Fractal Geometry has been widely used for the description of irregular phenomena in various scientific fields recently. In the subjects concerning fracture system characterization, fractals represent the fracture surfaces in two or 3D problems. In the last few years the fractal geometry of crack networks in damaged materials has been statistically characterized by two power laws, respectively, describing the spatial distribution of crack barycenters, and the crack length distribution. In this article, we explore the potential of the latter power-law. Merely using such statistical model to describe the population of cracks, besides providing a theoretical basis for explaining lower limits to the b-value both in seismicity and in acoustic emission (AE) tests, we find a simple relation between b and the fractal dimension D of the crack network. As a result, the b-value analysis in AE monitoring tests permits evaluation of the dimension D of the damaged domain. This method of evaluating D is herein applied to a concrete specimen in compression, subjected to AE monitoring, loaded up to failure. In this test, the characterization of the fracture process through analysis of AE signals emerging from the growing cracks has been performed in a post-processing environment, using two different procedures. In fact, besides the two-point correlation algorithm introduced by Grassberger and Procaccia, the damage process has been evaluated through the b-value analysis. Both procedures make it possible to evaluate the dimension D of the damaged domain, i.e., the fractal dimension of the crack network. The obtained results are consistent with our understanding of damage phenomenon.

In the literature, several approximate approaches have been proposed to analyse the lateral loading distribution of external loads in high‐rise buildings; in this paper, a general method is proposed for the analysis of the lateral loading distribution of three‐dimensional structures composed of any kind of bracings (frames, framed walls, shear walls, closed and/or open thin‐walled cores and tubes) under the customary assumption of floor slabs being undeformable in their planes. This general formulation allows analyses of high‐rise structures by taking into account the torsional rigidity of the elements composing the building without gross simplifications, even in the case of very complex shapes and with the contemporary presence of different kinds of bracing. The method is aimed at gaining an insight into the force flow in the structure, in order to understand how the building response is governed by decisive structural parameters and to compare preliminary calculations with other approaches such as the structural finite element analysis. Copyright © 2009 John Wiley & Sons, Ltd.

Extensive research and studies on concrete fracture and failure by means of the acoustic emission (AE) technique have shown that fracture and damage growth can be characterized through a single synthetic parameter, namely the b-value, which changes systematically during the different stages of the failure process, as shown by several AE tests carried out from the specimen to the structural scale [Sammonds PR, Meredith PG, Murrel SAF, Main IG. Modelling the damage evolution in rock containing porefluid by acoustic emission. In: Proceedings of the Eurock’94; 1994; Colombo S, Main IG, Forde MC. Assessing damage of reinforced concrete beam using “b-value” analysis of acoustic emission signals. J Mater Civil Eng ASCE 2003;15:280–6; Carpinteri A, Lacidogna G, Niccolini G. Critical behaviour in concrete structures and damage localisation by Acoustic Emission. Key Eng Mater 2006;312:305–10]. This parameter can be linked to the value of the exponent α of the power-law distribution of the crack size in a damaged structure. In this paper, we propose a statistical interpretation for the variation of the b-value during the evolution of damage, based on a treatment originally proposed by [Carpinteri A. Mechanical damage and crack growth in concrete: plastic collapse to brittle fracture. Dordrecht: Martinus Nijhoff Publishers; 1986; Carpinteri A. Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. Int J Solid Struct 1989;25:407–29; Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials. Int J Solid Struct 1994;31:291–302]. The proposed model captures the transition from the condition of criticality, in which α=3, to that of imminent failure, characterized by α=2, in terms of damage localisation.

Aim of this paper is to present a new fractal approach linking the macroscopic mechanical properties of micro- and nano-structured materials with the main parameters: composition, grain size and structural dimension, as well as contiguity and mean free path. Assuming the key role played by the interfaces, the proposed fractal energy approach unifies the influences of all the above parameters, through the introduction of a fractal structural parameter (FSP), which represents an extension of the Gurland’s structural parameter. This modeling approach is assessed through an extensive comparison with experimental data on poly crystalline diamond (PCD) and WC–Co alloys. The results clearly show that the theoretical fractal predictions are in a fairly good agreement with the experiments on both hardness and toughness. This new synthetic parameter is thus proposed to investigate, design and optimize new micro- and nano-grained materials. Eventually, FSP-based optimization maps are developed, that allow to design new materials with high hardness and toughness.

The size-scale effects on the mechanical properties of materials are a very important topic in engineering design. Three different modeling approaches have been proposed and analyzed at least, i.e. the statistical, the energetical and the fractal one. Aim of this paper is to revisit the fractal approach and to reject the most recurrent criticisms against it. Moreover, we will show that it is wrong to set the fractal approach to size-scale effects against the statistical one, since they are deeply connected. More in detail, by analyzing a fractal distribution of micro-cracks in the framework of Extreme Value theory, we will obtain a scaling law for tensile strength characterized, in the bi-logarithmic plot, by the slope -1/2. Conversely, by considering a fractal grain size distribution in a grained material, we will obtain a scaling law or fracture energy characterized, in the bi-logarithmic plot, by the positive slope 1/2. These slopes are the natural consequence of perfect self-similarity of the flaw (or grain) size distribution. Eventually, the theoretical results regarding the link between fractals and statistics will be confirmed by numerical simulations.

The so-called Complexity Sciences are a topic of fast growing interest inside the scientific community. Actually, researchers did not come to a definition of complexity, since it manifests itself in so many different ways [1]. This field itself is not a single discipline, but rather a heterogeneous amalgam of different techniques of mathematics and science. In fact, under the label of Complexity Sciences we comprehend a large variety of approaches: nonlinear dynamics, deterministic chaos theory, nonequilibrium thermodynamics, fractal geometry, intermediate asymptotics, complete and incomplete similarity, renormalization group theory, catastrophe theory, self-organized criticality, neural networks, cellular automata, fuzzy logic, etc. Aim of this paper is at providing insight into the role of complexity in the field of Materials Science and Fracture Mechanics [2-3]. The presented examples will be concerned with the snap-back instabilities in the structural behaviour of composite structures (Carpinteri [4-6]), the occurrence of fractal patterns and selfsimilarity in material damage and deformation of heterogeneous materials, and the apparent scaling on the nominal mechanical properties of disordered materials (Carpinteri [7,8]). Further examples will deal with criticality in the acoustic emissions of damaged structures and with scaling in the time-to-failure (Carpinteri et al. [9]). Eventually, results on the transition towards chaos in the dynamics of cracked beams will be reported (Carpinteri and Pugno [10,11]).

Understanding and predicting the process of material fracture and failure in heterogeneous materials such as rocks, concrete, ceramics and other composites is an extremely challenging scientific problem, central to a large number of applications and crucial in predicting the long-term structural integrity of structures. The complex phenomena occurring at the micro- and mesoscale can be monitored by using acoustic emission techniques that reveal some universal features in the damage evolution and fracture. In particular, extensive research works and studies on concrete fracture and failure have shown that fracture growth and damage evolution can be characterized through a single synthetic parameter, namely the b-value of the Gutenberg-Richter law, which changes systematically with the different stages of fracture growth. In this paper, we propose two different interpretations for the variation of the b-value during the evolution of damage, focused on the spatial development of cracks in the material. The first one, called the self-similarity approach, is based on fractal geometry and the statistical characterization of the cracks inside a material by means of two power-law distributions: one for the spatial arrangement of crack barycentres, and the second for the crack length distribution. The second interpretation is based oil the Yule process, originally proposed to explain the power-law size distribution of biological taxa. Both modelling idealizations capture the transition from the condition of criticality, in which b = 1.5, to that of imminent failure, characterized by b = 1.0, in terms of damage localisation. As a case study we present the method used by the authors to determine the conditions of the materials and the crack patterns in the structures of the Syracuse Cathedral, built in the 17(th) century on the structures of the ancient Greek "Temple of Athena" (5(th) century B.C.). In particular, the acoustic emission technique was used to evaluate the onset of critical conditions in a monitored pillar, which is part of the vertical load-bearing structures. The b-value trends are shown by several acoustic emission tests carried out on specimens of different dimensions extracted from the pillar, In addition, these results are compared to the acoustic emission data obtained from the in situ monitored pillar; it is shown that the b-value can be used both in the laboratory specimens and in the in situ measurements as a reliable indicator of the structural integrity.

Since the pioneering paper by Mandelbrot (Nature, 308:721–722, 1984) on the fractal character of the fracture surfaces in metals, the fractal aspects in the deformation and failure of materials have been investigated by several Researchers (see the reviews by Bouchaud (J Phys Condens Matter, 9:4319–4344) and Carpinteri et al. (Appl Mech Rev, 59:283–305, 2006)) and the attempts to apply fractals to fracture have grown exponentially. Aim of this paper is 2-fold: on one hand, it summarizes in a detailed yet concise fashion the major results of the fractal approach to the scaling of mechanical properties in solid mechanics; on the other hand, it reports some recent results concerning the size effect in the failure of reinforced concrete (RC) beams. These recent findings clearly show that the picture of the size-scale effects is much more complex when interaction among different collapse mechanisms occurs. The consequences on the size-scale effects are discussed in detail.

In this paper, we present a fracture-mechanics based model, the so-called bridged crack model (Carpinteri, A., 1981, “A Fracture Mechanics Model for Reinforced Concrete Collapse,” Proc. of IABSE Colloquium on Advanced Mechanics of Reinforced Concrete, Delft, I.A.B.S.E., Zürich, pp. 17–30; Carpinteri, A., 1984, “Stability of Fracturing Process in R.C. Beams,” J. Struct. Engng. (A.S.C.E.), 110, pp. 544–558) for the analysis of brittle matrix composites with discontinuous ductile reinforcements under the condition of repeated bending loading. In particular, we address the case of composites with very high number of reinforcements (i.e., fiber-reinforced composites, rather than conventionally reinforced concrete). With this aim, we propose a new iterative procedure and compare it to the algorithm recently proposed by Carpinteri, Spagnoli, and Vantadori (2004, “A Fracture Mechanics Model for a Composite Beam with Multiple Reinforcements Under Cyclic Bending,” Int. J. Solids Struct., 41, pp. 5499–5515), showing the advantages in terms of computational efficiency. Furthermore, we analyze the combined effects of crack length, brittleness number, and fiber number on the cyclic behavior of the composite beam, showing the conditions enhancing the energy dissipation in the composite system. Eventually, we analyze crack propagation and propose, consistently with the model premises, a fracture-mechanics-based crack propagation criterion that allows one to simulate cyclic bending tests under the fixed grip condition.

Extensive research and studies on concrete fracture and failure have shown that concrete should be viewed as a quasi-brittle material having a size-dependent behavior. Numerous experimental techniques have been employed to evaluate fracture processes, and a number of modeling approaches have been developed to predict fracture behavior. A non-destructive method based on the Acoustic Emission (AE) technique has proved to be highly effective, especially to assess and measure the damage phenomena taking place inside a structure subjected to mechanical loading. In this paper, comparing AE frequency-magnitude statistics in solids subjected to damage processes with defect size distributions for disordered materials, critical parameters defining instability conditions for monitored structures are found. In addition, an experimental investigation conducted on concrete and RC structures by means of the AE technique is described. Experimental results confirm the described theories.

The present paper proposes a double-multiplicative penalty strategy for constrained optimization by means of genetic algorithms (GAs). The aim of this research is to provide a simple and efficient way of handling constrained optimization problems in the GA framework without the need for tuning the values of penalty factors for any given optimization problem. After a short review on the most popular and effective exterior penalty formulations, the proposed penalty strategy is presented and tested on five different benchmark problems. The obtained results are compared with the best solutions provided in the literature, showing the effectiveness of the proposed approach.

The size-scale effects on concrete strength are a very important topic in engineering design. In recent years, the scientific community dedicated great efforts in order to gain a precise description of this phenomenon and to highlight the physical mechanisms that lie behind it. Aim of this paper is to review the fundamentals of the fractal approach (Carpinteri, 1994b), which has recently been at the centre of a scientific debate, and to revisit in detail its connections with statistics. The most recurrent criticisms against the fractal interpretation of the size-scale effects will be rejected and theoretical results regarding the link between fractals and statistics will be also confirmed by numerical simulations.

Aim of this letter to the Editor is at replying to the criticisms raised by Bažant and Yavari [Bažant ZP, Yavari A. Is the cause of size effect on structural strength fractal or energetic – statistical? Engng Fract Mech 2005;72:1–31] against the fractal approach to the size-scale effects on the mechanical properties of materials and the concept of the Multi-Fractal Scaling Law presented by Carpinteri [Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials. Int J Solids Struct 1994;31:291–302]. These criticisms will be analysed thoroughly, showing how they also contain some mistakes and misunderstandings. The presented elucidations should redirect the discussion to a more correct scientific debate.

The so-called Complexity Sciences are a topic of fast growing interest inside the scientific community. Aim of this paper is to provide an insight into the role of complexity in the field of Materials Science and Fracture Mechanics. The paper is divided into two parts, which deal with the two opposite natural trends of composite systems: order and structure emerging from large, complicated systems and the route towards randomness and chaos arising from simple nonlinear rules. The former trend has been illustrated in the companion paper (Part I); on the other hand, this part will focus on the latter trend. The first example concerns the snap-back instabilities in the structural behaviour of composite structures, which are illustrated in the framework of Catastrophe Theory; the second application deals with the transition towards chaos in the dynamics of cracked beams.

The present paper is a review of research carried out on scaling laws and multiscaling approach in the mechanics of heterogeneous and disordered materials in the last two decades, especially at the Politecnio di Torino. The subject encompasses theoretical, numerical and experimental aspects. The research followed two main directions. The first one concerns the implementation and the development of the cohesive crack model, which has been shown to be able to simulate experiments on concrete like materials and structures. It is referred to as the dimensional analysis approach, since it succeeds in capturing the ductile-to-brittle transition by increasing the structural size owing to the different physical dimensions of two material parameters: the tensile strength and the fracture energy. The second research direction aims at capturing the size-scale effects of quasibrittle materials, which show fractal patterns in the failure process. This approach is referred to as the renormalization group (or fractal) approach and leads to a scale-invariant fractal cohesive crack model. This model is able to predict the size effects even in tests where the classical approach fails, e.g., the direct tension test. Within this framework and introducing the fractional calculus, it is shown how the Principle of Virtual Work can be rewritten in its fractional form, thus obtaining a scaling law not only for the tensile strength and the fracture energy, but also for the critical strain.

The so-called Complexity Sciences are a topic of fast growing interest inside the scientific community. Actually, researchers did not come to a definition of complexity, since it manifests itself in so many different ways [1]. This field itself is not a single discipline, but rather a heterogeneous amalgam of different techniques of mathematics and science. In fact, under the label of Complexity Sciences we comprehend a large variety of approaches: nonlinear dynamics, deterministic chaos theory, nonequilibrium thermodynamics, fractal geometry, intermediate asymptotics, complete and incomplete similarity, renormalization group theory, catastrophe theory, self-organized criticality, neural networks, cellular automata, fuzzy logic, etc.