# Analysis of Transient Dynamic Behaviour of Spherical Cavity in Viscoelastic Soil Medium

Authors: J. P. Dwivedi; V. P. Singh; Radha Krishna Lal
Journal: ISRN Mechanical Engineering
 (2013)

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## Abstract

Stress, displacement, and pore pressure of a partially sealed spherical cavity in viscoelastic soil condition have been obtained in Laplace transform domain. Solutions of axisymmetric surface load and fluid pressure are derived.


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## Body

## 1. Introduction

Biot [1, 2] presented the propagation theory of elastic waves and the general solutions for fluid-saturated porous viscoelastic medium. Akkas and Zakout [3] discussed the solution for the transient response for an axisymmetric and nontorsional load of an infinite, isotropic, elastic medium containing a spherical cavity with and without thin elastic shell embedment. In this paper, considering a viscoelastic model presented by Eringen [4], the transient response of a spherical cavity with a partially sealed shell embedded in viscoelastic soil is investigated. The solutions of stresses, displacements and pore pressure induced by axisymmetric surface load and fluid pressure are derived in Laplace transform domain. Durbin’s [5] inverse Laplace transform is used to analyze the influence of partial permeable property of boundary and relative rigidity of shell and soil on the transient response of the spherical cavity. The solutions of permeable and impermeable boundary without shell are considered as two extreme cases.


## 2. Basic Equations and Solutions

In infinite viscoelastic saturated soil, a thin elastic shell shown in Figure1 with inner radius a, outer radius b, and thickness h=b-a, has been bored. (r,θ,ϕ) are the spherical coordinates, where θ and ϕ are the meridional and circumferential angles, respectively, σr, σθ, σϕ are nonvanishing components of stress tensor in case of an axisymmetric nontorsional load, that is, independent of θ and ϕ acting on the shell surface.Figure 1
Geometry of the problem.In spherical coordinate system(r,θ,ϕ), the equilibrium equation for soil mass is
(1)∂σr∂r+2σr-σθ-σϕr=∂2∂t2(ρur+ρfwr),
where ur and wr are radial displacement of soil skelton and displacement of pore fluid with respect to soil skelton, respectively; ρ=(1-n)ρs+ηρf, the density of soil; ρf and ρs are densities of fluid and soil grains respectively; n is porosity.The pore fluid equilibrium equation is given by(2)-∂p∂r=∂2∂t2(ρfur+ρfnwr)+η0kd∂wr∂t,
where p is excess pore pressure; η0 is the fluid viscosity, and kd is the intrinsic permeability of soil.Soil is not an ideal medium. Due to overcoming the interior friction of soil, a part of energy of the propagation wave is changed into heat energy during the propagation. This property is known as damping of material. Assuming that the viscoelastic property of soil may be simulated by Kelvin-Voigt model. Following Eringen [4], the stress-strain relationship is expressed as
(3)σr=λe+2G∂ur∂r+λ′∂e∂t+2G′∂∂t(∂ur∂r)-αp,(4)σθ=σϕ=λe+2Gurr+λ′∂e∂t+2G′∂∂t(urr)-αp,(5)p=Mξ-αMe,
where  e=∂ur/∂r+2ur/r and ξ=-(∂wr/∂r+2wr/r), dilations of solid and fluid, respectively; λ and G are Lame constants of the bulk material; λ′ and G′ are the dilatant and shear constant of the viscoelastic soil; α and M are the compressibility parameters of the two phase medium, 0≤α≤1, 0≤M≤∞ and M→∞, α→1 for a material with incompressible constituents.Substituting (3), (4), and (5) into (1) and (2), the governing equations of the transient response of a spherical cavity in viscoelastic solid condition can be reduced as
(6)(λ+2G+α2M)∂e∂r+(λ′+2G′)∂∂t(∂e∂r)-αM∂ξ∂r=∂2∂t2(ρur+ρfwr),(7)αM∂e∂r-M∂ξ∂r=∂2∂t2(ρfur+ρfnwr)+η0kd∂wr∂t.
Now, displacements ur and wr are assumed to be of the forms
(8)ur=∂∂r(U(r)cosωtr),(9)wr=∂∂r(W(r)cosωtr),
for solving (6) and (7). Consider
(10)e=∂ur∂r+2urr,∂e∂r=∂2ur∂r2+2r∂ur∂r-2r2ur,ur=∂∂r(U(r)cosωtr),ur=1r∂∂r(U(r)cosωt)-U(r)cosωtr2,∂ur∂r=1r∂2∂r2(U(r)cosωt)-2r2∂∂r(U(r)cosωt)+2r3U(r)cosωt,∂2ur∂r2=1r∂3∂r3(U(r)cosωt)-3r2∂2∂r2(U(r)cosωt)+6r3∂∂r(U(r)cosωt)-6r4U(r)cosωt.
Then,
(11)∂e∂r=1r∂3∂r3(U(r)cosωt)-1r2∂2∂r2(U(r)cosωt)=∂∂r(1r∂2∂r2(U(r)cosωt))=∂∂r(1r∇2(U(r)cosωt)).
Similarly,
(12)∂ξ∂r=-∂∂r(1r∂2∂r2(W(r)cosωt))=-∂∂r(1r∇2(W(r)cosωt)).
Substituting (8), (9), (11), and (12), into (6), we obtain
(13)(λ+2G+α2M)∂∂r(1r∇2(U(r)cosωt))+(λ′+2G′)∂∂t(∂∂r(1r∇2(U(r)cosωt)))-αM∂∂r(1r∇2(W(r)cosωt))=∂2∂t2(ρ∂∂r(1r∇2(U(r)cosωt))+ρf∂∂r(1r∇2(W(r)cosωt))),∂∂r[∂∂t1r(λ+2G+α2M)∇2U(r)cosωt+(λ′+2G′)∂∂t∇2U(r)cosωt+αM∇2W(r)cosωt]=∂∂r[1r∂2∂t2(ρU(r)cosωt+ρfW(r)cosωt)].
Integrating w.r.t. r, we have
(14){(λ+2G+α2M)+(λ′+2G′)∂∂t}∇2U(r)cosωt+αM∇2W(r)cosωt=∂2∂t2(ρU(r)cosωt+ρfW(r)cosωt),[(λ+2G+α2M)∇2+(λ′+2G′)∂∂t∇2-ρ∂2∂t2]×U(r)cosωt+(αM∇2-ρf∂2∂t2)W(r)cosωt=0,
where ∇2=∂2/∂r2.Substituting (8) to (12), in (7), we obtain
(15)αM∂∂r(1r∇2U(r)cosωt)+M∂∂r(1r∇2W(r)cosωt)=∂2∂t2{ρf∂∂r(1rU(r)cosωt)+ρfn∂∂r(1rW(r)cosωt)}+η0kd∂∂t(∂∂r(1rW(r)cosωt)),∂∂r[1r(αM∇2U(r)cosωt+M∇2W(r)cosωt)]=∂∂r[1r(∂2∂t2ρfU(r)cosωt+∂2∂t2ρfnW(r)cosωt+η0kd∂∂tW(r)cosωt)].
Integrating w.r.t. r, we have
(16)αM∇2U(r)cosωt+M∇2W(r)cosωt=∂2∂t2ρfU(r)cosωt+∂2∂t2ρfnW(r)cosωt+η0kd∂∂tW(r)cosωt,[αM∇2-ρf∂2∂t2]U(r)cosωt+[M∇2-ρfn∂2∂t2-η0kd∂∂t]W(r)cosωt=0.Taking Laplace transform in (14) yields
(17)[(λ+2G+α2M)∇2(ss2+ω2)+(λ′+2G′)∇2(-ω2s2+ω2)-ρ(-ω2ss2+ω2)]U(r)+(αM∇2(ss2+ω2)-ρf(-ω2ss2+ω2))W(r)=0,[(λ+2G+α2M)∇2-ω2s(λ′+2G′)∇2+ρω2]U(r)ss2+ω2+[αM∇2+ρfω2]W(r)ss2+ω2=0.Taking Laplace transform in (14) yields
(18)[αM∇2(ss2+ω2)-ρf(-ω2ss2+ω2)]U(r)+[M∇2(ss2+ω2)-ρfn(-ω2ss2+ω2)-η0kd(-ω2s2+ω2)]×W(r)=0,(αM∇2+ω2ρf)U(r)ss2+ω2+[(M∇2+ρfnω2)+η0kdω2s]×W(r)ss2+ω2=0.In a great range of vibration frequencies, viscoelastic damp coefficient of rock and soft soil may be assumed as a constant. The dimensionless damp coefficientη is considered as
(19)η=λ′λ=G′G.
Also,
(20)λ*=λG,M*=MG,ρ*=ρfρ,b*=η0a′kdρG,ρG~1
are nondimensional Lame constant, compressibility parameter, fluid density, and permeability coefficient of soil respectively. b is the radius of spherical shell, a′=b-h/2=a+h/2.Using (19) and (20) in (17) yields
(21)[((λ*+2)+α2M*)∇2-ω2s(ηλ′+2η)∇2+ω2]U(r)ss2+ω2+[αM*∇2+ρ*ω2]W(r)ss2+ω2=0,[{(λ*+2)(1-ω2sη)+α2M*s}∇2+ω2]U(r)ss2+ω2+(αM*∇2+ρ*ω2)W(r)ss2+ω2=0.
Similarly, using (19) and (20) in (18) yields
(22)(αM*∇2+ρ*ω2)U(r)ss2+ω2+[(M*∇2+ρ*ω2n)+b*ω2s]W(r)ss2+ω2=0,(αM*∇2+ρ*ω2)U(r)ss2+ω2+[M*∇2+(ρ*n+b*s)ω2]W(r)ss2+ω2=0.
Solving (21) and (22) yields
(23)[{((λ*+2)(1-ω2sη)+α2M*)∇2+ω2}×{M*∇2+(ρ*n+b*s)ω2}-{(αM*∇2+ρ*ω2)}2(1-ω2sη)]×[U(r)ss2+ω2,W(r)ss2+ω2]=0,(24)[(λ*+2)(1-ω2sη)M*∇4+{((λ*+2)(1-ω2sη)+α2M*)(ρ*n+b*s)+M*(1-2αρ*)((λ*+2)(1-ω2sη)+α2M*)}ω2∇2+{(ρ*n+b*s)-(ρ*)2}ω4(1-ω2sη)](U,W)(ss2+ω2)=0.
Equation (24) can be written as
(25)(∇2-γ12)(∇2-γ22)(U,W)(ss2+ω2)=0,
where γ1 and γ2 are the complex wave number of two dilation waves, that is,
(26)γ12=α1-α1-4α22,γ22=α1+α1-4α22,
with
(27)α1=[((λ*+2)(1-(ω2/s)η)+α2M*)×(ρ*/n+b*/s)+M*(1-2αρ*)((λ*+2)(1-(ω2/s)η)+α2M*)]ω2×((λ*+2)(1-(ω2/s)η)M*)-1,α2=[(ρ*/n+b*/s)-(ρ*)2]ω4(λ*+2)(1-(ω2/s)η)M*.The general solutions ofU(r)s/(s2+ω2) and W(r)s/(s2+ω2) in (25) are
(28)U(r)ss2+ω2=A1e-γ1r+A2e-γ2r+A3eγ1r+A4eγ2r,W(r)ss2+ω2=B1e-γ1r+B2e-γ2r+B3eγ1r+B4eγ2r.Considering the limitation property of radial displacement whenr→∞, that is,
(29)U(r)⟶∞asr⟶∞,W(r)⟶∞asr⟶∞.
In (28), we have, A3=A4=B3=B4,
(30)U(r)ss2+ω2=A1e-γ1r+A2e-γ2r,(31)W(r)ss2+ω2=B1e-γ1r+B2e-γ2r.
Constants A1, A2, B1, and B2 in (30) and (31) are linearly dependent and may be related by using (25) to obtain
(32)Bi=δiAi,i=1,2,
where  δi=-(αMγi2+ρ*ω2)/(M*γi2+(ρ*/n+b*/s)ω2).A
1, A2 can be obtained from boundary conditions. Now,
(33)ur=∂∂r(U(r)cosωtr).
The Laplace transformed solution of radial displacement ur, that is, u-r is given by
(34)u-r=∂∂r(U(r)sr(s2+ω2))=∂∂r1r(A1e-γ1r+A2e-γ2r),by(30)=-A1r(γ1+1r)e-γ1r-A2r(γ2+1r)e-γ2r.
Similarly, the Laplace transform solution of wr is w-r:
(35)wr=∂∂r(W(r)cosωtr),w-r=∂∂r(W(r)sr(s2+ω2))w-r=∂∂r1r(B1e-γ1r+B2e-γ2r),w-r=-1r(γ1+1r)e-γ1rB1-1r(γ2+1r)e-γ2rB2w-r=-δ1r(γ1+1r)e-γ1rA1w-r-δ2r(γ2+1r)e-γ2rA2,by(32).
Next, by (5),
(36)p=Mξ-αMe=-M(∂wr∂r+2wrr)-αM(∂ur∂r+2urr)PG=-M*(∂wr∂r+2wrr)-αM*(∂ur∂r+2urr).
Taking Laplace transform and using (34) and (35), we have
(37)(P-G)=-M*[∂∂r{-∑i=12δir(γi+1r)e-γirAi}+2r{-∑i=12δir(γi+1r)e-γirAi}]-αM*[∂∂r{-∑i=121r(γi+1r)e-γirAi}+2r{-∑i=121r(γi+1r)e-γirAi}]=∑i=12(α+δi)M*×Ai[-1r2(γi+1r)e-γir-1r3e-γir+2r(γi+1r)e-γir(-γi)+2r(γi+1r)e-γir],(P-G)=-∑i=12(α+δi)1rM*γi2e-γirAi.
Next, by (3),
(38)σrG=λ*(∂ur∂r+2urr)+2∂ur∂r+λ*η∂∂t(∂ur∂r+2urr)+2η∂∂t(urr)-αPG.
Taking Laplace transform of both sides
(39)(σ-rG)=λ*(∂u-r∂r+2u-rr)+2∂u-r∂r+λ*η(∂∂r(-ω2su-r)-2ω2su-rr)+2η∂∂r(-ω2su-r)-αP-G=(λ*+2)(1-ω2sη)∂u-r∂r+2λ*(1-ω2sη)u-rr-αP-G=∑i=12(λ*+2)(1-ω2sη)∂∂r(-1r(γi+1r)e-γir)+2λ*r(1-ω2sη)∑i=12(-1r(γi+1r)e-γirAi)+α∑i=12(α+δi)AiM*γi2e-γir,by(34)and(35)=-∑i=12(λ*-2)(1-ω2sη)×Ai{-1r2(γi+1r)e-γir-1r3e-γir+1r(γi+1r)e-γir(-γi)}-2λ*∑i=12(1-ω2sη)1r2(γi+1r)e-γirAi+α∑i=121r(α+δi)AiM*γi2e-γir=∑i=12(λ*+2)(1-ω2sη)e-γirAiγi2+2∑i=12(λ*+2)(1-ω2sη)e-γirAir2(γi+1r)-2∑i=12λ*(1-ω2sη)+e-γirAir2(γi+1r)+α∑i=121r(α+δi)AiM*γi2e-γir(σ-rG)=Ai∑i=12[1r(λ*+2)(1-ω2sη)γi24r2(1-ω2sη)×(γi+1r)+αrγi2(α+δi)M*(1-ω2sη)].
Similarly, by (4), (35), and (37), we have
(40)(σθG)=(σϕG)=λ*e+2urr+ηλ*∂e∂t+2η∂∂t(urr)-αpG=λ*(∂ur∂r+2urr)+2urr+ηλ*∂∂t(∂ur∂r+2urr)+2η∂∂t(urr)-αpG.
Then
(41)L(σθG)=(σ-θG)=λ*(∂u-r∂r+2u-rr)+2u-rr+ηλ*(-ω2s∂u-r∂r-2ω2su-rr)+2η(-ω2su-rr)-αp-G=λ*(1-ω2sη)∂u-r∂r+2(λ*+1)(1-ω2sη)u-rr-α(p-G)=-λ*(1-ω2sη)∑i=12∂∂r1r(γi+1r)e-γirAi-2(λ*+1)(1-ω2sη)×∑i=121r2(γi+1r)e-γirAi-α(p-G)=-λ*(1-ω2sη)×∑i=12[-1r2(γi+1r)-1r3-γir(γi+1r)]Aie-γir-2(λ*+1)(1-ω2sη)×∑i=121r2(γi+1r)e-γirAi-α(p-G)=∑i=12λ*r(1-ω2sη)γi2e-γirAi-∑i=12(1-ω2sη)×[-2λ*r2(γi+1r)+2λ*r2(γi+1r)+2r2]-α(p-G)=∑i=12[λ*r(1-ω2sη)γi2-2r2(1-ω2sη)(γi+1r)+αrγi2(α+δi)M*(1-ω2sη)]e-γirAi.


## 3. Solution of Shell Embedment and  Axisymmetric Loading

Dynamic loads applied on the surface of shell considered herein are an axially symmetric radial traction and axially symmetric fluid pressure with the step style shown in Figure2, where T* is the nondimensional step load time (T*=TG/ρ/b), T is actual step load time; (t*=tG/ρ/b), the nondimensional time; t is actual time; q0 is maximum of the step load. In the domain of Laplace transform, the load can be expressed as
(42)q-(s)=q0T*(1-e-T*ss2),r=b.Figure 2
Curve of radial displacement versus relative rigidity (fluid pressure).Here, the case of a thin, elastic shell embedded in infinite viscoelastic saturated soil subjected to axisymmetric surface load and fluid pressure is considered. The equation of motion of this shell under nontorsional axisymmetric loading is(43)2(1+μ1)μr′+γ02∂2μr′∂t2=q0(t)(a′)(1-μ12)E1h,
where γ02=c12/cp2;  c1=(λ+2G)/ρ and cp=E1/(ρf(1-μ2)) are the dilatational wave velocity and plate velocity, respectively; E1, μ1 are the modulus and Poission’s rate of shell, respectively; q0(t) is the net outward radial pressure. For a thin shell, the thick h/2 can be omitted without significant error. The interface shell and soil can be defined as r=a+h/2=a′. The stress and displacement condition at the interface is expressed as
(44)q0(t)=q(t)-σr,r=a+h2=a′,(45)ur=-ur′,r=a+h2=a′,
where q(t) is the radial stress applied at the inner surface of the shall; σr is the stress exerted by the soil on the shell and can be given by (39).In practical situation, the condition is frequently found in two extreme cases; permeable and impermeable.The partial permeable flow boundary condition is(46)∂p∂r=kpa′atr=a+h2=a′,
where k=(k1/kd)(1/log(b/a)) is a dimensionless permeability parameter that defines the flow capacity of the shell. The parameter k depends on the relative permeability of the shell and soil as well as the geometry of the shell, that is,(1)
when the spherical shell is impermeable, that is,k1=0, k tends to zero and(2)
when the shell is permeable, that is,k1=∞, k tends to infinity.Substituting (34), (39), (44) into (43) and (37) into (46), we obtain
(47)m1A1+m2A2=(a′)2(1-μ12)E1*hq-(s)G,n1A1+n2A2=0,
where
(48)mi=[2(1+μ1)+γ02ω2]1a′(γi+1a′)e-γia′+[1a′(λ*+2)(1-ω2sη)×γi24(a′)2(1-ω2sη)(γi+1a′)+αa′γi2(α+δi)M*](a′)2(1-μ12)E1*he-γia′ni=1a′(γi+k+1a′)(α+δi)M*γi2e-γia′,i=1,2,Ei*=E1G,A1=-n2n1A2,A2=-(n1n1m2-n2m1)(a′)2(1-μ12)E1*hq-(s)G.Under the fluid pressure on shell surface, the displacement and stress components are continuous at the kinematic interface between the spherical shell and soil. In this case, flow boundary conditions are(49)q0(t)=-σratr=a+h2=a′,(50)ur=-ur′atr=a+h2=a′,(51)∂p∂r=ka(p+q(t))atr=a+h2=a′.Substituting (34), (39), (49), (50) into (43) and (37) into (51) yields
(52)m1A1+m2A2=0,n1A1+n2A2=ka'q-(s)G,
or
(53)A1=-m1m2A2,A2=ka′(m1m1n2-m2n1)q-(s)G.
Using inverse Laplace transform and numerical computation, the final solutionin time domain can be obtained after determining A1 and A2.


## 4. Results and Discussion

In this paper, we will discuss the influences of partial permeable property of boundary and relative rigidity of shell and soil (defined asRR=El/Es) on the transient response of the spherical cavity. The numerical results are presented for the material and geometric parameters which are listed in Table 1.Table 1
Parameters used in computation.
Quantity
Notation
Value
Elastic modulus of soil (MPa)
E
s
7
Poisson rate of soil
μ
0.35
Poisson rate of shell
μ
l
0.09
Dimensionless shell density
ρ
l
*
1.4
Dimensionless shell thickness
h
/
a
0.04
Dimensionless fluid density
ρ
*
0.45
Compressible parameter of material
α
0.96
Dimensionless compressible parameter
M
*
18
Dimensionless permeability coefficient
b
*
9
Viscoelastic damp coefficient
η
0.35
Porosity
n
0.35
Gradually applied step load time
T
*
1


### 4.1. Solutions Corresponding to Fluid Pressure

The histories of dimensionless radial displacement under fluid pressure are shown in Figure2 when parameters k=0.5. It is noted that at a certain time instant as shown in Figure 2, there exists maximum displacement at the interface of shell and soil. With the increase of time, radial displacement decreased vibrationally and finally to an asymptotic value of zero. Radial displacement decreased obviously with increasing relative rigidity, and increased with increasing of parameter k (Figure 3).Figure 3
Curve of radial displacement versus parameterk (fluid pressure).The excess pore pressures induced by fluid pressure are shown in Figure4. However, the influence of parameter k (Figure 4) is significant. When the shell boundary became almost impermeable (k→0), almost no excess pore pressure existed, whereas with the increasing of time, the excess pore pressure at the interface equaled the fluid pressure (P=-q0) when the shell boundary became almost permeable (k→∞).Figure 4
Curve of pore pressure versus parameterk (fluid pressure).


### 4.2. Solutions Corresponding to Radial Load

The histories of dimensionless radial displacement at the interface of shell and soil induced by axially symmetric radial surface load are shown in Figure5 when the parameter k=0.5. With the increase of the dimensionless time  (t*=tG/ρ/a), radial displacement increases to maximum value, then decreases and is noted once again. Eventually, it tends to an asymptotic value. When relative rigidity RR=0, the shell is complete flexible, there is the maximum radial displacement   at the interface of shell and soil. The value of radial displacement decreases with the increase of relative rigidity. The influence of permeability parameter k on radial displacement is indicated in Figure 6. It can be seen that the influence of parameter k on radial displacement induced by axisymmetric radial surface load is not remarkable.Figure 5
Curve of radial displacement versus relative rigidity.Figure 6
Curve of radial displacement versus parameterk.The histories of dimensionless pore pressure are shown in Figure7 at the interface of shell and soil for the parameter k=0.5. Pore pressure is zero at t*=0 and increases rapidly with time in the interval 0<t*≤T* and reaches to its peak value nearly at t*=T*. Thereafter, it decreases with time and reaches to its maximum suction values. With increasing time the values of suction decreases and pore pressure is noted once again. On the other hand, the pore pressure decreases with the increase of parameter k (Figure 8). As a result, both the relative rigidity and parameter k have great influence on the pore pressure under the condition of axisymmetric radial surface load.Figure 7
Curve of pore pressure versus relative rigidity.Figure 8
Curve of pore pressure versus parameterk.


## 4.1. Solutions Corresponding to Fluid Pressure

The histories of dimensionless radial displacement under fluid pressure are shown in Figure2 when parameters k=0.5. It is noted that at a certain time instant as shown in Figure 2, there exists maximum displacement at the interface of shell and soil. With the increase of time, radial displacement decreased vibrationally and finally to an asymptotic value of zero. Radial displacement decreased obviously with increasing relative rigidity, and increased with increasing of parameter k (Figure 3).Figure 3
Curve of radial displacement versus parameterk (fluid pressure).The excess pore pressures induced by fluid pressure are shown in Figure4. However, the influence of parameter k (Figure 4) is significant. When the shell boundary became almost impermeable (k→0), almost no excess pore pressure existed, whereas with the increasing of time, the excess pore pressure at the interface equaled the fluid pressure (P=-q0) when the shell boundary became almost permeable (k→∞).Figure 4
Curve of pore pressure versus parameterk (fluid pressure).


## 4.2. Solutions Corresponding to Radial Load

The histories of dimensionless radial displacement at the interface of shell and soil induced by axially symmetric radial surface load are shown in Figure5 when the parameter k=0.5. With the increase of the dimensionless time  (t*=tG/ρ/a), radial displacement increases to maximum value, then decreases and is noted once again. Eventually, it tends to an asymptotic value. When relative rigidity RR=0, the shell is complete flexible, there is the maximum radial displacement   at the interface of shell and soil. The value of radial displacement decreases with the increase of relative rigidity. The influence of permeability parameter k on radial displacement is indicated in Figure 6. It can be seen that the influence of parameter k on radial displacement induced by axisymmetric radial surface load is not remarkable.Figure 5
Curve of radial displacement versus relative rigidity.Figure 6
Curve of radial displacement versus parameterk.The histories of dimensionless pore pressure are shown in Figure7 at the interface of shell and soil for the parameter k=0.5. Pore pressure is zero at t*=0 and increases rapidly with time in the interval 0<t*≤T* and reaches to its peak value nearly at t*=T*. Thereafter, it decreases with time and reaches to its maximum suction values. With increasing time the values of suction decreases and pore pressure is noted once again. On the other hand, the pore pressure decreases with the increase of parameter k (Figure 8). As a result, both the relative rigidity and parameter k have great influence on the pore pressure under the condition of axisymmetric radial surface load.Figure 7
Curve of pore pressure versus relative rigidity.Figure 8
Curve of pore pressure versus parameterk.


## 5. Conclusions

An extensive parameters study conducted to investigate the influence of therelative rigidity of shell and soil and permeability parameterk, showed that permeability parameter k depends on the relative permeability of the liner and soil as well as the geometry of the liner. Relative rigidity and parameter k have significant influences on the transient response of spherical cavity with a shell embedded in viscoelastic saturated soil. The solutions under permeable and impermeable boundary conditions are only two extreme cases. Thus partially sealed boundary condition and the relative rigidity of shell and soil in the designing and computation of spherical shell in viscoelastic saturated medium are remarkable.



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*Source: 186204-2013-01-29.xml*