Stress-Induced Anisotropy of Small Strain Shear Modulus
in Saturated and Unsaturated Cohesive Soils
Hirofumi Toyota1, Ngoc Bao Le 2, and Susumu Takada3
1

Professor, Dept. of Civil and Environmental Eng., Nagaoka University of Technology, Japan(E-mail: toyota@vos.nagaokaut.ac.jp)
2
Engineer, Kiso-Jiban Consultants Co., Ltd., Japan(E-mail: le.ngocbao@kiso.co.jp)
3
Technical staff, Dept. of Civil and Environmental Eng., Nagaoka University of Technology, Japan(E-mail: stakada@konomi.nagaokaut.ac.jp)

ABSTRACT
Cohesive soils in nature have been existed under anisotropic stress conditions:
Embankment produces an additional vertical load in the ground just under theembankment, while producing an additional horizontal load in the ground a little awayfrom the embankment. This study investigates stress-induced anisotropic effects on small
strain shear modulus in saturated and unsaturated cohesive soils. A triaxial apparatus
installed local small strain (LSS) measurement devices and bender elements (BEs) wasused to measure the small strain shear modulus. LSS and BE tests were conducted using
specimens normally consolidated under constant effective stress p’=300 kPa or net meanstress pnet=300 kPa with different stress ratios, represented as K= 'h/'v for saturated soils
and Knet = (h–ua)/(v–ua) for unsaturated soils. The results demonstrated that Kconsolidation under constant p’ produces large differences in the initial shear modulus G0
in saturated cohesive soil, but Knet slightly affects G0 in unsaturated cohesive soil becauseof the strong matric suction effects. Finally, G0 was normalized successfully considering
the effects of void ratio e and stress ratio K and Knet.
Key words: bender element, induced anisotropy, local small strain, shear modulus, unsaturated
soil

1. INTRODUCTION
The natural ground initially exhibits an anisotropic stress state of the K0 condition after thenatural sedimentation. Since then, the ground has had complicated stress histories through naturaland artificial processes: tectonic movement, landslides, and construction works. Stress-induced
anisotropy, which is induced by application of the anisotropic stress state, was first introduced by
Casagrande and Corrilo (1944). It has since been recognized as an important mechanical property,especially for cohesive soils.
The small strain shear modulus plays a crucial role for design tasks related to ground structures
under static and/or dynamic loading conditions. Regarding the cohesive soils, Rampello et al.
(1997), Chien and Oh (2002), and Hao and Lok (2008) reported that the shear modulus is greaterin an anisotropic stress condition than that with isotropic stress consolidation. Mitaritona et al.
(2014) considered the effects of both inherent anisotropy and induced anisotropy by investigating
the ratio Ghh/Gvh under different stress-paths  = q/p’, indicating  = 0 (isotropic) and   0(anisotropic). They demonstrated for clay that the ratio Ghh/Gvh increases from 1.0 at an isotropic
stress condition to 1.2 at  = 0.8. Ng et al. (2009) found shear modulus anisotropy for unsaturated
soils. They proved that G0(hh)/G0(vh), where G0(hh) and G0(vh) respectively represent the shear
modulus in the horizontal and vertical directions, decreased significantly from 1.03 to 0.87 when changed from 0 (isotropic condition) to 1 (anisotropic condition).
Although current achievements for the study of this kind are described above, morecomprehensive studies are recommended to clarify the mechanism of anisotropic shear modulus
induced by stress conditions. Therefore, a series of triaxial tests, including bender element (BE)tests and local small strain (LSS) tests were conducted using the saturated and unsaturated
specimens consolidated at various K='h/'v and Knet=(h–ua)/(v–ua) under the same meaneffective stress p’= ('v+2'h)/3 and net mean stress pnet= p–ua of 300 kPa, respectively.

2. EXPERIMENTS
2.1. Testing Materials and Specimen Preparation
Yoneyama silt, whose grain size distribution and index properties are presented in Fig. 1, wasused in the study. Dry Yoneyama silt was mixed with de-aired water with water content of 80 %to produce slurry. The vacuum was applied to remove air from the slurry in an acrylic cylindricalmold. Then, air pressure of 45 kPa was applied to the upper space of the piston in the mold. The
soil block, its water content of about 40 %, was extracted from the mold after 2 days of onedimensional consolidation. A cylindrical specimen of 125 mm height and 50 mm diameter was
trimmed from the extracted soil block.
The drying portion of soil-water characteristic curves obtained from the pressure plate method
is shown in Fig. 2. Constant isotropic confining stresses of pnet=100 kPa and pnet=300 kPa areapplied to the specimens during the drying using a triaxial apparatus. From the results, the air
entry values of the soil under pnet=100 kPa and pnet=300 kPa can be respectively detected as about
50 kPa and 150 kPa.

Yoneyama silt3
s=2.74 g/cm , wL=52 %, Ip=29

100

Degree of saturation, Sr (%)

Percentage finer by mass

100
80
60
40
20

Silt

Clay

0
0.001

0.01

Sand

0.1

Grain size (mm)1pnet=300 kPa

Yoneyama silt
Pressure plate method
Drying (dehydration)90

pnet=100 kPa

80

70

60

1

10

100

1000

Matric suction, s (kPa)

Figure 1. Grain size distribution curve and its Figure 2. Soil-water characteristic curves
physical properties(Toyota et al., 2017)

2.2. Testing Apparatus
A triaxial apparatus with integrated BEs and LSS measuring devices was used to measure theshear modulus at a small strain (e.g. Le et al., 2024). The LSS technique was developed to measure
the smallest strain of 10-4 % using the proximity transducers with 0.2 µm resolution. Two targets
were directly pasted onto the upper and the lower parts of specimen’s side surface with distanceof 80 mm for the vertical strain measurement. Two proximity transducers set on two individualcolumns respectively can be adjusted the distance between the transducers and the targets fromoutside of the triaxial cell. For horizontal strain measurements, a clamp device equipped with theproximity transducer and target was installed on the center of the specimen.
The BEs of 2.5 mm height, 12 mm length, and 1 mm thickness were attached to the top cap as
a transmitter and the base pedestal as a receiver. Then BE tests were conducted to follow thestandard of JGS 0544 (2020). Single sinusoidal waves with frequencies of 10, 15, 20 kHz at
voltage ±10 V were used as the input wave. The start-to-start and the tip-to-tip methods were used
to evaluate the shear wave velocity Vs.

2.3. Testing Methods
The stress paths are presented in Fig. 3. The testing procedure is described below.
1. Saturation and isotropic consolidation process: A specimen was saturated using the vacuum
saturation procedure (Rad and Clough, 1984). Then, the cell water was removed while negative
pore water pressure of 20 kPa was applied inside the specimen. The proximity transducers for
LSS tests were installed. After re-supplying the cell water to the triaxial cell, the specimen was
consolidated isotropically under confining pressure of 50 kPa. Back pressure of 200 kPa wasapplied to achieve a Skempton’s B-value greater than 0.97 for the saturated soil.

2. Drained q-loading process: Maintaining p’=50 kPa, the specimen was sheared at an axial strain
rate of 0.005%/min under a drained condition. The shearing continued until reaching a target
deviator stress q=’v’h, in which a certain K = ’h/’v of 0.35, 0.43, 0.6, 1.0, 1.5, 2.0, 3.0, and3.5 was achieved. The test case at K of 3.5 was only conducted in BE tests of saturated soil
because of testing apparatus limitations.
3. Drying process: q and Knet =(hua)/(vua) were maintained while matric suction (s = uauw)
of 400 kPa was applied to the specimen. This matric suction is greater than the air entry value(AEV) of the specimen (Fig. 2). During this process, pnet=50 kPa was also kept constant using
the axis-translation technique. This process was continued until the water drained from theceramic disc became less than 0.2 cm3 per day. This process was skipped in the case of saturated
soil.
4. K or Knet-consolidation process: K or Knet was kept constant with the same value set at the qloading process. Also, p’ or pnet was increased from 50 kPa to 300 kPa while maintaining K or
Knet. Toyota et al. (2001) demonstrated that when the specimen of Yoneyama silt, which isprepared by the same method as this study, is isotropically consolidated with 100 kPa, it shows
almost isotropic shear behavior. This indicates the structural anisotropy caused by the onedimensional consolidation is largely eliminated by the isotropic consolidation with twice asmuch as the vertical stress of one-dimensional pre-consolidation. Hence, the application of p’or pnet=300 kPa will eliminate the effects of one-dimensional pre-consolidation with the verticalstress of 45 kPa on the subsequent shear behavior. The degree of saturation for unsaturated
Yeneyama silt was about 80 % in this stage. Then, the BE test was conducted to obtain the initialshear modulus G0 based on Vs of the specimen. K or Knet-consolidation was maintained for 24 h
in all cases to make the aging effects even.

800
Stress path, K=h'/v'
Knet=(hua)/(vua)

q=vh (kPa)

600
400

K or Knet
0.35,
1.0,

0.43,
2.0,

0.6
3.0

K or Knet consolidation(Compressional)
Drained q-loadingp'=50 kPa

200
0
-200
-400

Drying, s=400 kPa

0

50

100

K or Knet consolidation(Extensional)

150

200

250

Triaxial
compression

300

350

p'=(v'+2h')/3 or pnet=(v+2h)/3ua (kPa)

Figure 3. Stress paths during testing400

5. Shearing process: Drained monotonic compression was conducted under constant cell pressure
with an axial strain rate of 0.0025%/min. Shearing was continued until the shear strain s
exceeded 0.1%.

3. RESULTS
3.1. Effects of K or Knet-consolidation on the Secant Shear Modulus
Figure 4 presents the secant shear modulus Gsec versus the shear strain in LSS tests of saturated
and unsaturated soils under triaxial compression. The value of Gsec is calculated as shown inequation (1).
∆𝑞
1
𝐺
3∆𝜀
Here, the shear strain s = 2/3(vh), v is vertical strain, and h is horizontal strain. In the saturated
soil: At the range of s <0.001%, there is a trend that smaller K consolidated soil has greater Gsec
(comprehensive discussion is conducted in the next section). However, at the greater shear strain
range, the trend is reversed, that is, smaller K consolidated soil has smaller Gsec. In the unsaturated
soil: At the range of s <0.001%, the difference in Gsec is slight (comprehensive discussion isconducted in the next section). At the greater shear strain range, the trend, in which greater K
consolidated soil has greater Gsec, is apparent. Those facts indicate that degradation of Gsec withshear strain is greater when K or Knet is small (the drained monotonic compression starts from acompressional stress state) but it is less when K or Knet is greater (the drained monotonic
compression starts from an extensional stress state).

150

300
Yoneyama silt, K-consolidation
Triaxial compression
p'=300 kPa

100(a) Saturated
Compresional consolidation
K=0.35
K=0.43
K=0.6
K=0.8
K=1.0
Extensional consolidation
K=1.5
K=2.0
K=3.0

50

0
0.0001

0.001

0.01

0.1

Shear strain, s (%)

Secant shear modulus, Gsec (MPa)Secant shear modulus, Gsec (MPa)

200

250

Yoneyama silt, Knet-consolidation
Triaxial compressionpnet=300 kPa, s=400 kPa

200
150

(b) Unsaturated
Compresional consolidation
Knet=0.35
Knet=0.43
Knet=1.0
Extensional consolidation
Knet=2.0
Knet=3.0100
50
00.0001

0.001

0.01

0.1

Shear strain, s (%)

Figure 4. Secant shear modulus in LSS tests (p’ or pnet= 300 kPa): (a) saturated and (b) unsaturated
cohesive soils (s =400 kPa)3.2. Effects of K or Knet-consolidation on G0

3.2.1. Experiment results
Figure 5(a) shows G0 of saturated cohesive soil obtained from both BE and LSS tests. The
results of G0 of saturated cohesive soil were similar for both LSS and BE tests. Compressional
consolidation (K<1.0): G0 increases concomitantly with the decrease of K from 1.0 to 0.43, after
which it declines at K=0.35. Consequently, the peak appears at around K=0.43, which iscoincident with K0 (=0.43) of Yoneyama silt, as reported in Toyota et al. (2014). K0 is just thethreshold K value, which divides positive (compression) horizontal strain region (K>K0) andnegative horizontal strain region (K<K0) during K-consolidation.
Extensional consolidation (K>1.0): G0 increases slightly with the increase of K from 1.0 to 2.0
and then declines at greater K (≥3.0). Therefore, G0 has two peaks at K0=0.43 and K=2.0.
Moreover, G0 in the compressional consolidation (K<1.0) is greater than that in the extensional
consolidation (K>1.0). Except for K≥3.0, G0 in the anisotropically consolidated cohesive soil (K1.0) is greater than that in isotropically consolidated soil (K=1.0) under the same p’ condition.
Regarding unsaturated cohesive soil presented in Fig. 5(b), G0 is almost independent with thevariation of net stress ratio Knet, although G0 declines slightly at Knet=0.35. It is noteworthy thatthe effects of K in G0 of saturated cohesive soil, as portrayed in Fig. 5(a), diminishes in that ofunsaturated cohesive soil because of the application of matric suction (Fig. 5(b)). The values of
G0 are very close between BE and LSS tests. Therefore, G0 obtained from BE is used as arepresentative G0 hereinafter.
220

150

Yoneyama silt, K-consolidation
Triaxial compression
p'=300 kPa

140

K=0.43 (=K0)(a) Saturated
BE test
LSS test

130
120
110
100
0.0

Extensional
consolidation

Compressional
consolidation

0.5

1.0

1.5

2.0

2.5

Stress ratio, K=h'/v'

3.0

3.5

Initial shear modulus, G0 (MPa)

Initial shear modulus, G0 (MPa)

160

210

(b) Unsaturated

Yoneyama silt, Knet-consolidation
Triaxial compressionpnet=300 kPa, s=400 kPa

200
190
180
170
160
0.0

Compressional
consolidation

0.5

BE test
LSS test

Extensional
consolidation

1.0

1.5

2.0

2.5

3.0

3.5

Net stress ratio, Knet=(hua)/(vua)

Figure 5. G0 from LSS and BE tests: (a) saturated and (b) unsaturated cohesive soils

3.2.2. Void ratio effects
G0 of unsaturated cohesive soils is about 30%40% greater than that of saturated soils becauseof the application of matric suction of 400 kPa (comparison between Figs. 5(a) and 5(b)). Voidratios of saturated and unsaturated cohesive soils vary respectively from 0.857 to 0.802 (about 7%
variation) and from 0.844 to 0.789 (about 5% variation). Densification occurs during the drying
process by the application of suction 400 kPa. Then, Knet-consolidation is conducted in theunsaturated specimen, which has stronger stiffness than the saturated soil. Consequently, the void
ratios of unsaturated cohesive soil were about 1.0% smaller than those of saturated soil (Le et al.,

2024). To remove the void ratio effects, G0 was normalized by a void ratio function. Le et al.
(2024) investigated the effectiveness for normalization using two kinds of void ratio functions(hyperbolic and exponential forms). The results indicated that the normalized G0 showed identicaltrend between two kinds of void ratio functions. Therefore, as the void ratio function, thehyperbolic form, represented by equation (2), was chosen. Value of a=2.973 was used in theequation based on the reports of Hardin and Black (1968) and Hardin and Black (1969).
𝑎 𝑒
2
1 𝑒
Figures 6(a) and 6(b) show the relation between “normalized G0” by the void ratio function andstress ratio K for saturated and net stress ratio Knet for unsaturated cohesive soils, respectively.
The normalized result of saturated soil still exhibits similar trend with Fig. 5(a). Therefore, theeffects of K on G0 are not eliminated merely by using the void ratio function. On the other hand,
the G0 of unsaturated soil is less sensitive to the stress condition (Knet-value) than that of saturated
soil.
𝐹 𝑒

100
(a) Saturated

Yoneyama silt, K-consolidation
BE test, p'=300 kPa

60

K=0.43 (=K0)

Normalized shear modulus, G0 (MPa)Normalized shear modulus, G0 (MPa)70

Normalized G0: G0/F(e)2F(e)=(2.973e) /(1+e)

50

40Compressional
consolidation

30
0.0

0.5

Extensional
consolidation

1.0

1.5

2.0

2.5

3.0

3.5

(b) Unsaturated

Yoneyama silt, Knet-consolidation
BE test, pnet=300 kPa, s=400 kPa

90Normalized G0: G0/F(e)2F(e)=(2.973e) /(1+e)

8070Extensional
consolidation

Compressional
consolidation

60
0.0

0.5

Stress ratio, K=h'/v'

1.0

1.5

2.0

2.5

3.0

3.5

Net stress ratio, Knet=(hua)/(vua)

Figure 6. Initial shear modulus normalized by the void ratio

3.2.3. Stress state effects
Hardin and Richart (1963) and Hardin and Drnevich (1972) proposed an empirical expression
for G0, accounting for the effects of effective stress p’ and void ratio e:
𝐺

𝐴𝐹 𝑒 𝑝𝑝𝑝3where pa is the atmospheric pressure of 100 kPa, A and n are fitting parameters, F(e) is the void
ratio function, and pa(p’/pa)n is the stress function. For unsaturated soils, pnet is used instead of p’.
Aside from the mean effective stress, the importance of individual stress for G0 was reported
by Hardin and Blandford (1989) and Jamiolkowski et al. (1995):
𝐺

𝑆 𝐹 𝑒 𝑂𝐶𝑅 𝑝

Therein, OCRk is the OCR function, and 𝑝

𝜎′ 𝜎′

𝜎

𝜎

4

is the stress function. k is an exponentof OCR that depends on plasticity index Ip. The value of k in Yoneyama silt has been discussed

in Le et al. (2024). Also, i and j are denoted respectively as the direction of the wave
propagation/polarization or applied stress direction. ni and nj are empirical exponents that
respectively account for vertical and horizontal effective stresses. Sij is the dimensionlesscoefficient. Stokoe et al. (1985), Hardin and Blandford (1989), and Bellotti et al. (1996) reported
ni and nj as both equal to 0.25. Yu and Richart (1984) also reported that the difference of ni and nj
between the extension test (σ’h>σ’v) and compression test (σ’v>σ’h) is extremely small. However,Roesler (1979) insisted that ni is smaller than nj under the relation of ni + nj = 0.5.
G0 in Fig. 6 is measured under the same p’ of 300 kPa for the saturated cohesive soil and thesame pnet of 300 kPa for the unsaturated cohesive soils, even the K or Knet-value is different. The
normalized G0 are varied with the different K-values in the saturated soil (Fig. 6(a)). However,the variation of this kind is slight in the unsaturated soil (Fig. 6(b)). Therefore, pnet can be regarded
as a good indicator for unsaturated cohesive soil to eliminate the effects of Knet using equation (3).
However, equation (4) is applied for the saturated soil because equation (3) is insufficient toeliminate the effects of K from the saturated soil.
Figures 7(a) and 7(b) present the normalized G0 by the void ratio function (2) and the stressfunction (4) for saturated and the stress function (3) unsaturated soils, respectively. Through thedifferent combinations of ni and nj, the best combination was chosen as ni=0.25 and nj =0.25 toeliminate the effects of K for saturated cohesive soil (Fig. 7(a)). By contrast, Fig. 7(b) depicts G0
normalized by equation (3) with n =0.5. Effects of Knet for unsaturated cohesive soil can be almost
removed. Therefore, individual stresses such as vertical and horizontal normal stresses presented
in equation (4) are less important for G0 of unsaturated cohesive soil. Consequently, to predict G0
under an anisotropic stress state, net mean stress is effective for unsaturated cohesive soil, while
individual normal stresses should be introduced for saturated cohesive soil.

400

600
(a) Saturated

Yoneyama silt, K-consolidation
BE test, p'=300 kPa

1ninjni

njNormalized G0: G0/(F(e)pav' h' )2F(e)=(2.973e) /(1+e), ni=0.25, nj=0.25

350
300
250
200
0.0

Compressional
consolidation

0.5

Extensional
consolidation

1.0

1.5

2.0

2.5

Stress ratio, K=h'/v'

3.0

3.5

Normalized shear modulus, G0 (MPa)Normalized shear modulus, G0 (MPa)

450

550

Yoneyama silt, Knet-consolidation
BE test, pnet=300 kPa, s=400 kPa(b) UnsaturatednNormalized G0: G0/(F(e)pa(pnet/pa) )2
F(e)=(2.973e) /(1+e), n=0.5

500
450
400
350
0.0

Compressional
consolidation

0.5

Extensional
consolidation

1.0

1.5

2.0

2.5

3.0

3.5

Net stress ratio, Knet=(hua)/(vua)

Figure 7. G0 normalized by the void ratio function and stress function: (a) saturated and (b)unsaturated cohesive soils

4. CONCLUSIONSAnisotropic shear modulus induced through K or Knet-consolidation was investigated using
saturated and unsaturated Yoneyama silts. The following conclusions can be drawn from thepresent study.
1. Very small shear strain: G0 is strongly affected by anisotropic stress conditions in the saturated
cohesive soil under the same p’. G0 under anisotropic consolidation is greater than that underisotropic consolidation, except for the case of K≥3.0. In addition, G0 in the compressional
consolidation (K<1.0) can have a greater value than that in the extensional consolidation(K>1.0). However, those anisotropic stress effects on G0 were not significant for theunsaturated cohesive soil because of the effects of matric suction.
2. Shear strain greater than 0.03%: The trends of Gsec are changed from the above-mentioned
trends of G0 through the increase of shear strain. With extensional consolidation (K and
Knet>1.0), Gsec is greater than that in isotropic consolidation (K and Knet=1.0). With
compressional consolidation (K and Knet<1.0), Gsec is less than that in isotropic consolidation.
The values of G0 under anisotropic stress conditions are normalized successfully using the void
ratio function (2) and the stress function (4) for saturated soil or the stress function (3) for
unsaturated soil. Therefore, G0 under different stress conditions can be inferred from a particularG0, e.g., in an isotropic stress condition, using those functions.

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