Spacing of edge fractures in layered materials

Opening-mode fractures developed from a free surface in a layered material often terminate at the interface that divides the fractured layer and the underlying layer. They also display regular spacing that is of the same order of magnitude as the thickness of the fractured layer. We have investigated the stress distribution between two adjacent edge fractures as a function of the ratio of fracture spacing to thickness of the fractured layer using a two-layer elastic model with a fractured top layer. The results show that when the ratio of fracture spacing to the layer thickness changes from greater than to less than a critical value the normal stress acting perpendicular to the fractures near the free surface changes from tensile to compressive. This stress state transition precludes further infilling of fractures unless they are driven by mechanisms other than a pure extension, or there are flaws that significantly perturb the local stress field between the fractures. Hence, the critical fracture spacing to layer thickness ratio defines a lower limit for fractures driven by extension, which also defines the condition of fracture saturation. The critical value of the fracture spacing to layer thickness ratio is independent of the average strain of the fractured layer, and it increases with increasing ratio of Young's modulus of the fractured layer to that of the underlying layer. The critical value increases with increasing Poisson's ratio of the fractured layer, but it decreases with increasing Poisson's ratio of the underlying layer. For the case with the same elastic constants for the fractured layer and the underlying layer, the critical spacing to layer thickness ratio is about 3.1. Delamination between the fractured layer and the underlying layer makes the critical spacing to layer thickness ratio much greater. Infilling fractures grow more easily from flaws located near the bottom of the fractured layer than from those located near the free surface when the spacing to layer thickness ratio is less than the critical value. The propagation of an edge flaw between adjacent edge fractures is unstable, but for the flaw to propagate to the interface, its height has to be greater than a critical size, that decreases with increasing fracture spacing to layer thickness ratio. The propagation behavior of an internal flaw with its lower tip at the interface depends on the edge fracture spacing to layer thickness ratio. The propagation is unstable, when the fracture spacing to layer thickness ratio is greater than a critical value; stable, when the fracture spacing to layer thickness ratio is less than another critical value; and first unstable, then stable, and/or unstable again, when the fracture spacing to layer thickness ratio is between these two critical values.