Skip to main content
SpringerLink
Log in

Symmetric Galerkin boundary element method in plasticity and gradient plasticity

  • Originals
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A boundary integral equation (BIE) formulation and a boundary element (BE) method which confer symmetry to key operators are concisely described with reference to quasi-static plasticity. This formulation is based on the combined use of static and kinematic sources, on Galerkin weighted residual enforcement of integral equations for displacements and tractions along the boundary and for stresses in the potentially yielding domain and on space discretizations in terms of generalized variables in Prager's sense. Typical theoretical results of computational interest not available in conventional nonsymmetric BE methods are surveyed. The subjectivity (mesh-dependence) implied by material instability is illustrated by examples. As a remedy, a symmetric BIE-BE formulation for nonlocal, gradient-dependent plasticity is developed and discussed on the basis of its variationally consistent discretization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Antes, H.; Panagiotopoulos, P. D. 1992: The Boundary Integral Approach to Static and Dynamic Contact Problems, Basel, Birkhäuser Verlag

    Google Scholar 

  • Ahn, B. H. 1983: Iterative methods for linear complementarity problems with upper bounds on primary variables, Math. Progr., 26: 295–315

    Google Scholar 

  • Balakrishna, C.; Gray, L. J.; Kane, J. H. 1994: Efficient analytical integration of symmetric Galerkin boundary integrals over curved elements: thermal conduction formulation. Comp. Meth. Appl. Mech. Engng., 111: 335–355

    Google Scholar 

  • Banerjee, P. K.; Henry, D. P.; Raveendra, S. T. 1989: Advanced inelastic analysis of solids by the boundary element method. Int. J. Mech. Sci., 31: 309–322

    Google Scholar 

  • Bazant, Z. P.; Pijaudier-Cabot, G. 1988: Nonlocal continuum damage, localization instability and convergence. J. Appl. Mech., 55: 287–293

    Google Scholar 

  • Bonnet, M., Bui, H. D. 1993: Regularization of the displacements and traction BIE for 3D elastodynamics using indirect methods. In: H.Kane, G.Maier, N.Tosaka and S. N.Atluri editors, Advances in Boundary Element Techniques, 1–29, Berlin, Springer-Verlag

    Google Scholar 

  • Brebbia, C. A.; Telles, J. C. F.; Wrobel, L. C. 1984: Boundary Element Techniques, Berlin, Springer Verlag

    Google Scholar 

  • Bui, H. D. 1990: Symétrie, dualité et variables conjuguées dans les équations intégrales de frontière, C. R. Acad. Sci., Paris, t. 311, Série II, 7–13

    Google Scholar 

  • Carini, A.; Diligenti, M.; Maier, G. 1991: Boundary integral equation analyses in linear viscoelasticity: variational and saddle point formulation, Comput. Mech., 8: 87–98

    Google Scholar 

  • Carini, A.; Diligenti, M.; Maier, G. 1991: Variational formulation of the boundary element method in transient heat conduction, Math. Computational Modelling, 15: 71–80

    Google Scholar 

  • Cen, Z.; Maier, G.; Novati, G. 1990: On boundary element inelastic analysis in the presence of softening. In: B. S.Annigeri and K.Tseng (eds): Boundary Element Methods in Engineering, pp. 300–309, Berlin, Springer-Verlag

    Google Scholar 

  • Comi, C.; Maier, G.; Perego, U. 1992: Generalized variable finite element modelling and extremum theorems in stepwise holonomic elastoplasticity with internal variables. Comp. Meth. in Appl. Mech. and Engng., 96: 133–171

    Google Scholar 

  • Comi, C.; Maier, G. 1992: Extremum, convergence and stability properties of the finite-increment problem in elastic-plastic boundary element analysis. Int. J. Solids and Structures., 29: 249–270

    Google Scholar 

  • Comi, C.; Perego, U. 1995a: A regularization technique for elastoplastic softening analyses based on generalized variables, Proc. COMPLAS 4, Barcelona, Spain, 3–9 April, 1995

  • Comi, C.; Perego, U. 1995b: A generalized variable formulation for gradient dependent softening plasticity, to appear, 1995

  • Corigliano, A.; Maier, G.; Pycko, S. 1995: Dynamic shakedown analysis and bounds for elastoplastic structures with nonassociated, internal variables constitutive laws. Int. J. Solids and Structures, to appear

  • Costabel, M. 1987: Symmetric methods for the coupling of the finite elements and the boundary elements. In: C. A.Brebbia, W. I.Wendland and G.Kuhn editors, Boundary Elements IX, pp. 411–420 Berlin, Springer-Verlag

    Google Scholar 

  • Cruse, T. A.; Polch, E. Z. 1986: Elastoplastic BIE analysis of cracked plates and related problems, Part I: formulation, Part 2: numerical results. Int. J. Num. Meth. Engng., 23: 429–437, 439–452

    Google Scholar 

  • DeBorst, R.; Sluys, L. J.; Mülhaus, H.-B.; Pamin, J. 1993: Fundamental issues in finite element analysis of localization of deformation. Engng. Comp., 10: 99–121

    Google Scholar 

  • DeFigueiredo, T. G. B.; Brebbia, C. A. 1991: A new variational boundary element model for potential problems. Engng. Analysis with BE, 8: 45–50

    Google Scholar 

  • deFreitas, J. A. T. 1988: Duality and symmetry in mixed integral methods of elastostatics. Int. J. Num. Meth. Engng., 28: 1161–1179

    Google Scholar 

  • Gebbia, M. 1891: Formule fondamentali della statica dei corpi elastici. Rend. Circ. Matematico di Palermo, 5: 320–323

    Google Scholar 

  • Guiggiani, M.; Krishnasamy, G.; Rizzo, F. J.; Rudolphi, T. J. 1991: Hypersingular boundary integral equations: a new approach to their numerical treatment. In: L.Morino and R.Piva (eds), Boundary Integral Methods, 211–220, Berlin, Springer-Verlag

    Google Scholar 

  • Guiggiani, M. 1994: Hypersingular formulation for boundary stress evaluation. Engng. Analysis, 13: 169–179

    Google Scholar 

  • Hartmann, F.; Katz, C.; Protopsaltis, B. 1985: Boundary elements and symmetry. Ing. Arch., 55: 440–449

    Google Scholar 

  • Holzer, S. (1993): How to deal with hypersingular integrals in the symmetric BEM. Communications in Num. Meth. in Engng., 9: 219–232

    Google Scholar 

  • Holzer, S. 1993: On the engineering analysis of 2D problems by the symmetric Galerkin boundary element method and coupled BEM/FEM. J. H.Kane, G.Maier, N.Tosaka and S. N.Atluri (eds), Advances In Boundary Element Techniques, 187–208, Berlin, Springer-Verlag

    Google Scholar 

  • Kane, J. H. 1994: Boundary Element Analysis in Engineering Continuum Mechanics. Englewood Cliffs, Prentice-Hall Inc

    Google Scholar 

  • Kane, J. H.; Balakrishna, C. 1993: Symmetric Galerkin boundary formulations employing curved elements. Int. J. Num. Meth. Eng., 36: 2157–2187

    Google Scholar 

  • Li, H.; Han, G.; Mang, H. A.; Torzicky, P. 1986: A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies. Comp. Meth. Appl. Mech. Engng., 54, 161–185

    Google Scholar 

  • Maier, G.; Diligenti, M.; Carini, A. 1991: A variational approach to boundary element elastodynamic analysis and extension to multidomain problems. Comp. Meth. Appl. Mech. Engng., 92: 193–213

    Google Scholar 

  • Maier, G.; Miccoli, S.; Novati, G.; Sirtori, S. 1993: A Galerkin symmetric boundary element method in plasticity: formulation and implementation. In: H.Kane, G.Maier, N.Tosaka and S. N.Atluri (eds), Advances in Boundary Element Techniques, 288–328, Berlin, Springer-Verlag

    Google Scholar 

  • Maier, G.; Nappi, A. 1984: On bounding post-shakedown quantities by the boundary element method. Engng. Analysis, 1: 223–229

    Google Scholar 

  • Maier, G.; Novati, G.; Cen, Z. 1993: Symmetric Galerkin boundary element method for quasi-brittle-fracture and frictional contact problems. Comput. Mech., 13: 74–89

    Google Scholar 

  • Maier, G.; Novati, G.; Sirtori, S. 1989: Symmetric formulation of indirect boundary element method for elastic-plastic analysis and relevant extremum properties. In: M. Tanaka and T. A. Cruse (eds), Boundary Element Methods in Applied Mechanics, 215–224, Oxford

  • Maier, G.; Novati, G. 1983: Elastic-plastic boundary element analysis as a linear complementarity problem. Applied Mathem. Modelling, 7: 74–82

    Google Scholar 

  • Maier, G.; Polizzotto, C. 1987: A Galerkin approach to boundary elements elastoplastic analysis. Comp. Meth. Appl. Mech. Engng., 60: 175–194

    Google Scholar 

  • Maier, G. 1983: On elastoplastic analysis by boundary elements. Mech. Research Comm., 10: 45–52

    Google Scholar 

  • Mason, J. 1985: Methods of Functional Analysis for Application in Solid Mechanics. Amsterdam, Elsevier

    Google Scholar 

  • Mülhaus, H.-B.; Aifantis, E. C. 1991: A variational principle for gradient plasticity. Int. J. Solids Struct., 7: 845–857

    Google Scholar 

  • Ortiz, M.; Quigley, J. J. 1990: Adaptive mesh refinement in strain localization problems. Comp. Meth. Appl. Mech. Eng., 90: 781–804

    Google Scholar 

  • Pamin, J. 1994: Gradient-dependent plasticity in numerical simulation of localization phenomena. Ph. D Thesis, Delft, Delt University press

    Google Scholar 

  • Pan, E.; Maier, G.; Amadei, B. 1995: A variational formulation of the boundary element method in transient poroelasticity. Proc. Conference of the Int. Ass. for Boundary Element Methods (IABEM), Hawaii, August 1995

  • Panzeca, T.; Polizzotto, C.; Zito, M. 1994: Dynamic plasticity analysis by boundary/interior elements. Engineering Analysis with BE, 14: 113–120

    Google Scholar 

  • Panzeca, T. 1992: Shakedown and limit analysis by the boundary integral equation method. Eur. J. Mech., A/Solids, 11: 685–699

    Google Scholar 

  • Polizzotto, C.; Borino, G.; Fuschi, P. 1992: A finite interval approach to evolutive elastic-plastic analysis. In: P.Ladevenze and O. C.Zienkiewicz, editors, New Advances in Computational Structural Mechanics, 23–27, Amsterdam, Elsevier

    Google Scholar 

  • Polizzotto, C.; Zito, M. 1994: Variational formulations for coupled BE/FE methods in elastostatics. Z. angew. Math. Mech., 74: 533–543

    Google Scholar 

  • Polizzotto, C.; Zito, M. 1990: A variational formulation of the BEM for elastic-plastic analysis. In: G.Kuhn and H.Mang, editors, Discretization Methods in Structural Mechanics, Berlin, Springer Verlag, 201–210

    Google Scholar 

  • Polizzotto, C. 1988: An energy approach to the boundary element method, part I: elastic solids. Comp. Meth. Appl. Mech. Eng., 69: 167–184

    Google Scholar 

  • Polizzotto, C. 1988: An energy approach to the boundary element method, part II: elastic-plastic solids. Comp. Meth. Appl. Mech. Eng., 69: 263–276

    Google Scholar 

  • Romano, G.; Rosati, L.; Marotti De Sciarra, F. 1993: A variationally theory for finite-step elastoplastic problems. Int. J. Solids Structures, 30: 2317–2334

    Google Scholar 

  • Simo, J. C.; Govindjee, S. 1990: Nonlinear B-stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity. Int. J. Num. Meth. Engn., 31: 151–176

    Google Scholar 

  • Simo, J. C.; Kennedy, J. G.; Govindjee, S. 1988: Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. Int. J. Num. Meth. Engng. 26: 2161–2185

    Google Scholar 

  • Sirtori, S.; Maier, G.; Novati, G.; Miccoli, S. 1992: A Galerkin symmetric boundary element method in elasticity: formulation and implementation. Int. J. Num. Meth. Engng., 35: 255–282

    Google Scholar 

  • Sirtori, S.; Miccoli, S.; Korach, E. 1993: Symmetric coupling of finite elements and boundary elements. In: H.Kane, G.Maier, N.Tosaka and S. N.Atluri (eds), Advances in Boundary Element Techniques, 407–427, Berlin, Springer-Verlag

    Google Scholar 

  • Sirtori, S. 1979: General stress analysis method by means of integral equations and boundary elements. Meccanica, 14: 210–218

    Google Scholar 

  • Telles, J. C. F.; Carrer, J. A. M. 1991: Implicit procedures for the solution of elasto-plastic problems by the BEM, Math. Computational Modelling, 15: 303–311

    Google Scholar 

  • Tralli, A. 1984: On the analysis of membranes stretched over a unilateral support by the BEM. Civ. Engn. Systems, 1: 311–318

    Google Scholar 

  • Willam, K. J.; Dietsche, A. 1990: Fundamental aspects of strain-softening descriptions, Fracture Mechanics of Concrete Structures, Ed. Z. P. Bazant, Elsevier Appl. Sci., pp. 227–230

  • Zienkiewicz, O. C.; Kelly, D. W. Bettess, P. 1977: The coupling of the finite element method and boundary solution procedures. Int. J. Num. Meth. Engn., 11: 355–375

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Structural Engineering and Mechanical Engineering, Technical University (Politecnico) of Milan, Italy

    G. Maier, S. Miccoli & U. Perego

  2. Department of Structural Mechanics and Design Automation, University of Trento, Trento, Italy

    G. Novati

Authors
  1. G. Maier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Miccoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. U. Perego
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Novati
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S. N. Atluri, 18 August 1995

Dedicated to J. C. Simo

Extended version of a key-note lecture at the 4th International Conference on Computational Plasticity, Barcelona, April 3–6, 1995.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maier, G., Miccoli, S., Perego, U. et al. Symmetric Galerkin boundary element method in plasticity and gradient plasticity. Computational Mechanics 17, 115–129 (1995). https://doi.org/10.1007/BF00356484

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00356484

Keywords

Search

Navigation