FE Verification using photoelasticity and constrained shape optimization

Common Project: FE verification using photoelasticity and constrained shape optimization

Summary

(i) In Problem 1, you will verify the results of your finite element simulation with a

photoelastic experiment results for a isotropic material.

(ii) In problem 2,3, you will design a composite component with maximum strength

a. In case 1, you will perform carbon fiber path optimization for a tensile test.

b. In case 2, you will optimize under shear loading.

Note: Computational exercise only, manufacturing and experiments are not required for

this project.

Details

1. Validation of FE model, comparison with photoelastic fringes:

We will first validate the finite element model. Consider a rectangular plate of uniform

thickness with a circular hole at the center. The plate is a homogeneous isotropic elastic

body (plexiglass) with Young's modulus E = 3.0 GPa and the Poisson ratio of 0.3. The

plate width is 20.85 mm, plate length is 64.9 mm and the diameter of the hole is 12.98

mm. Plate thickness is 4.98mm.

A uniform y-axis tension of 14.11 MPa is applied on the top surface. Model the system in

2D within plane stress approximation. Because of symmetry in the model and loading,

model only one quarter of the plate as shown in Fig 1.

An interesting comparison can be made with the stress pattern obtained by photoelasticity

in which fringes correspond to the maximum shear stress max 1 2

( ) / 2    . Here,

1 2 ,  are the principal stresses at each point in the material.

2 2

1 2 , ( )

2 2 xy x y x y

      

    

In photoelasticity, the stress optic law states

1 2 ( )

NF

t

  

where F 

= 14.35 N/mm/fringe is the material fringe value, N is the fringe order and t is

the thickness of the model.

6 4 .9

m m

20.85 mm

12 .9

8 m

m

x

y

Fig. 1: Geometry of the problem. One quarter of the plate is modeled due to symmetry

To simulate photoelastic results, one has to plot the contours corresponding to N. The

experimental result from photoelastic measurement of this problem is given in Fig. 2.

(i) Verify that the contour of N obtained by the finite element approximation are same as the fringe pattern obtained by photoelasticity as shown in figure 2.

Fig 2. Experimental results

1 of the fringe contours (N) for problem 1. Locations of the

lowest (N = 0) and highest (N = 24.5) fringe orders are also given.

1 M. M. Frocht, Photoelasticity, vol. 1, Wiley, New York, 1941.

2. The plate with hole geometry and loading in part (1) is now made using an epoxy matrix-carbon fiber composite with the following properties for the horizontal fiber (zero

degree) ply (moduli in GPa):

E1=145.880, G12=4.386, E2=13.312, G23=4.528,

v12=v13=0.263, v23=0.47

(a) What is the maximum von Mises stress for fibers oriented along the loading direction?

Von Mises stress is defined using the principal stresses as:

2 2 2

1 2 1 2

1 [( ) ]

2 vm

       

(b) Find the fiber topology that minimizes the maximum von Mises stress during y-axis tension. Use the function surfacegen to model the fiber topology. The function is

described in the powerpoint presentation. Also report the maximum allowable load in

the optimized specimen if failure stress is 144 MPa. What is the percentage

improvement in this maximum load compared to the straight fiber case in part 2(a).

(c) The y-axis tension in Fig 1 is replaced with an x-axis load (along positive x direction) of 14.11 MPa on the top surface. Report the optimal fiber topology in this case.