Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 227 damage development through a discrete element model. In this context, this paper presents a method of discrete elements formed by bars. This method consists essentially in representing the continuum through a set of uniaxial elements whose stiffness is equivalent to the solid that must be modeled, model masses are lumped in the nodes connecting the elements. The non-linear behavior is captured using a very simple bilinear constitutive law applied in each element. The area below this constitutive law is coherent with the energy balance of the system. The governing equations obtained as result of the spatial discretization are integrated in the time domain using an explicit scheme of integration (Central Difference Method). In the present paper two applications are presented. In the first one, a plate of Functionally Grade Material is studied when the variation of the critical crack path is simulated for different levels of heterogeneity in the mechanical properties. The Functionally Graded Materials (FGM) are a new generation of engineering composites characterized for having a smooth variation of mechanical/thermal, or electromagnetic properties. They are new advanced multifunctional materials, which are tailored to take advantage of their constituents, for example, in a ceramic/metal FGM, heat and corrosion resistance of ceramics works together with mechanical strength and toughness of metals. To carry out an application of FGMs, scientific knowledge of fracture and damage tolerance are important for improving their structural integrity. For this kind of problems, the Finite Element Method with cohesive interfaces is usually employed (Xu and Needleman [1]; Paulino and Zanhg [2]). In the second application, the lattice discrete element method here presented is applied in the simulation of subcritical crack growth in a heterogeneous material. In this example, the proposed strategy to measure crack propagation velocity is commented. Other approaches in the simulation of subcritical crack propagation due to fatigue can be quoted, among them the works of Xu and Yuan [3] and Unger et al [4], using the extended Finite Elements Method, and the works of Yang & Cheng [5], Yamaguchi et al [6] and Siegmund [7], that combine the Finite Elements Method with cohesive interface techniques. D ISCRETE ELEMENT MODEL DESCRIPTION he Discrete Element Method employed in this paper was proposed by Riera [8] and is based on the representation of a solid by means of a cubic arrangement of elements able to carry only axial loads. The discrete elements representation of the orthotropic continuum was adopted to solve structural dynamic problems by means of explicit numerical integration of the equations of motion, assuming the mass lumped at the nodes. Each node has three degrees of freedom, corresponding to the nodal displacements in the three orthogonal coordinate directions. Fig. 1a,b illustrates the basic bar arrangement used in this approach. The equations that relate the properties of the elements to the elastic constants of an isotropic medium are:       2 2 9 8 9 2     ,           ,          ,      4 8 18 24 3 l d A E L E A E L E                (1) in which E and  denote the Young’s modulus and the Poisson’s ratio, respectively, while l A and d A represent the areas of longitudinal and diagonal elements. The resulting equations of motion may be written in the well-known form:     0 r Mx C x F t P t           (2) in which x  represents the vector of generalized nodal displacements, M is the diagonal mass matrix, C is the damping matrix, also assumed diagonal,   r F t  is the vector of internal forces acting on the nodal masses and   P t  is the vector of external forces. Obviously, if M and C are diagonal, Eq. 2 is not coupled. Then the explicit central finite differences scheme may be used to integrate Eq. 2 in the time domain. Since the nodal coordinates are updated at every time step, large displacements can be accounted for in a natural and efficient manner. Non-linear constitutive model for material damage The softening law for quasi-brittle materials, proposed by Hillerborg [9], was adopted to handle the behavior of the materials simulated here by means of the triangular constitutive relationship for the LDEM bars, presented in Fig. 1c, which allows accounting for the irreversible effects of crack nucleation and propagation. The area under the force vs. strain curve (the area of the triangle OAB) is related to the energy density necessary to fracture the element area of influence. Thus, for a given point P on the force vs. strain curve, the area of the triangle OAP quantifies the energy density dissipated by damage. T