Edelen’s dissipation potentials and the visco-plasticity of particulate media

Goddard, J. D.

doi:10.1007/s00707-014-1123-3

Cited by 22 publications

(63 citation statements)

References 46 publications

(131 reference statements)

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“…Before formulating the reaction−diffusion problems, we recall that the potentials in eq 2 are (Legendre−Fenchel) convex conjugates, as noted elsewhere 14 (where it is also pointed out that for nondifferentiable potentials "max" is to be replaced by "sup" and partial derivatives by set-valued "sub-gradients"), with…”

Section: ■ Dissipative Reaction−diffusion Systemsmentioning

confidence: 99%

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Dissipation Potentials for Reaction-Diffusion Systems

Goddard

2014

Ind. Eng. Chem. Res.

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This work considers strongly dissipative reaction–diffusion systems with constitutive equations given by Edelen’s dissipation potentials, whose existence is tantamount to nonlinear Onsager symmetry. As the main goal of this work, it is shown that the associated extremum principle yields the relevant steady-state species balances and concentration fields for well-mixed reactors as well as for spatially inhomogeneous reactions accompanied by diffusion. The above principle is contrasted briefly with the popular notion of maximum entropy production. Potential applications of the theory to a wide range of reaction–diffusion systems are discussed, and questions are raised as to the possible breakdown of strong dissipation and nonlinear Onsager symmetry arising from reversible chemical or chemomechanical couplings.

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Section: ■ Dissipative Reaction−diffusion Systemsmentioning

confidence: 99%

“…14 In the present article, we focus mainly on force potentials of the type φ. Homogeneous Systems. We consider a chemical reaction involving n chemical species Whenever the quantity D = −μ i r i is non-negative definite, it represents the dissipation in the entropy balance (Clausius− Duhem) inequality.…”

Section: ■ Dissipative Reaction−diffusion Systemsmentioning

confidence: 99%

Dissipation Potentials for Reaction-Diffusion Systems

Goddard

2014

Ind. Eng. Chem. Res.

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“…To the extent that the first type of instability is strongly dissipative ("hyperdissipative"), it corresponds to loss of convexity of an inelastic potential, since one can show that rate-independent plasticity is marginally convex [56]. Hence, we discern two limiting forms of the loss of convexity in potential: the first involving a hyperdissipative potential and, in the second, a hyperelastic potential.…”

Section: -4 / Vol 66 September 2014mentioning

confidence: 99%

“…As done in much of the applied-mechanics literature, we often blur the distinction between generalized velocities and conjugate forces, e.g., deformation rate and stress, as elements of dual spaces [56], but strive whenever possible to distinguish them, respectively, by contravariant and covariant tensors indices.…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Continuum Modeling of Granular Media

Goddard

2014

Applied Mechanics Reviews

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This is a survey of the interesting phenomenology and the prominent regimes of granular flow, followed by a unified mathematical synthesis of continuum modeling. The unification is achieved by means of "parametric" viscoelasticity and hypoplasticity based on elastic and inelastic potentials. Fully nonlinear, anisotropic viscoelastoplastic models are achieved by expressing potentials as functions of the joint isotropic invariants of kinematic and structural tensors. These take on the role of evolutionary parameters or "internal variables," whose evolution equations are derived from the internal balance of generalized forces. The resulting continuum models encompass most of the mechanical constitutive equations currently employed for granular media. Moreover, these models are readily modified to include Cosserat and other multipolar effects. Several outstanding questions are identified as to the contribution of parameter evolution to dissipation; the distinction between quasielastic and inelastic models of material instability; and the role of multipolar effects in material instability, dense rapid flow, and particle migration phenomena.

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“…The survey is far from being complete; important results are missing from it; e.g., the works of Ziegler [6][7][8][9][10][11][12] and his influence [13][14][15][16][17], Biot [18,19] and Edelen [20][21][22][23][24][25][26][27][28][29][30][31] are omitted. A more skeptic website on the topics is The Azimuth Project [32].…”

Section: Introductionmentioning

confidence: 99%

Gyarmati’s Variational Principle of Dissipative Processes

Verhás

2014

Entropy

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Like in mechanics and electrodynamics, the fundamental laws of the thermodynamics of dissipative processes can be compressed into Gyarmati's variational principle. This variational principle both in its differential (local) and in integral (global) forms was formulated by Gyarmati in 1965. The consistent application of both the local and the global forms of Gyarmati's principle provides all the advantages throughout explicating the theory of irreversible thermodynamics that are provided in the study of mechanics and electrodynamics by the corresponding classical variational principles, e.g., Gauss' differential principle of least constraint or Hamilton's integral principle.

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Edelen’s dissipation potentials and the visco-plasticity of particulate media

Abstract: The phrase following Eq. ( 19), together with Eq. (20) and the sentence following Eq. (20) should be corrected to read: "where the obvious relation −1 = * requires the connection between {R, * } and {L, }:

Cited by 22 publications

References 46 publications

Dissipation Potentials for Reaction-Diffusion Systems

Dissipation Potentials for Reaction-Diffusion Systems

Continuum Modeling of Granular Media

Gyarmati’s Variational Principle of Dissipative Processes

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