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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