Issue34

L. E. Kosteski et alii, Frattura ed Integrità Strutturale, 34 (2015) 226-236; DOI: 10.3221/IGF-ESIS.34.24 229 1 f R Y a  (8) in which f R is the so-called failure factor, which may account for the presence of an intrinsic defect of critical size a . f R may be expressed in terms of critical fracture dimension of a , as presented in Eq. 8, where Y represents the dimensionless parameter that depends on both the specimen and the crack geometry. Notice that, if the characteristic dimension of the simulated body, L , is smaller than the intrinsic critical crack size, a , the collapse will be stable. On the other hand, if L > a , an instable global collapse is expected. Also, one could write the failure factor, f R , in terms of the fragility number, s , proposed by Carpinteri [11], this dimensionless parameter measures the fragility quality of a determined structure. In the expressions presented in Eq. 9, one can see the fragility number, s , in terms of the critical stress intensity factor, IC K , and the critical stress, p  ; or in terms of the critical specific fracture energy, f G , and the critical strain, p  ; or, using Eq. 7 and Eq. 8, in terms of the LDEM parameters, as shown in Eq. 10. f IC p p G E K s L E L     (9) 1 f a s Y L R L   (10) If two models are built with different materials and different dimensions, one can expect similar global mechanical behavior if both models have the same s value.; The element loses its load carrying capacity when the limit strain, r  , is reached (point B in Fig. 1c). This value must satisfy the condition that, upon failure of the element, the dissipated energy density equals the product of the element fracture area, f A , and the specific fracture energy, , f G divided by the element length. Hence:   2 0 2 r f f i r p i i G A K EA F d L        (11) 2 2 f f i r p i i G A K E A L                    (12) 2 2 f f i cr p i G A L E A                 (13) in which the sub index i is replaced by l or d depending on whether the element under consideration is longitudinal or diagonal. The coefficient r K is a function of the material properties and the element length i L . In order to guarantee the stability of the algorithm, the condition   1 r K  must be satisfied. In this sense, it is interesting to define the critical element length cr L (see Eq. 13).   3/ 22 f l l A A         (14)   3 /11 f d d A A         (15)