Isotropic Damage Mechanics

Constitutive Law

This MPM softening material is an isotropic, elastic material, but once it initiates damage, it evolves using isotopic damage mechanics methods. The constitutive law for this material is

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

where [math]\displaystyle{ \mathbf{C} }[/math] 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 to initiate and evolve damage according is to use a scalar metric. Once that is in place, material modeling can simply apply one-dimensional damage mechanics methods. Unfortunately, real-life damage is not one dimensional and it is likely the entire concept of isotropic damage mechanics is misguided. A better approach to extending 1D damage mechanics to 3D is to switch to anisotropic damage mechanics. Anisotropic damage mechanics replaces the scalar metric from 1D methods with evolution determined by 3 components of traction on a crack surface. Isotropic damage mechanics is made available here mostly for comparison purposes. It is likely not a realistic approach to modeling damage in materials, although it can be more stable to other damage mechanics methods (but stability alone is not enough to justify its use).

Stress Metric

One common metric is based on strain energy, which is implemented here using a stress metric:

      [math]\displaystyle{ \Phi(\mathbf{\sigma},D) = \sqrt{E \mathbf{\sigma}\cdot \mathbf{S}\mathbf{\sigma}} - F(D) }[/math]

Here [math]\displaystyle{ \mathbf{\sigma} }[/math] is current stress, [math]\displaystyle{ E }[/math] and [math]\displaystyle{ \mathbf{S} }[/math] are modulus and compliance tensor of the isotropic material, and [math]\displaystyle{ F(D) }[/math] is a strength model that gives the material's current strength as a function of the scalar damage variable. The modulus [math]\displaystyle{ E }[/math] is included to convert the energy metric to units of stress for comparison to the current strength. For example, under uniaxial loading in the [math]\displaystyle{ x }[/math] direction:

      [math]\displaystyle{ \sqrt{E \mathbf{\sigma}\cdot \mathbf{S}\mathbf{\sigma}} = \sqrt{ \sigma_{xx} E \varepsilon_{xx}} = {\rm abs}(\sigma_{xx}) }[/math]

or the current axial stress that can be compared to strength model in [math]\displaystyle{ F(D) }[/math] to check for failure.

Tensile Principle Stress Metric

A drawback of a the stress metric is that it includes energy due to both tension and compression. For example, the material would damage in unaixal compression when [math]\displaystyle{ {\rm abs}(\sigma_{xx})=F(D) }[/math]. A modification to avoid this drawback is to find principal stresses, [math]\displaystyle{ \bar{\mathbf{\sigma}} }[/math], and redefine the stress metric to a tensile principal stress metric:

      [math]\displaystyle{ \Phi(\mathbf{\sigma},D) = \sqrt{E \langle\bar{\mathbf{\sigma}}\rangle\cdot \mathbf{S}\bar{\mathbf{\sigma}}} - F(D) }[/math]

where [math]\displaystyle{ \langle\mathbf{\sigma}\rangle }[/math] means to include only tensile principal stress in the stress tensor. Now under uniaxial compression for example:

      [math]\displaystyle{ \sqrt{E \langle\bar{\mathbf{\sigma}}\rangle\cdot \mathbf{S}\bar{\mathbf{\sigma}}} = \sqrt{ \langle\sigma_{xx}\rangle E \varepsilon_{xx}} = 0 }[/math]

when [math]\displaystyle{ \sigma_{xx}\lt0 }[/math] and damage will only occur in tension.

Mixed-Mode Metric

A drawback of both the above metrics is that they make no distinction between tensile or shear failure. An option to model both tension and shear failure is to propose a failure surface in principal stress space that models tension and shear failure. The option implemented here is to assume damage evolves when maximum principal stress exceeds current tensile strength of the material ([math]\displaystyle{ F_I(D) }[/math]) or when maximum shear stress exceeds current shear strength of the material ([math]\displaystyle{ F_{II}(D) }[/math]). A plot of this failure criterion in principal stress state is given in the Isotropic Material Failure Surface initiation law. When using this approach, the metric, referred to as the mixed-mode metric, is defined (symbolically) as:

      [math]\displaystyle{ \Phi(\mathbf{\sigma},D) = d\bigl(\bar{\mathbf{\sigma}},F_I(D),F_{II}(D)\bigr) }[/math]

where [math]\displaystyle{ d() }[/math] represent a calculation of signed distance from the current principal stress state [math]\displaystyle{ \bar{\mathbf{\sigma}} }[/math] to the principal stress failure surface defined by current tensile and shear strengths ([math]\displaystyle{ F_I(D) }[/math] and [math]\displaystyle{ F_{II}(D) }[/math]). The sign is negative when stress is inside the failure surface and positive when it is outside.

Strength Models

Note that [math]\displaystyle{ F(D) }[/math], [math]\displaystyle{ F_I(D) }[/math], and [math]\displaystyle{ F_{II}(D) }[/math] strength models in the above metrics are actually recast in terms of associated damage parameter [math]\displaystyle{ \delta }[/math], [math]\displaystyle{ \delta_I }[/math], and [math]\displaystyle{ \delta_{II} }[/math] that correspond to crack-opening displacement in 1D damage mechanics. Strength models referred to as "linear" or "exponential" (etc.) mean they are linear or exponential in [math]\displaystyle{ \delta }[/math] variables not in D. This material can use any softening law defined in terms of [math]\displaystyle{ \delta }[/math] for use with anisotropic damage mechanics materials.

When using stress metric or tensile principal stress metric, only [math]\displaystyle{ F(D) }[/math] is used and it is specfied using this material's "tensile strength" model. When using mixed-mode metric, [math]\displaystyle{ F_I(D) }[/math], and [math]\displaystyle{ F_{II}(D) }[/math] and both used and entered as separate tensile and shear strength models.

Damage Evolution

Damage evolution occurs whenever the chosen metric from the previous section exceeds 0 (i.e., when the current stress state exceeds the strength). Whenever that state occurs, damage evolves by solving the consistency equation

      [math]\displaystyle{ \nabla\Phi(\mathbf{\sigma},dD)\cdot(d\mathbf{\varepsilon},dD)=0 }[/math]

where [math]\displaystyle{ d\mathbf{\varepsilon} }[/math] is the stress increment for the current time step and [math]\displaystyle{ dD }[/math] is the calculated damage evolution to solve the equation. For stress energy metric and linear softening, this equation can be solved in closed form. For most other cases, it requires a numerical solution, but the computational cost is small. The implementation here uses a bracketed, Illinois method that appears fast and very stable.

When D evolves to one (i.e., the current strength degrades to zero), the material point is failed and the decohesion is reported to the output file (or can be diverted to a global results file using the "Decohesion" quantity).

Material Properties

When this material is undamaged, it's response is identical to properties entered for the underlying isotropic material. Once those are specified, you have to select a damage metric and one or 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 = stress metric, 1 = tensile principal stress metrix, and 2 = mixed mode metric (see above).	none	0
sigmac	Critical stress for [math]\displaystyle{ F_I(D) }[/math] strength model (it is tensile strength when metric=2)	pressure units	infinite
SofteningI	Attach a softening law (by name or ID) for evolution of damage or for tensile failure when metric=2. Once attached, enter all required properties for that law by prefacing each property with "I-". [math]\displaystyle{ F_I(D) }[/math] is sigmac times this softening law.	none	Linear
tauc	Critical shear stress for [math]\displaystyle{ F_{II}(D) }[/math] strength model (it is only used when metric=2)	pressure units	infinite
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-". [math]\displaystyle{ F_{II}(D) }[/math] is tauc times this softening law.	none	Linear
coefVariation wShape wV0 statDistributionMode	See the corresponding properties defined for an Isotropic softening material.
(other)	Properties common to all materials	varies	varies

If needed, you can set set initial damage parameter for particles using particle boundary conditions.

History Variables

This material stores allocates all history variables used by its parent IsoSoftening material, but only the following ones are used:

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
     • 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.
2. δ isotropic damage variable (δ_I for metric=2)
3. δ_II (metric=2 only)
4. Cumulative dissipated energy (this material does not separate mode I and mode II energy)
5. D or the scalar damage variable. It varies from 0 to 1 where 1 is complete damage or failure.
6. not used
7. not used
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.
12. Relative strengths set by coefVariation and coefVariationMode properties or assigned with a PropertyRamp Custom Task.
13. Relative toughnesses set by coefVariation and coefVariationMode properties or assigned with a PropertyRamp Custom Task.

Variables 8-10 define the normal to the damage crack plane. For metric=0 or metric=1, the normal direction is not clear; for a substitute, these values will record the maximum principal normal stress direction at initiation. For metric=2, the normal will be in the initial direction; it will be normal to the failure surface at the time of initiation. 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
  sigmac 10
  SofteningI Linear
  I-Gc 10000
Done