Issue 10

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 10 (2009) 3-11; DOI: 10.3221/IGF-ESIS.10.01 10 distortions in the trajectory are consequences of the presence of the super- or subharmonics. Also in this case, the diagram is nonsymmetric as the spatial positions of the cracks. This method is able to catch the transition toward deterministic chaos, like the occurrence of a period doubling, as shown in the numerical examples and experimentally observed in the context of cracked beam by Brandon and Sudraud [46]. C ONCLUSIONS he so-called "Complexity Sciences" represent a subject of fast-growing interest in the Scientific Community. They have entered also our more circumscribed Communities of Material Science and Material Strength, as the proposed examples may confirm. The presented topics were concerned with the structural behaviour of composite structures with snap-back instabilities (an example of cusp catastrophe), the occurrence of fractal patterns and geometrically self-similar morphologies in deformation, damage and fracture of heterogeneous materials, the apparent scaling in the nominal mechanical properties of disordered materials, the acoustic emission criticality in progressive structural collapse, the route towards chaos in the dynamics of cracked structures. As shown in these examples, the most interesting behaviors and phenomena can be synthetically interpreted only through the use of new and refined conceptual tools in the framework of "Complexity Sciences". A CKNOWLEDGEMENTS he authors would like to gratefully acknowledge the contributions made to this work by all members of the research group led by the senior author at the Department of Structural Éngineering and Geotechnics of the Politécnico di Torino. In particular, the warmest thanks go to Giuseppe Ferro, Nicola Pugno, Pietro Cornetti and Giuseppe Lacidogna. Support by the European Community is gratefully acknowledged by the authors. Thanks are also due to the Italian Ministry of University and Research (MIUR). R EFERENCES [1] M.S. Garrido, R.Vuela Mendes, Complexity in physics and technology, World Scientific, Singapore, (1992). [2] A. Carpinteri, S. Puzzi, Strength, Fracture and Complexity, 4 (2006) 189. [3] A. Carpinteri, S. Puzzi, Strength, Fracture and Complexity, 4 (2006) 201. [4] A. Carpinteri, in Application of Fracture Mechanics to Cementitious Composites, edited by S.P. Shah, Martinus Nijhoff Publishers, Dordrecht, (1985) 287. [5] A. Carpinteri, J. Mech. Phys. Solids, 37 (1989) 567. [6] A. Carpinteri, Int. J. Frac., 44 (1990) 57. [7] A. Carpinteri, Mech. Mater., 18 (1994) 89. [8] A. Carpinteri, Int. J. Solids Struct., 31 (1994) 291. [9] A. Carpinteri, G. Lacidogna, N. Pugno, Engng. Frac. Mech., 74 (2007) 273. [10] A. Carpinteri, N. Pugno, J. Appl. Mech., 72 (2005) 511. [11] A. Carpinteri, N. Pugno, J. Appl. Mech., 72 (2005) 519. [12] R. Thom, Structural Stability and Morphogenesis: an Outline of a General Theory of Models. Benjamin (1975). [13] G. I. Barenblatt, J. Appl. Math. Mech., 23 (1959) 622. [14] D. S. Dugdale, J. Mech. Phys. Solids, 8 (1960) 100. [15] A. Carpinteri, P. Cornetti, F. Barpi, S. Valente, Engng. Frac. Mech., 70 (2003) 1809. [16] A. Hillerborg, M. Modeer, P.E. Petersson, , Cement Concr. Res., 6 (1976) 773. [17] A. Carpinteri, P. Cornetti, S. Puzzi, Appl. Mech. Rev., 59 (2006) 283. [18] A. Carpinteri, Int. J. Solids Struct., 25 (1989) 407. [19] A. Carpinteri, Engng. Frac. Mech., 32 (1989) 265. [20] Determination of the fracture energy of mortar and concrete by means of three-point bending tests on notched beams. Technical Report 18, Materials and Structures, R.I.L.E.M. (1985). [21] L. Biolzi, S. Cangiano, G.P. Tognon, A. Carpinteri, Mater. Struct., 22 (1989) 429. [22] B.B. Mandelbrot, The Fractal Geometry of Nature. New York: Freeman (1982). T T