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 4th rank damage tensor. The important questions for implementing this material are:
- When does damage initiate?
- Once damage is form, what damage tensor, D, should be used to describe the anisotropic response after failure?
- 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 defines the normal to the crack plane model by this damage mechanics material.
The second question is answered by using the damage tensor proposed by Chaboche[1]. This fourth rank tensor depends on two damage variables, which can be show to relate to mode I and mode II damage only the crack plane. The normal to the crack plane, as found from the initiation law, defines the natural axis system for the damage tensor.
The third question is answered by attaching softening laws to this material. Because an underlying isotropic material has two damage parameters, this material needs two softening laws. These laws handle mode I and mode II damage and the areas under the laws correspond to GIc and GIIc for the material.
In brief, this material models crack initiation and propagation through damage mechanics. By use of softening laws, the material properties are tied to toughness properties for the material. The scheme can handle interacting cracks (really interacting damage zones) and 3D cracks, although it is uncertain whether or not it can capture all the physics of models that use explicit cracks instead of damage mechanics. History variables can archive the extent of damage and the orientation of the damage planes. More details on implementation of this material are planned for a future paper.[2]
History Variables
Examples
References
- ↑ J. Chaboche (1979). 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.
- ↑ J. A. Nairn (2016), Numerical Implementation of Anisotropic Damage Mechanics, in preparation.