2.5 塑性力学—应变状态

个人专栏—塑性力学

1.1 塑性力学基本概念 塑性力学基本概念
1.2 弹塑性材料的三杆桁架分析 弹塑性材料的三杆桁架分析
1.3 加载路径对桁架的影响 加载路径对桁架的影响
2.1 塑性力学——应力分析基本概念 应力分析基本概念
2.2 塑性力学——主应力、主方向、不变量 主应力、主方向、不变量

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$应变分析$

• 应变与位移的关系

如图所示，由几何方程得：
$$\{\begin{array}{ll}{\epsilon }_x=\frac{\partial u}{\partial x}& {\gamma }_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\\ {\epsilon }_y=\frac{\partial v}{\partial y}& {\gamma }_{yz}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\\ {\epsilon }_z=\frac{\partial w}{\partial z}& {\gamma }_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\end{array}$$
剪应变张量表示为：
$$\{\begin{array}{l}{\epsilon }_{xy}=\frac{1}{2}{\gamma }_{xy}=\frac{1}{2}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\\ {\epsilon }_{yz}=\frac{1}{2}{\gamma }_{yz}=\frac{1}{2}\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\\ {\epsilon }_{zx}=\frac{1}{2}{\gamma }_{zx}=\frac{1}{2}\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\end{array}$$

• 一点的应变状态：知道一点的6个独立的应变分量：${\epsilon }_x,{\epsilon }_y,{\epsilon }_z,{\gamma }_{xy},{\gamma }_{yz},{\gamma }_{zx}$,任一方向的应变即可确定，称该点的应变情况为应变状态。

• 应变分量 $\varepsilon_x,\varepsilon_y,\varepsilon_z,\varepsilon_{xy},\varepsilon_{yz},\varepsilon_{zx} $构成应变张量

$$\begin{array}{c}{\epsilon }_{ij}=\left[\begin{array}{ccc}{\epsilon }_x& {\epsilon }_{xy}& {\epsilon }_{xz}\\ {\epsilon }_{yx}& {\epsilon }_y& {\epsilon }_{yz}\\ {\epsilon }_{zx}& {\epsilon }_{zy}& {\epsilon }_z\end{array}\right]=\left[\begin{array}{ccc}{\epsilon }_x& \frac{1}{2}{\gamma }_{xy}& \frac{1}{2}{\gamma }_{xz}\\ \frac{1}{2}{\gamma }_{yx}& {\epsilon }_y& \frac{1}{2}{\gamma }_{yz}\\ \frac{1}{2}{\gamma }_{zx}& \frac{1}{2}{\gamma }_{zy}& {\epsilon }_z\end{array}\right]\\ {\epsilon }_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})\end{array}$$

• 应变张量的三个不变量

$$\begin{array}{c}{\epsilon }_{ij}=\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}\end{array}\right]\\ {\epsilon }^3-I_1{\epsilon }^2+I_2{\epsilon }^2-I_3=0\\ I_1={\epsilon }_x+{\epsilon }_y+{\epsilon }_z={\epsilon }_1+{\epsilon }_2+{\epsilon }_3\\ I_2={\epsilon }_x{\epsilon }_y+{\epsilon }_y{\epsilon }_z+{\epsilon }_z{\epsilon }_x-{\epsilon }_{xy}^2-{\epsilon }_{yz}^2-{\epsilon }_{zx}^2={\epsilon }_1{\epsilon }_2+{\epsilon }_2{\epsilon }_3+{\epsilon }_3{\epsilon }_1\\ I_3={\epsilon }_x{\epsilon }_y{\epsilon }_z+2{\epsilon }_{xy}{\epsilon }_{yz}{\epsilon }_{zx}-{\epsilon }_x{\epsilon }_{yz}^2-{\epsilon }_y{\epsilon }_{zx}^2-{\epsilon }_z{\epsilon }_{xy}^2={\epsilon }_1{\epsilon }_2{\epsilon }_3\end{array}$$

• 应变偏张量的三个不变量
  $$\begin{array}{c}{\epsilon }_{ij}=\left[\begin{array}{ccc}{\epsilon }_m& 0& 0\\ 0& {\epsilon }_m& 0\\ 0& 0& {\epsilon }_m\end{array}\right]+\underset{e_{ij}}{\underbrace{\left[\begin{array}{ccc}{\epsilon }_{11}-{\epsilon }_m& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}-{\epsilon }_m& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}-{\epsilon }_m\end{array}\right]}}={\epsilon }_m{\delta }_{ij}+e_{ij}\\ {\epsilon }_m=\frac{{\epsilon }_x+{\epsilon }_y+{\epsilon }_z}{3}=\frac{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}{3}\\ {\epsilon }_x-{\epsilon }_m=\frac{2{\epsilon }_x-{\epsilon }_y-{\epsilon }_z}{3},{\epsilon }_y-{\epsilon }_m=\frac{2{\epsilon }_y-{\epsilon }_x-{\epsilon }_z}{3},{\epsilon }_z-{\epsilon }_m=\frac{2{\epsilon }_z-{\epsilon }_y-{\epsilon }_x}{3}\end{array}$$

体积应变 $\theta=\varepsilon_x+\varepsilon_y+\varepsilon_z=3\varepsilon_m $,只引起单元体的体积改变；剪切应变： $e_{ij} $只产生形状改变。

$$\begin{array}{rl}{I}_1^{{}^{\prime }}& =e_{11}+e_{22}+e_{33}=0I_3^{{}^{\prime }}=|e_{ij}|=e_1{e}_2{e}_3\\ I_2^{{}^{\prime }}& =\frac{1}{2}{e}_{ij}{e}_{ji}=\frac{1}{6}[(e_{11}-e_{22}{)}^2+(e_{22}-e_{33}{)}^2+(e_{33}-e_{11}{)}^2]+e_{12}^2+e_{23}^2+e_{31}^2\\ & =\frac{1}{6}[(e_x-e_y{)}^2+(e_y-e_z{)}^2+(e_z-e_x{)}^2]+\frac{1}{4}({\gamma }_{xy}^2+{\gamma }_{yz}^2+{\gamma }_{zx}^2)\\ & =\frac{1}{6}[({\epsilon }_1-{\epsilon }_2{)}^2+({\epsilon }_2-{\epsilon }_3{)}^2+({\epsilon }_3-{\epsilon }_1{)}^2]\end{array}$$

• 八面体剪应变：与三个应变主轴方向具有相同倾角平面上的剪应变
  $$\{\begin{array}{l}{\epsilon }_8=\frac{1}{3}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3)\\ {\gamma }_8=\frac{2}{3}\sqrt{({\epsilon }_1-{\epsilon }_2{)}^2+({\epsilon }_2-{\epsilon }_3{)}^2+({\epsilon }_3-{\epsilon }_1{)}^2}=\sqrt{\frac{8}{3}{I}_2^{{}^{\prime }}}\end{array}$$

• Lode应变参数

$${\mu }_{\epsilon }=\frac{2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3}{{\epsilon }_1-{\epsilon }_3}=2\frac{{\epsilon }_2-{\epsilon }_3}{{\epsilon }_1-{\epsilon }_3}-1$$

• 单向拉伸 ${\epsilon }_1=\epsilon ,{\epsilon }_2={\epsilon }_3=-0.5\epsilon ,{\mu }_{\epsilon }=-1$

• 单向压缩 $\varepsilon_3=-\varepsilon,\quad \varepsilon_2=\varepsilon_1=0.5\varepsilon,\quad \mu_{\varepsilon}=1 $

• 纯剪切 $\varepsilon_1=0.5\gamma,\quad \varepsilon_2=0,\quad \varepsilon_3=-0.5\gamma,\quad \mu_{\varepsilon}=0 $

• 等效应变

$$\begin{array}{rl}\bar{\epsilon }& =\sqrt{\frac{4}{3}{I}_2^{{}^{\prime }}}\\ & =\frac{\sqrt{2}}{3}\sqrt{(e_1-e_2{)}^2+(e_2-e_3{)}^2+(e_3-e_1{)}^2}\\ & =\sqrt{\frac{2}{3}}\sqrt{e_1^2+e_2^2+e_3^2}\end{array}$$

$应用示例$

• 已知位移分量 $u=(2x+y)a,v=(2y+x)a,w=-az $，求：应变张量并分解应变强度。

$$\begin{array}{c}{\epsilon }_x=\frac{\partial u}{\partial x}=2a,{\epsilon }_y=\frac{\partial v}{\partial y}=2a,{\epsilon }_z=\frac{\partial w}{\partial z}=-a\\ {\gamma }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=2a,{\epsilon }_{xy}=\frac{1}{2}{\gamma }_{xy}=a\\ {\epsilon }_{yz}=\frac{1}{2}(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}),{\epsilon }_{zx}=\frac{1}{2}(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z})\\ {\epsilon }_m=\frac{{\epsilon }_x+{\epsilon }_y+{\epsilon }_z}{3}=a\\ {\epsilon }_{ij}=\left[\begin{array}{ccc}2a& a& 0\\ a& 2a& 0\\ 0& 0& -a\end{array}\right]=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]+\left[\begin{array}{ccc}a& a& 0\\ a& a& 0\\ 0& 0& -2a\end{array}\right]\end{array}$$