Problems for Chapter 8
Theory and Implementation of the Finite Element Method
8.4. The Finite Element Method For Hyperelastic Materials
8.4.1. Write an input file for the demonstration hyperelastic finite element code described in Section 8.4.7 to calculate the stress in a Neo-Hookean tensile specimen subjected to uniaxial tensile stress. It is sufficient to model the specimen using a single 8 noded brick element. Use the code to plot a graph showing the nominal uniaxial stress as a function of the stretch ratio . Compare the finite element solution with the exact solution.
8.4.2. Extend the hyperelastic finite element code described in Section 8.4.7 to solve problems involving a Mooney-Rivlin material. This will require the following steps:
220.127.116.11. Calculate the tangent stiffness for the Mooney-Rivlin material
18.104.22.168. Modify the procedure called Kirchoffstress(…), which calculates the Kirchoff stress in the material in terms of , and modify the procedure called materialstiffness(…), which computes the corresponding tangent stiffness. Run the following tests on the code, using an 8 noded brick element to mesh the specimen:
· Subject the specimen to a prescribed change in volume, and calculate the corresponding stress in the element. Compare the FE solution with the exact solution.
· Subject the specimen to a prescribed uniaxial tensile stress, and compare the FE solution with the exact solution
· Subject the specimen to a prescribed biaxial tensile stress, and compare the FE solution to the exact solution.
8.4.3. Modify the hyperelastic finite element code described in Section 8.4.7 to apply a prescribed true traction to the element faces. To do this, you will need to modify the procedure that calculates the element distributed load vector, and you will need to write a new routine to compute the additional term in the stiffness discussed in Section 8.4.6. Test your code by repeating problem 1, but plot a graph showing true stress as a function of stretch ratio.
(c) A.F. Bower, 2008