PHYSICAL REVIEW B VOLUME 43, NUMBER 1 1 JANUARY 1991 Scale-invariant disorder in fracture and related breakdown phenomena Alex Hansen and Einar L. Hinrichsen Fysisk Institutt, Universitetet i Oslo, Postboks 1048, Blindern, N-0316 Oslo 3, Norway Stephane Roux* Centre d'Enseignement et de Recherche en Analyse des Materiaux, Ecole Nationale des Ponts et Chaussees, Central IV, 1 avenue Montaigne, F-93167 Noisy-le-Grand CEDEX, France (Received 26 July 1990) We introduce and discuss the concept of scale-invariant disorder in connection with breakdown and fracture models of disordered brittle materials. We show that in the case of quenched-disorder models where the local breaking thresholds are randomly sampled, only two numbers determine the scaling properties of the models. These numbers characterize the behavior of the distribution of thresholds close to zero and to infinity. We review brieAy some results obtained in the literature and show how they fit into this framework. Finally, we address the case of an annealed disorder, and show via a mapping onto a quenched-disorder model, that our analysis is also valid there. I. INTRODUCTION It has been long known that materia1 properties may be strongly influenced by the presence of disorder. Howev- er, the sensitivity to the disorder is widely different, ac- cording to which property one is interested in. Usually, transport properties, for example, conductance and elas- tic constants, are much less sensitive than breakdown properties such as material strength. In breakdown pro- cesses, such as the seemingly simple case of brittle fracture — i.e. , fracture that do not involve local plastic deformations — extreme sensitivity to rare events makes the problem very dificult to handle theoretica11y. This has led to transport properties having been much more studied. As a result, they are therefore today much better understood than the breakdown processes. How- ever, recently the breakdown problems have been ap- proached within the same statistical-physics framework as the transport problems, and a number of interesting re- sults have been found. ' In particular, severa1 recent numerical studies of the breakdown of networks of either elastic or electrical ele- ments have been done with the aim of investigating the relation between disorder and the global properties of the entire network, such as evolution of breaking stress or strain, the damage at peak stress, or the total damage at the breakdown point. At the outset, it was expected that these global properties of the networks should depend strongly on the type of disorder that was put into the models. However, this turned out not to be the case, but a rather puzzling picture emerged: For wide classes of disorders — here in the maximum loads each bond in the network is able to sustain before breaking down — the glo- bal properties are very little sensitive, in fact so little that one might suspect universality, in much the same way as has been found in connection with transport prop- erties in disordered materials. For other distributions, on the other hand, the dependence of the global proper- ties on the disorder was very strong. The universality manifests itself through scaling laws between the global properties of the networks and the size of the networks. These scaling laws are governed by nontrivial scaling exponents, and universality means that the exponents are independent of the details of the partic- ular breakdown model that is used. The suggestion of universality was based on numerical measurements of the exponents involved in three very different models, one based on an electrical network of random fuses, the second one based on a network of elastic beams, each having random breaking thresholds, and the third model consisting of a network of central-force springs — i.e. , springs that rotate frictionless about their end points. Also in this case the springs were assigned random max- imum loads they could sustain before breaking. In each of these three models several different distributions for the randomness of the breaking thresholds were investi- gated. The exponents describing the breakdown process- es seen in these models as a function of external load were found to be rather insensitive to the disorder (we will come back to this statement more precisely later) and even to which model was used. In Ref. 4, a study was made of the transition from an ohmic to a superconducting state of a network of super- conductors, as a current through it is lowered. In this case the disorder is introduced through the current thresholds of each bond in the network, below which the bond becomes superconducting. Also in this case, scaling behavior was found, with the corresponding exponents being insensitive to the distribution of thresholds. The appearance of such a universality in breakdown processes such as fracture would be quite important, as these are not scaling exponents describing a physical sys- tem near a critical point — i. e. , in a limited region of some parameter space — but rather describing the typical situa- tion, much in the same way that the Kolmogorov scaling theory is able to describe universal features in fully developed turbulence. It is the aim of this paper to investigate under what cir- 43 665 1991 The American Physical Society