Neutral inhomogeneities in conduction phenomena

\ PERGAMON

Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

Neutral inhomogeneities in conduction phenomena Y[ Benvenistea\\ T[ Milohb a

Department of Solid Mechanics\ Materials and Structures\ School of En`ineerin`\ Tel!Aviv University\ Ramat!Aviv\ Tel!Aviv 58867\ Israel b Department of Fluid Mechanics and Heat Transfer\ School of En`ineerin`\ Tel!Aviv University\ Ramat!Aviv\ Tel!Aviv 58867\ Israel Received 06 August 0887^ received in revised form 06 December 0887

Abstract A neutral inhomogeneity in heat conduction is de_ned as a foreign body which can be introduced in a host solid without disturbing the temperature _eld in it[ The existence of neutral inhomogeneities in conduction phenomena is studied in the present paper[ Both the inhomogeneity and the host body are assumed to be isotropic\ with the inhomogeneity being either less or more conducting than the surrounding body[ The property of neutrality is de_ned in this work with respect to an applied constant temperature gradient in the host solid[ It is achieved by introducing a non!ideal interface between the two media across which the continuity requirement of either the temperature _eld or the normal component of the heat ~ux is relaxed[ These interfaces are called {non!ideal interfaces| and represent a thin interphase of low or high conductivity^ they are characterized in terms of some scalar interface parameters which usually vary along the interface in order to ensure neutrality[ The conditions to be satis_ed by the _eld variables at a non!ideal interface with a variable interface parameter are _rst derived\ and closed form solutions are presented for the interface parameters at neutral inhomogeneities of various shapes[ In two!dimensional problems\ duality relations are established for composite media with non!ideal interfaces and variable interface parameters[ These are implemented in establishing general criteria for neutrality[ The terminology of heat conduction is used through! out in the paper but all the results can be directly transferred to the domains of electrical conduction\ dielectric behavior or magnetic permeability[ Þ 0888 Elsevier Science Ltd[ All rights reserved[ Keywords] Conductivity of composite media^ Imperfect interfaces^ Neutral inhomogeneities

 Corresponding author[ E!mail] benbenÝeng[tau[ac[il 9911!4985:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 9 9 1 1 ! 4 9 8 5 " 8 7 # 9 9 0 1 6 ! 5

0763 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

0[ Introduction Let us consider a homogeneous solid which is subjected to a certain loading on its external boundary[ The loading may be a distribution of tractions in the case of elastic deformations of the solid\ or a heat ~ux in the case of conduction phenomena in it[ It is well known that the introduction of a cavity or a foreign body\ called an inhomogeneity\ in the host medium will change the original distribution of the _eld parameter in the surrounding body[ The idea of making certain changes around the foreign body so that the original _eld in the host solid remains undisturbed\ can be traced back to a paper by Mans_eld "0842#[ In that paper Mans_eld showed that a hole in an elastic sheet can be reinforced in such a manner so that the stress _eld outside it is the same as that which was present in the uncut sheet[ That hole was called {neutral|[ This idea of a neutral hole was further elaborated in Richards and Bjorkman "0871#\ Budiansky et al[ "0882#\ and Senocak and Waas "0884# where other related references on the same topic can be found[ Very recently\ Ru "0887# studied the idea of neutrality in the context of inhomogeneities imbedded in a two!dimensional elastic body[ Ru makes the rightful point that although the idea of a neutral hole was the subject of several previous papers\ the study of neutral inhomogeneities has not received yet the deserved attention in the literature in spite of their relevance to the _eld of composite materials[ For the case in which the inhomogeneity is sti}er than the host medium\ Ru relaxed the displacement continuity condition between the inhomogeneity and the surrounding body\ and used instead a non!ideal interface condition which allows a jump in the displacement _eld[ He then showed that if the interface parameter which relates the discontinuity of the displacement to the tractions at the interface is properly chosen\ then a two!dimensional stress _eld in the original body will remain undisturbed after the introduction of the inhomogeneity[ Generally\ the interface parameter is found to vary from point to point on the interface[ In the present paper the existence of neutral inhomogeneities in conduction prob! lems is studied[ The conductivity setting is simpler than the one of elasticity\ so that one expects to be able to derive here a richer class of analytical results which are valid not only for three!dimensional problems but that also pertain to inhomogeneities which may be either less or more conducting than the host medium[ Neutrality is de_ned with respect to a constant temperature gradient in the host medium\ and is achieved by introducing between the matrix and the inhomogeneity a so!called {non! ideal interface|[ For the case in which the inhomogeneity is more conducting than the host solid\ a non!ideal interface representing an interphase of low conductivity is chosen[ If the inhomogeneity is less conducting than the host medium\ then an interface corresponding to an interphase of high conductivity is selected[ At weakly conducting interfaces the normal component of the heat ~ux is continuous\ whereas the temperature _eld undergoes a discontinuity which is proportional to the normal component of the heat ~ux[ This is the counterpart in heat conduction of the interface condition used by Ru "0887# in elasticity[ At highly conducting interfaces\ the tem! perature _eld is continuous\ whereas the normal component of the heat ~ux undergoes a discontinuity which is proportional to a certain di}erential expression of the tem!

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0764

perature which involves surface derivatives only[ It should be noted that although the topic of weakly conducting interfaces in composite media has been investigated in the literature for almost two decades\ the study of the counterpart of highly conducting interfaces is extremely recent\ see Torquato and Rintoul "0884#\ Cheng and Torquato "0886#\ Lipton "0886a\ b\ 0887#\ Miloh and Benveniste "0887#\ and Benveniste and Miloh "0887# where an extensive list of references on both kinds of non!ideal interfaces can also be found[ Scalar parameters a and b are used to characterize highly and weakly conducting interfaces\ respectively[ These parameters are related to the proper! ties of the thin interphase by a  kct\ and b  kc:t\ where kc denotes the isotropic conductivity of the interphase and t is its thickness[ For general shapes it is found that in order to ensure neutrality a variable a or b on the interface is needed^ for interphases of constant thickness this can be achieved by varying the conductivity kc along the interphase[ It is _nally pointed out that highly conducting interfaces can also _nd their counterpart in elastic composites as thin but sti} interlayers[ Such reinforcements appear however to have been explored until now only on the outside boundaries of an elastic body or around holes "see\ for example\ Schiavone and Ru "0887# and the references therein#[ Previous studies on non!ideal interfaces in heat conduction in composite media have assumed a constant interface parameter[ In Section 1\ we derive for the _rst time the interface conditions prevailing at non!ideal interfaces with variable interface parameters[ Although at interfaces of low conductivity "LC!type#\ the interface con! ditions have the same form both for the case of constant and variable interface parameters\ at interfaces of high conductivity "HC!type# they are basically di}erent with the interface conditions for a variable interface parameter being more involved[ In Section 2\ neutral inhomogeneities with LC!type interfaces are studied[ Explicit expressions for the LC!type interface parameter b which result from the solution of an algebraic equation are obtained for various inhomogeneity shapes[ It is shown that inhomogeneities of surprisingly various complex shapes may be neutral with a proper design of a variable interface parameter[ A criterion on the topological struc! ture of the inhomogeneity is derived for the condition of neutrality[ In Section 3\ neutral inhomogeneities with HC!type interfaces are studied[ It turns out the nature of the non!ideal interface condition in this case results in a partial di}erential equation for the interface parameter a which ensures neutrality[ In the _rst part of Section 3\ we study a triaxial ellipsoidal neutral inhomogeneity and provide a remarkable simple closed form solution for the variable interface parameter a[ This solution recovers\ as special cases\ the previously known constant values of a for the degenerate shapes of spheres and cylinders of circular cross section "see Torquato and Rintoul\ 0884^ Lipton\ 0886a\ b#[ The second part of Section 3 is concerned with two!dimensional problems[ Here\ the duality connections recently established by Lipton "0886a#\ between the _elds in composites with LC and HC!type interfaces with constant interface parameters\ are generalized for the case of variable interface parameters[ The derived relations allow us to transfer the results on LC!type interfaces in Section 2\ to HC!type interfaces and thus to establish a general criterion for neutrality for the latter case as well[

0765 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

1[ Non!ideal interfaces in heat conduction between two media Let us consider two isotropic media with conductivities k0 and k1 which are separated by an interface G[ An ideal interface is de_ned as one across which the temperature and the normal component of the heat ~ux are continuous\ i[e[\ f"0# = G  f"1# = G \ qi"0# ni = G  qi"1# ni = G \ qi"r#  −kr

1f"r# 1xi

r  0\ 1\

"1[0#

where f denotes the temperature\ qi is the heat ~ux vector satisfying qi\i  9\ xi are the coordinates of a Cartesian system\ and ni are the components of the unit normal to G\ taken for example\ from medium 1 to medium 0[ Two types of non!ideal interfaces will be now de_ned] "a# a non!ideal interface which models a thin interphase of low conductivity*this will be called an LC!type interface\ "b# a non!ideal interface which models a thin interphase of high conductivity*this will be called a HC!type interface[ The interface conditions at both kinds of interfaces involve a certain interface par! ameter which may be variable along the interface[ For the case of constant interface parameters these conditions have been derived by Sanchez!Palencia "0869#\ Pham Huy and Sanchez!Palencia "0863# and more recently by the present authors "Miloh and Benveniste\ 0887# where the curved nature of the interface has been explicitly incorporated in the framework of a proper curvilinear coordinate system[ This deri! vation will be generalized below to the case of non!constant interface parameters[ We will _rst state the interface conditions and then indicate brie~y how they can be obtained[ At an LC!type interface the normal component of the heat ~ux is continuous\ whereas the temperature _eld undergoes a discontinuity which is proportional to the normal component of the heat ~ux] qi"0# ni = G  qi"1# ni = G  b"f"1# = G −f"0# = G #\

"1[1#

where b\ allowed to be variable on G\ is the interface parameter of the LC!type interface de_ned by b  Lim = kt:9 "kc :t#\ :9

"1[2#

c

with kc and t denoting\ respectively\ the variable conductivity and the constant thick! ness of the thin interphase from which the interface G has resulted in a limiting sense[ A value of b :  represents an ideal interface "isothermal contact#\ whereas a value of b  9 stands for adiabatic "vacuous# contact[ At an HC!type interface\ the temperature _eld is continuous\ whereas the normal component of the heat ~ux undergoes a discontinuity] f"0# = G  f"1# = G \ qi"0# ni = G −qi"1# ni = G  a"DS f#= G ¦"9S f = 9S a#= G \

"1[3#

where a\ allowed to be non!constant on G\ is related to the parameters of the original thin interphase by a  Lim = kt:9 "kc t#[ : c

"1[4#

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0766

The quantities 9Sf\ 9Sa\ de_ned as surface gradients of f and a\ respectively\ are given by 9S f 

0 1f 0 1f 0 1a 0 1a u0 ¦ u1 \ 9S a  u0 ¦ u1 \ h0 1u0 h1 1u1 h0 1u0 h1 1u1

"1[5#

where h0 and h1 are the metric coe.cients of two orthogonal parametric curvilinear coordinates "u0\ u1# which describe the interface and "u0\ u1# denote the unit vectors of these curvilinear coordinates[ The quantity DSf is the surface Laplacian of the temperature de_ned as "see for example\ Van Bladel\ 0853#] DS f 

0

1

0

1

0 1 h1 1f 0 1 h0 1f ¦ [ h0 h1 1u0 h0 1u0 h0 h1 1u1 h1 1u1

"1[6#

An equivalent description of the surface gradient 9Sf\ and the surface Laplacian DSf in Cartesian coordinates is given by "see Gilbarg and Trudinger\ 0872^ Lipton 0886a#] 9S f  9f−"9f = n#n\ DS f 

0

1 1 −ni nk 1xi 1xk

10

11 f 1f 1f 11 f 1f 1nj −ni nj  −ni nj −ni [ "1[7# 1xi 1xj 1xi 1xi 1xi 1xj 1xi 1xj

1

Note that both DSf\ and 9Sf contain only derivatives tangential to G so that in view of "1[3# 0\ there is no ambiguity as to each side of G\ "1[3# 1 is to be evaluated[ Finally a value of a : 9 in "1[3# denotes an ideal interface condition whereas a value of a :  describes contact with a medium of in_nite conductivity[ The derivation of "1[3# given in Miloh and Benveniste "0887# for the special case of constant a consists essentially of two steps[ Firstly\ a thin interphase of constant thickness and constant conductivity is represented in the framework of a {parallel coordinate| system "see Rouse\ 0848\ for example# which is generated by the inner surface of the interphase "taken adjacent to phase 1# and the normals to it[ More speci_cally\ let this system be denoted by "u0\ u1\ g#\ where "u0\ u1# denote an orthogonal set of parametric curves on the inner surface\ and g is the distance measured between a given point in the interphase and the corresponding closest point on the inner surface "see Fig[ 0"a##[ With this convention\ the inner surface is obtained by setting g  9\ and the outer surface by setting g  t[ Let "s0\ s1\ g# denote the arc lengths along this coordinate system\ and "h0\ h1\ hg# denote its metric coe.cients\ with hg  0[ Secondly\ under the condition t : 9\ the temperature f"c# in the interphase at a generic point P of the outer surface\ is expressed in a Taylor series about the corresponding point Q on the inner surface as follows] f"c# = P  f"c# = Q ¦

1f"c# 1g

b

t¦O"t1 #\

"1[8#

Q

with a similar expansion being admitted for the product of the normal component of the heat ~ux and a surface element DS? on the outer surface of the interphase]

0767 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

Fig[ 0[ "a# An interphase between two media[ "b# The location of the inhomogeneity in the host medium[

qg"c# = P DS?  qg"c# = Q DS¦

1 "c# "qg DS# 1g

b

t¦O"t1 #\

"1[09#

Q

where DS denotes an element on the inner surface\ with O"zDS# ¾ O"t#[ As has been shown in Miloh and Benveniste "0887#\ eqs "1[8# and "1[09# allow us _rst to show that at an LC!type interface\ the normal component of the heat ~ux is continuous\ whereas at an HC!type interface the temperature is continuous[ These conclusions continue to be valid for a non!constant conductivity kc of the interphase\ and thus\ for a variable b or a of the interface[ In fact\ the full form of eqs "1[1# and "1[3# can be shown to originate from the expansions "1[8# and "1[09# for the case of a variable b or a[ To get "1[1#\ we write "1[8# in the form f"0# = P  f"1# = Q −qg"c# "t:kc #¦O"t1 #\

"1[00#

where the temperatures at the inner and outer surface of the interphase f"c# = Q\ f"c# = P have been replaced by the temperatures f"1# = Q\ f"0# = P in the media 1 and 0\ respectively[ Letting t : 9\ with kc : 9\ while de_ning b by "1[2# leads to "1[1#[

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0768

To get "1[3#\ eqn "1[09# is _rst written in the form qg"0# = P "DS?:DS#  qg"1# = Q ¦

0 1 "c# "qg DS# DS 1g

b

t¦O"t1 #\

"1[01#

Q

where the normal component of the heat ~ux at the inner and outer surfaces of the interphase qg"c# = Q \ qg"c# = P \ have been replaced by the corresponding quantities in media 1 and 0[ The balance of the heat ~ux demands on the other hand that

b

b

0 1 "c# 0 1 "c# 0 1 "c# "q0 DS0 # ¦ "q1 DS1 # "qg DS# ¦ DS 1g DS0 1s0 DS1 1s1 Q Q

b

 9\

"1[02#

Q

where "DS0\ DS1# and "s0\ s1# are\ respectively\ the in_nitesimal elements of area and the arc lengths of the "u0\ u1# coordinate system[ This last equation can be further developed into 0 1 "c# "qg DS# DS 1g

0 1 h1 1f"c# 0 1 h0 1f"c# ¦ h0 h1 1u0 h0 1u0 h0 h1 1u1 h1 1u1

b $ 

Q

0

1 $

¦

0

1%b

kc

Q

0 1f"c# 1kc 0 1f"c# 1kc ¦ h01 1u0 1u0 h11 1u1 1u1

%b

\

"1[03#

Q

where use has been made of the fact that hg  0\ as well as of the equalities DS0  h1 du1 dg\ DS1  h0 du0 dg\

1 0 1 1 0 1  \  [ 1s0 h0 1u0 1s1 h1 1u1

"1[04#

Let now DS?  DS"0¦kt# in "1[01#\ where k is the mean curvature of the inner surface\ and assume that kt ð 0[ It is easy to check that the substitution of "1[03# into "1[01# and making use of the de_nition of a in "1[4#\ readily yields the sought interface condition "1[3#[ This section is concluded by giving a simple physical explanation to the above described behavior of the temperature and normal component of the heat ~ux across LC! and HC!type interfaces[ The only way the temperature _eld may be discontinuous across an interface\ is if the temperature gradient is su.ciently large within the thin interphase modeled by that interface[ On the other hand\ given a _nite heat ~ux across the interphase\ the temperature gradient in it will be large only if that interphase has a su.ciently low conductivity ðin view of a relation of the type in "1[0# 2Ł[ Therefore\ at LC!type interfaces the temperature _eld is discontinuous\ whereas at HC!type interfaces it is continuous[ As to the behavior of the normal component of the heat ~ux\ the only way this component may be discontinuous across the interface is when it is compensated by a suitable tangential heat ~ux along that interface "the heat ~ux balance within an element of the thin interphase can then be achieved#[ However\ for a very thin interphase\ the resultant tangential heat ~ux will not vanish only if the interphase has a su.ciently high conductivity[ Thus\ at LC!type interfaces the normal component of the heat ~ux is continuous\ whereas at HC!type interfaces it is dis! continuous[

0779 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

2[ Neutral inhomogeneities with low conductivity "LC!type# interfaces Consider a homogeneous and isotropic medium with conductivity k0[ Let this medium be subjected on its boundary to an intensity Hi  −"1f:1xi# which is uniform in the x0!direction\ resulting throughout in a temperature _eld in the form f"0# "x#  −H9 x0 [

"2[0#

We wish to introduce somewhere in this medium an isotropic inhomogeneity of a given shape and of conductivity k1\ with k1 − k0\ and have the _eld "2[0# remain undisturbed in the host medium[ It will now be shown that this may be possible for certain inhomogeneity shapes provided that a properly designed LC!type interface is chosen\ possibly with a variable b on G[ These inhomogeneities will be called neutral[ The temperature _eld inside such inhomogeneities can be readily determined in the course of the derivation[ Consider an inhomogeneity of a given shape\ and introduce it in the host medium with a temperature _eld "2[0#\ so that a given point P in it coincides with the center of a Cartesian coordinate system located in the host medium "see Fig[ 0"b##[ Furthermore\ let the orientation of the inhomogeneity\ with respect to the axes of that coordinate system\ be also prescribed[ After the introduction of the inhomogeneity\ let the temperature in the host medium be still given by "2[0#\ and the temperature inside the inhomogeneity be described by f"1# "x#  −H9 "k0 :k1 #x0 −AH9 \

"2[1#

where A is a constant to be determined[ It is observed that since f"1# is linear in x0\ it trivially satis_es Laplace|s equation\ so that the only condition which needs to be ful_lled is the satisfaction of the interface conditions "1[1# on G[ It will now be shown that the implementation of these interface conditions will lead to the proper choice of b and the determination of the constant A in "2[1#[ First\ it is noted that the structure of the _rst term in "2[1# automatically insures the continuity of the normal ~ux across G[ The implementation of the remaining part of the interface condition in "1[1# on the other hand leads to b"−H9 x0 ¦H9 "k0 :k1 #x0 ¦AH9 #  −k0 H9 n0 \

"2[2#

where n0 is the x0!component of the outside unit normal to G[ This equation provides b  k0 n0 :"ð0−"k0 :k1 #Łx0 −A#[

"2[3#

For an inhomogeneity of a given shape and orientation\ the constant A needs to be chosen such that b − 9 is insured everywhere on G[ Clearly\ in view of its de_nition in "1[2#\ this parameter needs always to be non!negative[ For certain inhomogeneities this will be possible either with a unique or non!unique choice of A[ There are inhomogeneities for which no such choice for A is possible[ It is best to illustrate the implementation of these conditions by some examples[

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0770

2[0[ An ellipsoidal inhomo`eneity Consider an ellipsoidal inhomogeneity and let it be required that its center coincide with the origin of a Cartesian coordinate system with its axes being in the x0!\ x1! and x2!directions "Fig[ 1"a##[ It is then readily veri_ed that the choice of A  9 is the only one which would insure b − 9 throughout on G[ In fact\ with this choice of A\ the parameters n0 and x0 are always of the same sign on the ellipsoidal surface\ thus implying b − 9[ On the other hand\ if it is required that the inhomogeneity be slanted at a certain angle to the coordinate axes "Fig[ 1"b##\ then there is no possible value for A\ for which b − 9 throughout G[ Let it now be required to position the ellipsoid such that its origin coincides with the point "x¹0\ x¹1\ x¹2# in space and let its axes be parallel to the "x0\ x1\ x2# axes "Fig[ 1"c##[ In order to have b − 9\ A is now chosen such that ð0−"k0 :k1 #Łx0 −A  ð0−"k0 :k1 #Ł"x0 −x¹0 #\

"2[4#

yielding

Fig[ 1[ "a#\ "c# An ellipsoidal inhomogeneity oriented properly with respect to the intensity in the host medium for the condition of neutrality[ "b# An ellipsoidal inhomogeneity slanted with respect to the intensity in the host medium*a con_guration for which neutrality is not possible[

0771 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

A  ð0−"k0 :k1 #Łx¹0 [

"2[5#

Substitution of "2[5# in "2[2# provides the following value for b] b  k0 n0 :"ð0−"k0 :k1 #Ł"x0 −x¹0 ##[

"2[6#

It is thus noted that the distribution of b on G is the same for the ellipsoids in Figs 1"a# and "c#[ The temperature _eld within the ellipsoids in both cases di}er by a constant\ whereas the intensity remains the same[ The above observations show that the temperature _eld in the host medium will remain undisturbed if many ellipsoids are positioned throughout in space with the condition that their axes be parallel to the "x0\ x1\ x2# axes\ and that b be given by "2[6#\ where x¹0 refers now to the x0! coordinate of the center of the respective ellipsoid[ Finally\ it is interesting to observe that for a spherical inhomogeneity with radius a\ since there is n0  "x0−x¹0#:a\ the interface parameter becomes constant throughout on G and is given by b  "k0 :a#:ð0−"k0 :k1 #Ł\

"2[7#

a result which is in agreement with the previous _ndings of Torquato and Rintoul "0884#\ and Lipton and Vernescu "0885#[ 2[1[ Neutral inhomo`eneities of complex shapes The above analysis for the case of ellipsoidal inhomogeneities leads the way to some general criteria concerning neutral inhomogeneities of complex shapes[ Consider an inhomogeneity of an arbitrary shape\ and let it be required to position it at a certain location and orientation in the host medium without disturbing the ambient temperature _eld "2[0#[ The preceding discussion helps us to establish the following condition on the topological structure of the inhomogeneity in order that it be neutral for a constant intensity along the x0!direction in the host medium] there has to be at least one point "x?0 \ x?1 \ x?2 # in space such that for any point "x0\ x1\ x2# on G\ "x0 −x?0 # and n0 are of the same sign[ Once the shape of the inhomogeneity obeys this criterion\ the interface parameter b can be designed according to the formula b  k0 n0 :"ð0−"k0 :k1 #Ł"x0 −x?0 ## − 9[

"2[8#

The temperature _elds outside and inside the inhomogeneity are then\ respectively\ given by "2[0# and "2[1#\ with the constant A being now A  ð0−"k0 :k1 #Łx?0 [

"2[09#

Examples of several admissible shapes with the corresponding choices for the point "x?0 \ x?1 \ x?2 # are given in Fig[ 2[ For clarity in the illustration\ we have shown only two!dimensional shapes^ the same criterion also holds for three!dimensional shapes[ It is important to emphasize that the shapes are selected for a condition of neutrality under an initial intensity in the host medium which is in the x0!direction[ The same shapes may or may not be neutral for an intensity in another direction[ It is worth noticing that the shapes in Figs 2"a# and "d# admit only a single choice of x?0 whereas the shapes in Figs 2"b# and "c# admit several choices for this variable[ Thus\ there is

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0772

Fig[ 2[ Several admissible inhomogeneity shapes for neutrality with an LC!type interface[

a single choice for the parameter b in the inhomogeneities of 2"a# and "d# whereas there exists several such choices for the shapes in 2"b# and "c#[ Furthermore\ note that corners have been admitted on G for some of the neutral shapes[ This may result in a discontinuous b at these locations[ Finally\ some shapes which cannot be neutral for a ~ux in the x0!direction are illustrated in Fig[ 3[ As an illustration for the design of the interface parameter b for neutral complex shapes\ let us choose the case of a cube whose center is located at the origin of the coordinate system "Fig[ 4"a##[ One choice for x?0 is clearly x?0  9[ This yields a value of b] b  k0 :"ð0−"k0 :k1 #Ł"a:1##\

"2[00#

for the two faces of the cube perpendicular to the x0!axis\ and b  9 for the rest of the faces[ It can be readily veri_ed that with these choices of the interface parameter\ the solution f"0#  −H9 x0 \ f"1#  −H9 "k0 :k1 #x0 \ qi"0#  qi"1#  k0 H9 di0 \

"2[01#

with dij being the Kronecker delta\ ful_ls all the interface conditions on the cube surface G[ It should be indicated however that other choices for b are also possible[ For example\ let x?0  a:1 in eqn "2[8#[ This choice yields b :  for the right face of the cube at x0  a:1\ whereas b  k0 :"ð0−"k0 :k1 #Ł"a##\

"2[02#

0773 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

Fig[ 3[ Examples of inhomogeneity shapes with an LC!type interface for which neutrality is not possible[ "Correction added in proofs] The shape in Fig[ 3b is neutral for an intensity in the x0!direction and not neutral for an intensity in the x1!direction[#

Fig[ 4[ Two neutral inhomogeneities with an LC!type interface for which explicit expressions for the interface parameter are derived[

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0774

for the left face x0  −a:1\ and b  9 for all the other faces[ The solution with these choices of b is now summarized below] f"0#  −H9 x0 \ f"1#  −H9 "k0 :k1 #x0 −H9 ð0−"k0 :k1 #Ł"a:1#\ qi"0#  qi"1#  k0 H9 di0 [

"2[03#

Further choices of b are also admissible\ as a consequence of choosing a value of x?0 which obeys the relation −a:1 ¾ x?0 ¾ a:1[

"2[04#

This section is concluded by presenting another example of a three!dimensional inhomogeneity consisting of two cones of height L which are attached at their common basis of radius a\ see Fig[ 4"b#[ The interface parameter for this inhomogeneity is given by b  k0 ða:zL1 ¦a1 Ł:"ð0−"k0 :k1 #Ł=x0 =#[

"2[05#

The temperature and ~ux _eld are the same as in "2[01# and can be veri_ed to ful_l the interface conditions on G throughout[

3[ Neutral inhomogeneities with high conductivity "HC!type# interfaces As in Section 2 we consider an isotropic medium of conductivity k0 with a constant ~ux _eld[ We wish to introduce somewhere in this medium an isotropic inhomogeneity of a given shape and conductivity k1\ this time with k1 ¾ k0\ and have the temperature _eld in this host medium remain undisturbed[ We will show that this can in principle be achieved for certain inhomogeneity shapes by properly choosing a variable HC! type interface condition on its surface G[ Yet\ it turns out that this time the condition which will determine the interface parameter a will be a partial di}erential equation and not an algebraic one as in the case of b\ and the problem becomes an order of magnitude more di.cult[ The present section consists of two parts[ For three!dimensional inhomogeneities of arbitrary shapes\ a general solution being out of hand\ we concentrate in the _rst part on a triaxial ellipsoidal inhomogeneity and give a closed form solution for the interface parameter a for the case of an intensity in the host medium which is along either of the principal axes of the ellipsoid[ The second part deals with two!dimensional neutral inhomogeneities in which case general criteria for neutrality are established[ 3[0[ A neutral ellipsoidal inhomo`eneity 2 Consider a tri!axial ellipsoid Si0 "xi :ai # 1  0 such that a0 − a1 − a2\ with "x0\ x1\ x2# representing a Cartesian coordinate system[ Introduce this ellipsoidal inhomogeneity in a host medium in which there is a constant intensity _eld parallel to either of the principal axis of the ellipsoid[ Let also the conductivity of the inhomogeneity k1 be smaller than the conductivity k0 of the host medium[ The derivation starts by post!

0775 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

ulating that the temperature _elds inside and outside the inhomogeneity "f# "i#\ due to an intensity in the xi!direction are given by "f"0# # "i#  "f"1# # "i#  −H9 xi \ i  0\ 1\ 2[

"3[0#

Laplace|s equation is satis_ed throughout\ thus the only conditions to be met are the interface conditions on G[ The _rst interface condition in "1[3# on the continuity of the temperature _eld is trivially ful_lled[ The implementation of the second interface condition is conveniently handled in terms of a triply!orthogonal ellipsoidal coor! dinate system "r\ m\ n# where the three surfaces r  constant\ m  constant and n  constant denote\ respectively\ ellipsoids\ hyperboloids of one sheet and hyper! boloids of two sheets[ They form a spatial triply orthogonal coordinate system[ In this system the canonical ellipsoid is simply given by r  a0[ The metric coe.cients in this system are "Hobson\ 0844\ p[ 343#] hr1 

"r1 −m1 #"r1 −n1 # "r1 −h1 #"r1 −k1 #

\ hm1 

"r1 −m1 #"m1 −n1 # "m1 −h1 #"k1 −m1 #

\ hn1 

"r1 −n1 #"m1 −n1 # "h1 −n1 #"k1 −n1 #

\

"3[1# where k1  a01 −a21 \ h1  a01 −a11 \

"3[2#

 − r1 − k1 \ k1 − m1 − h1 \ h1 − n1 − 9[

"3[3#

with

The transformation between ellipsoidal and Cartesian coordinates is further rep! resented by xi  li E 0i "r#E 0i "m#E 0i "n#

i  0\ 1\ 2

"3[4#

with l0  "hk# −0 \ l1  "hzk1 −h1 # −0 \ l2  "kzk1 −h1 # −0 \

"3[5#

E 00 "t#  t\ E 01 "t#  z"t1 −h1 #\ E 02 "t#  z"t1 −k1 #\

"3[6#

and

which denote the Lame functions of the _rst kind of order one and of degree i  0\ 1\ 2[ Since use is made of the triply!orthogonal ellipsoidal system\ it is important to identify surface gradients and the surface Laplacian operators of "1[5# and "1[6# in this context[ To write these operators on the ellipsoidal surface it is most convenient to choose the parametric curves "u0\ u1# which are the intersections of the m and n surfaces with the canonical ellipsoid "r  a0#[ These curves\ according to Dupin|s theorem\ are also the principal lines of curvature of the ellipsoidal surface and constitute an orthogonal system[ The metric coe.cients in "1[5# and "1[6# are then identi_ed as h0  hm\ h1  hn\ so that eqn "1[3# takes now the form

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0776

k1

1f"1# 1xi

b

ni −k0

G

1f"0# 1xi

b

ni 

G

$ 0 1 0 1%

1 hn 1f a hm hn 1m hm 1m ¦

1 hm 1f 1n hn 1n

¦

0 1f 1a 0 1f 1a ¦ 1 [ 1 1m 1m hm hn 1n 1n

"3[7#

Introducing "3[0# in "3[7#\ and making use of ni "1:1xi#  "1:1n#  "0:hr#"1:1r# pro! vides "k1 −k0 #

$ 0

1 0

0 1 hn 1xi hm 1xi 0 1xi 1  a ¦ a hr 1r hn hm 1m i hm 1m 1n i hn 1n

1%

\

"3[8#

where ai denotes the interface parameter a\ for an intensity in the xi!direction[ In order to solve the partial di}erential eqn "3[8# for ai "r\ m\ n#= ra0 \ we _rst note that since xi is harmonic\ then

0

1 0

1 0

1

1 hr hm 1xi 1 hn hm 1xi 1 hr hn 1xi ¦ ¦  9[ 1m hm 1m 1n hn 1n 1r hr 1r

"3[09#

Thus\ taking advantage of the separable structure of "3[1# and "3[4#\ and using the Laplacian "3[09#\ suggests that the solution of "3[8# can be sought in the form ai "m\ n#  Ai

"k0 −k1 # hr "r\ m\ n# a0

b

i  0\ 1\ 2\

"3[00#

ra0

with Ai being constants to be determined[ Substituting "3[00# in "3[8# and making use of "3[09# yields Ai  a0

0 1>$ 0 1 %

1xi hm hn 1r hr

1 hm hn 1xi [ 1r hr 1r

"3[01#

Introducing now "3[4# and "3[1# in "3[01# shows in fact that the variables m and n drop out\ so as to yield the following expression for the constants Ai] Ai  a0

1E 0i z"r1 −h1 #"r1 −k1 # 1r

6

1 1E 0i z"r1 −h1 #"r1 −k1 # 1r 1r

7 >6 $ ra0

%7

[

ra0

"3[02# Making use of "3[2# and "3[6# in "3[02# yield the following simple expressions for Ai] A0 

a11 a21

\ A1  1

a11 ¦a2

a01 a21

\ A2  1

a01 ¦a2

a01 a11 a01 ¦a11

[

"3[03#

Finally\ the expression in "3[00# for ai simpli_es as well if use is made of "3[1# 0 and of the identity]

0777 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

"a01 −m1 #"a01 −n1 #  "a0 a1 a2 # 1

$

x01 3 0

a

¦

x11 3 1

a

¦

x21 a23

%

\

"3[04#

which follows from "3[4#Ð"3[6#[ One _nally gets from "3[1# 0\ "3[00#\ and "3[04#\ the following closed form solution for the interface parameter ai] ai "x0 \ x1 \ x2 #  Ai "k0 −k1 #

$

x01

x11

x21

a0

a1

a23

¦ 3

¦ 3

0:1

%

i  0\ 1\ 2\

"3[05#

with the constants Ai being given by "3[03#[ Clearly\ substitution of "3[05# in the Cartesian forms of "1[7# and eventually in "1[3# 1 together with "3[0#\ shows that "1[3# 1 is identically satis_ed[ We _nd it extraordinary that the solution of the partial di}erential "3[8# equation admits this extremely simple form[ Several limiting cases are readily obtained from "3[05#[ A spherical inhomogeneity is obtained by letting ai  a\ and results in a constant interface parameter a given by a  "k0 −k1 #a:1\

"3[06#

in agreement with the previous results of Torquato and Rintoul "0884# and Lipton "0886a\ b#[ The case of an in_nite cylindrical inhomogeneity\ with an elliptical cross section\ is produced by letting the longest axis go to in_nity[ This results in

0

a0 "x0 \ x1 #

a1 "x0 \ x1 #

a11

1 01 

a01

"k0 −k1 #

x01

x11

a0

a13

$

¦ 3

0:1

%

[

"3[07#

The case of a cylinder with a circular cross!section is readily deduced from "3[07# as a0  a1  "k0 −k1 #a[

"3[08#

It should _nally be emphasized that if the ellipsoidal inhomogeneity were slanted with respect to the direction of the intensity in the host medium\ as in Fig[ 1"b#\ then the form of "3[00# for the interface parameter is not admissible\ and the question of neutrality could not be easily answered in this case[ It is emphasized that in order to have neutrality it is not enough to solve the partial di}erential equation for a\ but to ensure in addition that the obtained solution for this interface parameter is non! negative everywhere on G\ as in "3[05#[ Generally\ for three!dimensional inhom! ogeneities of complex shapes\ a numerical solution for interface parameter seems to be the most natural means to test neutrality[ 3[1[ Two!dimensional neutral inhomo`eneities with LC!type interfaces This subsection is concerned with neutral inhomogeneities in two!dimensional composite media with HC!type interface conditions between the constituents[ The two!dimensional context allows us to derive general results for inhomogeneities of arbitrary shapes[ This is achieved by means of certain duality connections which relate the two!dimensional _elds in a composite with LC!type interfaces\ and those in a composite with HC!type interfaces[ Such duality relations have been recently estab!

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0778

lished by Lipton "0886# where the constancy of the interface parameters a and b along the interfaces in both media have been assumed[ In this section we show that duality connections of the type given by Lipton continue to prevail for the case of non! constant interface parameters[ A precise de_nition of the duality relations\ together with brief derivation of the generalization to the case of variable interface conditions\ emphasizing special features additional to the analysis of Lipton\ will be given below[ The derived connections are then used to transfer results on two!dimensional neutral inhomogeneities with LC!type interfaces\ to neutral inhomogeneities with HC!type interfaces[ Let us consider a composite medium with isotropic constituents and a two!dimen! sional microgeometry in the "x0\ x1# plane[ This microgeometry may describe either a planar sheet!like structure\ or a three!dimensional heterogeneous body with cylindrical boundaries between the constituents[ For simplicity\ we consider a two!phase body with isotropic constituents which are separated by an LC!type interface G with a non! constant b[ Multiphase media could be considered with no additional di.culty[ Let this medium be subjected to a temperature _eld on its external boundary S] f"S#  f9 "x0 \ x1 #\

"3[19#

and let the induced temperature\ intensity and ~ux _elds be described as f"r#\ H"r#\ q"r#  krH"r#\ with r  0\ 1\ where f"r# satis_es the boundary condition "3[04#\ H"r# is curl!free\ q"r# is divergence!free\ and the boundary conditions "1[1# on G are ful_lled[ This is de_ned as Problem I[ De_ne now new two!dimensional _eld quantities given by E"r#  aRq"r# \ j"r#  bRH"r# \ where R is a 1×1 matrix de_ned by

"3[10#

$

9 −0

%

\ and a and b are arbitrary constants 0 9 introduced for concerns of physical dimensions[ Note that the e}ect of R on the two! dimensional vector _eld is a counterclockwise rotation by 89>[ It can be readily veri_ed that\ due to the two!dimensional structure\ E"r# is curl!free\ whereas j"r# is divergence! free[ Thus\ one can de_ne a new potential _eld x"r# by E i"r#  −

1x"r# [ 1xi

"3[11#

Furthermore\ introduce the new set of material parameters by s"r#  "b:a#"k"r# # −0 \ a"x0 \ x1 #  "b:a#ðb"x0 \ x1 #Ł −0 [

"3[12#

The duality theorem states that if the _elds f"r#\ H"r#\ q"r# are the solution of Problem I\ then the _elds x"r#\ E"r#\ j"r#\ with j"r#  s"r#E"r# are the solution of another problem\ de_ned as Problem II\ in which one has the same microgeometry\ new material parameters given in "3[12#\ interface conditions of HC!type on G\ and _nally with the boundary condition on S being given by

0789 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

"j = m# = S  b

1f9 "x0 \ x1 # \ 1s

"3[13#

where f9 "x0\ x1# was de_ned as the temperature on the boundary S in Problem I\ m denotes the normal to S\ and s is an arc!length along S measured counterclockwise from m[ Since we have already seen that E"r#\ j"r# are curl!free and divergence!free\ respectively\ it only remains to show that the _eld quantities x"r#\ E"r#\ j"r# ful_l the interface conditions of HC!type on G\ and that "3[13# is ful_lled[ In order to show the ful_lment of the HC!type boundary conditions it is convenient to decompose the _elds "H"r#\ q"r##\ "E"r#\ j"r## at a generic point of G as H"r#  H n"r# n¦H t"r# t\ q"r#  qn"r# n¦qt"r# t E"r#  E n"r# n¦E t"r# t\ j"r#  jn"r# n¦jt"r# t\

"3[14#

with "n\ t# denoting the unit normal and tangential vectors\ respectively\ to G\ and H n"r#  −

1f"r# 1f"r# 1x"r# 1x"r# \ H t"r#  − \ E n"r#  − \ H t"r#  − \ 1n 1s 1n 1s

"3[15#

where s is the arc!length running along G[ Using the transformation "3[10# in the representation "3[14# and "3[15# yields 1x"r# 1f"r# 1f"r# 1x"r#  aqt"r# \  −aqn"r# \ jn"r#  b \ jt"r#  −b [ 1n 1s 1s 1n

"3[16#

From eqn "3[16# 1 it is seen that since qn"r# was continuous across G in Problem I\ it follows that x"r# is continuous across G in Problem II[ This establishes the ful_lment of the _rst part of the HC!type interface condition in Problem II[ The ful_lment of the second part is achieved by taking the derivative of "1[1# with respect to s\ so as to obtain dqn"0# dqn"1# db "1# df"1# df"0# \   "f −f"0# #¦b − ds ds ds ds ds

0

1

"3[17#

where the use of the total derivative sign instead of the partial derivative is justi_ed by interpreting qn"r# and f"r# as the {restriction| of the variables to G[ Substituting "f"1#−f"0## from "1[1# into "3[17#\ and making use of the connections in "3[16# yields jn"0# −jn"1# 

b 0 db dx 0 b d1 x [ − b a ds1 a b1 ds ds

01

01

"3[18#

Finally\ since from "3[12# 1\ there is

01

da b 0 db − \ ds a b1 ds eqn "3[18# takes its _nal form

"3[29#

Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781 0780

jn"0# −jn"1#  a

d1 x ds1

¦

da dx ds ds

"3[20#

which is the two!dimensional form of the HC!type boundary condition "1[3# 1 in Problem II[ Finally\ "3[16# 2\ leads immediately to "3[13#[ It should be noted that the implementation of the duality connections from LC! type to HC!type interfaces necessitates a variation b on G with well!de_ned _rst derivatives of this parameter along the interface^ a value of a obtained thus from "3[12# will have well!de_ned _rst derivatives and the HC!type interface condition can be implemented with no di.culty[ The derived duality connections for the case of non!ideal interfaces with variable interface parameters a or b allows the transfer of results from two!dimensional neutral inhomogeneities of LC!type to neutral inhomogeneities of HC!type and vice versa[ Let us illustrate this procedure for the case of an elliptical inhomogeneity[ The solution for the parameter b for a cylindrical inhomogeneity with an elliptical cross section in the "x0\ x1# plane\ and k1 − k0 is readily obtained from "2[6# by letting n0 

x0 :a01 ð"x01 :a03 #¦"x11 :a13 #Ł 0:1

\

"3[21#

and taking x¹0  9\ as we locate the center of the inhomogeneity at the origin of the Cartesian coordinate system[ This yields a value for the interface parameter b given by b

0 1$

0 k0 k1 k1 −k0 a01

x01

x11

a0

a13

¦ 3

−0:1

%

\

"3[22#

which insures neutrality for an inhomogeneity with LC!type interface\ k1 − k0\ under an intensity in the x0!direction[ Using the duality connections for the material par! ameters given in "3[12#\ provides] a  "s0 −s1 #"a01 #

$

x01

x11

a0

a13

¦ 3

0:1

%

[

"3[23#

In view of the transformations described in "3[10#\ this a parameter insures neutrality for a ~ux in the x1!direction[ Since intensity and ~ux _elds are constant in the host medium\ clearly\ this is equivalent to neutrality under the boundary condition x"S#  −E9x1[ The expression for a in "3[23# is thus seen to be identical to that given in "3[07# "with the conductivities being now denoted by si# and obtained directly without duality considerations[ The above analysis leads to the following general criterion for neutral inhom! ogeneities with HC!type interfaces in two!dimensions] if a certain two!dimensional! inhomogeneity with k1 − k0 is neutral for an intensity in the host medium which is in a given direction l\ and the corresponding value of the LC!type interface parameter b has well de_ned _rst derivatives along G\ then the same inhomogeneity\ this time

0781 Y[ Benveniste\ T[ Miloh : Journal of the Mechanics and Physics of Solids 36 "0888# 0762Ð0781

with k0 − k1 and with an HC!type interface parameter a dictated by the duality relations "3[12#\ will be neutral for an intensity in a direction perpendicular to l[

Acknowledgement The authors are grateful to Graeme W[ Milton for fruitful discussions[

References Benveniste\ Y[\ Miloh\ T[\ 0887[ Composites with superconducting interfaces[ In] Inan\ E[\ Markov\ K[Z[ "Eds[#\ Continuum Models and Discrete Systems "CMDS8#[ Proceedings of the 8th International Symposium\ 18 JuneÐ2 July 0887\ Istanbul\ Turkey[ World Scienti_c Publishing Co[\ in press[ Budiansky\ B[\ Hutchinson\ J[W[\ Evans\ A[G[\ 0882[ On neutral holes in tailored sheets[ J[ Appl[ Mech[ 59\ 0945Ð0947[ Cheng\ H[\ Torquato\ S[\ 0886[ E}ective conductivity of dispersion of spheres with a superconducting interface[ Proc[ Roy[ Soc[ London A 342\ 0220Ð0233[ Gilbarg\ D[\ Trudinger\ N[S[\ 0872[ Elliptic Partial Di}erential Equations of Second Order[ Springer! Verlag\ Berlin\ p[ 280[ Hobson\ E[W[\ 0844[ The Theory of Spherical and Ellipsoidal Harmonics[ Chelsea\ New York\ Ch[ 00[ Lipton\ R[\ 0886a[ Reciprocal relations\ bounds\ and size e}ects for composites with highly conducting interface[ SIAM J[ Appl[ Math[ 46\ 236Ð252[ Lipton\ R[\ 0886b[ Variational methods\ bounds\ and size e}ects for composites with highly conducting interface[ J[ Mech[ Phys[ Solids 34\ 250Ð273[ Lipton\ R[\ 0887[ In~uence of interfacial surface conduction on the DC electrical conductivity of particle reinforced composites[ Proc[ Roy[ Soc[ London A 343\ 0260Ð0271[ Lipton\ R[\ Vernescu\ B[\ 0885[ Composites with imperfect interface[ Proc[ Roy[ Soc[ London A 341\ 218Ð 247[ Miloh\ T[\ Benveniste\ Y[\ 0887[ On the e}ective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces[ Proc[ Roy[ Soc[ London A\ in press[ Mans_eld\ E[H[\ 0842[ Neutral holes in plane sheet!reinforced holes which are equivalent to the uncut sheet[ Quart[ J[ Mech[ Appl[ Math[ 5\ 260Ð267[ Pham Huy\ H[\ Sanchez!Palencia\ E[\ 0863[ Phenomenes de transmission a travers des couches minces de conductivite elevee[ J[ Math[ Anal[ and Appl[ 36\ 173Ð298[ Richards Jr[\ R\ Bjorkman Jr[\ G[S[\ 0871[ Neutral holes!theory and design[ J[ of the Eng[ Mech[ Div[ ASCE 097\ 834Ð859[ Rouse\ H[\ 0848[ Advanced Mechanics of Fluids[ John Wiley and Sons\ Appendix[ Ru\ C[Q[\ 0887[ Interface design of neutral elastic inclusions[ Int[ J[ Solids Struct[ 24\ 446Ð461[ Sanchez!Palencia\ E[\ 0869[ Comportement limite d|un probleme de transmission a travers une plaque faiblement conductrice[ Comp[ Rend[ Acad[ Sci[ Paris Ser[ A 169\ 0915Ð0917[ Schiavone\ P[\ Ru\ C[Q[\ 0887[ Integral equation methods in plane!strain elasticity with boundary reinforce! ment[ Proc[ R[ Soc[ London A 343\ 1112Ð1131[ Senocak\ E[\ Waas\ A[\ 0884[ Neutral cutouts in laminated plates[ Mech[ of Comp[ Mat[ and Struct[ 1\ 60Ð 78[ Torquato\ S[\ Rintoul\ M[D[\ 0884[ E}ect of the interface on the properties of composite media[ Physical Review Letters 64\ 3956Ð3969[ Van Bladel\ J[\ 0853[ Electromagnetic Fields[ McGraw!Hill\ New York\ Appendix 1[