Cavity Expansion in Sands

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 16, 3-23 (1992)

CAVITY EXPANSION I N SANDS UNDER DRAINED LOADING CONDITIONS

I. F. COLLINS, M. J. PENDER AND WANG YAN

School of Engineering, University of Auckland, Private Bag, Auckland, New Zealand

SUMMARY

Solutions for the expansion of cylindrical and spherical cavities in sands are presented. The sand is modelled using recently proposed critical-state models in which the values of the friction and dilation angles depend on the deformation history. Similarity solutions are obtained which enable the limit pressure to be calculated as a function of the initial conditions. Comparisons with existing perfectly plastic theories are made and consequences for the interpretation of cone penetrometer measurements are indicated.

1. INTRODUCTION

The expansion of cylindrical or spherical cavities in an infinite medium is one of the basic boundary value problems of applied mechanics.'-'3 The solutions to such problems have been of particular interest in geomechanics where they have been used to develop approximate analyses of the stresses and deformations induced by driven piles, to interpret the results of pressuremeter and cone penetrometer tests as well as modelling the opening and closure of tunnel^.'^-^' In the cone penetrometer and driven pile applications the cavity initially has zero radius and the resulting strains are large, whilst the initial radius is finite and the induced strains are normally small in the pressuremeter problem. Norbury and Wheeler2' have shown formally that under certain material restrictions the cavity expansion solution is the leading term in the asymptotic expansion of the solution to the wedge or cone penetration problem.

In this paper we are concerned with the large strain solutions for cavities growing from zero initial radius in cohesionless elastic/plastic materials chosen to model the real behaviour of sands. Such solutions which grow in a geometrically self-similar manner can also be viewed as the asymptotic solutions, valid at large times, for cavities which are expanded from a finite starting radius.

The starting point of the present analysis are the studies by Carter, Booker and Yeung? Collins and Wang Yan,' and Bigoni and Landiero3 who solved the problem of the expansion of spherical and cylindrical cavities in an elastic/plastic material, yielding according to the Mohr-Coulomb yield condition, but with a non-associated flow rule. The dilation and internal friction angles are hence in general different. These studies include both small and large strain solutions for materials with either zero or non-zero cohesion. In all cases, however, the material parameters, such as the internal friction and dilation angle were taken as constants and independent of the deformation history of the material element. Carter and Yeung' have used finite elements techniques to study the expansion of a cylindrical cavity in a shear hardening/softening material. Our concern here is with materials in which the changes in volume rather than shear strains govern the material properties.

0363-906 1/92/01oO03-23$11 S O 0 1992 by John Wiley & Sons, Ltd.

Received 29 January 1991 Revised 8 May 1991

4 1. F. COLLINS, M. J. PENDER AND WANG YAN

The pressure in the cavity is constant, when the cavity is expanded from zero radius. This is not a consequence of the assumption of constant material properties, but will always be true as long as the problem has no characteristic length scale, so that the expansion occurs in a geometrically self-similar manner. The ratio of the radii of the elastic/plastic boundary and cavity wall will also remain constant during such an expansion. This constant pressure can also be viewed as the limit pressure attained asymptotically as a finite sized cavity is expanded to infinity. The dilation of the material is an essential part of these solutions so that the simplifications in the analysis of cavity expansions that can be made for incompressible materials are not applicable here.

The point of departure of the present analysis from that given in reference^^.^,' is that here we abandon the perfectly plastic assumption, with constant material parameters, and allow the angles of internal friction and dilation to depend on the deformation history. In plasticity theory such models would normally be referred to as hardening or softening but in a cohesionless material there is no material strength parameter with the dimensions of stress which can increase or decrease, yielding is governed by the magnitudes of the ratios of stress components. The material model used is that recently used by a number of in which the material response depends on a state parameter, which depends both on the current specific volume and mean stress. This model has been used to successfully explain the markedly different observed behaviour of sheared specimens of sand initially at the same voids ratio but at different confining pressures and vice versa.

2. THE STATE PARAMETER MODEL

One of the basic assumptions of the model is the existence of a critical state at which the sand deforms without any plastic volume change so that the dilation angle is zero. In recent years the term steady state has gained in popularity.2z The differences, if any, between the steady-state and critical-state lines are certainly small and have no effect on the ideas discussed here. We prefer the term critical state as it more properly describes a constitutive property. The critical-state line is a straight line in the v-ln p diagram as illustrated in Figure 1, where v is the specific volume and p,

- P = P1 In (P/Pl)

Figure 1. Critical-state line and state parameter

CAVITY EXPANSION IN SANDS 5

is the mean effective pressure. The equation of the critical-state line is hence

v + Aln(p/pl) = rl (1) rl is the intercept on the p = pl, or In(p/p,) = 0 axis, and hence its value depends on the choice of the non-dimensionalising reference pressure pl . In analysing test data it is usual to take p1 to be 1 kPa. However in formulating and solving boundary value problems it is preferable to non- dimensionalise the pressure in (1) by a stress representative of the particular problem under study, such as for a field pressure or an elastic modulus. If the non-dimensionalising stress is changed from p1 to p2, the intercept is changed from rl to

r2 = l-1 + W P l / P , ) (2) The basic constitutive assumption is now made that the behaviour and properties of the sand prior to the achievement of a critical state, depend both on the specific volume and the mean pressure but through the single composite state ~ a r a m e t e r : ~ ~ - ~ *

6 = v + 121n(p/pl) - rl (34 Thus in Figure 1 if Q(v, p) represents the current state of the sand, 5 is the amount by which the specific volume must be decreased to reach the critical state at the same mean efective pressure p. The state parameter is zero on the critical-state line and lines of constant 5 are parallel to the critical-state line. If this line through Q(v, p) intersects the u-axis at uA, then

5 = V L - rl (3b) Note if the non-dimensionalising stress p1 is changed to p 2 , the whole figure is translated laterally through a distance ln(p,/p,), but the value of the state parameter is unaltered. States above ( 5 > O)/below (6 < 0) the Critical-state line are termed loose/dense, respectively since the specific volumes are greaterfless than that which the material deforms at constant plastic volume for the given confining pressure.

The plastic volume change will be positive (compaction) in a loose state but will be negative (dilation) in a state on the dense side of critical. This model allows specimens at the same specific volume to exhibit loose or dense behaviour, since increasing the confining pressure at constant u eventually transforms a dense state into a loose one.

It is of interest to note in passing that although they are widely accepted in critical-state soil mechanics, there is a fundamental objection to working with constitutive equations which involve stresses non-dimensionalised by arbitrarily chosen reference pressures as in (3a). Whilst this objection applies equally to cohesive and cohesionless models, it is particularly graphic in the latter, Suppose we wish to propose an yield function for an isotropic, cohesionless material, appropriate for plane strain or uniaxial compression, where there are two active, effective principal stress components oi, and o;, say, and which in addition depends on the specific volume. Since the material is cohesionless, there is no material yield stress so that on dimensional grounds the yield function must be of the form

f @i/o;,U) = 0 (44 In a general three-dimensional stress state, the stress ratio would be replaced by a dimensionless combination of stress invariants (e.g. Lade and Duncan29 use Z;/i3 to fit their experimental data, whilst Matsuokas mobilised shear plane mode130 predicts the combination I , Z2/i3). However in the critical-state models it is supposed that the yield condition is of the form

f (o i /4 , t ) = 0 (4b)

6 I. F. COLLINS, M. J. PENDER AND WANG YAN

where the state parameter t, given by (3a) involves the mean stress and not the stress ratio aJo; in violation of (4a). The use of the reference pressure p1 in (3a) to non-dimensionalise the arguments of the logarithm does not solve this paradox, since if it were permitted to introduce reference pressures in this way, (4a) could be rewritten as

(W f (a;lPl? U P l , 0) = 0 and the dependence on a; and aj would be arhitrary, making a nonsense of the dimensional analysis argument.

The way out of this paradox is to replace the arbitrary reference pressure p1 in the definition of the state parameter 5 by a quantity with dimensions of stress which actually has a definite physical significance. One candidate for such a stress would be the elastic shear modulus G- although this is open to the objection that its physical significance has nothing to do with the plastic yielding of the material and would not be present in a rigid/plastic model.

For clays at least the form of the state parameter 5 is determined by the nature of the normal consolidation curve. The objections to this relation on dimensional grounds do not apply when the consolidation equation is expressed in differential form, i.e.

but only appear upon integration, when some reference pressure must be introduced. This relation is only valid over the intermediate range of voids ratios. The (u, p) relation must deviate from (5) at both sufficiently low and sufficiently high specific volumes. In this intermediate interval the crushing strength of individual grains may perhaps be the most appropriate choice of reference pressure. has recently developed a theory similar to that used here in which the deformation is regarded as the sum of that produced by the rearrangement of the granules and of that due to the distortion of the individual grains. The latter is governed by the hardness of the grains and hence introduces a reference stress describing their strength. Whilst the assignment of a physical significance to the reference pressure is necessary when comparing the behaviour of two types of material, it is not needed when using experimental data to describe a particular material.

3. YIELD CONDITION AND FLOW RATE

Been and JefferiesZ4 show that by comparing data from a diversity of sand types there is a good correlation when the difference between the angle of internal friction at a given state and the corresponding angle at the critical state is plotted against the state parameter-see their Figure 16 reproduced here as Figure 2. For the purposes of the calculations presented here this variation is represented by the curve

cp - cpc =f(t) = ACexp( - a - 11 (4) where cpc is the internal angle of friction at the critical state and A is a parameter in the range 04-0-95 depending on the type of sand (the angles are measured in radians).

The corresponding flow rule is defined by specifying the relationship between the dilation angle + and the friction angle cp or the state parameter

CAVITY EXPANSION IN SANDS

24.

7

I I 1 j o Hokksund sand (NG I ) Monlerey no. 0 sand (Lade, 1972)

4 8 r 0 I

J e ~ e 1 1 ~ ~ has recently shown that this law is effectively identical with Rowes stress dilatancy equation and also Taylors plane strain saw-tooth model up to dilation angles of 20. The values of the friction and dilation angles in plane strain differ from those pertaining under triaxial conditions. The relationship between these two sets of angles depends on the assumed form of the full three-dimensional yield function and plastic potential. The differences however are small, for example WrothIg has argued, on the basis of Matsuokas3 failure criteria, that the ratio of the plane strain to triaxial compression friction angles is approximately 9: 8. Similarly, Moust Jacobsen3 estimates that this ratio is of the order of 1.1 : 1-a figure which is recommended in the Danish Code of Practice for Foundation Engineering. In view of this small difference and the scatter in the data represented by (6), it was decided that it would not be meaningful to attempt to distinguish between the plane strain and triaxial values of cp and $ in the present study.

4. THE CAVITY EXPANSION PROBLEM

The basic formalism and notation is based on that used by Carter et a14 We shall consider both cylindrical and spherical cavities and use the parameter k which is put equal to 1 for a cylindrical cavity and 2 for a spherical cavity. The cavity is expanded from zero radius in a medium initially subject to the hydrostatic stress state po and at a uniform specific volume v o . In this problem the sand is either supposed to be dry so that we do not have to distinguish between effective and total stresses, or expanded under drained conditions so that the pore pressure is effectively constant and can be subtracted out of the analysis. In the latter case all stresses should be interpreted as effective stresses. (There is hence no need to continue with the dashed notation for effective stresses.) At time t, the radius of the cavity is a and that of the elastic/plastic boundary is R. Thus for r > R the deformation is purely elastic, but for a < r c R there are both elastic and plastic strains. In both regions the radial and hoop stresses satisfy the equilibrium equation

8 I. F. COLLINS. M. J. PENDER AND WANG YAN

with or = oc on the cavity wall and or + p o as r + 00 . For a spherical cavity o, = og by symmetry so that each material element is subject to triaxial compression as in the standard triaxial test. Thus test data obtained from such a test are directly applicable to the spherical cavity problem, but assumptions have to be made to apply such data to the plane strain analysis as discussed in the previous section.

If w is the radial velocity component, the principal components of the rate of deformation (rate of strain) tensor are

where compressive strains are taken to be positive. For spherical symmetry e, = e, and in plane strain e, = 0.

The linear elastic stress-strain relations can be written in matrix form:

i.e. CT = Lce where (E;, .$) are the elastic (small) strain components. G is the elastic shear modulus and v is Poissons ratio, the small strains being defined by

u being the radial displacement. In the elastic zone the strains are small and the strain rate in (9) can be regarded as the local time derivatives of the strains in (1 1). However, this is not true in the elastic/plastic regime where the strains can be very large. The relation between e, and 8, etc has been discussed in this context by Collins and Wang Yan.7 In the elastic/plastic regime the constitutive equation is formulated in rate form in terms of the strain rates in (9), so that the elastic law (10) becomes

V Q = Leg ( W

V where G denotes the Jaumann stress rate. However, since in this problem the spin is everywhere zero this stress rate reduces to the ordinary material derivative:

where 6 is the local derivative evaluated at a fixed position r. (The convected part of this stress rate was neglected in the analysis of Carter et uL4-as was shown in Reference 7 this can produce differences in the prediction of the limit pressures of up to 15 per cent.)

The angle of internal friction is defined by

so that the stress ratio

N = (o,/o,) = (1 + sin cp)/(l - sin cp) (15) The corresponding flow rule parameter is

M = - (kei/er) = (1 + sin@)/(l - sin$) (16)


where ep are the plastic strain rates and Ifi is the dilation angle. It is important to note that N and M will both be functions of the state parameter 5 using (4) or (5 ) and hence will vary with the deformation. The rate law (12a) can be rewritten in terms of plastic strain rates as

6 = L(e - ep) ( 1 2b) The flow rule (16) can then be used to eliminate the plastic strain rates between the two equations in (12b) to yield the single equation

aw k w - + -- = .- [A(M)&, + B(M)&,]/2G ar M r

where (9) has been used to express the total strain rates in terms of velocity components, whilst A and B, which are functions of M and hence of the state parameter 5, are given by

(18) A ( M ) = [(l - 2 ~ ) + kv(M - l)/M]/[l + (k - l ) ~ ] B ( M ) = [k(l - v) /M - kv]/[l + (k - l ) ~ ]

Since N also depends on the state parameter 5, the rate form of the yield condition (15) is b, = N&,g + ( N ' / N ) i o ,

t = t,,; + 5,,i

(19) where N' = dN/d


5. SOLUTION PROCEDURE

5.1. The solution in the elastic region

In the outer elastic zone (8), (10) and (11) can be solved to find the elastic stress components which must be superposed upon the existing isotropic stress state p o . The only solution for a displacement field which remains bounded as r -+ 00 is u = Br-k, which has an associated zero volume strain. The specific volume of a material element is hence unchanged by the elastic deformation and is still uo when a material particle enters the plastically deforming region. Following Carter et al. and Hughes et al.36 we shall write this displacement in the elastic region as

U = E ~ ( ; ) k R

where E~ is the circumferential strain at the elastic/plastic boundary ( r = R) . The radial and circumferential stress components are given by

(27) so that the mean pressure, and hence the state parameter 5, are both unaltered by the elastic deformation. At the elastic/plastic boundary ar/co = N o where N o = N ( 5 , ) corresponds to the initial value of 5, so that from (27) we deduce that

6, = P o -k 2 G k & ~ ( R / r ) ~ + ' ; Uo = P o - 2 G & ~ ( R / r ) ~ + l

&R = ( N O - l)pO/(NO + k)2G

OR = f1 + k ) N O p O / ( N O + k ,

(28)

(29)

whilst the radial stress at the elastic/plastic boundary is

These provide the outer boundary conditions for the solution in the elastic/plastic region.

5.2. Similarity solution in the elasticlptastic zone

If the initial radius of the cavity is zero, the problem has no characteristic length, since the problem-defining parameters are the dimensionless variables 1, rl, v and u,; and G and p o , both of which have dimensions of stress. The deformation must hence proceed in a geometrically self- similar manner and the ratio of the radius of the elastic/plastic boundary to the cavity wall radius @/a) must remain constant, its value depending on the above problem defining parameters. The velocity, stress components and state parameter must hence depend on r and t through the dimensionless radial co-ordinate

= r /R = r/Wt (30) where W = R is the speed of expansion of the elastic/plastic boundary. W can be taken to be a constant since the elastic/plastic constitutive equations are rate independent. We could equally well choose to non-dimensionalise r by the cavity wall radius a, but it proves to be com- putationally more convenient to choose R since more information is known about the dependent variables at the elastic/plastic boundary than is known at the cavity wall. The velocity and stress components are non-dimensionalised similarly:

w = w / w , 5 = a/p, (31)

(32)

At the elastic/plastic boundary

C,(1) = (1 + k)No/(No + k )

CAVITY EXPANSION IN SANDS 1 1

from (29). Differentiating (26) with respect to t and putting r = R the speed of a material particle currently on the elastic/plastic boundary is seen to be (k + l)+fi, so that

$(I) = ( k + 11% = [ ( N o - l)(k + 1MNo + k)I (Po/~G) (33) On the cavity wall the material particle speed is ci. However, by geometrical similarity, u/R = a/R so that the cavity wall is located at the point where

@(?) = ? (34) All the various derivatives which occur in the governing equations can now be expressed in terms of d/dq, using (30):

ajar = ( a / a q ) / ~ ; ( ' ) = - wq(a/a?)/~; ( = w ( ~ - q ) ( a / a q ) / ~ (35) The last relation comes from non-dimensionalising the standard relation between material and local derivatives as used in (13) for example. We now have a system of ordinary differential equations to solve. In terms of these dimensionless variables the equilibrium equation become

whilst the constitutive equation (24) and state parameter evolutionary equation (25) become

and

respectively, where A ( M ) and B ( M ) are given by (18) and

C(t) = [l - AN'/(N + k ) ] / N ; D(< , U) = uN' /NZ and

E ( t ) = [l + AkN'/N(N + k ) ] Equations (36)-(37) were solved as the system

for the solution vector xT = At each stage of the solution the specific volume in (38) is calculated from

9, 5 ) using a standard NAG library differential equation solver.

u = t + rl - Aln[~,(p,/p,)(N +' k)/N(l + k ) l (40) which follows from (3a) and (15). Note that there are two independent problem-defining stress ratios which can be formed from the initial and reference pressures and the elastic shear modulus occurring in (39). The starting values for the dependent variables pertaining at the elastic/plastic boundary q = 1 are given by (32) and (33) for zr and iij, whilst the initial value of the state parameter is

t o = vo - rl + WPO/Pl) (41)


The solution is halted when the cavity wall is reached and W = i j [cf. (34)]. In order to effect this stopping condition it proved convenient to replace the independent variable q by 8 = W - q, so that d/dq = d/dO(dW/dq - 1) and the position of the cavity wall is now at the &origin.

An important point to note about these equations is that the derivative of the state parameter becomes infinite at the cavity wall, wheie W = q, in equation (38) unless dG/dq + kW/q vanishes. This is consistent with the constitutive equation [(17) or (37)] only if M = 1, which means that the material has reached the critical state and t = 0. This result can be understood physically as follows. Since the ratio of the radius of the cavity wall to that of the elastic/plastic boundary remains constant during the deformation, a material element on the cavity wall must remain at the same state during the deformation, Since the cavity pressure and state parameter are hence both constant, the specific volume must also remain fixed. The deformation of a material element at the cavity wall hence occurs at constant volume. However the elastic component of the total volumetric strain rate is zero since the pressure is constant. It follows therefore that the plastic part of the volumetric strain rate is also zero. The material at the cavity wall is hence at the critical state.

6. DISCUSSION OF RESULTS

6.1. Distribution of stress and voids ratio through plastic annulus

Curves showing the variation of the dimensionless mean pressure p/2G and the voids ratio e, starting at the initial values (po/2G, e,) at the elastic/plastic boundary and ending up on the critical state line at the cavity wall are shown in Figure 3. Although these graphs are presented in dimensionless form, it must be remembered that the solution also depends on the ratio of G/p, . Initially we are concerned with the qualitative nature of the solutions which can adequately be discussed with reference to Figure 3, in which the elastic shear modulus is held constant at 25000 kPa, and the slope A of the critical state line in the [u , ln(p/p,)] plane is 0.029. Since the solutions obtained represent geometrically self-similar deformations, these curves can either be thought of as describing the variation of p/2G and e through the plastically deforming region at a

Figure 3. Variation of voids ratio e and dimensionless mean effective stress (p/2G) through the elastic/plastic annulus for various initial conditions ( A = 0.029, G = 25,000 kPa), (a) cylindrical cavity (b) spherical cavity


$xed time, or as the variation of these quantities associated with a given material particle as it moves through the expanding plastically deforming annulus from the elastic interface to the cavity wall.

The mean pressure increases monotonically from the elastic/plastic boundary to the cavity wall, so that the elastic part of the volumetric strain rate will always be compressive. The plastic part of the volumetric strain rate is of course dilational, for states on the dense side of critical but compressive on the loose side. The voids ratio must therefore always decrease in loose states, but can vary either way on the dense side of critical depending on the relative magnitudes of the elastic and plastic volumetric strain rates.

As can be seen from the figure, when the initial state is sufficiently far into the dense region, the total volumetric strain is initially effectively zero until some critical point is reached at which the voids ratio starts to increase until the critical-state line is reached. Hence in the outer part of the elastic/plastic annulus the elastic compaction and plastic dilation effectively cancel each other out. This is because in this region the deformation is still constrained by the outer elastic material which is deforming at constant volume. As the material particle approaches the cavity wall, this influence diminishes and the material is now free to expand plastically. However for initially dense states closer to critical, the elastic strains initially dominate. If e, - 0.7 the material element reaches a critical state before getting to the cavity wall, crosses into the loose zone, but then returns to the critical-state line at a lower voids ratio. By contrast, for an initially more compacted material, with e, - 0.6, the initial elastic compression is later swamped by the plastic dilation and the material element ends up a critical state with a voids ratio appreciably greater than e,.

If the initial state is on the loose side of critical, the voids ratio decreases until the critical state line is attained. If this occurs before the cavity wall is reached the state of the element crosses into the dense regime and the elastic compression is ultimately dominated by the plastic dilation and the ultimate critical state is attained at an even smaller voids ratio. Even if the sand is initially at the critical state it is possible for a material particle to go into the dense region before returning to the critical-state line at a lower voids ratio.

6.2. Comparison with perfectly plastic solution

The importance of including the variation of the internal friction and dilation angles in the cavity expansion model can be assessed by comparing the predictions of the cavity wall pressure from the present theory with those given by the perfectly plastic solutions given by Carter et aL4 or Collins and Wang. A representative comparison is made in Figure 4. The shear modulus is set at 25,000 kPa and the initial specific volume at 1.6. The initial values of the state parameter to, internal friction angle cp, and dilation angle Jl, are then determined by the value of p o . The values of the cavity wall pressure calculated from the perfectly plastic model in which cp and $ are held fixed at their initial values are seen to overestimate the values predicted by the present critical- state theory. The percentage error being larger for smaller values of initial effective pressure [more than 100 per cent for values of (p0/2G) less than Very much better agreement is obtained if the average values of cp and $ are used in the perfectly plastic computation. Since the cavity wall is known to be at the critical state, these mean values are simply (cp, + cpJ2 and $,/2, respectively.

6.3. Influence of the variation of the elastic shear modulus

In the above discussion it was assumed that the value of G can be chosen independently of the other problem-defining parameters. However the elastic shear modulus is known to have a strong dependence on the values of the voids ratio and mean effective pressure. A widely used empirical


.ooOol .ooO1 ,001 .Ol .I .OOOOl .ooO1 ,001 .01 . I

POIZG POIZG

Figure 4. Comparison of predicted variation of cavity wall pressure with far field hydrostatic pressure using (1)-present critical-state model (2). . . perfectly plastic model with initial values of friction and dilation angles, and (3)-.- .- . perfectly plastic model but with averaged values of friction and dilation angles. (a) cylindrical cavity (b) spherical cavity

formula for dry sands under isotropic confinement has been discussed by Richart et aL3:

G = S [ ( e , - e)2/(1 + e)]p12 (42) where e, is a dimensionless constant with values ranging from 2.17 for round-grained sands to 2.97 for sands with angular grains. The corresponding range for the constant S is 6.90-3-23 x lo3 (kPa)12. The simplest micromechanical models based upon hertzian contact predict a p13 dependency for G. Modificitions to these models which include the effects of buckling of particle chains have recently been made by G ~ d d a r d . ~ ~ These theories predict the observed square-root pressure dependency at least at low pressures.

Relation (42) has been used to calculate Go, the initial value of the elastic shear modulus in the elastic region, and to update the value of this parameter as a particle moves through the elastic/plastic annulus when solving the governing system of equations (39). The ratio of the values of the shear modulus at the cavity wall to that at the elastic/plastic boundary is typically around 10. The main effects of allowing for the variation in the value of G is to decrease the magnitudes both of the elastic and plastic volume strains-this effect being more marked for spherical rather than cylindrical cavities-and also to decrease slightly the value of a/R for a given set of initial conditions (see Figure 5).

The variation of G with e and p has little effect on the e-ln p plots as previously illustrated in Figure 3. Representative e-ln p plots for two particular sands are shown in Figure 6. The values of the critical-state parameters are taken from data given by Been and J e f f e r i e ~ ~ ~ and Been et ~ l . * ~ and reproduced here in Table I. Kogyuk and Monterey No. 0 sands represent the two extreme values of &-the slope of the critical-state line. The variation of the elastic shear modulus has been calculated from (42).

The general form of these figures is similar to the dimensionless plots in Figure 3. The far-field conditions have been chosen so as to give initial values of the state parameter of - 1.0, - 05,00 and 05. The shape of the e-ln p curves is seen to depend not only on the initial values of the state parameter but also on the initial voids ratio (or effective pressure). As explained below this fact has important consequences for the interpretation of cone penetrometer measurements. It should also be noticed that even if the sand is initially at a critical state, the state of the material element


volumeuic strain

0.05 ~

i -0.10 .001 .o 1 .1 1

r/R

volumetric strain "."J

0.00

-0.05

-0.10

-0.15 .001 .01 .1 1

r/R

Figure 5. Variation of positive elastic and negative plastic volumetric strains through elastiq'plastic annulus; (1) keeping G constant; (2) varying G using equation (42): (a) cylindrical cavity (b) spherical cavity (Monterey No 0 sand, e, = 0 5 ,

p o = los kPa)

deviates away from critical as it moves through the expanding plastic annulus and returns to the critical-state line but at a lower voids ratio when it finally reaches the cavity wall. At low pressures this deviation is into the loose side of critical, but at higher pressures ( 2 lo5 kPa) the reverse occurs and the material element moves through states which are denser than critical. Such higher pressures are of course unlikely to be of interest in practice.

6.4. Relevance of results to interpretation of penetrometer tests

A detailed discussion of the relevance of the results of these cavity expansion calculations to the interpretation of cone penetrometer tests will be given in a future paper. Only a number of preliminary results will be discussed here.

Of particular relevance is the value of oc/po, the ratio of the cavity wall pressure to the far-field mean effective stress. This ratio corresponds to the normalised tip resistance in a penetrometer


e

1

e

(b) Figure 6. a, b (Continued)

test normally written as (qc - p')/p' for dry sands and (qc - p) /p ' for drained tests on saturated sands, qc being the tip resistance. In the notation used in this paper p' = p o , the effective stress at infinity, whilst p is equal to po plus the constant pore pressure. In practice, p' and p are small compared with qc, so that in both situations the normalised tip resistance is effectively equal to 4clP'.


e

0.3 t - * ..-- - ..----I - - -..... = =.--I = - ..-* ....-. - -*.I

n 7 .

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- -...\

0.7 m

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0.5 4 -u\

0.4

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avity

cat

egor

y C

1

c2

c3

c4

Mon

tere

y no

. 0 sa

nd

0029

1.

878

32

0.83

Su

brou

nded

k

=l

2

31

8~

10

~ -0.7

66

0313

6.

017

k=

2

9-38

2 x l

o7 - 0.

938

0374

7.

426

Hok

ksun

d sa

nd

0054

1.

934

32

080

Suba

ngul

ar

k=

l

1.2

63

~1

0~

-075

6 02

96

5.60

9 k

=2

4.

279

x lo

7 - 0.

918

0.34

9 6.

915

Kog

yuk

sand

0.

066

1.84

9 31

0.

75

Suba

ngul

ar

k=

l

5.1

52

~1

0~

-073

2 0.

296

5397

k

=2

2.

247

x lo

7 - 0

929

0.37

4 6.

945

Otta

wa

sand

00

28

1.75

4 28

.5 0.

95

Rou

nded

k

=l

4.0

68

~1

0~

-080

4 0.

361

6835

k

=2

2.

663

x lo

* - 1

'012

0.

450

8.65

8

Rei

d B

edfo

rd s

and

0065

2.014

32

063

Suba

ngul

ar

k=

l 5.

039

x lo

5 - 0

.711

02

65

4.99

6 k

=2

1.

539 x

10'

- 0

.871

0.

314

6.20

7

Tic

ino

sand

00

56

1.98

6 31

06

0 Su

brou

nded

k

=l

5.

453

x lo

5 - 0

702

0.26

8 5.

142

k=

2

2.0

12

~1

0~

-0.8

75

0326

64

81

F

4

U e Q 4 -e 4


resistance-state parameter relationship was not unique but varied systematically with the mean stress level.

Any two of the initial parameter values vo, po or to are sufficient to define the initial conditions, the three parameters being related by (3a). If vo , po are chosen as the base parameters, the results illustrated in Figure 7 can be approximated by relations of the form

exp ( - c4v0) (43) (CZ +CJUO) b,/Po = ClPO The values of the four constants c1-c4 for different sands are given in Table I. The corresponding relations applicable when to and either uo or p o are used to define the initial conditions can be simply obtained from (3a).

7. CONCLUSIONS

The important conclusions to be drawn from this investigation are:

1. The use of similarity assumptions, which are appropriate when cavities are expanded from zero initial radius, enables problems which incorporate more realistic, deformation dependent, constitutive models to be readily tractable.

2. The material elements at the cavity wall are always at the critical state. Even if the sand is initially at a critical condition, it will compact and end up at a denser critical state.

3. Allowing for the variation of the friction and dilation angles through the expanding plastic annulus reduces the predicted values of the cavity wall pressure significantly. Much better agreement with perfectly plastic models can be obtained if average values of these two angles are used.

4. The value of the ratio of the cavity wall pressure to the initial effective stress, which corresponds to the normalised tip resistance in a cone penetrometer test, depends not only on the initial value of the state parameter, but also on the initial value of the voids ratio (or effective pressure).

In this paper we have concentrated on presenting the theory and the general results of the analysis. More detailed comparisons of the predictions of this theory with experimental cone penetrometer measurements on particular sands will be presented elsewhere.

Acknowledgements

The authors are grateful to Transit New Zealand for financial support of this project.

NOTATION

radius of cavity wall voids ratio radial and circumferential strain-rate components plastic components of radial and circumferential strain-rate components elastic shear modulus equal to 1 for a cylindrical cavity and 2 for a spherical cavity matrix of elastic moduli [ = - keg/ef = (1 + sin $)/(1 - sin $)I [ = cs,/ao = (1 + sin rp)/(l - sin rp)]


8 cp cpC

$

mean effective pressure initial pressure in undisturbed sand reference pressure in critical-state model tip resistance in cone penetrometer tests radius of elastic/plastic boundary time radial displacement component specific volume ( = 1 + e) radial velocity component speed of expansion of elastic/plastic boundary ( = R / t ) non-dimensional radial velocity ( = w/ W) radial and circumferential strain components circumferential strain at elastic/plastic boundary [ = ( N o - l)po/(No + k ) 2 G ] slope of critical-state line Poissons ratio dimensionless radial coordinate [ = ( r / R ) ] radial and circumferential effective stress components radial stress at elastic/plastic boundary [ = N,(1 + k)po / (No + k ) ] dimensionless stress components [ = or/po, 6, = oe/po] local effective stress rate material effective stress rate Jaumann effective stress rate state parameter intercept of critical-state line on specific volume axis where the pressure is equal to the reference pressure modified independent variable in similarity equations ( = W - q ) angle of internal friction angle of internal friction at critical state dilation angle

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