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Nonsmooth Mechanics. Convex and Nonconvex Problems

  • Jaroslav Haslinger
  • Markku Miettinen
  • Panagiotis D. Panagiotopoulos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 35)

Abstract

Nonlinear, multivalued and possibly nonmonotone relations arise in several areas of mechanics. A multivalued or complete relation is a relation with complete vertical branches. Boundary laws of this kind connect boundary (or interface) quantities. A contact relation or a locking mechanism between boundary displacements and boundary tractions in elasticity is a representative example. Material constitutive relations with complete branches connect stress and strain tensors, or, in simplified theories, equivalent stress and strain quantities. A locking material or a perfectly plastic one is represented by such a relation. The question of nonmonotonicity is more complicated. One aspect concerns nonmonotonicity of a constitutive or a boundary law. Certainly, at a local microscopic level a nonmonotone relation corresponds to an unstable material or boundary law. Examples from damage or fracture mechanics may be presented. On a macroscopic level the complete mechanical behaviour of structural components can be described with such nonmonotone and possibly multivalued relations. A typical example of this kind is the delamination process of a composite structure, where local delaminations, crack propagation and interface or crack contact effects lead to a sawtooth overall load-displacement relation (see Panagiotopoulos and Baniotopoulos, 1984, Mistakidis and Stavroulakis, 1998, Li and Carlsson, 1999). The latter relation is adopted here as a constitutive law for the study of the structure at a macroscopic level. Another reason for nonmonotonicity is the large displacement or deformation effects. Let us consider that there exists a convex deformation energy potential which is a function of some appropriate strain quantity. In a kinematically nonlinear mechanical theory the geometric compatibility relation, which connects strains with displacements of the structure, is nonlinear. Therefore the same potential energy, considered as a composite function of a convex function with a nonlinear relation, is, in general, nonconvex in the displacement variables.

Keywords

Variational Inequality Hemivariational Inequality Convex Case Unilateral Contact Strain Energy Density Function 
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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References

  1. Adly, S., Goeleven, D., and Théra, M. (1996). Recession mappings and non-coercive variational inequalities. Nonlinear Analysis Theory Methods and Applications, 26 (9): 1573–1604.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Dem’yanov, V. F., Stavroulakis, G. E., Polyakova, L. N., and Panagiotopoulos, P. D. (1996). Quasidifferentiability and nonsmooth modelling in mechanics, engineering and economics. Kluwer Academic, Dordrecht.Google Scholar
  3. Duvaut, G. and Lions, J. L. (1972). Les inéquations en méchanique et en physique. Dunod, Paris.Google Scholar
  4. Ekeland, I. and Temam, R. (1976). Convex analysis and variational problems. North-Holland, Amsterdam.zbMATHGoogle Scholar
  5. Goeleven, D. (1997). A bifurcation theory for nonconvex unilateral laminated plate problem formulated as a hemivariational inequality involving a potential operator. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 77 (1): 45–51.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Goeleven, D. and Mentagui, D. (1995). Well-posed hemivariational inequalities. Numerical Functional Analysis and Optimization, 16 (7–8): 909–921.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Goeleven, D., Miettinen, M., and Panagiotopoulos, P. D. (1999). Dynamic hemivariational inequalities and their applications. to appear in J. Opt. Theory Appl.Google Scholar
  8. Goeleven, D., Stavroulakis, G. E., and Panagiotopoulos, P. D. (1996). Solvability theory for a class of hemivariational inequalities involving copositive plus matrices. Applications in robotics. Mathematical Programming Ser. A, 75 (3): 441–465.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Goeleven, D. and Théra, M. (1995). Semicoercive variational hemivariational inequalities. Journal of Global Optimization, 6: 367–381.zbMATHCrossRefGoogle Scholar
  10. Haslinger, J., Baniotopoulos, C. C., and Panagiotopoulos, P. (1993). A boundary multivalued integral “equation” approach to the semipermeability problem. Applications of Mathematics, 38: 39–60.MathSciNetzbMATHGoogle Scholar
  11. Li, X. and Carlsson, L. A. (1999). The tilted sandwich debond (TSD) specimen for face/core interface fracture characterization. Journal of Sandwich Structures and Materials, 1: 60–75.CrossRefGoogle Scholar
  12. Miettinen, M. and Panagiotopoulos, P. D. (1999). On parabolic hemivariational inequalities and applications. Nonlinear Analysis, 35: 885–915.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Mistakidis, E. S. and Stavroulakis, G. E. (1998). Nonconvex optimization in mechanics. Smooth and nonsmooth algorithms, heuristics and engineering applications by the F.E.M. Kluwer Academic Publisher, Dordrecht, Boston, London.Google Scholar
  14. Moreau, J. and Panagiotopoulos, P. D., editors (1988). Nonsmooth mechanics and applications, volume 302 of CISM Lect. Notes, New York. Springer.Google Scholar
  15. Moreau, J. J., Panagiotopoulos, P. D., and Strang, G., editors (1988). Topics in nonsmooth mechanics, Basel-Boston. Birkhäuser.Google Scholar
  16. Motreanu, D. and Panagiotopoulos, P. (1998). Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Kluwer Academic Publisher, Dordrecht, Boston, London.Google Scholar
  17. Motreanu, D. and Panagiotopoulos, P. D. (1995). An eigenvalue problem for a hemivariational inequality involving a nonlinear compact operator. Set Valued Analysis, 3.Google Scholar
  18. Naniewicz, Z. and Panagiotopoulos, P. D. (1995). Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New York.Google Scholar
  19. Panagiotopoulos, P. D. (1983). Nonconvex energy functions. Hemivariational inequalities and substationary principles. Acta Mechanica, 42: 160–183.MathSciNetGoogle Scholar
  20. Panagiotopoulos, P. D. (1985). Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Birkhäuser, Basel, Boston, Stuttgart.Google Scholar
  21. Panagiotopoulos, P. D. (1988). Nonconvex superpotentials and hemivariational inequalities. Quasidifferentiability in mechanics. In Moreau, J. J. and Panagiotopoulos, P. D., editors, Nonsmooth Mechanics and Applications, number 302 in CISM Lect. Notes, New York.Google Scholar
  22. Panagiotopoulos, P. D. (1993). Hemivariational inequalities. Applications in mechanics and engineering. Springer, Berlin, Heidelberg, New York.Google Scholar
  23. Panagiotopoulos, P. D. (1995). Variational principles for contact problems including impact phenomena. In Raous, M., Jean, M., and Moreau, J., editors, Contact Mechanics,pages 431–440. Plenum.Google Scholar
  24. Panagiotopoulos, P. D. and Baniotopoulos, C. C. (1984). A hemivariational inequality and substationarity approach to the interface problem. Theory and prospects of applications. Engineering Analysis, 1: 20–31.CrossRefGoogle Scholar
  25. Panagiotopoulos, P. D. and Glocker., C. (1998). Analytical mechanics. addendum i: Inequality constraints with elastic impacts. the convex case. Zeitschrift far angewandte Mathematik und Mechanik (ZAMM), 78 (4): 219–229.MathSciNetCrossRefGoogle Scholar
  26. Panagiotopoulos, P. D. and Koltsakis, E. (1987). Interlayer slip and delamination effect. Proc. Canadian Soc. Mech. Eng., 11: 43–52.Google Scholar
  27. Panagiotopoulos, P. D. and Liolios, A. (1989). On the dynamic of inelastic shocks. a new approach. In Greek-German Seminar on Structural Dynamics and Earthquake Engineering,pages 12–18. Hellenic Society of Theoretical and Applied Mechanics.Google Scholar
  28. Panagiotopoulos, P. D. and Stavroulakis, G. E. (1988). Variational-hemivariational inequality approach to the laminated plate theory under subdifferential boundary conditions. Quarterly of Applied Mathematics, pages 409–430.Google Scholar
  29. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Panagiotopoulos, P. D. and Stavroulakis, G. E. (1992). New types of variational principles based on the notion of quasidifferentiability. Acta Mechanica, 94: 171–194.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Pfeiffer, F. and Glocker, C. (1996). Multibody dynamics with unilateral contacts. John Wiley, New York.zbMATHCrossRefGoogle Scholar
  32. Pop, G. and Panagiotopoulos, P. D. (1998). On a type of hyperbolic variationalhemivariational inequalities. preprint.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Jaroslav Haslinger
    • 1
  • Markku Miettinen
    • 2
  • Panagiotis D. Panagiotopoulos
    • 3
  1. 1.Charles UniversityCzech Republic
  2. 2.University of JyväskyläFinland
  3. 3.Aristotle UniversityGreece

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