Nonconvex Superpotentials and Hemivariational Inequalities. Quasidifferentiability in Mechanics

  • P. D. Panagiotopoulos
Part of the International Centre for Mechanical Sciences book series (CISM, volume 302)


These lectures concern the study of mechanical problems involving non-convex and possibly nondifferentiable energy functions, the superpotentials. First superpotentials connected with the notion of generalized gradient are considered, then the V-superpotentials are defined and finally the quasidifferential. After the defintion and formulation of several classes of mechanical problems involving superpotentials we study the questions of existence and approximation of the solution of a variationalhemivariational inequality resulting in the delamination theory of von Kármán laminated plates. Then a general semicoercive hemivariational inequality is studied and necessary and sufficient conditions are derived. The last section is devoted to the new notion of quasidifferentiability and its application to the study of mechanical problems.


Variational Inequality Generalize Gradient Hemivariational Inequality Sublinear Function Convex Hausdorff Topological Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Mem. Accad. Naz. Lincei, VIII 7 (1964), 91–140.MATHMathSciNetGoogle Scholar
  2. [2]
    Fichera, G.: Existence Theorems in Elasticity. In: Encyclopedia of Physics (ed. by S. Flügge) Vol. VIa/2, Springer-Verlag, Berlin 1972.Google Scholar
  3. [3]
    Lions, J.L. and G. Stampacchia: Variational Inequalities, Comm. pure and applied Math., XX (1967), 493–519.CrossRefMathSciNetGoogle Scholar
  4. [4]
    Moreau, J.J.: La notion de sur-potentiel et les liaisons unilatérales en élastostatique, C.R. Acad. Sc. Paris, 267A (1968), 954–957.MATHMathSciNetGoogle Scholar
  5. [5]
    Panagiotopoulos, P.D.: Non-convex Superpotentials in the Sence of F.H. Clarke and Applications, Mech. Res. Comm, 8 (1981), 335–340.MATHMathSciNetGoogle Scholar
  6. [6]
    Panagiotopoulos, P.D.: Nonconvex Energy Functions. Hemivariational Inequalities and Substationarity Principles, Acta Mechanica, 42 (1983), 160–183.MathSciNetGoogle Scholar
  7. [7]
    Duvaut, G. and J.L. Lions: Les inéquations en Mécanique et en Physique, Dunod, Paris 1972.MATHGoogle Scholar
  8. [8]
    Baiocchi, C. and A. Capelo: Variational and Quasivariational Inequalities, J. Wiley, New York 1984.Google Scholar
  9. [9]
    Kinderlehrer, D. and G. Stampacchia: An Introduction to Variational Inequalities and their Applications, Academic Press, New York 1980.MATHGoogle Scholar
  10. [10]
    Friedman, A.: Variational Principles and Free Boundary Problems, J. Wiley, New York 1982.Google Scholar
  11. [11]
    Rodriguez, J.F.: Obstacle Problems in Mathematical Physics, North Holland, Amsterdam 1987.Google Scholar
  12. [12]
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Boston-Basel 1985 (Rus. Transl. M/R Pub., Moscow 1987 ).Google Scholar
  13. [13]
    Clarke, F.H.: Optimization and Nonsmooth Analysis, J. Wiley, New York 1983.Google Scholar
  14. [14]
    Rockafellar, R.T.: Generalized Directional Derivatives and Sub-gradients of Non-convex Functions, Can. J. Math., XXXII (1980), 257–280.MathSciNetGoogle Scholar
  15. [15]
    Rockafellar, R.T.: La théorie des sous-gradients et ses applications à 1’optimization. Fonctions convexes et non-convexes, Les Presses de l’Université de Montréal, Montréal 1979.Google Scholar
  16. [16]
    Hamel, G.: Theoretische Mechanik, Springer-Verlag, Berlin 1967.MATHGoogle Scholar
  17. [17]
    Floegl, H. and H.A. Mang: Tension Stiffening Concept Based on Bond Slip, ASCE, ST 12, 108 (1982), 2681–2701.Google Scholar
  18. [18]
    Panagiotopoulos, P.D.: Hemivariational Inequalities in Frictional Contact Problems and Applications. In: Mech. of Material Interfaces (ed. by A.P.S. Selvadurai, G.Z. Voyatzis ), Elsevier Sc. Publ., Amsterdam 1986.Google Scholar
  19. [19]
    Pires, E.B. and J.T. Oden: Analysis of Contact Problems with Friction Under Oscillating Loads,Comp. Meth. Appl. Mech. Eng., 39 (1983), 337–362.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    Green, A.E. and P.M. Naghdi: A General Theory of an Elastic-Plastic Continuum, Arch. Rat. Mech. Anal., 18 (1965), 251–281.CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    Halphen, B. and N.Q. Son: Sur les Matériaux Standards Généralisés, J. de Mécanique, 14 (1975), 39–63.MATHGoogle Scholar
  22. [22]
    Germain, P.: Cours de Mécanique des milieux continus I, Masson, Paris 1973.MATHGoogle Scholar
  23. [23]
    Panagiotopoulos, P.D. and E.K. Koltsakis: Interlayer Slip and Delaminàtion Effect: A Hemivariational Inequality Approach, Trans. of the C.S.M.E., 11 (1987), 43–52.Google Scholar
  24. [24]
    Baniotopoulos, C.: Analysis of Structures with Complete Stress-strain Laws, Doct. Thesis, Aristotle Univ., Dept. Civil Eng., Thessaloniki 1985.Google Scholar
  25. [25]
    Panagiotopoulos, P.D. and C. Baniotopoulos: A Hemivariational Inequality and Substationarity Approach to the Interface Problem: Theory and Prospects of Applications, Eng. Anal., 1 (1984), 20–31.CrossRefGoogle Scholar
  26. [26]
    Holister, G.S. and C. Thomas: Fibre Reinforced Materials, Elsevier, London 1966.Google Scholar
  27. [27]
    Green, A.K. and W.H. Bowyer: The Testing and Analysis of Novel Top- Hat Stiffener Fabrication Methods for Use in GRP Ships, Proc. 1st Int. Conf. on Composite Structures (ed. by I.H. Marshall ), Applied Science Publishers, London 1981.Google Scholar
  28. [28]
    Williams, J.G. and M.D. Rhodes: Effect of Resin on Impact Damage Tolerance of Graphite/Epoxy Laminates, Proc. 6th Int. Conf. on Composite Materials, Testing and Design, ASTM STP 787 (ed. by I.M. Daniel ), ASTM, Philadelphia 1982.Google Scholar
  29. [29]
    Schwartz, M.M.: Composite Materials Handbook, McGraw-Hill, New York 1984.Google Scholar
  30. [30]
    Baniotopoulos, C.C. and P.D. Panagiotopoulos: A Hemivariational Inequality Approach to the Analysis of Composite Material Structures, Proc. Conf. Composite Structures Patras, Omega Press, London 1987.Google Scholar
  31. [31]
    Dunford, N. and J. Schwartz: Linear Operators, Part I: General Theory, Interscience Publ., New York 1966.Google Scholar
  32. [32]
    Sheikh, A. and S.M. Uzumeri: Analytical model for concrete confinement in tied columns, J. Struct. Div. ASCE, 108 (ST12) (1982), 2703.Google Scholar
  33. [33]
    Emori, K. and W.C. Schnobrick: Inelastic behavior of concrete frame-wall structures, J.Struct. Div. ASCE, 107 (ST1) (1981), 145.Google Scholar
  34. [34]
    Gilbert, J. and R.F. Warner: Tension stiffening in reinforced concrete slabs, J. Struct. Div. ASCE, 104 (ST12) (1978), 1885.Google Scholar
  35. [35]
    Hutchinson, J.W.: Micromechanics of Damage in Deformation and Fracture, Edition of Solid Mech. Dept., TU, Denmark 1987.Google Scholar
  36. [36]
    Panagiotopoulos, P.D.: Ioffe’s Fans and Unilateral Problems: A New Conjecture, Proc. 3nd Conf. Unil. Problems CISM, Springer-Verlag, Wien 1987.Google Scholar
  37. [37]
    Klarbring, A.: Contact Problems in Linear Elasticity, Linköping Studies in Science and Technology, Dissertation No. 133, Linköping 1985.Google Scholar
  38. [38]
    Panagiotopoulos, P.D.: Dynamic and Incremental Variational Inequality Principles, Differential Inclusions and their Applications to Co-Existent Phases Problems, Acta Mechanica, 40 (1981), 85–107.CrossRefMATHMathSciNetGoogle Scholar
  39. [39]
    Nitsiotas, G.: Die Berechnung statisch unbestimmter Tragwerke mit einseitigen Bindungen, Ing. Archiv, 41 (1971), 46–60.MATHGoogle Scholar
  40. [40]
    Panagiotopoulos, P.D.: A Variational Inequality Approach to the Inelastic Stress-Unilateral Analysis of Cable-Structures, Comp. and Struct., 6 (1976), 133–139.CrossRefMATHGoogle Scholar
  41. [41]
    Mitsopoulou, E.: Unilateral Contact, Dynamic Analysis of Beams by a Time-stepping Quadratic Programming Procedure, Meccanica, 18 (1983), 254–265.CrossRefMATHGoogle Scholar
  42. [42]
    Doudoumis, I.N.: Modelling “infill finite elements” with unilateral Interface conditions and general nonlinear constitutive laws, Doct. Thesis, Aristotle Univ., Dept. Civil Eng., Thessaloniki 1988 (to appear).Google Scholar
  43. [43]
    Mitsopoulou, E.N. and I.N. Doudoumis: A Contribution to the Analysis of Unilateral Contact Problems with Friction, S.M. Archives, 12 (1987), 165–186.MATHGoogle Scholar
  44. [44]
    Crisfield, M.A. and J. Wills: Solution Strategies and Softening Materials, Comp. Meth. Appl. Mech. Eng., 66 (1988), 267–289.CrossRefMATHGoogle Scholar
  45. [45]
    Strodiot, J.J. and V.H. Nguyen: On the Numerical Treatment of the Inclusion OEaf(x). In: Topics in Nonsmooth Mechanics (ed. by J.J. Moreau, P.D. Panagiotopoulos, G. Strang ), Birkhäuser Verlag, Basel 1988.Google Scholar
  46. [46]
    Panagiotopoulos, P.D.: Nonconvex Problems of Semipermeable Media and Related Topics, ZAMM, 65 (1985), 29–36.ADSCrossRefMATHMathSciNetGoogle Scholar
  47. [47]
    Panagiotopoulos, P.D.: Hemivariational Inequalities and Substationarity Principles in the Static Theory of von Kârmân Plates, ZAMM, 65 (1985), 219–229.ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    Panagiotopoulos, P.D. and G. Stavroulakis: A Variational-Hemivariational Inequality Approach to the Laminated Plate Theory under Subdifferential Boundary Conditions, Q. of Appl. Math. (to appear).Google Scholar
  49. [49]
    Panagiotopoulos, P.D.: A Nonlinear Programming Approach to the Unilateral Contact-and Friction-Boundary Value Problem in the Theory of Elasticity, Ing. Archiv, 44 (1975), 421–432.MATHMathSciNetGoogle Scholar
  50. [50]
    NeEas, J., Jaruek, J. and J. Haslinger: On the Solution of the Variational Inequality to the Signorini Problem with Small Friction, Bulletino U.M.I., 17B (1980), 796–811.Google Scholar
  51. [51]
    Panagiotopoulos, P.D.: Variational Inequalities and Multilevel Optimization Techniques in the Theory of Plasticity, Comp. and Struct., 8 (1978), 649–650.CrossRefMATHGoogle Scholar
  52. [52]
    Koltsakis, E.K.: Doct. Thesis, Aristotle Univ., Dept. Civil Eng., Thessaloniki (to appear).Google Scholar
  53. [53]
    Panagiotopoulos, P.D. and E.K. Koltsakis: Hemivariational Inequalities for Linear and Nonlinear Elastic Materials, Meccanica, (1987), (to appear).Google Scholar
  54. [54]
    Jones, R.: Mechanics of composite materials, McGraw Hill, New York 1975.Google Scholar
  55. [55]
    Stavroulakis, G. and P.D. Panagiotopoulos: Laminated Orthotropic Plates under Subdifferential Boundary Conditions. A Hemivariational Inequality Approach, ZAMM, (to appear).Google Scholar
  56. [56]
    Panagiotopoulos, P.D. and G. Stavroulakis: A Hemivariational Inequality Approach to the Delamination Effect in the Theory of Layered Plates, Archives of Mechanics, (to appear).Google Scholar
  57. [57]
    Naniewicz, Z. and C.Z. Wozniak: On the quasi-stationary models of debonding processes in layered composites, Ing. Archiv, (to appear).Google Scholar
  58. [58]
    Naniewicz, Z.: On some nonmonotone subdifferential boundary conditions in Elastostatics, J. Eng. Math., (to appear).Google Scholar
  59. [59]
    Naniewicz, Z.: On some Nonconvex Variational Problems related to Hemivariational Inequalities Nonlinear Analysis and Applications, (to appear).Google Scholar
  60. [60]
    Chang, K.C.: Variational Methods for Non-differentiable Functionals and their Applications to Partial Differential Equations, J. Math. Anal. Appl., 80 (1981), 102–129.CrossRefMATHMathSciNetGoogle Scholar
  61. [61]
    Rockafellar, R.T.: Convex Analysis, Princeton Univ. Press, Princeton 1970.Google Scholar
  62. [62]
    Ekeland, I. and R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam and American Elsevier, New York 1976.Google Scholar
  63. [63]
    Panagiotopoulos, P.D. and G.E. Stavroulakis: The Delamination Effect in Laminated v.Kârmân Plates under Unilateral Boundary Conditions. A Variational-Hemivariational Inequality Approach, J. of Elasticity, (to appear).Google Scholar
  64. [64]
    Berger, M.S. and P.C. Fife: Von Kârmân’s Equations and the Buckling of a Thin Elastic Plate II. Plate with General Edge Conditions, Comm. Pure Appl. Math., XXI (1968), 227–241.MathSciNetGoogle Scholar
  65. [65]
    Strauss, W.: On weak solutions of semilinear hyperbolic equations, An. Acad. Brasil. Ci., 42 (1970), 645–651.MATHMathSciNetGoogle Scholar
  66. [66]
    Rauch, J.: Discontinuous semilinear differential equations and multiple valued maps. Proc. A.M.S., 64 (1977), 277–282.CrossRefMATHMathSciNetGoogle Scholar
  67. [67]
    Aubin, J.P.: Applied Functional Analysis, J. Wiley, New York 1979.Google Scholar
  68. [68]
    Landesman, E.M. and A.C. Lazer: Nonlinear Perturbations of Linear Elliptic Boundary Value Problems at Resonance, J. Math. Mech., 19 (1970), 609–623.MATHMathSciNetGoogle Scholar
  69. [69]
    McKenna, P.J. and R. Rauch: Strongly Nonlinear Perturbations of Nonnegative Boundary Value Problems with Kernel, J. Diff. eq., 28 (1978), 253–265.ADSCrossRefMATHMathSciNetGoogle Scholar
  70. [70]
    Warga, J.: Derivate Containers, Inverse Functions and Controllability, In: Calculus of Variations and Control Theory (ed. by D.L. Russell ), Acad. Press, New York 1976.Google Scholar
  71. [71]
    Demyanov, V.F. and A.M. Rubinov: On Quasidifferentiable Mappings, Math. Operationsforsch. u.Statist. Ser. Optimization, 14 (1983), 3–21.CrossRefMATHGoogle Scholar
  72. [72]
    Demyanov, V.F. and L.C.W. Dixon (eds): Quasidifferential Calculus, Math. Progr. St., 29 (1986)Google Scholar
  73. [73]
    Demyanov, V.F. and L.V. Vasil’ev: Nondifferentiable Optimization, Optimization Software Inc., New York 1985.CrossRefMATHGoogle Scholar
  74. [74]
    Demyanov, V.F., Polyakova, L.N. and A.M. Rubinov: Nonsmoothness and Quasidifferentiability, Math. Progr. Study, 29 (1986), 1–19.CrossRefMATHMathSciNetGoogle Scholar
  75. [75]
    Shapiro, A.: On Optimality conditions in quasidifferentiable optimization, SIAM J. Control Opt., 22 (1984), 610–617.CrossRefMATHGoogle Scholar
  76. [76]
    Demyanov, V.F. and A.M. Rubinov: On quasidifferentiable functionals, Soviet Math. Doklady, 21 (1980), 13–17.Google Scholar
  77. [77]
    Demyanov, V.F., Polyakova, L.N. and A.M. Rubinov: On one generalization of the concept of subdifferential. In: All Union Conf. on Dynamic Control, Abstracts and Reports, Sverdlovsk 1979.Google Scholar
  78. [78]
    Polyakova, L.N.: On minimizing the sum of a convex function and a concave function, Math. Progr. Study, 29 (1986), 69–73.CrossRefMATHMathSciNetGoogle Scholar
  79. [79]
    Kiwiel, K.C.: A linearization method for minimizing certain quasi-differentiable functions, Math. Proar. Study, 29 (1986), 85–94.CrossRefMATHMathSciNetGoogle Scholar
  80. [80]
    Râdström, H.: An embedding theorem for spaces on convex sets, Proc. AMS, 3 (1952), 165–169.MATHGoogle Scholar
  81. [81]
    Ball, J.M.: Convexity Conditions and Existence Theorems in Nonlinear Elasticity, Arch. Rational Mech. Anal., 63 (1977), 337–403.MATHGoogle Scholar
  82. [82]
    Ciarlet, P.G. and J. Necas: Unilateral Problems in Nonlinear Three-Dimensional Elasticity, Arch. Rational Mech. Anal., 87 (1985), 319338.Google Scholar
  83. [83]
    Ciarlet, P.G. and J. Necas: Injectivity and Self-Contact in Nonlinear Elasticity, Arch. Rational Mech. Anal., 97 (1987), 171–188.MathSciNetGoogle Scholar
  84. [84]
    Demyanov, V.F.: Quasidifferentiable functions: Necessary conditions and descent directions, Math. Progr. Study, 29 (1986), 20–43.CrossRefMATHMathSciNetGoogle Scholar
  85. [85]
    Rubinov, A.M. and A.A. Yagubov: The space of star-shaped sets and its applications in nonsmooth optimization, Math. Progr. Study, 29 (1986), 176–202.CrossRefMATHMathSciNetGoogle Scholar
  86. [86]
    Germain, P.: La méthode de puissances virtuelles en mécanique des milieux continus, 1ère partie. Théorie du second gradient, J. de Mécanique, 12 (1973), 235–274.MATHMathSciNetGoogle Scholar
  87. [87]
    Salencon, J. and A. Tristan-Lopez: Analyse de la stabilité des talus en sois cohérents anisotropes, C.R. Acad. Sc. Paris, 290B (1980), 493–496.Google Scholar

Copyright information

© Springer-Verlag Wien 1988

Authors and Affiliations

  • P. D. Panagiotopoulos
    • 1
    • 2
  1. 1.Aristotle UniversityThessalonikiGreece
  2. 2.R.W.T.H.AachenGermany

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