Materials with memory

In continuum physics, materials with memory, also referred as materials with hereditary effects are a class of materials whose constitutive equations contains a dependence upon the past history of thermodynamic, kinetic, electromagnetic or other kind of state variables.

Historical notes

The study of these materials arises from the pioneering articles of Ludwig Boltzmann and Vito Volterra, in which they sought an extension of the concept of an elastic material. The key assumption of their theory was that the local stress value at a time t depends upon the history of the local deformation up to 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 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, 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 equation:

${\boldsymbol {T}}({\boldsymbol {x}},t)={\boldsymbol {G}}_{0}({\boldsymbol {x}}){\boldsymbol {E}}({\boldsymbol {x}},t)+\int _{0}^{+\infty }\!\!\!\!{\boldsymbol {G}}^{\prime }({\boldsymbol {x}},s){\boldsymbol {E}}({\boldsymbol {x}},t-s)\mathrm {d} s$

See also

• Biomaterial
• Biomechanics
• Dielectric relaxation
• Hysteresis
• Rheology
• Viscosity
• Viscoelasticity
• Viscoplasticity

References

Articles

• Boltzmann, Ludwig (8 October 1874), "Theorie der elastischen Nachwirkung" [Theory of elastic after effects], Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Classe. Abteilung IIA, Mathematik, Physik, Chemie, Mechanik, Meteorologie und Astronomie (in German), 70: 275–306, hdl:2027/hvd.32044090836305, JFM 06.0648.02, published also as Boltzmann, Ludwig (1876), "Zur Theorie der elastischen Nachwirkung" [On the theory of elastic after effects], Annalen der Physik und Chemie, Neue Folge (in German), Suppl. Band VII: 624–655, JFM 08.0639.01.
• Boltzmann, Ludwig (1878), "Zur Theorie der elastischen Nachwirkung" [On the theory of elastic after effects], Annalen der Physik und Chemie, Neue Folge (in German), Band V (11): 430–432, Bibcode:1878AnP...241..430B, doi:10.1002/andp.18782411107, JFM 10.0670.01 – via Gallica.
• Volterra, Vito (1912), "Sur les équations intégro-différentielles et leurs applications" [On integro-differential equations and their applications], Acta Mathematica (in French), 35: 295–356, doi:10.1007/BF02418820, ISSN 0001-5962, JFM 43.0431.03 (online version ISSN 1871-2509).
• Coleman, Bernard D.; Noll, Walter (April 1961), "Foundations of Linear Viscoelasticity", Reviews of Modern Physics, 33 (2): 239–249, Bibcode:1961RvMP...33..239C, doi:10.1103/RevModPhys.33.239, ISSN 0034-6861, MR 0158605, Zbl 0103.40804.
• Graffi, Dario (1936), "Una teoria ereditaria dell'effetto Lussemburgo" [An hereditary theory of the Luxemburg effect], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 7: 36–54, JFM 62.0546.02, Zbl 0015.13705. 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.
• Coleman, Bernard D. (1964), "Thermodynamics of materials with memory", Archive for Rational Mechanics and Analysis, 17 (1): 1–46, Bibcode:1964ArRMA..17....1C, doi:10.1007/BF00283864, ISSN 0003-9527, MR 0171419, S2CID 123022911 (online version ISSN 1432-0673).
• Graffi, Dario (December 1974), "Sull'espressione dell'energia libera nei materiali viscoelastici lineari" [On the formula of free energy in linear viscoelastic materials], Annali di Matematica Pura ed Applicata, IV Serie (in Italian), 98: 273–279, doi:10.1007/BF02414027, ISSN 0373-3114, MR 0345497, S2CID 120967727, Zbl 0281.73027 (online version ISSN 1618-1891). In this paper, Graffi introduces the free energy now called Graffi–Volterra free energy after him.
• Bordoni, Piero Giorgio (1990), "Memoria e dissipazione nei solidi", in Grioli, Giuseppe (ed.), Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986) [Round table on the topic: continua with memory (Rome, 9 May 1986)], Contributi del Centro Linceo Interdisciplinare "Beniamino Segre" (in Italian), vol. 81, Rome: Accademia Nazionale dei Lincei, pp. 13–24, ISSN 0394-0705. English: "Memory and dissipation in solids" is a short survey on the determination of the response functions of materials with memory by acoustical measurements.
• Fichera, Gaetano (1990), "Sul principio della memoria evanescente", in Grioli, Giuseppe (ed.), Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986) [Round table on the topic: continua with memory (Rome, 9 May 1986)], Contributi del Centro Linceo Interdisciplinare "Beniamino Segre" (in Italian), vol. 81, Rome: Accademia Nazionale dei Lincei, pp. 57–69, ISSN 0394-0705. English: "On the principle of fading memory" is a note on fading memory principle.
• Graffi, Dario; Fabrizio, Mauro (1989), "Sulla nozione di stato per materiali viscoelastici di tipo "rate"" [On the notion of state for "rate" type viscoelastic materials], Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian), 83 (1): 201–208, MR 1142459, Zbl 0732.73023.
• Graffi, Dario (1992), "L'opera di Vito Volterra sui fenomeni ereditari e alcune sue conseguenze", in Amaldi, E.; Amerio, L.; Fichera, G.; Gregory, T.; Grioli, G.; Martinelli, E.; Montalenti, G.; Pignedoli, A.; Salvini, Giorgio; Scorza Dragoni, Giuseppe (eds.), Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990) [International congress in memory of Vito Volterra (8–11 October 1990)], Atti dei Convegni Lincei (in Italian), vol. 92, Roma: Accademia Nazionale dei Lincei, pp. 39–76, ISSN 0391-805X, MR 1783028, Zbl 0977.01022, archived from the original on 2017-01-07, retrieved 2017-03-29. "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.
• Truesdell, Clifford (1992), "Functionals in the modern mechanics of continua", in Amaldi, E.; Amerio, L.; Fichera, G.; Gregory, T.; Grioli, G.; Martinelli, E.; Montalenti, G.; Pignedoli, A.; Salvini, Giorgio; Scorza Dragoni, Giuseppe (eds.), Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990) [International congress in memory of Vito Volterra (8–11 October 1990)], Atti dei Convegni Lincei, vol. 92, Roma: Accademia Nazionale dei Lincei, pp. 39–76, ISSN 0391-805X, MR 1783036, Zbl 1043.74501, archived from the original on 2017-01-07, retrieved 2017-03-29. An introduction to modern rational continuum mechanics by one of its leading contributors.
• Del Piero, Giampietro; Deseri, Luca (1997), "On the concepts of state and free energy in linear viscoelasticity", Archive for Rational Mechanics and Analysis, 17 (1): 1–35, Bibcode:1997ArRMA.138....1P, CiteSeerX 10.1.1.571.4752, doi:10.1007/s002050050035, ISSN 0003-9527, MR 1463802, S2CID 3593361, Zbl 0891.73024 (online version ISSN 1432-0673). G. Del Piero, L. Deseri (1997); "", Arch. Rational Mech. Anal. 138, 1–35.

Books

• Alberty, Robert A.; Silbey, Robert J. (2001) [1948], Physical Chemistry, New York: John Wiley & Sons, pp. 969, ISBN 978-0-471-38311-6, OCLC 959753597.
• Allen, Samuel M.; Thomas, Edwin L. (1999), The Structure of Materials, The MIT Series in Materials Science & Engineering, New York: John Wiley & Sons, pp. xvi+464, ISBN 978-0-471-00082-2.
• Amendola, Giovambattista; Fabrizio, Mauro; Golden, John Murrough (2012), Thermodynamics of Materials with memory. Theory and Applications, New York: Springer Verlag, pp. 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).
• Bloom, Frederick (1981), Ill-posed problems for integrodifferential equations in mechanics and electromagnetic theory, SIAM Studies in Applied Mathematics, vol. 3, Philadelphia: SIAM, pp. IX+222, ISBN 978-0-89871-171-4, MR 0635277, Zbl 0465.45010.
• Crandall, Stephen H.; Dahl, Norman C. Dahl; Lardner, Thomas J., eds. (1972), An introduction to the mechanics of solids (2nd ed.), New York: McGraw-Hill Book Company, pp. XII+628, ISBN 978-0-07-013441-6, Zbl 0433.73001.
• Dimitrienko, Yuriy I. (2011), Nonlinear continuum mechanics and large inelastic deformations, Solid Mechanics and Its Applications, vol. 174, Dordrecht: Springer Verlag, pp. 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).
• Fabrizio, Mauro; Morro, Angelo (1992), Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Philadelphia PA: SIAM, pp. ix+203, ISBN 978-0-89871-266-7, MR 0153021, Zbl 0753.73003.
• Fabrizio, Mauro; Morro, Angelo (1992), Electromagnetism of continuous media. Mathematical modelling and applications, Oxford Science Publications, Oxford: Oxford University Press, pp. xvii+668, doi:10.1093/acprof:oso/9780198527008.001.0001, ISBN 978-0-19-852700-8, MR 1996323, Zbl 1027.78001.
• Lemaitre, Jean; Chaboche, Jean Louis (1990), Mechanics of solid materials, translated by Shrivastava, B., with a foreword to the French edition by Paul Germain and a foreword to the English edition by Fred Leckie, Cambridge–New York–Melbourne–Madrid–Cape Town–Singapore–Sao Paulo–Delhi: Cambridge University Press, pp. xxv + 556, doi:10.1017/CBO9781139167970, ISBN 978-0-521-32853-1, Zbl 0743.73002
• Volterra, Vito (1913), Leçons sur les fonctions de lignes, Collection de Monographies sul la Théorie des Fonctions (in French), recueil et rèdigèes par Joseph Pérès, Paris: Gauthier-Villars, pp. vi+230, JFM 44.0458.01 (see also JFM 44.0410.01).
• Volterra, Vito (1959) [1930], Fantappiè, Luigi (ed.), Theory of functionals and of integral and integro-differential equations, with a foreword of G. C. Evans, a biography of Vito Volterra and a bibliography of his published works by Sir Edmund Whittaker (2nd ed.), New York: Dover Publications, pp. [39]+XIV+226, JFM 56.1010.01, MR 0100765, Zbl 0086.10402 (see also JFM 55.0814.01: the JFM reviews refer to the original 1930 English edition).

Congress proceedings and collections of articles on materials with memory

• C.I.M.E., ed. (1979), Materials with memory – Bressanone, 2–11 Giugno [Materials with memory – Brixen, 2–11 Giugno], Corsi del C.I.M.E. (in Italian), vol. I Ciclo 1977, Napoli: Liguori Editore, p. 294, ISBN 978-88-207-0756-9
• Grioli, Giuseppe, ed. (1990), Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986) [Round table on the topic: continua with memory (Rome, 9 May 1986)], Contributi del Centro Linceo Interdisciplinare "Beniamino Segre" (in Italian), vol. 81, Rome: Accademia Nazionale dei Lincei, p. 73, ISSN 0394-0705