Finite Element Implementation of Finite Viscoelasticity for Compressible Transversely Isotropic Two-dimensional Solids

This paper presents a computational framework based on the quasi-linear viscoelastic (QLV) theory to model the finite viscoelastic response of compressible transversely isotropic solids under two-dimensional (2D) conditions. Soft biological tissues and engineering structures are known to exhibit time-dependent finite deformations necessitating that the compressibility and directional anisotropy must be addressed for realistic modeling. Adopting the QLV framework, the time-dependent stress is obtained by convolving a reduced relaxation function with the instantaneous elastic stress of a compressible transversely isotropic hyperelastic material. This formulation leads to a computationally efficient approach that remains physically realistic for finite viscoelasticity problems. The model is implemented using the high-order Lagrangian finite element method (FEM), with particular attention to derivation of the algorithmic tangent stiffness matrix, to achieve locking-free behavior and high convergence rates. The framework is validated through four well-known benchmark problems demonstrating its ability to accurately capture stress relaxation, creep and hysteresis phenomena. Results show that the proposed approach effectively models the rate-dependent stiffness and energy dissipation under sustained loading conditions. Thus, a robust and efficient tool is provided for simulating time-dependent anisotropic behavior in two-dimensional geometries with direct applications in biomechanics and soft engineering structures.