望“弹性力学”个源码

'''弹性力学''' (Theory of Elasticity)，也叫'''弹性理论'''，是[[固体力学]]个一隻分支，研究[[弹性体]]因为受到外力作用、边界约束或温度改变咾啥个原因发生个[[应力]]、[[应变（物理学）|应变]]、[[位移]]问题。

弹性力学是[[力学]]学科个一隻重点，是[[塑性力学]]、[[计算力学]]搭仔其他交叉学科个基础。

== 基本概念同基本假设 ==

=== 基本概念 ===
作用拉物体丄个外力可分成功体积力（body force）搭仔[[表面力]] （surface force）。体积力是作用拉物体内部体积丄个外力，简称体力，譬如[[重力]]、[[惯性力]]、[[电磁力]]咾啥个。表面力是作用拉物体表面丄个外力，简称面力，譬如[[流体压力]]、接触力等。

[[弹性体]]是一种特殊个变形体，伊个特征是：外力作用下头，物体变形，当外力弗超过某一限度个辰光，去脱外力以後物体会得恢复原状样。去脱外力以後，物体个残余变形咾小个辰光，一般性会得拿伊看成功弹性体。

=== 基本假设 ===
# 连续性：假定物体是连续个，也就是整個物体个体积侪被组成箇個物体个介质填没脱，呒没任何空隙，勒拉整个变形过程当中保持伊个连续性。
# 完全弹性：假定物体是完全弹性个，也就是物体勒拉去除脱引起形变个外力以後会得完全恢复伊一开始个形状搭仔尺寸，物体个形变同其所受个外力有一一对应个函数关系。
# 均匀性：假定物体是均匀个，也就是整個物体个所有部分侪具有相同个弹性性质。
# 各向同性：既定物体是各向同性個，也就是物体个弹性性质勒拉所有个方向侪一式一样，同考察方向呒没任何关系。
# 小变形：假定物体受力以後个位移同形变是微小个，整個物体所有个点个位移侪远比物体本生个尺寸小，而且应变同转角侪远小于1。
# 无初应力：假定物体勒拉弗受外力或者温度变化咾啥个作用之前，伊里向没应力。
符合上述前4项假定个物体叫理想弹性体，是弹性力学个主要研究对象。

== 基本方程 ==

=== 平衡方程 ===
==== 应力形式个静力平衡方程 ====
<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> <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>

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

=== 几何方程 ===
==== 应变同位移个关系式 ====
<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> <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>

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

=== 本构方程(虎克定律) ===
{{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><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> 

== 平面问题 ==
=== [[平面应力]]问题 ===
<math> \sigma_z = \tau_{zx} = \tau_{zy} = 0 </math> <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>
=== [[平面应变]]问题 ===
<math> \epsilon_z = \gamma_{zx} = \gamma_{zy} = 0 </math> <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> 

== 同连续介质力学个关系 ==
{| class="wikitable" border="1"
| rowspan="4" |[[连续介质力学]]：研究连续介质个力学
| rowspan="2" |[[固体力学]]：研究[[固体]]连续介质（不受力个辰光形状固定）个力学
| colspan="2" |[[弹性力学]]：固体受到[[应力]]作用后，会得恢复原来个形状
|-
|[[塑性力学]]：固体勒拉受到相当大个应力以后，产生个永久变形
| rowspan="2" |[[流变学]]：研究外力作用情况下头，物事个变形同流动
|-
| rowspan="2" |[[流体力学]]：研究[[流体]]连续介质（形状随容器变化）个力学
| |[[非牛顿流体]]
|-
| colspan="2" |[[牛顿流体]]
|}

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

== 参考 ==
{{Reflist}}

[[Category:固体力学]]