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Interaction of two elliptic inclusions
Int. ,7. Solids Structures Vol. 32, No. 3/4,pp. 451~466, 1995
~
Copyright© 1994ElsevierScienceLtd Printedin Great Britain.All rightsreserved 002~7683/95 $9.50 + .00
Pergamon 0020-7683(94) 00130-8
I N T E R A C T I O N O F TWO E L L I P T I C I N C L U S I O N S MARK J. M E I S N E R Department of Civil Engineering, University of Arizona, Arizona, U.S.A.
and D E M I T R I S A. K O U R I S Department of Mechanical and Aerospace Engineering, Arizona State University, Arizona, U.S.A. (Received
23
June
1994)
Abstract--The paper presents the solution of a boundary value problem involving two interacting elliptical inhomogeneities in an infinite elastic body. The loading cases considered include a far-field biaxial tension and a thermally induced residual field that is modeled by uniform eigenstrains sustained by the inhomogeneities. The mathematical formulation is based upon the Papkovich-Neuber displacement approach. Certain interesting aspects of the solution and the effects of perfectly bonded and slipping interfaces are discussed in some detail.
STATEMENT OF THE P R O B L E M A N D THE M A T H E M A T I C A L M O D E L
Consider an infinite region with two elliptic inhomogeneities, ~1 and ~'~2, with centers at O1 and 02, respectively. Let the centers be at the origins of the Cartesian coordinates (xl,Yl) and (x2,y2), and the Xl, X2-axis be the center line, as illustrated in Fig. 1. If the central distance O102 = ~, then Xl = X2 "]- ~,
yl=Y2
•
(1)
It is convenient to use a coordinate system such that the boundaries involved in the problem correspond to a constant value of one coordinate. The elliptical coordinates (c~,r ) will be used in the present investigation. The elliptic coordinates are obtained from the coordinate transformation
xi = Ccosh~icosfli, yi = Csinheesinfli,
(2)
where i = 1, 2 and C is the eccentricity of the inclusions. The total displacement vector u may be represented by
u = Ul +u2,
(3)
where Ul and u2 are the displacement vectors corresponding to the coordinate centers, O 1 and 02, respectively. Utilizing the Papkovich-Neuber displacement formulation (Papkovich, 1932 ; Neuber, 1934), the displacement fields, Ul and u2, are given by
grad [q001-~-Xl q)ll
+ Y l (/)21] - - (g-~- 1)[(Pl 1,~021]
(4)
2Gu2 = grad [(P02-~-x2(P12~-Y2(])22] -- (g'-t- 1)[(p12,q~22],
(5)
2GUl =
and
where G is the shear modulus, tc is the Kolosov constant, and (Pu are arbitrary harmonic 451
M. J. Meisner and D. A. Kouris
452
Y2
Yl
0~0
> X2
X1
I I
b I Y
~
~
0x
Fig. 1. Geometry of the problem and the elliptical coordinate system.
functions. [q~, ~0mn]corresponds to ¢Pu for the x-component and (49mnfor the y-component of the displacement vector. In order to satisfy the boundary conditions along f~l (el = c~0), it is necessary to express eqn (5) in terms of the O, coordinate system. Substitution of eqn (1) into eqn (5) leads to
2Gu2 =
grad [@02- - ~(/912 ~-Xl (~012~-Yl (D22] - - (KT-~ 1)[q~,2, ~022].
For the boundary conditions along f~2 (0~2
=
(6)
(~0), a similar procedure yields
2Gul = grad [~Ool+ ~o,1 --~x2(D11 "}-Y2~21] -- (/¢-~- 1)[~011, ~921]"
(7)
Thermal loadin 9
The residual field due to thermal loading is modeled by a pair of uniform eigenstrains (e*, e*). These eigenstrains (Mura, 1987) are proportional to the temperature change and
Interaction of two elliptic inclusions
453
the mismatch of the inclusion/matrix thermal expansion coefficients. Consequently, residual displacements are introduced in the inhomogeneities according to a~* = ~*x,
b/y-*= e*y.
(8)
Their corresponding components in elliptical coordinates are
C2h zT*= ~ sinh 2c~[(1 + cos 2fl)e*+ (1 - cos 2fl)e*], C2h ~7~'= ~ - - sin 2/3[(1 + c o s h 2c0e.*+ (1 - c o s h 2:0e*],
(9)
where 1
h = C(cosh 2c~-cos 2--1/2 /s) '
(10)
When the inhomogeneities are perfectly bonded, the boundary conditions along the elliptical interfaces (e, = c¢0)are u ~ = a ~ + - u~, *
us = a~ +a~,
a s = 6-~, and
z~n = %a.
(11)
Here, (u~, us, o-~,z~) denote elastic quantities in the matrix, (a~, ~Tp,~ , %a) denote elastic quantities in the inclusion, and (u~, u~) indicate the contribution of the thermal field. In the case of a slipping interface, the boundary conditions become
u~=u~+u~,
o-~=g-~,
z~p=%p=0,
(12)
due to the assumption of perfect slip, which implies that no shear tractions can be sustained by the interface.
Biaxial tension In the case of biaxial tension (Z~, Ty) at infinity, the applied load is represented by the following potential set : 1 ~Oo = ~ ( r ~ - 75)(~ + 1)(x 2 _ y2),
~o,
1 =
-
;lvx,
Z--
0,2 = - ~ r~y.
(13)
The boundary conditions along the two interfaces are given by u~ + u ° = a~,
u s + u~ = us~, o-~+ o-° = ~ ,
and
"c~ + z°n = ~n
(perfect bonding), (14)
and by
454
M.J. Meisner and D. A. Kouris
u ~ + u ° = a~,
a~ +Cr° = ~ ,
"c~p+~°p = ~B = 0
(perfect slip).
(15)
The quantities u~°,u~,°cry°,and ~o denote the contribution of the applied tension and are derived from the potential functions given in eqn (13).
SOLUTION
OF
THE
BOUNDARY
VALUE
PROBLEM
If a set of harmonic displacement potentials can be determined such that the boundary conditions along the interfaces and the far field are satisfied, a unique solution can be obtained. Based upon geometric considerations, the Papkovich-Neuber displacement potentials for the matrix (~ > c%) are chosen as
(fl0i ~ P 0
•
i
i -ncc,cosnfli Ane 1
@l i ~- P0
n=l
Bn e n~,cos nfl~
go2i = 0,
(16)
where po is a normalizing factor related to the applied load (thermal or mechanical). Similarly, the displacement potentials for the two inhomogeneities (c~i< c~0)are chosen as
qSoi = Po ~ A-incosh nc~icos nfii n=l (Pli =
P0 ~,, /~/coshnei cosnfli n=l
02i = O.
(17)
The potential functions given in eqns (16)-(17) represent the disturbance of the elastic field in the matrix and the inclusions, respectively, due to the presence of the inclusions. As a result, their contribution decays away from the inhomogeneities. The boundary conditions along the two elliptical interfaces provide the means for the evaluation of the unknown coefficients (i.e. F~, A~,,B;, A~, and/~,). By enforcing the boundary conditions, eight equations are obtained for the interfaces ~'~1(~1 = ~0) and ft2(ct2 = c~0). However, since the inhomogeneities are identical and a symmetric load is applied, only one of the interfaces needs to be considered, provided that the relations presented in Appendix A are appropriately utilized and that the following symmetry relations among the coefficients are taken into account when al -- c~2= e0 : Fo = F~o = F~ A.
=
A 1 =(
B.
=
B1
=
--
.._., l'lnA2
l ' l n + l R:,,, 2 (--=,
A . = A- .1 = ( - 1 ) ' z { . :
B n = B -. 1 = ( - -
1)n+lB2.
(18)
The boundary conditions in the case of eigenstrain loading are presented below; the equations are identical to the ones corresponding to the mechanical load, with the exception of the forcing terms. Consequently, the continuity of the normal displacement given by u~ = fi~+ a* at el -- e0 yields
Interaction of two elliptic inclusions
455
C
ro6n,o-A.V~,- -~[Bn_,U.,+ ~.+ lUg:]
C ~, Bm[(l-(~.,i)dm,n-lUc2-1-(1-Ono)dmn+lUc3] m=l
'
1
C
'
_
F C 2
4 sinh 2ao[(e* +
e*)6.,o + (e~ - ey)6.,z] (n = O, 1,2 .... ).
an+
For the displacements in the tangential direction, up =
c
(19)
~ at e, = eo gives
[
}
A.VA, q--4 [B,,_IVBI +B,,+IVB2]-1-(1--6na ) FoZ.-I- ~ (Am-l-~Bm)d,.,n Vc, m=l
C
-
-
C
_
L B,.[(1-6. Odin._, Vc2-t-dm.+, Vc3] - ~[A.VA,-4- ~(B._, ~-t-B.+ 17m)]
4 m = l
,
,
,
I
C 2
4 [(e* + e*) + (e* - e*) cosh 2~o]6.,2
(n = 1,2,...).
(20)
where F = 6/G. The requirement for continuity of normal and shear tractions yields
1
1
2 Fo sinh 2~o6.,0- ~
C
16 [Bn- 3SB' -
4 FoZ._2+
[An_2SA1--AnSA2--~-An+2SA3]
B._ laB2 -- Bn+ 1SB3~t-Bn+ 3SB4]
rn=l
(Am+~B.,)dm,,,_2 S c l + ~
m=l
FoZ.+
1
(Am+~Bm)d,.,. Sc2
"
C co + -~ E Bm[dm,n- 3gc4--dm,n- lScs-drn,n+ lSc6"~-dm,n+3Sc7] m=l
+ +
and
1
C "~ It)
[~--3~1--~--1~2--~+1~3"~-~+3~4]
=
0
(n =
0, 1 , 2 , 3 , . . )
(21)
M. J. Meisner and D. A. Kouris
456 1
1
-- -~ Fo3,, 2 -- 4 [An-zTA1-- AnTAz + An+ 2TA3] C
16 [B,_ 3Tm -- Bn_ I TB2-- Bn+ I TB3 + B,+ 3TB4]
+ ] [FoZ,_ 2 +
+ 4 F°Zn+2-'l-
(Am+~Bm)dm,n+2Tc3 m=l
C o0 16 y' Om[dm'n-3Tc4--dm'n-lTc5--dm'n+lTc6-l-dm'n+3Tc7] m=l
1
4 [.'~,,_ 2TA,---4,,TA2 +-'~,,+ 2TA3] C 16 [/~,_3Ts,-B,-,Ts2-B,+lT83+,0,+3TB4] = 0
(n = 1 , 2 , 3 , . . ) ,
(22)
respectively. 6i,j denotes Kronecker's delta; U~j, V,j, S Uand T Uare known functions of n, c~0, x and ~ (Appendix B). After solving the system of eqns (19)-(22) for the unknown coefficients F0, A,, B,, An, and B,, stresses and displacements anywhere in the elastic body can be determined. The series converge as n increases and no more than 10 terms are required for matching the boundary conditions to a level of three significant figures. The mathematical formulation for the case of perfect slip is similar to the one presented above [using the conditions (12) or (15)].
RESULTS AND DISCUSSION Based upon the formulation presented above, the solution can be obtained for any combination of the eigenstrains (ex, ey) and the far-field tension (T~, Ty). Results are presented for the cases of uniform thermal expansion e* = e* and allaround tension Tx = Ty. Poisson's ratios of the matrix and the inhomogeneities have been assumed equal (v = 7 = 0.3). Variations of the inclusion shape, shear moduli ratio and relative distance between the inhomogeneities are introduced through the parameters s, F and 2, respectively. These dimensionless parameters are defined by a
s=~,
F=~
G
and
2-2b.
(23)
Figures 2-5 illustrate the results in the case of uniform eigenstrains e* = e*. By keeping the aspect and shear moduli ratios constant, the dependence of the normal stress a= along the interface on the inclusion relative distance can be determined (Fig. 2). As the relative distance 2 decreases, the stress concentration increases, particularly along the slipping interface. For values of 2 greater than 2.5 (separation distance is greater than three times the major semi-axis of the elliptic inhomogeneity), the interaction effects are no longer present and the solution for the single inclusion in an infinite medium is recovered. In analyzing the normal stress along the interfaces, variations of the relative stiffness F (G/G) have also been considered. As the inclusions become stiffer relative to the matrix (F increases), the absolute values of the matrix interfacial stresses increase. In both cases (perfect bond and sliding) the normal stress o-~ assumes an absolute maximum at fl = 0, with the higher value corresponding to the case of perfect slip. The effect of the shear
Interaction of two elliptic inclusions -0.25 '
I
s = 0.50 F=2
iir
-0.50 '
11%_
\
-0.75 ' -1.00
po
457 7
'
/~
-1.25'
;:1""--
~.= 1.125
¢
;',, = 1.25
II
Z= 1.5
I~
L=2.5-~
-k.
t
-1.50 ' //
t
-1.75 ' /
-2.00 ' 0
45
0.0'
= o.~o r=21 .,5 - ~ / ~ ' -
90
135
180
135
180
ll~w
-0.5 '
-1.0
Po
'
-1.5 '
-2.0 '
-2.5 '
/
-3.0 '
0
45
90
Fig. 2. Eigenstrain loading : normal stress along the interface as a function of the relative distance between the inclusions for (a) perfect bonding, (b) sliding.
moduli ratio F (relative stiffness) on the hoop stress o-8 is more pronounced in the case of perfect bonding. Nevertheless, higher stress concentrations are again obtained at fl = 0. When the interface is free to slip, the hoop stress crp shown in Fig. 3 increases considerably at fl = 0 with decreasing 2. The stress in the y-direction along the inclusion central line is shown in Figs 4 and 5. In the first case (Fig. 4), the relative stiffness varies while the aspect and distance ratios are kept constant. It is clear that the variation of ay in the inclusions and the matrix, as well as the discontinuity at the interface, are dependent upon the condition of the interface (bonded vs slipping interface). The examination of the aspect ratio effect yields the stress distribution shown in Fig. 5. In the case of perfect slip along the interface, the discontinuity of O-yat the interface increases drastically as s ~ 0. The results for s = 0.99 (a ~ b -- 2), correspond to the solution of two circular inclusions given by Kouris and Tsuchida (1991). Finally, the effects of various shear moduli and aspect ratios on the interfacial matrix stresses are investigated. It is found that the normal matrix stress ax at the interface remains compressive for all values of the relative distance 2 between the inclusions. This is not the case, however, for the normal stress ~y. It can be observed that the distributions of ay for perfect bonding and sliding are drastically different. When the effect of different aspect
458
M.J. Meisner and D. A. Kouris
0.95
(a)
i
s = 0.50
F=2
/
i
0.90
1.125 L= 1.25 Z=
0.85 8 B
0.80
Po
L=2.5 - ~ /
e
0.75
0.70
0.65
0.60 0
(b)
45
90
135
3.0
180
t s = 0.50 r=2
I
2.5
L=1.125 2.0
$
L=l.~
u
k=l.5
o
~=2.5-~
1.5
Po
0.5' 0.0' 0
45
90
135
180
Fig. 3. Eigenstrain loading : hoop stress along the interface as a function of the relative distance between the inclusions for (a) perfect bonding, (b) sliding.
Interaction of two elliptic inclusions
(a)
459
1 , 2 . sL= = 0.501.5 1.0-
0,8-
i
0.6 L
ff_.L PO
~
} tr
0.4-
F=I F=2 F=5 F=10
•
-=
|
0.2-
~
oo"
i i
-0.2 ~
-
'
i
~
J
~
-0,4 -0.6 0,0
0,5
2.5,.~=o.5o
(b)
2.0--L
1.0
=
1
1.0
.
~
t
5
1.5
t
!
0.5;
ff_L Po
I
i........}i
2.0
2.5
.....
.I
I
I
~
*
F=2
~
F=5
3.0
I
22i -1,5 -2.0 -"
-2.5" -3.0 '" ' -3.5 '" 0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
Fig. 4. Eigenstrain loading: stress distribution along the inclusion central line for various F, (a) perfect bonding, (b) sliding.
460
M . J . Meisner and D. A. Kouris
(a)
1.00'
F=2 L = 1.5 1
0.75'
s = 0.10 0.50'
m
s = 0.50 S = 0.99
II
Cry Po
0"00 'l.~-~-~ ~H~-~H~-~~
.
.
.
,,.
-z~ -~-~ o,~-t
-
-0.50 j ~i u , _ _~.~. ._~_ ~m-m -0.75 .i 0.0
50 '
(b)
.
,
0.25'
-0.25
.
40'
i
!
;
]
0.5
i
1.0
1.5
i
~
2.0
~
2.5
3.0
F =9
~,= 1.5
:
'
~
i
'
30 .........................
s=0.10 s =0.50
m
s = 0.99
~._L PO
20.
i
10'
.
0 ~i. . . . . . . . .
-10' 0.0
i
i_ _
.
0.5
.
_
.
.
.
1.0
.
.
.
.
.
1.5
7
2.0
1
2.5
3.0
x
Fig. 5. Eigenstrain loading : stress distribution along the inclusion central line as a function of the inclusion shape, (a) perfect bonding, (b) sliding.
Interaction o f two elliptic inclusions
(a)
461
1.750 S = 0,50 ~=1.5 j 1.625
Nk~
1.500
.N~
!F=2 F=I
--
i'=5
"'
1.375
PO
~
~
V = 10 .......
1,250"
1.125 - ~
1.000 ~ ~
0.875 ' 0
45
90
135
180
2.25
(b)
s 2.00:
F=2
m O
~--~-~
0.50
/
F=I 4
1.75 -
=
~
F=5 "
F=10
1.50"
Po 1.25 -
1,00 •
0.75 45
90
135
180
Fig. 6. Biaxial tension : normal stress along the interface as a function of F for (a) perfect bonding, (b) sliding.
5AS 3 2 : 3 / 4 - H
M. J. Meisner and D. A. Konris
462 1.125"
(a)
1
I 1 f
s = 0.50 L = 1.5
0.875'
po
F=I
7
1.000'
i
J
1-
.I.
!
F=5 0.750'
\
!---+
0.625'
W'T#
F = 10
+
0.500'
I
45
(b)
F=2 .1
I
90
135
180
1.125'
I
1.000
1
0.875 r 0.750 r
0.625'
xx
//if
po 0.500
F=I F=2 F=5 F=I0
I []
0.375 -
~
0.250 0.125 0
45
90
135
180
Fig. 7. Biaxial tension: hoop stress along the interface as a function of F for (a) perfect bonding, (b) sliding.
ratios is considered, a loss of the interfacial bonding corresponds to very high tensile values of ay. As the value of s decreases, the differences between perfect bonding and sliding become more pronounced• In the case of all-around tension (Tx = Ty), the concentration of the normal stress a~ along the interface is proportional to the relative stiffness F (Fig. 6). However, the opposite is true in the case of the matrix stress a~ (Fig. 7). As the relative distance between the inhomogeneities decreases, the matrix stresses as and a~ increase (Fig. 8). CONCLUSIONS The present study analyzes the interaction of two interacting inhomogeneities subjected to thermo-mechanical loading. The problem was formulated using the Papkovich-Neuber displacement potentials and analytical solutions were obtained for the cases of perfectly bonded and slipping interfaces• Unlike Eshelby's (1957) result for the single inclusion, the stress field inside the interacting inhomogeneities is no longer uniform. The local elastic field is determined in
Interaction of two elliptic inclusions
463
1.375"
(a)
t S =
0.50
F=2 1.325" 1.275"
L=I.I~
~=1.25 L=1.5 ~=2.5-~
i
1.225-
,a,
#'
-~
P0 1.t75"
i]2_
1.125 •
#.-
1.075" 45
90
180
35
0.875
(b)
!
_
i
l
s = 0.50 .... r
°.8°5 L_iili 0.8550.845 ' ~
......
y/
PO 0.835' " ~ J t ~
o825
~= 1.5
• •
~
~.= 2.5-
\
i] ............
~
!
I
0.815 [ 0
i
45
90
135
180
Fig. 8. Biaxial tension : normal and hoop stresses along the interface as a function of the relative distance between the inclusions in the case of perfect bonding.
the form of infinite series and is dependent upon the relative distance, the aspect ratio, and the elastic properties of the inhomogcneities. In addition, loss of the interfacial bond yields high stress concentrations. A number of special cases can be explored by considering the appropriate limits of the shear moduli ratio F and the aspect ratio s. As s approaches unity, the solution for two circular inclusions are obtained. In the case of s--* 0, the solution corresponds to the problem of two thin, interacting, inserts in an infinite body. The crack and anticrack solutions can then be obtained by setting F --* 0 and F --* o% respectively. Acknowledgements--The authors are pleased to acknowledge support from the Air Force Office of Scientific Research through Award AFOSR 90-0235.
REFERENCES Cooke, J. C. (1959). Some relations between Bessel and Legendre functions. Q. J. Math. (Oxford Ser.) 12, 322328. Erdelyi, A. (1954). Tables oflntegral Transforms, Vol. 1. McGraw-Hill, New York. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. Ser. A 241, 376-396. Kouris, D. and Tsuchida, E. (1991). On the elastic interaction between two fibers in a continuous fiber composite under thermal loading. Mech. Mater. 12, 131-146.
464
M. J. Meisner and D. A. Kouris
Mura, T. (1987). Micromechanics of Defects in Solids. Martinus Nijhoff, The Hague. Neuber, H. (1934). Ein neuer Ansatz zur L6sung raumlicher Probleme der Elastizitgtstheorie. Z. Angew. Math. Mech. 14, 203-212. Papkovich, P. F. (1932), Solution g6n&ale des 6quations diff6rentielles fondamentales d'61asticit6, exprim6e par trois fonctions harmarmoniques. C. R. Acad. Sci. 195, 513-515.
APPENDIX A The following relations between the elliptic harmonic functions are utilized in the expressions for the boundary conditions along f~(el = co) and f~2(e2 = Co) :
O~I = ~ concoshrlo~2 cosnfl2, n=O
e2 = ~ Z . coshncq cosn~l , n=0
e -m% cosm31 = ~ (-- 1)"din,. coshnc~2 cosn/~2, n=0
e ' ~ cos mfl2 = ~ (-- 1)mdm,,cosh naa cos nflt.
(A1)
n=0
The coefficients co. and Z . are given by
g ( - 1 )
~ .
coo = Z0 = c~a+ 22,__1- - - - ~ a 2 ~ . 0
(A2)
and 2 & (-1) m Z. = co.(-1)" = - - e "=~+ 2. d2m, n n
m=l
m
where n = 1, 2 . . . . . and
C~d= cosh (~).
(A3)
The coefficients din., can be obtained by starting with (Cooke, 1959)
e -m~, cosm/~l = m f o / , . ( 2 c ) 2 - 1 e - ~ l cos2yl d2
(A4)
e -~x2 cos2yz = ~ e , ( - 1)"I,(2C) eoshnc~2 cosn/~2,
(A5)
and
n=0
where x2> 0, m = 1, 2 . . . . . and
{1 °=0 ~"=
2
n=l,2 ....
(A6)
F r o m the relations (1), eqns (A5) and (A6) can be transformed into
e -"~, cosm/~l = m
fo
I m ( 2 C ) 2 - ' e -~* ~ e . ( - 1)"I.(2C) coshm~2cosn32 d2. n = 0
According to the definition of din.. given in eqn (A1), the coefficients dm,. are obtained by
(A7)
Interaction of two elliptic inclusions
465
d~.. = e.m l~lm(2C)I.(2C) e-a( ,q,-1 d2. ~o
(A8)
The evaluation of the integral in eqn (A8) is discussed in Erdelyi (1954). APPENDIX B The functions Uo, Vu, Su, and T,j are the following : UA1
=
n e -"~o,
Unl = (n-- 1 + to) e -("-2)'o + (n -- 1 -- x) e - ' o ,
Un2 = ( n + 1 + x ) e-"~o+ ( n + 1 --to) e -~n+2)~o, Ucl = n sinhn%,
Uc2 = ( n - l + x) sinh ( n - 2)c~o + ( n - 1 - x) sinh n~o, GA1 = nsinhnct0,
Uc3 = ( n + l + x ) s i n h n ~ o + ( n + l - ~ c ) s i n h ( n + 2 ) ~ o ,
Unl = ( n - 1 + ~) sinh ( n - 2)c~o + ( n - 1 - ~) sinh nc~o, /.782 = (n + 1 + t~) sinh no% + (n + 1 - ~) sinh (n + 2)cto,
VB1 = (n-- l - - x ) {e-("-2~o+e-"~o},
VAI = n e - " %
VB2 = ( n + 1 +tc){e-"~o+e-("+2)~o},
Vcl = ncoshnct0,
Vcz = (n - 1 - x) { cosh (n-- 2)C~o+ cosh n~o }, Vc3 = (n+ l +x){coshm% +cosh (n+ 2)cto},
I?A1 = ncoshm%,
l?m = (n -- 1 -- ~) { cosh (n-- 2)cto + cosh n~o }, ITBz = ( n + 1 + ~ ) { coshnc% + c o s h ( n + 2 ) % } , SAI = ( n - 2 ) ( n - 3 ) SA2 = n { ( n + l ) e
e -~"-2~o, ("-2~'o+(n-1)e-("+2)~o}-2e~o6.,1,
SA3 = ( n + 2 ) ( n + 3 ) e - ( " + 2 ) %
Snl = ( n - 3) {(n - 2) e -("-4~'o + (n - 3 - re) e-("-2~o}, Sn2 = ( n - 1){(n+2) e -("-4)~o - 2 ( x - 2 )
e -("-2~o
+ 4e -"~o + ( n - 1 - to) e -("+ 2)~o} - (x + 1)6.,2, Sn3 = ( n + 1){(n+ 1 + x ) e -~"-2)~. - 4 e -"~o + 2 ( x - 2 ) e (,,+2)~o+ ( n - 2 ) e -C"+4~0} - 2 { ( x + 1) e -~o +e-3"o}6., l,
&4 = (n + 3) { (n + 3 + tc) e-{"+ =)=°+ (n + 2) e-{"+4~=°}, Scl= (n-2)(n-3)
cosh ( n - 2 ) % ,
Sc2 = n { (n + 1) cosh ( n - 2)~0 + ( n - 1) cosh (n + 2) c%}, Sc3 = ( n + 2 ) ( n + 3 ) cosh (n+2)Cto, Sc4 = (/'/-- 3) ( (n -- 2) cosh (n-- 4)c~0 + (n -- 3 -- x) cosh (n-- 2)~o }, Sc~ = (n-- 1){(n + 2 ) cosh ( n - 4 ) ~ o - 2 ( x -
2) cosh ( n - 2 ) ~ o
+ 4 cosh n~o + ( n - 1 - x) cosh (n + 2)C~o} - 0 c + 1)6..2,
Sc~ = (n + 1) {(n + 1 + to) cosh ( n - 2)% - 4 cosh n~o + 2 ( x - 2) cosh (n + 2)Cto + (n - 2) cosh (n + 4)cto} - 2 { ( x + 1)e -~o + e - ~o}6.. ~,
Sc7 = (n + 3) { (n + 3 + x) cosh (n + 2)ct 0 + (n + 2) cosh (n + 4)ct o}, S~1 = ( n - 2 ) ( n - 3 )
cosh ( n - 2 ) 7 o ,
Saz = n { (n + 1) cosh (n - 2)c~0 + ( n - 1) cosh (n + 2)C~o} - 2 cosh Cto6..~, Sa3 = ( n + 2 ) ( n + 3 ) cosh (n+2)c~ o, ~nl = ( n - 3 ) { ( n - 2 ) cosh ( n - 4 ) ~ o + ( n - 3 - ~ ) cosh ( n - 2 ) ~ o }, Snz = (n - 1){(n + 2 ) cosh ( n - 4 ) % - 2 ( R -
2) cosh ( n - 2)~o
+ 4 cosh n~o + ( n - 1 - R) cosh (n + 2)~o } - (~ + 1)6., 2, S~3 = ( n + 1){(n+ I +R) cosh ( n - 2)Cto - 4 c o s h nc~o- 2 ( R - 2 )
cosh (n + 2)~o
+ ( n - 2) cosh (n + 4)C~o} - 2 { (R + 2) cosh c~0+ cosh 3C~o} an, 1, Sg4 = (/'/+ 3) {(n + 3 + ~) cosh (n + 2)% + (n + 2) cosh n~o },
466
M . J . Meisner and D. A. Kouris
TAI = ( n - - 2 ) ( n - - 3 ) e -('-2)%, TA2 = n{(n+l)e--('-2)~°+(n--l)e ('+2)~'°}+2e%6 .... TA3 = ( n + 2 ) ( n + 3 ) e -("+2)~o, Tnl = (n--3){(n--4) e -("-4)~o + (n-- 3 -- to) e (,-2)~o}, TB2 = ( n - - 1 ) { n E
(" 4)%--2(x--2)e-(" 2)~o+(n--l--x)e-("+z)~o}+{(x+l)+Ze-2~o}b,,z,
TB3 = ( n + 1){(n+ 1 +to) e-('-2)~o + 2(tc-- 2) e -('+z)"o + n e -("+ 4)%} + 2{(~c+ 2) + 3 e-3~o}6,,~, T84 = (n+ 3){(n+ 3+~:)e-('+2)~°+(n+4)e-("+4)~°}, Tcl = ( n - - 2 ) ( n - - 3 ) sinh ( n - 2 ) a o , Tc2 = n{ (n + 1) sinh (n--2)ao + ( n - 1) sinh (n + 2)ao }, Tc3 = (n + 2) (n + 3) sinh (n + 2)ao,
Zc4
( n - 3){ ( n - 4) sinh ( n - 4)c% + ( n - 3 - x ) sinh ( n - 2)~o ),
Tc5 = (n - 1) {n sinh (n - 4) o~o- 2(x - 2) sinh (n - 2) c%+ (n - 1 - to) sinh (n + 2) ao } + { (x + 1) + 2e- 2~o}6., 2, Tc6 = (n + 1) { (n + 1 + x) sinh (n - 2)~o + 2 (x - 2) sinh (n + 2) 0%+ n sinh (n + 4)c~o} + 2 { (x + 2) + 3 e - 3~o} 5..~, Tc7 = (n + 3) { (n + 3 + x) sinh (n + 2)~o + (n + 4) sinh (n + 4)c~o}, TA~= (n - 2)(n - 3) sinh ( n - 2)c~o, TA2= n { (n + 1) sinh (n - 2) c~o + (n - 1) sinh (n + 2) ~o } + 2 sinh ~o 5.. 1,
TA3
(n + 2 ) ( n + 3) sinh (n+2)O~o, (n - 3) { (n - 4) sinh ( n - 4)% + ( n - 3 - ~) sinh (n - 2)O~o},
TB2
(n - 1) {n sinh (n - 4)cq - 2(~ - 2) sinh (n - 2)~o + (n - 1 - ~) sinh (n + 2)~o } + 2 sinh 2ao,5., 2
TB3
(n + 1) { (n + 1 + ~) sinh (n - 2)C~o+ 2 ( ~ - 2) sinh (n + 2)~o + n sinh (n + 4)~o }
TB4
( n + 3){(n+ 3 + k ) sinh (n + 2)c~o + ( n + 4 ) sinh (n +4)ao}.
+ 2 ~(~ + 2) sinh c~o+ 3 sinh 3c~o) 6., ~,
~
Copyright© 1994ElsevierScienceLtd Printedin Great Britain.All rightsreserved 002~7683/95 $9.50 + .00
Pergamon 0020-7683(94) 00130-8
I N T E R A C T I O N O F TWO E L L I P T I C I N C L U S I O N S MARK J. M E I S N E R Department of Civil Engineering, University of Arizona, Arizona, U.S.A.
and D E M I T R I S A. K O U R I S Department of Mechanical and Aerospace Engineering, Arizona State University, Arizona, U.S.A. (Received
23
June
1994)
Abstract--The paper presents the solution of a boundary value problem involving two interacting elliptical inhomogeneities in an infinite elastic body. The loading cases considered include a far-field biaxial tension and a thermally induced residual field that is modeled by uniform eigenstrains sustained by the inhomogeneities. The mathematical formulation is based upon the Papkovich-Neuber displacement approach. Certain interesting aspects of the solution and the effects of perfectly bonded and slipping interfaces are discussed in some detail.
STATEMENT OF THE P R O B L E M A N D THE M A T H E M A T I C A L M O D E L
Consider an infinite region with two elliptic inhomogeneities, ~1 and ~'~2, with centers at O1 and 02, respectively. Let the centers be at the origins of the Cartesian coordinates (xl,Yl) and (x2,y2), and the Xl, X2-axis be the center line, as illustrated in Fig. 1. If the central distance O102 = ~, then Xl = X2 "]- ~,
yl=Y2
•
(1)
It is convenient to use a coordinate system such that the boundaries involved in the problem correspond to a constant value of one coordinate. The elliptical coordinates (c~,r ) will be used in the present investigation. The elliptic coordinates are obtained from the coordinate transformation
xi = Ccosh~icosfli, yi = Csinheesinfli,
(2)
where i = 1, 2 and C is the eccentricity of the inclusions. The total displacement vector u may be represented by
u = Ul +u2,
(3)
where Ul and u2 are the displacement vectors corresponding to the coordinate centers, O 1 and 02, respectively. Utilizing the Papkovich-Neuber displacement formulation (Papkovich, 1932 ; Neuber, 1934), the displacement fields, Ul and u2, are given by
grad [q001-~-Xl q)ll
+ Y l (/)21] - - (g-~- 1)[(Pl 1,~021]
(4)
2Gu2 = grad [(P02-~-x2(P12~-Y2(])22] -- (g'-t- 1)[(p12,q~22],
(5)
2GUl =
and
where G is the shear modulus, tc is the Kolosov constant, and (Pu are arbitrary harmonic 451
M. J. Meisner and D. A. Kouris
452
Y2
Yl
0~0
> X2
X1
I I
b I Y
~
~
0x
Fig. 1. Geometry of the problem and the elliptical coordinate system.
functions. [q~, ~0mn]corresponds to ¢Pu for the x-component and (49mnfor the y-component of the displacement vector. In order to satisfy the boundary conditions along f~l (el = c~0), it is necessary to express eqn (5) in terms of the O, coordinate system. Substitution of eqn (1) into eqn (5) leads to
2Gu2 =
grad [@02- - ~(/912 ~-Xl (~012~-Yl (D22] - - (KT-~ 1)[q~,2, ~022].
For the boundary conditions along f~2 (0~2
=
(6)
(~0), a similar procedure yields
2Gul = grad [~Ool+ ~o,1 --~x2(D11 "}-Y2~21] -- (/¢-~- 1)[~011, ~921]"
(7)
Thermal loadin 9
The residual field due to thermal loading is modeled by a pair of uniform eigenstrains (e*, e*). These eigenstrains (Mura, 1987) are proportional to the temperature change and
Interaction of two elliptic inclusions
453
the mismatch of the inclusion/matrix thermal expansion coefficients. Consequently, residual displacements are introduced in the inhomogeneities according to a~* = ~*x,
b/y-*= e*y.
(8)
Their corresponding components in elliptical coordinates are
C2h zT*= ~ sinh 2c~[(1 + cos 2fl)e*+ (1 - cos 2fl)e*], C2h ~7~'= ~ - - sin 2/3[(1 + c o s h 2c0e.*+ (1 - c o s h 2:0e*],
(9)
where 1
h = C(cosh 2c~-cos 2--1/2 /s) '
(10)
When the inhomogeneities are perfectly bonded, the boundary conditions along the elliptical interfaces (e, = c¢0)are u ~ = a ~ + - u~, *
us = a~ +a~,
a s = 6-~, and
z~n = %a.
(11)
Here, (u~, us, o-~,z~) denote elastic quantities in the matrix, (a~, ~Tp,~ , %a) denote elastic quantities in the inclusion, and (u~, u~) indicate the contribution of the thermal field. In the case of a slipping interface, the boundary conditions become
u~=u~+u~,
o-~=g-~,
z~p=%p=0,
(12)
due to the assumption of perfect slip, which implies that no shear tractions can be sustained by the interface.
Biaxial tension In the case of biaxial tension (Z~, Ty) at infinity, the applied load is represented by the following potential set : 1 ~Oo = ~ ( r ~ - 75)(~ + 1)(x 2 _ y2),
~o,
1 =
-
;lvx,
Z--
0,2 = - ~ r~y.
(13)
The boundary conditions along the two interfaces are given by u~ + u ° = a~,
u s + u~ = us~, o-~+ o-° = ~ ,
and
"c~ + z°n = ~n
(perfect bonding), (14)
and by
454
M.J. Meisner and D. A. Kouris
u ~ + u ° = a~,
a~ +Cr° = ~ ,
"c~p+~°p = ~B = 0
(perfect slip).
(15)
The quantities u~°,u~,°cry°,and ~o denote the contribution of the applied tension and are derived from the potential functions given in eqn (13).
SOLUTION
OF
THE
BOUNDARY
VALUE
PROBLEM
If a set of harmonic displacement potentials can be determined such that the boundary conditions along the interfaces and the far field are satisfied, a unique solution can be obtained. Based upon geometric considerations, the Papkovich-Neuber displacement potentials for the matrix (~ > c%) are chosen as
(fl0i ~ P 0
•
i
i -ncc,cosnfli Ane 1
@l i ~- P0
n=l
Bn e n~,cos nfl~
go2i = 0,
(16)
where po is a normalizing factor related to the applied load (thermal or mechanical). Similarly, the displacement potentials for the two inhomogeneities (c~i< c~0)are chosen as
qSoi = Po ~ A-incosh nc~icos nfii n=l (Pli =
P0 ~,, /~/coshnei cosnfli n=l
02i = O.
(17)
The potential functions given in eqns (16)-(17) represent the disturbance of the elastic field in the matrix and the inclusions, respectively, due to the presence of the inclusions. As a result, their contribution decays away from the inhomogeneities. The boundary conditions along the two elliptical interfaces provide the means for the evaluation of the unknown coefficients (i.e. F~, A~,,B;, A~, and/~,). By enforcing the boundary conditions, eight equations are obtained for the interfaces ~'~1(~1 = ~0) and ft2(ct2 = c~0). However, since the inhomogeneities are identical and a symmetric load is applied, only one of the interfaces needs to be considered, provided that the relations presented in Appendix A are appropriately utilized and that the following symmetry relations among the coefficients are taken into account when al -- c~2= e0 : Fo = F~o = F~ A.
=
A 1 =(
B.
=
B1
=
--
.._., l'lnA2
l ' l n + l R:,,, 2 (--=,
A . = A- .1 = ( - 1 ) ' z { . :
B n = B -. 1 = ( - -
1)n+lB2.
(18)
The boundary conditions in the case of eigenstrain loading are presented below; the equations are identical to the ones corresponding to the mechanical load, with the exception of the forcing terms. Consequently, the continuity of the normal displacement given by u~ = fi~+ a* at el -- e0 yields
Interaction of two elliptic inclusions
455
C
ro6n,o-A.V~,- -~[Bn_,U.,+ ~.+ lUg:]
C ~, Bm[(l-(~.,i)dm,n-lUc2-1-(1-Ono)dmn+lUc3] m=l
'
1
C
'
_
F C 2
4 sinh 2ao[(e* +
e*)6.,o + (e~ - ey)6.,z] (n = O, 1,2 .... ).
an+
For the displacements in the tangential direction, up =
c
(19)
~ at e, = eo gives
[
}
A.VA, q--4 [B,,_IVBI +B,,+IVB2]-1-(1--6na ) FoZ.-I- ~ (Am-l-~Bm)d,.,n Vc, m=l
C
-
-
C
_
L B,.[(1-6. Odin._, Vc2-t-dm.+, Vc3] - ~[A.VA,-4- ~(B._, ~-t-B.+ 17m)]
4 m = l
,
,
,
I
C 2
4 [(e* + e*) + (e* - e*) cosh 2~o]6.,2
(n = 1,2,...).
(20)
where F = 6/G. The requirement for continuity of normal and shear tractions yields
1
1
2 Fo sinh 2~o6.,0- ~
C
16 [Bn- 3SB' -
4 FoZ._2+
[An_2SA1--AnSA2--~-An+2SA3]
B._ laB2 -- Bn+ 1SB3~t-Bn+ 3SB4]
rn=l
(Am+~B.,)dm,,,_2 S c l + ~
m=l
FoZ.+
1
(Am+~Bm)d,.,. Sc2
"
C co + -~ E Bm[dm,n- 3gc4--dm,n- lScs-drn,n+ lSc6"~-dm,n+3Sc7] m=l
+ +
and
1
C "~ It)
[~--3~1--~--1~2--~+1~3"~-~+3~4]
=
0
(n =
0, 1 , 2 , 3 , . . )
(21)
M. J. Meisner and D. A. Kouris
456 1
1
-- -~ Fo3,, 2 -- 4 [An-zTA1-- AnTAz + An+ 2TA3] C
16 [B,_ 3Tm -- Bn_ I TB2-- Bn+ I TB3 + B,+ 3TB4]
+ ] [FoZ,_ 2 +
+ 4 F°Zn+2-'l-
(Am+~Bm)dm,n+2Tc3 m=l
C o0 16 y' Om[dm'n-3Tc4--dm'n-lTc5--dm'n+lTc6-l-dm'n+3Tc7] m=l
1
4 [.'~,,_ 2TA,---4,,TA2 +-'~,,+ 2TA3] C 16 [/~,_3Ts,-B,-,Ts2-B,+lT83+,0,+3TB4] = 0
(n = 1 , 2 , 3 , . . ) ,
(22)
respectively. 6i,j denotes Kronecker's delta; U~j, V,j, S Uand T Uare known functions of n, c~0, x and ~ (Appendix B). After solving the system of eqns (19)-(22) for the unknown coefficients F0, A,, B,, An, and B,, stresses and displacements anywhere in the elastic body can be determined. The series converge as n increases and no more than 10 terms are required for matching the boundary conditions to a level of three significant figures. The mathematical formulation for the case of perfect slip is similar to the one presented above [using the conditions (12) or (15)].
RESULTS AND DISCUSSION Based upon the formulation presented above, the solution can be obtained for any combination of the eigenstrains (ex, ey) and the far-field tension (T~, Ty). Results are presented for the cases of uniform thermal expansion e* = e* and allaround tension Tx = Ty. Poisson's ratios of the matrix and the inhomogeneities have been assumed equal (v = 7 = 0.3). Variations of the inclusion shape, shear moduli ratio and relative distance between the inhomogeneities are introduced through the parameters s, F and 2, respectively. These dimensionless parameters are defined by a
s=~,
F=~
G
and
2-2b.
(23)
Figures 2-5 illustrate the results in the case of uniform eigenstrains e* = e*. By keeping the aspect and shear moduli ratios constant, the dependence of the normal stress a= along the interface on the inclusion relative distance can be determined (Fig. 2). As the relative distance 2 decreases, the stress concentration increases, particularly along the slipping interface. For values of 2 greater than 2.5 (separation distance is greater than three times the major semi-axis of the elliptic inhomogeneity), the interaction effects are no longer present and the solution for the single inclusion in an infinite medium is recovered. In analyzing the normal stress along the interfaces, variations of the relative stiffness F (G/G) have also been considered. As the inclusions become stiffer relative to the matrix (F increases), the absolute values of the matrix interfacial stresses increase. In both cases (perfect bond and sliding) the normal stress o-~ assumes an absolute maximum at fl = 0, with the higher value corresponding to the case of perfect slip. The effect of the shear
Interaction of two elliptic inclusions -0.25 '
I
s = 0.50 F=2
iir
-0.50 '
11%_
\
-0.75 ' -1.00
po
457 7
'
/~
-1.25'
;:1""--
~.= 1.125
¢
;',, = 1.25
II
Z= 1.5
I~
L=2.5-~
-k.
t
-1.50 ' //
t
-1.75 ' /
-2.00 ' 0
45
0.0'
= o.~o r=21 .,5 - ~ / ~ ' -
90
135
180
135
180
ll~w
-0.5 '
-1.0
Po
'
-1.5 '
-2.0 '
-2.5 '
/
-3.0 '
0
45
90
Fig. 2. Eigenstrain loading : normal stress along the interface as a function of the relative distance between the inclusions for (a) perfect bonding, (b) sliding.
moduli ratio F (relative stiffness) on the hoop stress o-8 is more pronounced in the case of perfect bonding. Nevertheless, higher stress concentrations are again obtained at fl = 0. When the interface is free to slip, the hoop stress crp shown in Fig. 3 increases considerably at fl = 0 with decreasing 2. The stress in the y-direction along the inclusion central line is shown in Figs 4 and 5. In the first case (Fig. 4), the relative stiffness varies while the aspect and distance ratios are kept constant. It is clear that the variation of ay in the inclusions and the matrix, as well as the discontinuity at the interface, are dependent upon the condition of the interface (bonded vs slipping interface). The examination of the aspect ratio effect yields the stress distribution shown in Fig. 5. In the case of perfect slip along the interface, the discontinuity of O-yat the interface increases drastically as s ~ 0. The results for s = 0.99 (a ~ b -- 2), correspond to the solution of two circular inclusions given by Kouris and Tsuchida (1991). Finally, the effects of various shear moduli and aspect ratios on the interfacial matrix stresses are investigated. It is found that the normal matrix stress ax at the interface remains compressive for all values of the relative distance 2 between the inclusions. This is not the case, however, for the normal stress ~y. It can be observed that the distributions of ay for perfect bonding and sliding are drastically different. When the effect of different aspect
458
M.J. Meisner and D. A. Kouris
0.95
(a)
i
s = 0.50
F=2
/
i
0.90
1.125 L= 1.25 Z=
0.85 8 B
0.80
Po
L=2.5 - ~ /
e
0.75
0.70
0.65
0.60 0
(b)
45
90
135
3.0
180
t s = 0.50 r=2
I
2.5
L=1.125 2.0
$
L=l.~
u
k=l.5
o
~=2.5-~
1.5
Po
0.5' 0.0' 0
45
90
135
180
Fig. 3. Eigenstrain loading : hoop stress along the interface as a function of the relative distance between the inclusions for (a) perfect bonding, (b) sliding.
Interaction of two elliptic inclusions
(a)
459
1 , 2 . sL= = 0.501.5 1.0-
0,8-
i
0.6 L
ff_.L PO
~
} tr
0.4-
F=I F=2 F=5 F=10
•
-=
|
0.2-
~
oo"
i i
-0.2 ~
-
'
i
~
J
~
-0,4 -0.6 0,0
0,5
2.5,.~=o.5o
(b)
2.0--L
1.0
=
1
1.0
.
~
t
5
1.5
t
!
0.5;
ff_L Po
I
i........}i
2.0
2.5
.....
.I
I
I
~
*
F=2
~
F=5
3.0
I
22i -1,5 -2.0 -"
-2.5" -3.0 '" ' -3.5 '" 0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
Fig. 4. Eigenstrain loading: stress distribution along the inclusion central line for various F, (a) perfect bonding, (b) sliding.
460
M . J . Meisner and D. A. Kouris
(a)
1.00'
F=2 L = 1.5 1
0.75'
s = 0.10 0.50'
m
s = 0.50 S = 0.99
II
Cry Po
0"00 'l.~-~-~ ~H~-~H~-~~
.
.
.
,,.
-z~ -~-~ o,~-t
-
-0.50 j ~i u , _ _~.~. ._~_ ~m-m -0.75 .i 0.0
50 '
(b)
.
,
0.25'
-0.25
.
40'
i
!
;
]
0.5
i
1.0
1.5
i
~
2.0
~
2.5
3.0
F =9
~,= 1.5
:
'
~
i
'
30 .........................
s=0.10 s =0.50
m
s = 0.99
~._L PO
20.
i
10'
.
0 ~i. . . . . . . . .
-10' 0.0
i
i_ _
.
0.5
.
_
.
.
.
1.0
.
.
.
.
.
1.5
7
2.0
1
2.5
3.0
x
Fig. 5. Eigenstrain loading : stress distribution along the inclusion central line as a function of the inclusion shape, (a) perfect bonding, (b) sliding.
Interaction o f two elliptic inclusions
(a)
461
1.750 S = 0,50 ~=1.5 j 1.625
Nk~
1.500
.N~
!F=2 F=I
--
i'=5
"'
1.375
PO
~
~
V = 10 .......
1,250"
1.125 - ~
1.000 ~ ~
0.875 ' 0
45
90
135
180
2.25
(b)
s 2.00:
F=2
m O
~--~-~
0.50
/
F=I 4
1.75 -
=
~
F=5 "
F=10
1.50"
Po 1.25 -
1,00 •
0.75 45
90
135
180
Fig. 6. Biaxial tension : normal stress along the interface as a function of F for (a) perfect bonding, (b) sliding.
5AS 3 2 : 3 / 4 - H
M. J. Meisner and D. A. Konris
462 1.125"
(a)
1
I 1 f
s = 0.50 L = 1.5
0.875'
po
F=I
7
1.000'
i
J
1-
.I.
!
F=5 0.750'
\
!---+
0.625'
W'T#
F = 10
+
0.500'
I
45
(b)
F=2 .1
I
90
135
180
1.125'
I
1.000
1
0.875 r 0.750 r
0.625'
xx
//if
po 0.500
F=I F=2 F=5 F=I0
I []
0.375 -
~
0.250 0.125 0
45
90
135
180
Fig. 7. Biaxial tension: hoop stress along the interface as a function of F for (a) perfect bonding, (b) sliding.
ratios is considered, a loss of the interfacial bonding corresponds to very high tensile values of ay. As the value of s decreases, the differences between perfect bonding and sliding become more pronounced• In the case of all-around tension (Tx = Ty), the concentration of the normal stress a~ along the interface is proportional to the relative stiffness F (Fig. 6). However, the opposite is true in the case of the matrix stress a~ (Fig. 7). As the relative distance between the inhomogeneities decreases, the matrix stresses as and a~ increase (Fig. 8). CONCLUSIONS The present study analyzes the interaction of two interacting inhomogeneities subjected to thermo-mechanical loading. The problem was formulated using the Papkovich-Neuber displacement potentials and analytical solutions were obtained for the cases of perfectly bonded and slipping interfaces• Unlike Eshelby's (1957) result for the single inclusion, the stress field inside the interacting inhomogeneities is no longer uniform. The local elastic field is determined in
Interaction of two elliptic inclusions
463
1.375"
(a)
t S =
0.50
F=2 1.325" 1.275"
L=I.I~
~=1.25 L=1.5 ~=2.5-~
i
1.225-
,a,
#'
-~
P0 1.t75"
i]2_
1.125 •
#.-
1.075" 45
90
180
35
0.875
(b)
!
_
i
l
s = 0.50 .... r
°.8°5 L_iili 0.8550.845 ' ~
......
y/
PO 0.835' " ~ J t ~
o825
~= 1.5
• •
~
~.= 2.5-
\
i] ............
~
!
I
0.815 [ 0
i
45
90
135
180
Fig. 8. Biaxial tension : normal and hoop stresses along the interface as a function of the relative distance between the inclusions in the case of perfect bonding.
the form of infinite series and is dependent upon the relative distance, the aspect ratio, and the elastic properties of the inhomogcneities. In addition, loss of the interfacial bond yields high stress concentrations. A number of special cases can be explored by considering the appropriate limits of the shear moduli ratio F and the aspect ratio s. As s approaches unity, the solution for two circular inclusions are obtained. In the case of s--* 0, the solution corresponds to the problem of two thin, interacting, inserts in an infinite body. The crack and anticrack solutions can then be obtained by setting F --* 0 and F --* o% respectively. Acknowledgements--The authors are pleased to acknowledge support from the Air Force Office of Scientific Research through Award AFOSR 90-0235.
REFERENCES Cooke, J. C. (1959). Some relations between Bessel and Legendre functions. Q. J. Math. (Oxford Ser.) 12, 322328. Erdelyi, A. (1954). Tables oflntegral Transforms, Vol. 1. McGraw-Hill, New York. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. Ser. A 241, 376-396. Kouris, D. and Tsuchida, E. (1991). On the elastic interaction between two fibers in a continuous fiber composite under thermal loading. Mech. Mater. 12, 131-146.
464
M. J. Meisner and D. A. Kouris
Mura, T. (1987). Micromechanics of Defects in Solids. Martinus Nijhoff, The Hague. Neuber, H. (1934). Ein neuer Ansatz zur L6sung raumlicher Probleme der Elastizitgtstheorie. Z. Angew. Math. Mech. 14, 203-212. Papkovich, P. F. (1932), Solution g6n&ale des 6quations diff6rentielles fondamentales d'61asticit6, exprim6e par trois fonctions harmarmoniques. C. R. Acad. Sci. 195, 513-515.
APPENDIX A The following relations between the elliptic harmonic functions are utilized in the expressions for the boundary conditions along f~(el = co) and f~2(e2 = Co) :
O~I = ~ concoshrlo~2 cosnfl2, n=O
e2 = ~ Z . coshncq cosn~l , n=0
e -m% cosm31 = ~ (-- 1)"din,. coshnc~2 cosn/~2, n=0
e ' ~ cos mfl2 = ~ (-- 1)mdm,,cosh naa cos nflt.
(A1)
n=0
The coefficients co. and Z . are given by
g ( - 1 )
~ .
coo = Z0 = c~a+ 22,__1- - - - ~ a 2 ~ . 0
(A2)
and 2 & (-1) m Z. = co.(-1)" = - - e "=~+ 2. d2m, n n
m=l
m
where n = 1, 2 . . . . . and
C~d= cosh (~).
(A3)
The coefficients din., can be obtained by starting with (Cooke, 1959)
e -m~, cosm/~l = m f o / , . ( 2 c ) 2 - 1 e - ~ l cos2yl d2
(A4)
e -~x2 cos2yz = ~ e , ( - 1)"I,(2C) eoshnc~2 cosn/~2,
(A5)
and
n=0
where x2> 0, m = 1, 2 . . . . . and
{1 °=0 ~"=
2
n=l,2 ....
(A6)
F r o m the relations (1), eqns (A5) and (A6) can be transformed into
e -"~, cosm/~l = m
fo
I m ( 2 C ) 2 - ' e -~* ~ e . ( - 1)"I.(2C) coshm~2cosn32 d2. n = 0
According to the definition of din.. given in eqn (A1), the coefficients dm,. are obtained by
(A7)
Interaction of two elliptic inclusions
465
d~.. = e.m l~lm(2C)I.(2C) e-a( ,q,-1 d2. ~o
(A8)
The evaluation of the integral in eqn (A8) is discussed in Erdelyi (1954). APPENDIX B The functions Uo, Vu, Su, and T,j are the following : UA1
=
n e -"~o,
Unl = (n-- 1 + to) e -("-2)'o + (n -- 1 -- x) e - ' o ,
Un2 = ( n + 1 + x ) e-"~o+ ( n + 1 --to) e -~n+2)~o, Ucl = n sinhn%,
Uc2 = ( n - l + x) sinh ( n - 2)c~o + ( n - 1 - x) sinh n~o, GA1 = nsinhnct0,
Uc3 = ( n + l + x ) s i n h n ~ o + ( n + l - ~ c ) s i n h ( n + 2 ) ~ o ,
Unl = ( n - 1 + ~) sinh ( n - 2)c~o + ( n - 1 - ~) sinh nc~o, /.782 = (n + 1 + t~) sinh no% + (n + 1 - ~) sinh (n + 2)cto,
VB1 = (n-- l - - x ) {e-("-2~o+e-"~o},
VAI = n e - " %
VB2 = ( n + 1 +tc){e-"~o+e-("+2)~o},
Vcl = ncoshnct0,
Vcz = (n - 1 - x) { cosh (n-- 2)C~o+ cosh n~o }, Vc3 = (n+ l +x){coshm% +cosh (n+ 2)cto},
I?A1 = ncoshm%,
l?m = (n -- 1 -- ~) { cosh (n-- 2)cto + cosh n~o }, ITBz = ( n + 1 + ~ ) { coshnc% + c o s h ( n + 2 ) % } , SAI = ( n - 2 ) ( n - 3 ) SA2 = n { ( n + l ) e
e -~"-2~o, ("-2~'o+(n-1)e-("+2)~o}-2e~o6.,1,
SA3 = ( n + 2 ) ( n + 3 ) e - ( " + 2 ) %
Snl = ( n - 3) {(n - 2) e -("-4~'o + (n - 3 - re) e-("-2~o}, Sn2 = ( n - 1){(n+2) e -("-4)~o - 2 ( x - 2 )
e -("-2~o
+ 4e -"~o + ( n - 1 - to) e -("+ 2)~o} - (x + 1)6.,2, Sn3 = ( n + 1){(n+ 1 + x ) e -~"-2)~. - 4 e -"~o + 2 ( x - 2 ) e (,,+2)~o+ ( n - 2 ) e -C"+4~0} - 2 { ( x + 1) e -~o +e-3"o}6., l,
&4 = (n + 3) { (n + 3 + tc) e-{"+ =)=°+ (n + 2) e-{"+4~=°}, Scl= (n-2)(n-3)
cosh ( n - 2 ) % ,
Sc2 = n { (n + 1) cosh ( n - 2)~0 + ( n - 1) cosh (n + 2) c%}, Sc3 = ( n + 2 ) ( n + 3 ) cosh (n+2)Cto, Sc4 = (/'/-- 3) ( (n -- 2) cosh (n-- 4)c~0 + (n -- 3 -- x) cosh (n-- 2)~o }, Sc~ = (n-- 1){(n + 2 ) cosh ( n - 4 ) ~ o - 2 ( x -
2) cosh ( n - 2 ) ~ o
+ 4 cosh n~o + ( n - 1 - x) cosh (n + 2)C~o} - 0 c + 1)6..2,
Sc~ = (n + 1) {(n + 1 + to) cosh ( n - 2)% - 4 cosh n~o + 2 ( x - 2) cosh (n + 2)Cto + (n - 2) cosh (n + 4)cto} - 2 { ( x + 1)e -~o + e - ~o}6.. ~,
Sc7 = (n + 3) { (n + 3 + x) cosh (n + 2)ct 0 + (n + 2) cosh (n + 4)ct o}, S~1 = ( n - 2 ) ( n - 3 )
cosh ( n - 2 ) 7 o ,
Saz = n { (n + 1) cosh (n - 2)c~0 + ( n - 1) cosh (n + 2)C~o} - 2 cosh Cto6..~, Sa3 = ( n + 2 ) ( n + 3 ) cosh (n+2)c~ o, ~nl = ( n - 3 ) { ( n - 2 ) cosh ( n - 4 ) ~ o + ( n - 3 - ~ ) cosh ( n - 2 ) ~ o }, Snz = (n - 1){(n + 2 ) cosh ( n - 4 ) % - 2 ( R -
2) cosh ( n - 2)~o
+ 4 cosh n~o + ( n - 1 - R) cosh (n + 2)~o } - (~ + 1)6., 2, S~3 = ( n + 1){(n+ I +R) cosh ( n - 2)Cto - 4 c o s h nc~o- 2 ( R - 2 )
cosh (n + 2)~o
+ ( n - 2) cosh (n + 4)C~o} - 2 { (R + 2) cosh c~0+ cosh 3C~o} an, 1, Sg4 = (/'/+ 3) {(n + 3 + ~) cosh (n + 2)% + (n + 2) cosh n~o },
466
M . J . Meisner and D. A. Kouris
TAI = ( n - - 2 ) ( n - - 3 ) e -('-2)%, TA2 = n{(n+l)e--('-2)~°+(n--l)e ('+2)~'°}+2e%6 .... TA3 = ( n + 2 ) ( n + 3 ) e -("+2)~o, Tnl = (n--3){(n--4) e -("-4)~o + (n-- 3 -- to) e (,-2)~o}, TB2 = ( n - - 1 ) { n E
(" 4)%--2(x--2)e-(" 2)~o+(n--l--x)e-("+z)~o}+{(x+l)+Ze-2~o}b,,z,
TB3 = ( n + 1){(n+ 1 +to) e-('-2)~o + 2(tc-- 2) e -('+z)"o + n e -("+ 4)%} + 2{(~c+ 2) + 3 e-3~o}6,,~, T84 = (n+ 3){(n+ 3+~:)e-('+2)~°+(n+4)e-("+4)~°}, Tcl = ( n - - 2 ) ( n - - 3 ) sinh ( n - 2 ) a o , Tc2 = n{ (n + 1) sinh (n--2)ao + ( n - 1) sinh (n + 2)ao }, Tc3 = (n + 2) (n + 3) sinh (n + 2)ao,
Zc4
( n - 3){ ( n - 4) sinh ( n - 4)c% + ( n - 3 - x ) sinh ( n - 2)~o ),
Tc5 = (n - 1) {n sinh (n - 4) o~o- 2(x - 2) sinh (n - 2) c%+ (n - 1 - to) sinh (n + 2) ao } + { (x + 1) + 2e- 2~o}6., 2, Tc6 = (n + 1) { (n + 1 + x) sinh (n - 2)~o + 2 (x - 2) sinh (n + 2) 0%+ n sinh (n + 4)c~o} + 2 { (x + 2) + 3 e - 3~o} 5..~, Tc7 = (n + 3) { (n + 3 + x) sinh (n + 2)~o + (n + 4) sinh (n + 4)c~o}, TA~= (n - 2)(n - 3) sinh ( n - 2)c~o, TA2= n { (n + 1) sinh (n - 2) c~o + (n - 1) sinh (n + 2) ~o } + 2 sinh ~o 5.. 1,
TA3
(n + 2 ) ( n + 3) sinh (n+2)O~o, (n - 3) { (n - 4) sinh ( n - 4)% + ( n - 3 - ~) sinh (n - 2)O~o},
TB2
(n - 1) {n sinh (n - 4)cq - 2(~ - 2) sinh (n - 2)~o + (n - 1 - ~) sinh (n + 2)~o } + 2 sinh 2ao,5., 2
TB3
(n + 1) { (n + 1 + ~) sinh (n - 2)C~o+ 2 ( ~ - 2) sinh (n + 2)~o + n sinh (n + 4)~o }
TB4
( n + 3){(n+ 3 + k ) sinh (n + 2)c~o + ( n + 4 ) sinh (n +4)ao}.
+ 2 ~(~ + 2) sinh c~o+ 3 sinh 3c~o) 6., ~,