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Rock formations are very often characterized by the presence of fracturesthat have grown subcritically over geological time scales and under evolvingstress fields. In mechanically layered systems, such fractures can eitherbecome layer-bound or penetrate into adjacent strata. The growth of fracturesin brittle materials is generally dependent on the energy release rate, howeverno closed form analytical solutions exist for these when a crack tip is in theproximity of a material interface. In this study, we present new empiricalformulas for calculating the energy release rate at the tip of a crackperpendicular to a material interface in a symmetric 3-layer system. In theseformulas, the normalized energy release rate is expressed as the product of abase term that integrates the normalized stiffness modulus over the cracklength, and a correction factor that accounts for the presence of a materialinterface. The latter is assumed to be dependent on two quantities: the ratioof the crack length to the inner layer thickness, and the contrast in materialstiffness between the inner and outer layers. The correction factors areobtained by fitting the parameters of carefully chosen expressions to a set offinite element solutions in order to yield predictions that are accurate towithin one percent of the numerical results.

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