WEEK 9
Soil Behaviour at Small Strains: Part 2
13. Stiffness anisotropy and influencing factors
13-1. General and cross-anisotropic elasticity
We have already reviewed a compliance matrix (i.e. the inverse of a stiffness matrix) of an
isotropically elastic material in Week 2 (the prime indicating the effective stress is omitted
here for simplicity).
∆
∆
∆
∆
∆
∆
+
+
+
−−
−−
−−
=
∆
∆
∆
∆
∆
∆
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
E
τττσσσ
νν
ννν
νννν
γγγεεε
)1(200000
0)1(20000
00)1(2000
0001
0001
0001
1
For isotropic elasticity, only 2 moduli are independent (either two of E, G, K, ν, etc.).
If you look up the entry “shear modulus” in Wikipedia, a quite useful table for conversion
between the moduli is available: http://en.wikipedia.org/wiki/Shear_modulus
(From Wikipedia)
However, soils are rarely isotropic. They are normally anisotropic, reflecting internal
structure (remember? Structure = fabric + bonding, according to Mitchell’s (1976) definition)
developed during and after sedimentation. If a soil is deposited uniformly under the gravity,
we have a good reason to assume that it is cross-anisotropic; i.e. it is anisotropic in vertical
cross-sections but isotropic in horizontal cross-sections (see next page). If such
environment is not the case, a soil may be completely anisotropic (i.e. general elasticity).
1
Independent number of moduli in elastic material:
Without any consideration, number of the elastic moduli seems to be 36 (= 6 x 6).
However, a thermodynamic consideration (for example, p.99, Love, 1934) requires that the
matrix is symmetric. This condition reduces number of independent moduli in general
elasticity to 21.
Considering a cross-anisotropically elastic medium, let us assume that the z-direction is the
axis of symmetry. The distinction between the x- and y-axes needs to disappear. Then,
∆
∆
∆
∆
∆
∆
=
∆
∆
∆
∆
∆
∆
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
CC
CC
τττσσσ
γγγεεε
6661
1611
LL
MOM
MOM
LL
∆
∆
∆
−−
−−
−−
∆
∆
∆
y
x
vvhhhhh
vvhhhhh
y
x
EEE
EEE
EEE
σσσ
νννννν
εεε
0001
0001
0001
Here you find 7 moduli, but the following relationships exist, reducing number of
independent moduli in cross-anisotropic media to 5.
Note that the notation adopted here is for general continuum mechanics. In soil mechanics,
the elastic constitutive relationships can also be expressed in terms of effective stress. The
elastic moduli defined in terms of total and effective stresses are not generally identical.
See Appendix and Assignment 3.
2
∆
∆
∆
∆
−−=
∆
∆
∆
∆
zx
yz
xy
z
vh
vh
hh
vhhvhhv
zx
yz
xy
z
G
G
G
EEE
τττσνν
γγγε
100000
010000
00/1000
0001
hhvvvh EE // νν =
)1(2/ hhhhh EG ν+= This is the same expression of shear modulus
as in isotropic elasticity (see the table in the previous
page). This is because isotropy holds in the horizontal
plane.
Cross-anisotropic fabric in sand
A pioneering study by Oda (1972) looked into microscopic arrangements of sand grains in
pluviated sand. The example shown here is from a more recent study by Yang et al. (2008).
Note the coincidence of the vertical axis and the axis of cross-anisotropy. Such coincidence
is often observed but not always true.
Horizontal
Vertical
Horizontal
The right figure shows an example confirming
granular soils’ isotropy in a horizontal plane
(Hoque et al., 1996).
A microscopic particle-by-particle study is
difficult in clays given its scale (although
we can visualise their microstructure).
However, anisotropy is usually observed also
in clay in terms of stiffness, strength and
permeability.
3
Vertical cross-section: Anisotropic Horizontal cross-section: (lLargely) Isotropic
Distribution of long-axis of Toyoura sand’s particles after deposition (Yang et al., 2008)
What does this equation of cross-anisotropy mean?
Let us think cases of uniaxial compression (change in only one normal stress), assuming
that the cross anisotropy’s symmetry axis coincides with the vertical axis (z-axis), following
the example in the previous page.
∆
∆
∆
∆
∆
∆
−−
−−
−−
=
∆
∆
∆
∆
∆
∆
zx
yz
xy
z
y
x
vh
vh
hh
vhhvhhv
vvhhhhh
vvhhhhh
zx
yz
xy
z
y
x
G
G
G
EEE
EEE
EEE
τττσσσ
νννννν
γγγεεε
100000
010000
00/1000
0001
0001
0001
zσ∆z
v
zE
σε ∆=∆1
xσ∆
x
h
hvz
Eσ
νε ∆−=∆
For simple shear,
4
Uniaxial compression in x-direction
z
v
vhx
Eσ
νε ∆−=∆ z
v
vhy
Eσ
νε ∆−=∆
xσ∆
x
h
xE
σε ∆=∆1
x
h
hhy
Eσ
νε ∆−=∆
Uniaxial compression in z-direction
)( hy
)( vz
)( hx
hh
xy
xyG
τγ
∆=∆
Shear: xy-direction
vh
zxzx
G
τγ
∆=∆
Shear: zx-direction
vh
yz
yzG
τγ
∆=∆
Shear: yz-direction
13-2. Measuring anisotropic stiffness
For the moment let us assume cross-anisotropic soil with the axis of symmetry coinciding
with the vertical, and think how each modulus can be measured by different testing
methods.
(i) Triaxial compression
Same as the previous example of uniaxial
compression. By (locally) measuring the
vertical and horizontal strains against vertical
loading (∆σz), Ev and νvh are obtained.
(ii) Triaxial extension
Three independent moduli
appear, for measurement of
two variables (vertical and
horizontal strains), so none
of the moduli can be obtained.
zσ∆
z
v
vhx
Eσ
νε ∆−=∆ z
v
vhy
Eσ
νε ∆−=∆
z
v
zE
σε ∆=∆1
hhh
xE
σν
ε ∆−
=∆1
z
h
hvz
Eσ
νε ∆−=∆
2
hh σν
ε ∆−
=∆1
)( yxh σσσ ∆=∆=∆
What if a specimen is set in the
cell laid horizontally?
(iii) Simple shear / torsional shear
Hollow cylinder torsion shear apparatus
tests a specimen which has an inner
cavity in a triaxial cell. By applying
torque and measuring the torsion,
Gvh is obtained.
In this apparatus, the inner and outer
pressures and the axial force can
also be controlled. With suitable
measurement of corresponding strains,
this apparatus allows determining
all the five cross-anisotropic moduli
(e.g. Zdravkovic, 1996; Gasparre et al.,
2007).
5
hσ∆
h
h
xE
σε ∆=∆h
h
hhy
Eσε ∆=∆
hσ∆
vhτ∆
vhvhvh G/τγ ∆=∆
(iv) Bender elements and cross-/down-hole methods
Vhh: Horizontally propagating,
horizontally polarised wave velocity
Vvh : Vertically propagating,
horizontally polarised wave velocity
Vhv : Horizontally propagating,
vertically polarised wave velocity
In field,
Cross-hole method
Down-hole method
IncidentallyG if Ghh is obtained from bender element tests, (Page 2).
ρρ // hvvhhvvh GGVV ===
ρ/hhhh GV =
hvhh GG ,
vhG
12/ −= hhhhh GEνIncidentallyG if Ghh is obtained from bender element tests, (Page 2).
Combined with triaxial extension, Eh and νhv are determined. So if a triaxial specimen is
fitted with vh & hh or hv & hh bender elements, all the five cross-anisotropic moduli are
determined (Kuwano et al., 2000; Lings et al., 2000; Gasparre et al., 2007). But this
complicated technique remains mainly in the research sphere so far.
In practice, it is important to remember that, when you encounter data showing different
magnitudes of shear modulus for a same soil, it may not be solely due to experimental
errors. It may be because of anisotropy. You need to check the detail of the employed
methods.
6
12/ −= hhhhh GEν
13-3. Factors influencing soil stiffness
Among many factors influencing soil stiffness, the following three has been identified from
early times; stress, density and over-consolidation ratio (OCR). Now, let us look at some
equations for expressing soils’ stiffness. Let us forget about anisotropy for a moment.
Traditionally, Gvh among other moduli has been under extensive research, as it is directly
relevant to horizontal seismic motions.
More generally, these equations are expressed as
Where pa is the atmospheric pressure (needed just to make the equation unit-independent).
So the three factors raised above are incorporated in this single equation.
5.02
)[psi] (1
)17.2(2630[psi] v
e
eG σ ′
+−
=(Hardin
& Richart, 1963)
nn
a
k pOCRefSG )()( 1 σ ′⋅⋅= −
5.02
)[psi] (1
)97.2(1230[psi] v
e
eG σ ′
+−
=
Round-grained granular soils:
Angular-grained granular soils:
5.02
)[psi] (1
)97.2(1230[psi] v
kOCRe
eG σ ′
+−
=Clays: (Hardin & Black, 1969)
So the three factors raised above are incorporated in this single equation.
Some researchers argue (e.g. Rampello et al., 1994; Viggiani & Atkinson, 1995), however,
that use of all the three parameters in a stiffness equation ignores the fact that they are not
totally independent but related to each other. Looking at the idealised compression curves,
if you determine either two of the three, the rest is automatically determined. So, the
argument goes, the above expression has redundancy. They proposed a following form,
removing f(e);
where pr is the unit pressure. Think about pros
and cons about their argument.
7
σ ′ln
e
cσ ′σ
cOCR σσ ′′= /
*1* )(* nn
r
m ppOCRSG ′⋅= −
13-4. Inherent anisotropy and induced anisotropy
There are two components in soils’ anisotropy (not only in stiffness, but also in strength,
permeability, etc.); inherent and induced components.
Inherent anisotropy: Anisotropy deriving from its inherent structure
Induced anisotropy: Anisotropy deriving from anisotropically applied stress (i.e. stress-
induced anisotropy) or anisotropically developed strain (strain-induced
anisotropy)
It is not always easy or meaningful to separate these. But in any case, let us start with
stress-induced anisotropy first.
(i) Stress-induced anisotropy in shear moduli
Roesler (1979) noted that shear modulus Gij is
dependent on the effective normal stresses in
the i- and j-directions only. Hardin & Blandford
(1989) then proposed;
jiji n
j
n
i
nn
a
k
ijij pOCRefSG )()()(1 σσ ′′= −−
ij
ij
ijG
τγ
∆=∆
Shear: ij-direction
j
k
i
Stress-inducedInherent
Many studies (Hardin & Blandford, 1989;
Jamiolkowski et al., 1994; Belotti et al., 1996)
indicate or assume that ni = nj.
Examples:
Six natural Italian clays: ni = nj. = 0.20-0.29
(Jamiolkowski et al., 1994)
Ticino sand: ni = nj. = 0.224-256
(Belotti et al., 1996)
8
Example of the Pisa Clay
(Jamiolkowski et al., 1994):
As void ratio function,
is used.
)6.12.1( )( −== − xeef x
(ii) Stress-induced anisotropy in Young’s moduli
In a similar manner as for the shear moduli, the stress-dependency of Young’s moduli is
proposed by Tatsuoka and his co-workers as
where the subscript 0 indicates a reference state.
Shown here are the triaxial test data on
large, prism-shaped Ticino Sand specimens
by Hoque et al. (1996)
vn
v
vvv EE
′′
=0
0 σσ hn
h
hhh EE
′′
=0
0 σσ
9
σv’ [kPa]
Ev/ f(
e) [MPa]
Eh
/ f(
e) [MPa]
σh’ [kPa]
Further examples of stress-induced anisotropy
The results shown were obtained by HongNam and Koseki (2005), who conducted triaxial
and hollow cylinder tests on the Toyoura Sand. Note the following
- Ev (expressed as Ez in their notation) is proportional to (m = 0.44-0.48)
- Ev is independent of
- Gvh (expressed as Gvθ in their notation) is proportional to (n = 0.444-0.495)
- Gvh is independent of
m
v )(σ ′
n
hv
5.0)( σσ ′⋅′vhτ
vhτ
10
(HongNam and Koseki; 2005)
Inherent anisotropy
The term “inherent anisotropy” is often mentioned in soil mechanics, but its meaning is not
always clear, because it is difficult to agree upon what the “inherent state” should be. The
soil we see in the ground has already been consolidated to some extent, thus experiencing
straining in the geological time-scale (having strain-induced anisotropy). In addition, the in-
situ stress is generally not isotropic, whether K0-conditions apply or not (thus having stress-
induced anisotropy). Then, where can we encounter the inherent anisotropy?
A realistic view would be to consider inherent anisotropy as that seen at isotopic stress
states (that is, to include the strain-induced component as part of inherent one). In most
soils, strong stiffness anisotropy is seen even at isotropic stress states.
The example shown here is by Jovicic and
Coop (1997), who conducted bender element
tests to measure Gvh and Ghh in the London Clay.
11
Natural samples Reconstituted samples
(Jovicic and Coop, 1998)
In-situ anisotropy
In-situ stiffness anisotropy reflects combined effect of ‘inherent’ and stress-induced
anisotropy.
Example of the London Clay at Heathrow Terminal 5 (Gasparre et al., 2007):
Note how heavy over-consolidation and high K0-values (see Page 9, Week 7) has led to
large ratios of Ghh / Gvh and Eh / Ev .
40
30
20
10
0
Dep
th b
elo
w g
roun
d level [m
]
0 200 400
Young's Moduli [MPa]
Ev' (TX)
Ev' (HCA)
0 100 200
Shear moduli [MPa]
Gvh (BE)
Ghh (BE)
0 100 200
Bulk modulus, K [MPa]
London Clay
Gravel
12
50
Ev' (HCA)
Eh' (TX)
Eh' (HCA)
Ghh (BE)
Gvh (RC)
Gvh (Static)
50
40
30
20
10
0
De
pth
be
low
gro
un
d le
ve
l [m
]
-0.5 0 0.5 1 1.5
Poisson's ratios
TX
νvh'
νhh'
νhv'
HCA
νvh'
νhh'
νhv'
1 2 3
Modulus ratios
Eh'/Ev' (TX)
Eh'/Ev' (HCA)
Ghh/Gvh (TX)
Ghh/Gvh (HCA)
References
Bellotti, R., Jamiolkowski, M., Lo Presti, D.C.F. and O’neill, D.A. (1996) “Anisotropy of small
strain stiffness in Ticino sand,” Geotechnique 46(1) 115-131.
Gasparre, A., Nishimura, S., Anh-Minh, N., Coop, M.R. and Jardine, R.J. (2007) “The
stiffness of natural London Clay,” Geotechnique 57(1) 33-47.
Hardin, B.O. and Richart, Jr., F.E. (1963) “Elastic wave velocities in granular soils,” Journal
of the Soil Mechanics and Foundation Division, ASCE 89(SM1) 33-65.
Hardin, B.O. and Black, W.L. (1969) Closure to “Vibration modulus of normally consolidated
clay,” Journal of the Soil Mechanics and Foundation Division, ASCE 95(SM6) 1531-1537.
Hardin, B.O. and Blandford, G.E. (1989) “Elasticity of particulate materials,” Journal of the
Geotechnical Engineering Devision, ASCE 115(GT6) 788-805.
HongNam, N. and Koseki, J. (2005) “Quasi-elastic deformation properties of Toyoura Sand
in cyclic triaxial and torsional loadings,” Soils and Foundations 45(5) 19-38.
Hoque, E., Tatsuoka, F. and Sato, T. (1996) “Measuring anisotropic elastic properties of
sand using a large triaxial specimen,” Geotechnical Testing Journal 19(4) 411-420.
Jamiolkowski, M., Lancellotta, R. and Lo Presti, D.C.F. (1994) “Remarks on the stiffness at
small strains of six Italian clays,” Proceedings of the 1st International Conference on Pre-
failure Deformation Characteristics of Geomaterials, Sapporo, Japan, Vol.1 817-836.
Jovicic, V. and Coop, M.R. (1998) “The measurement of stiffness anisotropy in clays with
bender element tests in the triaxial apparatus,” Geotechnical Testing Journal 21(1) 3-10.
Kuwano, R, Connolly, T.M. and Jardine, R.J. (2000) “Anisotropic stiffness measurements in
a stress-path triaxial cell,” Geotechnical Testing Journal, GTJODJ 23(2) 141-157.
Lings, M.L., Pennington, D.S. and Nash, D.F.T. (2000) “Anisotropic stiffness parameters Lings, M.L., Pennington, D.S. and Nash, D.F.T. (2000) “Anisotropic stiffness parameters
and their measurement in a stiff natural clay,” Geotechnique 50(2) 109-125.
Love, A.E.H. (1934) “”The mechanical theory of elasticity,” Fourth Edition, Cambridge
University Press.
Mitchell, J.K. (1976): “Fundamentals of soil behavior,” John Wiley & Sons, Inc.
Oda, M. (1972)” Initial fabrics and their relations to the mechanical properties of granular
materials,” Soils and Foundations 12(1) 17-36.
Rampello, S. Viggiani, G. and Silvestri, F. (1994) “The dependence of G0 on stress state
and history in cohesive soils,” Pre-failure Deformation Characteristics of Geomaterials,
Sapporo, Japan, Vol.1, 1155-1160.
Roesler, S.K. (1979) “Anisotropic shear modulus due to stress anisotropy,” Journal of the
Geotechnical Engineering Division, ASCE 105(GT7) 871-880.
Viggiani, G. and Atkinson, J.H. (1995) “Stiffness of fine-grained soil at very small strains,”
Geotechnique 45(2) 249-265.
Yang, Z. X., Li, X.S. and Yang, J. (2008) “Quantifying and modelling fabric anisotropy of
granular soils,” Geotechnique 58(4) 237-248.
Zdravkovic, L. (1996) “The stress-strain-strength anisotropy of a granular medium under
general stress conditions,” PhD Thesis, Imperial College, University of London.
13