Pattern Formation and Selection in Quasistatic Fracture

Abstract

Fracture in quasi-statically driven systems is studied by means of a discrete spring-block model. Developed from close comparison with desiccation experiments, it describes crack formation induced by friction on a substrate. The model produces cellular, hierarchical patterns of cracks, characterized by a mean fragment size linear in the layer thickness, in agreement with experiments. The selection of a stationary fragment size is explained by exploiting the correlations prior to cracking. A scaling behavior associated with the thickness and substrate coupling, derived and confirmed by simulations, suggests why patterns have similar morphology despite their disparity in scales.PACS numbers: 05.65.+b, 62.60.Mk, 46.50.+a Nature is full of fascinating patterns [1]. A ubiquitous yet relatively less explored class of patterns is that produced by the fracture of solids [2], as often seen in cracked structures, battered roads and dried out fields. Although there have been early observations and characterizations [1,3,4], only rather recently has it been studied systematically [5][6][7][8][9]. The mathematical problem of a network of interacting cracks [11] is a formidable one, in view of the difficulties facing a lone crack propagating in a homogeneous medium [10]. One way to make progress is to turn to a statistical description. This view has been pursued extensively in investigations of the fracture of disordered media [12], borrowing concepts such as percolation and universality from studies of phase transitions. For crack patterns, an immediate observation is the geometrical similarities of patterns over a wide range of scales, from microns [5] to kilometers [3]. This suggests some universal mechanism is at work and microscopic details may be unimportant. Hence, analogous to phase transitions, a mesoscopic, coarse-grained description may be sufficient and more useful than a microscopic one, provided essential features are captured.Crack patterns often arise from slow physical (e.g., embrittlement, contraction) or chemical (e.g., oxidation) variations, or their combination, in the material properties of an overlayer on top of an inert substrate. The overlayer may fail in different ways [14], such as decohesion, buckling, spalling and in-plane cracking. In this paper, we address the pattern-selection aspects of quasi-static, in-plane cracking by identifying the dominant mechanism and control parameters that determine the ensuing patterns. By understanding a simplified, coarse-grained model, we hope to achieve the same for the phenomena in general.A Bundle-Block Model -In a mesoscopic approach, the grains in the overlayer are represented by an array of blocks. Each pair of neighboring blocks is connected by a bundle of H bonds (coil springs) each of which has spring constant k ≡ 1 and relaxed length l. Initially, the blocks are randomly displaced by r = (x, y), where | r| ≪ l, about their mean positions on a triangular lattice. For slow cracking on a frictional substrate, the motion of the grains is overdamped, the ...

Fragmentation of drying paint layers

“…Since we controlled the layer thickness by the amount of solvent placed in the tube, in Figure 1a we use D to quantify the layer thickness up to a proportionality constant. This data proves to be in good agreement with previously published results [9], confirming once again the claim that the average fragment size exhibits a power-law-type dependence as a function of the layer thickness. Our experiments suggest a scaling exponent of 1.72, in excellent agreement with previous results in the literature [9].…”

“…Recent theoretical and simulation studies proved that a discrete spring-block model can elegantly reproduce the main qualitative and topological features of the cracks and some experimentally observed scaling laws for the fragment-size statistics [9,10]. The same model-family was successfully applied for studying crack propagation in glass plates [11] and spiral crack formation in drying precipitates [12].…”

“…Up to now, the majority of the spring-block type simulations were realized in quasi-three-dimensional approach. In order to obtain a computationally efficient model, the thickness of the material has been forced in a two-dimensional spring-block topology, where the blocks are allowed to move on the surface of a two dimensional plane and the third dimension (thickness) is taken into account by using multiple spring layers between the blocks [9,10].…”

Understanding self-assembled nanosphere patterns

“…We focus on an experimentally simple case, when 2D selfassembled arrays of nanometer-sized polystyrene spheres will form from a colloidal suspension which is drying on a substrate. Some characteristic patterns obtained as a result of this phenomenon are present in Figure 1.The model is rather similar with the spring-block stickslip model successfully used for describing fragmentation structures obtained in drying granular materials in contact with a frictional substrate [12,13]. The new feature of the present model is that a predefined lattice is not considered anymore.…”

“…Considering a classical molecular dynamics simulation for relaxation would be very time-consuming. Following the method used for simulating drying processes in granular media [12,13], we choose thus a simplified dynamics, where the connection with real time is lost, but the relaxation remains realistic:…”