View source for Isotropic Softening Material

== Constitutive Law ==

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\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<sup>th</sup> rank damage tensor appropriate for damage in isotropic materials, and <math>\mathbf{\varepsilon}_{res}</math> is any residual strain (such as thermal or solvent induced strains).

An appropriate damage tensor was first proposed by Chaboche<ref name="chaboche"/>, and was implemented in mpm using this material typel for complete modeling of anisotropy caused by 3D damage evolution<ref name="dmref"/>. 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 Laws|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 [[Setting Forces and Fluxes#Initial Particle Damage|initial particle damage]] using [[Particle-Based Boundary Conditions|particle boundary conditions]].

== Damage Evolution ==

Damage evolution is determined by [[Softening Laws|softening laws]] to predict degradation of normal and shear tractions across the crack plane. You need to attach two [[Softening Laws|softening laws]] to this material. These two laws handle tensile and shear damage and the areas under the laws correspond to fracture toughnesses ''G<sub>Ic</sub>'' and lumped ''G<sub>IIc</sub>''/''G<sub>IIIc</sub>'' 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<ref name="dmref"/>. A second paper shows that damage mechanics and fracture mechanics can give identical results provided damage parameters are appropriately chosen<ref name="dmFM"/>.

== Damage Generalizations ==

[[File:FailureSurfacePlots.png|400px|right]]

A third paper generalizes anisotropic damage mechanics including a method to couple damage parameters for more realistic modeling.<ref name="dmGen"/> This coupling is based on choice of a traction failure surface and the three options are pictured on the right. These surfaces show failure surface in terms of current normal and shear tractions on the crack surface (they are for crack opening or tension on the crack surface; the changes for compression are described below). The response for each are as follows:

* Cuboid (0): This option provides uncoupled damage mechanics. Each direction evolves damage independently. If any direction exceeds the surface, it returns to an evolved surface in a direction normal to the surface. Although this approach is common in damage and cohesive zone modeling, the lack of coiupling means that damage in one direction does not influence loading in other directions. Imagine loading a material in tension such that it causes much damage but does not cause failure. If that material was unloaded and then reloaded in shear, the use of uncoupled damage parameters means it would act like a virgin material with no damage. Because this response is likely unrealistic, the other two surface add coupling options.
* Elliptical Cylinder (1): This option couples shear directions only. If traction exceeds the surface, it returns normal to the cylinder to an evolved state, but that state will evolve damage in both shear directions. The tensile damage is still decoupled and occurs when traction exceeds top of the cylinder. The next surface couples in tension as well.
* Ovoid (2): This option couples all directions. If the crack surface traction exceeds this surface, it returns to an evolved surface by following a path in the direction of the origin. By this approach, the tractions in all directions simultaneously reach zero at failure. In effect, damage variables are coupled such that d<sub>n</sub> = d<sub>xy</sub> = d<sub>xz</sub> = d, but the damage crack opening variables (&delta;<sub>n</sub>, &delta;<sub>xy</sub>, and &delta;<sub>xz</sub> parameters) will remain different.
* Coupled Cuboid (3): This option uses the cuboid failure surface but combines that with the full coupling of the ovoid surface. In brief, if an traction direction exceeds the cuboid surface, the damage evolves by returning to a new cuboid surface by a path to the origin rather than by a path normal to the cuboid surface. This approach keeps the damage parameters coupled (<i>i.e.</i>, d<sub>n</sub> = d<sub>xy</sub> = d<sub>xz</sub> = d) and guarantees all components of traction simultaneously reach zero at failure.

This picture does not show behavior when crack surfaces are in contact. The cuboid and elliptical cylindrical surface extend their shapes to the compression half plane with no limit (<i>i.e.</i>, no normal direction failure in compression). The ovoid surface uses an infinite elliptical cylinder in compression but damage evolution follows a path to the origin rather then normal to the surface. In other words, shear damage that occurs in compression evolves tensile damage parameters as well to keep all directions coupled.

These failure surfaces are selected with the <tt>tractionFailureSurface</tt> material property and ovoid is currently the default failure surface.

== White Noise Statistical Variations ==

Damage mechanics is often more realistic when the strength and/or toughness of the material points have statistical variations. The recommended way to model these statistical variations is to use Gaussian random fields, but that takes more work. A simple option in this material is to just randomly assigned strengths and/or toughness to material points at the start of the simulation. This method is a special case of a Gaussian random field with spatial correlation function a step function equal to the particle radius. In other words, this method will depend on mesh size and cannot vary spatial effects to match real materials. Nevertheless, simulations with damage mechanics materials are often improved by adding some white noise statistical variations. This option is especially helpful in problems where boundary conditions cause premature damage initiation.

Statistical variations can use a normal distribution or a Weibull distribution. The chosen distribution is used to assign relative strength or toughness for each material point. The values are relative to strength in the [[Damage Initiation Laws|damage initiation law]] or toughness in the [[Softening Laws|softening law]] for this material. The relative values can apply to strength alone, toughness alone, or to both strength and toughness. The one that are varied are set using the <tt>statDistributionMode</tt> property. The statistical values can be visualized by viewing [[#History Variables|history 12 or 13]] (which remain constant throughout the simulation).

=== Normal Distribution ===

When using a normal distribution, the cumulative distribution for probability, <math>p</math>, that a relative value is less then <math>r</math> is

&nbsp;&nbsp;&nbsp;&nbsp;
<math>p = \frac{1}{2}\left[1 + {\rm erf}\left( \frac{r-1}{C_v\sqrt{2}}\right)\right]</math>

where <math>C_v</math> is coefficient of variation of the distribution (entered with <tt>coefVariation</tt> parameter). When using this distribution, the simulations starts by randomly picking <math>p</math> (over the interval 0.001 to 0.999) and then numerical solving for <math>r</math>. This calculation is done for each particle.

=== Weibul Distribution ===

When using

<math>w = 1 - \exp\left(-V_r\left(\frac{\sigma}{\langle\sigma\rangle}\Gamma\left(1+\frac{1}{\alpha}\right)\right)^\alpha\right)</math>

== Material Properties ==

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

{| class="wikitable"
|-
! Property !! Description !! Units !! Default
|-
| ([[Isotropic Material#Material Properties|Isotropic Properties]]) || Enter all properties needed to define the underlying isotropic material response || varies || varies
|-
| Initiation || Attach [[Damage Initiation Laws|damage initiation law]] by name or ID that is compatible with isotropic materials. Once attached, enter all required material properties for that law. || none || [[Isotropic Material Failure Surface|IsoFailure]]
|-
| SofteningI || Attach a [[Softening Laws|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 Laws|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 || Selects traction failure surface and coupling methods as disucssed above [[#Damage Generalizations|above]]. The options are 0 = cuboid, 1 = elliptical cylinder, 2 = ovoid, and 3 = coupled cuboid. For backward compatibility, old files with the [[#Deprecated Material Properties|deprecated shearFailureSurface]] property are accepted and its rectangular and elliptical options are converted to cuboid (which is identical to prior rectangular option) and the new elliptical cylinder (which corrects prior elliptical option) methods. || none || 0
|-
| coefVariation || This property assigns a coefficient of variation to failure properties. The property that is affected is determined by the <tt>coefVariationMode</tt> 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
|-
| ([[Common Material Properties|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.

{| class="wikitable"
|-
! Property !! Description !! Units !! Default
|-
|rowspan="2"|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
|-
| <b>Reason</b>: 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:

# 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
#** 0.9: damage initiated by tensile failure
#** 1.0: damage initiation by both in-plane principle stresses exceeding tensile strength
#** 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<sub>I</sub>/G<sub>total</sub> = h[1]-2 or 2 is pure mode II failure, 3 is pure mode I failure, and anything between is mixed-mode failure.
# δ<sub>n</sub> or the maximum normal cracking strain.
# δ<sub>xy</sub> or the maximum x-y shear cracking strain.
# This variable has two options:
#* For 3D when using cuboid surface: δ<sub>xz</sub> or the maximum x-z cracking strain.
#* For all other cases: G<sub>I</sub> or cumulative mode I dissipated energy.
# d<sub>n</sub> or damage variable for normal loading. It varies from 0 to 1 where 1 is complete damage or failure.
# d<sub>xy</sub> or damage variable for x-y shear loading. It varies from 0 to 1 where 1 is complete damage or failure.
# This variable has two options:
#* For 3D when using cuboid surface: d<sub>xz</sub> or damage variable for x-z shear loading (from 0 to 1 where 1 is complete damage or failure).
#* For all other cases: G<sub>II</sub> or cumulative mode II dissipated energy.
# For 2D it is cos(θ), but for 3D it is Euler angle α.
# For 2D it is sin(θ), but for 3D it is Euler angle β.
# For 2D it is not used, but for 3D it is Euler angle γ.
# ''A<sub>c</sub>''/''V<sub>p</sub>'' where ''A<sub>c</sub>'' is crack area within the particle and ''V<sub>p</sub>'' is particle volume.
# Relative strength derived at the start by <tt>coefVariation</tt> and <tt>coefVariationMode</tt> properties.
# Relative toughness derived at the start by <tt>coefVariation</tt> and <tt>coefVariationMode</tt> 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 [[MPM Archiving Options|<tt>damagenormal</tt> 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<sub>c</sub>''/''V<sub>p</sub>'' (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  [[MPM Archiving Options|<tt>plasticstrain</tt> archiving option]]. The strain is archived in the global axis system. If you also [[MPM Archiving Options|archive the <tt>damagenormal</tt>]], 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

== References ==

<references>
<ref name="chaboche">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 Behavior of Anisotropic Solids / Comportment M ́echanique des Solides Anisotropes, pages 737–760. Springer Netherlands.</ref>

<ref name="dmref">J. A. Nairn, C. C. Hammerquist, and Y. E. Aimene, "Numerical Implementation of Anisotropic Damage Mechanics," Int. J. for Numerical Methods in Engineering, 112(12), 1846-1868 (2017). [http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MPMSoftening.pdf PDF]</ref>

<ref name="dmFM">J. A. Nairn, "Direct Comparison of Anisotropic Damage Mechanics to Fracture Mechanics of Explicit Cracks," Eng. Fract. Mech., 203, 197-207 (2018). [http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/ADAMvsFM.pdf PDF]</ref>

<ref name="dmGen">J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," Int. J. for Numerical Methods in Engineering, in press (2022).</ref>
</references>