A Principles of Hyperplasticity Part 9 doc

It can be shown that for transient and cyclic loading, the continuous plastic formulation of the model produces stress-strain behaviour identical to that of a classical plasticity model

It can be shown that for transient and cyclic loading, the continuous plastic formulation of the model produces stress-strain behaviour identical to that of a classical plasticity model with an infinite number of kinematic harden-ing elliptical yield surfaces, each one with the associated flow rule and Ziegler’s translation rule

hyper-9.5.5 Concluding Remarks

The purpose of this section has been to demonstrate that a model for the small strain behaviour of soils, previously presented by Puzrin and Burland (1998) within classical plasticity concepts, together with some additional rules for han-dling stress reversals, can be expressed within a rigorous thermomechanical formulation Modelling of the small strain non-linearity of soils is one of the key challenges of current theoretical soil mechanics The establishment of a theoreti-cal framework within which such models can be achieved is regarded as an im-portant step towards a fuller understanding of soil behaviour A particular chal-lenge is to combine the modelling of soil behaviour at small and large strains, where the latter has been very successfully achieved within the context of critical state soil mechanics In the next chapter, we present a model which addresses this issue

Figure 9.5 Normalized stress-strain curves for uniform proportional cyclic loading

Applications in Geomechanics: Plasticity and Friction

10.1 Critical State Models

Critical State Soil Mechanics (Schofield and Wroth, 1968) is arguably the single most successful framework for understanding the behaviour of soils In particu-lar, within this approach, complete mathematical models have been formulated

to describe the behaviour of soft clays The most widely used of them is the

“Modified Cam-Clay” of Roscoe and Burland (1968) Such models are highly successful in describing the principal features of soft clay behaviour: yield and relatively large strains under certain stress conditions and small, largely recov-erable, 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 consoli-dation conditions Nor do they fit well the behaviour of heavily overconsolidated clays Most importantly, they do not describe a wide range of phenomena that occur due to non-linearities within the conventional yield surface Treatment of these phenomena within the hyperplastic framework will be presented in the next section; in this section, we show how the classical Modified Cam-Clay plas-ticity model can be derived

10.1.1 Hyperplastic Formulation of Modified Cam-Clay

In the following, we use the triaxial effective stress variables p qc,  defined in Section 9.2 Because all stresses discussed are effective stresses, the mean effec-

tive stress will be written simply p rather than pc Conjugate to the stresses are

the strains H H For the plastic strains, we use v, s D Dp, q, and the generalised stresses conjugate to them are F Fp, q or F Fp, q

The Gibbs free energy is expressed as g g p q , ,D D , and then it follows that v, s

v

g p

d

w

F

The dissipation function must be a homogeneous first-order function of the

internal variable rates D Dp,q

Alternatively, if the yield surface is specified in the form

q q

y

w

D /wF

where / is an undetermined multiplier We use / for the plastic multiplier here

rather than O, because the latter is usually used as one of the material

parame-ters in the Modified Cam-Clay model

Formally, these equations, together with the condition that

F Fp, q F Fp, q are all that is needed to specify completely the constitutive

behaviour of a plastic material

The Modified Cam-Clay model can be expressed conveniently within the

hy-perplastic approach by defining the following functions The main parameters

are defined in Figure 10.1, where V is the specific volume (total volume divided

by volume of solids) The values of p, q, and p at the reference (zero) values of x

strain and plastic strain are p , 0, and o p xo, respectively The (constant) shear

modulus is G The following equations can be simplified by noting the definition

The Gibbs free energy is chosen as

The first two terms define the elastic behaviour (the unusual first term results

in a bulk modulus proportional to pressure) The third term ensures that the

internal variable plays the role of the plastic strain (see Table 5.2 in Section 5.5)

The final term defines the hardening of the yield surface From Equations (10.1)

and (10.2), it follows that

g q

ic N CL

ON

Critica

l st ateline

Figure 10.1 Definitions of modified Cam-clay parameters

These equations may be combined to obtain the yield function, which of course

can alternatively be taken as the starting point:

2 2

wF

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

define incremental behaviour consistent with the usual formulation of

Modi-fied Cam-Clay The only exception is that consolidation and swelling lines are

considered straight in ln ,lnp V space rather than ln ,p V space The result 

is that O and N have slightly different meanings from their usual ones and that

the (variable) elastic bulk modulus is given by K p N rather than the more

usual K pV N 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 addressed briefly by Collins and Houlsby (1997) and is discussed more fully

in the following section

10.1.2 Non-uniqueness of the Energy Functions

Collins and Houlsby (1997) demonstrated that the modified Cam-Clay model

can be derived from either of two different pairs of Gibbs free energy and

dissi-pation functions This raises the interesting concept that, because the same

stitutive behaviour can be derived from different energy functions, then

con-versely the energy functions are not uniquely determined by the constitutive

behaviour The energy functions are not therefore objectively observable

quanti-ties The case discussed by Collins and Houlsby is a special case of the following

more general result

Consider a model specified by g g1V D and ,  d d1V D D It follows , , 

that H wg1 wV , F wg1 wD , and F wd1 wD Using F F gives

F w wD  w wD and F wd1 wD  w g2 wD However, using F F again

gives wg1 wD  wd1 wD  0 Thus identical constitutive behaviour is given by the

two models Of course, the models are acceptable only if both d1V D D !, , 0 and

d V D D  w g wD D ! for all D For typical forms of the dissipation

func-tion, it often proves possible to find a function g that satisfies this condition 2

The particular model described above may alternatively be derived from the

10.2 Towards Unified Soil Models

One of the major limitations of the basic critical state models is that they do not

describe a wide range of phenomena that occur due to the irreversibility of

strains and non-linearities within the conventional yield surface which may take

place at a very small strain level In Sections 9.5 and 10.1, we demonstrated how

both large-scale yielding and small strain plasticity can be formulated

(sepa-rately) within a hyperplastic and continuous hyperplastic framework This

sec-tion presents a unified formulasec-tion, where both large and small strain plasticity

are described within a single, unified, continuous hyperplastic model

10.2.1 Small Strain Non-linearity: Hyperbolic Stress-strain Law

In Section 9.5, we presented a continuous hyperplastic formulation of small

strain non-linearity based on the Puzrin and Burland (1996) logarithmic

func-tion It provides realistic fitting of the typical “S-shaped” curves of secant shear

stiffness against the logarithm of shear strain observed for soils (see

Fig-ure 10.2b) For the purposes of this section, however, the simple hyperbolic form

is sufficient to illustrate the principles involved As in the logarithmic model,

a one-dimensional hyperbolic model can be defined by two potential functionals:

1 0

w

wD K

so that the field of yield functions is given by

ˆ

g

d E

Combining Equations (10.24), (10.26), and (10.27) allows the stress-strain curve

for monotonic loading (from zero initial plastic strain) to be expressed as

*

0 ˆ

k d

Differentiation of Equation (10.28) twice with respect to V (using standard

re-sults for the differential of a definite integral in which the limits are themselves

variable) leads to the important result

2 2

1ˆ

d

d

HV

so that the plastic modulus function hˆ K is uniquely related to the second

de-rivative of the initial “backbone” curve H V (see Section 8.10)

The hyperbolic stress-strain curve (Figure 10.2a) is given by (see Section 8.7)

E h

a

 K

K

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 10.2 (a) Hyperbolic stress strain curve (for a = 5); (b) typical stiffness - log strain curve for soil

10.2.2 Modified Forms of the Energy Functionals

The continuous hyperplastic formulation presented in Chapter 8 considered

a Gibbs free energy of the form,

where gc indicates the derivative with respect to the function ˆD By defining ij

ˆ

ij ij

ˆˆ

ij ij ij ij ij ij

g H V  F D  F ³ K D K K  Td s (10.41) and the definition of D as a constraint equation, ij

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

an augmented energy expression g / , where c c / is a multiplier to be de-c

ˆˆ

ˆˆ

d

w

F

Given particular forms of the functions, the above equations are sufficient to

determine the constitutive behaviour

10.2.3 Combining Small-strain and Critical State Behaviour

As mentioned in earlier sections, a major criticism of critical state models is

the fact that they describe the behaviour of soils inadequately at small strains

Coupled to this is poor performance with respect to modelling cyclic loading,

and no modelling of the effects of immediate past history To remedy this

situation, the benefits of the simple hyperplastic framework for describing

kinematic hardening of an infinite number of yield surfaces is combined with

the Modified Cam-Clay model The following expressions are suggested for the

have been introduced

Note in the above that the stiffness factors for the kinematic hardening of the

yield surfaces [the integral term in Equation (10.48)] have been made

propor-tional to preconsolidation pressure rather than pressure 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

preconsoli-dation pressure in these expressions considerably complicates the derivatives of

the Gibbs free energy functional In the following, a linearised version of the

continuous hyperplastic Modified Cam-Clay is therefore explored, although the

derivation from Equations (10.48) and (10.49) is set out in Section 10.2.6 This

avoids some of the coupling terms and will serve as an example to illustrate the

main features of the model The suggested Gibbs free energy and dissipation

ˆ 3 ˆ1

Dw

h

M

Dw

2

ˆˆ

To derive the incremental constitutive behaviour, the above are differentiated

further to give the following, which also use ˆF F and ˆ F F [see Equation 0

(10.47)]:

p K

H    D

3

q G

Although it would be attractive to carry out calculations using the functions of K directly, at least with the presently available software, it is necessary first to dis-cretise these functions in terms of a finite set of values The internal function

ˆ

D K is therefore represented by a set of n internal variables ˆD , i i ! The 1 n

result is that the field of an infinite number of yield surfaces is thus approximated

by a finite number of yield surfaces In the calculations presented below, the field

is represented by 10 yield surfaces For the purposes of numerical calculation, the continuous hyperplasticity 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 adopting more sophisticated numeri-cal representations of the function A detailed implementation of the above equa-tions for numerical calculations is addressed in Section 10.2.5

10.2.4 Examples

Some example calculations illustrate the features of the model described above All the following calculations use the constants G 2000, K 2000, h 200, 1

M , and a 1.1 The units are arbitrary, but note that for a model of a specific

soil, the constants G, K, and h have the dimensions of stiffness, whilst M and a

are dimensionless

Figure 10.3 shows the behaviour of the soil in an isotropic consolidation test, first loaded to p 100, then unloaded to p 20, and then reloaded It can be seen that the unloading line is curved, as is the reloading line When the precon-solidation pressure is reached, there is no sharp yield, but instead the reloading curve merges smoothly with the virgin consolidation line This type of behaviour

is a feature of many soils The curvature of the unloading and reloading lines (and hence the openness of the hysteresis loop) is controlled by the form chosen

for the kernel function in the expression for the Gibbs free energy [equivalent to

h K in Equation (10.22)] In this particular model, the curvature is controlled

by the value of the parameter a As the curvature is reduced, the

preconsolida-tion point on reloading becomes more sharply defined in the response

Figure 10.4 shows (bold line) the stress path for a sample which is first

consoli-dated isotropically to p 100 and then sheared undrained The undrained stress path is typical of a normally consolidated clay Also shown in the figure are the positions of the 10 yield surfaces used in the calculation Note that the yield sur-faces overlap There is a widely held misconception that multiple yield surfaces

should be “nested,” i e non-overlapping This condition is usually attributed to

Prevost (1978), but there are no strong reasons why this needs to be the case, see action 6.5

Note two features of the field of yield surfaces Firstly, the small surfaces are

“dragged” by the stress point, and therefore provide coding of the past stress history Secondly, note that the largest yield surface has never been engaged by the stress point This latter effect is due to a subtlety of the interaction between the yield surfaces As plastic volumetric strain occurs on the inner surfaces, the size of all yield surfaces increases The result is that during isotropic consolida-tion, once a sufficient number of the inner surfaces have been engaged, the sur-faces expand at a sufficient rate that the outer surfaces are never encountered by the stress point

Although it would be attractive to carry out calculations using the functions of K directly, at least... conventional yield surface which may take

place at a very small strain level In Sections 9. 5 and 10.1, we demonstrated how

both large-scale yielding and small strain plasticity can be... hyperplastic formulation presented in Chapter considered

a Gibbs free energy of the form,