  We propose a new material viscoelastic model and mathematical solution to
simulate relaxation modulus and viscoelastic response. The model formula of
relaxation modulus is extended from sigmoidal function considering nonlinear
strain hardening and softening. Its physical mechanism can be interpreted by a
spring network viscous medium model with only five parameters in a simpler
format than the molecular-chain based polymer models to represent general
materials. We also developed a three-dimensional finite-element method and
robust numerical algorithms to implement this model for solving partial
differential equations. We validate the model through both experimental data
and numerical simulations on a broad range of materials including bitumen,
shape-memory polymer, spider-inspired silk, hydrogel, biomaterials and bone. By
satisfying the 2nd law of thermodynamics in the form of Calusius-Duhem
inequality, the model is able to simulate creep and sinusoidal deformation, and
energy dissipation. As compared to Prony series, the most general model being
used often with a large number of model parameters, the proposed model has
improved accuracy in fitting experimental data and predicting modulus outside
of the experimental range, and the latter one is especially useful for material
design. The new model also has higher numerical accuracy while competitive
numerical stability and computation speed for convergence.
