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NonlinearHoffman

Orthotropic Hoffman Material

References

  1. Constitutive Modelling Cookbook

Parameter Order

All orthotropic models take nine parameters to define the elasticity.

Using a standard notation (11, 22, 33, 12, 23, 13), the nine parameters needed are listed as follows.

  1. Six moduli: \(E_{11}\), \(E_{22}\), \(E_{33}\), \(G_{12}\), \(G_{23}\), \(G_{13}\). Note \(G_{ij}=G_{ji}\).
  2. Three Poisson's ratios: \(\nu_{12}\) \(\nu_{23}\) \(\nu_{13}\). Note \(\nu_{ij}\neq\nu_{ji}\).

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.

C++
1
2
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