From progressive to finite deformation and back

Abstract

It is widely accepted that any finite strain recorded in the field may be interpreted in terms of the simultaneous combination of a pure shear component with one or several simple shear components. To predict strain in geological structures, approximate solutions may be obtained by multiplying successive small increments of each elementary strain component. A more rigorous method consists in achieving the simultaneous combination in the velocity gradient tensor, but solutions already proposed in the literature are valid for some special cases only and cannot be used, e.g., for the general combination of a pure shear component and six elementary simple shear components. In this paper, we show that the combination of any strain components is very simple, both analytically and numerically. The finite deformation tensor is given by D = exp (LΔt), where LΔt is the time‐integrated velocity gradient tensor. This method makes it possible to predict finite strain for any combination of strain components. Reciprocally, LΔt = ln (D), which allows us to unravel the simplest deformation history that might have generated a given finite deformation. Given the strain ellipsoid only, it is still possible to constrain the range of compatible deformation tensors and thus the range of strain component combinations. Interestingly, certain deformation tensors, though geologically sensible, have no real logarithm and so cannot be explained by a deformation history implying strain rate components with a common time dependence. This implies significant changes of stress field or material rheology during deformation.