Issue 29

S. de Miranda et alii, Frattura ed Integrità Strutturale, 29 (2014) 293-301; DOI: 10.3221/IGF-ESIS.29.25 295 debonding of tile flooring (see e.g. [6]). With this notation, the stiffness coefficients of the springs modelling the adhesive are expressed as * , a a t a E k t  and , a a l a G k t  for the transverse and the longitudinal springs respectively, while, as regards the substrate, it follows * s s s E k t  . No vertical loads are applied to the flooring, self-weight is neglected and constant substrate shrinkage 0  is imposed. The assumed axial deformed configuration of the flooring due to substrate shrinkage is shown in Fig. 3. The imposed substrate shrinkage 0  induces a displacement 0 u along the substrate height, which, in turn, induces a tile compression acting at the interface between adhesive and tile (i.e. eccentric with respect to tile axis). This compression is equal to   * , 0 a l k u u  , where * u denotes the longitudinal displacement at the bottom position of the tile. According to the classical Euler-Bernoulli hypothesis, * u can be expressed as follows: * ' 2 t t u u v   (1) where ' v denotes the derivative of v with respect to tile axis direction. Of course, the above assumption leads to a coupled axial-flexural response of the tile. In this regard, it should be noted that a beam on a Pasternak foundation has been already used by Cocchetti et al. [6] to model tile debonding with a simple but effective mathematical formulation which does not account for the inherent eccentricity due to the geometrical configuration of the system. In the present work, by considering the eccentric position of the compression transferred by the adhesive to the tile, the coupled axial-flexural response of the tile is recovered. Furthermore, the assumption of flexible substrate, modeled as a layer of vertical springs interconnected by a shear layer, allows to enrich the model still preserving its formulation relatively simple. Figure 3 : Longitudinal interaction forces at tile-adhesive interface due to substrate shrinkage. The potential energy  Π of the tile-adhesive-substrate system can be expressed as: T A S Π Π Π Π     (2) where T A S Π , Π , Π are the functionals of the potential energy associated to the tile, the adhesive and the substrate, respectively: