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 two Hill plastic yield criteria available are HillStyle=1:

      [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]

and HillStyle=2:

      [math]\displaystyle{ 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 = \bigl(H(\alpha)\bigr)^2 }[/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 reason for two styles is discussed below. 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. These yields stresses are given by:

      [math]\displaystyle{ {1\over \sigma_{Y,xx}^2} = G+H \qquad {1\over \sigma_{Y,yy}^2} = F+H \qquad {1\over \sigma_{Y,zz}^2} = F+G }[/math]

Clearly only one of F, G, and H can be negative (otherwise a square yield stress would be negative). The yield surface also has to describe a contained elastic region. This condition is satisfied by rewriting the yield surfaces as

      [math]\displaystyle{ \sqrt{\vec \sigma\mathbf{A}\vec \sigma} = H(\alpha) \quad({\rm style=1}) \qquad{\rm and}\qquad \vec \sigma\mathbf{A}\vec \sigma = H(\alpha)^2 \quad({\rm style=2}) }[/math]

where [math]\displaystyle{ \mathbf{A} }[/math] (when using Voight notation for stresses) is

      [math]\displaystyle{ \mathbf{A} = \left( \begin{array}{cccccc} G+H & -H & -G & 0 & 0 & 0 \\ -H & F+H & -F & 0 & 0 & 0\\ -G & -F & F+G & 0 & 0 & 0\\ 0 & 0 & 0 & 2L & 0 & 0 \\ 0 & 0 & 0 & 0 & 2M & 0 \\ 0 & 0 & 0 & 0 & 0 & 2N \end{array} \right) }[/math]

For a valid yield surface, the Eigenvalues of matrix [math]\displaystyle{ \mathbf{A} }[/math] must be non-negative. In a more practical form, given any two yield stresses related by some ratio:

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

and we order i and j such that [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]

Example Yield Strengths

Some examples in various types of materials follow. If one direction is prevented from yielding by setting its yield strength to ∞, then

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

In other words, the other two directions must have the same yield stress. If two directions have the same yield strength, then the third direction must be greater than half that value:

      [math]\displaystyle{ \sigma_{Y,jj} = \sigma_{Y,ii}\quad\implies\quad R = 1 \quad\implies\quad \sigma_{Y,kk} \ge {\sigma_{Y,ii}\over 2} }[/math]

If two directions are similar, such as in in-plane, random fiber composites (e.g., medium density fiber board, particle board, fiberglass composites, or paperboard), the yield strength in the thickness direction must still be:

      [math]\displaystyle{ \sigma_{Y,jj} \sim \sigma_{Y,ii} \quad\implies\quad R \sim 1 \quad\implies\quad \sigma_{Y,zz} \gtrapprox \frac{1}{2}\min(\sigma_{Y,xx},\sigma_{Y,yy}) }[/math]

Because the thickness direction may be much weaker than the in-plane direction, this limitation limits modeling thickness-direction plasticity. Either Hill plasticity needs to be replaced with another anisotropic yielding model or it is realistic that thickness direction of such materials has very little plasticity. They may still have plasticity in shear.

Hardening Laws

There currently available hardening laws:

      [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) = \left\{ \begin{array}{ll} 1 + \frac{K}{\beta}(1-\exp(-\beta\alpha)) & \alpha \le \alpha_{max} \\ 1 + K^*\alpha & \alpha \gt \alpha_{max}\end{array} \right. }[/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 law becomes unstable for n less than about 0.7.

The parameter [math]\displaystyle{ \alpha_{max} }[/math] is to prevent exponential hardening from devolving into elastic-plastic deformation. When this parameter is used, the hardening is converted to linear hardening once [math]\displaystyle{ \alpha }[/math] exceeds [math]\displaystyle{ \alpha_{max} }[/math] and the slope of the linear hardening, [math]\displaystyle{ K^* }[/math], is set equal to the slope of the exponential hardening law at [math]\displaystyle{ \alpha_{max} }[/math]:

      [math]\displaystyle{ K^* = K \exp(-\beta\alpha_{max}) }[/math]

Thus, both the hardening law and its slope are continuous at [math]\displaystyle{ \alpha_{max} }[/math]. One way to pick [math]\displaystyle{ \alpha_{max} }[/math] is to choose a minimum relative slope compared to initial slope and then set:

      [math]\displaystyle{ \alpha_{max} = -\frac{\ln s_{rel}}{\beta} \qquad{\rm where}\qquad s_{rel} = \frac{K^*}{K} }[/math]

Why Two Hill Criteria

The two Hill criteria described above can be written with a symmetric, fourth-rank tensor as

      [math]\displaystyle{ \sqrt{\sigma\cdot \mathbf{A} \sigma } = H(\alpha) \quad {\rm and} \quad \sigma\cdot \mathbf{A} \sigma = \bigl(H(\alpha)\bigr)^2 }[/math]

where elements of [math]\displaystyle{ \mathbf{A} }[/math] are derived from material properties F through N. A discussion with implementation details for HillStyle=1 (the first one with a square root) can be found in Simo and Hughes. Unfortunately, that derivation has a step that multiplies by [math]\displaystyle{ \mathbf{A}^{-1}\mathbf{A} }[/math] but the tensor for Hill plasticity is singular. Despite this inconsistency, the overall implementation seems to work well.

To avoid this problem, the criteria can use HillStyle=2 instead (with squared terms). Implementation by this approach does not need to use [math]\displaystyle{ \mathbf{A}^{-1} }[/math]. It is now the default style for this material type.

If you want to use 2D plane stress analysis, the modeling must use HillStyle=2.

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 law is used; if n<0, the Nonlinear 2 law is used with abs(n).	dimensionless	1
exphard	Hardening law β parameter in the exponential hardening law. Use of this parameter also switches to using exponential hardening	dimensionless	0
alphaMax	Optional [math]\displaystyle{ \alpha_{max} }[/math] where exponential hardening converts to linear hardening	dimensionless	∞
HillStyle	Use 1 for square-root criterion or 2 for squared criterion. Style 2 is required when using 2D plane stress anlaysis.	dimensionless	2
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 provided 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.