Isotropic Damage Mechanics

Constitutive Law

This MPM 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 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 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 using 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.

A 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.

A drawback of a the above strength metric is that it includes energy due to both tension and compression. A modification to avoid this drawback is to find principle stresses, [math]\displaystyle{ \bar{\mathbf{\sigma}} }[/math], and redefine the stress metrivc 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 principle stress in the stress tensor.

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 principle 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 principle stress state is given in the Isotropic Material Failure Surface initial 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 as signed distance from the current principle stress state [math]\displaystyle{ \bar{\mathbf{\sigma}} }[/math] to the principle 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.

Note that [math]\displaystyle{ F_I(D) }[/math] and [math]\displaystyle{ F_{II}(D) }[/math] strength models are actually recast in terms of an associated damage parameter [math]\displaystyle{ \delta }[/math] that corresponds 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] and not in D. This material can used an softening law defined for use with [Isotropic Softening Material|anisotropic damage mechanics]] combine with tensile and shear strengths.

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},D)\cdot(d\mathbf{\varepsilon},D)=0 }[/math]

where [math]\displaystyle{ d\mathbf{\varepsilon} }[/math] is the stress increment for the current time step. For stress energy metric and linear softening, the equation can be solved in closed form. For most other cases, it requires a numerical solution, but the computation cost is not large. The implementation here uses a bracketed, Illinois method that appears fast and very stable.

When D evolves to one (i.e., the current strengths 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 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 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 energy, 1 = tensile principle stress energy, and 2 = mixed mode failure surface (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	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:

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
3. not used
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 strength derived at the start by coefVariation and coefVariationMode properties.
13. 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
  sigmac 10
  SofteningI Linear
  I-Gc 10000
Done