Statistical Physics of Fracture Surfaces Morphology

Abstract

Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, succeeding to reproduce the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up with the proposition of new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.It is a privilege to dedicate this paper to Pierre Hohenberg and Jim Langer who contributed, respectively, decisive ideas to scaling concepts in statistical physics and to the study of fracture. It is our hope that the marriage of these two issues in the present paper may give them some degree of pleasure.

Random and correlated roughening in slow fracture by damage nucleation

“…We believe that this is an important aspect in the success of model A in reproducing a correlated roughening, quantitatively close to the experimental observations. It also explains why models which used the linear approximation of a straight crack plus a perturbed K II achieved scaling exponents different from those observed in experiments [15,16].…”

Effect of disorder and notches on crack roughness

“…Although multiscaling of p(∆h(ℓ)) was argued in Ref. [28], it has been shown in references [29,30] that p(∆h(ℓ)) follows a self-affine monoscaling relation given by Eq. 1 and that multiscaling is an artefact that results at small scales due to the removal of crack profile overhangs.…”

“…The scaling properties of the crack profiles h(x) can also be studied using the probability density distribution p(∆h(ℓ)) of the height differences ∆h(ℓ) = [h(x + ℓ) − h(x)] of the crack profile between any two points on the reference line (x-axis) separated by a distance ℓ. Recently, there has been a debate over the scaling of this p(∆h(ℓ)) distribution [28,29,30], i.e., whether the scaling properties of p(∆h(ℓ)) can be described by a single scaling exponent ζ or multiple scaling exponents are required to describe the scaling of p(∆h(ℓ)). The self-affine property of the crack profiles implies that the probability density distribution p(∆h(ℓ)) follows the relation…”

Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models

“…Nevertheless, as will be shown below, under conditions for strong anisotropy any choice of two orthogonal directions will lead to the same set of two different exponents αx,y , which guarantees the generality of Ansatz (6). In the case of fracture experiments, alternative choices for anisotropic scaling Ansätze are also available, in which, e.g., either an auxiliary dynamics is postulated [23,32] or expansions of observables over appropriate functional bases are performed that exploit the fact that isotropic materials often have anisotropic fracture surfaces only because of the breaking of isotropy by the initial conditions [33,34].…”