Minimum Free Energies for Materials with Finite Memory

Fabrizio, Mauro; Golden, Murrough

doi:10.1007/1-4020-2308-1_23

Cited by 9 publications

(9 citation statements)

References 16 publications

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“…Relation (2.1) 1 was the basis of the early papers on this topic [14,13,17]. However, in more recent work, for example [18,30,19,31], formulae based on the relative history have been used because they yield explicitly positive free energies. Also, various quantities have better convergence properties in the frequency domain.…”

Section: Basic Relationships and Free Energiesmentioning

confidence: 99%

“…This formula follows from a new physical hypothesis, which results in the assignment of explicit weights to each member of a family of free energies, each of these being associated with a particular factorization of the function H(ω) used in earlier work [14,13,17,20,19,18]. Once the free energy is known, the rate of dissipation can also be determined without difficulty.…”

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A proposal concerning the physical rate of dissipation in materials with memory

Golden¹

2005

Quart. Appl. Math.

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Abstract. It has been known for several decades that the free energy and entropy of a material with memory is not in general uniquely determined, nor are the total dissipation in the material over a given time period and the rate of dissipation. The dissipation in a material element would in particular seem to be a quantity that has immediate physical objectivity. It must be seen therefore as a significant weakness in the thermodynamics of materials exhibiting memory effects, that a quantity as basic as the rate of dissipation cannot be predicted in terms of the constitutive parameters.The objective of the present work is to propose a formula for the physical free energy of a linear scalar viscoelastic material in terms of a family of free energies, each of which can be regarded as an estimate of the physical quantity. This formula follows from a new physical hypothesis of Maximum Parametric Symmetries, which states that the physical free energy and dissipation have the closest possible level of symmetry among the parameters of the theory to that of the work function. This results in the assignment of explicit weights to all members of the family of free energies, each of these being associated with a particular factorization of a quantity closely related to the loss modulus of the material. It is interesting that the final formula proposed for the physical free energy can be expressed in simple, closed form. Once the free energy is known, the corresponding physical rate of dissipation can also be determined without difficulty.It is shown that non-trivial equivalence classes of states, in the sense of Noll, exist only if the material has a relaxation function derivative, the Fourier transform of which has only isolated singularities in the complex frequency plane. The members of the family of free energies used to determine the physical free energy are all functions of such an equivalence class. The derivation of their form is a generalization of work reported in Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. 60, 341 -381 (2002 1]-[18]). A consequence of this is that the total dissipation in the material over a given time period and the rate of dissipation are not uniquely determined either.Thus, we may have different estimates of the total dissipation over a time interval or, alternatively, the rate of dissipation at a given time, in a material with specified constitutive relations, undergoing a prescribed deformation history (in the isothermal case, for example). Now, in contrast perhaps to free energy or entropy, the rate of dissipation in a material element would seem to be a quantity that has immediate physical objectivity. In particular, for isothermal conditions, the rate of dissipation, due to a specified mechanical process, is simply the rate of heat production in the element, by that process.It must be seen therefore as a significant weakness in the thermodynamics of materials exhibiting memory effects that a quantity as basic as the rate of dissipation cannot...

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Section: Basic Relationships and Free Energiesmentioning

confidence: 99%

mentioning

confidence: 99%

A proposal concerning the physical rate of dissipation in materials with memory

Golden¹

2005

Quart. Appl. Math.

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“…To this end, in the spirit of Graffi's work [32], they suggested that any free energy functional endows the history space with a natural norm [17,18]. In this direction, starting from [3,10], many other papers proposed new analytic expressions of the maximum and minimum free energies [12,15,19,20,27]. The first and well-known expression of the Helmholtz potential in linear viscoelasticity is the so called Graffi-Volterra free energy density…”

Section: 2mentioning

confidence: 99%

A New Approach to Equations with Memory

Fabrizio

Giorgi

Pata

2010

Arch Rational Mech Anal

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113

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Abstract. We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of a state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G :we consider an abstract version of the evolution equationarising from linear viscoelasticity. Contents

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“…The unique factorization of H is given by (4.23), (4.24) and (4.27). We shall use the formalism for relative histories, defined by (2.1) 2 , as in [9,7,8,10] rather than in [2,4]. Taking the Fourier transform of (3.8) yields …”

Section: The Minimum Free Energymentioning

confidence: 99%

“…Detailed, explicit expressions for the minimum free energy and related quantities were given in [2,4] for discrete spectrum materials, namely those for which the relaxation function is a sum of exponentials. The minimum free energy of compressible viscoelastic fluids was determined in [9] while materials with finite memory were considered in [7]. These results are used in the context of the viscoelastic Saint-Venant problem in [10].…”

Section: Introductionmentioning

confidence: 99%

The Minimum Free Energy for Continuous Spectrum Materials

Deseri¹,

Golden²

2007

SIAM J. Appl. Math.

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Abstract. A general closed expression is given for the isothermal minimum free energy of a linear viscoelastic material with continuous spectrum response. Two quite distinct approaches are adopted, which give the same final result. The first involves expressing a positive quantity, closely related to the loss modulus of the material, defined on the frequency domain, as a product of two factors with specified analyticity properties. The second is the continuous spectrum version of a method used by Breuer and Onat [1] for materials with relaxation function given by sums of exponentials. It is further shown that minimal energy states [6] are uniquely related to histories and the work function is the maximum free energy with the property that it is a function of state.

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Minimum Free Energies for Materials with Finite Memory

Cited by 9 publications

References 16 publications

A proposal concerning the physical rate of dissipation in materials with memory

A proposal concerning the physical rate of dissipation in materials with memory

A New Approach to Equations with Memory

The Minimum Free Energy for Continuous Spectrum Materials

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