Deformation of the crack front during propagation in some disordered medium: Theoretical and experimental studies

In heterogeneous disordered materials, a straight crack front experiences toughness fluctuations during its propagation that generate geometric fluctuations. Their long time statistical behavior has been studied by Lazarus et al. (JMPS, 2008) using Bueckner-Rice weight function theory. In particular, the evolution of the auto-correlation function, power spectrum and variance of the front fluctuations have been derived. The aim here is to compare these results to some experiments performed on transparent plexiglas blocks with the same apparatus as in Schmittbuhl and Maloy (PRL, 1997) by measuring the amplitude evolution of the crack front fluctuations in addition to the self-affinity roughness parameters. In a perfectly ideal homogeneous material, an initial straight crack front remains straight during propagation. But in an heterogeneous disordered materials, it becomes rough. The aim of the present paper is to derive an analytical description of the evolution of this roughness and to compare it to experimental results. The assumption of quasi-static brittle crack propagation will be done. Among the experimental works, one may cite on the one hand, the pioneer work of Daguier et al. [2] in which the crack front is obtained postmortem, the crack surface being marked by ink and on the other hand, the works of Delaplace, Maloy and Schmittbuhl [8, 3] in transparent plexiglas in which the crack front can be observed in situ during its evolution. They deal mainly with the universal self-affine character of the crack front. The roughness exponent £ was measured between 0.5 and 0.6. Here, we have again used the experimental framework of [8, 3] to measure the time evolution of the fluctuations in addition to its roughness. All the theoretical studies of the statistical properties of the crack front performed in quasi-static, use Bueckner[1]-Rice[7] weight function theory, also called line elastic models, to evaluate the stress intensity factors along the perturbed crack front. Among them, one may distinguish two groups depending on the type of the advance law used. The first ones [9, 10, 6] deal with crack advance governed by brittle fracture Irwin's criterion with a slightly heterogeneous toughness. This criterion is a threshold type one: the crack propagates only if the stress intensity factor becomes equal to the local toughness. In particular, by a first order analysis the roughness exponent was derived and found to be ζ = 0.37 or ζ = 0.5 depending on the papers. This apparent discrepancy will be considered further. The second group deals with crack advance governed by Paris' law (fatigue or sub-critical fracture). It is a time dependent type criterion: the rate of crack advance is proportional to a power law of the stress intensity factor. Lazarus, Leblond and coauthors have performed the study of a tensile tunnel-crack [4] and of a tensile semi-infinite interfacial crack [6]. Contrary to the case of threshold advance law, their first order study in crack advance was not sufficient to obtain the crack front roughness. However Adda-Bedia and Katzav [5] performed the second order study for a semi-infinite crack and obtained ζ = 0.5. Here, the work of Pindra, Lazarus and Leblond [6] is applied to experiments made with the same framework as Delaplace, Maloy and Schmittbuhl [8, 3]. For Irwin's advance law, using Bueckner-Rice formulation for a semi-infinite crack subjected to line loading on its faces, the evolution of the variance and power spectrum, so as the roughness exponent are derived and compared to previous theoretical results of Schmittbuhl, Vilotte and coauthors [9, 10]. Then, comparison with experiments are performed, not solely on the roughness exponent as in previous papers but also on the evolution of the crack front amplitude.