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Transcript of unsat1adfad2
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Derivation of Relationships for Volume Change
Requirements for Volume Change Theory
Representation of deformation behavior requires constitutive
equations relating a deformation state variable related to
two independent stress state variables.
Stress State Variables
The required stress state variables are:
1) (m - ua);
2) (ua - uw);
3) (1
- 3).
where m is the mean or average of the three principal
stresses.
Alternatively, the stress state variable (m - uw) could
be substituted for (m - ua) above.
Proposed constitive equations for the soil structure and
water phase are:
)()()( 31232221 ddCdudCduduCV
dVamwa
++=
)()()( 31131211 ddCdudCduduCV
dV
amwa
w++=
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Deformation State Variables
The required deformation state variables are:
1) void ratio, e, used to represent the volume
change behavior of the soil structure;
2) water content, w, or degree of saturation used
to represent the volume change behavior of
the water.
Constitutive Relationships
To define a constitutive relationship, it is necessary to
show that a unique surface can be formed when
plotting one of the two deformation state
parameters against two of the stress state
parameters.
For unsaturated soil, it may be possible to obtain
unique surfaces for monotonic loading, i. e. for one
cycle of either wetting or drying.
To verify that uniqueness is valid, soil is tested at
various state conditions, including isotropic and
anisotropic states of stress.
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Requirements from Continuum Mechanics
Continuity
Continuity is required if mass is to be conserved, i.e. no
gaps in space not occupied by mass will exist.
Total Volume Change
The total change in volume is equal to the sum in change
of volume of the air, water and solid phases.
Assuming that the solids are incompressible, then volume
changes as the air is compressed or the fluids flow.
Volume change occurs as the soil structure responds tostress and as the pore water responds to pressure
gradients.
Volume changes should be expressed as the change in
volume per unit volume.
There are three forms of constitutive relations that are
used for volume change behavior of unsaturated soils;
elasticity, compressibility and volume-mass form.
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Linear Elasticity
Expressions relating stress and strain are obtained
assuming:
1) normal strains are due to normal stresses;
2) shear strains are due to shear stresses;
3) superposition is valid.
Applying Hooke's Law to unsaturated soil:
xx a
y z aa w
y
y a
x z aa w
zz a
x y aa w
dd u
E Eu
d u u
H
dd u
E E ud u u
H
dd u
ud u u
d
d
d
=
+ +
=
+ +
=
+ +
( )( )
( )
( )( )
( )
( )( )
( )
2
2
2
xy xy yz yz zx zxdd G d dG d dG = =; ; =
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Expressions for Volume Change
The following expressions satisfy continuity:
1) Volume change of soil structure:
dv = dx + dy + dz
v mean a a wd u d u u 3 d3
H=
+
( )( ) (
1 2)
2) Volume change of fluids (drained conditions):
d dV
V
dV
V
dV
Vv
v
o
w
o
a
o
= = +
Using a relationship for water continuity:
d V 3E
dw
wo
mean aa w
wVu d u u
H= + ( ) ( )
Constitutive Surfaces
(m - ua)
(ua uw)
v
V
V
o o
w
V
V
(m - ua)
(ua uw)
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Compressibility Form (Soils Alternative to Linear Elasticity)
Replace material parameters from elasticity (E, and H)with compressibility coefficients, 1 2,
m .
1) Volume change of soil structure:
v mean a au d u u wd m d m 1s
2s= + ( ) ( )
2) Volume change of fluids (drained conditions):
d V m d m
w1w
2w
o
mean a a w
Vu d u u= +( ) ( )
Important facts regarding use of coefficients of volume
change for constitutive relations include:
1) coefficients are negative for stable soils;
2) constitutive relations are nonlinear but can be taken
as linear for a small increment of stress or strain;
3) the relations depend upon the actual loading
conditions, i.e. uniaxial, triaxial, plane strain, etc.;
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Volume-Mass Form
Equivalent expressions are obtained using de in place
of dv and coefficients at and am, and using dw inplace of dVw/Vo and coefficients bt and bm.
)(a)(dade mt uuu waamean d +=
)(b)(dbw mt uuu waamean dd +=
Verification of Volume Change Theory
Verification is approached by showing uniqueness of the
constitutive surfaces.
Uniqueness Hypothesis
There is only one relationship (constitutive surface)
between the deformation and stress state variables
provided the changes in stress state variables results
in deformations (strains) which are always of thesame direction.
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( ua)
(ua uw)
v
Complete uniqueness requires that you would arrive at the
same volume regardless of the path followed, as shownabove.
Hysteresis of soil behavior does not allow for complete
uniqueness.
Uniqueness can exist if the loading occurs in one direction
only, i.e both net normal stress and matric suction are
either increased or decreased.
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Corollary to Uniqueness Hypothesis - Hysteresis Effects
Hysteresis effects will result in nonuniqueness.
Hysteresis due to soil structure:
VoidRatio
et Normal Stress
Hysteresis due to contractile skin:
Water
Content
Matric Suction
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Experimental Verification of Uniqueness Hypothesis
Verification consists of testing either three identical
specimens or one specimen and:
1) increase the net normal stress and maintain the
matric suction constant and compute
)(
/
)(
1
1
am
oww
am
vs
ud
VdVm
ud
dm
=
=
2) increase the matric suction and maintain the
net normal stress constant and compute
)(
/
)(
2
2
wa
oww
wa
vs
uud
VdVm
uuddm
=
=
3) increase both the matric suction and the net
normal stress and predict the total net normal
stress and the water volume changes using the
computed coefficients.
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Results of Verification Studies
Results of testing indicate good agreement for
predictions of the total volume change but lessagreement for predictions of water volume
change which was attributed to test difficulties.
For stable soil, the volume will increase (expand)
with decreases in matric suction.
For metastable soil, the volume will decrease
(collapse) with decreases in matric suction.
The conclusions should be valid for both stable and
metastable soil.
Addition to the Uniqueness Hypothesis
The final deformation state is independent of the stress
path followed.
Nonlinearity of Constitutive Surface
The constitutive surface is nonlinear because the soil
stiffness increases as both the net normal stress and
the matric suction pressures increase.
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Mechanics of Unsaturated Soils
Homework Assignment