Crack paths in three-dimensional elastic solids. i: two-term expansion of the stress intensity factors—application to crack path stability in hydraulic fracturing

Abstract

The aim of this work is to lay theoretical foundations for the prediction of crack paths within the theory of quasistatic LEFM under the most general hypotheses: arbitrary three-dimensional geometry, arbitrary loading. This objective requires to derive the expression of the stress intensity factors along the crack front after an arbitrary infinitesimal propagation. Only the first two terms of their expansion in powers of the crack extension length δ, proportional to δ 0 = 1 and δ fn1 fn1 The aim of the introduction of the multiplicative decomposition of δ ( s′) in the form ε η ( s′) is precisely to allow for expansions in terms of a single parameter ε instead of the function δ ( s′), which can be characterized only by an infinite number of parameters. fn2 fn2 Note that L∗ still depends on C and Γ, however, because the SIFs just after the kink depend on those just before it which, when expressed as functions of the loading T, depend on the curvatures of the surface and the front of the initial crack. , are considered in this paper. Fully general formulae for these terms are obtained by combining arguments of dimensional analysis (scale changes) and regularity properties (continuity, differentiability) of the stresses at a fixed, given point with respect to δ for δ = 0 derived from the Bueckner–Rice weight function theory. This notably allows to extend the Cotterell–Rice criterion for stable rectilinear propagation of a mode I crack under plane strain conditions to the three-dimensional case. As an application, a penny-shaped crack induced by hydraulic fracturing is considered. Conclusions concerning the influence of the orientation and depth of such a crack upon the stability of its coplanar propagation seem to be compatible with experimental evidence.