Ogden类超弹性本构推导

应变能定义为：
$$W=\frac{\mu }{2}(I_1-3-2\ln J)+\frac{\lambda }{2}(\ln J{)}^2$$
其中$I_1=\mathrm{t}\mathrm{r}(\mathbf{C})$，
在推导之前需要知道的张量运算(如果之前从来没见过，最好自己推一遍，这里不给出证明)
$$\frac{\partial \mathrm{t}\mathrm{r}(\mathbf{C})}{\partial \mathbf{C}}=\mathbf{I}\frac{\partial J}{\partial \mathbf{C}}=\frac{J}{2}{\mathbf{C}}^{-1}\frac{\partial {\mathbf{C}}^{-1}}{\partial \mathbf{C}}=-\mathbf{C}\odot \mathbf{C}=-\frac{1}{2}(C_{ik}^{-1}{C}_{jl}^{-1}+C_{il}^{-1}{C}_{jk}^{-1})$$

考虑right Cauchy-Green变形张量 $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$ 和对应的Green-Lagrange应变张量 $\mathbf{E}=\frac{1}{2}({\mathbf{F}}^T\mathbf{F}-\mathbf{I})=\frac{1}{2}(\mathbf{C}-\mathbf{I})$.
按照定义，可以写出second Piola-Kirchhoff stress:
$$\begin{array}{rl}\mathbf{S}& =\frac{\delta W}{\delta \mathbf{E}}=\frac{2\partial W}{\partial \mathbf{C}}\\ & =2\{\frac{\mu }{2}(\frac{\partial I_1}{\partial \mathbf{C}}-\frac{2}{J}\frac{\partial J}{\partial \mathbf{C}})+\frac{\lambda }{2}(2\ln J)\frac{1}{J}\frac{\partial J}{\partial \mathbf{C}}\}\\ & =2\{\frac{\mu }{2}(\mathbf{I}-\frac{2}{J}\cdot \frac{J}{2}{\mathbf{C}}^{-1})+\lambda \ln J\cdot \frac{1}{J}\cdot \frac{J}{2}{\mathbf{C}}^{-1}\}\\ & =\mu (\mathbf{I}-{\mathbf{C}}^{-1})+\lambda \ln J{\mathbf{C}}^{-1}\end{array}$$
如果使用参考构型，请务必将其转换为1st Piola-Kirchhoff stress:
$$\mathbf{P}=\frac{\partial W}{\partial \mathbf{F}}=\mathbf{F}\mathbf{S}$$
如果是使用当前构型，需要将其转换为Cauchy stress: $\sigma =J^{-1}\mathbf{F}\mathbf{S}{\mathbf{F}}^T$。

接下来在上诉公式的基础上进行本构推导：
$$\begin{array}{rl}\mathrm{J}\mathrm{a}\mathrm{c}& =\frac{\delta \mathbf{S}}{\delta \mathbf{E}}=\frac{2\partial \mathbf{S}}{\partial \mathbf{C}}\\ & =2\mu \cdot (-\frac{\partial {\mathbf{C}}^{-1}}{\partial \mathbf{C}})+2\lambda \cdot \frac{1}{J}\frac{\partial J}{\partial \mathbf{C}}\otimes {\mathbf{C}}^{-1}+2\lambda \ln J\cdot \frac{\partial {\mathbf{C}}^{-1}}{\partial \mathbf{C}}\\ & =2\mu {\mathbf{C}}^{-1}\odot {\mathbf{C}}^{-1}+2\lambda \frac{1}{J}\cdot \frac{J}{2}{\mathbf{C}}^{-1}\otimes {\mathbf{C}}^{-1}+2\lambda \ln J(-{\mathbf{C}}^{-1}\odot \mathbf{C})\\ & =2\mu {\mathbf{C}}^{-1}\odot {\mathbf{C}}^{-1}+\lambda {\mathbf{C}}^{-1}\otimes {\mathbf{C}}^{-1}-2\lambda \ln J{\mathbf{C}}^{-1}\odot {\mathbf{C}}^{-1}\end{array}$$