Stress State Variables
• Definition
Stress variables are used to determine the state of stress in a
soil that is then used to model mechanical behavior.
These stress state variables cannot be dependent on the
properties of the soil.
• Saturated Soils
The primary stress state variable used to characterize
saturated soil behavior is the effective stress.
σ' = (σ - uw)
Example:
The shear strength of a saturated soil can be given in
terms of the effective stress for a given soil (MohrCoulomb).
No material properties are required in the expression of
effective stress.
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• Unsaturated Soils
It is not possible to express the effective stress in terms of the
pore water pressure (negative) because the pore water is
discontinuous and thus does not act throughout the soil.
It is more difficult to characterize the behavior of
unsaturated soils because both liquid and air phases occur
requiring at least two independent stress state variables.
Various forms of effective stress equations that include a
parameter(s) that is a material property.
Examples:
σ' = (σ - β' uw)
σ' = (σ - ua) + χ (ua - uw)
σ' = (σ + ψ p'')
σ' = (σ + β p'')
σ' = (σ - ua) + χm(um - ua) + χs(us - ua)
σ' = σ + χm p'' + χs p''
The problem is that none of the parameters are single-valued.
Put another way, it is not possible to express quantitatively
the effect of the negative pore water pressure that acts
throughout the soil which adds to the soil effective stress.
See handout from Lambe and Whitman (1969)
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• Proposed Solution
Two independent stress state variables can be used to
characterize unsaturated soil behavior.
Examples:
1) (σ - ua) and (ua - uw)
2) (σ - uw) and (ua - uw)
3) (σ - ua) and (σ - uw)
These stress state variables can be used to define constitutive
relations to define shear strength and volume change
behavior and hydraulic conductivity.
To understand this, it is necessary to review concepts from
mechanics;
a) representative stress element;
b) equilibrium equations.
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• Theoretical Treatment of Force Equilibrium at a Point
Saturated Soil uses the surface traction forces and the
gravitational forces acting on a representative elementary
volume.
(See Figure 3-2.)
Stresses can be either total or effective stresses.
Unsaturated Soil
Equilibrium equations can be written in terms of the total
equilibrium and the equilibrium of the independent
phases: water phase equilibrium; air phase equilibrium and
contractile skin equilibrium using the respective porosities
and densities of each phase. (See Appendix B of the text.)
The set of equations can be written in terms of two
independent tensors that include 1) total stress and air
pressure or total stress and water pressure and 2) matric
suction.
Therefore, you need two independent stress state
parameters for unsaturated soil behavior.
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Special Conditions of Stress State Variables
• (σ - ua)
Under normal conditions, the pore air pressure ua is
equivalent to atmospheric pressure with ua in gage
pressure ( ua = 0) rather than absolute pressure.
Therefore this stress state variable is equal to the total soil
stress.
A critical condition occurs when ua exceeds σ. A case that
would not occur under normal circumstances.
• (σ - uw)
For unsaturated soils, uw is less than atmospheric pressure or
is given as negative using gage pressure.
This stress state variable is not equal to the effective soil
stress because the pore water does not completely occupy
the pore void spaces.
A critical condition occurs when uw is positive and exceeds
σ in which case the soil becomes quick.
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• (ua - uw)
For unsaturated soils uw is less than atmospheric pressure.
For (ua - uw) = 0 the soil is saturated or the air is occluded
(i.e., air is not continuous throughout the soil and is
usually assumed to be immobile).
This stress state variable is a measure of the soil suction
pressure and is usually required for constitutive
relationships. It is referred to as the soil water matric
suction.
Example constitutive relationship:
0.6
0.5
Θw
0.4
0.3
0.2
0.1
0
0.1
1
10
102
103
104
105
106
Matric Suction, kPa
Figure 3.1 Soil moisture characteristic curve.
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Concepts Involved in Determining Stress State Variables
• Axis Translation
Problem:
It is not possible to measure the high negative pore water
pressures (e.g., ua - uw = 106 kPa) that are used to
define soil moisture characteristic relationships
because:
a) cavitation (dissolved air and water vapor comes out
of solution as the water pressure approaches the
vapor pressure) occurs in water which interferes
with the measurement of pressure;
b) for a perfect vacuum, the negative pressure is
zero (10-4 kPa) on the absolute scale.
Concept:
Using a specially designed apparatus, it is possible to
vary the air pressure and the water pressure (both
positive) in soil until the soil pore water comes to
equilibrium.
The difference between the two positive pressures is
the soil moisture suction.
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• Verification of Stress State Variables
Question:
Is it possible to use the three stress state variables
defined previously; i.e., (σ - ua), (σ - uw), and
(ua - uw); to characterize soil behavior?
1) Null tests:
If all the component stresses of the stress state
variables are varied uniformly then the stress
state variables will remain constant and therefore;
a) the volume will remain constant, and
b) degree of saturation will remain constant.
2) Shear strength tests:
If σ3, ua and uw are varied uniformly then (σ3 - ua)
and (ua - uw) will remain constant and therefore;
a) the shear strength will remain constant, and
b) the stress strain behavior will remain constant.
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Extension of Mechanics of Saturated Soils
• Soil Stress
The determination of in situ stress remains the same except:
1) total vertical stress is referred to as net normal stress
and given as (σ - ua);
2) the soil water matric suction varies seasonally so the
effective stress (unknown) varies;
3) the horizontal stresses are computed using an at rest
lateral earth pressure coefficient that is given in terms
the net stresses instead of the effective stresses, i.e.,
K=
(σ h − u a)
(σ v − u a)
where K ≠ Ko;
4) the soil is comparable to an overconsolidated soil;
5) other soil properties which depend upon effective soil
stress (such as shear strength, compressibility and
permeability) vary considerably.
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• Analysis of Stress
Mohr's circles can be used to represent the state of stress at a
point using the net vertical and horizontal stresses as the
principal stresses, and the matric soil suction pressure as a
third dimension.
Extended Mohr’s circles are used to characterize unsaturated
soil by using a third axis for matric suction.
(ua – uw)
(σ1-σ3)/2
τ
(σ – ua)
(σ3-ua)
(σ1+σ3)/2-ua
(σ1-ua)
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Stress Invariants
Expressions that are given in terms of principal stresses and
matric suction are constants valued in space (do not
depend upon direction) are thus called stress invariants.
Three stress invariants each having two independent stress
tensors are obtained from the determinant of the
equilibrium equations. The determinant must be equal to
zero in order to give solutions not equal to zero so:
S3 – [(σ1-ua)+ (σ2-ua) + (σ3-ua)]S2
+ [(σ1-ua )(σ2-ua)+ (σ2-ua )(σ3-ua) + (σ3-ua)(σ1-ua)]S
– [(σ1-ua)(σ2-ua)(σ 3-ua)] = 0
where S in the normal stress on a plane. (See handout by
Timoshenko and Goodier.)
Three stress invariants of the first tensor are given in terms of
the net normal stress from the above equation.
Three stress invariants of the second stress tensor are given in
terms of the matric suction.
Four stress invariants are required to characterize the state of
stress in unsaturated soil, three invariants of the first stress
tensors and one invariant of the second stress tensor.
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Stress paths
Stress paths can be used to represent changes in the state of
stress the mean stress (p), one half the deviator stress (q)
and the matric suction (r).
(σ 1 + σ 3 )
− ua
2
(σ − σ 3 )
q= 1
2
p=
r = (u a − u w )
• Osmotic Suction
The total soil suction is
ψ = (ua - uw) + π
where π is the osmotic suction due to soil ions.
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Mechanics of Unsaturated Soils
Homework Assignment
The following results were obtained using the axis translation
technique. Complete the following table and plot volumetric
water content versus matric suction. The water pressure was
equal to atmospheric pressure so the matric suction is equivalent
to the applied air pressure. (Use Gs = 2.7 for the calculations.)
Table 3.2 - Pressure Plate Test Results, Direct Measurement of Water Content
Applied
Grav.
Wet
Dry
Volum.
Air
Water
Unit
Unit
Water
Press.
Content Weight Weight Content
kPa (psi)
(%)
(g/cc) (g/cc)
(%)
0.0
21.3
2.17
35. (5)
19.9
2.17
69. (10)
17.7
2.08
138. (20)
16.7
2.06
207. (30)
17.0
2.01
276. (40)
16.2
2.08
345. (50)
15.1
2.09
414. (60)
15.4
2.01
345. (50)
15.9
2.04
276. (40)
17.2
1.93
207. (30)
16.2
2.03
138. (20)
16.4
2.00
69. (10)
16.0
2.09
35. (5)
16.3
2.11
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Degree
Void
of
Ratio Porosity Saturation
n
(%)
Use Fig. 2 and Fig. 3 below to plot the Extended Mohr Diagram
and Stress Paths. For the tests, the air pressure was kept at
atmospheric and the suction was developed by osmotic pressure
so the net normal stress is equivalent to the normal stress.
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