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5. 弹性理论的微分提法、解法及一般原理

Course NotesEngineering Mechanics for Civil EngineeringAbout 2 minAbout 612 words

5.1 弹性力学问题的微分提法

平衡 (运动) 方程

\labeleq:5.1σij,j+fi=0(=ρu¨i) \begin{equation} \label{eq:5.1} \sigma_{ij, j} + f_i = 0 \quad (= \rho \ddot{u}_i) \end{equation}

几何方程

\labeleq:5.2εij=12(ui,j+uj,i) \begin{equation} \label{eq:5.2} \varepsilon_{ij} = \frac{1}{2} \pqty{u_{i,j} + u_{j,i}} \end{equation}

应变协调方程

\labeleq:5.3εij,kl+εkl,ij−εik,jl−εjl,ik=0 \begin{equation} \label{eq:5.3} \varepsilon_{ij,kl} + \varepsilon_{kl,ij} - \varepsilon_{ik,jl} - \varepsilon_{jl,ik} = 0 \end{equation}

本构方程
应变-应力公式

\labeleq:5.4εij=1+νEσij−νEσkkδij \begin{equation} \label{eq:5.4} \varepsilon_{ij} = \frac{1 + \nu}{E} \sigma_{ij} - \frac{\nu}{E} \sigma_{kk} \delta_{ij} \end{equation}

应力-应变公式

\labeleq:5.5σij=2Gεij+λεkkδij \begin{equation} \label{eq:5.5} \sigma_{ij} = 2 G \varepsilon_{ij} + \lambda \varepsilon_{kk} \delta_{ij} \end{equation}

基本方程组
第一组

\labeleq:5.6平衡方程\eqrefeq:5.1本构方程\eqrefeq:5.5几何方程\eqrefeq:5.2 \begin{equation} \label{eq:5.6} \begin{gathered} \text{平衡方程} \eqref{eq:5.1} \\ \text{本构方程} \eqref{eq:5.5} \\ \text{几何方程} \eqref{eq:5.2} \end{gathered} \end{equation}

第二组

\labeleq:5.7协调方程\eqrefeq:5.3本构方程\eqrefeq:5.4平衡方程\eqrefeq:5.1 \begin{equation} \label{eq:5.7} \begin{gathered} \text{协调方程} \eqref{eq:5.3} \\ \text{本构方程} \eqref{eq:5.4} \\ \text{平衡方程} \eqref{eq:5.1} \end{gathered} \end{equation}

边界条件
力边界

σjiνj=X‾i \sigma_{ji} \nu_j = \overline{X}_i

位移边界

ui=u‾i u_i = \overline{u}_i

5.2 位移解法

位移法定解方程 / 拉梅方程 / L-N 方程

Warning

要求平面情况推导

Gui,jj+(λ+G)uj,ji+fi=0(i=1,2,3) G u_{i,jj} + \pqty{\lambda + G} u_{j,ji} + f_i = 0 \quad (i = 1, 2, 3)

常规形式为

G∇2u+(λ+G)\pdvθx+f1=0G∇2v+(λ+G)\pdvθy+f2=0G∇2w+(λ+G)\pdvθz+f3=0 \begin{align*} G \laplacian{u} + \pqty{\lambda + G} \pdv{\theta}{x} + f_1 & = 0 \\ G \laplacian{v} + \pqty{\lambda + G} \pdv{\theta}{y} + f_2 & = 0 \\ G \laplacian{w} + \pqty{\lambda + G} \pdv{\theta}{z} + f_3 & = 0 \end{align*}

力边界条件

\bqtyG(ui,j+uj,i)+λuk,kδijνj=X‾i \bqty{G \pqty{u_{i,j} + u_{j,i}} + \lambda u_{k,k} \delta_{ij}} \nu_j = \overline{X}_i

5.3 应力解法

应力协调方程 / 贝尔脱拉密-密乞尔方程 / B-M 方程

∇2σx+11+ν\pdv[2]Θx=−2\pdvf1x−ν1−ν(\pdvf1x+\pdvf2y+\pdvf3z)∇2σy+11+ν\pdv[2]Θy=−2\pdvf2y−ν1−ν(\pdvf1x+\pdvf2y+\pdvf3z)∇2σz+11+ν\pdv[2]Θz=−2\pdvf3z−ν1−ν(\pdvf1x+\pdvf2y+\pdvf3z)∇2τyz+11+ν\pdvΘyz=−(\pdvf2z+\pdvf3y)∇2τzx+11+ν\pdvΘzx=−(\pdvf3x+\pdvf1z)∇2τxy+11+ν\pdvΘxy=−(\pdvf1y+\pdvf2x) \begin{gather*} \laplacian{\sigma_x} + \frac{1}{1 + \nu} \pdv[2]{\Theta}{x} = - 2 \pdv{f_1}{x} - \frac{\nu}{1 - \nu} \pqty{\pdv{f_1}{x} + \pdv{f_2}{y} + \pdv{f_3}{z}} \\ \laplacian{\sigma_y} + \frac{1}{1 + \nu} \pdv[2]{\Theta}{y} = - 2 \pdv{f_2}{y} - \frac{\nu}{1 - \nu} \pqty{\pdv{f_1}{x} + \pdv{f_2}{y} + \pdv{f_3}{z}} \\ \laplacian{\sigma_z} + \frac{1}{1 + \nu} \pdv[2]{\Theta}{z} = - 2 \pdv{f_3}{z} - \frac{\nu}{1 - \nu} \pqty{\pdv{f_1}{x} + \pdv{f_2}{y} + \pdv{f_3}{z}} \\ \laplacian{\tau_{yz}} + \frac{1}{1 + \nu} \pdv{\Theta}{y}{z} = - \pqty{\pdv{f_2}{z} + \pdv{f_3}{y}} \\ \laplacian{\tau_{zx}} + \frac{1}{1 + \nu} \pdv{\Theta}{z}{x} = - \pqty{\pdv{f_3}{x} + \pdv{f_1}{z}} \\ \laplacian{\tau_{xy}} + \frac{1}{1 + \nu} \pdv{\Theta}{x}{y} = - \pqty{\pdv{f_1}{y} + \pdv{f_2}{x}} \end{gather*}

5.7 圣维南原理

由作用在物体局部表面上的自平衡力系 (即合力与合力矩为零的力系) 所引起的应力和应变, 在远离作用区 (距离远大于该局部区域)