Evolution Laws For Internal Variables Research Articles

A Review of Physically Based and Thermodynamically Based Constitutive Models for Soft Materials

Abstract In this paper, we review constitutive models for soft materials. We specifically focus on physically based models accounting for hyperelasticity, visco-hyperelasticity, and damage phenomena. For completeness, we include the thermodynamically based viscohyperelastic and damage models as well as the so-called mixed models. The models are put in the frame of statistical mechanics and thermodynamics. Based on the available experimental data, we provide a quantitative comparison of the hyperelastic models. This information can be used as guidance in the selection of suitable constitutive models. Next, we consider visco-hyperelasticity in the frame of the thermodynamic theory and molecular chain dynamics. We provide a concise summary of the viscohyperelastic models including specific strain energy density function, the evolution laws of internal variables, and applicable conditions. Finally, we review the models accounting for damage phenomenon in soft materials. Various proposed damage criteria are summarized and discussed in connection with the physical interpretations that can be drawn from physically based damage models. The discussed mechanisms include the breakage of polymer chains, debonding between polymer chains and fillers, disentanglement, and so on.

Evolution of internal variables in an expanding hollow cylinder at large plastic strains.

An efficient method for calculating the evolution of internal variables in an expanding hollow cylinder of rigid/plastic material is proposed. The conventional constitutive equations for rigid plastic, hardening material are supplemented with quite an arbitrary set of evolution laws for internal variables assuming that the material is incompressible. No restriction is imposed on the hardening law. The problem is solved in Lagrangian coordinates. This significantly facilitates a numerical treatment of the problem. In particular, the initial/boundary value problem is reduced to a system of equations in characteristic coordinates. A finite difference scheme is used for solving these equations. An illustrative example is presented assuming that the internal variables are the equivalent plastic strain and a damage parameter.

A large deformation framework for shape memory polymers: Constitutive modeling and finite element implementation

Shape memory polymers commonly experience both finite deformations and arbitrary thermomechanical loading conditions in engineering applications. This motivates the development of three-dimensional constitutive models within the finite deformation regime. In the present study, based on the principles of continuum thermodynamics with internal variables, a three-dimensional finite deformation phenomenological constitutive model is proposed taking its basis from the recent model in the small strain regime proposed by Baghani et al. (2012). In the constitutive model derivation, a multiplicative decomposition of the deformation gradient into elastic and inelastic stored parts (in each phase) is adopted. Moreover, employing the mixture rule, the Green–Lagrange strain tensor is related to the rubbery and glassy parts. In the constitutive model, the evolution laws for internal variables are derived during both cooling and heating thermomechanical loadings. Furthermore, we present the time-discrete form of the proposed constitutive model in the implicit form. Using the finite element method, we solve several boundary value problems, that is, tension and compression of bars and a three-dimensional beam made of shape memory polymers, and investigate the model capabilities as well as its numerical counterpart. The model is validated by comparing the predicted results with experimental data reported in the literature that shows a good agreement.

A thermodynamically-consistent 3 D constitutive model for shape memory polymers

The ever increasing applications of shape memory polymers have motivated the development of appropriate constitutive models for these materials. In this work, we present a 3D constitutive model for shape memory polymers under time-dependent multiaxial thermomechanical loadings in the small strain regime. The derivation is based on an additive decomposition of the strain into six parts and satisfying the second law of thermodynamics in Clausius–Duhem inequality form. In the constitutive model, the evolution laws for internal variables are derived during both cooling and heating thermomechanical loadings. The viscous effects are also fully accounted for in the proposed model. Further, we present the time-discrete form of the evolution equations in the implicit form. The model is validated by comparing the predicted results with different experimental data reported in the literature. Finally, using the finite element method, we solve two boundary value problems e.g., a 3D beam and a medical stent made of shape memory polymers.

A constitutive model for shape memory polymers with application to torsion of prismatic bars

In this article, satisfying the second law of thermodynamics, we present a 3D constitutive model for shape memory polymers. The model is based on an additive decomposition of the strain into four parts. Also, evolution laws for internal variables during both cooling and heating processes are proposed. Since temperature has considerable effect on the shape memory polymer behavior, for simulation of a shape memory polymer–based structure, it is required to perform a heat-transfer analysis. Commonly, an experimentally observed temperature rate–dependent behavior of shape memory polymers is justified by a rate-dependent glassy temperature, but using the heat-transfer analysis, it is shown that the glassy temperature could be considered as a constant material parameter. To this end, implementing the constitutive model within a nonlinear finite element code, we simulate torsion of a shape memory polymer rectangular bar and a circular tube. Moreover, we compare the predicted results with experimental data recently reported in the literature, which shows a good agreement.

Lq‐almost solvability in viscoplasticity theory

AbstractIn this work we discuss some existence issues for elasto/visco‐plastic models of monotone type with evolution laws for internal variables governed by non‐coercive nonlinearities (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)