A continuous hyperplasticity model for sands under cyclic loading

Puzrin and Houlsby 2001b present a simple model for the undrained behaviour of a cohesive material, essentially by gen-eralising the von Mises model.. The task of develop-ing an equivale

1 INTRODUCTION

Plasticity theory is established as the most important

framework for describing the behaviour of soils

un-der monotonic loading It is capable of describing

most of the salient features of soil behaviour, and

many soil models based on plasticity have been

de-veloped since its first application to constitutive

modelling of soils in the 1950’s However, it has

been less successful in describing soil behaviour

un-der repeated (cyclic) loading The principal problem

is that, as is now well known empirically, soils

ex-hibit elastic behaviour at only the smallest of strains,

and as the amplitude of strain cycling is increased,

the secant stiffness steadily reduces and the damping

increases This pattern of behaviour does not match

simply with plasticity theory, in which a finite

elas-tic region is a fundamental part of the theory

Plasticity has been modified in a variety of ways

to cope with this problem, with the two main

ap-proaches being multi-surface plasticity and bounding

surface plasticity Of these multi-surface plasticity

has more justification, since bounding surface

plas-ticity cannot describe the well-established effects of

immediate stress history Multi-surface plasticity

can, however, be rather cumbersome

An alternative is the “continuous hyperplasticity”

approach, Puzrin and Houlsby (2001a) This may be

thought of as a variant of the multi-surface approach,

in which the process is taken to its logical conclu-sion and an infinite number of surfaces are used Continuous variations of stiffness and damping can

be modelled An advantage of the continuous hyper-plasticity approach is that it is relatively compact mathematically The entire constitutive response is specified through just two scalar functionals, thus

avoiding the plethora of ad hoc assumptions that are

often encountered in complex soil models

The purpose here is to develop a simple model for the behaviour of an idealised frictional material (e.g

a sand) under cyclic loading Puzrin and Houlsby (2001b) present a simple model for the undrained behaviour of a cohesive material, essentially by gen-eralising the von Mises model The task of develop-ing an equivalent simple model for frictional behav-iour is complicated by the fact that no standard model for frictional behaviour (with the same level

of acceptance that the von Mises model for cohesive behaviour enjoys) is currently available

In this paper we limit our attention to triaxial stress states, and so we do not address the shape of the yield surface in the octahedral plane Neverthe-less, even with this simplification, there is no firmly established frictional model The principal difficulty lies in the treatment of the dilation of the sand Al-though a fixed friction and dilation angle could be

A continuous hyperplasticity model for sands under cyclic loading

G.T Houlsby

Department of Engineering Science, Oxford University, UK

G Mortara

Department of Mechanics and Materials, University of Reggio Calabria, Italy

ABSTRACT: Soils exhibit truly elastic behaviour at only very small strains, so that cycling at small to moderate strains involves hysteretic behaviour As the amplitude of cycling increases the secant modulus decreases and the damping ratio increases These facts are well established experimentally, but theories that successfully describe this behaviour are less well developed We present here a simple model for the behaviour of sand under cyclic loading, that is able to capture the main features of small-strain cycling An essential part of the model is that volume changes (or effective stress changes in the case of undrained loading) are modelled realistically The model is described using the “continuous hyperplasticity” framework Essentially this involves an infinite number of yield surfaces, thus allowing smooth transitions between elasticity and plasticity The framework allows soil models to be developed in a relatively succinct mathematical form, since the entire constitutive behaviour can be determined through the specification of two scalar functionals Dilation and compression is incorporated through the use of kinematic constraints, and dilation is accompanied by the development of anisotropy in the sand

used, we set ourselves here the more ambitious task

of creating a unified model for the same sand at

dif-ferent densities, so that variable dilation, variable

friction and the approach to the critical state must all

be modelled This inevitably leads to a model of

some complexity, although we have attempted to

minimise this here as far as possible The model

de-veloped is therefore intended as a basic model, to be

used as a starting point for more sophisticated

ap-proaches In particular we do not include here

pres-sure-dependent stiffness or (as noted above) the

gen-eralisation of the model in the octahedral plane

2 MODEL DESCRIPTION

The model described here is an extension of a

previ-ous single surface model (Houlsby, 1992) within the

continuous hyperplasticity framework This

ap-proach employs an infinite number of yield surfaces,

which are expressed in terms of an internal

coordi-nate η (see Puzrin and Houlsby, 2001a,b) In

prac-tice, however, the infinite number of surfaces have

to be replaced by a finite number N of surfaces We

label each surface n (1≤n≤ N), and the factor n N

plays the same role as η In the following we

pre-sent the model directly in terms of the finite number

of surfaces, as this requires less sophisticated

mathematics and leads more directly to the

imple-mentation It should be borne in mind, however, that

the underlying model involves an infinite number of

surfaces, and this can be obtained by replacing n N

by η, and by replacing summations by integrals

The model is formulated in terms of triaxial stress

and strain variables:

3

1

3 1 3

1

3

2 2

3

'

2

'

'

ε

− ε

= ε ε

+

ε

=

ε

σ

− σ

= σ

+

σ

=

q p

q p

(1)

Volumetric and deviatoric plastic strains related to

the n plastic mechanism are indicated as th α(n p) and

)

(n

q

α respectively The specification of two scalar

functions, a Gibbs energy function g or,

alterna-tively, a Helmholtz free energy function f

( ( 1 ) ( ) ( 1 ) ( ))

, ,

p

g

) ,

, ,

( p q (p1) (p N) (q1) (q N)

f

and a dissipation function d

) ,

, ,

,

,

(

) ( ) 1 ( ) ( )

1

(

) ( ) 1 ( ) ( ) 1 (

N q q

N p p

N q q

N p p

q

p

d

d

α α α α

α α α α

=

&

K

&

&

K

&

K K

(4)

is sufficient to define completely the constitutive be-haviour The following two functions are used here:

∑

=

⎟

⎠

⎞

⎜

⎝

+

+

−

−

=

N

n

n q n n

q n

p N

G

q K

p g

1

2 ) ( ) ( )

( ) (

2 2

2

3 '

1

6 2

'

(5)

∑

=

=

α

=

n

n q N

n

N

n M N

d N

d

1

) ( 1

' 1

1

where H (n) is the hardening modulus related to the

th

n mechanism and M is the value that the stress

ratio q / p' attains at critical state conditions The normalisation term 1/N in (5) and (6) makes the formulation independent on the number of surfaces

In the previous model (Houlsby, 1992) the energy function used was the Helmholtz free energy func-tion, while the dissipation function was formulated

in terms of strains However, the two energy func-tions are linked by the Legendre transformation

q

p q p g

f = + ε + ε and either g and f can be used

for describing the behaviour of material

The definition of appropriate constrains enables the introduction of dilation as well as anisotropy into the model (Houlsby, 1992):

C d n & p n c &q n d&q n (7)

0 1

1

) ( 1

)

⎠

⎞

⎜

⎝

−

=

=

N

n

n q N

n

n q

N a

The first constraint specifies that dilation is made up from isotropic (compressive) and anisotropic (dila-tive) parts given by functions β and c β respec-d tively The second constraint specifies the evolution

of the anisotropy parameter a, which varies between

1 + and 1− for positive and negative shearing The rate of evolution of anisotropy is determined by the constant A With dissipation specified, it is possible

to obtain the yield functiony(n) associated with each set of plastic strains through the degenerate special case of the Legendre transformation of d (n), which

is homogeneous of degree 1 in the rates:

0 ) ( ) ( ) ( ) ( ) ( ) ( )

q n q n p n p n n

d

where by definition:

) (

) ( ) ( )

( ) (

) (

) ( ) ( )

( ) (

n q

a a N

i d i d n

q

n q

n p

a a N

i d i d n

p

n p

C C

d

C C

d

α

∂

∂ Λ + α

∂

∂ Λ + α

∂

∂

= χ

α

∂

∂ Λ + α

∂

∂ Λ + α

∂

∂

= χ

∑

∑

=

=

&

&

&

&

&

&

(10)

ln v

ln p'

ln B

ln Γ

ln D

1 λ

ln B

ln Γ

ln D

M n

N (1- a sgn(α. (n) q))

. (n)

N a sgn(αq )

where Λ(n d) and Λ are Lagrangean multipliers a

From the constraints (7) and (8) it follows that

( )

p

n

Λ and Λa =0 The generalised stresses are:

) ( ) ( )

(

)

(

) (

)

(

3 n q n

n

q

n

q

n

p

n

p

H q g

p g

α

−

= α

∂

∂

−

=

χ

= α

∂

∂

−

=

χ

(11)

and Ziegler’s orthogonality condition χ(n) =χ(n)

leads to the yield function in terms of stress:

sgn

N

n M H

⎠

⎞

⎜

⎝

= α

The yield surfaces exhibit kinematic hardening,

given by the term H(n)αn q

3 , where the expression for the variation of the hardening modulus is:

b n

N

n

h

⎠

⎞

⎜

⎝

⎛ −

= 1

)

(

(13)

with h and b being parameters of the model To

in-troduce the difference between compression and

ex-tension, the critical stress ratio M is given by:

sgn 2

q e

c e

M

where M e =r ec M c, and r is the ratio between the ec

critical stress ratios in extension M and compres- e

sion M As in the previous model (Houlsby, 1992), c

the values of β and c β depend on the state of the d

soil, defined by the distance between the current

specific volume and the critical state line, which is

assumed to be linear in a bi-logarithmic plot:

Γ

−

Γ

− β

=

β

+

β

−

−

−

=

β

λ λ

ln ln

ln ln

ln ln

ln ln

max

D

v N n

D B

D v

N

n

M

d

c

c

(15)

where βmax is the maximum rate of dilation while

λ

v , B , Γ and D are the specific volumes at a

refer-ence mean pressure p' ref for current, loosest, critical

and densest states respectively Thus, according to

constraint (7), the rate of dilation is given by

( )

max

) (

sgn ln

ln

ln ln

sgn 1

ln ln

ln ln

n q

n q n

q

n

p

a D

v N

n

a D B

D v

N

n

M

α Γ

−

Γ

− β

−

+ α

−

−

−

=

α

α

λ

λ

&

&

&

&

(16)

where the first term refers to compression and the second to dilation It is worth noting that for vλ =Γ the second term is always zero while the first one is null only when the term 1−asgn(α(q n)) vanishes Figure 1 shows the graphical interpretation of contractive and dilative terms in (16)

Figure 1 Density constants and graphical interpretation of compression and dilation rules (equation 16)

3 EXAMPLE ANALYSES

We illustrate the model by example analyses of ide-alised tests Although we do not compare these here with specific data sets, the patterns of behaviour cor-respond to those that are well-established empiri-cally The example calculations are carried out using the parameter values given in Table 1

Figure 2 shows a set of drained constant mean pressure tests on sands with different initial densi-ties As the index of density v increases (looser λ samples) the strength reduces and the samples change from being strongly dilative to contractive Although not apparent in Figure 2, the denser sands show a mild peak in the stress-strain response

Figure 3 shows the results for drained cycling over a constant range of strain The upper plots are

for a loose sample, which exhibits an accumulation

of compressive strain The resulting densification

causes a slight increase in stiffness of the response to

the cycles The lower plots show the equivalent for a

dense sand This time the sand dilates during the

and compression

0.8 Table 1: Example parameters for model

deviatoric strain εq (%) 0

40

80

120

160

deviatoric strain εq (%) -2

-1

0

1

εp

Figure 2 Response of the model in drained compression tests

on sands with different initial densities

deviatoric strain εq (%) -120

-80 -40 0 40 80 120

deviatoric strain εq (%) 0

2 4 6 8

εp

deviatoric strain εq (%) -150

-100 -50 0 50 100 150

deviatoric strain εq(%) -6

-4 -2 0 2

εp

p '= 100 kPa

vλ = 1.85

p '= 100 kPa

vλ = 1.65

p '= 100 kPa

vλ = 1.85

p '= 100 kPa

vλ = 1.65

Figure 3 Effect of relative density on drained cyclic tests

cycling, resulting in a slight reduction of stress in the

cycles as the material loosens

Stress-strain curves and effective stress paths for

undrained monotonic tests on samples of different

densities are shown in Figure 4 The loose sands

show a reduction in effective stress, whilst dense

sands show a strong increase in effective stress

Sands of medium density show a slight reduction of

mean effective stress before the increase This

pattern of behaviour is well known

Undrained cycling over a constant stress

ampli-tude is shown in Figure 5 for two densities of sand

A loose sand (upper plots) shows an initial reduction

in effective stress, after which the p ,′ plot settles q

into a characteristic “butterfly” shape, and the strain

amplitude also becomes constant There is a large

amount of hysteresis in each cycle The dense sand

shows a similar pattern, but the “butterfly” plot is

narrower, and the response both stiffer and with less

hysteresis

Figure 6 shows undrained cycling on a dense

sand at a higher stress range Although a stable

“butterfly” pattern is developed, note that this time

the dilation during each cycle means that there is a

net increase rather than a decrease of mean effective

stress

Finally, figure 7 shows the effect of the strain

amplitude in on the stress-paths of a loose sample in

constant strain amplitude undrained cycling For the

larger amplitude (∆εq =±2%) stabilization of stress

is achieved while for the other (∆εq =±1%) cyclic

liquefaction is obtained after just two cycles

4 CONCLUSIONS

A model for the cyclic behaviour of sand under

triaxial conditions has been presented The model

successfully describes typical trends of behaviour

for undrained and drained cycling, including typical

variation of volumetric behaviour for sands of

different densities

REFERENCES

Houlsby, G.T (1992) "Interpretation of Dilation as a

Kinematic Constraint", Proceedings of the Workshop on

Modern Approaches to Plasticity, Horton, Greece, June

12-16, ISBN 0-444-89970-7, pp 19-38

Puzrin, A.M and Houlsby, G.T (2001a) "A

Thermomechani-cal Framework for Rate-Independent Dissipative Materials

with Internal Functions", Int Jour of Plasticity, Vol 17, pp

1147-1165

Puzrin, A.M and Houlsby, G.T (2001b) "Fundamentals of

Kinematic Hardening Hyperplasticity", Int Jour of Solids

and Structures, Vol 38, No 21, May, pp 3771-3794

deviatoric strain εq (%) 0

100 200 300

mean stress p' (kPa) 0

100 200 300 400

deviatoric strain εq (%) 0

100 200 300 400

Figure 4 Response of the model in undrained compression tests on sands with different initial densities

-4 -3 -2 -1 0 1 2 deviatoric strain εq (%) -80

-40

0 40

80

mean stress p' (kPa) -80

-40

0 40 80

deviatoric strain εq (%) -80

-40

0 40

80

mean stress p' (kPa) -80

-40

0 40 80

deviatoric strain εq (%) -120

-80 -40 0 40 80 120

80 100 120 140 160 mean stress p' (kPa) -120

-80 -40 0 40 80 120

0 20 40 60 80 100 mean stress p' (kPa) -80

-40 0 40 80 120

0 20 40 60 80 100 mean stress p' (kPa) -80

-40 0 40 80 120

∆εq = ± 2%

∆εq = ± 1%

Figure 5 Simulation of stress controlled undrained cyclic tests

for loose and dense sand

Figure 6 Large stress controlled undrained cyclic tests for dense sand showing increase of mean effective stress

Figure 7 Effect of cyclic strain amplitude on the behaviour of loose samples