Int. Z Engng Sci. Vol. 33, No. II, pp. 1633-1662, 1995 ... · either t---~ 0 + or t--* +00 and ... influence of non-classical interface phenomena including friction ... where 8ij

Pergamon Int. Z Engng Sci. Vol. 33, No. II, pp. 1633-1662, 1995

Copyright © 1995 Elsevier Science Ltd 0020-7225(95)00031-3 Printed in Great Britain. All rights reserved

0020-7225/95 $9.5{) + 0.00

O N T H E M E C H A N I C S O F A R I G I D D I S C I N C L U S I O N E M B E D D E D IN A F L U I D S A T U R A T E D P O R O E L A S T I C

M E D I U M

ZHONG QI YUE Pavements Laboratory, Institute for Research in Construction, National Research Council of Canada,

Montreal Road (M20), Ottawa, Ontario, Canada K1A 0R6

A. P. S. SELVADURAI Department of Civil Engineering and Applied Mechanics, McGili University, Montreal, Quebec,

Canada H3A 2K6

Abstract--The paper utilizes the classical integral transform techniques to develop the systems of Fredholm inl:egral equations of the second kind, in the Laplace transform domain, governing the generalized displacement or loading of a rigid disc inclusion embedded in either permeable or impermeable bonded contact with a fluid saturated poroelastic infinite space. The generalized displacements correspond to an axial displacement, a rotation about the axial axis, a rotation about a diametral axis and an in-plane translation. The coupled integral equations in the Laplace transform domain are solved in a numerical fashion to generate results of technological interest. The numerical procedures focus on quadrature schemes for the solution of the integral equations and a procedure used for the inversion of Laplace transforms. The closed-form solutions of the integral equations as either t---~ 0 + or t--* +00 and as v--~ v, are also obtained. The numerical results are presented for the time-dependent displacements and rotations of the inclusion subjected to Heaviside-step function type loads and moments, and for the time-dependent relaxation of the force and moment resultants in an inclusion subjected to Heaviside-step function type displacements and rotations. In particular, the influence of the compressibility of the pore fluid on the time dependent responses of the inclusion is documented.

I N T R O D U C T I O N

The mechanics of inclusions embedded in deformable media has always occupied an important position in applied mechanics and in the theory of materials. The notion of an inclusion, three-dimensional or otherwise can be used to describe an inhomogeneity or defect which is encountered in composite and multiphase materials. Inclusions are introduced as a strengthen- ing mechanism for otherwise brittle materials which are susceptible to fracture and damage due to both mechanical and environmental loads. Classical studies in elasticity related to three-dimensional inclusions embedded in bonded contact with an elastic medium are given by Goodier [1], Dewey [2], Edwards [3], Robinson [4], Eshelby [5], and Lur'e [6]. Since these seminal works, :~tudies related to inclusion problems have been extended to cover: (i) the influence of non-classical interface phenomena including friction separation and slip, (ii) the influence of anisotropic materials, (iii) dynamic and wave propagation effects, (iv) interactions with nuclei of strain and (v) the influence of boundaries and neighbouring inclusions. Extensive reviews of the subject of inclusions embedded in elastic media are given in the review articles by Sternberg [711, Willis [8], Walpole [9], Mura [10] and in the texts by Gladwell [11] and Mura [12].

The disc inclusion is a particular limiting case of the general class of three-dimensional inclusions with an ellipsoidal or spheroidal shape. Despite its highly idealized configuration, the mechanics of disc inclusions has been the subject of a sustained research effort. In particular, the idealization of the inclusion, as a defect or inhomogeneity of infinitesimal thickness enables the formulation of such inclusion problems as appropriate mixed boundary value problems associated with a halfspace region. This aspect has been successfully exploited in the analysis of disc inclusions embedded elastic media. Collins [13] examined the problem of axial loading of a

1633

1634 Z . Q . YUE and A. P. S. SELVADURA1

rigid circular disc inclusion embedded in an elastic medium. The in-plane translation of a rigid circular disc inclusion embedded in an elastic infinite space was examined by Keer [14]. Since these early studies, the disc inclusion problem associated with an elastic medium has been extensively studied to include: (i) flexibility of the inclusion, (ii) delamination at the inclusion-elastic medium interface, (iii) interaction of the inclusion with cracks, (iv) interaction with external forces, (v) interaction with neighbouring inclusions, (vi) influence of anisotropic materials, (vii) dynamic loads and wave propagation effects and (viii) elliptical configuration of the disc inclusion. Accounts of these developments are given by Mura [10], Yue [15] and in a forthcoming article by Selvadurai [16].

Many of the current studies related to disc inclusion problems have been motivated by their potential application to studies in mechanics of multiphase media reinforced by low concentrations of such inclusions. In the area of geomechanics, the embedded disc inclusion also serves as a model for the examination of the compliance of ground anchors [17, 18] and for the study of in situ testing devices used to determine the properties of geomaterials [19]. When dealing with geomaterials in particular, the assumptions of elastic behavior places a restriction on the applicability of such results to media which exhibit time-dependent phenomena. Geomaterials can exhibit time-dependent constitutive responses as a result of consolidation and/or creep effects. The objective of the current investigation is to extend the studies related to embedded disc inclusions to include effects of poroelasticity. In particular, the generalized theory of poroelasticity proposed by Biot [20,21] is used to examine the time-dependent behavior of embedded rigid disc inclusions which are subjected to axial, rotational or in-plane translational displacements or the equivalent generalized forces and moments.

The classical theory of poroelasticity has provided the basis for the solutions of many problems of practical interest to mechanics of fluid saturated geomaterials. The consideration of an ideal fluid and elastic behavior of the deformable porous solid is regarded as a useful first approximation for the treatment of geomaterials in which the applied loads do not induce irreversible deformations in the porous solid. The extension of Biot's basic formulations to include such irreversible phenomena can be contemplated only as exercises in computational modelling [22, 23]. Even within the context of Biot's formulation, the majority of investigations have focused on the analysis of poroelastic media which are saturated with incompressible fluids. With poroelastic soils and rocks which are saturated with pore fluids which contain dissolved or distributed air voids or other gaseous phases, it becomes important to consider the influence of undrained compressibility of the medium. This aspect of undrained compressibility has been investigated by Rice and Cleary [24] who develop certain fundamental solutions for stress diffusion in isotropic porelastic media. Bishop and Hight [25] have also developed estimates for Skempton's pore pressure parameter B for instances where the poroelastic medium exhibits undrained compressibility effects.

The present study focuses on the application of Biot's theory for a poroelastic medium, saturated with a compressible pore fluid, to the examination of the mechanics of an embedded rigid circular disc inclusion. At first, the paper focuses for the development of the integral equations governing the inclusion problems. Fourier and Laplace transforms techniques are applied to generate the systems of Fredholm integral equations of the second-kind governing the basic displacment modes of the embedded inclusion which can exhibit impermeable or permeable interface conditions. Secondly, numerical solutions for the integral equations governing the three modes of displacement of the embedded rigid disc inclusion will be presented by adopting a numerical scheme based on the Laplace transform inversion. Finally, the time-dependent results for the translational and rotational stiffness and compliance of the embedded rigid inclusion are evaluated for various choices of the governing constitutive parameters of the porelastic medium.

Mechanics of a rigid disc inclusion 1635

G O V E R N I N G E Q U A T I O N S

In the ensuing we shall present a brief account of the governing equations referred to a Cartesian tensor notation. The constitutive equations governing the quasi-static response of a poroelastic medium, which consists of an isotropic poroelastic soil skeleton saturated with a compressible pore fluid take the forms

2itV2v l~kk~ij + 2 i te ; / - 3(v,, - v) o'ij 1 - B(1 - 2v)(1 + v,,) pSij

2itB2(1 - 2v)(1 + v.) 2 2itB(1 + v.) P = 9--~,,S_~,)~_--~v ~ cv 3 ( 1 - 2 % , ) ekk ( la)

where 8ij is the Kronecker delta; p is the pore fluid pressure; ¢o is the volumetric strain in the pore fluid; tr u is the total stress tensor; e u are the soil skeleton strains defined by

1 eij = "~ (Ui,j -F Uj.i) ( lb)

u~ are the corresponding displacement components and the comma denotes a partial derivative with respect to a spatial variable (i, j = x, y or z). In the absence of body forces, the quasi-static equations of equilibrium take the forms

O'ij,j : O. (Ic)

The equations governing quasi-static fluid flow are defined by Darcy's law which takes the form

vi = - Kp,i ( ld)

where vi is the specific discharge vector in the pore fluid. The continuity equation associated with quasi-static fluid flow is

0¢v - - + vi.i = 0. ( le) 0t

The governing equations are characterized by the five independent material parameters which are the following: the drained and undrained Poisson's ratios v and v, respectively, the shear modulus It, Skempton's pore pressure coefficient B, and the K (= k/7~, where k is the coefficient of permeability and 7w is the unit weight of pore fluid). Considering requirements for a positive definite strain energy potential, it can be shown that the material parameters have the following thermodynamic constraints: It > 0; 0-< B -< 1; - 1 < < v,, <-0.5; K > 0 [24].

Applying the theory of Fourier integral transforms [26], it can be shown that the following sets of solution :representations exist for the field variables in a linear, isotropic, poroelastic halfspace saturated with a compressible pore fluid. Considering the cylindrical polar coordinate systems (r, 0, z) and (p, q~, z), we have, in either the temporal domain or the Laplace transform domain,

l f f f f = u(r, 0, z, t) ---- ~ II~wK dq~ dp,

lfi f;" Tz(r, 0, z, t) := ~ I I~YzK p d~ dp,

v(r, 0, z, t) := ~ II~gKp 2 dq~ dp,

fi°f2" p(r, O, z, t) := p~Kp d~ dp,

w(p, ~, z, t) - ~ Jo Jo II*~uK*r dO dr

Yz(P, ~, z, t) - ~ 30 Jo [I*~TzK*r dO dr

O(p, ~, z, t) = ~ II*~vK*r dO dr

1 f~ f2= P ~ ( P ' ¢ ' Z ' t ) = ~ J o Jo pK*rdOdr (2a)

1636 Z.Q. YUE and A. P. S. SELVADURAI

where 0 + --- z < % or -00 < z -< 0- , 0 < t < 0% and the integrals are interpreted in the sense of a Cauchy principal value. The vectors in (2a) are defined by u = (Ur, UO, U~) "r, T , = (o',~, o'o~, 0"~) "r, V = (Or, 110, VZ) T, W = (Wi, W2, W3) T, Yz = (Zl, 'l"2, z3) T, and 0 = (01, ~"~2, 193) T" The superscript T stands for the transpose of matrix. K* and II* are, respectively, the complex conjugates of the Fourier matrix kernel functions K and Ha defined by

[ i s i n ( 0 + ~ p ) icos(O+~p) 0 ~ K = e iprsin(°+*), I I a = i i c o s ( O + ~ p ) . - i s in(O+ ~p) ~ ) . (2b)

\ o 0

By assuming the initial condition ~vl,=o = 0, the governing equations can be re-writen as two sets of first-order ordinary differential equations in the Fourier and Laplace transform domains; i.e.

dz Vo(p, ~o, z, s) = pCvVo(p, tp, z, s), V,(p, 9, z, s) = pC,,V,,(p, ~p, z, s ) (3a)

where

and

( 1 4. Vu = Wl, 1~3, ~ ¢3, ¢1' fiw, 2~Ka d ,

0 - 1

v 0

1 - v

C,, = 0 0

1 0

1 - - v

0 0

3,2 _ 1 0

2/zB2(1 - v)(1 + v,)2K v,, - v

c = 9 ( v , - v ) ( 1 - v , ) ' v k - ( l _ v ) ( l _ v , , ) ,

0 2 0

1 - 2v 0 vk

1 - v

0 1 0

--V 0 vk

1 - v

0 0 0

1 - 3, 2 0 _ 3,2

B(1 + v,,) ad -- 3(1 -- %) '

0

0

0

0

- 1

0

1 s + 3, = ~/c~- ~ 1

(3b)

and the superscript ^ stands for the Laplace transform with respect to t, and s is the Laplace transform parameter.

Solution representations governing the field variables in the halfspace regions (0+~ < z < ~) and ( - o ¢ < z ~<0-) can be further obtained by solving the ordinary differential equations together with the four regularity conditions required as z---> +oo and z-->-o% respectively. It can be shown that there are four independent boundary equations which govern the eight variables at the boundary surface of the halfspace region: (i) For the halfspace region (0 + --- z < oo), we have

( 1 1 1 1 0 i / 1 1 (1 1)V~(O +) = 0; 1 - ~ - 1 -~ - 1 V.(O +) = (4a)

1 1 3 -4v , , 3 - 4 % , - a ~ a ~ )


(ii) For the halfspace region (-o0 < z -< 0-), we have (_1_11 (1 -1)Vv(O-) =0 ; - 1 -__11 1 1 1

Y Y

-1 1 4v , , - 3 3 -4v , , a~ a~

where V~(O ±) = Vv(p, ~o, 0 ±, s); V.(O ±) = V.(p, ~, 0 ±, s); and

4(v. - v) a . = ( 1 _ v) (y z - 1)"

V,,(O-) = (4b)

(4c)

THE DISC I N C L U S I O N P R O B L E M S

We consider the generalized displacements of a disc inclusion in a poroelastic medium saturated with a compressible pore fluid (Fig. 1). The generalized displacements induced at the inclusion-poroelastie medium interface in the r, 0 and z directions are denoted by, U(r, O, t)= (Ur, Uo, Uz) T. The associated force and moment resultants acting on the plane inclusion are denoted by P( t )=(Px , Py, Pz) T and M ( t ) = (Mx, My, Mz) T, respectively. The interface between the saturated poroelastic medium and the inclusion is assumed to be either impermeable or permeable. By considering the symmetry or asymmetry of the embedded disc inclusion problems it can be shown that all of these generalized loading problems can be formulated in relation to mixed boundary value problems associated with the halfspace regions occupying z E [0 ÷, +oo) and z ~ (-oo, 0-]. The notation 0 ÷ and 0- are intended to designate the plane surfaces of the halfspace regions corresponding to z t> 0 and z-< 0 respectively. For convenience we introduce two regions .4 and ,4 which correspond to the regions on z = 0 occupied respectively by the inclusion and its exterior. It may be noted that fi~ t2 A = A where A is the entire plane z = 0 containing the plane inclusion and A N .4 = {0}, implying that there is no separation, interpenetration or cracking induced by the loading of the embedded inclusion for any time t e (0, +~). Considering the generalized displacements we have,

u(r, 0, 0 ÷, t) = u(r, 0, 0-, t) = U(r, 0, t), for r, 0 E .4. (5a)

The pore pressure boundary conditions at the interface are either

p(r,O,O+,t)=p(r,O,O-,t)=O, for r, OcTt (5b) or

Vz(r, O, 0 +, t) = vz(r, 0, 0-, t) = 0, for r, 0 e .4 (5c)

Fig. 1. A di:~c inclusion embedded in bonded contact with a poroelastic medium saturated with a compressible pore fluid.

1638 z.Q. YUE and A. P. S. SELVADURAI

which correspond to permeable and impermeable interfaces respectively. The definition of the boundary conditions can be made complete with the specification of continuity conditions associated with the region ,4. We have, in general,

u(r, 0, 0 ÷, t) = u(r, 0, 0-, t),

p(r, O, 0 ÷, t) = p(r, O, 0- , t),

Tz(r, O, 0 +, t) = Tz(r, O, 0- , t), for r, 0 E ,4

Vz(r, O, 0 +, t) = v~(r, O, 0- , t), for r, 0 E ,4. (Sd)

The force and moment resultants acting on the inclusion can be expressed in terms of the contact at the interfaces as follows:

P(t) = - f y IIb[Tz(r, O, 0 +, t) - Tz(r, O, 0- , t)]r dO dr A

M(t) = - f y lI~[T~(r, O, 0 +, t) - Tz(r, O, 0 - , t)]r 2 dO dr A

(5e)

where

cos0 - s i n 0 ! )

I I b = ~ S i o 0 COS/90 ' (i ° II~= 0 1

(5f)

2-D INTEGRAL EQUATIONS GOVERNING A PLANE INCLUSION

Considering the continuity conditions for variables u and p at the plane containing the permeable inclusion (z = 0, 0 -< r < +oo, 0 -< 0 < 2g), we have fi(r, 0, 0 ±, s) = fi(r, 0, 0, s) and p(r, O, O±, s) = fi(r, O, O, s). The corresponding conditions in the transform domain are @(p, ~p, 0 ±, s) = @(p, ~p, 0, s)( = @(0)) and t~w(p, q~, 0 ±, s) =fiw(p, ~, O, s)( = flw(0)). Using equations (2) and the boundary conditions (5a, 5b, 5d), we obtain the following system of 2-D Fourier transform based integral equations in the Laplace transform domain:

2--~ Jo J0 IIaff(0)K d~p dp = 0(r, 0, s),

2zr pw(O)Kp d~ dp = O,

fo~/"2= ,, + 2.1 Jo IL[¥~(O ) - ~'~(0-)]Kp d,p dp = 0,

1 f f fo2" 2~ [~3(o +) - ~ ( O - ) ] K p 2 d¢ dp = O,

r, O c fii

r, O e fii

r, O e A

r, 0 e.4. (6)

Using equations (4), the variables Yz(0±)( = ~'z(P, ~P, 0±, s)) and ~3(0±)( = O3(P, ~P, 0 ±, s)) can be expressed in terms of @(0) and pw(0): i.e.

(i~3(0 ±) \ [ b,,('y) ~bl2('Y) b13('y) 0 \ / ~ , ( 0 ) \ ~,(°~) l = I='=b~'(~') b~(~,) =,=b~(~,) 0 |{,~(o) l

¢~(o ~-) / o ~ : ~ / \ ,~2(o) /

(7)


where the coefficients bo(y ) (i, j = 1, 2, 3) are given in Appendix A. Substituting the expressions (7) for ¢1'~(0 ±) and 03(0±), into the equations (6), we decouple

the system of 2-D integral equations (6) as two systems in terms of ~3(0) and @p(0) (= [~(0), ~2(0), fiw(0)]T), respectively. The first system is

&3(0)K d~p dp = Uz(r, O, s); r, O e A

bt2(Y)~3(O)Kp dq~ dp= 0; r, 0 e A (8a)

and the second system is

lrf - -~z Jo -to IIa%(O)K dq~ dp = Oh(r, O, S); r, O e A

2--~ IIdBa@p(O)Kp d~o dp = O; r, 0 • fi, (8b)

where Oh = (Or, ~ro, 0), and

IId = / ' i COS(~ + 0) - i sin(~ + 0) 0 , Ba = ~ •

0 p \b3,(y) 0 b33(Y)/

(8c)

Similarly, considering the continuity conditions for variables u and vz at the plane containing the impermeable inclusion ( z = 0 , 0 < - r < + ~ , 0 - < 0 < 2 z ) , we have @(p,q~,0 ± , s )= @(p, ~p, 0, s)(= ~(0)) and 03(p, 9, 0 ±, s) = 03(p, q~, 0, s)(= 03(0)). Using equations (2) and the boundary conditions (5a, 5c, 5d), we obtain the following system of 2-D integral equations in the Laplace transtorm domain:

l r l TM

~z Jo Jo IIa@(0)K d~p dp= 0(r, 0, s), r, O e fit

2z 03(O)Kp z dq~ dp = O, r, 0 E fit

1 r~ f2~r ^

2-'-z J0 Jo lIa[Yz(0+)-'~'z(0-)IKp d~pdp=0, r, OeTt

~z [/gw(O +) - /~(O-)]Kp d~o dp= O, r, 0 e A. (9)

The variables ~'z(0+)(=Viz(p, ~p, 0±,s)) and pw(o±)(=pw(p, ¢,O±,s)) can be expressed in terms of ~(0) and 03(0) by solving the equations (4), i.e.

¢,(o ±) = / ~c~,(r) ~ ( T ) c~(r) ~ ( o ) | ;:c32(3') ~:c33(3') /

~, t~(o ±) o o ~:~ ~2(o) /

(lO)

where the coefficients c~ (i, j = 1, 2, 3) are given in Appendix A. Substituting the: expressions (10) of ¢/~(0 ±) and pw(0 ±) into equations (9), we decouple the


system of 2-D integral equations as two systems in terms of @~(0)(= [~3(0), ~3(0)] w) and @~(0)(= [,3~(0), ,32(0)]T), respectively. The first system is

Jo J0 A,f~(0)K d~p do = U,(r, 0, s); r, 0 E .4

1 f~ f2,'r ^ ~ J o Jo Cawv(O)Kp dcp dp = O; r, O E fit (11a)

and the second system is

f,;f? 2-~ neCn(O)Kp d¢ do = Ub(r, O, S); r, 0 E ,4

2---Z IIeCb@h(O)K O d~o do = 0; r, 0 E ,4 ( l lb)

where Ida = (0z, 0) T, ~Jb = (Or, Uo) T, and

( i s i n 0 p + 0 ) i c o s ( ¢ + 0 ) ] Aa = ( 1 0 IIe=

\icos(cp + 0) --i sin(q~ + 0)]' \0 p2]

0) Ca = \pC32(,y ) pC33(T)], Cb= \ 0 /Z " ( l lc)

The 2-D integral equations governing the plane inclusion problems are now decoupled into two systems associated with the vertical and horizontal displacements of the inclusions, respectively. In the Laplace transform domain, the non-zero force and moments acting on the permeable or impermeable plane inclusion associated with the vertical displacements can be expressed as follows: (,,s,,

~ x ( s ) | = - 2 j . ~ rsin G~(r. O, O, s)r dO dr (12a) ](/y(s) / \ r cos

where the normal stress is given by

G~(r, O, O, s) - ~ Jo Jo ¢3(P, ¢, 0, s)Kp d,p dp. (12b)

The unknown variable '~3(P, ~, 0, S) is obtained by the solution of the following systems of 2-D integral equations, i.e. (i) For a permeable plane inclusion, we have

TZJo Jo a~(y)~3(P'~°'O's)gd~°dO=Oz(r'O's)' r, OeTt

1 ~ 2'~ ~ ~o ~o ~3(P, ~, 0, s)gp dq~ do = 0, r, 0 E A. (13a)

(ii) For an impermeable plane inclusion, have

Jo Jo Aa¢~a(P' ~o, O, s)K d~ dp = [Ja(r, 0, s ) , r, 0 E ffi

1 (~ f2;,r^ Jo Jo Ya(P, ~o, O, s)Kp d~o dp = O, r, 0 e ffi (13b) 2---z

where ¢/o(p, ~, 0, s) = [~3 (0 ) , f l w ( 0 ) ] T, and

( a 1 ( 7 ) a2(7) ~ (13c) A,, = \pEa3(7) p2a4(7)].


The coefficients ai(y ) (j = 1, 2, 3, 4) are defined in Appendix A. In the Laplace transform domain, the non-zero forces and moment acting on either the

permeable or the impermeable plane inclusion associated with the horizontal displacement can be expressed as follows:

,( Px(s) ) !f ff coso -sinO\ 0 cos (14a) "si- ' 0 ^ |[O'rz(r, 0,0,

~l~lz(s ) r ] \tr°z(r'O'O'

where the contact shear stresses are governed by,

",z(r,O,O,s)~ 1 ~ ~'r aoz(r, O, O, s ) /= ~ IleClb(p, ~o, O, s)Kp d~0 dp. (14b)

The unknown variables ~'b(P, 9, 0, S)(= [t~(0), t2(0)] "r) are governed by the following systems of 2-D integral equations, i.e. (i) For a permeable plane inclusion, we have

IL f= f2,~ ^ 2~'Jo Jo HdBhYh(P'~°'O'sld~°dP=[Jh(r'O's)' r, OEfil

1 f= f2~ ^ ~ J o Jo HdYh(P' ~' O' s)p d~ dp= O' r, O e fl. (15a)

(ii) For an impermeable plane inclusion, 1 f= f2tr ^

~rJo Jo HeBbVb(p'~°'O's)Kd~°do=Ob(r'O's)' r , O ~ a

m fo~ "C '~ i).-~ ~ Ile¢ib(p, ~o, O, s)Kp d~ do = O, r, 0 • fi, (15b)

where ~'h(p, ~0, 0, s) = (f,(0), f2(0), 1~3(0)) T, and

, Oo ba(y) 0 b4(~t) J

n b =

The coefficients bj(y) (j = 1, - , 6) are defined in Appendix A.

(15c)

I N T E G R A L E Q U A T I O N S G O V E R N I N G A R I G I D D I S C I N C L U S I O N

We consider tile particular case of the displacements of a rigid disc inclusion which is embedded in bonded contact with a poroelastic medium of infinite extent (Fig. 2). The

Fig. 2. The fundamental loading and displacement modes of a rigid circular disc inclusion in a saturated poroelastic medium.

1642 Z . Q . YUE and A. P. S. SELVADURAI

generalized displacement vector for the inclusion, in the Laplace transform domain, has components

f_l~(r, O, s) = IDz(S) + ~y(s)r cos 0, /Jr(r, 0, s) =/)x(s)cos 0, f-]o(r, 0, s) = -/)~(s)sin 0+ r~z(S ) (16)

where 0 -< r -< a, 0 -< 0 < 2Jr, a is the radius of rigid disc,/5~(s) and IDz(s) are, respectively, the horizontal and vertical translations at the centre of rigid disc inclusion, and ~y(s) and ~z(s) are, respectively, the central and axial rotations of the rigid disc inclusion about the y and z axes.

Substituting the vertical displacements into the systems of 2-D integral equations (13), we can find, with the help of Fourier series expansions, the following solutions for the unknown variables.

~'3(P, qO, 0, S) = ~3O(P, S) -- 2i sin ~0~31(P , S)

p,,(p, ~o, O, s) = Pwo(p, s) - 2i sin ¢p,,,(p, s). (17a)

The force and the moments acting on the inclusion can be expressed in the following simplified form:

Pz(s) = - 4 x a ~?3o(P, s)Jl(pa) dp; fitx(s) = o; fitr(S) = -47ra2 t3,(P, S)Jz(pa) do

(17b)

and the normal stress at the interface plane of the inclusion are,

foo f0 6"zz(r, O, O, s) = ~3o(P, S)Jo(pr)p do + 2 cos 0 ~31(P, s)J,(or)p do. (17c)

The unknown variables ~3j(O, s) and P,4(O, s) (j = O, 1) in equations (17) are governed by the following sets of integral equations. (i) For the permeable inclusion problem, we have

o°al(y)~3j(p,s)Jj(pr)dp = Dj(r,s), O<-r<-a

fo°~'3j(p, s)Jj(pr)p do a r (18a) O, < < O0

r ^

where j = 0, 1; Do(r, s ) = Dz(s); h i ( r , s ) = ~f~y(s); and Jj(pr) is the Bessel function of order j.

(ii) For the impermeable inclusion problem, we have

[Ab¢6(O, s)Jj(or) dp s), r a 0 <

f, °CZj(O,s~(or)p do a r (18b) O, < <

wherej = 0, 1; ~rj(O, s) = [f3j(O, s), Pwj(O, s) l ' ; f~o(r, s) = [bz(s), 01T; and I), (r, s) = fir(S), 0

Substituting the boundary condition for the horizontal displacements given in equations (16) into the systems of 2-D integral equations (15), w e can use the Fourier series expansion technique to find the following solutions for the unknown variables.

f,(O, ~o, 0, s) = -i[l~l(o, s) + fit(O, s)]sin ~o

t ~(t,, ~, o, s) = ~ ~,,(o, s) + [t?(o, s) - fit(o, ~)]cos ~o

~3(P, ~o, 0, s) = -2i~3~(p, s)sin ~o. (19a)

Mechanics of a rigid disc inclusion

The forces and moment acting on simplified forms:

Px(s) = -4~ta l~l(p, s)J,(pa) do, Py(s) = O, J

the rigid inclusion can be expressed

1643

in the following

f hT/z(s) = -4~ra2 ¢2o(P, s)J,(pa) dp.

(19b)

Similarly, the associated shear stresses at the interface plane of the inclusion are given by

d~z ( , , O, O, s )= cos o [ ft~ l~l (p, S )Jo(pr )p dp - ft~'~l (P' s )J2(pr )p dp ]

[fo o f ] 6"ez(r, 0, 0, s) = - s in 0 IV(p, s)Jo(pr)p dp + .~l (p, s)J2(pr)p dp

f + e2o(p, s)J,(pr)p do. (19c)

The unknown varJiables/V(p, s) and ,(/(p, s) associated with the in-plane translation of the rigid inclusion are governed by the following sets of integral equations:

where (i) for the permeable [/)~(s), 0, 0] T, and

fo~IIoBoYd(p, S) d p = I~o(S),

fo~IIoYd(p, s)p dp = O,

inclusion,

O<_r<_a

a < r < ~ (20a)

i ' , (p,s)= T, n o :

/ bs(T) b6(~)

Bo = I b6(T) bs(T) l\~b3(~) l b b4(~/) ] pJl(pr)

(20b)

and (ii) for the impermeable inclusion, ¢~d(P, S) = [~/(p, S), /f/(p, S)] T, Ho = [bx(s), 0] T, and

= (b5('y) b6(T)~, Bo \b6(3') bs(')')]

(Jo(pr) 0 \ Ill, = / (20c)

k 0 J2(pr)'/

The unknown variable ¢2o(p, s) associated with the axial rotation of a rigid disc inclusion with both permeable and impermeable interfaces is governed by the following dual integral equations:

ft~ ~20(P, s)J l(pr) dp r a 0

fo~2o(p, s)Jl(pr)p dp = 0, a < r < oo. (21)

From the above results, it is evident that the integral equations governing the four basic displacement (loacling) modes of the rigid disc inclusion are decoupled. Consequently, there is only one loading mode Pz(t) (My(t), Px(t), or Mz(t)) corresponding to the basic displacement mode D~(t) (fly(t), Dx(t) or flz(t)). In particular, it is found that the dual integral equations (21) for the axial rotation of the rigid inclusion can be solved in exact closed-from. The

1644 Z . Q . YUE and A. P. S. SELVADURAI

relationships among the axial moment, the associated rotation and the contact shear stress are given by

Mz(t ) = 32/za3 f~z(t), troz(r, t) -4tzr 3 - r r ~ f~z(t)" (22)

This solution is identical to that obtained for the analogous problems in classical elasticity [27].

FREDHOLM INTEGRAL EQUATIONS GOVERNING THE RIGID DISC INCLUSION PROBLEMS

The sets of integral equations (18) and (20) are singular integral equations with unknown singularities. Such singularities require detailed consideration in the numerical evaluation of the sets of ~integral equations. For example, Small and Booker [28] have numerically evaluated this type of singular integral equations for the axisymmetric translation of a rigid disc inclusion by assuming the unknown variables have the same order singularity as that in the elastostatic problem. Further simplifications and reductions have to be made to eliminate and separate the unknown singularities from the integral equations. It is found that all the sets of integral equations can be reduced into the systems of Fredholm integral equations of the second kind. Based on the procedure given by Sneddon [26], we define the following solution representations for the unknown variables

fo ~3o(p, s) = ~o(X, s)cos(ox) dx,

t31(O, S) = ~,(X, s )s in(px) dx,

R(o, s) = ~(x, s)cos(ox) dx,

P~o(P, s) = P So(X, 0)sin(px) dx

. Bf ° p~,(p, s) = ~/2 V~p Jo ~,(x, s)J3;2(px) dx

~/ 2 )o ~(x, s)J3/z(px) dx

l~31(P , S) = KB (O(X, s ) s in(px) dx (23) P

where t~o(0, s) = 0 and limx_,0 ~ ~j(x, s) = 0. It is worth noting the following integral relationships play a key role in the derivations

[26, 29-31]:

F(b)21-h phr a ,,fi" (r Exa-b+ l J~(pr ) - -S-~_b J~_b(pX ) dx (24a)

where Re(b) >0, Re(a - b) > -1.

fo c 2 b + 1 -a xbr-a pb-a+lj~(pr)Jb(px) dp r(a - b) (r 2 - x2) b+~-a H(r - x) (24b)

where R e ( a ) > R e ( b ) > - I with r # x or Re(a )>Re(b + 1 ) > 0 with r=x, and H(x) is a Heaviside step function.

~/~ f~ f(x) ~/~ d fti~ yg(y) g(y) = ~ dx, f(x) = ~ ~ dy. (24c)

This is a pair of integral equations of the Abel type. The sets of integral equations (18) and (20) can now be reduced to the integral equations of

the Abel type. The Fredholm integral equations of the second kind in the Laplace transform domain for the inclusion problems can be obtained from the integral equations of the Abel type.


(a) Fredholm integral equations governing the axial translation of the rigid disc inclusion

Introducing the non-dimensional variables

P~(3 - 4v) ~,., . r p~() p ~ ( ) X = - $ ~ - $ I a

a2s -P~ ~ q,a(X, s,) s,=--,c o(r,s) ^

1645

(25)

(b) Fredholm integral equations governing the central rotation of the rigid disc inclusion

Introducing the following non-dimensional variables

~y(s) My(3 - 4v) Xy(x, s~) , /l~/y(s) = My~'y(s~) - 64a3tz(1 ---v)

~ , ( r , s ) = ~ b ( x , s , ) , ~,(r,s) =-M>'~xsl 87raSh ~n(x, s,) (27)

we can then obtain the non-dimensional forms of the integral equations as follows: (i) for a permeable inclusion, we have

6b(X. S,) + K,(X, y. s,)6b(Y, s,) dy = xff.~(s,) (28a)

and (ii) for an intpermeable inclusion, we have

(~b(x,s,)] ~'[R,(x,y, st) Rz(x,y,s,)](~n(y, sl)]dy f(ro(S,) qJbtX, S,)/ + \R3(x.y, st) g,(x,y,s,)/\~bh(y,s,)/ =(X ) (28b)

t$ $3-11-1t

(26e) fo ~a(Y' Sl ) d y = ~ ( S I ) .


~a(X, Sl) "4" •l(X, y. S,)Zb.(y, S,) dy = X~(s,) (26a)

and (ii) for an impermeable inclusion, we have

, : , (x ,y ,s , ) f + dy (26b) ~b.tx, s,)/ Jo \R3(x, y. s,) R4(x, y. s,)/\•.(y, Sl)/ \ 0 /

where 0 - - -x - 1; and the non-dimensional kernel functions are

[¢,(x, y, s,) = -a21,(x, y, Sl) , g 2 ( x , y, s,) = -a3V~sl [Iz(x, y, s,) - n(y - x)]

R . ( x , y, sO = 13(x, y, s .) , R3(x, y, s . ) = -a.V-S~S~ [12(y, x, s . ) - H ( x - Y)I (26c)

where

_2sl f~cos(px)cos(py) d t2(x, y , s , ) 2s~ f~ c___.os(__px)sin(py) I,(x, y, s,) - rc -Io (P + VP-Y-~I) 2 p' = - ~ Jo p(p + p2x/~-~s~)2 dp

_ 2s, f~ sin(__px)sin(py) d (260) 13(x. y , s , ) - tc Jo P(P + V~+--~I) p"

For both permeable and impermeable cases, we have the following non-dimensional expression for the applied axial load:

1646 Z. Q. YUE and A. P. S. SELVADURAI

where 0 5 x 5 1; and the non-dimensional kernel functions are

&(x, y, s1) = -4(x, Y, SJ, .R*(x,Y,sl)=-ff,~[z*(x,Y,sl)-~H(y -41

kl(X, Y, Sl) = 4(x, Y9 Sl)> g,(~,,,,,)=-,,~[IZ(Y,*,s,)-~~(x-Y)] (2&)

where

Z,(x, y, s,) = 3 I a sin(px

?r 0 [p + m]2dp’

sintw 1

z2(x,Y,J1)=? I

m sin(w) [ ~ - COSCPY)

PY 1 0 p[p +A@T-Q dp

sin(px) Ww 1 &(x, y, s,) = 2

I

co [ ~ - cos(px)

PX I[ - - MPY) PY 1 dp. (284

0 P[P+G=il

For both permeable and impermeable cases, we have the following non-dimensional expression for the applied central moment:

I

1

&(Y, SJY dy = I;. VW 0

(c) Fredholm integral equations governing the in-plane translation of the rigid disc inclusion

Introducing the following non-dimensional variables:

B,(s) = P’(7 - 8~)

64ap(l- Y) K(% Sl), kx(s> = P,~x,(Sl)

44, s> = 2 AAx, sd, $(r, s) = 2 8(r, s) = 2 4% sd (29)


K,(x, Y, s1) K2(X, Yt sd K,(-% Y, Sl)

&(X, y, s1) &(X, Y? Sl) &(x7 Y, Sl)

&(x, Y, SJ &(4 Y9 Sl) &(x7 Y, Sk)

) X (k$?) dy = ($%I (304

and (ii) for an impermeable inclusion, we have

(,‘, (job)

where 0 5 x < 1; and the non-dimensional kernel functions are

K,(x, y, s*) = -41(x, Y9 s,), &(X, y, Sl) = G-6 Y, 3,)

K2(X, y, Sl) = 4&(x, Y, Sl) - 12(x, Y, SJI - YH(Y -XI

K,(x, y, Sl) = 44(x9 Y9 Sl) + 6 WY -x)1

K&y, Sl) = %M& Y9 4 - I2(Y,& sdl -yw -Y)

where


Ks(x, y, s,) = -a6[l,(x, y, s,) - Iz(x, y, s,) - 12(y, x, s,) + 13(x, y, sl)]

K6(x, y, sl)=a,[x-~sl ls(x, y , s , ) - 214(x, y , s , ) - X/'~s~ H ( y - x ) ]

g7(x, y, Sl) = - -a4[14(y , X, $1) + H(x - y)]

K8(x, y ,s , )= -a4 I5(x, y, s l ) - I4(y,x, s l ) - - - -~ -H(x- y)

1647

(30c)

ll(X, y, sl) 2sl

12(x, y, Sl) 2s~ /[

~ [fl (P, $1 ) "t- A(P' S I )]COS(OX)COS(OY) do

f: ; [fl(O, sl) + f2(o, s,)lcos(px)sin(py) do

13(x, y, sl) 2sl ~pl-~xy = ~ [fl(P, sl) + f2(O, sl)]sin(ox)sin(py) do

14(x, y, sl) = T f3(P, s.)cos(px)sin(py ) do,

fo Is(x, y, $1 ) =--2Sl fl(P, sl)sin(px)sin(py) do (30d)

1 p 2 - sl - pX/p2 + Sl f,(o,s,) o2+sl+o~+s, A(o, sl) o2+sl+o,c,o~+s,

P f2(P, $1) = 2p(p 2 + sl) + (2p 2 + sl)~/p 2 + sl"

Again for both permeable and impermeable cases, we have the following non-dimensional expression for the applied in-plane load:

f01 s,) dy = (30e) q~c(y, L(s,).

The material constants a t (] = 1 ~ 8) in the above equations are defined in Appendix A. For completeness, we shall record here the stresses acting on (0 -< x < 1, 0 - 0 < 2~); i.e.

~rzz(X, O, O, t ) - -Pz F ~b.(1,/) "- 4xa 2 L lx/"~-Z~- x z

crx~(x, O, O, t) - -Px [ ~bc(1, t) - 47ra 2 L 1vT-z-~- x 2

a single face of the inclusion

' 1 d f' dpb(y,t) dy ~ Odp.(y,t) dy ] My ay 4/r-a3dx Jx ~ cos O

f~ lOdA'(y't----~)Oy ~dyl +xd ' Oc(y,t) . cos20] f - - - - y y2X/-~Z-~_x2Oy

[ d f f ~c(y,t) ] cryx(x,O,O,t) --Px-4zca 2 x -~ y y2X/-~-~_x2dY sin20. (31)

L I M I T I N G R E S P O N S E S

It is instructive to record certain limiting results obtained via the generalized Fredholm integral equations which govern the mechanics of a bonded rigid circular disc inclusion in a poroelastic medium, with either a permeable or an impermeable surface. The limiting cases pertain to the initial and final responses of the poroelastic medium as t - * 0 + and t---~ +o0


respectively, as well as the response of the poroelastic medium as v ~ v,. The limiting response of the rigid inclusion as t---,0 ÷ and t ~ +o0 can be obtained in exact closed-form from the Fredholm integral equations by using the Tauberian theorems for Laplace transforms. For both permeable and impermeable rigid inclusions, we have

~bo(x, T~)= 8/~(1 - Z ) D tT.~ ck~(x, T a ) - 8p.(1 -Z)f~r(T~)x ' zr(3 - 4Z) z, o,, 7r(3 - 4X)

6(x, T~) - 16/~(1 - Z) D iT. ~ @(x, T~) = 0 (32a) r(7- 8z) x, o,,

and the corresponding stiffnesses of the rigid disc inclusions are given by

- 64/z(1 - z )a . . . . pz(T~) 32/z(1 X)a oz (T , ) ' My (Ta )_641z a3 (1 -Z )~y (Ta ) , Px(T~)- - 3 - 42' 3(3 - 4Z) 7 Z 8"Z uxt~")

(32b)

where (i) for t ~ 0 ÷, we have T~ = 0 + and X = v,; (ii) for t---~ +0% we have T~ = +oo and X = v; (iii) for v ~ v, , we have T~ = t and X = v = v,.

These closed-form results are in agreement with elastostatic solutions for a rigid disc inclusion embedded in an elastic medium [13, 14, 18, 32-34].

N U M E R I C A L SOLUTION OF THE I N T E G R A L EQUATIONS

An examination of the structure of the kernel functions occurring in these systems of Fredholm integral equations indicates that they are not amenable to exact solutions [35-38]. Consequently, it becomes necessary to adopt accurate and efficient techniques for the numerical solution of these integral equations. In this paper, we adopt the following numerical scheme for the evaluation of time-dependent solutions of the integral equations in the Laplace transform domain. First, we separate the complex variables in the integral equations into their real and imaginary parts. We then obtain new systems of Fredholm integral equations of the second kind associated with real variables. Secondly, we divide the integration interval [0, 1] into N segments with ends defined by rk = (k - 1)/N; k = 1, 2, 3 . . . . . (N + 1). The collocation points are Xk = (rk + rk+])/2, k = 1, 2, 3 , . . . , N. Consequently we can convert the integral equations into the systems of linear algebraic equations. These linear algebraic equations can be written in the generalized matrix form

L

[A/j]{Xj} = {B,} (33) j=]

where l = 1, 2, 3 . . . . . L, L = 2N + 2 for equations (26a) and (27a); L = 4N + 2 for equations (26b), (27b), and (30b); and L = 6N + 2 for equations (30a).

Equation (33) can be solved numerically to generate the unknown variables .~'j(st) in the Laplace transform domain. The time-dependent results of the variables Xj(ct /a 2) can be evaluated by using numerical inversion of Laplace transforms.

The numerical techniques adopted here involve three computational steps which are essentially based on repeated numerical integrations. The first step involves the numerical integration of the infinite integrals, which occur in the kernel functions. It can be shown that these infinite integrals are absolutely convergent. However, there are two unusual aspects in the numerical integrations. One refers to the infinite limit and the other is associated with the integrands. As the values of the complex variable s] become large, the integrands will become rapidly oscillatory functions. This property of the integrands will show the convergenee of the numerical integration and render the numerical integration procedure unstable. In the


numerical inversion of Laplace transforms, large values of the complex variable s~ are associated with small values of the time factor ct/a 2. The technique adopted in this study overcomes these two numerical problems. The details of this technique are documented in Appendix B. This particular technique consists of an approximation technique and a proceeding limit technique. In the approximation technique, the integrands are separated into two parts. The first relates to their asymptotic behavior as st/p2---> O. The second relates to the difference between the integrands and their asymptotic functions. Closed-form results can be obtained for the infinite integrals associated with the asymptotic functions. The proceeding limit technique, which is based on an adaptively iterative Simpson's quadrature, is used for the evaluation of the infinite integrals associated with the remaining terms.

The second repeated numerical integration involves systems of Fredholm integral equations, the numerical so]mion of which have been well investigated by many researchers [37, 38]. These studies show that with the increase in the segment number N it is possible to obtain more accurate and readily convergent solutions for the integral equations.

The last computational step involves the numerical inversion of Laplace transforms. A modified algorithm is proposed and verified for the numerical inversion of Laplace transforms (see Appendix C). Due to the accumulation of errors in each repeated numerical integration, however, it is necessasry to evaluate and test the convergence and accuracy of the time-dependent results for the systems of Fredholm integral equations of the second kind in the Laplace transform domain. The details of these calculations are documented by Yue [15]. It is concluded from this verification that the numerical scheme and techniques adopted in this study provide highly stable and accurate solutions in the time domain for the systems of Fredholm integral equation.,; of the second kind in the Laplace transform domain and in particular, these procedures overcome the numerical problems customarily associated with the initial stages ( 1 0 - 4 ----- ct/a 2 <- 113, -2) of the consolidation of the poroelastic medium.

G E N E R A L I Z E D R E S P O N S E S OF A R I G I D D I S C I N C L U S I O N

The inclusion can be subjected to at least two types of loading conditions which occur quite frequently in problems in geomechanics. The first category of loading is one where the time-dependent 'variation of either the axial force Pz(t)(=PzYz(t)) or the central moment My(t)(= MyYy(t)) or the in-plane force Px(t)(= PxYx(t)) applied on the inclusion are known a priori. In the second category the time-dependent behaviour of either the axial translation Dz(t)(= D~Xz(t)) or the central rotation f~y(t)(= f~yXy(t)) or the in-plane translation Dx(t)( = DxX~(t)) of the rigid disc inclusion is prescribed. (It may be noted here that owing to the linearity of the d]ifferential equations governing a problem in poroelasticity, it is also possible to impose combinations of these generalized loadings.)

The most basic: case in the first category of loading is a constant imposed generalized loading of the type

Pz(t) = PzH(t), My(t) = MyH(t), Px(t) = PxH(t) (34)

where H( ) is a Heaviside step function. For this type of loading, the resulting generalized displacements and rotation take the forms

alxD~(ct/a 2) 3 - 4v Xz(ct/a 2)

Pz 32(1 - v)

a3~f~,(ct[a 2) _ 3 - 4v)v Xy(ct/a z) My 64(1 -

al~Dx(ct/a z) 7 - 8v Xx(ct[a2).

P~ 64(1 - v) (35a)


We define the "Degree of Consolidation", associated with these three states of loading, as follows:

Oz(ct /a 2) - O~(0 +) X~(ct /a 2) - 38 U~(c'/a2)= ~ z ( ~ ) - - - - D z " ~ - 1 - a s

Uy(ct/a2 ) = f y ( c t / a z) - ~y(0 +) = Xy(c t /a 2) - 338 fy(OO) - fy(0 +) 3 - 338

U~(ct/a2 ) = D~(ct/a 2) - / 9 . ( 0 + ) _ X~(ct /a 2) - 39 (35b) Ox(~) - / 9 , ( 0 +) 1 - - O~ 9

where X,~(ct/a 2) (or =x , y, z) are non-dimensional functions numerically evaluated from the equations (33) together with the applied constant forces and moment defined by equations (34). The functions X,~(ct/a 2) include only the material constants of the drained and undrained values of Poisson's ratios. The constants as and 39 are dependent on v and v,, (see Appendix A).

The second basic category of loading is a constant imposed generalized displacement of the type

Dz(t ) = DzH( t ), ~"]y(t) = ~)yH(t), D.(t) = D.H(t ) . (36)

For this type of displacements and rotation, the induced forces and moment take the forms

Pz(ct/a 2) 32(1 - v)y~(ct /a2 ) alzD z 3 - 4v

My(ct/a 2) 64(1 - v) Yy(ct/a2 ) a 3 t X f y 3 -- 4v

P,(ct /a 2) 64(1 - / z ) atxDx 7-----8v Y*(ct/a2)" (37a)

The corresponding expressions for the "Degree of Relaxation" associated with the loadings are given by

Vz(ct/a2 ) = Pz(ct/a 2) - Pz(O+)_ aSYz(Ct/a 2) - 1 Pz(~) - P~(O +) as - 1

Vy( ct / a2 ) = My( ct / a 2) - My(0+)= 338 Yy( Ct / a 2) - 1 My(oo) - My(0 +) 38 - 1

V~(ct/a2 ) = P.(ct /a 2) - P.(0+)_ agYx(ct /a 2) - 1 (37b) Px(~) - Px(O +) a9 - 1

where Y , ( c t / a : ) ( a = x , y , z) are non-dimensional functions numerically evaluated from the equations (33) together with the applied load of either constant displacements or constant rotation of the rigid inclusion (36). The functions X~(ct /a 2) include only the material constants of the drained and undrained values of Poisson's ratios.

NUMERICAL RESULTS

The time-dependent behavior of an embedded rigid disc inclusion is represented by the relationship between the fundamental loading mode Pz(t) (My(t), Px(t), or Mz(t)) and the corresponding fundamental displacement mode Dz(t ) (~'~y(/), D,(t) , or f z ( t ) ) . The time- dependent relationship between the axial moment M~(t) and the axial rotation f~(t) of the

embedded inclusion can be simply expressed in exact closed-form, i.e. Mz(t ) = 32~-a3 fz(t). The 3

time dependent response of the associated fundamental mode, i.e. D~(t) (fly(t), D,(t) , Pz(t)


My(t), or Px(t)) can be examined by considering separately the influence of the interface drainage conditions and the five material parameters (l~, v, v,,, B, and K). The influence of the three material pa:rameters (/x, B, and x) on the behavior of the rigid inclusion can be easily examined by using the equations (35) and (37). This is due to the fact that the non-dimensional functions X~(ct/a 2) or Y~(ct/a 2) (a = x, y, z) depend only the drained and undrained Poisson ratios (v, v,) and the interface drainage conditions. In the ensuing, we shall present the numerical results which will clearly illustrate the time dependent behavior of the rigid disc inclusion by taking account the influences of interface drainage conditions and the drained and undrained Poisson ratios v and v,, respectively.

The conventional time factor ct/a 2 is utilized in the presentation of the numerical results. It is noted that this conventional time factor depends on the drained and undrained Poisson's ratios [see equations (3b)]. Therefore, the influence of the drained and undrained Poisson ratios on the behavior of the rigid disc inclusion can be further understood by using a modified time factor Cmt/a 2 (= 2tzB2Kt[a2). This modified time factor Cmt/a 2 eliminates the dependency on Poisson's ratios v and vu.

For the first category of loading, where the loads are prescribed in equation (34), the results corresponding to equations (35) are plotted against the conventional time factor ct/a 2 (Figs 3-5). For the second category of loading [equation (36)], the results corresponding to equations (37) are plotted against the conventional time factor ct/a 2 (Figs 6-8). Some general qualitative and specific results associated with the time-dependent behavior of the rigid disc inclusion are further summarized in Tables 1 and 2. These results are applicable to the four basic displacement modes and the four basic loading modes. It is noted that (i) in Table 1, AU(ct/a 2) refers to the difference in the degree of consolidation induced displacements between the permeable and the impermeable inclusions and (ii) in Table 2, AV(ct/a 2) refers to the difference in the degree of consolidation induced load relaxations between the permeable and the impermeable inclusions.

CONCLUSIONS

The paper develops the systems of coupled integral equations governing quasi-static behavior of a rigid circular disc inclusion embedded in bonded contact with a poroelastic infinite medium saturated with a compressible pore fluid. These coupled integral equations are developed for disc inclusions which display either permeable or impermeable interface conditions. The generalized displacements of the rigid disc inclusion include an axial displacement, rotation about two axes and an in-plane translation. Due to their symmetry or asymmetry, these inclusion problems can be formulated as mixed boundary value problems related to a halfspace region and referred to variables in the Laplace transform domain. Through the generalized treatment presented in the paper, it is established that the modes of deformation associated with the translation and rotation of the inclusion give rise to independent sets of transient problems. It is shown that all sets of integral equations governing the quasi-static response of the rigid disc inclusion can be reduced to the standard systems of Fredholm integral equations of the second kind in the Laplace transform domain.

The numerical evaluations of the systems of Fredholm integral equations of the second kind are used to demonstrate the time-dependent behavior of the inclusion which is subjected to either a step function-type loading or a step function-type displacement. It is shown that the kernel functions in the systems of Fredholm integral equations contain infinite integrals with rapidly oscillatory integrands. The evaluation of these infinite integrals is based on the proceeding limit technique and an approximation scheme. The computation schemes also involve Laplace transform inversion which is achieved via a modified algorithm. These numerical procedures offer accurate and efficient techniques for solution of this class of coupled systems of integral equations of the Fredholm type in the Laplace transform domain.

1652 Z . Q . Y U E and A. P. S. $ E L V A D U R A I

U~(ct/,,') 0.0

0.2

0,4

0.6

0.8

l.O t 0.0001

~ - - - - - - - - - , ~ ,,..o.5. ,,.o.o I ....... ~,=0.5. ~=0.3 I . . . . t%=0.5, t.=0.49 [

i

I I

0.001 0.01 0.1 1 10 ct/a'

Uz(et 0.0

0.2

0.4

0.6

0.8

l.O

/ a ' )

0.0001 0.001

~ - - - - - - - - ~ ~ ..... :::oO:55: ~:g:o - % . ~ \ ~ . . . . ~..o~ ~=o,~

O.Ot 0.1 l tO tO0 1000 ct/a'

Ux(et 0.0

0.2

0.4

0.6

0.8

/a ')

l.O 0.0001 O.OOl

~~,~,. ( e )

_ _ ~ . = 0 . 5 , ~ = 0 . 0 '~ " ~ " ....... ~ , :0 .5 . ~ :0 .3

, , , , , , , , . O.Ol 0.1 1 tO 100 1000

Ct//82

u~(ct/a') 0.0 ~ . . . . ~.:O.Ot.~:O.O

I ~ "-"~.. -~ ....... v,:O.3. ~=0.0 _. I ,, :o.o

0.6 i

0.8

| . 0 I I I I I I I l [ I I I I I I I I I I I l l l l l l I I I I I I I I 1 [ I I I I I l l l

0.000! O.OOI 0.01 O.l l tO ct/a 2

Uz(et/a ~) 0.0 _ ~ , , : . . . . t,,,=O.Ol,'v=O.O

" ~ ~.."b ....... ~.=0.3. ~=0.0 0.2 ,,"~"~,~~~ __ ~.:0.5. ~:0.0

0.80"00'4 " ~ ' e ~ ' ~ o ~ ~ e ( d )

1.0 J ~ , v , , l ,t,,II~ v f,H,, o.oool o.oot o.ot o . l t to too 1ooo

ct/a'

Ux(ct/a ~) o.o

- ~ ~ , 0.2

0.4 " ~ ; ( f )

~ \ \ ~ . 0.6

. . . . 'v, fO.Ol,'v=O.O \ ~ . 0.8 ....... :..:o~ :_:oO:O

| . 0 f f Jlrt l l~ I I i l i l l q i I I I I f f l r I I l i i t i q J t i t r t t t [ I i l i l i l q J | i l l i i

o.oool o.ooi o.ot o.t t 1o loo looo ct/a 2

Fig. 3. Degree of consolidation induced displacements of a rigid inclusion vs the time factor ct/a2: effect of interface drainages and Poisson's ratios.

M e c h a n i c s o f a r i g i d d i s c i n c l u s i o n 1653

lOaapflv(ct/aZ)/Mv 0 . 0 0 . 9

1.0 1.0

1.1 . . . . . . I.I

1.4 1.4

0 .0001 0 .001 0 .01 0 . I I I0

ct/a*

1 0 a / ~ D z ( c t / a ' ) / P z o.8 -,

I

o8° . . . . . . .

~: :g :~ : : : :o°:: " ......... - . . . . . . , _ _ . . . . . . . . ~ . : 0 . 5 , ~ : 0 . 2 \ . . . . ~ , : 0 . 5 , ~ : 0 . 3 ~ - ~ - ~ . . . _ _

0 .9 _ - _ ~ , : 0 . 5 . ~ : 0 . 4 ~ . . . . . . - - _ ~ , = 0 . 5 . ~ : 0 . 4 9

I"'l l i l i l l I i i l t l l lq I IgilUUl I i lUlUlil~ I IlilUll I I Ullllll[ I llUltil

0 .0001 0 .001 0 .01 0 .I ! 10 I 0 0 I 0 0 0

ct/s ~

I Oa/~Dx( c t / a e ) / P x 0 .02 '

o . .

: . . . . . . 1 . 0 0 ( e ) .

~, o.5. ~.o.o \ \ ........ 1.04 . _ _ _ ~ , , : 0 . 5 , v : O . | ~ , \ \ .........

....... ~ , = 0 . 5 , ~ = 0 . 2 \ ",~ .......... . . . . ~ , = 0 . 5 . ~ = 0 . 3 ~ "--. . . .~ . _ _ ~ . = 0 . 5 , ~ = 0 . 4 ~ " ~ . . . . . . . .

1.08 _ _ ' u , : O . 5 , ' v , : 0 . 40

I l l l l l l q [ l~'l'lllq I l l l l l l q I l l l l l l q I IH [ l lq I l l l l l l q I I I I I I I I 0 ,0001 0 .001 0 .01 '0,1 1 10 100 1000

ct/a'

lOaS~zr~y(ct/az)/My

- - _ ~ , : 0 . 0 1 , ~ : 0 . 0 _ _ _ v , : O . l , ~ : 0 . 0 . . . . v , : O . 2 , ~ : 0 . 0 . . . . . . . ~ . : 0 . 3 , ~ = 0 . 0 ~ _ ~ , : 0 . 4 , 1~:0.0

~ , : 0 . 5 , ~ : 0 . 0

. . . . . ~ (b)

. . . . .

o . o o o i o . o o i o .ox o . I I lO

c t / a ~

lOa~Dz(ct/a')/Pz 0.6 ........

_ _ ~ , = 0 . 0 1 . ~ = 0 . 0 _ _ _ ~ , = 0 . 1 . t,=O.O . . . . ~ , = 0 . 2 . t,=O.O . . . . . . . v . = 0 . 3 , t,=O.O

0.7 ~ ........ ~ , = 0 . 4 , v=O.O I J s = O . ~ , 1)----0.0

. . . . . . . . . " ~ ' - - . (d)

I I I I I I I I I I I I I I I I1[ I [ l l l l l l I I I I I I I I 1 I I I I I I I q l - i l - I I I l ~ I I I I I I I I I o.oool o .oo i o .o i o.s I IO 1oo IOOO

ct/a ~

I Oa~Dx(ct/aS)/Px o.82

_ _ ~ , = O . O I N = O . O - ~ ~ _ _ _ ~ , = 0 . 1 , ~ = 0 . 0

~ . . . . . ~ , = 0 . 2 . v=O.O 0 . 9 6 - - ~ . . . . . . . t~,=0.3, 1,'=0.0

. . . . . . ~ , = 0 . 4 . ~ = 0 . 0 X _ _ ~v,=0.5. ~ = 0 . 0

1,00 - - . . . . . . . . . . . . . . . . ~ ............ " ' - ~ X . . . . . . . . . . . . . -'-" ~-., , \ (r)

"1 I I I I 1 ~ I I I I I I1~ | 11111]~ I I I l ] l l q I I I I I I1~ I I I I I I1~ ''1' I I I I I I 0,0001 0.00| 0.01 0.1 1 10 100 1000

C ~ 8 2

Fig. 4. The non-dimensional displacements of a permeable and rigid inclusion vs the time factor ct/a2: effect of Poisson's ratios.

1654 Z . Q . Y U E and A. P. S. S E L V A D U R A I

l O a " p O ~ ( c t / a ' ) / M ~ 0.9

1.0 --

I.I "

1.2 ~

1.3 ~

1.4 -

O.OOOl

. . . . . ~ ) . = 0 . 5 , 'u:O. l \ \ . . . . . . . . . . . . . . . . . . . . . . . . ,~ .=0.5 , ~ : O . e \ \

__ ~ 0 . 5 , , o : 0 . ~ \ ~ ...................... _ _ _ u , = 0 . 5 , ~ : 0 . 4 _ _ ~ v = 0 . 5 , ~ : 0 . 4 9

I I I I I I I I I I I I l l l l l I I I I I I I I I I I I l l l l l l I I I l l l l l l 0.001 0 .01 0.1 1 10

c t / a e

1 0 a / ~ D z ( c t / a e ) / P z o.0

- ~-C-~.-.

~) .=0.5, v : O . O \ \ """ .... . . . . ,).--0._~, ~=0_.! \ \ . . . . . . . . . . . ..... ~ . - -0 .5 . v = O . e \ "~... " ........... . . . . z,~o.~, ~=o.a \ ............

0 . 9 _ _ _ , . , . = o . ~ ~ = o . 4 ~ .................................. _ - - ~ . : 0 . 5 , ~ = 0 . 4 9

; l l l i l l l I I i i l l l lq i l i l i l i i I i l i l t l lq I ]nfi iq I IIIIIIq I I l l l l i t o.ooo~ o oot o.o~ o.~ ~ ,o ~oo ~ooo

ct/a ~

1 0 a # D x ( c t / a a ) / P x 0 .92

1 . 0 0 ~ 91 l imiq

O.OOOl 0 .001 0 . 0 | 0.1 1 ct/a z

10 lOO 1000

1 0 a S / ~ y ( c t / a z ) / / M y

o.g

1 .0

1 . 1 - -

1 . 2 - -

1 . 3 - -

1.4

_ _ ~ . : O . O l , ~ : O . O ~ . : 0 . 1 , ~ :O .O

. . . . ~ . : 0 . : ~ , ~ ) :0 .0 . . . . . . . z*.:O.3, 'u:O.O . . . . . ~ . : 0 . 4 , ~):O.O _ _ u . : O . 5 , ~ ) :0 .0

f I I I I I I I I 0 . 0 0 0 1 0 . 0 0 1

I I I I l ; l i I I I I I I I I I I I I I I I I I I I I I I I I l l l 0.01 0.1 1 10

ct/a 8

2 1 0 a / z D z ( c t / a ) / P z 0.0

_ _ v . = O . O l . ~ ) = O . O _ _ _ v . = 0 . 1 , ~):O.O . . . . ~ . = 0 . 2 , ~)=0.0 ....... u . : 0 . 3 , I ) : 0 . 0

0 .7 ~ ............ ',.=0.4. 'u :O.O _ _ 'p~.=0.5. 'o=O.O

................................... ............................................. ( d ) .................... iiiiii-iiii:,

I IJlilll I I IIIIIII I I ]l l l l lq I IIllllq I I l l l l lq ] IIIIII]] I [ l l l l l 0 . 0 0 O l 0 . 0 0 l 0 . 0 1 0 .1 1 10 1OO 10O0

c t / a 2

lOa#Dx(ct/a~)/Px 0 .92

_ _ v,,--O.Ol,v--O.O - ~ v,,=O. 1, ~p=O.O

- _ - _ - v , = o . ~ , ~ = o . o 0 , 9 0 ~ . . . . . . . . v . = 0 . 3 , u = O . O

.............. ~ . = 0 . 4 , v=O.O _ _ v . = 0 . 5 , ~)=O.O

1.00 ............................................... (r )

1 904

1 .00 . . . . . . . . . . I i i l l i i l l i l i _ i i @ i ~ ~ i l i ~ l I I I I ~ - " ' ' " ' " ' " ~ ~ ; ~ o.oooi o.ooi o.oi o.1 x IO 1oo 1ooo

c t / a z

Fig. 5. The non-dimensional displacements of an impermeable and rigid inclusion vs the time factor ct/a2: effect of Poisson's ratios.

M e c h a n i c s o f a r ig id disc inc lus ion 1655

V~(ct/a ~)

' " ~ ~ ~°"~ _ _ ,.=o.~. ,,=..o . . . . . . . v,=O.S, v=0 .3

- <' v ,=0 .5 , v=0 .49 0.2 . . . .

o.,~ ,~,,,, x~::,g (a)

• % \,,,,, \,,,, 0.6

• , , • ,, x - , • ,, ,,

• , , , , , , , ,

, . 8 " ," ","

1 , 0 I I I l l l l l I I I l l l l l l i I I l l l l l l i I I I I l l l l [ I I I I I I I I

0 . , , 0 1 0.001 O.OI 0.1 1 10 ct/a ~

V,(ct/a z)

0.0 _ ~ . . ~ . . . . . . v,=O.Ol,v=O.O ~':S_-."., ~'~,~-, . . . . . . . v .=0 ,3 , v=O.O -.I ~ ' . - . . "~'k,~ ~ _ _ ,~,:o.5, ,~=o.o 0.;~

0.4 (b)

1.0 t I I I I I I I l i I II i l ? T l l l 0 . , , 0 1 0.001 O.Oi 0.1 1 10

ct/a z

V~(ct/2) V,(ct/2)

" ' ~ . . v.=o.5, v=o., o.o ~ ~ ,~ . . . . . v,=O.Ol,v=o.o i ~-..~'~,,:.. ~, ....... .,.,,--o.5. ,.,--o.a I . -~ . ' , . . ~,%-., ~ ....... ,.,,,--0.3. ~,--o.o

. . . . o : o , . - t _ , , : o o

o.4 • (c ) o.4 ~ ~",,',,"~':,, ~' (d )

, . 6 - - ~ , ' , ~ " ~ - ",,~ .;,, o.6 , ,

' . 8 ' " ~ ~ " ~ 0.8 "~':~.": ~

1 . ' I I I IIIIq I I l i l l l q I I I II l l l i I I i l l l l l [ I I IIIIIq I 111111 ! I I IIIIII 1 . 0 I I IIIIIIJ I I IIIIIIJ I I IIIIIIJ I I IIIIIIJ I I IIIIIIJ I 1111111 I I IIIII

0 .0 ' 01 0.001 0.01 0.1 1 I0 100 1000 ct/a z

0.00,1 0.001 0.01 0.1 1 10 100 1000 e t / a z

Vx(ct/a z) Vx(ct/a ~) 0.0 ~ . 0.0

0.2

, . 4

0.6

0.8

v,=0.5 , v=O.O I

0.01 , . 1 1 !0 100 1000 ct/a e

1.o I 0.000 ! 0.001

0.2

0.4

0.6

0.8

1.0

0.0001 0.001

::__ ~::0:0~.~:o:o _ _ v . = 0 . 5 , v=O.O

I I I I I I Iq I I I I I I Iq I I l i l l l l I I 1111111 I I I I I I Iq I IIIIIII1 I I IHf l l I

0.01 0.1 1 10 100 000 ct/a z

Fig. 6. D e g r e e o f c o n s o l i d a t i o n i n d u c e d l o a d r e l a x a t i o n s in a r ig id inc lus ion vs t he t ime f a c t o r ct/a2: effect o f i n t e r f a c e d r a i n a g e a n d P o i s s o n ' s ra t ios .

1656 Z . Q . YUE and A. P. S. S E L V A D U R A I

My(ct/aS)/(1Oa=/~Dy)

0 .96 ~ : , , ' - . . - . _

. . . . . . . . v , = 0 . 5 , v = O . l ~ ' ~ . . " . . . . . . . . . . . . . . 0 . 7 5 - ~ . . 2 ~ ~ ....................

. . . . v , = O . 5 , v : O . 3 1 ~ , : 0 . 5 , v : O . 4

_ _ ~ , = 0 . 5 . p = 0 . 4 9

0 . 6 5 I I I I I I I I I I I I I I I I 1 [ I I I I I I l l I I I I I I I I I I I I I I I I

0.0001 0.001 0.01 0.1 l 10 ct/a ~

1 . 0 5 - -

0 . 9 5

0 . 0 5

0 . 7 5

_ _ v , = O . O l . v = O . O _ _ _ v , = O . l , t~=O.O

~ _ . . . . v , = 0 . 2 , i~=O.O . . . . . . . v , = O . 3 , v=O .O .......... v , = 0 . 4 , v=O .O _ _ v , = 0 . 5 , v=O .O

. . . . . . . . _ -_- -_ -_- :_-: : : . - . ~ . ~ ~

0 , 6 5 I I I I I I I I I I I I l l l l l I I I I I I I I I I I I I I I I I I I I I I I I I I I

0.0001 O.O01 0.01 0.1 1 lO ct/a ~

Pz(ct/aS)/(10a/~Dz)

1 . 6 0

1,50 --

1.40 -~

1.30 f

1.20 f

l.lO f

(c) ~>.".. - ......

, , . : ~ . ,,:o.? ~ , 2 ......... ............ " - = 9 . . 5. v = 9 . ! ~'-- ....................... ~ . ~ ~_-o.~ ~.~" ....................

_ _ v . = 0 . 5 , v = 0 . 4 v , , = 0 . 5 , v = 0 . 4 8

1.00 i i l l l n ] [ I I I I I I I I i I I I I f l l l i I I I I I I l l i i i l i i i i l I i l i l i . i i i { H i l i l

O.O00 l 0 . 0 0 1 0 . 0 1 0 .1 1 10 1 0 0 1 0 0 0

ct/a z

Pz(ct/a2)/(lOa/~Dz)

1 .60 - -

1 .50 f

1.40 f

1.30 f

1.20 f

1.10 f

1 . 0 0

0.000

_ _ ~ , = O . O l . v = O . O _ _ _ l ~ , = O . l . ~ = O . O . . . . I J ,=O.2 , ~ = 0 . 0 . . . . . . . v , = O . 3 , ~ = 0 . 0 .......... • , . = 0 . 4 . v=O.O _ _ ~ , = 0 . 5 , v=O .O

I

, , I , I , , l , I , , , , q , , , , , , q , , , , , m [ ' " ' " " I ' " " " ! i , , , , , , , l 0.001 0.01 0.1 1 10 100 100O

ct/a ~

Px(ct/aS)/( I 0a/~Dx) Px(ct/~)/( I oa~,Dx)

1 . 0 6 - -

1 . 0 2

0 . 0 0

0 . 9 4

0 . 9 0

O.OOO! 0 . 0 0 1

= ~... ~... .

( e ) ~ . ' - . ' . . - - - -

__ ~.=o.5 ~=o.o \~'" ........ . . . . . . . . . . . . . . I~,:0.5,~':0"5' ~:0.2~:0"1 . . . . . . . . . ' \ . . . . . . . . . . . . .

~o,~: ~--~ ~-----....-...........: ~o°'16 I l l l l l l l I I I I I I I I I i I I I I I I I q I I I I I I I1~ I I I I I I I I I I I I I I I I 1 [ I I I I I l l

0.01 O.l 1 I0 100 1 0 0 0 ct/a z

1 . 0 6

1 . 0 2

0 . 9 6

0 . 9 4

0 . 9 0 i

0.000 ! 0.001

__ ~ ~:-0:0~.~-o:o . . . . v , = 0 . 2 , v=O .O

: : : : : ~:---o°i~: ~_=o% ° _ _ ~ , : 0 . 5 , t~:O.O

O.Ol 0.1 1 10 100 1000 ct/a ~

Fig. 7. The non-dimensional force and moment resultants in a permeable and rigid inclusion vs the time factor ct/aZ: effect of Poisson's ratios.


M y ( c t / a S ) / ( 1 0 a S # . f } ~ . )

1 . o 6 -

0 . 9 5

( 0.85

0.75

0.65

0.0001

. . . . . . . .

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 [ I I I I I I I I I

0.001 0.01 O.l 1 10

c t / - ~

~y(et/a~)/(lOa~O~)

1 . 0 5 - -

0 . 9 5

0 . 8 5

0 . 7 5

_ _ ~ . : 0 . 0 1 , ~ : 0 . 0 _ _ _ ~ . : 0 . 1 , ~ : 0 . 0 . . . . , ~ . : 0 . 2 , ~ : 0 . 0 . . . . . . . ~ , = 0 . 3 . ~ : 0 . 0 .......... , ~ . = 0 . 4 , , ~ = 0 . 0 _ _ ~ . = 0 . 5 . ~ = 0 . 0

0 . 6 5 t I I I I I l l I I I I l l l l l I I I I l l l l l I I I I I I I l l I I I I l l l l l

0.0001 0.001 0.01 0.1 1 10

ct/a =

P z ( c t / a S ) / ( 1 0 a , D z )

1 . 6 0 - -

1 . 5 0 --

1.40 -

1.30 --

1.20 -

1.10 --

1 . 0 0

0.0001 0.001

to) %',, - .......

_°._-0.5.:_0.0 ............. . . . . . . . . . ~ . = 0 . 5 , ) = O . i \ \ ~ . . . . . . . . . . . . . . . . . . . . . . ~ . = 0 . 5 , 1 ) = 0 . 2 ~ " ~ - . . . " . . . . . . . . . .

. . . . • v , , = 0 . 5 . 1 , ,=0 .3 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . _ _ _ ' u . = 0 . 5 , ~.,'=0.4 ~ _

_ _ ~ . = 0 . 5 , ~ : 0 . 4 9

I I I I l l l l [ I I l l ] I l l [ I I I I I I Iq I I l l l l l~ I I I I I I Iq I I I IHI l i [ I I I I

0 . 0 ! 0 .1 1 I0 100 000

ct/a ~

Pz(ct/al)/(10a,Dz)

1 . 6 0 - -

1 . 5 0 --

1.40 £

1.30 £

1.20 --

1.10 £

1 . 0 0

0.0001 0.001

_ _ ~ . = 0 . 0 1 , ~ = 0 . 0 _ _ _ ~ . = 0 . 1 , ~ = 0 . 0 . . . . , ~ . = 0 . 2 , ~ = 0 . 0

~ . . . . . . . . ~ . = 0 . 3 , ~ = 0 . 0 ........... ~ . = 0 . 4 , 1~=0 .0 _ _ ~ . = 0 . 5 , ~ = O . O

....................... ~..\ (d)

I I I I I I I I t I I I I I I Iq I I l l l l l l I t I l l l t lq I I l t t l l l I I I I I I l l q I I I I l l t l

0.01 0.1 1 10 100 1000 ct/a z

Px(ct/aZ)/( l Oa,Dx)

1 . 0 6 -

1 . 0 2

0 . 9 8

0 . 9 4

0 . 9 0

O.O001 0.001

~,--o:5: ;Z-G:; ~"-.. " ................. ....... , ~ , = 0 . 5 . ~ = 0 . 2 ~'~.--...

. . . . v . : O . 5 . ~ : 0 . 3 ~ ..................... _ _ _ t 1 . : 0 . 5 , ~ : 0 . 4

_ _ ~ . = 0 . 5 , ~ v = 0 . 4 9

I I I I I I Iq I I I I I I t l I I I I I I I Iq I I l l l l l q I I I I I I Iq I I I I I I I I I I I l l l l l

0 .0! 0.1 1 tO IO0 1000 ct/a z

Px(ct/a ~)/(lOa~Dx)

1 . 0 6

1 . 0 2

0 . 9 8

0 . 9 4

0 . 9 0

0.0001 O.OOt

_ _ ~ . = 0 . 0 1 , ~ = 0 . 0 - - ~ _ _ _ v . = O . l , v = O . O

. . . . ~ . = 0 . 2 , v = O . O . . . . . . . v . = O . 3 , v = O . O

~ .............. 1I,,,,=0.4 , 1 3 = 0 . 0 .................................................... ~ _ _ ~ . = 0 . 5 , ~ = 0 . 0

~ m , I ~ J H m q ~ H m q ~ m m I ~ H m q ~ t . m q ~ . m , O.Ot 0.1 1 10 100 tO00

e t / a I

Fig. 8. T h e n a n - d i m e n s i o n a l f o r c e a n d m o m e n t r e s u l t a n t s in a n i m p e r m e a b l e a n d r ig id i nc lu s ion vs t h e t i m e f a c t o r ct/a2: e f f ec t o f P o i s s o n ' s r a t io s .

1658 Z . Q . Y U E a n d A . P. S. S E L V A D U R A I

T a b l e 1. T h e t i m e - d e p e n d e n t b e h a v i o r o f a r ig id d i sc i nc lus ion in a s a t u r a t e d p o r o e l a s t i c m e d i u m . C a s e ( A ) : c o n s o l i d a t i o n d i s p l a c e m e n t d u e to a n a p p l i e d c o n s t a n t l o a d

Displacement modes

Conditions and phenomena

Solution Time for consolidation process

consolidation rate Drainage magni tude

max imum AU(ct/a z) The four parameters p., r , B, a

initial displacement

Poisson's final displacement ratios v, v,,

consolidation displacement

Vertical displacement modes Horizontal displacement modes

axial translation central rotation al.LD :(t)/ P: a~l~g~v(t)/ Mv

numerical results f rom integral equations, N > 1 / ' V ~ - ~ longer shorter much longer

significant more significant limited no effect on the initial and final values of displacement

17.3% 21.2% 4.8% consolidation t ime is inversely proportional to 2~KBZ/a z

3 - 4v u 3(3 - 4vu) 7 - 8v . 32(1 - v,,) 64(I - v . ) 64(1 - v . )

3 - 4 v 3 ( 3 - 4v) 7 - 8 v 32(1 - v) 64(1 - v) 64(1 - v)

(V,, -- v)Uz(ct/a 2) 3(v . - v)Uv(ct/a 2) (v u - v)V~(ct/a 2) 32(1 - v ) ( l - v.) 64(1 - v ) ( l - v . ) 64(1 - v) ( l - v . )

in-plane translation axial rotation

ap.D.(t)/p ~ a" Izf~: (t)/M~ exact

shortest 0 no effect

no effect no effect

3 64

3 64

very limited effect by using the time f a c t o r ct/a 2

(1-- v)(l + v u ) 2 . consolidation t ime significant effect due to % = m c = vhc, . no effect 9(v,, - v)(1 - v,,)

by using the modified t ime factor cmt/a 2

T a b l e 2. T h e t i m e - d e p e n d e n t b e h a v i o r o f a r ig id d i sc i nc lu s ion in a s a t u r a t e d p o r o e l a s t i c m e d i u m . C a s e (B) : f o r c e r e l a x a t i o n d u e to a n a p p l i e d c o n s t a n t d i s p l a c e m e n t

Displacement modes

Conditions and phenomena

Solution T ime for relaxation process

relaxation rate Drainage magni tude

max imum AV(ct/a 2) The four parameters p,, K, B, a

Vertical displacement modes Horizontal displacement modes

axial force central momen t in-plane force axial momen t 3 3 Pz(t)/(ap.D:) My(t)/(a I.t~v) p~(t)/(ap.Dx) M~(t)/(a ixf~z)

numerical results f rom integral equations. N > 1/ctV~/a 2 exact longer shorter much longer shortest 0

significant more significant limited no effect no effect on the initial and final values of forces

20.0% 22.6% 5.0% no effect relaxation time is inversely proport ional to 2p.rB2/a 2 no effect

initial force 32(1 - v,,) 64(1 - v.) 64(1 - v,,) 64 (3 - 4v,,) 3(3 - 4 v . ) 7 - 8v,, 3

final force 32(1 - v) 64(1 - v) 64(1 - v) 64 3 - 4 v 3(3 - 4v) 3 - 4v 3

relaxation force 32(v - v.)Vz(ct/a 2) 64(v - vu)Vv(ct/a 2) 64(v - vu)V.~(ct/a 2) 0 (3 - 4v)(3 - 4 v . ) 3(3 - 4v)(3 - 4v,,) (7 - 8v)(7 - 8v . )

degree by ct[a 2 significant effect limited effect degree by Cmt/a 2 significant due to the counteracting effects of v, v,, no effect

in % and in the integral equations

Poisson's ratios v, v.

The time-dependent behavior of the rigid disc inclusion has been examined in detail by taking into account the influences of the interface drainage conditions and the five material parameters governing classical poroelasticity.

For an axisymmetric rotation about the z-axis, in particular, the state of stress induced in the poroelastic medium corresponds purely to shear stresses and the solution for the poroelasticity problem reduces to that for the analogous problem in classical elasticity. It is of interest to note that in the case where the embedded inclusion is subjected to an in-plane translation, the traction orthogonal to the direction of translation and within the inclusion region, are non-zero for any arbitrary time t. These tractions however reduce to zero as t---~0 and t---, +oo. The limiting responses as t---~0, t---~ +~, and v - - - , v,, can also be recovered as special cases of the generalized systems of Fredholm integral equations of second kind. These closed-form results are in agreement with the appropriate solutions for disc inclusions embedded in elastic media.


Acknowledgements--The work described in this paper was supported by a NSERC Research Grant A3866 awarded to A. P. S. Selvadurai. Z. Q. Yue would like to acknowledge the award of the Research Scholarship provided by Carleton University.

R E F E R E N C E S

[1] J. N. GOODIER, J. Appl. Mech. 55, 39 (1933). [2] J. M. DEWEY, J. Appl. Phys. 18, 528 (1947). [3] R. H. EDWARD:5, J. Appl. Mech. 18, 19 (1951). [4] K. ROBINSON, Y. Appl. Phys. 22, 1045 (1951). [5] J. D. ESHELBY, Proc. R. Soc. Lond., Ser. A 241, 376 (1957). [6] A. I. LUR'E, Mech. Tverdogo Tela 1, 80 (1967). [7] E. STERNBERG, Appl. Mech. Rev. 11, 1 (1958). [8] J. R. WILLIS, V~xiational and related methods for the overall properties of composites. In Advances in Applied

Mechanics (Edited by C.-S. YIH), Vol. 21, pp. 1-78. Academic Press, New York (1981). [9] L. J. WALPOLF, Elastic behaviour of composite materials: theoretical formulation. In Advances in Applied

Mechanics (Edited by C.-S. YIH), Vol. 21, pp. 169-242. Academic Press, New York (1981). [10] T. MURA, AppL Mech. Rev. 41, 15 (1988). [11] G. M. L. GLADWELL, Contact Problems in the Theory of Elasticity. Sijthoff and Noordhoff, The Netherlands

(1980). [12] T. MURA, Micromechanics of Defects in Solids. Martinus Nijhoff, The Hague (1982). [13] W. D. COLLINS, Proc. R. Soc. A 203, 359 (1962). [14] L. M. KEER, Int. J. Solids Struct. 1, 257 (1965). [15] Z. Q. YUE, Mechanics of rigid disc inclusions in fluids saturated poroelastic media. Thesis submitted in partial

fulfillment of the requirements for Ph.D. degree, Carleton University, Ottawa, Ontario, Canada (1992). [16] A. P. S. SELVADURAI, Disc inclusion problems in mechanics (unpublished) (1995). [17] S. C. HUNTER and D. GAMBLEN, J. Mech. Phys. Solids 2 2 , 371 (1974). [18] A. P. S. SELVADURAI, Geotechnique 26, 603 (1976). [19] A. P. S. SELVADURAI and T. J. NICHOLAS, Proc. 3rd Int. Conf. Num. Anal. Meth. Geomech., Aachen, Vol.

2, p. 1245 (1979). [20] M. A. BIOT, J. Appl. Phys. 12, 155 (1941). [21] M. A. BIOT, J. Appl. Mech. 23, 91 (1956). [22] J. C. SMALL, J. R. BOOKER and E. H. DAVIS, Int. J. Solids Struct. 12, 431 (1976). [23] R. W. LEWIS and B. A. SCHREFLER, The Finite Element Methods in the Deformation and Consolidation of

Porous Media. Wiley, New York (1987). [24] J. R. RICE and M. P. CLEARY, Rev. Geophys. Space Phys. 14, 227 (1976). [25] A. W. BISHOP and D. W. HIGHT, Geotechnique 27, 369 (1977). [26] I. N. SNEDDON, The Use of Integral Transforms. McGraw-Hill, New York (1972). [27] I. N. SNEDDON, Proc. Glasgow Math. Ass. 7, 136 (1966). [28] J. C. SMALL ancl J. R. BOOKER, Int. J. Numer. Anal. Meth. Geomech. 11, 269 (1987). [29] G. N. WATSON, A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge (1962). [30] C. J. TRANTER, Bessel Functions with Some Physical Applications. Hart, New York (1968). [31] I. S. GRADSHTEYN and I. M. RYZHIK, Table of Integrals, Series, and Products. Academic Press, New York

(1980). [32] R. P. KANWAL and D. L. SHARMA, J. Elasticity 6, 405 (1976). [33] A. P. S. SELVADURAI, Int. J. Numer. Anal. Meth. Geomech. 4, 121 (1980). [34] A. P. S. SELVADURAI, Int. J. Engng Sci. 18, 979 (1980). [35] K. ATKINSON, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the

Second-Kind. So¢. Industrial and Appl. Math., Philadelphia, PA (1976). [36] C. ATKINSON, Appl. Mech. Rev. 32, 123 (1979). [37] C. T. H. BAKER, The Numerical Treatment of Integral Equations. Clarendon Press, Oxford (1977). [38] L. M. DELVES and J. L. MOHAMED, Computational Methods for Integral Equations. Cambridge Univ. Press,

Cambridge (19851). [39] K. S. CRUMP, J. Ass. Comp. Machin. 23, 89 (1976). [40] J. R. MACDONALD, J. Appl. Phys. 10, 3034 (1964).

(Revision received 25 April 1994; accepted 2 March 1995)

A P P E N D I X A The coefficient matrices {bi](y)} and {eli(y)} (i, j = 1, 2, 3) are defined by

b]l = b22 = ~ [4v. - 2 + av(1 - ~,)],

8~(v. - 1) b12 = b21 Db '

b l 3 = _ b 2 3 = ot.r(~ / - 1)," ¢tdO b

b31 = _ b 3 2 8/,I, K a d ( 1 - v . ) ( 1 - y ) Db

K b33 = ~ [(8v. - 6)V + ay(1 - v)l,

D b = 8v. - 6 + ay(1 - y),

c , , = c22 = ~-~ [a~,(1 - y) - 2(1 - 2v,,)y] --c

el2 = c21 = 8/z(v, - 1)~, Dc

at(1 - y) C13 = --C2 3 = K~dD c

ca1 = -c32 = 8/~a(1 - v,)(1 - y) oc

1 c33 = ~D-~[6 - 8v, + a~(3, - 1)]

Dc = (8v, - 6)y + a~(1 - y). (A1)

1660 Z . Q . YUE and A. P. $. S E L V A D U R A I

T h e coe f f i c i ent s as(*/) ( j = 1, 2, 3, 4 ) a n d b / ( y ) ( j = 1 . . . . . 6) are d e f i n e d b y

3 - 4v al(Y) 4/z(1 - v) [l + azkl(y)] , a3(y) ffi 2KBa4k2(y), a4(',/) = K[1 + kz(y)l

a~3(3 - - 4V) a2(y) ~ _-- ~-~ [1 + k ,(Y)], b3(y) = 2Ba4k4(y), b4(y) = 1 [1 + k4(y)]

a.~(7 - 8v) 7-8v +a,k3(~)]+~. b~(~,)-- 8---~-~-~)11+k.~(~)1 b,(v) = 4/z(1 - v)[1

b s ( Y ) = ½ [ b l ( y ) - l ] , ( 7 - 8 v ) , bb(Y) = 8/J,(1 - v ) [Ot l - a 6 k 3 ( y ) ]

1 - y 2 1 - y , = = - - ( A 2 ) k~(y) = 1 + y k2(y) = y - 1, k3('Y) (1 + y )y - 1, k4(y ) Y

The constants a s (j = 1 ~ 6, 8, 9) are defined by

1 v,, - v 3(v,, - v) (3 - 4 v . ) ( 1 - v) a l = 7 - 8 v ' a2 ( 1 - v.)(3 - 4v) ' a3 =(1 + v.)(3 - 4v) ' a ' = ( 3 - 4v)(1 - v.)

1 + v,, 6 ( v . - v ) v . - v (7 - 8 v . ) ( 1 - v ) a4 6(1 - v . ) ' a5 (1 + v.)(7 - 8v ) ' ot~ = (1 - v,,)(7 - 8v ) ' a~ = (7 - 8v)(1 - v,,)" (A3)

A P P E N D I X B

Evaluation o f the Infinite Integrals in the Kernel Functions

In general, the infinite integrals in the kernel functions can be written in terms of the three independent variables x, y, and s; i.e.

l,j( x, y, s ) = 2 ~ fj(p, s )cos(px )cos(py ) dp

12j( x, y, s ) = 2 ~ l fl(p, s )sin(px )cos(py ) dp

where O<-x, y<-l, s=s.+is~, s . > 0 , - ~ < s i < + ~ , i = V ' - ~ , and the functions f j ( p , s ) ( j = l , 2 , 3 , 4 , 5 ) ( j # 2 for 12s(x, y, s) and Iaj(x, y, s)) are

$ $ s fl(p,s) <p+ p2V~-~s)2, f2tp, S )=p t p+ p2V~-~s ) , fatp, S ) = p 2 + s + p V p 2 + s

ps = p2 - s - pX/p2 + s f4(p,s) 2p(pZ+s)+(2p2+s) p2X/-~-~s, A(p , s ) p 2 + s + p p2X/-p~ s. (B2)

The functions fj(p, s) ( j = I, 2, 3, 4, 5) have the approximations a/(p, s) as Isl/p2 ---, 0, where

s 2s 2s al(p,s)=4p2 + 2 s , a2(p,s)=4p2 + s, a3(p,s)=402 + 3 s

s -9s a4(p,s) =4pz + 4s, as(p,s)=]2pz+lOs. (B3)

We can then re-write the infinite integrals (B1) in the following forms:

°

l,s(x,y,s)=~t [fi(P,s)-astp, s)]cos(px)cos(.y)dp+ l'l'l(x,y,s)

2 ~1 121(x, y, s) = ~r ~ p [f/(p' s) - as( p, s ) ]sin(px )cos(py ) dp + l~<x, y, s)

6s(x, y, s) = ~ [~(o, s) - aj(o, s)lsin(ox)sin(oy) do + r~j(x, y, s) (B4)


where fi-<x, y -< 1, Re(s) >0 , and

lj~x, y, s) = aj(p, s)cos(px)cos(py) dp

2 f~l 17/(x, y, s) = ~ Jo -~a/(p, s)sm(px)eos(py) dp

I~j{x, y, s) = ~ aj(p, s)sin(ox)sin(oy) dp. (B5)

The infinite integrals in (B5) are integrated in exact closed-form with the help of the following results:

• -

I , ~ s l n ( p z ) do = ~(1 - e - x ~ ) (a6)

where X >- 0, Re(s) > 0. The adjusted infinile integrals in (B4) can be evaluated by using the proceeding limit technique; i.e.

S; S? r" S? +' F(p,x ,y ,s)dp ~- F(p,x ,y ,s )dp+ F ( p , x , y , s ) d p + " "+ F(p,x ,y ,s)dp (B7) "SAt) n

where 0 < Ao < A i • • • < A ,+ i is a sequence of numbers that approaches infinity. Each finite integral on the r ight-hand side is proper and can be calculated by using the Simpson's quadrature based adaptively iterative integration. The limits of A0, A1 . . . . . A,÷~ are chosen as the zeros of the oscillatory part of the integrands, i.e., ei ther sin p(x ± y ) or cos p(x ±y). The evaluation of these proceeding finite integrals is automatically terminated provided the following condition is satisfied

f~ an+l F(p) dp <- e~ (B8) An

where ~,. is an assigned absolute error. As a result, the adjusted infinite integrals in equations (B4) can be evaluated with greater accuracy and faster

convergence, using the Simpson's quadrature based proceeding limit technique. With the above procedure, we can accommodate the problems associated with infinite integrals which contain rapidly oscillatory integrands.

A P P E N D I X C

,Algorithm for the Numerical Inversion of Laplace Transforms

Let f(t) be a real function of t, with f(t) = 0 for t < 0; the Laplace transform and its inversion formula are defined as follows:

, ( s ) = f ; f ( t ) e - " d t , f ( t )=2~f f f i~ f (s )eS 'ds (CI)

where s = a + ito, a > a 0 > 0, f(t) = O(e ~') as t--~ ~, a 0 is a constant that exceeds the real part of all singularities of j?(s), and the integrals are in the sense of the Cauchy principal value. In the ensuing discussion, it is therefore assumed that equations (C1) are defined for Re(s )>-a > 0.

The following algorithm for numerical inversion of Laplace transforms was proposed by Crump [39]. The basic formula for approximating the inverse Laplace transform was derived by expanding the integral in the form of a Fourier series; i.e.

at 1 ~ e - _ . m r i . 4 , f(t) T {2J~(a)+ Re[ ~,=l.~(a+l T )e ]} (C2) where 0<t - -<2T, ~b ::n't/T.

The epsilon technique suggested by MacDonald [40] was incorporated in the algorithm for a faster convergent approximation of the infinite summation in (C2), i.e.

where

,4',7,> +,- Re[ ~ k(a+'~]e '°<,] L n - - I . 1 / J

. ~ ' - - o, .~" = s., = Re ,~ a + , ¥ / e - ]

1 ~(~l) g(rn+ I) p+,=.p_, 4 ep('+°-etp"')' m = 1 , 2 , 3 . . . . . 2 M + l .

The constant parameter a in equation (C2) was obtained from error analysis as follows:

In(e.) a = t~, 2T

where • o is a relative error and a, is a number larger than max{Re(P); where P is a pole of •(s)}.

(c3)

(C4)

(c5)

E$ 3 3 - 1 1 - I

1662 Z . Q . YUE and A. P. S. S E L V A D U R A I

In order to apply the above algorithm to the study of poroelastic media, we made two modifications as follows. The first modification is implemented to reduce the round off error induced in the direct calculation of the partial sums of the Fourier series in Crump's algorithm. By defining,

b,,,+2=b,,,+l=O, b,,= a+t--~ + 2 c o s ~ b b n + l - b n + 2, n=m,m-1 . . . . . 1 (C6)

the partial sums can be evaluated more accurately by the following equation

S,, = Re[bl(cos ~b + i sin ~b) - b2]. (C7)

The second modification is suggested for overcoming Gibbs phenomena in Crump's algorithm. Instead of increasing the total number 2M + 1 of the partial sums for convergence, we fix the total number 2M + 1 and use a time dependent T as T = a~.t, where a~. is a constant. Many functions such as the Heaviside step function, the exponential function and the error function have been employed to verify the accuracy and efficiency of the algorithm for inverting Laplace transforms. In this study, it was observed that by choosing the constants as ~ a = 1 0 - 6 ~ 1 0 - 8 ; M = 5 - 1 5 ; a, = 0.9 ~ 1.95; a I = 0, the adopted algorithm gives optimal results in the time interval of 10 -5 -< t -< 103. This range of non-dimensional time factor meets the requirements applicable to the study of a variety of fluid saturated poroelastic media.

Documents

Int. Z Engng Sci. Vol. 33, No. II, pp. 1633-1662, 1995 ... · either t---~ 0 + or t--* +00 and ... influence of non-classical interface phenomena including friction ... where 8ij