1 ElasticityElasticity 3 Summarize 1-1 Movement differential equations of elastic objects 11-2 Without rotating wave and equal volume wave 11-3 Transverse wave and vertical wave 11-4 Spherical wave Chapter 11 Elastic Wave 4 概述 1-1 弹性体的运动微分方程 11-2 无旋波与等容波 11-3 横波与纵波 11-4 球面波 第十一章 弹性波 5 Summarize: When elastic object bears loads in static force equilibrium conditions, not all the parts of object has displacement, distortion and stress. At the beginning of the loads, the parts which are more far from the loads have no impacts . After then ,the displacement , distortion and stress caused by loads transmit to other places in a finite speed of wave. This wave is called elastic wave. This chapter will first give movement differential equations of elastic objects, then introduce some conceptions of elastic wave and simplify the equations according to different elastic waves, at last give the speed transmitting formulas of wave in infinite elastic objects. 6 概述 当静力平衡状态下的弹性体受到荷载作用时，并不是在 弹性体的所有各部分都立即引起位移、形变和应力。在作用 开始时，距荷载作用处较远的部分仍保持不受干扰。在作用 开始后，荷载所引起的位移、形变和应力，就以波动的形式 用有限大的速度向别处传播。这种波动就称为弹性波。 本章将首先给出描述弹性体运动的基本微分方程，然后 介绍弹性波的几个概念，针对不同的弹性波，对运动微分方 程进行简化，最后给出波在无限大弹性体中传播速度公式。 7 11-1 Movement differential equations of elastic objects The two assumptions are equal to the basic assumptions when we discuss static force questions. So the physic and geometry equations and elastic equations where stress component is expressed by displacement component , still are the same with movement equations at any instantaneous time. The only difference is that the equilibrium differential equations of static questions must be substituted by movement differential equations . This chapter we still adopt the assumptions: （1） Elastic objects are ideal elastic objects. （2） The displacement and distortion are tinny. 8 11-1 弹性体的运动微分方程 上述两条假设，完全等同于讨论静力问题的基本假 设。因此，在静力问题中给出的物理方程和几何方程， 以及把应力分量用位移分量表示的弹性方程，仍然适用 于讨论动力问题的任一瞬时，所不同的仅仅在于，静力 问题中的平衡微分方程必须用运动微分方程来代替。 本章仍然采用如下假设： （1） 弹性体为理想弹性体。 （2） 假定位移和形变都是微小的。 9 Toward any tiny object , when we apply dAlembert theory , we must consider stress , body force and the inertia force of elastic objects caused by acceleration . In space right-angle coordinate system, the x, y, z directions component of inertia force of every unite volume are: Where is the density of elastic objects. 10 对于任取的微元体，运用达朗伯尔原理，除了 考虑应力和体力以外，还须考虑弹性体由于具有加 速度而产生的惯性力。每单位体积上的惯性力在空 间直角坐标系的x,y,z方向的分量分别为: 其中为弹性体的密度。 11 Because of the equilibrium relations ,we simplify them and get: The above formulas are called movement differential equations of elastic objects. They and geometry equations and physic equations are the basic equations of movement questions of elasticity mechanics 12 由平衡关系，并简化后得： 上式称为弹性体的运动微分方程。它同几何方程和物理方程 一起构成弹性力学动力问题的基本方程。 13 Note 1: geometry equations: 14 注1：几何方程 15 Note2: physic equations 16 注2：物理方程 17 Because the displacement component is difficult to be expressed by stress and its derivative, so movement equations of elasticity mechanics are usually solved according to the displacement. Substitute the elasticity equations where stress components are expressed by displacement component into movement differential equations, and we let: Then we get: 18 由于位移分量很难用应力及其导数来表示，所以弹 性力学动力问题通常要按位移求解。将应力分量用位移 分量表示的弹性方程代入运动微分方程，并令： 得： 19 These are the basic differential equations of movement equations solved by displacement. They are also called Lame equations. We need boundary conditions to solve Lame equations. Besides these we still need original conditions , because displacement components are the function of time variable . In order to simplify calculation , usually we neglect body force. Now the movement differential equations of elastic objects can be simplified as : 20 这就是按位移求解动力问题的基本微分方程，也称 为拉密（Lame)方程。 要求解拉密方程，显然需要边界条件。除此之外， 由于位移分量还是时间变量的函数，因此求解动力问题 还要给出初始条件。 为求解上的简便，通常不计体力，此时弹性体的运 动微分方程简化为： 21 11-2 Without rotating Wave and equal volume wave 1. Without rotating waves Without rotating wave means that in elastic objects , the distortion caused by waves is not rotating . That means rotating values of three vertical coordinates at any point in the elastic object are zero. Where is potential function of displacement. This displacement is called without rotating displacement, and the elastic wave corresponds to the displacement are called without rotating wave. Suppose the displacement of elastic objects can be expressed by : u ,v , w 22 11-2 无旋波与等容波 一、无旋波 所谓无旋波是指在弹性体中，波动所产生的变形不存在旋 转，即弹性体在任一点对三个垂直坐标轴的旋转量皆为零。 假定弹性体的位移u,v,w可以表示成为： 其中 是位移的势函数。这种位移称为无旋位 移。而相应于这种位移状态的弹性波就称无旋波。 23 proving：at any point of elastic object ,the rotating value of z axis is : So the rotating values of the three coordinates at any point of the elastic object are zero. substitute into the formula , we can get: Similarly: