Internal springs distribution for quasi brittle fracture via Symmetric Boundary Element Method

Abstract

Abstract In this paper the symmetric boundary element formulation is applied to the fracture mechanics problems for quasi brittle materials . The basic aim of the present work is the development and implementation of two discrete cohesive zone models using Symmetric Galerkin multi-zone Boundary Elements Method . The non-linearity at the process zone of the crack will be simulated through a discrete distribution of nodal springs whose generalized (or weighted) stiffnesses are obtainable by the cohesive forces and relative displacements modelling. This goal is reached coherently with the constitutive relation σ − Δ u that describes the interaction between mechanical and kinematical quantities along the process zone. The cracked body is considered as a solid having a “particular” geometry whose analysis is obtainable through the displacement approach employed in [Panzeca, T., Salerno, M., 2000. Macro-elements in the mixed boundary value problems. Comp. Mech. 26, 437-446; Panzeca, T., Cucco, F., Terravecchia, S., 2002b. Symmetric Boundary Element Method versus Finite Element Method . Comput. Meth. Appl. Mech. Engrg. 191, 3347-3367] by some of the present authors in the ambit of the Symmetric Galerkin Boundary Elements Method (SGBEM). In this approach the crack edge nodes are considered distinct and the analysis is performed by evaluating all the equation system coefficients in closed form [Guiggiani, M., 1991. Direct evaluation of hypersingular integrals in 2D BEM. In: Proceedings of the 7th GAMM Seminar on Numerical Techniques for Boundary Element Methods. Kiel, Germany; Gray, L.J., 1998. Evaluation of singular and hypersingular Galerkin boundary integrals: direct limits and symbolic computation. In: Sladek, J., Sladek V. (Eds.), Singular Integrals in Boundary Element Methods, Computational Mechanics Publications, Southampton; Panzeca, T., Fujita Yashima, H., Salerno, M., 2001. Direct stiffness matrices of BEs in the Galerkin BEM formulation. Eur. J. Mech. A/Solids 20, 277–298; Terravecchia, S., 2006. Closed form in the Symmetric Boundary Element Approach. Eng. Anal. Bound. Elem. Meth. 30, 479–488]. Some examples show the goodness of the methodology proposed through a comparison with other formulations [Barenblatt, G.I., 1962. Mathematical theory of equilibrium cracks in brittle fracture . Adv. Appl. Mech. 7, 55–129; Saleh, A.L., Aliabadi, M.H., 1995. Crack growth analysis in concrete using Boundary Element Method. Eng. Fract. Mech. 51, 533–545; Aliabadi, M.H., Saleh, A.L., 2002. Fracture mechanics analysis of cracking in plain and reinforced concrete using boundary element method. Eng. Fract. Mech. 69, 267–280]. In these examples the applied loads and the length of the process zone are a priori given and kept fixed during the analysis in order to check the constitutive behavior along the process zone.