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