Boundary element formulation for 3D dynamic problems of anisotropic viscoelasticity and isotropic poroviscoelasticity

Abstract

The dynamic behavior of anisotropic viscoelastic and poroviscoelastic solids is considered. A poroviscoelastic formulation is based on the Biot model of fully saturated poroelastic media. The elastic-viscoelastic correspondence principle is applied to describe the viscoelastic properties of a porous material skeleton. The system of differential equations of the full Biot model in Laplace transforms and formulas for elastic modules are given. A solution to the original problem is constructed using Laplace transforms, and numerical inversion yields the solution in the time domain. The system of direct boundary integral equations is introduced, and the system of regularized boundary integral equations is considered. Discrete analogues are obtained through mixed boundary element discretizations. The collocation points of the boundary integral equation coincide with the interpolation nodes of unknown boundary functions. Anisotropic fundamental solutions are represented as a sum of static and dynamic parts expressed in terms of integrals over the unit circle and the half of the unit sphere, respectively. Numerical inversion of Laplace transform is realized at the nodes of the Runge-Kutta scheme using the time-step method. The boundary element analysis is conducted to demonstrate the influence of the viscoelastic properties of poroviscoelastic and anisotropic viscoelastic materials on the amplitude and shape of responses during the transition from instantaneous to equilibrium moduli. A numerical solution to the problem of force acting on a prismatic anisotropic viscoelastic solid is given. The problem of growing pressure in a spherical cavity inside a poroviscoelastic solid is also numerically solved.