In this paper we utilize the principles of linear elastic fracture mechanics (LEFM), and examine the formation and stability of a system of parallel cracks under a combination of bending and hydrostatic compression, as a possible mechanism for the formation of steeply dipping joint sets. In particular, we develope relationships between the length and spacing of joints, in response to changes in tectonic stresses and strains, and changes in rock fracture toughness. Our results indicate closer joint spacing in rocks with a lower fracture toughness, and wider joint spacing in joints formed at deeper depths. Both of these findings are supported by field observations.
1 INTRODUCTION
Steeply dipping joint sets are found in virtually all geologic settings and rock types, yet the mechanisms responsible for their formation and characteristic dimensions are not well understood at the present time. In general, these joints are characterized by subparallel fractures of lengths and spacings ranging from centimeters to hundreds of meters. The fractures can have uniform lengths and spacings, as found in bedded deposits of sandstones and shales, or nonlinear distributions of lengths and spacings, as found in large homogeneous rock masses such as granite (Price, 1966). The relative displacements across the fractures are predominantly dilatational, and the estimated depth below the earth's surface at the time of joint formation ranges from .5 to 15 km. It is generally agreed that the joints formed in a linear elastic regime, under conditions of predominantly tensile stress, and the fractures extended in a quasistatic manner (Segall and Pollard, 1982). Mechanisms proposed for the origins of tensile stresses include tectonic uplift, thermal contraction, fluid pressure, sediment removal, and others (Nut, 1982). Proposals for mechanisms that control joint length, depth, and spacing are only preliminary and tentative at present, and include regional strain energy, joint interaction, bedding geometry, and others. The principal purpose of this paper is to, analyze the relationships between the length and spacing of joints, in response to changes in tectonic stresses and strains, and changes in rock fracture toughness. We utilize the principles of linear elastic fracture mechanics (LEFI/), and examine the formation and stability of a system of parallel cracks under a combination of bending and hydrostatic compression, as a possible mechanism for the formation of steeply dipping joint sets. In particular, we calculate preferred crack lengths and crack spacings for a two dimensional beam containing parallel cracks and subjected to the fixed displacement boundary conditions of a uniform rotation 6 and a compressional strain. As noted in the works of Berry (1960), Segali (1984), and Kemeny and Cook (1985), fixed displacement boundary conditions are necessary in order to produce stable cracks of finite length (total available energy is fixed). Our boundary conditions are also consistent with the process of tectonic uplift at some depth below the surface (i.e., a rock layer impinged from below by a cylinrical plug with a constant radius of curvature and frictionless surface, and compressed by gravity). Bending along with gravity loading as a mechanism for the formation of joints has also been studied by Price (1966), Nur (1982), and others.
