Nonsmooth Mechanics II

Variational Formulations using Quasidifferentiability
  • Vladimir F. Dem’yanov
  • Georgios E. Stavroulakis
  • Ludmila N. Polyakova
  • Panagiotis D. Panagiotopoulos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 10)


In the present Chapter we derive model variational problems in mechanics by using the notion of the quasidifferential. The corresponding problems are systems of variational inequalities. The link with the hemivariational inequalities, in the sense of Panagiotopoulos, which are based on the generalized gradient operator of Clarke, is briefly discussed. For static problems the classical minimum propositions for the potential and the complementary energy are extended to analogous inf—stationarity propositions. For more details the reader is referred to [31], [33], [34], [35], [36], [41] [37], [38], [42], [44], [28], [26], [27], [23], [25], [2], [47], [48], [49], [16], [17], [9], [1], [46].


Variational Inequality Variational Formulation Inequality Problem Duality Pairing Laminate Plate 
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  1. 1.
    Adly S. (1995), Analyse variationnelle appliquee aus problemes unilateraux, These. Faculte des Sciences de Limoges.Google Scholar
  2. 2.
    Antes H. and Panagiotopoulos P.D. (1992), The boundary integral approach to static and dynamic contact problems. Equality and inequality methods, Birkhauser Verlag, Basel, Boston.CrossRefzbMATHGoogle Scholar
  3. 3.
    Berger M.S. (1967), On von Karmans Equations and the buckling of a thin elastic plate. I. The clamped plate, Comm. Pure Appl. Math., XX, 687–719.CrossRefGoogle Scholar
  4. 4.
    Berger M.X. and Fife P.C. (1968), Von Karmans equations and the buckling of a thin elastic plate. II. Plate with general edge conditions, Comm. Pure Appl. Math., XXI, 227–241.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brezis H. (1972), Problemes unilateraux, J. Math, pures et Appl. 51, 1–168.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brezis H. (1973), Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland Publ. Co., Amsterdam and American Elsevier Publ. Co., New York.Google Scholar
  7. 7.
    Ciarlet P.G. (1980), A justification of the von Karman equations. Arch. Rat. Mech. Anal., 73, 349–389.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clarke F.H. (1983), Optimization and nonsmooth analysis, Wiley, New York.zbMATHGoogle Scholar
  9. 9.
    Dinca G., Panagiotopoulos P.D. and Pop G. (1995), Inegalites hemi-variationnelles semi-coercives sur des ensembles convexes. C.R. Acad. Sci. Paris, Serie I, 320, 1183–1186.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Duvaut G., and Lions J.L. (1971), Un Probleme d’ elasticity avec Frottement, J. de Mecanique 10, 409–420.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Duvaut G., and Lions J.L. (1972), Les Inequations en mecanique et en physique, Dunod, Paris.zbMATHGoogle Scholar
  12. 12.
    Duvaut, G. (1980), Equilibre d’ un solide elastique avec contact unilateral et frottement de Coulomb. C.R. Acad. Sc. Paris, 290, 263–265.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fichera G. (1972), Boundary value problems in elasticity with unilateral constraints. In: Encyclopedia of Physics (ed. by S. Fliigge) Vol. VI a/2, Springer-Verlag, Berlin.Google Scholar
  14. 14.
    Girkmann K. (1963), Flachentragwerke. Springer-Verlag, Wien.CrossRefGoogle Scholar
  15. 15.
    Jones R. (1975), Mechanics of composite materials. McGraw Hill, New York.Google Scholar
  16. 16.
    Karamanlis I. (1991), Buckling problems in composite von Kar“an plates. Doct. Thesis, Aristotle University Dept. of Civil Engineering, Thessaloniki.Google Scholar
  17. 17.
    Karamanlis I. and Panagiotopoulos P.D. (1992), The eigenvalue problem in hemi-variational inequalities and its application to composite plates. Journal of the Mech. Behaviour of Materials, Freund Publ. House, Tel Aviv (to appear).Google Scholar
  18. 18.
    Kufner A., John O. and Fucik S (1977), Function spaces. Nordhoff International Publ., Leyden.zbMATHGoogle Scholar
  19. 19.
    Lanczos C. (1966), The variational principles of mechanics. University of Toronto Press, Toronto.Google Scholar
  20. 20.
    Mistakidis E. and Panagiotopoulos P.D. (1993), On the numerical treatment of non-monotone (zigzag) friction and adhesive contact problems with debonding. Approximation by monotone subproblems, Computers and Structures, 47, 33–46.CrossRefzbMATHGoogle Scholar
  21. 21.
    Mistakidis E. and Panagiotopoulos P.D. (1994), On the approximation of nonmono-tone multivalued problems by monotone subproblems. Computer Methods in Applied Mechanics and Engineering, 114, 55–76.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Moreau J.J. and Panagiotopoulos P.D.(eds) (1988), Nonsmooth mechanics and applications, CISM Vol. 302, Springer Verlag, Wien.Google Scholar
  23. 23.
    Moreau J.J., Panagiotopoulos P.D. and Strang G.(eds) (1988), Topics in nonsmooth mechanics, Birkhauser Verlag, Basel, Boston.zbMATHGoogle Scholar
  24. 24.
    Moser K. (1992), Faserkunststoffverbund, VDI Verlag, Diisseldorf.Google Scholar
  25. 25.
    Motreanu D. and Panagiotopoulos P.D. (1993), Hysteresis: The eigenvalue problem for hemivariational inequalities. In: Models of hysteresis, ed. by A. Visintin, Pitman Research Notes in Mathematics, Longman, Harlow.Google Scholar
  26. 26.
    Naniewicz Z. (1989), On some nonconvex variational problems related to hemivariational inequalities. Nonlin. Anal. 13, 87–100.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Naniewicz Z. and Wozniak C.Z. (1989) On the quasi-stationary models of debonding processes in layered composites. Ing. Archiv, 60, 31–40.CrossRefzbMATHGoogle Scholar
  28. 28.
    Naniewicz Z. (1988), On some nonmonotone subdifferential boundary conditions in elastostatics. Ing. Archiv, 58, 403–412.CrossRefzbMATHGoogle Scholar
  29. 29.
    Naniewicz Z. and Panagiotopoulos P.D. (1995), Mathematical theory of hemivariational inequalities and applications. Marcel Dekker.Google Scholar
  30. 30.
    Naniewicz Z. (1995) Hemivariational inequalities with functional which are not locally Lipschitz. Nonlinear Analysis, Theory, Methods and Applications, 25 (12), 1307–1320.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Panagiotopoulos P.D. (1981), Non-convex superpotentials in the sense of F.H. Clarke and applications. Mech. Res. Comm., 8, 335–340.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Panagiotopoulos P.D. (1982), Non-convex energy functionals. application to non-convex elastoplasticity. Mech. Res. Comm., 9, 23–29.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Panagiotopoulos P.D. (1983), Nonconvex energy functions. Hemivariational inequalities and substationarity principles. Acta Mechanica, 42, 160–183.MathSciNetGoogle Scholar
  34. 34.
    Panagiotopoulos P.D. (1985), Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Birkhauser Verlag, Basel, Boston, Russian translation MIR Publ. Moscow 1989CrossRefzbMATHGoogle Scholar
  35. 35.
    Panagiotopoulos P.D. and Koltsakis E.K. (1987), Interlayer slip and delamination effect: A hemivariational inequality approach. Canadian Society for Mech. Engineering, IX, 43–52.Google Scholar
  36. 36.
    Panagiotopoulos P.D. (1987), Ioffe’s fans and unilateral problems: A new conjecture. In: Unilateral problems in structural analysis 2, eds. G. del Piero, F. Maceri, CISM Courses and Lectures 304, Springer Verlag, Wien, N.York.Google Scholar
  37. 37.
    Panagiotopoulos P.D. and Stavroulakis G.E. (1988), A variational hemivariational inequality approach to the laminated plate theory under subdifferential boundary conditions, Quarterly of Applied Mathematics, XLVI, 409–430.Google Scholar
  38. 38.
    Panagiotopoulos P.D. (1989), Semicoercive hemivariational inequalities. On the de-lamination of composite plates. Quart, of Appl. Math., XLVII, 611–629.Google Scholar
  39. 39.
    Panagiotopoulos P.D. (1985), Nonconvex problems of semipermeable media and related topics. ZAMM, 65, 29–36.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Panagiotopoulos P.D. and Koltsakis E.K. (1990), The nonmonotone skin effects in plane elasticity obeying to linear elastic and subdifferential material laws, ZAMM, 70, 13–21.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Panagiotopoulos P.D. (1988), Nonconvex superpotentials and hemivariational inequalities. Quasidiferentiability in mechanics, in Moreau, J.J. and Panagiotopoulos, P.D. (eds.), Nonsmooth Mechanics and Applications, CISM Lect. Nr. 302, Springer Verlag, Wien - New York.Google Scholar
  42. 42.
    Panagiotopoulos P.D. and Stavroulakis G.E. (1990), The delamination effect in laminated von Karman plates under unilateral boundary conditions. A variational–hemivariational inequality approach. Journal of Elasticity, 23, 69–96.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Panagiotopoulos P.D. (1991), Coercive and semicoercive hemivariational inequalities. Nonlinear Analysis. Theory Methods and Applications, 16, 209–231.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Panagiotopoulos P.D. and Stavroulakis G.E. (1992), New types of variational principles based on the notion of quasidifferentiability, Acta Mechanica 94, 171–194.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Panagiotopoulos P.D. (1993), Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag, Berlin - Heidelberg - New York.CrossRefzbMATHGoogle Scholar
  46. 46.
    Panagiotopoulos P.D. (1995), Hemivariational inequalities and fan-variational inequalities. New applications and results. Atti Sem. Mat. Fis. Univ. Modena, XLIII, 159–191.Google Scholar
  47. 47.
    Stavroulakis G.E. (1991), Analysis of structures with interfaces. Formulation and study of variational-hemivariational inequality problems, Ph.D. Dissertation, Aristotle University, Thessaloniki.Google Scholar
  48. 48.
    Stavroulakis G.E. (1993), Convex decomposition for nonconvex energy problems in elastostatics and applications, European Journal of Mechanics A/Solids 12 (1), 1–20.MathSciNetzbMATHGoogle Scholar
  49. 49.
    Stavroulakis G.E. and Panagiotopoulos P.D. (1993), Convex multilevel decomposition algorithms for non-monotone problems, Int. J. Num. Meth. Engng. 36, 1945–1966.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tonti E. (1973), A systematic approach to the search for Variational Principles. In: Variational Methods in Engineering, eds. C.A. Brebbia and H. Tottenham, Vol. I, Southampton Univ. Press, Southampton.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Vladimir F. Dem’yanov
    • 1
  • Georgios E. Stavroulakis
    • 2
  • Ludmila N. Polyakova
    • 1
  • Panagiotis D. Panagiotopoulos
    • 3
    • 4
  1. 1.Department of MathematicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Lehr- und Forschungsgebiet für Mechanik; Lehrstuhl C für MathematikRWTHAachenGermany
  3. 3.Department of Civil EngineeringAristotle UniversityThessalonikiGreece
  4. 4.Faculty of Mathematics and PhysicsRWTHAachenGermany

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