Continuum Field Description of Crack Propagation

Abstract

We develop continuum field model for crack propagation in brittle amorphous solids. The model is represented by equations for elastic displacements combined with the order parameter equation which accounts for the dynamics of defects. This model captures all important phenomenology of crack propagation: crack initiation, propagation, dynamic fracture instability, sound emission, crack branching and fragmentation. PACS: 62.20.Mk, 46.50.+a, 02.70.Bf The dynamics of cracks is the long-standing challenge in solid state physics and materials science [1,2]. The phenomenology of the crack propagation is wellestablished by recent experimental studies [3][4][5][6][7][8][9]: once a flux of energy to the crack tip passes the critical value, the crack becomes unstable, it begins to branch and emits sound. Although this rich phenomenology is consistent with the continuum theory, it fails to describe it because the way the macroscopic object breaks depends crucially on the details of cohesion on the microscopic scale [10].Significant progress in understanding of fracture dynamics was made by large-scale (about 10 7 atoms) molecular dynamics (MD) simulations [11,12]. Although limited to sub-micron samples, these simulations were able to reproduce several key features of the crack propagation, in particular, the initial acceleration of cracks and the onset of dynamic instability. However, detailed understanding of the complex physics of the crack propagation still remains a challenge [13].The uniform motion of the crack is relatively wellunderstood in the framework of the continuum theory [14]. Most of the studies treat cracks as a front or interface separating broken/unbroken materials and propagating under the forces arising from elastic stresses in the bulk of material and additional cohesive stresses near the crack tip [15][16][17][18]. Although these investigations revealed some features of the oscillatory crack tip instability, they are based on built-in assumptions, e.g., on specific dependence of the fracture toughness on velocity, structure of the cohesive stress etc. To date there is no continuum model capable to describe in the same unified framework the whole phenomenology of the fractures, ranging from crack initiation to oscillations and branching.In this Letter we present a continuum field theory of the crack propagation. Our model is the wave equations for the elastic deformations combined with the equation for the order parameter, which is related to the concentration of material defects. The model captures all important phenomenology: crack initiation by small perturbation, quasi-stationary propagation, instability of fast cracks, sound emission, branching and fragmentation.Model. Our model is a set of the elasto-dynamic equations coupled to the equation for order parameter ρ.We define the order parameter as the relative concentration of point defects in amorphous material (e.g., microvoids). Outside the crack (no defects) ρ = 1 and ρ = 0 inside the crack (all the atomic bonds are broken). At the crack surf...

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