Transversely Isotropic Softening Material

Constitutive Law

This MPM Material is a transversely isotropic, elastic material, but once it fails, it develops anisotropic damage and will become orthotropic. The material is available only in OSParticulas and still in development. 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 transversely isotropic material and D is an anisotropic 4^th rank damage tensor appropriate for damage in transversely isotropic materials, and [math]\displaystyle{ \mathbf{\varepsilon}_{res} }[/math] is any residual strain (such as thermal or solvent induced strains).

The MPM implementation of softening isotropic materials is described in Nairn, Hammerquist, and Aimene^[1] This extension to transversely isotropic will be in a future publication.

TransIsoSoftening 1 and 2

TransIsoSoftening 1 and TransIsoSoftening 2 give identical materials but with different initial orientations. TransIsoSoftening 1 has the unrotated axial direction along the z (or θ if axisymmetric) axis while TransIsoSoftening 2 has unrotated axial direction along the y (or Z if axisymmetric) axis. You can change the unrotated direction to any other orientation when defining material points by selecting rotation angles. For 2D analyses, the two options allow for axial direction in the x-y (or R-Z if axisymmetric) analysis plane (TransIsoSoftening 2) or normal to that plane (TransIsoSoftening 1). For 3D analyses, only TransIsoSoftening 1 is allowed (and it in the only one needed).

Damage Process

The current implementation limits crack formation such that the crack normal is normal to a symmetry direction of the material. This approach means cracks either have the normal in the axial direction and perpendicular to the axial direction. Crack growth in material symmetry direction within these two planes suggest a material defined by three critical toughness values:

G_AT,c - toughness for crack growth in plane with normal in the axial direction. Because this crack plane is the material's isotropic plane, the toughness should be the same for all crack growth directions within that plane.
G_TA,c and G_TT,c - toughness for crack growth in plane with normal in the transverse direction. Within this plane, the crack may propagate in the axial direction (by toughness G_TA,c) or in the transverse direction (by toughness G_TT,c).

In other words, the first letter in the toughness subscript is the normal to the crack plane while the second is the crack propagation direction within that plane. Each of these crack paths make be loaded in tension (mode I) or shear (mode II) leading to six toughness values. This material seeks to model all these crack options. Because a material point is either failed or not failed, it is not possible to separate the two mode I paths in the transverse plane. We are thus left with five fracture process and five toughness values:

G_AT,c^(I) - mode I crack growth in plane with normal in axial direction
G_AT,c^(II) - mode II crack growth in plane with normal in axial direction
G_T,c^(I) - mode I crack growth in plane with normal in transverse direction (an average of all direction is that plane)
G_TA,c^(II) - mode II crack growth in plane with normal in transverse direction for crack propagating in axial direction
G_TT,c^(II) - mode II crack growth in plane with normal in transverse direction for crack propagating in transverse direction

Damage Initiation and Propagation

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 crack plane is calculated depending on type of failure. 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. The only damage surface currently allowed for a transversely isotropic softening material is the TIFailure initiation law. Damage propagation and evolution is determined by softening laws laws to predict degradation of normal and shear tractions across the crack plane.

Full definition of a transversely isotropic requires specification of a strength and a softening law corresponding to each of the five modeled crack propagation modes described above:

G_AT,c^(I) - strength is sigmacA, softening law is softeningAI (i.e., for axial failure in mode I)
G_AT,c^(II) - strength is taucT, softening law is softeningTII
G_T,c^(I) - strength is sigmac, softening law is softeningI
G_TA,c^(II) - strength is taucA, softening law is softeningAII
G_TT,c^(II) - strength is tauc, softening law is softeningII

You need to attach five strengths and five softening laws to this material. In brief, this material models crack initiation and propagation through damage mechanics. The softening law properties tie the damage mechanics to toughness properties for the material. The scheme can handle all model failure modes, interacting cracks (which become interacting damage zones), and 3D cracks.

The damage evolves during a simulation, but, if desired, predamage on any particle at the start of a simulation can be set using initial particle damage using particle boundary conditions.

Material Properties

When the material is undamaged, its response is identical to properties entered for the underlying transversely isotropic material. Once those are specified, you have to attach one damage initiation law and five softening laws, and all properties needed by those laws. These properties determine how the material responds after initiates and propagates damage.

Property	Description	Units	Default
(Transversely Isotropic Properties)	Enter all properties needed to define the underlying transversely isotropic material response	varies	varies
Initiation	Attach damage initiation law by name or ID that is compatible with this material. Once attached, enter all required strength properties for that law.	none	TIFailure
SofteningAI	Attach a softening law (by name or ID) for propagation of tensile damage by G_AT,c^{I crack growth. Once attached, enter all required properties for that law by prefacing each property with "AI-".}	none	Linear
SofteningTII	Attach a softening law (by name or ID) for propagation of shear damage by G_AT,c^{II crack growth. Once attached, enter all required properties for that law by prefacing each property with "TII-".}	none	Linear
SofteningI	Attach a softening law (by name or ID) for propagation of tensile damage by G_T,c^{II crack growth. Once attached, enter all required properties for that law by prefacing each property with "I-".}	none	Linear
SofteningAII	Attach a softening law (by name or ID) for propagation of shear damage by G_TA,c^{II crack growth. Once attached, enter all required properties for that law by prefacing each property with "AII-".}	none	Linear
SofteningII	Attach a softening law (by name or ID) for propagation of shear damage by G_TT,c^{II crack growth. Once attached, enter all required properties for that law by prefacing each property with "II-".}	none	Linear
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
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, but the feature is not yet implemented.	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

History Variables

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

0, 0.75, 0.8, 0.85, 0.95, 1.05, 1.15, 1.25, or 1 higher than previous 7 to indicate undamaged (0), damage propagation (0.75, 0.8, 0.85, 0.95, 1.05, 1.15, 1.25), or post failure (decohesion) state of the particle. The specific values indicate the failure mode that initiated the damage:
- 0.75, 1.25, 1.75, 2.25: Transverse Tension
  - 0.75, 1.75: Material axial direction along z axis in the crack axis system
  - 1.25, 2.25: Material axial direction along y axis in the crack axis system
- 0.95, 1.95: Axial Tension
  - Material axial direction along x axis in the crack axis system
- 0.80, 1.15, 1.80, 2.15: Axial Shear
  - 0.80, 1.80: Material axial direction along z axis in the crack axis system
  - 1.15, 2.15: Material axial direction along y axis in the crack axis system
- 1.05, 2.05: Transverse Shear
  - Material axial direction along x axis in the crack axis system
- 0.85, 1.85: Rolling Shear (or shear failure in isotropic plane)
  - Material axial direction along z axis in the crack axis system
δ_n or the maximum normal cracking strain.
δ_xy or the maximum x-y shear cracking strain.
δ_xz or the maximum x-z cracking strain (zero for 2D).
d_n or damage variable for normal loading. It varies from 0 to 1 where 1 is complete damage or failure.
d_xy or damage variable for x-y shear loading. It varies from 0 to 1 where 1 is complete damage or failure.
d_xz or damage variable for x-z shear loading. It varies from 0 to 1 where 1 is complete damage or failure (zero for 2D).
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. 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

This example can be a starting point for modeling of wood

 Material "wood","Douglas fir","TransIsoSoftening"&#type$
   EA 12000
   ET 900
   GA 800
   nuT .4
   nuA .33
   alphaA 0
   alphaT 40
   rho 0.5
   largeRotation 1
   Initiation "TIFailure"
   sigmac 10
   tauc 3
   sigmacA 100
   taucA 10
   taucT 30
   strengthCoefVariation 0.3
   SofteningEA Linear
   SofteningGA Linear
   SofteningET Linear
   SofteningGT Linear
   EA-Gc 800
   GA-Gc 600
   ET-Gc 200
   GT-Gc 400
 Done

References

↑ 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). PDF

[dmref-1] 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). PDF

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