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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

K. A. Lurie, "Introduction to the Mathematical Theory of Dynamic Materials," Advances in Mechanics and Mathematics, 15, Springer, New York, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. A. Maugin, "Material Inhomogeneties in Elasticity," Applied Mathematics and Mathematical Computation, 3, Chapman & Hall, London, 1993.  Google Scholar

[7]

G. A. Maugin, "Configurational Forces. Thermomechanics, Physics, Mathematics, and Numerics," CRC Series: Modern Mechanics and Mathematics, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[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.  Google Scholar

[9]

D. E. Neuenschwander, "Emmy Noether's Wonderful Theorem," Johns Hopkins University Press, Baltimore, MD, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

A. I. Vesnitskii and A. V. Metrikine, Transition radiation in mechanics, Physics-Uspekhi, 39 (1996), 983-1007. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

K. A. Lurie, "Introduction to the Mathematical Theory of Dynamic Materials," Advances in Mechanics and Mathematics, 15, Springer, New York, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. A. Maugin, "Material Inhomogeneties in Elasticity," Applied Mathematics and Mathematical Computation, 3, Chapman & Hall, London, 1993.  Google Scholar

[7]

G. A. Maugin, "Configurational Forces. Thermomechanics, Physics, Mathematics, and Numerics," CRC Series: Modern Mechanics and Mathematics, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[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.  Google Scholar

[9]

D. E. Neuenschwander, "Emmy Noether's Wonderful Theorem," Johns Hopkins University Press, Baltimore, MD, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

A. I. Vesnitskii and A. V. Metrikine, Transition radiation in mechanics, Physics-Uspekhi, 39 (1996), 983-1007. Google Scholar

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