NonlinearHoffman
Orthotropic Hoffman Material
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.
| C++ |
|---|
| virtual double compute_k(double) const = 0;
virtual double compute_dk(double) const = 0;
|
History Layout
| location |
parameter |
initial_history(0) |
accumulated plastic strain |