Classical and Unilateral Contact Analysis in Statics and Dynamics

  • Georgios E. Stavroulakis
  • Heinz Antes
Part of the International Centre for Mechanical Sciences book series (CISM, volume 440)


Classical and unilateral contact analysis problems in statics and dynamics are considered. Emphasis is posed on the combination of effective tools from the general theory of nonsmooth mechanics with more classical boundary and finite element methods in statics and dynamics. Unilateral contact interaction belongs to the class of unknown, load-dependent nonlinearity. The main difficulty is the either contact -or separation nature of this nonliner joint. It is shown that elements of nonsmooth analysis, which in the most simple case leads to complementarity relations, can be used for the modelling and the subsequent numerical solution of the mechanical problem. Similar nonlinearities arise in frictional contact problems (in this case, the either-or decision modells the stick-slip effects) or in adhesive contact problems (with applications on delamination modelling, ductile crack modelling etc). Boundary or interface nonlinearities are suitable for boundary element methods. This explains the choice of this method in this chapter. Discussion of further models and extensions, as well as references to appropriate literature, are included.


Variational Inequality Boundary Element Boundary Element Method Contact Problem Complementarity Problem 
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. Adly, S. and Goeleven, D. (2000): A discretization theory for a class of semi-coercive unilateral problems. Numerische Mathematik, 87: 1–34.MathSciNetMATHGoogle Scholar
  2. Al- Fahed, A.M., Stavroulakis, G.E., and Panagiotopoulos, P.D. (1991): Hard and soft fingered robot grippers. ZAMM, Z. Angew. Math. Mech., 71: 257–266.MathSciNetMATHGoogle Scholar
  3. Al-Fahed, A.M., Stavroulakis, G.E., and Panagiotopoulos, P.D. (1992): A linear complementarity approach to the frictionless gripper problem. International Journal of Robotics Research, 11: 112–122.Google Scholar
  4. Al- Fahed, A.M., and Stavroulakis, G.E. (2000): A complementarity problem formulation of the frictional grasping problem. Computer Methods in Applied Mechanics and Engineering, 190: 941–952.MATHGoogle Scholar
  5. Antes, H. (1998): Anwendungen der Methode der Randelemente in der Elastodynamik und der Fluiddy-namik. B.G. Teubner Verlag, Stuttgart.Google Scholar
  6. Antes, H., and Panagiotopoulos, P.D. (1992): The Boundary Integral Approach to Static and Dynamic Contact Problems. Equality and Inequality Methods. Birkhäuser, BaselBoston- Berlin.MATHGoogle Scholar
  7. Antes, H., and Steinfeld, B. (1991): Unilateral contact with friction by time domain BEM. In P. Wriggers and W. Wagner, editors, Nonlinear Computational Mechanics—State of the Art, pages 193–211, Berlin Heidelberg, Springer Verlag.Google Scholar
  8. Antes, H., and Steinfeld, B. (1992): A boundary formulation study of massive structures static and dynamic behaviour. In 14th Boundary Element International Conference. Sevilla, Spain.Google Scholar
  9. Antes, H., Steinfeld, B., and Tröndle, G. (1991): Recent developments in dynamic stress analyses by time domain BEM. Engineering Analysis with Boundary Elements, 8: 176–184.Google Scholar
  10. Baggio, C. (2000): Collapse behaviour of three-dimensional brick-block systems using non-linear programming. Structural Engineering and Mechanics 10: 181–195.Google Scholar
  11. Baiocchi, C., and Capelo, A. (1984): Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. J. Wiley and Sons, Chichester.MATHGoogle Scholar
  12. Barber, J.R., and Ciavarella, M. (2000): Contact mechanics. International Journal of Solids and Structures, 37: 29–43MathSciNetMATHGoogle Scholar
  13. Bathe, K. J. (1996): Finite Element Procedures. Prentice- Hall, New Jersey.Google Scholar
  14. Bathe, K.J., and Bouzinov, P.A. (1997): On the constraint function method for contact problems. Computers and Structures, 64: 1069–1086.MATHGoogle Scholar
  15. Billups, S.C., and Murty, K.G. (2000): Complementarity problems. Journal of Computational and Applied Mathematics, 124: 303–318, 2000.MathSciNetGoogle Scholar
  16. Blanze, C., Champaney, L., Cognard, J.-Y., and Ladeveze, P. (1996): A modular approach to structure assembly computations. Application to contact problems. Engineering Computations, 1: 15–32.MathSciNetGoogle Scholar
  17. Bolzon, G., Ghilotti, D., and Maier, G. (1997): Parameter identification of the cohesive crack model. In H. Sol and C.W.J. Oomens, editors, Material identification using mixed numerical experimental methods, pages 213–222, Dordrecht, Kluwer Academic Publishers.Google Scholar
  18. Bolzon. G., and Maier, G. (1998): Identification of cohesive crack models for concrete on the basis of three-point-bending tests. In R. deBorst, N. Bicanic, H. Mang, and G. Meschke, editors, Computational Modelling of Concrete Structures, pages 301–310, Dordrecht, Kluwer Academic Publishers.Google Scholar
  19. Bolzon, G., Maier, G. and Tin-Loi, F. (1997): On multiplicity of solutions in quasibrittle fracture computations. Computational Mechanics, 19:511–516.MATHGoogle Scholar
  20. Brogliato, B. (1999): Nonsmooth Mechanics. Models, Dynamics and Control. Springer Verlag, Berlin, second edition.MATHGoogle Scholar
  21. Chabrand, P., Dubois, F., and Raous, M. (1998): Various numerical methods for solving unilateral contact problems with friction. Mathematical and Computer Modelling, 28: 109–120.MathSciNetGoogle Scholar
  22. Champaney, L., Cognard, J.Y., Rureisseix, D., and Ladevèze, P. (1997): Large scale applications on parallel computers of a mixed domain decomposition method. Computational Mechanics, 19: 253–263.MATHGoogle Scholar
  23. Christensen, P.W., and Klarbring, A. (1999): Newton’s method for frictional contact problems. In P. Wunderlich, editor, ECCM’99 European Conference on Computational Mechanics August 31 September 3, Munich Germany, Technical University of Munich.Google Scholar
  24. Christensen, P.W., Klarbring, A., Pang, J.S., and Strömberg, N. (1998): Formulation and comparison of algorithms for frictional contact problems. International Journal for Numerical Methods in Engineering, 42: 145–173.MathSciNetMATHGoogle Scholar
  25. Christensen, P.W. (2000): Computational Nonsmooth Mechanics: Contact, Friction and Plasticity. Linköping University, Sweden.Google Scholar
  26. Clarke, F.H. (1983): Optimization and Nonsmooth Analysis. J. Wiley, New York.MATHGoogle Scholar
  27. Coda, H.B., Venturini, W.S., and Aliabadi, M.H. (1999): A general 3D BEM/FEM coupling applied to elastodynamic continua/frame structures interaction analysis. International Journal for Numerical Methods in Engineering, 46: 695–712.MATHGoogle Scholar
  28. Czekanski, A., and Meguid, S.A. (2001): Solution of dynamic frictional contact problems using nondifferentiable optimization. Intern. Journal of Mechanical Sciences, 43: 1369–1386.MATHGoogle Scholar
  29. Dafermos, S. (1980): Traffic equilibrium and variational inequalities. Transportation Science 14 (1): 42–54.MathSciNetGoogle Scholar
  30. 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.MATHGoogle Scholar
  31. Demyanov, V., and Rubinov, A. (2000): (Eds.), Quasidifferentibility and Related Topics, Kluwer Academic Publishers, Dordrecht.Google Scholar
  32. Dominguez, J. (1993): Boundary Elements in Dynamics. Computational Mechanics Publications and Elsevier Applied Science, Shouthampton and New York.MATHGoogle Scholar
  33. Doudoumis, I., Mitsopoulou, E., and Charalambakis, N. (1995): The influence of the friction coefficients on the uniqueness of the solution of the unilateral contact problem. In M. Raous, M. Jean, and J.J. Moreau, editors, Contact Mechanics, pages 79–86, New York and London, Plenum Press.Google Scholar
  34. Ekeland, I., and Temam, E. (1976): Convex Analysis and Variational Problems. NorthHolland, Amsterdam.MATHGoogle Scholar
  35. Facchinei, F., Fischer, A., and Kanzow, C. (1996): Inexact Newton methods for semismooth equations with applications to variational inequality problems. In G. DiPillo and F. Giannessi, editors, Nonlinear Optimization and Applications, pages 125–149, New York, Plenum Press.Google Scholar
  36. Ferris, M.C., and Tin-Loi, F. (2001): Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints. International Journal of Mechanical Sciences, 43: 209–224.MATHGoogle Scholar
  37. Fichera, G. (1972): Boundary value problems in elasticity with unilateral constraints. Enzyclopedia of Physics, Ed. S. Flügge, Vol. VI a/2, Springer Verlag, Berlin.Google Scholar
  38. Fischer, A. (1992): A special Newton-type optimization method. Optimization, 24: 269–284, 1991.Google Scholar
  39. Fremond, M. (2002): Nonsmooth Thermomechanics. Springer Verlag, Berlin.Google Scholar
  40. Fukushima, M. (1992): Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming, 53: 99–110.MathSciNetMATHGoogle Scholar
  41. Gao, D.Y. (2000): Duality Principles in Nonconvex Nonsmooth Systems: Problems, Methods and Solutions. Kluwer Academic Publishers, Dordrecht.Google Scholar
  42. Gao, D.Y., Ogden, R., and Stavroulakis, G.E. (Eds.) (2001): Nonsmooth\Nonconvex Mechanics: Modeling, Analysis and Numerical Methods. Kluwer Academic Publishers, Dordrecht, Boston, London.Google Scholar
  43. Glocker, Ch. (2001): Set-valued Force Laws: Dynamics of Non-smooth Systems. Springer Verlag, Berlin.Google Scholar
  44. Goeleven, D. Stavroulakis, G.E., Salmon, G., and Panagiotopoulos, P.D. (1997): Solvability theory and projection methods for a class of singular variational inequalities. Elastostatic unilateral contact appli-cations. Journal of Optimization Theory and Applications, 95 (2): 263–294.MathSciNetMATHGoogle Scholar
  45. Gonzalez, J.A., and Abascal, R. (2000): Solvind 2D rolling problems using NORM-TANG iteration and mathematical programming. Computers and Structures, 78: 149–160.Google Scholar
  46. Harker, P.T., and Pang, J.S. (1990): Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming, 48: 161–220.MathSciNetMATHGoogle Scholar
  47. Haslinger, J., Miettinen, M., and Panagiotopoulos, P.D. (1999): Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht.MATHGoogle Scholar
  48. Hilding, D., Klarbring, A., and Petersson, J. (1999): Optimization of structures in unilateral contact. ASME Applied Mechanics Review, 52 (4): 139–160.Google Scholar
  49. Jean, M. (1999): The non-smooth contact dynamics method. Computer Methods in Applied Mechanics and Engineering, 177: 235–257.MathSciNetMATHGoogle Scholar
  50. Johnson, K.L. (1987): Contact Mechanics. Cambridge University Press.Google Scholar
  51. Kanzow, C. (1994): Some equation-based methods for the nonlinear complementarity problem. Optimization Methods and Software, 3: 327–340.Google Scholar
  52. Kanzow, C. (1996): Nonlinear complementarity as unconstrained optimization. Journal of Optimization Theory and Applications, 88: 139–155.MathSciNetMATHGoogle Scholar
  53. Kikuchi, N. (1988): Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia.MATHGoogle Scholar
  54. Kim, J.O., and Kwak, B.M. (1996): Dynamic analysis of two-dimensional frictional cotnact by linear complementarity problem formulation. International Journal of Solids and Structures, 33: 4605–4624.MathSciNetMATHGoogle Scholar
  55. Kim, J.O., and Kwak, B.M. (1996): Numerical implementation of dynamic contact analysis by a complementarity formulation and applications to a valve-cotter system in motorcycle engines. Mechanics of Structures and Machines, 28: 281–301.Google Scholar
  56. Klarbring, A. (1986): A mathematical programming approach to three— dimensional contact problems with friction. Comput. Meth. Appl. Mech. Engng., 58(2):175— 200.Google Scholar
  57. Klarbring, A. (1990a): Derivation and analysis of rate boundary-value problems. European Journal of Mechanics A/Solids, 9 (1): 53–86.MathSciNetMATHGoogle Scholar
  58. Klarbring, A. (1990b) Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction. Ingenieur Archiv, 60: 529–541.Google Scholar
  59. Klarbring, A. (2000): Contact, friction, discrete mechanical structures and mathematical programming. In P. Wriggers and P. Panagiotopoulos, editors, New Developments in Contact Problems, CISM Courses and Lectures No. 384, pages 55–100. Springer, Wien.Google Scholar
  60. Klarbring, A., and Björkman, G. (1998): A mathematical programming approach to contact problem with friction and varying contact surface. Computers and Structures, 30: 1185–1198.Google Scholar
  61. Klisch, T. (1999): Contact mechanics in multibody systems. Mechanism and Machine Theory, 34: 665–675.MathSciNetMATHGoogle Scholar
  62. Kortesis, S., and Panagiotopoulos, P.D. (1993): Neural networks for computing in structural analysis. Methods and prospects of applications. International Journal for Numerical Methods in Engineering, 36: 2305–2318.MATHGoogle Scholar
  63. Kosior, F., Guyot, N., and Maurice, G. (1999): Analysis of frictional contact problem using boundary element method and domain decomposition. International Journal for Numerical Methods in Engineering, 46: 65–82.MATHGoogle Scholar
  64. Ladèveze, P. (1995): Mecanique des Structures Nonlineaires. Hérmes, Paris.Google Scholar
  65. Ladeveze, P., Lemoussu, H., and Boucard, P.A. (2000): A modular approach to 3-D impact computation with frictional contact. Computers and Structures, 78: 45–51.Google Scholar
  66. Laursen, T.A. (2002): Computational Contact and Impact Mechanics. Springer Verlag, Berlin.MATHGoogle Scholar
  67. Leblond, J.-B. (2000): Basic results for elastic fracture mechanics with frictionless contact between the crack lips. European Journal of Mechanics, A/Solids, 19: 633–647.MATHGoogle Scholar
  68. Leung, A.Y.T., Guoqing, Ch., and Wanji, Ch. (1998): Smoothing Newton method for solving twoand three-dimensional frictional contact problems. International Journal for Numerical Methods in Engineering, 41: 1001–1027.MathSciNetMATHGoogle Scholar
  69. Luo, Z. Q., Pang, J. S., and Ralph, D. (1996): Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge.Google Scholar
  70. Martins, J.A.C., Marques, M.D., and Gastaldi, F. (1994): On an example of non-existence of solution to a quasistatic frictional contact problem. European Journal of Mechanics A/Solids, 13: 113–133.MathSciNetMATHGoogle Scholar
  71. Mijar, A.R., and Arora, J.S. (2000a): Review of formulations for elastostatic frictional contact problems. Structural and Multidisciplinary Optimization, 20: 167–189.Google Scholar
  72. Mijar, A.R., and Arora, J.S. (2000b): Study of variational inequality and equality formulations for elasto-static frictional contact problems. Archives of Computational Methods in Engineering, 7 (4): 387–449.MathSciNetMATHGoogle Scholar
  73. Mistakidis, E.S., and Panagiotopoulos, P.D. (1998): A multivalued boundary integral equation for adhesive contact problems. Engineering Analysis with Boundary Elements, 21 (4): 317–327.MathSciNetMATHGoogle Scholar
  74. Mistakidis, E.S., and Stavroulakis, G.E. (1998): Nonconvex Optimization in Mechanics. Algorithms, Heuristics and Engineering Applications by the F.E.M. Kluwer Academic Publishers, Dordrecht.Google Scholar
  75. Moreau, J.J. (1999): Numerical aspects of the sweeping process. Computer Methods in Applied Mechanics and Engineering, 177: 329–349.MathSciNetMATHGoogle Scholar
  76. Murty, K. G. (1988): Linear Complementarily, Linear and Nonlinear Programming. Heldermann, Berlin.Google Scholar
  77. Nagurney, A. (1994): Variational inequalities in the analysis and computation of multisector, multi-instrument financial equlibria. Journal of Economic Dynamics and Control, 18 (1): 161–184.MathSciNetMATHGoogle Scholar
  78. Nagurney, A., and Siokos, S. (1999): Dynamic multi-sector, multi-instrument financial networks with futures: Modeling and computation. Networks 33 (2): 93–108.MathSciNetMATHGoogle Scholar
  79. Nappi, A., and Tin-Loi, F. (2001): Numerical model for masonry implemented in the framework of a discrete formulation. Structural Engineering and Mechanics, 11(2):171— 184.Google Scholar
  80. Oden, J.T., and Kikuchi, N. (1980): Theory of variational inequalities with applications to problems of flow through porous media. International Journal of Engineering Science, 18 (10): 1173–1284.MathSciNetMATHGoogle Scholar
  81. Outrata, J., Kocvara, M., and Zowe, J. (1998): Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications, and Numerical Results. Kluwer Academic Publishers, Dordrecht.Google Scholar
  82. Pagano, S., and Alart, P. (1999): Solid-solid phase transition modelling: relaxation procedures, configura- tional energies and thermomechanical behaviours. Journal of Engineering Science 37: 1821–1840.MathSciNetMATHGoogle Scholar
  83. Panagiotopoulos, P.D. (1985): Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel Boston - Stuttgart.MATHGoogle Scholar
  84. Panagiotopoulos, P.D., and Stavroulakis, G.E. (1992): New type of variational principles based on the notion of quasidifferentiability. Acta Mechanica, 94: 171–194.MathSciNetMATHGoogle Scholar
  85. Panagiotopoulos, P.D. (1993): Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer, Berlin - Heidelberg - New York.MATHGoogle Scholar
  86. Panagiotopoulos, P.D., and Glocker, Ch. (1998): Analytical mechanics. Addendum I: Inequality constraints with elastic impact. The convex case. ZAMM 78 (4): 219–230.MathSciNetGoogle Scholar
  87. Pang, J. S., and Trinkle, J.C. (1996): Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction. Mathematical Programming, 73: 199–226.MathSciNetMATHGoogle Scholar
  88. Pang, J.-S., and Stewart, D.E. (1999): A unified approach to discrete frictional contact problem. Interna-tional Journal of Engineering Science 37: 1747–1768.MathSciNetMATHGoogle Scholar
  89. Park, J. K., and Kwack, B.M. (1994): contact analysis using the homotopy method. ASME Journal of Applied Mechanics, 61: 703–709.Google Scholar
  90. Pfeiffer, F., and Hajek, M. (1992): Stick-slip motion of turbine blade dampers. Phil. Trans. of the Royal Society of London, Ser. A, 338: 503–517.Google Scholar
  91. Pfeiffer, F., and Glocker, Ch. (1996): Multibody Dynamics with Unilateral Contacts. John Wiley, New York.MATHGoogle Scholar
  92. Pfeiffer, F. (1998): Robots with unilateral constraints. Annual Reviews in Control, 22: 121–132.Google Scholar
  93. Pfeiffer, F., and Glocker, Ch. (2000): Contacts in multibody systems. Journal of Applied Mathematics and Mechanics, 64: 773–782.Google Scholar
  94. Polyakova, L.N., and Stavroulakis, G.E. (2000): QD and DC optimization for pseudoelastic modeling of Shape Memory Alloys. In: Quasidifferentibility and Related Topics, Chapter 9, pp. 215–233, Eds. V. Demyanov, A. Rubinov, Kluwer Academic Publishers.Google Scholar
  95. Refaat, M.H., and Meguid, S.A. (1994): On the elastic solution of frictional contact problems using variational inequalities. International Journal of Mechanical Science, 36: 329–342.MATHGoogle Scholar
  96. Rohde, A., and Stavroulakis, G.E. (1997): Genericity analysis for path-following methods. Application in unilateral contact elastostatics. ZAMM, 77: 777–790.MathSciNetMATHGoogle Scholar
  97. DeSaxcé, G., and Feng, Z.Q. (1998): The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms. Mathematical and Computer Modelling, 28: 225–245.MathSciNetGoogle Scholar
  98. Sewell, N.J. (1987): Maximum and Minimum Principles. A Unified Approach with Applications. Cambridge University Press, Cambridge.MATHGoogle Scholar
  99. Simunovic, S., and Saigal, S. (1992): Frictionless contact with BEM using quadratic programming. ASCE Journal of Engineering Mechanics 118: 1876–1891.Google Scholar
  100. Song, P., Kraus, P., Kumar, V., and Dupont, P. (2001): Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. ASME Journal of Applied Mechanics, 68: 118–128.MATHGoogle Scholar
  101. Stamos, A.A., and Beskos, D.E. (1995): Dynamic analysis of large 3D underground structures by the BEM. Earthquake Engineering and Structural Dynamics, 24: 917–934.Google Scholar
  102. Stavroulakis, G.E. (1993): Convex decomposition for nonconvex energy problems in elastostatics and applications. European Journal of Mechanics A I Solids, 12: 1–20.MathSciNetMATHGoogle Scholar
  103. Stavroulakis, G.E. (1995): Optimal prestress of cracked unilateral structures: finite element analysis of an optimal control problem for variational inequalities. Computer Methods in Applied Mechanics and Engineering, 123: 231–246.MathSciNetMATHGoogle Scholar
  104. Stavroulakis, G.E. (1999): Impact-echo from a unilateral interlayer crack. LCP- BEM modelling and neural identification. Engineering Fracture Mechanics, 62: 165–184.Google Scholar
  105. Stavroulakis, G.E., and Antes, H. (1999): Nonlinear boundary equation approach for inequality 2- D elastodynamics. Engineering Analysis with Boundary Elements, 23: 487–501.MATHGoogle Scholar
  106. Stavroulakis, G.E., and Antes, H. (2000): Nonlinear equation approach for inequality elastostatics. A 2-D BEM implementation. Computers and Structures, 75 (6): 631–646.Google Scholar
  107. Stavroulakis, G.E., Panagiotopoulos, P.D., and Al-Fahed, A.M. (1991): On the rigid body displacements and rotations in unilateral contact problems and applications. Computers and Structures, 40: 599–614.MATHGoogle Scholar
  108. Stavroulakis, G.E. (2000): Inverse and Crack Identification Problems in Engineering Mechanics. Kluwer Academic Publishers, Dordrecht. Habilitation Thesis, Technical University of Braunschweig, Germany, 2000.Google Scholar
  109. Stavroulakis, G.E., and Antes, H. (1997): Nondestructive elastostatic identification of unilateral cracks through BEM and neural networks. Computational Mechanics, 20: 439–451.MATHGoogle Scholar
  110. Stavroulakis, G.E., and Antes, H. (2000): Unilateral crack identification. a filter-driven, iterative, boundary element approach. Journal of Global Optimization, 17: 339–352.MathSciNetMATHGoogle Scholar
  111. Stavroulakis, G.E., Antes, H., and Panagiotopoulos, P.D. (1999): Transient elastodynamics around cracks including contact and friction. Computer Methods in Applied Mechanics and Engineering, 177: 427–440.Google Scholar
  112. Telega, J.J. (1988): Topics on unilateral contact problems in elasticity and inelasticity. In: Nonsmooth Mechanics and Applications, Eds. J.J. Moreau and P.D. Panagiotopoulos, CISM Lecture Notes No. 302, pp. 341–462, Springer Verlag, Wien.Google Scholar
  113. Tin-Loi, F., and Li, H. (2000): Numerical simulations of quasibrittle fracture processes using the discrete cohesive crack model. International Journal of Mechanical Sciences, 42: 367–379.MATHGoogle Scholar
  114. Tin-Loi, F., and Xia, S.H. (2001): Nonholonomic elastoplastic analysis involving unilateral frictionless contact as a mixed complementarity problem. Computer Methods in Applied Mechanics and Engineering, 190: 4551–4568.MATHGoogle Scholar
  115. Trinkle, J. C., Pang, J. S., Sudarsky, S., and Lo, G. (1997): On dynamic multi-rigid-body contact problems with Coulomb friction. Zeitschrift fir Angewandte Mathematik und Mechanik (ZAMM), 77: 267–280.MathSciNetMATHGoogle Scholar
  116. Tzaferopoulos, M.Ap., and Panagiotopoulos, P.D. (1993): Delamination of composites as a substationarity problem: Numerical approximation and algorithms. Computer Methods in Applied Mechanics and Engineering 110: 63–86.MathSciNetMATHGoogle Scholar
  117. Vola, D., Pratt, E., Jean, M., and Raous, M. (1998): Consistent time discretization for a dynamical frictional contact problem and complementarity techniques. Revue Européenne des éléments Finis, 7: 149–162.MATHGoogle Scholar
  118. Wolfsteiner, P., and Pfeiffer, F. (2000): Modeling, simulation and verification of the transportation process in vibratory feeders. ZAMM, 80: 35–48.MATHGoogle Scholar
  119. Wriggers, P. (1995): Finite element algorithms for contact problems. Archives of Computational Methods in Engineering, 2: 1–49.MathSciNetGoogle Scholar
  120. Zavarise, G., Wriggers, P., and Schrefler, B.A. (1998): A method for solving contact problems. Interna-tional Journal for Numerical Methods in Engineering, 42: 473–498.MATHGoogle Scholar
  121. Zhong, Z.-H. (1993): Finite Element Procedures for Contact-Impact Problems. Oxford University Press, Oxford.Google Scholar
  122. Zozulya, V.V., and Gonzalez-Chi, P.I. (2000): Dynamic fracture mechanics with contact interaction at the crack edges. Engineering Analysis with Boundary Elements, 24: 643–659.MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Georgios E. Stavroulakis
    • 1
    • 2
  • Heinz Antes
    • 2
  1. 1.Department of MathematicsUniversity of IoanninaGreece
  2. 2.Department of Civil EngineeringTechnical University of BraunschweigGermany

Personalised recommendations