Instability of an anisotropic power-law fluid in a basic state of plane flow

Abstract

Initiation of cylindrical structures by buckling or necking in an anisotropic power-law fluid is treated for general plane flow. The principal axis of anisotropy, x′, in the stiffest direction in shortening or extension may be viewed as the trace of a foliation or lamination. Plane-flow constitutive relations between components of rate of deformation, D ′ x x and D ′ x y , and of deviatoric stress, s ′ x x and s ′ x y , for the fluid are D ′ x x = B ( Y ′ 2 ) [ ( n − 1 ) / 2 ] s ′ x x and D ′ x y = a 2 B ( Y ′ 2 ) [ ( n − 1 ) / 2 ] s ′ x y , where Y ′ 2 = ( s ′ x x ) 2 + a 2 ( s ′ x y ) 2 is an anisotropic invariant, a 2 is the anisotropy parameter, and n is the stress exponent. We determine the rate of amplification of wavelength components in the deflection of the foliation, θ, from a mean orientation parallel to x. Linearly independent, or non-interacting normal modes have a periodic, band-like form θ ( x , y ) ≅ ∂ ζ / ∂ x = − ( λ A ) sin [ λ ( x − v y ) ] , where ζ is the height of a foliation trace above its mean plane, v=tan β, where β is the angle between the normal to mean foliation and the axial surface, positive clockwise, and L=2π/ λ is the foliation-parallel wavelength. Evolution of a component may be followed through a finite bulk deformation provided θ remains ≪1. The growth rate of slope, λA, is independent of L. Components with axial plane normal to the foliation ( β=0) are strongly amplified in foliation-parallel shortening. If n>>1, internal necking (boudinage) occurs in foliation-parallel extension for components with axial plane inclined at a large angle to the foliation normal. In combined shortening and shear, the most rapidly growing component has an axial plane that dips steeply in the direction of shear. For n>1, maximum instability occurs for combined foliation-parallel shear and shortening rather than pure shortening. Weak instability is present in foliation-parallel shear. This anisotropic nonlinear fluid approximates the behavior of an isotropic power-law medium containing preferentially oriented but anastomosing slip surfaces, or that of a rock in which a stiffer component of lenticular form is embedded in a softer matrix.