NonlinearHoffman

Orthotropic Hoffman Material

References

1. Constitutive Modelling Cookbook

Theory

The NonlinearHoffman defines an orthotropic material using Hoffman yield criterion and associative plasticity.

The yield surface is defined as

$$\begin{array}{c}F(\sigma ,{\overline{\epsilon }}_p)=C_1({\sigma }_{11}-{\sigma }_{22}{)}^2+C_2({\sigma }_{22}-{\sigma }_{33}{)}^2+C_3({\sigma }_{33}-{\sigma }_{11}{)}^2+\hfill \\ \hfill C_4{\sigma }_{12}^2+C_5{\sigma }_{23}^2+C_6{\sigma }_{13}^2+C_7{\sigma }_{11}+C_8{\sigma }_{22}+C_9{\sigma }_{33}-K^2({\overline{\epsilon }}_p)\end{array}$$

with $\sigma =[{\sigma }_{11}{\sigma }_{22}{\sigma }_{33}{\sigma }_{12}{\sigma }_{23}{\sigma }_{13}{]}^{\mathrm{T}}$ is the stress, $C_1$ to $C_9$ are material constants. $K({\overline{ϵ}}_p)$ is the isotropic hardening function.

The constants are defined as follows.

$$\begin{array}{rl}{C}_1& ={\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{{\sigma }_{11}^t{\sigma }_{11}^c}}+{\displaystyle \frac{1}{{\sigma }_{22}^t{\sigma }_{22}^c}}-{\displaystyle \frac{1}{{\sigma }_{33}^t{\sigma }_{33}^c}}),\\ C_2& ={\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{{\sigma }_{22}^t{\sigma }_{22}^c}}+{\displaystyle \frac{1}{{\sigma }_{33}^t{\sigma }_{33}^c}}-{\displaystyle \frac{1}{{\sigma }_{11}^t{\sigma }_{11}^c}}),\\ C_3& ={\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{{\sigma }_{33}^t{\sigma }_{33}^c}}+{\displaystyle \frac{1}{{\sigma }_{11}^t{\sigma }_{11}^c}}-{\displaystyle \frac{1}{{\sigma }_{22}^t{\sigma }_{22}^c}}),\\ C_4& ={\displaystyle \frac{1}{{\sigma }_{12}^0{\sigma }_{12}^0}},C_5={\displaystyle \frac{1}{{\sigma }_{23}^0{\sigma }_{23}^0}},C_6={\displaystyle \frac{1}{{\sigma }_{13}^0{\sigma }_{13}^0}},\\ C_7& ={\displaystyle \frac{{\sigma }_{11}^c-{\sigma }_{11}^t}{{\sigma }_{11}^t{\sigma }_{11}^c}},C_8={\displaystyle \frac{{\sigma }_{22}^c-{\sigma }_{22}^t}{{\sigma }_{22}^t{\sigma }_{22}^c}},C_9={\displaystyle \frac{{\sigma }_{33}^c-{\sigma }_{33}^t}{{\sigma }_{33}^t{\sigma }_{33}^c}}.\end{array}$$

The Hoffman function allows different yield stresses for tension and compression. To recover the original Hill yield function, simply set ${\sigma }_{ii}^t={\sigma }_{ii}^c$ for $i=1,2,3$.

The hardening function $K({\overline{\epsilon }}_p)$ can be user defined. It shall be noted that $K(0)=1$. The following method shall be implemented.

virtual double compute_k(double) const = 0;
virtual double compute_dk(double) const = 0;

History Layout

location	parameter
initial_history(0)	equivalent plastic strain
initial_history(1:7)	plastic strain