Hypocontinua

Capriz, Gianfranco; Giovine, Pasquale

doi:10.1007/978-3-319-28241-1_3

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…In recent years, some papers were published with the aim of formulating a dynamics of continua for which the largely tacit primal axiom of absolute persistence of material elements fails. () The glaring instance of such a body is a gas though the occasional placing of its dynamics within a course of lectures as a chapter of standard continuum mechanics seems to imply the opposite. We quote for all from Truesdell and Muncaster, page 3: “The kinetic gas …is a continuous medium …Not only is the kinetic theory a field theory, but also the kinetic gas is endowed with the very same field descriptors as continuum mechanics …” and in a footnote … “For an elementary introduction to the basic concepts and assumptions of continuum mechanics the reader may consult the book of C.Truesdell, A First Course in Rational Mechanics …” There, the mentioned primal axiom is glossed over (page 36) by a seemingly innocent Axiom of Impenetrability: “In continuum mechanics, contrarily, the mappings

χ false(\cdot, t false) : scriptB \to χ_{Ω} false(scriptB, t false)

is assumed bijective,” professed already also in Truesdell and Toupin .…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, when pursuing analogies with the standard theory, we could introduce a mesoscopic displacement gradient in 2 ways: either as the double vector G by mimicking strictly the standard case via integration of

Ġ = B G,

or by using a distinct differential equation for a double vector G ∗ , which includes the effects of mass gain or loss

Ġ_{*} = B G_{*} - \frac{1}{2} σ G_{*}

. The alternative choice calls for quite different stipulation of the Piola time derivatives of vectors and tensors; we follow here the first choice opposite of that followed in earlier papers,() as details of analysis here become simpler.…”

Section: Introductionmentioning

confidence: 99%

“…They are bound together as in the theories of quasi‐rigid bodies

K = Y B^{T};

thus, it is B that descends from Y and K and satisfies the (usually named) balance of moment of inertia

\frac{+ \partial Y}{\partial τ} + false(grad Y false) v = B Y - Y B^{T},

which instead it is a kinematic identity due to the relation of Y with the measures of meso‐strain (see comments on page 164 of Capriz et al and equation 29 of Capriz and Giovine).…”

Section: Introductionmentioning

confidence: 99%

“…We refer readers to Capriz and Mariano for more specific information. We quote only the set of equations 32 to 36 of Capriz and Giovine on which the theory rests: balance of mass, of momentum, of moment of momentum, of Reynolds tensor:…”

Section: Introductionmentioning

confidence: 99%

“…Formally, the right‐hand side of is identical to one accepted for bodies with affine microstructure and for multipolar continua . In , ρ J , Z , and third‐order tensor j account, respectively, for external or intimate bulk compulsions and contact compulsion (see, eg, the closure equations for eddy viscosity models in the dynamics of fluids,() other than, in general, Capriz and Capriz and Giovine).…”

Section: Introductionmentioning

confidence: 99%

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Classes of ephemeral continua

Capriz

Giovine

2017

Math Methods in App Sciences

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The qualifier “ephemeral” was proposed for continuous models of bodies, such as gases, for which the generally tacit axiom of permanence of material elements fails to apply. Consequently, to their scrutiny, a Eulerian (local) approach is mandatory, such as one adopted, eg, in molecular dynamics. Within the scheme of ephemeral continua, we discuss here 3 essential subclasses of bodies: (1) those undergoing energy‐preserving processes (in this sense hyperelastic), (2) hypoelastic bodies inspired by a type proposed by Truesdell, and (3) a number of minor ones. We re‐examine the essential issues of the general format focusing on the proposal of appropriate concepts of strains and strainings.

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χ false(\cdot, t false) : scriptB \to χ_{Ω} false(scriptB, t false)

is assumed bijective,” professed already also in Truesdell and Toupin .…”

Section: Introductionmentioning

confidence: 99%

Ġ = B G,

or by using a distinct differential equation for a double vector G ∗ , which includes the effects of mass gain or loss

Ġ_{*} = B G_{*} - \frac{1}{2} σ G_{*}

Section: Introductionmentioning

confidence: 99%

“…They are bound together as in the theories of quasi‐rigid bodies

K = Y B^{T};

thus, it is B that descends from Y and K and satisfies the (usually named) balance of moment of inertia

\frac{+ \partial Y}{\partial τ} + false(grad Y false) v = B Y - Y B^{T},

which instead it is a kinematic identity due to the relation of Y with the measures of meso‐strain (see comments on page 164 of Capriz et al and equation 29 of Capriz and Giovine).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classes of ephemeral continua

Capriz

Giovine

2017

Math Methods in App Sciences

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A Multi-Scale Continuum View on Granular Flows

Mariano

2020

Advances in Mechanics and Mathematics

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the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Multi-scale kinetic description of granular clusters: invariance, balance, and temperature

Capriz

Mariano

2017

Continuum Mech. Thermodyn.

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Hypocontinua

Cited by 5 publications

References 12 publications

Classes of ephemeral continua

Classes of ephemeral continua

A Multi-Scale Continuum View on Granular Flows

Multi-scale kinetic description of granular clusters: invariance, balance, and temperature

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