利用刚度矩阵法求解多层弹性半空间体的温度应力

Thermo-stress in multi-layered elastic half space solved with stiffness matrix method

从热弹性力学的基本方程出发,采用Hankel 积分变换和Laplace 积分变换等数学手段,首先推导出了单层弹性半空间轴对称体的温度应力问题的刚度矩阵,然后按传统有限元的方法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程组,再对其进行Hankel 和Laplace 积分逆变换就可解出在外荷载和温度联合作用下多层弹性半空间轴对称问题的解析解。由于刚度矩阵的元素中不含有正指数项,计算时不会出现溢出的现象,从而克服了传递矩阵法的缺点。在推导过程中,因不用事先人为的选择应力函数,使得问题的求解更加合理,同时也为进一步研究这类问题如湿度场、动力学等奠定了理论基础。最后,文中还给出了计算实例来证明推导结果的正确性。

In the paper, thermo-stress in multilayered elastic half space is presented. Firstly, the stiffness matrix for a layer is derived based on the fundamental elasticity equations and some mathematic methods such as Hankel integral transformation. Then the global stiffness matrix is established for multilayered elastic half space using the finite element concepts in which layers are completely contacted. Therefore, explicit solution for axisymmetrical problems in multilayered elastic half space is obtained from the solution of the algebraic equation formed by global stiffness matrix and the inverse Hankel integral transformation. Because positive exponential function is not included in the element of matrix, the calculation is not overflowed. Therefore, the shortages of transfer matrix method are overcome. This method is clear in concept, and the corresponding formulas given in the paper are not only simple but also convenient for application. More important is that this method can be used to solve other problems of multilayered elastic half space such as thermo field and dynamics. An example of road surface deflection is presented to prove the calculated results.