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 appropriate for damage in isotropic materials.

An appropriate damage tensor was first proposed by Chaboche, but was modified in this material for more complete modeling of anisotropy caused by 3D damage evolution. This fourth rank tensor depends on two damage variables, which can be shown to relate to tensile and shear damage along the crack plane. These two damage variables can be related to mode I and lumped mode II/III fracture mechanics failure modes.

Damage Initiation

Damage initiation is controlled 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 normal is need to find the anisotropic D tensor (which involves rotating analysis into the crack axis system where the x axis is aligned with the crack normal.

Damage Evolution

Damage evolution is determined by softening laws laws to predict degradation of normal and shear tractions across the crack plane. Because an underlying isotropic material has two damage parameters, you need to attach two softening laws to this material. These two laws handle tensile and shear damage and the areas under the laws correspond to fracture toughnesses G_Ic and G_IIc for the material.

In brief, this material models crack initiation and propagation through damage mechanics. The softening laws properties tie the damage mechanics to toughness properties for the material. The scheme can handle interacting cracks (really interacting damage zones) and 3D cracks. MPM modeling using this material is described in a recent 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 that is compatible with isotropic materials. Once attached, enter all required material properties for that law.	none	MaxPrinciple
SofteningI	Attach a softening law (by name or ID) for propagation of tensile 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 shear damage. Once attached, enter all required properties for that law by prefacing each property with "II-".	none	Linear
shearFailureSurface	Select failure surface assumed when modeling shear damage in 3D calculations. Use 1 for an elliptical failure criterion based on current degraded shear strengths. Use 0 for a rectangular failure surface that encloses the elliptical failure criterion. The elliptical surface is preferred, but rectangular is more efficient. || none || 1
(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. 0, 0.9, 1.1, 1.9, or 2.1 to indicate undamaged (0), damage propagation (0.9 or 1.1), or post failure (decohesion) state of the particle (1.9 or 2.1). 0.9 and 1.9 indicate the failure initiated by tensile strength while 1.1 and 2.1 indicate failure initiated by shear strength.
2. δ_n or the maximum normal cracking strain.
3. δ_xy or the maximum x-y shear cracking strain.
4. δ_xz or the maximum x-z cracking strain (zero for 2D).
5. d_n or damage variable for normal loading. It varies from 0 to 1 where 1 is complete damage or failure.
6. d_xy or damage variable for x-y shear loading. It varies from 0 to 1 where 1 is complete damage or failure.
7. d_xz or damage variable for x-z shear loading. It varies from 0 to 1 where 1 is complete damage or failure (zero for 2D).
8. For 2D it is cos(θ), but for 3D it is Euler angle α.
9. For 2D it is sin(θ), but for 3D it is Euler angle β.
10. For 2D it is not used, but for 3D it is Euler angle γ.
11. A_c/V_p where A_c is crack area within the particle and V_p is particle volume.

Variables 8-10 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 save enough information for plotting the normal.

This material also tracks the cracking strain which can be saved 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.