Li , X. S. & Dafalias, Y. F. (2000). Gc!U/(!Chllicjl{(! 50, No.4. 449-460

Dilatancy for cohesionless soils

X . S. L1 * and Y. F. DAFALIASt

Dilatancy is often considered a unique function of the stress ratio 1/ = q / pi, in terms of the triaxial stress variables q and p'. With this assumption, the direction of plastic flow is uniquely related to 1/, irrespective of the material internal state. This obviously contradicts the facts. Consider two specimens of the same sand, one is in a loose state and the other in a dense state. Subjected to a loading from the same 1/, the loose specimen contracts and the dense one dilates. These two distinctly different responses are associated with a single 1/ but two different values of dilatancy, one positive and the other negative. Treating the dilatancy as a unique function of 1/ has developed into a major obstacle to unified modelling of the response of a cohesionless material over a full range of densities and stress levels (before particle crushing). A theory is presented that treats the dilatancy as a state-dependent quantity within the framework of critical state soil mechanics. Micromechanical analysis is used to justify and motivate a simple macroscopic constitutive fra­mework. A rudimentary model is presented, and its simula­tive capability shown by comparison with experimental data of the response of a sand under various initial state and loading conditions.

KEYWORDS: constitutive relations; plasticity; sands.

INTRODUCTION The concept of critical state (Roscoe et al. , \958) has been successfully applied to modelling the behaviour of cohesive soils. However, sand modelling has not always been formulated within the critical state framework, and this is because the behaviour of sand is somehow different from clay. Consider the stress-strain response of a sand along a path of constant stress ratio 17 = q/ pi, where pi = (01 + 20':.)/3 is the effective con­fining pressure and q = 0 I - 03 is the deviatoric stress in a triaxial setting. First, unlike clay, sand does not possess a unique relationship between the void ratio e and pi for a particular '7. In fact, the density of a typical sand in the pressure range before particle crushing cannot be altered considerably by a constant t7 compression, either isotropic (17 = 0) or anisotropic (17 #- 0). Secondly, when the '7 of a sand reaches its limiting valuc M (the critical stress ratio) during plastic loading, it does not necessarily follow that the sand is at a critical state. The stress path can actually move along the '7 = M line, as for example in an undrained dilative shear path up to ultimate failure. These differences suggest that the well-established framework for clay modelling should not be directly trans­planted to sand without a careful examination.

One of the fundamental issues in modelling the stress- strain behaviour of a soil is to correctly describe its dilatancy d, the ratio of plastic volumetric strain increment to plastic deviatoric strain increment in the triaxial space : d = dE~ / ld E l~ l, where dE \" = dEl + 2dE3, dEq = 2(dfl - dE, )/3, and the superscript' p ' stands for 'plastic' (Roscoe & Burland, 1968; Nova & Wood, 1979; Wood, 1990; Wood el al., 1994; Vardoulakis & Sulem, 1995). The second law of thermodynamics shows that '7 and d are interrelated at a very fundamental level (Vardoulaki s &

Manuscript received 18 October Il)l)l); revised manuscript accepted 28 January 2000 Discussion on thi s paper closes 26 November 2000. * The Hong Kong University of Science and Technology. "I" University of California at Davis.

La dilatance est souvent consideree comme une fonction unique du rapport d'effort 1] = q/ pi, en termes de variantes de contrainte triaxiales q et p'. Avec cette hypothese, la direction de l'ecoulement plastique est Iiee uniquement it '1),

quel que soit I'etat interne du materiau. Ceci contredit les faits de maniere evidente. Prenons deux specimens d'un meme sable, Pun meuble et I'autre dense. Soumis it un chargement du meme '1), Ie specimen meuble se contracte et Ie specimen dense se dilate. Ces deux reponses bien differ­entes sont associees it un seul '1) mais it deux valeurs de dilatance differentes, une positive et I'autre negative. Le fait de traiter la dilatance comme fonction unique de '1) est devenu un obstacle majeur it la creation de modeles unifies de la reponse d'un materiau non cohesif sur toute une gamme de den sites et de niveaux de contraintes (avant Ie broyage des particules). Cet expose presente une theorie qui traite la dilatance comme une quantite dependante de Petat dans Ie cadre de travail de la mecanique de sol it Petat critique. Nous utilisons une analyse micromecanique pour justifier et motiver un cadre de travail constitutif macrosco­pique simple. Nous presentons ensuite un modele rudimen­taire et nous montrons sa faculte simulative par une comparaison avec les donnees experimentales de la reponse d'un sable dans divers etats initiaux et sous diverses condi­tions de charge.

Sulem, 1995). Taylor (1948) proposed 17 + d = constant, based on the hypothesis that a constant ' effective' friction coefficient exists. Rowe (1962) showed, based on the theory of least rate of internal work, that d could be expressed as a function of the stress ratio and the true angle of friction between the mineral surfaces of the particles. Although the particular forms proposed by Taylor & Rowe were different, both of them suggested that the dilatancy d was a unique function of the stress ratio I]:

449

d = d(17 , C) (I)

where C is a se t of intrinsic material constants. Equation (I) worked quite satisfactorily for cohesive soils.

For example, d in two versions of Camclay models (Roscoe & Schofield , 1963; Roscoe & Burland, 1968) were given by d = M - '1 and d = (M 2

- 172 )/ 217, respectively, where M is the critical stress ratio, an intrinsic material constant. In agree­ment with the concept of critical state, these models make sure that the soil yielding at '1 = M is coincident with d = 0; that is , the material being modelled reaches its ultimate failure whenever a plastic deformation takes place at '1 = M.

In contrast, it was soon found, based on experimental evidence, that the applicability of equation (I) to granular soils depends on the density. Observing the divergence between the proposed theory and the test results, Rowe (1962) pointed out that a variable depending on the sample density and the stress history should be added to the stress - dilatancy relationship that he had derived earlier. Rowe attributed the divergence to a rearrangement of particle packing, a fact that was ignored when his stress- dilatancy relationship was derived .

Rowe 's \Nork was followed in many later inves tigations (e.g. Nova & Wood, 1979; Pastor el al. , 1990; Wood, 1990; Jefferies, 1993 ; Wood el al. , 1994) on sand modelling, in which, however, the dependence of d on the material internal state was consid­ered insignificant and thus dropped. This simplification leads to the cOl11l11on practice that treats a sand with different initial densities as different material s and results in multiple sets or

450 Ll AND DAFALlAS

parameters for a single sand and thus does not have a good control over changes in the material state during loading.

Attempts have been made in recent years to tackle this issue from the perspective of dilatancy. With the concept of critical state as basis, Been & Jefferies (1985) introduced a scalar quantity Vi called the state parameter, which measures the difference between the current and critical void ratios at the same p' . Kabilamany & Ishihara (1990) provided experimental evidence showing that d + 'q increases as shear deformation increases. Manzari & Dafalias (1997) presented a sand model in which a linear dependence of the phase transformation, or dilatancy stress ratio (the stress ratio at which the response changes from contractive to dilative) on Vi was introduced. Li (1997) investigated the response of sand at the ultimate stress ratio and explicitly pointed out that the dilatancy d is not related only to the stress ratio but is also a function of plastic volumetric strain. More recently, Wan & Guo (1998) proposed a model with its dilatancy modified from Rowe's stress-dilatancy equation . The modified dilatancy equation includes the density dependence with the critical void ratio as a reference. Cubrinovski & Ishihara (1998) also showed a dilatancy relation­ship that depends on the material state represented by cumula­tive plastic shear strain. Li et 01. (1999) introduced a state­dependent dilatancy .into an existing hypoplasticity sand model (Wang et al., 1990), resulting in a successful simulation of the responses of Toyoura sand to both drained and undrained triaxial loading over a wide range of densities and pressures.

In the present paper a number of issues on this subject are discussed, starting from some microscopic analytical considera­tions and ending with the presentation of a simple macroscopic constitutive framework and modelling, the simulative capability of which is shown by comparison with experimental data of the response of a sand under various initial state and loading conditions.

MICROSCOPIC OBSERVATION ON DILATANCY Consider a pack of uniform rigid rods, packing A, as shown

in Fig. I. Rowe showed that

0; tan(¢fJ + (3)

02(\ + dV / VEt} tanf3 (2)

where (J I and 02 are the major and minor principal stresses, respectively; EI is the strain in 0; direction (tak ing compression as positive for both stress and strain); V is the volume of the pack ; ¢p is the angle of friction between the surfaces of the

L~. --II~

L, --II~

i. \

I L .

Fig. I. Regular packing A of a uniform rod

1\ = 60

particles; and f3 is the deviation of the tangent at the contact points from the major principal direction. The quantity D = I + d V / VEl is clearly a form of dilatancy measure.

As shown in Appendix I, resulting from the equilibrium condition at rod contacts, the stress ratio R = 0; / 02 is uniquely related to f3 as follows:

0; R = - = tan(q')" + (3) tanf3 (3) 02 ...

and resulting from kinematical compatibility condition, the void ratio

e = 8 sin f3 cos f3 _ 1 :rr

(4)

By combining equations (2)-(4) with f3 as an implicit variable, a unique relation between Rand D is established, with e or f3 as an implicit dependent variable. That is , Rand D are uniquely related, but the value of R (or D) depends on the volume of the packing. This dependence is due to the equilibrium and kine­matic constraints imposed by the given packing.

Now consider a different packing B, as shown in Fig. 2. For this packing equation (2) is still valid but equations (3) and (4) are not. As shown in Appendix I, for this packing

0; 2sinf3 --, = tan(¢fJ + (3) I + 2 f3 (5) 02 cos

and

e= 8 sin f3( I + 2 cos (3)

3:rr (6)

It shows again that there exists a unique relationship between R and D, with e or (3 as an implicit variable. However, the relationship for this packing is different from that for the packing A.

Rowe applied equation (2) to random mass of irregular particles based on the hypothesis that the rate of internal work done is a minimum. This hypothesis yielded f3 = 45° - ¢fJ/2. Thus,

(7)

Equation (7) is the well-known stress- dilatancy equation. It can be seen that the minimization procedure makes the stress ratio R uniquely related to D and independent of the packing of the particles and the volume of the mass, in contradiction to the

L L,

DILATANCY FOR COHESIONLESS SOILS 451

L?

L2

Fig. 2. Regular packing B of a uniform rod

exact analytical conclusion reached by the sets of equations (2)-(4) and (2), (5), and (6), for the two examples of different packing arrangements.

Figure 3 shows the relationships between (a; - aD/ (a; +02) and -dV/Vfl (a form of plots similar to 1] versus d plot in the triaxial setting) for the packings A and B, and the stress- dilatancy equation (equation (7) , respectively. Fig. 3 indicates that the relationships between the stress ratio and the dilatancy depend on microscopic constraints. At a given stress ratio, a particular packing is associated with a particular void ratio, reflecting the internal microscopic constraints. Therefore, the dilatancy depends not only on the stress ratio, but also on the void ratio.

This density dependence is not reflected in equation (7), this being the result of an unconstrained minimization of the rate of internal work. In the derivation, the rate of work was obtained from microscopic observations on regular packings of particles sliding in a given direction, but the rate minimization was done by zeroing its derivative with respect to the sliding direction. This approach implicitly treated a particulate system as a continuum without considering the static and kinematical con­straints at the particle contacts. This treatment captured the main feature of dilatancy and led to a unique relationship between the dilatancy and the stress ratio . However, as it does not take into account the microconstraints. which vary as the

10 r----------------- - --.-- - --,

O~; . ...........

+ .. 0'6

~, 't; 0-4

I -- F'ael--in9 A

0·2 -- Packing B

Equation Ci!

C' not specHI 'cl

<! 0: (1 ,261

2

·· · dVi V~;

·1

Fig. 3. Deviation from the stress-dilatancy equation due to con­straints at contacts

L2

Lz

I .. .

.Lt.

L

L1 L1

material state changes, the theory also shows notable deviations from experimental observations. It is commonly observed that at low deviatoric strains both dense and loose samples show contractive behaviour and the dilatancy at that stage is not so related to the material state . However, as shearing increases, the deviation between the dilatancy of loose and dense specimens becomes increasingly pronounced. This phenomenon can also be seen in Fig. 3. As any microscopic constraints resulting in a deviation from the hypothetical sliding direction f3 = 45° -cjJfJ/2 will increase the rate of internal work, these constraints tend to increase d. In Fig. 3, at the same (a; - 02)/(0; + aD, the stress-dilatancy equation (equation (7)) yields the lowest value of -dV / VEl'

PROBLEMS WITH UNIQUE RELATIONSHIP BETWE EN d AND 1/ Although the dependence of dilatancy on the internal state of

the material was noted decades ago, it has been largely ignored in subsequent developments. This fact is, however. not trivial, as a major obstacle to unified modelling of sand behaviour arises from this ignorance.

When subjected to shear, loose sand contracts and dense sand dilates. According to critical state soil mechanics, a loose or dense state is defined not only in terms of density but also of the confining pressure. This is because such a definition is relative to the critical state line in the e versus p' space. For a given e, for example, the sand will behave like dense for a sufficiently low and like loose for a sufficient ly high p'. Furthermore, for a sand that initially is either in the loose or dense state, there is an ultimate state of failure at which the volumetric strain rate is zero. This ultimate state is the well­known critical state (Roscoe el al. , 1958) characterized by a unique combination of p', q and critical void ratio ec in a triaxial setting.

Consider two specimens of the same sand. One is in a loose state and the other in a dense state, accounting for both density and pressure . Subjected to a shear loading increment from the same '] , the loose specimen contracts and the dense specimen dilates, as shown in Fig 4 (data from Verdugo & Ishihara (\996» in terms of undrained stress path in q- p' space. These two distinctly different responses are associated with a single rl but two different values of dilatancy, one positive and the other negative. However, if d were a unique function of 17. the direction of plastic flow, and hence of the undrained stress path, would be uniquely related to 17, irrespective of the material

452 Ll AND DAFALIAS

0:;

2;: 0-

~ ~ 0 ro ;; Q)

0

1GOO

1400

1200

1000

800

600

400

200

o ."" o

" " 0" ,,,"1 ')0/0//0/

200

...........

400 600 800 1000 Effective mean normal stress . p ' : kPa

(a )

1200

0:;

2;: '0' ui <fl Q)

~ 0 ro ;; <ll 0

'.000 r-------------------,,-----,

1500

1000

500

e = 083:3 Dr = 3'7 '9%

Dense state, d < 0

1500 Effective mean normal stress. p' : kPa

(b )

2000

Fig. 4. Variation in dilatancy with material state (data from Verdugo & Ishihara (1996» . Undrained response of a sand with (a) different densities and (b) the same density but under different confining pressures

state. This assumption obviously contradicts the fact as de­scribed.

Now consider a sand in a dense state subjected to an undrained shear. As shear proceeds, '7 passes a so-called 'phase transformation state' at which '7 = Md and d = 0 (Ishihara et at., 1975) and then approaches the critical state at which '7 = M and d = O. If equation ( I) held true , Md wou ld be equal to M , because d is equal to zero at both the phase transforma­tion state and the critical state . As the critical stress ratio M is considered an intrinsic material property, independent of the in itial material state, the logical outcome M" = M from equa­tion ( I) would render the phase transformation intrinsic too , resulting in a unique phase transformation line for a particu lar sand at which the response of the sand would change from contractive to dilative , irrespective of its density and stress level. However, tests show that the phase transformation phe­nome non can be seen only when the material is in a dense state, and M" is in general a variab le quantity not eq ual to M. As sand becomes 'looser' , Md becomes hi gher. This is clearly corroborated by test data such as those shown in Fig. 5 (Verdugo & Ishihara, 1996), where MrI is identified by the dark

. circles on the q- pI path where the tangent is parallel to the q

4000 ,------------------------,

~ ~IOOO

Ul

~ "O()(1 tn L. I

C> (ii ;; 2; 1000 ·

Va rieci pl l ase tidflsior lTIatiC!n

GOO 1000 1500 200\) 2500

Efiecl ive mean narrn,]1 sir ' 5S, p ' kP,] 3000

fig. 5. Variation in the phase transformation stress ratio with material state (data from Verdugo & Ishihara (1996»

axis. Eventually, when the sand becomes too loose, the phase transformation phenomenon totally disappears. The assumption that d is uniquely related to 1J again contradicts the observation .

Furthermore, undrained tests on dense sand often show that the q- pI stress path eventually converges with a line of more or less constant 1J = M towards an ultimate state (Figs 5 and 6). At the ultimate state, both the stresses and the plastic volumetric strain stop changi ng, as shown in Fig. 6 (see also Figs 10- 12). Since the stresses do not change, neither does the elastic volumetric strain . By definition , this is a critical state where dpl = dq = dE" = 0 while dEq =f. O. Along the approxi­mately '7 = M path and before the critical state is reached, the fact that Ee tends towards a constant value implies that the dilatancy d tends towards a zero va lue . If d were a unique function of 'I , however, along this path d wou ld be essenti ally a constant, which means that, as shear proceeds, pI would in­crease continuou sly as a result of the undrained constraint of zero total vo lumetric change, and the critical state would never be reached. One may argue that along this path the stress ratio 'I only approaches M asymptotica ll y, and correspondingly the evolu tion of d from a non-zero value towards zero would be a result of the tiny deviations in '7 frolll M . However, this argument on ly facilitates a mathematical description that barely makes equation ( I) not violate the concept of a critical state. Considering the uncertaint ies involved in soil testing, one would not be able to physically identify and quantify sllch tiny deviations , if any, in a meaningful manner. On the other hand, wit h the hypothesis that d and '] are not uniquely re lated. such an argument becomes unnecessary.

The above observations lead to the concl usion that a sand model with its dilatancy fo llowing equatio n (I) works well on ly when the change in the material internal state is minor.

GENERAL EXPRESSION FOR STATE-DEPENDENT DILATANCY Based on the aforementioned observat ions and accounting for

the critical state constitutive framework , one Illay propose a general expression for the dilatancy:

d = dUI , e, Q, C ) (8)

where Q and C, as collective terms, denote internal state variables other than the vo id ratio (! (e.g. the evolving tensor of

DILATANCY FOR COHESIONLESS SOILS 453

Ultimate/critical slate

i ,1 -~ 0)

dL P

q (d = 0) ... . .. .. .. ........ . . -... ... .... .. .. .... .. .. .. ...... .. ...... .. -.. _. _._. -.. . -. -- .. _. _. _. - .:.;-..... "-; .. ~ ..

(d;i 0)

Phase tr8ns forrnation

(8) (b)

Ultimate/critical states

( q

Fig. 6. Illustration of the dilative shear on the failure surface: (a) stress path; (b) stress-strain response

anisotropy (Dafalias, 1986)) and intrinsic material constants, respectively. Equation (8) expresses the dependence of d on the state variables, which consist of the external variable '7 and the internal variables e and Q. Hence, equation (8) defines a state­dependent dilatancy. Strictly speaking, a dilatancy expressed by equation (I) is also related to material state via 1]. However, as discussed above, such a relationship is not unique because it is not complete. The term 'state-dependent dilatancy ' introduced here signifies and emphasizes the need to define state depen­dence on both '7 and e and Q. With this additional dependence, d is now uniquely related to an existing state, a combination of the external stress state expressed via '1] , and the internal material state expressed via e and Q. A subtle point here is that, although it may appear that no explicit dependence of d on p' is introduced in equation (8) , the dependence on e and Q may in fac t introduce indirectly such p' dependence.

There are certain requirements to be satisfied in formulating d within the framework of equation (8) . First , the dilatancy must be zero at a critical state; that is, d = 0 when II = tv! and e = e" (the vo id ratio at the critical state) si multaneously:

dUI = M , e = ee, Q, C) = 0 (9)

In other words, the condition '7 = M alone does not guarantee d = O. It is important to emphasize in relation to equation (9) that when 17 and e attain their critical values M and ec it is not necessary for Q to reach a corresponding critical value. It is entirely possible to reach a critical state with different values of Q.

Secondly, it is possible for sand to have a so-called ' phase transformation state' at which d = 0 but 17 i- M and e i- ee, as discussed earlier. Analytically thi s means that the equation

( 10)

can be lI sed to specify the combination of 'I , e , and Q that defines a phase transformation state . Conversely, one may use an ([ priori experimental knowledge of phase transformation states to specify via equation (10), together with equation (9) , appropriate forms of the dilatancy function in equation (8).

PARTICULAR EXPRESSION OF STATE-DEPENDENT DILATANCY To obtain d within the framework of equation (8) subjected

to the requirement of equation (9) for the critical state response , one needs to quanti fy the dependence on e and the variables Q. As the state of a material depends not only on its density (void ratio e) but also on p', the aforementioned quantification should be able to describe adequately the physical conditions of a material, including both its density and its confining pressure .

Attempts have been made to describe the state, on which d depends, with a single scalar quantity, which of course implies the assumption of isotropy, since otherwise the use of tensor­valued quantities is necessary. Been & Jefferies (1985) defined a state parameter 1/' = e - ee, where e is the current void ratio and ee is the critical void ratio on the critical state line in the e-- p' plane corresponding to the current p', as shown in Fig. 7. Here V, is a measure of how far the material state is from the critical state in terms of density. Bolton (1986) proposed a scalar parameter h , called the 'relative dilatancy index', that also combined the influence of density and confining pressure. Ishihara (1993) introduced a scalar quantity Is, called the 'state index', that takes some characteristic states other than critical state in the e- p' plane as references .

u B >

f

(pip):

ClllTen l sl<l te ;0

Il' > 0 (contractive)

Criticl:I l stale jin '

Fig. 7. Critical state line and state parameter IfJ

454 Ll AND DAFALIAS

In the present study, 1/.' was chosen to be the state va ri able that, in conjunction with '7, affects d. For improving the fitting with experimental data of certain sands, V.' is represented by

(II)

where er, Ac and £ are the material constants determining the critical state line in the e- p' plane (Li & Wang, 1998), and Pu is the atmospheric pressure for normalization. The dependence of don e occurs via its dependence on V, = e - ec( p') (observe introduction of explicit p' dependence via ec), while any other dependence on Q is suppressed. Hence, an equation of the following form is proposed:

d = d(rJ , l/J , C) (12)

which, according to equation (9), must satisfy the condition d(17 = M, 1/J = 0, C) = O.

To illustrate analytically the effect of dilatancy on the stress­strain relationship, consider undrained triaxial loading during which dev = de~ + de~ = O. With deq = de~ + de~ and the elas­tic relations de~ = dp'IK , de~ = dql3G in terms of the elastic bulk and shear moduli K and G, respectively, the condition dev = 0 yields deq/dq = (l/3G)-(dp'/dq)IKd , recalling that d = dee! de~. When the stress path reaches and moves along a line of more or less constant stress ratio 17 = M (failure surface ; see Fig. 6(a», one has dp' /dq = II M, and hence the above relationship becomes (Li , 1997)

(deq) _~ _ _ I dq 11 =114- 3G KMd

( 13)

Equation (13) portrays an analytical conclusion for a general elastoplasticity class of constitutive setting, not related to a specific model. This equation states that the deq/dq, while 17 = M, is controlled primarily by the dilatancy d = deC/de~, which defines the direction of plastic strain increment in the volumetric-deviatoric plane. Equation (13) also shows that (de q /dq)'7=M is independent of plastic hardening within the approximation implied by setting dp'/dq = II M .

Figure 6(b) illustrates a typical response described by equa­tion (13) for medium-to-dense sand. The shear stress- strain curve is characterized by a slope dq I df q which, for lower q values, keeps increasing, but as q reaches higher values begins to decrease as the stress-strain curve bends over, eventually leading to the critical state a t which dp = dq = dE" = 0 while dEq =I O. Along the path to the critical state, while 17 is constant, d changes tending towards a zero value. Recalling that the state parameter 1/' enters equation (12) for d, and that d is the main variable, which according to equation (13) can be used to describe the curve of Fig . 6(b), one may propose a form of equation (12) for 17 = M:

d = do((' 111 '/' - I ) ( 14)

in which do and 111 are two positive modelling parameters. Observe that d and 1/' have the same sign and d = 0 when II' = 0, which sati sfies the relationship between the dilatancy and the internal material state at the critical state.

While equation (14) may describe the dilatancy for I} = M, one needs to generalize it for 17 f M . In this investigation , the specific form of the lJI dependent dilatancy according to equa­tion ( 12) subjected to y ielding (equation (14» when '7 = M is obtained from a modification to the dilatancy function in the original Camclay model , as follows:

( 15)

It can be seen that the Camclay dilatancy d = M - 1} is a special case of equation (15) (111 = 0 and d o = M). Note that at I} = M equation (15) is reduced to equation (14). More impor­tantly, at a critical state, VI = 0 and '7 = M s imultaneously, and hcnce equation (15) yields a zero dilatancy, satisfying the requireme nt set by equation (9).

Applying now equation (10) to the particular forlll of d, (equation ( 15)). one obtains the phase transformation stress ratio 17 = M" = Me ""1'. This yields the following interesting interpre­tation to the last member of equation (15) for d. It postulates that the dilatancy d depends on the difference of the current stress ratio 1} from a reference stress ratio Me 1111/', which is similar to Rowe's stress- dilatancy theory but with the reference stress ratio varying with 1/J instead of being fixed. Based on equation ( 10) this reference stress ratio represents the variable with 1/J phase transformation line. This is exactly the concept described by Manzari & Dafalias (1997), who used the linear dependence Md = M + m1/!. The same concept was recently used by Li et al. (1999) to improve the performance of the hypoplasticity model proposed by Wang et al. (1990). In other words, one could have started with the hypothesis that Md = Me 1111/', corroborated by data, such as those shown in Fig. 5 where the variation in the phase transformation line can be clearly seen, and then define d by d = (dol M)(M d - rJ), ac­cording to the classical stress- dilatancy framework . It follows now that 1/' < 0 (dense states) implies Md < M, 1/' > 0 (loose states) implies Md> M, and l/J = 0 (phase transformation states) implies Md = M.

A SIMPLE MODEL FOR TRIAXIAL COMPRESSION Within the scope of this paper, it is assumed that plastic

deformation occurs whenever 17 exceeds its historic maximum and a constant rJ path induces no plastic deformation. This is, of course, only approximately true, but it is still a good approximation in many cases since under nonnal levels of confining pressures of interest a constant 17 path induces only a relatively small plastic volume change in sands, before grain­crushing levels of pressures are reached as corroborated experi­mentally by Poorooshasb et al . ( 1966, 1967). However, for a fully fledged model where the plastic deformations under con­stant rJ are to be considered, additional mechanisms, such as a pi controlling cap, can be added with ease (Wang et al., 1990).

Under the above assumption, the yield criterion can be written as

r = q -lIP' = 0 (16)

By the theory of plasticity Wafalias, 1986). a loading index L can be defined as

I (Of , of ) L =- -,dp + -::----dq Kp up ()q

dq - '7 dp' P'd17

Kp Kp (17)

where Kp is a plastic hardening modulus. With the dilatancy d = d E~)/ d t~;, the plastic strain increment can then be written as

{ d f~ } _ L{ 1 } _ { p' dl}1 Kp } dE~ - d - d p ' dlJ l Kp

( 18 )

Therefore. for L > 0 :

_ c p _ dq p' elI7 _ df q - d Ep + d Eq - 3G + -;z;:- ( I 1) '7 -+- dq --dp'

3G Kp Kp

( 19)

c dp' ) d ( I d17) de\'= dE \, + dE~ =- +d df~=-dq+ ~-- dp ' K Kp 1\ Kp

(20)

Equations (19) and (20) establish the relationship between the stress and strain increments. They can be inverted by a straight­forward algebraic manipulation, and expressed in a matri x form as:

DILATANCY FOR COHESIONLESS SOILS 455

{:;, } =

[( 30G 0) h(L) ( 9C

2

K Kp + 3C - K1}d 3KCd

(21 )

where h(L) is a Heaviside function with heLl = I for L > 0 and h(L) = 0 otherwise . Note that the extension of the foregoing relationship to account for reverse loading requires some addi­tional mechanism such as a back-stress (Manzari & Dafalias, 1997), or a memory of the reversal stress ratio point (Wang el 01., 1990). These aspects are not addressed within the restricted scope of this paper.

For the model to be completed, in addition to the dilatancy d, which is defined in equation (15), the moduli C, K, and Kp must also be defined. The elastic shear modulus G can be expressed by the following empirical equation (Richart el 01., 1970):

(2.97 - e)2 ~ G = Go V P Pa

I + e (22)

where Go is a material constant, and e is used instead of initial void ratio employed in Richart et af. (1970). Based on elasticity theory, the elastic bulk modulus K is equal to

K = G _2_(1_+_v_) 3(1 - 2v)

(23)

where v is the Poisson's ratio. In this model, 11 is considered as a material constant independent of pressure and density.

For the plastic modulus Kp the following constitutive relation is proposed:

(24)

where hand 11 are two positive model parameters; the state parameter 1/1 is calculated from equation (II) . Equation (24) is a modified version of the plastic modulus in a bounding surface hypoplasticity model (Wang et aI., 1990). The modification is intended to model the peak stress ratio response and softening of dense sands, the lack of such a response for loose sands, and the failure at a residual stress ratio M at the critical state for all densities. Based on the last member in equation (24), it follows that Kp depends on the difference of the current stress ratio 'I}

from a 'virtual' peak stress ratio MO = Me- llIl' attainable at the

current state defined by 1/'. Such peak stress ratio is variable with 1/' in a way that yields MO > M for 1/' < 0 (dense states), MO < MfaI' 1/ ' > 0 (loose states), and Mb = M for l/J = 0 (critical states) . The idea of having a virtual peak stress ratio varying with 1/' in order to address the issue of peak stress and subsequent softening of dense sand in drained conditions was proposed by Wood ef al. (1994). The idea was followed by Manzari & Dafalias (1997), where a slightly modified version of the Wood ef 01. (1994) linear relation Mb = M - m/' was introduced instead of the present M h = Me - 111/' .

Observe that it follows from equation (18) that Kp = p' dll ldf~ , which requires the Kp function (equation (24» to satisfy the following conditions:

(a) Kp = 00 at 17 = 0, because dE~ = 0 in response to a non­zero dl7 at J7 = 0 (the material IS assumed isotropic without previous loading);

(h) Kp = 0 at the critical states (I) = M and 1/, = 0) because at a critical state dl7 /dE~ = 0;

(e) Kp = 0 at drained peak stresses M b, because at those peaks

d}}ldE~ = O.

For the hardening and softening responses before and after the peaks , Kp is positive and negative as 17 < Mb and 17 > MO, respectively. It can be seen that conditions (a) and (b) are

automatically satisfied by equation (24). Condition (e) will be met at a particular material state 1/1 < 0 (a dense state), for which 1} = Mb = Me - lIl

l', depending on the parameter 11 . In addition to the above conditions, it also follows from equation (18) that dEel P' dry = d l Kp. As dEe/d17 = 0 at 17 = 0 is nor­mally observed, it is necessary to have d I Kp = 0 at '/1 = O. By combining equations ( 15) and (24), it can be seen that this condition is also satisfied automatically, since Kp = 00 at I} = O.

In equation (24), G serves as a normalizing factor of h. It was found that a variable h with density fitted the experimental data better. In the present investigation, the simple linear dependence

(25)

was used, where hI and h2 are two material constants, and e is the current void ratio.

As shown in equation (19), the shear stress-strain response is controlled by Kp. Substitution of equation (24) into equation (19) with a constant p' (a drained condition) and 11 = 0 for simplicity, yields

dEq = (3~ + ;J dq = C~ + hG(q~ _ q») dq (26)

where qr = Mp' is the value of q at failure . Integration of both sides of equation (26) yields

q (I I I )-1 - = -----In(1 - r) Eq 3G hG hGr .

(27)

where r = q I qr. Equation (27) can be converted into a normal­ized modulus reduction curve (secant shear modulus normalized to its maximum value Gmax versus shear strain) with h as a parameter. Fig. 8 shows a family of such curves together with the curve based on the hyperbolic stress-strain relationship ql Eq = 3Gmax /( I + EIEqr) , where Gmax and Eqr are two material constants. It can be seen that the Kp function used here allows more flexibility in calibrating the shear stress-strain response than does using the hyperbolic stress-strain response.

CALIBRATION OF MODEL CONSTANTS There are eleven material constants in the model, shown

according to their functions in separate columns in Table I. A systematic procedure can be followed to calibrate all these parameters, based on triaxial data, as follows.

The critical state constants consist of M, the critical state stress ratio, and the parameters er , ..1. c , and'; of equation (II). These four constants can be determined by directly fitting the

12 ,------,--- --,--- --,---,----,-----,.----,------,

- 11 :: 6 (5

~ (Hi -+-- h :::3

~ ~ 0 4 - --- h::: 1 2 ::; D ~ 02 - ·· ... ·· 11 =06

....... h :~ 0 3

0 03 0 1 0 3 10

Fig. 8. Modulus reduction as a function of the hardening parameter

"

456 LI AND DAFALIAS

Table I. Model parameters calibntted for Toyoura sand

Elastic Critical state Dilatancy Hardening parameters parameters parameters parameters

Gn = 125 M = 1·25 do = 0·88 hi = 3· 15 v = 0·05 er = 0·934 III = 3.5 h2 = 3·05

Xe = 0·0 19 n = 1·1 ,; = 0·7

test data for the critical stress ratio and the critical state line in the e- p' plane.

The parameter 111 can be determined by equation (15) at a phase transformation state, at which d = O. Hence,

I Md m=-In-

ljJd M (28)

where l/Jd and Md are the values of 1/) and 17 at the phase transformation state, measured from drained or undrained test results .

The parameter 11 can be determined by equation (24) at a drained peak stress state, at which Kp = O. Hence,

I M n =-In - (29)

l/) b Mb

where l/Jb and Mb are the values of l/J and 17 at the drained peak stress state, measured from test results .

Next, consider the drained tests. Ignoring the smal1 elastic deformations,

dEy ~ dE~ = d = do (e 1111/' _ !l..) dEq dE~ M

(30)

The parameter do can then be calibrated based on the Ev- Eq curves.

By combining equations (19) and (24) for the drained tests, with either the conventional test (dp' = dq/3) or the constant p' test (dp' = 0), one has

dq ~ dq _ Kp dEq ~ dE~ - 1 - aJ7

{(2.97 - e )2 '; p' Pa[(M /'ll- ell./'J}

= hGo (I + e)( 1- m71

(31 )

in which the parameter (J is either equal to 1/ 3 (for conven­tional tests) or to zero (for constant p' tests) . As al l the material constants in the brackets have been predetermined, the COl1l­

bined parameter hGo ca n be calibrated independently based on the experimental q - ELI curves. It may be found during calibra­tion that the quantity hGo varies with density. Fitting these values of hGn into equation (25) yie lds the constants hi and h2 (aftcr Go has been determincd).

Now let us turn to undrained (constant volume) tests . For dE" = 0, equation (20) yields :

~=l - !.J..=1 _ 3(1 - 2v) {hGLl[(M/lll - ell'/'J} (32) dp' 7 Kd 7 2Go( 1 + v) d

As al1 the material constants in the brackets have been prede­termined, the combined parameter 2Go( I + v)/3( I - 2v) be­comes the only means at this stage of adjusting the undrained p' - q responses of the model. By matching these responses with their experimental counterparts, the value of 2Go( I + v) / 3( I - 2v) can be determined.

Finally, one needs to separate the parameter Go from hand v . If shear stiffness at small strains is important , Go should be determined by independent small strain tests, such as resonant column tests or bendcr element tests , through fitting the test data into equation (22). However, if Go is high, the value of Poisson 's ratio ]I is reduced based on the already calibrated

value of 2Go( I + 1')/ 3(1 - 2v), and could be negative. A micro­mechanics study (Chang & Misra, 1990) has shown that the Poisson 's ratio of an assembly of particles is predominantly control1ed by the ratio of the shear stiffness to the normal stiffness at particle contacts. The value of the Poisson's ratio of the assembly could be much lower than that for the particle material itself. If this stiffness ratio is high, v could be negative. Even though a negative v is theoretically justifiable and affects nothing but the volumetric strain at extremely low strain ievel (pure elastic range) , if it is encountered and disliked, as an alternative one may pick a v value first and then calculate Go. As this alternative approach does not guarantee the accuracy of the elastic shear response, it should be used only when the shear stiffness in the elastic range is unimportant or when accurate values of G are unavailable. Once Go has been determined, h, and therefore hI and h2, can finally be found from equations (25) and (31).

Last, but not least, one can obtain the undrained deviatoric stress-strain response by substituting dp' = K dE~ = - K dE~ = -Kd dE~ into equation (32) and accounting for dq = 3G dE~. The relationship is as follows:

dq

dEq ( I I) -I dq -+ ~- - K -17Kd 3 G Kp - 1] Kd dE~ - p

(33)

As all the model parameters have already been determined, the simulated undrained q-Eq curves can be used against their experimental counterparts to examine the quality of the calibra­tion based on equation (33). If the fit is not satisfactory, one or more parameters are to be fine tuned and the calibration can be repeated until an optimal result is obtained. Note that, after the undrained stress path approximately converges with 1] = M, the Kp value is still not zero as long as ljJ i= 0 (equation (24» . Hence, the second member of equation (33) does not yield equation (13) until Kp becomes very small and is neglected. This shows the approximate character of equation (13) in reference to an actual model such as the one presented here.

It should be pointed out that the set of model parameters calibrated is for one material over a wide range of densities and pressures. There is no need to do the calibration again for the same material when the initial state changes.

SIMULATION BY THE MODEL Verdugo & Ishihara ( 1996) presented a sequence of triaxial

test results on Toyoura sand whic h are particularly suited to demonstrating the simulative capability of a critical-state-based sand model, since some of the data provide a definite identifica­tion of the critical state line in (! - p' space, which is of cardinal importance for the determination of 1/.'. The sand is described as uniform fine sand consisting of subrounded to subangular particles. The maximum and minimum void ratios are 0·977 and 0·597. respectively. Verdugo & Ishihara reported a total of 17 shear loading tests in their paper. The tests include both drained and undrained triaxial compression tests. The density ranged from e = 0·735 (relative density Dr = 63·7%) to e = 0·996 (Dr ~ 0%). The initial confining pressure p ' for the tests ranged from 100 kPa to 3000 kPa. This set of test results covers comprehens ively the behaviour of the Toyoura sand under monotonic triaxial compression loading conditions.

All 17 tests were simulated using the simple model described earlier with the unified set of model parameters listed in Table I. Figs 9- 14 show the experimental results for all the 17 tests, as well as the results of the simulations obtained with the model. It can be seen that the model simulations broadly match the experimental results, indicating the effect iveness of the critical-state framework in conj unction with the state-dependent dilatancy.

To compare with others, the discrepancies between the model simulation and the test results for drained responses at very low densities (initial void ratios eo = 0·96 and eo = 0·996; or rela­tive densities Dr = 4·5% and Dr ~ 0%) are more notable. A simple modification of equation (25) by replacing the void ratio

DILATANCY FOR COHESIONLESS SOILS 457

OY5 r----------------------------------------------,

0-90

~ 085 ,9 E 'D

g 080

• Expllrirnental 075

Effective mean stress , p': MPa

Fig. 9. Experimental data and theoretical fitting of the critical state line in e- p' space for Toyoura sand

5000r---------------------------------------------~

co 4000 2z '0-~ 3000

~ E 2000 co :;; Q)

o 1000

-- Model simulation

.......... Test results

500 1000 1500 2000 Effective rneoln normal stress. p': kPa

(a )

2500 3000

5000r---------------------------------------------~

4000 co 2z u; JOO()

~ .u;.

E 2000 cr·

.~

o

.. ~ ..... --.. - .--.

./ ···· ····~ .. Plj' : i 00 kPa

.-- '" Pu' 1000 kPa

p '- : 3000 kPa

10 15 /-\x ial strclin, . ',' (::~

(tJ)

20 2[; 30

Fig. 10. Comparison between undrained triaxial compression test results and model simulations for Dr = 63'7 (e = 0'735)

e with the initial void ratios eo (i.e. h = hi - 112eo instead of h = hi - h2 e) brought the model simulations much closer to the experimental counterparts (Fig. 15). This modification also slightly improved other drained simulations, but has no effect on the undrained response because I:' = eo under undrained conditions. This 1:'0 dependence may be due to the influence of material fabric at very low densities, since thc void ratio of a sand without a significant shearing history eo is strongly correlated to the packing structure of the sand (see Figs I and 2). However, since the stress and material state corresponding to a given eo are not always clearly and objectively defined, introducing eo into a constitutive equation as a general para­meter needs further investigation ,

2500 r---------------------------------------------.

ro 2000 2z 'C7

u; (f,)

~ Ui a ro '~

o

co 2z '0-

~ e: Ui a ro ':;; Q)

0

2500

2000

1500

00

-- Mod ":! simulation

.......... Test resulls

500 1000 1500 2000 2500 3000 Effective mean normal stress, p' : kPa

(a)

Po" 3000 kPa

---::::::::-;:::-.:.-...

Po': 1000 kPa

Po' : 2000 kPa

5 10 15 20 25 30 Axial strain , 1' 1: a: :n

(b)

Fig. 11. Comparison between undrained triaxial compression test results and model simulations for Dr = 37·9% (e = 0·833)

1200

1000 ro 2z :.:1- SOO

~ e: )00 "th (5 n,;

400 :; Q;

0

200

12ClO

1000 ro c: C'" 800 UJ if;

2 7T:

.9 \"1:-.;;: CJ 0

U [)

-- Model simulation

--... ----. Test results

5

Effectiv rm,,(ln norrn81 siress. [J ' kF\ :l

(8)

2000 kPa

10 15 20 P.Xi81 ,;trail1 , I' . ~:i,

(Il)

2500 :lOOO

30

Fig. 12. Comparison between undrained triaxial compression test results and model simulations for Dr = 18·5% (e = 0·907)

458

3'.in

300

ru

~ 250 .&

~ 200

~ 150 g in 0:; 100 Q)

0

50

0 0·8

350

300

m 22 250 0-

~ ~ 0 m s Q)

0

0 0

Fig. 13.

-- IvlocJ el sirnulil lion

Test results

085 09 Void ratio. e

(a )

095

10 15 20 Axi al strain , {' 1: %,

(b)

Comparison between drained triaxial results and model simulations for Pn = 100 kPa

1600 Model simulation

Test results

~ 1 200 .;,L •

"c t~ ~ BOO

c 1] > (!)

0 -100

0 Of) () 85 09

Void "li'o, e la)

1600

~ 1200

0-~: rr,

~ 800 '>J)

0 m ;; Q

r.J 400

0 0 10 15 20

Axial strain I, 0·' : \;

(t))

Fig. 14. Comparison between drained triaxial results and model simulations for Pn = 500 kPa

LI AND DAFALIAS

10

25 30

compression test

095

25 30

compression test

CONCLUSIONS The classical stress dilatancy theory in its exact form ignored

the extra energy loss due to the static and kinematic constraints at particle contacts. While this hypothesis leads to a unique relationship between the stress ratio and dilatancy, it obstructs unified modelling of the behaviour of cohesionless soils over a full range of densities and stress levels .

To remove this obstacle, additional dependence of dilatancy on the internal state of a material is needed. The concept of state-dependent dilatancy was introduced, in conjunction with the basic concepts of critical-state soil mechanics. The general expression and basic requirements for dilatancy were addressed .

It has been shown that the state parameter, the difference between the current void ratio and the critical state void ratio corresponding to the current confining pressure, is an effective means of measuring how far the material state is from the critical state. With the state parameter as the state variable, a particular form of state-dependent dilatancy was proposed, and was shown to be equivalent to an interpretation whereby the phase transformation stress ratio is variable with the state parameter, an idea introduced by Manzari & Dafalias (1997).

With this fonn of dilatancy and a state-parameter-dependent plastic modulus , for which an interpretation is again possible whereby the peak stress ratio depends on the state parameter as proposed by Wood et al. (1994), a simple model in the triaxial space as well as a systematic calibration procedure was intro­duced. It was shown that this simple model has the ability to simulate data successfully, with a single set of model constants, for a suite of 17 triaxial tests , both drained and undrained, of Toyoura sand over a relative density range of around 0-64% subjected to a confining pressure range of 100- 3000 kPa.

ACKNOWLEDGEMENTS The financial support provided to X. S. L. by the Research

Grants Council (RGC) of Hong Kong through Grant HKUST721 /96E is gratefully acknowledged. Y. F. D. acknowl­edges the support from the National Science Foundation, Grant No. CMS-9800330, of the programme directed by Dr Cliff Astill.

APPENDIX. DILATANCY OF TWO REGULAR PACKINGS OF UNIFORM RIGID RODS Packing II (Fig. I) Following Rowe (1962). when sliding takes place, one has the ratio of the loads

(34)

where L; is the load per rod in direction i, 1)" is the angle of friction between the surfaces of the rods in contact, and (J is the deviation of the tangent at the contact points from the direction I. The size of a basic cell in the packing is characterized by

I, = 4rsin/j

and

I" = 4reos/i

[35a)

(35b)

Therefore, the ratio of the major principal st ress to the minor principal stress is

0; /01 = L,I, / L" I" = tan/3tan(¢" +/1 ) (36)

The ratio of the strain rates in directions I and 2 is

E" -dii/ v-f, ( dV) i"l, ') -= =- 1+- =~ = -tan- II E, E, VEl I,I?

(37)

Therefore, the ratio of the work done per unit volume by the major principal stress to the work done on the minor principal stress is

aiEl 0; tan(<j.l,,+13) OlE? 02(1 -I- d/>/VE ,) tan 13

(38)

The void ratio of the packing is

I:' = 1,1" - 2m'?' = 16r~ sin/3cosj) _ I = 8sinl3cos(J _ I 2:71'~ 2;r r ~ ::r

(39)

DILATANCY FOR COHES ION LESS SOILS 459

m ~ t:i-

~ ~ 0 rn ',> Q)

0

ru ~ t:i-

~ e! (jj

0 rn ',> Q]

0

350

300

250

200

150

100

50

0 08

350

300

250

200

150

100

50

0 0

-- MO{jE:1 sirnu l3tio [l

•••••• Test results

084 0,88 0·92

10

Void ralio, e

Axial strain , E;: % (a)

0,96 10

-.-..-.

20 30

iGOO

~ 1200

Q' vi <1)

~ 800 (jj

0 rn '> Q)

0 400

0

1600

~ 1200

t:i-g ~ 800 (jj

0 rn ',> Q) 400 0

0

08

0

Moclel simulati on

Test results

084 0·88 Void ratio, e

092 096

-_._ .. _----.. _ .. - .... - ...

eQ =0'810

10 20 Axial strain. E1: ~/O

(b)

30

Fig. 15. Simulations of drained tests with h varying with eo instead of e: (a) pu = 100 kPa, It = hI - "zeo; (b) Po = 500 kPa, h = hI - "zen

Packing B (Fig. 2) At each contact point, sliding takes place when the ratio of thc loads

LI ' (3' Ld2 = tan(if>J1 + , )

Thc size of a basic cell in this packing is characterized by

I I = 4rsin/~

and

12 = 21' + 4,.cos/3 = 2,.0 + 2cos(J)

(40)

(41 a)

(4Ib)

Thereforc, the ratio of the major principal stress to the minor principal stress is

0; 2L I/ I J 2sin/3 01 = L212 = tan(r/J p +/) 1+ 2cos(J (42)

The ratio of the strains in directions I and 2 is

h ( di; ) i2/1 2sin2/J ~ = - I +~ = il/2 =- - (I + 2 cos (3)cos/J

(43)

Thercfore, the ratio of the work done per unit voluille by the major principal strcss to the work done on the minor principal stress is equal to

tJ; EI 0; tan(if>f l + /-1) tJ2E2 02(1 +dil / vEI) tan/3

The void ratio of the packing is

11/2 -3m·2 e==

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8,.2 sin /J( I + 2 cos /3) 3m·2

(44)

8sin/3( 1 + 2 cos (3) 1= - I

3;r (45)

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460 LI AN D DAFALIAS

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