Void Formation and Roughening in Slow Fracture

TL;DR

This paper develops an analytic model using iterated conformal maps to understand void formation and crack roughening in slow fracture, providing estimates of roughness exponents and exploring the effects of multiple voids.

Contribution

It introduces a novel conformal mapping approach to model void-induced crack roughening and extends the theory to include multiple voids ahead of the crack tip.

Findings

The roughening exponent in 2D is estimated around 0.65.

Allowing one void simplifies the model and shows robustness of the roughening exponent.

Including two voids complicates the stress field analysis, indicating need for improved plasticity modeling.

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. Post mortem analysis of the resulting fracture surfaces of ductile and brittle materials on the $\mu$m-mm and the nm scales respectively, reveals self-affine cracks with anomalous scaling exponent $\zeta\approx 0.8$ in 3-dimensions and $\zeta\approx 0.65$ 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.