Critical State Models and Small-Strain Stiffness

8/2/2019 Critical State Models and Small-Strain Stiffness

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Critical State Models and Small-Strain Stiffness

G.T. HoulsbyOxford University, UK

Keywords: plasticity, hyperplasticity, critical state, small strain, stiffness

ABSTRACT: Critical state models are highly successful in reproducing the behaviour of soft claysas far as the major differences between elastic and plastic behaviour are concerned. However, theyare not able to model the non-linear behaviour of soils at small strains. The latter is now wellunderstood in terms of empirical evidence, but less satisfactory progress has been made ontheoretical modelling. The purpose of this paper is to draw together two diverse topics in soilmechanics: small strain stiffness and critical state modelling. This is achieved within a

thermodynamically based formulation of plasticity theory, using the continuous hyperplasticityapproach recently developed by Puzrin & Houlsby.

1 INTRODUCTION

Critical State Soil Mechanics (Schofield & Wroth 1968) is arguably the single most successfulframework for understanding the behaviour of soils. In particular, within this approach, completemathematical models have been formulated to describe the behaviour of soft clays. Of these themost widely used is the Modified Cam-Clay of Roscoe & Burland (1968). Such models arehighly successful in describing the principal features of soft clay behaviour: yield and relativelylarge strains under certain stress conditions and small, largely recoverable, strains under other

conditions. The coupling between volume and strength changes is properly described.The basic critical state models of course have their deficiencies. They do not, for instance,describe the anisotropy developed under one-dimensional consolidation conditions. Nor do they fitwell the behaviour of heavily overconsolidated clays. Perhaps most importantly they do notdescribe a wide range of phenomena which occur due to nonlinearities within the conventionalyield surface.

It is appropriate that this paper should be presented at this Symposium, since John Bookerhimself worked brie fly on the problem of integrating a realistic cyclic response within the yieldsurface of a Modified Cam-Clay model. The resulting paper (Carter et al. 1982) describes a modelwhich fits broadly into the bounding surface category. It incorporates a rule by which, on elasticunloading, the yield surface contracts slightly, and on elastic reloading it stays fixed. The mainresult is that cycles of constant stress amplitude produce small amounts of plastic strain in each

cycle. The model has, however, some drawbacks too, and does not seem to have been pursuedfurther.

2 HYPERPLASTICITY

Based on the work of Ziegler (1977), an approach to plasticity theory was developed by Houlsby(1981, 1982) in which models can be derived within the context of thermodynamics. In thisapproach two scalar potentials provide a complete description of the material, with no additionalassumptions being necessary. The potential functions were (a) the Helmholtz free energy and (b) a


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dissipation function, and were expressed in terms of kinematic variables (strains and internalvariables which are in fact identical to the plastic strains). From these functions the incrementalstress-strain response could be derived for particular models. Houlsby (1996) generalised this resultto derive the incremental response for any model specified this way (subject to a restriction on theform of the dissipation function).

Collins & Houlsby (1997) made extensive use of Legendre Transformations to reformulate theabove approach in terms of stresses (using the Gibbs free energy), and also demonstrated that theyield surface is a degenerate Legendre Transformation of the dissipation function. Houlsby &Puzrin (2000) introduced temperature and entropy to the formulation and explored more generalrelationships between internal energy, enthalpy, Gibbs free energy and Helmholtz free energy.They developed a number of alternative formulations, and demonstrated that the incrementalresponse can always be determined automatically if an energy function and the yield surface arespecified. More recently this approach has been termed hyperplasticity by analogy withhyperelasticity (Fung 1965) in which elastic behaviour is derived entirely from a potential.

In the following, plasticity models are derived within the above framework. The examples givenare entirely in terms of the stresses attainable in the conventional triaxial test, and are expressed in

terms of the usual triaxial effective stress variables ( )q p , . Because all stresses discussed areeffective stresses the mean effective stress will be written simply p rather than p . Conjugate to thestresses are the strains ( ),v . For the plastic strains we use q p , rather than p pv , , as theformer notation is consistent with that used in Collins & Houlsby (1997) and Houlsby & Puzrin(2000). In hyperplasticity (unlike conventional plasticity) use is made of quantities conjugate to theplastic strains q p , . These are called generalised stresses and are q p , or q p , . Infact the and variables are always numerically equal, but for some formal purposes have to betreated as separate variables.

The Gibbs free energy is expressed as q pq pgg = ,,, , and then it follows that:

pg

v

= , qg

= , p pg

= , qqg

= (1.a-d)

If the dissipation function is used it is expressed in the form q pq pq pd d = && ,,,,, , andthen:

p p

d = & ,

qq

d = & (2.a,b)

The dissipation function must a homogeneous first order function of the internal variable rates.

q p && , .Alternatively, if the yield surface is specified in the form 0,,,,, == q pq pq p y y , then:

p p

y=& ,

qq

y=& (3.a,b)

where is an undetermined multiplier.Formally these equations , together with the condition that q pq p = ,, are all that are

needed to specify completely the constitutive behaviour of a plastic material. The condition = is equivalent to Ziegler's "orthogonality condition", and is based on considerations of thermodynamics. A discussion of orthogonality is not appropriate here, but relevant issues areaddressed by Ziegler (1977), Collins & Houlsby (1997) and Houlsby and Puzrin (2000).


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The Modified Cam-Clay model can conveniently be expressed within the hyperplastic approachby defining the following functions. The main parameters are defined on Figure 1, where V is thespecific volume (total volume divided by volume of solids). The values of p , q and x p at the

reference (zero) values of strain and plastic strain are o p , 0 and xo p respectively. The (constant)shear modulus is G . The following equations can be simplified by noting the definition

( ) = p xo x p p exp .The Gibbs free energy is:

( ) ( )

++

= p xoq p

o pq p

Gq

p p

pg exp6

1log2

(4)

The first two terms define the elastic behaviour (the unusual first term resulting in bulk modulusproportional to pressure). The third term ensures that the internal variable plays the role of theplastic strain (Collins & Houlsby 1997). The final term defines the hardening of the yield surface. It

follows that:

po p

p pg

v +

=

= log (5)

qGq

qg +=

=3

(6)

== p xo

p p p p

gexp (7)

qgq

q == (8)

The dissipation function may be specified as:

222exp q p p

xo M pd +

= && (9)

which leads to:

q

p

M

ln(p )

ln(V )I s o t

r o p i c N C L

p x p x

C r i t i c a l s t a t e l i n e

Figure 1: Definitions of Modified Cam-Clay parameters


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222exp

q p

p p xo

p p

M p

d

+

==

&&

&

& (10)

222

2

expq p

q p xo

qq

M

M pd

+

== &&

&& (11)

Alternatively the yield function may be specified as:

0exp2

2

22 =

+= p xoq

p p M

y (12)

from which it follows that:

p

p

p y =

= 2& (13)

qq

q M

y ==

22& (14)

The derivation of the Modified Cam-Clay model from the above functions is not pursued here,but it can readily be verified that the above equations do indeed define incremental behaviourconsistent with the usual formulation of Modified Cam-Clay. The only exception is thatconsolidation and swelling lines are considered as straight in ( )V p ln,ln space rather than ( )V p ,ln space. The result is that and have slightly different meanings from their usual ones, and thatthe (variable) elastic bulk modulus is given by = pK rather than the more usual = pV K .

Butterfield (1979) argues that the modified form is more satisfactory.Note that the above choice of the energy functions is not unique: this topic was addressedbriefly by Collins and Houlsby (1997), and is discussed more fully in Appendix A.

Finally note that a particularly unsatisfactory feature of the original form of the Modified Cam-Clay model was the use of a constant shear modulus. A slight improvement can be achieved byintroducing a shear modulus proportional to pressure. This can readily be achieved by modifying

the term in Gq 62 in equation 4 to gpq 62 , where the parameter g is equal to pG on theisotropic axis. On differentiation a number of extra coupling terms appear in the equations, and itcan be verified that this form implies stress-induced anisotropic incremental elastic response fornon-isotropic stress states. This form implies a constant Poissons ratio ( ) ( ) + = gg 2623 onthe isotropic axis.

3 CONTINUOUS HYPERPLASTICITY

In the above example it is seen that a single set of internal variables (plastic strains) is associatedwith a single yield surface. It is straightforward to extend this approach to include two or more setsof plastic strains, each associated with an individual yield surface. Such models, expressed inconventional plasticity terms, have become popular as means of capturing approximately theeffects of stress history and of small strain stiffness. Typically models employ three yield surfaces,see for example Stallebrass and Taylor (1997). Puzrin and Houlsby (2000a) take this process to itslogical conclusion, generalising the concept of multiple yield surfaces to that of a field of an


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infinite number of yield surfaces each contributing an infinitesimal component of the plastic strain.In this approach the finite set of plastic strain variables is replaced by a continuous plastic strainfunction (or internal function). The Gibbs free energy and the dissipation, which had previouslybeen functions of the internal variable, now become functionals of the internal function. Looselyspeaking a functional is a function of a function. In the following, functionals are denoted by useof square brackets: [ ]K f . An approach using energy functionals, which has been termedcontinuous hyperplasticity allows the piecewise-linear incremental responses predicted bymultisurface models to be replaced by continuous smooth responses.

In the hyperplastic method, much emphasis is placed on the derivation of the constitutivebehaviour by differentiation of the energy and dissipation functions with respect to stresses,internal variables and internal variable rates. The concept of differentiation of a function withrespect to a variable can be extended to that of differentiation of a functional with respect to afunction by use of the Frechet differential. The application of the Frechet differential in this contextis discussed in the appendix to Puzrin and Houlsby (2000a).

Puzrin and Houlsby (2000b) use the continuous hyperplasticity method to formulate kinematichardening models. They consider in particular models in which the Gibbs free energy and

dissipation can be written in the form [ ] ( )( ) =1

0

,, d gg and [ ] ( )( ) =1

0

,, d d d && , where

is the internal coordinate. In the following any quantity which is a function of is denoted bya "hat" notation, e.g . ( ) above. For this case they showed that there are generalised stressfunctions ( ) ( )= which play an exactly analogous role to the generalised stresses = , andthat furthermore ( ) ( )= g and ( ) ( )= & d . A simple example of a one-dimensional model will suffice here to demonstrate the principle. Consider first a one-dimensionalmodel with a single yield surface and linear kinematic hardening. This can be expressed throughthe two functions:

22

22+= h

E g (15)

= &k d (16)The generalisation of this model to the case of an infinite number of yield surfaces proves to be

straightforward, and such a model can be defined by two potential functionals:

[ ] ( ) ( ) ( )( ) +=1

0

21

0

2

2

2, d

hd

E g (17)

[ ] ( ) = 10

d k d && (18)

The comparison of equations 15 and 16 with 17 and 18 reveals the relationship between the twomodels. The detailed stages of the development of this type of model are given by Puzrin andHoulsby (2000b). The dissipative generalised stress function ( ) is obtained as:

( )( )

( )( )== &sgn

k

d (19)


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so that the field of yield functions is given by

( ) ( )( ) 0 222 == k y (20)The generalised stress is defined by:

( )( )

( ) ( )==

hg

(21)

and the strain is given by

( ) +==

1

0

d E

g(22)

Combining equations 19, 21 and 22 allows the stress-strain curve for monotonic loading (from

zero initial plastic strain) to be expressed as:

( )

+=*

0

d h

k E

(23)

where * specifies the largest yield surface that has yet been activated, such that( ) ( ) === *** k .

Differentiation of equation 23 twice with respect to (using standard results for the differentialof a definite integral in which the limits are themselves variable) leads to the important result:

( )k khd

d

=

12

2(24)

so that the hardening function ( )h is uniquely related to the second derivative of the initial back-bone curve ( ) . For example, the hyperbolic stress-strain curve given by

( )( )( )

( )( )

+==

c E a

E c E ac 212

, where 50 E E a = , E is the initial stiffness and 50 E is the

secant stiffness to 2c= (see Figure 2a), is generated by the function

( )( )( )3

2

2

2 121

=

= k E

k a

d

d k kh

, so that ( ) ( )( )121 3

=

a E

h .

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 E / c

/ c

a 1

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-3 -2 -1 0 1 2

log10( E / c )

E s e c a n

t /

E

b

Figure 2: (a) Hyperbolic stress strain curve (for a = 5), (b) typical stiffness - log strain curve for soil


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Puzrin et al. (1999) discuss more complex forms of ( )h which result in realistic fitting of thetypical S-shaped curves of secant shear stiffness against the logarithm of shear strain as areobserved for soils (see Figure 2b). For the purposes of this paper, however, the simple form issufficient to illustrate the principles involved. The curve shown in Figure 2b is in fact derived from

the hyperbolic form above with 5=a .

4 MODIFIED FORMS OF THE ENERGY FUNCTIONALS

Puzrin and Houlsby(2000a,b) considered a Gibbs free energy of the form:

[ ] ( )( ) ( )

= d gg ijijijij ,,, (25)

and a dissipation function of the form:

[ ]( )

( )( )

=1

0

,,,, d d d ijijijijijij

&& (26)

They derived the following result from the Frechet differential of the energy functional:

[ ] ( )( )

( )

= d d

gd g ij

ij

ijijijijij

,,, (27)

where g indicates the derivative with respect to the function ij . By defining:

( ) ( ) ( ) =

&&&& sd g ijijijij (28)

an expression for the generalised stress function is obtained:

( )=ij

ijg

(29)

Similarly Puzrin and Houlsby (2000a) obtained:

( )=ij

ijd

&

(30)

In the cases considered below g takes a rather more complicated form that can be written as:

[ ] ( ) ( ) ( )( ) ( )ijijijijijijijijij

gd gggg ++= 41

0 321,,, (31)

where for convenience a variable =1

0

d ijij is also introduced. From the above it can be shown

that:

[ ] ( ) ( )( )( )

=

1

0

32

,,, d d

ggd g ij

ij

ijijijijijij (32)


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and:

( )( )ij

ijij

ijij

gd g

gg

+

+= 4

1

03

2 (33)

If the time rate of change of the Gibbs free energy is written:

( ) ( ) = &&&& sd g ijijijijijij1

0

(34)

and the definition of ij as a constraint equation:

( ) ( )( ) ( ) 01

0

1

021 ==+= d d ccc ijijijij (35)

it is then possible to define the generalised stresses in terms of the derivatives of an augmented

energy expression cg c+ , where c is a multiplier to be determined:

( ) ( )( )( )

( )

=ij

ijc

ij

ijijij

cgg

, 232 (36)

( )( )ij

cij

ijij

ijijcg

d gg

= 14

1

03

2 (37)

The dissipation is considered to be of the form:

[ ] ( ) ( )( ) =1

021 ,,,, d d d d ijijijijijij

&& (38)

from which the Frechet differential leads to the result:

( ) ( )( )

=ij

ijijijd

d &

,, 21 (39)

and it also follows that:

0==

ijij

d & (40)

5 COMBINING SMALL-STRAIN AND CRITICAL STATE BEHAVIOUR

A major criticism of critical state models is the fact that they describe inadequately the behaviourof soils at small strains. Coupled to this is a poor performance with respect to modelling cyclicloading, and no modelling of the effects of immediate past history. In order to remedy this situationthe benefits of the simple hyperplastic framework for describing kinematic hardening of an infinitenumber of yield surfaces is combined with the Modified Cam-Clay model. Following a similarapproach to the development from equations 15 and 16 to 17 and 18, the suggested expressions are:


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( )

( )

( )

( )( )

x

q x p x

q po

pd gp p

a

q pgp

q p p

pg

++

++

=

1

0

223

2

2

3

12

1

61log

(41)

( ) +=1

0

222 d M pd q p x&& (42)

where the definitions =1

0

d and

= p xo x p p exp have been introduced.

Note in the above that the stiffness factors for the kinematic hardening of the yield surfaces (theintegral term in equation 25) have been made proportional to preconsolidation pressure rather thanpressure. This has the advantage of avoiding elastic -plastic coupling, which alters the meaning of the internal variable (Collins and Houlsby, 1997). However, the presence of the preconsolidationpressure in these expressions considerably complicates the derivatives of the Gibbs free energyfunctional. In the following a linearised version of the continuous hyperplastic Modified Cam-Clayis therefore explored, although the derivation from equations 41 and 42 is set out in Appendix B.This avoids some of the coupling terms, and will serve as an example to illustrate the main featuresof the model. The suggested Gibbs free energy and dissipation expressions are:

( ) ( )( ) 22

3

121

62

21

0

22322 pq pq p

hd

GK

aq p

Gq

K p

g

++

++= (43)

( ) +=1

0

222 d M hd q p p&&

(44)

Comparison of equation 38 with equation 31 givesG

qK

pg

62

22

1 = , 12 =g ,

( )( ) 2

3

121

223

3q p GK

ag

+= ,

2

2

4 ph

g

= . From these it follows that

pK p

p

gv +=

= 1 , qG

qq

g +==

31 (45.a,b)

( )( ) cp p p

pcp

p p a

K cg +=

=

121

323 (46)

( )( ) cqqq

qcq

qq a

Gcg +=

= 12

13

323 (47)

cp p p

pcp

p p h p

cg p =

= 14 (48)


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cqq

qcq

qq q

cgq =

= 4 (49)

( ) 222

q p

p p

p p

M hd

+

== &&

&

& (50)

( )222

2

q p

q p

qq

M

M h

d

+

=

=

&&

&

& (51)

The yield surface is therefore:

( )22

22 p

q p h

M y

+= (52)

To derive the incremental constitutive behaviour the above are differentiated further to give thefollowing, which also make use of = and 0== (see equation 40):

pK p

v += &&

& , qGq += &&

&3

(53.a,b)

( )( ) p p p

ha

K p

= &&&& 12

13

(54)

( )( ) qq a

Gq

= &&& 12

133

(55)

Finally the derivatives of the yield surface and the consistency condition are required:

p p

p y == 2& , q

qq

M

y == 2

2

& (56.a,b)

( ) p pqq

p p h M

y

+= &&

&& 22

2 22

(57)

Although it would be attractive to carry out calculations using the functions of directly, atleast with the presently available software it is necessary first to discretize these functions in termsof a finite set of values. The internal function ( ) is therefore represented by a set of n internalvariables i , ni K1= . The result is that the field of an infinite number of yield surface is thusapproximated by a finite number of yield surfaces. In the calculations presented below the field isrepresented by 10 yield surfaces. For the purposes of numerical calculation the continuoushyperplasticity models therefore have much in common with multi-surface plasticity. There is,however, a significant difference in that the internal variables clearly play the role of approximating the underlying internal function. This opens up possibilities in the future of adoptingmore sophisticated numerical representations of the function.

The detailed implementation of the above equations for numerical calculations is addressed inAppendix B.


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6 EXAMPLES

Some example calculations illustrate the features of the model described above. All the followingcalculations use the constants 2000=G , 2000=K , 200=h , 1= M , 1.1=a . The units arearbitrary, but note that for a model for a specific soil the constants G , K and h each have thedimensions of stiffness, whilst M and a are dimensionless.

Figure 3 shows the behaviour of the soil in an isotropic consolidation test, first loaded to100= p , then unloaded to 20= p and then reloaded. It can be seen that the unloading line is

curved, as is the reloading line. When the preconsolidation pressure is reached there is no sharpyield, but instead the reloading curve merges smoothly with the virgin consolidation line. This typeof behaviour is of course a feature of many soils. The curvature of the unloading and reloadinglines (and hence the openness of the hysteresis loop), is controlled by the form chosen for thekernel function in the expression for the Gibbs free energy (equivalent to ( )h in equation 17). Inthis particular model the curvature is controlled by the value of the parameter a . As the curvature isreduced the preconsolidation point on reloading becomes more sharply defined in the response.

Figure 4 shows (bold line) the stress path for a sample which is first consolidated isotropicallyto 100= p and then sheared undrained. The undrained stress path is typical of a normallyconsolidated clay. Also shown on the figure are the positions of the ten yield surfaces used in thecalculation. Note that the yield surfaces overlap. There is a widely held misconception that multipleyield surfaces should be "nested", i.e. non-overlapping, with this condition usually being attributedto Prevost (1978), but in fact there are no strong reasons why this needs to be the case.

Note two features of the field of yield surfaces. Firstly the small surfaces are "dragged" by thestress point, and therefore in essence provide a coding of the past stress history. Secondly note thatthe largest yield surface has never been engaged by the stress point. This latter effect is due to asubtlety of the interaction between the yield surfaces. As plastic volumetric strain occurs on theinner surfaces, the size of all the yield surfaces increases. The result is that during isotropicconsolidation, once sufficient of the inner surfaces have been engaged, the surfaces expand at asufficient rate that the outer surfaces are never encountered by the stress point.

Figure 5 shows the undrained deviator stress against deviator strain curve for the same test as inFigure 4, but continued with two unload-reload cycles. Not only is there hysteresis on unloadingand reloading, but there is also some accumulation of shear strain during the cycles.

One of the most important features of the model described here is its ability to capture the

0

1

2

3

4

5

6

7

8

90 20 40 60 80 100 120 140 160

Mean stress p

V o

l u m e

t r i c s t r a

i n v

Figure 3: Consolidation curve for linearised continuous hyperplastic Modified Cam-Clay


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effects of recent stress history. Figure 6 gives the definitions of a series of stress points. In thefollowing we will consider the results of four undrained tests from point B, which is at anoverconsolidation ratio of 2. However, each of the tests will have been preceded by a differentimmediate past history, so that point B will have been approached by a drained loading from thedirection of point C, A, D or E respectively in the four tests. Such a group of tests was used byAtkinson et al. (1990) to demonstrate the importance of immediate past stress history. Also shown

on the figure is the position of the yield surface that would exist for a simple Modified Cam-Claymodel: all the stress histories lie within this surface, so that the simple model would predict purelyelastic response for all four cases.

Figure 7 shows the positions of the yield surfaces after the four different stress histories. Thedragging of the surfaces behind the stress point is clear. Figure 8 shows the resulting undrainedstress-strain curves when the sample is sheared from point B. The four responses are quitedifferent. The sample which has just suffered the most severe stress reversal (c) shows the stiffestresponse, whilst the sample in which the stress path is a smooth continuation of the previous path(d) shows the most flexible behaviour. The two cases where the stress path turns through a 90 o

-6 0

-4 0

-2 0

0

20

40

60

0 20 40 60 80 100 120 p

q

Figure 4: Undrained stress path and field of yield surfaces

0

5

10

15

20

25

30

35

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Deviator strain

D e v

i a t o r s t r e s s q

Figure 5: Undrained stress-strain curves


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angle show an intermediate response. At large strain the response for case (b) is softest because of the swelling that has occurred during the unloading to point A.

The above behaviour is shown more clearly in Figure 8, which shows the same data in terms of the normalised secant stiffness GG s , where = 3qG s against the deviatoric strain (on alogarithmic scale). Each of the tests shows the characteristic "S-shaped" curve which is commonly

q

p A B C

E

D

O

Figure 6: Stress paths for clay with OCR = 2

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120 p

q(a)

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120 p

q(b)

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120 p

q(c)

-60

-40

-20

0

20

40

60

0 20 40 60 80 100 120 p

q (d)

Figure 7: Fields of yield surfaces after stress paths (a) OACB, (b) OACAB, (c) OACBDB, (d) OACBEB


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observed. The high stiffness is maintained longest by the sample with the complete stress reversal.The much lower stiffness for the sample with a smoothly continuing stress path is apparent.

The past stress history affects not only the stiffness of the sample. Figure 10 shows theundrained stress paths for the four tests. In this case the most notable differences are between case(a), where the immediate past history involved a reduction of mean stress, and case (b), where itinvolved an increase of stress. In the first case the subsequent undrained path first involves anincrease in mean stress, and in the second it involves a reduction. Thus the model predicts that

effective stress paths (and hence pore pressures) during undrained behaviour would depend onimmediate past history. This behaviour is exactly as observed by Stallebrass and Taylor (1997)The above examples illustrate that a model based on the continuous hyperplasticity approach is

capable of capturing many of the important features of soil behaviour at small strains, whilst beingconsistent with the ideas of critical state soil mechanics for larger strains. Models which achievethis have of course been published before, and often make use of the multiple surface plasticityconcept. The benefits of the continuous hyperplastic approach lie, however, in the extremelycompact representation of the models. All the results presented in figures 3 to 10 arise from amodel which is specified entirely by two equations (43 and 44) together with standard procedures.No additional ad hoc assumptions or rules are required.

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4deviator strain

d e v i a

t o r s t r e s s q

c

(b)

d

a

Figure 8: Stress-strain curves for clay at OCR of 2 after different immediate past stress histories

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0001 0.001 0.01 0.1 1Deviatoric strain (log scale)

G / G 0

d

(b)a c

Figure 9: Normalised stiffness against log(strain) for clay at OCR of 2 after different immediate past stresshistories


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7 CONCLUSIONS

A model for soil in triaxial space has been described in which realistic small-strain behaviour iscombined with a critical state model at medium to large strains. The model is formulated usingcontinuous hyperplasticity in which the entire constitutive behaviour is derived from two scalarfunctionals. The model effectively uses an infinite number of elliptical yield surfaces. It is defined,however, using only five material parameters ( K , G , M , h , a ) (plus two arbitrary parameters o p and xo p defining the reference state). The emphasis here is, however, on the mathematicalframework used to define the model, and not on the specific details.

8 ACKNOWLEDGEMENTS

This work arises directly from many fruitful discussions with Dr Sasha Puzrin of the Technion,Haifa, Israel.

9 REFERENCES

Atkinson, J.H., Richardson, D. and Stallebrass, S.E. 1990. Effect of recent stress history on the stiffness of overconsolidated soil. Gotechnique , 40(4): 531:540

Butterfield, R. 1979. A natural compression law for soils (an advance on e-log p '). Gotechnique , 29(4): 469-480

Carter, J.P., Booker, J.R. & Wroth, C.P. 1982. A Critical State Model for Cyclic Loading, in Soil Mechanics -Cyclic and Transient Loads , Pande, G.N. and Zienkiewicz, O.C. (Eds), Chichester: John Wiley, 219-252

Collins, I.F. & Houlsby, G.T. 1997. Application of Thermomechanical Principles to the Modelling of Geotechnical Materials, Proc. Royal Society of London, Series A , 453: 1975-2001

Fung, Y.C 1965. Foundations of Solid Mechanics . New Jersey: Prentice Hall.Houlsby, G.T. 1981. A Study of Plasticity Theories and Their Applicability to Soils , Ph.D. Thesis, University

of CambridgeHoulsby, G.T. 1982. A Derivation of the Small-Strain Incremental Theory of Plasticity from

Thermomechanics, Proc. Int. Union of Theoretical and Applied Mechanics (IUTAM) Conf. on Deformation and Flow of Granular Materials , Delft, Holland, August 28-30: 109-118

0

5

10

15

2025

30

35

40

0 20 40 60 80p

q

(d)(c)ab

Figure 10: Stress paths for clay at OCR of 2 after different immediate past stress histories


16/18

16

Houlsby, G.T. 1996. Derivation of Incremental Stress-Strain Response for Plasticity Models Based onThermodynamic Functions, Proc. Int. Union of Theoretical and Applied Mechanics (IUTAM) Symp. on

Mechanics of Granular and Porous Materials , Cambridge, 15-17 July, Kluwer Academic Pub.: 161-172Houlsby, G.T. & Puzrin, A.M. 2000. A Thermomechanical Framework for Constitutive Models for Rate-

Independent Dissipative Materials, International Journal of Plasticity , in press.Prevost, J.H. 1978. Plasticity theory for soil stress-strain behaviour, Proc. ASCE, Jour. Eng. Mech. Div ., 104

(EM5): 1177-1194.Puzrin, A.M. & Houlsby, G.T. 2000a. A Thermomechanical Framework for Rate-Independent Dissipative

Materials with Internal Functions, International Journal of Plasticity , in press.Puzrin, A.M. & Houlsby, G.T. 2000b. Fundamentals of Kinematic Hardening Hyperplasticity, International

Journal of Solids and Structures , in press.Puzrin, A.M., Houlsby, G.T. & Burland, J.B. 1999. Thermomechanical Formulation of a Small Strain Model

for Overconsolidated Clays, Submitted to Proc. Royal Society of London, Series A. Roscoe, K.H. & Burland, J.B. 1968. On the Generalised Behaviour of Wet Clay, in Engineering Plasticity ,

Heyman, J. and Leckie, F.A. (Eds), Cambridge University Press, 535-610Schofield, A.N & Wroth, C.P. 1968. Critical State Soil Mechanics , London: McGraw Hill.Stallebrass, S.E. and Taylor, R.N. 1997. The development and evaluation of a constitutive model for the

prediction of ground movements in overconsolidated clay. Gotechnique , 47(2): 235-254Ziegler, H. 1977. An Introduction to Thermomechanics . Amsterdam: North Holland (2nd edition 1983).

APPENDIX A: NON-UNIQUENESS OF THE ENERGY FUNCTIONS

Collins and Houlsby (1997) demonstrated that the Modified Cam-Clay model could be derivedfrom either of two different pairs of Gibbs free energy and dissipation functions. This raises theinteresting concept that, since the same constitutive behaviour can be derived from different energyfunctions, then conversely the energy functions are not uniquely determined by the constitutivebehaviour. The energy functions are not therefore objectively observable quantities. The casediscussed by Collins and Houlsby is a special case of the following more general result.

Consider a model specified by ( )= ,1gg and ( )= &,,1d d . It follows that = 1g ,= 1g and = &1d . Using = gives 011 =+ &d g .Now consider a model in which ( ) ( )+= 21 , ggg and ( ) ( )= && 21 ,, gd d . In this

case we again have = 1g , but this time = 21 gg and= 21 gd & . However, using = again gives 011 =+ &d g . Thus identical

constitutive behaviour is given by the two models. Of course the models are only acceptable if both( ) 0,,1 > &d and ( ) ( ) 0,, 21 > && gd for all & . For typical forms of the dissipation

function in fact it often proves possible to find a function 2g which satisfies this condition.

APPENDIX B: DERIVATION OF CONTINUOUS HYPERPLASTICITY MODIFIED CAM-CLAY

From equations 41 and 42 the following can be obtained:

po gp

q p p

v +

=

2

2

6log , qgp

q +=3

(58.a,b)


17/18

17

( )( ) cp p

x p

p

a+

= 12

13

(59)

( )( ) cqqq gpa +

= 312

1

3

(60)

( )( )

( )( ) cp x

q x p x p pd

gp p

a p

+

=

1

0

223

2

3

121

(61)

cqq q = (62)

222

q p

p x p

M p

+

=

&&

&

(63)

222

2

q p

q xq

M

M p

+

=

&&

&

(64)

from which the field of yield surfaces can be derived as:

( ) 0

22

22 =

+= x

q p p

M y (65)

From the above the following can be derived:

( )( )

( )( )

( )( )

+ = p x xq x p x p pa pd

gp pa

p 12

12

312

1 31

0

223(66)

( )( ) qq

gpa

q = 3

121

3(67)

The derivation of the incremental form of the model is complicated by the presence of theintegral term in equation 66, but this can nevertheless be accommodated by a method very similarto that adopted in Appendix C.

APPENDIX C: A NOTE ON IMPLEMENTATION OF THE CALCULATIONS

The calculations for section 6 are carried out using the linearised form of the incremental equations.These can conveniently be cast in matrix form as described below. The method employed is farfrom the most efficient computationally, as it involves inversion of an unnecessarily large matrix.However, for development purposes it proves to be very convenient as it involves the minimum of mathematical manipulation. For moderate numbers of yield surfaces the computation time perincrement is small enough not to be inconvenient.

The matrix equation is expressed in the form:


18/18

18

=

0

00

0

00

0

0

1000000000

10

0000

0

0000000

0

00000000

0000100000

000010

0

000000

0

000000000

00000000

00000000

000000000

1

3

1

1

1

1

11

23

2

24

2

111

1

1

111

21

312

24

211

21

221

sn

s

en

n

n

n

e

nn

T

n

T

n

T n

T n

n

n

n

n

T T T T

y

y

C

y

d

d

y

d

d d

d

d

A A

y y y y

I gg I

y I

A A

y y y y

I gg

I

y I

I I I

I I g

C C

MM

L

L

L

L

MMMMOMMMMMMM

L

L

L

L

L

L

L

(68)

The first row shown symbolically (in fact two rows) define the control of stress or strain increment:suitable choice of 1C , 2C and 3C allow any permissible combination of stress or strain incrementsto be applied.

The second symbolic row (again in fact two rows) implements equations 53.a,b, and the thirdsymbolic row (again in fact two rows) implements the constraint equation 35.

The next four symbolic rows appear n times, once for each of the yield surfaces used in thecalculation. The first of these (again in fact two rows) implements the flow rule, equations 56.a,b.The second (again in fact two rows) implements equations 54 and 55 for the rate of change of thegeneralised stress. The third (a single row) implements the change in the value of the yieldfunction, equation 57: si y is the value of the i-th yield condition at the start of the increment and

ei y is the value at the end. The fourth row (a single row) implements the switch between elasticity

and plasticity. If 1=i A then 0 =i and the i-th yield surface is not active: this state is onlypermissible if 0 ei y . If 0=i A then 0 =ei y and the i-th yield surface is active: this state is only

permissible if 0 i . The calculation proceeds iteratively: at first it is assumed that all the yieldsurfaces are inactive and the increment is purely elastic. The end values of the yield surfaces arechecked, and the i A values adjusted to "switch on" active yield surfaces. Yield surfaces also laterbe switched off if the plastic multiplier goes negative. In all the cases explored here, a simpleiterative procedure converges very rapidly to a solution that can satisfy all the necessary criteria.

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Critical State Models and Small-Strain Stiffness