Finite element based dynamic analysis of viscoelastic solids using the approximation of Volterra integrals

• A mathematical description of viscoelastic solids based on the generalized Kelvin-Voigt model is presented and it is applied to modeling vibrations in a solid.
• Two finite element methods for either creep or relaxation functions as the input data are proposed for the analysis of viscoelastic materials.
• The longitudinal wave propagation subject to a step and a sinusoidal excitation is studied and compared in an elastic and viscoelastic media.
• Analytical solutions based on the Laplace transform method are provided for the case studies.
• The accuracy of the proposed computational methods were estimated in terms of their error variation and mesh size.

Two new finite element based methods for time-domain dynamic analysis of viscoelastic solids are proposed in this research. The viscoelastic property is given by either the relaxation or creep functions and is simulated by the conventional generalized Kelvin–Voigt model. The viscoelastic behavior during the dynamic response is taken into account by the Volterra integral. This avoids the difficulties associated with the need for high order time-derivatives used in differential models of viscoelasticity. An accurate numerical approximation for the Volterra integrals is provided. It is used for implementation of the finite-element procedure in the time domain by the introduction of additional terms to the mass matrix (or the stiffness matrix) and the force vector. The additional terms are functions of calculations at the previous time step. In the two provided finite element formulations, one uses the relaxation and the other employs the creep compliance function. This makes it unnecessary to calculate the creep function from the given relaxation modulus or vice versa which are cumbersome operations. As a case study, the wave propagation in a one- and three-dimensional viscoelastic rod subject to a step and sinusoidal load is formulated and solved by the proposed two finite element methods. The computations were validated by direct solutions based on the Laplace transform method.