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Materials with memory

Topic: Physics

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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[1][2] and Vito Volterra,[3][4] in which they sought an extension of the concept of an elastic material.[5] 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,[4] 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

[math]\displaystyle{ \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

Notes

  1. ↑ (Boltzmann 1874).
  2. ↑ (Boltzmann 1878).
  3. ↑ (Volterra 1912)
  4. ↑ 4.0 4.1 (Volterra|1959}}|Volterra 1930).
  5. ↑ See (Coleman Noll)

References

Articles

Books

  • Alberty, Robert A.; Silbey, Robert J. (2001), Physical Chemistry, New York: John Wiley & Sons, pp. 969, ISBN 978-0-471-38311-6, OCLC 959753597, https://archive.org/details/physicalchemistr00silb_0/page/969 .
  • 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, https://books.google.com/books?id=JgtpQgAACAAJ .
  • 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  (also available as e-book with ISBN:978-1-4614-1692-0).
  • Bloom, Frederick (1981) (in en), Ill-posed problems for integrodifferential equations in mechanics and electromagnetic theory, SIAM Studies in Applied Mathematics, 3, Philadelphia: SIAM, pp. IX+222, ISBN 978-0-89871-171-4, https://books.google.com/books?id=ZTGd1pkDfO0C .
  • 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 .
  • Dimitrienko, Yuriy I. (2011), Nonlinear continuum mechanics and large inelastic deformations, Solid Mechanics and Its Applications, 174, Dordrecht: Springer Verlag, pp. xxiv + 721, doi:10.1007/978-94-007-0034-5, ISBN 978-94-007-0033-8  (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, 12, Philadelphia PA: SIAM, pp. ix+203, ISBN 978-0-89871-266-7, https://books.google.com/books?id=YOTuQbpBzs8C .
  • 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 .
  • Lemaitre, Jean; Chaboche, Jean Louis (1990), Mechanics of solid materials, 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 
  • Volterra, Vito (1913) (in fr), Leçons sur les fonctions de lignes, Collection de Monographies sul la Théorie des Fonctions, recueil et rèdigèes par Joseph Pérès, Paris: Gauthier-Villars, pp. vi+230, https://archive.org/details/leonssurlesfon00voltuoft  (see also JFM 44.0410.01).
  • Volterra, Vito (1959), 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  (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) (in it), Materials with memory – Bressanone, 2–11 Giugno, Corsi del C.I.M.E., I Ciclo 1977, Napoli: Liguori Editore, pp. 294, ISBN 978-88-207-0756-9 
  • Grioli, Giuseppe, ed. (1990) (in it), Tavola rotonda sul tema: continui con memoria (Roma, 9 maggio 1986), Contributi del Centro Linceo Interdisciplinare "Beniamino Segre", 81, Rome: Accademia Nazionale dei Lincei, pp. 73, ISSN 0394-0705 



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