Stress and Deformation Response of Viscoelastic Solids in the Gravitational Field

In this paper, the stress response and time dependence of viscoelastic materials under gravity are studied in state space. The Laplace integral transformation method is used to transform the governing equation under the Lagrangian system into the Hamiltonian system, so that the separation of variables method can be implemented successfully. Therefore, the complete solution space including zero eigensolution and non-zero eigensolution is found, and the orthogonal normalization relationship between the basic solutions is established. In addition, the specific methods for solving various nonhomogeneous problems are given by using the orthogonal normalization relation.