A mathematical model for fitting and predicting relaxation modulus and simulating viscoelastic responses

Abstract

We propose a mathematical model for relaxation modulus and its numerical solution. The model formula is extended from sigmoidal function considering nonlinear strain hardening. Its physical meaning can be interpreted by a macroscale elastic network-viscous medium model with only five model parameters in a simpler format than the molecular-chain-based polymer models to represent general solid materials. We also developed a finite-element (FE) framework and robust numerical algorithm to implement this model for simulating responses under both static and dynamic loadings. We validated the model through both experimental data and numerical simulations on a variety of materials including asphalt concrete, polymer, spider silk, hydrogel, agar and bone. By satisfying the second law of thermodynamics in the form of Calusius-Duhem inequality, the model is able to simulate creep and sinusoidal deformation as well as energy dissipation. Compared to the Prony series, the widely used model with a large number of model parameters, the proposed model has improved accuracy in fitting experimental data and prediction stability outside of the experimental range with competitive numerical stability and computation speed. We also present simulation results of nonlinear stress-strain relationships of spider silk and hydrogels, and dynamic responses of a multilayer structure.

Idealized force decay of orthodontic elastomeric chains follows Nutting Equation

“…These authors presented the Nutting equation in the form of ε = ψ × σ β × t − n , where ε = strain, σ = stress, t = time, and ψ , β and n are material constants, with n being related to the damping constant from dynamic mechanical measurements. Recently, Xu and Engquist (2018) proposed a generalized Maxwell (GM) model for fitting the viscoelastic responses of polymers. This GM model is an extension of the Maxwell–Weichert model and contains a larger number of individual Maxwell elements.…”

“…These authors noted that the physical meaning of the GM model corresponds to a macroscale elastic network–viscous medium polymer. There are five parameters for the Xu and Engquist (2018) GM model, and these authors note that while incorporation of more Maxwell elements in the GM model may yield a better fit to the experimental data, the process of mathematically fitting the experimental data becomes cumbersome. The empirical Nutting equation, which can be considered as an approximate outcome of the GM model, has the benefit of simplicity for comparing the idealized in vitro force decay behaviour of orthodontic elastomeric chains.…”

Elucidating the short and long-term mechanical response of the cell nucleus with a hybrid-viscoelastic model

“…Hence, to simulate this increment in time of the contractility generated by the actomyosin apparatus (until it reaches another stable tension level in response to the increment of the substrate stiffness), it is assumed that behaves as an accumulation function of a logistic f distribution. This is justified by the fact that it was reported that this kind of S-shape function may capture adequately the stress-strain response of a macroscale elastic network in a viscous medium (like the cytoskeleton) at short and long time ranges (thresholds) 119 . So, the plot presented in the Figure 2B-ii is possible to observe the rapid increment of the protein concentration until reaching a constant value which means that the tensión exerted on the NE reaches an equilibrium state.…”

Viscoelastic Cell Microenvironment: Hydrogel‐Based Strategy for Recapitulating Dynamic ECM Mechanics

“…Taking full advantage of computational material science, it helps to develop theoretical models (e.g., friction beads–spring model, tube model, arm‐retraction model, Maxwell or generalized Maxwell model, fractional derivative model, and other nonlinear viscoelastic models) for modeling of viscoelastic behaviors of the materials. [ 237–242 ] Such a prediction of the viscoelasticity from theoretical models can make considerable contributions in the design of hydrogel biomaterials with elaborately controlled viscoelastic properties for engineering cell mechanical microenvironment, which is an emerging area. Therefore, development of advanced hydrogels with precisely tunable viscoelastic properties will greatly contribute to elucidate more connections between dynamic matrix mechanics and cellular responses.…”