OO22-5096193
$5.00+0.00
Pergamon
Press
Ltd
J. Me& P/i~v. Solids Vol. 41, No. 1. pp. 117~142, 1993.
Printed
inGreat
Britain.
INTERFACIAL
INSTABILITY
OF DENSITY-STRATIFIED
TWO-LAYER
SYSTEMS UNDER INITIAL
STRESS
N. TRIANTAFYLLIDIS
Department
of Aerospace
Engineering,
University
of Michigan
Ann Arbor,
MI 48109-2140,
USA
and
F. K. LEHNER
Koninklijke/Shell
Exploratie
(Rrceiced
en Produktie
8 Muy 1991
Laboratorium,
2288 GD Rijswijk Z.H., The Netherlands
; in recised ,form I3
February
1992)
ABSTRACT
THE ONSET of interfacial
instability in two coherent semi-infinite layers of different properties
and the
corresponding
critical wavelength are found by solving a static bifurcation
problem in finite plane strain.
Subsequently,
the stability ofperturbations
of any wavelength is determined from the appropriate
linearized
equations
of motion. For gravitationally
stable or unstable density stratifications,
the critical stress at
which the interface is destabilized
is shown to depend on the wavelength
of a perturbation;
it is also
determined in a complex manner by initial stress gradients perpendicular
to the layer interface and by layer
stiffness, as is illustrated here in detail for the examples of a hyperelastic solid and an elastoplastic
solid,
both resting on an inviscid fluid of different density. The very large wavelength that is predicted for the
gravitational
instability of a semi-infinite elastic solid on a buoyant fluid substratum
brings forward the
essential role of pre-stress and associated stiffness reduction as well as that of a finite layer thickness in
destabilizing geological and geophysical two-layer systems.
1.
INTRODUCTION
A VARIETY of problems lead to the question of the precise conditions under which an
initially plane interface between two layers of different properties may be destabilized
by layer-parallel
stresses, counteracted
or aided in their effect by body forces. An
example that has been of particular
interest to geologists and geophysicists
is the
instability
of density-stratified
two-layer systems in a gravitational
field [see e.g.
RAMBERC
(1981)] and the same question has motivated
the present investigation.
Earlier theoretical
studies of this problem have mostly been concerned
with the
Rayleigh-Taylor
model of two superimposed
viscous fluids, in which gravity provides
the only destabilizing
force. In the classical stability analysis of this problem (TAYLOR,
~~~~;CHANDRASEKHAR,
1955, I~~I;DANES, 1964; SELIG, 1965)theinitialgrowth
of
a small interfacial
perturbation
is studied. Although all wavelengths
are found to
satisfy the linearized perturbation
equations,
the presence of viscosity implies the
existence of a dominant wavelength that possesses the fastest growing amplitude and
is therefore expected to characterize the evolution of the system. However, an essential
limitation of fluid dynamical models of interfacial stability has been their inability to
IN‘S
1992Shell Research B.V.
117
118
N. TKIANTAFYLLIIXS
and F. K. LEHNER
for the effects of layer stiffness and the stabilizing or destabilizing
role of
anisotropic
and nonhydrostatic
states of stress. An entirely different approach to the
problem of surface as well as interfacial instability in laterally compressed elastic and
viscoelastic materials was pioneered by BIOT (1963a, b. c ; 1965), who treated the onset
of interfacial instability as a bifurcation
problem.
This paper seeks to extend Biot’s analysis by considering
the combined effects of
gravity and nonhomogeneous
states of initial stress on interfacial instability;
its aim
is to establish conditions for the existence of a nontrivial solution to the incremental
equilibrium
equations of two solid material half-spaces of differing densities that are
separated by an initially planar interface normal to the direction of gravity and are
allowed to undergo finite deformation
in plane strain. An analytical condition will be
given that relates the critical wavelength for a bifurcation
to the material properties
and stress states in each half-space. The modes of interfacial
instability
that are
obtained from the following analysis exclude an entire range of wavelengths and-for a fixed load--imply
a single admissible critical wavelength.
A second aim of this paper is to study the stability of the layer interface under the
assumed general conditions.
The stability of interfacial
disturbances
of arbitrary
wavelength will be characterized
by means of a perturbation
analysis of the system’s
linearized equations
of motion. A perturbation
is characterized
as unstable if its
amplitude increases with time; periodic perturbations
of constant amplitude are to
be considered stable for reasons explained in Section 2. Finally, perturbations
which,
according to a linear analysis, exhibit neutral stability with time-independent
perturbation modes are shown to correspond to the critical wavelength. The connection
between bifurcation
and stability is thereby established in a quantitative
way.
For the case of a gravitationally
unstable density stratification
and in the absence
of a gradient in prestress, perturbation
modes below the critical wavelength are found
to be stable while those above the critical wavelength are found to be unstable as
expected. Indeed, perturbation
modes below a certain wavelength require more work
than can be extracted from the system by lowering its potential energy; these modes
are therefore inhibited. The critical wavelength is that which balances the two energies
and allows the incremental
equilibrium
equations for the corresponding
perturbation
mode to be satisfied. The situation
is considerably
complicated
by a gradient in
prestress, as a consequence
of which the results become more sensitive to the nonlinearity of the constitutive
relation.
Following the development
of the general theory, an application
will be made to
the case where the !ower half-space is occupied by an inviscid fluid and the upper by
a solid of different density, both materials being taken as incompressible.
The effects
of non-linear
constitutive
behaviour
will be explored by contrasting
the stability
behaviour of a rubber-like solid with that of a rate-independent
elastoplastic material
whose stiffness decreases with deformation.
account
2.
GOVERNING
EQUATIONS
The instability
of a pre-stressed heavy layer resting on a substratum
of different
density will be analysed in the following as a static bifurcation
phenomenon.
The
Interfacial instability
119
onset of the bifurcation,
i.e. the instance of loss of uniqueness of the trivial principal
solution for a flat interface, will be studied first. This is a natural question to address,
for it provides the relation between the characteristic
wavelength of the bifurcation
mode on one hand and the corresponding
layer parameters, such as stresses, tangent
moduli and densities, on the other. A theoretical
framework
for the analysis of
bifurcation
instabilities
has been established for elastic solids by KOITER (1945) and
for rate independent
elastoplastic
solids by HILL (1958) and the same subjects have
been reviewed comprehensively
by BUDIANSKY (1974) and HUTCHINSON (1974).
For a general formulation
of the type of problem considered here, the governing
equations
will be stated first in a full Lagrangian
form. Accordingly,
the material
points of a body of volume V and bounding
surface dV in the undeformed
configuration are identified by their initial Cartesian coordinates _I’,,while the components
of the displacement
vector are denoted by u,. A superimposed
dot-such
as on ti,designates
the derivative
of a field variable with respect to some monotonically
increasing
time-like parameter
that traces the evolution
of stress and deformation states during a loading process. The comma notation
u,,, is used to denote
partial differentiation
with respect to X,. Einstein’s summation
convention
over
repeated indices is adopted throughout
this work, with Latin indices ranging from
1 to 2 in plane strain problems and from 1 to 3 in more general three-dimensional
formulations.
The prebifurcation
state of the body or material region that will be considered is
characterized by a planar interface between the two layers and a horizontally
uniform
stress state. This fundamental solution can be completely specified as a function of a
monotonically
increasing scalar quantity A, the loadparameter.
At every stage of the
deformation
process one may seek the incremental
response of the solid layer-in
terms of the rates 7-&jand I_.&
of the first Piola-Kirchhoff
stress 7~,~and displacement
vector u, in the interior of the layer--for
a given I\ that specifies a boundary traction
increment i; or boundary displacement
increment ti,.
The equation of equilibrium
for the solid, written in its variational
(or weak) form,
is
where F,, denote the components
of the deformation
gradient tensor, p the density of
the solid per unit reference volume and bj the components
of the acceleration
of
gravity. Assuming that p and b, are time-independent
constants for each layer, the
condition of continuing
equilibrium,
i.e. the rate-form of eqn (I), becomes
Suppose now that at some value of A a bifurcation
in the
of the solid becomes posssible, so that for a given increment n
solutions tiyi, zi: and ti(:, I$. Let A(.) = (*)” - (*)” denote the
such solutions for any field quantity (*). At every boundary
deformational
response
there exist two different
difference between two
point either Api = 0, if
120
N. TKIANTAFYLLIDIS
and
F. K. LEHNER
tractions are prescribed, or Sti, = 0, if displacements
are prescribed. The difference
between any two incremental
equilibrium
solutions x and /I to eqn (2) must therefore
satisfy
s
I
Ati,i SF,, d V =
si c
Ai; 6u, dA = 0.
(3)
An integration
by parts now furnishes both the pointwise incremental
equilibrium
or Euler-Lagrange
equation,
as well as the corresponding
boundary
and interface
conditions. The latter are fully determined only by an additional adherence condition,
which will be assumed to hold for the material on either side of the overburden/
substratum
boundary.
One obtains
(A%,),,= 0
[Ati,,N,]
= 0,
(at interior
[Ati)] = 0
points of V),
(at the interface).
(4)
where IV, are the components
of the unit normal to the material interface and the
customary
bracket notation
i.. .I) is employed to denote a jump in a field quantity
across that boundary. The second jump condition expresses the assumption of perfect
bonding of the two materials. To complete the formulation,
further conditions must
be imposed on Ati, and Ap!, at infinite distances from the interface;
these will be
considered later in the context of specific applications.
The rate-independent
materials that are to be considered will be assumed to obey
the following constitutive
equations :
%,, =
(compressible
L,,dh
r-t/, = &&/’
-P(S,,
+ u,.,)
’ _1 det (6,, + Q)
= 1
(incompressible
solid).
solid),
(5)
where the second equation involves the pressure increment @. The components
Llih,
of the incremental
modulus tensor generally depend on the current stress state as well
as on the stress path from the reference to the current state. Having selected one or
the other form of (5), one may now seek a solution to the incremental boundary value
problem, such that 7i,, and ti, are found in the interior of Vfor a given increment A or
corresponding
increments in i; and ti, along a V.
Equations (4) and (5) provide a basic set of equations for studying the bifurcation
problem that is associated with the stability of the overburden/substratum
interface.
They are written in terms of a fixed reference configuration,
corresponding
to a full
Lagrangian formulation.
For problems, such as the present one, that are characterized
by simple prebifurcation
states, it is often convenient to select the current configuration
as the reference configuration
and accordingly,
to take U, = 0, F,, = S,,, but ti, # 0,
6, # 0. This updated Lagrangian formulation
will be used in this paper, its advantage
lying in the use of Cauchy stress and in the fact that the mode shapes are measured
in the current configuration
at the onset of the bifurcation.
Next, in considering
the stability of an initially flat interface, it will suffice to
investigate the behaviour of the system when subjected at time t = 0 to a perturbation
Interfacial
121
instability
of small amplitude E but arbitrary wavelength about the equilibrium
state of interest.
This is achieved by studying the solution of the linearized dynamical equations for
the system. The starting point for this analysis is the weak form of the system’s
equations of motion
s s
nl,
&-ii,)
6F,, d V =
T,&,dA,
&,dV+
uj,&givenat
t = 0.
s iY
I'
1'
(6)
All field quantities
for the perturbed
system can be written as a sum of their corresponding
fundamental
(unperturbed)
values, which are denoted by a superscript
(‘), plus a term that depends on the amplitude t: of the initial disturbance. By expanding
these field quantities with respect to the initial disturbance
amplitude, one has for the
first Piola-Kirchhoff
stress, the displacement
and pressure fields
7-r,j= 71,:+&E,j+O(&2),
24;= uI~+Flli+O(&2),
$7=p0+&@+o(&2).
(7)
It is worth mentioning
at this point that all O(1) field quantities associated with the
fundamental
solution are functions of A and are independent
of time, unlike all the
O(C) (p >, 1) dependent terms in the perturbation
which do depend on time.
A straightforward
linearization
of the equations of motion obtained by introducing
(7) into (6) and by keeping only the terms of O(E) gives the linearized equations of
motion
(72i/ ).r
=
(at interior
PG,
[CijN,] = 0,
Moreover,
yields
(at the material
[L&J = 0
the same linearization
points
procedure
applied
;
7111= 4&.h
interface).
G,,, = 1
(8)
to the constitutive
(compressible
Zii = L,,ii,ii,,k-$,,,
of V),
equation
(5)
solid),
(incompressible
solid).
(9)
The above two sets of equations (8) and (9), complemented
by the initial conditions
in Wiand di which are given at t = 0 completely specify the O(E) term in the expansion
of the perturbed system.
Following
the standard assumption
adopted in linearized stability analysis, it is
assumed that the omitted O(c2) terms are negligible over the time interval of interest
and that the first order terms adequately characterize the motion of the system. Since
the coefficients in the system of linearized equations of motion do not depend on time,
the solution to (8) and (9), subject to the aforementioned
initial conditions
of given
displacement
and velocity, takes the form
B(X,,XZ,XJ)
d(x,rx2,x3,t)
22,(x,,x2,
ftj(xl,
Upon introducing
x2,
x3,
x3~
t)
t,
1i
=
e@
2 2 ,(x l
,x 2 ,
(10)
x3) .
72tj(xI,x2,X3)
this in (8) one arrives at the following
eigenvalue
problem
122
N. TRIANTAFYLLIDIS
and F. K. LEHNER
jl,;., +pptij
[6,iN,]
where the relation
of (IO) into (9)
(at interior
= 0
= 0,
between
[ti,] = 0
points of Y),
(at the interface),
7tii, 6, and 6 can be easily found following
(compressible
n/i = L,&.n
711’= L,,xG&,I;-$,I,
c,., = 1
a substitution
solid),
(incompressible
solid).
(12)
Notice that 2’ is the linear eigenvalue of the above system of equations (1 I), (12).
A more recognizable
form of the eigenvalue problem, which is the weak formulation
of the above equations,
is obtained by introducing
(12) into (1 I), multiplying
the
resulting equation by 6u, and subsequently
integrating over the volume V of the solid
s
‘
L,,,kziA,,6u,,,dV=
tz
When the incremental
moduli possess the symmetry property L,,,k = L,k,, as will be
in the case of applications
considered here, all eigenvalues 5’ are real. Consequently
if the minimum
eigenvalue
[,zZ> 0, it follows that the solution (10) must remain
bounded in time so that the interface is stable (recall that no dissipation mechanism
is included in the system; the presence of dissipation-inevitable
in reality--will
result in the decay of the solution’s amplitude).
For <,f, < 0 (10) will permit solutions
that increase without bound, implying that the system is unstable. It is also seen that
if TZ = 0 for a certain perturbation,
then eqns (1 I), (12) and (4), (5) coincide and the
eigenmode ti; becomes the bifurcation
eigenmode (z&= A&).
3.
_
MODEL FORMULATION. BIFURCATION CRITERION AND LINEARIZED STABILITY
In the interest of analytical simplicity, only the plane strain bifurcation and stability
problem for the interface between two solid half-spaces will be analysed here. The
constitutive
description
will be kept as simple as possible in terms of a rate and
pressure insensitive, incompressible
and orthotropic
solid that has one axis of orthotropy oriented parallel to the interface. As was shown by BIOT (1965), the constitutive
equation of such a solid may then be stated as follows :
*
V
O,, = 2nti,,, --@.
V
03 = 2&L> -6,
&z = $2, = /I(ti,,?-t?i,,)
(14)
in terms of the objective Jaumann rate 8, of the Cauchy stress, the hydrostatic pressure
p and the two incremental
moduli ,Uand $ that characterize the material. ‘The relation
between the rate of the first Piola-Kirchhoff
stress and the Jaumann
rate of the
Cauchy stress for an incompressible
material is [see HILL and HUTCHINSON(1975)]
ti,, = x,,Equations
;a,&.&, - C,,k) - :(&
(14) and (15) therefore give
+ &,,)~A,.
(15)
123
Interfacial instability
(16)
b-r deriving (16) from (14) and (I 5), the fact has been used that the prebifurcation
stress state and the material are orthotropic
with respect to the same axes, so that
u,~ = 0 [see BKIT (1965) for further details]. Moreover,
the condition
of incompressibility
det (F;,) = 1, when expressed in the updated Lagrangian
formulation,
requires that
ti,,, +z&
= 0.
(17)
One can easily verify that the incremental
constitutive
equation (16) is of the general
form (5)?, in agreement with the assumption
made in the previous section.
A tacit assumption
underlying
the subsequent use of (14) is that it is applicable to
the initial constitutive
response on the bifurcated
equilibrium
branch as well (i.e.
the unloading
that might occur in the postbifurcation
solution is not considered in
formulating the bifurcation problem). Indeed, as was shown by HILL (1958), providing
the principal solution satisfies the condition of plastic loading everywhere (as is true
for the present problem), the tangent moduli p and 131
of the plastic loading response
can be safely employed in the bifurcation
analysis, since the resulting critical load is
less than or equal to the actual bifurcation
load. Working on the postbifurcation
response of elastoplastic
solids, HUTCHINSON (1974) has further shown that the use
of the tangent moduli p and ; of the plastic loading response gives the correct
bifurcation
load, provided that the prebifurcation
solution satisfies plastic loading
everywhere. Consideration
of unloading
is required for the calculation
of the postbifurcated solution.
Consider now two incompressible
orthotropic
half-spaces of densities p” and ph,
respectively, perfectly bonded along the interface x1 = 0 as shown in Fig. 1. The axes
of orthotropy
of each half-space are aligned with the coordinate axes x,, I? and their
constitutive response is given by (14). The prebifurcation
stress state is also orthotropic
with respect to the same axes. Gravity acts in the negative x2 direction so that h, = 0,
h2 = -g in (1). Although the incremental
moduli p and G depend on the history of
deformation,
for pressure insensitive materials and assuming that proportional
loading is a reasonable
approximation
of the loading history, these moduli depend on
(~~~-cr,,( only, so that
In = /J(r),
;; = Z(r);
z = (cJ2*-cJI,)/2,
o- = (CT,,+fJJ/2.
(18)
Henceforth omission of the subscript identifying a layer in an equation will imply that
the relevant quantities are either defined for both layers or else are to be associated
with one particular layer that will be recognized from the context.
124
FIG. I. Coherent
interface
x2 = 0 between
In order to keep the subsequent
prebifurcation
stress state is assumed
two semi-infinite
analysis
media in a gravitational
as simple
as possible,
field g
the following
The constant k as well as the stresses G, and g2 at the interface will in general take on
different values in each half-space, except for the condition LT’;= 0’; that is demanded
by equilibrium
across the interface. The parameter
k plays an important
role in
that it determines the gradient in the stress deviator within each layer according to
ds/ds, = kpg, as may be seen from (18) and (19). Since t governs the constitutive
response of the material, its gradient must have a controlling
influence on any bifurcation instability.
Hence, if k = 0, there will be no stress gradient effect on the
bifurcation
behaviour. As discussed in the previous section, an updated Lagrangian
formulation
of the field equations will be used for which the reference configuration
coincides with the current configuration
at the onset of bifurcation
when U, = 0 while
tij # 0. The condition
of incompressibility
(17) ensures the existence of a potential
function $, satisfying
Ati, = G.2,
When (14) and (20) are introduced
one obtains
K2k,)k,2--~1,,
Ati2 = -$.,.
into the incremental
(20)
equilibrium
+[(~+-)~.22-(~--~~,1,1.2
[(2~-~,,)~.,,+~~1,2+[(iu-5)11/.,,
equations
(4),,
= 0,
-t/J--M,221.1 = 0.
(21)
Interfacial
The corresponding
~1,= di2, are
jump
conditions
instability
across
125
the interface
x2 = 0, with unit normal
The first two of these relations are a consequence
of the continuity
requirement
(4)3
for the displacement
rate, while the last two express the traction rate continuity
(4)2.
In addition to (21) and (22) one requires conditions at infinity for I//. The validity of
the present double-half-space
model as an approximation
for the more realistic problem
of the instability
of a finite-thickness
layer resting on a half-space depends upon an
insensitivity
of the interfacial response to the exact free-surface boundary conditions.
Accordingly
only those eigenmodes AZ.&that decay away from the interface will be
studied ; from (20) the required condition is that (GI,,,$,2 -+ 0 as (xz( + co.
Next, one eliminates the pressure from the governing equations
(21) and (22) to
obtain
and the interface
conditions
The coefficients in eqn (23) and the boundary
conditions
(24) are independent
of
x, (the aforementioned
boundary value problem being translationally
invariant with
respect to x,). This suggests the use of the Fourier transform $ = Y[$(x,, x2), x, -+
w] for obtaining
an ordinary
differential
equation in x2. Also, the new variable
y = --wxz will henceforth be used. Consequently,
and in view of equations (18) and
(19), the governing equation (23) assumes the form
;;;
i
The corresponding
interface
conditions
at y = 0 are
+ (1 + Z//L)$
= o.
(25)
N. TKIANTAFYLLIDIS
and F.
K.
LEHNEK
= 0.
Equations
(25) and (26) are complemented
by the condition
$- 0 as 1,~)-+ CU.
The amplitude of the eigenmode is expected to decay exponentially
away from the
interface.
It will be assumed that the coefficients appearing
in (25) and (26) do
not vary significantly
within the decay distance of the eigenmode so that they may
be approximated
by their values at y = 0. The solution to (25) of interest is then
given by
Re(z,)
$(w,J:)
taking
= A,((r))e’I?‘+.4,(w)e’2’,
<: 0,
ify > 0,
(half-space
h),
Re (z,) > 0,
if t’ < 0,
(half-space
u),
(27)
cu > 0. Here z,, z2 (z, # z2) are roots of the fourth order polynomial
Since the admissible modes are required to decay to zero away from the interface, it
is the range of parameters for which the above equation has two roots with a positive
(negative) real part in half-space u(h) that is of interest, for eqn (23) will be elliptic in
this case. Notice also, that substitution
of z = i(n,/nz) in (20) yields the characteristic
equation
of (23), (n,,n2) being the unit normal to the characteristic
lines. Equation (23) is elliptic in character,
if it possesses no real characteristics.
The loss of
ellipticity of (23)--for
certain values of its coefficients--entails
the possibility of
discontinuous
solutions, that is strain discontinuities
in the solid in the form of shear
bands. However, as discussed by HILL and HUTCHINSON (1975), YOUNG (1976) and
RICE (1976), such discontinuities
will always appear at higher stress levels than a
bifurcation
with eigenmodes that are varying smoothly in space. The bifurcation
is
thus expected to occur. when at the interface the material parameters
lie in the
elliptic range.
introducing
(27) into the boundary conditions
(26), one has
Interfacial
127
instability
=
0.
The continuity
of A, + A2 and z, A, + z,A,
use of the following convenient notation
Aq+A”, = Ab;+A\
Moreover,
z C,,
across the interface
z:Aq +&A;
enables
= zb,A; +z$A;
(29)
one to make
3 X2.
(30)
one has the identities
z:A, +z:A,
= (z, +z$:,-z,z,C,
z:A, f&4,
= (z:+z,z~+z;&-z,zz(z,
+zJC,,
(31)
=o,
(32)
in terms of which (29) can be written
C,J2+C,,~,
=o,
C2*C,+C:!,~,
with
c,* = UP(l +rlW1
c,,
= [P(l +rIP)(l
c22 =
+z*)&
-z&],
(4;ill-l+tllL)-(1+5/~)(Z:+Z,z~+z:)+k~
po,(dr
*+1
>
(2 +z )
’
2 11’
11
.
One trivial solution of the above system is Z:, = C2 = 0 which, on account
implies that I,&= $ = 0, thus excluding bifurcation.
A bifurcation
becomes
only when a non-zero solution for $ can be found, that is if
c,2c,,-c,,c22
=
0.
of (27),
possible
(33)
In this bifurcation condition the matrix coefficients C, are given by (32) while the zYare
appropriate roots of the fourth order polynomial in (28). Condition (33) can be considered as an implicit equation in o, since the C, and zg are functions of o. The critical
wavenumber
0,. that will permit the first appearance of a bifurcation at the interface
of the two half-spaces can thus be found as a function of the densities, the stresses
and the incremental moduli as well as their derivatives, all evaluated at the interface.
Needless to say that, in general, 0,. will only be obtainable by solving (33) numerically.
Attention
is now focused on determining
the stability of an initially flat interface,
based on the sign criterion for the minimum eigenvalue of (1 l), (12) as discussed in
128
N. TRIANTAFYLLIDIS and F. K. LEHNEH
the previous
section. The perturbation
fi, in the displacement
field again satisfies the
condition of incompressibility
C ,, , + z?>,>= 0 which is satisfied identically by a stream
function
$ such that til = $,> and ii1 = - tj, , Upon making use of this representation
in (9), , while substituting for ti,, a constitutive relation of the form (15), one arrives at
K2;-mkl?-/%
+[(~++2)~.22-(~--)~,111.2+P~2~,2
[(2~-~22)~.12+d1,3+[(~--Z)~.,,
and, after elimination
-(P--M,221.1
-k2$,,
=
0.
=
0
(34)
of b,
~~~---z)~,~1-~~--~~,221.,1+~~~~~-~~~,~21.~2+~~cl+~>~.22-~~-~>~.,~1,22
+P5’(Ic1.l
The relations pertaining
to the interface
derivation of (24), resulting in
[*,I]
are obtained
through
I +$.22)
= 0.
steps that parallel
(35)
the
= 0,
[ti,?] = 0,
~(P+w.22-$J
pK$+tM,,21.1
A Fourier
transformation
=
0,
+~~~+~.)~~.22-~,,1~1,2+~22.2~,11
applied
+P5’$,2]
= 0.
(36)
to (34) and (35) yields
in complete correspondence
with (25), recalling
interface conditions at J’ = 0 are
that y = -~cc)x2. The corresponding
[G] = 0,
d’g
d6
dy - (I+ T/P) dy3
+k;~($+l)($++~$]J=o.
(38)
Interfacial
129
instability
Since the amplitude of the perturbation
is expected to decay exponentially
away from
the interface, it will be assumed that $ -+ 0 as 1y( --t 00 and that the coefficients
appearing in (37) and (38) may be approximated
by their values at y = 0. The solution
to (37) of interest is therefore again of the form (27), z, and z2 being the roots with
positive (negative) real part in halfspace a(b) of the fourth order polynomial
-2kg
‘$+I
(
Introducing
one obtains
(27) into the interface
conditions
z~+(~+T/&~
= 0.
(39)
>
(38), with z,,zz
extracted
from (39),
[h+A2]=0,
[%4,
+Z24]
=
[PU
+~IPL)(Z:A,
0,
+&,+A,
+A211
=
0,
These conditions may once again be cast in the form (32) upon making use of (30)
and (31), the coefficients C, ,, C,2, CzI remaining the same, but C2* becoming
c22
=
i
P
K
4&l+r/ii-5
1
-(I+r/~)(2:+z,z2+z:)
+kE
(“::
+ l)(z,
+zz)]].
(41)
Equation
(33) may now be solved numerically
for the minimum
value trS, with
coefficients C,, and roots zYbeing determined by (32) and (41), respectively. For given
material properties and a given state of stress it is thus possible to say whether or
not an interfacial perturbation
with wavenumber
o will be unstable (5’ < 0). The
bifurcation
wavenumber
o, corresponds
to ;“i = 0, as is evident from the above
relations. Therefore, a graph of the relationship
between the critical wavenumber
(or
wavelength)
and the prestress at the interface will separate regions of stability and
instability in the wavenumber-prestress
plane, as will be seen in the following.
4.
SPECIAL CASES
Although the bifurcation
and stability
critical wavenumber
W, and the minimum
criteria that have been derived
eigenvalue ei to be determined
allow the
from the
130
N.
TRIANTAFYLLILXS
and
F. K. LEHNEK
solution of a highly complicated implicit equation, it is possible to give explicit analytic
results for certain special cases of interest, two of which will now be discussed. In each
case an elastoplastic
half-space is considered to rest on a fluid. The stiffness of the
lower half-space is thus ignored, i.e. ph = ;” = 0 and one can put ,u = ,LL”,
$ = 2.
The case k = 0. This corresponds
to the situation in which the principal stress
difference in each half-space remains constant and independent
of x2. Equation (28)
then becomes a biquadratic
equation in z, with roots
if 4$(p-c)
3 r’.
(42)
The above results apply to the half-space a, where Re (z,) > 0 according to (27). If
the lower half-space is of interest and is modelled as a solid, one simply reverses the
signs of the roots in (42).
The roots (42) do in fact allow an explicit determination
of the critical wavenumber
Q,.. Thus, after computation
of the matrix coefficients C, ,, Clz, C?,. Cz2 in (32), the
bifurcation
criterion (33) yields
(43)
where the z,,~ are given by (42) and Ap = pii-ph. Here p as well as $ are of course
taken to be functions of z. Also, the superscript u has been deleted, it being understood
that--with
the exception of Apg/o,.--in
(43) and subsequent
expressions all indexfree variables and functions are defined on the side of the interface belonging to halfspace a.
Since for the bifurcation
of interest the half-space a has to be in the elliptic regime,
as explained in the discussion following (28), the denominator
S in (43) is always
strictly positive; moreover, for r = 0 the term within brackets becomes constant and
equal to 2. These observations
imply that for a denser upper half-space (Ap > 0)
a bifurcation
is always possible, while for a denser lower half-space
(Ap < 0) a
hydrostatically
stressed overburden
cannot bifurcate. For large enough values of (~1
the right-hand side of (43) can vanish. The vanishing of the numerator
in (43) yields
BIOT'S(1965) condition for the surface bifurcation
of an (orthotropic)
half-space that
is subjected to a lateral stress (-6,)
and it confirms Biot’s prediction
that the
corresponding
critical wavelength is zero.
For r = 0, G(O) = ~(0) =-G,
yields the interesting result
where
G is the linear
Apg~=1.
2P(O)U,.
elastic
shear
modulus,
(43)
(44)
Interfacial
131
instability
This shows that when a hydrostatically
stressed elastic half-space overlies a less dense,
incompressible
fluid, the critical wavelength for a bifurcation
is likely to be very large.
Thus, if the quantity (- az)/Pg is taken as a measure of overburden
thickness in a
geological setting, then the ratio critical wavelength/overburden
thickness will be of
the order of 2~/( - g2) or between lo2 and 104, when k is an elastic shear modulus.
As was established in the general analysis of the previous section, the graph of the
critical wavelength
(43) contains
all neutral equilibrium
points (5: = 0) in the
curve” therefore
separates
a
(Apg/2pto) - (z/p) space. This “critical-wavelength
region of stable perturbation
wavelengths from a region of unstable perturbation
wavelengths, which in the following will be referred to as “stable” and “unstable”
regions,
respectively.
When k = 0, a simple way of discriminating
between the stable and unstable side
of the critical-wavelength
curve is by calculating pc2/yw2 for points on the ALpg/2po
axis in the neighbourhood
of the point (z/p = 0, Apg/2p(O)o = 1). Since c(O)/p(O) = 1
by assumption,
the roots of interest of (39) are
z, = 1,
22 = (I-/I)“2,
/1 = p52//LLo2.
(45)
Using these in (32) and (33) and recalling that C22 must be determined from (41) for
the stability analysis, one arrives at the following equation for /? at points with z/p = 0,
Apg/2p(O)o = 1 +E, where 1.~1<< 1 :
( >
1-i;
An asymptotic
solution
fl+l+&--(l-/I)
~:++$+
(46)
for /II is of the form
fl = -j&+0(&2),
(47)
indicating that for E > 0 (Apg/2p(O)w > 1) the interface is unstable. This result does
not come as a surprise since the gravitational
potential energy released by perturbation
wavelength beyond the critical exceeds the concomitant
elastic energy stored in the
solid layer. Along with this determination
of fl one must of course ensure that all
other eigenvalues-found
by solving (32), (39), (41) numerically-are
in fact larger.
The case jkpg/pw( << 1. This short-wavelength
limit has the following significance.
From (IS), (19) and the constitutive relation (14) it is easily seen that the gradient of
the normal strain E, , in the (vertical) x2 direction is given by da1 ,/dx2 = -kpg/2$(z).
The condition may therefore be expressed as \2j?(da, ,/dx2)/pw( << 1. Moreover, since
2$~ = O(1) for most applications,
this is equivalent
to ((da, ,/dx2)LJ << 1, where
L = 271/o is the wavelength of the mode in the x1 direction. Hence, with
(48)
the asymptotic analysis for /I-I << 1 applies when strain gradients in the x2 direction
are much smaller than the inverse of the eigenmode’s wavelength in the x, direction.
The roots zl,z2 of (28) that are required for evaluating
(32) and the bifurcation condition (33), are found, in the case )I\ <c 1, with the aid of a straightforward
132
N. TKIANTAFYLLWIS and F.
asymptotic
expansion
K. LEHNEK
of (28) with respect to iL
(49)
where the zT are given by (42). Upon entering these into (32) and evaluating (33) by
collecting terms of like order in i, one obtains, after lengthy but straightforward
algebraic manipulations
and use of (42), the following expression for the critical
wavenumber
w,, correct to the first order in 1. :
I( 1- dP)l(l + T/P)1
‘,‘(2h +4PL)-T/P
Aw
2pt0, = ~~l-[k~(l-p”/p”)][2d;/dt-d~/d~+(d~/dz+1)(2+[(I-r/~)~(l+r~~)]’~’)J)~
where S is defined as in (43). It should be kept in mind, however, that for large
prebifurcation
strains t/p cannot be neglected and the correction terms,d,$dz
and
d/l/dz can also become important.
5.
DISCUSSION OF RESULTS FOR Two
PARTICULAR MATERIALS
The above expressions for the critical wavenumber
o,.--eqns
(28), (32) and (33)
for the general case as well as (43) and (50) for the two special cases- are valid for
any incompressible,
pressure insensitive, orthotropic material that obeys a constitutive
relation of the general form of equation (14). It is nevertheless clear that numerical
values for the critical wavenumber
will strongly depend on the particular form of the
functions p(t) and ;(r) in (1X). Of the many different possibilities,
two relatively
simple models will be explored.
The first material to be studied is a Mooney-Rivlin
hyperelastic material, which,
under plane strain conditions satisfies the stress-strain
relation
r/G = sinh (cz2-c:,
,).
(51)
Here G is the material’s initial shear modulus and E, ,, ,zz2are the logarithmic strains
in the X, and x2 directions, respectively. The pure shear stress-strain
curve for the
Mooney-Rivlin
material is depicted in Fig. 2.
The plane strain incremental
moduli p(z) and 1*;(z)defined by (14) are given by
p(z) =
Z(T)= G[l
+ (z/G)‘] “’
(52)
for this material.
The second material to be considered is a hypoelastic St&-en-Rice
material [see
ST~REN and RICE (1975)], the stress--strain
relation of which is given by the discontinuous
power law
Interfacial
133
instability
30
-2
20
10
i
-9
0
P
-10
-20
IT-30,
-4
-2
0
2
4
%2-El1
FIG.
2. Stress-strain
relation
for Mooney-Rivlin
material.
(53)
Here m denotes a hardening modulus (m > l), while ?J is the yield stress in pure shear.
This type of stress-strain
relation is shown in Fig. 3.
The corresponding
expressions for the incremental
moduli are given by
The fundamental
difference
FIG.
in the behaviour
3. Stress-strain
relation
of the two materials
for Stiiren-Rice
material.
is that the functions
134
N. TKIANTAFYLLIDIS and
F. K.
LEHNEK
and G(r) increase with t for the Mooney-Rivlin
material (52), but decrease with
z for the St&en-Rice
power law material (54).
Taking either (52) or (54) as material law, the critical wavenumber
w, may now be
calculated from (33) or-in
the limit Ii/ << I--from
(50) upon substitution
for p(t)
and E(T) from either (52) or (54). For a given gradient dz/dx, = kpg of the stress
deviator and a given density ratio #/pa, eqn (33) is solved numerically
with the aid
of a straightforward
bisection method. The exact results obtained in this manner are
presented
together with the asymptotic
results in Fig. 4, for the MooneyyRivlin
material, and Figs 5 and 6 for a StiirenRice
material with the hardening exponents
n? = 2 and m = 4, respectively. In all figures a dimensionless
critical wavelength has
been plotted against a dimensionless
stress deviator for various values of the parameter
k. assuming the fixed density ratio p”/p” = 0.8. Thus, k = 0, -0.03, 0.03 in Figs 4aa
c, while k = 0, -0.015,
0.015 in Figs 5aPc (m = 2) and Figs 6a-c (m = 4). For the
MooneyyRivlin
material the dimensionless
wavelength and stress deviator are defined
as Apg/2Gw,. and z/G, respectively,
while for the StiirenRice
material the corresponding
quantities
are defined as Apg/2t,w,
and s/z,., respectively.
In all calculations performed for the StiirenRice
material the value t,/G = 10 ’ was assumed
for the yield strain in simple shear.
Some general remarks on the results shown in Figs 46 are now in order. First, the
determination
of the character
of the equilibrium
states adjacent to the criticalwavelength curves, i.e. the delineation
of regions of stability and instability,
is based
on the sign criterion for the minimum eigenvalue ti of the linearized perturbation
problem; these results were computed numerically for a number of interior points in
each region. Thus, for a given state of stress in the unperturbed
half-spaces, perturbations
with wavelengths that lie on one side of the critical curve are stable, that
is will not grow with time, but decay in the presence of the slightest amount of viscous
dissipation,
while perturbations
with wavelengths
on the other side of the critical
curve are unstable and will tend to grow, although there remains the possibility of the
perturbed system evolving towards a new, postbifurcation
equilibrium
state that will
be characterized by a finite-amplitude
disturbance of the layer interface. The existence
of such equilibrium states in the postbifurcation
regime may be expected for solid/fluid
two-layer systems from the fact that the solid possesses an elastic range and finite
strength. Such states are of considerable
interest, for example in geological or geotechnical studies of saltdoming
instabilities,
but the conditions
for their existence
as well as their exact nature have been studied only little.
Secondly, we recall that negative values of Apg/2Go, or Apg/2z,.(ti,. correspond
to
a gravitationally
stable density stratification
(Ap z p”-pph < 0). For this case and
under the additional restriction of a vanishing stress gradient ds/dxz = 0 (k = 0) WC
shall recover certain earlier results by BITT (1963b, 1965).
Finally, it will be clear that an unstably stratified two-layer system of irzfinitelateral
extent can indeed be destabilized
by any prevailing
stress state. Nevertheless,
the
results of this study can be meaningfully
applied to overburden/substratum
interfaces
of finite extent that delimit the range of critical wavelengths by their width. Thus, for
finite systems, the quantities of interest will be the critical load required for activating
the maximum available wavelength or, alternatively,
the critical wavelength associated
with a given state of stress.
p(z)
Interfacial
instability
STABLE
UNSTABLE
5-
-5
0
5
T/G
FIG. 4(a)-(c). Critical wavelength versus initial stress at interface between Mooney-Rivlin
material and
inviscid fluid, showing effect of different stress gradients k; dashed line is asymptotic
solution (50); +
marks limit of decaying solutions.
(4
400
200
f&STABLE
UNSTABLE
i
:
;
:
:
:
:
;
STABLE
-2oo-
-4ooI
I
I
i
;
:
:
’
i
;
STABLE
I
I
I
I
I
I
(b)
40-
20x
y”
3
c-4
.
M
o--
_
a”
-2o-
_
-4o-
UN-
i jSTA8LE
I
1
1
1
1
jjxq
Cc)
40-
UNSTABLE
20-
3”
--
STABLE
n
-___
STABLE
,
,)
-30
-20
-10
0
IO
20
30
71Zy
FIG. S(a)-(c).
Critical wavelength versus initial stress at interface between Storen-Rice
material (m = 2)
and inviscid fluid, showing effect of different stress gradients k; dashed line is asymptotic solution (50) ;
0 marks loss of ellipticity of eqn (23).
(a)
400
UNSTABLE
0
7/“cy
(b) ‘0
UNSTABLE
5
gz
Y
o
7
M
a
STABLE
-5
UNSTABLE
‘,
I
-10
L
I
I
-4
I
0
-2
4
:
I
4
2
717,
Cc)
‘C
(k-o.0151
UNSTABLE
c
x
p”
3
c-4
.
M
a
c
I_
STABLE
;
,I
_e--
:
_.*-
_c
.
STABLE
.’
‘\_.’
UNSTABLE
-1C
j
f
-4
-2
0
I
2
I
4
7 / zy
FIG. 6(a)-(c). Critical wavelength versus initial stress at interface between Sti%en-Rice material (m = 4)
and inviscid fluid, showing effect of different stress gradients k ; dashed line is asymptotic
solution (50) ;
l marks loss of ellipticity of eqn (23).
138
N. TRIANTAFYLLIIHSand
F. K. LEH~K
Consider
now the behaviour
of a half-space
composed
of a Mooney-Rivlin
material. Figure 4a gives the dimensionless
critical wavelength as a function of the
dimensionless
stress deviator at the interface when the latter remains uniform throughout the half-space (k = 0). The stable states lie below and the unstable states above
the curve. The zero-wavelength
intercept on the stress axis determines
the stress
level required for inducing a pure surface bifurcation
mode. This surface bifurcation
was discussed by BITT (1963b, 1965) in his study of a gravitationally
stable halfspace; his results fork = 0 and AQ d 0 are reproduced by the corresponding
segment
of the critical curve in Fig. 4a. For Ap > 0, the zero-stress intercept of the critical
curve determines the critical wavelength that will be destabilized by gravity alone, all
shorter wavelengths requiring some compression
for their appearance.
The stress deviator s/G affects the critical wavelength in a twofold manner. First,
if 7 is positive (negative), through the progressive destabilization
(stabilization)
of the
system with increasing (decreasing) T/G or increased (decreased) lateral compression
of the layer. This is a purely geometric effect that is responsible for the instability of
solids under compression,
with examples ranging from Euler columns and plate
buckling in structures to surface instabilities on half-spaces arising from surface-parallel
compression
in finitely strained solids (BIOT,196313, 1965). The second effect is the
progressive stiffening of the material, i.e. the increase of its incremental
moduli, with
increasing
IzJ/G. The two effects cooperate when 7 is negative, but counteract
each
other for positive z when indeed the destabilizing
influence of lateral compression
becomes dominant
at z > z,, where 5,_ is Biot’s critical stress for the appearance of
a surface instability on a weightless half-space. For the Mooney-Rivlin
material, the
asymmetric
geometric effect of stabilizing
lateral tension and destabilizing
lateral
compression dominates the symmetric constitutive stiffening effect ; the resulting graph
of critical wavelength versus stress deviator is therefore skew symmetric with respect
to the zero stress axis. For the further interpretation
of Fig. 4 it is therefore best to
think about the critical curve in terms of the critical compressive stress required to
destabilize a given wavelength,
if z > 0 and in terms of the critical tensile stress
required to stabilize a given wavelength, if z < 0.
The effect of a negative stress deviator gradient (k = -0.03) on the stability of a
half-space of Mooney-Rivlin
material is apparent from Fig. 4b. Three regions of the
critical wavelength versus stress deviator graph can be distinguished,
according to the
prevailing stabilization
or destabilization
mechanism.
For Ap > 0 and T < T* < 0.
where T* is defined by w, (z* ; k = - 0.03) = o, (z *., 0), the material stiffness increases
away from the interface due to the increase in 1~1.The increase in the interface-parallel
tensile stress provides a geometric stabilization.
Due to these stabilization mechanisms,
the critical wavelength for a given value of z is higher than its counterpart
for k = 0.
The interfacial stability is thus increased in this region by a negative stress gradient
li. For Ap > 0 and z = 0, the decrease in r (or increase in lateral tension) away from
the interface stiffens the material and reduces the critical mode’s decay length. This
in turn reduces the critical wavelengths in the region 5* < z < rI around z = 0 relative
to the wavelengths for k = 0. If A/, < 0 and when T exceeds T,*., it decreases away from
the interface and the associated geometric stabilization
requires more compression for
destabilizing a given wavelength than is needed fork = 0. Again, therefore, the interface
is stabilized by a negative k. It will further be noted that there is good agreement
Interfacial instability
139
between the exact numerical
solution of (32) and (33) and the asymptotic
results
derived from (50), due to the fact that the value of Ii\ = (kpg/p(z)w,l
remains
sufficiently small over a wide range of z-values.
Figure 4c illustrates the effect of a positive stress deviator gradient (k = 0.03).
Three regions may again be distinguished.
For Ap > 0 and z < T* < 0, where
W,.(T*,k = 0.03) = oC(z*, 0), the material
becomes softer with increasing distance from
the interface, since Iz\ decreases. The lateral tension required for stabilizing a given
wavelength is higher than that for k = 0. Equivalently,
the critical wavelength for a
given value of z is lower than the corresponding
wavelength for k = 0. The interfacial
stability is thus decreased in this region by a positive stress gradient k. In the small
region r* < 5 < z, around the point z = 0, the increase of T away from the interface
increases the critical mode’s decay length and has a stabilizing effect on the system
leading to higher critical wavelengths. Eventually the destabilizing effect of the increasing lateral compressive stress dominates and the critical wavelength vanishes at 7,.
For Ap < 0 and z > z,, T increases away from the interface and the associated
geometric effect of increased axial compression
of the half-space permits a smaller
positive z to destabilize a given wavelength,
than would be required for k = 0. The
asymptotic results obtained for this case from (50) are in good agreement with the
results obtained from (32) and (33) near zg, where the critical wavelength is small,
and at large absolute value of z where the incremental
moduli acquire large values,
resulting in small values of (A/ = jkpg/p(z)w,(
in these cases.
Apart from the solution branch that passes through z, and is expected from the
results for k = 0, -0.03, a second branch appears near z = 0, if the density stratification of the system is stable (Ap < 0). The reason for this instability is the following.
The increase in lateral compression
away from the interface destabilizes the system
and there exists a mode with an adequately
large decay length that releases more
energy than the potential energy required to lift the stably stratified interface. Any
wavelength
that is larger than this critical wavelength will thus be unstable, as is
verified by a linearized stability analysis. This second branch of the critical curve has
a vertical asymptote at some positive value of z close to z = 0, which reflects the
stabilizing influence of material stiffening; the branch ends at some value z < 0 that
marks the emergence of solutions with infinite decay length.
Figures 5 and 6 give the dimensionless
critical wavelength Apg/2z,o, as a function
of the dimensionless
stress deviator z/z, for the case of two variations of the StbrenRice hypoelastic material, one with a higher hardening (m = 2 in Fig. 5) and the other
with lower hardening (m = 4 in Fig. 6) in their uniaxial stress-strain
response. In all
the graphs presented here the results are given ou:side the elastic zone (T/T,./ < 1. Due
to the discontinuity
of the incremental
modulus p(s) at ITJT~~= 1 and the subsequent
rapid decline of both the incremental
moduli p(z) and p(t) [see (54)] for ]T/T,.\
> 1
there is a difference of the order of G/T,, between the critical wavelength values inside
and outside the elastic zone. Indeed, for \z/zYl d 1 the wavelengths agree essentially
with the corresponding
wavelengths
for the Mooney-Rivlin
material in the neighbourhood of T = 0, and for stresses of the order of z,/G the effect of the difference in
constitutive
response is negligible. The calculations
are terminated
at values of Z/Z,
such that the governing equations become hyperbolic [cf. the discussion of eqns (23)
and (28)]. This loss of ellipticity represents an additional
feature of a StGren-Rice
140
N. TRIANTAFYLLIDIS
and F. K. LEHNER
material which not only is destabilized
in both tension and compression,
but also
admits shear band modes of localized deformation
at appropriate
levels of stress, as
was discussed by ST~REN and RICE (1975). By contrast, it can easily be seen that for
a Mooney-Rivhn
material (23) will remain elliptic at all stress levels.
In Fig. 5a a plot of the critical wavelength versus the stress deviator is shown for a
Storen-Rice
material with hardening
exponent m = 2 for the case of a vanishing
gradient of z in the x2 direction (k = 0). As discussed previously, an increase in T will
destabilize the layer, while a decrease will have the opposite effect. This asymmetric
effect is independent
of the constitutive
law and controlled
by the sign of T. The
constitutive
response of the layer is symmetric in Z, since it depends upon jz( only and
the rapid decrease in stiffness of a Storen-Rice
material with increasing
1~1 has a
destabilizing
effect that increases with the hardening exponent m. The behaviour of
the critical wavelength
as a function of the stress deviator as shown in Fig. 5a is
clearly dominated by the symmetric constitutive
response. As a consequence,
a zerowavelength surface instability appears both under compressive as well as tensile lateral
stress at 7: > 0 and t,; < 0, respectively. The asymmetric effect of z or g, (assuming
cr? remains fixed) expresses itself in the steeper decline of the positive branch of the
critical curve, resulting in It?+] < IT; 1 for the surface instability.
The graph of the
dimensionless
critical wavelength
remains
continuous
across the elastic range
- 1 < z/T,. c 1, where Apg/2z,.o,. % G/z?. = 103.
The effect of a negative gradient in T on the stability of the half-space is shown in
Fig. 5b. The graph of the critical wavelength
has lost its symmetry,
despite the
persistence of zero-wavelength
surface instabilities
in tension and compression
at the
previous critical loads z;/z,. and T~/T),respectively. Thus, in the case of gravitationally
unstable density stratification
(Ap > 0) there exists a finite critical wavelength for all
zC;, < z < -z,., but only within a certain interval z, 6 z < z:. For Ap < 0 the z 3 T,
branch reaches some maximum value of r outside the range of the figure, the complex
behaviour of the critical curve in this quadrant depending sensitively upon the value
of m as a comparison
with Fig. 6b will show. It is of interest to note that the two
branches of the asymptotic solution shown in Fig. 5b for positive z are separated by
a vertical asymptote (not shown here). The behaviour of the critical curve near the
origin may be understood
in terms of a hypothetical
material defined by a smooth
continuation
of the power-law relation in (54) to zero stress, where such a material
will become infinitely stiff. As the asymptotic result (50) shows, the conditions dp/dz,
d:/dt --f x, as z + 0 implying a vanishing critical wavelength.
Associated with this
hypothetical
material will be a smooth continuation
of the critical curve across the
yield stress + zYand through the coordinate origin of Figs 5b, c and 6b, c. This shows
that the sharp discontinuity
exhibited by the actual critical curve at the yield stress
originates from the discontinuity
in the tangent modulus of the Stiiren--Rice material.
The behaviour
of the critical curve in Fig. 5b can be explained in terms of the
combined effects of the interface-parallel
stress (for a fixed normal stress) and the rate
of decrease k of the normal stress difference 5 away from the interface. When T is
negative and small in absolute value, the latter will increase away from the interface
and add a further stabilizing influence, due to the stabilizing effect of a tensile lateral
stress; this leads to critical wavelengths that increase with /tl within some range of
negative T values. However, beyond a certain absolute value of z the destabilizing
Interfacial instability
141
stiffness reduction associated with high values of Iz( again dominates the stabilizing
effects of interface-parallel
tension and increasing stiffness away from the interface,
leading eventually to a surface bifurcation
instability mode in tension at 7:) which is
unaffected by k and coincides with the critical tensile stress for k = 0 of Fig. 5a.
At positive values of z the destabilizing
effect of interface-parallel
compression
is
counteracted
by the decrease in \z( away from the interface (k < 0). As a consequence,
for z >, z~, the critical wavelength for a gravitationally
unstable stratification
Ap > 0
is much larger than its counterpart
value for -T (outside the scale of the graph),
except for values of z close enough to the critical load ~2 where the influence of k
ceases to be felt. Notice also that for z 2 7Y any sufficiently large wavelength perturbation
of a gravitationally
stable interface (Ap < 0) will be unstable as a consequence of increasing layer-parallel
compression
away from the interface.
Figure 5c illustrates the effects on interfacial stability of a decrease in the stress
deviator away froin the interface. As expected, the graph represents roughly a mirror
image of Fig. 5b.
Analogous results for the more rapidly softening St&en-Rice
material (m = 4) are
shown in Fig. 6.
6.
CONCLUSIONS
The instability of a solid/solid or solid/fluid interface is determined by a complex
interplay of the effects of initial or pre-stress and gravity and is distinguished
from
the classical Rayleigh-Taylor
instability in stratified fluids by the existence of a finite
critical wavelength.
Equivalently,
there exists a critical stress at which an interfacial
perturbation
of a given wavelength will become unstable.
For an unstressed elastic half-space with a shear modulus G and a density exceeding
that of a fluid substratum
by some positive Ap, the critical (unstable) wavelength is
given by 4nG/Apy. For typical values of G z 109/10” MPa, Ap z 500 kg rn~ ‘, g =
10 m s- 2 one finds L z 106/108 m. The fact that these values are far in excess of any
relevant wavelength
even for geological systems illustrates the importance
of the
destabilizing
effects of pre-stress and material softening, both of which are found to
be capable of reducing critical wavelengths drastically. In addition, there is the effect
of a finite layer thickness, which will have to be taken into account whenever critical
wavelengths reach comparable
magnitudes.
The instability of the interface between an elastoplastic layer and a buoyant fluid
substratum
will be promoted both by lateral compression
and extension, if the layer
stiffness is thereby reduced.
The post-bifurcation
behaviour of a destabilized, evolving interface in a two-layer
solid/fluid
or solid/solid
system remains to be studied. A particularly
important
question concerns the nature of stable finite-amplitude
disturbances and the conditions
under which they will exist in layers of finite stiffness.
ACKNOWLEDGEMENTS
This paper was published with the permission of Shell Research B.V. We thank Dr Y. Leroy
for helpful discussions
and suggestions.
N. TRIANTAFYLLIDIS
and F. K. LEHNER
142
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