A simple constitutive model for granular soils: Modified stress-dilatancy approach

Computers and Geotechnics, Vol. 22, No. 2, pp. 10%133, 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII:S0266-3252X(98)00004-4 0266-352X/98/%19.00+ 0.00 ELSEVIER

A Simple Constitutive Model for Granular Soils: Modified Stress-Dilatancy Approach R. G. Wan & P. J. Guo The University of Calgary, Calgary, Alberta, Canada, T2N lN4 (Received

25 February

1997; revised version received accepted 10 February 1998)

28 January

1998;

ABSTRACT The dependencies of granular soil behaviour on void ratio and stress are modelled within plasticity theory enriched by a modtjied stress-dilatancy law. Simplicity has been kept in the development of the constitutive law for easy future implementation in aJinite element code. By using a void ratio dependent factor which measures the deviation of the current void ratio from the critical one, Rowe’s stress dilatancy equation is modtjied. This modtfication indeed corresponds to a new energy dissipation equation for a granular assembly which sustains kinematical constraints under the action of stresses. The proposed constitutive law predicts in a very consistent manner the response of sands in monotonic loading conditions for a large range of initial void ratios and confining pressures without the need to make any adjustments to the material parameters. Well known published data for the drained triaxial compression of Sacramento River sand was successfully modelled. 0 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION The characterization of the stress-strain and failure behaviour of granular soils is complex due to their particulate nature. There is a number of micromechanical models available in the constitutive relations literature for addressing the discrete particle to particle description of granular soils as introduced by Bathurst and Rothenberg [l] and Jenkins and Strack [2]. However, these models are still considered as very idealized, and cannot be directly used to analyze general boundary value problems. On the other hand, a continuum approach which consists of using a macroscopic constitutive 109

110

R. G. Wan and P. J. Guo

model based on plasticity theory, together with some stress dilatancy rule, still remains the most commonly used framework. This is due to its flexibility towards finite element modelling. Some of the earlier classical constitutive models worth mentioning here are those proposed by Drucker et al. [3], Roscoe et al. [4], Poorooshasb et al. [5], Roscoe and Burland [6] and Lade and Kim [7]. Other models address more complicated issues such as cyclic loading and strain localization [8-141. The crucial factor that governs the behaviour of granular soils is stress-dilatancy, the soil’s ability to increase in volume due to shear stresses and geometrical effects. Rowe [ 15, 161developed a basic theory which explains how the geometrical interlocking of the particles influences the strength of the material, and provided a simple relationship between stress ratio and dilatancy factor which basically quantifies the geometrical effect. Nova and Wood [17] later on proposed a stress invariant form of the relationship for addressing general stress and strain conditions. Within the framework of plasticity theory, Rowe’s stressdilatancy relationship represents in fact a flow rule which determines the direction of plastic strains. Although Rowe’s theory [15,16] is a powerful concept, obstacles are met towards properly capturing density and pressure sensitivities in granular soil response. Subtle behavioural features such as stress induced anisotropy as well as shear dilation and contraction evolution with respect to initial void ratio are not addressed. On the other hand, in the more recent literature [I 8-211 focus has been made principally on aspects of stress induced anisotropy, cyclic loading and principal stress axes rotation with respect to stress-dilatancy. A review of the literature reveals that the issue of density and pressure dependencies has not been dealt with in an explicit manner, although some consideration to the issue has been made in constitutive models such as described in Refs [22-261. However, to the authors’ knowledge, the only explicit incorporation of density and pressure dependencies can be found in Bauer and Wu [27], and Gudehus [28] who base their models on hypoplasticity which was first introduced by Kolymbas [29]. It is essentially assumed that the rate of the Cauchy granular stress in a soil is related to its deformation rate and void ratio through a functional. As such, the model deviates from the classical plasticity theory in that there is no need to introduce concepts of yield surfaces or flow rules. Both density and pressure sensitivities, coined by Gudehus [28] as pyknotropy and barotropy respectively, are included by choosing appropriate factors involving current void ratio and mean stress. This type of hypoplastic model can capture most aspects of soil behaviour including cyclic loading. Its implementation into finite elements is possible, but at the expense of complicated loading and unloading material functions. The above brief, non-exhaustive review reveals that advanced constitutive models exist, but their complexities seem to generally present a major obstacle towards successful and robust implementation into finite element

A simple constitutive model for granular soils

111

codes for the analysis of boundary value problems. Hence, the main objective of this paper is to develop a comprehensive, but yet simple, constitutive model with relatively few material parameters for easy implementation into a finite element program. The authors propose a model which hinges upon concepts of coupled plastic hardening-softening and stress dilatancy in order to address both the void ratio and pressure dependency behaviour of granular soils. In particular, the main contributions of this paper are the development of a stressdilatancy equation which embeds both pyknotropy and barotropy, and the introduction of a single void ratio dependent plastic evolution law which can describe hardening and softening. The importance of these developments which are based on experimental evidence and physical arguments will become apparent later in the paper. Other salient features include a failure surface governed by Mohr-Coulomb or Matsuoka-Nakai [30] criterion, a family of shear yield surfaces which assume the same form as the failure surface, a vertical cut-off cap for compaction or consolidation, and a special flow rule. The first part of the paper presents the framework of the proposed model followed by the determination of material parameters. Finally, the model performance is illustrated with reference to available published experimental data. While developments in this paper exclusively deal with monotonic loading, the proposed model is currently being extended to cyclic loading.

FRAMEWORK

OF CONSTITUTIVE

MODEL

For the development of the constitutive model, non-linear elasticity and flow theory of plasticity are used to characterize elastic and plastic deformations, respectively. It is assumed that plastic deformations are due to shear and hydrostatic compaction mechanisms at the macroscopic level. Hence, the model is comprised of two yield surfaces, each of which addresses shear and compaction mechanisms. The plastic strains are controlled by a hardening law combining both plastic volumetric and shear strains in such a way that the evolutions of the two yield surfaces are intimately coupled. Moreover, plastic strain directions due to shearing mechanisms are controlled by a modified stress-dilatancy equation which will be developed in the later sections. The following sections summarize the major components of the formulation in which all stresses referred to are effective stresses. Also, compressive stresses and strains are taken as positive. Deformation split

The total strain increment in the material when loaded is divisible into elastic and plastic strain components so that de = de” + d&p with dae as the elastic

112

R. G. Wan and P. J. Guo

strain increment calculated by generalized Hooke’s law, and dcJ’ as the total plastic strain increment determined from the plastic constitutive law. Two plastic strain mechanisms are assumed to contribute to the total plastic strain increment: one due to compaction or consolidation (de:), and the other one due to dilatant or compressive shearing (d&f) such that d&f’ = dc{ + d&f‘. The elastic strain increments which are recoverable upon unloading are calculated from a generalized Hooke’s law using a non linear shear modulus G. It is well recognized that the shear modulus of a soil varies principally with its density (or void ratio e) and mean effective stress p. Many empirical relationships of the above dependence have been experimentally deduced for a variety of sand and clays. In this paper, the expression suggested by Hardin and Richart [31] and which works for most sandy soils is used, i.e. G

=

Go

c2-17 - e>2 l+e

fi

where Go is a material constant referring to the shear modulus at very small amplitudes of shear strains (approximately 1 x 10e6). It is further assumed that the Poisson’s ratio v remains constant so that the Young’s modulus E varies non-linearly as implied by the empirical relationship Eqn (1). Failure criterion

For describing the shear deformation mechanism, the failure condition at both peak and residual states are assumed to be path independent and governed by a Mohr-Coulomb criterion written in terms of ~1 and ~3 which are the major and minor principal stresses respectively. Thus, the expression of the failure surface is simply given as

(2) in which the frictional angle q can assume values of vf or qcVdepending on whether peak or residual (constant volume) states are described, respectively. It is worth mentioning that, for the consideration of 3-D stress conditions and intermediate principal stress, a generalization of Eqn (2) can also be used such as the Matsuoka-Nakai failure criterion [30] which represents a smooth three dimensional failure surface in the principal stress space. In stress invariant form, the Matsuoka-Nakai surface is expressed as F=ZiZ2-

(9+8tan2q)Z3

=0

(3)

A simple constitutive model for granular soils

where stress invariants +f_Tp3;

I3

=

113

are given as 11 = 01 + a;! + as; 12 = ata2 + ata3

OlDp3.

Yield surfaces

Yield surfaces can be viewed as surfaces representing either contours of equal plastic work or states of plastic yielding in the stress space [5,7]. A two-surface model which is composed of shear and compaction yield envelopes will be formulated for simplicity. For the shearing mechanism, it is assumed that the yield surface takes the same form as the failure surface. Moreover, it is assumed that plastic shear yielding corresponds to a continuous mobilization of the frictional angle which results into an isotropic expansion of the yield surface in the stress space [l I]. At a given yielding state, the expression of the shear yield surface is FS = (61 -

~3)

-

(al

+

03)

sinvm = 0

(4)

in which 4pmis the mobilized frictional angle. With regards to describing the compaction mechanism, a vertical cut-off cap surface which also moves in the stress space is used for simplicity. The functional form of the cap is the same as in Vermeer [32], i.e. F,=p-po=O

(5)

with po referring to an equivalent consolidation pressure. The latter is made function of current plastic volumetric strains which include both compaction and shear induced components. Consequently, both the cap and shear yield surface movements are coupled. For example a purely deviatoric stress path will cause plastic volumetric strains which in turn induces movement of the cap even though there is no change in mean stress. The converse situation which involves a purely hydrostatic stress path is also true. The formulation of the above two yield surface model for describing coupled plastic deviatoric and hydrostatic responses is relatively simple but comprehensive. However, shear strains developed during stress paths, involving only changes in mean stresses, cannot be modelled unless the vertical cut-off cap is replaced by a curved one. Flow rules

In the multi-surface plasticity, the plastic strain increments from non-associated flow rules, i.e.

are calculated

114

R. G. Wan and P. J. Guo

(6) in which dAi and Qi are plastic multipliers and plastic potential functions, respectively. The subscript i varies such that a hydrostatic mechanism is represented by i = c, and a shearing mechanism by i = s. In this paper, an associated flow rule is assumed for the hydrostatic consolidation process, i.e. QC = F, = p -PO. However, for the, shearing process, the plastic potential function can be mathematically derived from some plastic energy dissipation equation [6] or a stress-dilatancy law [16]. A more expedient way of achieving the same latter result is to assume that the plastic potential takes the same form as the failure surface [l 11,i.e. Qs

=

(~1

-

03)

-

(01

+

03)

sin +m

(7)

in which the parameter $m refers to the mobilized dilation angle at a certain stress level. The dilation angle relates to the negative sign of the ratio of plastic volumetric (c$) to deviatoric ($) strain increments, i.e. sin$m = -d 0 corresponds to a shear induced increase in volume and $m -C0 to shear induced contraction. The dilatancy angle evolves with deformation history following Rowe’s stress dilatancy law [15,16] which is herein modified in order to account for void ratio and pressure dependencies. Figure 1 illustrates the evolution of the yield and plastic potential surfaces together with the direction of plastic strain increment vector with respect to different stress paths. Modified Rowe’s stress-dilatancy

equation

The original Rowe’s stress dilatancy equation [15,16] couples the dilatancy factor, D = 1 - de/d4 to effective stress ratio R = 0,/q, so as R = K,,D. The factor Kc, = tan* ($ + y) is a material constant derived from energy dissipation considerations, and pCVrepresents the friction angle at constant volume deformations. When written in terms of mobilized dilation and frictional angles, Rowe’s stress dilatancy equation takes the classical form [ 111: sin $rm =

sin qrn - sin 4pCV 1 - sin 4pmsin qocV

As discussed earlier in the paper, Rowe’s equation cannot describe the density or void ratio dependency during the deformational process, neither the

A simple constitutive model for granular soils

115

Fig. 1. Plastic strain directions together with yield, failure and plastic potential surfaces.

difference between pre-peak and post-peak regimes for dense sands. In order to address the above deficiencies, a modification of the original Rowe’s equation is proposed based on the value of the current void ratio e relative to the critical void ratio ecr [33], i.e. sin $m =

sin corn- (e/e,,)“sin qocV 1 - (e/e,,)“sin q, sin (pCV

(9)

in which cxis a parameter to be determined. The validation of Eqn (9) hinges upon a modified energy dissipation equation which embeds a so-called state parameter related to the current and critical void ratios. In fact, the energy based factor KCyin the original Rowe’s equation is made void ratio sensitive so that the modified stress dilatancy relationship becomes

The angle (p* corresponds to the friction angle mobilized along a certain macroscopic plane which evolves during deformation history. This mobilized friction angle will tend to qCVas the current void ratio approaches the critical void ratio. Also, it is obvious that void ratio dependent g, reverts to the

R. G. Wan and P. J. Guo

116

original Kc, when the void ratio is fixed to the critical void ratio, i.e. (e/e,,)” = 1. Figure 2 refers to the modification of the original Rowe’s equation in a stress dilatancy plot and illustrates the initial void ratio dependency. Also, any dependency of dilatancy on fabric change and stress path can be actually represented by the parameter cxwhich can be linked to some fabric tensor, plastic deviatoric strains and the effect of intermediate principal stress measured by parameter b = (02 - crs)/(c~t - ~3). For example, in plane strain conditions, experimental data shows that the value of a! is zero as demonstrated in Ref. [34]. Hardeningsoftening

laws

The evolution laws for both shear and compaction mechanisms will be discussed. As plastic flow takes place, both the shear yield surface and the cap are constrained to grow according to some evolution law based on two conveniently chosen internal plastic variables, < the plastic volumetric strain and Vp the plastic deviatoric strain defined as follows

(11) in which 9 is the deviatoric part of the plastic strain rate tensor 9 such that # = S - ,+‘1/3; 1, = Sg = Kronecker delta.

original Rowe’s

1

Fig. 2. Modification

of Rowe’s theory to account

D= l- d.c,PId$

for initial void ratio dependency.

A simple constitutive model for granular soils

117

Coupled hardening-softening for shearing mechanism

During deviatoric loading history, it is postulated that both volumetric strains (void ratio) and plastic deviatoric strains control hardening and softening according to a simple function given as [33]: sin pm = ssin

pCV;fd(e) = (e/e,,)+

(12)

with a, /3 as constants, and qpcVis the friction angle at constant volume. Equation (12) basically represents a hyperbolic variation of mobilized friction angle with plastic shear as well as total volumetric strains (deviator+ and hydrostatic pressure induced) by virtue of the void ratio functionfd. It is also evident that, depending upon whether the current void ratio e is denser or looser of critical, the factorfd will adjust the functional representation of sinCOrnso as to make it evolve into either a softening or hardening trend, as shown in Fig. 3. Here, softening refers to material weakening within the , sin Q, m

Fig. 3. Mobilization

=

yPfd(e) sing, a+y’

cy

of friction angle as a function of initial void ratio.

118

R. G. Wan and P. J. Guo

homogeneous deformation regime. Softening which arises from material instability such as strain localization is not addressed in this paper. The prediction of strain localization into a shear band is discussed in Ref. [35]. The following discussion further motivates the physical significance of Eqn (12). Due to the inherent discontinuities existing in a granular soil, the deformation process involves both the rearrangement of particles and the local sliding among particles. The former mechanism is responsible for the production of dilation and compaction for dense and loose sands, while the latter mechanism leads to local failures and thus strain softening. Hence it is possible to have both strain hardening and softening occurring simultaneously during the deformation process. These are all represented macroscopically in Eqn (12) via the evolutions of yp andfd. Cap hardening

For describing the plastic volumetric strain components due to compaction, the cap surface grows (hardens) isotropically in the stress space with increasing irrecoverable volumetric plastic strains. An intermediate step towards calculating the plastic volumetric strains due to compaction is the introduction of an exponential relationship linking the compaction related void ratio to mean stress p, i.e.

e = eoew[-(p/h)“] in which hl is a modulus and m an exponent. strains due to compaction eVcc)is

(13) Hence the total volumetric

E (c)= eov - exPMJlh)ml) Y I + e0

(14)

Since the movement of the cap is prescribed by plastic volumetric strains as mentioned earlier in the Yield surfaces section, Eqn (14) must be supplemented with shear induced and elastic volumetric strains. It is clear that shear induced plastic volumetric strains are calculated from the modified stress-dilatancy, while elastic volumetric strains are obtained from generalized Hooke’s law. Dependence of critical void ratio on stress level

It is generally accepted that the critical void ratio is not a constant, but it varies with stress level as shown in Fig. 4 for the case of Toyoura sand for illustration purposes [36]. The same trend has also been observed for other

A simple constitutive model for granular soils

119

Toyoura sand

0.01

1

0.1

10

mean stress p (MPa) Fig. 4. Relationship

between

critical

void ratio and confining Ishihara [36]).

pressure

(after Verdugo

and

sands. In fact, the achievement of a critical void ratio by a random arrangement of grains largely depends on the applied confining pressure and the geometrical shape of the grains. At the macroscopic level, an appropriate functional description of the critical void ratio dependency on stress level is given as

c(>l ncr

ecr = ecd exp -

P

h CT

(15)

where ecd is the critical void ratio at very small confining stress, ecr is an exponent number and h,, some parameter. The effect of the particle geometrical shape can be actually included in the parameter eC,+ For practical purposes, the value of ecrOcan be usually regarded as the maximum void ratio of a granular soil.

MODEL PARAMETERS There is a total number of 11 parameters needed to describe the proposed model. These may be classified into the following categories: elastic parameters, shear dilatancy parameters and compaction parameters. The elasticity parameters include the shear modulus parameter Go and the Poisson ratio v which can be obtained from conventional tests. The intrinsic properties of the material are described by the initial void ratio eo, the critical void

120

R. G. Wan and P. J. Guo

ratio ecd at very low confining pressure, herein approximated to emax, parameter h,,, exponent number ncr, an d the friction angle at constant volume vpcV. Other parameters a, a! and ,!lgovern the influence of density on the shear dilatancy, contraction in the deviatoric mode, as well as hardening and softening. The cap parameters hl and m describe the compaction behaviour in the hydrostatic compression mode. The determination of the above material parameters from standard tests is quite straight forward except for those related to hardening-softening of the shear and cap yield surfaces. Amongst all of the above parameters, it is worth explaining the procedure used to determine the two hardening parameters: a and /3. Referring to Eqn (12) the peak stress conditions in a contest can be described by imposing ventional triaxial compression mathematically sin qPm= sin cpfand d(sin qm)/@ = 0. Thus, sin ‘pr $ -=-efpP sin 40,~ a + $ 0 ecr

(16)

and

2sin Lax

S(l + eO)ecr ($) ef - S(l + eO)ecr$

a=

sinhax

(17)

where ( .)f represents the argument at peak-state, eo = initial void ratio, and sin emax = (-~&$/dvp), Since the plastic volumetric strains, the shearing strains and the friction angle at peak stress states can all be measured, the specific values of a and #I can be ultimately obtained by solving Eqn (16) and Eqn (17) numerically. A large number of high quality experiments has been carried out in the 1960s. The published experimental data for a number of sands such as Brasted sand [37], Ham River sand [38] and Sacramento River sand [39] were used to determine parameters a and j% It was found that these parameters can be regarded as material constants since they remain invariant at normal stress level (~3 < 2 MPa), irrespective of initial void ratio of the sand. At very high stress levels, i.e. 03 > 2 MPa, the parameters a and /? may not be constant because grain crushing may take place and lead to a different deformation mechanism. In the next sections, the evaluation of the proposed constitutive model is made for Sacramento River sand. Material parameters have been deduced from conventional triaxial compression test data for Sacramento River sand and published by Lee and Seed [39]. These are summarized in Table 1.

A simple constitutive model for granular soils

121

TABLE 1

Material parameters for Sacramento River Sand Go = 3900 kPa a = 0.011 m = 0.8324

V = 0.25 a! = 1.2828 h,, = 22,139 MPa

ecfi = 1.03 /I = 1.25 nCr= 0.7075

(p,, = 340 hl = 63.90 MPa

MODEL ASSESSMENTS The simulations undertaken in this study attempt to reproduce the experimental data of Lee and Seed [39] for both dense and loose Sacramento River sand over a large range of confining pressures in drained triaxial compression. These data have been used by numerous researchers in the past for validating their constitutive models, for example, Bardet [22]. The following discussions demonstrate the capability of the proposed constitutive model to capture pressure and initial void ratio dependencies in a consistent manner using only relatively few material parameters, as listed in Table 1. The two initial void ratios tested were eo = 0.61 and 0.87, while confining pressures ranged from 100 kPa to 2 MPa. Stress-strain-volumetric

responses

Figure 5(a) and (b) shows the comparison of the model prediction with experimental data for the case of dense Sacramento River sand at an initial void ratio of 0.61. The numerical simulations for both stress difference and volumetric strains are in very close agreement with the experimental data, and the model consistently captures suppression of dilation at a high confining pressure of 2 MPa. Using the same set of material parameters given in Table 1, the model gives excellent agreement with experimental data for the same sand tested at an initial void ratio of 0.87 (looser state) as shown in Fig. 6(a) and (b). The volumetric responses of the sand at different confining pressures are mainly contractant, except for the confining pressure at 100 kPa, where dilation is correctly captured despite the sand’s initially loose state. A closer examination of Figs 5 and 6 indicates that the model predicts greater compaction at small strains, and also more dilation at high strains. These discrepancies are due to non-uniformities introduced by the standard end plates used in the tests. In fact, Bishop and Green [38] showed that in the small, uniform deformation range, both compression and dilation would be underestimated unless lubricated end plates are used. The situation gets even worse at large strains where localized deformations within the sample are inevitable. The void ratio of the soil sample is no longer uniformly distributed

R. G. Wan and P. J. Guo

122 (a)

7000 6000

g d 8 8 ti 3 3 g

5000 4000 3000 2000 1000 0 0

(b)

-0.2

c solid line: calculated symbols: measured-

-0.15 .9 z ,” ._ b

-0.05

g -?

0

0.1 axial strain

-0.1 .,,TT~T~~~~~~

2000 kPa 0.05 0.1

t,,,,,.,,,,,,,,,,,,,, 0

Fig. 5. (a) Stress

0.05

0.1 axial strain

0.15

0.2

difference vs axial strains: dense Sacramento River sand. strains vs axial strains: dense Sacramento River sand.

(b) Volumeric

throughout, with most of the dilation taking place in the localized zone. Since the experimental data is based on an average void ratio less than the void ratio in the localized zone, the model consistently gives higher volumetric predictions. The hardening and softening behaviour of the sand can be better illustrated if principal stress ratio al/as are plotted against axial strains as shown in Fig. 7(a) and (b). Figure 7(a) corresponds to Sacramento River sand at an initial void ratio of 0.61. It is shown that strain softening can be

A simple constitutive model for granular soils

123

6000 CT3 = 2.9 n =

.r.

‘op”

1270 kPa

450 kPa

0.1

0.15

0.2

0.25

0.3

0.2

0.25

0.3

axial strain

solid line: calculated symbols: measured

0.1 0

0.05

0.1

0.15 axial strain

Fig. 6. (a) Stress difference vs axial strain: loose Sacramento River sand. (b) Volumeric vs axial strain: loose Sacramento River sand.

strain

suppressed at high confining pressure, here 2 MPa, even though the sand is initially dense. All curves tend asymptotically to a value equal to tan2(n/4 + qCy/2) with y_~”= 34O, which corresponds to a state of constant volume. The small discrepancy between calculated and measured curves is due to the effect of strain localization into a shear band in the near pre-peak and post peak regions. In fact, the appearance of a shear band in the experiment leads to a higher rate of softening than in the calculated one which made no provision for strain localization. For the case of an initially

124

R. G. Wan and P. J. Guo

Dense, e, = 0.61

I 0

I1

.. measured ,,I,,,/,

I,

0.05

0.1

0.15

0.2

axial strain (b)

“I- a.=lOOkPa.. 3

..

2000 kPa

Loose, e, = 0.87

1

1, L

0

1

I

I

1

0.05

I

I

I

0.1

I

I

:.I-, -I

0.15 axial strain

I

0.2

calculated measured 1

,

I

0.25

I

1

0.3

Fig. 7. (a) Principal stress ratio vs axial strain for dense Sacramento River sand. (b) Principal stress ratio vs axial strain for loose Sacramento River sand.

loose sand (es = 0.87), strain softening is less prominent confining pressures in the neighbourhood of 100 kPa.

except at lower

Maximum mobilized friction angle variation with confining pressure

The achievement of a maximum (peak for strain softening cases) mobilized friction angle during deformation history is normally a function of both density and stress levels. In this model, the evolution of peak or maximum mobilized friction angle with density and stress level is naturally captured

A simple constitutive model for granular soils

125

due to the use of a special hardening-softening law, Eqn (12). Figure 8 shows the variation of calculated maximum friction angle with confining pressure for two different void ratios as compared to experimental data. It is found that there is a good agreement between calculated and experimental maximum friction angles for confining pressures less than 2MPa, especially for dense sand. However, for confining pressures above 2 MPa, the experimental data seems to indicate that there is a slight drop in mobilized maximum friction angle until approximately 5MPa, after which the mobilized friction angle increases again ultimately achieving a constant value at very large stresses. The drop in friction angle may be due to grain crushing which was not included in the model. This explains the discrepancy between the calculated and measured friction angles at high stress levels above 2 MPa. Figure 9 shows a surface defining states of maximum mobilized friction angle for all possible values of initial void ratio and confining pressure. Any triplet (e0, a3,Vmax ) plotted below this surface corresponds to a state of yielding producing either contraction or dilation. It is clear that there is a general tendency for maximum mobilized friction angle to decrease with increasing initial void ratio especially for the low confining pressure range. At high confining pressures, say 4MPa, any initial void ratio will make the material response achieve a maximum mobilized friction angle equal to pCpcv, state at which there is no dilation. Also, for initially very loose states, no dilation will be obtained for any confining pressure as the maximum mobilized friction angle will coincide with t,~,,,.Therefore, the deviation of the maximum mobilized friction angle surface from the horizontal plane corresponding to cpCV = 34” reflects the influence of stress-dilatancy.

measured, loose

5

u

calculated, dense

u

calculated, loose

7.5

10

Confining pressure erg (MPa) Fig. 8. Influence

of confining

pressure

on maximum

mobilized

friction

angle.

R. G. Wan and P. J. Guo

126 46

maximum 42 mobilized 4. friction angle

confining pressure kPa 0

Fig. 9. Variation

Stress-dilatancy

of maximum

mobilized

-0.6

initial void ratio

friction angle with initial void ratio and confining pressure.

plots

The evolution of stress-dilatancy with deformation history is plotted in Fig. 10(a) and (b) for the two initial void ratios discussed earlier. Here, only numerical predictions are shown because it is difficult to get stress-dilatancy plots for comparison from Lee and Seed’s original paper [39]. Referring to the initially dense sand, Fig. 10(a) shows a consistent increase in mobilized stress ratio with dilatancy factor D. The rate of dilatancy is maximum for the lowest confining pressure, i.e. 03 = 100 kPa, while for ~3 larger than 2 MPa, the volumetric strains tend to be mainly contractant (D < 1) with less dilation. If the confining pressure is further increased, say above 4MPa, the tendency for the sand to dilate will be completely suppressed even though the sample was initially very dense (eo = 0.61). The stressdilatancy plots for initially loose sand (q, = 0.87) are shown in Fig. 10(b). They reveal mainly contractant behaviour except for confining pressures greater than 50 kPa. Furthermore, it is interesting to note that under the same confining pressure, for example 03 = 100 kPa, the initially dense sand has a much higher maximum dilatancy rate and stress ratio than the initially loose sand. In general, the maximum ratio R is influenced by the dilatancy factor; in order to sustain an increasing dilatancy rate, the stress ratio has to increase. After reaching a peak dilatancy rate, the stress ratio will decrease gradually to approach Critical State with zero dilation rate (D = 1). In conclusion, we emphasize that the dilatancy plots obtained from the model indeed describe the complete stress-dilatancy history of sand, an aspect which the original Rowe’s stress-dilatancy relationship (R = KJI)

A simple constitutive model for granular soils (a)

127

6

Dense sand e, = 0.61

1

0 0

0.5

1

1.5

2

2.5

3

2

2.5

3

D = l- d&“/d&,

(b)

0

0.5

1

1.5 D = l- d&“/de,

Fig. 10. (a) Stress-dilatancy pressures. (b) Stress-dilatancy

plots for dense Sacramento plots for loose Sacramento pressures.

River sand at different River sand at different

confining confining

could not capture adequately. Figure 11 shows a plot of maximum mobilized stress ratio values against the corresponding dilatancy factor. The points normally do not lie on the KCyline as implied by the original Rowe’s stressdilatancy equation, but they fall on a line found in between the Kc, ((pcy = 34O) and K p ( pp = 24O, see [39]) lines. It can also be seen from Fig. 11 that the proposed model gave predictions with higher dilatancy than the measured values; the reason for this has been discussed earlier.

128

R. G. Wan and P. J. Guo

0

0.5

1

2

1.5

2.5

3

D = 1 - (dev/de,), Fig. 11. Maximum

stress ratio vs maximum

dilatancy

factor plot for Sacramento

River sand.

Prediction for conventional triaxial extension (CTE) stress path

In order to test the performance of the model in different stress conditions, the behaviour of Sacramento River sand was investigated in both conventional triaxial compression (CTC) and conventional triaxial compression extension (CTE). It is reminded that in CTE tests, the radial stress a, is gradually increased while keeping axial stress a, constant so that a, = ~1 = ~2, a, = ~3. Therefore, for the same ~3, the mean stress in CTE is greater than that in CTC as p = (201 + a3)/3. The material parameters used in the simulation are the same as those given in Table 1. Also, the stress path dependent parameter o was kept the same as the one used in CTC due to lack of experimental data in CTE. In Fig. 12(a), the stress-strain responses are compared for dense Sacramento River sand (eo = 0.61) in both CTC and CTE stress paths. Only experimental data in CTC are plotted since the CTE data was not available. There is good agreement between model prediction and experimental data for the CTC stress path at all confining pressures. For the CTE tests, a smaller stress difference, hence friction angle, is mobilized at peak in comparison to the CTC tests. The maximum mobilized friction angle is mainly influenced by the current and critical void ratios [Eqn (12)]. Each one of these void ratios depends on mean stress p following separate rules mainly controlled by parameters m, n,,, hr and h,,. For the set of parameters used (Table 1) with a constant LY,a larger p in CTE led to a smaller maximum mobilized friction angle, hence maximum stress difference, than in CTC. The calculation of a smaller maximum mobilized friction angle pf in CTE is

A simple constitutive model for granular soils

(4

7000

129

L

6000 5000 4000 3000 2000 1000 0

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.2

0.15

axial strain -0.15

-0.1

-0.05

0

0.05

” LI I ’ -0.2 -0.15

I “I -0.1

I ‘I ‘I I “‘I -0.05

/ ‘I ’ I 1 “‘c 0

0.05

0.1

I “I’

1 ‘I ’

0.15

0.2

axial strain

Fig. 12. (a) Stress difference

vs axial strain for CTE and CTC tests. (b) Volumeric axial strain for CTE and CTC tests.

strain vs

consistent with experimental findings such as reported in Vaid and Sasitharan [40], for example. It is also interesting to note that the model consistently predicts different vf in CTC and CTE by virtue of its dependency on stress and void ratio levels as described in the section Maximum mobilized friction angle variation with confining pressure, above. If the value of a was properly chosen, it would help towards explaining why in some cases, qf in CTE is greater than in CTC, as found in Green [41].

130

R. G. Wan and P. J. Guo

Figure 12(b) shows the volumetric responses. As expected, the volume change curves for the CTC case compare well with the experimental data. On the other hand, it is found that more compaction is generally obtained in the CTE case than in CTC due to a higher mean principal stress in the former. This observation is again consistent with the findings of Vaid and Sasitharan [40] regarding the influence of stress path on stress dilatancy of sands.

SUMMARY

AND

CONCLUSIONS

This paper demonstrates that a constitutive model based on simple concepts can be used to describe successfully the mechanical behaviour of granular soils for different densities and stress levels. All these parameters in the model have very clear physical meaning and can be obtained from standard tests. By introducing a void ratio dependent term into the original Rowe’s stressdilatancy equation [15,16] and a special hardening rule, we can replicate various aspects of sand behaviour. In particular, the stress and volumetric responses exhibited by sand at different stress levels together with initial void ratios can be properly modelled using a unique set of material parameters. As such, it was found that numerical results are in good agreement with the experimental data published by Lee and Seed [39] for Sacramento River sand. Results for the modelling of Sacramento River sand indicate the following: (1) In support of experimental results, the model describes accurately stress-dilatancy characteristics of sand at different confining pressures and initial void ratios using only one set of material parameters. Amongst others, the model can capture subtle features such as the suppression of dilation and strain softening at high confining pressures even though the sand was initially dense. (2) Using a unique value of 34’ for qpcy,it is possible to obtain physically consistent results such as the achievement of the same critical void ratio at the same confining pressure for sands. These were initially at totally different void ratios. (3) The calculated variation of maximum mobilized friction angle with maximum dilatancy factor agrees very closely to experimental data. This could not be obtained if the original Rowe’s stress dilatancy law was used. (4) In drained conventional triaxial extension stress paths, the model seems to give results consistent with experimental findings, i.e. more compaction is obtained than in conventional triaxial compression.

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A simple constitutive model for granular soils

In closing, we would like to point out that no attempts have been made to describe other important issues such as stress-induced anisotropy, cyclic loading and 3-D stress conditions in the model. These aspects are presently being addressed and results will be presented in forthcoming publications.

ACKNOWLEDGEMENTS The authors are grateful to funding received from the Natural Science and Engineering Research Council of Canada and the National Energy Board of Canada. REFERENCES 1. Bathurst, R. J. and Rothenberg, granular assemblies with linear Mechanics,

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