Impulsive displacement of a quasi-linear viscoelastic material through accurate numerical inversion of the Laplace transform

An analytical model for the one-dimensional, impulsive displacement of a quasi-linear viscoelastic material has been developed. The quasi-linear model of Fung [1] has been used successfully for a wide range of soft biological tissues. Due to the integral definition of linear viscoelastic materials, solutions are conveniently performed in the Laplace transform plane. Complex kernels like the quasilinear model are challenging to invert back to the real plane. Here, the method of Gaver [2] and Stehfest [3] is used to numerically carry Laplace space solutions to the real plane. Parametric results for a basic impulsive disturbance problem are presented. Results indicate that stress wave propagation is weakly dependent on the fast time, slow time ratio and more strongly dependent on the logarithmic damping parameter. Limitations of the numerical inversion method in the face of discontinuities are discussed as well using asymptotic methods. As an alternative to the numerical/polynomial-based Gaver-Stehfest method, a semianalytical regularization function useful near large gradient regions method is developed. A composite method that utilizes both the fully numerical and semianalytical convolution-based method is also described. The composite model provides improved results in terms of reducing computational undershoot and overshoot (wiggles) which limit both the fully numerical and the semianalytical models alone.