Isotropic Softening Material

Constitutive Law

This MPM Material is an isotropic, elastic material, but once it fails, it develops anisotropic damage. The material is available only in OSParticulas.The constitutive law for this material is

      [math]\displaystyle{ \mathbf{\sigma} = (\mathbf{I} - \mathbf{D}) \mathbf{C} \mathbf{\varepsilon} }[/math]

where C is stiffness tensor for the underlying isotropic material and D is an anisotropic 4^th rank damage tensor. The important questions for implementing this material are:

1. When does damage initiate?
2. Once damage is form, what damage tensor, D, should be used to describe the anisotropic response after failure?
3. How does damage evolve?

The first question is answered by attaching a damage initiation law to the material. This laws define a failure envelop. Once the response reaches the envelop, the damage process is initiated and the normal to the envelop define the normal to the crack plane model by this damage mechanics material.

The second question is answered by using the damage tensor proposed by Chabocch

Chaboche, J. (1982). Le concept de contrainte effective appliqu ́e a` l’ ́elasticit ́e et a` la viscoplasticit ́e en pr ́esence d’un endommagement anisotrope. In Boehler, J.-P., editor, Mechanical Behav- ior of Anisotropic Solids / Comportment M ́echanique des Solides Anisotropes, pages 737–760. Springer Netherlands.