A geometric approach to two-dimensional finite strain compatibility: interpretation and review

The purpose of this paper is to collect, clarify, augment and modify the authors' previous work on the subject of finite strain compatibility. The derivations of the fundamental equations are reviewed so that the geometric meaning of each step can be explained. Besides providing a basis for the geological interpretations of the equations, these derivations also lead to a useful new form of the strain compatibility equations.We begin by showing that compatibility is a geometric property of continuous and smooth coordinate grids, and we derive and explain the coordinate grid compatibility equations. We then use the fact that every finite deformation may be described by two coordinate grids to derive finite strain compatibility equations in principal coordinates and Cartesian coordinates. The resulting strain compatibility equations are not easily solved for general strain fields in any coordinate system. Nonetheless, we show that many common geological strain patterns have simple geometries for which the compatibility equations can be interpreted. For example, if a deformation has constant strain in one direction, as most shear zones do, then compatibility provides an iterative method for determining the strain throughout the deformed region if the strain is initially known at any one point. Some of the other strain geometries to which we apply compatibility in this paper include simple shear, inhomogeneous pure shear, parallel and similar folding.