Issue 47

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 329 a) b) c) Figure 6 : Scheme used for approximating the softening branch of the shear stress-slip law: a) phase 1 * 1 1 ; i e s s s s   ; b) phase 2 zone 1: * 1 1 ; i e s s s s   ; zone 2 - zone 3: * 1 1 ; i e s s s s   ; c) phase 3 zone 1: * 1 1 ; i e s s s s   ; zone 2 - zone 3: * 1 1 ; i e s s s s   ; zone 4 - zone 5: * 1 1 ; i e s s s s   . The analytical solution of this system depends on four constants of integration determined by introducing suitable boundary conditions. In particular, considering the attainment of the slip s 1 at the lower interface (generally, the lower interface exhibits higher values of shear stresses than the upper interface), the following conditions are indeed enforced:         1 0 0 0 0 0 p e c e c i L s L s        (14) The subsequent phase is then characterized by the upper interface in the pre-peak stage and the lower interface in the post-peak stage (Fig. 6.b). This phase ends when the upper interface attains the slip s 1 at the loaded end. The occurrence of this condition identifies the number of steps approximating the descending branch of the shear stress-slip law of the lower interface involved in this phase. The number of equations governing the problem depends on the number of the zones in which the systems is divided and, thus, on the number of steps considered in the approximate descending branch of the interface constitutive law. For instance, considering the case of the Fig. 6.b, it is supposed that there are three zones characterizing the behavior of the lower interface, which include two steps approximating the descending branch. Consequently, the differential equations composing the system are the following: - zone 1: 2 1 1 1 1 2 2 2 1 1 2 1 2 2 0 0 - ( ) 0 i e e i i i e e e d s K G s G s dx x L a b d s d s K G s dx dx                           (15) - zone 2: 2 2 1 2 2 2 2 2 2 2 2 2 2 2 0 - ( ) - 0 i e e i e e e d s K G s dx L a b x L b d s d s K G s dx dx                            (16) upper interface lowe interface (1) (2) (3) (1) (2) (3) (4) (5)