Issue 30

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 30 (2014) 145-152; DOI: 10.3221/IGF-ESIS.30.19 146 marble after environmental exposition decreases due to grain decohesion. For instance, Royer-Carfagni [3] showed that thermal action produces self-equilibrated stress states at calcite grain (whose size ranges typically between 100 and 500  m) interfaces, which are responsible of progressive damage in the material leading to initiation and propagation of intergranular cracks. In the present paper, following some recent works by the authors [10-11], a theoretical model to estimate the progressive bowing and the thermal fatigue of marble slabs submitted to temperature cycles is briefly recalled and applied to a Carrara marble sample whose microstructure is experimentally analysed in details. The model, developed within the framework of LEFM, takes into account the mechanical microstructural characteristics of the marble as well as the actual cyclic temperature field in the material. The slabs are subjected to a thermal gradient along their thickness as well as to thermal fluctuation on the two sides of the slab due to daily and seasonal temperature excursions. This thermal action causes a stress field which can locally determine microcracks due to decohesion of grains. Stress intensification near the cracks occurs and leads to crack propagation in the slab. Such crack propagation under thermal actions is evaluated and the corresponding deflection (bowing) is calculated. A Monte Carlo simulation, where the orientation distribution of grain optic axis along the slab thickness is varied randomly, is performed in order to quantify the scatter of bowing evolution. T HE MECHANICS OF THERMAL DEGRADATION one-dimensional theoretical model of a marble slab is considered. The thickness of the slab is h, while x and z are the through-thickness and longitudinal axis, respectively (x = 0 corresponds to the inner surface of the slab). The kinematic assumptions are those of beam theory. The material is mechanically linear elastic, homogeneous and isotropic, while the thermal expansion is heterogeneous along the panel thickness. The resulting longitudinal normal stress is ( ) (1 ) ( , ) ( , ) ( , ) (1 2 )(1 ) (1 2 ) z z z x E E x t x t T x t               (1) where  T(x, t) is the temperature function of time t and space x. Considering the relevant case of a slab with hinged ends, we have ( , ) ( ) ( ) z x t A t x B t    where A and B, functions of time, are determined by the condition of axial force and bending moment being equal to zero at the slab ends. As has been mentioned in the Introduction section, the thermal expansion of calcite grains is anisotropic. In the present model, where thermal expansion is assumed to be heterogeneous and hence the coefficient z  is a function of the through-thickness coordinate x, the thermal expansion heterogeneity is linked to the aforementioned thermal anisotropy of calcite grains. Now let us assume that the thermal expansion of calcite grains is orthotropic (1-2 are the material thermal expansion axes, characterized by the coefficients 1  and 2  , respectively), the longitudinal thermal expansion coefficient along the z-axis z  is obtained from 4 4 1 2 ( ) cos ( ) sin ( ) z x x x        (2) where  is the angle formed by the material thermal expansion axes with the longitudinal axis z. In the light of the above, the thermal expansion heterogeneity (see ( ) z x  ) is due to the different orientation (see ( ) x  ) of the material thermal expansion axes of each grain. Therefore ( ) z x  is hereafter assumed to be a stepwise varying function where a jump in such a function occurs at each calcite grain boundary. Assuming a geometrically simple grain arrangement for our model, we might have a stack of layers with different values of z  in each layer, where the layer thickness might be taken as the calcite grain mean dimension d . In the following, we consider a random distribution of thermal axis orientation in each grain with some specific statistical characteristics. Under environmental exposure a diffuse cracking, mainly developing at the calcite grain boundaries, can take place at the external surface of the marble slab. Such a diffuse cracking can be incorporated in the present model to calculate the consequent convex bowing of the slab due to thermal loading. Intergranular cracking due to decohesion of calcite grains are treated as equivalent multiple edge cracks. To this end a crack density parameter n , corresponding to a number of equivalent external edge cracks, can be defined. Assuming hence that intergranular cracking occurs, the crack density A