# Weak form for elastoplastic wave propagation

I am trying to simulate elastoplastic seismic wave propagation using Fenics Solid Mechanics Application.

The app. provides some quasi-static demos to show elastoplastic behaviour in a cube/ beam/ square. The weak form they implement is taken from page 199 of this paper written by the developers themselves, and is given as:

All the terms are defined in a paragraph below the equation in the paper. The equation is very similar to the weak form obtained for linear elasticity. It must be noted that in computational plasticity, return mapping algorithms (Simo and Hughes 1998) are popular and hence the application uses a 'closest point projection' return mapping algorithm along with Newton method. The stress (sigma) in the linear form 'L' is the residual stress (or stress from previous iteration). In the LHS (or bilinear form 'a') the variable 'C' is the consistent elastoplastic tangent moduli. Other variables are Body force 'f'; traction 'h'; test function 'v'; and trial function 'u'.

So far so good. Now, for elastic linear wave propagation the weak form implemented is:

Here, the test function 'w' is the same as the test function 'v' in equation (6) and (7). The only new term is the 'intertia term', where $/rho$ is the mass density of material and $/ddot{u}$ is the acceleration.

Considering the above 2 equations, is it fair to say that I can implement an elastoplastic wave using the following weak form?

Here $C_{ep}$ is the same as $C$ in equation (6).

• You can ask FEniCS-specific question at fenicsproject.org/qa. – Christian Clason Jun 24 '16 at 17:06
• Because of complications that arise when time derivatives are integrated over a domain, the problem is typically split into FE for space and finite differences for time. Also, the closest point projection is in a space that is scaled so that it is the energy norm (see p. 188 in Simo & Highes). – Biswajit Banerjee Jun 28 '16 at 0:23