View source for Materials with memory

{{Continuum mechanics}}
In [[continuum physics]], '''materials with memory''', also referred as '''materials with hereditary effects''' are a class of materials whose [[constitutive equation]]s contains a dependence upon the past history of [[Thermodynamics|thermodynamic]], [[Kinetics (physics)|kinetic]], [[electromagnetic]] or other kind of [[state variable]]s.

==Historical notes==
The study of these materials arises from the pioneering articles of [[Ludwig Boltzmann]]<ref>{{harv|Boltzmann|1874}}.</ref><ref>{{harv|Boltzmann|1878}}.</ref> and [[Vito Volterra]],<ref name="Vol1912">{{harv|Volterra|1912}}</ref><ref name="Func1930">([[#{{harvid|Volterra|1959}}|Volterra 1930]]).</ref> in which they sought an extension of the concept of an [[Cauchy elastic material|elastic material]].<ref>See {{harv|Coleman|Noll|1961|p=239}}</ref> The key assumption of their theory was that the local stress value at a time {{mvar|t}} depends upon the history of the local deformation up to {{mvar|t}}. In general, in materials with memory the local value of some constitutive quantity (stress, heat flux, electric current, polarization and magnetization, etc.) at a time {{mvar|t}} depends upon the history of the state variables (deformation, temperature, electric and magnetic fields, etc.). The hypothesis that the remote history of a variable has less influence than its values in the recent past, was stated in modern continuum mechanics as the [[fading memory principle]] by [[Bernard Coleman]] and [[Walter Noll]].
This assumption was implicit in the pioneer works: when restricted to cyclic hystories, it traces back to the closed cycle principle stated by Volterra,<ref name="Func1930" /> which leads to a constitutive relation of integral convolution type.
In the linear case, this relation takes the form of a Volterra equation.

===Constitutive relations of materials with memory===
In the linear case, this relation takes the form of a [[Volterra integral equation|Volterra equation]]:
:<math>\boldsymbol{T}(\boldsymbol{x},t)=\boldsymbol{G}_0(\boldsymbol{x})\boldsymbol{E}(\boldsymbol{x},t)+\int_0^{+\infty}\!\!\!\!\boldsymbol{G}^\prime(\boldsymbol{x},s)\boldsymbol{E}(\boldsymbol{x},t-s)\mathrm{d}s</math>

==See also==

*[[Biomaterial]]
*[[Biomechanics]]
*[[Dielectric relaxation]]
*[[Hysteresis]]
*[[Rheology]]
*[[Viscosity]]
*[[Viscoelasticity]]
*[[Viscoplasticity]]

==Notes==
{{Reflist|30em}}

==References==
{{refbegin}}

===Articles===
*{{Citation
|last =Boltzmann
|first =Ludwig
|author-link =Ludwig Boltzmann
|title =Theorie der elastischen Nachwirkung
|trans-title=Theory of elastic after effects
|journal =[[Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Classe. Abteilung IIA, Mathematik, Physik, Chemie, Mechanik, Meteorologie und Astronomie]]
|volume =70
|pages =275–306
|date =8 October 1874
|language =de
|jfm =06.0648.02
|hdl =2027/hvd.32044090836305
}}, published also as {{Citation
|last =Boltzmann
|first =Ludwig
|author-link =Ludwig Boltzmann
|title =Zur Theorie der elastischen Nachwirkung
|trans-title=On the theory of elastic after effects
|journal =[[Annalen der Physik und Chemie]]
|series=Neue Folge
|volume =Suppl. Band VII
|pages =624–655
|year =1875
|language =de
|jfm =08.0639.01}}.
*{{Citation
|last =Boltzmann
|first =Ludwig
|author-link =Ludwig Boltzmann
|title =Zur Theorie der elastischen Nachwirkung
|trans-title=On the theory of elastic after effects
|journal =[[Annalen der Physik und Chemie]]
|series=Neue Folge
|volume =Band V
|issue =11
|pages =430–432
|year =1878
|language =de
|url =http://gallica.bnf.fr/ark:/12148/bpt6k152498/f438.image.r=Boltzmann
|doi =10.1002/andp.18782411107
|bibcode =1878AnP...241..430B
|jfm =10.0670.01
|via=[[Gallica]]}}.
*{{Citation
|last =Volterra
|first =Vito
|author-link =Vito Volterra
|title =Sur les équations intégro-différentielles et leurs applications
|trans-title=On integro-differential equations and their applications
|journal =[[Acta Mathematica]]
|volume =35
|pages =295–356
|year =1912
|language =fr
|doi =10.1007/BF02418820
|jfm =43.0431.03
|issn = 0001-5962
|doi-access =free
}} (online version {{issn|1871-2509}}).
*{{Citation
|last1 =Coleman
|first1 =Bernard D.
|author-link =Bernard D. Coleman
|last2 =Noll
|first2 =Walter
|author2-link =Walter Noll
|title =Foundations of Linear Viscoelasticity
|journal =[[Reviews of Modern Physics]]
|volume =33
|issue =2
|pages =239–249
|date =April 1961
|issn =0034-6861
|doi =10.1103/RevModPhys.33.239
|bibcode =1961RvMP...33..239C
|mr =0158605
|zbl =0103.40804
}}.
*{{Citation
|last =Graffi
|first = Dario
|author-link =Dario Graffi
|title =Una teoria ereditaria dell'effetto Lussemburgo
|trans-title=An hereditary theory of the Luxemburg effect
|journal =[[Rendiconti del Seminario Matematico della Università di Padova]]
|volume =7
|pages =36–54
|year =1936
|language =it
|url =http://www.numdam.org/item?id=RSMUP_1936__7__36_0
|zbl =0015.13705
|jfm =62.0546.02}}. In this paper, written only few years after the discovery of the effect itself, Dario Graffi proposes a theory of the Luxemburg effect based on Volterra's theory of hereditary phenomena.
*{{Citation
|last =Coleman
|first =Bernard D.
|author-link =Bernard D. Coleman
|title =Thermodynamics of materials with memory
|journal =[[Archive for Rational Mechanics and Analysis]]
|volume =17
|issue =1
|pages =1–46
|year =1964
|doi =10.1007/BF00283864
|bibcode =1964ArRMA..17....1C
|issn =0003-9527
|mr =0171419
|s2cid =123022911
}} (online version {{issn|1432-0673}}).
*{{Citation
|last =Graffi
|first =Dario
|author-link =Dario Graffi
|title =Sull'espressione dell'energia libera nei materiali viscoelastici lineari
|trans-title=On the formula of free energy in linear viscoelastic materials
|series = IV Serie
|journal =[[Annali di Matematica Pura ed Applicata]]
|volume =98
|pages =273–279
|date =December 1974
|language =it
|doi = 10.1007/BF02414027
|issn=0373-3114
|mr =0345497
|zbl =0281.73027
|s2cid =120967727
|doi-access =free
}} (online version {{issn|1618-1891}}). In this paper, Graffi introduces the [[Thermodynamic free energy|free energy]] now called [[Graffi–Volterra free energy]] after him.
*{{Citation
|last =Bordoni
|first =Piero Giorgio
|author-link =Piero Giorgio Bordoni
|editor-last =Grioli
|editor-first =Giuseppe
|editor-link =Giuseppe Grioli
|contribution=Memoria e dissipazione nei solidi
|title =Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986)
|trans-title=Round table on the topic: continua with memory (Rome, 9 May 1986)
|language = it
|place = Rome
|publisher = [[Accademia Nazionale dei Lincei]]
|year = 1990
|series = Contributi del Centro Linceo Interdisciplinare "Beniamino Segre"
|volume = 81
|pages = 13–24
|issn = 0394-0705
}}. {{lang-en|italic=yes|"Memory and dissipation in solids"}} is a short survey on the determination of the response functions of materials with memory by [[Acoustics|acoustical measurements]].
*{{Citation
|last =Fichera
|first =Gaetano
|author-link =Gaetano Fichera
|editor-last =Grioli
|editor-first =Giuseppe
|editor-link =Giuseppe Grioli
|contribution=Sul principio della memoria evanescente
|title =Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986)
|trans-title=Round table on the topic: continua with memory (Rome, 9 May 1986)
|language = it
|place = Rome
|publisher = [[Accademia Nazionale dei Lincei]]
|year = 1990
|series = Contributi del Centro Linceo Interdisciplinare "Beniamino Segre"
|volume = 81
|pages = 57–69
|issn = 0394-0705
}}. {{lang-en|italic=yes|"On the principle of fading memory"}} is a note on [[fading memory principle]].
*{{Citation
|last1 =Graffi
|first1 =Dario
|author-link =Dario Graffi
|last2 =Fabrizio
|first2 =Mauro
|author2-link =Mauro Fabrizio
|title =Sulla nozione di stato per materiali viscoelastici di tipo "rate"
|trans-title=On the notion of state for "rate" type viscoelastic materials
|journal = [[Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali]]
|series=Serie VIII
|volume =83
|issue =1
|pages =201–208
|year =1989
|language =it
|url =http://www.bdim.eu/item?id=RLIN_1989_8_83_1_201_0
|mr =1142459
|zbl =0732.73023 
}}.
*{{Citation
 |first            = Dario
 |last             = Graffi
 |author-link      = Dario Graffi
 |editor-last      = Amaldi
 |editor-first     = E.
 |editor-link      = Edoardo Amaldi
 |editor2-last     = Amerio
 |editor2-first    = L.
 |editor2-link     = Luigi Amerio
 |editor3-last     = Fichera
 |editor3-first    = G.
 |editor3-link     = Gaetano Fichera
 |editor4-last     = Gregory
 |editor4-first    = T.
 |editor5-last     = Grioli
 |editor5-first    = G.
 |editor5-link     = Giuseppe Grioli
 |editor6-last     = Martinelli
 |editor6-first    = E.
 |editor6-link     = Enzo Martinelli
 |editor7-last     = Montalenti
 |editor7-first    = G.
 |editor8-last     = Pignedoli
 |editor8-first    = A.
 |editor8-link     = Antonio Pignedoli
 |editor9-last     = Salvini
 |editor9-first    = Giorgio
 |editor9-link     = Giorgio Salvini
 |editor10-last    = Scorza Dragoni
 |editor10-first   = Giuseppe
 |editor10-link    = Giuseppe Scorza Dragoni
 |contribution     = L'opera di Vito Volterra sui fenomeni ereditari e alcune sue conseguenze
 |title            = Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990)
 |trans-title      = International congress in memory of Vito Volterra (8–11 October 1990)
 |url              = http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862
 |language         = it
 |series           = Atti dei Convegni Lincei
 |volume           = 92
 |year             = 1992
 |pages            = 39–76
 |place            = Roma
 |publisher        = [[Accademia Nazionale dei Lincei]]
 |issn             = 0391-805X
 |mr               = 1783028
 |zbl              = 0977.01022
 |access-date      = 2017-03-29
 |archive-url      = https://web.archive.org/web/20170107005930/http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862
 |archive-date     = 2017-01-07
 |url-status       = dead
}}. "''The work of Vito Volterra on hereditary phenomena and some of their consequences''" is an ample technical survey paper on the research work of Vito Volterra on hereditary phenomena in mathematical physics.
*{{Citation
|first =Clifford
|last =Truesdell
|author-link =Clifford Truesdell
|editor-last =Amaldi
|editor-first =E.
|editor-link =Edoardo Amaldi
|editor2-last =Amerio
|editor2-first =L.
|editor2-link =Luigi Amerio
|editor3-last =Fichera
|editor3-first =G.
|editor3-link =Gaetano Fichera
|editor4-last =Gregory
|editor4-first =T.
|editor5-last =Grioli
|editor5-first =G.
|editor5-link =Giuseppe Grioli
|editor6-last =Martinelli
|editor6-first =E.
|editor6-link =Enzo Martinelli
|editor7-last =Montalenti
|editor7-first =G.
|editor8-last =Pignedoli
|editor8-first =A.
|editor8-link =Antonio Pignedoli
|editor9-last =Salvini
|editor9-first =Giorgio
|editor9-link =Giorgio Salvini
|editor10-last =Scorza Dragoni
|editor10-first =Giuseppe
|editor10-link =Giuseppe Scorza Dragoni
|contribution =Functionals in the modern mechanics of continua
|title =Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990)
|trans-title =International congress in memory of Vito Volterra (8–11 October 1990)
|url =http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862
|series =Atti dei Convegni Lincei
|volume =92
|year =1992
|pages =39–76
|place =Roma
|publisher =[[Accademia Nazionale dei Lincei]]
|issn =0391-805X
|mr =1783036
|zbl =1043.74501
|access-date =2017-03-29
|archive-url =https://web.archive.org/web/20170107005930/http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862
|archive-date =2017-01-07
|url-status =dead
}}. An introduction to modern [[rational continuum mechanics]] by one of its leading contributors.
*{{Citation
|last1 =Del Piero
|first1 =Giampietro
|author-link =Giampietro Del Piero
|last2 =Deseri
|first2 =Luca
|title =On the concepts of state and free energy in linear viscoelasticity
|journal =[[Archive for Rational Mechanics and Analysis]]
|volume =17
|issue =1
|pages =1–35
|year =1997
|doi =10.1007/s002050050035
|bibcode =1997ArRMA.138....1P
|issn =0003-9527
|mr = 1463802
|zbl = 0891.73024
|citeseerx =10.1.1.571.4752
|s2cid =3593361
}} (online version {{issn|1432-0673}}). G. Del Piero, L. Deseri (1997); "", Arch. Rational Mech. Anal. 138, 1–35.

===Books===
*{{Citation
|last1 =Alberty
|first1 =Robert A.
|last2 =Silbey
|first2 =Robert J.
|title =Physical Chemistry
|place =New York
|publisher =[[John Wiley & Sons]]
|orig-year =1948
|year =2001
|pages =[https://archive.org/details/physicalchemistr00silb_0/page/969 969]
|url =https://archive.org/details/physicalchemistr00silb_0/page/969
|isbn =978-0-471-38311-6
|oclc =959753597
|url-access =registration
}}.
*{{Citation
|last1 =Allen
|first1 =Samuel M.
|last2 =Thomas
|first2 =Edwin L.
|title =The Structure of Materials
|place =New York
|publisher =[[John Wiley & Sons]]
|series =The MIT Series in Materials Science & Engineering
|year =1999
|pages =xvi+464
|url =https://books.google.com/books?id=JgtpQgAACAAJ
|isbn =978-0-471-00082-2
}}.
*{{Citation
|last1 =Amendola
|first1 =Giovambattista
|last2 =Fabrizio
|first2 =Mauro
|author2-link =Mauro Fabrizio
|last3 =Golden
|first3 =John Murrough
|title =Thermodynamics of Materials with memory. Theory and Applications
|place = New York
|publisher =[[Springer Verlag]]
|year =2012
|pages =xv+574
|doi =10.1007/978-1-4614-1692-0
|isbn =978-1-4614-1691-3
|mr =2856615
|zbl =1237.80001
}} (also available as [[e-book]] with {{ISBN|978-1-4614-1692-0}}).
*{{Citation
|last =Bloom
|first =Frederick
|title =Ill-posed problems for integrodifferential equations in mechanics and electromagnetic theory
|place =Philadelphia
|publisher =[[Society for Industrial and Applied Mathematics|SIAM]]
|series =SIAM Studies in Applied Mathematics
|volume =3
|year =1981
|pages =IX+222
|language =en
|url =https://books.google.com/books?id=ZTGd1pkDfO0C
|isbn =978-0-89871-171-4
|mr =635277
|zbl =0465.45010}}.
*{{Citation
|editor-last =Crandall
|editor-first =Stephen H.
|editor2-last =Dahl
|editor2-first =Norman C. Dahl
|editor3-last =Lardner
|editor3-first =Thomas J.
|title =An introduction to the mechanics of solids
|place =New York
|publisher =[[McGraw-Hill Book Company]]
|year =1972
|edition =2nd
|pages =XII+628
|isbn =978-0-07-013441-6
|zbl =0433.73001
}}.
*{{Citation
|last =Dimitrienko
|first =Yuriy I.
|title =Nonlinear continuum mechanics and large inelastic deformations
|place =Dordrecht
|publisher =[[Springer Verlag]]
|series =Solid Mechanics and Its Applications
|volume =174
|year =2011
|pages =xxiv + 721
|doi =10.1007/978-94-007-0034-5
|isbn =978-94-007-0033-8
|mr =3025695
|zbl =1209.74002
}} (available also as [[e-book]] with {{ISBN|978-94-007-0034-5}}).
*{{Citation
|last1 =Fabrizio
|first1 =Mauro
|author-link =Mauro Fabrizio
|last2 =Morro
|first2 =Angelo
|title =Mathematical Problems in Linear Viscoelasticity
|place =Philadelphia PA
|publisher =[[Society for Industrial and Applied Mathematics|SIAM]]
|series =SIAM Studies in Applied Mathematics
|volume =12
|year =1992
|pages=ix+203
|url =https://books.google.com/books?id=YOTuQbpBzs8C
|isbn =978-0-89871-266-7 
|mr =153021
|zbl =0753.73003}}.
*{{Citation
|last1 =Fabrizio
|first1 =Mauro
|author-link =Mauro Fabrizio
|last2 =Morro
|first2 =Angelo
|title =Electromagnetism of continuous media. Mathematical modelling and applications
|place =Oxford
|publisher =[[Oxford University Press]]
|series =Oxford Science Publications
|year =1992
|pages =xvii+668
|doi =10.1093/acprof:oso/9780198527008.001.0001
|isbn =978-0-19-852700-8
|mr =1996323
|zbl =1027.78001
}}.
*{{Citation
|last1 =Lemaitre
|first1 =Jean
|last2 =Chaboche
|first2 =Jean Louis 
|translator-first=B.
|translator-last=Shrivastava
|title =Mechanics of solid materials
|others=with a foreword to the French edition by [[Paul Germain (Mathematician)|Paul Germain]] and a foreword to the English edition by Fred Leckie
|place =Cambridge–New York–Melbourne–Madrid–Cape Town–Singapore–Sao Paulo–Delhi
|publisher =[[Cambridge University Press]]
|year =1990
|pages =xxv + 556
|doi =10.1017/CBO9781139167970
|isbn =978-0-521-32853-1
|zbl =0743.73002}}
*{{Citation
|last =Volterra
|first =Vito
|author-link =Vito Volterra
|others =recueil et rèdigèes par [[Joseph Pérès]]
|title =Leçons sur les fonctions de lignes
|place =Paris
|publisher =[[Gauthier-Villars]]
|series =Collection de Monographies sul la Théorie des Fonctions
|year =1913
|pages =vi+230
|language =fr
|url =https://archive.org/details/leonssurlesfon00voltuoft
|jfm = 44.0458.01}} (see also {{JFM|44.0410.01}}).
*{{Citation
|last =Volterra
|first =Vito
|author-link =Vito Volterra
|editor-last=Fantappiè
|editor-first=Luigi
|editor-link=Luigi Fantappiè
|others=with a foreword of [[Griffith Conrad Evans|G. C. Evans]], a biography of Vito Volterra and a bibliography of his published works by Sir [[Edmund Whittaker]]
|title =Theory of functionals and of integral and integro-differential equations
|place =New York
|publisher =[[Dover Publications]]
|orig-year =1930
|year =1959
|edition =2nd
|pages =[39]+XIV+226
|mr =0100765
|zbl =0086.10402
|jfm = 56.1010.01}} (see also {{JFM|55.0814.01}}: the [[Jahrbuch über die Fortschritte der Mathematik|JFM]] reviews refer to the original 1930 English edition).

===Congress proceedings and collections of articles on materials with memory===
*{{Citation
|editor-last =C.I.M.E.
|editor-link =Centro Internazionale Matematico Estivo
|title =Materials with memory – Bressanone, 2–11 Giugno
|trans-title=Materials with memory – [[Brixen]], 2–11 Giugno
|language = it
|place =Napoli
|publisher = [[Liguori Editore]]
|year = 1979
|series = Corsi del C.I.M.E.
|volume = I Ciclo 1977
|pages = 294
|isbn = 978-88-207-0756-9
}}
*{{Citation
|editor-last =Grioli
|editor-first =Giuseppe
|editor-link =Giuseppe Grioli
|title =Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986)
|trans-title=Round table on the topic: continua with memory (Rome, 9 May 1986)
|language = it
|place = Rome
|publisher = [[Accademia Nazionale dei Lincei]]
|year = 1990
|series = Contributi del Centro Linceo Interdisciplinare "Beniamino Segre"
|volume = 81
|pages = 73
|issn = 0394-0705
}}

{{refend}}

[[Category:Materials science]]
[[Category:Non-Newtonian fluids]]
[[Category:Rubber properties]]
[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Classical mechanics]]
[[Category:Electrodynamics]]