Prolegomena to studies on dynamic materials and their space-time homogenization

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December 2013, 6(6): 1599-1608. doi: 10.3934/dcdss.2013.6.1599

Prolegomena to studies on dynamic materials and their space-time homogenization

Gerard A. Maugin ^1, and Martine Rousseau ^1,

1.	Université Pierre et Marie Curie, Institut Jean Le Rond d'Alembert, UMR CNRS 7190, Case 162, Tour 55, 4 place Jussieu, 75252 Paris Cedex 05, France, France

Received June 2012 Revised September 2012 Published April 2013

Abstract
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This short contribution aims at introducing the notion of dynamic materials (as initiated by Blekhman and Lurie) and the corresponding allied techniques of homogenization and asymptotic analysis. Main role is played by the canonical conservation laws of energy and wave momentum - the latter most often ignored in the field of continuum mechanics - as follows from an application of the celebrated theorem of E. Noether.

Keywords: space-time homogenization., inhomogeneity, Linear waves, dynamic materials.

Mathematics Subject Classification: Primary: 74E05, 74H10, 74J05, 74Q10; Secondary: 74H4.

Citation: Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599

References:

[1]	I. I. Blekhman and K. A. Lur'e, On dynamic materials, (Russian) Doklady Akademii Nauk, 371 (2000), 182-185. doi: 10.1134/1.171720.
[2]	V. L. Ginzburg and V. N. Tsytovich, Several problems of the theory of transition radiation and transition scattering, Physics Reports, 49 (1979), 1-89; Original Russian in Usp. Fiz. Nauk, 126 (1978), 553-563. doi: 10.1016/0370-1573(79)90052-8.
[3]	K. A. Lurie, "Introduction to the Mathematical Theory of Dynamic Materials," Advances in Mechanics and Mathematics, 15, Springer, New York, 2007.
[4]	K. A. Lurie, On homogenization of activated laminates in 1D-space and time, Zeit. Angew. Math. Mech., 89 (2009), 333-340. doi: 10.1002/zamm.200800185.
[5]	K. A. Lurie, D. Onofrei and S. L. Weekes, Mathematical analysis of the waves propagation through a rectangular material structure in space-time, J. Math. Analysis and Applications, 355 (2009), 180-194. doi: 10.1016/j.jmaa.2009.01.031.
[6]	G. A. Maugin, "Material Inhomogeneties in Elasticity," Applied Mathematics and Mathematical Computation, 3, Chapman & Hall, London, 1993.
[7]	G. A. Maugin, "Configurational Forces. Thermomechanics, Physics, Mathematics, and Numerics," CRC Series: Modern Mechanics and Mathematics, CRC Press, Boca Raton, FL, 2011.
[8]	G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9), 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002.
[9]	D. E. Neuenschwander, "Emmy Noether's Wonderful Theorem," Johns Hopkins University Press, Baltimore, MD, 2011.
[10]	M. Rousseau, G. A. Maugin and M. Berezovski, Elements of study on dynamic materials, Arch. Appl. Mechanics, 81 (2011), 925-942. doi: 10.1007/s00419-010-0461-4.
[11]	E. Sánchez-Palencia and A. Zaoui, eds., "Homogenization Techniques for Composite Media," Papers from the course held in Udine, July 1-5, 1985, Lecture Notes in Physics, 272, Springer-Verlag, Berlin, 1987. doi: 10.1007/3-540-17616-0.
[12]	C. A. Truesdell and R. A. Toupin, Classical theory of fields, in "Handbuch der Physik, Vol. III/1" (ed. S. Flügge), Springer-Verlag, Berlin, 1960.
[13]	A. I. Vesnitskii and A. V. Metrikine, Transition radiation in mechanics, Physics-Uspekhi, 39 (1996), 983-1007.

show all references

References:

[1]	I. I. Blekhman and K. A. Lur'e, On dynamic materials, (Russian) Doklady Akademii Nauk, 371 (2000), 182-185. doi: 10.1134/1.171720.
[2]	V. L. Ginzburg and V. N. Tsytovich, Several problems of the theory of transition radiation and transition scattering, Physics Reports, 49 (1979), 1-89; Original Russian in Usp. Fiz. Nauk, 126 (1978), 553-563. doi: 10.1016/0370-1573(79)90052-8.
[3]	K. A. Lurie, "Introduction to the Mathematical Theory of Dynamic Materials," Advances in Mechanics and Mathematics, 15, Springer, New York, 2007.
[4]	K. A. Lurie, On homogenization of activated laminates in 1D-space and time, Zeit. Angew. Math. Mech., 89 (2009), 333-340. doi: 10.1002/zamm.200800185.
[5]	K. A. Lurie, D. Onofrei and S. L. Weekes, Mathematical analysis of the waves propagation through a rectangular material structure in space-time, J. Math. Analysis and Applications, 355 (2009), 180-194. doi: 10.1016/j.jmaa.2009.01.031.
[6]	G. A. Maugin, "Material Inhomogeneties in Elasticity," Applied Mathematics and Mathematical Computation, 3, Chapman & Hall, London, 1993.
[7]	G. A. Maugin, "Configurational Forces. Thermomechanics, Physics, Mathematics, and Numerics," CRC Series: Modern Mechanics and Mathematics, CRC Press, Boca Raton, FL, 2011.
[8]	G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9), 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002.
[9]	D. E. Neuenschwander, "Emmy Noether's Wonderful Theorem," Johns Hopkins University Press, Baltimore, MD, 2011.
[10]	M. Rousseau, G. A. Maugin and M. Berezovski, Elements of study on dynamic materials, Arch. Appl. Mechanics, 81 (2011), 925-942. doi: 10.1007/s00419-010-0461-4.
[11]	E. Sánchez-Palencia and A. Zaoui, eds., "Homogenization Techniques for Composite Media," Papers from the course held in Udine, July 1-5, 1985, Lecture Notes in Physics, 272, Springer-Verlag, Berlin, 1987. doi: 10.1007/3-540-17616-0.
[12]	C. A. Truesdell and R. A. Toupin, Classical theory of fields, in "Handbuch der Physik, Vol. III/1" (ed. S. Flügge), Springer-Verlag, Berlin, 1960.
[13]	A. I. Vesnitskii and A. V. Metrikine, Transition radiation in mechanics, Physics-Uspekhi, 39 (1996), 983-1007.

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