A Xi Symmetric Fundamental Solutions

http://www, zju. edu. cn/jzus; E- mail: jzus @ zju. edu. cn ISSN 1009 - 3095 Journal of Zhejiang University SCIENCE V. 4, No. 4, P. 393 - 399, July - A u g . , 2003

Axisymmetric fundamental solutions for a finite layer with impeded boundaries"

CHENG Ze-hai ( ~ - ~ ) * , LING D a o - s h e n g ( ~ J ~ ) ,

CHEN Y u n - m i n ( ~ ; f / ~ ) TANG X i a o - w u ( N ~ )

( College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310027, China ) *E-mail : czhl002 @ 163. com Received Sept. 12,2002 ; revision accepted Dec. 2,2002 Abstract: Axisymmetric fundamental solutions that are applied in the consolidation calculations of a finite clay layer with impeded boundaries were derived. Laplace and Hankel integral transforms were utilized with respect to time and radial coordinates, respectively in the analysis. The derivation of fundamental solutions considers two boundary-value problems involving unit point loading and ring loading in the vertical. The solutions are extended to circular distributed and strip distributed normal load. The computation and analysis of settlements, vertical total stress and excess pore pressure in the consolidation layer subject to circular loading are

presented. Key w o r d s : Consolidation, Integral transform, Finite layer, Impeded boundaries Document code : A C L C n u m b e r : TB121

INTRODUCTION Blot' s consolidation theory was proposed in 1941 from continuous medium fundamental equations. The theory taking into account the coupling between the solid strains and pore pressure dissipation is called "coupled consolidation theory". The solution of governing equations of a porous solid is more complicated than those involving ideal elastic solids. McNamee et al. (1960a) presented a solution for plane and axially symmetric problems in terms of two displacement functions. The general solution for displacement functions is obtained through the application of Laplace and Fourier integral transforms for plane problems and Laplace and Hankel integral transforms for axisymmetric problems. Gibson et a l . (1970) obtained a solution for plane strain and axially symmetric consolidation of a clay layer on a smooth impervious base. Puswewala et a l . (1988) presented axisymmettic fundamental solutions for a completely saturated porous elastic solid. Gu et a l . (1992) presented a solution for an axisymmetric vertical loaded multi-layer base. Huang et a l . ( 1 9 9 6 )

also obtained an analytical solution for finite layer through expanding the field of solution. All the above investigations assumed that the consolidation layer surface is fully permeable or impermeable, but that the surface might be affected by geotechnical engineering impediments such as sand cushion in preloading, replacement layer of embankment impeding permeation of the consolidation layer. The impeded layer can be simplified impeded boundaries of the sub-consolidation layer. Xie ( 1 9 9 6 ) studied one dimensional consolidation of layered soils with impeded boundaries. However results for two-dimensional problem with impeded boundaries have not been reported. This paper deals with the derivation of axisymmetric fundamental solutions that are applied in the consolidation calculations of a finite layer with impeded boundaries. The derivation of fundamental solutions considers two boundary-value problems involving unit point loading and ring loading in the vertical direction. The solutions are extended to circular loading and strip loading. The calculation and analysis of settlements, vertical total stress and excess pore pressure in

* Project(No. 500117) supportedby the NationaI NaturaI Science Foundationof Zhejiang Province

394

CHENG Zehai, CHEN Yunmin et al.

the consolidation layer subject to circular loading are p r e s e n t e d .

GOVERNING EQUATIONS AND GENERAL SOLUTION

z,t), e=(r,z,t) and e z r ( r , z , t ) are bulk strain components. Following McNamee et a l . ( 1 9 6 0 b ) , displacements Ur ( r , Z, t ) and u, ( r , z , t ) and pore pressure pf ( r , z , t ) are expressed in terms of two functions r and ~b in the following form:

a~

Following Biot ( 1941 ) , the displacement equations for axisymmetric deformations of a completely saturated isotropie and homogeneous poroelastie solid in a cylindrical coordinate system are expressed in the following: (V 2 -

r

Ur --

03r Uz -

03e 1 03pf 0 V2Ur "at+ (2r/ - 1) ~zz + G 03z -

03e

cV2e

3)

-- 03t

034

2 G ( 03q~ -

2)

(8)

03r + Z Orr - r

pf =

03e 1 03pf )Ur + (2r/ -- 1) Yrr + G 03r - - 0 1)

034

03r + z 03r

aT-

(9)

V2~b)

(10)

~

It is convenient to dedimensionalize all quantities with respect to length and time by selecting a certain length " R " as u n i t y , and " R 2 / c '' as a unit of t i m e , r e s p e c t i v e l y . The substitution of E q s . ( 8 ) - ( 1 0 ) in E q s . ( 1 ) - ( 3 ) y i e l d the following governing equations for functions 9~ and

so: Where 032

V 2 -

1 03

-t- - - - -

03r 2

r

-t-

03r

V4@

02 03z 2

V2~

1 [

( ) O-~-_rUr_ + r 03u~1 e = -7- 03r -ff-z-zl r/ = (1 - v ) / ( 1 - 2 v ) c = 2 Grlk In the above equations, Ur ( r , z , t ) and u+ ( r , z , t ) denote displaeements in r and z directions, respectively; pf ( r , z , t ) is the exeess pore p r e s s u r e ; G and v denote shear modulus and Poisson' s ratio of the bulk material ; k is the eoeffieient of permeability of the m e d i u m ; e is the dilatation of the bulk material. The stress-strain relations are expressed as CYrr

2G

= err + ( q - -

1)e

+

Pf 2G

=

(4)

V2

03~

03t

= 0

(11) (12)

Laplace and H a n k e l transforms of E q s . ( 11 ) and ( 1 2 ) result in the following two ordinary differential equations. Define ~hl and ~bhl as zeroorder Hankel transform and Laplace transform of funetions ~ and @, respectively. 032 -- ~2

032

(0@2- ~2)~bhl=0

(14)

where p is Laplace transform p a r a m e t e r ;

~ is

H a n k e l transform p a r a m e t e r ; u = ~/~e + P . Solution of the above two equations are @hl = A( ~ , p ) e -z~ + B ( ~ , p ) e -~7 +

O'00

2G ~ zz

2G

= e00 + ( r / -

Ps

1)e +-2G

=e=+(rl-1)e+--

(5)

9~hl =

Pi 2G

(6)

O'zr

2 G - e+,

(7)

Where arr(r,z,t), aoo(r,z,t), a=(r,z, t ) and azr ( r , z , t ) are non-zero stress components of the bulk material; e r r ( r , z , t ) ,

C((,p)e

eoo(r,

~ + D(~,p)e

*r

E ( ~ ) e -"~ + F ( ~ ) e ++~

(15) (16)

Based on E q s . ( 4 ) - ( 1 6 ) , general solutions for d i s p l a c e m e n t s , excess pore pressure and stresses for axisymmetric deformations of a completely saturated porous elastic solid can be written as follows :

g+ = f+ ~ J o ( r ( ) E ~'Ae -+'g + ?+,Be-++? _ ~'CeZ~ _ Jo

Axisymmetric fundamental solutions for a finite layer with impeded boundaries $ D e ~7 - ( z ~ + 1 ) E e - ~ + ( z ~ - 1 ) F e Z ~ d ~ (17)

idation layer are derived in the following forms. ( z = O)

(22)

~rz ----0

(Z = O)

(23)

a==poa(r)/(2=r)

(z=O)

(24)

k2 3~-z s

= .I(7 ~ 2 J l ( r ~ ) [ A e - Z r

+ Be-Z7

pf

=

2G

~Ee - ~ -

~FeZ~]d~

e=

fo

= 2G

(18)

~Jo(r~)[rlpBe-ZZ

0

rlpDe~7

+

(19)

~J0(r~)[-

0

+

~Ae-Z~ _

~Ce z~ - ~De ~7 + ( z ~ 1) F e " ; ~d~

kl ~Pf

+ CeZ; +

De ~ - zEe -~ - zFeZ;]d~

In the m e a n t i m e , it is assumed that the base of the consolidation layer rests on a smooth, rigid and impervious m e d i u m , so that, over this plane,

~Be-*r _

+ 1 ) E e -z~ + ( z ~ (20)

Orz = 0

(z=h)

(25)

Opf _ 0 3z

(z=h)

(26)

Ltzz = 0

(z = h)

(27)

oo

~zr

=

2G

X 0

~2Jl(r~)

[ -

~A e - ~r _

y B e - z7 +

~ C e zr + y D e ~7 + z ~ E e - ~ - z ~ F e ~ ] d ~

(21)

where the superposed bar denotes the Laplace transform of the relevant quantity. Combining the above general solutions with different boundary conditions, fundamental solutions can be o b t a i n e d .

395

By Laplace transform and the zero order Hankel transform, the boundary conditions are as follows : z = 0)

(28)

~rrz = 0

a = 0)

(29)

~zz = p o / ( 2 7 v p )

z = 0)

(30)

FUNDAMENTAL SOLUTIONS

arz = 0

z = h)

(31)

3pf _ 0 Oz

z = h)

(32)

1. Fundamental solutions subject to point loading

u= = 0

z= h)

(33)

H

Impeded layer k~

P0

F

0 Consolidation layer k2,c,G,v

Z Fig. 1

Axisynnnetric point loading

3pf

k2 ~ - z

=

Pf

1 H-

Base on E q s . ( 1 7 ) - ( 3 3 ) , the fundamental solutions in the Laplace domain can be obtained in the following form. ( 1 ) Displacement in the radial direction: ur = -

P0 16Gp~

.I: ~Jl(r~)e-Z(:~+g)-h(~'+2r

a 1 + e4hga2) (e 2z7 + e 2hz )

eZT+2(h+ z)g( In Fig. 1, H denotes the thickness of the imp e d e d layer; h is the thickness of the consolidation layer; k l denotes the coefficient of permeability of the i m p e d e d layer; k2 is the coefficient of permeability of the consolidation. It is assumed that permeation only takes place vertically in the impeded layer not considering the vertical deformation of the impeded layer and shearing stress over the plane ( z = 0 ) . Following Terzaghi one dimensional consolidation t h e o r y , the boundary conditions for the surface of the consol-

k

7 a 2 --

( ~e*r

eZT+2h~( u

_

2e2hr +

a3Pl ) + eZT+2h(7+.~) (

+

~3/~1) 4yOL1 4-

014/~1) 4 " e 2 h ( 7 + ~')+z(7+2~) ( -- yo~ 2 4" 0~4/~1) 4" e2hT+ zy +4h'~ ( _ ~OL2 4" p Z ~ 4

) 4"

eZT+4h~( 70~2

--

PZT"~3 ) -- eZ(7+2g) ( ~,0~1 4" pz~c~3 ) 4" e2hT+ zT + 2zg ,

( 7a 1 4" p z ~ t 4 ) ) / ( c o s h ( h u

( - Hk2 ~3 +

2 h k l p ~ ? + Hk2 ~ 3 c o s h ( 2 h ~ ) + kl ( ~2 + p~?).

sinh(2h~) ) + ysinh(hu

( - 2k1 ~ c o s h ( h ~ ) 2 +

Hk2 (2hp~? + ( - ~2 + p r ] ) s i n h ( 2 h ~ ) ) ) ) d ~ (34) 2) Displacement in the vertical direction:

396

uz -

CHENG Zehai, CHEN Yunmin et al. PO

I=

4Gp~Jo

Jo(r~)

e-

~( r + ~)

5) Excess pore-water pressure : (7~e~(e

eZh7 ) ( 2 H k 2 (e2h(( _ a l _ ot2eghg )

( - 7"~

+

2*r-

+ p~3

- a3/3~ ~ ) + e ~+2h~r+ ~ ( ) ' ( ~ l -

pria4 + a 4 f l l ( )

- e zz+2(h+~)~ ( )'(a2 - pr/a3 -

a3/91~)

+

e2h(7+r )+'(1+2~)

a 4 p 1 g ) -I- e "(r+2r e2h)'+")'+2zr e ~7+4h~ ( u

( - u

( " ) " ~ a 2 -- pr]a 4

( )'ga 1 + pa3(

-- u

-- 1 + z g )

eZh~)(ot3eZT(e

e 2h~)

+

_

ot4e*Y+ZhT(e 2z~

e2 h r ) ) / ( c o s h ( h u

(-- Hk2 ~3 + 2 h k l p ~ r ] +

Hk2~3cosh(2h~)

+ kl(~:

+

7sinh(hu

+ pr/)sinh(2h~))

Hke(2hp~r/ + (_

(2hp~r] +

+

( - 2kl ~ c o s h ( h ~ ) 2 + ~2 + p r / ) s i n h ( 2 h ~ ) ) ) ) d ~

(38)

+ e2hT) "

( - 1 + e 2hg )2 ~3 + 4eh(7+2~) 7 s i n h ( h u

+

e2h~) _ aleZ~(e2Z~ + e zhT) + a2eZ~+Zh~(e 2~z +

r]) +

2 -- p a 3 ( 1 + z ~ ) r]) + e 2h7+ zT+4h'~o

( _ ~2 + p r / ) s i n h ( 2 h ~ ) ) )

2z~

1 +

--

-- p a 4 ( -- 1 + Z ~ ) r ] )

+ pa4(1 + z~)rl))/(Hk2((1

Po [ = e-.,( 7+ ~')- h(}'+2~') 87rJo r/~J~ ( r ~ ) (-

PU -

eZ7 +2h~o

where

+ k l ( ( 1 - e 2 h T ) (1 +

e2hg) 2 )'~ + 4 e h(7+2g) c o s h ( h ) ' ) ( 2 h p ~ r I + ( ~2 +

a 1 =-

pq) sinh(2h~) ) ) )d~

a2 = kl + Hk2

(35)

a 3 =-

3) Vertical total compressive stress:

a4 =

k I + Hk2(

kl

+ Hk2u

k 1 + Hk2u

fll = P r l ( 2 h

Po I = ~(7+ ~) a= - 2pTrJo ~Jo ( r ~ ) e ( ~2eZr (e2Zl + -

- z)

e 2h7 ) ( - - 2 H k 2 ( e 2h'~ + a 1 + a2 e4hg ) -eZT+2h~ .

2. Fundamental solution subject to ring loading (~/~0~ 1 - - p r l o t 3 + Ot3~ 1 ~ )

PW4

+ a4/~l ~ )

- I - e ~7+2h(7+g) ( u

+ eZT+2(h+z) g ( ")'~~

1 --

-- P W 3

--

a3/~ 1 ~ ) + e2h(7+ ~)+~(7+2~) ( _ y ~ z 2 + pr]o~4 +

a4/31 g) - e=(r+ir + p a 3 ( - 1 + z ' ~ ) r I) + e2hT+zT+2z'~(y~0r 1 + pa4( -- 1 + z'~):7) + e "7+4h~. (u

- p a 3 ( 1 + z~) r/) + eZhr+~r+4hg( -- u

The system under consideration is shown in Fig. 2. Application of Laplace transform and Hankel transform yields the following boundary conditions over the plane ( z = 0) :

+

per4(1 + z ~ ) r ] ) ) / ( H k 2 ((1 + e2hr)(1 + eZhr )2 ~3 + 4eh(~+2r y s i n h ( h u (2hp~r] + ( _ ~2 + p 7 ) s i n h ( 2 h ~ ) ) ) + kl((1 -eZhr)(1 + e2h~)2u + 4 e h ( Z + Z ~ ) c o s h ( h u pr]) s i n h ( 2 h g ) ) ) ) d ~

(2hp~r] + ( ~2 +

f

Impeded layer k,

H

r

(36) Consolidation layer

4) Shearing stress: a= -

(u

k2,c,G,v

8p~Jo 2~r - e2h~)( -- 2Hk2 ~'e2t4" + al + a2 e4~') +

e ~ +ah~ ( 7 a l

Fig.2

+ a 3 f l l ) + e ~ + 2 ( ~ + ")~ ( ) ' a z - a3/~1 ) -

Axisymmetric ring loading

eZ7 + 2h( 7 + [) . ( 70t 1 -t- a 4/~1 ) -I- e TM 7 + ~) + z(7 + 2~) .

( - ~

+ ~4/~1) + e ~ + z ~ §

-p~4~)

-

e zl+4h'~ ( "),'a2 -- pzria 3 ) - e z(7+2g) ( ya~ + pzrla3 ) +

e2hT+ z7 + 2zg ( rot1 + pzT.]~ 4 ) ) / ( c o s h ( h u

~rz

( - Hk2 ~3 + 2 h k l p ~ r ] + Hk2 ~ 3 c o s h ( 2 h ~ ) +

k l ( ~2 + p r ] ) s i n h ( 2 h ~ ) )

k2 Opf 3z -

+ ysinh(hu

----

kl p f H

(39)

0

(40)

a= = R J I ( R ~ )

(41)

Only change Po/27r in the fundamental solution for point load into RJo ( R ~ ) , and the fundamental solution for ring load can be obtained.

( - 2kl ~ c o s h ( h ~ ) 2 + Hk2 (2hp~r] + ( _ ~2 + p r / ) s i n h ( 2 h ~ ) ) ) ) d ~ (37)

Axisymmetric fundamental solutions for a finite layer with impeded boundaries

SOLUTIONS FOR O T H E R DISTRIBUTED LOAD

.........

1. Solutions for circular loading

q

H

~

h

9M(x,z)

Strip loading

All the above solutions should be evaluated numerically. In the numerical quadrature s c h e m e , all infinite integrals with r e s p e c t to the H a n k e l transform p a r a m e t e r ~ are e v a l u a t e d using the Simpson rule ; and time domain solutions are c o m p u t e d using the a p p r o x i m a t e L a p l a c e inverse formula suggested by S c h a p e r y ( 1 9 6 2 ) .

Consolidation layer kvc, G,v

Axiymmetric circular loading

kl Pf -H

-

x

( C I R C U L A R LOADING)

I Z

k2 ~ P f 3z

Consolidation layer _ Vz Fig .4

V

-\\

OR i

Fig. 3

....

VVV

y

,r162

d

VVvVvv _ R

b

N U M E R I C A L COMPUTATION AND ANALYSIS

Impeded layer k~

'

q

0

The system u n d e r consideration is shown in Fig. 3 . Use of L a p l a c e transform and H a n k e l transform yields the following b o u n d a r y conditions over ( z = O) :

1

Impeded layer b

397

(42)

r rz = 0

(43)

a= = q R J I ( R [ ) / ( p ~ )

(44)

0 n l y c h a n g e Po/2~r in the f u n d a m e n t a l solution for point loading into qRJ1 ( R ~ ) I ~ , and the f u n d a m e n t a l solution for ring loading c a n be obtained. 2 . Solutions for strip loading

The system subject to strip loading is shown in F i g . 4 . B a s e d on the f u n d a m e n t a l solutions for point loading, the infinite integration in the y direction and finite integration in the x direction of the f u n d a m e n t a l solutions yields the solution of point M in the m e d i u m for strip l o a d i n g . (~)H=5 ~) H=2 (~) H=I @ H=O.1

-0.5 ~ -0.6

1. The settlement at the center of the loaded area

As shown in Fig. 3 , for k l / k 2 = 1, R = h = l m , q = 1 N / m 2, v = 0 , H f r o m 0 m t o 5 m , the effect of the i m p e d e d l a y e r thickness on the progress of settlement at the c e n t e r of the loaded area is shown in Fig. 5 . The results h a v e b e e n plotted in the form of curves of 2 G u = / q R against c t / R 2. It is a p p a r e n t that the i m p e d e d layer t h i c k n e s s has an important i n f l u e n c e . W h e n the i m p e d e d l a y e r thickness is z e r o , the b o u n d a r y is fully p e r m e a b l e . 2 . The top surface settlement of consolidation layer

For k l / k 2 =

1, R = h = l m ,

H= lm,

q=

1 N / m 2 , v = O, when the i m p e d e d layer thickness is l m , the top surface s e t t l e m e n t of the consoli-

R

a<

s

..~ -0.7 o

9 -0.8 r

-0.9

,

0.001

,

%

0.01 0.1 T=ct/R2 (a)

1

10 (b)

Fig.5 (a) Effect of the impeded layer thickness on settlement at the center of the loaded area; ( b ) sketch map for the position of calculation point

398

CHENG Zehai, CHEN Yunmin et al.

dation l a y e r develops with time as shown in Fig. 6. T h e r e are some slight u p h e a v a l s beyond the loaded a r e a . -0.2r 0

X//

.

0.4~,#/~1[

:000l

m T:O.O1

o.81 _ / /

,

T:I

1~![ ] 1.2 ~

x 9

T=10 T:100

3 . Total compressive stress at 0 . 5 m under the center of the loaded area

For k l / k 2 = 1, R = h = l m , q = 1 N / m 2 , v = 0 , the changing with time of vertical total c o m p r e s s i v e stress at O. 5 m u n d e r the c e n t e r of the loaded area during consolidation is shown in the F i g . 7 . The results have b e e n plotted in the form

Radial (r/R) Fig.6 The development with time of the top surface settlement of the consolidation layer

of

curves

of

a=/q

against

ct/R

2 .

The

t h i c k e r the i m p e d e d l a y e r , the smaller the c h a n g e range of the total stress. It is clearly shown in Fig. 8 that the change of vertical total stress is related to the c h a n g e of excess pore pressure.

@ H=0 | H=O.1

@

1.06 1.05

f ' , , @H=I ./ _\ @H=2

.w 1.04 ~"~ 1.03 1.02

,m?:

1.01 0.00001

0.001

0.1

0

10

T=ct/R =

z

(a)

(b)

Fig.7 ( a ) Effect of the impeded layer thickness and time on vertical total compressive stress; ( b ) sketch map for the position of calculation point

plotted in the form of curves of p f / q

4. Excess pore pressure

For k l / k 2 = 1, R = h = l m , q = 1N/m2, v = 0 , the changing with time of excess pore p r e s sure at O. 5 m u n d e r t h e c e n t e r o f t h e l o a d e d a r e a is s h o w n

in the Fig. 8

. The

results

have

@ a=5 | H:2 |

0.5 0.4 -~ 0.3 0.2

against c t /

R 2 . M a n d e l - C r y e r effect is obvious for H = 0 .

M a n d e l - C r y e r effect is w e a k e n e d with t h i c k e n i n g of the i m p e d e d l a y e r .

been

R

H=O.1 r

0

0.1 0 0.00001

0.001

0.1

1 10

T=ct/R 2

(a)

(b)

Fig. 8 (a) Effect of the impeded layer thickness and time on excess pore pressure; ( b ) sketch map for the position of calculation point

Axisymmetric fundamental solutions for a finite layer with impeded boundaries

399

References

CONCLUSIONS E x p l i c i t s o l u t i o n s for L a p l a c e t r a n s f o r m s of d i s p l a c e m e n t s , t r a c t i o n s , pore p r e s s u r e ware d e r i v e d for a f i n i t e l a y e r with i m p e d e d b o u n d a r i e s a n d s u b j e c t e d to a x i s y m m e t r i c p o i n t l o a d s a n d ring loads. The Laplace and Hankel transforms u t i l i z e d i n this p a p e r are e f f i c i e n t . T h e s o l u t i o n s are e x t e n d e d to c i r c u l a r d i s t r i b u t e d a n d strip d i s tributed normal loadings. gent and satisfactory.

T h e r e s u l t is c o n v e r -

A s a r e s u l t of the c a l c u l a t i o n a n d a n a l y s i s , it is a p p a r e n t that the t h i c k n e s s of the i m p e d e d layer has an important influence on settlement, v e r t i c a l total stress a n d pore p r e s s u r e of t h e c o n s o l i d a t i o n l a y e r . T h e v e r t i c a l total stress is t i m e d e p e n d a n t during consolidation. The thicker the i m p e d e d l a y e r i s , t h e s m a l l e r the c h a n g i n g r a n g e of the v e r t i c a l total stress a n d t h e w e a k e r the Mandel-Cryer effect. The i m p e d e d layer delays the

settlement

layer.

progress

of

the

consolidation

Biot, M . A . , 1941. General theory of three-dimensional consolidation. J . Appl . Phys , 12,155. Gibson, R. E . , Schiffman, R . L . and Pu, S . L . , 1970. Plane strain and axially symmetric consolidation of a clay layer on a smooth impermeable Base. Q . J . Mech. Appl. Math, 23:505 - 519. Gu,Y. Z. and Jin, B . , 1992. Bolt consolidation analytical solutions for multi-layer base subject to axisymmetric loading. J . Geotechnical Engineering, 20 : 17 - 21. Huang, C.Z. and Xiao, Y . , 1996. Analytical solutions for two dimensional consolidation problems. J . Geotechnical Engineering, 18:47 - 54. McNamee, J. and Gibson, R. E . , 1960a. Plane strain and axially symmetric problems of the Consolidation of a Semi-infinity Clay Stratum. Q . J . Mech . Appl . Math, 13:210 - 227. McNamee, J. and Gibson, R. E. ~ 1960b. Displacement function and linear transforms applied to diffusion Through Porous Elastic Media. Q . J . Mech. Appl. M a t h . , 13: 8 9 - 111. Puswewala, U. G. A.and Rajapakse, R. K. N. D. ,1988. Axisymmetric fundamental solutions for a completely saturated porous elastic solid. Int. J . Engng. Sci,26 (5) :419 - 436. Schapery, R . A . , 1962. Approximate methods of transform inversion for viscoelastic stress analysis. Proc. 4th U. S.Nat. Cong.on Appl. Mech, p.1075. l i e , K . H . , 1996. One dimensional consolidation analysis of layered soils with impeded boundaries. J . Zhejiang University, 30( 5 ) : 567 - 575 ( in Chinese).

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