Buckling stability of a viscous single-layer system, oblique to the principal compression

Abstract

A new theory is developed for single-layer buckling, where the layer is not parallel to the principal stresses. The model chosen consists of a single layer with Newtonian viscosity η embedded in an infinite matrix of viscosity η 1. The layer lies at an angle θ to the bulk principal compressive stress in the embedding medium. It is deformed in equal-area plane strain, with the direction of no strain and the third principal bulk stress, parallel to the layer; hence the obliqueness to the principal stresses is only in two dimensions. It is shown that stress refraction is a necessary condition for this system, and an expression is derived for its value in terms of η, η 1 and θ. Buckling stability equations are completely developed which satisfy the Navier-Stokes equilibrium equations for the buckling layer, and the condition of stress continuity at the layer-embedding medium interface. The dominant wavelength of the buckles is shown to be independent of θ, but the stress required increases with θ. The results of this work have an important bearing on natural folds, since there is no evidence that rock layers are initially parallel to the stresses which fold them, an assumption made in former buckling theories. It is suggested that refraction of stresses and the resulting incremental strains gives rise to the finite structure of cleavage refraction so common in deformed rocks, and that the progressive development of folds in layers oblique to the principal bulk stresses gives rise to asymmetry.