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Anisotropically elasto-plastic solution to undrained cylindrical cavity expansion in K0-consolidated clay
Computers and Geotechnics 73 (2016) 83–90
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Technical Communication
Anisotropically elasto-plastic solution to undrained cylindrical cavity expansion in K0-consolidated clay Lin Li a,b, Jingpei Li a,b,⇑, De’an Sun c a
Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China c Department of Civil Engineering, Shanghai University, 149 Yanchang Road, Shanghai 200072, China b
a r t i c l e
i n f o
Article history: Received 15 May 2015 Received in revised form 5 September 2015 Accepted 26 November 2015
Keywords: K 0 -MCC model Anisotropy Cylindrical cavity Undrained expansion Semi-analytical solution
a b s t r a c t This paper presents an anisotropically elasto-plastic solution to the undrained expansion of a cylindrical cavity in K 0 -consolidated clay. The elasto-plastic constitutive relationship following the soil yielding process is described by the K 0 -based modified Cam-clay (K 0 -MCC) model, which can properly reflect the anisotropic effects on the soil behaviour. Following the large strain deformation theory, the problem is reduced to solving a system of first-order ordinary differential equations in the plastic region. The semi-analytical solutions to the radial, tangential and vertical effective stresses are obtained using the Lagrangian method and the elastic–plastic (EP) boundary conditions. In addition, based on the semianalytical results, an approximate closed-form solution is presented for practical purposes. Extensive comparisons with the isotropic constitutive model-based solutions have been performed to illustrate the effects of the initial stress anisotropy and initial stress-induced anisotropy on the cavity expansion and the stress distributions. The present solution incorporates the anisotropic properties of the natural K 0 -consolidated clay, thereby providing a more realistic theoretical basis for the practical engineering problems such as the pile installation and the pressuremeter tests. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The cavity expansion theory has been widely applied to the modelling of complex geotechnical problems, such as the prediction of end-bearing and shaft capacities of a driven pile [1–6], as well as the interpretation of pressuremeter or cone penetration tests for determination of the soil properties in-situ [7–15]. Over the past few decades, the progress of cavity expansion theory has been mainly focused on the development of new solution techniques and the application of more realistic yield criterions or constitutive models to the cavity expansion problems [16–27]. Hill [24] first presented a comprehensive large strain solution to the expansion of the spherical and cylindrical cavities in an isotropic Tresca material. Subsequently, Vesic [8], Carter et al. [2,17], and Yu and Houlsby [18] presented a closed-form solution for cylindrical and spherical cavities by assuming the isotropic soil behaves as a non-associated Mohr–Coulomb material. Collins et al. [6], Collins and Stimpson [25], and Collins and Yu [26] proposed a similarity ⇑ Corresponding author at: Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China. Tel.: +86 021 65982773. E-mail address: [email protected] (J. Li). http://dx.doi.org/10.1016/j.compgeo.2015.11.022 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.
solution technique for zero initial radius cavity expansion problems, and the MCC model was adopted to describe the elastoplastic behaviours of the initial isotropic soil. Yu [27] summarised and presented a full account of the cavity expansion solutions in isotropic soils using a variety of critical state models for both drained and undrained loading conditions. Cao et al. [20] derived closed-form solutions for both spherical and cylindrical cavities in isotropic MCC soils by assuming the ultimate deviator stress distributions in the plastic zone. More recently, Chen and Abousleiman [21,22] derived semi-analytical solutions for the expansion of drained and undrained cylindrical cavities in anisotropic clay. However, the isotropic assumption-based MCC model was still adopted to derive the elasto-plastic constitutive matrix during the cavity expansion process. Although the existing cylindrical cavity expansion solutions have been widely used to solve practical geotechnical problems, most of the solutions are based on the assumptions of isotropic in-situ stress and elastic perfectly plastic. It is well known that the initial in-situ stress of the natural clay is likely to be anisotropic due to the change of sediment and consolidation environments [23], and the initial stress anisotropy and initial stress-induced anisotropy have pronounced effects on the soil behaviour. However, anisotropically elasto-plastic solutions to the undrained cylindrical
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cavity expansion in K 0 -consolidated anisotropic clay are not currently available. In this paper, to account for the effects of the initial stress anisotropy and the stress-induced anisotropy on the cylindrical cavity undrained expansion process, an anisotropically elasto-plastic semi-analytical solution was derived by employing the K 0 -MCC model [28,29]. In addition, an approximate anisotropically elastoplastic analytical solution is presented for practical purposes. Both of the proposed solutions, the semi-analytical and the approximate analytical solutions, overcome the shortcomings of the isotropic assumption, as a result, these solutions can model the cylindrical cavity expansion process in K 0 consolidated anisotropic clay properly and accurately. 2. Mechanical model for the cylindrical cavity expansion
r2x r2x0 ¼ a2 a20
ð1Þ
In both the elastic and plastic regions, the equilibrium equation can be written as
@ r0r @u r0r r0h þ þ ¼0 @r @r r
ð2Þ
where r0r and r0h are the effective radial and tangential stresses, respectively; u is the pore water pressure. 2.2. Constitutive relation for K0-consolidated clays According to Hooke’s law, the elastic stress–strain relationship of the K 0 -consolidated clay can be expressed in the incremental form as follows
1 þ v0 0 v0 0 drij drmm dij E E
2.1. Basic assumptions and definition of the problem
deeij ¼
The analytical framework for the cylindrical cavity expansion in K 0 -consolidated anisotropic clay is based on the following basic assumptions:
where E ¼ 2Gð1 þ v 0 Þ is the elastic modulus. In the K 0 -MCC model, the shear modulus G is defined as follows [30]
(a) The K 0 -consolidated anisotropic clay is saturated and homogeneous regarding the following properties: the in-situ effective horizontal stress r0h0 , vertical stress r0v 0 , and the hydrostatic pore-water pressure u0 . (b) The pore water is incompressible, and the compressive stresses and strains are taken as positive. (c) The cylindrical cavity expands under plane strain and undrained conditions; thus, the volume of the soil remains constant during the cavity expansion process. (d) The soil is assumed to be linearly elastic for small deformation in the elastic state, whereas the soil is modelled by the K 0 -MCC model and large deformation theory after yielding. Fig. 1 shows the expansion of a cylindrical cavity in an infinite K 0 -consolidated clay: as the internal cavity pressure gradually increases from rh0 to ra , the cavity expands from the initial radius a0 to the current radius a. If the cavity pressure ra increases further, then the soil on the cavity wall yields firstly, and then a plastic zone is formed from the current cavity radius a to the EP boundary rp . The symbol rp0 refers to the initial location of a soil particle that instantly becomes plastic. For the undrained condition, the conservation of volume condition gives the relationship between the current radius rx , which was initially at rx0 , and the current and initial radii of the cavity as follows
σh0 Elastic zone
G¼
3ð1 2v 0 Þtp0 2ð1 þ v 0 Þj
a
rp
a0
σh0 σa
o
ð4Þ
where j is the slope of swelling line; t is the specific volume, and m0 is effective Poisson’s ratio. The natural clay was usually consolidated under anisotropic K 0 conditions, and thus the soil has initial anisotropic stresses, i.e., the in-situ vertical stress is not equal to the horizontal stress in the K 0 consolidated clay. Because the mechanical properties of the soil associate closely with the stress states, the behaviour of the K 0 consolidated clay is significantly affected by the initial anisotropic stresses, and the K 0 -consolidated clay exhibits so-called initial stress-induced anisotropy properties during shearing [28,29]. To consider the effects of the initial stress-induced anisotropy on the mechanical behaviour of-K 0 consolidated clay, a relative stress ratio g and a relative stress ratio at critical state M are adopted in the K 0 -MCC model instead of the stress ratio gð¼ q=p0 Þ and the stress ratio at critical state M used in the MCC model. The yield function ðf Þ, which for associative plasticity serves also as a plastic potential function ðg Þ, of the K 0 -MCC model can be expressed as follows [28,29]
f ¼1þ
g
M
2
p0c ¼0 p0
ð5Þ
where u0 is the effective friction angle; p0c is the mean effective yielding pressure; the stress parameter p0 , the relative stress ratio g and the relative stress ratio at critical state M are, respectively defined as follows
p0 ¼
Plastic zone
1 0 r 3 ii
ð6aÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 g ¼ gij gij0 gij gij0 2
rx0 rx rp0 rp σh0
ð3Þ
M ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 g20
ð6bÞ ð6cÞ
with
gij ¼ σh0 Fig. 1. Expansion of a cylindrical cavity in K 0 -consolidated anisotropic clay.
r0ij p0 dij p0
;
3ð1 K 0 Þ 2K þ 1
g0 ¼
0
gij0 ¼
r0ij0 p00 dij p00
ð7aÞ
ð7bÞ
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M¼
6 sin u0 3 sin u0
ð7cÞ
where dij is Kronecker’s delta; r0ij0 and g0 are the value of r0ij and g at the end of the anisotropic consolidation, respectively. After taking the strain hardening rule into consideration, the yield function or the plastic potential function can be given as follows:
! 2
kj p0 kj g f ¼ ln 0 þ ln 1 þ 1 þ e0 p0 1 þ e0 M
ev ¼ 0 p
ð8Þ
where k and j are the slopes of compression and swelling lines, respectively; e0 is the initial void ratio of soil for p0 ¼ p00 , and epv is the plastic volumetric strain which is used as a hardening parameter.
3.1. Elastic analysis According to Eqs. (2) and (3) and the small strain theory, the solutions for the stress and displacement in the elastic region can be easily derived as follows [20–22]
rr
rh ¼ rh0 rrp rh0
r p 2 r
rz ¼ rv Ur ¼
rrp r 2G0
2 h0 r p
r
Du ¼ 0 where
3 p0 1 þ 2K 0 0
ð19Þ
According to Eq. (12), one can obtain
U rp ¼ r xp r x0 ¼
r0rp r0r0 2G0
r xp
ð9Þ
rxp
2G a 0 ¼ 2G0 r0rp r0h0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 a 2 x 0 þ 1 a a
ð21Þ
where r xp denotes the position of a specific particle that comes into the plastic state instantly. Furthermore, the current location of the EP boundary can be obtained by equating both rxp and rx to rp [21]
ð22Þ
3.3. Elasto-plastic analysis In the plastic region around the cavity, the total strain increment deij can be decomposed into elastic and plastic strain increments, i.e.,
deij ¼ deeij þ depij ð10Þ
ð20Þ
Substituting Eq. (1) into Eq. (20) gives
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 u 0 1 rp u a ¼u r0 r0 2 t a rp h0 1 1 2G0
3. Anisotropically elasto-plastic semi-analytical solution
r p 2 ¼ rh0 þ rrp rh0 r
r0zp ¼ r0v 0 ¼
ð23Þ
where the elastic strain increment deeij can be still calculated using Eq. (3), and the plastic strain increment depij in the K 0 -MCC model
ð11Þ
can be derived by taking the associated plastic flow rule as follows
ð12Þ
depij ¼ K
ð13Þ
where the scalar multiplier K is given as follows
ð24Þ
0
K¼
rrp denotes the total radial stress at the EP boundary °.
@f @ r0ij
ð@f =@p0 Þdp þ ð@f =@ g Þdg ð@f =@ epv Þ @f =@ r0ii
ð25Þ
3.2. Elastic–plastic boundary analysis
where
As shown in Eqs. (9)–(11) and (13), the mean effective stress, p0p , at the EP boundary can be given as
@f kj 1 ¼ @p0 1 þ e0 p0
ð26aÞ
p0p ¼ p00
@f kj 2g ¼ 2 @g 1 þ e0 M þ g2
ð26bÞ
@f ¼ 1 @ epv
ð26cÞ
@f @f @p0 @f @ g ¼ 0 þ 0 0 @ rij @p @ rij @ g @ r0ij
ð26dÞ
ð14Þ
Substituting Eq. (14) into Eq. (5), the relative stress ratio at the EP boundary, gp , can be obtained as follows
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gp ¼ M OCR 1
ð15Þ
where OCR is the overconsolidation ratio, defined as p0c0 =p00 , and the pressure, p0c0 , is the maximum mean preconsolidation stress. According to Eq. (6b) gp can also be written as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 g ¼ gijp gij0 gijp gij0 2 p
in which
ð16Þ
where gijp is the value of the stress ratio gij at the EP boundary. According to Eqs. (15) and (16), and taking the in-situ stresses into consideration, the effective radial, tangential and vertical stresses at the EP boundary can be derived as follows
p0 g r0rp ¼ r0h0 þ p0 ffiffiffip
ð17Þ
p0 g r0hp ¼ r0h0 p0 ffiffiffip
ð18Þ
3
3
@p0 dij ¼ @ r0ij 3
ð27aÞ
o @ g 1 n ¼ 0 3 gij gij0 gmn ðgmn gmn0 Þdij 0 @ rij 2g p
ð27bÞ
Substituting Eqs. (26a)–(26c), with Eqs. (27a) and (27b) into Eq. (26d) gives
h i3 2 3 g g g ð g g Þd ij ij ij0 mn mn mn0 @f cp dij 5 ¼ 4 þ @ r0ij p0 3 M 2 þ g2
ð28Þ
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2
3 @f cp 4 3gmn ðgmn gmn0 Þ5 ¼ 1 @ r0ii p0 M2 þ g2
ð29Þ
c31 ¼
where cp ¼ ðk jÞ=ð1 þ e0 Þ. Next, the scalar multiplier K can be obtained as follows
dij dr0 ij 3 gij gij0 K¼ dr0ij þ 3 M2 g2
c12 ¼
ð30Þ
Furthermore, the detailed relationship between the plastic strain increment depij and the stress increment dr0ij can be expressed
c22 ¼ c32 ¼
as follows
h i3 2 3 g g g ð g g Þd ij ij ij0 mn mn mn0 @f cp d 5 4 ij þ ¼
2 @ r0ij 3 2 3gmn ðgmn gmn0 Þ þ g M 0 p 1 M2 þg2 ð Þ h i9 8
2 2 8 9 1 þ bar der > E > > > < = 6 6 v deh ¼ 6 6 E þ bar ah > > > > : ; 4 dez vE þ baz ar
vE þ bar ah 1 E
þ
2 bah
vE þ baz ah
3 9 vE þ bar az 8 dr0 > r> > 7> 7< 0 = d r vE þ bah az 7 7> h > 5> : 0> ; 2 1 drz þ ba z E
ð32Þ
ar ¼
1 ½3ðgr gr0 Þ gmn ðgmn gmn0 Þ þ 3 M 2 þ g2
ð33aÞ
ah ¼
1 ½3ðgh gh0 Þ gmn ðgmn gmn0 Þ þ 3 M 2 þ g2
ð33bÞ
az ¼
1 ½3ðgz gz0 Þ gmn ðgmn gmn0 Þ þ 3 M 2 þ g2
ð33cÞ
cp
1
p0
3gmn ðgmn gmn0 Þ ðM2 þg2 Þ
ð33dÞ
Taking the inverse calculation to Eq. (32), and adopting the undrained condition and large deformation theory, the constitutive matrix for the cylindrical cavity expansion problem can be further reduced to a set of first-order ordinary differential equations as follows[21]
dr0r c11 c12 ¼0 dr nr
ð34aÞ
dr0h c12 c22 ¼0 dr nr
ð34bÞ
0 z
dr c31 c32 ¼0 dr nr 1 1 v 02 þ Ea2h b þ 2Ev 0 ah az b þ Ea2z b 2 E
E
1h 2
E
i Ear ðah þ v 0 az Þb þ v 0 1 þ v 0 Eah az b þ Ea2z b
ð35bÞ
i Ear ðv 0 ah þ az Þb þ v 0 1 þ v 0 Eah az b þ Ea2h b
ð35cÞ
1 2
E
1 E2
1 v 02 þ Ea2r b þ 2Ev 0 ar az b þ Ea2z b
v 0 þ v 02 þ Ev 0 a2r b Eah az b Ev 0 ar ðah þ az Þb
1 þ v0
"
ð35dÞ
# 1 þ v 0 þ 2v 02 þ Ebð1 þ v 0 Þ a2r þ a2h þ a2z 2Ev 0 bðah az þ ah ar þ ah az Þ
E3
ð35eÞ
ð35fÞ
As the values of r0rp ; r0hp ; r0zp and r xp are given in Eqs. (17)–(19) and (22), respectively, the ordinary differential Eqs. (34a)–(34c), can be solved by the Lagrangian analysis method as an initial value problem with r starting at rxp using a numerical method. Next, the excess pore pressure at an arbitrary location in the plastic zone, Durx , can be obtained by integrating Eq. (2) from the EP boundary r p to the point rx
Durx ¼ r0rp r0rx þ
Z
rp rx
r0r r0h r
dr
ð36Þ
The anisotropically elasto-plastic semi-analytical solution presented above gives an exact and realistic framework for the undrained expansion of the cylindrical cavity in the K 0 consolidated anisotropic clay. However, the calculation procedures are complicated; thus an approximate analytical solution in terms of total stress will be derived below for practical purposes. Note that the previous basic assumptions made for the semi-analytical solution are still valid in deriving the approximate analytical solution. According to the undrained condition, the total volumetric strain increment, which consists of the elastic and plastic parts in the plastic region, is equal to zero. That is
j
dp0 dp0 þ ðk jÞ 0c ¼ 0 tp0 tpc
p0c ¼ p0 OCR
ð35aÞ
0 kkj p p00
ð38Þ
Substituting Eq. (38) into Eq. (5), the relationship between the mean effective stress and the relative stress ratio in the plastic region can be given as follows
"
g ¼M
ð34cÞ
ð37Þ
Integrating Eq. (37) and taking EP boundary conditions into consideration give
2
where
c11 ¼
2
4. Approximate anisotropically elasto-plastic analytical solution
where
b¼
n¼
1h
2
# 0 q1 p OCR 0 1 p0
ð39Þ
where q is the plastic volumetric strain ratio, defined as (1 j=kÞ. For the K 0 -MCC model, the effective stress and the plastic volumetric strain remain constant when the soil reaches the critical state, i.e.
h i cp M2 þ g2 3gmn ðgmn gmn0 Þ @f dij ¼ ¼0 @ r0ij p0 M 2 þ g2
ð40Þ
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Substituting Eqs. (34a)–(34c), into Eq. (40), one obtains
5
d r0r þ r0h 2r0z 3bð2v 1Þða1 a2 Þ½ðrr þ rh 2rz Þ=p þ 6ð1 K 0 Þ=ð2K 0 1Þ ¼ ¼0 dr Enr M2 þ g2
Theoretical distribution
(σr'-σθ') /p0'
ð41Þ
where the only possible zero term on the right side of Eq. (40) is ½ðr0 r þ r0 h 2r0 z Þ=p0 þ 6ð1 K 0 Þ=ð2K 0 1Þ, and thus the relationship of the principal effective stresses in the critical zone can be given as follows
r0rf þ r0hf 2r0zf ¼
6ðK 0 1Þ 0 p 2K 0 þ 1 f
OCR=10, K0=2
4 Assumed distribution
3 2
OCR=3, K0=1
1
OCR=1.2, K0=0.625
0
ð42Þ
1
10
r/a
From Eq. (40), the critical state condition of the K 0 -MCC model can also be derived as
M ¼ gf
ð43Þ
where gf ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2
gijf gijf denotes the stress ratio at the critical state.
Substituting Eqs. (42) and (43) into Eq. (40), the relative stress ratio at the critical state, gf , can be given as follows 2 2 g2 f ¼ M þ g0
18ðK 0 1Þ2 ð2K 0 þ 1Þ2
Fig. 2. Theoretical and assumed distributions of deviator stresses around the cavity.
Based on the above assumption, the total stresses in the plastic zone can be derived from Eqs. (46)–(51) as follows
¼
p00
q OCR 2
ð45Þ
q OCR 2
ð46Þ
qf ¼ Mp00
3p0f
ð47Þ
2K 0 þ 1
r0rf ¼
3K 0 b p0 þ p0 2K 0 þ 1 f 2 f
ð48Þ
r0hf ¼
3K 0 b p0 p0 2K 0 þ 1 f 2 f
ð49Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i where b ¼ 2 3 M2 ð2K 0 þ 1Þ2 9ð1 K 0 Þ2 ½3ð2K 0 þ 1Þ. Based on Eq. (2), the total radial stress in the plastic region, rrx , can be given as
Z
rrx ¼ rrp
rx
rp
ð52Þ
b 2
2 G0 a a20 bp0f r0rp r0r0 r 2x
ð53Þ
rzx
" # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4q2f 3b2 p02 b 0 G0 a a20 f 1 ¼ rrp þ pf ln 0 2 rrp r0r0 r 2x 2 ð54Þ
Combining Eqs. (42), (45) and (46), the effective stress in the critical region can be derived as follows
r0zf ¼
2 G0 a a20 r0rp r0r0 r 2x
rhx ¼ rrp þ p0f ln
deviator stress, qf , in the critical region can be obtained as follows:
p0f
b 2
rrx ¼ rrp þ p0f ln
ð44Þ
According to Eqs. (39) and (44), the mean stress, p0f , and the
100
r0r r0h r
dr
Durx ¼ pfx p0f p0 p00
q p00 gp 3K 0 OCR þ pffiffiffi ¼ p00 2 1 þ 2K 0 3 " # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4q2f 3b2 p02 f b 0 G0 a a20 1 þ pf ln 0 2 rrp r0r0 r2x 6
ð55Þ
It is interesting to note here that, for the isotropic clay (K 0 =1), the approximate anisotropically elasto-plastic analytical solution presented in this paper reduces to the solution obtained by Cao et al. [20], which demonstrates that the solution obtained by Cao et al. [20] is just a special case of the presented solution.
ð50Þ 5. Discussions and verifications
where the plastic radius rp can be simplified by ignoring higher . r0rp r0r0 2G0 in Eq. (22) as follows
order terms of
r 2 a 2 G0 p 0 ¼ 0 1 0 a rrp rr0 a
In Eq. (54), the plus sign is taken when K 0 6 1; conversely, the minus sign is taken when K 0 > 1. Furthermore, the excess pore pressure in the plastic region can be derived by the effective stress principle
ð51Þ
Because the term r0r r0h is related to the location r x , Eq. (50) cannot be integrated directly. To obtain the closed-form solutions, the following additional assumption is necessary: The term r0r r0h in the plastic zone is assumed to be equal to r0rf r0hf , because the theoretical distribution of r0r r0h in the plastic region, which is obtained from the above semi-analytical solution, is very close to the assumed distribution, as shown in Fig. 2.
In this section, the effects of the initial stress-induced anisotropy on the cylindrical cavity undrained expansion will be first discussed by comparing the presented solution with the Chen and Abousleman’s solution. Next, the approximate anisotropically elasto-plastic analytical solution will be verified by comparison with the exact semi-analytical solution presented in this paper. All the soil parameters selected for the analysis are cited from Chen and Abousleman [21]. Table 1 summarises the properties of the three typical K 0 -consolidated clays. 5.1. Comparison with Chen and Abousleman’s solution The comparisons of Chen and Abousleman’s solution with the presented solution on the stress distributions around the cavity
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Table 1 Properties of the Boston Blue clay [21]. K0
OCR
rv 0 (kPa)
1.5 Critical region G0 (kPa)
t0
u0 (kPa)
1.2
4348 4113 3756
2.09 1.97 1.8
100 100 100
0.9
σ ' /p0'
M = 1.2, k = 0.15, j = 0.03, m = 0.278 0.625 1.2 160 1 3 120 2 10 72
Present study Chen & Abousleiman ( 2012）
σr'
0
Plastic region
0.6
σz'
0.3
σθ'
Elastic region
5.2. Cavity expansion process analysis The normalised excess pore pressure, Dua =p00 , and the internal cavity pressure, ra =p00 , at the cavity wall during the cavity expansion process are plotted against the normalised instant cavity radius, a=a0 , in Figs. 5 and 6, respectively. From Figs. 5 and 6, it can be seen that, the excess pore pressure and the internal cavity pressure are affected by the initial stress-induced anisotropy obviously when a/a0 > 2, and the results from the MCC model based solution overestimated the cavity pressure and excess pore pressure at the cavity wall during the expansion process except for the isotropic initial stress condition (K 0 ¼ 1). In addition, Fig. 6 shows that the discrepancy of the cavity pressures calculated from the two models increases with the degree of initial stress anisotropy. 5.3. Comparison between the exact semi-analytical solution and the approximate closed-form solution The approximate analytical solution is verified in Fig. 7 by comparing the stress distributions with the exact semi-analytical solution. In Fig. 7, The normalised excess pore pressure, Dua =p00 , total
0.0
1
10
100
r/a
(a) OCR=1.2, K0=0.625 3.0 Present study Chen & Abousleiman ( 2012 ）
Critical region
2.5
σz'
2.0
Plastic region
σ '/p0'
1.5
σr'
1.0
Elastic region
0.5
σθ'
0.0 -0.5
1
10
100
r/a
(b) OCR=3, K 0=1 8
σr' 6
σz'
4
σ '/p0'
when a/a0 ¼ 2 are shown in Figs. 4 and 5, where all the stresses have been normalised with respect to p00 , and the radial axis, r, to the current cavity radius, a. Note that, although the in-situ initial anisotropic stresses were considered in Chen and Abousleman’s MCC model-based solution, the impacts of the initial stress-induced anisotropy on the cavity expansion process cannot be effectively reflected by the MCC model. Therefore, comparisons with the Chen and Abousleman’s solution will reflect the effects of the initial stress-induced anisotropy on the cavity expansion. From Fig. 3, it can been seen that the effective radial and vertical stresses are overestimated in the plastic region by the MCC model based solutions of Chen and Abousleman in all cases other than K 0 ¼ 1, and the tangential stress is underestimated when K 0 < 1 whereas overestimated when K 0 > 1. It is also interesting to note in Fig. 3 that in general, the vertical effective stress is not equal to the mean of the radial and tangential effective stresses unless K 0 ¼ 1, which can be well explained by Eq. (42). As for the case of the isotropic in-situ stress state (K 0 ¼ 1), the K 0 -MCC model becomes the MCC model, and the solutions presented in this paper reduce to the MCC model-based solutions obtained by Chen and Abousleman [21] (Figs. 3(b) and 4(b). This demonstrates that Chen and Abousleman’s solution is just a special case of the solution presented in this paper. It is seen from Fig. 4 that both the mean effective stress and the deviator stress are overestimated by Chen and Abousleman’s solution, and the excess pore pressure is overestimated near the cavity wall but underestimated with increasing radial distance. In addition, from Figs. 3 and 4, note that the increase in the initial stress anisotropy results in the increase of the discrepancy between the results calculated from the two solutions, which demonstrates that the initial stress-induced anisotropy has a pronounced effect on cylindrical cavity expansion.
σθ'
Critical region Present study Chen & Abousleiman ( 2012）
Plastic region Elastic region
2 0 -2
1
10
r/a
100
(c) OCR=10, K0=2.0 Fig. 3. Comparison of the effective radial, tangential and vertical stresses distributions around the cavity computed by the presented solution and those from Chen and Abousleman’s solution.
radial stress, ðrr rh0 Þ=p00 , and total tangential stress, ðrh rh0 Þ=p00 , are plotted against the normalised radial distance r=a0 . It is seen from Fig. 7 that the results from the approximate analytical solutions match with the semi-analytical solutions fairly well, especially near the cavity wall, which demonstrates the assumption for the approximate analytical solution is reasonable and valid. There is a discrepancy only near the EP boundary due to the assumption that r0r r0h is equal to r0rf r0hf in the whole plastic region. However, for the practical engineering problems, such as the pile installation and the piezocone tests, care is often taken near the cavity wall, and thus the stresses at the cavity wall are more important. Therefore, the approximate analytical solution is sufficient to interpret and solve such geotechnical problems.
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14
2.5 Critical region
2.0
10
q
0.5
p'
8
σa /p'0
Plastic region
1.0
OCR=3, K0=1
6
OCR=1.2, K0=0.625
4 Elastic region
Present study (K0-MCC based )
2
0.0
MCC model based solution
0
-0.5
OCR=10, K0=2
12
Δu
1.5
Stress /p0'
Present study Chen & Abousleiman ( 2012）
1
10
1
2
3
4
5
r/a (a) OCR=1.2, K0=0.625
6
7
8
9
10
a/a0
100
Fig. 6. Variation of normalised cavity pressure with cavity radius during the cavity expansion process.
3.0 Critical region
2.5 2.0
Stress /p0'
Present study Chen & Abousleiman ( 2012 ）
Δu
Plastic region
p'
1.5
2.5
q Elastic region
0.5 0.0 1
10
100
r/a (b) OCR=3, K0=1 Critical region
Δu
0.5 0.0
σq
-0.5 1
10
(a) OCR=1.2, K0=0.625 5
Elastic region
0
Δu/p'0
-2
(σ-σh0)/p'0;
1
EP boundary
10
100
r/a (c) OCR=10, K0=2
σr
3 2
OCR=3 K0=1
Δu
1 0 σq
-1 -2
Fig. 4. Comparison of the effective mean stress, deviator stress and excess pore pressure around the cavity computed by the presented solution and those from Chen and Abousleman’s solution.
1
10
(b) OCR=3, K0=1 EP boundary Closed - form solution Theoritical distrubution
6
2 OCR=1.2, K0=0.625
1 0
Δu/p'0
OCR=10, K0=2 OCR=3, K0=1
4
(σ-σh0)/p'0;
4 3
100
r/a
8
Δua /p'0
Closed - form solution Theoritical distrubution
4
Δu
2
σr
OCR=10 K0=2
σq
0 Δu
-2 Present study (K0-MCC based)
-1 -2
100
r/a
Plastic region
p'
2
OCR=1.2 K0=0.625
1.0
Present study Chen & Abousleiman (2012）
q
4
Closed - form solution Theoritical distrubution
σr
1.5
-1.0
6
Stress /p0'
(σa-σh0)/p'0;
Δua/p'0
1.0
-0.5
EP boundary
2.0
-4
MCC model based solution
1
2
3
4
5
6
7
8
9
10
a/a0 Fig. 5. Variation of normalised excess pore pressure with cavity radius during the cavity expansion process.
1
101
00
r/a
(c) OCR=10, K0=2 Fig. 7. Comparison of the stress distributions calculated from the exact semianalytical solution and the approximate analytical solution.
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6. Conclusions Based on the K 0 -MCC model, an anisotropically elasto-plastic semi-analytical solution and an approximate analytical solution to the cylindrical cavity undrained expansion in saturated anisotropic clay were derived in this paper. The effects of the initial stress anisotropy and initial stress-induced anisotropy on the cylindrical cavity expansion were discussed in detail. The main conclusions are as follows: (a) Both the anisotropically elasto-plastic semi-analytical solution and analytical solution presented here improve upon the conventional solutions by considering the anisotropic properties of natural clay and can be reduced to the MCC model based solutions when K 0 =1. The presented solutions can yield more realistic stress distributions induced by cylindrical cavity expansion in natural K 0 -consolidated clay. (b) The initial stress-induced anisotropy has a significant influence on the stress distribution around the cavity. The discrepancy of the stress distributions calculated from the proposed solutions and previous MCC model-based solutions increases with the degree of anisotropy. (c) The vertical stress is not equal to the mean of the radial stress and tangential stress in the critical region unless the clay is initial isotropy. The cavity pressure and excess pore pressure at the cavity wall during the cavity expansion process are overestimated by the MCC model-based solutions except for the isotropic initial stress condition.
Acknowledgements The authors are grateful for the support provided by the National Natural Science Foundation of China (Grant No. 41272288) for this research work. The anonymous reviewers’ comments have improved the quality of this paper and are also greatly acknowledged. References [1] Randolph MF, Carter JP, Wroth CP. Driven piles in clay-the effects of installation and subsequent consolidation. Géotechnique 1979;29(4):361–93. [2] Carter JP, Randolph MF, Wroth CP. Stress and pore pressure changes in clay during and after the expansion of a cylindrical cavity. Int J Numer Anal Meth Geomech 1979;3(4):305–22.
[3] Randolph MF, Dolwin J, Beck R. Design of driven piles in sand. Géotechnique 1994;44(3):427–48. [4] Randolph MF. Science and empiricism in pile foundation design. Géotechnique 2003;53(10):847–75. [5] Yu HS, Houlsby GT. Finite cavity expansion in dilatants soils: loading analysis. Géotechnique 1991;41(2):173–83. [6] Collins IF, Pender MJ, Wang Y. Cavity expansion in sands under drained loading conditions. Int J Numer Anal Meth Geomech 1992;16(1):3–23. [7] Gibson RE, Anderson WF. In-situ measurement of soil properties with the pressuremeter. Civ Eng Public Works Rev 1961;56:615–8. [8] Vesic AC. Expansion of cavities in infinite soil mass. J Soil Mech Found Div 1972;98:265–90. [9] Hughes JMO, Wroth CP, Windle D. Pressuremeter tests in sands. Géotechnique 1977;27(4):455–77. [10] Houlsby GT, Withers NJ. Analysis of the cone pressuremeter test in clay. Géotechnique 1988;38(4):575–87. [11] Yu HS, Mitchell JK. Analysis of cone resistance: review of methods. J. Geotech Geoenviron Eng 1998;124(2):140–9. [12] Yu HS, Schnaid F, Collins IF. Analysis of cone penetration tests in sand. J. Geotech Geoenviron Eng 1996;122(8):623–32. [13] Yu HS. Interpretation of pressuremeter unloading tests in sands. Géotechnique 1996;46(1):17–31. [14] Yu HS. Cavity expansion theory and its application to the analysis of pressuremeters [Ph.D. thesis]. Oxford University; 1990. [15] Chang MF, Teh CI, Cao LF. Undrained cavity expansion in modified Cam clay II: application to the interpretation of the piezocone test. Géotechnique 2001;51 (4):335–50. [16] Chadwick P. The quasi-static expansion of spherical cavity in metals and ideal soils. Quart J Mech Appl Math 1959;12:52–71. [17] Carter JP, Booker JR, Yeung SK. Cavity expansion in cohesive frictional soils. Géotechnique 1986;36(3):349–53. [18] Yu HS, Houlsby GT. Finite cavity expansion in dilatants soils: loading analysis. Géotechnique 1991;41(2):173–83. [19] Yu HS. Expansion of a thick cylinder of soils. Comput Geotech 1992;14:21–41. [20] Cao LF, Teh CI, Chang MF. Undrained cavity expansion in modified Cam clay I: theoretical analysis. Géotechnique 2001;51(4):323–34. [21] Chen SL, Abousleiman YN. Exact undrained elasto-plastic solution for cylindrical cavity expansion in modified Cam Clay soil. Géotechnique 2012;62(5):447–56. [22] Chen SL, Abousleiman YN. Exact drained solution for cylindrical cavity expansion in modified Cam Clay soil. Géotechnique 2013;63(6):510–7. [23] Zhou H, Liu HL, Kong GQ. Analytical solution of undrained cylindrical cavity expansion in saturated soil under anisotropic initial stress. Comput Geotech 2014;55:232–9. [24] Hill R. The mathematical theory of plasticity. London: Oxford University Press; 1950. [25] Collins IF, Stimpson JR. Similarity solutions for drained and undrained cavity expansion in soils. Géotechnique 1994;44(1):21–34. [26] Collins IF, Yu HS. Undrained cavity expansion in critical state soils. Int J Numer Anal Methods Geomech 1996;20(7):489–516. [27] Yu HS. Cavity expansion methods in geomechanics. Kluwer Academic Publishers; 2000. [28] Sekiguchi H, Ohta H. Induced anisotropy and time dependency in clays. In: Proceedings of the 9th ICSMFE, vol. 2; 1977. p. 229–38. [29] Wheeler SJ, Naatanen A, Karstunen M, et al. An anisotropic elastoplastic model for soft clays. Can Geotech J 2003;40(2):403–18. [30] Matsuoka H, Sun DA. The SMP concept-based 3D constitutive models for geomaterials. Taylor & Francis; 2006.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
Anisotropically elasto-plastic solution to undrained cylindrical cavity expansion in K0-consolidated clay Lin Li a,b, Jingpei Li a,b,⇑, De’an Sun c a
Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China c Department of Civil Engineering, Shanghai University, 149 Yanchang Road, Shanghai 200072, China b
a r t i c l e
i n f o
Article history: Received 15 May 2015 Received in revised form 5 September 2015 Accepted 26 November 2015
Keywords: K 0 -MCC model Anisotropy Cylindrical cavity Undrained expansion Semi-analytical solution
a b s t r a c t This paper presents an anisotropically elasto-plastic solution to the undrained expansion of a cylindrical cavity in K 0 -consolidated clay. The elasto-plastic constitutive relationship following the soil yielding process is described by the K 0 -based modified Cam-clay (K 0 -MCC) model, which can properly reflect the anisotropic effects on the soil behaviour. Following the large strain deformation theory, the problem is reduced to solving a system of first-order ordinary differential equations in the plastic region. The semi-analytical solutions to the radial, tangential and vertical effective stresses are obtained using the Lagrangian method and the elastic–plastic (EP) boundary conditions. In addition, based on the semianalytical results, an approximate closed-form solution is presented for practical purposes. Extensive comparisons with the isotropic constitutive model-based solutions have been performed to illustrate the effects of the initial stress anisotropy and initial stress-induced anisotropy on the cavity expansion and the stress distributions. The present solution incorporates the anisotropic properties of the natural K 0 -consolidated clay, thereby providing a more realistic theoretical basis for the practical engineering problems such as the pile installation and the pressuremeter tests. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The cavity expansion theory has been widely applied to the modelling of complex geotechnical problems, such as the prediction of end-bearing and shaft capacities of a driven pile [1–6], as well as the interpretation of pressuremeter or cone penetration tests for determination of the soil properties in-situ [7–15]. Over the past few decades, the progress of cavity expansion theory has been mainly focused on the development of new solution techniques and the application of more realistic yield criterions or constitutive models to the cavity expansion problems [16–27]. Hill [24] first presented a comprehensive large strain solution to the expansion of the spherical and cylindrical cavities in an isotropic Tresca material. Subsequently, Vesic [8], Carter et al. [2,17], and Yu and Houlsby [18] presented a closed-form solution for cylindrical and spherical cavities by assuming the isotropic soil behaves as a non-associated Mohr–Coulomb material. Collins et al. [6], Collins and Stimpson [25], and Collins and Yu [26] proposed a similarity ⇑ Corresponding author at: Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China. Tel.: +86 021 65982773. E-mail address: [email protected] (J. Li). http://dx.doi.org/10.1016/j.compgeo.2015.11.022 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.
solution technique for zero initial radius cavity expansion problems, and the MCC model was adopted to describe the elastoplastic behaviours of the initial isotropic soil. Yu [27] summarised and presented a full account of the cavity expansion solutions in isotropic soils using a variety of critical state models for both drained and undrained loading conditions. Cao et al. [20] derived closed-form solutions for both spherical and cylindrical cavities in isotropic MCC soils by assuming the ultimate deviator stress distributions in the plastic zone. More recently, Chen and Abousleiman [21,22] derived semi-analytical solutions for the expansion of drained and undrained cylindrical cavities in anisotropic clay. However, the isotropic assumption-based MCC model was still adopted to derive the elasto-plastic constitutive matrix during the cavity expansion process. Although the existing cylindrical cavity expansion solutions have been widely used to solve practical geotechnical problems, most of the solutions are based on the assumptions of isotropic in-situ stress and elastic perfectly plastic. It is well known that the initial in-situ stress of the natural clay is likely to be anisotropic due to the change of sediment and consolidation environments [23], and the initial stress anisotropy and initial stress-induced anisotropy have pronounced effects on the soil behaviour. However, anisotropically elasto-plastic solutions to the undrained cylindrical
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L. Li et al. / Computers and Geotechnics 73 (2016) 83–90
cavity expansion in K 0 -consolidated anisotropic clay are not currently available. In this paper, to account for the effects of the initial stress anisotropy and the stress-induced anisotropy on the cylindrical cavity undrained expansion process, an anisotropically elasto-plastic semi-analytical solution was derived by employing the K 0 -MCC model [28,29]. In addition, an approximate anisotropically elastoplastic analytical solution is presented for practical purposes. Both of the proposed solutions, the semi-analytical and the approximate analytical solutions, overcome the shortcomings of the isotropic assumption, as a result, these solutions can model the cylindrical cavity expansion process in K 0 consolidated anisotropic clay properly and accurately. 2. Mechanical model for the cylindrical cavity expansion
r2x r2x0 ¼ a2 a20
ð1Þ
In both the elastic and plastic regions, the equilibrium equation can be written as
@ r0r @u r0r r0h þ þ ¼0 @r @r r
ð2Þ
where r0r and r0h are the effective radial and tangential stresses, respectively; u is the pore water pressure. 2.2. Constitutive relation for K0-consolidated clays According to Hooke’s law, the elastic stress–strain relationship of the K 0 -consolidated clay can be expressed in the incremental form as follows
1 þ v0 0 v0 0 drij drmm dij E E
2.1. Basic assumptions and definition of the problem
deeij ¼
The analytical framework for the cylindrical cavity expansion in K 0 -consolidated anisotropic clay is based on the following basic assumptions:
where E ¼ 2Gð1 þ v 0 Þ is the elastic modulus. In the K 0 -MCC model, the shear modulus G is defined as follows [30]
(a) The K 0 -consolidated anisotropic clay is saturated and homogeneous regarding the following properties: the in-situ effective horizontal stress r0h0 , vertical stress r0v 0 , and the hydrostatic pore-water pressure u0 . (b) The pore water is incompressible, and the compressive stresses and strains are taken as positive. (c) The cylindrical cavity expands under plane strain and undrained conditions; thus, the volume of the soil remains constant during the cavity expansion process. (d) The soil is assumed to be linearly elastic for small deformation in the elastic state, whereas the soil is modelled by the K 0 -MCC model and large deformation theory after yielding. Fig. 1 shows the expansion of a cylindrical cavity in an infinite K 0 -consolidated clay: as the internal cavity pressure gradually increases from rh0 to ra , the cavity expands from the initial radius a0 to the current radius a. If the cavity pressure ra increases further, then the soil on the cavity wall yields firstly, and then a plastic zone is formed from the current cavity radius a to the EP boundary rp . The symbol rp0 refers to the initial location of a soil particle that instantly becomes plastic. For the undrained condition, the conservation of volume condition gives the relationship between the current radius rx , which was initially at rx0 , and the current and initial radii of the cavity as follows
σh0 Elastic zone
G¼
3ð1 2v 0 Þtp0 2ð1 þ v 0 Þj
a
rp
a0
σh0 σa
o
ð4Þ
where j is the slope of swelling line; t is the specific volume, and m0 is effective Poisson’s ratio. The natural clay was usually consolidated under anisotropic K 0 conditions, and thus the soil has initial anisotropic stresses, i.e., the in-situ vertical stress is not equal to the horizontal stress in the K 0 consolidated clay. Because the mechanical properties of the soil associate closely with the stress states, the behaviour of the K 0 consolidated clay is significantly affected by the initial anisotropic stresses, and the K 0 -consolidated clay exhibits so-called initial stress-induced anisotropy properties during shearing [28,29]. To consider the effects of the initial stress-induced anisotropy on the mechanical behaviour of-K 0 consolidated clay, a relative stress ratio g and a relative stress ratio at critical state M are adopted in the K 0 -MCC model instead of the stress ratio gð¼ q=p0 Þ and the stress ratio at critical state M used in the MCC model. The yield function ðf Þ, which for associative plasticity serves also as a plastic potential function ðg Þ, of the K 0 -MCC model can be expressed as follows [28,29]
f ¼1þ
g
M
2
p0c ¼0 p0
ð5Þ
where u0 is the effective friction angle; p0c is the mean effective yielding pressure; the stress parameter p0 , the relative stress ratio g and the relative stress ratio at critical state M are, respectively defined as follows
p0 ¼
Plastic zone
1 0 r 3 ii
ð6aÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 g ¼ gij gij0 gij gij0 2
rx0 rx rp0 rp σh0
ð3Þ
M ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 g20
ð6bÞ ð6cÞ
with
gij ¼ σh0 Fig. 1. Expansion of a cylindrical cavity in K 0 -consolidated anisotropic clay.
r0ij p0 dij p0
;
3ð1 K 0 Þ 2K þ 1
g0 ¼
0
gij0 ¼
r0ij0 p00 dij p00
ð7aÞ
ð7bÞ
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L. Li et al. / Computers and Geotechnics 73 (2016) 83–90
M¼
6 sin u0 3 sin u0
ð7cÞ
where dij is Kronecker’s delta; r0ij0 and g0 are the value of r0ij and g at the end of the anisotropic consolidation, respectively. After taking the strain hardening rule into consideration, the yield function or the plastic potential function can be given as follows:
! 2
kj p0 kj g f ¼ ln 0 þ ln 1 þ 1 þ e0 p0 1 þ e0 M
ev ¼ 0 p
ð8Þ
where k and j are the slopes of compression and swelling lines, respectively; e0 is the initial void ratio of soil for p0 ¼ p00 , and epv is the plastic volumetric strain which is used as a hardening parameter.
3.1. Elastic analysis According to Eqs. (2) and (3) and the small strain theory, the solutions for the stress and displacement in the elastic region can be easily derived as follows [20–22]
rr
rh ¼ rh0 rrp rh0
r p 2 r
rz ¼ rv Ur ¼
rrp r 2G0
2 h0 r p
r
Du ¼ 0 where
3 p0 1 þ 2K 0 0
ð19Þ
According to Eq. (12), one can obtain
U rp ¼ r xp r x0 ¼
r0rp r0r0 2G0
r xp
ð9Þ
rxp
2G a 0 ¼ 2G0 r0rp r0h0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 a 2 x 0 þ 1 a a
ð21Þ
where r xp denotes the position of a specific particle that comes into the plastic state instantly. Furthermore, the current location of the EP boundary can be obtained by equating both rxp and rx to rp [21]
ð22Þ
3.3. Elasto-plastic analysis In the plastic region around the cavity, the total strain increment deij can be decomposed into elastic and plastic strain increments, i.e.,
deij ¼ deeij þ depij ð10Þ
ð20Þ
Substituting Eq. (1) into Eq. (20) gives
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 u 0 1 rp u a ¼u r0 r0 2 t a rp h0 1 1 2G0
3. Anisotropically elasto-plastic semi-analytical solution
r p 2 ¼ rh0 þ rrp rh0 r
r0zp ¼ r0v 0 ¼
ð23Þ
where the elastic strain increment deeij can be still calculated using Eq. (3), and the plastic strain increment depij in the K 0 -MCC model
ð11Þ
can be derived by taking the associated plastic flow rule as follows
ð12Þ
depij ¼ K
ð13Þ
where the scalar multiplier K is given as follows
ð24Þ
0
K¼
rrp denotes the total radial stress at the EP boundary °.
@f @ r0ij
ð@f =@p0 Þdp þ ð@f =@ g Þdg ð@f =@ epv Þ @f =@ r0ii
ð25Þ
3.2. Elastic–plastic boundary analysis
where
As shown in Eqs. (9)–(11) and (13), the mean effective stress, p0p , at the EP boundary can be given as
@f kj 1 ¼ @p0 1 þ e0 p0
ð26aÞ
p0p ¼ p00
@f kj 2g ¼ 2 @g 1 þ e0 M þ g2
ð26bÞ
@f ¼ 1 @ epv
ð26cÞ
@f @f @p0 @f @ g ¼ 0 þ 0 0 @ rij @p @ rij @ g @ r0ij
ð26dÞ
ð14Þ
Substituting Eq. (14) into Eq. (5), the relative stress ratio at the EP boundary, gp , can be obtained as follows
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gp ¼ M OCR 1
ð15Þ
where OCR is the overconsolidation ratio, defined as p0c0 =p00 , and the pressure, p0c0 , is the maximum mean preconsolidation stress. According to Eq. (6b) gp can also be written as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 g ¼ gijp gij0 gijp gij0 2 p
in which
ð16Þ
where gijp is the value of the stress ratio gij at the EP boundary. According to Eqs. (15) and (16), and taking the in-situ stresses into consideration, the effective radial, tangential and vertical stresses at the EP boundary can be derived as follows
p0 g r0rp ¼ r0h0 þ p0 ffiffiffip
ð17Þ
p0 g r0hp ¼ r0h0 p0 ffiffiffip
ð18Þ
3
3
@p0 dij ¼ @ r0ij 3
ð27aÞ
o @ g 1 n ¼ 0 3 gij gij0 gmn ðgmn gmn0 Þdij 0 @ rij 2g p
ð27bÞ
Substituting Eqs. (26a)–(26c), with Eqs. (27a) and (27b) into Eq. (26d) gives
h i3 2 3 g g g ð g g Þd ij ij ij0 mn mn mn0 @f cp dij 5 ¼ 4 þ @ r0ij p0 3 M 2 þ g2
ð28Þ
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L. Li et al. / Computers and Geotechnics 73 (2016) 83–90
2
3 @f cp 4 3gmn ðgmn gmn0 Þ5 ¼ 1 @ r0ii p0 M2 þ g2
ð29Þ
c31 ¼
where cp ¼ ðk jÞ=ð1 þ e0 Þ. Next, the scalar multiplier K can be obtained as follows
dij dr0 ij 3 gij gij0 K¼ dr0ij þ 3 M2 g2
c12 ¼
ð30Þ
Furthermore, the detailed relationship between the plastic strain increment depij and the stress increment dr0ij can be expressed
c22 ¼ c32 ¼
as follows
h i3 2 3 g g g ð g g Þd ij ij ij0 mn mn mn0 @f cp d 5 4 ij þ ¼
2 @ r0ij 3 2 3gmn ðgmn gmn0 Þ þ g M 0 p 1 M2 þg2 ð Þ h i9 8
2 2 8 9 1 þ bar der > E > > > < = 6 6 v deh ¼ 6 6 E þ bar ah > > > > : ; 4 dez vE þ baz ar
vE þ bar ah 1 E
þ
2 bah
vE þ baz ah
3 9 vE þ bar az 8 dr0 > r> > 7> 7< 0 = d r vE þ bah az 7 7> h > 5> : 0> ; 2 1 drz þ ba z E
ð32Þ
ar ¼
1 ½3ðgr gr0 Þ gmn ðgmn gmn0 Þ þ 3 M 2 þ g2
ð33aÞ
ah ¼
1 ½3ðgh gh0 Þ gmn ðgmn gmn0 Þ þ 3 M 2 þ g2
ð33bÞ
az ¼
1 ½3ðgz gz0 Þ gmn ðgmn gmn0 Þ þ 3 M 2 þ g2
ð33cÞ
cp
1
p0
3gmn ðgmn gmn0 Þ ðM2 þg2 Þ
ð33dÞ
Taking the inverse calculation to Eq. (32), and adopting the undrained condition and large deformation theory, the constitutive matrix for the cylindrical cavity expansion problem can be further reduced to a set of first-order ordinary differential equations as follows[21]
dr0r c11 c12 ¼0 dr nr
ð34aÞ
dr0h c12 c22 ¼0 dr nr
ð34bÞ
0 z
dr c31 c32 ¼0 dr nr 1 1 v 02 þ Ea2h b þ 2Ev 0 ah az b þ Ea2z b 2 E
E
1h 2
E
i Ear ðah þ v 0 az Þb þ v 0 1 þ v 0 Eah az b þ Ea2z b
ð35bÞ
i Ear ðv 0 ah þ az Þb þ v 0 1 þ v 0 Eah az b þ Ea2h b
ð35cÞ
1 2
E
1 E2
1 v 02 þ Ea2r b þ 2Ev 0 ar az b þ Ea2z b
v 0 þ v 02 þ Ev 0 a2r b Eah az b Ev 0 ar ðah þ az Þb
1 þ v0
"
ð35dÞ
# 1 þ v 0 þ 2v 02 þ Ebð1 þ v 0 Þ a2r þ a2h þ a2z 2Ev 0 bðah az þ ah ar þ ah az Þ
E3
ð35eÞ
ð35fÞ
As the values of r0rp ; r0hp ; r0zp and r xp are given in Eqs. (17)–(19) and (22), respectively, the ordinary differential Eqs. (34a)–(34c), can be solved by the Lagrangian analysis method as an initial value problem with r starting at rxp using a numerical method. Next, the excess pore pressure at an arbitrary location in the plastic zone, Durx , can be obtained by integrating Eq. (2) from the EP boundary r p to the point rx
Durx ¼ r0rp r0rx þ
Z
rp rx
r0r r0h r
dr
ð36Þ
The anisotropically elasto-plastic semi-analytical solution presented above gives an exact and realistic framework for the undrained expansion of the cylindrical cavity in the K 0 consolidated anisotropic clay. However, the calculation procedures are complicated; thus an approximate analytical solution in terms of total stress will be derived below for practical purposes. Note that the previous basic assumptions made for the semi-analytical solution are still valid in deriving the approximate analytical solution. According to the undrained condition, the total volumetric strain increment, which consists of the elastic and plastic parts in the plastic region, is equal to zero. That is
j
dp0 dp0 þ ðk jÞ 0c ¼ 0 tp0 tpc
p0c ¼ p0 OCR
ð35aÞ
0 kkj p p00
ð38Þ
Substituting Eq. (38) into Eq. (5), the relationship between the mean effective stress and the relative stress ratio in the plastic region can be given as follows
"
g ¼M
ð34cÞ
ð37Þ
Integrating Eq. (37) and taking EP boundary conditions into consideration give
2
where
c11 ¼
2
4. Approximate anisotropically elasto-plastic analytical solution
where
b¼
n¼
1h
2
# 0 q1 p OCR 0 1 p0
ð39Þ
where q is the plastic volumetric strain ratio, defined as (1 j=kÞ. For the K 0 -MCC model, the effective stress and the plastic volumetric strain remain constant when the soil reaches the critical state, i.e.
h i cp M2 þ g2 3gmn ðgmn gmn0 Þ @f dij ¼ ¼0 @ r0ij p0 M 2 þ g2
ð40Þ
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L. Li et al. / Computers and Geotechnics 73 (2016) 83–90
Substituting Eqs. (34a)–(34c), into Eq. (40), one obtains
5
d r0r þ r0h 2r0z 3bð2v 1Þða1 a2 Þ½ðrr þ rh 2rz Þ=p þ 6ð1 K 0 Þ=ð2K 0 1Þ ¼ ¼0 dr Enr M2 þ g2
Theoretical distribution
(σr'-σθ') /p0'
ð41Þ
where the only possible zero term on the right side of Eq. (40) is ½ðr0 r þ r0 h 2r0 z Þ=p0 þ 6ð1 K 0 Þ=ð2K 0 1Þ, and thus the relationship of the principal effective stresses in the critical zone can be given as follows
r0rf þ r0hf 2r0zf ¼
6ðK 0 1Þ 0 p 2K 0 þ 1 f
OCR=10, K0=2
4 Assumed distribution
3 2
OCR=3, K0=1
1
OCR=1.2, K0=0.625
0
ð42Þ
1
10
r/a
From Eq. (40), the critical state condition of the K 0 -MCC model can also be derived as
M ¼ gf
ð43Þ
where gf ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2
gijf gijf denotes the stress ratio at the critical state.
Substituting Eqs. (42) and (43) into Eq. (40), the relative stress ratio at the critical state, gf , can be given as follows 2 2 g2 f ¼ M þ g0
18ðK 0 1Þ2 ð2K 0 þ 1Þ2
Fig. 2. Theoretical and assumed distributions of deviator stresses around the cavity.
Based on the above assumption, the total stresses in the plastic zone can be derived from Eqs. (46)–(51) as follows
¼
p00
q OCR 2
ð45Þ
q OCR 2
ð46Þ
qf ¼ Mp00
3p0f
ð47Þ
2K 0 þ 1
r0rf ¼
3K 0 b p0 þ p0 2K 0 þ 1 f 2 f
ð48Þ
r0hf ¼
3K 0 b p0 p0 2K 0 þ 1 f 2 f
ð49Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i where b ¼ 2 3 M2 ð2K 0 þ 1Þ2 9ð1 K 0 Þ2 ½3ð2K 0 þ 1Þ. Based on Eq. (2), the total radial stress in the plastic region, rrx , can be given as
Z
rrx ¼ rrp
rx
rp
ð52Þ
b 2
2 G0 a a20 bp0f r0rp r0r0 r 2x
ð53Þ
rzx
" # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4q2f 3b2 p02 b 0 G0 a a20 f 1 ¼ rrp þ pf ln 0 2 rrp r0r0 r 2x 2 ð54Þ
Combining Eqs. (42), (45) and (46), the effective stress in the critical region can be derived as follows
r0zf ¼
2 G0 a a20 r0rp r0r0 r 2x
rhx ¼ rrp þ p0f ln
deviator stress, qf , in the critical region can be obtained as follows:
p0f
b 2
rrx ¼ rrp þ p0f ln
ð44Þ
According to Eqs. (39) and (44), the mean stress, p0f , and the
100
r0r r0h r
dr
Durx ¼ pfx p0f p0 p00
q p00 gp 3K 0 OCR þ pffiffiffi ¼ p00 2 1 þ 2K 0 3 " # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4q2f 3b2 p02 f b 0 G0 a a20 1 þ pf ln 0 2 rrp r0r0 r2x 6
ð55Þ
It is interesting to note here that, for the isotropic clay (K 0 =1), the approximate anisotropically elasto-plastic analytical solution presented in this paper reduces to the solution obtained by Cao et al. [20], which demonstrates that the solution obtained by Cao et al. [20] is just a special case of the presented solution.
ð50Þ 5. Discussions and verifications
where the plastic radius rp can be simplified by ignoring higher . r0rp r0r0 2G0 in Eq. (22) as follows
order terms of
r 2 a 2 G0 p 0 ¼ 0 1 0 a rrp rr0 a
In Eq. (54), the plus sign is taken when K 0 6 1; conversely, the minus sign is taken when K 0 > 1. Furthermore, the excess pore pressure in the plastic region can be derived by the effective stress principle
ð51Þ
Because the term r0r r0h is related to the location r x , Eq. (50) cannot be integrated directly. To obtain the closed-form solutions, the following additional assumption is necessary: The term r0r r0h in the plastic zone is assumed to be equal to r0rf r0hf , because the theoretical distribution of r0r r0h in the plastic region, which is obtained from the above semi-analytical solution, is very close to the assumed distribution, as shown in Fig. 2.
In this section, the effects of the initial stress-induced anisotropy on the cylindrical cavity undrained expansion will be first discussed by comparing the presented solution with the Chen and Abousleman’s solution. Next, the approximate anisotropically elasto-plastic analytical solution will be verified by comparison with the exact semi-analytical solution presented in this paper. All the soil parameters selected for the analysis are cited from Chen and Abousleman [21]. Table 1 summarises the properties of the three typical K 0 -consolidated clays. 5.1. Comparison with Chen and Abousleman’s solution The comparisons of Chen and Abousleman’s solution with the presented solution on the stress distributions around the cavity
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Table 1 Properties of the Boston Blue clay [21]. K0
OCR
rv 0 (kPa)
1.5 Critical region G0 (kPa)
t0
u0 (kPa)
1.2
4348 4113 3756
2.09 1.97 1.8
100 100 100
0.9
σ ' /p0'
M = 1.2, k = 0.15, j = 0.03, m = 0.278 0.625 1.2 160 1 3 120 2 10 72
Present study Chen & Abousleiman ( 2012）
σr'
0
Plastic region
0.6
σz'
0.3
σθ'
Elastic region
5.2. Cavity expansion process analysis The normalised excess pore pressure, Dua =p00 , and the internal cavity pressure, ra =p00 , at the cavity wall during the cavity expansion process are plotted against the normalised instant cavity radius, a=a0 , in Figs. 5 and 6, respectively. From Figs. 5 and 6, it can be seen that, the excess pore pressure and the internal cavity pressure are affected by the initial stress-induced anisotropy obviously when a/a0 > 2, and the results from the MCC model based solution overestimated the cavity pressure and excess pore pressure at the cavity wall during the expansion process except for the isotropic initial stress condition (K 0 ¼ 1). In addition, Fig. 6 shows that the discrepancy of the cavity pressures calculated from the two models increases with the degree of initial stress anisotropy. 5.3. Comparison between the exact semi-analytical solution and the approximate closed-form solution The approximate analytical solution is verified in Fig. 7 by comparing the stress distributions with the exact semi-analytical solution. In Fig. 7, The normalised excess pore pressure, Dua =p00 , total
0.0
1
10
100
r/a
(a) OCR=1.2, K0=0.625 3.0 Present study Chen & Abousleiman ( 2012 ）
Critical region
2.5
σz'
2.0
Plastic region
σ '/p0'
1.5
σr'
1.0
Elastic region
0.5
σθ'
0.0 -0.5
1
10
100
r/a
(b) OCR=3, K 0=1 8
σr' 6
σz'
4
σ '/p0'
when a/a0 ¼ 2 are shown in Figs. 4 and 5, where all the stresses have been normalised with respect to p00 , and the radial axis, r, to the current cavity radius, a. Note that, although the in-situ initial anisotropic stresses were considered in Chen and Abousleman’s MCC model-based solution, the impacts of the initial stress-induced anisotropy on the cavity expansion process cannot be effectively reflected by the MCC model. Therefore, comparisons with the Chen and Abousleman’s solution will reflect the effects of the initial stress-induced anisotropy on the cavity expansion. From Fig. 3, it can been seen that the effective radial and vertical stresses are overestimated in the plastic region by the MCC model based solutions of Chen and Abousleman in all cases other than K 0 ¼ 1, and the tangential stress is underestimated when K 0 < 1 whereas overestimated when K 0 > 1. It is also interesting to note in Fig. 3 that in general, the vertical effective stress is not equal to the mean of the radial and tangential effective stresses unless K 0 ¼ 1, which can be well explained by Eq. (42). As for the case of the isotropic in-situ stress state (K 0 ¼ 1), the K 0 -MCC model becomes the MCC model, and the solutions presented in this paper reduce to the MCC model-based solutions obtained by Chen and Abousleman [21] (Figs. 3(b) and 4(b). This demonstrates that Chen and Abousleman’s solution is just a special case of the solution presented in this paper. It is seen from Fig. 4 that both the mean effective stress and the deviator stress are overestimated by Chen and Abousleman’s solution, and the excess pore pressure is overestimated near the cavity wall but underestimated with increasing radial distance. In addition, from Figs. 3 and 4, note that the increase in the initial stress anisotropy results in the increase of the discrepancy between the results calculated from the two solutions, which demonstrates that the initial stress-induced anisotropy has a pronounced effect on cylindrical cavity expansion.
σθ'
Critical region Present study Chen & Abousleiman ( 2012）
Plastic region Elastic region
2 0 -2
1
10
r/a
100
(c) OCR=10, K0=2.0 Fig. 3. Comparison of the effective radial, tangential and vertical stresses distributions around the cavity computed by the presented solution and those from Chen and Abousleman’s solution.
radial stress, ðrr rh0 Þ=p00 , and total tangential stress, ðrh rh0 Þ=p00 , are plotted against the normalised radial distance r=a0 . It is seen from Fig. 7 that the results from the approximate analytical solutions match with the semi-analytical solutions fairly well, especially near the cavity wall, which demonstrates the assumption for the approximate analytical solution is reasonable and valid. There is a discrepancy only near the EP boundary due to the assumption that r0r r0h is equal to r0rf r0hf in the whole plastic region. However, for the practical engineering problems, such as the pile installation and the piezocone tests, care is often taken near the cavity wall, and thus the stresses at the cavity wall are more important. Therefore, the approximate analytical solution is sufficient to interpret and solve such geotechnical problems.
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14
2.5 Critical region
2.0
10
q
0.5
p'
8
σa /p'0
Plastic region
1.0
OCR=3, K0=1
6
OCR=1.2, K0=0.625
4 Elastic region
Present study (K0-MCC based )
2
0.0
MCC model based solution
0
-0.5
OCR=10, K0=2
12
Δu
1.5
Stress /p0'
Present study Chen & Abousleiman ( 2012）
1
10
1
2
3
4
5
r/a (a) OCR=1.2, K0=0.625
6
7
8
9
10
a/a0
100
Fig. 6. Variation of normalised cavity pressure with cavity radius during the cavity expansion process.
3.0 Critical region
2.5 2.0
Stress /p0'
Present study Chen & Abousleiman ( 2012 ）
Δu
Plastic region
p'
1.5
2.5
q Elastic region
0.5 0.0 1
10
100
r/a (b) OCR=3, K0=1 Critical region
Δu
0.5 0.0
σq
-0.5 1
10
(a) OCR=1.2, K0=0.625 5
Elastic region
0
Δu/p'0
-2
(σ-σh0)/p'0;
1
EP boundary
10
100
r/a (c) OCR=10, K0=2
σr
3 2
OCR=3 K0=1
Δu
1 0 σq
-1 -2
Fig. 4. Comparison of the effective mean stress, deviator stress and excess pore pressure around the cavity computed by the presented solution and those from Chen and Abousleman’s solution.
1
10
(b) OCR=3, K0=1 EP boundary Closed - form solution Theoritical distrubution
6
2 OCR=1.2, K0=0.625
1 0
Δu/p'0
OCR=10, K0=2 OCR=3, K0=1
4
(σ-σh0)/p'0;
4 3
100
r/a
8
Δua /p'0
Closed - form solution Theoritical distrubution
4
Δu
2
σr
OCR=10 K0=2
σq
0 Δu
-2 Present study (K0-MCC based)
-1 -2
100
r/a
Plastic region
p'
2
OCR=1.2 K0=0.625
1.0
Present study Chen & Abousleiman (2012）
q
4
Closed - form solution Theoritical distrubution
σr
1.5
-1.0
6
Stress /p0'
(σa-σh0)/p'0;
Δua/p'0
1.0
-0.5
EP boundary
2.0
-4
MCC model based solution
1
2
3
4
5
6
7
8
9
10
a/a0 Fig. 5. Variation of normalised excess pore pressure with cavity radius during the cavity expansion process.
1
101
00
r/a
(c) OCR=10, K0=2 Fig. 7. Comparison of the stress distributions calculated from the exact semianalytical solution and the approximate analytical solution.
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L. Li et al. / Computers and Geotechnics 73 (2016) 83–90
6. Conclusions Based on the K 0 -MCC model, an anisotropically elasto-plastic semi-analytical solution and an approximate analytical solution to the cylindrical cavity undrained expansion in saturated anisotropic clay were derived in this paper. The effects of the initial stress anisotropy and initial stress-induced anisotropy on the cylindrical cavity expansion were discussed in detail. The main conclusions are as follows: (a) Both the anisotropically elasto-plastic semi-analytical solution and analytical solution presented here improve upon the conventional solutions by considering the anisotropic properties of natural clay and can be reduced to the MCC model based solutions when K 0 =1. The presented solutions can yield more realistic stress distributions induced by cylindrical cavity expansion in natural K 0 -consolidated clay. (b) The initial stress-induced anisotropy has a significant influence on the stress distribution around the cavity. The discrepancy of the stress distributions calculated from the proposed solutions and previous MCC model-based solutions increases with the degree of anisotropy. (c) The vertical stress is not equal to the mean of the radial stress and tangential stress in the critical region unless the clay is initial isotropy. The cavity pressure and excess pore pressure at the cavity wall during the cavity expansion process are overestimated by the MCC model-based solutions except for the isotropic initial stress condition.
Acknowledgements The authors are grateful for the support provided by the National Natural Science Foundation of China (Grant No. 41272288) for this research work. The anonymous reviewers’ comments have improved the quality of this paper and are also greatly acknowledged. References [1] Randolph MF, Carter JP, Wroth CP. Driven piles in clay-the effects of installation and subsequent consolidation. Géotechnique 1979;29(4):361–93. [2] Carter JP, Randolph MF, Wroth CP. Stress and pore pressure changes in clay during and after the expansion of a cylindrical cavity. Int J Numer Anal Meth Geomech 1979;3(4):305–22.
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