Depinning Dynamics of Crack Fronts

Abstract

We investigate experimentally and theoretically the dynamics of a crack front during the microinstabilities taking place in heterogeneous materials between two successive equilibrium positions. We focus specifically on the spatio-temporal evolution of the front, as it relaxes to a straight configuration, after depinning from a single obstacle of controlled strength and size. We show that this depinning dynamics is not controlled by inertia, but instead, by the rate dependency of the dissipative mechanisms taking place within the fracture process zone. This implies that the crack speed fluctuations around its average value vm can be predicted from an overdamped equation of motion (v −vm)/v0 = (G−Gc(vm))/Gc(vm) involving the characteristic material speed v0 = Gc(vm)/G c (vm) that emerges from the variation of fracture energy with crack speed. Our findings pave the way to a quantitative description of the critical depinning dynamics of cracks in disordered solids and open up new perspectives for the prediction of the effective failure properties of heterogeneous materials.Woods, nacre, bones or rationally designed artificial materials, are all heterogeneous solids, with mechanical properties far exceeding those of their constitutive components. Understanding the role of microscale heterogeneities on the macroscale fracture behavior of solids still remains a query. This becomes especially relevant now, as rapid developments in microfabrication techniques allow the tailoring of microstructures at ever smaller scales, yielding new types of composites, known as meta-materials, with unprecedented mechanical properties [1][2][3][4][5][6]. Recently, significant progresses were made for weakly heterogeneous brittle solids where models describing a crack front as a deformed interface pinned by tough obstacles have been successfully applied [7][8][9][10][11]. The homogenized fracture properties can be computed exactly within the so-called weak pinning limit [12], where the elastic energy release rate G balances the fracture energy G c at any time and any position along the front. This approach holds for weak variations of toughness along the propagation direction. The crack evolution is then smooth and can be properly approximated by a continuous succession of equilibrium front configurations [13,14]. This approach was successfully used to design weakly heterogeneous systems with improved and new macroscopic failure properties [15][16][17][18].However, most natural and engineered materials have a microstructure composed of discontinuous heterogeneities which cannot be described within the weak pinning regime. The strong pinning regime that predominates for large toughness gradients challenges standard homogenization approaches. Crack propagation is not quasi-static but proceeds by intermittent and local micro-instabilities. Further, for a disordered distribution of obstacles, crack growth takes place close to the socalled depinning critical transition [19][20][21], so that the h s d = 141 �m, C = 1.2, v m = 24 �m/s FIG. 1. (a...

Effective toughness of disordered brittle solids: A homogenization framework

“…The last missing ingredient is the kinetic law that relates the local crack velocity v to G and G c . For brittle materials, this kinetic law can be derived from Griffith (1921)'s criterion by accounting for the variations of the toughness with crack speed (Kolvin et al, 2015;Chopin et al, 2018). It reads :…”

“…is a characteristic velocity of the material tied to the rate-dependency of its toughness, and [•] + the positive part function. This equation of motion has been largely used in the literature (see for example (Gao and Rice, 1989;Ramanathan et al, 1997;Ponson and Bonamy, 2010)) and was recently shown to capture quantitatively the relaxation dynamics of a crack depinning from a single obstacle (Chopin et al, 2018).…”

Observation of cavitation governing fracture in glasses

“…The multiple discrete cavitation-dominated cracking process is reminiscent of the discontinuous percolation process of shear transformation zones during the formation of a shear band in MGs (52), the depinning transition of cracks in disordered solids (53), and the stick-slip behavior of frictional instability (54). All these discontinuous processes in disordered systems have the nature of far from equilibrium and perhaps follow at the root of some common physical principles.…”

“…From this viewpoint, it is promising to tailor the heterogeneities of glasses to avoid the occurrence of cavitation-induced cracking and thus achieve glasses with sufficient ductility and damage tolerance, such as the recently reported highly ductile amorphous oxide with a dense and flaw-free atomic arrangement (3,5). From a theoretical perspective, the intrinsic heterogeneities of glasses invalidate the classical continuum understanding of dynamic crack propagation (14,15,47,53), which is based on the continuum concept of matter and is valid for the description of generic fracture on the macroscale (4,11). Therefore, further fundamental fracture theory taking into account the heterogeneities of amorphous solids needs to be undertaken to describe the nanoscale cavitation-dominated crack propagation of glasses.…”

Universal Scaling of the Velocity Field in Crack Front Propagation

“…A 5mm thick plate of PMMA is detached from a thick PDMS substrate using the beam cantilever geometry depicted in Fig. 1 [29]. To introduce disorder, we print obstacles of diameter d 0 = 100 µm with a density of 20 % on a commercial transparency that is then bonded to the PMMA plate before bringing it in contact with the PDMS.…”