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Adv. Theor. Appl. Mech., Vol. 4, 2011, no. 4, 189 - 198
Stress, Displacement and Pore Pressure
of Partially Sealed Circular Tunnel
Surrounded by Viscoelastic Medium
J. P. Dwivedi
Department of Mechanical EngineeringInstitute of Technology
Banaras Hindu UniversityVaranasi-221005, India
V. P. Singh
Department of Mechanical EngineeringInstitute of Technology
Banaras Hindu UniversityVaranasi-221005, [email protected]
Radha Krishna Lal
Department of Mechanical EngineeringInstitute of Technology
Banaras Hindu UniversityVaranasi-221005, India
Abstract
The dynamic response of a partially sealed circular tunnel in vis-coelastic soil condition has been studied. By introducing two scalarpotential functions, in Laplace transform domain, the analytical solu-tions of stresses, displacements and pore pressure are derived.
Keywords: Dynamic response, circular tunnel, viscoelastic soil, stresses,displacements and pore pressure
190 J. P. Dwivedi, V. P. Singh and Radha Krishna Lal
1 Introduction
It is important to note that the generation and propagation of stress-wavesin an elastic medium containing a cavity due to arbitrary dynamic loadingapplied on the cavity is considerable in the fields of seismology, geophysicalprospecting, underground tunnels and deeply buried pipelines, particularly asa model of an earthquake source.
Dynamic response of buried pipelines are studied by several researcherslike Ben-Menahem and Cisternas [1], Thiruvenkatachar and Viswanathan [8],Karakostas and Manolis [5] etc.
In this paper, dynamic response of a circular tunnel subjected to axiallysymmetric transient radial traction and fluid pressure in viscoelastic fluid -saturated soil is studied. The soil is described with the model of Kelvin-Voigt, and the partial sealing boundary condition [6] for circular tunnel isadopted. The analytical solutions of stresses, displacements and pore pressureare derived in Laplace transform domain parallel to Kang-he Xiea, Gan-binLiua and Zu-yuan Shib [4] considering two different scalar potentials U(r)sinωtand W (r)sinωt .
2 Base equations and general solution
A circular tunnel, as shown in figure, is assumed to be bored in a viscoelas-tic saturated soil at a deep depth with inner and outer radius a and b, re-spectively. The thickness of liner (h = b − a) is assumed to be so smallwith respect to the radius of tunnel that there is no need to distinguishwhether a load is applied at r = a or r = b. It is also assumed that thetunnel is long enough so that the dynamic response of the circular tunnelin the soil can be considered as a plane strain axially symmetric problem.
Stress, displacement and pore pressure 191
The equilibrium equation for soil mass in polar coordinate system can bewritten as:
∂σr
∂r+
στ − σθ
r=
∂2
∂t2(ρuτ + ρfwr) (1)
where, ur and wr are, respectively, radial displacement of soil skeleton and theone of pore fluid with respect to the soil skeleton; ρf and ρs are densities offluid and soil grain, respectively; ρ = (1− n)ρs + nρf ; the density of soil; n isthe porosity.
The pore fluid equilibrium equation is given by:
−∂P
∂r=
∂2
∂t2
(ρfuτ +
ρfnwr
)+
η0ks
∂wr
∂t, (2)
where, P is excess pore water pressure; η0, the fluid viscosity; and ks, theintrinsic permeability of soil. Because soil is not an ideal medium, a part ofenergy of the wave will be changed into heat energy during the propagationdue to overcoming the interior friction of soil. This is so called damping ofmaterial. Assuming that the viscoelastic property of soil may be simulated byKelvin-Voigt model, the stress - strain relationship can be expressed by ([3],[9]):
σr = λe+ 2G∂ur
∂r+ λ′∂e
∂t+ 2G′ ∂
∂t
(∂ur
∂r
)− αP (3)
σθ = λe+ 2Gur
r+ λ′∂e
∂t+ 2G′ ∂
∂t
(ur
r
)− αP (4)
P = Mξ − αMe (5)
where
e =∂ur
∂r+
ur
rand ξ = −
(∂wr
∂r+
wr
r
)the dilatations of solid and fluid, respectively; λ and G are the Lame constantsof the bulk material; λ′ and G′ are the dilatant constant and shear constantof the viscoelastic soil; α and M are the compressibility parameters of thetwophase medium. It is noted that 0 ≤ α ≤ 1 and 0 ≤ M ≤ ∞, and M → ∞,α → 1 for a material with incompressible constituents.
Substituting Eq.(3), (4) and (5), into Eqs. (1) and (2), the governingequations of dynamic response of circular tunnel in viscoelastic saturated soilcan be obtained:
(λ+ 2G+ α2M)∂e
∂r+ (λ′ + 2G′)
∂
∂t
(∂e
∂r
)− αM
∂ξ
∂r=
∂2
∂t2(ρur + ρfwr) (6)
αM∂e
∂r−M
∂ξ
∂r=
∂2
∂t2
(ρfur +
ρfnwr
)+
η0ks
∂wr
∂t(7)
192 J. P. Dwivedi, V. P. Singh and Radha Krishna Lal
Equations (6) and (7) can be written as
(λ+2G+α2M)∇21ur+(λ′+2G′)
∂
∂t
(∇2
1ur
)+αM∇2
1wr =∂2
∂t2(ρur + ρfwr) (8)
αM∇21ur +M∇2
1wr =∂2
∂t2
(ρfur +
ρfnwr
)+
η0ks
∂wr
∂t(9)
where ∇21 =
∂2
∂r+
1
r
∂
∂r− 1
r2.
To solve Equations (8) and (9), the following two scalar potentials U(r) sinωtand W (r) sinωt are introduced:
ur =∂
∂r(U(r) sinωt) (10)
wr =∂
∂r(W (r) sinωt) (11)
Substituting Equations (10) and (11) into (8) and (9) yields[(λ+ 2G+ α2M)∇2 + (λ′ + 2G′)
∂
∂t∇2 − ρ
∂2
∂t2
]∂
∂r(U(r) sinωt)
+
(αM∇2 − ρf
∂2
∂t2
)∂
∂r(W (r) sinωt) = 0
(12)
(αM∇2 − ρf
∂2
∂t2
)∂
∂r(U(r) sinωt)+
(M∇2 − ρf
n
∂2
∂t2− η0
ks
∂
∂t
)∂
∂r(W (r) sinωt) = 0
(13)
where ∇2 =∂2
∂r2+
1
r
∂
∂r.
After simplification, above equation reduce into[(λ+ 2G+ α2M)∇2 + (λ′ + 2G′)
∂
∂t∇2 − ρ
∂2
∂t2
](U(r) sinωt)
+
(αM∇2 − ρf
∂2
∂t2
)(W (r) sinωt) = 0
(14)
(αM∇2 − ρf
∂2
∂t2
)(U(r) sinωt)+
(M∇2 − ρf
n
∂2
∂t2− η0
ks
∂
∂t
)(W (r) sinωt) = 0
(15)Applying of the Laplace Transform to equation (14) and (15), we get[
(λ+ 2G+ α2M)∇2
(ω
s2 + ω2
)+ (λ′ + 2G′)∇2
(ωs
s2 + ω2
)− ρ
(−ω3
s2 + ω2
)]U(r)
+
(αM∇2
(ω
s2 + ω2
)− ρf
(−ω3
s2 + ω2
))W (r) = 0
(16)
Stress, displacement and pore pressure 193
[αM∇2 − ρf
(−ω3
s2 + ω2
)]U(r)+
(M∇2 − ρf
n
(−ω3
s2 + ω2
)− η0
ks
(ωs
s2 + ω2
))W (r) = 0
(17)
[(λ+ 2G+ α2M)∇2 + s(λ′ + 2G′)∇2 + ρω2
]U(r)
(ω
s2 + ω2
)+(αM∇2 + ρfω
2)W (r)
(ω
s2 + ω2
)= 0
(18)
[αM∇2 + ρfω
2]U(r)
(ω
s2 + ω2
)+
[M∇2 +
ρfnω2 +
η0s
ks
]W (r)
(ω
s2 + ω2
)= 0
(19)
[(λ+ 2G+ α2M)∇2 + s(λ′ + 2G′)∇2 + ρω2
]U(r) +
(αM∇2 + ρfω
2)W (r) = 0
(20)
[αM∇2 + ρfω
2]U(r) +
[M∇2 +
ρfnω2 +
η0s
ks
]W (r) = 0 (21)
Lou[7] reported that the viscoelastic damping coefficient of rock and softsoil could be assumed as a constant in a great range of vibration frequency.This is also assumed in this paper, ie.
λ′
λ=
G′
G= η (22)
where, η is a constant that denotes the damping cofficient of the viscoelasticsaturated soil.
In term of all dimensionless variables in equations (20), (21) and by use ofequation (22), we obtain{(
(λ∗ + 2)(1 + ηs) + α2M∗)∇2 + ω2}U +
(αM∗∇2 + ρ∗ω2
)W = 0 (23)
(αM∗∇2 + ρ∗ω2
)U +
(M∗∇2 +
ρ∗
nω2 + b∗s
)W = 0 (24)
where λ∗ =λ
G, M∗ =
M
G, ρ∗ =
ρfρ, b∗ =
η0ks
b√ρG
,P
G≈ 1
the non-dimensional Lame constant, compressibility parameter fluid densityand permeability coefficient of soil respectively; b is the radius of circular tun-nel.
194 J. P. Dwivedi, V. P. Singh and Radha Krishna Lal
Solving (23) and (24), we have[{((λ∗ + 2)(1 + ηs) + α2M∗)∇2 + ω2
}{M∗∇2 +
ρ∗
nω2 + b∗s
}−(αM∗∇2 + ρ∗ω2
)2(U,W )
]= 0
[M∗ ((λ∗ + 2)(1 + ηs) + α2M∗)∇4 +
{((λ∗ + 2)(1 + ηs) + α2M∗)(ρ∗
nω2 + b∗s
)}∇2
+M∗ω2∇2 +
(ρ∗
nω2 + b∗s
)ω2 − α2M∗2∇4 − ρ∗2ω4 − 2αM∗ρ ∗ ω2∇2
](U,W ) = 0
[{M∗(λ∗ + 2)(1 + ηs)}∇4 +
{((λ∗ + 2)(1 + ηs) + α2M∗)(ρ∗
nω2 + b∗s
)+M∗ω2 − 2αM∗ρ∗ω2
}∇2 +
{(ρ∗
nω2 + b∗s
)w2 − (ρ∗ω2)2
}](U,W ) = 0
(25)
The equation (25) can be written as(∇2 − γ2
1
) (∇2 − γ2
2
)(U,W ) = 0 (26)
where γ1 and γ2 are the complex wave numbers of two dilatational waves i.e.
γ21,2 =
α1 ∓√α21 − 4α2
2
with α1 =((λ∗ + 2)(1 + ηs) + α2M∗)
(ρ∗
nω2 + b∗s
)+M∗ω2 − 2αM∗ρ∗w2
(λ∗ + 2)(1 + ηs)M∗ and
α2 =
(ρ∗
nω2 + b∗s
)ω2 − (ρ∗ω2)2
(λ∗ + 2)(1 + ηs)M∗ .
The solution of U(r) in equation (26) is given by
U(r) = A1I0(γ1r) +D1K0(γ1r) + A2I0(γ2r) +D2K0(γ2r) (27)
where I0(x) and K0(x) are the modified Bessel functions of the first and secondkind of zero order, respectively.
It is noted that I0(x) → ∞ when x → ∞, so constants A1 and A2 shouldequal to zero. The solution of W (r) can be obtained in the same way.
As a result, the general solution of U(r) and W (r) can be written as
U(r) = D1K0(γ1r) +D2K0(γ2r) (28)
W (r) = E1K0(γ1r) + E2K0(γ2r) (29)
Stress, displacement and pore pressure 195
in which, the four constants D1, D2, E1 and E2 are linearly dependent, andcan be related by using Eqs (24), (28) and (29) as
Ei = δiDi (i = 1, 2) (30)
where
δi =−αM∗γ2
i + ρ∗s2
M∗γ2i −
ρ∗s2
n− b∗s
D1 and D2 are the variables that can be obtained from boundary conditions.The radial displacement ur, wr can be obtained from Eqs. (10), (11), (28)
and (29), i.e.
ur = − [γ1K1(γ1r)D1 + γ2K1(γ2r)D2] sinωt (31)
wr = − [δ1γ1K1(γ1r)D1 + δ2γ2K1(γ2r)D2] sinωt (32)
Using (5) and (31) & (32), general solution of pore pressure and its Laplacetransform can be obtained as
P
G= −M∗
(∂wr
∂r+
wr
r
)− αM∗
(∂ur
∂r+
ur
r
)= −M∗ [−δ1γ
21k0(γ1r)D1 − δ2γ
22k0(γ2r)D2
]sinωt
−αM∗ [−γ21k0(γ1r)D1 − γ2
2k0(γ2r)D2
]sinωt
=[(α+ δ1)M
∗γ21k0(γ1r)D1 + (α+ δ2)M
∗γ22k0(γ2r)D2
]sinωt
L
[P
G
]=
P̃
G=
[(α+ δ1)M
∗γ21k0(γ1r)D1 + (α+ δ2)M
∗γ22k0(γ2r)D2
]( w
s2 + w2
).
Radial stress by (5) is given by
σr = λ
(∂ur
∂r+
ur
r
)+ 2G
∂ur
∂r+ λ′ ∂
∂t
(∂ur
∂r+
ur
r
)+ 2G′ ∂
∂t
(∂ur
∂r
)− αP
σr
G= λ∗
(∂ur
∂r+
ur
r
)+ 2
∂ur
∂r+ ηλ∗ ∂
∂t
(∂ur
∂r+
ur
r
)+ 2η
∂
∂t
(∂ur
∂r
)− α
P
G
= λ∗ [−γ21k0(γ1r)D1 − γ2
2k0(γ2r)D2
]sinωt
+[−γ2
1k0(γ1r)D1 − γ22k0(γ2r)D2 + γ2
1k2(γ1r)D1 + γ22k2(γ2r)D2
]sinωt
+ηλ∗ [−γ21k0(γ1r)D1 − γ2
2k0(γ2r)D2
]ω cosωt
+η[−γ2
1k0(γ1r)D1 − γ22k0(γ2r)D2 + γ2
1k2(γ1r)D1 + γ22k2(γ2r)D2
]ω cosωt
−α[(α+ δ1)M
∗γ21k0(γ1r)D1 + (α+ δ2)M
∗γ22k0(γ2r)D2
]sinωt
= − [{(λ∗ + 1) + α(α+ δ1)M∗} k0(γ1r)− k2(γ1r)] γ
21D1 sinωt
− [{(λ∗ + 1) + α(α+ δ2)M∗} k0(γ2r)− k2(γ2r)] γ
22D2 sinωt
− [(λ∗ + 1)ηk0(γ1r)− ηk2(γ1r)] γ21D1ω cosωt
− [(λ∗ + 1)ηk0(γ2r)− ηk2(γ2r)] γ22D2ω cosωt
196 J. P. Dwivedi, V. P. Singh and Radha Krishna Lal
L[σr
G
]=
σ̃r
G= − [{(λ∗ + 1) + α(α+ δ1)M
∗} k0(γ1r)− k2(γ1r)] γ21D1
(ω
s2 + ω2
)− [{(λ∗ + 1) + α(α+ δ2)M
∗} k0(γ2r)− k2(γ2r)] γ22D2
(ω
s2 + ω2
)− [(λ∗ + 1)ηk0(γ1r)− ηk2(γ1r)] γ
21D1
(sω
s2 + ω2
)− [(λ∗ + 1)ηk0(γ2r)− ηk2(γ2r)] γ
22D2
(sω
s2 + ω2
)
σ̃r
G= −
(ω
s2 + w2
)[{(λ∗ + 1)(1 + ηs) + α(α+ δ1)M
∗} k0(γ1r)− (1 + ηs)k2(γ1r)] γ21D1
−(
ω
s2 + w2
)[{(λ∗ + 1)(1 + ηs) + α(α+ δ2)M
∗} k0(γ2r)− (1 + ηs)k2(γ2r)] γ22D2.
Circumferential stress by (4) also can be obtained by
σθ
G= λ∗
(∂ur
∂r+
ur
r
)+ 2
(ur
r
)+ ηλ∗ ∂
∂t
(∂ur
∂r+
ur
r
)+ 2η
∂
∂t
(ur
r
)− α
P
G
= λ∗ (−γ21k0(γ1r)D1 − γ2
2k0(γ2r)D2
)sinωt
−2(γ1rk1(γ1r)D1 +
γ2rk2(γ2r)D2
)sinωt
+ηλ∗ ∂
∂t
(−γ2
1k0(γ1r)D1 − γ22k0(γ2r)D2
)ω cosωt
−2η∂
∂t
(γ1rk1(γ1r)D1 +
γ2rk2(γ2r)D2
)ω cosωt
−α[(α+ δ1)M
∗γ21k0(γ1r)D1 + (α+ δ2)M
∗γ22k0(γ2r)D2
]sinωt
= λ∗ (−γ21k0(γ1r)D1 − γ2
2k0(γ2r)D2
)sinωt
−[γ21 {k2(γ1r) + k0(γ1r)}D1 + γ2
2 {k2(γ2r) + k0(γ2r)}D2
]sinωt
−ηλ∗ (γ21k0(γ1r)D1 + γ2
2k0(γ2r)D2
)ω cosωt
−η[γ21 {k2(γ1r) + k0(γ1r)}D1 + γ2
2 {k2(γ2r) + k0(γ2r)}D2
]ω cosωt
−((α+ δ1)M
∗γ21k0(γ1r)D1 + (α+ δ2)M
∗γ22k0(γ2r)D2
)sinωt
= − [{1 + λ∗ + αM∗(α+ δ1)} k0(γ1r) + k2(γ1r)] γ21 sinωtD1
− [{1 + λ∗ + αM∗(α+ δ2)} k0(γ2r) + k2(γ2r)] γ22 sinωtD2
− [(1 + λ∗)k0(γ1r) + k2(γ1r)] ηγ21ω cosωtD1
− [(1 + λ∗)k0(γ2r) + k2(γ2r)] ηγ22ω cosωtD2
Stress, displacement and pore pressure 197
L[σθ
G
]=
σ̃θ
G= − [{1 + λ∗ + αM∗(α+ δ1)} k0(γ1r) + k2(γ1r)] γ
21
(ω
s2 + ω2
)D1
− [{1 + λ∗ + αM∗(α+ δ2)} k0(γ2r) + k2(γ2r)] γ22
(ω
s2 + ω2
)D2
− [(1 + λ∗)k0(γ1r) + k2(γ1r)] ηγ21
(sω
s2 + ω2
)D1
− [(1 + λ∗)k0(γ2r) + k2(γ2r)] ηγ22
(sω
s2 + ω2
)D2
σ̃θ
G= −
(ω
s2 + w2
)[{(λ∗ + 1)(1 + ηs) + α(α+ δ1)M
∗} k0(γ1r) + (1 + ηs)k2(γ1r)] γ21D1
−(
ω
s2 + w2
)[{(λ∗ + 1)(1 + ηs) + α(α+ δ2)M
∗} k0(γ2r) + (1 + ηs)k2(γ2r)] γ22D2
where, K1(x) and K2(x) are the modified Bessel function of second kind ofzero and two order respectively and
k1(x)
x=
1
2[k2(x)− k0(x)].
If neglection the viscoelastic damping of medium, that is, η = 0 then theabove solution become
P̃
G=
[(α+ δ1)M
∗γ21k0(γ1r)D1 + (α+ δ2)M
∗γ22k0(γ2r)D2
]( w
s2 + w2
).
σ̃r
G= −
(ω
s2 + w2
)[{(λ∗ + 1) + α(α+ δ1)M
∗} k0(γ1r)− k2(γ1r)] γ21D1
−(
ω
s2 + w2
)[{(λ∗ + 1) + α(α+ δ2)M
∗} k0(γ2r)− k2(γ2r)] γ22D2.
σ̃θ
G= −
(ω
s2 + w2
)[{(λ∗ + 1) + α(α+ δ1)M
∗} k0(γ1r) + k2(γ1r)] γ21D1
−(
ω
s2 + w2
)[{(λ∗ + 1) + α(α+ δ2)M
∗} k0(γ2r) + k2(γ2r)] γ22D2.
3 Conclusions
Numerical results in time domain can be obtained by inverse Laplace transformpresented by Durbin[2] and used to analyze the influences of partial permeableproperty of boundary and viscoelastic damping coefficient of soil on dynamic
198 J. P. Dwivedi, V. P. Singh and Radha Krishna Lal
response of the tunnel. It can be shown that the attenuation of radial displace-ment appeared with the increase of viscoelastic damping coefficient of soil, andrelative rigidity of liner and soil, and the influence of partial sealing property ofboundary on stresses, displacements and pore pressure is remarkable. There-fore, it is of significance to consider the partially sealed boundary conditionand the damping property of soil as well as relative rigidity of liner and soil inthe designing and computation of circular tunnel under dynamic condition.
Acknowledgment. Authors are grateful to Prof. P. C. Upadhyay, Ex.Head, Department of Department of Mechanical Engineering, Institute ofTechnology,Banaras Hindu University, Varanasi, India, for his kind help andcooperation in preparation of this paper.
References
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[3] AC. Eringen, In: R Krveqer, editor. Mechanics of comtinua, New York:Huntington Press, 1980, 385-406.
[4] Xie Kang-he, Liu Gan-bin and Shi Zu-yuan , Dynamic response of par-tially sealed circular tunnel in viscoelastic saturated soil, Soil Dynamicsand Earthquake Engineering, 2004, 24, 1003-1011.
[5] CZ Karakostas, GD. Manolis, Dynamic response of unlined tunnels in soilwith random properties, Eng. Struct. 2000,22, 1013-1027.
[6] X. Li, Stress and displacement field around a deep circular tunnel withpartial sealing, Comput. Geotech, 1999,4, 125-140.
[7] ML Lou, G. Lin, The effect of wave refection of artifical boundaries inviscoelastic media, China ShuiLi XueBao 1986,6,20-30.
[8] V. Thiruvenkatachar, K. Viswanathan, Dynamic response of an elastichalf-space with cylinder cavity to time-dependent surface tractions overthe boundary of the cavity, J. Math. Mech., 1965,14, 541-572.
[9] CJ Xu , YQ Cai . Dynamic response of spherical cavity in viscoelasticsaturated soils., Chin. Civil. Eng. J., 2001,34(4), 88-92.
Received: April, 2011
