The geometrical significance of strain trajectory curvature

Abstract

We derive the following two-dimensional equations for the curvature of intersecting strain trajectories: K 1 = λ 2(k 1 − f 1); K 2 = λ 1(k 2−f 2) The subscripts refer to one or the other trajectory; K expresses curvature in the undeformed state; k, curvature in deformed state; λ is the stretch (final length divided by original length); and f is the flexure, a logarithmic strain gradient defined for each trajectory as f 1 = − ∂ε 1/ ∂s 2 and f 2 = ∂ε 2/ ∂s 1, where ε = ln λ is logarithmic stretch and s is true arc length along a trajectory. The curvature equations are simple forms of the equations of strain compatibility. We derive them from first principles in the Appendix. In the main text, we illustrate and verify the curvature equations using six theoretical examples of strain fields and an experimental one. Each theoretical example is deliberately chosen to illustrate the contribution of one or more terms in the curvature equations. The first four examples are single fans, where trajectories are polar co-ordinates. The other three examples are double fans from more complex strain fields. The equations and examples sometimes uphold and sometimes contradict the convergence hypothesis, in which strain intensity is assumed to increase as trajectories converge. The equations also show that strain gradients can exist even if both trajectories have no curvature and there are no volume changes. In practice, if a geologist knows only the trajectory curvatures at one point in the deformed state, he cannot determine strain gradients, because there are too many unknowns in the curvature equations.