变边界粘弹性轴对称问题的复变函数法

COMPLEX VARIABLE METHOD FOR VISCOELASTIC AXISYMMETRIC PROBLEM INVOLVING TIME-DEPENDENT BOUNDARY REGIONS

对边界几何形状、位置随时间变化的变边界结构,给出了用复变函数求解粘弹问题的解析方法.文中用拉普拉斯变换结合平面弹性复变方法,对内外边界变化时粘弹性轴对称问题进行求解.引入两个与时间、空间相关的解析函数,给出了变边界情况下应力、位移以及边界条件与解析函数的关系.当解析函数形式部分确定,则可用边界条件求解其中与时间相关的待定函数.求解待定函数的方程一般情况下为一系列积分方程,特殊情况可求得解析解.对轴对称问题中应力边值问题、位移边值问题以及混合边值问题,分别利用边界条件求得相关系数,从而得到了应力与位移的解析表达.当取Boltzmann粘弹模型时,进行不同边值问题的分析.分析显示,应力、位移的形态与大小均与边界变化过程相关,与固定边界粘弹性问题有较大不同.本文解答可用于粘弹性轴对称问题内外边界任意变化及各种边值问题的力学分析.此外,该法可进一步进行荷载非对称、复杂孔型变边界问题的求解.

This paper presents the complex variable method for viscoelastic problem whose boundaries are var led with time. Laplace transformation is introduced to complex variable method to solve the axisyrnmetric problem of viscoelasticity. Stress and displacement fields, and boundary conditions are expressed by two analytical functions in terms of time and space, and coefficients in analytical functions can be determined by the boundary conditions. The equations about the coefficients are generally in integral form, but analytical solutions can be obtained in special cases. For the axisymmetric problems of stress, displacement or mixed boundary, the corresponding coefficients are determined exactly by boundary conditions and analytical solutions of displacement and stresses are given also. As an application example, Boltzmann viscoelastic model is employed. The solutions show that stresses and displace- ments are correlated with boundary variation process. The method in this paper can be applied to axisymmetric problems with random variation of inner or outer radius. In addition, problems with asymmetric load or non-circular section can be solved similarly.