A Principles of Hyperplasticity Part 8 pdf

ij 9.4 Dependence of Stiffness on Pressure In recent years, a considerable amount of experimental research has been ried out to investigate the mechanical behaviour of soils undergoing

and the Second Law is written in the form derived from the Clausius-Duhem

The mechanical dissipation can now be split into the dissipation related to the

deformation of the soil skeleton and the part that results from the flow of pore

water It is clear that we can identify the term w uc k ,k w as the input power that is

dissipated in the flow of pore water, so we can write

where d is the dissipation associated with deformation of the soil skeleton

The Second Law becomes

, ,w k k 0

So we see the close analogy between the energy dissipated by the heat flux q k

and the water flux w Although the Second Law requires only that the sum of k

the three terms in inequality (9.10) be non-negative, we again assume that each

individual term is non-negative and therefore that dt 0

From Equation (9.9), we obtain w u k ,ck wq k k, T s d , so that the First Law

becomes

ij ij

and the analysis proceeds exactly as in Section 4.1, but with V replacing ij V ij

and d replacing d throughout Thus we see that the development in Chapter 4,

which was in terms of total stresses (in the absence of pore pressure), is equally

applicable in terms of the conventionally defined effective stresses for a

satu-rated granular material under the usual assumption of incompressible grains

and pore fluid

Houlsby (1997) developed expressions for the power input to an unsaturated

granular material (i e one composed of grains, liquid, and gas), subject to some

simplifying assumptions An analysis similar to that presented above can be

applied to the unsaturated case Appropriate definitions for effective stress can

be made so that the development in earlier chapters is applicable, but additional

terms appear in the equations, related to the degree of saturation and the

differ-ence between pore water and pore air pressures The application to unsaturated

materials is not, however, straightforward, and further work is required in this

area A more detailed study of the processes of soil deformation and flow in

porous media is presented in Chapter 12

In the remainder of this chapter and in Chapter 10, we work entirely in terms

of effective stresses, and for brevity we use V and d instead of ij V and d ij

9.4 Dependence of Stiffness on Pressure

In recent years, a considerable amount of experimental research has been ried out to investigate the mechanical behaviour of soils undergoing very small strains, for which the response is usually assumed to be reversible The interpre-tation of the data shows that the initial soil stiffness (or small strain stiffness) is

car-a non-linecar-ar function of the stress (specificcar-ally the mecar-an effective stress) The stiffness is also affected by other variables, such as the voids ratio, and/or the preconsolidation pressure, which we address in Section 9.4.3

The small strain tangent stiffness depends on the stress level, and typically the elastic moduli vary as power functions of the mean stress Simple elastic or hypoelastic models of this non-linearity can result in behaviour that violates the laws of thermodynamics To ensure that an elasticity model is thermodynami-cally acceptable, we use the hyperelastic approach here to derive models which allow for variation of elastic moduli as power functions of mean stress

Analysis of many geotechnical problems depends on a realistic representation

of the non-linear dependence of the initial stiffness on stress, and we first plore how this has often been achieved in the past The usual approach is to adopt “hypoelastic” formulations (Fung, 1965) in elastic-plastic models, in which varying tangent moduli are defined For instance, it is common to adopt the following procedure to calculate elastic moduli for the modified Cam-clay

ex-model The bulk modulus K is usually defined through the pressure-dependent

expression p1 e

N , and the shear modulus G is then obtained by

assum-ing a constant Poisson ratio v Such a model leads to a non-conservative tic” response (Zytynski et al., 1978) Instead we adopt the “hyperelastic” ap-

“elas-proach, which naturally leads to a conservative elastic response, guaranteed to obey the first law of thermodynamics

It is worth remarking here that it is well-recognised that the soil stiffness is also significantly dependent on the strain amplitude This raises more difficult prob-lems of hysteresis and energy loss and is addressed later in Sections 9.5 and 9.6 Several models have been developed to reproduce the reversible behaviour of

soils, and these are reviewed by Houlsby et al (2005), who also review briefly

some typical experimental observations of the small strain stiffness of soils and their semi-empirical interpretations We present here a hyperelastic, isotropic energy potential capable of accounting for the non-linear dependence of elastic stiffness on stress We first develop this for triaxial conditions and later for more general stresses

9.4.1 Linear and Non-linear Isotropic Hyperelasticity

Experimental evidence suggests that the small strain bulk and shear stiffnesses

of soils can be well represented as power functions of the mean effective stress,

and we write them in the following forms:

n

p K = k

n s

p G

where k and g are dimensionless constants; s p is a reference pressure, often r

conveniently taken as atmospheric pressure; and n is an exponent 0d d The n 1

possibility that the exponent n could be different for the bulk and shear moduli

might be considered, but is not pursued further here, as this leads to

consider-able additional complexity in the mathematical development

In the following, we will focus our attention on deriving stress-dependent

stiffness from potentials that are expressed as functions of invariants either of

the strain or of the stress tensor, so that the material behaviour described will be

fundamentally isotropic, although it will be seen below that “stress-induced”

anisotropy is predicted under certain conditions

Expressed in terms of the triaxial variables, the Helmholtz free energy f (also

equal to the elastic strain energy) is written as a function of the strains, i e

G = w = w

Furthermore, it can be shown that off-diagonal terms may in general appear

in the incremental stiffness matrix,

v s

When J is non-zero, the material behaves incrementally in an anisotropic

manner, even though f is an isotropic function of the strains This is a case of

stress-induced anisotropy

Although elastic behaviour can be derived by differentiation, as in Equations

(9.15) and (9.17), this has certain disadvantages The resulting expressions for

the moduli are in terms of the strains, which can be inconvenient, because

moduli expressed as functions of stress are usually of more practical use

There-fore, it is useful to use instead the Gibbs free energy function g (which in the

context of elasticity theory is also the negative complementary energy):

The Helmholtz and Gibbs free energy expressions for linear elasticity (which

corresponds to n in (9.6) and (9.7)) are each quadratic in form: 0

with k and g dimensionless constants From the above, it is straightforward to s

derive p = k p rH , v q = g p3 s rH , s K = k p , r G = g p , and s r J = 0

The expressions that give non-linear elasticity (i e Kvp n) under purely

iso-tropic stress conditions (i e without q and H terms) can also be established s

unambiguously For n z , the expressions for f and g must be 1

n n r

For n = , the above expressions become singular A difficulty also arises 1

when n = that, if the volumetric strain is taken as zero at 1 p , then it is infi-0

nite for all finite stresses This problem can be avoided by shifting the reference

point for zero volumetric strain from the origin (p ) to 0 p p r This is

achieved by changing (9.27) and (9.28) to

but note that this does not effect the expression for stiffness in terms of

pres-sure The asymptotic expressions for n = are 1

Equations (9.25) and (9.26) apply for n for any triaxial stress states, whilst 0

(9.31) and (9.32) [or (9.35) and (9.36) for n ] apply when 1 n z but only on 0

the isotropic axis It is our purpose in the following to obtain more general

ex-pressions which apply both to any triaxial stress states and for n z , and reduce 0

to each of the above equations in the appropriate special cases

This generalisation can be done in a variety of ways We first consider (for

simplicity) the case where the reference point for volumetric strain is at p 0

Three possible ways of generalising Equations (9.27) and (9.28) are as follows:

(a) f is of the form,

m

where A, B and m are constants It can be shown that no simple form of g

exists for this case

It can be shown that no simple form of f exists in this case Einav and Puzrin

(2004a) pursue this form of energy function and show [following Houlsby

(1985)] how it also has the disadvantage that it introduces a limiting stress

ratio, implying that some stress states are unattainable Both the inability to

derive the f function and the inaccessibility of certain stress states are

signifi-cant drawbacks of this form of function

All the forms described above exhibit a constant Poisson's ratio under tropic stress conditions At least at present, available experimental data are in-sufficient to distinguish definitively between the above three approaches, so the selection is based on the simplicity of the mathematics The approach that proves most versatile is (b), so this is adopted in the following

iso-9.4.2 Proposed Hyperelastic Potential

Triaxial Formulation

Following approach (b) above, the generalisation of Equations (9.25) and (9.31)

that we seek for the function f must consist of a quadratic function of H and *v

s

H , raised to an appropriate power Inspection of the forms of Equations (9.25) and (9.31), followed by some calculation to determine some appropriate con-stant factors, shows that the required general expression is

*

31

12

It can be shown (after some manipulation) that the Gibbs free energy sion which is the Legendre transform of the expression in Equation (9.43) is

11

s r

n o n r

Although Equations (9.43) and (9.44) may appear complex, it can be seen that

they have the basic structure of (9.40) and (9.41) (allowing for the shift of origin

for strain) The particular forms of the functions are chosen so that, after

differ-entiation, the moduli reduce to simple expressions It follows from the above

that

1

11

n z , but this is not the case for (9.43)–(9.45) Noting that for n , 1 p o , the p

asymptotic values of the compliances for n are 1

2

11

3 s

kq c

g p

 The asymptotic expressions for n = replacing 1(9.43)–(9.45) are

23exp

1ln

The implications of the above choice of free energy (and hence

complemen-tary energy) are as follows:

On the isotropic axis simple modulus expressions may be obtained, and 0

J = On this axis, it is also possible to define Poisson’s ratio

The shapes of shear strain and volumetric strain contours are given directly

by Equations (9.45), (9.46), and (9.51) Within the range of stress ratios of est, the volumetric strain contours are similar (but not identical to) parabolae symmetrical at about the p -axis and convex toward the origin Shear strain

inter-contours are curves, convex upward in the region of the p q plot accessible , 

for reasonable soil properties For n = , the shear strain contours become 1straight lines radiating from the origin

Some undesirable features of the model proposed by Houlsby (1985) and

ex-tended by Borja et al (1997), in particular the crossing of volumetric strain

con-tours, are absent

Figure 9.1 shows contours of shear and volumetric strains for n 0.5 Note that the volumetric strain contours correspond to undrained stress paths for elastic behaviour For nz , the approximately parabolic undrained stress paths indicate 0that (other than on the isotropic axis) the response of the soil is incrementally anisotropic This stress-induced anisotropy arises as a natural consequence of the hyperelastic formulation, and corresponds well to observations of soil behaviour

p' 0

200 400 600 800

Figure 9.1 Example of volumetric (solid lines) and shear (dashed lines) strain contours for n = 0.5

Many studies have presented evidence that the stiffness of sands can be expressed

as a power function of stress level, but a special feature of the hyperelastic

ap-proach adopted here is that it predicts the related effect of the curvature of strain

contours Figure 9.2 is reproduced from Shaw and Brown (1988), who follow the

approach of Pappin and Brown (1980) in plotting “resilient” shear and volumetric

strain contours derived from an extensive series of cyclic tests on granular

mate-rial Importantly, they show that the volumetric strain contours are approximately

parabolic and curved approximately as in Figure 9.1 The shear strain contours

(which Shaw and Brown assume to be straight) are also very similar to those in

Figure 9.1 Comparable data for clays were presented by Borja et al (1997)

General Stress Formulation

The results described above can be generalised to other than triaxial stress

states, if the free energy f is written as a function of the strains H and the com-ij

plementary energy g as a function of the stresses V In this case, the normal ij

expressions for the stresses and strains, as derivatives of the free energies, are

used

The stiffness matrix is

ij ijkl

f = =

and the compliance matrix is

Figure 9.2 Shear and volumetric strain contours presented by Shaw and Brown (1988), based on

experimental data on crushed limestone

ij ijkl

g = =

The latter form is also applicable for n It can also be shown that the stiffness 1

matrix can be expressed as

It can easily be shown that tests performed in triaxial systems equipped with vertically fitted bender elements allow measurement of the stiffness component

1212 2

d G, where the vertical and radial directions are 1 and 2, respectively Under triaxial stress conditions, the above expressions reduce to those obtained earlier

The form of the stiffness matrix in Equation (9.59) has two important quences Firstly and obviously, the terms depend on the stresses (not just the mean stress) This means that the stiffness can be determined only by reference

conse-to the complete stress system Secondly and less obviously, it can be shown that (other than on the isotropic axis) the incremental stiffness cannot be expressed just in terms of isotropic stiffnesses, that is to say, Equation (9.59) does not sim-

ply imply stress-dependent values of the parameters K and G The response can

be represented only by anisotropic elasticity This is an example of induced” anisotropy: it has nothing to do with the fundamental structure of the material, which is isotropic, but is induced by the stress field Of course, real soils may also exhibit structural anisotropy Extension of the concepts discussed here to structurally anisotropic materials is not addressed here

“stress-The compliances or stiffnesses expressed in Equations (9.58) and (9.59) can

be used directly, for instance, in a finite element program for general stress states, ensuring fully conservative elastic behaviour when the moduli are func-

tions of pressure The expressions require just three dimensionless constants k,

s

g , and n

9.4.3 Elastic-plastic Coupling in Clays

Incorporation of the density and/or loading history dependency of elastic ness into the hyperplastic framework implies even more subtlety in the elastic response Changes in density are normally expressed through the volumetric plastic strain, and the loading history is reflected in the overconsolidation ratio (OCR) and preconsolidation pressure, which can again be expressed through the volumetric plastic strain In this case, the specific Helmholtz free energy

stiff- ij, ij

f H D and the specific Gibbs free energy gV Dij, ij depend on the matic internal variable tensor D , most often associated with plastic strain, so ijthat in the following we adopt p

kine-ij ij

D { H (see Chapter 4) The incremental elastic stress-stain response is in this case affected by the plastic strain increment as well This is called “elastic-plastic coupling.” In this case, great care is required

to make a careful distinction between the “plastic” and “irreversible” nents of strain Collins and Houlsby (1997) discuss elastic-plastic coupling within the hyperplastic framework and address its consequences for the overall behaviour of soils

compo-An important consequence of a plastic-strain-dependent complementary ergy relates to the decomposition of the elastic strain tensor, as shown by

experimental data on crushed limestone

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Although Equations (9.43) and (9.44) may appear complex, it can be seen that

they have the basic structure of (9.40) and (9.41) (allowing for the shift of origin... (9.7)) are each quadratic in form: