Strain-softening Damage Modeling UsingBoundary Element Method

Abstract

A plasticity model with yield limit degradation is implemented in a boundary element program to study the fracture behavior of quasi-brittle materials. A special integration method is applied to deal with the singular integrations encountered in the volume integrals over internal cells. Strain-softening damage localizations are investigated. It is found that the damage tends to localize into a zone of one cell wide which leads to an incorrect result when different cell meshes are used in the analysis. Some sort of localization limiter has to be incorporated into the analysis in order to achieve meaningful results. INTRODUCTION In the simulation of fracture behavior of quasi-brittle materials such as concrete or ceramics, it is essential to take into account the evolution of a relative large fracture process zone forms in front of the crack tip. The process zone consists of hundreds of microcracks and its overall behavior can generally be simulated either by discrete-cracking models such as the fictitious crack model [4, 5] or smeared-cracking models such as strain-softening damage models [2, 3]. In this paper the fracture process zone is simulated by a plasticity model with yield limit degradation. When the stresses near the crack tip reach the yield surface, the yield surface will contract rather than expand. This results in a decrease in stresses during an increase in strains, i.e., a strain softening. The decrease in stresses simulates a material damage caused by mainly the formation of microcracks. Difficulty has been encountered in the finite element analysis involving strain-softening behavior. As different meshes are used in an analysis, Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 304 Boundary Elements different results are obtained. The results are not objective with respect to finite element meshes. Also, the energy dissipation of a structure at failure decreases with the refinement of the finite element mesh. It implies that the structure will fail at near zero energy dissipation when a very fine mesh is used. This is in contradiction with the reality. All these problems are considered due to damage localization [1,2,3]. In this paper, the plasticity model with yield limit degradation is incorporated into a boundary element program to study the damage localization problems. Many publications on the boundary element analysis of elastoplastic problems [7-14] have been seen since the first paper on the problem published in 1971 by Swedlow and Cruse [6]. However, the literature on the study of damage localization problems using boundary element approach is rarely seen. The initial stress boundary element method is applied to deal with the nonlinear damage problems, For localization problems, damage tends to localize into a small region of the body, and thus the internal cells required for integration purposes are needed only in a small region rather than the entire body. A special semi-analytical integration method for triangular cell is used to carry out the volume integration over the cell that contains a singular node. A rectangular panel with three different cell meshes are analyzed. It is shown that the results are not objective to the cell meshes even though all other conditions are kept the same. BOUNDARY ELEMENT FORMULATION FOR ELASTOPLASTIC ANALYSIS For elastoplastic problems based on small strain theory, the basic equations in incremental form are as follows: The governing equations are da^ + db* = 0 (1) with boundary conditions du;(x) = (x) x e I (2) cR (x) x e I I + T^T and total strain increment de^ = de^ 4de^= (du + dUjJ/2 (3) where de^ and de^ are elastic and plastic strain increments. T is the boundary of body 0 under consideration, da^ ,db; ,du;, dr; are the stress, body force (per unit volume), displacement, and traction increment respectively. du ± and df j are prescribed displacements and tractions on boundary I and r%. Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 305 The yield criterion is F(a%,k)= f(a$ <Kk) = 0 (4) where F and f are yield functions, $ is a work hardening function, and k is the work hardening parameter. The constitutive equations are de'u) = d<^ do, = DV^U (5) where D^= &pvl(l-2v)] 6fo + M<Wji + W