查看“︁弹性力学”︁的源代码

{{NoteTA|G1=Physics}}
'''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。

== 基本概念与基本假设 ==

=== 基本概念 ===
{{See also|力|应力|应变 (物理学)|形變|位移|弹性 (物理学)}}
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。

=== 基本假设 ===
# 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。

== 基本方程 ==

=== 平衡方程 ===
==== 應力形式的靜力平衡方程式 <ref> {{Cite book | author = Stephen Timoshenko and J. N. Goodier | title = Theory of Elasticity | url = https://archive.org/details/theoryofelastici00timo | location = New York | publisher = McGraw-Hill | date = 1951 | pages = [https://archive.org/details/theoryofelastici00timo/page/n251 229] | language = en }} </ref>====
<math>
\begin{align}
\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{align}
</math>

==== 张量形式 ====
<math>
\nabla \cdot \boldsymbol{\sigma} + \boldsymbol{f} = \boldsymbol{0}
</math>

=== 几何方程 ===
==== 應變與位移關係式 <ref> {{Cite book | author = Stephen Timoshenko and J. N. Goodier | title = Theory of Elasticity | url = https://archive.org/details/theoryofelastici00timo | location = New York | publisher = McGraw-Hill | date = 1951 | pages = [https://archive.org/details/theoryofelastici00timo/page/n45 23] | language = en }} </ref>====
<math>
\begin{align}
\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{align}
</math>

==== 张量形式(向量) ====
<math>
\boldsymbol{\epsilon} = \frac{1}{2} \left( \boldsymbol{u} \nabla + \nabla \boldsymbol{u} \right)
</math>

=== 等向性材料的應力與應變關係式(虎克定律)（本构方程） <ref> {{Cite book | author = Stephen Timoshenko and J. N. Goodier | title = Theory of Elasticity | url = https://archive.org/details/theoryofelastici00timo | location = New York | publisher = McGraw-Hill | date = 1951 | pages = [https://archive.org/details/theoryofelastici00timo/page/n47 25] | language = en }} </ref> ===
{{See also|弹性模量|泊松比|剪切模量}}
<math>
\begin{align}
\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{align}
</math>

== 平面问题 ==
=== [[平面應力]]問題 <ref> {{Cite book | author = Stephen Timoshenko and J. N. Goodier | title = Theory of Elasticity | url = https://archive.org/details/theoryofelastici00timo | location = New York | publisher = McGraw-Hill | date = 1951 | pages = [https://archive.org/details/theoryofelastici00timo/page/n33 11] | language = en }} </ref>===
<math> \sigma_z = \tau_{zx} = \tau_{zy} = 0 </math>
=== [[平面應變]]問題 <ref> {{Cite book | author = Stephen Timoshenko and J. N. Goodier | title = Theory of Elasticity | url = https://archive.org/details/theoryofelastici00timo | location = New York | publisher = McGraw-Hill | date = 1951 | pages = [https://archive.org/details/theoryofelastici00timo/page/n33 11] | language = en }} </ref> ===
<math> \epsilon_z = \gamma_{zx} = \gamma_{zy} = 0 </math>

== 参见 ==
* [[胡克定律]]
* [[材料力学]]
* [[结构力学]]
* [[有限单元法]]

== 参考文献 ==
{{reflist|30em}}

== 外部链接 ==

{{连续介质力学}}
[[Category:固体力学]]