  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.
