Symmetries and Hamiltonian Formalism for Complex Materials

Capriz, Gianfranco; Mariano, Paolo Maria

doi:10.1023/b:elas.0000018775.44668.07

Cited by 33 publications

(28 citation statements)

References 6 publications

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“…(13) is the Doyle-Ericksen formula for a porous body. It is a particular case of a generalized version of the Doyle-Ericksen formula obtained in Capriz and Mariano (2003) where the microstructural term here presented governs the transfer of energy from substructural level to macroscopic level (Capriz and Mariano, 2003). Moreover we obtain on the boundary that…”

Section: Balance Of Momentum For Phase Onementioning

confidence: 78%

Electrically polarized mixtures of porous bodies

Magnarelli

2010

International Journal of Solids and Structures

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a b s t r a c tIn this paper we propose a mathematical model of a mixture between a porous material and a polarized fluid in the mechanics of complex bodies. We use a variational approach to derive the global and microstructural balance laws for each phase of the mixture and for the mixture as a whole. Such balances differ from those generally proposed in theory of mixtures. We consider an open system and we account for external sources. Moreover, via the Coleman and Noll procedure we point out that the transport equation for the matter in a polarized mixture due to electrical field is governed by the Nernst-Planck equation.

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Section: Balance Of Momentum For Phase Onementioning

confidence: 78%

Electrically polarized mixtures of porous bodies

Magnarelli

2010

International Journal of Solids and Structures

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“…Eq. (5c) is the hyperelastic constitutive law (4). Particular forms of the strain energy density function W used in models are discussed in Section 2.3.…”

Section: Equations Of Motionmentioning

confidence: 99%

“…In particular, the Lie algebra of generators of point symmetries admitted by a given system of equations is coordinate-independent, i.e., invariant under point transformations acting on the variables of the system (including transformations to curvilinear coordinates, as well as more general transformations). From a more formal point of view, Lie symmetries constitute a guide in the Lagrangian and Hamiltonian formalisms in continuum mechanics, especially for complex materials endowed with a microstructure [4]. For models that admit a variational formulation, there is a oneto-one correspondence between variational Lie symmetries and local conservation laws through Noether's theorem; for non-variational models, this relation generally does not hold [5].…”

Section: Introductionmentioning

confidence: 99%

Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

Cheviakov

Ganghoffer

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tNonlinear dynamic equations for isotropic homogeneous hyperelastic materials are considered in the Lagrangian formulation. An explicit criterion of existence of a natural state for a given constitutive law is presented, and is used to derive natural state conditions for some common constitutive relations.For two-dimensional planar motions of Ciarlet-Mooney-Rivlin solids, equivalence transformations are computed that lead to a reduction of the number parameters in the constitutive law. Point symmetries are classified in a general dynamical setting and in traveling wave coordinates. A special value of traveling wave speed is found for which the nonlinear Ciarlet-Mooney-Rivlin equations admit an additional infinite set of point symmetries. A family of essentially two-dimensional traveling wave solutions is derived for that case.

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“…This rate equation is postulated a priori in a manner which guaranties the reversibility of the applied dynamical process and is interpreted as a balance equation between the standard and the substructural interactions, when the microstress tensor [see Capriz and Mariano, 2003;de Fabritiis and Mariano, 2005;Mariano, 2008b;Yavari and Marsden, 2009] is null. The integrability conditions of this rate equation follow naturally by means of Frobenius theorem [e.g., see Bishop and Goldberg, 1980, pp.…”

Section: Introductionmentioning

confidence: 99%

A Phenomenological Constitutive Model of Non-Conventional Elastic Response

Panoskaltsis

Soldatos

2013

Int. J. Appl. Mechanics

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In this paper, a general model of elastic (non-dissipative) behavior is developed. This model belongs to a class of models, developed for the description of complex bodies, in which the local state is assumed to be determined not only by the deformation, but also by a family of additional material parameters. The latter, unlike some additional structures used in the mechanics of complex bodies (e.g., directors, order parameters, internal degrees of freedom), are not considered as interactions of microscopic nature; rather they are considered as variables of macroscopic nature that describe the internal structure of the material, while their rates describe the evolution of the internal structure in the course of deformation. Accordingly, these variables are assumed to evolve continuously with time in a manner that guaranties the reversibility of the applied dynamical process. A covariant theory for the continuum in question is derived by means of invariance properties of the global form of the spatial energy balance equation, under the superposition of arbitrary spatial diffeomorphisms. In particular, it is shown that the assumption of spatial covariance of the equation of balance of energy yields the standard conservation and balance laws of classical mechanics but it does not yield the standard Doyle-Ericksen formula. In fact, the "Doyle-Ericksen formula" derived in this work, has some extra terms in it, which are related directly to the internal structure of the material, as the latter is controlled by the additional parameters. In a similar manner, by assuming the absolute temperature as an additional state variable and by employing the invariance properties of the local form of the spatial balance of energy under superimposed spatial diffeomorphisms, which also include a temperature rescaling, a nonisothermal covariant constitutive theory is naturally obtained. A formal comparison of the proposed elastic material with the standard hyperelastic (Green elastic) solid is also presented.

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Symmetries and Hamiltonian Formalism for Complex Materials

Cited by 33 publications

References 6 publications

Electrically polarized mixtures of porous bodies

Electrically polarized mixtures of porous bodies

Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

A Phenomenological Constitutive Model of Non-Conventional Elastic Response

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