 Problems for Chapter 8

Theory and Implementation of the Finite Element Method

8.5.  The Finite Element Method For Plastic Materials

8.5.1.        Set up the demonstration viscoplastic finite element code described in Section 8.5.7  to calculate the stress-strain relation for the viscoplastic material under uniaxial tension.  Mesh the specimen with a single 8 noded brick element, using the mesh shown in the figure.   Apply the following boundary constraints to the specimen:  at node 1;  at node 2,  at nodes 3 and 4.  Apply a uniform traction whose magnitude increases from 0 to 20 (arbitrary units) in time of 2 units on face 2 of the element.

8.5.1.1.               Run a simulation with the following material parameters:  and plot a graph showing the variation of traction with displacement on the element

8.5.1.2.              Modify the boundary conditions so that only the constraint  is enforced at node 2.  Run the code to attempt to find a solution (you will have to abort the calculation).  Explain why the Newton iterations do not converge.

8.5.1.3.              Repeat 1.1 with  , the correct boundary conditions, and with a maximum traction of 18 units.  In this limit the material is essentially rate independent.  Compare the predicted traction-displacement curve with the rate independent limit.

8.5.2.        Modify the viscoplastic finite element program to apply a constant (nominal) uniaxial strain rate to the specimen described in the preceding problem, by imposing an appropriate history of displacement on the nodes of the mesh.  Test the code by plotting a graph showing uniaxial stress-v-strain in the specimen, for material parameters  (in this limit the material is essentially a power-law creeping solid with constant flow stress) with an applied strain rate of  .  Compare the numerical solution with the exact solution. 8.5.3.        Modify the viscoplastic finite element program so that instead of using a power-law function to represent the variation of flow-stress  with accumulated plastic strain  , the flow stress is computed by interpolating between a user-defined series of points, as indicated in the figure (the flow stress is constant if the plastic strain exceeds the last point).  Test your code by using it to calculate the stress-strain relation for the viscoplastic material under uniaxial tension.  Use the mesh described in Problem 1, and use material parameters  (this makes the material essentially rigid and rate independent, so the stress-strain curve should follow the user-supplied data points).

8.5.4.        Modify the viscoplastic finite element code described in Section 8.5.7 to solve problems involving a rate independent, power-law isotropic hardening elastic-plastic solid, with incremental stress-strain relations    and a yield criterion  Your solution should include the following steps:

8.5.4.1.              Devise a method for calculating the stress  at the end of a load increment.  Use a fully implicit computation, in which the yield criterion is exactly satisfied at the end of the load increment.  Your derivation should follow closely the procedure described in Section 8.5.4, except that the relationship between  and  must be calculated using the yield criterion, and you need to add a step to check for elastic unloading.

8.5.4.2.              Calculate the tangent stiffness  for the rate independent solid, by differentiating the result of 4.1.

8.5.4.3.              Implement the results of 4.1 and 4.2 in the viscoplastic finite element code.

8.5.4.4.              Test your code by using it to calculate the stress-strain relation for the viscoplastic material under uniaxial tension.  Use the mesh, loading and boundary conditions described in Problem 1, and use material properties  8.5.5.        Modify the viscoplastic finite element code described in Section 8.5.7 to solve problems involving a rate independent, linear kinematic hardening elastic-plastic solid, with incremental stress-strain relations    and a yield criterion and hardening law  Your solution should include the following steps:

8.5.5.1.              Devise a method for calculating the stress  at the end of a load increment.  Use a fully implicit computation, in which the yield criterion is exactly satisfied at the end of the load increment.  Your derivation should follow closely the procedure described in Section 8.5.4, except that the relationship between  and  must be calculated using the yield criterion and hardening law, and you need to add a step to check for elastic unloading.

8.5.5.2.              Calculate the tangent stiffness  for the rate independent solid, by differentiating the result of 8.5.5.1.

8.5.5.3.              Implement the results of 8.5.5.1 and 8.5.5.2 in the viscoplastic finite element code.

8.5.5.4.              Test your code by using it to calculate the stress-strain relation for the viscoplastic material under uniaxial tension.  Use the mesh, loading and boundary conditions described in Problem 1, and use material properties  8.5.6.        In this problem you will develop a finite element code to solve dynamic problems involving viscoplastic materials.  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  8.5.6.1.              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  .

8.5.6.2.              To implement the finite element method, it is necessary to calculate the stress  in the solid.  Idealize the solid as a viscoplastic material with constitutive equations described in Section 8.5.1.  Since very small time-steps must be used in an explicit dynamic calculation, it is sufficient to integrate the constitutive equations with respect to time using an explicit method, in which the plastic strain rate is computed based on the stress at the start of a time increment.   Show that the stress  at time  can be expressed in terms of the stress  at time t, the increment in total strain  during the time interval  and material properties as  8.5.6.3.              The 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 viscoplastic code and the Newmark elastodynamic code provided. 8.5.6.4.              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.

(c) A.F. Bower, 2008
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