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Lobo, Miguel; Perez, Eugenia

Asymptotic behaviour of an elastic body with a surface having small stuck regions. (English) Zbl 0659.73006

RAIRO, Modélisation Math. Anal. Numér. 22, No. 4, 609-624 (1988).
The authors examine the title problem where “stuck region” refers to parts of the surface of an elastic body which are attached to a plane. If these stuck areas have diameter \(\epsilon\), then it is shown that a “critical distance” of separation between stuck areas is of the order \(\epsilon^ 2\). When stuck areas are separated by either significantly larger or smaller distances, the surface is analogous to a totally free or totally stuck surface.
The paper is carefully written in the notion of modern elasticity theory. It should be of greatest interest to theoreticians concerned with boundary value problems.
Reviewer: R.L.Huston

Cited in 1 Review
Cited in 23 Documents

MSC:

74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
35B40 Asymptotic behavior of solutions to PDEs

Keywords:

critical distance of separation between stuck areas; surface partially stuck to fixed plane; limit behaviour; boundary condition which is intermediate between the perfect stuck and unstuck cases; boundary homogenization problems

Citations:

Zbl 0615.73009
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Full Text: DOI EuDML

References:

[1] H. ATTOUCH, Variational convergence for functions and operators, Pitman. London (1984). Zbl0561.49012 MR773850 · Zbl 0561.49012
[2] A. BRILLARD, M. LOBO, E. PEREZ, Homogénéisation de frontières par épi-convergence en élasticité linéaire. (A paraître). Zbl0691.73013 · Zbl 0691.73013
[3] D. CIORANESCU, F. MURAT, Un terme étrange venu d’ailleurs. Collège de France Seminar, Research Notes in Mathematics, Pitman, London, (1982)n^\circ 60, p. 98-138, n^\circ 70, pp. 154-178. Zbl0496.35030 · Zbl 0496.35030
[4] J. DENY, Sur la convergence de certaines intégrales de la théorie du potentiel. Arch. Math. Vol. V, (1954), p. 367-371. Zbl0057.33104 MR66513 · Zbl 0057.33104 · doi:10.1007/BF01898378
[5] G. DUVAUT, J. L. LIONS, Les inéquations en mécanique et en physique. Dunod. Paris (1972). Zbl0298.73001 MR464857 · Zbl 0298.73001
[6] W. ECKHAUS, Asymptotic Analysis and Singular Pertubations, North-Hoïland, Amsterdam (1979). Zbl0421.34057 MR553107 · Zbl 0421.34057
[7] O. A. LADYZHENSKAY, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach. London (1969). Zbl0184.52603 MR254401 · Zbl 0184.52603
[8] L. LANDAU, E. LIFCHITZ, Théorie de l’Elasticité. Mir. Moscou (1967). Zbl0166.43101 · Zbl 0166.43101
[9] J. L. LIONS, E. MAGENES, Problèmes aux limites non homogènes et applications. Vol. I. Dunod, Paris (1968). Zbl0165.10801 · Zbl 0165.10801
[10] M. LOBO, E. PEREZ, Comportement asymptotique d’un corps élastique dont une surface présente de petites zones de collage. C.R. Acad. Se. Paris, t. 304, Série II n^\circ 5, 1987. Zbl0602.73019 MR977600 · Zbl 0602.73019
[11] R.C. Mac CAMY, E. STEPHAN, Solution Procedures for Three-Dimensional Eddy Current Problems. J. Math. Anal. Appl. 101 (1984), 348-379. Zbl0563.35054 MR748577 · Zbl 0563.35054 · doi:10.1016/0022-247X(84)90108-2
[12] F. MURAT, Neumann’s Sieve. Proceedings of the meeting on variational methods in nonlinear analysis, Isle of Elba 1983. Research Notes in Mathematics,^\circ 127 Pitman, London, 1985. Zbl0586.35037 MR807534 · Zbl 0586.35037
[13] C. PICARD, Analyse limite d’équations variationnelles dans un domaine contenant une grille. Thèse d’Etat. Université de Paris-Sud. Orsay (1984).
[14] E. SANCHEZ-PALENCIA, Boundary value problems in domains containing Perforated walls. In Nonlinear Differential Equations, Collège de France Seminar, Vol. III, Research Notes in Mathematics, 70, p. 309-325, Pitman, London (1982). Zbl0505.35020 MR670282 · Zbl 0505.35020
[15] J. SANCHEZ-HUBERT, E. SANCHEZ-PALENCIA, Acoustic fiuid flow through holes and permeability of perforated walls. Jour. Math. Anal. Appl., 87, (1982) p. 427-453. Zbl0484.76101 MR658023 · Zbl 0484.76101 · doi:10.1016/0022-247X(82)90133-0
[16] I.N. SNEDDON, Fourier transforms, McGraw-Hill. London (1951). Zbl0038.26801 MR41963 · Zbl 0038.26801
[17] R. TEMAM, Problèmes mathématiques en plasticité. Gauthier-Villars. Paris (1983). Zbl0547.73026 MR711964 · Zbl 0547.73026
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