Issue 12

M. Angelillo et alii, Frattura ed Integrità Strutturale, 12 (2010) 63-78 ; DOI: 10.3221/IGF-ESIS.12.07 65 state to another. Though it seems natural to look at “gradient flows”, so that the material moves downhill in the steepest direction, in studying initiation (that is the onset of cracks in a previously flawless body) the crack cannot open up in any “nearby” optimal direction without having the energy to increase somewhere. A way out to restore physical plausibility is to assume that the system can overcome “small” energy barriers, where the term “small” refers both to the amount of the energy increase before the descent begins and to the length of the path leading to the point of descent. Such a “smallness” could be measured by introducing a convenient barrier-norm to be compared with a given threshold specific of the material considered. Since an increase in energy always requires the work of external active forces, these energy jumps should be actually forbidden in a quasi-static evolution and can be justified only admitting the presence of finite disturbances in the geometry and in the data. A NALYTICAL AND NUMERICAL FORMULATION OF THE PROBLEM ur approach is reported in details i n [2] and it will be briefly summarized for the reader’s convenience. Preliminaries Consider a two–dimensional body B occupying in the original configuration a bounded plane domain  and undergoing small deformations. Let  be the infinitesimal strain tensor and u the displacement field defined over  . The boundary of  is partitioned into two parts where displacements u and tractions p are given. We admit that u may be discontinuous on a set  assumed to belong to an admissible set of cracks S(  ). The crack pattern  is the most relevant unknown in variational fracture problems and the theorems of Ambrosio [12], valid in particular for Griffith’s type variational formulations based on global minimizers, ensure that  is sufficiently regular to be approximated by sets composed by regular arcs, such as line elements. Therefore  is a 1d closed set and the unit normal n and tangent t exist a.e. along it. The jump of u across  , denoted [[ u ]] = u + − u - , is resolved into two components relative to n , t [[ u ]] = a n + b t (1) If a is positive the corresponding points on the two opposite sides of the interface  separate and determine a vacancy. Finally we assume that u belongs to H 1 (  \  ), that is u has the regularity required in linear elasticity away from  . Volume and surface energy densities The material is assumed to be elastic and isotropic, that is characterized by the elastic energy density ) 2 ) (*( 2 1 )( 2           tr (2) a scalar field defined over  \  . In (2)  and  are the elastic moduli for generalized plane stress. To model brittle fracture, we introduce on  the interface energy density 0 , 0, 0 ( , ) , 0 , 0 0 & 0 c a b a b G a a or a b              (3) where G c is the thoughness of the material and a , b are the components of the displacement jump along  . Typical Boundary Value Problems We consider quasi–static BVP’s. A typical example is represented by the plane strip of height H and width B shown in Fig. 1 t o which we refer for notations. On the lateral sides of the rectangle the traction condition p = 0 is enforced and on the bases a combination of relative displacements u is given. We assume u to be time dependent. At any time t of the loading process we seek equilibrium states of the strip as points of local minima of the functional O