The extremum rule of angular variation of finite deformation and its application in structural geology

Abstract

This paper is devoted to finite deformation theory and more specifically to the dispersion of cones of lines about an axis in the neighbourhood of a point. The paper has been fully verified that when a small angle Δθ in an initial sphere changes to Δθ in the strain ellipsoid, the limit ratio δθ δΘ has extrema, provided Δθ is within the second principal plane and contains the first or third principal stretch axis. The extremum rule of angular variation is applied for principal stretch axes. For uniaxial tensile, pure shear, slide and simple shear, the author has derived the formulae of the principal stretch axes, and illustrated the equivalence relation of the rotation angle of the principal axes to the mean rotation angle of all line elements passing through a point. To determine the principal stretch axes of a fossil deformed in two dimensions, the author has used the curve of θ against Θ, i.e. the orientation angle of a mark line in the deformed fossil against the one in the undeformed fossil. In the last part of this paper the analytical expressions of extreme stretch trajectories in a heterogeneous shear zone with the distribution of parabolic displacement are obtained.