10 It is important to point out that for ν = 0.25, the correspondence between the equivalent discrete solid and the isotropic continuum is complete. On the other hand, discrepancies appear in the shear terms for values of ν ≠ 0.25. These discrepancies are small and may be neglected in the range 0.20≤ ν ≤0.30. For values outside this range, a different array of elements for the basic module should be used (see Nayfeh and Hefzy, 1978). It is interesting to note that while no lattice model can exactly represent a locally isotropic continuum, it can also be argued that no perfect locally isotropic continuum exists in practical engineering applications. Isotropy in solids is a bulk property that reflects the random distribution of the constituent elements orientation. The equations of motion are obtained from equilibrium conditions of all forces acting on the nodal masses, resulting in a system of equations of the form: ( ) ( ) + + - 0 t t = Mx Cx F P && & (6) in which x, x& and x&& denote vectors containing the nodal displacements, velocities and accelerations, respectively, while M and C are the mass and damping matrices, both are diagonals and the damping matrix is proportional only to the mass. The vectors ( )t F and P (t) contain the internal and external nodal load. Following the Courant-Friedrichs-Lewy criterion (see Bathe, 1996), the stability of the integration scheme is ensured by limiting the size of the time step. For the present implementation, the elements in the worst condition (this is, those requiring the smallest Δt) are the diagonal ones. Thus, considering the relationships in Equations (4) and (5), the limitation to the time increment is: 0.6L t Cρ Δ ≤ (7) where Cρ is the longitudinal wave speed, / C E ρ ρ = (8) The truss-like DEM has a natural ability to model cracks. They can be introduced into the models as pre-existent features and as the irreversible effect of crack nucleation and propagation. Pre-existent cracks are modeled using a simple strategy which consists in duplicate the nodes located on the crack surface together with the elimination of the elements connecting the material on both sides of the crack. This way, the DEM discretization is allowed to “open” along the crack locus, and pre-existent cracks are integrated seamlessly into the DEM formulation. Crack nucleation and propagation make use on non-linear constitutive models for material damage which allow the elements to break when they attain a critical condition. The details about the formulation and implementation of these non-linear constitutive models are given in the next section. 3.1 Non-linear constitutive models for material damage Rocha et al. (1991) extended the lattice method here implemented (DEM) to model quasi brittle materials. To this end, they introduced the bilinear constitutive relationship illustrated in Figure 2. This constitutive law aims to capture the irreversible effects of crack nucleation and propagation by accounting for the reduction in the element load carrying capacity. The area under the force versus strain curve (the area of the triangle OAB in Figure 2) is the energy density necessary to fracture the area of influence of the element. Thus, for a given point P on the force vs. strain curve, the area of the triangle OPC represents the reversible elastic energy density stored in the element, while the area of the triangle OAP is the dissipated fracture energy density. Once the dissipated energy density equals the fracture energy, the element fails and loses its load carrying capacity. On the
RkJQdWJsaXNoZXIy MjM0NDE=