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′xx and D′xy, and of deviatoric stress, s′xx and s′xy, for the fluid are D′xx = B(Y′2[(n-1)/2]s′xx and D′xy = a2 B(Y′2 [(n-1)sol;2]s′xy, where Y′2 = (s′xx)2 + a2(s′xy)2 is an anisotropic invariant, a2 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 - vy)], where ζ is the height of a foliation trace above its mean plane, v=tanβ, 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.
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