A Novel Approach to the Description of Constitutive Relations

Implicit rate-type constitutive relations utilising discontinuous functions provide a novel approach to the purely phenomenological description of the inelastic response of solids undergoing finite deformation. However, this type of constitutive relation has so far been considered only in the purely mechanical setting, and the complete thermodynamic basis is largely missing. We address this issue, and we develop a thermodynamic basis for such constitutive relations. In particular, we focus on the thermodynamic basis for the classical elastic–perfectly plastic response, but the framework is flexible enough to describe other types of inelastic response as well.

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“…where H denotes the Heaviside step function (5) and T y is a constant. This formula is motivated by the one-dimensional model (4).…”

Section: Thermodynamicsmentioning

confidence: 99%

“…For example [3] and [4] have shown that the novel constitutive framework (2) can be gainfully exploited in the modelling of the response of rubber. (See [5] and [6] for recent reviews of the applications of this concept in the mechanics of solids. )…”

Section: Introductionmentioning

confidence: 99%

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

Cichra

Průša

2020

Mathematics and Mechanics of Solids

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“…As the first constitutive model, we generalize the basic second-grade fluid model proposed in Mctigue and others (1985), assuming that the characteristic constitutive parameters depend on the pressure p , the mean normal stress (Rajagopal and others (2012, 2015); Rajagopal (2015); Rajagopal and Saccomandi (2016)): We thus consider 2 where tr( T + p I ) = 0. As stated earlier, we assume that the coefficients characterizing the constitutive Eqn (6) of the fluid may depend on p so that the constitutive model is implicit.…”

Section: The Modelmentioning

confidence: 99%

Constitutive equations with pressure-dependent rheological parameters for describing ice creep

et al. 2019

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Experimental data from creep tests on polycrystalline ice samples highlight not only the non-Newtonian behavior of ice but also suggest a critical dependence of the various rheological parameters upon the applied hydrostatic pressure. We propose a new modeling framework, based on implicit theories of continuum mechanics, that generalizes two well-known constitutive models by taking into account the effect of the pressure in the description of ice in creep. To ascertain the validity of the proposed models, we fit the physical parameters with experimental data for the elongational flow of ice samples. The results show good agreement with the experimental creep curves. In particular, the proposed generalized models reproduce the increase of the creep rate due to the presence of hydrostatic pressure.

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“…A generalization of Cauchy and Green elasticity has been initiated by Rajagopal and his co-authors in a series of papers [14][15][16][17][18][19][20][21][22], in which the authors describe the implicit constitutive models that have a strong tenor of thermodynamic framework. The response of an elastic body 1 is best described by the implicit constitutive relations between Cauchy stress and deformation gradient tensors [23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

Yoon

Mallikarjunaiah

2021

Mathematics and Mechanics of Solids

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It is well known that the linearized theory of elasticity admits the logically inconsistent solution of singular strains when applied to certain naive models of fracture while the theory is a first-order approximation to finite elasticity in the asymptotic limit of infinitesimal displacement gradient. Meanwhile, the strain-limiting models, a special subclass of nonlinear implicit constitutive relations, predict uniformly bounded strain in the whole material body including at the strain-concentrator such as a crack tip or reentrant corner. Such a nonlinear approximation cannot be possible within the standard linearization procedure of either Cauchy or Green elasticity. In this work, we examine a finite-element discretization for several boundary value problems to study the state of stress–strain in the solid body of which response is described by a nonlinear strain-limiting theory of elasticity. The problems of notches, oriented cracks, and an interface crack in anti-plane shear are analyzed. The numerical results indicate that the linearized strain remains below a value that can be fixed a priori, therefore, ensuring the validity of the nonlinear model. In addition, we find high stress values in the neighborhood of the crack tip in every example, thereby suggesting that the crack tip acts as a singular energy sink for a stationary crack. We also calculate the stress intensity factor (SIF) in this study. The computed value of SIF in the nonlinear strain-limiting model is corresponding to that of the classical linear model, and thereby providing a tenet for a possible local criterion for fracture. The framework of strain-limiting theories, within which the linearized strain bears a nonlinear relationship with the stress, can provide a rational basis for developing physically meaningful models to study a crack evolution in elastic solids.

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A Novel Approach to the Description of Constitutive Relations

Cited by 23 publications

References 59 publications

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

Constitutive equations with pressure-dependent rheological parameters for describing ice creep

A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

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