Asymptotic Analysis of Stratified Elastic Media in the Space of Functions with Bounded Deformation

We consider a heterogeneous elastic structure which is stratified in one direction. We derive the limit problem under the sole assumption that the Lame coefficients and their inverses weakly* converge to some Radon measures.

[1]  Mohammed Hnid,et al.  Étude de transmission à travers des inclusions minces faiblement conductrice de «codimension un» : homogénéisation et optimisation des structures , 1990 .

[2]  Guy Bouchitté,et al.  HOMOGENIZATION OF ELLIPTIC PROBLEMS IN A FIBER REINFORCED STRUCTURE. NON LOCAL EFFECTS , 1998 .

[3]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[4]  G V Sandrakov,et al.  Homogenization of elasticity equations with contrasting coefficients , 1999 .

[5]  Björn Gustafsson,et al.  Γ‐convergence of stratified media with measure‐valued limits , 2000 .

[6]  Michel Bellieud,et al.  Analyse asymptotique de milieux élastiques stratifiés dans les espaces de fonctions à déformation bornée , 2016 .

[7]  Ronald F. Gariepy.  FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs) , 2001 .

[8]  Sergio Gutiérrez,et al.  Laminations in Linearized Elasticity: The Isotropic Non-very Strongly Elliptic Case , 1998 .

[9]  Piotr Hajłasz,et al.  United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency on Approximate Differentiability of Functions with Bounded Deformation , 2022 .

[10]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[11]  Roger Temam,et al.  Functions of bounded deformation , 1980 .

[12]  F. Murat,et al.  Compacité par compensation , 1978 .

[13]  Guy Bouchitté,et al.  Convergence of Sobolev spaces on varying manifolds , 2001 .

[14]  Michel Bellieud.  Torsion Effects in Elastic Composites with High Contrast , 2009, SIAM J. Math. Anal..

[15]  Luc Tartar.  Memory Effects and Homogenization , 1990 .

[16]  Jacqueline Mossino,et al.  H-convergence and regular limits for stratified media with low and high conductivities , 1995 .

[17]  B. Heron,et al.  Homogénéisation de structures stratifiées avec faible et forte conductivités , 1992 .

[18]  Michel Bellieud,et al.  A Notion of Capacity Related to Elasticity: Applications to Homogenization , 2012 .

[19]  R. Kohn,et al.  New estimates for deformations in terms of their strains , 1979 .

[20]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[21]  Pierre M. Suquet,et al.  Un espace fonctionnel pour les équations de la plasticité , 1979 .

[22]  Michel Bellieud.  Problèmes capacitaires en viscoplasticité avec effets de torsion , 2013 .

[23]  J. Moreau.  Application of convex analysis to the treatment of elastoplastic systems , 1976 .

[24]  Georges Griso,et al.  Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps , 2008, Multiscale Model. Simul..

[25]  G. Bouchitté,et al.  Singular perturbations and homogenization in stratified media , 1996 .

[26]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[27]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[28]  Kirill Cherednichenko,et al.  Two-Scale Γ-Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic Composites , 2012 .

[29]  P. Seppecher,et al.  Determination of the Closure of the Set of Elasticity Functionals , 2003 .

[30]  Guy Bouchitté,et al.  Homogenization of a soft elastic material reinforced by fibers , 2002 .

[31]  Umberto Mosco,et al.  Composite media and asymptotic dirichlet forms , 1994 .

[32]  Grigory Panasenko.  Multicomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions , 1995 .

[33]  V. V. Zhikov,et al.  Weighted Sobolev spaces , 1998 .

[34]  Gianni Dal Maso,et al.  Fine Properties of Functions with Bounded Deformation , 1997 .

[35]  Mc Connell,et al.  On the approximation of elliptic operators with discontinuous coefficients , 1976 .

[36]  Mariano Giaquinta,et al.  Existence of the displacements field for an elasto-plastic body subject to Hencky's law and Von Mises yield condition , 1980 .

[37]  Pierre Seppecher,et al.  A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium , 1997 .

[38]  Shane Cooper,et al.  Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect , 2017, SIAM J. Math. Anal..

[39]  O. Oleinik,et al.  Mathematical Problems in Elasticity and Homogenization , 2012 .

[40]  Michel Bellieud.  Homogenization of stratified elastic media with high contrast , 2015 .

[41]  Michel Bellieud,et al.  Homogenization of Stratified Elastic Composites with High Contrast , 2017, SIAM J. Math. Anal..

[42]  Isabelle Gruais,et al.  Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non-local effects. Memory effects , 2005 .

[43]  Michel Bellieud.  Homogeneisation de problemes elliptiques avec effets non locaux , 1997 .

[44]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[45]  Irene Fonseca,et al.  A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION , 2000 .

[46]  V. V. Zhikov,et al.  On gaps in the spectrum of the operator of elasticity theory on a high contrast periodic structure , 2013 .

[47]  Shane Cooper,et al.  Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions , 2013, 1304.7043.

[48]  G. Bouchitte,et al.  Représentation intégrale de fonctionnelles convexes sur un espace de mesures , 1987, ANNALI DELL UNIVERSITA DI FERRARA.

[49]  A. Chambolle.  An approximation result for special functions with bounded deformation , 2004 .

[50]  Jacqueline Mossino,et al.  Homogenization of some quasilinear problems for stratified media with low and high conductivities , 1994 .

[51]  Björn Gustafsson,et al.  Stratified materials allowing asymptotically prescribed equipotentials , 1994 .

[52]  Luc Tartar,et al.  The General Theory of Homogenization: A Personalized Introduction , 2009 .

[53]  Björn Gustafsson,et al.  Compensated compactness for homogenization and reduction of dimension: The case of elastic laminates , 2006, Asymptot. Anal..

[54]  G. Bellettini,et al.  Compactness and lower semicontinuity properties in $SBD(\Omega)$ , 1998 .

[55]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[56]  Luc Tartar.  Remarks on Homogenization , 1986 .

[57]  B. Heron,et al.  H-convergence pour des structures stratifiées avec faible et forte conductivités , 1993 .

[58]  K. D. CHEREDNICHENKO,et al.  Spectral Analysis of One-Dimensional High-Contrast Elliptic Problems with Periodic Coefficients , 2015, Multiscale Model. Simul..

[59]  Luc Tartar,et al.  The General Theory of Homogenization , 2010 .

[60]  Giuseppe Geymonat,et al.  Asymptotic Analysis of a Linear Isotropic Elastic Composite Reinforced by a Thin Layer of Periodically Distributed Isotropic Parallel Stiff Fibres , 2016 .

[61]  Graeme W Milton,et al.  On modifications of Newton's second law and linear continuum elastodynamics , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[62]  V. P. Smyshlyaev,et al.  Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization , 2009 .

[63]  R. Temam,et al.  Problèmes mathématiques en plasticité , 1983 .