Issue 10

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 10 (2009) 3-11; DOI: 10.3221/IGF-ESIS.10.01 4 I NTRODUCTION omplexity, as a discipline, generally refers to the study of large-scale systems with many interacting components, in which the overall system behaviour is qualitatively different from (and not encoded in) the behaviour of its components. Complex systems lie somehow in between perfect order and complete randomness –the two extreme conditions that occur only very seldom in nature– and exhibit one or more common characteristics, such as: sensitivity to initial conditions, pattern formation, spontaneous self-organization, emergence of cooperation, hierarchical or multiscale structure, collective properties beyond those directly contained in the parts, scale effects. Complexity has two distinct and almost opposite meanings: the first goes back to Kolmogorov's reformulation of probability and his algorithmic theory of randomness via a measure of complexity, now referred to as Kolmogorov Complexity [1]; the second to the Shannon's studies of communication channels via his notion of information. In both cases, complexity is a synonym of disorder and lack of a structure: the more random a process is, the more complex it results to be. The second meaning of complexity refers instead to how intricate, hierarchical, structured and sophisticated a process is. Associated with these two almost opposite meanings, are two natural trends of composite systems, and two corresponding questions: how does order and structure emerge from large, complicated systems? And, conversely, how do randomness and chaos arise from systems with only simple constituents, whose behaviour does not directly encode randomness? The former case is typical of all those phenomena which could be described through the concepts of scale invariance, phase transition, and with the use of power laws. The latter case is that of instability and bifurcations and of dynamical systems showing chaotic attractors and transition to chaos. In this paper, several fracture mechanics applications will be shown, in which both trends are present. T HE NONLINEAR COHESIVE CRACK MODEL : SNAP - BACK INSTABILITY AS A CUSP CATASTROPHE he first example dates back to the 1980's, when the senior author [4-6] approached the snap-back instability of cracked bodies with a Cohesive Crack model, which can be interpreted in the general framework of Catastrophe Theory (Thom [12]) . This first section is thus devoted to a brief review of the ductile-to-brittle transition in the mechanical behaviour of cracked solids, described by means of the Cohesive Crack model. The Cohesive Crack Model was initially proposed b y Barenblatt [13] and Dugdale [14]. Subsequently, Dugdale's model was reconsidered by several other Authors (for a review see [15]); Hillerborg et al. [16] proposed the Fictitious Crack Model in order to study crack propagation in concrete. The cohesive crack model is based on the following assumptions ([4,15]): 1. The cohesive fracture zone (plastic or process zone) begins to develop when the maximum principal stress achieves the ultimate tensile strength  u . 2. The material in the process zone is partially damaged but still able to transfer stress. Such a stress is dependent on the crack opening displacement w . The energy G F necessary to produce a unit crack surface is given by the area under the  w diagram. The real crack tip is defined as the point where the distance between the crack surfaces is equal to the critical value of crack opening displacement w c and the normal stress vanishes. On the other hand, the fictitious crack tip is defined as the point where the normal stress attains the maximum value and the crack opening vanishes (Fig. 1). With some modifications, the cohesive crack model has been applied to model a wide range of materials and fracture mechanisms, most prominently concrete. Regarding this material, there is a very large literature; for a review, the reader is referred to the review papers by Carpinteri and co-workers [15,17]. Now, let us quantify the ductile-to-brittle transition by showing synthetically the numerical results for concrete elements in Mode I conditions (Three Point Bending Test - TPBT), based on the cohesive model, obtained using the Finite Element Code FR.ANA. (FRacture ANAlysis Carpinteri [5,18,19]) . Extensive series of analyses were carried out from 1984 to 1989 by A. Carpinteri and co-workers. The experimental results can be found in the RILEM report [20]. The cases described in the reference papers regard three slenderness ratios, and four initial crack depths, and a concrete-like material. Fig. 2a refers to the case of an initially uncracked beam, whilst Fig. 2b r eports results for the case of an initially cracked beam with relative crack depth equal to 0.5. For each ratio, the response was analyzed for different values of the brittleness number, S E [4]. As can be seen from the diagrams, by increasing S E the behaviour of the element changes from brittle to ductile. Generally speaking, the specimen behaviour is brittle (snap-back) for low S E numbers, i.e., for low fracture toughness, G F , high tensile strengths,  u , and/or large sizes, h . In particular, in the case of uncracked beam, for S E 10.45x10 -5 , the P –δ curve presents positive slope in the C T