Abstract
In this paper the integral equation approach is developed to describe elastic-damaging materials. An isotropic damage model is implemented to study nonlinear structural problems involving localisation phenomena. Especially for the cases that exhibit stress or strain concentrations, an integral approach can be recommended. Besides, the technique is able to represent well high gradients of stress/strain. The governing integral equations are discretised by using quadratic isoparametric elements on the boundary and quadratic continuous/discontinuous cells in the zone where the nonlinear phenomenon occurs. Two numerical examples are presented to show the physical correctness and efficiency of the proposed procedure. The results are compared with the local theory and they turn out to be free of the spurious sensitivity to cell mesh refinement.
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Alessandri C, Mallardo V, Tralli A (2000) Nonlocal continuum damage: a B.E.M. formulation. In: Atluri SN, Brust FW (eds) Advances in computational engineering & sciences, vol II. Tech Science Press, USA, pp 1293–1298
Aliabadi MH (2002) The boundary element method, vol 2: applications in solids and structures. Wiley, London
Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech ASCE 128: 1119–1149
Benallal A, Botta AS, Venturini WS (2006) On the description of localization and failure phenomena by the boundary element method. Comput Methods Appl Mech Eng 195: 5833–5856
Benvenuti E, Borino G, Tralli A (2002) A thermodynamically consistent nonlocal formulation for damaging materials. Eur J Mech A/Solids 21: 535–553
Bonnet M (1995) Boundary integral equation methods for solids and fluids. Wiley, London
Borino G, Failla B, Parrinello F (2003) A symmetric nonlocal damage theory. Int J Solids Struct 40: 3621–3645
Brebbia CA, Telles JCF, Wrobel LC (1984) Boundary element techniques. Springer-Verlag, NY
Bui HD (1978) Some remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations. Int J Solids Struct 14: 935–939
Comi C, Perego U (2001) Numerical aspects of nonlocal damage analyses. Eur J Finite Elem 10: 227–242
Crisfield MA (1991) Nonlinear finite element analysis of solids and structures, vol 1. Wiley, Chichester
Garcia R, Florez-Lopez J, Cerrolaza M (1999) A boundary element formulation for a class of non-local damage models. Int J Solids Struct 36: 3617–3638
Guiggiani M, Gigante A (1990) A general algorithm for multidimensional Cauchy Principal Value integrals in the Boundary Element Method. J Appl Mech 57: 906–915
Jirásek M (1998) Nonlocal models for damage and fracture: comparison of approaches. Int J Solids Struct 35: 4133–4145
Jirásek M (1999) Computational aspects of nonlocal models. In: Proceedings of ECCM ’99—European conference on computational mechanics, August 31–September 3, Munchen, Germany
Jirásek M, Patzak B (2002) Consistent tangent stiffness for nonlocal damage models. Comput Struct 80: 1279–1293
Jirásek M, Rolshoven S (2003) Comparison of integral-type nonlocal plasticity models for strain-softening materials. Int J Eng Sci 41: 1553–1602
Krajcinovic D (1996) Damage mechanics. North-Holland, Amsterdam
Krishnasamy G, Rizzo FJ, Rudolphi TJ (1992) Hypersingular boundary integral equations: their occurrence, interpretation, regularization and computation. In: Banerjee PK, Kobayashi S (eds) Developments in boundary element method, vol 7. Advanced dynamic analysis, chapter 7. Elsevier, Amsterdam
Lachat JC (1975) A further development of the boundary integral techniques for elastostatics. Ph.D. Thesis, University of Southampton
Lin FB, Yan G, Bažant ZP, Ding F (2002) Nonlocal strain-softening model of quasi-brittle materials using boundary element method. Eng Anal Bound Elem 26: 417–424
Mallardo V (2004) Localisation analysis by BEM in damage mechanics. In: Leitão VMA, Aliabadi MH (eds) Advances in boundary element techniques V. EC Ltd., UK, pp 155–160
Mallardo V, Alessandri C (2004) Arc-length procedures with BEM in physically nonlinear problems. Eng Anal Bound Elem 28: 547–559
Salvadori A (2002) Analytical integrations in 2D BEM elasticity. Int J Numer Methods Eng 53: 1695–1719
Sfantos GK, Aliabadi MH (2007a) A boundary cohesive grain formulation for modelling intergranular microfracture in polycrystalline brittle materials. Int J Numer Methods Eng 69: 1590–1626
Sfantos GK, Aliabadi MH (2007b) Multi-scale boundary element modelling of material degradation and fracture. Comput Methods Appl Mech Eng 196: 1310–1329
Sládek J, Sládek V (1983) Three-dimensional curved crack in an elastic body. Int J Solids Struct 19: 425–436
Sládek J, Sládek V., Bažant Z.P. (2003) Nonlocal boundary integral formulation for softening damage. Int J Numer Methods Eng 57: 103–116
Zhang X, Zhang X (2004) Exact integration for stress evaluation in the boundary element analysis of two-dimensional elastostatics. Eng Anal Bound Elem 28: 997–1004
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Mallardo, V. Integral equations and nonlocal damage theory: a numerical implementation using the BDEM. Int J Fract 157, 13–32 (2009). https://doi.org/10.1007/s10704-008-9297-0
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DOI: https://doi.org/10.1007/s10704-008-9297-0