unloading

Vickers Hardness

The Vickers procedure known in metals testing can also be applied to plastics. A square-based pyramid with an angle of 136° between opposite surfaces serves as the indenter. Loading has to be adapted to the particular geometrical and morphological circumstances; testing is generally done at small forces 5 N. The length of the indentational diagonal is defined as measured and the mean indentation diagonal is calculated which is required for calculating Vickers hardness HV according to Eq. 4.159.

2d

F 8544.1

A

F HV (4.159)

HV Vickers hardness value in N mm-2 F test load (force) in N A area of indentation in mm2 d arithmetic mean of the two diagonals in mm

The diagonals are normally measured after unloading by light microscopy; however, they can also be measured under load [4.79]. In this case, the specimen surface and with it the indentation diagonals are observed through the diamond indenter; this allows to make statements on creep behavior virtually in real-time.

The Vickers procedure is not standardized for plastics, but has achieved special significance as a procedure for micro and low-load hardness testing (cf. Section 4.7.3).

186 4 Mechanical Properties of Polymers

Knoop Hardness

The Knoop procedure is similar in principle to the Vickers procedure, but exhibits two fundamental differences. For one thing, a strongly anisotropical rhombic-based pyramid with a diagonal ratio of 7.114 : 1 is used as the indenter; for another, Knoop hardness HK is calculated with the aid of the projected area of indentation, in contrast to Vickers hardness, in which indentation surface is used for calculation. HK is calculated using the length of the long indentation diagonal (Eq. 4.160).

2l

F 23.14

A

F HK (4.160)

HK Knoop hardness in N mm-2 F test load (force) in N A projected area of indentation surface in mm2 l length of the long indentation diagonal in mm

Since indentation depth is only approx. 1/30 of the long diagonal, this method is especially suited for testing very thin small assembly units and narrow, near-edge areas as well as plastic foils and coatings. It should be noted that the area tested has to be extremely flat to accomodate indenter geometry. Knoop hardness is especially suited for detecting material anisotropies by examining the directional dependence of the hardness values obtained. Figure 4.80 shows the influence of orientation on indentation geometry, illustrating the difference between Knoop and Vickers procedures. In oriented materials, Vickers indentation is no longer symmetric. The resulting long diagonal lies perpendicular, the short one parallel to the direction of orientation. This anisotropy develops after unloading, since the stresses under the indenter are greater in the direction of orientation than perpendicular to it, which

orientation

Vickers

Knoop

Fig. 4.80: Influence of orientation on indentation geometry in Vickers and Knoop hardness tests

4.7 Hardness Test Methods 187

is why this direction rebounds more strongly. Thus higher hardness values are obtained in the direction of orientation than perpendicular to it.

Other conditions are present in the unsymmetrical Knoop pyramid, since the strain field under the indenter is no longer symmetrical; instead, higher strains occur in the direction of the short main axis [4.80]. If the long main axis lies parallel to orientation, maximum strain lies perpendicular to the preferred orientation of macromolecules, leading to increased indentation, i.e., lower hardness values. If the long main axis is oriented perpendicular to the preferred orientation of the macromolecules, maximum strain occurs parallel to molecular orientation, resulting in reduced indentation. Due to these relationships, Knoop hardness reacts more sensitively to material anisotropies than Vickers hardness.

Rockwell Hardness Testing (scales R, L, M, E, K)

Following the wide-spread Rockwell hardness testing of metallic materials, balls with various diameters (scale R: 12.7 mm, scales L and M: 6.35 mm, scales E and K: 3.175 mm) are preloaded with F0. The indentation depth thus achieved serves as a reference level. Due to preloading, surface effects are reduced and defined conditions are obtained for the contact between indenter and specimen, or test indentation. Following a preload exposure time of 10 s, additional test load F1 is applied and, subsequent to a holding time of 15 s, removed. The remaining indentation depth h under effective preload is measured and Rockwell hardness HR can be determined according to the definition given in Eq. 4.161.

mm 002.0/h130HR (4.161) h indentation depth in mm

Rockwell tests using scales R, L, M, E and K cover a wide range of hardness, thereby registering only the remaining deformation component. Unfortunately, the results obtained using different scales lack comparability.

4.7.2.2 Test Methods for Determining Hardness Values under Load

Ball Indentation Hardness

This procedure uses a hardened steel ball 5 mm in diameter which, after being preloaded, is loaded with additional test loads of 49 N, 132 N, 358 N or 961 N (Fig. 4.81). The resulting indentation depth has to range from 0.15 to 0.35 mm in order to ensure a nearly linear relationship between indentation diameter and

188 4 Mechanical Properties of Polymers

1 2

3

4

specimen

support

steel ball

dial gauge

load step

frameF

h h (

t)

0

F0 F0 + F

D Fig. 4.81: Ball indentation procedure (D - ball diameter, F0 - preliminary test load (preload),

h0 - indentation depth after preloading, F - additional test load, h - indentation depth)

indentation depth, i.e., identical surface pressure. Ball compression hardness HB is determined following a holding time of 30 s:

0.04)(h

F HB (4.162)

prefactor: = 0.0535 mm-1 F total test load (force) in N h actual penetration in mm

The test result is presented in the form HB 132 / 30 = 20 N mm-2, in which the numerical values are stated in this order: additional test load in N, holding time in s and hardness value.

Ball indentation hardness is a measurement method under acting test load that includes elastic and plastic deformation components and is suited for testing inhomogenous and/or anisotropic materials due to its relatively large test indentations.

Table 4.9 lists ball indentation hardness values for various molding compounds.

Ball Indentation Hardness IRHD

This test method was specially developed for soft and medium hard rubber and very soft thermoplastics such as PVC-P. A fundamental distinction has to be made in ball indentation hardness IRHD between method N (normal test, method H (test for high hardness), method L (test for low hardness) and method M (microhardness). Essentially, theses methods differ with respect to the diameter of the indenting ball and load level, whereby the parameters selected have to suit the particular case of application. Ball diameters range from 2.5 mm in method N, 1 mm in method H,

4.7 Hardness Test Methods 189

5 mm in method L and 0.395 mm in method M. After preload has been applied, additional test load is applied to a ball-shaped indenter to create a reference plane: 5.4 N in methods N, H and L and 0.145 N in method M. After 30 s, the additional indentation depth is measured under total test load.

For the particular indentation depth achieved, the corresponding International Rubber Hardness Degree (IRHD) can be read from tables. The hardness scale is selected such that "0" corresponds to the hardness of a material with a Young's modulus of zero and "100" to the hardness of a materials with an infinitely large modulus.

- Rockwell Hardness

In contrast to the Rockwell hardness procedures described above, indentation depth is measured in this procedure under total test load (F0 + F1), thus registering elastic and plastic deformation components. It follows from the definition of -Rockwell hardness HR (Eq. 4.163) and the permissible range of indentation depth of up to 0.5 mm, that at indentation depths > 0.3 mm, negative hardness values are obtained that are permissible.

mm 002.0/h150HR (4.163) h depth of indentation measured under total test load in mm

Due to the 12.7 mm (1/2 in.) ball diameter used, expansive specimen areas are covered.

Shore Hardness

In this procedure, a truncated cone (Shore A) or a truncated cone with a spherical cap (Shore D), is forced into the specimen by a spring (Fig. 4.82). Indentation depth serves as a measure of hardness, whereby Shore hardness is defined as the difference

Shore A

a

Δ l

h

F F

Shore D

a

Δ l

h

Fig. 4.82: Shore A and Shore D hardness tests

190 4 Mechanical Properties of Polymers

Table 4.9: Hardness values of plastics according to VDI/VDE 2616 (> materials hardness is greater than measurable with this method; < materials hardness is smaller than measurable with this method)

Shore hardness H

(N mm-2) A D HR Barcol hardness

PS 145 to 195 > 80 100 to 110 20 to 30

PMMA 185 to 210 > 87 to 88 110 50 to 55

PC 115 to 135 > 82 to 85 95 to 100 10 to 20

PVC-U 95 to 145 > 75 to 80 75 to 95 to 10

ABS 95 to 120 > 75 to 80 85 to 95 < to 15

PE-LD 10 to 25 95 to > 40 to 50 < to -110 <

PE-HD 40 to 65 > 50 to 70 25 to 55 <

PP 40 to 80 > 65 to 75 30 to 70 <

POM 135 to 175 > 79 to 82 95 to 105 < to 15

PA 66 120 > 80 95 8

PA 610 90 > 78 80 <

PA 612 105 to 120 > 75 to 80 95 <

PA 66/GF 230 > 85 115 40

PP/GF 75 to 115 > 70 to 75 65 to 90 <

UP/GF 300 to 475 > > > 57 to 77

between the number 100 and indentation depth under total test load in mm divided by the scale value 0.025. Shore A is used for soft rubbers and very soft plastics such as plastized PVC; Shore D for hard rubber and thermoplastics such as PTFE (examples in Table 4.9). One advantage of the Shore procedure is its mobile application capability, since hand-held devices are often used.

Barcol Hardness

Barcol hardness is especially suited for testing fiber reinforced thermosets and hard thermoplastics. The Barcol test device is designed exclusively as a hand-held device for use both in laboratories and in any position, including overhead. Test load is applied to the indenter (truncated ball of hardened steel) by a spring. From the indentation depth under load registered by a dial gauge, Barcol hardness is calculated according to Eq. 4.164.

mm 0076.0/h100hardnessBarcol (4.164) h indentation depth measured under load in mm

4.7 Hardness Test Methods 191

Compared to the Shore D hardness test, this procedure has the advantage that it can test plastics of even greater hardness.

4.7.2.3 Special Testing Methods

In addition to the test methods of hardness measurement after unloading and under load, a number of special methods have been developed for engineering applications in which the hardness value is partially or completely determined using different physical parameters.

VDI/VDE guideline 2616 lists more than 30 special methods related to hardness tests on thin layers and coatings in the aerospace and automobile industries. These include pendulum methods for determining “pendulum hardness”, scratch methods for determining “scratch hardness”, as well as special indentation methods. The disadvantage of these methods lies in the fact that they provide no standardized hardness values.

Wide acceptance has been gained by scratch hardness test procedures which, by using a gouging or scratching principle, continuously record the indentation depth of a needle or hardened ball in translational motion either visually or via the measurement of force and indentation depth. Loading is either set or altered in steps. Among the test methods with subsequent visual evaluation are the scratch test with an Erichsen hardness test bar and the cross-cut test. Indentation depth is continuously recorded by special auxiliary equipment in material testing machines or external testing devices (scratch-indenter tester).

The UCI (Ultrasonic Contact Impedance) procedure is another special method for hardness testing. Here a piezoelectric transducer excites a staff-shaped resonator with a Vickers diamond to oscillate freely at a particular frequency. During indentation, the staff no longer oscillates freely, and the occurring resonance shift represents a quantity for the contact surface. The softer the material, the larger the indentation surface and the greater the frequency change. The E modulus and Poisson ratio of the material to be tested and the diamond must be known in order to measure a hardness value. The measured hardness value represents a reference value that has to be identified by its method of measurement whenever stated.

4.7.2.4 Comparability of Hardness Values

As shown by the examples mentioned, hardness measurement methods for plastics differ with respect both to the indenters, test loads, preloads and test times used, as well as to the indentation sizes (under load or after unloading). In order to perform

192 4 Mechanical Properties of Polymers

S h

o re

A

120

80

40

0 100 200 300 400

HB (Nmm ) -2

100

80

60

40

20 0 10 20 30 40 50

Shore D

ba

H R

�

Fig. 4.83: Diagram of the relationships between ball indentation hardness HB and Rockwell hardness HR (a) as well as Shore A and Shore D (b)

material comparisons, as well as to save time and expense, it is often necessary to convert the hardness values obtained by one particular method into another hardness scale. This is also the case when available data banks are used for selecting materials and designing components without performing tests on the material. Due to the viscoelastic behavior of plastic materials, two hardness values obtained with different methods can be converted to each other given the following conditions [4.81]:

• Both hardness values have to be determined either under load or after unloading. • The same indentation depth–load functions must apply to both indenters under

the given geometric dimensions. • Loading times must be approximately equal.

In place of identical indentation depth–load functions, similar indentation depth– surface functions can suffice for conversion. Empirical conversion to different hardness scales is possible on this basis.

Between ball indentation hardness and Rockwell hardness relate to each other by (cf. Fig. 4.83a) [4.81, 4.82]:

23.1)HR150(

18279 HB (4.165)

Shore A and Shore D relate to one another as follows (cf. Fig. 4.83b) [4.82, 4.83]:

12.2D Shore

1409 1.116A Shore (4.166)

4.7 Hardness Test Methods 193

4.7.3 Instrumented Hardness Test

4.7.3.1 Fundamentals of Measurement Methodology

To enhance the information gained from hardness measurements on plastics, it is necessary to record both the force required by the indenter to penetrate the specimen and the indentation depth over the entire indenting process [4.84]. For this purpose the indentation process is recorded and information on the viscoelastic-plastic behavior of the polymer is derived by evaluating loading and unloading curves. The testing cycle can be performed either load or indentation depth controlled, or at a constant indentation strain rate (dh/dt )/h. Various indenters are used: rectangular- based Vickers or Knoop pyramids, triangular-based Berkovich pyramids or so-called “cube corners”, conical tips or even specially rounded indenters.

In addition to the ease of automating the procedure, the advantage of instrumented hardness tests lies especially in the comparability of all materials within one hardness scale. Figure 4.84 illustrates the gradation of load ranges and the relationship between Martens hardness and indentation depth for various material groups.

Hardness values, indentation modulus, strain hardening exponents and viscoelastic properties can be measured with the instrumented indentation test. Also measurable are the fracture toughness of brittle materials as well as the influence of residual stress in solid material or thin layers, or the elastic behavior (spring constant) of miniaturized components. The presence of orientations can also be detected [4.85].

10 0

10 -1

10 -2

10 -3

10 -4

10 -5

10 -6

10 0

10 1

10 2

10 3

10 4

HM (MPa)

h (

m m

)

polymers

non-ferrous- metals steels

hard metals ceramics

rubber

2 N > F and h > 200 nm

h < 200 nm

2 N < F < 30 kN

macrohardness

microhardness

nanohardness

0.02 N

lo a d

F

Fig. 4.84: Definition of test load ranges for instrumented hardness tests

194 4 Mechanical Properties of Polymers

indenter

specimen

load cell adapter

measurement

traverse adapter

specimen

frame

load cell

indenter socket

support

distance

Fig. 4.85: Instrumented hardness measuring devices : for installing in a materials testing machine (left

hand) and a self-contained unit (right hand)

Expanding hardness testing into the area of smallest test loads and indentation depths (h < 200 nm), the so-called nano region, provides experimental access to structural elements and their interfaces with the aim of creating quantitative morphology– hardness correlations. Section 12.3 treats the experimental possibilities for demonstrating interface adhesion by nanoindentation testing.

Figure 4.85 shows a structural diagram of a device for instrumented hardness testing in the microhardness range, which can either be installed in a material testing machine for high stiffness, or is commercially available as a self-contained unit (e.g., Fischerscope®, Fig. 4.86). For the nano range, industrial-size devices, so-called nanoindenters, have been developed. Their schematic structure is comparable with that of microhardness test devices, but the demands placed on their force and indentation depth resolution are significantly higher.

With the instrumented hardness testing devices illustrated in Fig. 4.85, the following functional dependencies can be measured:

Fig. 4.86: Fischerscope® H100C XYp microhardness test station

4.7 Hardness Test Methods 195

• Load as a function of indentation depth during load increase, • Load and indentation depth as functions of time for determining relaxation and

creep behavior • Elastic recovery during reloading.

This enables the separation of the plastic and elastic components of total deformation during hardness measurement.

4.7.3.2 Material Parameters Derived from Instrumented Hardness Tests

There are various approaches for evaluating load–indentation depth curves, all with the goal of describing material behavior precisely and/or acquiring characteristic values [4.86]. Quantities to be measured include: maximum load Fmax and maximum indentation depth hmax from the load curve, point of intersection of the tangent to curve b with the indentation depth-axis hr and indentation modulus, as well as deformation energy components from the load–indentation depth curve (Fig. 4.87). The area between indentation function and h-axis is the total deformation energy of indentation Wtotal.

hr

hmaxhp

S

h

F

Fmax

Wplast

a

b

hc

Welast

Fig. 4.87: Load-indentation depth curve (loading curve a, unloading curve b)

Due to plastic deformation, the unloading function does not pass through the origin, so that there is a difference between indentation and unloading, i.e., plastic energy Wplast. The elastic energy is the difference: Welast = Wtotal – Wplast.

Martens Hardness

Martens hardness is measured under applied test load F and contains the elastic and plastic deformation energy of indentation. It is defined for Vickers and Berkovich

196 4 Mechanical Properties of Polymers

indenters. Martens hardness HM is the quotient of test load F and the contact area calculated from the corresponding indentation depth h:

2h43.26

F HM (4.167)

F test load in N h indentation depth under applied test load in mm

Plastic Hardness and Indentation Hardness

Plastic hardness Hplast and indentation hardness HIT are measured using maximum load and applying tangents to the unloading curve. They are a measure for resistance to permanent deformation or damage.

2 r

max plast

h43.26

F H (4.168)

Fmax maximum load in N hr point of intersection of the tangent to unloading curve b at Fmax with the indentation depth-axis in mm

At the transition to smaller indentation depths, the contact area changes continuously and with it the contact stiffness dF /dh. This necessitates a correction made by introducing the so-called projected contact area Ap. So-called indentation hardness is the quotient of maximum acting test load Fmax and projected contact area Ap between indenter and specimen.

p

max IT

A

F H (4.169)

Fmax maximum applied load in N Ap projected (cross-sectional) area of contact between indenter and specimen determined from the load–

indentation depth curve and the area function of the indenter in mm2

The projected contact area Ap is a function of contact depth hc (Eq. 4.170) and presumes knowledge of the indenter area function.

rmaxmaxc hhhh (4.170)

hc depth of indenter contact with specimen at Fmax in mm correction factor, dependent on indenter geometry (Vickers and Berkovich: = 0.75)

For indentation depths h > 6 μm, a first approximation to the projected area, Ap, is given by the theoretical shape of the indenter. For a standard Vickers indenter, that is:

2cp h50.24A (4.171)

4.7 Hardness Test Methods 197

For indentation depths h < 6 μm, the area function of the indenter cannot be assumed to be that of the theoretical shape, since all pointed indenters will have some degree of rounding at the tip and spherically-ended indenters (spherical and conical) are unlikely to have a uniform radius. The determination of the exact area function for a given indenter is required for indentation depths < 6 μm, but is beneficial for larger indentation depths.

Elastic Indentation Modulus

The elastic indentation modulus EIT is measured from the slope in the tangent used for calculating indentation hardness.

i

2 i

r

2 S

IT

E

1

E

1

1 E (4.172)

dh

dF

A2 E

p

r (4.173)

S Poisson´s ratio of the specimen

i Poisson´s ratio of the indenter (for diamond 0.07) Er reduced modulus of indentation contact Ei modulus of indenter (for diamond 1.14 106 N mm

-2) Ap projected contact area, value of indenter area function at contact depth

Due to differences in the type of loading and measuring methods, there is no correspondence with the E modulus from the tensile test. There is additional influence on measurement results when bulges and sink holes develop in the material surrounding indentations.

Plastic and Elastic Components of Indentation Work

The total mechanical work of indentation Wtotal is expended only partially for plastic deformation Wplast. The remainder is released during the unloading process as elastic reverse deformation work of indentation Welast. The correlation

IT = Welast /Wtotal 100 % (4.174)

contains material information suitable for characterizing deformation behavior. The plastic component Wplast /Wtotal follows as 100 % – IT .

198 4 Mechanical Properties of Polymers

4.7.3.3 Examples of Applications

The parameters of the instrumented hardness test describe basic material behavior and are indicators of changes in materials. Their structural sensitivity is illustrated by the example of PP with two different crystalline structures. A semicrystalline, isotactic PP serves as the model material in which the monoclinic phase and a trigonal phase arise parallel during solidification. The phase exhibits low stiffness and hardness, as well as higher ductility than the phase. Toughness is increased under quasi-static as well as impact loading [4.87], i.e., this opens up new areas of application for nucleated PP materials. However, it must be noted that -PP melts at lower temperatures, and its heat-distortion resistance is also lower than that of

-PP. The load–indentation depth curves presented in Fig. 4.88 illustrate their differences in mechanical behavior. Greater indentation depth and indentation depth increase under maximum load as well as a less slope of the unloading curve of

0 100 200 300 0

0.05

0.10

0.15

0.20

h (nm)

�

F (m

N )

�

�

�

100 μm

Fig. 4.88: Load–indentation depth curve gathered from the and modification of PP with holding

time in maximum loading; spherulitic supermolecular structure of PP: the phase appears bright due to its negative birefringence

the phase are indicators of lower hardness, stronger creep tendency and lower stiffness compared to the phase. This may be due to greater chain mobility, as could be shown by investigations of mechanical loss factor tan [4.86]. The values for indentation hardness HIT and indentation modulus EIT determined from load– indentation depth curves are listed in Table 4.10.

4.7 Hardness Test Methods 199

Table 4.10: Indentation hardness HIT and indentation modulus EIT for - and -PP

HIT (MPa) EIT (MPa)

-phase 108 9 2024 54

-phase 98 11 1943 147

The supermolecular structure of thermoplastics is strongly influenced by their processing conditions and subsequent heat treatments. This is especially the case for semicrystalline polymers. That is why changes in the crystalline phase caused by heat treatment (tempering) and their effects on mechanical properties are quite interesting from an engineering point of view. In order to measure the influence of tempering temperature on crystalline structure, PP was subjected to temperatures of Ta = 80, 100, 120, 140 and 150 °C for one hour. The lamellae thickness distributions taken from melt curves using the Alberola method [4.88] (Fig. 4.89b) are suitable for describing changes occurring in the crystalline phase. The distribution peak shifts from approx. 18 to 19 nm in the original state to approx. 21 nm for PP tempered at140 and 150 °C. Simultaneously, the segment of small lamellae clearly decreases, since the lamellae melt and the molten material absorbs onto the remaining, thicker lamellae. The increased chain movement in the amorphous phase effects additional

H (M

P a )

0 50 100 150 100 1500

2000

2500

3000

T (°C)a

5 10 15 20 25

initial state

140 °C

150 °C

I (nm)theo

E

H IT

120

140

160

180

200

IT

b

a

E (M

P a )

IT

IT

Fig. 4.89: Correlation between parameters gathered in instrumented indentation tests, and lamellae thickness distribution in PP: indentation hardness HIT and indentation modulus EIT as functions of temper temperature Ta (a) and lamellae thickness distributions measured by DSC [4.88] at various temper temperatures (b)

200 4 Mechanical Properties of Polymers

growth of smaller lamellae. The lamellae thicknesses ltheo of PP tempered at 140 °C are distributed bimodally (Fig. 4.89b), at Ta = 150 °C the distribution curve is relatively narrow. Hardly any lamellae thicknesses less than 14 nm occur, whereas at Ta = 140 °C and in the original state, thinner lamellae are present. Indentation modulus and hardness do not increase until temperatures rise above 100 °C (Fig. 4.89a), i.e., only tempering temperatures that effect a change in crystalline structure can lead to changes in characteristic mechanical property values.

4.7.4 Correlating Microhardness with Yield Stress and Fracture Toughness

Information on the relationship between hardness and other mechanical parameters, such as strength, E modulus and toughness is of enormous practical importance both from the point of view of testing and for understanding macroscopic material behavior. Experimentally discovered empirical relationships enable efficient quality assurance for materials and components. However, it must be noted that these empirical correlations are only valid within particular classes of materials. An estimation of macroscopic yield stress or yield point is known in the hardness testing of metallic materials using the Tabor relation [4.89]. For theoretical polymer material behavior, linear proportionality has the form:

� (MPa)y

H V

(M P

a )

PVC + 35 % DOP PE-LD

PTFE

PE-HD

PP

CA

ABS

ABS

PVC

PC

PPO POM

PS POM-Co

SAN PMMA

250

200

150

100

50

0 0 20 40 60 80 100

HV 2.33 �y

P V

C +

2 5

% D

O P

P A

6 ;

9 %

H

O 2

P A

6 ;

3 %

H O

2

PA6; 0.4 % H O2

Fig. 4.90: Relation between Vickers hardness (test load 2 N) and yield stress derived by tensile test

4.7 Hardness Test Methods 201

C pH

Y

m

Y

(4.175)

pm Indenter/specimen indentation pressure acting perpendicular to contact area (pm = 1.08 HV for a Vickers pyramid)

C Proportionality factor (C 3)

The Tabor equation is the fundamental relation for presenting the relationship between hardness and yield point. Weiler [4.82] determined the empirical relationship between separately measured Vickers hardness and yield stress under tensile load for a number of thermoplastics (Fig. 4.90). The relation HV 2.33 y is derived for the range of materials listed in Fig. 4.90.

When considering such relationships, fundamental methodological and material- related aspects have to be kept in mind. Equation 4.175 defines the relationship between compressive stress at yield and hardness, i.e., correlations between yield stresses from the tensile test and hardness values must lead to deviations from C = 3, since deformation behavior changes due to the emergence of a hydrostatic component under compression loading. This is illustrated in Fig. 4.91 using stress– strain diagrams in tensile and compression tests on the example of E/P copolymers.

� (MPa)y

H (M

P a

)

60

40

20

0 0 20 40 60 80 100

60

40

20

0 0 10 20 30

150

100

50

0 0 20 40 60

� (%)

H = 3.05 �y

H = 1.75 �y

tension

compression

0 mol.-% ethylene 4 mol.-% ethylene 6 mol.-% ethylene 8 mol.-% ethylene

� (M

P a

)

� (M

P a

)

IT

� (%)

a b

c

0 mol.-% ethylene 4 mol.-% ethylene 6 mol.-% ethylene 8 mol.-% ethylene

Fig. 4.91: Tensile stress–tensile strain (a) and compression stress–compression strain diagrams (b) for

E/P copolymers with differing ethylene content; correlation between indentation hardness HIT and yield stress or compressive stress at yield y (c)

202 4 Mechanical Properties of Polymers

a 0.050

0.045

0.040 0 2 4 6 8

ethylene content (mol.-%) H / E IT

3.0

2.5

2.0

1.5

1.0 0.044 0.046 0.048 0.050

b

IT

H

/ E

IT IT

J

(

N m

m )

IdS T

-1

in c re

a s e

o f

p la

s ti c it y

Fig. 4.92: Dependence of quotient HIT/EIT on ethylene content (a) and relationship between resistance

to unstable crack propagation JId ST and quotient HIT/EIT (b) for E/P copolymers

The corresponding values for yield stress and compressive stress at yield are presented in Fig. 4.91c in correlation to indentation hardness. Clear differences become obvious between tensile load (HIT = 3.05 y) and compression load (HIT = 1.75 y) that can be found in the literature [4.85] for PE materials as well. In cases of elastic–plastic material behavior, the relation to the E modulus must also be considered in addition to the correlation between hardness and yield point. It is generally the case that smaller H/E values mean higher plasticity connected to higher toughness values. In evaluation of the results presented in Fig. 4.91, Fig. 4.92a shows that the quotient HIT /EIT of statistical E/P copolymers decreases with increasing ethylene content. For the JId

ST values (cf. Section 5.4.2.4) measured by instrumented notched Charpy impact test under impact loading, an increase in toughness is observed with increasing ethylene content. In connection with Fig. 4.92a, this means that the smaller the HIT /EIT, the greater the resistance to unstable crack propagation (Fig. 4.92b) [4.86].

A relation has been proposed by Studman [4.99] (Eq. 4.176), based on a model by Johnson [4.91], that enables evaluation of yield stress values with the aid of experimentally measured hardness and E modulus under compression load. This model has proven advantageous when no values can be obtained from the compression stress–compression strain curve, or when these are not experimentally accessible. Equation 4.176 makes clear that yield stress is essentially determined by hardness value; the E modulus is effective only as a corrective factor in the logarithmic term.

4.8 Friction and Wear 203

� (MPa)y

H V

(M P

a )

0 0 20 40 60 80

PE-HD

PP PVC

PMMA

200

150

100

50

PS

Fig. 4.93: Dependence of Vickers hardness under load HV on compression yield stress y

yy

m

3

tanE ln1

3

2 5.0

p (4.176)

contact angle between specimen and indenter ( = 19.7° for a Vickers indenter)

Figure 4.93 illustrates the relationship between hardness under load from the instrumented hardness test [4.92] and yield stress determined by Eq. 4.176 and given the E modulus under compression load for selected thermoplastics. The relationship between hardness and yield stress is described with reference to Eq. 4.175 independent of the material by the relation HV/ y = 2.5.

For calculating yield stress values, the physically relevant values from instrumented hardness tests are to be preferred over those measured conventionally, due to visco- elastic-plastic material behavior that can lead to considerable scattering among conventionally measured hardness values [4.92].

4.8 Friction and Wear

4.8.1 Introduction

Polymers are used increasingly for tribologically stressed components, whereby metallic bearings, gear wheels or sliding elements are replaced by plastic components. The fact that plastics are often rather economical to produce, especially in very complex shapes with good functional integration, explains this trend.

204 4 Mechanical Properties of Polymers

These polymeric materials are utilized mostly as functionally optimized, i.e., filled composite materials. In order to provide the required properties, reinforcements in the form of particles or fibers and internal lubricants, such as graphite or polytetrafluoroethylene (PTFE), have to be compounded into the polymeric material. If to the application involves abrasive loading, it may necessary to add hard ceramic particles as fillers to the material. The type of material modification varies widely depending on the ultimate application.

Among polymers in tribologically loaded components, a fundamental distinction is made between slide bearings made from plastics and components with additional, tribologically optimized properties. Plastics slide bearings are subdivided into polymer-coated bearings with metallic supports and solid plastic bearings. Examples for the use of polymers in slide bearing applications for the automotive industry include bearings for shock absorbers, grooved belt wheels in components such as ignitions, alternators or diesel injection pumps. In all these areas, high wear resistance is required at low friction coefficients and ever higher surrounding temperatures. Fundamentally different demands are made on materials that are used, for example, as roller coatings in paper machines or calenders. To be sure, tribological loading is again involved, but here the goal is rather to obtain abrasive wear resistance. This is also the case when designing lubricated pump bearings that must continue to function under extremely abrasive conditions.

The variety of final applications mentioned for plastics and composite materials in tribological applications is an indication of the resulting complexity among the required test methods. No single test method is capable of providing information for prediction and comparison under all different use conditions. Instead, the test conditions for these tribological applications have to be adapted as closely as possible to the ultimate use conditions. Such adaptation has to be done with respect to various criteria. First, the ambient medium must be the same as in actual use. Many plastics are utilized dry and unlubricated; other applications work in the presence of oil or water. This has to be considered when specifying test conditions. Another essential influencing factor is the material and surface structure of bodies encountered by the component. Yet another important point is the mechanical load collective, i.e., the pressure on the bearing material and the occurring sliding velocity. The result of both is an increase in thermal loading in the slide contact area. Besides the slide velocity of counterbodies, the type of relative motion has to be considered. In many cases, continuous slide is involved, such as for a PC ventilator bearing; in other cases, an oscillating relative motion may be involved, such as when used in a shock absorber. The last overriding influencing factor in tribological material testing is the mesh geometry of both friction surfaces and the contact geometry resulting from it. When a spherical surface touches a plane surface, point contact must be presumed for a first

4.8 Friction and Wear 205

approximation. If a cylinder slides on a plane, line contact takes place. When two planes slide against each other, planar contact is made. There is no clear distinction between these fundamental conditions of meshing. For instance, line contact transitions into planar contact in the case of hole faces of a slide bearing. Actual contact area changes with increasing wear.

The following provides a list of the possible material modifications and the various available tribological methods for testing and evaluating. Corresponding test standards are listed at the end.

4.8.2 Fundamentals of Friction and Wear

The science of friction and wear, including lubrication, concerns itself with surfaces acting on each other in relative motion and can be subsumed under the concept of tribology. Physical and chemical processes as well as mechanical and design aspects are involved. It must be remembered that friction and wear properties cannot simply be assigned to a material, but that their properties are dependent on the particular overall system (tribosystem). By tribosystem, we mean all the technical systems in which friction and wear processes take place [4.93]. These are mainly characterized by their conditions under use. For polymeric materials, bearing load, slide velocity, temperature in use and counterbodies have special significance [4.94, 4.95].

Besides system parameters, the tribological behavior of a polymer material is also strongly influenced by its microstructure. This includes molecular structure and degree of crystallinity (in thermoplastics) on one hand, and process-related structure features (morphology) on the other. Moreover, factors such as fiber orientation, filler content and filler distribution can have effects on tribological properties when various fillers and reinforces are added to a polymer matrix [4.96 – 4.100].

Due to the variety of influencing factors, the behavior of one tribosystem usually cannot be extrapolated to another. Thus, if no measurement values are available for the specific application conditions, the tribological behavior of a material can only be estimated using test results obtained under the same or similar conditions. Reliable statements can only be made by testing the case of application [4.101].

4.8.2.1 Frictional Forces

Frictional force is defined as the force that counteracts the relative motion of bodies in contact with one another. In order to maintain the motion of bodies against each other, a force FR is required for overcoming friction. According to Amonton and

206 4 Mechanical Properties of Polymers

Coulomb, FR is independent of contact area, but proportional to the acting normal force FN at which both bodies press against each other (Eq. 4.176).

NR FF (4.177)

coefficient of friction

As normal force increases, the pressing bodies hook into each other, thus increasing the frictional force. This so-called law of friction holds in principle for plastics as well, regardless of whether a system with one or two plastic friction couples is involved. Nonetheless, analysis is confronted by considerable problems, since heat and deformations, as well as further ambient influences on the frictional process, such as moisture and oxidation, can hardly be disentangled. The relation, however, is considered a good approximation in all cases [1.17, 4.101].

4.8.2.2 Temperature Increase Resulting from Friction

During the frictional process, the work of friction is partially transformed into heat energy (frictional heat). The increase in heat content of the bodies leads to a rise in temperature. This temperature increase T is especially dependent on the relative velocity between the base body and its counterbody (slide velocity), as well as on normal force FN. It can be estimated using the following relation 4.93, 4.101 :

RvFT N (4.178)

whereby R represents a thermal resistance parameter. This is determined as a function of cross-sectional surface A of the heat transfer paths n, their lengths l and their specific thermal conductivity :

1in1i i )l/()A/1(R (4.179)

Frictional heat can lead to the softening of materials, subsequent creep and even to surface melting. This mechanism can be readily observed on polymeric materials. The mechanical properties of polymeric materials, especially thermoplastics, can change substantially with increasing temperature. Moreover, temperature increases are responsible for changes in hardness under tribological loading, i.e., hardness decreases considerably with increasing temperature. On the other hand, friction- dependent temperature increase in surface areas is itself dependent on hardness. In consequence of temperature increases, the morphology and/or structure of polymeric materials can change too.

4.8 Friction and Wear 207

4.8.2.3 Wear as a System Characteristic

By wear we generally mean the progressive loss of material from the surface of a solid body occurring as a result of physical-chemical processes generated by contact and motion relative to a solid, fluid or gaseous counterbody. This can change the shape and mass of a body.

Measurable wear quantities can be subdivided into direct, specific and indirect measured quantities. The specific measurable quantities include, among others, “linear abrasion rate”, also known as (linear) “wear depth” or “depth wear rate”, and specific wear rate.

It must be noted that friction and wear values represent loss quantities that generally cannot simply be assigned to one material, but always have to be considered in relationship to the overall system. By contrast, typical material parameters, such as E modulus, tensile strength, yield point or fracture toughness can be assigned to a material and be directly transferred to the same material in another system. Under tribological loading, however, such a transfer of test results is possible only within very narrow limits. Wear is therefore termed a system characteristic and not a material property [4.93 – 4.95].

4.8.2.4 Wear Mechanisms and Formation of Transfer Film

Wear of polymeric materials can be distinguished by various wear mechanisms explained in the following [4.95, 4.101, 4.102].

In adhesion, the material from one friction partner sticks to the surface of the other partner and is subsequently separated from its base body. Adhesion takes the form of fretting, pitting, cusps and materials transfer. It is the mechanism that appears most often when the counterbody is not particularly rough.

Abrasion means that micro-roughnesses of the harder counterbody plow through softnesses on the other, removing material by microcutting or microcracking. Scratches, grooves, troughs or waves result. Abrasive wear thus occurs especially by rough counterbody surfaces.

Surface fatigue or degradation is local fatigue due to repeated contact with the counterbody and subsurface deformations. Due to repeated counterloading, defects begin to appear on the surface and cracks or dimples develop until wear particles are finally removed.

Due to the frictional process, so-called tribochemical reactions (e.g., corrosion, oxidation, chemical degradation) can be triggered in which reaction products (layers,

208 4 Mechanical Properties of Polymers

particles) arise, leading to material failure. Such reactions proceed faster under tribological loading than in a static state. Depending on the type and adhesion of reaction products to the surface, either wear intensifying or reducing effects can take place.

Often one of the mechanisms mentioned is dominant and responsible for momentary wear. Any change in slide conditions, however, can lead to a change of mechanism. Then the different mechanisms influence each other reciprocally. For example, hard particles or fiber fragments removed by adhesion can act abrasively (when remaining as a third body in the contact region).

Pure two-body contact is rare in a tribological system. During the wear process, an interim layer forms between the contacting surfaces that sticks to the friction surfaces in the form of compacted wear debris, or it may collect at the plane of tangency in the form of loose wear particles. Such an interim layer separates the friction partners, reduces the real contact area between the bodies and simultaneously supports some of the load. For one, it can function as temporary surface wear protection, reducing friction like a solid lubricant (e.g., PTFE transfer film). On the other hand, if such an interim layer contains hard particles, it can also act abrasively.

It is important to note that different wear mechanisms are activated depending on whether load is uniformly constant, periodically alternating or impacting [4.101].

4.8.3 Wear Tests and Wear Characteristics

Many different wear tests are performed in order to do tribo-technological tasks in research and industry. These range from complex and expensive investigations of complete machines under actual operating conditions down to laboratory tests on simple specimen geometries. Various tasks of wear testing are listed as follows with reference to [4.103]:

• Optimization of components or tribo-technical systems to realize a specified, wear-determined service life

• Determination of wear-determined influences on overall machine function • Monitoring wear-determined functionality of machines • Collection of data for establishment of intervals for inspection and maintenance • Preselection of materials and lubricants for practical application cases • Quality assurance of materials and lubricants • Simulation of wear on tribologically loaded components with the aid of substitute

systems • Wear research and mechanism oriented wear testing.

4.8 Friction and Wear 209

In order to perform these tasks, there must be a basis for decision-making in the form of wear characteristics determined by friction and wear measurements.

Wear tests are classified according to their transferability to real application cases [4.103]. The categories range from the operational test (category I), in which the original tribosystem is tested under real use conditions, down to model tests on simple specimens (category VI). In between lie gradations with a step-by-step reduction in test complexity down to the model test. This reduction of the original tribosystems is accompanied by a reduction in the transferability of results. On the other hand, it is easier to investigate the influence of individual parameters on wear, e.g., in laboratory or component tests, where the loading collective is well-known and controllable. Effort and costs are generally highest in operational tests and smallest in model tests. That is why friction and wear studies usually begin by performing model tests.

The point of departure for every wear test is the tribological system analysis based on which, for example, suitable materials can be preselected, as well as what type (e.g., sliding, rolling) and category (e.g., model test, component test) of wear test can be performed.

Polymers and polymer-composite materials generally exhibit good wear properties. By modifying them with aramid-, glass- or carbon-fibers and/or solid lubricants such as PTFE and molybdenumdisulfide (MoS2), friction and wear properties can be further improved and make it possible to realize dry-running and maintenance-free components for tribological loads. In the following sections, we will deal mainly with non-lubricated wear tests often used for testing plastics. Of course, plastics are also used in lubricated tribosystems. However, due to the variety of different methods, tribological loads and types of wear, an exemplary selection had to be made.

4.8.3.1 Selected Model Wear Tests

The many different wear test methods used in practice are all based on corresponding types of tribological loading, such as sliding, rolling, sliding with rolling or oscillating sliding. Sliding wear test methods, such as pin-on-disc, block-on-ring and “thrust washer“ tests are used widely in the wear testing of plastics. Figure 4.94 illustrates the testing principles of the pin-on-disk and block-on-ring test. In both tests, a specimen is pressed against a rotating ring or a rotating disk made from the selected counterbody material. The “thrust washer” test uses a ring-shaped specimen.

Wear due to vibrations can be investigated with a fretting wear testing machine. The testing principle of such a design is sketched in Fig. 4.95, in which the counter- body often is ball-shaped and g uided over the specimen in oscillating motion.

210 4 Mechanical Properties of Polymers

FN

specimen

test principle: block-on-ring test principle: pin-on-disc

F

specimen

counter partcontinuous rotation

wear track

wear track

a b

N

v

v

continuous rotation counter part

Fig. 4.94: Test principles of block-on-ring (a) and pin-on-disk (b) wear tests

The counterbody is pressed onto the specimen at a defined normal force FN. The contact conditions in such tests are not constant. At the start, there is point contact between the test body and the counterbody; the contact area grows with continuing wear. With different counterbodies, different contact geometries, such as line contact or planar contact can be realized.

Temperature greatly influences wear of plastics. That is the reason why many wear testing machines have a test chamber for tempering the counterbody. Moreover, a closed test chamber makes it possible to introduce technical gases or create a special climate (humidity and air temperature) with the aid of a climate conditioner.

When wear tests are performed to research wear mechanisms, the loaded test bodies and counterbodies are microscopically examined, since the surface topography often permits inferences to be made as to the essential wear mechanisms. With the aid of surface measurement methods such as profilometry or interferometry, worn-out

F N

specimen

test principle: cyclic wear

counter part oscillation

wear track

dot contact

line contact

area contact

Fig. 4.95: Testing principles of oscillating wear experiments

4.8 Friction and Wear 211

surfaces can be measured three-dimensionally. Thus, precise values can be obtained for the dimensions and depth of wear markings or tracks and it also enables a roughness analysis of the surface.

A comprehensive picture of a tribosystem under specific test conditions is created by the wear parameters measured, subsequent microscopic evaluation of wear surfaces and, in some cases, roughness analyses. All together, this information provides a basis for evaluating different materials and optimizing the selection of materials.

4.8.3.2 Wear Parameters and Their Determination

Several different parameters are in practical use for describing materials with respect to their wear resistance. Almost all wear parameters are based on the measurement of weight loss Wm, material worn away WV , or one of the proportional quantities related to it, e.g., linear wear factor Wl, , based on a change in length. These quantities are called wear factors. Figure 4.96 contains a diagram of wear factor W for two different specimen geometries.

When wear factors are derived according to their reference quantities, such as loading path s or test period t, the results are the so-called wear rates. Specific wear rate involves, in addition to wear path, the load on the specimen. The following equations describe the most commonly used wear rates:

• Wear rate

(derivation of wear factor according to loading time)

dt

dW W lt/l (m h

-1), or in terms of weight dt

dW W mt/m (kg h

-1) (4.180)

W l : linear wear value wear area A V

WV = Wl ·AV volumetric wear value

WV = Wq ·l

volumetric wear value

l

W q : planimetric wear value

Fig. 4.96: Diagram of linear, planimetric and volumetric wear factors [4.103]

212 4 Mechanical Properties of Polymers

• Wear-path ratio

(derivation of wear factor according to loading path)

ds

dW W ls/l (m m

-1), or in terms of weight ds

dW W ms/m (kg m

-1) (4.181)

• Specific wear rate

(derivation of material worn away according to loading path and loading force)

N

V 2

F,s/V Fs

W W (m3 (Nm)-1) (4.182)

FN, p, A normal force, planar compression, plane (FN = p A) s wear path (s = v t) v slide velocity t test period

WV/s,F is often abbreviated as Ws.

4.8.3.3 Wear Parameters and Their Presentation

When materials are selected, corresponding wear factors or wear rates are determined for the different materials for one or more test-parameter sets. They can be used directly for selecting materials. However, if it is desirable to present a selected material

static load limit

p-v line at defined stationary wear rate

p-v limit

thermal limit

log v

linear wear rate �p v

lo g

p

Fig. 4.97: p–v diagram for dry-running slide bearings [4.104]

4.8 Friction and Wear 213

and its tribological efficiency, this is often done with p–v values or p–v diagrams. Limiting p–v values provide a value in excess of which wear begins to increase disproportionally (Fig. 4.97). The p–v factor states the loading at which a defined wear rate, e.g., 0.5 μm h-1 can be measured.

A p–v diagram (Fig. 4.97) holds only for one particular tribological system. The provided curves constitute the limits within which a polymer can be utilized.

4.8.4 Selected Experimental Results

4.8.4.1 Counterbody Influence

Given the material combination polymer/steel present in many engineering applications (e.g., in slide bearings), tribological behavior is strongly influenced by the surface topography of the metallic friction partner. In general, the friction coefficients measured are higher for very smooth (polished) steel surfaces than for median roughness depths. At higher roughnesses, however, friction coefficients increase. Figure 4.98 illustrates this for PE-HD.

Increasing wear is generally found with increasing counterbody roughness, but here, too, it is possible that there is a range of minimal value. An explanation for the formation of such a minimum may lie in the transition from predominantly adhesive wear at low roughness depths to predominantly abrasive wear at higher roughnesses.

0 0.5 1.0 1..5 0

0.2

0.3

0.4

0.5

μ

R (μm)

1E-7

1E-6

1E-

1E-4

1E-310 -3

10 -4

10 -5

10 -6

10 -7

friction coefficient

specific wear rate

p = 1.4 MPa

v = 1 m/s

R (μm)a

0.1

W (m

m /N

m )

s 3

Fig. 4.98: Influence of counterbody roughness Ra on the friction coefficient and specific wear rate Ws

of PE-HD [4.105]

214 4 Mechanical Properties of Polymers

Feinle [4.106] investigated the influence of counterbody roughness on friction coefficients and linear pin-wear rate on friction-paired glass-fiber reinforced polyphenylenesulfide (PPS) and steel 100 Cr 6. In the investigated roughness range of 0.05 to 2.5 μm, friction and wear exhibited opposite behaviors. As the friction coefficient decreased with increasing roughness, a clear increase took place in the linear pin-wear rate. A minimum or optimum roughness – such as discovered for unreinforced PE-HD (Fig. 4.98) – could not be detected.

Besides roughness, the orientation of counterbody grooves also influences the wear behavior of plastics. Grooves in the direction of slide produce less wear than grooves perpendicular to it [4.107].

4.8.4.2 Influencing of Fillers

Although many unfilled polymers exhibit very good tribological properties, the use of appropriate fillers can further improve wear and friction coefficient to suit the particular tribosystem and its loading parameters. To reduce wear, polymer materiales are often enhanced with fibers made from glass (GF) or carbon (CF). They increase stiffness and strength while reducing creep tendency. Low adhesion between friction partners is useful for achieving a lower coefficient of friction . This is realized by using internal lubricants such as PTFE, MoS2 or graphite. Figure 4.99 shows that favorable wear and friction properties can be realized by combining high- performance polymers with fillers made from lubricants and reinforcing materials (range of concentric shading).

The influence of PTFE filler on the tribological values of a PEEK/steel friction pair is illustrated in Fig. 4.100. Both friction coefficient and specific wear rate Ws pass

high- performance

polymer

internal lubricants (PTFE, graphite, ...)

reinforcements (glass fibers, carbon-fibers)

Fig. 4.99: Components for tribologically optimization of a high-performance polymer