Fluid-driven cracks in an elastic matrix in the toughness-dominated limit

Ching-Yao Lai; Zhong Zheng; Emilie Dressaire; Howard A Stone

doi:10.1098/rsta.2015.0425

Fluid-driven cracks in an elastic matrix in the toughness-dominated limit

Philos Trans A Math Phys Eng Sci. 2016 Oct 13;374(2078):20150425. doi: 10.1098/rsta.2015.0425.

Authors

Ching-Yao Lai¹, Zhong Zheng¹, Emilie Dressaire², Howard A Stone³

Affiliations

¹ Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA.
² Department of Mechanical and Aerospace Engineering, New York University Tandon School of Engineering, Brooklyn, NY 11201, USA.
³ Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA hastone@princeton.edu.

Abstract

The dynamics of fluid-driven cracks in an elastic matrix is studied experimentally. We report the crack radius R(t) as a function of time, as well as the crack shapes w(r,t) as a function of space and time. A dimensionless parameter, the pressure ratio Δpf/Δpv, is identified to gauge the relative importance between the toughness (Δpf) and viscous (Δpv) effects. In our previous paper (Lai et al. 2015 Proc. R. Soc. A 471, 20150255. (doi:10.1098/rspa.2015.0255)), we investigated the viscous limit experimentally when the toughness-related stresses are negligible for the crack propagation. In this paper, the experimental parameters, i.e. Young's modulus E of the gelatin, viscosity μ of the fracturing liquid and the injection flow rate Q, were chosen so that the viscous effects in the flow are negligible compared with the toughness effects, i.e. Δpf/Δpv≫1. In this limit, the crack dynamics can be described by the toughness-dominated scaling laws, which give the crack radius R(t)∝t(2/5) and the half maximum crack thickness W(t)∝t(1/5) The experimental results are in good agreement with the predictions of the toughness scaling laws: the experimental data for crack radius R(t) for a wide range of parameters (E,μ,Q) collapse after being rescaled by the toughness scaling laws, and the rescaled crack shapes w(r,t) also collapse to a dimensionless shape, which demonstrates the self-similarity of the crack shape. The appropriate choice of the viscous or toughness scaling laws is important to accurately describe the crack dynamics.This article is part of the themed issue 'Energy and the subsurface'.

Keywords: fluid–structure interactions; geophysical and geological flows; thin films.