| Author | Message | ||
     Ananda Wijesinghe (ananda) New member Username: ananda Post Number: 1 Registered: 05-2006 |
I am attempting to calculate elasto plastic material deformation problems (in 2D) with FlexPDE. Typically, these problems involve an incremental formulation in which the strains are decomposed additively into elastic and plastic contributions. The elastic strains are related to the stresses through Hookes Law. The plastic contribution is non-zero only if the material has yielded and the stress state lies on the yield surface. Continued loading beyond yield requires the plastic stress increments to remain on the yield surface. The plastic strains are obtained through a flow rule and the yield "consistency" condition. Usually these problems are posed in incremental form, which is not possible with FlexPDE which requires the problem to be posed as differential equations supplemented by non-incremental analytical expressions. To overcome this limitation, I converted the incremental elasto-plasticity equations to ODE's with respect to time (as in visco-plasticity formulations) and tried to solve them as ODE's in time together with the equilibrium PDEs which do not have time derivatives. However, this failed to work, because FlexPDE did not like the combination of ODE's in time with PDE's in space. Has anyone solved elasto-plastic deformation problems with FlexPDE? Thank you for your responses. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 607 Registered: 06-2003 |
FlexPDE applies timestep control logic only to those equations that contain time derivatives. If none of your equations have time derivatives, there will be no timestep control, and the solution will run amok. 1) You might try using a manual timestep. SELECT FIXDT and set the timestep in the TIME section. You will need to choose a timestep small enough to survive the intervals of most rapid change. 2) alternatively, you could add a variable and equation that is time dependent (an energy equation, perhaps) that can serve as a basis for timestep control. | ||
| Ananda Wijesinghe (ananda) New member Username: ananda Post Number: 2 Registered: 05-2006 |
Thank you very much, Robert. Actually, I have selected the velocities and stresses to be independent solution variables and have written the stress-strain constitutive equations as ODEs wrt to time to relate the stress rates and velocity gradients. I use the initial stress state as the initial conditions for the stresses. I solve for the velocities and not the displacements to avoid time derivatives on both sides of the ODEs; I get the displacements subsequently using the "tintegral" function to integrate the velocities in time. These ODE constitutive equations are intended to enable time step control as you suggested in your second tip, and are tagged as the governing equations for the stress components. In addition, there are two equilibrium equations without time-derivatives that are obtained by setting the divergence of the stress tensor equal to zero. They are tagged as the governing equations for the two velocity components. The code took a long time to evaluate the Jacobian. Because the rate form of the constitutive equations had already linearized the non-linear elasto-plastic constitutive functions, I speeded up the Jacobian evaluation by declaring the linearization coefficients to be "Passive" functions to prevent further differentiation. I am now using manual time stepping with SELECT FIXDT as you suggested in your first tip, but the script still does not work because of a divide by zero error. I will debug the script to identify the source of this and will post a simplified version for you to look at, if it still does not work. Thanks again. |