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 formed, 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. These 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 modeled by this damage mechanics material.

The second question is answered by using the anisotropic damage tensor proposed by Chaboche. This fourth rank tensor depends on two damage variables, which can be shown to relate to mode I and mode II damage along the crack plane. The normal to the crack plane, as found from the initiation law, defines the crack 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 G_Ic and G_IIc 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.

Material Properties

When the material is undamaged, it response is identical to properties entered for the underlying isotropic material. Once those are specified, you have to attach one damage initiation law and two softening laws to define how the material responds after initiation of damage.

Property	Description	Units	Default
(Isotropic Properties)	Enter all properties needed to define the underlying isotropic material response	varies	varies
Initiation	Attach damage initiation law by name or ID. Once attached, enter all required material properties for that law	none	MaxPrinciple
SofteningI	Attach a softening law (by name or ID) for propagation of mode I damage. Once attached, enter all required properties for that law by prefacing each property with "I-".	none	Linear
SofteningII	Attach a softening law (by name or ID) for propagation of mode II damage. Once attached, enter all required properties for that law by prefacing each property with "II-".	none	Linear
(other)	Properties common to all materials	varies	varies

History Variables

This material stores several history variables that track the extent of the damage and orientation of the damage plane:

1. δ_n or the current normal crack opening displacement.
2. δ_t or the current tangential crack opening displacement.
3. d_n or damage variable for normal loading. It varies from 0 to 1 where 1 is complete damage or failure.
4. d_t or damage variable for tangential loading. It varies from 0 to 1 where 1 is complete damage or failure.
5. For 2D it is cos(θ), but for 3D it is Euler angle α.
6. For 2D it is sin(θ), but for 3D it is Euler angle β.
7. For 2D it is not used, but for 3D it is Euler angle γ.
8. A_c/V_p where A_c is crack area within the particle and V_p is particle volume.

Variables 4-6 define the normal to the damage crack plane. For 2D, θ is the counter clockwise angle from the x axis to the crack normal. For 3D, (α, β, γ) are the three Euler angles for the normal direction using a Z-Y-Z rotation scheme. You can use the damagenormal archiving option to say enough information for plotting the normal.

This material also tracks the cracking strain by using the plasticstrain archiving option. The strain is archived in the global axis system. If you also archive the damagenormal, you will be able to plot a vector along the crack-opening displacement vector.