3. 应变理论

3.1 位移和应变

3.1.1 位移的描述

拉格朗日描述法

$$u_i=x_i(a_1,a_2,a_3)-a_i$$

3.1.2 变形的描述

格林应变张量

$$E_{ij}=\frac{1}{2}\text{pqty}\text{pdv}{x}_m{a}_i\text{pdv}{x}_m{a}_j-{\delta }_{ij}=\frac{1}{2}\text{pqty}\text{pdv}{u}_i{a}_j+\text{pdv}{u}_j{a}_i+\text{pdv}{u}_m{a}_i\text{pdv}{u}_m{a}_j$$$$\text{vb}\ast u\text{grad}=\text{pdv}{u}_i{a}_j\text{vb}\ast e_i\text{vb}\ast e_j\text{qc}\text{grad}\text{vb}\ast u=\text{pdv}{u}_j{a}_i\text{vb}\ast e_i\text{vb}\ast e_j$$$$\text{vb}\ast E=\frac{1}{2}\text{pqty}\text{vb}\ast u\text{grad}+\text{grad}\text{vb}\ast u+\text{grad}\text{vb}\ast u\cdot \text{vb}\ast u\text{grad}$$

长度比

$${\lambda }_{\nu }=\sqrt{1+2\text{vb}\ast \nu \cdot \text{vb}\ast E\cdot \text{vb}\ast \nu }$$

线元变形后的夹角变化

$$\cos (\text{vb}\ast {\nu }^{\prime },\text{vb}\ast t^{\prime })=\text{pqty}\text{vb}\ast \nu \cdot \text{vb}\ast t+2\text{vb}\ast \nu \cdot \text{vb}\ast E\cdot \text{vb}\ast t\frac{1}{{\lambda }_{\text{vb}\ast \nu }{\lambda }_{\text{vb}\ast t}}$$

3.2 小应变张量

3.2.1 小应变张量的定义和意义

柯西应变张量 / 小应变张量, 应变位移公式 / 几何方程

$$\begin{gathered} {\epsilon }_{ij}=\frac{1}{2}\text{pqty}{u}_{i,j}+u_{j,i} \\ \text{vb}\ast \epsilon =\frac{1}{2}\text{pqty}\text{vb}\ast u\text{grad}+\text{grad}\text{vb}\ast u \end{gathered}$$

长度比

$${\lambda }_{\text{vb}\ast \nu }=1+\text{vb}\ast \nu \cdot \text{vb}\ast \epsilon \cdot \text{vb}\ast \nu$$

工程正应变

$${\epsilon }_{\text{vb}\ast \nu }=\text{vb}\ast \nu \cdot \text{vb}\ast \epsilon \cdot \text{vb}\ast \nu$$

两线元间的夹角变化

$$\cos (\text{vb}\ast {\nu }^{\prime },\text{vb}\ast t^{\prime })=\text{pqty}1-{\epsilon }_{\text{vb}\ast \nu }-{\epsilon }_{\text{vb}\ast t}\text{vb}\ast \nu \cdot \text{vb}\ast t+2\text{vb}\ast \nu \cdot \text{vb}\ast \epsilon \cdot \text{vb}\ast t$$

工程剪应变

$${\gamma }_{\text{vb}\ast \nu \text{vb}\ast t}=2{\epsilon }_{\text{vb}\ast \nu \text{vb}\ast t}=2\text{vb}\ast \nu \cdot \text{vb}\ast \epsilon \cdot \text{vb}\ast t=2{\epsilon }_{ij}{\nu }_i{t}_j$$

若 $\text{vb}\ast \nu ,\text{vb}\ast t$ 为坐标轴方向的单位矢量, 则

$${\gamma }_{ij}=2{\epsilon }_{ij}(i\ne j)$$

3.2.2 小应变张量的性质

转轴公式

$${\epsilon }_{m^{\prime }{n}^{\prime }}={\beta }_{m^{\prime }i}{\beta }_{n^{\prime }j}{\epsilon }_{ij}$$

主应变

$$\text{pqty}\text{vb}\ast {\epsilon }_{ij}-{\epsilon }_{\nu }{\delta }_{ij}{\nu }_j=0$$

${\epsilon }_{\nu }$ — 主值, 沿主方向 $\text{vb}\ast \nu$ 的主应变

特征方程

$${\epsilon }_{\nu }^3-{\theta }_1{\epsilon }_{\nu }^2+{\theta }_2{\epsilon }_{\nu }-{\theta }_3=0$$

第一、第二和第三应变不变量

$$\begin{array}{rl}{\theta }_1& ={\epsilon }_{ii}={\epsilon }_1+{\epsilon }_2+{\epsilon }_3\\ {\theta }_2& =\frac{1}{2}\text{pqty}{\epsilon }_{ii}{\epsilon }_{jj}-{\epsilon }_{ij}{\epsilon }_{ij}={\epsilon }_1{\epsilon }_2+{\epsilon }_2{\epsilon }_3+{\epsilon }_3{\epsilon }_1\\ {\theta }_3& =e_{ijk}{\epsilon }_{1i}{\epsilon }_{2j}{\epsilon }_{3k}={\epsilon }_1{\epsilon }_2{\epsilon }_3\end{array}$$

3.3 刚体转动

转动张量

$$\frac{1}{2}\text{pqty}\text{vb}\ast u\text{grad}-\text{grad}\text{vb}\ast u=-\text{vb}\ast \mathrm{\Omega }$$

$\text{vb}\ast \mathrm{\Omega }$ — 转动张量, 反对称, 只有三个独立分量 ${\mathrm{\Omega }}_{12},{\mathrm{\Omega }}_{23},{\mathrm{\Omega }}_{31}$. 记

$$\begin{array}{rl}{\mathrm{\Omega }}_{12}& ={\omega }_3=\frac{1}{2}\text{pqty}\text{pdv}{u}_2{x}_1-\text{pdv}{u}_1{x}_2\\ {\mathrm{\Omega }}_{23}& ={\omega }_1=\frac{1}{2}\text{pqty}\text{pdv}{u}_3{x}_2-\text{pdv}{u}_2{x}_3\\ {\mathrm{\Omega }}_{31}& ={\omega }_2=\frac{1}{2}\text{pqty}\text{pdv}{u}_1{x}_3-\text{pdv}{u}_3{x}_1\end{array}$$

张量 $\text{vb}\ast \mathrm{\Omega }$ 的反偶矢量

$$\text{vb}\ast \omega ={\omega }_1\text{vb}\ast e_1+{\omega }_2\text{vb}\ast e_2+{\omega }_3\text{vb}\ast e_3$$$$-\text{vb}\ast \mathrm{\Omega }\cdot \text{dd}\text{vb}\ast x=\text{vb}\ast \omega \times \text{dd}\text{vb}\ast x$$

3.4 应变协调方程

几何方程

$${\epsilon }_{ij}=\frac{1}{2}\text{pqty}\text{pdv}{u}_i{x}_j+\text{pdv}{u}_j{x}_i$$

应变协调方程

$${\epsilon }_{ij,kl}+{\epsilon }_{kl,ij}-{\epsilon }_{ik,jl}-{\epsilon }_{jl,ik}=0$$$$\text{pdv}[2]{\epsilon }_{x}y+\text{pdv}[2]{\epsilon }_{y}x-\text{pdv}{\gamma }_{xy}xy=0$$

实体符号形式

$$\text{curl}\text{vb}\ast \epsilon \times \text{grad}=0$$

3.6 由应变求位移

几何方程

$$\begin{array}{rlrl}{\epsilon }_{11}& =\text{pdv}{u}_1{x}_1;& {\gamma }_{12}& =\text{pdv}{u}_1{x}_2+\text{pdv}{u}_2{x}_1\\ {\epsilon }_{22}& =\text{pdv}{u}_2{x}_2;& {\gamma }_{23}& =\text{pdv}{u}_2{x}_3+\text{pdv}{u}_3{x}_2\\ {\epsilon }_{33}& =\text{pdv}{u}_3{x}_3;& {\gamma }_{31}& =\text{pdv}{u}_3{x}_1+\text{pdv}{u}_1{x}_3\end{array}$$

3.6.1 线积分法

3.6.2 直接积分法

3.7 正交曲线坐标系中的几何方程

柱坐标系中的几何方程

$$\begin{array}{rlrl}{\epsilon }_r& =\text{pdv}ur;& {\gamma }_{r\theta }& =\frac{1}{r}\text{pdv}u\theta +\text{pdv}vr-\frac{v}{r}\\ {\epsilon }_{\theta }& =\frac{1}{r}\text{pdv}v\theta +\frac{u}{r};& {\gamma }_{\theta z}& =\text{pdv}vz+\frac{1}{r}\text{pdv}w\theta \\ {\epsilon }_z& =\text{pdv}wz& {\gamma }_{zr}& =\text{pdv}wr+\text{pdv}uz\end{array}$$

球坐标系中的几何方程

$$\begin{array}{rl}{\epsilon }_r& =\text{pdv}{u}_{r}r\\ {\epsilon }_{\theta }& =\frac{1}{r}\text{pdv}{u}_{\theta }\theta +\frac{u_r}{r}\\ {\epsilon }_{\phi }& =\frac{1}{r\sin \theta }\text{pdv}{u}_{\phi }\phi +\frac{u_r}{r}+\frac{\text{ctg}\theta }{r}{u}_{\theta }\\ {\gamma }_{r\theta }& =\frac{1}{r}\text{pdv}{u}_r\theta +\text{pdv}{u}_{\theta }r-\frac{u_{\theta }}{r}\\ {\gamma }_{\theta \phi }& =\frac{1}{r\sin \theta }\text{pdv}{u}_{\theta }\phi +\frac{1}{r}\text{pdv}{u}_{\phi }\theta -\frac{\text{ctg}\theta }{r}{u}_{\phi }\\ {\gamma }_{\phi r}& =\text{pdv}{u}_{\phi }r+\frac{1}{r\sin \theta }\text{pdv}{u}_r\phi -\frac{u_{\phi }}{r}\end{array}$$