PSI - Issue 79

America Califano et al. / Procedia Structural Integrity 79 (2026) 306–312

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radial coordinate R radius of the rigid sphere stress on the single fibril ̇ first derivative of the stress on the single fibril threshold stress level for the single fibril SVD Standard Viscoelastic Solid time (variable) 0 initial time instant of contact 1 first time instant of re-attachment 0 first time instant of detachment i-th time instant 2. Methodology

The considered system is based on a rigid sphere, of radius R , which indents a carpet of discrete viscoelastic springs, represented in Fig. 1a. The material behaviour of each spring can be modelled considering the Standard Viscoelastic Solid (SVD) model, which consists 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), see Fig. 1b, and whose constitutive equation is Eq. 1 + 2 ̇ = 1 + ( 1 + 2 ) 2 ̇ (1) where is the tension on the fibril and is its displacement. Introducing the instantaneous ( K ∞ ) and the relaxed ( K 0 ), eq. 1 becomes + ( ∞ − 0 ) ̇ = 0 + ∞ ( ∞ − 0 ) ̇ (2) Imposing a harmonic displacement law for the sphere ( ) = + ( + ) (3) where = ( ) = 2 2 is the quantity that introduced the dependence of the displacement from the radial coordinate, r . By integrating eq. (2), the following solution is obtained, considering r (and, thus, ) fixed for the moment: ( ) = − 1 ( − ) ( )+ 0 [1− − 1 ( − ) ]+ √ 12 02 + 2 ∞2 12 + 2 [ ( + ) − ( + ) − 1 ( − ) ] (4) where = √( 1 0 ) 2 + ( ∞ ) 2 , = + ∞ 1 0 , = − 1 = + ∞ 1 0 − 1 . Two different cases may be identified, thus leading to the particularization of eq. 4, still considering r fixed: 1) the sphere and the fibril are in contact/adhesion. In this case the sphere and the fibril move simultaneously, namely ( ) = ( ) . Therefore eq. 4 becomes: ( ) = − 1 ( − ) { ( )+∫ 1 ( − ) [ 1 0 ( ) + ∞ ̇ ( )] }= ( ; , ) (5) where 1 = ( ∞ − 0 ) ;