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We propose a new material viscoelastic model and mathematical solution tosimulate relaxation modulus and viscoelastic response. The model formula ofrelaxation modulus is extended from sigmoidal function considering nonlinearstrain hardening and softening. Its physical mechanism can be interpreted by aspring network viscous medium model with only five parameters in a simplerformat than the molecular-chain based polymer models to represent generalmaterials. We also developed a three-dimensional finite-element method androbust numerical algorithms to implement this model for solving partialdifferential equations. We validate the model through both experimental dataand numerical simulations on a broad range of materials including bitumen,shape-memory polymer, spider-inspired silk, hydrogel, biomaterials and bone. Bysatisfying the 2nd law of thermodynamics in the form of Calusius-Duheminequality, the model is able to simulate creep and sinusoidal deformation, andenergy dissipation. As compared to Prony series, the most general model beingused often with a large number of model parameters, the proposed model hasimproved accuracy in fitting experimental data and predicting modulus outsideof the experimental range, and the latter one is especially useful for materialdesign. The new model also has higher numerical accuracy while competitivenumerical stability and computation speed for convergence.

A mathematical and Physical Model Improves Accuracy in Simulating Solid Material Relaxation Modulus and Viscoelastic Responses