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Theory of elasticity and consolidation for a porous
anisotropic solid
Maurice A. Biot
To cite this version:
Maurice A. Biot. Theory of elasticity and consolidation for a porous anisotropic solid. Journal of
Applied Physics, American Institute of Physics, 1955, 26 (2), pp.182-185. �10.1063/1.1721956�. �hal-
01368659�
Theory of Elasticity and Consolidation for a Porous Anisotropic Solid
M. A. Biot
Citation: Journal of Applied Physics 26, 182 (1955); doi: 10.1063/1.1721956
View online: http://dx.doi.org/10.1063/1.1721956
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/26/2?ver=pdfcov
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JOURNAL
OF
APPLIED
PHYSICS
VOLUME
26,
NUMBER
2
FEBRUARY,
1955
Theory
of
Elasticity
and
Consolidation
for
a Porous
Anisotropic
Solid
M. A.
BIOT*
Shell
Development
Company,
New
York
City,
New
York
(Received
May
5, 1954)
The
author's previous theory of elasticity and consolidation for isotropic materials
U.
App!. Phys.
12,
155-164 (1941)J is extended to the general case of anisotropy. The method of derivation is also different
and
more direct.
The
particular cases of transverse isotropy and complete isotropy are discussed.
1.
INTRODUCTION
T
HE
theory of consolidation deals with the settle-
ment under loading of a porous deformable solid
containing a viscous fluid.
In
a previous publication!
a consolidation theory was deVeloped for isotropic
materials.
The
purpose
of
the present paper
is
to extend
the theory to the most general case of anisotropy. The
method
by
which the theory is derived
is
also more
general and direct. The same physical assumption
is
introduced,
that
the skeleton
is
purely elastic and con-
tains a compressible viscous fluid.
The
theory may
therefore also be considered as a generalization of the
theory
of
elasticity to porous materials.
It
is
applicable
to the prediction of the time history of stress and strain
in a porous solid in which fluid seepage occurs. The
general equations derived in
Sec. 2 are applied
to
the
case of transverse isotropy in
Sec.
3.
This
is
a case of
particular interest in the application
of
the theory to
soils
and
natural rock formations, since transverse iso-
tropic
is
the type of symmetry usually acquired
by
rock under the influence
of
gravity. For an isotropic
material the equations reduce to a simple form given
in
Sec. 4. They are shown to coincide with the equa-
tions derived in reference
1.
Application of the theory
to specific cases was made
previously,2-4
and
it was
shown
that
the operational calculus offers a very power-
ful tool for the solution of consolidation problems in
which a load
is
applied to the material
at
a given
instant
and
the time history of the settlement is to be
calculated. These methods are directly applicable to the
more general nonisotropic case. More general solutions
of the equations have been developed and
will
be pre-
sented in a forthcoming publication.
2. GENERAL EQUATIONS FOR
THE
ANISOTROPIC CASE
sample of bulk volume
Vb,
It
is
understood
that
the
term
"porosity" refers as is customary
to
the effective
porosity, namely,
that
encompassing only the inter-
communicating void spaces as opposed to those pores
which are sealed
off.
In
the following, the word "pore"
will
refer to the effective pores while the sealed pores
will be considered as
part
of
the solid.
It
will be noted
that
a property
of
the porosity f is
that
it
represents
also a ratio of areas
(2.2)
i.e., the fraction S p occupied
by
the pores in
any
cross-
sectional area
Sb
of the bulk material.
It
must be-
assumed, of course,
that
the pores are randomly dis-
tributed in location
but
not
necessarily in direction.
That
this relation holds may be ascertained
by
in-
tegrating Sp/Sb over a length
of
unity in a direction
normal to the cross section
Sb.
The
value of this integral
then represents the fraction
f of the volume occupied
by
the pores.
It
is seen
that
the ratio S
pi
S b is also inde-
pendent of the direction of the cross section.
The
stress tensor in the porous material
is
(2.3)
with the symmetry property
Uij"=Uji.
The partial components of this tensor do not have
the conventional significance.
If
we
consider a cube of
unit size of the bulk material,
U represents the total
normal tension force applied to the fluid
part
of the
faces
of
the cube. Denoting
by
p the hydrostatic pres-
sure of the fluid in the pores
we
may write
u=-fp·
(2.4)
The
remaining components U
xx,
U "'II, etc., of the tensor
are the forces applied to
that
portion of the cube faces
occupied
by
the solid.
Let
us consider an elastic skeleton with a statistical
distribution of interconnected pores. This porosity is
usually denoted
by
We shall now call our attention to this system of fluid
and solid as a general elastic system with conservation
properties. The solid skeleton
is
considered to have com-
(2.1) pressibility
and
shearing rigidity,
and
the fluid
may
be
where V p
is
the volume of the pores contained
III
a compressible. The deformation of a unit cube
is
as-
sumed to be completely reversible.
By
deformation is
* Consultant.
1 M.
A.
Biot, J. App!. Phys. 12,155-164 (1941).
meant
here
that
determined
by
both
strain tensors in
2 M.
A.
Biot,
J.
App!. Phys. 12,426--430 (1941). the solid
and
the fluid which will now be defined. The
aM.
A.
Biot and F. M. Clingan,
].
App!. Phys. 12,
578--581
d'
Itt
f
th
I'd'
d .
(1941). average
ISP
acemen componen
so.
e
so
I
IS
eSlg-
• M.
A.
Biot and F. M. Clingan,
J.
App!.
Phys.13,
35-40 (1942). nated
by
U"" U
II
,
U.,
and
that
of the
flUId
by
U:x,
U
II,
U z.
182
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TH
EOR
Y
OF
CONSOLI
DA
TI
ON
FOR
AN
I
SOTROP
I C
MA
TERI
ALS
183
The strain components for the solid and the fluid,
respectively, are
au,.
(au
y
au,.)
(2.5)
e
xx
=--
!e
zlI
=!
--+--
etc.
ax
ox oy
au"
.~fXy=!(aUli
+
au,,)
fzz=--
etc.
ax
ax ay
By
a generalization of the procedure followed in the
classical theory of elasticity
(5)
we
may write for the
elastic potential energy
V the expression
2V =
uZxe.rx+ulIlIellY+u
zzezz+ull.eyz
+u
zxezx+u
XyeXY+u€,
(2.6)
with
If
we
assume
that
the seven stress components are
linear functions
of
the seven strain components the
expression 2
V
is
a homogeneous quadratic function
of
the strain. This function
is
a positive definite form with
twenty-eight distinct coefficients. The stress com-
ponents are given
by
the partial derivatives
of
V as
follows:
a
v/ae
xx
=
ITxx
av/aexy=u
XY
, etc., (2.7)
av/ae=u.
This
is
written
u
xx
C11C12C13C14C16C16C171
e"x
u
IIY
C22C2S
C
24
C
25
C
ZS
C
27
e
llll
u
..
Caz
C
34
C
S5CZ6
C
S7
e
zz
u
liZ
C44
C
45
C
46
C
47
e
yz
(2.8)
u
xx
CSSC56
C
S7
e
zx
u Z1/
C66
C
67
l
:X1/
u
Cn
Because the matrix
of
coefficients
is
that
of
a quadratic
form
we
have the symmetry property
(2.9)
The total stress field (2.3)
of
the bulk material satisfies
the equilibrium equations
au
""
alJ'
Z1/
a
-+-+-(uzz+u)+pz=o,
oX
oy
OZ
(2.10)
where
p
is
the mass density
of
the bulk material and
X,
Y,
Z, the body force per unit mass. Substituting in
(2.10) the stress components as functions of the strains
from (2.8)
we
obtain three equations for the six un-
known displacement
U
x
'
••
U
x'
•
'.
Three further equa-
tions between these unknowns are obtained
by
intro-
ducing
tlie law governing the
flow
of a fluid in a porous
material.
We
introduce here a generalized form
of
Darcy's
law for a nonisotropic material
r
-ap/ax+PIX]
lkXX
k
xy
k:XZ]
lUX-
U
"]
-ap/ay+PIY
= k
1lx
k
yy
k
llz
~1J-~1I'
(2.11)
-op/az+Plz
k.
z
kZll
kzz
Uz-u.
where
PI
is
the mass density
of
the fluid. The matrix k
ij
constitutes a generalization
of
Darcy's constant if
we
include in
it
the viscosity coefficient. The average
velocities
of
the fluid and solid are denoted
by
U
x'
•
,u
x
' •
'.
The symmetry
of
the coefficients
(2.12)
results from the existence
of
a dissipation function
such
that
the rate
of
dissipation of the energy in the
porous material
at
rest
is
expressed by the positive
definite quadratic form
ij
2D= L k;jUiU
j
•
(2.13)
If
we
multiply Eq. (2.11) by f and take (2.4) into ac-
count
we
obtain
[
au/ox+p1X]
[bxx
b
xy
bxz]
[Ux-ux]
aa/ay+p
1Y = b
yx
b
yy
b
yz
qY-~1J'
alJ'/8z+P
1
Z b
zx
b
Zll
b
zz
Uz-u
z
(2.14)
with
Pl=pt/=the
mass
of
fluid per unit volume
of
bulk
material. The three equations obtained
by
combining
(2.10) and (2.8) in addition to the three Eqs. (2.14)
determine the six unknown displacement components
for the
fluid and the solid.
3.
THE
CASE OF TRANSVERSE ISOTROPY
The above equations are valid for the most general
case
of
a symmetry.
In
practice, however, materials
will
be either isotropic or exhibit a high degree
of
sym-
metry which greatly simplifies the equations.
Let
us
consider first the case of a material which is axially
symmetric about the
z axis. This type
of
symmetry
is
referred to
by
LoveS
as transverse isotropy (page 160).
The expression for the strain energy in this case
is
2 V = (A + 2N)
(e
xx
+c
yy
)2+Ce
z}+
2F
(eyy+cxx)e
zz
+ L(ell/+e
z
/)+
N(e
x1l
2
-4e
xx
eyy)
+2M(exx+eYYh+2Qczz~+RE2.
(3.1)
This expression
is
invariant under a rotation aroulltl the
z axis.
It
is written in such a way as to bring
out
expres-
sions such as
cxy2-4e"",ellY
and exx+c
yy
which are in-
variant under a rotation about the
z axis. The coefficient
A+2N
is written this way for reasons of conformity.
6
A.
E. H. Love, A Treatise On
the
Mathematical Theory
of
Elasticity (Dover Publications, New York, 1944).
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184
M.
A.
BlOT
Since A does
not
appear in
any
other term, the quan-
tity
A +
2N
is
an
independent coefficient which could
have been written as
P [see
(4.5)]'
The
stress-strain
relations derived from (2.7) and (3.1) are
U
xx
=
2N
exx+A
(exx+eyy)+Fezz+M
t;
U
yy
= 2Ne
yy
+A
(exx+eyy)+Fezz+M
t;
U zz= Cezz+F(exx+eyy)+Qt;
uyz=Le
llz
;
u
zx=
Le
zx
;
u
xlI
=Ne"'lI;
u=M(exx+eyy)+Qezz+Rt.
(3.2)
There are therefore in this case eight elastic coefficients·
The
equations of
flow
contain two coefficients
of
permeability, one in the Z direction, the other in the
x, y plane,
and
may
be written
au/
ax+Plx
=
bxxCO
x-ux);
au/
ay+
PlY
=bxx(O
II-Uy);
au/
az+Plz
= b zz(O
z-u
z
).
(3.3)
These equations along with the stress-strain relation
(3.2)
and
the equilibrium relations (2.10) yield six
equations for the six displacement components in the
case
of
transverse isotropy.
4.
THE
CASE
OF
ISOTROPY
In
the case
of
complete isotropy the strain energy
function (3.1) becomes
2V=
(A+2N)
(exx+eyy+ezz)2
+N(eYZ2+ezx2+eXy2-4eyyc
..
We shall assume
that
there
is
no body force and
put
X=Y=Z=O.
Substitution
of
expression (4.3) into
the equilibrium Eq.
(2.10) for the stresses
and
the
flow
Eq. (4.4) yield the six equations
NV'2
U
+(P-N+Q)
grade+(Q+R)
gradt=O
grad
(Qe+Rt)
=b(a/
at)
CO
-u).
(4.5)
We have
put
P=A+2N.
Taking the divergence
of
the second equation
we
may
also write
NV'2U+(P-N+Q)
grade+(Q+R)
gradt=O
QV'
2
e+
RV'2
t =
b(a/
at)
(t-
e).
(4.6)
In
the previous theory
(1)
we
had
obtained these
equations
by
a different method
and
in a different form.
To
show their equivalence
we
write the stress-strain
relations
by
eliminating t from Eqs. (4.3)
uxx=2Nexx+(
A-
~)e+
~
u;
U
yy
=
2,Ve
yy
+ (
A-
~)e+
~
U;
(4.7)
-4ezzexx-4ezzeyy) Substituting these in the equilibrium relation (2.10)
+2Q(exx+eyy+ezz)t+Rt2. (4.1)
we
find
We
put
The
stress-strain relations derived from (2.7) are
u xx=
2N
e
xx
+
Ae+Qt;
u yy=
2N
e
yy
+
Ae+Qt;
u zz=
2N
ezz+Ae+Qt;
(4.2)
NV"2u+[P-N-Q2/RJ
grade
+
(Q+R)/R
gradu=O. (4.8)
We also derive from (4.4)
a b au
Q+R
ae
V'
2
u=b-(t-e)=---b---.
at R at R at
(4.9)
U
yz
=
1\T
e
yz
;
uzx=l\'e
zx
;
(J'xy=Nc
xy
;
(4.3) Equations (4.8)
and
(4.9) are in the form obtained in
reference
1.
We note
that
the significance
of
u in
that
reference is equivalent to
-u/
j in our present notation.
u=Qe+Rt.
There are in this case four elastic constants,
and
this
checks with the result obtained in reference
1.
The
equations
of
flow
contain a single coefficient
b.
They
are
written
au/
ax+P1x
=b(O
x-uz);
au/ay+PlY
=bCOy-u
y
);
au/
iJz+P1Z=bCO
z-u
z
).
(4.4)
Consider now the case
of
an
incompressible material.
This corresponds to the condition
e(l-
f)+
Jt=O.
(4.10)
Since this
must
be satisfied for all values of u
we
derive
from the last
relatipn (4.3)
that
both
Rand
Q are
infinite with the condition
Q/R=
(1-
j)/
f.
(4.11)
Since
A-Q2/R=S
must
remain finite the stress strain
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