(10.73)
Taking A Ͻ Ͻ 1, we use the approximations:
(10.74)
We seek a solution accurate to terms proportional
to A. In the derivation, if terms proportional to
(A)
2 or higher are encountered, they are deleted.
We cannot hope to simulate the development of a
large-amplitude mullion (Fig. 10.10a) with this
approximation. Instead, we study mullion initiation and the inception of an asymmetric lobe-andcusp structure.
For the wavy interface, the basic-state solution
no longer satisfies all boundary conditions at the
interface to the desired accuracy, as we now
demonstrate by introduction of a correction term
in the form of additional sets of velocity, stress,
and rate of deformation components. The velocity components in the two media are the sum of
the respective components from the basic state
and from this corrective state, so they have the
form:
(10.75)
The stress and rate of deformation components
are composed of similar sums. Suppose that the
interface is welded, so velocity components of
adjacent particles across it are equal:
(10.76)
Since the basic-state velocity components already
satisfy these conditions, (10.76) reduce to:
(10.77)
The boundary conditions on the normal and
shear components of the traction on the interface
are:
(10.78)
The tractions will be evaluated for each medium
in terms of the interface geometry and the comt (1)
n (x, ) ϭ Ϫt (2)
n (x, )
t (1)
s (x, ) ϭ Ϫt (2)
s (x, )
v
~ (1)
y (x, ) ϭ v
~ (2)
y (x, )
v
~ (1)
x (x, ) ϭ v
~ (2)
x (x, )
v (1)
y (x, ) ϭ v (2)
y (x, )
v (1)
x (x, ) ϭ v (2)
x (x, )
v (2)
x ϭ v (2)
x ϩ v
~ (2)
x , v (2)
y ϭ v (2)
y ϩ v
~ (2)
y
v (1)
x ϭ v (1)
x ϩ v
~ (1)
x , v (1)
y ϭ v (1)
y ϩ v
~ (1)
y
cos Х 1 Ϫ 2 ր2 Х 1
sin Х Ϫ 3 ր6 Х Х Ϫ(A) sin x
tan Х ϩ 3 ր3 Х Х Ϫ (A) sin x
tan ϭ Ѩ րѨx ϭ Ϫ( A) sin x
ponents of stress (Fig. 10.11). We might also have
used the components with respect to the coordinate axes x and y. The present choice is more generally useful; for example, if the interface were
frictionless, the shear component of the traction
would be required to vanish. It would then be necessary to write the velocity boundary conditions
in terms of normal and tangential components
also, since slip on a frictionless interface would
exclude a condition being set on the tangential
component of velocity.
Expressing the tractions in terms of the stress
components referred to the local coordinate axes,
n and s:
(10.79)
In terms of the stress components, the boundary
conditions are:
(10.80)
These conditions must be written in terms of the
components of stress referred to the (x, y)-coordinates used in the field equations, in which the
solution will be expressed. Using the transformation law we have:
(10.81)
Using (10.74), and retaining only terms up to those
proportional to A, we have:
(10.82)
No correction to the basic-state solution is necessary if the interface is perfectly planar (A ϭ 0);
that is, the perturbing solution is identically zero.
Therefore we postulate that the components of
stress and velocity in the perturbing solution are
themselves proportional to A. Since
we
have
and the term xy in the first of
(10.82) can be discarded because it is a product of
two terms, both of which are proportional to A.
Applying these conditions, and the forms (10.74),
the stress boundary conditions (10.82) become:
xy ϭ
~ xy
xy ϭ 0,
ns Х Ϫ( xx Ϫ yy ) ϩ xy
nn Х Ϫ2 xy ϩ yy
ns ϭ Ϫ( xx Ϫ yy ) sin cos ϩ xy ( cos 2 Ϫ sin 2 )
nn ϭ xx sin 2 Ϫ 2 xy sin cos ϩ yy cos 2
(1)
ns (x, ) ϭ (2)
ns (x, )
(1)
nn (x, ) ϭ (2)
nn (x, )
t (1)
s ϭ (1)
ns , t (2)
s ϭ Ϫ (2)
ns
t (1)
n ϭ (1)
nn , t (2)
n ϭ Ϫ (2)
nn
10.4 VISCOUS FLOW IN LAYERS: MULLIONS AND FOLDS
399
Taking A Ͻ Ͻ 1, we use the approximations:
(10.74)
We seek a solution accurate to terms proportional
to A. In the derivation, if terms proportional to
(A)
2 or higher are encountered, they are deleted.
We cannot hope to simulate the development of a
large-amplitude mullion (Fig. 10.10a) with this
approximation. Instead, we study mullion initiation and the inception of an asymmetric lobe-andcusp structure.
For the wavy interface, the basic-state solution
no longer satisfies all boundary conditions at the
interface to the desired accuracy, as we now
demonstrate by introduction of a correction term
in the form of additional sets of velocity, stress,
and rate of deformation components. The velocity components in the two media are the sum of
the respective components from the basic state
and from this corrective state, so they have the
form:
(10.75)
The stress and rate of deformation components
are composed of similar sums. Suppose that the
interface is welded, so velocity components of
adjacent particles across it are equal:
(10.76)
Since the basic-state velocity components already
satisfy these conditions, (10.76) reduce to:
(10.77)
The boundary conditions on the normal and
shear components of the traction on the interface
are:
(10.78)
The tractions will be evaluated for each medium
in terms of the interface geometry and the comt (1)
n (x, ) ϭ Ϫt (2)
n (x, )
t (1)
s (x, ) ϭ Ϫt (2)
s (x, )
v
~ (1)
y (x, ) ϭ v
~ (2)
y (x, )
v
~ (1)
x (x, ) ϭ v
~ (2)
x (x, )
v (1)
y (x, ) ϭ v (2)
y (x, )
v (1)
x (x, ) ϭ v (2)
x (x, )
v (2)
x ϭ v (2)
x ϩ v
~ (2)
x , v (2)
y ϭ v (2)
y ϩ v
~ (2)
y
v (1)
x ϭ v (1)
x ϩ v
~ (1)
x , v (1)
y ϭ v (1)
y ϩ v
~ (1)
y
cos Х 1 Ϫ 2 ր2 Х 1
sin Х Ϫ 3 ր6 Х Х Ϫ(A) sin x
tan Х ϩ 3 ր3 Х Х Ϫ (A) sin x
tan ϭ Ѩ րѨx ϭ Ϫ( A) sin x
ponents of stress (Fig. 10.11). We might also have
used the components with respect to the coordinate axes x and y. The present choice is more generally useful; for example, if the interface were
frictionless, the shear component of the traction
would be required to vanish. It would then be necessary to write the velocity boundary conditions
in terms of normal and tangential components
also, since slip on a frictionless interface would
exclude a condition being set on the tangential
component of velocity.
Expressing the tractions in terms of the stress
components referred to the local coordinate axes,
n and s:
(10.79)
In terms of the stress components, the boundary
conditions are:
(10.80)
These conditions must be written in terms of the
components of stress referred to the (x, y)-coordinates used in the field equations, in which the
solution will be expressed. Using the transformation law we have:
(10.81)
Using (10.74), and retaining only terms up to those
proportional to A, we have:
(10.82)
No correction to the basic-state solution is necessary if the interface is perfectly planar (A ϭ 0);
that is, the perturbing solution is identically zero.
Therefore we postulate that the components of
stress and velocity in the perturbing solution are
themselves proportional to A. Since
we
have
and the term xy in the first of
(10.82) can be discarded because it is a product of
two terms, both of which are proportional to A.
Applying these conditions, and the forms (10.74),
the stress boundary conditions (10.82) become:
xy ϭ
~ xy
xy ϭ 0,
ns Х Ϫ( xx Ϫ yy ) ϩ xy
nn Х Ϫ2 xy ϩ yy
ns ϭ Ϫ( xx Ϫ yy ) sin cos ϩ xy ( cos 2 Ϫ sin 2 )
nn ϭ xx sin 2 Ϫ 2 xy sin cos ϩ yy cos 2
(1)
ns (x, ) ϭ (2)
ns (x, )
(1)
nn (x, ) ϭ (2)
nn (x, )
t (1)
s ϭ (1)
ns , t (2)
s ϭ Ϫ (2)
ns
t (1)
n ϭ (1)
nn , t (2)
n ϭ Ϫ (2)
nn
10.4 VISCOUS FLOW IN LAYERS: MULLIONS AND FOLDS
399