WEEK 10

Soil Behaviour at Medium Strains

14. Onset of yielding

14-1. What is yielding: Remember?

We have already studied what yielding means; if you don’t remember, refer back to Week 2.

It is defined by generation of plastic strain. According to the conventional elasto-plastic

concept that we studied in Weeks 3-5, soil behaviour was assumed to be elastic within the

yield surface. In many realistic cases, however, this is a wild idealisation. If the assumption

is true, soil would always exhibit elastic responses against cyclic loadings with a constant

stress amplitude. This week we will study real soil behaviour at medium strains, which are

larger than those to which the elastic stiffness is relevant, but smaller than those at which

the large-scale yielding leads to failure of soil.

In the previous weeks, we studied that

the stress-strain non-linearity appears

against very small strain (and hence stress)

increments. Does this signify yielding?

Remember, non-linearity per se does not

necessarily mean yielding.

q

Isotropic hardening

necessarily mean yielding.

p′

Stress path

1

(Addenbrook et al., 1997)

Example of

Evu – εa curves of

London Clay

Inside: Elastic?

14-2. Yielding at ‘relatively small’ to medium strains

The answer to the question seems, “yes”. See how the stress-strain relationships form a

loop even for small strain amplitudes (meaning irrecoverability of strains; hysteresis).

The τ – γ data shown below (from Iwasaki et al., 1978) were obtained with hollow cylinder

torsion shear apparatus (refer back to Week 9). A qualitatively same view is obtained for

the q – εq relationship that is obtained with triaxial apparatus, although the τ – γ and q – εqrelationships may be quantitatively different due to anisotropy.

q

p′

τor Cyclic loading with gradually

increasing stress-strain amplitude

q

qε

τor

γor

2

(Iwasaki et al., 1978)Definition of damping ratio

Change of hysteresis as strain

amplitude increases

Another effect of plasticity is dependency of the stress-strain relationship on loading

histories. Shown here are the (shear) strain contours drawn for loadings from different

origins along the K0-unloading path, obtained for the Magnus Till (Jardine, 1992). Note how

the strain contours align themselves along the unloading path. A change in the loading

direction generally leads to stiffer responses.

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(Jardine, 1992)

Another example by Atkinson et al. (1990) on reconstituted and over-consolidated London

Clay

Influence of stress-path direction changes on subsequent q – εq relationshps

4

(Atkinson et al., 1990)

Influence of stress-path direction changes on subsequent p’ – εp relationshps

14-3. Multiple yield surface and kinematic hardening: Concept

To interpret the yield at relatively small to medium strains, it is necessary to prepare more

than one yield surface. This concept of multi-surface plasticity is not limited for soils (for

example, Iwan, 1967). It is often combined with kinematic hardening rule.

The idea is to limit the elastic region to very small size and describe multiple distinct stages

of yielding by adopting multiple yield surfaces.

Kinematic yield

surface

q

p′

Boundary

surface

q

q

p′

Boundary

surface

(Yield surface)

q

Plasticity immediately

appears for

reloading/unloadingElasticity

5

qε qε

End of

elasticity

Onset of ‘large-scale’ yield

Critical StateEnd of

elasticity

&Onset of ‘large-scale’ yield

Critical State

qεlog

secG

0G

qεlog

secG

0GRealistic non-

linearity at small

strains may be

described

15. Behaviour under cyclic loading and principal stress axis rotations

15-1. Cyclic loading and accumulation of volumetric strain or excess pore water pressure

The cyclic and dynamic behaviour of soils will be discussed in detail in the lecture course

provided by Professor Miura, so this lecture allocates space for this topic less than it

deserves. The soil behaviour discussed under this topic is not necessarily limited to the

range of ‘medium’ strains, however you define it. For example, liquefaction eventually leads

to extremely large strains. However, the processes to reach such an ultimate state are

dominated by a sequence of medium-scale yielding, so probably it is appropriate to discuss

them here.

(i) Sands

Let us start with sands, with which liquefaction under cyclic loadings is always a great

concern (the significance of liquefaction phenomena will be discussed in detail in the other

e3 post-graduate course by given the lecturer, “Disaster Mitigation Geotechnology”).

The data shown here were obtained

for the Toyoura Sand in drained

cyclic simple shear, performed

in hollow cylinder apparatus.

See how volumetric strain

accumulates over number of

loading cycles.loading cycles.

How can we interpret, or how

can we not interpret this

behaviour from what we have

learnt previously?

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Shahnazari & Towhata (2002)

Simple shear in hollow cylinder

apparatus

In undrained cyclic simple shear,

the volumetric strain is forced to be

zero, but this time the pore water

pressure cannot be controlled. As a

result, the pore water pressure

increases and p’ decreases.

The ultimate state can be

liquefaction.

How can we interpret the behaviour

under drained and undrained conditions

in a unified way?

Actually, you already know the (or, ‘an’)

answer; remember what you studied

in Week 3?

Undrained conditions:

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Effective stress-paths and shear

stress-strain curves of loose and

dense Fuji River Sand (Ishihara, 1985;

reproduced after Iai et al., 1991)

e

p′

Drained conditions:

∆e due to plastic straining

Undrained conditions:

Because of plastic straining due

to the cyclic loading, e wants to

decrease, but it cannot (∆e must

be zero). So p’ is forced to

decrease instead.

(ii) Clays

In Clays, cyclic loading also leads

to excess pore water generation

and hence reduction in p’.

It is not common, however, for p’

to reach zero and attain a state of

liquefaction. Literature does not

cite liquefaction in clays.

Towhata (2008) notes some

similarity between behaviour of

dense sands and clays.Pore water pressure changes during

cyclic simple shear of Kaoline

(Ohara & Matsuda,1988)

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Effective stress-path and shear stress-strain relationships of Eastern Osaka Clay

(Adachi et al., 1995)

15-2. Soil behaviour under rotation of principal stress axes

First of all, get familiar with rotations of principal stress axes in 2D.

σ

τ( )

xyx τσ ,

( )xyy τσ ,

1σ3σy

x

yσ

xσxyτ

PD (Pole with regard

to direction)1σ

1σ

σ

τ

( )xyx τσ ,

σσy

yσ

xσxyτ

1σ

If the stress state (i.e. σx, σy and τxy) is changed in such a way that the Mohr’s stress circle’s

centre (= (σx + σy )/2) and radius (= [[(σx - σy )/2]2+[τxy]2]0.5) are not changed, a pure rotation of the principal stress axes occurs. It means that only the directions of σ1 and σ3 change, but not their magnitudes.

If σ1 and σ3 do not change, q, t, p, s, etc.do not change either (ignore σ2 for themoment). So, according to the models

based on these ‘invariant’ quantities

would not predict any change in state.

You stay where you are. Is this realistic?

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σ

( )xyy τσ ,

1σ3σy

x

1σ

q or t

p or s

In what situations do rotations of the principal stress axes matter particularly?

In quite many a situation, actually.

Examples:

(Bjerrum,1973)

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Rotation of principal stress directions

due to embankment construction

(Jardine & Smith., 1991)

Cyclic rotation of principal stress

directions in seabed due to wave loading

(Ishihara & Towhata., 1983)

The rotation of principal stress

directions is accompanied by

increases in p’ and q.

The rotation of principal stress

directions occurs almost with constant

p’ and q.

15-2. Soil behaviour under rotation of principal stress axes

If we assume elastic behaviour,

(see p.11, Week 2)

or,

A strain increment is parallel to

the stress path.

However, the drained hollow

cylinder test results by Gutierrez

et al. (1991) indicate significant

plastic strains for any ‘rotational’

stress path. Towhata and Ishihara

(1985) demonstrated that even

liquefaction can be triggered by Increase of q, with p’ and principal stress

directions fixed

∆−∆

=

∆

∆−∆

xy

yx

xy

yx

G

G

τσσ

γεε 2/)(

/10

0/1

2/)( θσσ −= zX

2/θτ zY =z

θ

∆−∆

=

∆

∆−∆

xy

yx

xy

yx

G τσσ

γεε 2/)(1

liquefaction can be triggered by

pure rotation of the principal stresses.

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directions fixed

Rotation of principal stress directions, with p’

and q fixed

(Gutierrez et al.,1993)

References

Adachi, T., Oka, F., Hirata, T., Hashimoto, T., Nagaya, J., Mimura, M. and Pradhan, T.B.S.

(1995) “Stress-strain behavior and yielding characteristics of Eastern Osaka Clay,” Soils

and Foundations, 35(3) 1-13.

Addenbrooke, T.I., Potts, D.M. and Puzrin, A.M. (1997) “The influence of pre-failure soil

stiffness on the numerical analysis of tunnel construction,” Geotechnique 47(3) 693-712.

Atkinson, J.H., Richardson, D. and Stallebrass, S.E. (1990): “Effect of recent stress history

on the stiffness of overconsolidated soil,” Geotechnique 40(4) 531-540.

Bjerrum, L. (1973) “Problems of soil mechanics and construction on soft clays and

structurally unstable soils (collapsible, expansive and others),” Proceedings of 8th

International Conference on Soil Mechanics and Foundation Engineering, Moscow, Vol.3,

111-159.

Gutierrez, M., Ishihara, K. and Towhata, I. (1991): ”Flow theory for sand during rotation of

principal stress direction,” Soils and Foundations 31(4) 121-132.

Iai, S., Matsunaga, Y. and Kameoka, T. (1992): “Strain space plasticity model for cyclic

mobility,” Soils and Foundations 32(2) 1-15.

Ishihara, K. (1985) “Stability of natural deposits during earthquakes,” Proceedings of 11th

International Conference on Soil Mechanics and Foundation Engineering, San Francisco,

Vol.1, 327-376.

Ishihara, K. and Towhata, I. (1983) “Sand response to cyclic rotation of principal stress

directions as induced by wave loads,” Soils and Foundations 23(4) 11-26.

Iwan, W.D. (1967): “On a class of models for the yielding behavior of continuous and

composite systems,” Journal of Applied Mechanics 34(E3) 612-617.composite systems,” Journal of Applied Mechanics 34(E3) 612-617.

Iwasaki, T., Tatsuoka, F. and Takagi, Y. (1978): “Shear modulus of sands under cyclic

torsional shear loading,” Soils and Foundations 18(1) 39-56.

Jardine, R.J. (1992): “Some observations on the kinematic nature of soil stiffness,” Soils

and Foundations 32(2) 111-124.

Jardine, R.J. and Smith, P.R. (1991) “Evaluating design parameters for multi-stage

construction,” Proceedings of the International Conference on Geotechnical Engineering

for Coastal Development, Geo-coast ‘91, Vol.1, 197-202.

Ohara, S. and Matsuda, H. (1988) “Study on the settlement of saturated clay layer induced

by cyclic shear,” Soils and Foundations, 28(3) 103-113.

Shahnazari, H. and Towhata, I. (2002) “Torsion shear tests on cyclic stress-dilatancy

relationship of sand,” Soils and Foundations 42(1) 105-119.

Towhata, I. (2008) “Geotechnical earthquake engineering,” Springer-Verlag Berlin

Heidelberg.

Towhata, I. and Ishihara, K. (1985) “Undrained strength of sand undergoing cyclic rotation

of principal stress axes,” Soils and Foundations 25(2) 135-147.

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