Problems for Chapter 8
Theory and Implementation of the Finite Element Method
8.6. Advanced Element Formulations
8.6.1. Volumetric locking can be a serious problem in computations involving nonlinear materials. In this problem, you will demonstrate, and correct, locking in a finite element simulation of a pressurized hypoelastic cylinder.
18.104.22.168. Set up an input file for the example hypoelastic finite element code described in Section 8.3.9 to calculate the deformation and stress in a hypoelastic pressurized cylinder deforming under plane strain conditions. Use the mesh shown in the figure, with appropriate symmetry boundary conditions on and . Apply a pressure of 20 (arbitrary units) to the internal bore of the cylinder and leave the exterior surface free of traction. Use the following material properties: . Plot a graph of the variation of the radial displacement of the inner bore of the cylinder as a function of the applied pressure. Make a note of the displacement at the maximum pressure.
22.214.171.124. Edit the code to reduce the number of integration points used to compute the element stiffness matrix from 9 to 4. (modify the procedure called `numberofintegrationpoints’). Repeat the calculation in 126.96.36.199. Note the substantial discrepancy between the results of 188.8.131.52 and 184.108.40.206 this is caused by locking. The solution in 5.2, which uses reduced integration, is the more accurate of the two. Note also that using reduced integration improves the rate of convergence of the Newton-Raphson iterations.
8.6.2. Modify the hypoelastic finite element code described in Section 8.3.9 to use selective reduced integration. Check your code by (a) repeating the calculation described in problem 8.6.1; and (b) running a computation with a mesh consisting of 4 noded quadrilateral elements, as shown in the figure. In each case, calculate the variation of the internal radius of the cylinder with the applied pressure, and plot the deformed mesh at maximum pressure to check for hourglassing. Compare the solution obtained using selective reduced integration with the
8.6.3. Run the simple demonstration of the B-bar method described in Section 8.6.2 to verify that the method can be used to solve problems involving near-incompressible materials. Check the code with both linear and quadratic quadrilateral elements.
8.6.4. Extend the B-bar method described in Section 8.6.2 to solve problems involving hypoelastic materials subjected to small strains. This will require the following steps:
220.127.116.11. The virtual work principle for the nonlinear material must be expressed in terms of the modified strain measures and defined in Section 8.6.2. This results in a system of nonlinear equations of the form
which must be solved using the Newton-Raphson method. Show that the Newton-Raphson procedure involves repeatedly solving the following system of linear equations for corrections to the displacement field
where is defined in Section 8.6.2.
18.104.22.168. Modify the hypoelastic code provided to compute the new form of the stiffness matrix and element residual. You will find that much of the new code can simply be copied from the small strain linear elastic code with the B-bar method
22.214.171.124. Test the code by solving the problems described in 8.6.1 and 8.6.2.
8.6.5. Extend the B-bar method described in Section 8.6.2 to solve problems involving hypoelastic materials subjected to small strains, following the procedure outlined in the preceding problem.
8.6.6. In this problem you will extend the B-bar method to solve problems involving finite deformations, using the hyperelasticity problem described in Section 8.4 as a representative example. The first step is to compute new expressions for the residual vector and the stiffness matrix in the finite element approximation to the field equations. To this end
· New variables are introduced to characterize the volume change, and the rate of volume change in the element. Define
Here, the integral is taken over the volume of the element in the reference configuration.
· The deformation gradient is replaced by an approximation , where n=2 for a 2D problem and n=3 for a 3D problem, while J=det(F).
· The virtual velocity gradient is replaced by the approximation
The virtual work equation is replaced by
126.96.36.199. Verify that
188.8.131.52. In calculations to follow it will be necessary to calculate . Find an expression for .
184.108.40.206. The virtual work equation must be solved for the unknown nodal displacements by Newton-Raphson iteration. Show that, as usual, the Newton-Raphson procedure involves repeatedly solving the following system of linear equations for corrections to the displacement field
and derive expressions for and .
(c) A.F. Bower, 2008