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

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.