considered to be a function of the moisture level
rameter of eq 5 were directly related to the unfro-
zen water content, as follows: 1) wug, expressed
expressed as the degree of saturation in the sample
as a decimal, normalized to a unit unfrozen water
and, when a range of data were available, the dry
content, w0, of 1.0 (i.e., as calculated with eq 4);
density. Thus, the general equation becomes
and 2) the volumetric unfrozen water content,
Mr = K1[ f (σ)]K2
(10)
wuv, expressed as a decimal, normalized to a unit
unfrozen water content, w0, of 1.0. The volumet-
which includes the term
ric unfrozen water content was determined with
the equation:
K1 = C0 (S / S0 )C1
(11)
wu- v = wu- g γ d
(7)
or
where γd
= dry density, Mg/m3. The resulting equa-
K1 = C0 (S / S0 )C1 (γ d / γ o )C2
(12)
tions with these terms substituted as the govern-
ing parameter were:
where f(σ) = a stress parameter normalized to a
unit stress of 6.9 kPa (1.0 lb/ft2)
Mr = K1(wu-g / w0 )K2
(8)
C0, C1, C2 = constants
S = the degree of saturation (%)
and
S0 = a unit saturation (1.0 %)
Mr = K1 (wu- v / w0 )K2
(9)
γo = a unit density (1.0 Mg/m3).
In analyzing the frozen resilient modulus data,
Three stress parameters were investigated to
the value of the governing parameter (wug/wt,
help characterize the stress dependence of the ma-
wug/w0, wug/w0) at each test point was deter-
terials tested. These included J1, the bulk stress
mined from the temperature (and total water con-
(or first stress invariant); toct, the octahedral shear
tent, if necessary). Then, regression analysis was
stress; and J2/τoct, the ratio of the second stress
conducted to determine the relationship between
invariant to the octahedral shear stress. In our re-
these values and the measured resilient modulus.
peated-load triaxial test, where s2 = s3 and s1 = s3
Data from the thawed, undrained state (assigned
+ sd, the functions are given as:
to be at a temperature just barely below freezing)
J1 = 3σ3 + σd
were analyzed along with the frozen data. Regres-
sion analysis was then conducted to determine the
relationship between these values and the mea-
2
τoct =
σd
sured resilient modulus (Table 1a).
3
The left-hand graphs in Figure 14 compare the
and
frozen resilient modulus data (solid circles) with
9σ32 + 6σ3σd
predictions from the regression equations with the
J2 / τoct =
2σd
three different governing parameters. The figure
shows that the frozen modulus does vary prima-
where
rily as a function of the unfrozen water content. A
minor amount of variation results from the vari-
J1 = σ1 + σ2 + σ3
ous stress combinations acting on the specimens,
J2 = σ1σ2 + σ2σ3 + σ1σ3
as shown from the vertical spread in the data at
.
any particular temperature. The predictive equa-
τoct = 1 2 (σ1 - σ2 ) 2 + (σ2 - σ3 )2 (σ1 - σ3 ) 2
tions without normalization to total water appear
to pass nearer to the center of the range of data
collected at temperatures warmer than 2.0C.
We found that the bulk stress parameter, J1, provid-
ed the best fit to the data for the class 6 base
Thawed/never frozen. To analyze data from the
material; the ratio J2/toct best fit the data of the
thawed or never frozen specimens, the governing
three subbases: class 3, class 4, and class 5; and toct
parameter in the general form equation (eq 5) was
set to be a stress function. The constant K1 was
best characterized the clay subgrades (Table 1b).
22
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