unsat1adfad2

7/29/2019 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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Unsat12 - 12

Mechanics of Unsaturated Soils

Homework Assignment

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