Integral equations and nonlocal damage theory: a numerical implementation using the BDEM. (English) Zbl 1308.74158

Summary: 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.


74S15 Boundary element methods applied to problems in solid mechanics
74R99 Fracture and damage
74C99 Plastic materials, materials of stress-rate and internal-variable type


Full Text: DOI


[1] 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
[2] Aliabadi MH (2002) The boundary element method, vol 2: applications in solids and structures. Wiley, London
[3] Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech ASCE 128: 1119–1149
[4] 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 · Zbl 1122.74051
[5] Benvenuti E, Borino G, Tralli A (2002) A thermodynamically consistent nonlocal formulation for damaging materials. Eur J Mech A/Solids 21: 535–553 · Zbl 1038.74006
[6] Bonnet M (1995) Boundary integral equation methods for solids and fluids. Wiley, London
[7] Borino G, Failla B, Parrinello F (2003) A symmetric nonlocal damage theory. Int J Solids Struct 40: 3621–3645 · Zbl 1038.74509
[8] Brebbia CA, Telles JCF, Wrobel LC (1984) Boundary element techniques. Springer-Verlag, NY
[9] Bui HD (1978) Some remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equations. Int J Solids Struct 14: 935–939 · Zbl 0384.73008
[10] Comi C, Perego U (2001) Numerical aspects of nonlocal damage analyses. Eur J Finite Elem 10: 227–242 · Zbl 1056.74055
[11] Crisfield MA (1991) Nonlinear finite element analysis of solids and structures, vol 1. Wiley, Chichester · Zbl 0809.73005
[12] 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 · Zbl 0940.74075
[13] 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 · Zbl 0735.73084
[14] Jirásek M (1998) Nonlocal models for damage and fracture: comparison of approaches. Int J Solids Struct 35: 4133–4145 · Zbl 0930.74054
[15] Jirásek M (1999) Computational aspects of nonlocal models. In: Proceedings of ECCM ’99–European conference on computational mechanics, August 31–September 3, Munchen, Germany
[16] Jirásek M, Patzak B (2002) Consistent tangent stiffness for nonlocal damage models. Comput Struct 80: 1279–1293
[17] Jirásek M, Rolshoven S (2003) Comparison of integral-type nonlocal plasticity models for strain-softening materials. Int J Eng Sci 41: 1553–1602 · Zbl 1211.74039
[18] Krajcinovic D (1996) Damage mechanics. North-Holland, Amsterdam
[19] 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
[20] Lachat JC (1975) A further development of the boundary integral techniques for elastostatics. Ph.D. Thesis, University of Southampton
[21] 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 · Zbl 1024.74048
[22] 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
[23] Mallardo V, Alessandri C (2004) Arc-length procedures with BEM in physically nonlinear problems. Eng Anal Bound Elem 28: 547–559 · Zbl 1130.74467
[24] Salvadori A (2002) Analytical integrations in 2D BEM elasticity. Int J Numer Methods Eng 53: 1695–1719 · Zbl 1041.74076
[25] 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 · Zbl 1194.74503
[26] Sfantos GK, Aliabadi MH (2007b) Multi-scale boundary element modelling of material degradation and fracture. Comput Methods Appl Mech Eng 196: 1310–1329 · Zbl 1173.74459
[27] Sládek J, Sládek V (1983) Three-dimensional curved crack in an elastic body. Int J Solids Struct 19: 425–436 · Zbl 0512.73086
[28] Sládek J, Sládek V., Bažant Z.P. (2003) Nonlocal boundary integral formulation for softening damage. Int J Numer Methods Eng 57: 103–116 · Zbl 1062.74644
[29] 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 · Zbl 1130.74478
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.