Void formation and roughening in slow fracture

Abstract

Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent zeta approximately = 0.8 in 3 dimensions and zeta approximately = 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

Random and correlated roughening in slow fracture by damage nucleation

“…This process is associated with a length scale ξ c characteristic of the distance of the damage element from the tip. A plausible physical picture for such a process was proposed in [5,6]. The idea is to identify ξ c with the size of the plastic zone that develops near the crack tip due to the large stresses concentrated there.…”

“…Model A was studied in Refs. [5,6] . In the absence of precise knowledge of the activation process we adopted reasonable probability distribution functions f (P (r)− P c ) and demonstrated that the cracks generated by the model were self-affine with ζ = 0.66 ± 0.03 irrespective of the specific form of f (P (r) − P c ).…”

“…This curve defines the elastic-plastic boundary. It was further shown [5,6] that the hydrostatic tension P , defined as…”

“…Refs. [5,6] demonstrated that the rupture lines generated by this model are self-affine rough graphs with a correlated scaling exponent ζ ≈ 0.66. Not only the value of the roughness exponent is found to be significantly above the random walk exponent ζ = 0.5, it also appears close to the measured one [2,3,4].…”

“…In recent work a model was proposed in 1+1 dimensions for such slow crack propagation via damage nucleation and coalescence ahead of the crack tip [5,6]. A crucial aspect of this model is the existence of a typical length scale ξ c ahead of the crack tip where damage nucleation can take place [7].…”