Anisotropic, Elastic-Plastic Material

Constitutive Law

This MPM material is identical to an orthotropic material in the elastic regime, but can plastically deform according to a built-in, anistropic Hill yielding criterion. The Hill plastic yield criterion is:

      [math]\displaystyle{ \sqrt{F(\sigma_{yy}-\sigma_{zz})^2 + G(\sigma_{xx}-\sigma_{zz})^2 + H(\sigma_{yy}-\sigma_{xx})^2 + 2L\tau_{yz}^2 + 2M\tau_{xz}^2 + 2N\tau_{xy}^2} = 1 + K\varepsilon_p^n }[/math]

where σ and τ are normal and shear stresses in the material axis system after rotation from the anaysis coordinates, K and n, are dimensionless hardening properties, and ε_p is cumulative plastic strain. The remaining constants are determined by the yield stresses:

      [math]\displaystyle{ F = {1\over 2}\left({1\over \sigma_{Y,yy}^2} + {1\over \sigma_{Y,zz}^2} - {1\over \sigma_{Y,xx}^2}\right) \qquad\qquad L = {1\over 2\tau_{Y,yz}^2} }[/math]

      [math]\displaystyle{ G= {1\over 2}\left({1\over \sigma_{Y,xx}^2} + {1\over \sigma_{Y,zz}^2} - {1\over \sigma_{Y,yy}^2}\right) \qquad\qquad M = {1\over 2\tau_{Y,xz}^2} }[/math]

      [math]\displaystyle{ H = {1\over 2}\left({1\over \sigma_{Y,xx}^2} + {1\over \sigma_{Y,yy}^2} - {1\over \sigma_{Y,zz}^2}\right) \qquad\qquad M = {1\over 2\tau_{Y,xy}^2} }[/math]

where σ_Y and τ_Y are yield stresses for loading in the indicated direction. The yield stresses have to be selected such that the plastic potential is positive semidefinite. Analysis shows that all tensile yield stress must satisfy:

      [math]\displaystyle{ \left({1\over \sigma_{Y,ii}^2} - {1\over \sigma_{Y,jj}^2} \right)^2 \le {1\over \sigma_{Y,kk}^2} \le \left({1\over \sigma_{Y,ii}^2} + {1\over \sigma_{Y,jj}^2} \right)^2 }[/math]

where i, j, and k are any combination or x, y, and z. In more practical terms, if two yield stresses are related by some ratio:

      [math]\displaystyle{ R = {\sigma_{Y,ii}\over \sigma_{Y,jj}} }[/math]

then the third yield stress is bracketed by:

      [math]\displaystyle{ R = {\sigma_{Y,ii}\over |1-R|} \le \sigma_{Y,kk} \le {\sigma_{Y,ii}\over |1+R|} }[/math]

Two extreme example are:

      [math]\displaystyle{ {\rm if\ }R = 0, \ \sigma_{Y,kk} = \sigma_{Y,ii},\ \sigma_{Y,jj} = \infty \qquad {\rm and}\qquad {\rm if\ }R\to\infty, \ \sigma_{Y,kk} = \sigma_{Y,jj},\ \sigma_{Y,ii} = \infty }[/math]

In other words, if an axial direction is prevented from yielding by setting its yield strength to ∞ the other two direction must have the same yield stress.

Material Properties

Property	Description	Units	Default
yldxx	Yield stress for axial loading in the x direction	pressure units	∞
yldyy	Yield stress for axial loading in the y direction	pressure units	∞
yldzz	Yield stress for axial loading in the z direction	pressure units	∞
yldxy	Yield stress for shear loading in the x-y plane	pressure units	∞
yldxz	Yield stress for shear loading in the x-z plane	pressure units	∞
yldyz	Yield stress for shear loading in the y-z plane	pressure units	∞
khard	Hardening law K paraemeter	dimensionless	0
nhard	Hardening law paraemeter	dimensionless	1
(other)	All other properties are identical to the properties for an othotropic material.	varies	varies

History Variables

The one history variable is the cummulative equivalent plastic strain. This variable can be archived as history variable 1.