Analytical solutions and boundary element analysis for holes and cracks in anisotropic viscoelastic solids via time-stepping method

In this study the time-stepping method, which can convert the problem of anisotropic viscoelasticity into two sets of elastic-like system, is employed for both analytical solutions and boundary element method (BEM) to solve the two-dimensional problems of holes and cracks in anisotropic viscoelastic solids. Based upon this method and the available solutions for the corresponding elastic problems, some analytical solutions for the two-dimensional problems of holes and cracks in anisotropic viscoelastic solids are presented, such as the one with an elliptical hole, a straight crack, two collinear cracks, or collinear periodic cracks under uniform load at infinity. We also present the fundamental solution for the problems with an elliptical hole, and use this solution to construct a special boundary element method (SBEM). Further extension of SBEM is made to develop the boundary-based finite element method (BFEM) to deal with the problems with multiple holes and cracks. The advantage of SBEM and BFEM is that no meshes are required on the boundaries of holes and cracks. To show the accuracy and efficiency of the proposed method, several representative examples are implemented and compared with those obtained by the finite element method and the elastic-viscoelastic correspondence principle. The results show that the solutions obtained by different methods are in well agreement with each other, and the ones obtained by the time-stepping method are around 3 to 36 times faster than the others depending on the problems and methods.