弹性力学

弹性力学 (Theory of Elasticity)，也称弹性理论，是固体力学的一个分支，研究弹性体由于受外力作用、边界约束或温度改变等原因而发生的应力、形变和位移问题。 作用于物体的外力可分为体积力（body force）和表面力 （surface force）。体积力是作用在物体内部体积上的外力，简称体力，例如重力、惯性力、电磁力等。表面力是作用在物体表面上的外力，简称面力，例如流体压力、接触力等。 连续性：假定物体是连续的，即整个物体的体积都被组成这个物体的介质所填满，不留下任何空隙，并且在整个变形过程中保持其连续性。 完全弹性：假定物体是完全弹性的，即物体在引起形变的外力去除后能完全恢复其初始的形状和尺寸，物体的形变与其所受外力具有一一对应的函数关系。 均匀性：假定物体是均匀的，即整个物体的所有部分具有相同的弹性性质。 各向同性：既定物体是各向同性的，即物体的弹性性质在所有各个方向都相同，与考察方向无关。 小变形：假定物体受力后的位移和形变是微小的，整个物体所有个点的位移都远小于物体原来的尺寸，且应变与转角都远小于1。 对符合上述前4项假定的物体，称为理想弹性体。 ∂ σ x ∂ x + ∂ τ y x ∂ y + ∂ τ z x ∂ z + X = 0 ∂ τ x y ∂ x + ∂ σ y ∂ y + ∂ τ z y ∂ z + Y = 0 ∂ τ x z ∂ x + ∂ τ y z ∂ y + ∂ σ z ∂ z + Z = 0 {\displaystyle {\begin{aligned}{\frac {\partial \sigma _{x}}{\partial x}}+{\frac {\partial \tau _{yx}}{\partial y}}+{\frac {\partial \tau _{zx}}{\partial z}}+X=0\\{\frac {\partial \tau _{xy}}{\partial x}}+{\frac {\partial \sigma _{y}}{\partial y}}+{\frac {\partial \tau _{zy}}{\partial z}}+Y=0\\{\frac {\partial \tau _{xz}}{\partial x}}+{\frac {\partial \tau _{yz}}{\partial y}}+{\frac {\partial \sigma _{z}}{\partial z}}+Z=0\\\end{aligned}}} ∇ ⋅ σ + f = 0 {\displaystyle \nabla \cdot {\boldsymbol {\sigma }}+{\boldsymbol {f}}={\boldsymbol {0}}} ϵ x = ∂ u ∂ x , γ y z = 1 2 ( ∂ w ∂ y + ∂ v ∂ z ) ϵ y = ∂ v ∂ y , γ z x = 1 2 ( ∂ u ∂ z + ∂ w ∂ x ) ϵ z = ∂ w ∂ z , γ x y = 1 2 ( ∂ v ∂ x + ∂ u ∂ y ) {\displaystyle {\begin{aligned}\epsilon _{x}={\frac {\partial u}{\partial x}},\quad \gamma _{yz}={\frac {1}{2}}\left({\frac {\partial w}{\partial y}}+{\frac {\partial v}{\partial z}}\right)\\\epsilon _{y}={\frac {\partial v}{\partial y}},\quad \gamma _{zx}={\frac {1}{2}}\left({\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}\right)\\\epsilon _{z}={\frac {\partial w}{\partial z}},\quad \gamma _{xy}={\frac {1}{2}}\left({\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}\right)\\\end{aligned}}} ϵ = 1 2 ( u ∇ + ∇ u ) {\displaystyle {\boldsymbol {\epsilon }}={\frac {1}{2}}\left({\boldsymbol {u}}\nabla +\nabla {\boldsymbol {u}}\right)} ϵ x = 1 E [ σ x − ν ( σ y + σ z ) ] , γ y z = 1 G τ y z ϵ y = 1 E [ σ y − ν ( σ z + σ x ) ] , γ z x = 1 G τ z x ϵ z = 1 E [ σ z − ν ( σ x + σ y ) ] , γ x y = 1 G τ x y {\displaystyle {\begin{aligned}\epsilon _{x}={\frac {1}{E}}\left[\sigma _{x}-\nu \left(\sigma _{y}+\sigma _{z}\right)\right],\quad \gamma _{yz}={\frac {1}{G}}\tau _{yz}\\\epsilon _{y}={\frac {1}{E}}\left[\sigma _{y}-\nu \left(\sigma _{z}+\sigma _{x}\right)\right],\quad \gamma _{zx}={\frac {1}{G}}\tau _{zx}\\\epsilon _{z}={\frac {1}{E}}\left[\sigma _{z}-\nu \left(\sigma _{x}+\sigma _{y}\right)\right],\quad \gamma _{xy}={\frac {1}{G}}\tau _{xy}\\\end{aligned}}} σ z = τ z x = τ z y = 0 {\displaystyle \sigma _{z}=\tau _{zx}=\tau _{zy}=0} ϵ z = γ z x = γ z y = 0 {\displaystyle \epsilon _{z}=\gamma _{zx}=\gamma _{zy}=0} 胡克定律 材料力学 结构力学 有限单元法 Stephen Timoshenko and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill. 1951: 229 （英语）.  Stephen Timoshenko and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill. 1951: 23 （英语）.  Stephen Timoshenko and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill. 1951: 25 （英语）.  Stephen Timoshenko and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill. 1951: 11 （英语）.  Stephen Timoshenko and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill. 1951: 11 （英语）.