619

Quasi-static undrained expansion of a cylindrical cavity in clay inthe presence of shaft friction and anisotropic initial stresses

C. Sagaseta a,∗, G.T. Houlsby b, H.J. Burd b

a Department of Materials and Ground Engineering, University of Cantabria, Av. de Los Castros, s/n, 39005 Santander, Spainb Department of Engineering Science, University of Oxford, Oxford, UK

Abstract

Solutions for cylindrical cavity expansion in an infinite incompressible medium are presented. The solutions account forlarge strain in both the plastic and elastic regions. They account also for the influence of an axial shear stress at the surfaceof the cavity, and also for the possibility of unequal vertical and horizontal stresses. The solutions are presented principallyin closed-form, although they make some use of a series solution and of integrals which must be evaluated numerically.The solutions have application to the determination of the stresses around axially loaded piles and around in-situ testingdevices such as penetrometers.

Keywords: Cavity expansion; Elastoplastic; Incompressible material; Analytical solution; Von Mises; Large strains;Eulerian formulation

1. Introduction

In the paper we present an analytical solution for thequasi-static expansion of a cylindrical cavity in an incom-pressible elastoplastic soil. There are many other publishedsolutions of the problem, either in closed form (for instance[1]) or numerically [2,4]. The novelty here lies in the factthat we account for the possibility of (a) shaft frictionin the axial direction, and (b) in-situ horizontal stress notnecessarily equal to in-situ vertical stress.

The soil is treated as incompressible elastic–perfectplastic, with the Von Mises yield condition. The proper-ties are defined by the parameters: G (incompressibilityrequires that Poisson’s ratio ν = 0.5), and su = (σ1 −σ3)/2in triaxial conditions (σ2 = σ3). The shear modulus may beexpressed by defining the rigidity index, Ir = G/su .

The cavity is vertical, and its radius, a, expands steadily,starting from zero. The expansion of a cavity with fi-nite initial radius a0 can be recovered from the solutionsgiven below, simply by tracking the stresses at radius

r =√

a20 −a2. As stated above, the analysis concentrates on

the influence of the following factors, not considered in theprevious solutions:

∗ Corresponding author. Tel.: +34 (942) 201813; Fax: +34 (942)201821; E-mail: sagasetac@unican.es

– The action of a longitudinal uniform shear traction atthe cavity wall, τa . This is relevant for driven piles andfor cone pressuremeters.

– An anisotropic, but uniform, initial state of stresses(σv0 �= σh0).

– Large strains in both the plastic and elastic zones (usu-ally, small strains are assumed in the elastic region).The first two points imply the presence of shear stresses,

additional to the effect of the cavity expansion. This has aninfluence on the extent of the plastic zone.

2. Problem formulation

The problem is formulated in cylindrical coordinates(r ,θ , z). The kinematic description of the motion, taking thecavity radius, a, as the time variable, is fully defined by theconditions of axial symmetry, the infinite length of cylinderand soil incompressibility:

r = (r 2

0 +a2)1/2

θ = θ0

z = z0 +w(r ,a)

(1)

The cavity axis, OZ, is not a principal direction, so thedeformation is not strictly one of plane strain. However, thecondition remains that all the variables are constant with z,and that the axial strain rate is zero.

2003 Elsevier Science Ltd. All rights reserved.Computational Fluid and Solid Mechanics 2003K.J. Bathe (Editor)

620 C. Sagaseta et al. / Second MIT Conference on Computational Fluid and Solid Mechanics

The components of the strain rate tensor are:

ε̇rr = −∂vr

∂r= − ∂

∂r

(∂r

∂a

)= a

r 2

ε̇θθ = −vr

r= −1

r·(∂r

∂a

)= − a

r 2

ε̇zz = −∂vz

∂z= − ∂

∂z

(∂z

∂a

)= 0

γ̇rz = −(∂vz

∂r+ ∂vr

∂z

)= − ∂

∂r

(∂z

∂a

)− ∂

∂z

(∂r

∂a

)

= − ∂

∂a

(∂z

∂r

)= − ∂

∂a

(∂w

∂r

)

(2)

where (vr = ∂r/∂a;vθ = 0;vz = ∂z/∂a) are the soil veloci-ties.

The strain rates are related to the material derivatives ofthe stress through the elastic–plastic constitutive equations:

ε̇i j = ε̇ei j + ε̇ p

i j = Cijkl · dσkl

da+λ · ∂ f

∂σi j(3)

where f = 0 is the Von Mises yield condition:

f = (σrr −σθθ )2 +(σθθ −σzz )2 +(σzz −σrr )2 +6τ 2rz −8s2

u (4)

From dimensional analysis, the stresses must dependon the radial coordinate r only through the ratio η =a/r . Hence, the material derivatives of the stresses can beexpressed in terms of the spatial radial derivative as:

dσi j

da= ∂σi j

∂r· ∂r

∂a+ ∂σi j

∂a=(a

r− r

a

)· ∂σi j

∂r(5)

The equilibrium equations are:

∂σr

∂r+ σr −σθ

r= 0

∂τrz

∂r+ τrz

r= 0

(6)

We assume that the shear stress at the inner boundary isconstant, so that the boundary conditions are:

for r = a : τrz = τa

for r = ∞ : σr = σθ = σh0; σz = σv0; τrz = 0(7)

3. Solution procedure

The stresses are decomposed into their isotropic (p) anddeviatoric (si j ) parts:

σi j = p · δi j + si j (8)

where δi j is the Kronecker delta (unit tensor). In the plasticzone, the following variables are used, generalized from

Lode’s parameters:

srr = 4

3· su · cosζ · sin

(ψ+ π

6

)

sθθ = −4

3· su · cosζ · sin

(ψ− π

6

)

szz = −4

3· su · cosζ · cosψ

srz = 2√3

· su · sinζ

(9)

The equations for stresses and strains in the vertical andhorizontal directions are only partially coupled. The veloci-ties and strain rates in any horizontal plane, (vr ,vθ , ε̇rr , ε̇θθ )are defined from Eqs. (1) and (2). On the other hand, thecorresponding shear stress τrz is fully defined by the secondequilibrium Eq. (6), both in the elastic and plastic zones.

This decoupling allows a stepped solution. First, theshear stress, τrz , is obtained from the second equilibriumEq. (6). Then, the system is integrated for the normalstresses, starting from the infinite boundary, and assumingelastic behavior, until the condition f = 0 is reached. Thisdetermines the position of the elastic–plastic boundary,η = ηR. Then, the stresses and strain rates are integrated inthe plastic zone, and the normal pressure at the cavity wallis obtained.

4. Results

In the presentation of results, use is made of the follow-ing non-dimensional parameters:– α, shaft shear factor (0 ≤ α ≤ 1):

τrz(r=a) = τa = 2√3α · su (10)

– ∆0, initial stress ratio:

∆0 = σv0 −σh0

2 · su(11)

4.1. Stresses in the elastic zone

We express the stresses in the form:

σrr = G ·Λ (η) +σh0

σθθ = G · [Λ(η)+2ln(1−η2)]+σh0

σzz = G · [Λ(η)+ ln(1−η2)]+σv0

τrz = τa ·η

(12)

where the function Λ(η) can be expressed as a powerseries:

Λ(η) =∑(

1

n2·η2n

)= η2 + 1

4η4 + 1

9η6 + . . . (13)

Note that this solution differs very slightly from theconventional solution in the elastic region in which smallstrains are assumed.

C. Sagaseta et al. / Second MIT Conference on Computational Fluid and Solid Mechanics 621

4.2. Elastic–plastic boundary (η = ηR)

The position of the boundary is given by:

ln(1−η2

R

)= − 2√3

· 1

Ir·√

1−∆20 −α2η2

R (14)

For α = 0 this reduces to:

η2R = 1−exp

(− 2√

3· 1

Ir·√

1−∆20

)(15)

4.3. Stresses in the plastic zone

Deviatoric stresses:The deviatoric stresses may be expressed by the follow-

ing functions which should be substituted into Eq. (9):

sinζ = α ·η1+ sinψ

cosψ= 1+ sinψR

cosψR

×(

cosζ +cos ζa

cosζR +cos ζa· cosζR −cos ζa

cosζ −cosζa

) √3

2 ·Ir · 1cosζa

(16)

For α = 0 these reduce to:

ζ = 0

1+ sinψ

cosψ= 1+ sinψR

cosψR·(

1−η2R

1−η2

)√3

2 ·Ir(17)

and for α = 1:

sinζ = η

1+ sinψ

cosψ= 1+ sinψR

cosψR· exp

[−√

3 · Ir

(1

cosζR− 1

cosζ

)](17′)

Isotropic pressure:The mean stress is expressed as:

p = pR − 4

3· su ·

[cosζ · sin

(ψ+ π

6

)

−cosζR · sin(ψR + π

6

)]+ 4√

3· su ·

η∫ηR

1

η· cosζ · sinψ ·dη

(18)

In all the expressions, the subscripts R and a mean thevalues at the elastic–plastic boundary (η = ηR) and at thecavity wall (η = 1), respectively. The integral in Eq. (18)needs to be evaluated numerically as a closed-form integra-tion is not available.

Wall pressure:The pressure at the cavity wall, σa , is the value of σrr

for η = 1. For the case of isotropic initial stresses (∆0 = 0)this can be obtained as a closed form expression:

σa = σh0 +G ·Λ(ηR)+ 4√3

· su

×(√

1−α2 −√

1−α2η2R − lnηR − ln

1+√1−α2

1+√1−α2η2R

)

(19)

which, in the absence of shaft friction (α = 0) reduces to:

σa = σh0 +G ·Λ(ηR)− 4√3

· su · lnηR

Expanding ηR (15) and � (13) into power series and takingonly their first term, leads to:

η2R

∼= 2√3

· 1

Ir

σa∼= σh0 + 2√

3· su ·

(1+ ln

√3

2Ir

) (20)

which coincides with the existing solutions for small strainsin the elastic region.

4.4. Velocities and strain rates

The velocities and strain rates in the horizontal plane(vr ,vθ , ε̇rr , ε̇θθ ) are defined from Eqs. (1) and (2). The ver-tical velocity, vz , becomes infinite, due to a logarithmicterm which necessarily appears in two-dimensional elastic-ity problems containing resultant forces [3]. This is usuallyovercome by assuming that the displacements vanish atsome arbitrary distance, such as the “magical radius” usedby Randolph and Wroth [4].

The velocity vz and the shear strain rate, γ̇rz , are givenby:

vz = ∂w

∂a

γ̇rz = − ∂

∂a

(∂w

∂r

) (21)

with:

w = we +w p

we (elastic) = τaa

Ir

[ln

rm

r− 1

6

(η2 −η2

m

)]

w p(plastic)

= 0 if η < ηR

=r=r∫

r=R

a=ar∫a=0

γ prz da dr if η ≥ ηR

(22)

γ̇ prz = 1

Ir· α

2a2(1−∆2

0

)3/2 (r 2 −a2

)r 3[r 2 −α2a2

(1−∆2

0

)]+8α

a2(1−∆2

0

)1/2sinψ

r 2[r 2 −α2a2

(1−∆2

0

)]1/2

In the above expressions, rm(ηm = a/rm ) is a “magicalradius” (in theory, rm → ∞, but in practice it can be takenas some appropriate multiple of the axial length over which

622 C. Sagaseta et al. / Second MIT Conference on Computational Fluid and Solid Mechanics

the cavity expansion occurs). The integration for w p mustbe performed numerically. The limit ar is the value of thecavity radius a for which the plastic zone reaches r , thepoint of calculation of w p , defined by the condition:

a2r

r 2= η2

r = 1−exp

[− 1

Ir

√(1−∆2

0

)(1−α2η2

r

)](23)

Acknowledgements

The main part of the work presented herein was per-formed during the visit of the first author to the Universityof Oxford, partially sponsored by a British–Spanish JointResearch project.

References

[1] Gibson RE, Anderson WF. In situ measurement of soilproperties with the pressuremeter. Civil Eng Pub Works Rev1978;56:615–618.

[2] Randolph MF, Wroth CP. An analysis of the deformation ofvertically loaded piles. J Geotech Eng Div ASCE 1978;104(GT12):1465–1488.

[3] Strack OE. Analytic Solutions of Elastic Tunneling Problems.Ph.D. Thesis, Delft University of Technology, Delft Univer-sity Press, 2002.

[4] Yu HS, Houlsby, GT. Finite cavity expansion in dilatantsoils: loading analysis. Géotechnique 1991;41(2):173–183.