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} = H(\alpha) }[/math]

where σ and τ are normal and shear stresses in the material axis system after rotation from the analysis coordinates, [math]\displaystyle{ H(\alpha) }[/math] is a hardening law, and [math]\displaystyle{ \alpha }[/math] is a plastic hardening variable. 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 N = {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 stresses must satisfy:

      [math]\displaystyle{ \left|{1\over \sigma_{Y,ii}^2} - {1\over \sigma_{Y,jj}^2} \right| \le {1\over \sigma_{Y,kk}^2} \le {1\over \sigma_{Y,ii}^2} + {1\over \sigma_{Y,jj}^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}^2\over \sigma_{Y,jj}^2} \quad\rm{or} \quad {1\over \sigma_{Y,jj}^2} = {R\over \sigma_{Y,ii}^2} }[/math]

and [math]\displaystyle{ R\le1 }[/math] (i.e., [math]\displaystyle{ \sigma_{Y,ii}\le\sigma_{Y,jj} }[/math]) then the third yield stress is bracketed by:

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

One extreme example is:

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

In other words, if one direction is prevented from yielding by setting its yield strength to ∞ the other two direction must have the same yield stress. Also note that if two directions are equal, the third direction must be:

      [math]\displaystyle{ {\rm if\ }R = 1, \ \sigma_{Y,kk} \ge {\sigma_{Y,ii}\over 2} }[/math]

Hardening Laws

There hardening laws are available:

      [math]\displaystyle{ {\rm Nonlinear\ 1:} \qquad H(\alpha) = 1 + K \alpha^n }[/math]

      [math]\displaystyle{ {\rm Nonlinear\ 2:} \qquad H(\alpha) = (1 + K \alpha)^n }[/math]

      [math]\displaystyle{ {\rm Exponential:} \qquad H(\alpha) = 1 + \frac{K}{k}(1-\exp(-k\alpha)) }[/math]

Note, that because [math]\displaystyle{ H'(\alpha) }[/math] in the first law is infinite for [math]\displaystyle{ \alpha=0 }[/math] when n is less than one. This steep slope causes this law to become unstable for n less than about 0.7.

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 parameter	dimensionless	0
nhard	Hardening law n parameter. If n>0, the Nonlinear 1 low is used; if n<0, the Nonlinear 2 loaw is used with abs(n).	dimensionless	1
exphard	Hardening law β parameter in the exponential hardening law	dimensionless	0
largeRotation	If used, this setting is ignored and material always uses 1	dimensionless	1 (fixed)
(other)	All other properties are identical to the properties for an othotropic material.	varies	varies

Although default tensile yield stresses are all infinite, one of them must be finite to use this material. Similarly, the combination of properties must satisfy conditions for positive definiteness described above. If not, an error message will appear and simulation will not run.

If only nhard is provided, the modeling will use Nonlinear 1 or Nonlinear 2 hardening law. If only exphard is provided, it will use the exponential hardening law. If both are given, the last one provide determines the hardening law.

History Variables

The one history variable is the value of plastic hardening variable [math]\displaystyle{ \alpha }[/math]. This variable can be archived as history variable 1.