MMC Chapitre 2 形变 Les Déformation

描述形变的矩阵

形变梯度张量 $\underline{\underline{\mathbb{F}}}$

$$\left[\begin{array}{l} \mathrm{dx}_1^{\prime} \\ \mathrm{dx}_2^{\prime} \\ \mathrm{dx}_3^{\prime} \end{array}\right]=\underbrace{\left[\begin{array}{lll} \frac{\partial \mathrm{x}_1^{\prime}}{\partial \mathrm{x}_1} & \frac{\partial \mathrm{x}_1^{\prime}}{\partial \mathrm{x}_2} & \frac{\partial \mathrm{x}_1^{\prime}}{\partial \mathrm{x}_3} \\ \frac{\partial \mathrm{x}_2^{\prime}}{\partial \mathrm{x}_1} & \frac{\partial \mathrm{x}_2^{\prime}}{\partial \mathrm{x}_2} & \frac{\partial \mathrm{x}_2^{\prime}}{\partial \mathrm{x}_3} \\ \frac{\partial \mathrm{x}_3^{\prime}}{\partial \mathrm{x}_1} & \frac{\partial \mathrm{x}_3^{\prime}}{\partial \mathrm{x}_2} & \frac{\partial \mathrm{x}_3^{\prime}}{\partial \mathrm{x}_3} \end{array}\right]}_{\underline{\underline{\mathrm{F}}}}\left[\begin{array}{l} \mathrm{dx}_1 \\ \mathrm{dx}_2 \\ \mathrm{dx}_3 \end{array}\right]$$ $$\underline{\underline{F}}=\left[\begin{array}{l|l|l}\frac{\partial \underline{\mathrm{x}^{\prime}}}{\partial \mathrm{x}_1} & \frac{\partial \underline{\mathrm{x}}^{\prime}}{\partial \mathrm{x}_2} & \frac{\partial \underline{\mathrm{x}}^{\prime}}{\partial \mathrm{x}_3}\end{array}\right]$$

位移梯度张量 $\underline{\underline{\mathbb{H}}}$

$$\left[\begin{array}{l}\mathrm{du}_1 \\\mathrm{du}_2 \\\mathrm{du}_3\end{array}\right]=\underbrace{\left[\begin{array}{lll}\frac{\partial \mathrm{u}_1}{\partial \mathrm{x}_1} & \frac{\partial \mathrm{u}_1}{\partial \mathrm{x}_2} & \frac{\partial \mathrm{u}_1}{\partial \mathrm{x}_3} \\\frac{\partial \mathrm{u}_2}{\partial \mathrm{x}_1} & \frac{\partial \mathrm{u}_2}{\partial \mathrm{x}_2} & \frac{\partial \mathrm{u}_2}{\partial \mathrm{x}_3} \\\frac{\partial \mathrm{u}_3}{\partial \mathrm{x}_1} & \frac{\partial \mathrm{u}_3}{\partial \mathrm{x}_2} & \frac{\partial \mathrm{u}_3}{\partial \mathrm{x}_3}\end{array}\right]}_{\underline{\mathrm{H}}=\underline{\underline{\operatorname{grad}}}(\mathrm{u})}\left[\begin{array}{l}\mathrm{dx}_1 \\\mathrm{dx}_2 \\\mathrm{dx}_3\end{array}\right]$$ $$\underline{\underline{H}}=\underline{\underline{F}}-\underline{\underline{I}}$$

柯西-格林膨胀张量

$$\underline{\underline{C}}=\underline{\underline{F}}^{\mathrm{t}} \underline{\underline{\mathrm{F}}}$$

格林-拉格朗日应变张量

$$\underline{\underline{E}}=\frac{1}{2}\left(\underline{\underline{F}}^{\mathrm{t}} \underline{\underline{F}}-\underline{\underline{I}}\right)=\frac{1}{2}(\underline{\underline{C}}-\underline{\underline{I}})$$ $$\underline{\underline{E}}=\frac{1}{2}\left(\underline{\underline{H}}+\underline{\underline{H}}^{\mathrm{t}}+\underline{\underline{H}}^{\mathrm{t}} \underline{\underline{H}}\right)$$

应变的种类

膨胀 Dilatation

$$\lambda(\overrightarrow{\mathrm{n}})=\frac{\left|\mathrm{d} \overrightarrow{\mathrm{x}}^{\prime}\right|}{|\mathrm{d} \overrightarrow{\mathrm{x}}|}=\sqrt{\underline{\mathrm{n}^{\mathrm{t}}} \underline{\underline{\mathrm{C}}} \underline{\mathrm{n}}}=\sqrt{1+2 \underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{E}} \underline{\mathrm{n}}}$$

法向相对应变 Allogement relatif

$$\varepsilon(\overrightarrow{\mathrm{n}})=\frac{\left|\mathrm{d} \overrightarrow{\mathrm{x}}^{\prime}\right|-|\mathrm{d} \overrightarrow{\mathrm{x}}|}{|\mathrm{d} \overrightarrow{\mathrm{x}}|}=\sqrt{\underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{\mathrm{C}}} \underline{\mathrm{n}}}-1=\sqrt{1+2 \underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{\mathrm{E}}}\underline{n}}-1$$

滑移角度 Angle de glissement

$$\begin{aligned}sin(\gamma(\vec n, \vec m)) = \frac{ {\underline{n^t}}\underline{\underline{C}}\underline{m}}{\sqrt{\underline{n^t}\underline{\underline{C}}\underline{n}}\sqrt{\underline{m^t}\underline{\underline{C}}\underline{m}}} & =\frac{2 \underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{\mathrm{E}}} \underline{\underline{\mathrm{m}}}}{\sqrt{1+2 \underline{\mathrm{n}}^{\mathrm{t}} \underline{\mathrm{E}} \underline{\mathrm{n}} \sqrt{1+2 \underline{\mathrm{m}}^{\mathrm{t}} \underline{\mathrm{E}} \underline{\mathrm{m}}}}} \\ & \end{aligned}$$

线性应变张量

在小形变$\frac{\partial \mathrm{u}{\mathrm{i}}}{\partial \mathrm{x}{\mathrm{j}}}<<1$状态下，忽略格林-拉格朗日应变张量中的二阶项，得到线性应变张量

线性应变张量

$$\underline{\underline{\varepsilon}}=\left[\begin{array}{lll}\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\\varepsilon_{12} & \varepsilon_{22} & \varepsilon_{23} \\\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}\end{array}\right]\ \ avec \ \ \varepsilon_{\mathrm{ij}}=\frac{1}{2}\left(\frac{\partial \mathrm{u}_{\mathrm{i}}}{\partial \mathrm{x}_{\mathrm{j}}}+\frac{\partial \mathrm{u}_{\mathrm{j}}}{\partial \mathrm{x}_{\mathrm{i}}}\right)$$ $$\underline{\underline{\varepsilon}}=\frac{1}{2}\left(\underline{\underline{H}}+\underline{\underline{H}}^{\mathrm{t}}\right)$$

小应变假设下的各种应变

$$\begin{aligned}& \lambda(\overrightarrow{\mathrm{n}})=\frac{|\mathrm{dx}|}{|\mathrm{dx}|}=1+\underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{\varepsilon}} \underline{\mathrm{n}} \\& \varepsilon(\overrightarrow{\mathrm{n}})=\frac{|\mathrm{dx}|-|\mathrm{dx}|}{|\mathrm{dx}|}=\underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{\varepsilon}} \underline{n} \\& \gamma(\overrightarrow{\mathrm{n}}, \overrightarrow{\mathrm{m}})=2 \underline{\mathrm{n}}^{\mathrm{t}} \underline{\underline{\varepsilon}} \underline{\mathrm{m}}\end{aligned}$$

应变分解

$$\underline{\varepsilon}=\frac{1}{3} \underbrace{\left[\begin{array}{ccc} \operatorname{tr}(\varepsilon) & 0 & 0 \\ 0 & \operatorname{tr}(\varepsilon) & 0 \\ 0 & 0 & \operatorname{tr}(\varepsilon) \end{array}\right]}_{\underline{\underline{\varepsilon_{\mathrm{s}}}} \text { partie sphérique }}+\underbrace{\underline{\underline{\varepsilon}-\varepsilon_{\mathrm{s}}}}_{\underline{\underline{\varepsilon_{\mathrm{d}}}}partie \ deviatorique}$$

主应变和主应变方向

与应力类似，对角化即可

旋转

应变可以分解旋转、变形和平移

旋转张量

$$\underline{\mathrm{du}}=\underbrace{\frac{1}{2}\left(\underline{\underline{H}}-\underline{\underline{H}}^t\right)}_{\underline{\underline{\omega}}} \underline{\mathrm{dx}}+\underbrace{\frac{1}{2}\left(\underline{\underline{H}}+\underline{\underline{H}}^t\right)}_{\underline{\underline{\varepsilon}}} \underline{\mathrm{dx}}$$ $$\underline{\underline{\omega}}=\left[\begin{array}{ccc}0 & \omega_{12} & \omega_{13} \\-\omega_{12} & 0 & \omega_{23} \\-\omega_{31} & -\omega_{23} & 0\end{array}\right] \omega_{\mathrm{ij}}=\frac{1}{2}\left(\frac{\partial \mathrm{u}_{\mathrm{i}}}{\partial \mathrm{x}_{\mathrm{j}}}-\frac{\partial \mathrm{u}_{\mathrm{j}}}{\partial \mathrm{x}_{\mathrm{i}}}\right)$$

• 这里要注意，旋转张量是反对称的

旋转向量

$$\vec{\Omega}=\left|\begin{aligned}-&\omega_{23}\\&\omega_{13}\\-&\omega_{12}\end{aligned}\right. \Rightarrow \underline{\Omega} \wedge \underline{\mathrm{dx}}=\underline{\underline\omega} \underline{\mathrm{dx}}$$

体积膨胀

相比于旋转，变形和平移，相对体积膨胀是一个不依赖于坐标系

$$\frac{\mathrm{V}^{\prime}-\mathrm{V}}{\mathrm{V}}=\frac{\Delta \mathrm{V}}{\mathrm{V}}=\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33}=\operatorname{trace}(\underline{\underline{\varepsilon}})$$

不可压性

$$\frac {\Delta V}V = 0$$

兼容性方程

CNS

$$\begin{cases}1 & \varepsilon_{11,22}+\varepsilon_{22,11}-2 \varepsilon_{12,12}=0 \\ 2 & \varepsilon_{11,23}+\left(\varepsilon_{23,11}-\varepsilon_{31,23}-\varepsilon_{21,13}\right)=0\end{cases}$$