Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions

References

• Fenchenko V, Khruslov E. Asymptotic behaviour or the solutions of differential equations with strongly oscillating and degenerating coefficient matrix. Dokl. Akad. Nauk. Ukrain. SSR Ser. A. 1980;4:26–30.
   Google Scholar
• EKhruslov. An averaged model of a strongly inhomogeneous medium with memoryUspekhi Mat. Nauk.. 199045197198
   Google Scholar
• GSandrakov. Homogenization of elasticity equations with contrasting coeffcientsSbornik Math.. 199919017491806
   Web of Science ®Google Scholar
• AAvila, GGriso, BMiara, ERohan. Multiscale modeling of elastic waves: theoretical justification and numerical simulation of band gapsMultiscale Model. Simul.. 20087121
   Web of Science ®Google Scholar
• Bellieud M. Torsion effects in elastic composites with high contrast (English summary). SIAM J. Math Anal. 2009/10;41:2514–2553.
   Google Scholar
• KCherednichenko. Two-scale asymptotics for non-local effects in composites with highly anisotropic fibresAsymptotic Anal.. 2006493959
   Web of Science ®Google Scholar
• KCherednichenko, VSmyshlyaev, VZhikov. Non-local homogenised limits for composite media with highly anisotropic periodic fibresProc. R. Soc. Edinb. A.. 200613687114
   Google Scholar
• GNguetseng. A general convergence result for a functional related to the theory of homogenizationSIAM J. Math Anal.. 198920608623
   Web of Science ®Google Scholar
• Allaire G. Homogenisation and two-scale convergence. SIAM J. Math. Anal. 1992;23: 1482–1518.
   Google Scholar
• VZhikov. On an extension of the method of two-scale convergence and its applicationsSbornik Math.. 20001919731014
   Web of Science ®Google Scholar
• Zhikov V. On spectrum gaps of some divergent elliptic operators with periodic coefficients. St.Petersburg Math. J. 2005;16:773–790.
   Google Scholar
• Jhikov V, Kozlov S., Oleinik O. Homogenization of differential operators and integral functionals: Springer-Verlag: Berlin Heidelberg; 1994.
   Google Scholar
• Bouchitte G, Felbacq D. Homogenization near resonances and artifical magnetism from dielectrics. C.R. Math. 2004;339:377–382.
   Google Scholar
• Pendry S. Metamaterials and the control of electromagnetic fields. In Conference on Coherence and Quantum Optics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper CMB2.
   Google Scholar
• VSmyshlyaev. Propagation and localisation of elastic waves in highly anisotropic periodic composites via two-scale homogenisationMech. Mater.. 200941434447
   Web of Science ®Google Scholar
• Kamotski I, Smyshlyaev V. Homogenisation of degenerating PDEs and applications to localisation of waves. Preprint. http://arxiv.org/abs/1309.4579
   Google Scholar
• NBakhvalov, MEglit. Averaging of the equations of the dynamics of composites of slightly compressible elastic componentsComput. Math. Math. Phys.. 199333939952
   Web of Science ®Google Scholar
• Panasenko G. Asymptotics of the solution of the elasticity theory system of equations in a weakly compressed bar. Russ. J. Math. Phys. 1996;4:112–116.
   Google Scholar
• Panasenko G. Averaged systems of equations of the theory of elasticity in a medium with weakly compressible inclusions. Moscow state Uni., Matematicheskie Zametki. 1992;51:126–133.
   Google Scholar
• Chechkin G, Piatnitski A, Shamaev A, Homogenisation: methods and applications. AMS, USA; 2007.
   Google Scholar
• Temam R. Navier-Stokes equations: theory and numerical analysis. AMS: AMS Chelsea Publishing North-Holland Publishing Company, Amsterdam, New York, NY, Oxford; 2000.
   Google Scholar
• Evans L. Partial differential equations, vol. 19 of Graduate studies in mathematics. AMS, USA; 2010.
   Google Scholar
• Ladyzhenskaya OA. On a relationship between the stokes problem and decompositions of the spaces and . (Russian) Algebra i Analiz. 2001;13:119–133; translation in. St. Petersburg Math. J. 2002;13:601–612.
   Google Scholar
• Duvaut G, Lions J. Inequalities in mechanics and physics: Springer-Verlag, Berlin Heidelberg New York; 1976.
   Google Scholar
• Reed M, Simon B. Methods of modern mathematical physics. Vol. 1: Functional analysis. Academic Press, London; 1980.
   Google Scholar