11 other hand, in the case of compressive loads the material behaves as linear elastic. Thus, the failure in compression is induced by indirect traction. This assumption is reasonable for quasi-brittle materials for which the ultimate strength in compression is usually from five to ten times larger than that in tension (see Kupfer and Gerstle, 1973). Figure 2. Triangular constitutive law adopted for DEM uni-axial elements. Constitutive parameters and symbols in Figure 2 are (see Rocha et al., 1991; and Riera and Rocha, 1991): • Force, F: the element axial force as a function of the longitudinal strain ε. • Element area, A: depending whether a longitudinal or a diagonal element is considered the values for Al or Ad, see equations (4) and (5), should be adopted. • Element stiffness: depending whether a longitudinal or a diagonal element is obtained multiplying the Young Modulus (E) by Al or Ad, should be adopted. • Length of the DEM module, L. • Specific fracture energy, Gf: the fracture energy per unit area, which is coincident with the material fracture energy, Gc. • Equivalent fracture area, f iA : this parameter enforces the condition that the energy dissipated by the fracture of the continuum material and its discrete representation are equivalent. With this purpose, a cubic sample with dimensions L×L×L is considered. The energy dissipated when a continuum sample fractures into two parts due to a crack parallel to one of its faces is 2 f f G G L Γ= Δ= (9) where Δ is the fracture area. By contrast, the energy dissipated when the DEM module fractures in two parts has to account for the contribution of five longitudinal elements (four coincident with the module edges and one internal one) and four diagonal elements, see Figure 1(a). Then, the energy dissipated by a DEM module can be written as follows 2 2 DEM 2 4 0.25 4 3 f A A A G c c c L ⎛ ⎞ ⎛ ⎞ Γ = + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (10) Damage energy, Udmg Elastic strain energy, Uel EAi F ε εp P O A C B εr
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