Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models
Identifieur interne : 006143 ( PascalFrancis/Checkpoint ); précédent : 006142; suivant : 006144Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models
Auteurs : L. I. Palade [France] ; P. Attane [France] ; R. R. Huilgol [Australie] ; B. Mena [Mexique]Source :
- International journal of engineering science [ 0020-7225 ] ; 1999.
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Abstract
Viscoelastic materials like amorphous polymers or organic glasses show a complex relaxation behavior in the softening dispersion region, i.e. from glass transition to the α relaxation zone. It is known that a uni-dimensional Maxwell model, modified within the conceptual framework of fractional calculus, has been found to predict experimental data in this range of temperatures. After developing a fully objective constitutive relation for an incompressible fluid, it is shown here that the fractional derivative Maxwell model results from the linearization of this objective equation about the state of rest, when some assumptions about the memory kernels are made. Next, it is demonstrated that the three dimensional, linearized version of the frame indifferent equation exhibits anomalous stability characteristics, namely that the rest state is neither stable nor unstable under exponential disturbances. Also, the material cannot support purely harmonic excitations either. Consequently, it appears that fractional derivative constitutive equations may be used to study a very limited category of flows in rheology rather than the whole spectrum.
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Attane</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>Laboratoire de Rhéologie, B.P. - 53 Domaine Universitaire</s1><s2>38041, Grenoble</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>2 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region">Auvergne-Rhône-Alpes</region><region type="old region">Rhône-Alpes</region><settlement type="city">Grenoble</settlement></placeName></affiliation></author><author><name sortKey="Huilgol, R R" sort="Huilgol, R R" uniqKey="Huilgol R" first="R. R." last="Huilgol">R. R. Huilgol</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Department of Mathematics and Statistics, Flinders University of South Australia, G.P.O. Box 2100</s1><s2>Adelaide, S. A. 5001</s2><s3>AUS</s3><sZ>3 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Adelaide, S. A. 5001</wicri:noRegion></affiliation></author><author><name sortKey="Mena, B" sort="Mena, B" uniqKey="Mena B" first="B." last="Mena">B. Mena</name><affiliation wicri:level="1"><inist:fA14 i1="03"><s1>Instituto de Investigaciones en Materiales, Departmento de Polimeros, U.N.A.M., Apartado Postal 70-360</s1><s2>Coyoacan 04510, Mexico, D.F.</s2><s3>MEX</s3><sZ>4 aut.</sZ></inist:fA14><country>Mexique</country><wicri:noRegion>Coyoacan 04510, Mexico, D.F.</wicri:noRegion></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">INIST</idno><idno type="inist">99-0103782</idno><date when="1999">1999</date><idno type="stanalyst">PASCAL 99-0103782 INIST</idno><idno type="RBID">Pascal:99-0103782</idno><idno type="wicri:Area/PascalFrancis/Corpus">006459</idno><idno type="wicri:Area/PascalFrancis/Curation">006B95</idno><idno type="wicri:Area/PascalFrancis/Checkpoint">006143</idno><idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">006143</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models</title><author><name sortKey="Palade, L I" sort="Palade, L I" uniqKey="Palade L" first="L. I." last="Palade">L. I. Palade</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>Laboratoire de Rhéologie, B.P. - 53 Domaine Universitaire</s1><s2>38041, Grenoble</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>2 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region">Auvergne-Rhône-Alpes</region><region type="old region">Rhône-Alpes</region><settlement type="city">Grenoble</settlement></placeName></affiliation></author><author><name sortKey="Attane, P" sort="Attane, P" uniqKey="Attane P" first="P." last="Attane">P. Attane</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>Laboratoire de Rhéologie, B.P. - 53 Domaine Universitaire</s1><s2>38041, Grenoble</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>2 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region">Auvergne-Rhône-Alpes</region><region type="old region">Rhône-Alpes</region><settlement type="city">Grenoble</settlement></placeName></affiliation></author><author><name sortKey="Huilgol, R R" sort="Huilgol, R R" uniqKey="Huilgol R" first="R. R." last="Huilgol">R. R. Huilgol</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Department of Mathematics and Statistics, Flinders University of South Australia, G.P.O. Box 2100</s1><s2>Adelaide, S. A. 5001</s2><s3>AUS</s3><sZ>3 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Adelaide, S. A. 5001</wicri:noRegion></affiliation></author><author><name sortKey="Mena, B" sort="Mena, B" uniqKey="Mena B" first="B." last="Mena">B. Mena</name><affiliation wicri:level="1"><inist:fA14 i1="03"><s1>Instituto de Investigaciones en Materiales, Departmento de Polimeros, U.N.A.M., Apartado Postal 70-360</s1><s2>Coyoacan 04510, Mexico, D.F.</s2><s3>MEX</s3><sZ>4 aut.</sZ></inist:fA14><country>Mexique</country><wicri:noRegion>Coyoacan 04510, Mexico, D.F.</wicri:noRegion></affiliation></author></analytic><series><title level="j" type="main">International journal of engineering science</title><title level="j" type="abbreviated">Int. j. eng. sci.</title><idno type="ISSN">0020-7225</idno><imprint><date when="1999">1999</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">International journal of engineering science</title><title level="j" type="abbreviated">Int. j. eng. sci.</title><idno type="ISSN">0020-7225</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Amorphous polymer</term><term>Constitutive equation</term><term>Experimental study</term><term>Fractional derivative</term><term>Glass transition</term><term>Material frame indifference</term><term>Memory effect</term><term>Modeling</term><term>Organic glass</term><term>Rheology</term><term>Three dimensional model</term><term>Viscoelasticity</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Viscoélasticité</term><term>Rhéologie</term><term>Verre organique</term><term>Polymère amorphe</term><term>Transition vitreuse</term><term>Modélisation</term><term>Equation constitutive</term><term>Dérivée fractionnaire</term><term>Effet mémoire</term><term>Modèle 3 dimensions</term><term>Indifférence matérielle</term><term>Etude expérimentale</term><term>4635</term><term>8350F</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Viscoelastic materials like amorphous polymers or organic glasses show a complex relaxation behavior in the softening dispersion region, i.e. from glass transition to the α relaxation zone. It is known that a uni-dimensional Maxwell model, modified within the conceptual framework of fractional calculus, has been found to predict experimental data in this range of temperatures. After developing a fully objective constitutive relation for an incompressible fluid, it is shown here that the fractional derivative Maxwell model results from the linearization of this objective equation about the state of rest, when some assumptions about the memory kernels are made. Next, it is demonstrated that the three dimensional, linearized version of the frame indifferent equation exhibits anomalous stability characteristics, namely that the rest state is neither stable nor unstable under exponential disturbances. Also, the material cannot support purely harmonic excitations either. Consequently, it appears that fractional derivative constitutive equations may be used to study a very limited category of flows in rheology rather than the whole spectrum.</div></front></TEI><inist><standard h6="B"><pA><fA01 i1="01" i2="1"><s0>0020-7225</s0></fA01><fA02 i1="01"><s0>IJESAN</s0></fA02><fA03 i2="1"><s0>Int. j. eng. sci.</s0></fA03><fA05><s2>37</s2></fA05><fA06><s2>3</s2></fA06><fA08 i1="01" i2="1" l="ENG"><s1>Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models</s1></fA08><fA11 i1="01" i2="1"><s1>PALADE (L. I.)</s1></fA11><fA11 i1="02" i2="1"><s1>ATTANE (P.)</s1></fA11><fA11 i1="03" i2="1"><s1>HUILGOL (R. 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Next, it is demonstrated that the three dimensional, linearized version of the frame indifferent equation exhibits anomalous stability characteristics, namely that the rest state is neither stable nor unstable under exponential disturbances. Also, the material cannot support purely harmonic excitations either. 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I." last="Palade">L. I. Palade</name></region><name sortKey="Attane, P" sort="Attane, P" uniqKey="Attane P" first="P." last="Attane">P. Attane</name></country><country name="Australie"><noRegion><name sortKey="Huilgol, R R" sort="Huilgol, R R" uniqKey="Huilgol R" first="R. R." last="Huilgol">R. R. Huilgol</name></noRegion></country><country name="Mexique"><noRegion><name sortKey="Mena, B" sort="Mena, B" uniqKey="Mena B" first="B." last="Mena">B. Mena</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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