Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 228 Once the damage energy density equals the fracture energy, the element fails and loses its load carrying capacity. On the other hand, under compression the material is assumed linearly elastic. Thus, failure in compression is induced by indirect tension. Figure 1 : LDEM discretization strategy: (a) basic cubic module, (b) generation of a prismatic body, (c) Bilinear constitutive law adopted for LDEM uniaxial elements. Constitutive parameters and symbols shown in Fig. 1c are defined below: the element axial force F depends on the axial strain ε. The area associated to each element was given in Eq. 1 for longitudinal and diagonal elements. An equivalent fracture area, f i A , of each element is defined in order to satisfy the condition that the energy dissipated by fracture of the continuum and by its discrete representation are equivalent. With this purpose, fracture of a cubic sample of dimensions L  L  L is considered. The energy dissipated by fracture of a continuum cube due to a crack parallel to one of its faces is presented in Eq. 3: 2 f f Г G G L    (3) 2 1 2 4 4 4 3 DEM f A A A Г G c c c L                       (4) in Eq. 3, Λ represents the actual fractured area, i.e., L 2 . On the other hand, the energy dissipated when a LDEM module of dimensions L  L  L fractures in two parts consists of the contributions of five longitudinal elements (four coincident with the module edges and one internal element) and four diagonal elements. Then, the energy dissipated by a LDEM module can be written as shown in Eq. 4 (Kosteski et al [10]). The first term between brackets, in Eq. 4, accounts for the four edge elements, the second term for the internal longitudinal element, while the third term represents the contribution of the four diagonal elements. The coefficient c A is a scaling parameter used to establish the equivalence between Γ and ΓDEM. Thus: 2 2 22 3 f f A G L G c L        (5) 2 2 3 4  ,   22 22 f f l d A L A L               (6) From Eq. 5 it follows that 3 / 22 A c  . Finally, the equivalent fracture areas of the longitudinal and the diagonal elements are presented in Eq. 6. The critical failure strain ( p  ) is defined as the largest strain attained by the element before damage initiation (point A in Fig. 1c). The relationship between p  and the specific fracture energy, f G , is given in terms of Linear Elastic Fracture Mechanics as:   2 1 f p f G R E     (7)