Issue 18

G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01 6 Other basic problems, such as the formation of a fracture in an initially unfractured body, were neglected for long time, and ony recently came to the attention of the scientific community. The use of variational techniques for energy minimization was introduced first for Griffith’s fracture model [5], and extended later to cohesive energy models [6, 7]. From the engineering side, it became clear that an appropriate choice of the analytic form of the cohesive energy may lead to a unified description of the phenomena of yielding, damage, and fracture [8, 9]. A very recent research trend consists in questioning the surfacic character of the fracture energy, and in exploring the alternative possibility of an energy diffused over the volume. An example is the paper [10], in which, in the context of a damage model, a rather detailed description of the process zone preceding the final rupture is obtained. In this communication a diffuse cohesive energy is considered, and fracture is regarded as the extreme stage of the localization of inelastic deformation. In a one-dimensional context, we assume that at every point of the bar the energy density is the sum of an elastic and a cohesive part. An incremental energy minimization is performed, under the only assumption that the cohesive energy is totally dissipative. With this simple model, a rather accurate description of the inelastic response of the bar, from the onset of the inelastic regime up to rupture, is obtained. The strain-softening case is not adequately described, but this inconvenience is repaired with the introduction of a non-local energy term of the gradient type. The traditional elements of elastic-plastic response, such as the yield condition, the flow law, the hardening law, the elastic unloading, come out of the model as necessary conditions for an energy minimum. We also find that a convex shape of the cohesive energy favorizes a uniform distribution of inelastic strain and a work-hardening response, while a concave shape is responsible of the localization of the inelastic strain, which, in turn, produces the phenomena of strain softening and necking . Rupture takes place when the localization becomes extreme. If the cohesive energy is approximated by a piecewise polynomial function, its expression becomes a function of a small number of material parameters. The first results of a series of numerical simulations show that an appropriate choice of these parameters provides a good approximation of the experimental response curves. This seems to confirm the efficiency and flexibility of the proposed model. T HE LOCAL MODEL onsider a bar of length l , with constant cross section, free of external loads, and subject to the axial displacements u (0) = 0 , u ( l ) =  l , (1) at the endpoints x  0 and x  l . The bar’s deformation is measured by the derivative u  of the axial displacement u . We assume that, at every point x of the bar’s axis, the deformation u  ( x ) can be split into the sum of an elastic part  ( x ) and of an inelastic part  ( x ): u  ( x ) =  ( x ) +  ( x ) , (2) and that the total energy of the bar has the form    l dx x x w E 0 ))) (( )) (( ( ) ,(      (3) where w and  are the volume densities of the elastic strain energy and of the cohesive energy , respectively. We also assume that w can be recovered, while  is totally dissipated. The last assumption requires that the cohesive power   (  ( x,t )) ),( tx   be non-negative for all x and at all instants t , and if we assume that  is an increasing function of  we get the dissipation inequality 0 ),(  tx   (4) to be satisfied for all x and at all t . In the spirit of the Calculus of Variations, the equilibrium configurations of the bar are identified with the stationary points of the energy 0 )) ( )) ((' )( )) ((' ( ) , , ,( 0     dx x x x x w E l           (5) C