《弹性力学》（双语版） 第八章 空间问题

Elasticity Spatial Problem

1 Elasticity

弹单性生力学 第八二间问题

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Chapter8 Space Problem p§8-1 Introduction D 88-2 The Basic Equation unde Rectangular Coordinate S88-3 The Axially Symmetric Problem of Space 88-4 The Spherical Symmetric Problem of Space

3 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

空间题 第八章间问题 §8-1概述 □§8-2直角坐标下的基本方程 §8-3空间轴对称问题 §8-4空间球对称问题

4 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

§8-1 Introduction 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 Z Axial Symmetric Problem Ball Symmetric Problem

5 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

空间题 §8-1概迷 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。 轴对称问题 球对称后题

6 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 球对称问题 x z y 轴对称问题 x z y P

88-2 Basic Equations under Rectangular Coordinate One Differential Equations of equilibrium Consider an arbitrary point inside the body and fetch a small parallel hexahedron, which 中 stress components on each side are shown as fs gure ay If ab denotes the line which joins the centers of two faces of the hexahedron, then from ∑ m=0 ve get d+x2a空 2 2 T+ dz ldxdv-=T dxdy=0 az 2 2 Canceling terms and neglecting higher order small variables, we get 7

7 §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 stress components 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 mab = 0 2 2 dy dxdz dy dy dxdz y yz yz yz    +           + 0 2 2 − =           − + dz dxdy dz dz dxdy z zy zy zy    Canceling terms and neglecting higher order small variables,we get

空间题 §8-2直角坐标下的基本方程 平衡微分方程 g,+ 在物体内任意一点P,取图 dr 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 0 心的直线为ab,则由∑mb=0得 gr+“a at dy dxdz +ty dxdz ay +h一x=0 az 化简并略去高阶微量,得 8

8 §8-2 直角坐标下的基本方程 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 mab = 0 得 2 2 dy dxdz dy dy dxdz y yz yz yz    +           + 0 2 2 − =           − + dz dxdy dz dz dxdy z zy zy zy    化简并略去高阶微量，得

Similarly we =y Here we prove the relation of the equality get of cross shears again fom∑x=0∑r=0∑Z=0 + +X=0 Ox List the equations. cancel terms we get +Y=0 These are differential equations of equilibrium a0 Otx+S under rectangular coordinate of space Ov+Z=0 Two. Geometric Equations For spatial problems, deformation components and displacement components should satisfy following geometric equations Ow Oy L OW y O 8= Ov au y 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 9

9 xy yx zx xz yz zy       = = = Similarly,we get Here we prove the relation of the equality of cross shears again from X = 0,Y = 0,Z = 0 List the equations,cancel terms,we get These are differential equations of equilibrium under rectangular coordinate of space          + =   +   +   + =   +   +   + =   +   +   0 0 0 Z z x y Y y z x X x y z z xz yz y z y xy x yx z x          Two. Geometric Equations For spatial problems, deformation components and displacement components should satisfy following geometric equations z w y v x u z y x   =   =   =    y u x v x w z u z v y w xy zx yz   +   =   +   =   +   =    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

空间题 同理可得z,==z=这只是又一次证明了剪应力的互等关系。 o、+0xm+O 出方程,经约后得0×3x。x=0 由∑x=0∑y=0.∑z=0 +Y=0 ax 这就是空间直角坐标下的0 aT + +Z=0 平衡微分方程。 二几何方程 在空间问题中,形变分量与位移分量应当满足下列6个几何 方程Ex Ow Oy ax Ox a yx=+一 az O 其中的第一式、第二式和第六式已在平面问题中导出,其余三式 可用相同的方法导出。 10

10 xy yx zx xz yz zy       = = 同理可得 = 这只是又一次证明了剪应力的互等关系。 由 X = 0,Y = 0,Z = 0 立出方程，经约简后得 这就是空间直角坐标下的 平衡微分方程。          + =   +   +   + =   +   +   + =   +   +   0 0 0 Z z x y Y y z x X x y z z xz yz y z y xy x yx z x          二 几何方程 在空间问题中，形变分量与位移分量应当满足下列 6 个几何 方程 z w y v x u z y x   =   =   =    y u x v x w z u z v y w xy zx yz   +   =   +   =   +   =    其中的第一式、第二式和第六式已在平面问题中导出，其余三式 可用相同的方法导出