PSI - Issue 79

America Califano et al. / Procedia Structural Integrity 79 (2026) 306–312 309 2) the sphere is detached from the fibril. In this case, as soon as a given detachment criterion (to be defined) is met, the sphere detaches from the fibril and, therefore, the fibril becomes unloaded: =0 . From the detachment on, the fibril starts to relax, according to the SVS material behaviour: ( ) = ( ) − 2 (( − )) = ( ; , ) (6) where 2 = 0 ∞ ( ∞ − 0 ) .

Fig. 1. (a) Rigid sphere and a carpet of discrete viscoelastic springs (or fibrils); (b) Standard Viscoelastic Solid (SVS) model consisting of a parallel between a spring (with K 1 stiffness) and a series of a spring (with K 2 stiffness) and a damper (with η damping coefficient) Let’s suppose that at the initial time, = 0 , the sphere and the single fibril are in contact. If the sphere is moving upwards, the fibril starts to elongate and, when the detachment criterion is met, it detaches. Imposing, in the first instance, the detachment criterion = (where needs to be chosen accordingly) and solving eq. 5, the following can be obtained = 0 + ( + ) − − 1 ( − 0 ) (7) where = 0 + ( ) ( 0 + ) and =√ 12 02 + 2 ∞2 12 + 2 . By solving eq. 7 in t , the detachment time, 0 , can be computed. From the time instant = d0 the fibril relaxes according to eq. 6, then the fibril and the sphere will re-attach at a given time instant, 1 . To compute it, it is necessary to solve, in t , the equation ( 0 )= ( 0 ) which can be rewritten as follows: ( 0 ) − 2 (( − 0 )) = [ + ( 0 + )] − 2 (( − 0 )) (8) By iterating the above logic, the time instants of detachment ( ) and re-attachment ( ) can be easily computed. 3. Results and discussion Based on the methodological approach described in the previous Section, preliminary results are herein presented. First of all, let’s consider r as fixed, namely the computation is carried out for a single fibril only. Once the sinusoid parameters ( , , ) , the material constants ( 0 , ∞ , ), the sphere’s radius ( R ) and the detachment criterion ( ) are fixed, supposing that the sphere and the fibril are in contact at the initial time-instant, by iterating the logic expressed through eq. 1-8, the quantities ( ), ( ) and ( ) may be evaluated. They are plotted in Fig. 2: the left-side y -axis refers to displacements, while the right-side y -axis refers to ( ) . At t = 0 the