Extend the sigmoidal function to a physical model for simulating stress relaxation and deformation of solids

We propose a new material viscoelastic model and mathematical solution to simulate the stress relaxation and deformation of solid materials within linear viscoelasticity. The model formula is extended from the sigmoidal function. As compared to the multi-molecular-chain based theories and models such as the spring-dashpot or friction-bead physical systems, the tube theory and arm reptation model, the proposed model can be physically represented by a two-phase thermodynamic system with only four model parameters. The simulated relaxation modulus adapts to the glass transition from the plateau modulus to the minimum value with a reaction rate to the time-dependent viscosity. We also developed a Galerkin finite element method and robust numerical algorithms to implement this model for accurate and fast computation of responses. We validate the model through both experimental tests of different materials (infrastructures, polymers, biomaterials, and tissues) and numerical simulations.