138_ftp

*Correspondence: to V. A. Osinov, Institute of Soil Mechanics and Rock Mechanics, University of Karlsruhe, Postfach6980; 76128 Karlsruhe, Germany.

Received 9 March 2000Copyright ( 2001 John Wiley & Sons, Ltd. Revised 8 September 2000

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2001; 25:473}495 (DOI: 10.1002/nag.138)

Theoretical investigation of the cavity expansion problem basedon a hypoplasticity model

V. A. Osinov* and R. Cudmani

Institute of Soil Mechanics and Rock Mechanics, University of Karlsruhe, Postfach 6980, 76128 Karlsruhe, Germany

SUMMARY

The problem of the symmetric quasi-static large-strain expansion of a cavity in an in"nite granular body isstudied. The body is assumed to be dry or fully drained so that the presence of the pore water can bedisregarded. Both spherical and cylindrical cavities are considered. Numerical solutions to the boundaryvalue problem are obtained with the use of the hypoplastic constitutive relation calibrated for a series ofgranular soils. As the radius of the cavity increases, the stresses and the density on the cavity surfaceasymptotically approach limit values corresponding to a so-called critical state. For a given soil, the limitvalues depend on the initial stresses and the initial density. A comparison is made between the solutions fordi!erent initial states and di!erent soils. Applications to geotechnical problems such as cone penetration testand pressuremeter test are discussed. Copyright ( 2001 John Wiley & Sons, Ltd.

KEYWORDS: cavity expansion problem; critical state; hypoplasticity; cone penetration test; pressuremetertest

1. INTRODUCTION

The problem of the expansion of a cavity in a continuous body has a variety of applications. Thepresent paper deals with the quasi-static large-strain expansion of a cavity in a granular soil. Thestudy is orientated towards geomechanical applications for the interpretation of cone penetrationtests and pressuremeter tests.

The cavity expansion problem is not a novel one and can be found in the literature. Inconnection with soil mechanics, there are two features that complicate the problem: largedeformations (the problem with small deformations is of less interest); and the state-dependentbehaviour of the soil including dilatancy or contractancy of the granular skeleton (this concernsmost problems in soil mechanics with large deformations). Large deformations require a properalgorithm that takes into account the translation of the particles and of the boundary. Thecomplex soil behaviour requires the use of a sophisticated constitutive theory.

In connection with the indentation of metals the problem of the spherical cavity expansion wassolved by Hill [1] for an elastic}plastic material with the Tresca yield condition. Chadwick [2]presented the solution for a spherical cavity with the Mohr}Coulomb yield condition and anassociated #ow rule. In both cases the solutions were obtained on the assumption of smalldeformations and incompressibility of the material in the plastic zone. In the solution of VesicH [3],also based on an elastic}plastic model, the material in the plastic zone was assumed to behave asa compressible plastic solid described by the Mohr}Coulomb shear strength parameters c (cohe-sion) and u (friction angle) and by the average volumetric strain. The determination of the limitpressure required an additional assumption for the value of the average volumetric strain in theplastic zone.

The solution with the Mohr}Coulomb yield condition and a non-associated #ow rule wasgiven by Carter et al [4]. The dilatancy angle and the internal friction angle were di!erent andwere taken to be constant during the deformation. The pressure}expansion relation was derivedfor the case of small strain. An explicit expression for the limit pressure was obtained assumingthat at large deformation a pseudo-steady state is approached in which the ratio of the size of theplastic region to the current cavity size remains constant. Based on a similar model, Yu andHoulsby [5] presented analytical solutions to the problem of the expansion of cylindrical andspherical cavities under large deformations. The limit pressure was obtained as a function of thedilatancy angle, the friction angle, the elastic moduli and the initial stresses.

Collins et al. [6] used an elastic}plastic critical-state model to calculate the expansion ofcylindrical and spherical cavities in cohesionless soils. The material response was governed bya state parameter which depends on the current void ratio and the current mean stress.A non-associated #ow rule enables the angle of internal friction and the angle of dilatancy to varywith the deformation as a function of the state parameter. The expansion of a cavity wasconsidered from zero initial radius. In this particular case the problem has no characteristiclength, and the ratio of the radius of the plastic zone to the cavity radius is constant. The pressureon the cavity surface remains constant as well and can be viewed as the limit pressure for aninitially "nite cavity. As follows from the solution, the material in the vicinity of the cavityasymptotically approaches a critical state as the radius of the cavity increases. The pressure andthe void ratio in the critical state depend on the initial stresses and the initial void ratio. Theincorporation of the pressure and density dependence of the friction and dilatancy angles into themodel considerably reduces the predicted limit pressure as compared with the values obtained byYu and Houlsby [5]. An elastic}plastic critical state model similar to that of Collins et al. [6] wasapplied by Shuttle and Je!eries [7] to the solution of the spherical cavity expansion problem insandy soils.

To solve the cavity expansion problem in sand under drained conditions, Ladanyi and Foriero[8] interpolated a set of experimental stress}strain curves obtained in drained triaxial compres-sion tests with constant mean pressure, and then used this interpolation instead of an analyticalconstitutive equation. The validity of such approach is doubtful because the stress}strainresponse of a granular material is path dependent. The strain path which a material elementfollows during the expansion of a cavity di!ers from the path in a triaxial test with constant meanpressure.

The present paper presents a large-strain analysis of the symmetric expansion of spherical andcylindrical cavities in a cohesionless granular soil. The constitutive behaviour of the soil isdescribed by a hypoplastic constitutive equation which has been calibrated for a number of soils.The constitutive theory is discussed in Section 2. The problem of the cavity expansion is described

474 V. A. OSINOV AND R. CUDMANI

Copyright ( 2001 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2001; 25:473}495

by a system of "rst-order partial di!erential equations for velocity, stresses and density. Thesystem is solved numerically by a "nite-di!erence technique. There are no limitations on the sizeof deformations as the positions of the material points are updated by integrating the velocity"eld. The spherical and the cylindrical problems are considered in Sections 3 and 4, respectively.Geotechnical applications are discussed in Section 5.

2. THE CONSTITUTIVE MODEL

Without taking into account such factors as grain cementation and structural anisotropy,a cohesionless soil can be considered as a &simple grain skeleton' [9], whose mechanical behaviouris governed by granulometric properties, density and stresses. Stress changes in a simple grainskeleton as a result of deformation are adequately modelled by the hypoplastic constitutiverelation [10}13]. This relation describes the irreversible plastic behaviour of granular soilsincluding dilatancy, contractancy and the dependence of sti!ness on stresses and density. Thegranulometric properties determine the parameters of the constitutive equation, while stressesand density are the state variables. Since the state variables enter into the constitutive equation aswell as the constitutive parameters, the response of the soil is governed both by the constitutiveparameters and by the current stresses and density. At the same time, the constitutive parametersare state independent and can be evaluated by the testing of remoulded samples in the laboratory[9,13].

As distinct from elastic}plastic models, hypoplasticity describes the plastic behaviour ofa material with the help of a single equation without resolution of the deformation into elastic andplastic parts and without introducing the notions of loading, unloading and yield surface. Thestate of a granular skeleton is determined by the current Cauchy stress tensor T and the void ratioe. Deformation is described in terms of the stretching tensor D (the symmetric part of the velocitygradient). Given T, e and rate of deformation D, the stress rate T0 is uniquely determined bya tensor-valued function H:

T0 "H(T, D, e) (1)

The condition of objectivity requires an objective stress rate in Equation (1) (the Jaumann stressrate is conventionally used in hypoplasticity). In the problems considered in this paper theobjective stress rate reduces to the material time derivative because of the symmetry of thevelocity gradient tensor.

The hypoplastic equation used in the present study was elaborated by Wol!ersdor! [12] and iswritten as

H"fbfe

1

tr(T) 2)[F2D#a2 tr(T) D)T)#f

da F (T)#T) *) DDDDD] (2)

where

T)"T

tr T, T) *"T)!

1

3I, DDDDD"Jtr(DD) (3)

and I is the unit tensor. This is a modi"cation of the relation given by Gudehus [10] and Bauer[11].

CAVITY EXPANSION PROBLEM 475


The coe$cients in (2) depend on the invariants of the stress tensor and on the void ratio. Thefunction H is homogeneous of degree one in D, and the behaviour of the material is thus rate-independent. The term DDDDD does not allow the function H to be linearized in the vicinity of D"0and to be written in an incrementally linear form.

The void ratio involved in the constitutive function is determined from the mass balanceequation

eR"(1#e) trD (4)

If the &direction' of deformation D/DDDDD is kept constant and there is no volume change (tr D"0),the stress tensor asymptotically approaches a certain value which depends on the initial stresses,density and the direction of deformation. In terms of soil mechanics, the state of the materialapproaches a critical state [14}16] de"ned as T0 "0, tr D"0 (DO0).

The void ratio e#in a critical state and the corresponding mean pressure !tr T/3 (tr T(0 for

compression) are assumed to be connected by the relation

e#"e

#0expC!A

!tr T

h4Bn

D (5)

where e#0

, h4, n are material constants. Since the usually assumed logarithmic relationship

between the critical void ratio and the mean pressure is known to fail at pressures higher thanabout 1 MPa [15,17,18], relation (5) was proposed by Bauer [11] to more realistically describethe critical void ratio for high as well as low pressures. For pressures much lower thanh4Equation (5) gives the power law proposed by Ohde [19]:

expC!A!trT

h4Bn

D+1!A!tr T

h4Bn

(6)

which is widely used in constitutive laws for granular materials [17,20}22].Besides the critical void ratio, two other characteristic void ratios are speci"ed as functions of

the mean pressure: the minimal possible void ratio, ed, and the void ratio in the loosest state, e

i.

The pressure dependence of these void ratios is postulated in the same form as for the critical-state void ratio:

ei

ei0

"

ed

ed0

"expC!A!tr T

h4Bn

D (7)

with the corresponding reference values ei0, e

d0for zero pressure (e

i0'e

#0'e

d0).

The factor a in (2) is determined by the friction angle u#in critical states

a"S3

8

(3!sin u#)

sinu#

(8)

The factor F is a function of T) *

F"S1

8tan2t#

2!tan2 t

2#J2 tant cos 3h!

1

2J2tant (9)



where

tant"J3 DDT) *DD, cos 3h"!J6tr(T) *3)

[tr(T) *2)]3@2(10)

The factors a and F determine the critical-state surface in the stress space. The factor

fd"A

e!ed

e#!e

dBa

(11)

where a is a material parameter, tends to unity as the state of the material approaches a criticalstate. The functions f

eand f

bare de"ned as

fe"A

e#eB

b(12)

fb"

hs

n A1#e

ieiBA

ei0

e#0BbA!trT

h4B1~n

C3#a2!J3 aAei0!e

d0e#0!e

d0BaD~1

(13)

where b is a material parameter. For a detailed discussion of the constitutive equation see theoriginal papers [10}12].

The calibration procedure for the determination of the constitutive parameters is described byHerle and Gudehus [9] and Herle [13]. The friction angle in critical states u

#is determined from

triaxial, simple shear or direct shear tests on loose samples, or can be taken equal to the angle ofrepose. The critical void ratio at zero pressure e

#0can be estimated by extrapolating the critical

state line obtained in the laboratory, for instance, using the technique proposed by Been et al.[15]. Alternatively, e

#0can be taken equal to e

.!9corresponding to the loosest state and

conventionally used in soil mechanics. The minimal void ratio at zero pressure ed0

can beevaluated by extrapolation from cyclic shear or triaxial tests with small amplitude of deformationat di!erent pressure levels. This parameter can also be taken equal to the void ratio e

.*/in the

densest state obtained in the laboratory by a standard technique. The void ratio in the looseststate e

i0for sand is taken as e

i0+1.15 e

#0.

The granular hardness h4

together with the exponent n can be determined by "tting thecompression curve obtained from an isotropic compression test. In the same way both para-meters can be estimated from an oedometric compression test with an initially loose specimenassuming a value of the pressure coe$cient K. The exponent a, which in#uences the tendency ofthe material to dilantancy, can be evaluated from a triaxial compression test on a dense specimen.The parameter b is responsible for the change in the compressibility of the grain skeleton with thevoid ratio at a given mean pressure, and is determined by comparing the sti!ness of loose anddense specimens at the same mean pressure and with the same loading path; oedometric orisotropic compression tests are usually used to "nd b.

For purposes of the present study the constitutive function (2) was calibrated for "ve soils withthe use of the available experimental data [17,23}26]. The constitutive parameters are presentedin Table I.

A quantity usually used as a characteristic of density of a granular soil is the relative density(also called the density index) de"ned as

ID"

e.!9

!e

e.!9

!e.*/

(14)



Table 1. Constitutive parameters.

Sand h4(MPa) n e

#0ed0

ei0

u#

a b

Ticino 250 0.68 0.94 0.59 1.11 31 0.11 1.0Toyoura 120 0.69 0.98 0.61 1.13 32 0.12 1.0L. Buzzard 6400 0.45 0.79 0.49 0.94 31 0.16 1.0Hokksund 150 0.70 0.87 0.53 1.01 31 0.09 1.0Monterey 8000 0.35 0.83 0.54 0.90 32 0.07 1.0

A generalization of the relative density (14) was proposed by Konrad [18], who used a nor-malized state parameter

tN"

e#!e

e.!9

!e.*/

(15)

with a bilinear compression curve for the pressure dependence of the void ratio.Neither I

Dnor t

Nimply pressure dependence of e

.*/and e

.!9. With (5) and (7) one can modify

ID

and introduce the pressure-dependent relative density

I*D"

e#!e

e#!e

d

(16)

As an alternative to void ratio, this quantity will be used below as a characteristic of density of thematerial in the initial state at non-zero pressure prior to the expansion of the cavity. The modi"edrelative density I*

Dis negative if the void ratio is higher than the critical void ratio for a given

pressure.Figures 1}4 present experimental results on the biaxial and triaxial compression of Toyoura

sand in comparison with the numerical calculations performed with the hypoplastic constitutiveequation (2). It is important to note that the constitutive equation was calibrated for Toyourasand independently of the results shown in Figures 1}4, with the use of experimental data of otherauthors [17,25]. The numerical and experimental tests shown in Figures 1}4 were performed withconstant lateral (horizontal) pressure p

3"const. (Since in the hypoplastic equation compressive

stresses are negative, in the "gures and below in this paper we write p for pressure which ispositive for compression.) The "gures show the deviatoric pressure q"(p

1!p

3)/2 and the

volumetric deformation e7

as functions of the vertical deformation e1

(taken to be positive) fordi!erent initial pressures and densities. The discrepancy between the numerical and the experi-mental curves after the peak, especially for biaxial compression, is a consequence of the in-homogeneous deformation of the sample and possibly of the shear band formation at largestrains, which is not modelled by the direct integration of the constitutive equation.

The deformation of the sample in a triaxial test is in a certain sense relevant to the deformationof a material element during the expansion of a spherical cavity. In both cases one principal axisof the stress tensor coincides with a principal axis of the deformation tensor, whereas the othertwo principal stresses and two principal deformations are equal and thus provide the cylindricalsymmetry of the deformation. The same correspondence holds between the biaxial tests and thesymmetric expansion of a cylindrical cavity under the plane strain conditions. However, the



Figure 1. Biaxial compression tests on loose Toyoura sand, I*D+0.2. Experimental data (Tatsuoka et al.

[32] on the left) versus calculations (present study, on the right).

circumferential pressure in the cavity expansion problem, which corresponds to the lateralpressure in the tests, is not constant and is unknown a priori.

3. SPHERICAL CAVITY

3.1. The boundary value problem

Consider a spherical cavity which expands quasi-statically and symmetrically in an in"nite or"nite body starting from an initial radius r0

a. If the body is "nite, it is bounded by an outer sphere

concentric with the cavity. The soil is assumed to be dry or fully drained so that the presence ofthe pore water can be disregarded. In the spherical coordinates r, h, u, the symmetric expansion ofa cavity is described by the velocity component v

r, the stress components ¹

rr, ¹hh"¹rr and the

void ratio e. All these quantities are functions of radius r and time t. The stretching tensor hasthree nonzero components D

rr"Lv

r/Lr and Dhh"Drr"v

r/r. For brevity, we will write ¹

r, ¹h, v

instead of ¹rr,¹hh, vr.



Figure 2. The same as in Figure 1 except for dense sand, I*D+0.9.

Under the assumed symmetry, the process of deformation is governed by a system of four "rst-order partial di!erential equations for four unknown functions v, ¹

r,¹h and e. The system consists

of the equilibrium equation

L¹r

Lr#

2

r(¹

r!¹h)"0 (17)

the constitutive equations (1)

L¹r

Lt#v

L¹r

Lr"H

rA¹r, ¹h,

Lv

Lr,v

r, eB (18)

L¹hLt

#vL¹hLr

"HhA¹r, ¹h,

Lv

Lr,v

r, eB (19)

and the mass balance equation (4)

Le

Lt#v

Le

Lr"(1#e)A

Lv

Lr#2

v

rB (20)



Figure 3. Triaxial compression tests on loose Toyoura sand, I*D+0.2. Experimental data (Fukushima and

Tatsuoka [33], on the left) versus calculations (present study, on the right).

Let ra(t) and r

b(t) be the radii of the inner and the outer spherical surfaces, respectively, which

bound the domain where the solution is sought. The initial radii are denoted by r0a

and r0b. The

case r0b"R corresponds to a cavity in an in"nite body.

The boundary value problem for system (17)}(20) is formulated as follows: given initialconditions

¹0r

(r), ¹0h (r), e0 (r) (21)

at t"0, "nd the solution v (r, t), ¹r(r, t), ¹h(r, t), e(r, t) for t*0 with a given velocity at the inner

boundary and a constant radial stress at the outer boundary, that is, with the boundaryconditions

r5a(t)"v

0'0 and ¹0

r(rb, t)"0 (22)

The initial stresses in (21) must satisfy the equilibrium equation (17). The initial states in thecalculations are taken to be hydrostatic with homogeneous density. In this case, as follows fromthe equations, the solution to the problem with r0

b"R and any r0

acan be obtained from the

solution with r0a"1 merely by rescaling the r-axis. Since the behaviour of the material is rate-

independent, the variable t plays the role of a loading parameter rather than of physical time. For



Figure 4. The same as in Figure 3 except for dense sand, I*D+0.9.

this reason, the value of v0

in (22) can be chosen arbitrarily. The second condition in (22) may bereplaced with v(r

b, t)"0, which gives the same result if r

b"R. However, in most calculations we

used the condition of constant pressure because this was found to give a better approximation toan in"nite body if r

bis large but "nite.

3.2. The numerical algorithm

The boundary value problem formulated above was solved numerically by a "nite-di!erencetechnique.

Di!erentiating the equilibrium equation (17) with respect to time and using the constitutiveequations (18), (19), we obtain the equation

LHr

Lr#

2

r(H

r!Hh)#

L¹r

Lr Av

r!

Lv

LrB"0 (23)

With respect to the velocity v(r, t) as a functions of r at a "xed t, this is a second-order ordinarydi!erential equation. Given ¹

r(r, t), ¹h(r, t), e(r, t) at a time t, the velocity v (r, t) as a function of

radius can be calculated by the integration of this equation with the use of two boundaryconditions expressed in terms of velocity and/or its gradient Lv/Lr. Namely, one needs either one



boundary condition at each point raand r

bto solve a boundary value problem, or two boundary

conditions (velocity and its gradient) at one point to solve the so-called Cauchy problem.The solution of the whole problem involves two integration procedures: (1) the integration of

Equation (23) from ra

to rb

with the use of the boundary conditions (22) in order to "nd thevelocity v(r, t) at a "xed time t; and (2) the integration of Equations (18)}(20) with respect to timein order to "nd the functions ¹

r(r, t), ¹h(r, t) and e(r, t). The integration of Equations (18)}(20)

does not require any boundary conditions.To integrate Equation (23), the interval [r

a, r

b] is divided into subintervals by introducing

discretization points r(j)

, j"0,2,N, where r(0)"r

a, r

(N)"r

b. These points are treated as

material points so that their co-ordinates r(j )

are changed with the deformation. Rather than useboundary conditions at r

aand r

bimmediately as dictated by (22), we prescribe two boundary

conditions at ra, namely velocity and its gradient g"Lv/Lr. To satisfy the boundary condition at

rb, the value of g at r

ais varied in order to "nd the right one. With the use of the Newton method

which is commonly used for the solution of nonlinear equations, only a few iterations are needed.The boundary value problem is thus reduced to a sequence of the Cauchy problems. Theadvantage of this approach is the following. To solve the boundary value problem immediately, itwould be necessary to solve a nonlinear system of N equations in N variables v

(1),2, v

(N). To

solve the Cauchy problem, we start from the point r(0)"r

aand proceed to the points r

(1), r

(2),2

so that at each step we have to solve only one non-linear equation in one variable (Equation (25)below).

Equation (23) can be viewed as that of the "rst order with respect to Hr. In order to obtain the

velocity v(j`1)

from given v(j)

and g(j)

and thus to perform the step-by-step integration withrespect to r, an implicit Euler scheme is applied to the function H

r:

Hr(j`1)

"Hr( j)

#

1

2GLH

rLr K

(j)

#

LHr

Lr K(j`1)

H*r(j)

(24)

where *r( j)"r

( j`1)!r

( j). The use of (23) gives

Hr( j`1)

!12)

(j`1)*r

( j)"H

r(j)#1

2)

( j)*r

(j)(25)

where we have denoted

)"

2

r(Hh!H

r)#

L¹Lr Ag!

v

rB (26)

After substituting

v(j`1)

"v( j)#1

2(g

(j)#g

( j`1))*r

( j)(27)

into )(j`1)

and then )( j`1)

into (25), we obtain a non-linear equation in one unknown variableg( j`1)

. This equation can easily be solved by the Newton method.To integrate Equations (18)}(20), the same implicit scheme is applied as for the spatial

integration (the upper index stands for time):

¹i`1r

"¹ir#1

2(¹Q i

r#¹Q i`1

r)*t (28)

¹i`1h "¹ih#12(¹Q ih#¹Q i`1h )*t (29)

ei`1"ei#12(eR i#eR i`1)*t (30)



Equations (28)}(30) are written for a given material point. The co-ordinates of the material pointsare updated by the integration of the velocity

ri`1"ri#12(vi#vi`1)*t (31)

The time derivatives in (28)}(30) are functions of ¹r, ¹h, g, v, r and e according to (18)}(20). The

system (28)}(31) can be written as

Fi`1"Fi#12MFQ (Fi, vi, gi)#FQ (Fi`1, vi`1, gi`1)N*t (32)

whereF denotes the column of the four quantities ¹r, ¹h, e, r. GivenFi, system (32) can be solved

for Fi`1 via successive approximations:

Fi`1*n`1+

"Fi#12MFQ (Fi, vi, gi)#FQ (Fi`1

*n+, vi`1

*n+, gi`1

*n+)N*t (33)

where the lower index in square brackets refers to the iteration number. Each iteration involvesthe integration of Equation (23) as described above.

Equation (23) from which the velocity is found was obtained by the di!erentiation of theequilibrium Equation (17) with respect to time. This actually means that, when solving theproblem, the condition of equilibrium for the stresses is replaced with the condition of equilibriumfor the stress rates. Analytically, if the initial stresses at t"0 and the stress rates at t*0 obeyequilibrium, the stresses will also obey equilibrium at t'0. However, during the numericalstep-by-step time integration, if only the stress-rate equilibrium is controlled, the residual in thestress equilibrium may accumulate and thus lead to an increase in the error of the solution. Inorder to avoid this accumulation, at each time step, before solving Equation (23), we calculate thestress residual, divide it by *t to obtain the time derivative and substitute this quantity with theopposite sign into the right-hand side of Equation (23).

The number of the discretization points in the calculations varied from 150 in the sphericalproblem to 300 in the cylindrical problem, with a higher degree of discretization in the vicinity ofthe cavity. One time step corresponds to a relative increment of 10~4 to 5]10~4 in the cavityradius.

3.3. Numerical solutions to the spherical problem

In the numerical calculations an in"nite body is modelled by taking r0bAr0

a. Apart from the degree

of spatial and time discretization, the accuracy of the approximation of an in"nite body dependson the size of the outer radius. Whether the outer radius is large enough can be judged from thechange in the circumferential stress ¹h at r

b(the radial stress ¹

ris constant according to the

boundary condition). The calculations performed for various soils showed that a twofold increasein the radius of the cavity is accompanied by a less than 0.1 per cent variation in the circumferen-tial stress at r

bif r0

b/r0

a*30 for loose soils (I*

D+0) and r0

b/r0

a*90 for dense soils (I*

D+0.9).

Figure 5 shows the pressure components pr, ph and the void ratio at the cavity wall during the

expansion from r0a"1 to r

a"3.3 calculated for Ticino sand. Di!erent curves correspond to

di!erent initial values of the relative density I*D, with the initial pressure being 100 kPa in all the

cases. As the cavity expands, both pressure components increase in magnitude and monotonicallyapproach their limit values. As seen from the curves, for the pressure and the density at the cavitywall to reach the limit values with su$cient accuracy, the cavity has to expand up to r

a+2r0

afor

a loose soil and up to ra+3r0

afor a dense soil.



Figure 5. Expansion of a spherical cavity in Ticino sand. The pressure components and the void ratio at thecavity wall as functions of the cavity radius.

Figure 6 shows the deviatoric stress q"(pr!ph)/2 and the void ratio at the cavity wall versus

the mean pressure p"(pr#2ph)/3. The points on the (p, q)-plane showing successive states

approach the critical-state line which is a straight line determined by the constitutive equation.From this it follows that the ratio of the limit radial pressure to the limit circumferential pressureis a constant for a given soil, regardless of the initial density and the initial pressure. Thee(p)-curves in Figure 6 also approach the critical-state line e

#(p) determined by the constitutive

equation (see Equation (5)). The changes in the void ratio are small if the soil is initially loose andthe void ratio is close to the critical one. If the soil is very loose, the limit state may be slightlydenser than the initial one. Note that the strain}stress paths during the cavity expansion are notapproximated by the paths in the triaxial compression tests with p

3"const or p"const. If the

cavity expands in an in"nite body and the initial state is hydrostatic and homogeneous, eachmaterial element follows the same strain-stress path as an element at the cavity wall.

Figure 7 presents an example of spatial distribution of the pressure components and the voidratio after the cavity has expanded from r0

ato r

a"3.3r0

a. At a distance of 20r0

athe change in the

pressure pramounts to only 4 per cent of the change at the cavity surface. The in#uence of the

cavity practically disappears at a distance of 40}50r0a. As the cavity begins to expand, the pressure

ph in each point "rst slightly decreases and the material contracts. This occurs at small strains



Figure 6. Expansion of a spherical cavity in Ticino sand. The q}p paths for p0"0.1 MPa (top left) and

p0"0.4 MPa (top right) and e}p paths (bottom) at the cavity wall.

when the change in the co-ordinate r of the point does not exceed 2}3 per cent. This range cannotbe seen at the scale of Figure 5 discussed above; however, this is seen from the Figure 7 where thepressure ph and the void ratio at r'10r0

aare slightly below their initial values.

The ratio of the limit values of prand ph to the initial pressure depends on the initial density: the

higher the density, the bigger the pressure change. Figure 8 shows the radial limit pressure pLS

asa function of the initial pressure p

0and the initial relative density I*

Dfor Ticino sand. The limit

pressure depends on the initial state in the same manner as the cone resistance in cone penetrationtests. Figure 9 compares the limit pressures calculated for "ve soils presented in Table I. The limitpressures for di!erent soils with the same initial state may di!er by a factor of 1.5.



Figure 7. Spatial distribution of the pressure components and the void ratio after the expansion ofa spherical cavity. I*

D"0.6, p

0"100 kPa.

Figure 8. The limit pressure pLS

for a spherical cavity versus the initial pressure p0for di!erent initial relative

densities I*D

calculated for Ticino sand.



Figure 9. The limit pressure pLS

for a spherical cavity versus the initial pressure p0for di!erent soils. I*

D"0.1

(on the left), I*D"0.9 (on the right).

4. CYLINDRICAL CAVITY

4.1. The boundary value problem

The mathematical formulation of the problem for a cylindrical cavity is analogous to that fora spherical cavity. In the cylindrical co-ordinates r, h, z, the symmetric expansion of a cavityunder plane strain conditions (in"nitely long cavity) is described by the velocity component v

r, the

stress components ¹rr, ¹hh, ¹zz

and the void ratio e. All these quantities are functions of radiusr and time t. The stretching tensor has two non-zero components D

rr"Lv

r/Lr and Dhh"v

r/r. For

brevity, we will write ¹r, ¹h, ¹z

, v instead of ¹rr, ¹hh, ¹zz

, vr.

The deformation is governed by the system of "ve equations: the equilibrium equation

L¹r

Lr#

1

r(¹

r!¹h)"0 (34)

the constitutive equations (1)

L¹r

Lt#v

L¹r

Lr"H

rA¹r,¹h,¹z

,Lv

Lr,v

r, eB (35)

L¹hLt

#vL¹hLr

"HhA¹r,¹h,¹z

,Lv

Lr,v

r, eB (36)

L¹z

Lt#v

L¹z

Lr"H

zA¹r,¹h,¹z

,Lv

Lr,v

r, eB (37)



and the mass balance equation (4)

Le

Lt#v

Le

Lr"(1#e)A

Lv

Lr#

v

rB (38)

The boundary value problem is formulated in the same way as in the spherical case: giveninitial conditions

¹0r

(r), ¹0h (r), ¹0z

(r), e0(r) (39)

at t"0, "nd the solution v(r, t), ¹r(r, t), ¹h(r, t), ¹z

(r, t), e (r, t) for t*0 with the boundary condi-tions (22). The initial stresses in (39) must satisfy the equilibrium equation (34). In the calculations,the initial stresses and density were taken to be homogeneous: ¹0

r"¹0h , ¹0

z"¹0

r/K with

a coe$cient K.The boundary value problem was solved by the same "nite-di!erence technique as in the

spherical case.

4.2. Numerical solutions to the cylindrical problem

The disturbance to the state of the soil caused by the expansion of a cylindrical cavity decreasesslowly with the distance from the cavity as compared to the spherical case. As a consequence,a larger radius of the outer surface is needed to model an in"nite body with the required accuracy.The minimum ratio r0

b/r0

awhich ensures a less than 0.1 per cent variation in the axial and

circumferential stresses at rbupon a twofold increase in the cavity radius was found to vary from

200 for loose soils (I*D+0) to 400 for dense soils (I*

D+0.9).

The solutions to the spherical and the cylindrical problems are qualitatively similar. Figure 10shows the pressure components and the void ratio at the cavity wall as functions of the cavityradius during the expansion from r0

a"1 to r

a"5, with a hydrostatic initial pressure of 100 kPa.

As the cavity expands, the pressure components increase in magnitude and approach limit values.As compared to the spherical case, this approach is slower so that a bigger change in the cavityradius is needed for the stresses to reach their limit values with the same accuracy; namely, thecavity has to expand up to r

a+3r0

afor a loose soil and up to r

a+4r0

afor a dense soil. For the

same initial state, the limit pressures are higher in the spherical case than in the cylindrical case.Figure 11 shows the deviatoric stress q"(p

r!ph)/2 and the void ratio at the cavity wall versus

the mean pressure p"(pr#ph#p

z)/3. The points on the (p, q)-plane approach the critical-state

line determined by the constitutive equation. As in the spherical case, the ratio between the limitvalues of the pressure components is a constant for a given soil and does not depend on the initialdensity and pressure. The e (p)-curves in Figure 11 approach the critical-state line e

#(p) deter-

mined by Equation (5). The soil becomes looser as the cavity expands except when the soil isinitially rather loose and the void ratio is close to the critical one. If the cavity expands in anin"nite body and the initial state is homogeneous, each material element follows the samestrain-stress path as an element at the cavity wall.

The spatial distribution of the pressure components and the void ratio after the expansion ofa cylindrical cavity from r0

ato r

a"5r0

ais shown in Figure 12. The curves are similar to those

shown in Figure 7 except that in the cylindrical case the in#uence of the cavity expansion spreadsfarther from the cavity.

The ratio of the limit radial pressure pLC

to the initial pressure p0

is higher for a denser soil, asseen from Figure 13. The in#uence of the coe$cient K"¹0

r/¹0

zon the limit pressure is shown in



Figure 10. Expansion of a cylindrical cavity in Ticino sand. The pressure components and the void ratio atthe cavity wall as functions of the cavity radius.

Figure 14. At the same initial mean pressure, the limit pressure is lower for a smaller K. Figure 15summarizes the results for "ve soils from Table I.

5. GEOTECHNICAL APPLICATIONS

The motivation for the present study was the application of the cavity expansion problem to theinterpretation of cone penetration tests (CPT) and pressuremeter tests (PMT). The results arebrie#y outlined below and will be presented in detail an a companion paper [27].

The interpretation of CPT and PMT consists in the evaluation of the state of the soil (stressesand density) from the quantities directly measured in the tests, namely, from the cone resistanceq#

in CPT and the limit pressure pL

in PMT, with the help of empirically or theoreticallyestablished relationships. The evaluation of the stress state implies the determination of the earthpressure coe$cient K; the vertical stress is determined by the weight of the overlying soil. Sincetwo independent quantities, q

#and p

L, can be measured in the "eld, it is in principle possible to

evaluate both the density and the earth pressure coe$cient. Below we discuss only the determina-tion of the density, assuming that K can be evaluated otherwise. Note that in the case of CPT the



Figure 11. Expansion of a cylindrical cavity in Ticino sand. The q}p paths for p0"0.1 MPa (top left) and

p0"0.4 MPa (top right) and e}p paths (bottom) at the cavity wall.

in#uence of K may be disregarded as it is much weaker than the in#uence of the pressure levelitself or the density.

Cone penetration and pressuremeter tests in large calibration chambers [23,24,28}31] allow usto "nd the dependence of q

#and p

Lon the initial state of the soil. Although this method is able to

provide reliable empirical relations, it is rather expensive and time-consuming. Another way oftackling the problem is to "nd the required relations theoretically with the use of an adequateconstitutive model, if available, calibrated for the soil of interest.

The deformation of the soil during PMT can be modelled directly by the solution of theproblem of the symmetric expansion of a cylindrical cavity if we neglect possible inhomogeneityof the deformation and assume the plane strain conditions. A series of solutions to the cavityexpansion problem for a given initial stress state and di!erent densities allows us to "nd the



Figure 12. Spatial distribution of the pressure components and the void ratio after the expansion ofa cylindrical cavity in Ticino sand. I*

D"0.6, p

0"0.4 MPa.

Figure 13. The limit pressure pLC

for a cylindrical cavity versus the initial pressure p0

for di!erent initialrelative densities I*

Dand K"1.0 calculated for Ticino sand.

relation between the limit pressure pLand the initial density. This relation can then be used for the

estimation of the density from the value of pL

measured in the "eld at a certain depth. Thisapproach has been thoroughly investigated by the authors [27] by the comparison of thecalculated limit pressures p

LCand the limit pressures p

Lmeasured in calibration chambers where

the initial state of the soil is known.




for a cylindrical cavity versus the initial pressure p0for K"0.4, 0.7 and 1.0

for I*D"0.2 (on the left), I*

D"0.8 (on the right) calculated for Ticino sand.


for a cylindrical cavity versus the initial pressure p0

for di!erent soils.I*D"0.1 (on the left), I*

D"0.9 (on the right).

The modelling of a cone penetration test with the help of the cavity expansion problem is not asstraightforward as the modelling of a pressuremeter test because the deformation of the soil in thevicinity of the cone tip does not correspond to the expansion of a spherical or a cylindrical cavity.However, the limit pressures p

LSand p

LCin the spherical and cylindrical cavity expansion



problems depend on the initial stresses and density in the same manner as the cone tip resistanceq#. The approach that we have adopted to relate q

#with p

LSconsists in introducing a &shape factor'

kq

so that q#

can be found as q#"k

qpLS

, where pLS

is the limit pressure for a spherical cavitycalculated with the same initial pressure and density. Alternatively, one can introduce a shapefactor to connect q

#with the limit pressure p

LCin the cylindrical problem. The dependence of the

shape factor on the pressure and density is reduced to a dependence on only one parameter,namely on the pressure-dependent relative density (16). The shape factor has been evaluatedfor "ve sands with the use of the experimental data on CPT in calibration chambers andthe corresponding solutions to the cavity expansion problem. It is found that all consideredsands exhibit a common shape factor as a function of the relative density. If the shape factor isknown, the way to evaluate the density of the soil consists in "nding I*

Dwhich satis"es the

equation

qc"k

q(I*

D) p

LS(p

0, I*

D) (40)

where p0

is the mean pressure at the depth where q#is measured.

6. CONCLUSION

The problem of the symmetric quasi-static large-strain expansion of a cavity in an in"nitegranular body has been solved numerically with the use of the hypoplastic constitutive relation.As the radius of the cavity increases, the stresses and the void ratio at the cavity wall asymp-totically approach limit values which correspond to a critical state of the soil. For a given soil,these limit values depend on the initial stresses and the initial density. The limit pressures forspherical and cylindrical cavities are calculated and presented as functions of the initial state for"ve sands for which the constitutive equation has been calibrated. The dependence of the limitpressure on the initial state of the soil in both spherical and cylindrical problems is similar to thedependence of the limit pressure in the pressuremeter tests and the cone resistance in the conepenetration tests. The comparison of the experimental results on cone penetration and pres-suremeter tests in large calibration chambers with the numerical solutions to the cavity expansionproblem allows us to develop a method of estimation of the state of the soil in the "eld from thecone penetration and pressuremeter tests.

ACKNOWLEDGEMENTS

The investigations on the cavity expansion problem were initiated by the "rst author within the frameworkof the project &Dynamic penetration' (Gu 103/41) supported by the Deutsche Forschungsgemeinschaft. Thepresent study is a part of the project &Reconstruction and stabilization of dumps and dump slopesendangered by settlement #ow' supported by the LMBV mbH (Lusatian and Central German MinesAdministration Company Ltd.) and by the BMBF (Federal Ministry of Culture, Science, Research andTechnology, Germany). The authors are grateful to Prof. D. Lo Presti and Dr. K. Been for providingexperimental results used for the calibration of the constitutive model, to Dr. I. Herle for the assistance in thedetermination of the hypoplastic parameters, to Prof. G. Gudehus for valuable discussions, and to Mr. R.Engel for the help in performing the numerical calculations.



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