Computer Methods in Mechanics pp 43-58

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 1) | Cite as

Nonconvex Inequality Models for Contact Problems of Nonsmooth Mechanics

  • Stanislaw Migórski
  • Anna Ochal


This review paper deals with selected nonsmooth and nonconvex inequality problems for dynamic frictional contact between a viscoelastic body and a foundation. The process is modeled by general nonmonotone possibly multivalued multidimensional Clarke subdifferential contact boundary conditions. The problems of frictional contact with both short and long memory, thermoviscoelastic frictional contact, bilateral frictional contact and bilateral contact for piezoelectric materials with adhesion are considered. The formulations and results on existence, and uniqueness of solutions are presented.


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  1. 1.
    Bartosz, K.: Hemivariational inequality approach to the dynamic viscoelastic sliding contact problem with wear. Nonlinear Anal. 65, 546–566 (2006)CrossRefGoogle Scholar
  2. 2.
    Bartosz, K.: Hemivariational inequalities modeling dynamic contact problems with adhesion. Nonlinear Anal. 71, 1747–1762 (2009)CrossRefGoogle Scholar
  3. 3.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Interscience, New York (1983)Google Scholar
  4. 4.
    Denkowski, Z., Migórski, S.: Sensitivity of optimal solutions to control problems for systems described by hemivariational inequalities. Control Cybernet. 33, 211–236 (2004)Google Scholar
  5. 5.
    Denkowski, Z., Migórski, S.: A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60, 1415–1441 (2005)CrossRefGoogle Scholar
  6. 6.
    Denkowski, Z., Migórski, S.: On sensitivity of optimal solutions to control problems for hyperbolic hemivariational inequalities. Lect. Notes Pure Appl. Math. 240, 145–156 (2005)Google Scholar
  7. 7.
    Denkowski, Z., Migórski, S., Ochal, A.: Existence and uniqueness to a dynamic bilateral frictional contact problem in viscoelasticity. Acta Appl. Math. 94, 251–276 (2006)CrossRefGoogle Scholar
  8. 8.
    Denkowski, Z., Migórski, S., Ochal, A.: Optimal control for a class of mechanical thermoviscoelastic frictional contact problems. Control Cybernet. 36, 611–632 (2007)Google Scholar
  9. 9.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)Google Scholar
  10. 10.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)Google Scholar
  11. 11.
    Duvaut, G., Lions, J.-L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972)Google Scholar
  12. 12.
    Frémond, M.: Adhérence des solides. J. Mécanique Théorique et Appliquée 6, 383–407 (1987)Google Scholar
  13. 13.
    Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society, International Press, Providence (2002)Google Scholar
  14. 14.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)Google Scholar
  15. 15.
    Kulig, A.: Evolution Inclusions and Hemivariational Inequalities for Unilateral Contact in Viscoelasticity. PhD Thesis, Jagiellonian University, Krakow (2009)Google Scholar
  16. 16.
    Li, Y., Liu, Z.: Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction. Nonlinear Analysis, Theory, Methods and Applications 71, 1414–1424 (2009)CrossRefGoogle Scholar
  17. 17.
    Liu, Z., Migórski, S.: Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete Contin. Dyn. Syst. Ser. B 9, 129–143 (2008)Google Scholar
  18. 18.
    Liu, Z., Migórski, S., Ochal, A.: Homogenization of boundary hemivariational inequalities in linear elasticity. J. Math. Anal. Appl. 340, 1347–1361 (2008)CrossRefGoogle Scholar
  19. 19.
    Migórski, S.: Homogenization technique in inverse problems for boundary hemivariational inequalities. Inverse Problems in Eng. 11, 229–242 (2003)CrossRefGoogle Scholar
  20. 20.
    Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669–699 (2005)CrossRefGoogle Scholar
  21. 21.
    Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim. 31, 505–533 (2005)CrossRefGoogle Scholar
  22. 22.
    Migórski, S.: Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comput. Math. Appl. 52, 677–698 (2006)CrossRefGoogle Scholar
  23. 23.
    Migórski, S.: Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete Contin. Dyn. Syst. 6, 1339–1356 (2006)Google Scholar
  24. 24.
    Migórski, S., Ochal, A.: Hemivariational inequality for viscoelastic contact problem with slip dependent friction. Nonlinear Anal. 61, 135–161 (2005)CrossRefGoogle Scholar
  25. 25.
    Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier-Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)CrossRefGoogle Scholar
  26. 26.
    Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elasticity 83, 247–275 (2006)CrossRefGoogle Scholar
  27. 27.
    Migórski, S., Ochal, A.: Existence of solutions for second order evolution inclusions with application to mechanical contact problems. Optimization 55, 101–120 (2006)CrossRefGoogle Scholar
  28. 28.
    Migórski, S., Ochal, A.: Vanishing viscosity for hemivariational inequality modeling dynamic problems in elasticity. Nonlinear Anal. 66, 1840–1852 (2007)CrossRefGoogle Scholar
  29. 29.
    Migórski, S., Ochal, A.: Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal. 69, 495–509 (2008)CrossRefGoogle Scholar
  30. 30.
    Migórski, S., Ochal, A.: Quasistatic hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)CrossRefGoogle Scholar
  31. 31.
    Migórski, S., Ochal, A.: An inverse coefficient problem for a parabolic hemivariational inequality. Appl. Anal. (in press, 2009)Google Scholar
  32. 32.
    Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008)CrossRefGoogle Scholar
  33. 33.
    Migórski, S., Ochal, A., Sofonea, M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal. 70, 3738–3748 (2009)CrossRefGoogle Scholar
  34. 34.
    Migórski, S., Ochal, A., Sofonea, M.: Weak solvability of a piezoelectric contact problem. Eur. J. Appl. Math. 20, 145–167 (2009)CrossRefGoogle Scholar
  35. 35.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York (1995)Google Scholar
  36. 36.
    Ochal, A.: Existence results for evolution hemivariational inequalities of second order. Nonlinear Anal. 60, 1369–1391 (2005)CrossRefGoogle Scholar
  37. 37.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985)Google Scholar
  38. 38.
    Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)Google Scholar
  39. 39.
    Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer, Berlin (2004)Google Scholar
  40. 40.
    Sofonea, M., Han, W., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman-Hall/CRC Press, New York (2006)Google Scholar
  41. 41.
    Sofonea, M., Matei, A.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Springer, New York (2009)Google Scholar
  42. 42.
    Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact, friction and wear. Internat. J. Solids Structures 33, 1817–1836 (1996)CrossRefGoogle Scholar
  43. 43.
    Wriggers, P.: Computational Contact Mechanics. Wiley, Chichester (2002)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Stanislaw Migórski
    • 1
  • Anna Ochal
    • 1
  1. 1.Institute of Computer Science, Faculty of Mathematics and, Computer ScienceJagiellonian UniversityKrakowPoland

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