Orthotropic Hoffman Material
References
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 σ = [ σ 11 σ 22 σ 33 σ 12 σ 23 σ 13 ] T \sigma=[\sigma_{11}~\sigma_{22}~\sigma_{33}~\sigma_{12}~\sigma_{23}~\sigma_{13}]^\mathrm{T} σ = [ σ 11 σ 22 σ 33 σ 12 σ 23 σ 13 ] T C 1 C_1 C 1 C 9 C_9 C 9 K ( ϵ ˉ p ) K(\bar\epsilon_p) K ( ϵ ˉ p )
The constants are defined as follows.
C 1 = 1 2 ( 1 σ 11 t σ 11 c + 1 σ 22 t σ 22 c − 1 σ 33 t σ 33 c ) , C 2 = 1 2 ( 1 σ 22 t σ 22 c + 1 σ 33 t σ 33 c − 1 σ 11 t σ 11 c ) , C 3 = 1 2 ( 1 σ 33 t σ 33 c + 1 σ 11 t σ 11 c − 1 σ 22 t σ 22 c ) , C 4 = 1 σ 12 0 σ 12 0 , C 5 = 1 σ 23 0 σ 23 0 , C 6 = 1 σ 13 0 σ 13 0 , C 7 = σ 11 c − σ 11 t σ 11 t σ 11 c , C 8 = σ 22 c − σ 22 t σ 22 t σ 22 c , C 9 = σ 33 c − σ 33 t σ 33 t σ 33 c . \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*} C 1 C 2 C 3 C 4 C 7 = 2 1 ( σ 11 t σ 11 c 1 + σ 22 t σ 22 c 1 − σ 33 t σ 33 c 1 ) , = 2 1 ( σ 22 t σ 22 c 1 + σ 33 t σ 33 c 1 − σ 11 t σ 11 c 1 ) , = 2 1 ( σ 33 t σ 33 c 1 + σ 11 t σ 11 c 1 − σ 22 t σ 22 c 1 ) , = σ 12 0 σ 12 0 1 , C 5 = σ 23 0 σ 23 0 1 , C 6 = σ 13 0 σ 13 0 1 , = σ 11 t σ 11 c σ 11 c − σ 11 t , C 8 = σ 22 t σ 22 c σ 22 c − σ 22 t , C 9 = σ 33 t σ 33 c σ 33 c − σ 33 t . The Hoffman function allows different yield stresses for tension and compression. To recover the original Hill yield function, simply set σ i i t = σ i i c \sigma_{ii}^t=\sigma_{ii}^c σ ii t = σ ii c i = 1 , 2 , 3 i=1,~2,~3 i = 1 , 2 , 3
The hardening function K ( ε ˉ p ) K(\bar\varepsilon_p) K ( ε ˉ p ) K ( 0 ) = 1 K(0)=1 K ( 0 ) = 1
Copy virtual double compute_k ( double ) const = 0 ;
virtual double compute_dk ( double ) const = 0 ; History Layout
location parameter equivalent plastic strain
