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Inplane deformation of a circular inhomogeneity with imperfect interface
ELSEVIER
theoretical and applied fracture mechanics Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
Inplane deformation of a circular inhomogeneity with imperfect interface M.A. Kattis a,., E. Providas h a Department of Civil Engineering University ofThessaly, Pedion Areos. Volos GR-383 34, Greece b Department of Mechanical and Industrial Engineering University t?fThessaly, Pedion Areos. Volos GR-383 34. Greece
Abstract
The plane elastic problem of a circular inhomogeneity with an imperfect interface of spring-constant-type is reduced to the solution of a Somigliana dislocation problem, when the solution for the corresponding problem with a perfect interface is known. The Burger's vector of the Somigliana dislocation is determined so that its components satisfy two interfacial conditions involving the traction components of the corresponding problem with a perfect interface. Employing complex variables, a two-phase potential solution to the Somigliana dislocation inhomogeneity problem is developed for a general form of the Burger's vector. Detailed results are reported for a uniform eigenstrain in the inhomogeneity, and for a remote uniform heat flow in the matrix. In the latter case, the inhomogeneity behaves as a void, when it begins to slide. © 1998 Elsevier Science Ltd. All rights reserved. Keywords: Inplane deformation; Circular inhomogeneity; Spring-constant-type;Somigliana dislocation; Burger's vector
1. Introduction
In real situations, the assumption of a perfect interface employed in the mechanics and micromechanics of composite materials is rarely fulfilled. At elevated temperatures, for example, the transition zone from an inhomogeneity to the matrix behaves like a viscous liquid loosing its capability of transmitting shear tractions and tangential displacements. This case is modelled by a sliding interface. Both perfectly bonded and sliding interfaces constitute special cases of a general type of interfacial conditions expressed by the traction continuity and a relationship between traction and displacement jump along the interface. This type of interfacial condi-
* Corresponding author.
tions corresponds to an imperfect interface, where the spring-constant-type interface is a special case. Also, an imperfect interface can be used to model the presence of a thin layer or coating enveloping the reinforced constituent in composite materials. In micromechanics of solids, elastic solutions to the inhomogeneity problem are often used to relate the overall deformation and strength properties of composite materials with the behaviour and interactions of their microscopic constituents. The effect of the imperfect interfaces on the mechanical behaviour of composites elastic solutions were obtained for a variety of inhomogeneity problems involving some highly idealised cases of imperfect interfaces. The most significant contribution in this subject is the recent works reported in Refs. [1-5]. The more realistic case of an imperfect interface modelled by spring-constant-type interfacial conditions has been
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M.A. Kattis, E. Prot,idas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
considered only for a few cases of circular and spheroidal inhomogeneities [6-10]. The present paper outlines a superposition method for calculating the elastic fields developed in an inhomogeneity-matrix system, when the mechanical behaviour of the interface is modelled by spring-constant-type interfacial conditions. The method utilises the elastic fields developed in the system for a perfect interface and those induced in the system by a Somigliana dislocation acting along the interface. The Burger's vector of the Somigliana dislocation satisfies a certain pair of interfacial conditions involving the interfacial traction components of the perfect bonding case. The fields corresponding to an imperfect interface in the system are taken by superposition. Thus, if the fields of a problem with a perfect interface are known, then the fields of the corresponding problem with an imperfect interface may be found by solving a Somigliana dislocation problem. In the present work, a solution to the inhomogeneity problem for a Somigliana dislocation acting along the interface is developed using the complex variable formalism of plane elasticity. The solution is expressed in terms of two potential functions for a general form of the Burger's vector. The analysis is developed for circular interfaces, but the obtained results are directly extensible to other curvilinear interfaces using conformal mapping. Instead of using the technique of analytic continuation, the present developments are based on the holomorphic transformation of a complex function regarding a circular boundary. The holomorphic transformation of a complex function f ( z , ?:) defined in the region Iz[ > R of the z plane is the holomorphic function f h ( z ) = f ( R 2 / z , z ) . Obviously, the holomorphic transformation carries a non-holomorphic function into a holomorphic one and it is the extension of the Kelvin (or hat) transformation to the case of a non-holomorphic function. The solution is worked out in detail for a uniform eigenstrain in the inhomogeneity and for a uniform remote heat flow in the matrix. For the uniform eigenstrain, the stress distribution within the inhomogeneity is given by a quadratic polynomial, whereas, for the uniform heat flow by a linear form. In the latter case, it is shown that the inhomogeneity behaves as a void when the inhomogeneity begins to slide.
b x
Fig. 1. Matrix and inhomogeneity.
In what follows, the formulation of the problem and the calculation of the elastic strain energy stored in the system are given. The solution to the Somigliana dislocation problem is then presented. The analysis developed is implemented to study the eigenstrain and the thermoelastic problem of a circular inhomogeneity.
2. Problem statement and general considerations Consider the inplane elastic deformation of the system shown in Fig. 1. The system consists of an infinite matrix containing a circular inhomogeneity, with different thermomechanical properties, occupying the region Izl < R of a z-plane (z = x + iy). The mechanical behaviour of the interface is described by elastic spring-constant-type conditions according to which radial and tangential displacement discontinuities are proportional to the respective traction components with interfacial tractions being continuous. Employing subscripts 1 and 2 for quantities referred to the matrix and inhomogeneity, respectively, the interfacial conditions with respect to polar coordinates p, 0 (z = pe i°) are
[uo]=
noR noR M ~1°°= M ~2~o,
[ ~p]
~ U l o - - R2p
(l) [uo] =
noR A4
no R = --ff- 2oo,
n o ] = U l O - - U20
(2)
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
where
215
interfacial conditions in Eqs. (1) and (2) may be written as 4p. 1
M ~1.],1 +
In Eqs. (1) and (2), /z is the shear modulus, K = 3 - 4 v for plane strain and K = (3 - v ) / ( 3 + v) for plane stress with v being the Poisson's ratio; n o and n o are the spring-constant-type material parameters chosen here in a dimensionless form. Zero values of these parameters provide the case of a perfectly bonded interface, whereas infinite values imply vanishing of interface tractions, and, therefore, a complete debonding of the two adjoining media. Any positive value of these parameters defines an imperfect interface. Furthermore, the case where n o = and n o = 0 represents the case of a sliding interface. The physical ground of the problem implies that the displacement jump in Eq. (1) cannot be negative, and therefore, these interfacial conditions are valid only for the case of tensile radial stresses. For interfacial regions with compressive radial stresses, the above conditions should be replaced by others, for example, those of a perfect interface.
2.1. D i s p l a c e m e n t j u m p
The solution to the problem stated above may be constructed as follows. The elastic deformation of the system is assumed to be coming from that of the perfect interface by imposing a Somigliana dislocation along the interface simulating the displacement jump ~'p = [Up],
R
no R
= --if-
+ % o ) = --if-
no
K1 "
~'o = [ % ]
(3)
where ~p and ~d0 are the Burger's vector components of the Somigliana dislocation. The elastic fields of the problem are then written tr,~t3 = o-~ + try/3,
u~ = u p + ,~
+ (5)
noR
= --M-
+
noR
p
+
(6) Thus, if the fields of an inhomogeneity problem with a perfect interface are known, then the fields of the corresponding problem with an imperfect interface are determined by solving the inhomogeneity problem subjected to a Somigliana dislocation along the interface, whose the components of the Burger's vector satisfy the conditions in Eqs. (5) and (6).
2.2. Strain energy
The relaxation effect of the imperfect interface can be evaluated by calculating the strain energy change from the perfectly bonded state to the imperfectly bonded state. This change is expressed by the elastic strain energy stored in the system due to the Somigliana dislocation imposed along the interface. It follows that 1 W s = ~ f n ( O ' x ~ e x x + O'yye~,y + 2 ~ r ~ . e x y ) d x d y
i f _ ( O'xxexx p p + O'v. .pv. pt~vv . + 2 t r ~pype x y ) d x d y
2 "ll
(7)
where 12 is the material domain corresponding to the whole z-plane. Using the divergence theorem, Eq. (7) is transformed to
(4)
where o-Pa and uP ( a , fl=p or 0) are the elastic fields of the perfectly bonded inhomogeneity; trot3, u~ are the additional fields caused by the Somigliana dislocation. With the aid of Eqs. (3) and (4), the
1
p
(8)
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M.A. Kattis. E Prov idas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
where /" is the circumference Izl=R. In deriving Eq. (8), the equilibrium equations and the condition u~ = 0 at infinity were used. Substitution of Eqs. (5) and (6) into Eq. (8) gives
n o RIo
w~
M
no RI o -< o M
conditions in Eq. (11) are equivalent to the following set of equations 1-13 'Wl(t ) -
(9)
1 f2n'(
'W2(t )
+ i +-
[,wd,) + ~2(t)]
y
where
lP= -2 Jo
l+a
+
s
2
°'lP°P+ °''m') pdO,
1 /'2zr. P s 2 (0"1 O0 + 0"100) pdO.
Io='~J 0
'~l(t)
l+a
~/(t,t)
1+]3 a÷]3 _ _ 1 + Ot '~'~2(t) ÷ l + a '~Y/'2(t) 2]3_ + --tW2(t l+a
(lO)
(12)
)
- ]3 ~(s)
Eq. (10) implies that the elastic strain energy in the imperfectly bonded inhomogeneity is lower than in the perfectly bonded inhomogeneity. This means that the imperfect interface causes a stress relaxation in the inhomogeneity system even though the stress increases locally.
l+a
d
t'( s) ds
1+ a
[tWo(t)+ ~r~(t~] t'(s)
ds (13)
where
3. The Somigliana dislocation problem
(14)
The Somigliana dislocation problem will be described. The two constituents of the system shown in Fig. 1 are perfectly fit over the interface, when they are two unconnected and undeformed parts. A relative displacement between the two opposite sides of the interface with Cartesian components U~ = Ux(x,y) and U, = U;(x,y) is given; they are then welded together by filing in material in the case of a gap, or taking out the extra material in the case of an overlap. The stressed deformation induced in the system will be analysed. According to the above statement, the following conditions should be satisfied across the interface
In Eqs. (12) and (13) and in the sequel, left and right primes denote indefinite integration and differentiation, respectively, with respect to the variable z; t represents an interracial point and s is an interfacial arc-coordinate measured from an arbitrarily chosen point; a and ]3 are the Dundurs parameters defined by
"~Jffl --'~'~6"r2= U~ ÷ i V ,
and
~9~, =.5~ 2
(11)
where ~ = ux+ iu~ with u x and u,. being the Cartesian displacement components and ~ ' is the force acting on an arc. The second condition expresses the traction continuity in its integrated form. Employing the Kolosov-Muskelishivili formalism of plane elasticity given in Appendix A, the interfacial
C ( l ÷ KI) -- (1 ÷ K2)
r ( l + ~,) + (1 + ,~) '
F(K,-
I ) - ( K 2 - 1)
F ( 1 + K,) + (1 + a:2)
8/x2 ~'= r ( 1 + KL) + (1 + K2) '
r=
~--2~ ~,
Eqs. (12) and (13) show that the KolosovMuskelishivili potentials of the matrix are expressible in terms of those of the inhomogeneity along the interface. This fact along with the holomorphic trans-
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
formation of a complex function defined in the introduction enables the construction of the general representations of the complex potentials of the two phases so that the interfacial conditions are satisfied. By extending the definition of ~'(z,~') to the whole z-plane, the following expressions for the complex potentials of the two phases are defined:
'w,(z) = 'Wo(z)
-
azWo
+/-/n~o ~
+m~'
z,
(15)
217
iarities introduced to the matrix potentials WI(z) and ~ f l ( z ) by the ~'-terms and they ensure vanishing stress at infinity. The latter means that the complex potentials Wl(z) and ~ l ( z ) should decay to zero as I zl tends to infinity. It should be noted that the two-phase potential functions may be defined in a variety of ways. For example, setting ( O / H )(R2/z)Wo(z) + ':gO'0(z) instead of Wo(z) in Eqs. (15)-(18), the following solution representation is defined: 'wl(z)
= 'W0(z)
z
(19) ' T f , ( z ) = '~go( z ) +
A'W o
g 2
=
-
R2
--%(z) Z
+( A + I2)--Wo( z )
R4
{ R2
Z
+T
d--~
-M
~
-M
--,z z
+----~' z dz
+A)'Wo(z )
1+/3'
g2=
(21) R2
A ) - - W o ( z ). z
(17)
(22.)
(18)
In the special case of a rigid inhomogeneity (P-2 _=_ac), the potentials of the matrix are obtained by setting A = I / H = K,, g2= - 1 in Eqs. (15) and (16) (or in Eqs. (19) and (20)). For P-2 = 0 ( a = A = 17-- -1), Eqs. (19) and (20) provide the complex potentials of a circular hole in an infinite plate with a prescribed displacement at the boundary. The equations of the latter case, in the absence of the ~'-terms, express the second circle theorem of plane elasticity established in Ref. [11].
for the inhomogeneity; A, 17 and g2 are auxiliary two-phase parameters given by 17=--
z,
' w 2 ( z ) = (l + / 7 ) ' W o ( z ) 'Tf2(z ) = (1 + H ) ' 7 / / ( z ) - (1 +
'~¢'2(Z) = (1 + H ) ' T f o ( Z )
1-/3'
+----~' z dz
(2O)
for the matrix, and
A = - -
--,z z
z, (16)
'W2(z ) = (1
~
1-/3
In the above equations, Wo(z) and ~Vo(Z) are two holomorphic functions, and Wo(R2/z), ~o(R2/z) are their Kelvin transformations; ~'(z, R2/z) and ~'(R2/z,z) are the holomorphic transformations of the complex functions ~ ( ~ , z ) and ~(z,~), respectively. The solution to the Somigliana dislocation problem is expressed by Eqs. (15)-(18) in terms of two holomorphic functions Wo(z) and 7//0(z), which constitute the two-phase potentials of the system. For a given Burger's vector ~', the two-phase potentials are determined so that they eliminate possible singu-
4. Applications
4.1. The eigenstrain problem The previous results are applied now to calculate the fields induced in the system, when the inhomo-
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M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
T geneity is subject to uniform eigenstrains e~, e,., and Exr r" The elastic fields of the corresponding problem with a perfect interface are listed in Appendix B. The interfacial stress components of this problem are given by
(15) and (16), the convergence of Wl(z) and 7fl(z) at infinity implies
1 I+A o'pPo= - -~ M ~ _ ~ F r
a 2 = a 4= ... = 0
MR2At al
MB t
1 - ~'~ '
a3
T '
(30)
1
b t =--MR2[ A t +4Bj + 3(Z +/2)B1],
2
1
+~M(I +lI)(Gre-2i°+Gre2i°) i o'pp = - -~M(1 + H ) ( G r e - 2 i ° -
GTe2i°).
(23) (24)
For the function ff(z,z), the following torm is assumed R2
8/( Z, ~.) = AtR2z +
TA-llZ
1 + -~z( B, z 2 +B-T~2)
b 2 = b 3= ... = 0 .
Having determined the two-phase potentials, the elastic field of the Somigliana dislocation problem are obtained by means of Eqs. (15)-(18) and Eqs. (Al), (A2), (A3) and (A4). The interfacial traction components are MR e 1 + A Ors
oo
2
(25)
R3 ( ~
1-
MR 2
-~t~ l -- - - ~ ' -
o
(I
--I-H)
4 ~ I + 3(A + a ) B l ] e -zi° MR 2
where Aj is a real constant and A l, B I are complex constants. The interracial components of the function are given by
4=atR +Tt T
(31)
- - -8( l + n ) X[AI+4B,+3(A+a)BI]e
+-~l)e-~,o
2i°
(32)
iMR 2
°°;
8
[6( 1 + A)B-T- ( 1 +/-/)(~-~ + 4~1)
+ ( a + 0)~1] e -z'° + T
~o-
+ BI e2iO
iR 3 4
(26)
iMR 2
+---Z[6(I +A)B, +(1 +//)(at +4Bt)
(a-Te-2i°-ate2i°)"
(27)
+ ( A + / 2 ) B t i e +2i°.
(33)
The constants At, A~ and B t will be calculated so that the interfacial conditions in Eqs. (5) and (6) are satisfied. The interfacial stresses o'o~ and %~ entering in these equations will be determined by solving the corresponding Somigliana dislocation problem. The two-phase potentials of this problem are sought in the form
Substituting the pairs of Eqs. (23) and (24), Eqs. (26) and (27) and Eqs. (32) and (33) into Eqs. (5) and (6) and comparing the coefficients of the same terms, it is found that
'W0(z) = a t z + a 2 z 2 + a3 z3 + . . .
A t = [3(l + ~ ) 2 n o n o + 2(1 + o r ) ( 1 - ~3)no]
'~rf'0(Z)
=
bl Z -t- b2 22 -~ b3 z3 -1- . . .
(28) (29)
where at, a2, a 3 . . . . and bj, b2, b 3 . . . . are complex constants. Substituting the appropriate functional forms of Eqs. (25), (28) and (29) into Eqs.
At = -
(1 + ~ ) n o 1 +a-2/3+(1
+a)n
o
( F T / R z)
(34)
× [2(1 + a ) ( 2 + / 3 ) ( n o + no) +3(1 + ~ ) 2 n o n o - 4 ( 1
-/32)]-'
(GT/R2) (35)
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M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
B, = [(1 + a ) ( l - f l ) ( n o - n o )
]
with those obtained in Ref. [9] for a spherical inhomogeneity with an imperfect interface.
× [2(1 + a ) ( 2 + fl)(n o + no) +3(1 + a)2nono-4(1 -/32)] - I ( G r / R 2 ) . (36) Thus, the solution of the Somigliana dislocation problem in terms of the two-phase potentials are 'W0(z) =
MREA1 1 - g2
MB1 z 3
(37)
2
MR 2 '~e0(z) = - - - ~ - ( a l + 4BI + 3( A + g2)Bi)z. (38) The elastic fields for the imperfectly bonded eigenstrain problem are obtained by superposing the fields for the Somigliana dislocation problem derived above on those of the perfectly bonded inhomogeneity problem given in Appendix C. The above results are in agreement with those obtained in Refs. [7,10]. As previously mentioned, the solution obtained for an inhomogeneity problem with the interfacial conditions in Eqs. (1) and (2) is accepted if it provides positive radial stress along the interface. Eq. (1) shows that a positive radial stress along the interface ensures a positive radial displacement jump and vice versa. The necessary and sufficient conditions, for which such a requirement is satisfied, are
liFr-12lGrl>Oand llFr +12lGrl>O
(39)
where (1 + ot)np l, = -
1 +a-2fl+(1
+a)np
(40)
4.2. The thermoelastic problem Considered as a second application is the thermoelastic problem of a circular inhomogeneity with an imperfect interface disturbing a remote temperature change of the reference temperature given by q0 T = ~-~lty+ Tc .
(42)
In Eq. (42), qo represents a uniform heat flow along the y-axis and T~ a uniform temperature change; k t denotes the thermal conductivity of the material. The thermal conditions along the interface are assumed to be those of a perfect thermal contact between the two phases. The interfacial traction for the corresponding problem with a perfect interface, as given in Ref. [12], is
M ( a , , - a2t)(1 + A)Tc tr°~ + itrp°P=
(1-O) - M(1 + A)RQo ei°
(43)
where 1( Q0 = "2
2a2t kl t 4- k2 t
air ) kh
qo
(44)
with a t being the coefficient of linear thermal expansion. Assuming the Burger's vector to be of the form
~-g'(Z,7.) = A2Rz + A2 R2 -+(BzZ+-~2~)Z z
(45)
its interfacial components are
12-- [1.5(1 + a)2nono 4- (1 + a ) ( 1 - fl)] × [2(1 + a ) ( 2 + f l ) ( n o + n 0 ) +3(1 + a)2npno- 4(1 -- [3)2] -1
(41)
The two-phase potentials, Eqs. (37) and (38), show that the stress field within the inhomogeneity is a quadratic polynomial. This field becomes uniform, when either np= n o or G r = 0 with F r 4=O. The latter case corresponds to a volumetric-type straining for which A~ = A 2 = 0. These results are consistent
iR2 ~ ' ° = - 2 (Azei°-"~2e-i°) where A2 is a real constant while complex constants.
(47) A2
and
B 2 are
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M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998)213-222
For the Somigliana dislocation problem, the twophase potentials and the traction components along the interface are obtained in the form
aM(1 + A ) 2 ( n o + n o)Qo
MA2R I_~z-M(A2+B2)z2,
'W0(Z) -
-
'~f0(z) = 0 R2M(1 + A) 2
1-/2
×[( A2 + B2)ei° + ('~2 + Bz)e-i° ]
(49)
RZM( 1 + A) i 2 × [ ( A 2 + B2) e i ° - ('~2 + Bz)e-i°] "
(50)
The unknown coefficients in Eq. (45) are determined as in the previous application. It is found that
(a,,-a2,) A2= 1 + a _ 2/3 + n o ( l + or)
~
T~
2(1-13) +(! +a)(no+n0)
(air -- a2t)Tc R
12q>_ 0
( a,t - a2t) Tc + h2q > 0 R where hj
(52)
2a2t h, - - -
i ( n p - no)(1 + or) 2(1-/3) +(1 +a)(np+n0)
Eqs. (54) and (55) along with those reported in Appendix C imply that a linear stress distribution is introduced to a circular inhomogeneity, when a remote uniform heat flow is applied to the matrix. A remarkable result regarding this field is the following: there is no stress distribution within the inhomogeneity in the case of a sliding interface. In other words, a sliding circular inhomogeneity embedded in a matrix behaves like a void when it disturbs a uniform heat flow. The necessary and sufficient conditions ensuring a positive radial displacement jump along the interface are
(51)
Qo
Q0-
(53)
The list of the elastic fields for the corresponding problem with a perfect interface is given in Appendix C. Some interesting aspects of the solution are associated with the stress distribution within the inhomogeneity. The part of this stress distribution corresponding to the Somigliana dislocation problem is calculated in the form M(1 + A) 2no(a . - a2t)Tc °'~°°+ icr~°°= - (1 - O)[1 - / 2 + C(1 + a)np]
r sin 0. (55)
hn
2ino(1 + o~)
B2 = -
2 + ( 1 + A)(np +no)
(48)
MA2(I + A) R 2
A2 = -
2M(1 + A ) 2 n o ( a , , - a2t)Tc (1 - g2)[1 - O + (1 + a ) n p ]
°2s°p + °2s° =
ah
kit + k2t
h2 =
2(1-fl) 1 + a-
Appendix ism
(56)
(57)
kit
+(1 +a)(n,,+no) 2/3+ (1 + ot)nt,
A. The Kolosov-Muskelishivili
(58)
formal-
4/.t(up + iUo) = e-i°[ K'W(Z) -- z W ( z ) - '~W( z)] (A1) crpp + %o = W(z) + W(~)
(A2)
2(o'po + i%o )
+
iM(I + A)Z(np + no)Qo
re io
2 + ( 1 + A)(no + no)
= W(z)
+
-
-
(A3)
Z
(54)
2 i,P~'= 'W(Z) + z W ( z ) + 'W(Z).
(A4)
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M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
Appendix B. The p-fields for the eigenstrain problem
(B 1)
2(o',°oo + io',Poo) I+A
1 7M(1 +l-I)'Gre -2i4' (B2)
= -M~_oFr+
2 ( a z t - a l t ) ( l + A)T¢ R 2 (1 + K , ) ( I - O ) pz +auTcP
ulP"=
I+A o'lPoo+ o'lP00= - M ~ Fr
(c3)
O'lPoO= 0%o0 = 0
2(a2t- a,t)(l +A)T~ u~° = (1 + Xl)(1-- g2) p + altTc P
(c5)
uPo= u2o = 0.
(C6)
For a temperature change -qo y/kit at infinity:
°'g,, + ~oo 1
=~M(1 + II)[Gre-2'°
R2
+ Gre2'*]-~-i
(B3)
41xlaltqoR
o'IPop =
(1 + tq)klt
2( o'~oo + io'~oo)
+]'+~
1+ A
R2
°',Poo=
R2/R]
R2
R2 2 (1 + / / ) ' 7
-1)7
ITIPp0
(1
R2
+
+ --
(B5)
l+k 4N2(u~p + iuS2o)
r G r = eyyr _ er + 2 i . r Eyy,
-1)7
KI)klt
(B6)
1-
l+k
) 1 psin0
6/.qaltqo(l + A ) [ 2a - - l i p sin 0 (1 + KI)klt ~ 1 + k (Cll)
Or2PpO~ --
For a uniform temperature change T~:
2pqauq0(1 + A ) [ 2a -- ljpcos 0 (1 +Ki)klt ~ l+k (C12) /
4/xl(alt- a2t)(1 + A)Tc R 2 (1 + KI)(1 - a ) p"
2ajtqo R2 [ 1 + K1 (C1)
(1 + K,)(1 - O)
(c9)
cos 0
7);
(Cl0)
(B7)
Appendix C. The p-fields for the thermoelastic problem
4 p . , ( a , t - a2t)(1 + A)T~
R3
1 +k
(1 + rl)klt
o-~o o =
trPoe=
crLP = °':'°° =
(c8)
sin 0
21xtaltqo(l + A) ( 2a
1 +A K2M = M l _ - i - - ~ F r - - - - ~ ( 1 +II)'Gre-2'e'p
O"lPPo= --O"100 =
1+
4p, laltqoR ~_+~_( 2a
M
1 X[KI"GTe -2i0 + (1 -- 7 ) Gre-4io[ . ,
R3
1 +k
(B4) l+k
7
(C7)
41zlauqoR ~ _ A ( 2a
4/x,(u~p + iu~o) 1+ A
7
1 - -~'] p ] sin 0
(1 + Kl)klt × [ G r e - 2 ' ° + 3 ( 1 - - ~ z ) G r e ai°]
=M~-oFr
1
l+k
R2
1
= - M ~ _ ~ F T 7 + ~-M(l + H ) V
F T = ~Tx +
(c4)
(C2)
u~o =
(1 + Kl)klt / X
+ I+K,
4 l+k
1
M.A. Kattis. E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
222
1_,( R2)
References
1--~- 5-
+2(l+k~-------S
,_, + 1 + k log
---1)o~} 1 +k
+
sin 0 (C13)
altqo R2 uPlo
1 + ~:l
(1 + K1)klt X
+
4
p2
1 +A
"~+
I+K I
1 -k
(
(
l+k
R2 )
2(i +~:)11 +k log
2a
7 ----
-
1]$"/cos0
1 +k
g
J
(C14)
2al__tqo_p2 u~o -
[ 1 + K,
[ ,+A(2a____)] l(2a )) (1 + K l ) k l ' /
X 1+
+2
l+KI
l+k
2altqop2
X l-}- lq-K1
sin~b
(c15)
4
___,)] 2a
l+k
2a I+k
1)g~} cos d~
where g"= 1 +A+(1
1
( 1 +Kj
I+A(
2
l+k
1 ~
( ~ + "~1 y-k~l, /
1(
4
-1- KI)~,~.
(C16)
[1] T. Mura, R. Furuhashi, The elastic inclusion with a sliding interface, J. Appl. Mech. 51 (1984)308-310. [2] E Tsuchida, T. Mura, J. Dundurs, The elastic field of an elliptic inclusion with a slipping interface, J. Appl. Mech. 53 (1986) 103-108. [3] I. Jasiuk, E. Tsuchida, T. Mura, The sliding inclusion under shear, int. J. Solids Struct. 23 (1987) 1373. [4] T. Mura, 1. Jasiuk, B. Tsuchida, The stress field of a sliding inclusion, Int. J. Solids Struct. 21 (1985) 1165-1179. [5] R. Furuhashi, J.H. Huang, T. Mura, Sliding inclusions and inhomogeneities with frictional interfaces, J. Appl. Mech. 59 (1992) 783-788. [6] Z. Hashin, Thermoelastic properties of fiber composite with imperfect interface, Mech. Mater. 8 (1990) 333-348. [7] A.J. Levy, The debonding of elastic inclusions and inhomogeneities, J. Mech. Phys. Solids 39 (1991) 477-505. [8] Z. Hashin, Thermoelastic properties of particulate composite with imperfect interface, J. Mech. Phys. Solids 8 (1991) 333-348. [9] Z. Hashin, The spherical inclusion with imperfect interface, J. Appl. Mech. 58 (1991) 444-449. [10] Z. Gao, A circular inclusion with imperfect interface: Eshelby's tensor and related problems, J. Appl. Mech. 62 (1995) 860-866. [11] L.M. Milne-Thomson, Plane Elastic Systems, SpringerVerlag, Berlin, 1968. [12] M.A. Kattis, S.A. Meguid, Two-phase potentials for the treatment of an elastic inclusion in plane thermoelasticity, J. Appl. Mech. 62 (1995) 7-12.
theoretical and applied fracture mechanics Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
Inplane deformation of a circular inhomogeneity with imperfect interface M.A. Kattis a,., E. Providas h a Department of Civil Engineering University ofThessaly, Pedion Areos. Volos GR-383 34, Greece b Department of Mechanical and Industrial Engineering University t?fThessaly, Pedion Areos. Volos GR-383 34. Greece
Abstract
The plane elastic problem of a circular inhomogeneity with an imperfect interface of spring-constant-type is reduced to the solution of a Somigliana dislocation problem, when the solution for the corresponding problem with a perfect interface is known. The Burger's vector of the Somigliana dislocation is determined so that its components satisfy two interfacial conditions involving the traction components of the corresponding problem with a perfect interface. Employing complex variables, a two-phase potential solution to the Somigliana dislocation inhomogeneity problem is developed for a general form of the Burger's vector. Detailed results are reported for a uniform eigenstrain in the inhomogeneity, and for a remote uniform heat flow in the matrix. In the latter case, the inhomogeneity behaves as a void, when it begins to slide. © 1998 Elsevier Science Ltd. All rights reserved. Keywords: Inplane deformation; Circular inhomogeneity; Spring-constant-type;Somigliana dislocation; Burger's vector
1. Introduction
In real situations, the assumption of a perfect interface employed in the mechanics and micromechanics of composite materials is rarely fulfilled. At elevated temperatures, for example, the transition zone from an inhomogeneity to the matrix behaves like a viscous liquid loosing its capability of transmitting shear tractions and tangential displacements. This case is modelled by a sliding interface. Both perfectly bonded and sliding interfaces constitute special cases of a general type of interfacial conditions expressed by the traction continuity and a relationship between traction and displacement jump along the interface. This type of interfacial condi-
* Corresponding author.
tions corresponds to an imperfect interface, where the spring-constant-type interface is a special case. Also, an imperfect interface can be used to model the presence of a thin layer or coating enveloping the reinforced constituent in composite materials. In micromechanics of solids, elastic solutions to the inhomogeneity problem are often used to relate the overall deformation and strength properties of composite materials with the behaviour and interactions of their microscopic constituents. The effect of the imperfect interfaces on the mechanical behaviour of composites elastic solutions were obtained for a variety of inhomogeneity problems involving some highly idealised cases of imperfect interfaces. The most significant contribution in this subject is the recent works reported in Refs. [1-5]. The more realistic case of an imperfect interface modelled by spring-constant-type interfacial conditions has been
0167-8442/98/$19.00 © 1998 Elsevier Science Ltd. All rights reserved. PII S01 67-8442(98)00006-8
214
M.A. Kattis, E. Prot,idas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
considered only for a few cases of circular and spheroidal inhomogeneities [6-10]. The present paper outlines a superposition method for calculating the elastic fields developed in an inhomogeneity-matrix system, when the mechanical behaviour of the interface is modelled by spring-constant-type interfacial conditions. The method utilises the elastic fields developed in the system for a perfect interface and those induced in the system by a Somigliana dislocation acting along the interface. The Burger's vector of the Somigliana dislocation satisfies a certain pair of interfacial conditions involving the interfacial traction components of the perfect bonding case. The fields corresponding to an imperfect interface in the system are taken by superposition. Thus, if the fields of a problem with a perfect interface are known, then the fields of the corresponding problem with an imperfect interface may be found by solving a Somigliana dislocation problem. In the present work, a solution to the inhomogeneity problem for a Somigliana dislocation acting along the interface is developed using the complex variable formalism of plane elasticity. The solution is expressed in terms of two potential functions for a general form of the Burger's vector. The analysis is developed for circular interfaces, but the obtained results are directly extensible to other curvilinear interfaces using conformal mapping. Instead of using the technique of analytic continuation, the present developments are based on the holomorphic transformation of a complex function regarding a circular boundary. The holomorphic transformation of a complex function f ( z , ?:) defined in the region Iz[ > R of the z plane is the holomorphic function f h ( z ) = f ( R 2 / z , z ) . Obviously, the holomorphic transformation carries a non-holomorphic function into a holomorphic one and it is the extension of the Kelvin (or hat) transformation to the case of a non-holomorphic function. The solution is worked out in detail for a uniform eigenstrain in the inhomogeneity and for a uniform remote heat flow in the matrix. For the uniform eigenstrain, the stress distribution within the inhomogeneity is given by a quadratic polynomial, whereas, for the uniform heat flow by a linear form. In the latter case, it is shown that the inhomogeneity behaves as a void when the inhomogeneity begins to slide.
b x
Fig. 1. Matrix and inhomogeneity.
In what follows, the formulation of the problem and the calculation of the elastic strain energy stored in the system are given. The solution to the Somigliana dislocation problem is then presented. The analysis developed is implemented to study the eigenstrain and the thermoelastic problem of a circular inhomogeneity.
2. Problem statement and general considerations Consider the inplane elastic deformation of the system shown in Fig. 1. The system consists of an infinite matrix containing a circular inhomogeneity, with different thermomechanical properties, occupying the region Izl < R of a z-plane (z = x + iy). The mechanical behaviour of the interface is described by elastic spring-constant-type conditions according to which radial and tangential displacement discontinuities are proportional to the respective traction components with interfacial tractions being continuous. Employing subscripts 1 and 2 for quantities referred to the matrix and inhomogeneity, respectively, the interfacial conditions with respect to polar coordinates p, 0 (z = pe i°) are
[uo]=
noR noR M ~1°°= M ~2~o,
[ ~p]
~ U l o - - R2p
(l) [uo] =
noR A4
no R = --ff- 2oo,
n o ] = U l O - - U20
(2)
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
where
215
interfacial conditions in Eqs. (1) and (2) may be written as 4p. 1
M ~1.],1 +
In Eqs. (1) and (2), /z is the shear modulus, K = 3 - 4 v for plane strain and K = (3 - v ) / ( 3 + v) for plane stress with v being the Poisson's ratio; n o and n o are the spring-constant-type material parameters chosen here in a dimensionless form. Zero values of these parameters provide the case of a perfectly bonded interface, whereas infinite values imply vanishing of interface tractions, and, therefore, a complete debonding of the two adjoining media. Any positive value of these parameters defines an imperfect interface. Furthermore, the case where n o = and n o = 0 represents the case of a sliding interface. The physical ground of the problem implies that the displacement jump in Eq. (1) cannot be negative, and therefore, these interfacial conditions are valid only for the case of tensile radial stresses. For interfacial regions with compressive radial stresses, the above conditions should be replaced by others, for example, those of a perfect interface.
2.1. D i s p l a c e m e n t j u m p
The solution to the problem stated above may be constructed as follows. The elastic deformation of the system is assumed to be coming from that of the perfect interface by imposing a Somigliana dislocation along the interface simulating the displacement jump ~'p = [Up],
R
no R
= --if-
+ % o ) = --if-
no
K1 "
~'o = [ % ]
(3)
where ~p and ~d0 are the Burger's vector components of the Somigliana dislocation. The elastic fields of the problem are then written tr,~t3 = o-~ + try/3,
u~ = u p + ,~
+ (5)
noR
= --M-
+
noR
p
+
(6) Thus, if the fields of an inhomogeneity problem with a perfect interface are known, then the fields of the corresponding problem with an imperfect interface are determined by solving the inhomogeneity problem subjected to a Somigliana dislocation along the interface, whose the components of the Burger's vector satisfy the conditions in Eqs. (5) and (6).
2.2. Strain energy
The relaxation effect of the imperfect interface can be evaluated by calculating the strain energy change from the perfectly bonded state to the imperfectly bonded state. This change is expressed by the elastic strain energy stored in the system due to the Somigliana dislocation imposed along the interface. It follows that 1 W s = ~ f n ( O ' x ~ e x x + O'yye~,y + 2 ~ r ~ . e x y ) d x d y
i f _ ( O'xxexx p p + O'v. .pv. pt~vv . + 2 t r ~pype x y ) d x d y
2 "ll
(7)
where 12 is the material domain corresponding to the whole z-plane. Using the divergence theorem, Eq. (7) is transformed to
(4)
where o-Pa and uP ( a , fl=p or 0) are the elastic fields of the perfectly bonded inhomogeneity; trot3, u~ are the additional fields caused by the Somigliana dislocation. With the aid of Eqs. (3) and (4), the
1
p
(8)
216
M.A. Kattis. E Prov idas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
where /" is the circumference Izl=R. In deriving Eq. (8), the equilibrium equations and the condition u~ = 0 at infinity were used. Substitution of Eqs. (5) and (6) into Eq. (8) gives
n o RIo
w~
M
no RI o -< o M
conditions in Eq. (11) are equivalent to the following set of equations 1-13 'Wl(t ) -
(9)
1 f2n'(
'W2(t )
+ i +-
[,wd,) + ~2(t)]
y
where
lP= -2 Jo
l+a
+
s
2
°'lP°P+ °''m') pdO,
1 /'2zr. P s 2 (0"1 O0 + 0"100) pdO.
Io='~J 0
'~l(t)
l+a
~/(t,t)
1+]3 a÷]3 _ _ 1 + Ot '~'~2(t) ÷ l + a '~Y/'2(t) 2]3_ + --tW2(t l+a
(lO)
(12)
)
- ]3 ~(s)
Eq. (10) implies that the elastic strain energy in the imperfectly bonded inhomogeneity is lower than in the perfectly bonded inhomogeneity. This means that the imperfect interface causes a stress relaxation in the inhomogeneity system even though the stress increases locally.
l+a
d
t'( s) ds
1+ a
[tWo(t)+ ~r~(t~] t'(s)
ds (13)
where
3. The Somigliana dislocation problem
(14)
The Somigliana dislocation problem will be described. The two constituents of the system shown in Fig. 1 are perfectly fit over the interface, when they are two unconnected and undeformed parts. A relative displacement between the two opposite sides of the interface with Cartesian components U~ = Ux(x,y) and U, = U;(x,y) is given; they are then welded together by filing in material in the case of a gap, or taking out the extra material in the case of an overlap. The stressed deformation induced in the system will be analysed. According to the above statement, the following conditions should be satisfied across the interface
In Eqs. (12) and (13) and in the sequel, left and right primes denote indefinite integration and differentiation, respectively, with respect to the variable z; t represents an interracial point and s is an interfacial arc-coordinate measured from an arbitrarily chosen point; a and ]3 are the Dundurs parameters defined by
"~Jffl --'~'~6"r2= U~ ÷ i V ,
and
~9~, =.5~ 2
(11)
where ~ = ux+ iu~ with u x and u,. being the Cartesian displacement components and ~ ' is the force acting on an arc. The second condition expresses the traction continuity in its integrated form. Employing the Kolosov-Muskelishivili formalism of plane elasticity given in Appendix A, the interfacial
C ( l ÷ KI) -- (1 ÷ K2)
r ( l + ~,) + (1 + ,~) '
F(K,-
I ) - ( K 2 - 1)
F ( 1 + K,) + (1 + a:2)
8/x2 ~'= r ( 1 + KL) + (1 + K2) '
r=
~--2~ ~,
Eqs. (12) and (13) show that the KolosovMuskelishivili potentials of the matrix are expressible in terms of those of the inhomogeneity along the interface. This fact along with the holomorphic trans-
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
formation of a complex function defined in the introduction enables the construction of the general representations of the complex potentials of the two phases so that the interfacial conditions are satisfied. By extending the definition of ~'(z,~') to the whole z-plane, the following expressions for the complex potentials of the two phases are defined:
'w,(z) = 'Wo(z)
-
azWo
+/-/n~o ~
+m~'
z,
(15)
217
iarities introduced to the matrix potentials WI(z) and ~ f l ( z ) by the ~'-terms and they ensure vanishing stress at infinity. The latter means that the complex potentials Wl(z) and ~ l ( z ) should decay to zero as I zl tends to infinity. It should be noted that the two-phase potential functions may be defined in a variety of ways. For example, setting ( O / H )(R2/z)Wo(z) + ':gO'0(z) instead of Wo(z) in Eqs. (15)-(18), the following solution representation is defined: 'wl(z)
= 'W0(z)
z
(19) ' T f , ( z ) = '~go( z ) +
A'W o
g 2
=
-
R2
--%(z) Z
+( A + I2)--Wo( z )
R4
{ R2
Z
+T
d--~
-M
~
-M
--,z z
+----~' z dz
+A)'Wo(z )
1+/3'
g2=
(21) R2
A ) - - W o ( z ). z
(17)
(22.)
(18)
In the special case of a rigid inhomogeneity (P-2 _=_ac), the potentials of the matrix are obtained by setting A = I / H = K,, g2= - 1 in Eqs. (15) and (16) (or in Eqs. (19) and (20)). For P-2 = 0 ( a = A = 17-- -1), Eqs. (19) and (20) provide the complex potentials of a circular hole in an infinite plate with a prescribed displacement at the boundary. The equations of the latter case, in the absence of the ~'-terms, express the second circle theorem of plane elasticity established in Ref. [11].
for the inhomogeneity; A, 17 and g2 are auxiliary two-phase parameters given by 17=--
z,
' w 2 ( z ) = (l + / 7 ) ' W o ( z ) 'Tf2(z ) = (1 + H ) ' 7 / / ( z ) - (1 +
'~¢'2(Z) = (1 + H ) ' T f o ( Z )
1-/3'
+----~' z dz
(2O)
for the matrix, and
A = - -
--,z z
z, (16)
'W2(z ) = (1
~
1-/3
In the above equations, Wo(z) and ~Vo(Z) are two holomorphic functions, and Wo(R2/z), ~o(R2/z) are their Kelvin transformations; ~'(z, R2/z) and ~'(R2/z,z) are the holomorphic transformations of the complex functions ~ ( ~ , z ) and ~(z,~), respectively. The solution to the Somigliana dislocation problem is expressed by Eqs. (15)-(18) in terms of two holomorphic functions Wo(z) and 7//0(z), which constitute the two-phase potentials of the system. For a given Burger's vector ~', the two-phase potentials are determined so that they eliminate possible singu-
4. Applications
4.1. The eigenstrain problem The previous results are applied now to calculate the fields induced in the system, when the inhomo-
218
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
T geneity is subject to uniform eigenstrains e~, e,., and Exr r" The elastic fields of the corresponding problem with a perfect interface are listed in Appendix B. The interfacial stress components of this problem are given by
(15) and (16), the convergence of Wl(z) and 7fl(z) at infinity implies
1 I+A o'pPo= - -~ M ~ _ ~ F r
a 2 = a 4= ... = 0
MR2At al
MB t
1 - ~'~ '
a3
T '
(30)
1
b t =--MR2[ A t +4Bj + 3(Z +/2)B1],
2
1
+~M(I +lI)(Gre-2i°+Gre2i°) i o'pp = - -~M(1 + H ) ( G r e - 2 i ° -
GTe2i°).
(23) (24)
For the function ff(z,z), the following torm is assumed R2
8/( Z, ~.) = AtR2z +
TA-llZ
1 + -~z( B, z 2 +B-T~2)
b 2 = b 3= ... = 0 .
Having determined the two-phase potentials, the elastic field of the Somigliana dislocation problem are obtained by means of Eqs. (15)-(18) and Eqs. (Al), (A2), (A3) and (A4). The interfacial traction components are MR e 1 + A Ors
oo
2
(25)
R3 ( ~
1-
MR 2
-~t~ l -- - - ~ ' -
o
(I
--I-H)
4 ~ I + 3(A + a ) B l ] e -zi° MR 2
where Aj is a real constant and A l, B I are complex constants. The interracial components of the function are given by
4=atR +Tt T
(31)
- - -8( l + n ) X[AI+4B,+3(A+a)BI]e
+-~l)e-~,o
2i°
(32)
iMR 2
°°;
8
[6( 1 + A)B-T- ( 1 +/-/)(~-~ + 4~1)
+ ( a + 0)~1] e -z'° + T
~o-
+ BI e2iO
iR 3 4
(26)
iMR 2
+---Z[6(I +A)B, +(1 +//)(at +4Bt)
(a-Te-2i°-ate2i°)"
(27)
+ ( A + / 2 ) B t i e +2i°.
(33)
The constants At, A~ and B t will be calculated so that the interfacial conditions in Eqs. (5) and (6) are satisfied. The interfacial stresses o'o~ and %~ entering in these equations will be determined by solving the corresponding Somigliana dislocation problem. The two-phase potentials of this problem are sought in the form
Substituting the pairs of Eqs. (23) and (24), Eqs. (26) and (27) and Eqs. (32) and (33) into Eqs. (5) and (6) and comparing the coefficients of the same terms, it is found that
'W0(z) = a t z + a 2 z 2 + a3 z3 + . . .
A t = [3(l + ~ ) 2 n o n o + 2(1 + o r ) ( 1 - ~3)no]
'~rf'0(Z)
=
bl Z -t- b2 22 -~ b3 z3 -1- . . .
(28) (29)
where at, a2, a 3 . . . . and bj, b2, b 3 . . . . are complex constants. Substituting the appropriate functional forms of Eqs. (25), (28) and (29) into Eqs.
At = -
(1 + ~ ) n o 1 +a-2/3+(1
+a)n
o
( F T / R z)
(34)
× [2(1 + a ) ( 2 + / 3 ) ( n o + no) +3(1 + ~ ) 2 n o n o - 4 ( 1
-/32)]-'
(GT/R2) (35)
219
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
B, = [(1 + a ) ( l - f l ) ( n o - n o )
]
with those obtained in Ref. [9] for a spherical inhomogeneity with an imperfect interface.
× [2(1 + a ) ( 2 + fl)(n o + no) +3(1 + a)2nono-4(1 -/32)] - I ( G r / R 2 ) . (36) Thus, the solution of the Somigliana dislocation problem in terms of the two-phase potentials are 'W0(z) =
MREA1 1 - g2
MB1 z 3
(37)
2
MR 2 '~e0(z) = - - - ~ - ( a l + 4BI + 3( A + g2)Bi)z. (38) The elastic fields for the imperfectly bonded eigenstrain problem are obtained by superposing the fields for the Somigliana dislocation problem derived above on those of the perfectly bonded inhomogeneity problem given in Appendix C. The above results are in agreement with those obtained in Refs. [7,10]. As previously mentioned, the solution obtained for an inhomogeneity problem with the interfacial conditions in Eqs. (1) and (2) is accepted if it provides positive radial stress along the interface. Eq. (1) shows that a positive radial stress along the interface ensures a positive radial displacement jump and vice versa. The necessary and sufficient conditions, for which such a requirement is satisfied, are
liFr-12lGrl>Oand llFr +12lGrl>O
(39)
where (1 + ot)np l, = -
1 +a-2fl+(1
+a)np
(40)
4.2. The thermoelastic problem Considered as a second application is the thermoelastic problem of a circular inhomogeneity with an imperfect interface disturbing a remote temperature change of the reference temperature given by q0 T = ~-~lty+ Tc .
(42)
In Eq. (42), qo represents a uniform heat flow along the y-axis and T~ a uniform temperature change; k t denotes the thermal conductivity of the material. The thermal conditions along the interface are assumed to be those of a perfect thermal contact between the two phases. The interfacial traction for the corresponding problem with a perfect interface, as given in Ref. [12], is
M ( a , , - a2t)(1 + A)Tc tr°~ + itrp°P=
(1-O) - M(1 + A)RQo ei°
(43)
where 1( Q0 = "2
2a2t kl t 4- k2 t
air ) kh
qo
(44)
with a t being the coefficient of linear thermal expansion. Assuming the Burger's vector to be of the form
~-g'(Z,7.) = A2Rz + A2 R2 -+(BzZ+-~2~)Z z
(45)
its interfacial components are
12-- [1.5(1 + a)2nono 4- (1 + a ) ( 1 - fl)] × [2(1 + a ) ( 2 + f l ) ( n o + n 0 ) +3(1 + a)2npno- 4(1 -- [3)2] -1
(41)
The two-phase potentials, Eqs. (37) and (38), show that the stress field within the inhomogeneity is a quadratic polynomial. This field becomes uniform, when either np= n o or G r = 0 with F r 4=O. The latter case corresponds to a volumetric-type straining for which A~ = A 2 = 0. These results are consistent
iR2 ~ ' ° = - 2 (Azei°-"~2e-i°) where A2 is a real constant while complex constants.
(47) A2
and
B 2 are
220
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998)213-222
For the Somigliana dislocation problem, the twophase potentials and the traction components along the interface are obtained in the form
aM(1 + A ) 2 ( n o + n o)Qo
MA2R I_~z-M(A2+B2)z2,
'W0(Z) -
-
'~f0(z) = 0 R2M(1 + A) 2
1-/2
×[( A2 + B2)ei° + ('~2 + Bz)e-i° ]
(49)
RZM( 1 + A) i 2 × [ ( A 2 + B2) e i ° - ('~2 + Bz)e-i°] "
(50)
The unknown coefficients in Eq. (45) are determined as in the previous application. It is found that
(a,,-a2,) A2= 1 + a _ 2/3 + n o ( l + or)
~
T~
2(1-13) +(! +a)(no+n0)
(air -- a2t)Tc R
12q>_ 0
( a,t - a2t) Tc + h2q > 0 R where hj
(52)
2a2t h, - - -
i ( n p - no)(1 + or) 2(1-/3) +(1 +a)(np+n0)
Eqs. (54) and (55) along with those reported in Appendix C imply that a linear stress distribution is introduced to a circular inhomogeneity, when a remote uniform heat flow is applied to the matrix. A remarkable result regarding this field is the following: there is no stress distribution within the inhomogeneity in the case of a sliding interface. In other words, a sliding circular inhomogeneity embedded in a matrix behaves like a void when it disturbs a uniform heat flow. The necessary and sufficient conditions ensuring a positive radial displacement jump along the interface are
(51)
Qo
Q0-
(53)
The list of the elastic fields for the corresponding problem with a perfect interface is given in Appendix C. Some interesting aspects of the solution are associated with the stress distribution within the inhomogeneity. The part of this stress distribution corresponding to the Somigliana dislocation problem is calculated in the form M(1 + A) 2no(a . - a2t)Tc °'~°°+ icr~°°= - (1 - O)[1 - / 2 + C(1 + a)np]
r sin 0. (55)
hn
2ino(1 + o~)
B2 = -
2 + ( 1 + A)(np +no)
(48)
MA2(I + A) R 2
A2 = -
2M(1 + A ) 2 n o ( a , , - a2t)Tc (1 - g2)[1 - O + (1 + a ) n p ]
°2s°p + °2s° =
ah
kit + k2t
h2 =
2(1-fl) 1 + a-
Appendix ism
(56)
(57)
kit
+(1 +a)(n,,+no) 2/3+ (1 + ot)nt,
A. The Kolosov-Muskelishivili
(58)
formal-
4/.t(up + iUo) = e-i°[ K'W(Z) -- z W ( z ) - '~W( z)] (A1) crpp + %o = W(z) + W(~)
(A2)
2(o'po + i%o )
+
iM(I + A)Z(np + no)Qo
re io
2 + ( 1 + A)(no + no)
= W(z)
+
-
-
(A3)
Z
(54)
2 i,P~'= 'W(Z) + z W ( z ) + 'W(Z).
(A4)
221
M.A. Kattis, E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
Appendix B. The p-fields for the eigenstrain problem
(B 1)
2(o',°oo + io',Poo) I+A
1 7M(1 +l-I)'Gre -2i4' (B2)
= -M~_oFr+
2 ( a z t - a l t ) ( l + A)T¢ R 2 (1 + K , ) ( I - O ) pz +auTcP
ulP"=
I+A o'lPoo+ o'lP00= - M ~ Fr
(c3)
O'lPoO= 0%o0 = 0
2(a2t- a,t)(l +A)T~ u~° = (1 + Xl)(1-- g2) p + altTc P
(c5)
uPo= u2o = 0.
(C6)
For a temperature change -qo y/kit at infinity:
°'g,, + ~oo 1
=~M(1 + II)[Gre-2'°
R2
+ Gre2'*]-~-i
(B3)
41xlaltqoR
o'IPop =
(1 + tq)klt
2( o'~oo + io'~oo)
+]'+~
1+ A
R2
°',Poo=
R2/R]
R2
R2 2 (1 + / / ) ' 7
-1)7
ITIPp0
(1
R2
+
+ --
(B5)
l+k 4N2(u~p + iuS2o)
r G r = eyyr _ er + 2 i . r Eyy,
-1)7
KI)klt
(B6)
1-
l+k
) 1 psin0
6/.qaltqo(l + A ) [ 2a - - l i p sin 0 (1 + KI)klt ~ 1 + k (Cll)
Or2PpO~ --
For a uniform temperature change T~:
2pqauq0(1 + A ) [ 2a -- ljpcos 0 (1 +Ki)klt ~ l+k (C12) /
4/xl(alt- a2t)(1 + A)Tc R 2 (1 + KI)(1 - a ) p"
2ajtqo R2 [ 1 + K1 (C1)
(1 + K,)(1 - O)
(c9)
cos 0
7);
(Cl0)
(B7)
Appendix C. The p-fields for the thermoelastic problem
4 p . , ( a , t - a2t)(1 + A)T~
R3
1 +k
(1 + rl)klt
o-~o o =
trPoe=
crLP = °':'°° =
(c8)
sin 0
21xtaltqo(l + A) ( 2a
1 +A K2M = M l _ - i - - ~ F r - - - - ~ ( 1 +II)'Gre-2'e'p
O"lPPo= --O"100 =
1+
4p, laltqoR ~_+~_( 2a
M
1 X[KI"GTe -2i0 + (1 -- 7 ) Gre-4io[ . ,
R3
1 +k
(B4) l+k
7
(C7)
41zlauqoR ~ _ A ( 2a
4/x,(u~p + iu~o) 1+ A
7
1 - -~'] p ] sin 0
(1 + Kl)klt × [ G r e - 2 ' ° + 3 ( 1 - - ~ z ) G r e ai°]
=M~-oFr
1
l+k
R2
1
= - M ~ _ ~ F T 7 + ~-M(l + H ) V
F T = ~Tx +
(c4)
(C2)
u~o =
(1 + Kl)klt / X
+ I+K,
4 l+k
1
M.A. Kattis. E. Providas / Theoretical and Applied Fracture Mechanics 28 (1998) 213-222
222
1_,( R2)
References
1--~- 5-
+2(l+k~-------S
,_, + 1 + k log
---1)o~} 1 +k
+
sin 0 (C13)
altqo R2 uPlo
1 + ~:l
(1 + K1)klt X
+
4
p2
1 +A
"~+
I+K I
1 -k
(
(
l+k
R2 )
2(i +~:)11 +k log
2a
7 ----
-
1]$"/cos0
1 +k
g
J
(C14)
2al__tqo_p2 u~o -
[ 1 + K,
[ ,+A(2a____)] l(2a )) (1 + K l ) k l ' /
X 1+
+2
l+KI
l+k
2altqop2
X l-}- lq-K1
sin~b
(c15)
4
___,)] 2a
l+k
2a I+k
1)g~} cos d~
where g"= 1 +A+(1
1
( 1 +Kj
I+A(
2
l+k
1 ~
( ~ + "~1 y-k~l, /
1(
4
-1- KI)~,~.
(C16)
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