×

zbMATH — the first resource for mathematics

Homogenization of reinforced periodic one-codimensional structures. (English) Zbl 0654.73017
The aim of the paper is to clarify the behaviour of periodic reinforced structures. The study is performed in the case of one-dimensional reinforced structures (stratified, alveolar). Such structures are characterized by three parameters: \(\epsilon\), the period of the structure, r, the thickness of the reinforced zone and \(\lambda\), the physical parameter (conductivity or elasticity coefficient) of the reinforced zone.
Asymptotic analysis of the constitutive equation made as \(\epsilon\to 0\), \(r\to 0\) with \((r/\epsilon)\to 0\), and \(\lambda\to \infty\), gives the macroscopic behavior of the material. The \(\Gamma\)-convergence method is used, the solutions being expressed as minimizers of the corresponding energy functionals. Results are available in any physical problem governed by a second order linear elliptic constitutive equation (conductivity, elasticity,...). The existence of a critical ratio \((\lambda^ r/\epsilon)\) is shown. Depending on the limit of \((\lambda^ r/\epsilon)\), the homogenized-reinforced material is characterized by its effective coefficients. These coefficients are computed for some particular structures.
Reviewer: Th.Lévy

MSC:
74E05 Inhomogeneity in solid mechanics
74E30 Composite and mixture properties
74S30 Other numerical methods in solid mechanics (MSC2010)
46S30 Constructive functional analysis
78A30 Electro- and magnetostatics
49J45 Methods involving semicontinuity and convergence; relaxation
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] H. Attouch , Variational Convergence for Functions and Operators . Pitman, Appl. Math. Ser. , Boston ( 1984 ). MR 773850 | Zbl 0561.49012 · Zbl 0561.49012
[2] D. Aze , Homogénéisation primale et duale par epi-convergence. Applications à l’élasticité . Publications A V AMAC 84 - 06 , Université de Perpignan, Perpignan ( 1984 ).
[3] N.S. Bahvalov - G.P. Panasenko , Averaged processes in periodic media . Moscow , Nauka 1984 (in russian). Zbl 0607.73009 · Zbl 0607.73009
[4] A. Bensoussan - J.L. Lions - G. Papanicolaou , Asymptotic Analysis for Periodic Structures . North-Holland , Amsterdam ( 1978 ). MR 503330 | Zbl 0404.35001 · Zbl 0404.35001
[5] G. Buttazzo , Su una definizione generale dei \Gamma -limiti . Boll. Un. Mat. Ital. , 14-B ( 1977 ), 722 - 744 . Zbl 0445.49016 · Zbl 0445.49016
[6] G. Buttazzo - G. Dal Maso , \Gamma -limits of integral functionals . J. Analyse Math. , 37 ( 1980 ), 145 - 185 . Zbl 0446.49012 · Zbl 0446.49012 · doi:10.1007/BF02797684
[7] G. Buttazzo - G. Dal Maso , Singular perturbation problems in the calculus of variations . Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 11 ( 1984 ), 395 - 430 . Numdam | MR 785619 · numdam:ASNSP_1984_4_11_3_395_0
[8] G. Buttazzo - G. Dal Maso , Integral representation and relaxation of local functionals . Nonlinear Anal ., 9 ( 1985 ), 515 - 532 . MR 794824 | Zbl 0527.49008 · Zbl 0527.49008 · doi:10.1016/0362-546X(85)90038-0
[9] D. Caillerie , Etude de la conductivité stationnaire dans un domaine comportant une répartition périodique d’inclusions minces de grande conductivité . RAIRO Anal. Numér. , 17 ( 1983 ), 137 - 159 . Numdam | MR 705449 | Zbl 0587.35041 · Zbl 0587.35041 · eudml:193413
[10] L. Carbone - C. Sbordone , Some properties of \Gamma -limits of integral functionals . Ann. Mat. Pura Appl. , 122 ( 1979 ), 1 - 60 . Zbl 0474.49016 · Zbl 0474.49016 · doi:10.1007/BF02411687
[11] E.H. Chabi , Homogénéisation de structures présentant de nombreuses inclusions (strates, fibres) fortement conductrices. Thèse , Université de Perpignan , Perpignan ( 1986 ).
[12] D. Cioranescu - S.J. Paulin , Reinforced and alveolar structures . Publication 85042 du Laboratoire d’Analyse Numérique Paris VI , Paris ( 1985 ).
[13] E. De Giorgi , Convergence problems for functionals and operators. Proceedings Recent Methods in Nonlinear Analysis”, Rome 1978 , edited by E. De Giorgi - E. Magenes - U. Mosco, Pitagora , Bologna ( 1979 ), 131 - 188 . MR 533166 | Zbl 0405.49001 · Zbl 0405.49001
[14] E. De Giorgi , G-operators and \Gamma -convergence. Proceedings of the International Congress of Mathematicians , Warszawa 1983 . Zbl 0568.35025 · Zbl 0568.35025
[15] E. De Giorgi - T. Franzoni , Una presentazione sintetica dei \Gamma -limiti generalizzati . Portugal. Math. , 41 ( 1982 ), 405 - 436 . Article | Zbl 0547.49006 · Zbl 0547.49006 · eudml:115512
[16] G. Dal Maso - L. Modica , A general theory of variational functionals. ”Topics in Functional Analysis 1980-81”, Scuola Normale Superiore , Pisa ( 1982 ), 149 - 221 . MR 671757 | Zbl 0493.49005 · Zbl 0493.49005
[17] J.L. Lions , Some Methods in the Mathematical Analysis of Systems and their Control . Science Press , Beijing (China); Gordon Breach , New York ( 1981 ). MR 664760 | Zbl 0542.93034 · Zbl 0542.93034
[18] P. Marcellini , Periodic solutions and homogenization of nonlinear variational problems . Ann. Mat. Pura Appl. , 117 ( 1978 ), 139 - 152 . MR 515958 | Zbl 0395.49007 · Zbl 0395.49007 · doi:10.1007/BF02417888
[19] E. Sanchez-Palencia , Non Homogeneous Media and Vibration Theory. Lecture Notes in Phys ., Springer-Verlag , Berlin ( 1980 ). Zbl 0432.70002 · Zbl 0432.70002
[20] C. Sbordone , Su alcune applicazioni di un tipo di convergenza variazionale . Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 2 ( 1975 ), 617 - 638 . Numdam | MR 417753 | Zbl 0317.49012 · Zbl 0317.49012 · numdam:ASNSP_1975_4_2_4_617_0 · eudml:83707
[21] E. Acerbi - G. Buttazzo - D. Percivale , Thin inclusions in linear elasticity: a variational approach , J. Reine Angew, Math. (to appear). MR 936993 | Zbl 0633.73021 · Zbl 0633.73021 · doi:10.1515/crll.1988.386.99 · crelle:GDZPPN002205440 · eudml:153018
[22] H. Attouch - G. Buttazzo , Homogenization of reinforced fibred structures . (Paper in preparation). · Zbl 0654.73017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.