Literature Review

Technical Note

Soil Compressibility Models for a Wide Stress Range Song-Hun Chong1 and J. Carlos Santamarina2

Abstract: Soil compressibility models with physically correct asymptotic void ratios are required to analyze situations that involve a wide stress range. Previously suggested models and other functions are adapted to satisfy asymptotic void ratios at low and high stress levels; all updated models involve four parameters. Compiled consolidation data for remolded and natural clays are used to test the models and to develop correlations between model parameters and index properties. Models can adequately fit soil compression data for a wide range of stresses and soil types; in particular, models that involve the power of the stress σ 0β display higher flexibility to capture the brittle response of some natural soils. The use of a single continuous function avoids numerical discontinuities or the need for ad hoc procedures to determine the yield stress. The tangent stiffness—readily computed for all models—should not be mistaken for the small-strain constant-fabric stiffness. DOI: 10.1061/(ASCE)GT.1943-5606.0001482. © 2016 American Society of Civil Engineers.

Author keywords: Soil compressibility models; Compression index; Remolded clays; Natural sedimentary clays; Tangential stiffness; Small-strain stiffness; Yield stress.

Introduction

Soils subjected to either isotropic compression or ko (where ko is under zero lateral strain condition) compression experience volume contraction. Contraction depends on soil type, formation history, diagenesis, prior stress history, porosity, and stress conditions.

A soil compressibility model used for settlement analysis often needs to justify the data in a relatively narrow stress range. How- ever, many geotechnical problems involve soils subjected to either extremely low or extremely high effective stress or a wide effective stress range. Examples include self-weight consolidation (Been and Sills 1981; Cargill 1984; Bartholomeeusen et al. 2002; Stark et al. 2005), seafloor engineering [suction casings (Houlsby et al. 2005) and skirted foundation (Bransby and Randolph 1998)], gradual movement of pipelines resting on the seafloor and lakebeds (Krost et al. 2011; Randolph et al. 2011), pile tips (Yang et al. 2010; Tsuha et al. 2012), blast loads (Wang et al. 2005), methane recovery by depressurization from hydrate bearing sediments, and filter-cake formation in drilling mud (Sherwood and Meeten 1997).

Soil compressibility models are sought in this study to analyze field conditions over a wide stress range. Models must be able to fit compressibility data for diverse soils, have physically correct asymptotic values at low stress σ 0 → 0 and high stresses σ 0 → ∞, and involve a small number of physically meaningful parameters (Ockham’s criterion).

Compressibility: Stress Regimes

Soil compressibility is briefly reviewed next. Three stress regimes are tentatively identified in reference to the standard stress

range in common geotechnical applications, namely 10 kPa < σ 0z < 1 MPa.

Low Stress Regime (σ 0z < 10 kPa)

Individual grains or flocs form a granular skeleton with a characteristic finite porosity that depends on grain geometry and pore fluid chemistry (Klein and Santamarina 2005; Palomino and Santamarina 2005). Compressibility at low stress reflects the for- mation fabric and postdepositional diagenetic changes triggered by preloading, moisture fluctuations, thermal history, fluid-mineral in- teraction, dissolution, and reprecipitation (Mitchell 1956; Burland 1990; Santamarina et al. 2001; Rinaldi and Santamarina 2008).

Intermediate Stress Regime (10 kPa < σ 0z < 1 MPa)

Soil compression in this stress regime remains affected by forma- tion conditions such as initial water content (Hong et al. 2010, 2012) and temperature (Campanella and Mitchell 1968; Baldi et al. 1988; Leroueil 1996; Sultan et al. 2002), and is affected by diage- netic processes such as cementation and aging (Mesri et al. 1975; Schmertmann 1983, 1984, 1991). This stress regime is of main in- terest to classical geotechnical practice; therefore, many studies have explored correlations between compressibility and index properties. In particular, the compression index Cc has a strong cor- relation with the liquid limit LL, or the void ratio at the liquid limit eLL (Skempton 1944; Terzaghi and Peck 1948; Burland 1990; Sridharan and Nagaraj 2000). Disturbance and/or remolding de- structures natural soils, and remolded soils exhibit a compression curve that plots at a lower void ratio than the undisturbed natural soil and with a less pronounced yield stress. Measured consolida- tion curves are affected by experimental procedures such as sam- pling disturbance (Casagrande 1936; Terzaghi and Peck 1948; Schmertmann 1955; Rochelle et al. 1981; Hight et al. 1992; Santagata and Germaine 2002), seating and boundary effects, and strain rate (Hanzawa 1989; Leroueil 1996; Leoni et al. 2008).

High Stress Regime (σ 0z > 1 MPa)

The void ratio decreases at a gradually lower rate at high stress (Athy 1930; Aplin et al. 1995), and the prevailing deformation mechanisms become particle compliance, pressure dissolution,

1Senior Researcher, High Speed Railroad Systems Research Center, Korea Railroad Research Institute, 176, Cheoldo bangmulgwan-ro, Uiwang-si, Gyeonggi-do 437-757, Republic of Korea (corresponding author). E-mail: [email protected]

2Professor, Earth Science and Engineering, King Abdullah Univ. of Science and Technology, Bldg. 5, Thuwal, Saudi Arabia 23955-6900.

Note. This manuscript was submitted on May 18, 2015; approved on December 3, 2015; published online on March 3, 2016. Discussion period open until August 3, 2016; separate discussions must be submitted for in- dividual papers. This technical note is part of the Journal of Geotechnical and Geoenvironmental Engineering, © ASCE, ISSN 1090-0241.

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crushing, and creep (Barden 1965; Mesri and Godlewski 1977; Mesri and Castro 1987). Past history loses relevance regardless of the natural or remolded origin of specimens (Terzaghi and Peck 1948; Chilingar and Knight 1960; Skempton 1969; Burland 1990; Hong et al. 2012).

Soil Compressibility Models

Classical e- logðσ 0Þ compressibility models and new functions are reviewed in this section. In all cases, the models are generalized to satisfy asymptotic void ratios eL as σ

0 → 0, and eH as σ 0 → ∞.

Semi-Logarithmic e- log σ 0 Models

The classical linear equation in logðσ 0Þ is the most common model used in geotechnical engineering (Terzaghi and Peck 1948; Schofield and Wroth 1968)

e ¼ eref − Cc log � σ 0

σ 0ref

� ð1Þ

where the void ratio eref corresponds to effective stress σ 0 ¼ σ 0ref.

The normalization stress σ 0ref is selected a priori for normalization, such as σ 0ref ¼ 1 kPa, and it is not a model parameter. Therefore, this function has two model parameters: eref and Cc; it fits normally consolidated soil data at intermediate stress levels, but predicts e → ∞ as σ 0z → 0 and e < 0 as σ 0z → ∞.

The lower soil compressibility at high stress requires higher or- der terms, such as the cubic polynomial suggested by Burland (1990) for remolded soil data

e ¼ eref − α · log � σ 0

σ 0ref

�

þ β · � log

� σ 0

σ 0ref

�� 3

ðstress range 10 kPa < σ 0 < 10 MPaÞ

ð2Þ in terms of three model parameters eref, α, and β. In addition, the asymptotic void ratio eL at low stress can be imposed as a plateau.

Alternatively, the classical semi-logarithmic Terzaghi model can be modified to satisfy asymptotic conditions: e → eL at low stress σ 0 → 0, and e → eH at high stress σ

0 → ∞

e ¼ ec − Cc log �

1 kPa σ 0 þ σ 0L

þ 1 kPa σ 0H

�−1 ð3Þ

where parameters ec and Cc determine the central trend, and void ratio asymptotes eL and eH define stresses σ

0 L and σ

0 H

σ 0H ¼ 10ðec−eHÞ=Cc · kPa when σ 0 → ∞ ð4Þ

σ 0L ¼ σ 0H

10ðeL−eHÞ=Cc − 1 when σ 0 → 0 ð5Þ

The generalized Terzaghi model in Eq. (3) involves four model parameters of clear physical meaning.

Models in Terms of e − σ 0β Power function: From gases to soils. Loosely packed small grains bear resemblance to the notion of a gas. Boyle-Mariotte’s law ignores the size of molecules, and concludes that pressure and volume are inversely related

PV ¼ constant → V ¼ a=P ð6Þ where α = constant. van der Waals corrected this expression to take into consideration the size of molecules and rewrote Boyle’s equation in terms of the contractible volume V 0, i.e., the total volume Vt minus the volume excluded Vex by the molecules, V 0 ¼ Vt − Vex. He also considered intermolecular interactions and the additional stress due to uncompensated attraction at the boundaries.

Following a parallel analysis, and taking into consideration electrical attraction and repulsion in fine-grained soils (σA−σR), a plausible equation for soil compressibility becomes:

ðσ 0 þ σA − σRÞðVt − VsÞ ¼ α ð7Þ If the volume of solids as Vs is assumed constant, this equation can be written in terms of void ratio e ¼ ðVt − VsÞ=Vs

e ¼ α 0

σ 0 þ σA − σR ð8Þ

This inverse relationship between void ratio e and effective stress σ 0 can be generalized as a four-parameter inverse power func- tion that accommodates the two void ratio asymptotes eL and eH

e ¼ eH þ ðeL − eHÞ � σ 0 þ σ 0c

σ 0c

�−β ð9Þ

When the applied effective stress equals the characteristic effective stress σ 0 ¼ σc and β ¼ 1, the predicted void ratio is the average of the asymptotes e ¼ ðeL þ eHÞ=2. Higher β-exponents cause higher early compressibility at lower stresses. Power- type equations have been suggested in the past (Hansen 1969; Butterfield 1979; Juárez-Badillo 1981; Houlsby and Wroth 1991; Pestana and Whittle 1995).

Exponential: The main characteristic of exponential functions y ¼ expðxÞ is that the rate of change dy=dx is defined by the cur- rent state. Sigmoidal and Gompertz functions are special examples (Gompertz 1825; Gregory et al. 2006). The four-parameter Gompertz function can be expressed in terms of stress and void ratio, and adapted to satisfy eL and eH as follows:

e ¼ eH þ ðeL − eHÞ · exp− � σ 0 σ 0c

� β

ð10Þ

When the exponent β ¼ 1, the simpler three-paremeter expo- nential expression is obtained (Cargill 1984); it predicts that the soil will experience 63% of the volume change eL − eH when the applied effective stress equals the characteristic effective stress σ 0 ¼ σ 0c.

Hyperbolic: The hyperbolic model is extensively used in geo- mechanics to capture the prepeak deviatoric stress versus strain data (Kondner 1963; Duncan and Chang 1970). This model has two parameters: one defines the initial rate of change dy=dxjo, and the other provides the asymptotic value of y as x → ∞. The model can be adapted to capture compressibility data in terms of σ 0 − e. The generalized four-parameter hyperbolic model is

e ¼ eL − ðeL − eHÞ 1 1 þ

� σ 0c σ 0 �β ð11Þ

The simpler hyperbolic model (β ¼ 1) predicts that the void ra- tio will reach the intermediate void ratio e ¼ ðeL þ eHÞ=2 when the applied effective stress equals the characteristic stress σ 0 ¼ σ 0c. Structuration and yield stress can be captured with higher values of the β-exponent.

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Arctangent: Other functions that provide S-shaped trends can be adapted to satisfy asymptotic conditions relevant to soil compres- sion data. For example, the arctangent function can be generalized to include the power of the stress σ 0β in order to fit more brittle soil responses

e ¼ eL þ 2

π ðeL − eHÞ arctan

� − � σ 0

σ 0c

� β �

ð12Þ

in terms of four model parameters eL, eH, the characteristic stress σ 0c, and the β-exponent. For β ¼ 1, the void ratio reaches e ¼ ðeL þ eHÞ=2 when the applied effective stress equals the character- istic stress σ 0 ¼ σ 0c.

Discussion

Examples

Compression data gathered for a wide stress range are fitted using these models in Fig. 1 for both remolded soils and for natural soils with distinct yield stress; fitting parameters are summarized in Table 1. Data points and fitted models are plotted in both log-linear and linear-linear plots; the last column in Fig. 1 shows computed trends for the tangent constrained modulus Mtan. The following can be observed:

• Remolded soils: All four-parameter models addressed here can adequately fit experimental data gathered for remolded soils [Fig. 1(a)]. The tangent stiffness computed with these models shows monotonic stiffening.

• Natural soils: Hyperbolic, arctangent, power, and exponential (albeit to a lesser extent) models written in terms of σ 0β approx- wimate the brittle response of natural structured soils better than logðσ 0Þ models [Fig. 1(b)]. The tangent stiffness com- puted with hyperbolic and arctangent models shows early softening until the yield stress, followed by compaction-driven stiffening. Furthermore, the S-shaped models presented earlier are well

suited to fit the response of overconsolidated soils as well. These observations are confirmed with multiple cases compiled from the literature.

Correlations—Low-Stress Void Ratio and Compressibility

A database of consolidation tests was compiled from the literature for the purposes of this study [The complete data- base can be found in Chong (2014).] The classical Terzaghi model was fitted to data gathered with remolded, normally consolidated soils within the available effective stress range. Results show that the void ratio e1 kPa is closely related to the void ratio at the liquid limit eLL ¼ GsLL=100 (assuming 100% saturation)

Fig. 1. Wide stress range 1D compression data (ko, zero lateral strain boundary condition) fitted with different soil compressibility models: (a) re- molded clay—sodium montmorillonite at 0.001 N and pH ¼ 7 (data from Mesri and Olson 1971); (b) natural clay—Bothkennar soil from 6.5 m (data from Burland 1990)

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e1 kPa ¼ 5

4 eLL ¼

1

80 Gs · LL

remolded soils ð28 cases; SD ¼ 0.36; R2 ¼ 0.92Þ ð13Þ

This equation compares well with the eLL versus e100 correlation in Burland (1990), as e1 kPa ¼ e100 kPa þ 2Cc. Furthermore, a direct link is obtained between the void ratio at 10 kPa and the void ratio at liquid limit

e10 kPa ¼ eLL remolded soils ð28 cases; SD ¼ 0.26; R2 ¼ 0.90Þ ð14Þ

The correlation between compressibility and liquid limit was recognized early on in the field (Skempton 1944; Terzaghi and Peck 1948; Burland 1990). The presented database shows a very similar trend

Cc ¼ 0.008 · ðLL − 5Þ remolded soils ð28 cases; SD ¼ 0.11; R2 ¼ 0.90Þ ð15Þ

Given the eLL − e1 kPa and Cc − LL correlations, this study ex- plored the relationship between model parameters e1 kPa and Cc [Fig. 2(a)]

e1 kPa ¼ 3.4 · Cc þ 0.48 remolded soils ð54 cases; SD ¼ 0.19; R2 ¼ 0.96Þ ð16Þ

A limited set of nine cases was identified to compare natural soils with the same soil after remolding (from Mitchell 1956; Mesri et al. 1975; Burland 1990; Hong et al. 2006; Wesley 2009). Structured natural soils pack at higher void ratio enatural1 kPa ¼ 1.6 · eremolded1 kPa and compress more Cnaturalc ¼ 1.7 · Cremoldedc than the remolded counterparts.

The link between void ratio at low stress and compressibility in logðσ 0Þ models emerges in other models too. For example, a mea- sure of compressibility in the hyperbolic model σ 0c=ðeL − eHÞ is linked to the asymptotic low-stress void ratio eL as [Fig. 2(b)]

eL ¼ 15.8 � σ 0c

1 kPa

eL − eH �−0.4

natural clays ð23 cases; SD ¼ 0.4; R2 ¼ 0.84Þ ð17Þ

Parameter Invertibility

Parameters were selected by least-square fitting the models to the data, min½Σðemi − epi Þ2�. The error surface about the optimal param- eter set is explored by varying one parameter at a time to assess the invertibility of each parameter. As anticipated, limited data at very low or very high stress results in poor convergence for eL or eH, respectively.

Model parameters can be constrained with a priori data, for ex- ample, mineralogy to bound eL, and geological data to restrict eH. Furthermore, the β-exponent is bounded (Table 1), and the low- stress void ratio eL and compressibility are correlated (Fig. 2). These constraints and correlations help to identify a self-consistent set of fitting parameters; furthermore, they suggest an effective model complexity lower than the four unknowns involved in these models.

Void ratio in sedimentary basins: Compressibility models [Eqs. (1)–(3), (9)–(12)] can be integrated to compute void ratio trends versus depth. The nonlinear decrease in void ratio with depth observed in sedimentary basins (e.g., data reported in Aplin et al. 1995) can be properly matched with all models reviewed here.

Tangent stiffness: Small-strain versus large-strain. Numerical solutions that use a tangent formulation involve the tangent constrained modulus M

M ¼ ∂σ 0

∂ε ¼ −∂σ 0 ∂e ð1 þ eÞ ð18Þ

The void ratio versus stress models discussed earlier can be ap- plied to isotropic loading conditions by replacing σ 0 → p 0 and e → v ¼ 1 þ e. Then, the tangent bulk stiffness K becomes

K ¼ dp 0

dεv ¼ −dp

0

dv v ð19Þ

Equations obtained for either M or K for all compressibility models [Eqs. (1)–(3) and (9)–(12)] can be found in the Supplemen- tal Data.

The tangent constrained modulus M or bulk modulus K com- puted using Eqs. (18) and (19) (Supplemental Data) are fundamen- tally different from the small strain stiffness measured at the same e − σ 0 state. The tangent stiffness is a mathematical concept that reveals the instantaneous rate of fabric change during a large strain test. By contrast, a small-strain perturbation (e.g., shear wave propagation) is a constant fabric measurement of stiffness and is determined by contact deformation. Therefore, the magnitude of tangent stiffness and the trends reported in Fig. 1 should not be associated to small-stress stiffness values.

Settlement computation: Estimation of yield stress: Standard settlement analyses assume recompression (er1 kPa, Cr) and normal compression (en1 kPa, Cc) segments before and after the yield stress σ 0y. Several ad hoc methods have been proposed to determine the yield stress or preconsolidation pressure (Casagrande 1936; Janbu 1969; Pacheco 1970; Sallfors 1975; Butterfield 1979; Becker et al. 1987; Oikawa 1987; Jose et al. 1989; Sridharan et al. 1991; Onitsuka et al. 1995; Grozic et al. 2003; Clementino 2005; Boone 2010; Ku and Mayne 2013). Single function models, such as those compiled and augmented here, can capture the complete compres- sion response and streamline computations.

Table 1. Fitting Parameters (Refer to Data Presented in Fig. 1)

Model Parameters Montmorillonite

0.001 N [Fig. 1(a)] Bothkennar [Fig. 1(b)]

Modified Terzaghi

eL 36.5 2.05 eH 0.20 0.52 ec 52 5.2 Cc 21.5 1.6

Power eL 35.1 2.03 eH 0.40 0.63 β 1.0 2.0

σ 0c (kPa) 48 700 Exponential eL 38.5 2.00

eH 1.40 0.72 β 0.55 1.0

σ 0c (kPa) 70 400 Hyperbolic eL 36.0 1.98

eH 0.40 0.65 β 0.9 1.32

σ 0c (kPa) 43 280 Arctangent eL 36.5 1.95

eH 0.30 0.72 β 0.7 1.3

σ 0c (kPa) 42 250

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Conclusions

Many geotechnical problems involve either very low, very high, or a wide range of effective stress. In this study, previously suggested compression models and other functions are modified to satisfy asymptotic conditions at low and high stress levels. These models are used to fit the response of normally consolidated remolded soils, structured natural soils, and overconsolidated soils. The fol- lowing salient observations can be made: • At least four parameters are required to fit soil compression data

gathered in a wide stress range. Physical insight, databases, and the high correlation between compressibility and the void ratio at low stress help to constrain the parameter space to determine a self-consistent set of fitting parameters;

• Models in terms of σ 0β (arctangent, hyperbolic, power, and exponential) show more flexibility to capture the compression response of structured natural soils with pronounced brittle transitions at the yield stress;

• The use of a single continuous function to capture soil com- pressibility data avoids numerical discontinuities, facilitates computing the tangent constrained modulus for numerical methods, and eludes ad hoc procedures to identify the yield stress; and

• The computed tangent constrained modulus should not be mistaken for the instantaneous small-strain modulus measured during the test. The tangent stiffness is a mathematical con- cept that reveals the instantaneous rate of fabric change during a large strain test. By contrast, a small-strain perturbation test gives a constant fabric stiffness that is determined by contact deformation.

Acknowledgments

This research was conducted by the authors while at the Georgia Institute of Technology. Support for this research was provided by the Department of Energy Savannah River Operations Office and the Goizueta Foundation. Additional support was provided by the Convergence R&D program of MSIP/NST (Convergence Research-14-2-ETRI) and the KAUST endowment.

Supplemental Data

Tables S1 and S2 are available online in the ASCE Library (www. ascelibrary.org).

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