Fractal Geometry in Contact Mechanics and Numerical Applications

  • P. D. Panagiotopoulos
  • O. K. Panagouli
Part of the International Centre for Mechanical Sciences book series (CISM, volume 378)


The contribution to the present volume deals with the study of the influence of fractal geometry on contact problems. After a short presentation of the new mathematical tools and methods used for the correct consideration of the fractal geometry we study unilateral contact and friction problems, adhesive contact problems in interfaces of fractal geometry and finally crack problems of fractal geometry. Numerical applications illustrate the uheory. This contribution contains also an advanced mathematical section concerning the nature of the forces on a fractal boundary.


Variational Inequality Crack Interface Besov Space Fractal Geometry Contact Mechanic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mandelbrot, B.: The Fractal Geometry of Nature, W.H. Freeman and Co, New York 1972.Google Scholar
  2. 2.
    Takayasu, H.: Fractals in the Physical Sciences, Manchester Univ. Press., Manchester 1990.MATHGoogle Scholar
  3. 3.
    Scholz, C.H., Mandelbrot, B.: Fractals in Geophysics, Birkhäuser, Boston 1989.Google Scholar
  4. 4.
    Le Méhauté, A.: Les Géométries Fractales, Lermes, Paris 1990.MATHGoogle Scholar
  5. 5.
    Barnsley, M. and Demko, S.: Chaotic Dynamics and Fractals, Academic Press, New York 1986.MATHGoogle Scholar
  6. 6.
    Feder, J.: Fractals, Plenum Press, New York 1988.CrossRefMATHGoogle Scholar
  7. 7.
    Artemiadis, N.: The geometry of fractals, Proc. Academy of Athens, 63 (1988), 479–500 (in Greek).Google Scholar
  8. 8.
    Barnsley, M.: Fractals Everywhere, Academic Press, New York 1988.MATHGoogle Scholar
  9. 9.
    Falconer, K.J.: The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge 1985.CrossRefMATHGoogle Scholar
  10. 10.
    Wallin, H.: Interpolating and orthogonal polynomials on fractals, Constr. Approx., 5 (1989), 137–150.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bell, J.F.: The experimental foundation of solid mechanics, in: Flügge, S. (ed.) Encyclopedia of Physics, Springer, Berlin 1973.Google Scholar
  12. 12.
    Jones, R.: Mechanics of Composite Materials, McGraw Hill, New York 1975.Google Scholar
  13. 13.
    Panagiotopoulos, P.D.: The mechanics of fractals, Proc. Academy of Athens, 65 (1990), 184–207 (in Greek).Google Scholar
  14. 14.
    Panagiotopoulos, P.D.: Fractals in Mechanics, in: Proc. 8th Confernece on the Trends in Applications of Pure Mathematics to Mechanics (STAMM 8 ), Longman Scientific and Technical Press, London 1990.Google Scholar
  15. 15.
    Panagiotopoulos, P.D.: On the fractal nature of mechanical theories, ZAMM, 70 (1990), 258–260.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Panagiotopoulos, P.D.: Fractals and Fractal Approximation in Structural Mechanics, in: Proc. of the Int. Conference on New Development in Structural Mechanics (in the memory of M. Romano), University of Catania, Catania 1990.Google Scholar
  17. 17.
    Hutchinson, J.F.: Fractals and selfsimilarity, Indiana Univ. J. of Math., 30 (1981), 713–747.MATHMathSciNetGoogle Scholar
  18. 18.
    Massopust, P.R.: Smooth interpolating curves and surfaces generated by iterated function systems, Zeitschrift für Analysis und ihre Anwendungen, 12 (1993), 201–210.MATHMathSciNetGoogle Scholar
  19. 19.
    Chadrasekhar, S.: Ellipsoidal Figures of Equilibrium, Dover Publ., N.York 1987.Google Scholar
  20. 20.
    Bell, J.F.: The experimental foundation of solid mechanics, in: Flügge, S. (ed.) Encyclopedia of Physics, Springer, Berlin 1973.Google Scholar
  21. 21.
    Panagiotopoulos P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Basel 1985. ( Russian Translation 1985: Moscow MIR Publ).CrossRefGoogle Scholar
  22. 22.
    Panagiotopoulos, P.D.: Nonconvex energy functions. Hemivariational inequalities and substationarity principles, Acta Mechanica, 42 (1983), 160–183.MathSciNetGoogle Scholar
  23. 23.
    Jonsson, A and Wallin, H.: Function Spaces on Subsets of Rn, Harwoord Acad. Publ. (Series: Math. Report Vol. 2), London 1984.Google Scholar
  24. 24.
    Wallin, H.: The trace to the boundary of Sobolev spaces on a snowflake, Rep. Dep. of Math. Univ. of Umea, (1989).Google Scholar
  25. 25.
    Panagiotopoulos, P.D., Panagouli, O.K. and Mistakidis, E.S.: Fractal geometry and fractal material behaviour in solids and structures, Archive of Applied Mechanics, 63 (1993), 1–24.CrossRefMATHGoogle Scholar
  26. 26.
    Jonsson, A. and Wallin, H.: The dual of Besov spaces on fractals, Res. Rep., 14 (1993), University of Umea, Dept. of Math. and studia Mathematica, 112 (1995), 285–300.MATHMathSciNetGoogle Scholar
  27. 27.
    Harrison, J. and Norton, A.: The Gauss–Green theorem for fractal boundaries, Duke Math. Journal, 67 (1992), 575–588.MATHMathSciNetGoogle Scholar
  28. 28.
    Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, Cambridge 1987.Google Scholar
  29. 29.
    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
  30. 30.
    Necas, J., Jarusek, J. and Haslinger, J.: On the solution of the variational inequality to the Signorini problem with small friction, Bulletino U.M.I., 17B (1980), 796–811.MATHMathSciNetGoogle Scholar
  31. 31.
    Panagiotopoulos, P.D.: A variational inequality approach to the friction problem of structures with convex energy density and application to the frictional unilateral contact problem, J. Struct. Mech., 6 (1978), 303–318.CrossRefGoogle Scholar
  32. 32.
    Zienkiewicz, O.C.: The Finite Element Method, McGraw Hill, London 1979.Google Scholar
  33. 33.
    Strang, G.: Introduction to Applied Mathematics, Wellesley-Cambridge Press, 1986.Google Scholar
  34. 34.
    Bisbos, C. and Mistakidis, E.: Application of the S.V.D. to the equilibrium and compatibility equations of matrix structural analysis, Proc. of the Aristotle University of Thessaloniki, 9 (1991).Google Scholar
  35. 35.
    Panagiotopoulos, P.D., Mistakidis, E.S. and Panagouli, O.K.: Fractal interfaces with unilateral contact and friction conditions, Computer Methods in Applied Mechanics and Engineering, 99 (1992), 395–412.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis, Prentice Hall, New Jersey 1982.Google Scholar
  37. 37.
    Antes, H. and Panagiotopoulos, P.D.: The Boundary Integral Approach to Static and Dynamic Contact Problems. Equality and Inequality Methods, Birkhäuser Verlag, Basel 1992.CrossRefMATHGoogle Scholar
  38. 38.
    Panagiotopoulos, P.D.: Hemivariational Inequalities and their Applications in Mechanics and Engineering, Springer Verlag, Berlin 1993.Google Scholar
  39. 39.
    Moreau, J.J., Panagiotopoulos, P.D. and Strang, G.: Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Basel 1988.MATHGoogle Scholar
  40. 40.
    Moreau, J.J. and Panagiotopoulos, P.D.: Nonsmooth Mechanics and Applications, in: Moreau, J.J. and Panagiotopoulos, P.D. (ed), CISM Lect. Notes, Vol. 302, Springer-Verlag, New York 1989.Google Scholar
  41. 41.
    Panagiotopoulos, P.D.: Coercive and semicoercive hemivariational inequalities, Nonlinear Analysis T.M.A., 16 (1991), 209–231.CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Pitman Publ. Ltd., London 1985.Google Scholar
  43. 43.
    Cruse, T.A.: Numerical solutions in three-dimensional elastostatics, Int. J. Solids and Struct., 5 (1969), 1259–1274.CrossRefMATHGoogle Scholar
  44. 44.
    Cruse, T.A. and Van Buren, W.: Three-dimensional elastic stress analysis of a fractured speciment with an edge crack, Intern. J. Fract. Mech., 7 (1971), 1–15.Google Scholar
  45. 45.
    Cruse, T.A.: Applications of the boundary integral equation method to three-dimensional stress analysis, Comp. and Struct., 3 (1973), 509–527.CrossRefGoogle Scholar
  46. 46.
    Blandford, G.E., Ingraffea, A.R. and Liggett, J.A.: Two dimensional stress intensity factor computations using the boundary element method, Int. J. for Numerical Methods in Engineering, 17 (1981), 387–404.CrossRefMATHGoogle Scholar
  47. 47.
    Pu, S.L., Hussain, M.A. and Lorensen, W.: International Journal of Numerical Methods in Engineering, 12 (1978), 1727–1742.MATHGoogle Scholar
  48. 48.
    Mistakidis, E.S. and Panagiotopoulos, P.D.: On the approximation of nonmonotone multivalued problems by monotone subproblems, Computer Methods in Applied Mechanics and Engineering, 114 (1994), 55–76.CrossRefMathSciNetGoogle Scholar
  49. 49.
    Panagiotopoulos,P.D.: Nonconvex Superpotentials and Hemivariational Inequalities. Quasidifferentiability in Mechanics, in: Moreau, J.J., Panagiotopoulos, P.D. (eds.) Nonsmooth Mechanics and Applications, CISM Lect. Notes 302, Springer, New York 1988.Google Scholar
  50. 50.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, J. Wiley, New York 1983.Google Scholar
  51. 51.
    Mistakidis, E.S. and Panagiotopoulos, P.D.: Numerical treatment of the non-monotone (zigzag) friction and adhesive contact problems with debonding. Approximation by monotone subproblems, Computer and Structures, 47 (1993), 33–46.CrossRefMATHGoogle Scholar
  52. 52.
    Fletcher, R.: Practical Methods of Optimization, 2nd edition; J. Wiley & Sons, Chichester-N.York-Brisbane-Toronto-Singapore 1990.Google Scholar
  53. 53.
    Duvaut, G. and Lions J.L.: Les Inéquations en Mécanique et en Physique, Dunod, Paris 1972.MATHGoogle Scholar
  54. 54.
    Theocharis, P.S., Panagiotopoulos, P.D.: On debonding effects in adhesively-bonded cracks. A boundary integral approach, Arch. of Appl. Mech., 61 (1991), 578–587.Google Scholar
  55. 55.
    Panagiotopoulos,P.D, Haslinger, J.: On the dual reciprocal variational approach to the Signorini-Fichera problem. Convex and nonconvex generalizations, ZAMM, 72 (1992), 497–506.CrossRefGoogle Scholar
  56. 56.
    Zaharav, D., Malah, Z. and Karnin, E.: Hierarchical Interpretation of Fractal Image Ccding and its Applications, in: Fractal Image Compression. Theory and Applications, Springer, New York 1994.Google Scholar

Copyright information

© Springer-Verlag Wien 1997

Authors and Affiliations

  • P. D. Panagiotopoulos
    • 1
    • 2
  • O. K. Panagouli
    • 1
  1. 1.Aristotle UniversityThessalonikiGreece
  2. 2.RWTHAachenGermany

Personalised recommendations