Implication of complex eigenvalues in homogeneous flows: a three dimensional kinematic analysis

We present an investigation of the kinematic properties of 3D homogeneous flow defined by complex eigenvalues. We demonstrate, using simple algebraic analysis, that the clear threshold between pulsating and non-pulsating fields, fixed for Wn > 1 and valid for planar flow, is not easily defined in a 3D flow system. In 3D flows, one of the three eigenvalues is always real and gives rise to an exponential flow, coexisting with a pulsating pattern defined by the other two complex conjugate eigenvalues. Due to this mathematical property, the existence of a stable or pulsating pattern depends strongly on the relative dominance of the real eigenvector with respect to the complex ones. As a consequence, the pattern of behaviour is not simply imposed by the kinematic vorticity numbers, but is also determined by both the amount of strain accumulation and the extrusion component. It is also shown that complex flow can occur locally within shear zones and can sustain some predictable hyperbolic strain paths. These results are applied to the kinematic analysis of some non-dilational and dilational monoclinic and triclinic flows. Some geological implications of this investigation, and the limit of applying these algebraic and kinematic results to real rocks fabric analysis, are briefly discussed.