Orthotropic Softening Material

Constitutive Law

This MPM softening material is an orthotropic, elastic material, but once it fails, it develops anisotropic damage. It will remain orthotropic, but properties in some dirrections will change. The constitutive law for this material is

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

where C is stiffness tensor for the underlying orthotopic material and D is an anisotropic 4^th rank damage tensor appropriate for damage in orthotropic 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. The extension to orthotropic materials is described in a paper on generalized damage mechanics methods.

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 will normal will always be in the x, y, or z direction of the material axes (these axes need not align with analysis axes). Crack growth in material symmetry directions within these three planes suggest a material defined by six cracking modes denoted by IJ where I and J are X, Y, or Z and I ≠ J. The first letter the the direction normal to the crack plane while the second letter is the direction of the crack growth. Each of these crack paths may be loaded in tension (mode I) or shear (mode II) leading to 12 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 of any plane. We are thus left with three mode I properties and six mode II properties that depend on the crack normal

1. Crack normal in the x direction
   • Mode I: initiation at sigmaXXc, toughness G_XX,c^(I)
   • Mode II: for propagation in y direction, initiation at tauXY-Yc, toughness G_XY-Y,c^(I)
   • Mode II: for propagation in z direction, initiation at tauXZ-Zc, toughness G_XZ-Z,c^(I)
2. Crack normal in the y direction
   • Mode I: initiation at sigmaYYc, toughness G_YY,c^(I)
   • Mode II: for propagation in x direction, initiation at tauXY-Xc, toughness G_XY-X,c^(I)
   • Mode II: for propagation in z direction, initiation at tauYZ-Zc, toughness G_YZ-Z,c^(I)
3. Crack normal in the z direction
   • Mode I: initiation at sigmaZZc, toughness G_ZZ,c^(I)
   • Mode II: for propagation in x direction, initiation at tauXZ-Xc, toughness G_XZ-X,c^(I)
   • Mode II: for propagation in y direction, initiation at tauYZ-Yc, toughness G_YZ-Y,c^(I)

Here tauIJ-Ic and tauIJ-Jc are strength values to initation failue in the IJ (I≠J) plane of the material. This failure will open a crack normal to the I (the common letter of the IJ pair) but propagating in the I or J direction (as indicate by -Ic or -Jc).

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 initiation law currently allowed for an orthotropic softening material is the OrthoFailure initiation law. Damage propagation and evolution is determined by softening laws that model degradation of normal and shear tractions across the crack plane.

Full definition of an orthotopic material requires specification of a strength and a softening law corresponding to each of the nine modeled crack propagation modes described in the previous section. Note that the six shear models indicate a shear plane with two indices following by a second index indicating crack growth direction. For example, tauXY-Xc is the strength to initiate failure by shear in the X-Y plane and that shear failure cases a crack to propagate the X direction, which corresponds to a crack with normal in the Y direction. The corresponding tauXY-Yc is failure in the same plane but resulting in a crack propagating in the Y direction, which corresponds to a crack with normal in the X direction. For X-Y shear loading of a given material, the shear crack formed will allows have its orientation defined by whichever shear strength is lower (min(tauXY-Xc,tauXY-Yc)). Thus, the larger shear value has no affect on shear failure initiation, but it can affect damage evolution. For example, if tauXY-Xc < tauXY-Yc, all shear cracks will have normal in the Y direction, but if tensile causes a crack with normal in the X direction, the value of tauXY-Yc will control subsequent shear damage evolution on that crack's surface in the Y direction. Similarly, shear failure in X-Z and Y-Z planes are controlled by two other pairs of shear strengths.

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 orthotropic material. Once those are specified, you have to attach one damage initiation law and nine softening laws, and all properties needed by those laws.

Property	Description	Units	Default
(Orthotropic Properties)	Enter all properties needed to define the underlying orthotropic 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	OrthoFailure
SofteningXX	Attach a softening law (by name or ID) for propagation of a tensile damage for a crack with normal in the x direction. Once attached, enter all required properties for that law by prefacing each property with "XX-".	none	Linear
SofteningYY	Attach a softening law (by name or ID) ffor propagation of a tensile damage for a crack with normal in the y direction. Once attached, enter all required properties for that law by prefacing each property with "YY-".	none	Linear
SofteningZZ	Attach a softening law (by name or ID) for propagation of a tensile damage for a crack with normal in the z direction. Once attached, enter all required properties for that law by prefacing each property with "ZZ-".	none	Linear
SofteningXYX	Attach a softening law (by name or ID) for propagation of shear damage for crack in with normal y direction caused by shear stress in the x-y plane. Once attached, enter all required properties for that law by prefacing each property with "XYX-".	none	Linear
SofteningXYY	Attach a softening law (by name or ID) for propagation of shear damage for crack in with normal x direction caused by shear stress in the x-y plane. Once attached, enter all required properties for that law by prefacing each property with "XYY-".	none	Linear
SofteningXZX	Attach a softening law (by name or ID) for propagation of shear damage for crack in with normal z direction caused by shear stress in the x-z plane. Once attached, enter all required properties for that law by prefacing each property with "XZX-".	none	Linear
SofteningXZZ	Attach a softening law (by name or ID) for propagation of shear damage for crack in with normal x direction caused by shear stress in the x-z plane. Once attached, enter all required properties for that law by prefacing each property with "XZZ-".	none	Linear
SofteningYZY	Attach a softening law (by name or ID) for propagation of shear damage for crack in with normal z direction caused by shear stress in the y-z plane. Once attached, enter all required properties for that law by prefacing each property with "YZY-".	none	Linear
SofteningYZZ	Attach a softening law (by name or ID) for propagation of shear damage for crack in with normal y direction caused by shear stress in the y-z plane. Once attached, enter all required properties for that law by prefacing each property with "YZZ-".	none	Linear
tractionFailureSurface	Select coupling method. Because all failure surfaces are assumed to be normal to material directions, the traction failure surface is always a cuboid surface. This parameter select how damage evolves when traction in any direction exceeds that surface. The options are: 0 = decoupled or directions evolve independently, 1 = not used, 2 or 3 = evolve in direction that returns the traction toward the origin. The later two are expected to be the most realistic because all tractions simultaneously reach zero at failure.	none	0
coefVariation wShape wV0 statDistributionMode	See the corresponding properties defined for an Isotropic softening material.
coeff	coefficient of friction for post-decohesion contact (default is 0 or frictionless)	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:

1. A flag to indicate damage process:
   • 0.1: undamaged material point
   • 1.0±0.25: damage initiated and evolving
   • 2.0±0.25: material point has failed (decohesion)
   • Within the above ranges for initiated and failed particles, the specific value tells how the damage initiated and the orientation crack or normal or the x direction in the crack axis system:
     • Crack normal in material z direction
       • 0.75 = ZZ tensile failure
       • 0.80 = XZ-X shear failure
       • 0.85 = YZ-Y shear failure
     • Crack normal in material x direction
       • 0.95 = XX tensile failure (2D)
       • 1.00 = XY-Y shear failure (2D)
       • 1.05 = XZ-Z shear failure
     • Crack normal in material y direction
       • 1.15= YY tensile failure (2D)
       • 1.20 = XY-X shear failure (2D)
       • 1.25 = YZ-Z shear failure
2. δ_n or the maximum normal cracking strain.
3. δ_xy or the maximum x-y shear cracking strain.
4. This variable has two options:
   • For 3D when using decoupled cuboid surface: δ_xz or the maximum x-z cracking strain.
   • For all other cases: G_I or cumulative mode I dissipated energy.
5. d_n or damage variable for normal loading. It varies from 0 to 1 where 1 is complete damage or failure.
6. d_xy or damage variable for x-y shear loading. It varies from 0 to 1 where 1 is complete damage or failure.
7. This variable has two options:
   • For 3D when using decoupled cuboid surface: d_xz or damage variable for x-z shear loading. It varies from 0 to 1 where 1 is complete damage or failure.
   • For all other cases: G_II or cumulative mode II dissipated energy.
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 3D it is Euler angle γ. In 2D has total dissipated energy by damage mechanics (in energy units)
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. 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 with x, y, and z corresponding to tangential, longitudinal, and radial directions"

 Material "wood","Douglas fir","OrthoSoftening"
   Ex 600
   Ey 12000
   Ez 900
   Gxy 800
   Gxz 80
   Gyz 800
   nuzx .4
   nuyx .2
   nuyz .24
   alphax 40
   alphay 0
   alphaz 40
   rho 0.5
   largeRotation 1
   strengthCoefVariation 0.3
   sigmaXXc 10
   sigmaYYc 100
   sigmaZZc 8
   tauXY-Xc 10
   tauXY-Yc 30
   tauYZ-Yc 30
   tauYZ-Zc 10
   tauXZ-Xc 2
   tauXZ-Zc 2
  Done