文档介绍：Spatial Problem
Chapter 8
Elasticity
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第八章空间问题
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Space Problem
Chapter8 Space Problem
§8-4 The Spherical Symmetric Problem of Space
§8-3 The Axially Symmetric Problem of Space
§8-2 The Basic Equation unde Rectangular Coordinate
§8-1 Introduction
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空间问题
第八章空间问题
§8-4 空间球对称问题
§8-3 空间轴对称问题
§8-2 直角坐标下的基本方程
§8-1 概述
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In this chapter we first give out the equations of equilibrium, the geometric equations and the physical equations under rectangular coordinate for spatial problems. For the analytic solutions of spatial problems can only be obtained under peculiar boundary conditions, we discuss the axial symmetric problems and the ball symmetric problems of space emphatically.
§8-1 Introduction
Ball Symmetric Problem
x
z
y
Axial Symmetric Problem
x
z
y
P
Space Problem
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本章首先给出空间问题直角坐标下的平衡方程、几何方程和物理方程。针对空间问题的解析解一般只能在特殊边界条件下才可以得到,我们着重讨论空间轴对称问题和空间球对称问题。
§8-1 概述
空间问题
球对称问题
x
z
y
轴对称问题
x
z
y
P
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§8-2 Basic Equations under Rectangular Coordinate
One. Differential Equations of Equilibrium
Consider an arbitrary point inside the body
and fetch a small parallel hexahedron, which ponents on each side are shown as figure.
If ab denotes the line which joins the centers of two faces of the hexahedron, then from we get
Canceling terms and neglecting higher order small variables,we get
Space Problem
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§8-2 直角坐标下的基本方程
空间问题
一平衡微分方程
在物体内任意一点 P,取图示微小平行六面体。微小平行六面体各面上的应力分量如图所示。
若以连接六面体前后两面中心的直线为ab,则由得
化简并略去高阶微量,得
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Similarly,we get
Here we prove the relation of the equality of cross shears again
from
List the equations,cancel terms,we get
These are differential equations of equilibrium under rectangular coordinate of space
Two. Geometric Equations
For spatial problems, ponents and ponents should satisfy following geometric equations
Of which the first two and the last have been obtained among plane problems, the other three can be led out with the same method.
Space Problem
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空间问题
同理可得
这只是又一次证明了剪应力的互等关系。
由
立出方程,经约简后得
这就是