Semicoercive hemivariational inequalities. On the delamination of composite plates
Author:
P. D. Panagiotopoulos
Journal:
Quart. Appl. Math. 47 (1989), 611-629
MSC:
Primary 73V25; Secondary 49J40, 73B27, 73K10, 73K20
DOI:
https://doi.org/10.1090/qam/1031680
MathSciNet review:
MR1031680
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper semicoercive hemivariational inequalities are studied in the framework of a concrete mechanical problem: the delamination effect of laminated plates. The interlaminar bonding forces are described by a nonmonotone multivalued law which may be written as the generalized gradient of a nonconvex superpotential in the sense of F. H. Clarke. Then necessary conditions are proved for the existence of the solution, as well as sufficient conditions using compactness and average value arguments.
- G. E. Stavroulakis and P. D. Panagiotopoulos, Laminated orthotropic plates under subdifferential boundary conditions. A variational-hemivariational inequality approach, Z. Angew. Math. Mech. 68 (1988), no. 6, 213–224 (English, with German and Russian summaries). MR 956248, DOI https://doi.org/10.1002/zamm.19880680610
- P. D. Panagiotopoulos, Nonconvex superpotentials in the sense of F. H. Clarke and applications, Mech. Res. Comm. 8 (1981), no. 6, 335–340. MR 639382, DOI https://doi.org/10.1016/0093-6413%2881%2990064-1
- P. D. Panagiotopoulos, Nonconvex energy functions. Hemivariational inequalities and substationarity principles, Acta Mech. 48 (1983), no. 3-4, 111–130. MR 715806, DOI https://doi.org/10.1007/BF01170410
- P. Panagiotopulos, Neravenstva v mekhanike i ikh prilozheniya, “Mir”, Moscow, 1989 (Russian). Vypuklye i nevypuklye funktsii ènergii. [Convex and nonconvex energy functions]; Translated from the English by I. R. Shablinskaya and R. A. Arutyunov; Translation edited and with a preface by V. F. Dem′yanov. MR 1024312
- G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. MR 0464857
- G. Duvaut and J.-L. Lions, Problèmes unilatéraux dans la théorie de la flexion forte des plaques. I. Le cas stationnaire, J. Mécanique 13 (1974), 51–74 (French, with English summary). MR 375885
- Michel Potier-Ferry, Problèmes semi-coercifs. Applications aux plaques de von Karmann, J. Math. Pures Appl. (9) 53 (1974), 331–346 (French). MR 374716
- Jean-Jacques Moreau, La notion de sur-potentiel et les liaisons unilatérales en élastostatique, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A954–A957 (French). MR 241038
- J.-J. Moreau, P. D. Panagiotopoulos, and G. Strang (eds.), Topics in nonsmooth mechanics, Birkhäuser Verlag, Basel, 1988. MR 957086
F. H. Clarke, Nonsmooth Analysis and Optimization, J. Wiley, New York, 1984
- Panagiotis D. Panagiotopoulos, Une généralisation non-convexe de la notion du sur-potentiel et ses applications, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 296 (1983), no. 15, 1105–1108 (French, with English summary). MR 720434
- P. D. Panagiotopoulos, Hemivariational inequalities and substationarity in the static theory of v. Kármán plates, Z. Angew. Math. Mech. 65 (1985), no. 6, 219–229 (English, with German and Russian summaries). MR 801713, DOI https://doi.org/10.1002/zamm.19850650608
L. D. Landau and E. M. Lifschitz, Theory of Elasticity, Pergamon Press, 1959
R. Jones, Mechanics of composite materials, McGraw Hill-Scripta Book Co., New York-Washington, 1975
- Jeffrey Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proc. Amer. Math. Soc. 64 (1977), no. 2, 277–282. MR 442453, DOI https://doi.org/10.1090/S0002-9939-1977-0442453-6
- Kung Ching Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102–129. MR 614246, DOI https://doi.org/10.1016/0022-247X%2881%2990095-0
- Philippe G. Ciarlet, A justification of the von Kármán equations, Arch. Rational Mech. Anal. 73 (1980), no. 4, 349–389. MR 569597, DOI https://doi.org/10.1007/BF00247674
- Melvyn S. Berger, On vonKármán’s equations and the buckling of a thin elastic plate. I. The clamped plate, Comm. Pure Appl. Math. 20 (1967), 687–719. MR 221808, DOI https://doi.org/10.1002/cpa.3160200405
G. Fichera, Existence theorems in elasticity, Encyclopedia of Physics (ed. by S. Flügge), Vol. VIa/2, Springer-Verlag, Berlin, 1972
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
- Jean-Pierre Aubin, Applied functional analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Translated from the French by Carole Labrousse; With exercises by Bernard Cornet and Jean-Michel Lasry. MR 549483
- J.-J. Moreau and P. D. Panagiotopoulos (eds.), Nonsmooth mechanics and applications, CISM International Centre for Mechanical Sciences. Courses and Lectures, vol. 302, Springer-Verlag, Vienna, 1988. MR 991345
P. D. Panagiotopoulos, Hemivariational inequalities in frictional contact problems and applications, Mechanics of Material Interfaces (ed. by A.P.S. Selvadurai and G. Z. Voyadjis), Elsevier Publ., Amsterdam, 1986 , pp. 25–42
P. D Panagiotopoulos and G. E. Stavroulakis; A Variational-hemivariational inequality approach to the laminated plate theory under subdifferential boundary conditions, Quart. Appl. Math. 43, 409–430 (1988)
P. D. Panagiotopoulos, Non-convex superpotentials in the sense of F. H. Clarke and applications, Mech. Res. Comm. 8, 335–340 (1981)
P. D. Panagiotopoulos, Non-convex energy functions. Hemivariational inequalities and substationarity principles, Acta Mechanica 42, 160–183 (1983)
P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Non-convex Energy Functions, Birkhäuser Verlag, Basel—Boston—Stuttgart, 1985 (Russian trans. MIR Publ. Moscow 1989)
G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (Engl, trans.: Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-Heidelberg-New York 1976)
G. Duvaut and J. L. Lions, Problèmes unilatéraux dans la théorie de la flexion forte des plaques (I). Le cas stationnarie, J. de Mécanique 13, 51–74 (1974)
M. Potier-Ferry, Problèmes semicoercifs. Applications aux plaques de von Kármán, J. Math. Pures et Appl. 53, 331–346 (1974)
J. J. Moreau, La notion de sur-potentiel et les liaisons unilatérales en élastostatique, C.R. Acad. Sc. Paris 267A, 954–957 (1968)
J. J. Moreau, P. D. Panagiotopoulos, and G. Strang, Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Boston—Basel, 1988
F. H. Clarke, Nonsmooth Analysis and Optimization, J. Wiley, New York, 1984
P. D. Panagiotopoulos, Une généralization non-convex de la notion du sur-potentiel et ses applications, C.R. Acad. Sc. Paris 296B, 580–584 (1983)
P. D. Panagiotopoulos, Hemivariational inequalities and substationarity in the static theory of von Kármán plates, ZAMM 65, 219–229 (1985)
L. D. Landau and E. M. Lifschitz, Theory of Elasticity, Pergamon Press, 1959
R. Jones, Mechanics of composite materials, McGraw Hill-Scripta Book Co., New York-Washington, 1975
J. Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proc. Amer. Math. Soc. 64, 277–282 (1977)
K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80, 102–129 (1981)
P. G. Ciarlet, A justification of the von Kármán equations, Arch. Rat. Mech. Anal. 73, 183–202 (1980)
M. S. Berger, On von Kármán’s equations and the buckling of a thin elastic plate, I. The Clamped plate, Comm. Pure Appl. Math. XX, 687–719 (1967)
G. Fichera, Existence theorems in elasticity, Encyclopedia of Physics (ed. by S. Flügge), Vol. VIa/2, Springer-Verlag, Berlin, 1972
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod/Gauthier-Villars, Paris, 1969
I. Ekeland and R. Temam, Convex Analysis and Variational problems, North-Holland, Amsterdam and American Elsevier, New York, 1976
J. P. Aubin, Applied Functional Analysis, J. Wiley, New York, 1979
J. J. Moreau and P. D. Panagiotopoulos (Eds.); Nonsmooth Mechanics and Applications, CISM Lect. Notes Vol. 302, Springer-Verlag, Wien—New York, 1988
P. D. Panagiotopoulos, Hemivariational inequalities in frictional contact problems and applications, Mechanics of Material Interfaces (ed. by A.P.S. Selvadurai and G. Z. Voyadjis), Elsevier Publ., Amsterdam, 1986 , pp. 25–42
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73V25,
49J40,
73B27,
73K10,
73K20
Retrieve articles in all journals
with MSC:
73V25,
49J40,
73B27,
73K10,
73K20
Additional Information
Article copyright:
© Copyright 1989
American Mathematical Society
