On the Stress Singularities at Multimaterial Interfaces and Related

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Marco Paggi1

e-mail: [email protected]

Alberto Carpinteri

Department of Structural and GeotechnicalEngineering,

Politecnico di Torino,Corso Duca degli Abruzzi 24,

10129 Torino, Italy

On the Stress Singularities atMultimaterial Interfaces andRelated Analogies With FluidDynamics and DiffusionJoining of different materials is a situation frequently observed in mechanical engineer-ing and in materials science. Due to the difference in the elastic properties of the con-stituent materials, the junction points can be the origin of stress singularities and apossible source of damage. Hence, a full appreciation of these critical situations is offundamental importance both from the mathematical and the engineering standpoints. Inthis paper, an overview of interface mechanical problems leading to stress singularities isproposed to show their relevance in engineering. The mathematical methods for theasymptotic analysis of stress singularities in multimaterial junctions and wedges com-posed of isotropic linear-elastic materials are reviewed and compared, with special at-tention to in-plane and out-of-plane loadings. This analysis mathematically demonstratesin a historical retrospective the equivalence of the eigenfunction expansion method, of thecomplex function representation, and of the Mellin transform technique for the determi-nation of the order of the stress singularity in such problems. The analogies betweenlinear elasticity and the Stokes flow of dissimilar immiscible fluids, the steady-state heattransfer across different materials, and the St. Venant torsion of composite bars are alsodiscussed. Finally, advanced issues for the stress singularities due to joining of angularlynonhomogeneous elastic wedges are presented. This review article contains 147references. DOI: 10.1115/1.2885134

IntroductionInterfaces between two phases are defined as bounding surfaces

here a discontinuity of some kind occurs. The property variationay be sharp or gradual. In general, the interface is an essentially

idimensional region through which material characteristics, suchs concentration of an element, crystal structure, elastic coeffi-ients, density, and coefficient of thermal expansion, change fromne side to another. This mismatch in the material properties is theeason for the occurrence of stress singularities.

Interfaces must typically sustain mechanical and thermoelastictresses without failure. Consequently, they exert an important,nd sometimes controlling, influence on the performance of theaterial. The interest in this research field is clearly demonstrated

y several workshops and special sessions in recent internationalonferences devoted to interface problems and interface modeling1–3. Moreover, this topic is characterized by interdisciplinaryspects since interface problems are one of the main concern inivil, mechanical, and electronic engineering, as well as in biome-hanics and in materials science see Ref. 4 for a wideverview.

In several boundary value problems of linear elasticity whereifferent materials are present, the stress field is found to haveingularities 5,6. Physically, these singular points correspond toegions of high stress in which some type of nonlinear behavior,.g., plastic flow or even fracture, is expected to relieve them.ence, from the mathematical point of view, they violate the basic

ssumptions of linear elasticity, although they do comply with allhe field equations. As a consequence, as stated by Dempsey andinclair 7, stress singularities are a real fact of the theory of

inear elasticity and must be taken into account in any elasticnalysis.

1Corresponding author.
Published online March 18, 2008. Transmitted by Editor Emeritus A. W. Leissa.
pplied Mechanics Reviews Copyright © 20

aded 17 Sep 2010 to 130.75.247.32. Redistribution subject to ASME

The wide literature in this field shows that different mathemati-cal techniques have been proposed and used for the characteriza-tion of the singular stress field. A long debate on their equivalencehas clearly produced a lack of standardization. Some problemshave been tackled using different mathematical methods, obtain-ing almost the same conclusions. Therefore, the main aim of thisarticle is to review, compare, and unify the main mathematicalmethods used for the analysis of stress singularities occurring ininterface problems. To keep the scope of the article within reason-able limits, the study is restricted to two-dimensional elasticityand to isotropic materials with or without homogeneous elasticproperties.

Compared to previous review articles on biharmonic problemsand stress singularities 8–10, the present study focuses on themathematical aspects and on the physical analogies of singulari-ties related to multimaterial interfaces. The paper is organized asfollows: In Sec. 2, an overview of some important interface me-chanical problems leading to stress singularities is presented. InSec. 3 the problem of multimaterial junctions subjected to in-plane loading is addressed. The mathematical methods are thencompared, and a unified matrix formulation is proposed. Thephysical analogy existing between elasticity problems and theflow of viscous fluids is also discussed. In Sec. 4, the problem ofmultimaterial junctions subjected to out-of-plane loading is ana-lyzed. Also in this case, the mathematical methods for the analysisof such problems are compared and unified. Physical analogieswith the steady-state heat transfer in wedges composed of differ-ent materials and with the St. Venant torsion of composite bars areput into evidence. In Sec. 5, the hypothesis of homogeneity of theconstituent materials is removed and the advanced issues concern-ing the singular stress field in nonhomogeneous materials are re-viewed. Finally, concluding remarks and future perspectives com-
plete the manuscript.
MARCH 2008, Vol. 61 / 020801-108 by ASME

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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Overview of Interface Mechanical Problems Lead-ng to Stress Singularities

An important class of mechanical problems leading to stressingularities are those related to two-dimensional contact mechan-cs in which displacements are imposed within the contact regionf the boundary of an elastic half-space. In these cases, the un-nown contact tractions can be computed by solving two coupledingular integral equations see Ref. 11 for a fully comprehen-ive description of the mathematical formulation.

In this framework, firstly Muskhelishvili Ref. 12, Sec. 115nd then Nadai 13 proposed the solution to the problem of thendentation of an elastic half-space by a frictionless rigid flatunch. They demonstrated that the stresses within the solid areingular near the edges of the punch with an order of stress sin-ularity equal to −1 /2; i.e., the stress components ij vary withhe inverse square root of the radial distance from the singularoint see Fig. 1a,

ij Kr−0.5 1

here K is the stress-intensity factor.On the basis of the fact that the order of the stress singularity

or the indentation of an elastic half-space by a frictionless rigidat punch is the same as that for a crack inside a homogeneousaterial, Giannakopoulos et al. 14 and Giannakopoulos anduresh 15 proposed a quantitative equivalence between contactechanics and fracture mechanics via asymptotic matching. This

nalogy, strictly valid for small-scale yielding and for either in-ompressible elastic substrates or rigid orthogonal punches withrictionless interfaces only, was then applied to life prediction inhe case of fretting fatigue.

Muskhelishvili Ref. 12, Sec. 114 also gave a solution to theroblem of normal indentation of the half-plane by a flat-endedough rigid punch acting over a finite contact region. This problems a typical mixed boundary value problem in which both compo-

ig. 1 Indentation of an elastic half-plane „a… by a frictionlessigid flat punch and „b… by an elastic punch with corner anglesther than /2

ents of displacements are specified over the contact region and

20801-2 / Vol. 61, MARCH 2008


stress-free conditions are specified over the remaining free sur-face. The stress singularity at the edges of the punch has a moreinvolved character than the previous one since a complex singu-larity was found. Denoting by the complex eigenvalue definingthe order of the stress singularity, the singular stress field is writ-ten as

ij KrRe −1 2

where K is referred to as the generalized stress-intensity factor16.

Starting from the principle appreciated by Boussinesq 17,which states that the pressure distribution between two elasticbodies, whose profiles are continuous through the boundary of thecontact area, falls continuously to zero at the boundary, it is pos-sible to generalize the above results concerning singularities incontact problems to other geometric scenarios. In fact, the mainconsequence of the above principle is that when one or both of thebodies has a discontinuous profile at the edge of the contact re-gion, i.e., the contact is said to be nonconforming, then stressconcentrations or even intensifications would be expected at thesesingular points.

Several researchers extended Muskhelishvili’s results to contactproblems involving punches that are also elastic as the half-planesubstrate and with a corner angle other than /2 see Fig. 1b.Different boundary conditions along the bimaterial interface werealso analyzed: Dundurs and Lee 18 treated the frictionless prob-lems; Gdoutos and Theocaris 19 and Comninou 20 took intoaccount friction in the formulation; Bogy 21 investigated thecase of a fully bonded interface.

This last case where two different elastic wedges are joinedtogether with a total wedge angle less than 2 is usually referredto as a bimaterial wedge. On the contrary, the terminology bima-terial junction implies that the total wedge angle formed by thetwo-material regions equals 2; i.e., the whole plane is occupiedby the elastic materials without any voids. Wedges and junctionsare very commonly observed in composite materials and receiveda great deal of attention in the past. The problem of bimaterialjunctions was firstly analyzed by Bogy and Wang 22 in 1971,and multimaterial junctions were addressed by Theocaris 23 in1974. Several configurations involving trimaterial junctions withperfectly bonded or debonded interfaces were also investigated byPageau et al. 24 in 1994. Bimaterial wedges firstly treated byBogy 21 were also addressed by Hein and Erdogan 25 in 1971,whereas a remarkable study on trimaterial wedges was due toInoue and Koguchi 26 in 1996.

In the field of composite materials, interfaces play a crucial roleduring both the fabrication process and the service conditions27,28. The geometry of the contact between material compo-nents during fabrication is, in fact, an important issue, which isusually studied in terms of the wettability concept 27. The termwettability has its origin in the fluid-mechanics literature, where itis used to describe the angle at which an isolated liquid islandmeets the solid substrate on which it rests see Fig. 2. The liquiddrop will spread and wet the surface completely only if this resultsin a net reduction of the system free energy. The same approach isroutinely used to describe morphologies where intersections ofsolid-solid interfaces occur. When the solids behave linear elasti-cally, the stress field diverges at the corner of the wedge formednear the triple line.

In 2001, Srolovits and Davis 29 investigated the effect of thesingular stress field on the wetting angle . In the absence ofelastic stresses, interfacial thermodynamics defines the equilib-rium angle in the configuration of Fig. 2 to satisfy the well-
known Young’s equation 30,
Transactions of the ASME


wvfj

easrieeedw

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tfaadsw

tsmmdtptgr

bt

Fsti

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cos =si − is

vi3

here vi, sv and is are, respectively, the vapor-island, substrate-apor, and island-substrate interfacial energies. When elastic de-ormation is present, the singular stress field at the trimaterialunction has the form reported in Eq. 2.

In order to determine whether the elastic field may change thequilibrium wetting angle, Srolovits and Davis 29 analyzed thesymptotic behavior of the interface tension and strain energyingularity. Calculating these two contributions within a circle ofadius R centered at the singular point, they found that the behav-or at the trimaterial junction is dominated by the interfacial en-rgy for Re 1 /2, and, in this case, the contribution of thelastic energy singularity is negligible. The two contributions arequally important when Re =1 /2, whereas the elastic energy isominant when Re 1 /2, with a possible modification of theetting angle predicted according to Young’s equation.Recently, experimental and theoretical studies have emphasized

he role of interfaces and grain boundaries in understanding theechanical properties of nanocrystalline materials 31,32. In

uch materials, having grain sizes less than 30 nm, a considerableraction of atoms reside in interfacial regions. As a consequence,anocrystalline materials often exhibit quite different propertieshan their conventional polycrystalline counterparts.

While for some properties it may be sufficient to consider onlyhe total volume fraction of the material associated with the inter-aces, it was recognized that other phenomena require a detailednalysis of the various contributing factors such as grain bound-ries, trimaterial junctions, and quadruple nodes. As a result, withecreasing grain size, an unexpected reduction of the materialtrength and of Young’s modulus was observed. This phenomenonas attributed to grain boundary and trimaterial junction effects.In 1996, Picu and Gupta 33 investigated stress singularities at

rimaterial junctions in single-phase polycrystals due to freelyliding grain boundaries. This study was motivated by the need ofodeling the nucleation of cracks and cavities in polycrystallineaterials. They found that the exponent of the singular stress field

epends on the grain boundary positions only, i.e., on the ampli-udes of the wedge angles formed by the grains at the singularoint see Fig. 3. Furthermore, supersingularities, i.e., singulari-ies with Re 0.5 in Eq. 2, were found, indicating criticaleometrical configurations of the microstructure, which can giveise to crack nucleation.

In 2001, Costantini et al. 34 performed a theoretical studyased on molecular-dynamics atomistic simulations of the struc-

Vapor

Island

Solid substrate

θ

γvi

γsvγis

r

O

ig. 2 Intersection of a solid or liquid island with a solid sub-trate. is the wetting angle and sv, is, and vi are, respec-ively, the interfacial tensions associated with substrate-vapor,sland-substrate, and vapor-island interfaces.

ure and energetics of a multiple-twin triple junction in silicon

pplied Mechanics Reviews


see Fig. 4. By analyzing the calculated excess line and volumeenergies, as well as the atomic-level stress, they showed that atriple junction can be considered as a true line defect. These criti-cal points can be a source of residual stresses that cumulate andconcentrate at the junction vertex.

In order to overcome the drawbacks due to an abrupt variationof the mechanical properties occurring at interfaces and materialjunctions, functionally graded materials FGMs have been re-cently designed. They are multicomponent materials with a con-tinuous inhomogeneity of composition or structure. They consistin one material on one side and in a second material on the other.The composition of the intermediate layer changes smoothly fromone material at one surface to the other material at the oppositesurface. FGMs arose from a unique idea for realization of inno-vative properties that cannot be achieved by conventional materi-als. The FGM concept was proposed in 1984 as a way of prepar-ing super-heat-resistant materials for spacecraft. Since thestructural components of the spacecraft are exposed to high heatload, the material has to withstand severe thermomechanical load-ing. To solve this problem, a FGM was produced by using heat-resistant ceramics on the high-temperature side and tough metalswith high thermal conductivity on the low-temperature side.

The gradient composition at the interface can effectively relaxthermal stress concentrations, which may occur between the twomaterials 35,36. A collection of technical papers that representscurrent research interests with regard to the fracture behavior ofFGMs has been recently proposed in Ref. 37. The possibility to

σrθ=0

Grain 3

O

Grain 1

Grain 2

σrθ=0

σrθ=0

Fig. 3 Scheme of the geometry of a grain triple junction. Thegrains are composed of the same material, and grain bound-aries are assumed to be freely sliding.

(a) (b)

Fig. 4 Atomic structure of the multiple-twin junction obtainedafter MD simulation „a…. Shaded rings indicate the grain bound-aries. Contour plot of the stress field in the vertical directionaround the triple junction „b…. The right scale stress are in units
of kbar „reprinted from Ref. †34‡….
MARCH 2008, Vol. 61 / 020801-3


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se such nonhomogeneous materials in multimaterial junctionsas recently explored by Carpinteri and Paggi 38, and the main

esults are summarized in Sec. 5.The characterization of stress singularities in anisotropic mate-

ials has been pursued in the literature with reference to bi- andrimaterial wedges and junctions with either perfectly bonded orebonded interfaces 39–48. The majority of these works, assum-ng plane strain or plane stress conditions, uses the powerfulekhnitskii–Eshelby-Stroh 49–52 complex variable formalismf anisotropic elasticity. Recent results on this topic have contrib-ted to a renewed interest in singularity problems involving iso-ropic materials since it has been demonstrated that the solutionso these problems may also be gained by using the mathematicalormalism of anisotropic elasticity applied to degenerate aniso-ropic materials see Refs. 47,53,54, among others. Hence, so-utions for isotropic materials can be particularly useful when were interested in checking the effectiveness of new approaches fornisotropic multimaterial junctions by comparison with limit iso-ropic solutions 47.

Finally, a number of studies on the order of stress singularitiest three-dimensional bimaterial interface corners have to be men-ioned. In these cases, the basic assumptions of plane elasticity doot hold anymore 55–61. The motivation, especially in earlierorks, concerned the study of the intersection of a crack with a

ree surface where a singularity different from the classical for arack occurs. In most approaches, the stress singularities are ob-ained as eigenvalues of a quadratic eigenvalue problem 60.hey depend on the elastic mismatch and on the asymptotic cor-er geometry. While several efforts have been directed towardomputing the order of the stress singularity for various cornereometries, only a few studies 57–59,62 have focused on theomplete determination of the stress field at these singular points.

In-Plane Loading

3.1 Statement of the Biharmonic Problem. The geometry ofplane elastostatic problem consisting of n dissimilar isotropic,

omogeneous sectors of arbitrary angles perfectly bonded alongheir interfaces converging at the same vertex O is shown in Fig.. Each of the material regions is denoted by i, with=0, . . . ,n−1, and it is comprised between the interfaces i andi+1. The first and last interfaces, defined by =0 and =2,oincide and are referred to as 0.

In plane elasticity, by neglecting body forces, stress and dis-lacement fields can be represented in terms of the Airy stressunction ir ,. In this case, in polar coordinates we have

r =1 i +

12

2i2 4a

Γ0=Γn

Γ1

Γ2

Γ3

Γ4

Γn -1

O

Ω1Ω2

Ω3

Ω4Ω5

Ωn -1

Ω0γ1

θ

Γ5

Fig. 5 Scheme of a multimaterial junction

r r r

20801-4 / Vol. 61, MARCH 2008


=2i

r2 4b

r = −1

r

2i

r+

1

r2

i

4c

ur

r=

1

2Gi1

r

i

r+

1

r2

2i

2 − 1 −1

mi2i 4d

u

r−

u

r+

1

r

ur

=

1

Gi−

1

r

2i

r+

1

r2

i

4e

where the constant mi depends on the Poisson ratio,

mi = 1 + i for plane stress

1/1 − i for plane strain 5

The strain compatibility equation expressed in terms of cylin-drical coordinates is

1

r2

2 r

2 +2

r2 −1

r

r

r+

2

r

r=

2

r

2r

r+

2

r2

r

6

When the stress-strain relations and Eqs. 4a–4c are introducedinto Eq. 6, the well-known biharmonic condition for the airystress function is derived 63,

4i = 0, ∀ r, i 7

3.2 Boundary Conditions. A unified treatment of elasticityproblems concerning stress singularities was provided by Rao 64in 1971. In his study, a general procedure for the identification ofstress singularities and concentrations at the intersection of two ormore interfaces in domains governed by harmonic and biharmonictypes of equations was presented see Fig. 6 for a sketch of someproblems involving interfaces investigated in Ref. 64. He listedsix types of boundary conditions for interfaces and five distincthomogeneous edge conditions along with their significance toproblems of plane stress extension, plane strain extension, andthin plate flexure.

Afterward, Dempsey and Sinclair 7 specialized on Rao’s re-sults for the single-material wedge and the n material wedge prob-lems under either plane strain or plane stress conditions see Fig.7. By denoting, respectively, with ur and u, and with r, , andr, the displacement and stress components in polar coordinateswith respect to point O, four types of homogeneous stress and/ordisplacement boundary conditions can be considered on a wedgeface 7. They can also be interpreted in the context of contactproblems and are summarized in Table 1. It has to be remarkedthat when the coefficient of friction f is equal to zero, Case 4 inTable 1 corresponds to the boundary condition associated with therigid lubricated punch.

Analogously, five types of interface conditions can be consid-ered along an interface between two dissimilar wedges and aresummarized in Table 2. The parameter f i denotes the coefficient offriction of the ith interface. From the engineering point of view,Cases B, C, D, and E differ from Case A in that one boundarycondition is required in place of a fourth matching condition.Cases C, D, and E are, in general, less frequent than Cases A andB.

Boundary conditions in the case of a perfect bonding are rep-resented by stress and displacement continuity conditions alongthe interfaces. For the ith interface, they are as follows:

i r,i+1 =

i+1r,i+1

ri r,i+1 = r

i+1r,i+1

uir,i+1 = ui+1r,i+1
r r


pi

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ui r,i+1 = u

i+1r,i+1 8

aying attention to the fact that for multimaterial junctions, thenterface 0 has to be defined by 0=0 for region 1 and by

Fig. 6 Some interface problems a†64‡…

ig. 7 Schemes of a two-dimensional single-material wedgea… and of a two-dimensional composite wedge „b…

Table 1 Boundary conditions in a wedge face from Ref. 7

CasePrescribedquantities

Physicalinterpretation

Contactinterpretation

1 =r=0 Stress-free Flexible lubricated punch2 =0, ur=0 Antisymmetry Flexible adhesive punch3 ur=u=0 Clamped Rigid adhesive punch4 u=0, r− f=0 — Rigid punch with friction



n+1=2 for region n.Furthermore, as argued by Chen and Nisitani 65, when the

problem is characterized by a geometric symmetry, it is possibleto subdivide the elastic field into a symmetric part and a skew-symmetric part, namely, into a part due to the Mode I deformationand a part due to the Mode II deformation see Fig. 8.

In the former case, the following symmetric conditions are ap-

plied at =0,:

ri r, = 0

essed by Rao „reprinted from Ref.

Table 2 Conditions on an interface from Ref. 7

CaseMatchedquantities

Boundarycondition

Contactinterpretation

A , r, ur, u — Perfectly bondedB , r, u r−i=0 No separation, slip in the pres

ence of frictionC , r, ur =0 Rough with separation, surfaces

just resisting one anotherD , ur, u ur=0 No separation, no displacement

in the radial directionE , ur, u u=0 No separation, no displacement

across the contact surface

O

Ω1

Ω2 Ω0

θ

γ1

(a) (b)

Γ3

Ω2

Ω3

Γ1

Γ3

Γ4

O

Ω1

Ω0

Γ2

θ

γ1

Γ2

Γ1

Γ4

Ω3

Fig. 8 Scheme of a multimaterial junction with a geometricsymmetry and subjected to symmetric „Mode I… „a… and to skew-

ddr

symmetric „Mode II… „b… deformation

MARCH 2008, Vol. 61 / 020801-5


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ui r, = 0 9

n the latter, skew-symmetric conditions are imposed,

i r, = 0

urir, = 0 10

n this way, it is possible to study half of the problem only, and theotal eigenequation is reduced to the product of two factors: theormer determines the eigenvalues corresponding to the Mode Ieformation, whereas the latter determines the eigenvalues corre-ponding to the Mode II deformation.

In the case of either a reentrant corner or a crack due to anmperfect bonding see Fig. 9, traction-free conditions along theaces have to be considered in addition to continuity conditions oftresses and displacements Eq. 8 at interfaces,

i r,i+1 = 0

ri r,i+1 = 0

i+1r,i+1 = 0

ri+1r,i+1 = 0 11

Summarizing, in the case of multimaterial junctions with a per-ect bonding without a geometric symmetry in the problem,oundary conditions are represented by Eq. 8 only. On the otherand, if a geometric symmetry exists, then Eqs. 8–10 can beonsidered. In the case of either a reentrant corner or a crack,oundary conditions are given by Eq. 8 for interfaces and by Eq.11 for stress-free surfaces.

The solution to this boundary value problem can be gained bysing one of the mathematical methods reviewed and compared inhe next subsection. For the sake of simplicity and without anyoss of generality, the comparison among the formulations will bearried out on the basis of the boundary conditions for a perfectlyonded multimaterial junction.

3.3 Mathematical Methods. Three different methods for thesymptotic analysis of stress singularities due to interface prob-ems in composite bodies consisting of dissimilar isotropic, homo-eneous, and elastic wedges were proposed in the literature,amely, the eigenfunction expansion method, the complex functionepresentation, and the Mellin transform technique. All of theseethods herein reviewed permit us to formulate an eigenproblemhose eigenvalues are directly related to the order of the stress

ingularity.

3.3.1 Eigenfunction Expansion Method. The eigenfunction ex-

Ω2

Γ1

Γ3

Γ4

O

Ω1Ω0

Γ2

θ

Γ0

Ω2

Γ1

Γ3

Γ4

O

Ω1

Ω0

Γ2

θ

Γ0

(a) (b)

Ω3 Ω3

ig. 9 Scheme of a multimaterial junction with a reentrant cor-er „a… and with a crack „b…. Case „a… is usually referred to as aultimaterial wedge.

ansion method was independently proposed by Wieghardt 66 in

20801-6 / Vol. 61, MARCH 2008


1907 and by Williams 67 in 1952 to the analysis of stress sin-gularities due to reentrant corners in plates in extension. Theydemonstrated that the solution can be generally written as a seriesof power functions in the radial coordinate originating from thesingular point. The eigenvalues obtained from the solution proce-dure determine the exponents of the radial coordinate. The eigen-functions describe the angular variation of stresses. Clearly, theeigenvalues and eigenfunctions for a composite wedge or junctiondepend on its geometry and material properties.

Numerous analytical solutions of the order of stress singularityand the stress fields for various material and geometric combina-tions have been derived in the literature. For example, Williams67 addressed the problem of stress singularities resulting fromvarious boundary conditions in angular corners of plates in exten-sion, Zak and Williams 68 considered the problem of a crackperpendicular to a bimaterial interface, and Williams 69 andRice and Sih 70 considered a crack along the interface betweentwo isotropic materials. Afterward, Fenner 71 extended the so-lutions by Zak and Williams 68 by considering different incli-nations of the crack meeting the bimaterial interface. This methodwas also used by Dempsey and Sinclair 72 in 1981 for the studyof the singular behavior at the vertex of a bimaterial wedge, withthe aim of unifying some previous findings obtained according toother mathematical techniques. In the sequel, the application ofthis technique to multimaterial junctions was mainly limited to thestudy of problems involving real eigenvalues, e.g., the antiplanestress field in trimaterial junctions 73.

Munz and Yang 74 applied this method to the computation ofreal eigenvalues resulting from singularities at the interface inbonded dissimilar materials under mechanical and thermal load-ings. Only very recently, Carpinteri and Paggi 75,76 applied theeigenfunction expansion method to the study of Mode I and ModeII stress singularities in bi- and trimaterial junctions with eitherperfectly bonded interfaces or interface cracks. Size-scale effectson the material strength due to dominant stress singularities werealso discussed in Ref. 77.

According to this method, it is possible to assume, for the ithmaterial region, the following separable form for the biharmonicstress function i:

ir, = j

rj+1f i,j, j 12

where j and f i,j are referred to as eigenvalues and eigenfunctions,respectively. Special cases where the separable form 12 requiresa modification are discussed in Sec. 3.5. The summation withrespect to the subscript j is introduced in Eq. 12, since it ispossible to have more than one eigenvalue for each problem, andthe superposition principle can be applied due to linear elasticity.

By substituting Eq. 12 into Eqs. 4a–4e, stress and dis-placement fields can be expressed in terms of the eigenfunction,

ri = r−1f i + + 1f i

i = r−1 + 1f i

ri = r−1− f i

uri =

r

2Gi− + 1f i +

1

mif i + + 12f i

ui =

r

2Gi− f i −

1

− 1mif i + + 12f i 13

Furthermore, the biharmonic condition requires f i,j to be of the
form


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ik

a

w

oceplcwtt

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f i,j, j = Ai,j sin j + 1 + Bi,j cos j + 1 + Ci,j sin j

− 1 + Di,j cos j − 1 14

here Ai,j, Bi,j, Ci,j, and Di,j are undetermined constants. To sim-lify the mathematical notation, the subscript j of the eigenvaluen Eq. 14 will be removed in the sequel.

Introducing Eq. 14 into Eq. 13, stresses and displacementsn polar coordinates can be explicitly written in terms of the un-nowns Ai,j, Bi,j, Ci,j, Di,j, and ,

ri = r−1 − Ai,j + 1sin + 1 − Bi,j + 1cos + 1

+ Ci,j3 − sin − 1 + Di,j3 − cos − 1

i = + 1r−1 Ai,j sin + 1 + Bi,j cos + 1

+ Ci,j sin − 1 + Di,j cos − 1

ri = r−1 − Ai,j + 1cos + 1 + Bi,j + 1sin + 1

− Ci,j − 1cos − 1 + Di,j − 1sin − 1

uri =

r

2Gi − Ai,j + 1sin + 1 − Bi,j + 1cos + 1

+ Ci,j4/mi − − 1sin − 1 + Di,j4/mi − − 1cos

− 1

ollowing Huth 84, presented in 1956 a method based on the



ui =

r

2Gi − Ai,j + 1cos + 1 + Bi,j + 1sin + 1

− Ci,j4/mi + − 1cos − 1 + Di,j4/mi + − 1sin

− 1 15

Introducing Eq. 15 into the interface matching conditions rep-resented by Eq. 8, a set of 4n equations in 4n+1 unknowns Ai,j,Bi,j, Ci,j, Di,j, and j can be symbolically written as

v = 0 16

where denotes the coefficient matrix that depends on the eigen-value and v represents the vector that collects the unknowns Ai,j,Bi,j, Ci,j, and Di,j.

The coefficient matrix in Eq. 16 is characterized by a sparsestructure,

= M1

0 − M1

1

M2

1 − M2

2

. . . . . .

Mi

i−1 − Mi

i

. . . . . .

Mn−1

n−2 − Mn−1

n−1

− M0=00 Mn=2

n−1

17

By letting += +1 and −= −1, the elementary matrix Mi

can be written as follows:

Mi =

+ sin+ + cos+ + sin− + cos−− + cos+ + sin+ − − cos− − sin−

−+

2Gisin+ −

+

2Gicos+ 4

mi− + sin−

2Gi 4

mi− + cos−

2Gi

−+

2Gicos+

+

2Gisin+ − 4

mi+ − cos−

2Gi 4

mi+ − sin−

2Gi

18

nd the components of the vector v are

v = v0,v1,v2, . . . ,vi, . . . ,vn−2,vn−1T 19

here

vi = Ai,j,Bi,j,Ci,j,Di,jT 20

A nontrivial solution of the equation system 16 exists if andnly if the determinant of the coefficient matrix vanishes. Thisondition yields an eigenequation, which has to be solved forigenvalues that are, in general, complex. For the present pur-oses, we are concerned only with those values of j, which mayead to singularities in the stress field. This fact, together with theondition of continuity of the displacement field at the vertexhere regions meet, implies that we are looking for eigenvalues in

he range 0Re j 1. Then, to find solutions to the eigenequa-ions, numerical techniques are usually employed7,44,47,71,78–82.

3.3.2 Complex Function Representation. The complex func-ion representation was mainly developed by Muskhelishvili 12nd applied to several problems of plane elasticity. Williams 83,

complex potentials for the study of a single-material wedge, du-plicating some of the findings previously obtained according tothe eigenfunction expansion method 67.

Then, with no claims of completeness, some noticeable studiesabout stress singularities carried out with this method have con-cerned reentrant corners of plates in extension 85,86, perfectlybonded bimaterial wedges, 78,87–89, and junctions 65. Theproblem of a crack between dissimilar materials 90,91 was alsoaddressed with this method. Moreover, Wang and Chen 92 ap-plied the complex function representation to the problem of acrack perpendicular to a bimaterial interface, reobtaining some ofthe results previously derived by Zak and Williams 68 using theeigenfunction expansion method.

The mathematical formulation for multimaterial junctions wasset up by Theocaris 23 in 1974 and was subsequently adopted byGdoutos and Theocaris 19 for the analysis of stress concentra-tions at the apex of a plane indenter acting on an elastic half-plane. This formulation was then revised by Pageau et al. 24 andapplied to trimaterial junctions with either perfectly bonded ordebonded interfaces. This mathematical technique was also re-cently adopted by Carpinteri et al. 93 for the analysis of reen-
trant corners symmetrically meeting a bimaterial interface.
MARCH 2008, Vol. 61 / 020801-7


at

wir

Swlh

wstG

w

a

osec

0

Downlo

According to Muskhelishvili 12 and England 5,94, the stressnd displacement fields in the ith material can be expressed inerms of the complex potentials i and i

ri + ir

i = iz + iz − ziz − zz−1iz

i − ir

i = iz + iz + ziz + zz−1iz

uri + iu

i =1

2Gie−iiiz − ziz − iz 21

here z=rei is a complex variable, Gi is the shear modulus of theth material, and i is the Kolosoff constant related to the Poissonatio i by

i = 3 − 4i for plane strain

3 −4i

1 + ifor plane stress 22

ymbols and in Eq. 21 denote, respectively, a derivativeith respect to z and the complex conjugate of the variable. Fol-

owing Refs. 85,89,95, the complex potentials are assumed toave the following form as z→0:

i = Ei,j + iFi,jzj

i = Gi,j + iHi,jzj 23

here Ei,j, Fi,j, Gi,j, and Hi,j are undetermined constants. By sub-tituting Eq. 23 into Eq. 21, stress and displacement fields inhe materials can be expressed in terms of the unknowns Ei,j, Fi,j,

i,j, Hi,j, and ,

Ei,j = Di,j 28

20801-8 / Vol. 61, MARCH 2008


ri = r−1 Ei,j3 − cos − 1 − Fi,j3 − sin − 1

− Gi,j cos + 1 + Hi,j sin + 1

i = r−1 Ei,j + 1cos − 1 − Fi,j + 1sin − 1

+ Gi,j cos + 1 − Hi,j sin + 1

ri = r−1 Ei,j − 1sin − 1 + Fi,j − 1cos − 1

+ Gi,j sin + 1 + Hi,j cos + 1

uri =

1

2Gir Ei,ji − cos − 1 − Fi,ji − sin − 1

− Gi,j cos + 1 + Hi,j sin + 1

ui =

1

2Gir Ei,ji + sin − 1 + Fi,ji + cos − 1

+ Gi,j sin + 1 + Hi,j cos + 1 24

Analogous to the eigenfunction expansion method, by consid-ering Eq. 24 along with the interface continuity conditions Eq.8, the matrix form of the equation system Eq. 16 can beobtained. By noting that the parameter i is related to mi by thefollowing relation:

i =4

mi− 1 25

i
the elementary matrix M can be written as
Mi =

− sin+ cos+ − + sin− + cos−cos+ sin+ − cos− − sin−sin+

2Gi−

cos+2Gi

− 4

mi− + sin−

2Gi 4

mi− + cos−

2Gi

cos+2Gi

sin+2Gi

4

mi+ − cos−

2Gi 4

mi+ − sin−

2Gi

26

here + and − denote, respectively, +1 and −1.According to this formulation, the components of the vector v

re

vi = Hi,j,Gi,j,Fi,j,Ei,jT 27

It is possible to show that the elementary matrix Eq. 26btained according to the Muskhelishvili complex function repre-entation coincides with the matrix derived according to theigenfunction expansion method in Eq. 18, once the followinghange of variables is considered:

Hi,j = −Ai,j

+ 1

Gi,j =Bi,j

+ 1

Fi,j = − Ci,j

The expressions of the eigenequations obtained by the compu-tation of det are influenced by this change of variables, whereasthe zeros of the eigenequations, which allow us to compute theeigenvalues, do not change. This result demonstrates that bothmathematical formulations can be used for the computation of theorder of the stress singularity in multimaterial junctions andwedges.

In 1974, Theocaris 23 observed that the form of the complexpotentials in Eq. 23 could not be used for determining realstresses and displacements when is complex. Noting that if C is one root of the eigenequation, then also its conjugate num-

ber, , is a root of the same equation, Theocaris assumed thefollowing alternative expression for the complex potentials 23:

i = Ii,jzj + Li,jz

j

i = Mi,jzj + Ni,jz

j 29

where Ii,j, Li,j, Mi,j, and Ni,j are unknown complex coefficients.After some algebra, by introducing Eq. 29 into Eq. 21, we

obtain



hgEcu

it

e

ts

arctt

A

Downlo

ri + ir

i = r−1Ii,jei−1 − Li,j2e−i−1 − Ni,je−i+1

+ 2Li,je−i−1 + r−1Li,jei−1 − Ii,j2e−i−1

− Mi,je−i+1 + 2Li,je−i−1 30a

i − ir

i = r−1Ii,jei−1 + Li,j2e−i−1 + Ni,je−i+1

+ r−1Li,jei−1 + Ii,j2e−i−1 + Mi,je−i+1

30b

uri + iu

i =r

2GiiIi,je

i−1 − Li,je−i−1 − Ni,je−i+1

+r

2GiiLi,je

i−1 − Ii,je−i−1 − Mi,je−i+1

30c

The expression for i can be simply obtained by taking one-

alf of Eq. 30b plus the conjugate of the same equation. Analo-ously, r

i can be computed as the difference of the conjugate ofq. 30b and Eq. 30b itself divided by 2i. The same procedurean also be applied to Eq. 30c to obtain the expressions of ur

i and

i

i =

1

2 r−1Mi,je

i+1 + Ni,je−i+1 + Ii,j + 1ei−1

+ Li,j + 1e−i−1 + r−1Mi,je−i+1 + Ni,jei+1

+ Ii,j + 1e−i−1 + Li,j + 1ei−1 31a

ri =

1

2i r−1Mi,jei+1 + Ni,je−i+1 + Ii,j − 1ei−1

− Li,j − 1e−i−1 + r−1− Mi,je−i+1 − Ni,jei+1

− Ii,j − 1e−i−1 + Li,j − 1e−i−1 31b

uri =

1

4Gi r− Mi,je

i+1 − Ni,je−i+1 + Ii,ji − ei−1

+ Li,ji − e−i−1 + r− Mi,je−i+1 − Ni,jei+1

+ Ii,ji − e−i−1 + Li,ji − ei−1 31c

ui =

1

4Gii rMi,je

i+1 − Ni,je−i+1 + Ii,ji + ei−1 − Li,ji

+ e−i−1 + r− Mi,je−i+1 + Ni,je

i+1 − Ii,ji

+ e−i−1 + Li,ji + ei−1 31d

Now, by substituting Eqs. 31a–31d into the interface match-ng conditions represented by Eq. 8, we observe that such rela-ions must hold for every value of the variable r. Hence, the co-

fficients multiplying r−1 and r−1 in Eqs. 31a and 31b and

hose multiplying r and r in Eqs. 31c and 31d should beeparately equated.

However, by recognizing that the expressions multiplying r

nd r−1 are simply the complex conjugates of those multiplying and r−1, only the coefficients multiplying r and r−1 need beonsidered and have to be retained in the equation system of theype of Eq. 16. After some algebra, the elementary matrix M

i is
hen found to be


Mi =

ei+

2

e−i+

2

+ei−

2

+e−i−

2

ei+

2i

e−i+

2i

−ei−

2i−

−e−i−

2i

−ei+

4Gi−

e−i+

4Gi

4

mi− +ei−

4Gi

4

mi− +e−i−

4Gi

ei+

4iGi−

e−i+

4iGi

4

mi+ −ei−

4iGi− 4

mi+ −e−i−

4iGi

32

with the following unknown vector:

vi = Mi,j,Ni,j,Ii,j,Li,jT 33

To demonstrate that the equation system characterized by thecomplex elementary matrix Eq. 32 has the same determinant asthat obtained according to the eigenfunction expansion method,we will show that M

i in Eq. 32 can be transformed into a realform, which matches exactly the elementary matrix in Eq. 18.

First of all, we have to remember that the governing equation ofthe problem is represented by the biharmonic condition Eq. 7for the Airy stress function. In the case of the eigenfunction ex-pansion method, this implies that the general expression of theeigenfunction must satisfy the following fourth order ordinary dif-ferential equation ODE:

f IV + 22 + 1f II + 2 − 12 = 0 34

It is easy to demonstrate that the general solution to Eq. 34 is

f = Ai,jei+1 + Bi,je

−i+1 + Ci,jei−1 + Di,je

−i−1

35

and it is also well known that Eq. 35 can be rewritten in anequivalent real form, which coincides with Eq. 14. Then, bysubstituting the complex expression Eq. 35 of f into Eq. 13,stress and displacement components can be computed, and it iseasy to verify that these expressions coincide with those in Eqs.31a–31d if the following change of variables is performed:

Mi,j = 2 + 1Ai,j

Ni,j = 2 + 1Bi,j

Ii,j = 2Ci,j

Li,j = 2Di,j 36

Keeping this result in mind, it is now clear that the elementarymatrix Eq. 32 just represents the complex form of the elemen-tary matrix obtained according to the eigenfunction expansionmethod. Thanks to the properties of the solution of the ODE inEq. 35, we have demonstrated that the complex function repre-sentation by Theocaris and the eigenfunction expansion methodby Williams are perfectly equivalent for the computation of theeigenvalues. This implies that both formulations can be trans-formed into the other without changing the zeros of theeigenequation of the problem.

3.3.3 Mellin Transform Technique. The Mellin transform tech-nique was introduced to the analysis of stress-singularities inplane elasticity by Tranter 96 in 1948. The potentials of thistechnique were also discussed by Godfrey 97 and Sneddon 98in their valuable articles and textbooks.

This mathematical method was mainly applied to the singular-

MARCH 2008, Vol. 61 / 020801-9


iaSippwl

si

w

A

Ff

Esq

w

Tof

ace

t

0

Downlo

ty analysis of bimaterial wedges subjected to a variety of bound-ry conditions by Bogy 21,99,100 and Hein and Erdogan 25.tresses in bonded materials with a crack perpendicular to the

nterface were obtained by Cook and Erdogan 101 in 1972. Ap-roximately in the same years, Bogy and Wang 22 tackled theroblem of stress singularities due to bimaterial junctions 22,hereas stress singularities due to trimaterial wedges were ana-

yzed with this technique only in 1996 26,102.According to this approach, the boundary value problem repre-

ented by Eq. 8 can be solved in a transformed plane by apply-ng the Mellin transform, which is defined as

Mf ,s =0

frrs−1dr 37

here s is a complex transform parameter.Consequently, the application of the Mellin transform to the

iry stress function reads

Mi,s = i*s, =

0

ir,rs−1dr 38

urthermore, the biharmonic condition requires i* to be of the

orm 96

i*s, = ai,jssins + bi,jscoss + ci,jssins + 2

+ di,jscoss + 2 39

ventually, taking the Mellin transform of r2 and r times thetresses and displacements in Eqs. 4a–4e, we obtain such
uantities in the transformed domain,
o the determinant of the coefficient matrix, D=det , allow us to

20801-10 / Vol. 61, MARCH 2008


Mr2ri ,s = − ai,jss + 1sins − bi,jss + 1coss − ci,js + s

+ 22sins + 2 − di,js + s + 22coss + 2

Mr2i ,s = ss + 1 ai,j sins + bi,j coss + ci,j sins + 2

+ di,j coss + 2

Mr2rri ,s = s + 1 ai,js coss − bi,js sins + ci,js + 2coss

+ 2 − di,js + 2sins + 2

Mruri ,s =

1

2Gi ai,js sins + bi,js coss + ci,js + 4/misins

+ 2 + di,js + 4/micoss + 2

Mrui ,s =

1

2Gi − ai,js coss + bi,js sins − ci,js + 2

− 4/micoss + 2 + di,js + 2 − 4/misins + 240

Applying the Mellin transform to the boundary conditions Eq.8 and then substituting Eq. 40 into them, a matrix form analo-gous to Eq. 16 can be determined. To be more specific, by ob-serving that the complex parameter s is related to by

s = − − 1 41

i
we obtain the following expression for the elementary matrix M:
Mi =

− + sin+ + cos+ − + sin− + cos−+ cos+ + sin+ − cos− − sin−+

2Gisin+ −

+

2Gicos+ − 4

mi− + sin−

2Gi 4

mi− + cos−

2Gi

+

2Gicos+

+

2Gisin+ 4

mi+ − cos−

2Gi 4

mi+ − sin−

2Gi

42

here + and − denote, respectively, +1 and −1.In this formulation, the components of the vector v are

vi = ai,j,bi,j,ci,j,di,jT 43

he elementary matrix in Eq. 42 coincides with that in Eq. 18btained by applying the eigenfunction expansion method if theollowing change of variables is made:

ai,j = − Ai,j

bi,j = Bi,j

ci,j = − Ci,j

di,j = Di,j 44

Also in this case, eigenequations derived from the condition ofvanishing determinant of the matrix are not influenced by this

hange of variables. The eigenvalues computed as the zeros of theigenequations are then the same in all the reviewed formulations.

3.4 Eigenequations for Limit Cases. Closed-form solutions

analytically investigate on the influence of both the junction ge-ometry and the mechanical parameters of the constituent materialson the shape of the eigenfunctions and their roots. Nonetheless,such closed-form solutions represent an important test for estab-lishing the accuracy of numerical procedures used for computingthe eigenvalues. Unfortunately, due to the involved character ofthe mathematical formulations, which yield nonlinear eigenvalueproblems, closed-form solutions of D for trimaterial junctions andwedges are available in the literature only for particular cases102.

As far as perfectly bonded bimaterial junctions are concerned, aclosed-form solution to the eigenfunction for any given modularratio E2 /E1 was provided by Bogy and Wang 22. Furthermore,they observed that for a limit material configuration involving aninfinitely stiff material region, the determinant D of the coefficientmatrix is given by see Fig. 10

DE1 → ,E2,1,2 = Dcl-cl2 Dfr-fr1

DE1,E2 → ,1,2 = Dcl-cl1 Dfr-fr2 45

where Dcl-cli and Dfr-fri denote, respectively, the eigenequa-
tions due to an angular corner of a plate with amplitude 2−i


wCi

lzatie

EcE

eoatr

tea

tatffcda

t

Ff

A

Downlo

ith either clamped or free edges see the scheme in Fig. 10.losed-form solutions to these problems were provided by Will-

ams 67 and are functions of the corner angle only.Therefore, the search of the zeros of the eigenequation for the

imit bimaterial junction problem reduces to the search of theeros of the eigenequations of two simpler subproblems whosenalytical solutions are well known. Furthermore, by observinghat the singular stress fields for the above subproblems exist onlyn the case of reentrant corners 67, i.e., only for i, theigenvalues are given by

zeros DE1 → ,E2,1,2 = zeros Dcl-cl2 if 2

zeros Dfr-fr1 if 1

no singularities if 1 = 2 =

zeros DE1,E2 → ,1,2 = zeros Dfr-fr2 if 2

zeros Dcl-cl1 if 1

no singularities if 1 = 2 =

46

On the other hand, the limit problem of a bimaterial wedge with1→0 correspondingly E2→0 can be simply treated as the limitase of a bimaterial junction with E2→ correspondingly,1→.Pageau et al. 24 proposed in 1994 a method for obtaining the

igenequation for a trimaterial junction when the elastic modulusf the third material tends to infinity. To be more specific, theyssumed that D for a trimaterial junction with E3→ is equal tohe eigenequation due to a bimaterial wedge composed of materialegions 1 and 2 with clamped edges,

DE1,E2,E3 → ,1,2,3 = Dcl-clE1,E2,1,2 47

Actually, this solution holds if and only if the wedge angle ofhe third material is less than , i.e., for 3. In addition, theigenequation corresponding to the limit trimaterial junction char-cterized by E3→0 was not addressed in the literature.

A complete solution can be obtained as an extension to trima-erial junctions of the formulation Eq. 46 provided by Bogynd Wang 22 for limit bimaterial junctions. Going into the de-ails, when the third material becomes infinitely stiff, the eigen-unction of this limit problem can be reduced to two factors. Theormer corresponds to the eigenequation of the bimaterial wedgeomposed of materials 1 and 2 with clamped edges. The latter isue to the eigenequation due to a reentrant corner with free edgesnd amplitude equal to 2−3 see the scheme in Fig. 11a.

Analogously, when the third material becomes infinitely soft,

θ1

Ε1→¶

Ε2 θ2 θ2

θ1

θ1

Ε1

Ε2→¶

θ2θ2

θ1

(a)

(b)

ig. 10 Subproblems for the evaluation of the eigenequationor a bimaterial junction with „a… E1\ and „b… E2\

he two factors correspond, respectively, to the eigenequation of



the bimaterial wedge formed by materials 1 and 2 with free edgesand the eigenequation due to a reentrant corner with clampededges and amplitude 2−3 see the scheme in Fig. 11b.

Noting that the eigenequations due to an angular corner in aplate can contribute to the singular stress field if and only if 3is greater than , we end up with the following compactexpressions:

DE1,E2,E3 → ,1,2,3 = Dcl-clE1,E2,1,2 Dfr-fr3

DE1,E2,E3 → 0,1,2,3 = Dfr-frE1,E2,1,2 Dcl-cl348

and

zeros DE1,E2,E3 → ,1,2,3

= zeros Dcl-clE1,E2,1,2 if 3

zeros Dcl-clE1,E2,1,2+ zeros Dfr-fr3 if 3

zeros DE1,E2,E3 → 0,1,2,3

= zeros Dfr-frE1,E2,1,2 if 3

zeros Dfr-frE1,E2,1,2+ zeros Dcl-cl3 if 3

49

where Dcl-clE1 ,E2 ,1 ,2 corresponds to the eigenequation for abimaterial wedge with clamped edges provided by Pageau et al.24. Dfr-frE1 ,E2 ,1 ,2 is the eigenequation for a bimaterialwedge with free edges provided by Bogy 21. Dfr-fr3 andDcl-cl3 correspond to the closed-form solutions to theeigenequations investigated by Williams 67 for a reentrant cor-ner in a plate with either free or clamped edges.

Some examples of application of these formulae to limit casesinvolving trimaterial junctions are illustrated in Table 3 see alsoFig. 12.

3.5 Eigenfunctions. Once the eigenvalues are computed, thesingular analysis is completed by the evaluation of the stress anddisplacement fields in the material regions. For this purpose, threemethods can be applied.

In the case of the eigenfunction expansion method, for each jwe can express the unknowns in the vector v in terms of the firstone, Ai,j. Then, the unknown constant Ai,j is normalized such that

θ3

(b)

Ε3→0

Ε2

θ3

Ε1θ1θ2

Ε2

Ε1θ1θ2

θ3

(a)

Ε3→¶

θ3

Ε2

Ε1θ1θ2

Ε2

Ε1θ1θ2

Fig. 11 Subproblems for the evaluation of the eigenequationfor a trimaterial junction with „a… E3\ and „b… E3\0

the following expression for the stress field can be written:

MARCH 2008, Vol. 61 / 020801-11


wrsfi

A

ma

sPtctps

gtcamsttano

btwa

Te

Fe

0

Downlo

ji = Kjr

j−1f i,j + O1 50

here O1 indicates nonsingular terms. This procedure can beepeated for each jth eigenvalue, allowing us to write an expres-ion where all the eigenvalues contribute to the singular stresseld near the vertex O,

i = j

Kjrj−1f i,j + O1 51

ccording to this method, stresses and displacements are deter-

ined to within multiplicative constants, Kj, which are referred tos generalized stress-intensity factors 16,103.

When the eigenvalues are complex, also the corresponding con-tants in the vector v are complex. Therefore, as pointed out byageau et al. 103, the direct application of Eqs. 4a–4e leads

o stresses and displacements that are complex valued which, ofourse, are not valid. In these cases, a valid solution can be ob-ained by considering the actual stresses and displacements, com-uted as a linear combination of the corresponding complextresses and displacements 103.

Eventually, particular attention has to be paid to materials andeometric configurations that correspond to the transition fromwo real roots to a complex conjugate pair. When the eigenvalueshange from real to complex at some characteristic openingngles of the junction geometry or in correspondence of certainaterial combinations, multiple eigenvalues corresponding to the

ame independent eigenfunction may occur. The analytical solu-ion of such a special case cannot be expressed as a single func-ion of the radial coordinate multiplied by a single function of thengular coordinate. These nonseparable solutions include termsot only with power functions, but also with logarithmic functionsf the radial coordinate.

The power-logarithmic stress singularities have been identifiedy Dempsey and Sinclair 7 for multimaterial wedges subjectedo homogeneous boundary conditions on the surfaces forming theedge vertex. Dempsey 104 examined the eigenvalues in the

symptotic solutions for isotropic composite wedges under homo-

able 3 Limit eigenequations and eigenvalues for three differ-nt tri material junctions sketched in Fig. 12

Figure E1 /E2 E3 /E2 limit D limit

12a 5 0 0.7389 Dcl-cl 30.8724 Dcl-cl 3

5 0.6736 Dfr-fr 30.9345 Dcl-cl E1 ,E2 ,1 ,2

12b 10 0 0.8589 Dfr-fr E1 ,E2 ,1 ,21.0000 Dfr-fr E1 ,E2 ,1 ,2

10 0.8147 Dcl-cl E1 ,E2 ,1 ,2

12c 10 0 0.6818 Dfr-fr E1 ,E2 ,1 ,20.8413 Dfr-fr E1 ,E2 ,1 ,2

10 0.7723 Dcl-cl E1 ,E2 ,1 ,2

(a)

Ε1Ε2

Ε3

0

90135

Ε1Ε2

Ε3

(c)

0

90

225

Ε2 Ε1

Ε3

(b)

0

90

180

ig. 12 Three examples of trimaterial junctions whose limit
igenvalues are reported in Table 3
20801-12 / Vol. 61, MARCH 2008


geneous boundary conditions and pointed out that the power-logarithmic function is more singular than the power function atthe vertex. Pageau et al. 24 investigated the power-logarithmicstress singularities in trimaterial junctions. Joseph and Zhang105 reviewed the studies on power-logarithmic stress singulari-ties for multiple material wedges. Angular variations of the dis-placement and stress fields are presented in Refs. 105–107 forwedges composed of isotropic materials.

Additional logarithmic singularities can be induced by inhomo-geneous boundary conditions on the surfaces forming the wedgevertex and by body forces 108. A related problem is a wedgeunder a concentrated couple studied by Sternberg and Koiter109. Dempsey 110 also obtained a solution for a single-material wedge at its critical angle, which includes a power-logarithmic stress term.

On the other hand, if the complex function representation isused, then the complex formulation outlined by Theocaris 23 hasto be adopted for the computation of stress and displacementfields in the case of real and complex eigenvalues. Also in thiscase, for each j we can express the unknowns in the vector v interms of the first, say, Ii,j. As a result, the eigenfunction can thenbe expressed in terms of the real and imaginary parts of this com-plex constant see Carpenter and Byers 78 for a detailed deriva-tion of the stress field in the case of a bimaterial wedge.

For the application of the Mellin transform, stresses and dis-placements can be computed by applying the inversion formula ofthe integral transform,

ri =

1

2ici−i

ci+i d2

d2 − si*r−s+2ds

ri + i

i =1

2ici−i

ci+i

s + 1 d

d+ is

i*r−s+2ds

uri + iu

i =− 1

4Gi

ci−i

ci+i d

d+ is

1 +

1 − i d

d− is d

d− is + 2

s + 1s + 2

i*r−s+1ds

uri

r+ i

ui

r=

1

4Gi

ci−i

ci+i

s + 1 d

d+ is

1 +

1 − i d

d− is d

d− is + 2

s + 1s + 2

i*r−s+2ds 52

where the parameters ci are such that rci−1i* are absolutely inte-

grable on 0,. The criterion for the choice of ci is then providedby the following regularity condition:

rs+m−1dm−1

i*

drm−1→ 0 as r → 0,, m = 1, . . . ,n 53

3.6 Stokes Flow Analogy. Two-dimensional Stokes flow of aviscous incompressible fluid can also be described as a problemgoverned by a biharmonic function. Stokes flow represents theflow of a fluid in the limit of vanishingly small Reynolds number,
where the viscous forces are much larger than the inertial forces.


Ada

wpa

c

t

wEd

c

t

Sc

ptatsHi

nvtt

A

Downlo

s a consequence, the equations describing Stokes flow can beeduced from the Navier–Stokes equations by setting the inertialnd body force terms equal to zero,

· v = 0

− p + 2v = 0 54

here p is the pressure, v=vrn+vt denotes the velocity field inolar coordinates, and = is the product between the density nd the kinematic viscosity of the fluid.

For two-dimensional incompressible flows, it is possible to re-ast the Navier–Stokes equations in terms of the stream function

and the vorticity . Recalling the identity

2v = · v − v 55

he Stokes equations Eq. 54 can be rewritten as

· v = 0

p = − 56

here =v is the vorticity. Taking the curl of the second ofq. 56, the equations of vorticity transport for a Stokes flow areerived,

· = 0

2 = 0 57

Introducing now a stream function such that the velocityomponents are given by

vr =1

r

58

v = −

r59

he vorticity can be expressed in terms of the stream function

= v = − 2k 60

ince the unit vector k perpendicular to the plane is constant, Eq.60 can be introduced into Eq. 57 obtaining the biharmonicondition for the stream function,

4 = 0, ∀ r, i 61

The relevance of the asymptotic analysis in fluid mechanicsroblems involving viscous fluids is clearly evidenced by the facthat there are many viscous flows in which the fluid must negoti-te corners see, e.g., the flow in a driven cavity or sector111–114 or the flow over rectangular cylinders or cones115,116. Contact-line problems 117,118, such as those relatedo meniscus problems 118, are usually treated as single-fluidystems in which the second fluid is regarded as a passive gas.owever, if the flow of both fluids is important, the flow of two

mmiscible fluids needs be considered 117,119.This problem can be generalized, in principle, to three or evenimmiscible fluids with different viscosities negotiating a corner

see Fig. 13. The boundary conditions at the interfaces imposeanishing velocities in the circumferential direction and the con-inuity of the velocities in the radial direction, as well as of theangential and circumferential stresses,

vi r,i+1 = v

i+1r,i+1 = 0

vrir,i+1 = vr

i+1r,i+1

i r,i+1 = i+1r,i+1
r r


i r,i+1 =

i+1r,i+1 62

These boundary conditions permit us to find out the local solu-tions to the problem and apply in the case of fluids characterizedby a small capillary number see, e.g., Ref. 119 for a compre-hensive discussion on this problem. When the capillary number isexactly equal to zero, all the aforementioned local boundary con-ditions are imposed, with the exception of the circumferential-stress boundary condition, which is identically satisfied. The cor-responding solutions are then referred to in the literature as partiallocal solutions since they provide an approximation of the veloc-ity field valid only for a zero capillary number 119.

As far as the boundary conditions at the wedge edges defined

by the angle are concerned, the boundary may be rigid walls onwhich the fluid velocity is prescribed, or surfaces on which thestress is prescribed with the pressure distribution to keep itplane. Three distinct categories of flow can be described by theseboundary conditions. In the first category, a nonzero velocity orstress is prescribed on one or both boundaries, and the flow isdescribed by a particular integral of the biharmonic equation sat-isfying appropriate inhomogeneous boundary conditions. One ofthese situations was addressed by Taylor 120, who described thecase when a viscous fluid is entrapped in a rigid plane, which isscraped along another at a constant velocity and inclination angle.

In the second category, either a velocity or the tangential stressvanishes on each boundary. The flow near the corner is induced bya general motion at a large distance from the corner. In this case,the boundary conditions are homogeneous and the problem seemsto have been first considered by Rayleigh 121. The flows in thiscategory all have a finite velocity at the corner vertex.

In the last category, perturbance is present near the origin, andthe velocity in the corresponding flow is infinite at r=0 but tendsto zero as r→. Such solution may describe the flow at a largedistance from the intersection of two planes when some steadydisturbance at r→0 takes place.

Considering the second category in more detail, the followingboundary conditions are usually imposed on the wedge edges at

= :

vrr, = 0

¯for a rigid surface 63

Ω0

Ω1

Ω2

Fig. 13 A scheme of a corner negotiated by three immiscibleviscous fluids

vr, = 0

MARCH 2008, Vol. 61 / 020801-13


gs

tfesafltf

hmc

rbettm

Fe

wptww

1aws

TaIm

cv

0

Downlo

vr, = 0

rr, = 0

r, = 0

for a free surface 64

In the case of free surfaces, the condition =0 can be ne-lected if the fluid has a zero capillary number, i.e., when we areeeking for partial local solutions.

From this preliminary discussion emerges the observation thathe flow of a viscous fluid is governed by the biharmonic equationor the stream function, in close analogy with the biharmonicquation for the Airy stress function in plane elasticity. As a con-equence, the eigenfunction expansion method can be profitablypplied, as firstly pursued by Rayleigh 121 for the study of theow of a single fluid inside a corner. Considering a stream func-

ion given as the product between a radial term and an angularunction,

ir, = j

rj+1f i,j, j 65

e derived an expression for f i,j analogous to that reported in Eq.14. However, at that time, he assumed that the exponent j +1ust be a multiple of /, where is the wedge angle. As a

onsequence of this assumption, he deduced that the expression off in Eq. 65 is possible only if is a multiple of ; i.e., heestricted to the values and 2. This assumption was removedy Dean and Montagnon 122 in 1948, who firstly applied theigenfunction expansion method to the analysis of the steady mo-ion of a viscous liquid around a corner with rigid surfaces. It haso be noticed that this application of the eigenfunction expansion

ethod precedes that by Williams 67 by four years.According to Eq. 65, the expressions for vr, v, and r are

vr = rf 66

v = + 1rf 67

r = r−1− 2 − 1f + f 68

rom these expressions, we readily recognize that Eqs. 66 and67 are analogous to those for the tangential and the circumfer-ntial stresses in plane elasticity reported in Eq. 13, i.e.,

vr = −r

el

v =r

el 69

here the superscript el denotes quantities related to biharmonicroblems in classical elasticity. Therefore, it is possible to statehat the problem of the flow of a viscous fluid in a single wedgeith rigid surfaces has its analogous counterpart in plane elasticityith the stress field of an angular plate with free surfaces.The eigenequation obtained by Dean and Montagnon 122 in

948 for the analysis of the steady motion of a viscous liquidround a corner with rigid surfaces is then coincident with theell-known expression derived by Williams in 1952 for the stress

ingularities in angular plates with free boundaries 67,

sin

= sin

70

he negative sign in Eq. 70 corresponds to Mode I deformationnd symmetric flow, whereas the positive sign is related to ModeI deformation and skew-symmetric flow see Fig. 14 for a sche-atic visualization of these flows.Another interesting point concerns the eigenvalues that can be

omputed as the roots of Eq. 70. Williams computed the eigen-
alues and plotted their real part as a function of the wedge
20801-14 / Vol. 61, MARCH 2008


angle, . For free-free boundary conditions, he found that is realand in the range 01 for obtuse angles, i.e., for plates withreentrant corners. No particular comments were provided by Wil-liams about the transition between real and complex eigenvaluessince this case occurs corresponding to 146 deg, i.e., forRe 1 leading to nonsingular stress fields. On the contrary, thistransition was carefully analyzed by Dean and Montagnon 122four years before Williams. This is because in the analogous prob-lem in fluid mechanics, the fluid velocity is always finite and thewhole range of variation of is of interest. Moffatt 123 in 1964showed that the complex eigenvalues observed for 146 degimply the existence of a flow near the corner consisting in a se-quence of eddies of decreasing size and rapidly increasing inten-sity see Fig. 15 for a visualization of such resistive eddies.

(a)

(b)

(c)

(d)

Fig. 14 Sketch of the stream lines for symmetric „a… and „b…and skew-symmetric „c… and „d… flows around a corner

Fig. 15 Sketch of the stream lines in corner eddies for=60 deg, reprinted from Ref. †123‡. The relative dimensions ofthe eddies are approximately correct and the relative intensi-
ties are indicated.


4

lc

pv

wd

a

we

Ii

Fp

A

Downlo

Out-of-Plane Loading

4.1 Statement of the Harmonic Problem. Out-of-planeoading due to antiplane shear Mode III on composite wedgesan lead to stresses that can be unbounded at the junction vertex Osee a scheme of this multimaterial problem depicted in Fig. 16

When out-of-plane deformations only exist, the following dis-lacements in cylindrical coordinates with the origin placed at theertex point O can be considered:

ur = 0

u = 0

uz = uzr, 71

here uz is the out-of-plane displacement. For such a system ofisplacements, the strain field becomes

r = = z = r = 0

rz =uz

r

z =1

r

uz

72

nd from Hooke’s law, the stresses reduce to

r = = z = r = 0

rz = Girz = Giuz

r

z = Giz =Gi

r

uz

73

here Gi is the shear modulus of the ith material component. Thequilibrium equations in the absence of body forces reduce to

rz

r+

1

r

z

+

1

rrz = 0, ∀ r, i 74

ntroducing Eq. 73 into Eq. 74, the harmonic condition upon uz

zx

Pr θ

Oy

i =1

i =2

Γ1

ig. 16 Scheme of a composite material subjected to anti-lane shear deformation

s derived,



2uz

r2 +1

r

uz

r+

1

r2

2uz

2 = 2uz = 0, ∀ r, i 75

In contrast to its biharmonic counterpart in elasticity, which hasbeen the subject of a considerable number of investigations, thecharacterization of the singularities of this harmonic problem incomposite regions appears to have received a minor attention.Antiplane shear problems for isotropic and anisotropic bimaterialwedges were addressed by Ma and Hour 124,125 in the late1980s using the Mellin transform technique. The antiplane sheardeformation problem of isotropic as well as anisotropic wedgeswas solved under different boundary conditions by Karganovinet al. 126 and Shahani 127, respectively. They found, using theMellin transform technique, analytical solutions for the displace-ment and stress fields and derived the order of the stress singular-ity at the wedge apex. Shahani and Adibnazari 128 consideredthe antiplane shear deformation problem of two-material junctionswith an interfacial crack.

Extensions to the computation of the eigenvalues and of theeigenfunctions for bonded and debonded trimaterial junctionswere provided by Pageau et al. 73 in 1995 using the eigenfunc-tion expansion method. More recent developments on this subjecthave concerned the use of cohesive laws instead of the classicalboundary conditions for reentrant corners in antiplane shear inorder to free them from stress singularities 129.

4.2 Boundary Conditions. Along the ith perfectly bondedinterface defined by the angle i+1, continuity conditions for stressand displacement have to be imposed,

uzir,i+1 = uz

i+1r,i+1

zi r,i+1 = z

i+1r,i+1 76

Moreover, in the case of either a reentrant corner or a crack at the

angle , the following classical stress or displacement boundaryconditions can be imposed along the edges:

zi r, = 0 for an unrestrained stress-free edge

uzir, = 0 for a fully restrained edge 77

When the problem is characterized by a geometric symmetry, itis worth noting that the problem can be reduced to the solution oftwo separate subproblems by considering the following boundary

condition along the symmetry line =0,:

zi r, = 0 for a symmetric deformation

uzir, = 0 for a skew-symmetric deformation 78

Cohesive-law boundary conditions have also been consideredby Sinclair 129 and were imposed on symmetry lines and onbimaterial interfaces instead of their classical counterparts. Theselaws admit to relative, albeit small, deflections between the twoconnected parts of the wedge, something not permitted with tra-ditional boundary conditions. To this aim, a constant stiffness Kfor a simplified linear version of a cohesive stress versus separa-tion law is introduced, and the following boundary condition canbe specified along the ith interface:

zr,i+1 = Kuzi+1r,i+1 − uz

ir,i+1 79

For the special case of stress-free single-material wedges ofangle in antiplane shear, Sinclair 129 demonstrated that theuse of traditional boundary conditions yields the following
asymptotic behavior r→0 of the tangential stresses:
MARCH 2008, Vol. 61 / 020801-15


Hstbs

tfm

wrpsmp

e

f

wms

pt

wiec

wva

s

0

Downlo

r2/−1 for a symmetric deformation

r/−1 for skew-symmetric deformation 80

ence, for reentrant corners, i.e., for /2, the skew-ymmetric stress field is singular. On the contrary, the adoption ofhe cohesive separation law along the symmetry line wedgeisector permits us to avoid the occurrence of these stressingularities.

4.3 Mathematical Methods

4.3.1 Eigenfunction Expansion Method. In the framework ofhe eigenfunction expansion method, the following separable formor the longitudinal displacement uz

i can be adopted for the ithaterial region:

uzir, =

j

rj f i,j, j 81

here j and f i,j are referred to as eigenvalues and eigenfunctions,espectively, as already introduced for the analysis of biharmonicroblems. Also in this case, the summation with respect to theubscript j is introduced in Eq. 81 since it is possible to haveore than one eigenvalue for each problem, and the superposition

rinciple can be applied.Introducing Eq. 81 into Eq. 73, tangential stresses can be

xpressed in terms of the eigenfunction and its prime derivative,

rzi = Gir−1f i,j

zi = Gir

−1f i,j 82

Furthermore, the harmonic condition requires f i,j to be of theollowing form:

f i,j, j = Ai,j sin j + Bi,j cos j 83

here Ai,j and Bi,j are undetermined constants. To simplify theathematical notation, the subscript j in Eq. 83 is dropped in the

equel.Then, introducing Eq. 83 into Eq. 82, the longitudinal dis-

lacement and the tangential stresses can be explicitly written inerms of the unknowns Ai,j, Bi,j, and ,

uzi = rAi,j sin + Bi,j cos

rzi = Gir−1Ai,j sin + Bi,j cos

zi = Gir−1Ai,j cos − Bi,j sin 84

Considering the problem consisting in a multimaterial junctionith n material regions, Eq. 84 has to be introduced into the

nterface matching conditions Eq. 76. In this way, a set of 2nquations in the 2n+1 unknowns Ai,j, Bi,j, and j can be symboli-ally written as:

v = 0 85

here denotes the coefficient matrix that depends on the eigen-alue, and v represents the vector that collects the unknowns Ai,jnd Bi,j.

The coefficient matrix in Eq. 85 is characterized by a sparse
tructure, as for the corresponding biharmonic problem,
20801-16 / Vol. 61, MARCH 2008


= M1

0 − M1

1

M2

1 − M2

2

. . . . . .

Mi

i−1 − Mi

i

. . . . . .

Mn−1

n−2 − Mn−1

n−1

− M0=00 Mn=2

n−1

86

The elementary matrix Mi is given by

Mi = sin cos

Gi cos − Gi sin 87

and the components of the vector v are

v = v0,v1,v2, . . . ,vi, . . . ,vn−2,vn−1T 88

where

vi = Ai,j,Bi,jT 89

A nontrivial solution of the equation system Eq. 85 exists ifand only if the determinant of the coefficient matrix vanishes. Thiscondition yields an eigenequation that has to be solved for theeigenvalues. For the present purposes, we are concerned only withthose values of j that may lead to singularities in the stress field.This fact, together with the condition of continuity of the displace-ment field at the vertex where regions meet, implies that we arelooking for eigenvalues in the range 0Re j 1.

4.3.2 Mellin Transform Technique. Ma and Hour 124,125tackled the harmonic problem represented by Eq. 75 by meansof the Mellin transform defined in Eq. 27. The application of thistechnique to the harmonic equation yields an ordinary differentialequation for the out-of-plane displacement uz

i* in the trans-formed plane s ,, the general solution of which is

uzi*s, = ai,jssins + bi,jscoss 90

Taking the Mellin transform of r times the tangential stresses,these quantities can be recast in the transformed domain

Mrrzi ,s = − Gisai,j sins + bi,j coss

Mrzi ,s = Gisai,j coss − bi,j sins 91

Applying the Mellin transform to the interface boundary condi-tions Eq. 76 and then introducing Eq. 91 into them, a matrixform analogous to Eq. 85 can be written. To be more specific,noting that the complex parameter s is related to by

s = − 92

we obtain the following expression for the elementary matrix Mi :

Mi = − sin cos

− Gi cos − Gi sin 93

In this formulation, the components of the vector v are

vi = ai,j,bi,jT 94

Hence, the elementary matrix in Eq. 93 coincides with that inEq. 87 obtained by applying the eigenfunction expansionmethod if the following change of variables is made:

ai,j = − Ai,j



Cice

trpac

wci

tsd

M

ps

t

cp

gh

w

is

tatscm

A

Downlo

bi,j = Bi,j 95

learly, the eigenequations derived from the condition of a van-shing determinant of the matrix are not influenced by thishange of variables. The eigenvalues computed as the zeros of theigenequations are then the same in both the formulations.

4.4 Steady-State Heat Transfer Analogy. The analogy be-ween steady-state heat transfer and antiplane shear in compositeegions was firstly addressed by Sinclair 130 in 1980. In bothroblems, the field equations for the longitudinal displacement uznd for the temperature T are harmonic. As a result, the followingorrespondences between these two problems can be considered:

2T = 0 ⇔ 2uz = 0 96

qr = − kiT

r⇔ rz = Gi

uz

r97

q =ki

r

T

⇔ z =

Gi

r

uz

98

here qr and q are, respectively, the heat flux in the radial andircumferential directions and ki is the thermal conductivity of theth material region.

Concerning the boundary conditions at the interfaces, the con-inuity of the longitudinal displacement and of the tangentialtress reported in Eq. 76 can be replaced by the continuity con-itions of temperature T and heat flux q

Tir,i+1 = Ti+1r,i+1

qi r,i+1 = q

i+1r,i+1 99

oreover, the stress or displacement boundary conditions im-

osed along the edges at = of reentrant corners reported in Eq.77 can be replaced by insulated boundary or temperature pre-cribed boundary conditions,

qi r, = 0 for insulated boundary

Tir, = 0 for temperature prescribed 100

When the problem is characterized by a geometric symmetry,he following boundary conditions analogous to those in Eq. 78an be imposed along the symmetry line at =0, to reduce theroblem to two separate subproblems:

qi r, = 0 for symmetric problems

Tir, = 0 for skew-symmetric problems 101

Cohesive-law boundary conditions in Eq. 79 have their analo-ous expression in the convective conditions for the steady-stateeat conduction problems 131,

qr,i+1 = hcTi+1r,i+1 − Tir,i+1 102

here hc denotes the convective heat transfer coefficient.Therefore, according to this analogy, the singularities character-

zing the heat fluxes qr and q are the same as those for the stressingularities in rz and r due to antiplane shear.

4.5 de Saint-Venant Torsion Analogy. The de Saint-Venantorsion problem of cylindrical bars and the related analogies have

longstanding fascinating history, which is reviewed in Refs.132–134. The mathematical formulation of this problem can beraced back to the middle of the 19th century 135,136. From thetudies on the torsional deformation of cylinders with generalross sections, the following fundamental properties can be sum-
arized.


1. The projection of each section perpendicular to the longitu-dinal axis on the x ,y plane rotates as a rigid body about thecenter of twist.

2. The amount of projected section rotation varies linearly withthe axial coordinate, i.e., z=z, where is defined as theangle of rotation between two cross sections having a uni-tary distance.

3. Plane cross sections do not remain plane after deformation,thus leading to a warping displacement.

The problem of stress singularity at the junction vertex of com-posite bars seems to have been firstly addressed by Rao in 1971using the eigenfunction expansion method 64. Polar coordinateswere introduced in that analysis, with the origin of the referencesystem, O, placed at the junction point, as usually performed inthe asymptotic analyses see Fig. 17a. However, at that stage,no attention was paid to the position of the center of twist, C,whose location might not coincide with the above-mentioned ori-gin of the reference system, O. If points O and C are more appro-priately distinguished in the analysis, then two different referencesystems need to be considered: the former is placed at the originO, where stress singularities are expected, whereas the latter isplaced at the center of twist, C. Then, the displacements of ageneric point P in polar coordinates with respect to the center oftwist are

uC = 0

Fig. 17 Scheme of the St. Venant torsion problem with a com-posite cylinder „a…. Relationship between the displacements inpolar coordinates computed in the reference systems with ori-gin O „singular point… and C „center of twist… „b….

r

MARCH 2008, Vol. 61 / 020801-17


wt

e

F

a

w

bu

dfsrNtt

ifbistdsbslasi

odb

0

Downlo

uC = zrC

uzC = uz

CrC,C 103

here rC and C are the polar coordinates of the generic point P inhe reference system centered in C see Fig. 17b.

These displacement components can be expressed in the refer-nce system centered in point O as follows:

ur = uC sin − C = zrC sin − C

u = uC cos − C = zrC cos − C

uz = uzr, 104

or such a system of displacements, the strain field becomes

r = = z = r = 0

rz =ur

z+

uz

r= rC sin − C +

uz

r

z =u

z+

1

r

uz

= rC cos − C +

1

r

uz

105

nd from Hooke’s law, the stresses reduce to

r = = z = r = 0

rz = Girz = GirC sin − C + Giuz

r

z = Giz = GirC cos − C +Gi

r

uz

106

here Gi is the shear modulus of the ith material component.The only nonvanishing equilibrium equation in the absence of

ody forces reduces to the well-known harmonic condition uponz, as previously derived for the antiplane shear deformation,

2uz

r2 +1

r

uz

r+

1

r2

2uz

2 = 2uz = 0, ∀ r, i 107

At this point, it is remarkable to mention that the above-escribed solution strategy of the torsion problem is usually re-erred to as displacement formulation, in which the problem isolved for the warp function uz. In this case, the field equation isepresented by the Laplace equation Eq. 107 with Dini-eumann traction boundary conditions imposed on the value of

he normal derivative of uz at the points pertaining to the border ofhe cross section.

Other solution strategies were also pursued in the literature. Fornstance, de Saint-Venant 136 postulated the existence of a stressunction such that the tangential stresses can be found from ity differentiation. However, the introduction of these expressionsn the compatibility equations written in terms of stresses, i.e., theo-called Beltrami–Michell equations 137,138, yields a Poisson-ype governing equation for . In this scheme, the boundary con-ition simplifies to a simpler Dirichlet boundary condition on thetress function; i.e., must be constant or must vanish at theorder of the cross section. According to this solution strategy, thetress function formulation allows us to interpret the torsion prob-em through mechanical analogies between the stress function nd the deformation of a membrane in tension 139, with thetream function of an ideal nonviscous fluid with constant vortic-ty 140 and with the equimomental lines of the electric field.

For the stress-singularity problem under consideration, the anal-gy has to be set between the antiplane shear deformation and theisplacement formulation for the de Saint-Venant torsion since in
oth cases the governing variable obeys the Laplace differential
20801-18 / Vol. 61, MARCH 2008


equation. Moreover, it has to be pointed out that this analogy isvalid for an asymptotic analysis only, i.e., for the study of thesingular stress field when r→0. In fact, a direct comparison be-tween Eqs. 106 and 73 shows that the tangential stresses forthe torsion problem are the sum of two components: a regularterm depending solely on and a possibly singular term, analo-gous to that for the antiplane shear deformation. Therefore, it ispossible to state that the St. Venant torsion and the antiplane sheardeformation problems can be considered analogous in theasymptotic sense, i.e., if and only if the analysis is restricted to thestress field near point O with r→0.

5 Advanced Issues for Nonhomogeneous MaterialsComposites frequently involve situations where nonhomoge-

neous materials are either present naturally or used intentionallyto attain a required mechanical performance. Functionally gradedmaterials FGMs are an illustrative example of two-phase synthe-sized materials designed in such a way that the volume fractionsof the constituents vary continuously along the thickness directionto give a predetermined composition profile 37,141–143. Thepotentials of these new material microstructures characterized bya given grading on the elastic modulus are under current investi-gation by the scientific community. In 1983, Erdogan 6 statedthat “…if the crack is embedded into a nonhomogeneous mediumwith smoothly varying elastic properties the square root nature ofthe stress singularity seems to remain unchanged.” The squareroot singularity was mathematically proven in 1987 by Eischen144 under the assumption that the elastic modulus varies bothradially and angularly with respect to the crack tip position at thesame time,

Er, = E01 + rE1 +r2

2E2 + Or3, r → 0,

108

where E0=const and E1 and E2 are smooth, bounded func-tions of . According to this expression, cracks perpendicular,parallel, and arbitrarily oriented with respect to the direction ofthe elastic gradient are situations where the nature of the squareroot singularity remains unchanged 141,145–147.

In 2005, Carpinteri and Paggi 38 extended the mathematicalformulation for the study of stress singularities in junctions be-tween different homogeneous materials to junctions composed ofangularly nonhomogeneous materials. It is, in fact, possible toassume that due to bonding, a smooth angular variation of theelastic constants takes place in the regions close to the interfacessee Fig. 18.

Hence, a general angular grading on Young’s modulus was al-lowed, i.e., Ei=Ei. This possibility was not contemplated in themathematical analysis proposed by Eischen 144. In fact, whenthe elastic modulus in Eq. 108 is independent of r, it turns out tobe equal to E0, which is a constant. Functions Ei are continu-ous, bounded, positive, and differentiable inside any materialregion.

When the stress-strain relations and Eqs. 4a–4e are substi-tuted into the strain compatibility equation Eq. 6, the followinggoverning equation for the Airy stress function in plane stressconditions is derived:

4i + 2

Ei2dEi

d2

−1

Eid2Ei

d2 1

r3

i

r+

1

r4

2i

2 −i

r2

2i

r2 +

1

Ei

dEi

d2i

r2

3i

r2−

2

r4

3i

3 −2

r3

2i

r

+1 + idEi−

22

3i2 +

23

2i −24

i = 0 109
Ei d r r r r r


iec

ot

wE

wwo

trfett

ti

Fot

A

Downlo

At this point, the eigenfunction expansion method can be prof-tably applied 144 by considering the leading term only in theigenfunction expansion, which contributes to the singular stressomponents as r→0,

ir, = rj+1f i,j 110

Upon substituting Eq. 110 into Eq. 109, the following fourthrder, linear, homogeneous ODE with nonconstant coefficients forhe unknown function f i,j is found:

f i,jIV − 2

Ei

Eifi,j + 2 j

2 + 1 + 2Ei

Ei2

−Ei

Ei f i,j + 2

Ei

Ei− j

2 + i

− 12 − 1 f i,j + j2 − 12 + j + 1 ji − 1− 2Ei

Ei2

+Ei

Ei f i,j = 0 111

here primes denote derivatives with respect to . Wheni=const, we obtain

f i,jIV + 2 j

2 + 1f i,j + j2 − 12f i,j = 0 112

hich corresponds to the fourth order, linear, homogeneous ODEith constant coefficients resulting from the biharmonic conditionn the Airy stress function in homogeneous materials.

The general solution of Eq. 111 has a more involved characterhan that of Eq. 112 since the coefficients multiplying the de-ivatives of f i,j depend on the elastic moduli, which are, in turn,unctions of the angular coordinate. It has to be noticed that if thelastic moduli have an exponential variation with respect to ,hen we have a fourth order ODE with constant coefficients. Inhat case, in fact, the ratios Ei /Ei and Ei /Ei are constants.

In any case, at least theoretically, once the angular variation ofhe elastic parameters inside the material regions is specified, the

ig. 18 Scheme of a FGM bimaterial junction „a… and variationf the elastic moduli in the two-material regions „b…. This varia-ion can be completely general, without any kind of symmetry.

ntegral of Eq. 111 can be determined within four unknown con-



stants, as for the homogeneous case see, e.g., Eq. 15. Then,stress and displacement components can be computed accordingto Eq. 7. Finally, interface matching conditions Eq. 9 for thefully bonded case can be specified. This procedure allows us toformulate an eigenvalue problem, which is completely analogousto that illustrated for homogeneous multimaterial junctions seeEq. 18. As a result, eigenvalues and eigenvectors can be com-puted in order to describe the state of stress at the singular point.

In summary, it has to be remarked that the effect of an angulargrading on the elastic moduli of the material regions leads to adifferent governing equation for the Airy stress function as com-pared to that for homogeneous materials. As a result, also in thecase of a vanishing elastic mismatch at interfaces, i.e., Eii=Ei+1i see Fig. 18, the state of stress inside the material re-gions depends on the specified grading, and it differs from that inhomogeneous materials. Carpinteri and Paggi 38 explored twoproblems for the special case of an exponential angular grading onYoung’s modulus: a crack inside an angular FGM material and atrimaterial junction with a single FGM transition wedge. In theformer case, they demonstrated that the order of the stress singu-larity is bounded and it cannot be higher than 0.5. In the latter, ithas been shown that the presence of a FGM intermediate materialis favorable since it significantly reduces the order of the stresssingularity as compared to the same trimaterial junction involvinghomogeneous different materials. This was particularly evidentwhen an interface crack is introduced and supersingularities canbe avoided.

6 Conclusions and Future PerspectivesThe state of the art of stress-singularity identification in classi-

cal elasticity includes a large variety of problems concerning in-plane loading of a plate, antiplane shear of a wedge, plate bend-ing, axisymmetric torsion of a cylinder, axisymmetric axialloading at a vertex, and axisymmetric axial loading at a cylindri-cal boundary. For all of these classes of configurations, theasymptotic expression of the singular stress field can be found inthe fundamental review article by Sinclair 9,10.

In this contribution, the specific problem of stress singularitiesarising at multimaterial interfaces in two-dimensional linear-elastic problems has been addressed. The relevance of this subjectand interdisciplinarity has contributed to the development of sev-eral research papers. They are mainly concerned with the compu-tation of the quantities associated to stress singularities, such asthe eigenvalues, the eigenfunctions, and the stress-intensity fac-tors, for a variety of junction and wedge problems with differentboundary conditions, geometrical configurations, and mechanicalparameters.

Besides this huge effort in the identification of stress singulari-ties for specific case studies, there has also been a considerabledevelopment of mathematical methods for the asymptotic analysisof stress singularities. The eigenfunction expansion method, firstlyintroduced by Wieghardt 66 in 1907, was then applied by Will-iams 67 in 1952 to the famous problem of reentrant corners ofplates in extension. Thereafter, this method was largely confinedto the analysis of problems admitting real eigenvalues. In the early1970s, the Mellin transform technique and the complex functionrepresentation were profitably applied to the problems of bimate-rial wedges and junctions. In 1974, Theocaris 23 firstly extendedthe complex function representation to the problem of multimate-rial wedges and proposed a modification of the complex potentialsto handle with complex singularities. In the late 1970s, Dempseyand Sinclair 7 reconsidered the eigenfunction expansion method“to remove any suggestion that it has a lacuna with respect to theMellin transform approach…to derive conditions for a logarithmicstress singularity.” However, only in 1996 it was shown by Pageauet al. 103 that “the eigenfunction expansion method can be usedto determine expressions for the displacement and stress fieldswhen the order of the stress-singularity is complex, although at
first glance the formulation may appear to be limited to the case of
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eal roots.” This long debate on the equivalence of the mathemati-al methods has clearly produced a lack of standardization wellvidenced by the large number of case studies duplicated in theiterature.

Hence, the main product of this review article was to provide,n a historical retrospective, a unification of the available math-matical techniques for the computation of the order of the stressingularity in two-dimensional interface problems. Presenting theroblem in an extremely synthetic matrix form, the equivalence ofhe eigenfunction expansion method, of the Muskhelishvili com-lex function representation and of the Mellin transform techniqueas been mathematically demonstrated. In fact, it has been shownhat all these mathematical methods lead to the same expression ofhe coefficient matrix of the nonlinear eigenvalue problem. As aesult, all these techniques can be used for the computation of realnd complex eigenvalues.

Concerning the in-plane loading problem in elasticity, the ex-sting analogy with the Stokes flow of dissimilar immiscible fluidsas been put into evidence. From the historical point of view, thexistence of this analogy implies that the eigenfunction expansionethod was independently applied in fluid dynamics even before

he fundamental paper by Williams 67. As regards the out-of-lane loading, the analogies with the steady-state heat transfercross different materials and with the St. Venant torsion of com-osite bars have also been discussed.

Finally, the recent advanced issues for the stress singularitiesue to joining of angularly nonhomogeneous elastic wedges haveeen presented. Clearly, much work remains to be done in thisesearch area in order to fully elucidate the potentials of the use ofGMs for the removal of stress singularities in junction problems.

cknowledgmentThe financial support of the European Union to the Leonardo da

inci Project I/06/B/F/PP-154069 “Innovative Learning andraining on Fracture ILTOF” is gratefully acknowledged.

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Marco Paggi was born in 1977 in Novara, Italy and received a combined BS-MS degree (summa cumlaude) in Civil Engineering from the Politecnico di Torino, Torino, Italy in 2001. He was appointed Ph.D.in Structural Engineering in 2005 for his work on interface mechanical problems in heterogeneous mate-rials and is currently an Assistant Professor of Structural Mechanics in the Department of Structural andGeotechnical Engineering of the Politecnico di Torino. He was recipient of the Optime Prize from theIndustrial Union of Torino in 2001. His research interests span various areas of Fracture Mechanics andContact Mechanics, namely, stress singularities in elasticity, contact mechanics of rough surfaces, interfaceconstitutive laws, finite element analysis in contact and fracture mechanics, fracture of functionally gradedmaterials, damage and fracture phenomena in cutting tools, and scaling laws in fatigue. He is the author ofmore than 55 publications, among which 20 are in international refereed journals and two are bookchapters. He is a fellow of the National Institution of Engineers and member of the Italian Group ofFracture (IGF) and of the European Structural Integrity Society (ESIS).

Alberto Carpinteri is a Professor of Structural Mechanics at the Politecnico di Torino, Torino, Italy since1986. He has been Director of the Department of Structural Engineering in the time period of 1989–1995,as well as a Founding Member and Director of the Graduate School in Structural Engineering since 1990.He is a Member of the Turin Academy of Sciences since 1995 and a Fellow since 2005. His intense activityin fracture mechanics has brought him to be the President of the European Structural Integrity Society(ESIS), 2002–2006, the President of the International Association of Fracture Mechanics for Concrete andConcrete Structures (IA-FraMCoS), 2004–2007, the Vice President of the International Congress on Frac-ture (ICF), 2005–2009, and a Member of the Congress Committee of the International Union of Theoreticaland Applied Mechanics (IUTAM), 2004–2008. He is a coeditor of the International Journal Strength,Fracture, and Complexity, and a Member of the Editorial Board of different international journals. Profes-sor Carpinteri has published more than 450 papers in refereed journals or proceedings, and 32 volumes. Hehas received numerous honors and awards, among which are the Robert l’Hermite International Prize fromRILEM (1982) and the JSME Medal (1993).



Documents

On the Stress Singularities at Multimaterial Interfaces and Related