Isotropic Softening Material

Constitutive Law

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

      [math]\displaystyle{ \mathbf{\sigma} = (\mathbf{I} - \mathbf{D}) \mathbf{C}( \mathbf{\varepsilon}- \mathbf{\varepsilon}_{res}) }[/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, and [math]\displaystyle{ \mathbf{\varepsilon}_{res} }[/math] is any residual strain (such as thermal or solvent induced strains).

An appropriate damage tensor was first proposed by Chaboche, and was implemented in mpm using this material typel for complete modeling of anisotropy caused by 3D damage evolution. This fourth rank tensor depends on three damage variables, which can be shown to relate to one tensile and two shear damage processes related to a crack plane. These three 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 needed 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.

Predamage on any particle at the start of a simulation can be set using initial particle damage using particle boundary conditions.

Damage Evolution

Damage evolution is determined by softening laws to predict degradation of normal and shear tractions across the crack plane. 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 lumped G_IIc/G_IIIc for the material.

In brief, this material models crack initiation and propagation through damage mechanics. The softening law's properties tie the damage mechanics to toughness properties for the material. The scheme can handle interacting cracks (which become interacting damage zones) and 3D cracks. MPM modeling using this material is described in a recent paper. A second paper shows that damage mechanics and fracture mechanics can give identical results provided damage parameters are appropriately chosen. A third paper generalizes anisotropic damage mechanics including a method to couple damage parameters for more realistic modeling.

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	IsoFailure
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
tractionFailureSurface	Select traction failure surface assumed that determines coupling between anisotropic damage mechanics parameters. The options are 0 = cuboid, 1 = cylindrical, and 2 = ovoid. For cuboid, the three damage parameters evolve independently. For cylindrical, the two shear parameters are coupled but tension and shear are independent. For ovoid, all three damage parameters are coupled. For backward compatibility, old files with the deprecated shearFailureSurface property are accepted and its rectangular and elliptical options are converted to the new cuboid (which is identical to prior rectangular option) and new cylindrical (which corrects prior elliptical option) methods.	none	2
coefVariation	This property assigns a coefficient of variation to failure properties. The property that is affected is determined by the coefVariationMode parameter. Each particle's relative property is set at the start of the simulation to have the same Gaussian distribution of values about their means, but will have no spatial correlations. A better approach to stochastic modeling would use Gaussian random fields with spatial correlation (see below).	none	0
coefVariationMode	The options are 1 = vary only strength, 2 = vary only toughness, and 3 = vary strength and toughness. Note that strength, toughness, and critical crack opening displacement (COD) are interrelated. Option 1 means COD will increase to keep toughness constant; 2 means COD will decrease to keep strength constant; 3 means COD will remain constant.	none	1
coeff	coefficient of friction for post-decohesion contact (default is 0 or frictionless) (experimental implementation in development in OSParticulas only)	none	0
(other)	Properties common to all materials	varies	varies

An alternative to randomly varying strength or toughness using coefVariation and coefVariationMode properties is to set the relative values using a PropertyRamp Custom Task. For example, a BMP image of a Gaussian random field could assign relative strengths or toughness with random variations that include spatial correlations.

Deprecated Material Properties

The following softening materials are no longer valid for the reasons given below.

Property	Description	Units	Default
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
	Reason: The two methods give similar results and some new theory showed that the elliptical coupling method did not couple shear damage properly. This property was deprecated January 6, 2020.	none	1

History Variables

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

1. The current damage state with the following possible values:
   • 0.1: indicates undamaged material. Note that undamaged value of 0.1 is to facilitate mapping of damage state to a grid such that undamaged regions can be distinguished by thresholding from empty regions (with zero damage).
   • 0.9, 1.0, or 1.1: indicates damage has initiated but particle has not yet failed. The three values are
     1. 0.9: damage initiated by tensile failure
     2. 1.0: damage initiation by both in-plane principle stresses exceeding tensile strength
     3. 1.1: damage initiated by shear failure
   • Between 2 and 3: particle has failure by decohesion. After failure, the fraction energy dissipated by mode I damage is G_I/G_total = h[1]-2 or 2 is pure mode II failure and 3 is pure mode I failure and anything else is mixed-mode failure.
2. δ_n or the maximum normal cracking strain.
3. δ_xy or the maximum x-y shear cracking strain.
4. This variable has two options:
   1. δ_xz or the maximum x-z cracking strain, but only for 3D when using cuboid surface.
   2. G_I cumulative mode I dissipated energy for all other cases.
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. This variable has two options:
8. d_xz or damage variable for x-z shear loading (from 0 to 1 where 1 is complete damage or failure), but only for 3D when using cuboid surface.
   1. G_II cumulative mode II dissipated energy for all other cases.
9. For 2D it is cos(θ), but for 3D it is Euler angle α.
10. For 2D it is sin(θ), but for 3D it is Euler angle β.
11. For 2D it is not used, but for 3D it is Euler angle γ.
12. A_c/V_p where A_c is crack area within the particle and V_p is particle volume.
13. Relative strength derived at the start by coefVariation and coefVariationMode properties.
14. Relative toughness derived at the start by coefVariation and coefVariationMode properties.

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. Although damaged normal is a unit vector, it is archived with magnitude equal to A_c/V_p (which gets another history variable archived and the value is used for some visualization options).

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.

Examples

Material "isosoft","Isotropic Softening Material",50
  E 1000
  nu .33
  a 60
  rho 1
  largeRotation 1
  Initiation MaxPrinciple
  sigmac 30
  tauc 20
  SofteningI Linear
  I-Gc 10000
  SofteningII Linear
  II-Gc 10000
Done