Isotropic Damage Mechanics

Constitutive Law

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

[math]\displaystyle{ \mathbf{\sigma} = (1-D) \mathbf{C}( \mathbf{\varepsilon}- \mathbf{\varepsilon}_{res}) }[/math]

where C is stiffness tensor for the underlying isotropic material, D is a scalar damage parameter, and [math]\displaystyle{ \mathbf{\varepsilon}_{res} }[/math] is any residual strain (such as thermal or solvent induced strains).

Damage Metric

The standard approach in isotropic damage mechanics is to evolve damage according to a scale metric. Once that is in place, material modeling can simple apply one-dimensional damage mechanics methods. Unfortunately, real-like damage is not one dimensional and it is likely the entire concept of isotropic damage mechanics is misguided. The solution extending 1D damage mechanics to 3D is to switch use anisotropic damage mechanics.

Damage Evolution

(to be written)

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 select a damage metric and or 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
metric	Choose the effective strain used in damage evolution with options 0 = Principal stress energy, 1 = Tensile stress energy, and 2 = mixed mode failure surface.	none	0
SofteningI	Attach a softening law (by name or ID) for evolution of damage or for tensile dame when metric=2. 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, but only used when metric=2. Once attached, enter all required properties for that law by prefacing each property with "II-".	none	Linear
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
(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.

History Variables

This material stores allocates all history variables used by its parent IsoSoftening material, but only some of them are used:

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 (metric=2 only)
  - 1.0: damage initiated (metric=0 or 1) or intiated by both in-plane principle stresses exceeding tensile strength(metric=2 only)
  - 1.1: damage initiated by shear failure (metric=2 only)
- After decohesion, this number adds 1 to one of the previous three values.
δ isotropic damage variable
not used
Cumulative dissipated energy (this material does not separate mode I and mode II energy)
D or the scalar damage variable. It varies from 0 to 1 where 1 is complete damage or failure.
not used
not used
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_c/V_p where A_c is crack area within the particle and V_p is particle volume.
Relative strength derived at the start by coefVariation and coefVariationMode properties.
Relative toughness derived at the start by coefVariation and coefVariationMode properties.

Variables 8-10 define the normal to the damage crack plane (only interesting for metric=2). 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 damage strain which can be saved by using the plasticstrain archiving option.

Examples

Material "isodam","Isotropic Damage Material",58
  E 1000
  nu .33
  a 60
  rho 1
  largeRotation 1
  SofteningI Linear
  I-Gc 10000
Done