A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies

Bustamante, R.; Rajagopal, Κ. R.

doi:10.1007/978-3-030-31547-4_7

Cited by 23 publications

(30 citation statements)

References 55 publications

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“…which is the expected result, since the use of Legendre transformation implies that H κ p(t) = (55) and (50)).…”

Section: Approach Based On Gibbs Free Energysupporting

confidence: 77%

“…A systematic study of constitutive relations of the type B = i(T) with a coherent explanation based on issues of causality for the choice of such constitutive relations was initiated by Rajagopal [4], and the number of works focused on this concept has been growing ever since. Regarding elastic (non-dissipative) solids, we refer the reader to a recent review by Bustamante and Rajagopal [50] and references therein, while some aspects of the description of non-elastic response such as plasticity are discussed in Rajagopal and Srinivasa [51,52] and Cichra and Průša [53] (a discussion and references on works dealing with the invertibility of the constitutive relations can be found in Sfyris and Bustamante [54]; one of the earliest studies of the same topic is that by Truesdell and Moon [55]). As an example of the benefits of the proposed change of perspective, we can mention especially the study by Muliana et al [56], who address the classical topic of the mathematical modeling of the mechanical response of rubber.…”

Section: Solidsmentioning

confidence: 99%

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Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

Průša

Rajagopal

2021

Fluids

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Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled using the same concepts as in the classical theory of nonlinear solids. Recently, new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of the elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy–Green tensor is expressed as a function of stress, is of interest. We show how to use this new perspective in the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalize to more complex settings wherein the classical approach might be impractical or even inapplicable.

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“…which is the expected result, since the use of Legendre transformation implies that H κ p(t) = (55) and (50)).…”

Section: Approach Based On Gibbs Free Energysupporting

confidence: 77%

Section: Solidsmentioning

confidence: 99%

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

Průša

Rajagopal

2021

Fluids

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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]. A crucial feature of Rajagopal's proposal is a prospect of obtaining a special hierarchy of "explicit" nonlinear relationship for the linearized strain as a nonlinear function of Cauchy stress.…”

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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“…In particular, it includes the simple case of the strain expressed as a function of the stress, which appears more consistent with Newtonian causality than standard constitutive relations since force causes displacement. Implicit constitutive relations have found a wide range of applications in the modelling of electro- and magneto-elastic bodies [5,6], fracture in brittle materials [7,8], gum metal [9] and many other materials (see [10] for further references).…”

Section: Introductionmentioning

confidence: 99%

On stretch-limited elastic strings

Rodriguez

2021

Proc. R. Soc. A.

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Motivated by the increased interest in modelling non-dissipative materials by constitutive relations more general than those from Cauchy elasticity, we initiate the study of a class of stretch-limited elastic strings : the string cannot be compressed smaller than a certain length less than its natural length nor elongated larger than a certain length greater than its natural length. In particular, we consider equilibrium states for a string suspended between two points under the force of gravity (catenaries). We study the locations of the supports resulting in tensile states containing both extensible and inextensible segments in two situations: the degenerate case when the string is vertical and the non-degenerate case when the supports are at the same height. We then study the existence and multiplicity of equilibrium states in general with multiplicity differing markedly from strings satisfying classical constitutive relations.

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A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies

Cited by 23 publications

References 55 publications

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

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

On stretch-limited elastic strings

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