Advertisement

Materially Nonlinear Analysis

  • Marc Bonnet
Part of the International Centre for Mechanical Sciences book series (CISM, volume 440)

Abstract

This chapter presents a review of domain-boundary element techniques for solving materially nonlinear solid mechanics problems. Quasi-static and dynamic formulations are addressed, with emphasis on the use of implicit constitutive integration techniques and the local and global consistent tangent operators. Symmetric Galerkin BEM formulations are also presented. A section is devoted to steady-state elastoplastic calculations for moving loads, and another to the simulation of abrasive wear, both types of nonlinear analyses being well-suited to D/BEM treatments. Another section deals with energy methods in fracture mechanics. Finally, a symmetric formulation for BEM-FEM coupling is presented as another way to use BEM, this time combined with FEM, for materially nonlinear analyses.

Keywords

Plastic Strain Boundary Element Boundary Element Method Boundary Integral Equation Tangent Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmad, S., and Banerjee, P. K. (1990). Inelastic Transient elastodynamic analysis of three-dimensional problems by BEM. International Journal for Numerical Methods in Engineering 29: 371–390.CrossRefMATHGoogle Scholar
  2. Aliabadi, M. H., and Martin, D. (2000). Boundary element hyper-singular formulation for elastoplastic contact problems. International Journal for Numerical Methods in Engineering 48: 995–1014.CrossRefMATHGoogle Scholar
  3. Balas, J., Sladek, J., and Sladek, V. (1989). Stress Analysis by Boundary Element Methods. Elsevier.Google Scholar
  4. Ballard, P., and Constantinescu, A. (1994). On the inversion of subsurface residual stresses from surface stress measurements. Journal of the Mechanics and Physics of Solids 42: 1767–1787.MathSciNetCrossRefMATHGoogle Scholar
  5. Banerjee, P. K., and Raveendra, S. T. (1986). Advanced boundary element analysis of two-and three-dimensional problems of elasto-plasticity. International Journal for Numerical Methods in Engineering 23: 985–1002.CrossRefMATHGoogle Scholar
  6. Banerjee, P. K., Henry, D. P. J., and Raveendra, S. T. (1989). Advanced inelastic analysis of solids by the boundary element method. International Journal of Mechanical Sciences 31: 309–322.CrossRefMATHGoogle Scholar
  7. Banerjee, P. K. (1994). The Boundary Element Method in Engineering (2nd. edition). McGraw Hill, London.Google Scholar
  8. Ben Mariem, J., and Hamdi, M. A. (1987). A new boundary finite element method for fluid-structure interaction problems. International Journal for Numerical Methods in Engineering 24: 1251–1267.CrossRefMATHGoogle Scholar
  9. Beskos, D. E. (1987a). Boundary element methods in dynamic analysis. Applied Mechanics Reviews 40: 1–23.CrossRefGoogle Scholar
  10. Beskos, D. E., ed. (1987b). Boundary Element Methods in Mechanics,volume 3 of Computational Methods in Mechanics. North Holland.Google Scholar
  11. Beskos, D. E. (1997). Boundary element methods in dynamic analysis, part. II (1986–1996). Applied Mechanics Reviews 50: 149–197.CrossRefGoogle Scholar
  12. Bhargava, V., Hahn, G. T., and Rubin, C. (1985a). An elastic-plastic finite element model of rolling contact. Part I: analysis of single contacts. ASME Journal of Applied Mechanics 52: 67–74.CrossRefGoogle Scholar
  13. Bhargava, V., Hahn, G. T., and Rubin, C. (1985b). An elastic-plastic finite element model of rolling contact. Part II: analysis of multiple contacts. ASME Journal of Applied Mechanics 52: 75–82.CrossRefGoogle Scholar
  14. Bhargava, V., Hahn, G. T., and Rubin, C. (1988). Analysis of rolling contact with kinematic hardening for rail steel properties. Wear 122: 267–283.CrossRefGoogle Scholar
  15. Bielak, J., MacCamy, R. C., and Zeng, X. (1995). Stable coupling method for interface scattering problems by combined integral equations and finite elements. Journal of Computational Physics 119: 374–384.MathSciNetCrossRefMATHGoogle Scholar
  16. Bonnet, M., and Bui, H. D. (1993). Regularization of the displacement and traction BIE for 3D elastodynamics using indirect methods. In Kane, J. H., Maier, G., Tosaka, N., and Atluri, S. N., eds., Advances in Boundary Element Techniques. Springer-Verlag. 1–29.Google Scholar
  17. Bonnet, M., and Mukherjee, S. (1996). Implicit BEM formulations for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept. International Journal of Solids and Structures 33: 4461–4480.CrossRefMATHGoogle Scholar
  18. Bonnet, M., and Xiao, H. (1995). Computation of energy release rate along a crack front using material differentiation of elastic BIE. Engineering Analysis with Boundary Elements 15: 137–150.CrossRefGoogle Scholar
  19. Bonnet, M., Maier, G., and Polizzotto, C. (1998a). Symmetric Galerkin boundary element method. Applied Mechanics Reviews 51: 669–704.CrossRefGoogle Scholar
  20. Bonnet, M., Poon, H., and Mukherjee, S. (1998b). Hypersingular formulation for boundary strain evaluation in the context of a CTO-based implicit BEM scheme for small strain elasto-plasticity. International Journal of Plasticity 14: 1033–1058.CrossRefMATHGoogle Scholar
  21. Bonnet, M., Burgardt, B., and Le Van, A. (1999). A regularized direct symmetric variational BIE formulation for three-dimensional elastoplasticity. In Burczyfiski, T., ed., IUTAMI1ABEMIIACM Symposium on Advanced mathematical and computational aspects of the boundary element method, 51–61. Kluwer.Google Scholar
  22. Bonnet, M. (1995a). Regularized BIE formulations for first-and second-order shape sensitivity of elastic fields. Computers and Structures 56: 799–811.MathSciNetCrossRefMATHGoogle Scholar
  23. Bonnet, M. (1995b). Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity. Engineering Analysis with Boundary Elements 15: 93–102.CrossRefGoogle Scholar
  24. Bonnet, M. (1999a). Boundary Integral Equations Methods for Solids and Fluids. John Wiley and Sons.Google Scholar
  25. Bonnet, M. (1999b). Stability of crack fronts under Griffith criterion: a computational approach using integral equations and domain derivatives of potential energy. Computer Methods in Applied Mechanics and Engineering 173: 337–364.MathSciNetCrossRefMATHGoogle Scholar
  26. Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C. (1984). Boundary Element Techniques. Springer-Verlag.Google Scholar
  27. Bui, H. D. (1977). An Integral Equation Method for Solving the Problem of a Plane Crack of Arbitrary Shape. Journal of the Mechanics and Physics of Solids 25: 29–39.MathSciNetCrossRefMATHGoogle Scholar
  28. Bui, H. D. (1978). Some remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations. International Journal of Solids and Structures 14: 935–939.CrossRefMATHGoogle Scholar
  29. Burgardt, B., and Bonnet, M. (2001). 3D elastoplasticity by symmetric Galerkin BEM and implicit constitutive integration. (in preparation).Google Scholar
  30. Burgardt, B. (1999). Contribution à l’étude des méthodes des équations intégrales et à leur mise en oeuvre numérique en élastoplasticité. Ph.D. Dissertation, Ecole centrale de Nantes, France.Google Scholar
  31. Castem 2000. (1999). see website http: //www. castem. org: 8001. CASTEM 2000 is a research FEM environment; its development is sponsored by the French Atomic Energy Commission (Commissariat à l’Energie Atomique - CEA).Google Scholar
  32. Chandra, A., and Mukherjee, S. (1997). Boundary Element Methods in Manufacturing. Oxford University Press, Oxford.Google Scholar
  33. Cisilino, A. P., and Aliabadi, M. H. (1999). Three-dimensional boundary element analysis of fatigue crack growth in linear and non-linear fracture problems. Engineering Fracture Mechanics 63: 713–733.CrossRefGoogle Scholar
  34. Cisilino, A. P., Aliabadi, M. H., and Otegui, J. L. (1998). A three-dimensional boundary element formulation for the elastoplastic analysis of cracked bodies. International Journal for Numerical Methods in Engineering 42: 237–256.CrossRefMATHGoogle Scholar
  35. Comi, C., and Maier, G. (1992). Extremum, convergence and stability properties of the finite-increment problem in elastic-plastic boundary element analysis. International Journal of Solids and Structures 29: 249–270.MathSciNetCrossRefMATHGoogle Scholar
  36. Dang Van, K., and Maitournam, M. H. (1993). Steady-state flow in classical elastoplasticity: applications to repeated rolling and sliding contact. Journal of the Mechanics and Physics of Solids 41: 1691–1710.MathSciNetCrossRefMATHGoogle Scholar
  37. De Borst, R., and Feenstra, P. (1990). Studies in anisotropic plasticity with reference to the Hill criterion. International Journal for Numerical Methods in Engineering 29: 315–336.CrossRefMATHGoogle Scholar
  38. Delorenzi, H. G. (1982). On the energy release rate and the J-integral for 3-D crack configurations. International Journal of Fracture 19: 183–194.CrossRefGoogle Scholar
  39. Destuynder, P., Djaoua, M., and Lescure, S. (1983). Quelques remarques sur la mécanique de la rupture élastique. Journal de Mécanique Théorique et Appliquée 2: 113–135.MATHGoogle Scholar
  40. Dong, C., and Bonnet, M. (2001). An integral formulation for steady-state elastoplastic contact over a coated half-plane. Computational Mechanics (to appear).Google Scholar
  41. Eringen, A. C., and Suhubi, E. S. (1975). Elastodynamics (vol II–linear theory). Academic Press. Foerster, A., and Kuhn, G. (1994). A field boundary element formulation for material nonlinear problems at finite strains. International Journal of Solids and Structures 31: 1777–1792.Google Scholar
  42. Frangi, A., and Maier, G. (1999). Dynamic elastic-plastic analysis by a symmetric Galerkin boundary element method with time-independent kernels. Computer Methods in Applied Mechanics and Engineering 171: 281–308.CrossRefMATHGoogle Scholar
  43. Frangi, A., and Novati, G. (1999). On the numerical stability of time-domain elastodynamic analyses by BEM. Computer Methods in Applied Mechanics and Engineering 173: 405–419.MathSciNetCrossRefGoogle Scholar
  44. Guiggiani, M. (1994). Hypersingular formulation for boundary stress evaluation. Engineering Analysis with Boundary Elements 14: 169–179.CrossRefGoogle Scholar
  45. Guiggiani, M. (1998). Formulation and numerical treatment of boundary integral equations with hyper-singular kernels. In Sladek, V., and Sladek, J., eds., Singular Integrals in Boundary Element Methods. Computational Mechanics Publications, Southampton. chapter 3, 85–124.Google Scholar
  46. Hellen, T. K. (1975). On the method of virtual crack extension. International Journal for Numerical Methods in Engineering 9: 187–207.CrossRefMATHGoogle Scholar
  47. Hills, D. A., Nowell, D., and Sackfield, A. (1993). Mechanics of Elastic Contacts. Butterworth-Heinemann Ltd.Google Scholar
  48. Hsiao, G. C. (1990). The coupling of boundary element and finite element methods. Zeitschrift für Angewandte Mathematik und Mechanik 70: 493–503.CrossRefMATHGoogle Scholar
  49. Huber, O., Dallner, R., Partheymuller, P., and Kuhn, G. (1996). Evaluation of the stress tensor in 3D elastoplasticity by direct solving of hypersingular integrals. International Journal for Numerical Methods in Engineering 39: 2555–2573.MathSciNetCrossRefMATHGoogle Scholar
  50. Israil, A. S. M., and Banerjee, P. K. (1992). Advanced development of boundary element method for two-dimensional dynamic elastoplasticity. International Journal of Solids and Structures 29: 1433–1451.CrossRefMATHGoogle Scholar
  51. Karabalis, D. L. (1991). A simplified 3D time-domain BEM for dynamic soil-structure interaction prob-lems. Engineering Analysis with Boundary Elements 8:139–145.Google Scholar
  52. Kleiber, M. e. a. (1997). Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations. J. Wiley and Sons, New York.Google Scholar
  53. Kontoni, D. P. N., and Beskos, D. E. (1992). Application of the DR-BEM to inelastic dynamic problems. In C. A. Brebbia, J. D., and Paris, F., eds., Boundary Elements XIV. Computational Mechanics Publications, Southampton.Google Scholar
  54. Lederer, G., Bonnet, M., and Maitournam, H. (1998). Modélisation par équations intégrales du frottement sur un demi-espace élasto-plastique. Revue Européenne des Elements Finis 7: 131–147.MATHGoogle Scholar
  55. Lederer, G. (1998). Modélisation tribo-mécanique du frottement en milieu agressif. Ph.D. Dissertation, Ecole Polytechnique, Palaiseau, France.Google Scholar
  56. Leu, L. J., and Mukherjee, S. (1993). Sensitivity analysis and shape optimization in nonlinear solid mechanics. Engineering Analysis with Boundary Elements 12: 251–260.CrossRefGoogle Scholar
  57. Li, S., Mear, M. E., and Xiao, L. (1998). Symmetric weak-form integral equation method for three-dimensional fracture analysis. Computer Methods in Applied Mechanics and Engineering 151: 435–459.MathSciNetCrossRefMATHGoogle Scholar
  58. Lorentz, E., Wadier, Y., and Debruyne, G. (2000). Brittle fracture in a plastic medium: definition of an energy release rate. Comptes Rendus à l’Académie des Sciences, série II 328: 657–662.MATHGoogle Scholar
  59. Maier, G., and Polizzotto, C. (1987). A Galerkin approach to boundary element elastoplastic analysis. Computer Methods in Applied Mechanics and Engineering 60: 175–194.CrossRefMATHGoogle Scholar
  60. Maier, G., Miccoli, S., Novati, G., and Sirtori, S. (1993). A Galerkin symmetric boundary-element method in plasticity: formulation and implementation. In Kane, J. H., Maier, G., Tosaka, N., and Atluri, S. N., eds., Advances in Boundary Element Techniques. Springer-Verlag. 288–328.Google Scholar
  61. Maier, G., Miccoli, S., Novati, G., and Perego, U. (1995). Symmetric Galerkin boundary element method in plasticity and gradient-plasticity. Computational Mechanics 17: 115–129.CrossRefMATHGoogle Scholar
  62. Mouhoubi, S., Bonnet, M., and Ulmet, L. (2001). Application of Symmetric Galerkin boundary integral equations to 3-D symmetric FEM-BEM coupling. European Conference on Computational Mechanics (Cracovie, Pologne, 26–29 juin 2001), CD-ROM proceedings.Google Scholar
  63. Mouhoubi, S. (2000). Couplage symétrique éléments finis-éléments de frontière en mécanique: formula-tion et implantation dans un code éléments finis. Ph.D. Dissertation, Université de Limoges.Google Scholar
  64. Mukherjee, S., and Chandra, A. (1987). Nonlinear solid mechanics. In Beskos, D. E., ed., Boundary element methods in mechanics., volume 1. North Holland. 285–332.Google Scholar
  65. Mukherjee, S., and Chandra, A. (1991). A boundary element formulation for design sensitivities in problems involving both geometric and material nonlinearities. Mathematical and Computer Modelling 15: 245–255.CrossRefMATHGoogle Scholar
  66. Nedelec, J. C. (1982). Integral equations with non integrable kernels. Integral Equations And Operator Theory 5: 562–572.MathSciNetCrossRefMATHGoogle Scholar
  67. Nguyen, Q. S., Pradeilles-Duval, and R. M., Stolz, C. (1989). Sur une loi régularisante en rupture et endommagement fragile. Comptes Rendus à l’Académie des Sciences, série II 309: 1515–1520.MATHGoogle Scholar
  68. Nguyen, Q. S., Stolz, C., and Debruyne, G. (1990). Energy methods in fracture mechanics stability, bifurcation and second variations. European Journal of Mechanics A/Solids 9: 157–173.MathSciNetMATHGoogle Scholar
  69. Nguyen, Q. S. (1977). On the elastic-plastic initial-boundary value problem and its numerical integration. International Journal for Numerical Methods in Engineering 11: 817–832.MathSciNetCrossRefMATHGoogle Scholar
  70. Partridge, P. W., Brebbia, C. A., and Wrobel, L. C. (1992). The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton.MATHGoogle Scholar
  71. Paulino, G. H., and Liu, Y. (1999). Implicit consistent and continuum tangent operators in elastoplastic boundary element formulations. Computer Methods in Applied Mechanics and Engineering 190: 2157–2179.CrossRefGoogle Scholar
  72. Peirce, A., and Siebrits, E. (1997). Stability analysis and design of time-stepping schemes for general elastodynamic boundary element models. International Journal for Numerical Methods in Engineering 40: 319–342.MathSciNetCrossRefGoogle Scholar
  73. Polizzotto, C. (1988). An energy approach to the boundary element method; Part II: Elastic-plastic solids. Computer Methods in Applied Mechanics and Engineering 69: 263–276.MathSciNetCrossRefMATHGoogle Scholar
  74. Poon, H., Mukherjee, S., and Bonnet, M. (1998a). Numerical implementation of a CTO-based implicit approach for the BEM solution of usual and sensitivity problems in elasto-plasticity. Engineering Analysis with Boundary Elements 22: 257–269.CrossRefMATHGoogle Scholar
  75. Poon, H., Mukherjee, S., and Fouad Ahmad, M. (1998b). Use of `simple solution’ in regularizing hypersin-gular boundary integral equations in elastoplasticity. ASME Journal of Applied Mechanics 65: 39–45.CrossRefGoogle Scholar
  76. Pradeilles-Duval, R. M. (1992). Evolution de systèmes avec surfaces de discontinuité mobiles: application au délaminage. Ph.D. Dissertation, Ecole Polytechnique, Palaiseau, France.Google Scholar
  77. Providakis, D. E., Beskos, D. E., and Sotiropoulos, D. A. (1994). Dynamic analysis of inelastic plates by the D/BEM. Computational Mechanics 13: 276–284.CrossRefMATHGoogle Scholar
  78. Schanz, M., and Antes, H. (1997). Application of `operational quadrature methods’ in time domain boundary element methods. Meccanica 32: 179–186.CrossRefMATHGoogle Scholar
  79. Serre, I., Bonnet, M., and Pradeilles-Duval, R. M. (2001). Modelling an abrasive wear experiment by the boundary element method. Comptes Rendus à l’Académie des Sciences, série II 329: 803–808.Google Scholar
  80. Serre, I. (2000). Contribution à l’étude des phénomènes d’usure par frottement en milieu marin. Ph.D. Dissertation, Ecole Polytechnique, Palaiseau, France.Google Scholar
  81. Simo, J. C., and Hugues, T. J. R. (1998). Computational Inelasticity. Springer-Verlag.Google Scholar
  82. Simo, J. C., and Taylor, R. L. (1985). Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering 48: 101–118.CrossRefMATHGoogle Scholar
  83. Stroud, A. H. (1971). Approximate Calculation of Multiple Integrals. Prentice-Hall.Google Scholar
  84. Suo, X. Z., and Combescure, A. (1989). Sur une formulation mathématique de la dérivée de l’énergie potentielle en théorie de la rupture fragile. Comptes Rendus à l’Académie des Sciences, série II 308: 1119–1122.MathSciNetMATHGoogle Scholar
  85. Swedlow, J. L., and Cruse, T. A. (1971). Formulation of boundary integral equations for three-dimensional elasto-plastic flow. International Journal of Solids and Structures 7: 1673–1681.CrossRefMATHGoogle Scholar
  86. Tada, H., Paris, P., and Irwin, G. (1973). The Stress Analysis of Cracks Handbook. Technical report, Del. Research Corporation, Hellertown, Pennsylvania, USA.Google Scholar
  87. Tanaka, M., Sladek, V., and Sladek, J. (1994). Regularization techniques applied to boundary element methods. Applied Mechanics Reviews 47: 457–499.CrossRefGoogle Scholar
  88. Telles, J. C. E, and Brebbia, C. A. (1981a). Boundary elements: new developments in elastoplastic analyses. Applied Mathematical Modelling 5: 376–382.MathSciNetCrossRefMATHGoogle Scholar
  89. Telles, J. C. F., and Brebbia, C. A. (1981b). Boundry element solution for half-plane problems. International Journal of Solids and Structures 17: 1149–1158.CrossRefMATHGoogle Scholar
  90. Telles, J. C. F., and Carrer, J. A. M. (1991). Implicit procedures for the solution of elastoplastic problems by the boundary element method. Mathematical and Computer Modelling 15: 303–311.CrossRefMATHGoogle Scholar
  91. Telles, J. C. F., and Carrer, J. A. M. (1994). Static and transient dynamic nonlinear stress analysis by the boundary element method with implicit techniques. Engineering Analysis with Boundary Elements 14: 65–4.CrossRefGoogle Scholar
  92. Telles, J. C. E, Carrer, J. A. M., and Mansur, W. J. (1999). Transient dynamic elastoplastic analysis by the time-domain BEM formulation. Engineering Analysis with Boundary Elements 23: 479–486.CrossRefMATHGoogle Scholar
  93. Wadier, Y., and Malak, O. (1989). The theta method applied to the analysis of 3D-elastic-plastic cracked bodies. In Proceedings of 10th International Conference on Structural Mechanics in Reactor Technology.Google Scholar
  94. Wei, X., Leu, L. J., Chandra, A., and Mukherjee, S. (1994). Shape optimization in elasticity and elastoplasticity. International Journal of Solids and Structures 31: 533–550.CrossRefMATHGoogle Scholar
  95. Zhang, Q., Mukherjee, S., and Chandra, A. (1992a). Design sensitivity coefficients for elasto-visco-plastic problems by boundary element methods. International Journal for Numerical Methods in Engineering 34: 947–966.CrossRefMATHGoogle Scholar
  96. Zhang, Q., Mukherjee, S., and Chandra, A. (1992b). Shape design sensitivity analysis for geometrically and materially nonlinear problems by the boundary element method. International Journal of Solids and Structures 29: 2503–2525.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Marc Bonnet
    • 1
  1. 1.Laboratory of Solid MechanicsEcole Polytechnique and CNRSFrance

Personalised recommendations