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

Raymond Charles Fletcher

Research output: Contribution to journalArticle

10 Citations (Scopus)

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′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.

Original languageEnglish (US)
Pages (from-to)1155-1167
Number of pages13
JournalJournal of Structural Geology
Volume27
Issue number7
DOIs
StatePublished - Jul 1 2005

Fingerprint

foliation
power law
fluid
anisotropy
boudinage
wavelength
buckling
lamination
deflection
amplification
matrix

All Science Journal Classification (ASJC) codes

  • Geology

Cite this

Fletcher, Raymond Charles. / Instability of an anisotropic power-law fluid in a basic state of plane flow. In: Journal of Structural Geology. 2005 ; Vol. 27, No. 7. pp. 1155-1167.
@article{8b0809cef4a3470887408bad593d5cf9,
title = "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′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.",
author = "Fletcher, {Raymond Charles}",
year = "2005",
month = "7",
day = "1",
doi = "10.1016/j.jsg.2004.08.002",
language = "English (US)",
volume = "27",
pages = "1155--1167",
journal = "Journal of Structural Geology",
issn = "0191-8141",
publisher = "Elsevier Limited",
number = "7",

}

Instability of an anisotropic power-law fluid in a basic state of plane flow. / Fletcher, Raymond Charles.

In: Journal of Structural Geology, Vol. 27, No. 7, 01.07.2005, p. 1155-1167.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Instability of an anisotropic power-law fluid in a basic state of plane flow

AU - Fletcher, Raymond Charles

PY - 2005/7/1

Y1 - 2005/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=23944439639&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23944439639&partnerID=8YFLogxK

U2 - 10.1016/j.jsg.2004.08.002

DO - 10.1016/j.jsg.2004.08.002

M3 - Article

AN - SCOPUS:23944439639

VL - 27

SP - 1155

EP - 1167

JO - Journal of Structural Geology

JF - Journal of Structural Geology

SN - 0191-8141

IS - 7

ER -