PSI - Issue 64

Rebecca Grazzini et al. / Procedia Structural Integrity 64 (2024) 1532–1539 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 In phase 2, 1 ≤ ( ) ≤ 1 , the CML of the external interface is in the softening branch and the CML of the internal interface is in the linear ascending branch; the joint is governed by two equations: 1 ′′( ) −2 1 1 ( ) = 0 ; 2 ′′( ) −(( 1 + 2 ) 2 ( ) + ) =0 1 ′(0) = 0 ; 1 ( ) = 1 ; 1 ( ) = 2 ( ) ; 1 ′( ) = 2 ′( ) (6) where a is the x value along the joint separating the two parts. In phase 3, 1 ≤ ( ) ≤ 0 , both interfaces experience the softening stage at the loaded end, hence, the joint is governed by three equations: 1 ′′( ) −2 1 1 ( ) = 0 ; 2 ′′( ) −(( 1 + 2 ) 2 ( ) + )=0 ; 1 ′(0)=0 ; 1 ( ) = 1 ; 1 ( ) = 2 ( ) ; 1 ′( ) = 2 ′( ) ; 2 ( ) = 3 ( ) ; 2 ′( ) = 3 ′( ) (7) where a and b are the x value along the joint separating the three parts. In phase 4, 0 ≤ ( ) ≤ , inner interface keeps in the the softening stage at the loaded end while external interfaces cannot provide a contribution anymore, hence, the joint is divided into four subparts ruled by a series of equations and boundary conditions reported in (7), plus the following: 4 ′′( ) − 2 4 ( ) + =0 ; 3 ( ) = 4 ( ) ; 3 ′( ) = 4 ′( ) (8) where c shows a similar meaning of a and b . In the last phase, ( ) > only the friction contribution provided by the internal interfaces is transmitted at the loaded end, while the rest of the joints is experiencing all the other phases at different abscissa values, i.e. it is assumed that the length is effective. Hence, in addition to the sets of equations (7, 8) a fourth abscissa value, g , separating the fifth differential slip function is added to solve the problem: 5 ′′( ) − =0 ; 4 ( ) = 5 ( ) ; 4 ′( ) = 5 ′( ) (9) 1537 6 Fig. 3. Slip, s(x), strain , s’(x) , and shear stress, (s(x)), along the joint representative of the Sst-L-D sample corresponding to the last phase and for which s(L) = 0.729 mm, P = 963 N, a = 100 mm, b = 116 mm, c = 135 mm and g = 164 mm. The local diagrams of the solution and the related derivatives are shown in Figure 3. It is worth mentioning that the values of abscissa where the change in the differential equation occurs, have to be searched numerically at each increment of the slip at the loaded end. The results of the estimations carried out according to this model, or according to the simplified one, i.e., bilinear for the coated specimens, are reported in Figure 4 and in Table 3. For dry textiles, lime mortar provided remarkably higher peak values of shear for the external interface CML, +37% and +89% compared to cement and gypsum, respectively. Peak stresses are comparable for cement and lime in the internal interface, while gypsum provides less than half the value. Gypsum, although providing the least bonding peak stress, offers the greatest displacement capacity, 3.57 and 5 times greater than lime and cement, respectively. The friction contribution provided by the gypsum matrix is 8 and 1.33 times greater than lime and cement mortars, respectively. Also, gypsum showed the least decrease in the values of the friction plateau. Indeed, although the peak shear values are much lower compared to the lime and cement matrices, the global peak load is analogous.