Issue 12

S. Marfia et alii, Frattura ed Integrità Strutturale, 12 (2010) 13-20 ; DOI: 10.3221/IGF-ESIS.12.02 14 phenomenon and to predict the possible decohesion of the FRP from the support material, special interface models are often adopted. The response of these special elements is mainly based on the damage and/or on plasticity models. On the other hand, the concrete as well as the masonry are quasi-brittle materials, whose mechanical response is characterized by damage with softening, which is due to the development of micro-cracks. Thus, two damage effects could be present in the quasi-brittle reinforced structural elements: the body damage, which develops inside the domain of the continuous body, and the interface damage, which occurs at the FRP – concrete or masonry interface. It has to be remarked that experimental evidences demonstrate that the detachment of the FRP from the support material occurs with the peeling of a thin layer from the external surface of the quasi-brittle material; this collapse behavior is due to the fact that the strength of the glue used to fix the FRP to the support is greater than the strength of the concrete or masonry support. From this observation, it can be deduced that the body damage and the interface damage cannot evolve independently one from the other, in other words, they are coupled. In particular, the interface damage has to depend on the distribution of the body damage. In the knowledge of the authors, the first paper in which a coupled body-interface damage model has been developed is due to Freddi and Frémond [1]. In the present paper a damage model is proposed for the quasi-brittle material. The damage evolution is governed by an equivalent strain variable. In order to prevent strain localization and strong mesh sensitivity of the solution, typical of cohesive heterogeneous materials subjected to damage and softening, an integral-type of nonlocal model based on the weighted spatial averaging of a strain-like quantity is developed. Regarding the interface, the damage is governed by the relative displacement occurring at bond. A suitable interface model which accounts for the mode I, mode II and mixed mode of damage is developed. The coupling between the body damage and the interface damage is performed ensuring that the interface damage is not lower than the body damage evaluated on the bond surface. An initial simple application is performed in order to assess the performances of the proposed model in reproducing the mechanical behavior of quasi-brittle material strengthened with external FRP materials. P OSITION OF THE PROBLEM mechanical system made of two bodies 1  and 2  , schematically illustrated in Fig. 1, is considered. The two bodies, subjected to body forces, surface forces and prescribed displacements, are connected by an adhesive interface 1 2     , which is assumed to have zero thickness. The problem is developed in the framework of small strain and displacement theory. The displacement fields of the two joined bodies are denoted as 1 u and 2 u , while the relative displacement at the typical point  x on the interface  is       1 2      s x u x u x . A local coordinate system on the interface   , T N x x is introduced. Figure 1 : Mechanical system of two bodies in adhesion. It is assumed that the body 1  and the interface  can develop damage, while the body 2  is assumed to behave as a linear elastic material. Thus, two damage variables are introduced, D  and D  , associated to the degradation state of the body 1  and of the interface  , respectively. In many physical problems, the interface failure is not due to the damage of the adhesive material joining the two bodies 1  and 2  , but it occurs because of the properties degradation of a thin layer of the body 1  ; in fact, the decohesion is A