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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:
8.4.2.1.
Calculate the tangent stiffness for the Mooney-Rivlin material
8.4.2.2.
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.
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