The compatibility equations and the pole to the Mohr circle

Abstract

We derive a new form of the compatibility equations for large deformations. These equations show that in a continuous inhomogeneous deformation, the strain gradients are related to the curvatures of the principal strain trajectories. In the case of uniform area strain, the equations express a direct relationship between the shape of the strain ellipse at a point and the curvatures of the principal trajectories. These relationships become particularly useful if the fabric in a deformed rock is taken as parallel to the principal strain trajectories. We demonstrate that the compatibility equations provide important strain information for several geologically interesting special cases: uniform area strain, compaction of a bed, fanned or axial planar cleavage, and some three-dimensional structures such as cylindrical folds. We also show that in a three-dimensional structure with one straight strain trajectory there will always be uniform area strain in the cross-section normal to the straight trajectory, as long as the volume strain in the structure is uniform. The pole of the finite strain Mohr circle is a unique point on the circle which graphically relates the state of strain in a body to its orientation in the physical plane. If a set of Mohr circles describes an inhomogeneous state of strain, then the curve connecting the poles of these circles is called the pole curve. We derive an exact analytical expression for the pole curve which applies to ductile deformation zones, refracted cleavage, and deformed stratigraphic sections; all with uniform area strain. For these special cases, the pole curve describes the entire strain field for the deformation as long as we know the strain at any one point in the deformed zone.