Problems for Chapter 8

Theory and Implementation of the Finite Element Method

8.3.  The Finite Element Method For Hypoelastic Materials

8.3.1.        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 50 (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 showing the variation of the radial displacement of the inner bore of the cylinder as a function of the internal pressure.  HEALTH WARNING: The example code uses fully integrated elements and will give very poor results for large values of n.

8.3.2.        Solve the following coupled nonlinear equations for x and y using Newton-Raphson iteration.

8.3.3.        Consider the simple 1D bar shown in the figure.  Assume that the boundary conditions imposed on the bar are identical to those for the 1D elastic bar discussed in Section 8.1.5, so that  is the only nonzero displacement component in the bar.  Assume that the material has a hypoelastic constitutive equation, which has a nonlinear volumetric and deviatoric response, so that the stress-strain relation has the form

where

8.3.3.1.              Show that the virtual work principle can be reduced to

8.3.3.2.              Introduce a 1D finite element interpolation for  and  following the procedure outlined in Section 8.1.5 to obtain a nonlinear system of equations for nodal values of .  Show that the nonlinear equations can be solved using a Newton-Raphson procedure, by repeatedly solving a system of linear equations

where  denotes a correction to the current approximation to .  Give expressions for  and .

8.3.3.3.              Extend the simple 1D linear elastic finite element code described in Section 8.1.5 to solve the hypoelasticity problem described here.

8.3.3.4.              Test your code by (i) calculating a numerical solution with material properties , n=2,  and loading b=0, .  and (ii) calculating a numerical solution with material properties , n=10,  and loading b=10, . Compare the numerical values of stress and displacement in the bar with the exact solution.

8.3.4.        Calculate the tangent moduli  for the hypoelastic material described in the preceding problem.

8.3.5.        In this problem you will develop a finite element code to solve dynamic problems involving a hypoelastic material with the constitutive model given in Section .  Dynamic problems for nonlinear materials are nearly always solved using explicit Newmark time integration, which is very straightforward to implement.  As usual, the method is based on the virtual work principle

8.3.5.1.              By introducing a finite element interpolation, show that the virtual work principle can be reduced to a system of equations of the form

and give expressions for .

8.3.5.2.              These equations of motion can be integrated using an explicit Newmark method, using the following expressions for the acceleration, velocity and displacement at the end of a generic time-step

A lumped mass matrix should be used to speed up computations.   Note that the residual force vector  is a function of the displacement field in the solid. It therefore varies with time, and must be re-computed at each time step.  Note that this also means that you must apply appropriate constraints to nodes with prescribed accelerations at each step. Implement this algorithm by combining appropriate routines from the static hypoelastic code and the Newmark elastodynamic code provided.

8.3.5.3.              Test your code by simulating the behavior of a 1D (plane strain) shown in the figure.  Assume that the bar is at rest and stress free at t=0, and is then subjected to a constant horizontal traction at  for t>0.  Fix the displacements for the node at  and apply  at .  Take the magnitude of the traction to be 2 (arbitrary units) and use material properties .  Take  in the Newmark integration, and use 240 time steps with step size 0.01 units.   Plot a graph showing the displacement of the bar at  as a function of time.  Would you expect the vibration frequency of the end of the bar to increase or decrease with n?  Test your intuition by running a few simulations.

(c) A.F. Bower, 2008
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