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柱坐标系下修正弹性应力波理论及数值分析
金红春, 刘占芳
重庆大学 航空航天学院, 重庆 400044
摘要:
指出了传统的波动理论存在的一些局限性和不足。给出了修正弹性应力波理论的建立推导过程。完成了波动方程从张量状态到柱坐标系下的转换过程,得到了柱坐标系下的波动方程。建立了波动变量在加载面上与载荷及速度的关系,应力波在自由面上发生反射时边界条件对波动变量的影响。通过使用Matlab编程计算了应力波在几何模型为z方向无限长的空心圆柱结构中的传播过程以及应力波在遇到边界时的反射情况。计算结果显示了应力波随时间在结构中的传播方式,应力波在加载面上会产生2个波,即体积波和形变波,遇到边界后会反射出2个波。同时体积波和形变波的第一部分复合在一起形成一个复合脉冲以相同的波速运动。
关键词:  应力波  柱面波  体积波  形变波  有限差分法  波动方程
DOI:10.11835/j.issn.1000-582X.2019.254
分类号:O347.4
基金项目:国家自然科学基金(11372365,11072276)。
Wave theory of modified elasticity under column coordinate system and numerical analysis
JIN Hongchun, LIU Zhanfang
College of Aerospace Engineering, Chongqing University, Chongqing 400044, P. R. China
Abstract:
In this paper, some boundednesses and inadequacies of the traditional fluctuation theory were pointed out. Then the establishment and derivation process of a modified elastic wave theory was given. The conversion process of wave equation from tensor state to column coordinate system was completed and the wave equation under column coordinate system was obtained. The relationship between the fluctuation variables and the load and velocity on the loading surface was established, revealing the influence of the boundary condition on the fluctuation variables when the stress wave reflection occurred on the free surface. By using Matlab programming, propagation process of the stress wave was calculated in the hollow cylindrical structure with an infinite length in the z direction of the geometric model and so was its reflection when encountering with the boundary. The calculation results showed the propagation process of stress wave in the structure with time, during which the stress wave would produce two waves on the loading surface, namely the volume wave and the deformation wave, and four waves would be reflected when encountering the boundary. At the same time, the volume wave and the first part of the deformation wave would be combined to form a composite pulse moving at the same wave speed.
Key words:  stress wave  cylindrical wave  volume wave  deformation wave  finite difference method  wave equation
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