粘弹介质中圆孔时变轴对称问题的解析分析

Analytical Research of Expanding Hole in Plane of Viscoelasticity

摘要:

对任意粘弹模型，用拉普拉斯变换法推导无限粘弹平面中圆孔半径任意时变时应力和位移的一般解析解。首先根据一般粘弹模型边界时变轴对称问题的基本方程，应用拉普拉斯变换得到拉氏空间中位移应满足的微分方程，并求得方程的通解，从而得到拉氏空间中位移、应力的一般表达式。对应力边界问题，将拉氏空间应力表达进行逆变换，再根据边界条件确定待定函数，最终得到应力和位移解答。本文的解答没有体积不可压缩的限制条件，并且适用于球量也具有粘弹效应的情况。作为应用，根据该解答求得H-Kelvin粘弹模型的解。算例显示，不同半径时变过程位移场的变化也不同。对线性时变过程，较慢的时变速度下位移变化平缓，但时变结束时刻的位移较大。

Abstract:

Analytical displacements and stresses during expanding hole in plane of viscoelasticity are studied in this paper. According to the basic equations, Laplace transform is introduced to deduce the differential equation of displacement in Laplace space. General expression of displacement and stresses in Laplace space is derived firstly. For stress boundary problem, inverse transforming of above stresses solutions, undetermined function in the solutions can be determined by boundary conditions, and final expressions of stresses and displacement are obtained. The solutions have no restrictive condition of volume incompressible, and are also suit for the cases that spherical tensor have the characteristic of viscoelasticity. The application of the method is given to the problem of H-Kelvin viscoelastic model. Comparison of displacement of different varying velocity shows that displacement is changed gently if radius varied slowly, but displacenmet is larger in end varying time.