Amin (hyperelasticity model)

In order to describe the softening effects at low stretches, Amin et al. (2002) suggested to add two coefficients ($C_4$ and $M$) for the strain energy density function in addition to that proposed by Yamashita and Kawabata (1993).

The strain energy density function for an incompressible Amin-Okui material

$W=C_1({\overline{I}}_1-3)+C_2({\overline{I}}_2-3)+C_3({\overline{I}}_1-3)exp(N+1)+C_4({\overline{I}}_1-3)exp(M+1),$

where $C_1$, $C_2$, $C_3$, $C_4$, N⩾1 and 0⩽M⩽1 are empirically determined material constants, and ${\overline{I}}_1$ and ${\overline{I}}_2$ are the first and the second invariant of $\overline{\mathit{B}}=(\det \mathit{B}{)}^{-1/3}\mathit{B}$ (the unimodular component of $\mathit{B}$):

$\begin{array}{rl}{\overline{I}}_1& =J^{-2/3}{I}_1,I_1={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2,\\ {\overline{I}}_2& =J^{-4/3}{I}_2,I_2={\lambda }_1^2{\lambda }_2^2+{\lambda }_2^2{\lambda }_3^2+{\lambda }_3^2{\lambda }_1^2\end{array}$

where $\mathit{F}$ is the deformation gradient and $J=\det (\mathit{F})={\lambda }_1{\lambda }_2{\lambda }_3$. For an incompressible material, $J=1$.

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