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{multline} F(\sigma,\bar\varepsilon_p)=C_1(\sigma_{11}-\sigma_{22})^2+C_2(\sigma_{22}-\sigma_{33})^2+C_3( \sigma_{33}-\sigma_{11})^2+\\[4mm] 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(\bar\varepsilon_p) \end{multline}

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. $K(\bar\epsilon_p)$ is the isotropic hardening function.

The constants are defined as follows.

$$\begin{align*} C_1&=\dfrac{1}{2}(\dfrac{1}{\sigma_{11}^t\sigma_{11}^c}+\dfrac{1}{\sigma_{22}^t\sigma_ {22}^c}-\dfrac{1}{\sigma_{33}^t\sigma_{33}^c}),\\[4mm] C_2&=\dfrac{1}{2}(\dfrac{1}{\sigma_{22}^t\sigma_{22}^c}+\dfrac{1}{\sigma_{33}^t\sigma_{33}^c}-\dfrac{1}{\sigma_ {11}^t\sigma_{11}^c}),\\[4mm] C_3&=\dfrac{1}{2}(\dfrac{1}{\sigma_{33}^t\sigma_{33}^c}+\dfrac{1}{\sigma_{11}^t\sigma_{11}^c}-\dfrac{1}{\sigma_ {22}^t\sigma_{22}^c}),\\[4mm] C_4&=\dfrac{1}{\sigma_{12}^0\sigma_{12}^0},\quad{}C_5=\dfrac{1}{\sigma_{23}^0\sigma_{23}^0},\quad{}C_6=\dfrac{1}{\sigma_ {13}^0\sigma_{13}^0},\\[4mm] C_7&=\dfrac{\sigma_{11}^c-\sigma_{11}^t}{\sigma_{11}^t\sigma_{11}^c},\quad{}C_8=\dfrac{\sigma_{22}^c-\sigma_ {22}^t}{\sigma_{22}^t\sigma_{22}^c},\quad{}C_9=\dfrac{\sigma_{33}^c-\sigma_{33}^t}{\sigma_{33}^t\sigma_{33}^c}. \end{align*}$$

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 $K(\bar\varepsilon_p)$ 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