Approximate formulas for the energy release rate of a crack perpendicular to a material interface

TL;DR

This work addresses the energy release rate $G$ of a mode-I crack perpendicular to a material interface in a symmetric 3-layer system. Using FE simulations, the authors derive two interface-correction functions, $f_1(\\Upsilon,\\lambda)$ for cracks in the inner layer and $f_2(\\Upsilon,\\lambda)$ for cracks that penetrate into outer layers, and express the normalized energy release rate as $\\Gamma = \\pi\\varepsilon_f^2\\lambda f_1(\\Upsilon,\\lambda)$ or $\\Gamma = \\pi\\varepsilon_f^2(\\Upsilon+\\lambda-1) f_2(\\Upsilon,\\lambda)$, respectively. These corrections are fitted with explicit closed-form expressions to within about 1\% of FE results, incorporating Koiter/Tada-type limiting behavior and nontrivial dependence on crack position and stiffness contrast. The resulting formulas enable efficient prediction of layer-bound vs through-cutting fracture growth in geological layering and may be extended to more complex multi-layer configurations and alternative fracture theories to address LEFM limitations near interfaces.

Abstract

Rock formations are very often characterized by the presence of fractures that have grown subcritically over geological time scales and under evolving stress fields. In mechanically layered systems, such fractures can either become layer-bound or penetrate into adjacent strata. The growth of fractures in brittle materials is generally dependent on the energy release rate, however no closed form analytical solutions exist for these when a crack tip is in the proximity of a material interface. In this study, we present new empirical formulas for calculating the energy release rate at the tip of a crack perpendicular to a material interface in a symmetric 3-layer system. In these formulas, the normalized energy release rate is expressed as the product of a base term that integrates the normalized stiffness modulus over the crack length, and a correction factor that accounts for the presence of a material interface. The latter is assumed to be dependent on two quantities: the ratio of the crack length to the inner layer thickness, and the contrast in material stiffness between the inner and outer layers. The correction factors are obtained by fitting the parameters of carefully chosen expressions to a set of finite element solutions in order to yield predictions that are accurate to within one percent of the numerical results.

Figures (15)

• Figure 1: Mechanically layered geological system with both layer-bound and through-cutting fractures, which may be either open or sealed. Instances of the latter can be seen in the lower-left and lower-right regions of the image.
• Figure 2: Infinite heterogeneous medium containing a finite central crack. We investigate two symmetric configurations: (a) one where the crack is confined to the inner layer, and (b) one where the crack tips are situated in the outer layers. Note that the cracks are generally vertical in real life, however as the current study ignores the effect of gravity, the equivalent analysis can also be performed on a model containing horizontal fractures.
• Figure 3: Computational domains with applied boundary conditions for (a) $a < L$, and (b) $a > L$. The prescribed upward displacement $\bar{u}_y$ is chosen such that a uniform strain $\varepsilon_{yy} = 0.1$ is realized at the top and bottom boundaries. Thus, $\bar{u}_y = 100 L$ when $a < L$, while $\bar{u}_y = 100 a$ when $a > L$.
• Figure 4: Meshing details for $a = 0.999L$. The full computational domain is shown at the left, together with a closeup of the crack-tip region. For clarity, the crack is highlighted in red, the material interface in blue, and the integration path for the $J$-integral in black. For all meshes, the closed path used to calculate the $J$-integral is constructed to lie fully within the material layer containing the crack tip, as illustrated in Figure \ref{['fig:compDomains']}.
• Figure 5: Variation in maximum principal stress in the crack tip vicinity for some combinations of the normalized crack length and material stiffness contrast: (a) $\Upsilon = 5$, $\lambda = 0.9$; (b) $\Upsilon = 0.2$, $\lambda = 0.9$; (c) $\Upsilon = 5$, $\lambda = 1.1$; (d) $\Upsilon = 0.2$, $\lambda = 1.1$.