Localized versus Diffuse Damage in Amorphous Materials D. De Tommasi, 1, * G. Puglisi, 1,† and G. Saccomandi 2,‡ 1 Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Bari, Via Re David 200, Bari, 70125, Italy 2 Dipartimento di Ingegneria Industriale, Universita ` degli Studi di Perugia, Via G. Duranti 67, Perugia 06125, Italy (Received 15 November 2007; published 29 February 2008) Based on a Griffith approach, we study the behavior of disordered media constituted at the microscale by distributions of elastic and breakable links with variable activation and fracture thresholds. Depending on the microscopic distribution properties, the material may be characterized by an unstable strain domain, which gives the possibilities of having homogeneous or localized damage. Our simple model delivers a theoretical scheme to describe main experimental effects observed at the microstructure and macroscopic scale in disordered materials undergoing damage and relates them to the inhomogeneity properties of the material. DOI: 10.1103/PhysRevLett.100.085502 PACS numbers: 62.20.M, 46.15.Cc, 61.41.+e, 71.55.Jv The damage and fracture behavior of materials charac- terized by microscopic structural disorder is determined by the distribution properties at the scale of the microstruc- ture. As these properties vary, the experiments show that the material changes its strategies of damage nucleation and propagation, showing diffuse or localized damage structures (shear bands, necking, etc.). These different damage evolutions lead to different material responses such as brittle, ductile, or rubberlike behavior. A typical important example is delivered by polymeric materials [1]. Analogous properties regulate the mechanical response of other disordered materials such as biological [2] and com- posite materials [3]. Thus the understanding of the mecha- nisms regulating macroscopic behavior in relation to the microscopic material distribution is not only important from a theoretical point of view, but it is also crucial for the technological interest on microstructural material optimization. The model considered in this Letter has been recently proposed in [4]. It is based on a prototypical scheme for the polymeric network of a vulcanized rubber which assumes rigid filler spheres connected by two kinds of links: elastic and breakable. To model the material disorder with poly- meric chains characterized by variable reference lengths and fracture limits, the fraction of breakable links is as- signed by a distribution of chains with different activation and breaking thresholds. The remaining fraction is an elastic foundation that delivers a residual strength for the fully damaged material (e.g., polymeric blend, composites matrix, or passive forces for skeletal muscles [2]). The assumption of distributions of material with variable con- stitutive properties is shared by this model with well known models proposed in the physics of fracture and damage such as fiber bundle models and fuse networks (see, e.g., [5,6]). In [4,7] the authors described a three-dimensional extension of this prototypical scheme, based on the as- sumption that at each material point of a continuous body there exists a distribution of materials with variable activation and breaking thresholds. They also showed the predictability of the model (i.e., distribution functions can be deduced by simple extension cyclic experiments) and its computational efficiency. Based on the described prototypical model, in this Letter we deduce a damage model for a bar of disordered mate- rial. Our main end is to show that the choice of the material distribution influences the damage evolution of the bar and, in particular, delivers the transition between homogeneous and localized damage and between ductile and fragile macroscopic response. Toy model.—Following [4], we begin by considering a simple toy model [see Fig. 1(a)] constituted by two rigid spheres connected by parallel links of two types. A fraction of the links is elastic with an elastic energy ’ ’ e ", and stress e " : d’ e "=d", where " is the strain. The remaining fraction 1 takes care of the continuous scission phenomenon associated to the alteration mecha- nism at the microscale. We assume [see Figs. 1(b) and 1(c)] that these links are elasto-fragile with an energy ’ ’ b " 8 > < > : 0; " " a ; E 2 " " a 2 ; " a <"<" b ; E 2 " b " a 2 ; " " b ; ; (1) where E is an elastic modulus and " a and " b represent the FIG. 1. (a) toy model; (b) energy-strain and (c) stress-strain relations of breakable links; (d) distribution of broken, active, and unloaded springs at given " and ^ " in the probability space. PRL 100, 085502 (2008) PHYSICAL REVIEW LETTERS week ending 29 FEBRUARY 2008 0031-9007= 08=100(8)=085502(4) 085502-1 2008 The American Physical Society