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3. 应变理论

Course NotesEngineering Mechanics for Civil EngineeringAbout 3 minAbout 940 words

3.1 位移和应变

3.1.1 位移的描述

拉格朗日描述法

ui=xi(a1,a2,a3)−ai 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

Eij=12(\pdvxmai\pdvxmaj−δij)=12(\pdvuiaj+\pdvujai+\pdvumai\pdvumaj) E_{ij} = \frac{1}{2} \pqty{\pdv{x_m}{a_i} \pdv{x_m}{a_j} - \delta_{ij}} = \frac{1}{2} \pqty{\pdv{u_i}{a_j} + \pdv{u_j}{a_i} + \pdv{u_m}{a_i} \pdv{u_m}{a_j}}

\vb∗u\grad=\pdvuiaj\vb∗ei\vb∗ej,\grad\vb∗u=\pdvujai\vb∗ei\vb∗ej \vb*{u}\grad = \pdv{u_i}{a_j} \vb*{e}_i \vb*{e}_j \qc \grad{\vb*{u}} = \pdv{u_j}{a_i} \vb*{e}_i \vb*{e}_j

\vb∗E=12(\vb∗u\grad+\grad\vb∗u+\grad\vb∗u⋅\vb∗u\grad) \vb*{E} = \frac{1}{2} \pqty{\vb*{u}\grad + \grad{\vb*{u}} + \grad{\vb*{u}} \cdot \vb*{u}\grad}

长度比

λν=1+2\vb∗ν⋅\vb∗E⋅\vb∗ν \lambda_{\nu} = \sqrt{1 + 2 \vb*{\nu} \cdot \vb*{E} \cdot \vb*{\nu}}

线元变形后的夹角变化

cos⁡(\vb∗ν′,\vb∗t′)=(\vb∗ν⋅\vb∗t+2\vb∗ν⋅\vb∗E⋅\vb∗t)1λ\vb∗νλ\vb∗t \cos(\vb*{\nu}', \vb*{t}') = \pqty{\vb*{\nu} \cdot \vb*{t} + 2 \vb*{\nu} \cdot \vb*{E} \cdot \vb*{t}} \frac{1}{\lambda_{\vb*{\nu}} \lambda_{\vb*{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

εij=12(ui,j+uj,i)\vb∗ε=12(\vb∗u\grad+\grad\vb∗u) \varepsilon_{ij} = \frac{1}{2} \pqty{u_{i,j} + u_{j,i}} \\ \vb*{\varepsilon} = \frac{1}{2} \pqty{\vb*{u}\grad + \grad{\vb*{u}}}

长度比

λ\vb∗ν=1+\vb∗ν⋅\vb∗ε⋅\vb∗ν \lambda_{\vb*{\nu}} = 1 + \vb*{\nu} \cdot \vb*{\varepsilon} \cdot \vb*{\nu}

工程正应变

ε\vb∗ν=\vb∗ν⋅\vb∗ε⋅\vb∗ν \varepsilon_{\vb*{\nu}} = \vb*{\nu} \cdot \vb*{\varepsilon} \cdot \vb*{\nu}

两线元间的夹角变化

cos⁡(\vb∗ν′,\vb∗t′)=(1−ε\vb∗ν−ε\vb∗t)\vb∗ν⋅\vb∗t+2\vb∗ν⋅\vb∗ε⋅\vb∗t \cos(\vb*{\nu}', \vb*{t}') = \pqty{1 - \varepsilon_{\vb*{\nu}} - \varepsilon_{\vb*{t}}} \vb*{\nu} \cdot \vb*{t} + 2 \vb*{\nu} \cdot \vb*{\varepsilon} \cdot \vb*{t}

工程剪应变

γ\vb∗ν\vb∗t=2ε\vb∗ν\vb∗t=2\vb∗ν⋅\vb∗ε⋅\vb∗t=2εijνitj \gamma_{\vb*{\nu} \vb*{t}} = 2 \varepsilon_{\vb*{\nu} \vb*{t}} = 2 \vb*{\nu} \cdot \vb*{\varepsilon} \cdot \vb*{t} = 2 \varepsilon_{ij} \nu_i t_j

若 \vb∗ν,\vb∗t\vb*{\nu}, \vb*{t} 为坐标轴方向的单位矢量, 则

γij=2εij(i≠j) \gamma_{ij} = 2 \varepsilon_{ij} \quad (i \neq j)

3.2.2 小应变张量的性质

转轴公式

εm′n′=βm′iβn′jεij \varepsilon_{m'n'} = \beta_{m'i} \beta_{n'j} \varepsilon_{ij}

主应变

(\vb∗εij−ενδij)νj=0 \pqty{\vb*{\varepsilon_{ij} - \varepsilon_{\nu} \delta_{ij}}} \nu_j = 0

εν\varepsilon_{\nu} — 主值, 沿主方向 \vb∗ν\vb*{\nu} 的主应变

特征方程

εν3−θ1εν2+θ2εν−θ3=0 \varepsilon_{\nu}^3 - \theta_1 \varepsilon_{\nu}^2 + \theta_2 \varepsilon_{\nu} - \theta_3 = 0

第一、第二和第三应变不变量

θ1=εii=ε1+ε2+ε3θ2=12(εiiεjj−εijεij)=ε1ε2+ε2ε3+ε3ε1θ3=eijkε1iε2jε3k=ε1ε2ε3 \begin{align*} \theta_1 & = \varepsilon_{ii} = \varepsilon_1 + \varepsilon_2 + \varepsilon_3 \\ \theta_2 & = \frac{1}{2} \pqty{\varepsilon_{ii} \varepsilon_{jj} - \varepsilon_{ij} \varepsilon_{ij}} = \varepsilon_1 \varepsilon_2 + \varepsilon_2 \varepsilon_3 + \varepsilon_3 \varepsilon_1 \\ \theta_3 & = e_{ijk} \varepsilon_{1i} \varepsilon_{2j} \varepsilon_{3k} = \varepsilon_1 \varepsilon_2 \varepsilon_3 \end{align*}

3.3 刚体转动

转动张量

12(\vb∗u\grad−\grad\vb∗u)=−\vb∗Ω \frac{1}{2} \pqty{\vb*{u}\grad - \grad{\vb*{u}}} = - \vb*{\Omega}

\vb∗Ω\vb*{\Omega} — 转动张量, 反对称, 只有三个独立分量 Ω12,Ω23,Ω31\Omega_{12}, \Omega_{23}, \Omega_{31}. 记

Ω12=ω3=12(\pdvu2x1−\pdvu1x2)Ω23=ω1=12(\pdvu3x2−\pdvu2x3)Ω31=ω2=12(\pdvu1x3−\pdvu3x1) \begin{align*} \Omega_{12} & = \omega_3 = \frac{1}{2} \pqty{\pdv{u_2}{x_1} - \pdv{u_1}{x_2}} \\ \Omega_{23} & = \omega_1 = \frac{1}{2} \pqty{\pdv{u_3}{x_2} - \pdv{u_2}{x_3}} \\ \Omega_{31} & = \omega_2 = \frac{1}{2} \pqty{\pdv{u_1}{x_3} - \pdv{u_3}{x_1}} \end{align*}

张量 \vb∗Ω\vb*{\Omega} 的反偶矢量

\vb∗ω=ω1\vb∗e1+ω2\vb∗e2+ω3\vb∗e3 \vb*{\omega} = \omega_1 \vb*{e}_1 + \omega_2 \vb*{e}_2 + \omega_3 \vb*{e}_3

−\vb∗Ω⋅\dd\vb∗x=\vb∗ω×\dd\vb∗x - \vb*{\Omega} \cdot \dd{\vb*{x}}= \vb*{\omega} \times \dd{\vb*{x}}

3.4 应变协调方程

几何方程

εij=12(\pdvuixj+\pdvujxi) \varepsilon_{ij} = \frac{1}{2} \pqty{\pdv{u_i}{x_j} + \pdv{u_j}{x_i}}

应变协调方程

εij,kl+εkl,ij−εik,jl−εjl,ik=0 \varepsilon_{ij,kl} + \varepsilon_{kl,ij} - \varepsilon_{ik,jl} - \varepsilon_{jl,ik} = 0

\pdv[2]εxy+\pdv[2]εyx−\pdvγxyxy=0 \pdv[2]{\varepsilon_x}{y} + \pdv[2]{\varepsilon_y}{x} - \pdv{\gamma_{xy}}{x}{y} = 0

实体符号形式

\curl\vb∗ε×\grad=0 \curl{\vb*{\varepsilon}} \times \grad = 0

3.6 由应变求位移

几何方程

ε11=\pdvu1x1;γ12=\pdvu1x2+\pdvu2x1ε22=\pdvu2x2;γ23=\pdvu2x3+\pdvu3x2ε33=\pdvu3x3;γ31=\pdvu3x1+\pdvu1x3 \begin{align*} \varepsilon_{11} & = \pdv{u_1}{x_1}; & \gamma_{12} & = \pdv{u_1}{x_2} + \pdv{u_2}{x_1} \\ \varepsilon_{22} & = \pdv{u_2}{x_2}; & \gamma_{23} & = \pdv{u_2}{x_3} + \pdv{u_3}{x_2} \\ \varepsilon_{33} & = \pdv{u_3}{x_3}; & \gamma_{31} & = \pdv{u_3}{x_1} + \pdv{u_1}{x_3} \end{align*}

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

εr=\pdvur;γrθ=1r\pdvuθ+\pdvvr−vrεθ=1r\pdvvθ+ur;γθz=\pdvvz+1r\pdvwθεz=\pdvwzγzr=\pdvwr+\pdvuz \begin{align*} \varepsilon_r & = \pdv{u}{r}; & \gamma_{r \theta} & = \frac{1}{r} \pdv{u}{\theta} + \pdv{v}{r} - \frac{v}{r} \\ \varepsilon_{\theta} & = \frac{1}{r} \pdv{v}{\theta} + \frac{u}{r}; & \gamma_{\theta z} & = \pdv{v}{z} + \frac{1}{r} \pdv{w}{\theta} \\ \varepsilon_z & = \pdv{w}{z} & \gamma_{zr} & = \pdv{w}{r} + \pdv{u}{z} \end{align*}

球坐标系中的几何方程

εr=\pdvurrεθ=1r\pdvuθθ+urrεφ=1rsin⁡θ\pdvuφφ+urr+ctgθruθγrθ=1r\pdvurθ+\pdvuθr−uθrγθφ=1rsin⁡θ\pdvuθφ+1r\pdvuφθ−ctgθruφγφr=\pdvuφr+1rsin⁡θ\pdvurφ−uφr \begin{align*} \varepsilon_r & = \pdv{u_r}{r} \\ \varepsilon_{\theta} & = \frac{1}{r} \pdv{u_{\theta}}{\theta} + \frac{u_r}{r} \\ \varepsilon_{\varphi} & = \frac{1}{r \sin{\theta}} \pdv{u_{\varphi}}{\varphi} + \frac{u_r}{r} + \frac{\textrm{ctg}{\theta}}{r} u_{\theta} \\ \gamma_{r \theta} & = \frac{1}{r} \pdv{u_r}{\theta} + \pdv{u_{\theta}}{r} - \frac{u_{\theta}}{r} \\ \gamma_{\theta \varphi} & = \frac{1}{r \sin{\theta}} \pdv{u_{\theta}}{\varphi} + \frac{1}{r} \pdv{u_{\varphi}}{\theta} - \frac{\textrm{ctg}{\theta}}{r} u_{\varphi} \\ \gamma_{\varphi r} & = \pdv{u_{\varphi}}{r} + \frac{1}{r \sin{\theta}} \pdv{u_r}{\varphi} - \frac{u_{\varphi}}{r} \end{align*}