Difference between revisions of "Orthotropic Softening Material"

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where '''C''' is stiffness tensor for the underlying orthotopic material and '''D''' is an anisotropic 4<sup>th</sup> rank damage tensor appropriate for damage in orthotropic materials, and <math>\mathbf{\varepsilon}_{res}</math> is any residual strain (such as thermal or solvent induced strains).
where '''C''' is stiffness tensor for the underlying orthotopic material and '''D''' is an anisotropic 4<sup>th</sup> rank damage tensor appropriate for damage in orthotropic materials, and <math>\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.<ref name="dmref">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). [http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MPMSoftening.pdf PDF]</ref> The extension to orthotropic materials is described in a paper on generalized damage mechanics methods.<ref name="genref">J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," Int. J. for Numerical Methods in Engineering, 123, 5072-5097 (2022). [https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/GenDamage.pdf PDF]</ref>
The MPM implementation of softening isotropic materials is described in Nairn, Hammerquist, and Aimene.<ref name="dmref"/> The extension to orthotropic materials is described in a paper on generalized damage mechanics methods.<ref name="dmGen">


== Damage Process ==
== Damage Process ==

Revision as of 10:39, 29 March 2026

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 4th 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.[1] The extension to orthotropic materials is described in a paper on generalized damage mechanics methods.Cite error: Closing </ref> missing for <ref> tag

[2]

</references>

  1. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named dmref
  2. ↑ J. A. Nairn, "Generalization of Anisotropic Damage Mechanics Modeling in the Material Point Method," Int. J. for Numerical Methods in Engineering, 123, 5072-5097 (2022). (See PDF)
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