Issue 12

S. Marfia et alii, Frattura ed Integrità Strutturale, 12 (2010) 13-20 ; DOI: 10.3221/IGF-ESIS.12.02 17 The derivatives of the free energies with respect to the observable and internal variables lead to the state equations. The derivative of the pseudo-potentials of dissipation with respect to the damage variable allows to derive the evolutionary law for the damage. Balance equation are recovered in the FF model using the virtual power principle. The following remarks regarding the FF model can be reported: o both the continuum damage model and the interface damage model are nonlocal; in particular, the nonlocal effect is accounted for by the presence of the squares of the damages gradient appearing in the free energies   and   ; o damage and strain localizations are avoided by the nonlocal damage effect and by the presence of the viscosity in the damage evolution law; o the interface-body damage coupling is due to the presence of the term   2 , k D D      in the definition of the free energy   ; o the interface damage and the body damage evaluated on the surface of adhesion are constrained, in the limit that , k    , to assume the same value; o because of the damage coupling and of the nonlocal damage diffusion, if the body damage evolves, then the interface damage occurs; analogously, the growth of the interface damage induces the body damage evolution. T HE NONLOCAL INTERFACE - BODY DAMAGE MODEL he mechanical system under investigation is composed by the body 2  , whose behavior is considered linear elastic, in adhesion with the body 1  , characterized by a quasi-brittle cohesive response, by means of a glue whose mechanical properties are much better that those of the support cohesive material. In such a case, the interface model describes the overall response of the glue material and of a thin layer on the surface of the support material. Experimental evidences show that the body damage strongly influences the degradation state of the interface, reducing the mechanical properties of the thin layer on the surface of the support material. On the contrary, the damage and also the failure of the interface does not induces damage diffusion inside the body 1  . In other words, if the body 1  is damaged the degradation of the interface occurs; but it is possible to induce damage and even failure of the interface without damaging the body, i.e. the body 2  can be detached from the body 1  without inducing degradation of the mechanical properties of the body 1  , unless of the thin layer of material which, indeed, is modeled by the interface. For this reason, the idea is to develop a interface-body damage model whose coupling is governed by a unilateral effect: D D    . A very simple body damage model is considered, developed in the two-dimensional framework. As a softening constitutive law is introduced for the body 1  , localization of the strain and damage could occur. In order to overcome this pathological problem and, also, to avoid strong mesh sensitivity in finite element analyses, nonlocal constitutive law is considered. In particular, an integral nonlocal model is adopted; in fact, the nonlocal value of the strain tensor evaluated as: (9) where    x is a weight function. It is assumed that the degradation of the cohesive material is due to the positive principal strain [11, 12]; thus, the equivalent strain is introduced: (10) where 1  and 2  are the nonlocal principal strains and the symbol  denotes the positive part of a number. An exponential damage evolution law is considered: (11) T         1 1 1 d d            ε x x y ε y x y 2 2 1 2 eq            0 0 max min 1, eq k eq history eq e D D D               