# 通过各种超弹性材料模型对测量数据进行拟合

2015年 6月 24日

### 各向同性、近似不可压缩超弹性

W_s = W_{iso}+W_{vol}

S = -p_pJC^{-1}+2\frac{\partial W_{iso}}{\partial C}

S = -p_pJC^{-1}+2\left(J^{-2/3}\left(\frac{\partial W_{iso}}{\partial \bar{I_{1}}}+\bar{I_{1}} \frac{\partial W_{iso}}{\partial \bar{I_{2}}} \right)I-J^{-4/3} \frac{\partial W_{iso}}{\partial \bar{I}_{2}} C -\left(\frac{\bar{I_{1}}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{1}} + \frac{2 \bar{I}_{2}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{2}}\right)C^{-1}\right)

\begin{align}P& = FS\\
\sigma& = J^{-1}FSF^{T}
\end{align}

S =\sideset{}{^3_{a=1}}
\sum S_{a} \hat{\mathbf{N}_{a}} \otimes \hat{\mathbf{N}_{a}}

C = \sideset{}{^3_{a=1}}
\sum\lambda_a^2 \hat{\mathbf{N}_a}\otimes\hat{\mathbf{N}_a}

S_a = \frac{-p_p J}{\lambda_a^2}+2\left(J^{-2/3}\left(\frac{\partial W_{iso}}{\partial \bar{I_{1}}}+\bar{I_{1}} \frac{\partial W_{iso}}{\partial \bar{I_{2}}} \right) -J^{-4/3} \frac{\partial W_{iso}}{\partial \bar{I}_{2}} \lambda_a^2 -\frac{1}{\lambda_a^2}\left(\frac{\bar{I_{1}}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{1}} + \frac{2 \bar{I}_{2}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{2}}\right)\right)

\lambda_1 = \lambda, \lambda_2 = \lambda_3 = \lambda^{-1/2}

\begin{array}{c} F = \\ \end{array} \left(\begin{array}{ccc} \lambda &0 &0 \\ 0 &\frac{1}{\sqrt{\lambda}} &0 \\ 0 &0 &\frac{1}{\sqrt{\lambda}}\end{array}\right)

(1)

S_{1} = 2\left(\frac{1}{\lambda} -\frac{1}{\lambda^4}\right) \left(\lambda \frac{\partial W_{iso}}{\partial \bar{I}_{1_{uni}}}+\frac{\partial W_{iso}}{\partial \bar{I}_{2_{uni}}}\right) ,\; P_1 = \lambda S_1\; \sigma_1 = \lambda^2 S_1,\;\;\;\;

\begin{align*}
\bar{I}_{1_{uni}} = \left(\lambda^2+\frac{2}{\lambda}\right) \\
\bar{I}_{2_{uni}} = \left(2\lambda + \frac{1}{\lambda^2}\right)
\end{align*}

\lambda_1 = \lambda_2 = \lambda, \; \lambda_3 = \lambda^{-2}

(2)

S_1 = S_2 = 2\left(1-\frac{1}{\lambda^6}\right)\left(\frac{\partial W_{iso}}{\partial \bar{I}_{1_{bi}}}+\lambda^2\frac{\partial W_{iso}}{\partial \bar{I}_{2_{bi}}}\right),\; P_1 = \lambda S_1,\; \sigma_1 = \lambda^2 S_1\;\;\;\;

\begin{align*}
\bar{I}_{1_{bi}} = \left( 2\lambda^2 + \frac{1}{\lambda^4}\right) \\
\bar{I}_{2_{bi}} = \left(\lambda^4 + \frac{2}{\lambda^2}\right)
\end{align*}

### 不可压缩超弹性材料模型中的应力与主拉伸

#### Neo-Hookean

Neo-Hookean 材料模型的总应变能密度为

W_s = \frac{1}{2}\mu\left(\bar{I}_1-3\right)+\frac{1}{2}\kappa\left(J_{el}-1\right)^2

\begin{align*}
P_{1_{uniaxial}} &= \mu\left(\lambda-\lambda^{-2}\right)\\
P_{1_{biaxial}} &= \mu\left(\lambda-\lambda^{-5}\right)
\end{align*}

#### Mooney-Rivlin，两个参数

\begin{align*}
P_{1_{uniaxial}} &= 2\left(1-\lambda^{-3}\right)\left(\lambda C_{10}+C_{01}\right)\\
P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\left(C_{10}+\lambda^2 C_{01}\right)
\end{align*}

#### Mooney-Rivlin，五个参数

\begin{align}\begin{split}
P_{1_{uniaxial}}& = 2\left(1-\lambda^{-3}\right)\left(\lambda C_{10} + 2C_{20}\lambda\left(I_{1_{uni}}-3\right)+C_{11}\lambda\left(I_{2_{uni}}-3\right)\\
P_{1_{biaxial}}& = 2\left(\lambda-\lambda^{-5}\right)\left(C_{10}+2C_{20}\left(I_{1_{bi}}-3\right)+C_{11}\left(I_{2_{bi}}-3\right)\\
\end{split}
\end{align}

#### Arruda-Boyce

\begin{align}
P_{1_{uniaxial}} &= 2\left(\lambda-\lambda^{-2}\right)\mu_0\sum_{p=1}^{5}\frac{p c_p}{N^{p-1}}I_{1_{uni}}^{p-1}\\
P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\mu_0\sum_{p=1}^{5}\frac{p c_p}{N^{p-1}}I_{1_{bi}}^{p-1}
\end{align}

#### Yeoh

\begin{align}
P_{1_{uniaxial}} &= 2\left(\lambda-\lambda^{-2}\right)\sum_{p=1}^{3}p c_p \left(I_{1_{uni}}-3\right)^{p-1}\\
P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\sum_{p=1}^{3}p c_p \left(I_{1_{bi}}-3\right)^{p-1}
\end{align}

#### Ogden

\begin{align}
P_{1_{uniaxial}} &= \sum_{p=1}^{N}\mu_p \left(\lambda^{\alpha_p-1} -\lambda^{-\frac{\alpha_p}{2}-1}\right)\\
P_{1_{biaxial}} &= \sum_{p=1}^{N}\mu_p \left(\lambda^{\alpha_p-1} -\lambda^{-2\alpha_p-1}\right)
\end{align}

2018-12-13