Analysis of Data Step 3

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CHAPTER 9 POROELASTICITY
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Figure 5.2. Illustration of an RVE for a
poroelastic medium.
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The theme for this chapter is contained in a
quote from the 1941 paper of M. A. Biot that
clearly describes an RVE Consider a small cubic
element of soil, its sides being parallel with
the coordinate axes. This element is taken to be
large enough compared to the size of the pores so
that it may be treated as homogeneous, and at the
same time small enough, compared to the scale of
the macroscopic phenomena in which we are
interested, so that it may be considered as
infinitesimal in the mathematical treatment.
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x m (matrix material), d (drained), or u
(undrained)
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(T, p) are forces (stresses) (E,?) are kinematic
measures (strains)
dW TdE pd?

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?? 1 - (Kd/Km)
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Derivation of
The loading of a porous object
is decomposed into two loadings
(a)
and
(b)
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Derivation of
The associated strain states are
and their sum is
thus
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Darcys law
?fv - H(p)?p(x, t), H(p) HT(p)
H ?foK/?? where K is the intrinsic Darcys law
permeability tensor, ?fo is a reference value of
the fluid density and ? is the fluid viscosity.
q (??f/?fo)v - (1/?)K?p(x, t), K KT
q (??f/?fo)v - (k/?) ?p(x, t), for isotropy
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Incompressible constituents
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Incompressible constituents
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Incompressible constituents

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Incompressible constituents
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Incompressible constituents
In the compressible case p is a thermodynamic
variable determined by an equation of state. In
the incompressible case p is a Lagrange
multiplier whose value is determined by the
boundary conditions.
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The undrained elastic coefficients (?? 0)
Incompressible case
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Isotropic case
Isotropic, incompressible case
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Anisotropic case
Isotropic compressible case
Incompressible case
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The basic equations of poroelasticity
An overview of the theory of poroelastic
materials can be obtained by considering it as a
system of eighteen equations in eighteen scalar
unknowns. This system of equations and unknowns,
a combination of conservation principles and
constitutive equations, is described in this
section. The eighteen scalar unknowns are the six
components of the stress tensor T, the fluid
pressure p, the fluid density rf, the variation
in fluid content ?, the six components of the
strain tensor E and the three components of the
displacement vector u. The eighteen scalar
equations of the theory of poroelastic solids are
the six equations of the strain-stress-pressure
relation (9.1), the six strain displacement
relations (3.49), 2E ((??u)T ??u), the three
equations of motion (4.37), the one fluid
content-stress-pressure relation (9.10) (or the
one fluid content-strain-pressure relation
(9.11)) and the one mass conservation equation
(9.46) and a relation between the fluid pressure
and the density p p(rf) which is not specified
here.
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conservation of mass
conservation of momentum
Changing the form of mass conservation, let ? ?
??f and divide by ?fo, thus
? ?? /?t ???q ?.
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(6)
2E ((??u)T ??u)
(6)
(3)
(1)
(1)
p p(rf)
(1)
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The parameters of a poroelasticity problem are
the drained effective elastic constants of porous
matrix material , the Biot effective stress
coefficients , , the fluid viscosity ?, the
intrinsic permeability tensor K, and the body
force b, which are all assumed to be known. If
the displacement vector u is taken as the
independent variable, no further equations are
necessary.
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Substituting for ???and q in ?? /?t ???q ? ,
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Substituting for ???and q in ?? /?t ???q ? ,
O r t h o t r o p i c
ca s e
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The boundary conditions on the pore pressure
field customarily employed in the solution of
this differential equation are (i) that the
external pore pressure p is specified at the
boundary (a lower pressure permits flow across
the boundary), (ii) that the pressure gradient
?p at the boundary is specified (a zero pressure
gradient permits no flow across the
boundary), (iii) that some linear combination of
(i) and (ii) is specified.
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Example 9.10.1 Formulate the differential
equations governing the problem of determining
the vertical surface settlement of a layer of
poroelastic material resting on a stiff
impermeable base subjected to a constant surface
loading. The layer, illustrated in Figure 9.2, is
in the x1, x2 plane and the x3 positive
coordinate is in the thickness direction and it
is pointed downward in Figure 9.2. The surface is
subjected to an applied compressive stress T33
-P(t), the only non-zero strain component is E33,
and there are no internal sources of fluid, ?
0. However, the free surface of the layer permits
the passage of fluid out of the layer.
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Figure 9.2. Illustration of a layer of
poroelastic material resting on a stiff
impermeable base subjected to a uniform time
varying surface loading.
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Solution First, since the free surface of the
layer permits the passage of fluid and the
supporting base of the layer is impermeable, the
boundary conditions on the pore pressure field
are p 0 at x3 0, ?p/?x3 0 at x3 L. Next,
using the fact that the only non-zero strain
component is E33, and that the applied
compressive stress T33 -P(t) is uniform
throughout the layer, the strain-stress-pressure
relations (9.1) specialize to the following
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The solutions of these equations for T11, T22,
trT and T33 are
and the single strain component E33, is given by
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From the stress equations of equilibrium
thus
Since trE E33,
substitution of E33 into the pressure diffusion
equation
yields
where
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where

when the matrix material and the fluid are
incompressible, c cI and W 1. For the special
cases of Kf Km it may be shown that 1 W
1/3 and ? c/cI 1. The two following examples
examine special solutions of these equations
which may then be specialized to these more
special assumptions by the appropriate selection
of c and W.
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Example 9.10.2 Determine the vertical surface
settlement of a layer of poroelastic material
resting on a stiff impermeable base subjected to
a constant surface loading T33 -P(t) -poh(t).
The layer, illustrated in Figure 9.2, is in the
x1, x2 plane and the x3 positive coordinate
direction is downward. The conditions for the
drainage of the layer are described in Example
9.10.1.
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Solution Substituting the surface loading P(t)
poh(t) into the pressure diffusion equation
then
then, introducing the notation p
the equation simplifies to a diffusion equation
in p
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The solution for p subject to the boundary
conditions
0 at x3 L, is
p 0 at x3 0,
and the solution for p is
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The total displacement u3(x3, t) must satisfy the
boundary condition u3(L, t) 0, and the desired
settlement of the free surface is given by u3(0,
t). In order to calculate u3(0, t), u3(x3, t)
must first be calculated. The displacement
u3(x3, t) is obtained from the pressure solution
obtained above and the equilibrium condition that
the derivative of the T33 stress component with
respect to x3 must vanish,
The total displacement u3(x3, t) of the points in
the layer is decomposed into two components, the
initial displacement at t 0, , and the
transient displacement thus
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The initial displacement at t 0, is determined
from the equilibrium condition for T33 above thus
The transient displacement is obtained from the
pressure solution for p obtained above and the
same equilibrium condition for T33, thus
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The desired settlement of the free surface u3(0,
t) may now be calculated. The sum of the initial
displacement at x3 0, and the transient
displacement at x3 0, is given by
where
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Figure 9.3. Illustration of the vertical
settlement of a layer of poroelastic material
resting on a stiff impermeable base subjected to
a constant surface loading. This is a plot of the
function g(t) against time. The curve underneath
the top curve is for a layer twice as thick as
the layer of the top curve. The two curves below
are for layers that are five and ten times as
thick, respectively, as the layer associated with
the top curve. Compare with Figure 2.22.
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Figure 2.22. Application of the Terzaghi model to
articular cartilage.
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Example 9.10.3 Determine the vertical surface
settlement of a layer of poroelastic material
resting on a stiff impermeable base subjected to
a harmonic surface loading T33 -P(t)
The layer, illustrated in Figure 9.2, is in the
x1, x2 plane and the x3 positive coordinate
direction is downward. The conditions for the
drainage of the layer are described in Example
9.10.1
Solution Substituting the surface loading P(t)
into the pressure diffusion equation in Example
9.10.1, the diffusion takes the form
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The solution is obtained by assuming that it is
of the form , thus the equation above reduces to
an ordinary differential equation,
after dividing through by . The boundary
conditions of the previous example, Example
9.10.2, are to be applied. The solution to this
equation then permits the following solution to
the original differential equation, thus
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The desired settlement of the free surface, u3(0,
t) is calculated following the method of Example
9.10.2 thus
A plot against frequency of the absolute value of
the function
determining the amplitude of the settlement of
the free surface u3(0, t) is shown in Figure 9.4.
At very large frequency the poroelastic layer
behaves as if it were undrained. At low
frequencies it behaves as if it were drained.
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Figure 9.4. Illustration of the vertical
settlement of a layer of poroelastic material
resting on a stiff impermeable base subjected to
a harmonic surface loading. A plot of the
absolute value of the function against frequency.
Example 9.10.3.
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Example 9.11.1 This example concerns the boundary
value problem that is a model for the unconfined
compression of a disk shaped specimen. The
material of the disk is assumed to be
transversely isotropic. In unconfined compression
tests a thin cylinder is compressed between two
parallel rigid and impermeable smooth plates, but
the specimen is assumed to have no frictional
contact with the end plates so that it is free to
expand radially thus there are assumed to be no
end effects (Figure 9.6). The passage of fluid is
free across the lateral boundaries (the curved
surfaces of the thin cylinder) while it is not
permitted across the boundaries forming the flat
end plates of the thin cylinder due to the two
parallel rigid and impermeable smooth plates that
are squeezing the thin cylinder. In this test
either a step in end plate displacement may be
applied, which would constitute a creep test for
the material of the thin cylinder, or a step in
total applied end force load may be applied,
which would constitute a stress relaxation test.
This test is used in the study of articular
cartilage.
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Figure 9.6 Schematic of an unconfined compression
test, From Cohen et al. (1998)
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The formulation of the mathematical problem
begins with assumptions concerning the functional
dependence of variables. Cylindrical coordinates
are employed and the displacement field
components, the pressure and the variation in
fluid content are assumed to have the functional
dependencies indicated below
If both and are transversely
isotropic with respect to a common axis, then
the formula for in cylindrical coordinates
is
where
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The axial force applied to the disk
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The isotropic incompressible case
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p(s) and q(s) are analytic at s ?n, p(?n) ? 0,
and ?n are simple poles of p/q
for 0 lt t lt to
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for t gt to
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Figure 9.7 Stress relaxation time history in an
unconfined compression test in response to a
ramped displacement for various values of the
dimensionless time variable cto/b2. The quantity
b2/c is a characteristic time of the disk
obtained from the ratio of the disk radius
squared, b2, to the diffusion constant c and
to is the ramp time in the ramped displacement .
The vertical scale is the non-dimensional load
intensity obtained by dividing P(t) by .From
Cohen et al. (1998)
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Figure 9.8 A typical stress-relaxation time
history in an unconfined compression test in
response to a ramped displacement. The ramped
displacement is , or 10 compressive strain at
131 seconds. The plots are curve fits of the same
data to both isotropic and transversely isotropic
models. From Cohen et al. (1998)
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