analysis of stress singularities in thin coatings bonded to a semi-infinite elastic substrate

International Journal of Solids and Structures 48 (2011) 1915–1926

Contents lists available at ScienceDirect

International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

Analysis of stress singularities in thin coatings bonded to a semi-infiniteelastic substrate

L. LanzoniDipartimento di Ingegneria Meccanica e Civile, University of Modena and Reggio Emilia, 41125 Modena, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 November 2010Received in revised form 28 February 2011Available online 9 March 2011

Keywords:Interfacial stressStress singularityChebyshev polynomialsIntegral equation

0020-7683/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2011.03.001

E-mail address: [email protected]

The interfacial stress arising between thin structures bonded to a 2D elastic substrate has been investi-gated in the present work. Chebyshev polynomials have been adopted to approximate the shear stressoccurring across the contact region. Perfect adhesion has been assumed among the coating structuresand the underlying substrate, leading to a singular integral equation which has been reduced to an alge-braic system. Thin bonded structures having several geometric configurations under different load con-ditions have been considered. In particular, the stress concentration in the neighborhood of the coatingedges and around the points of load application has been evaluated in detail.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

A variety of devices employed in high-tech industries, mainlyin microelectronics, electrochemistry, semiconductors and opticalelectronics, involves thin films. Micro–electro–mechanical sys-tems (MEMS), biomedical components, chemical reactors, inte-grated circuit, solar cells, flat panel displays and protectionsystems are typically based on thin film coatings technology. Inparticular, flip-chip microprocessor packages, high power tunablemicrowave devices, disk resonators, bulk ceramics-based phaseshifters, coplanar-plate capacitors and varactors represent somemodern systems involving coatings deposit and thin film layers(Gevorgian, 2009).

In most engineering applications, thin films and coatings havenot a primary structural function; nevertheless, the most part ofthin film-based devices are often subjected to high stress levelsand, in turn, to damaging phenomena like fractures and excessivepermanent deformations which can compromise the usefulnessand functionality of this kind of components (e.g. Freund, 2000;Hsueh, 2002). Thus, the design of thin film-substrate systems re-quires the accurate knowledge of the mechanical interaction be-tween the bonded structures and the underlying substrate, withparticular reference to the interfacial stress arising across the con-tact region (Freund and Suresh, 2003; Shen, 2010). In particular,the proper evaluation of the local effects, like stress and strainlocalizations, becomes essential in order to predict the functional-ity and performances of structures coated by thin films.

ll rights reserved.

The contact problem of thin elastic structures like ribs, bars androds welded to an elastic substrate was largely investigated bymany researchers. In particular, the work of Melan (1932) is oneof the first studies devoted to the evaluation of the mechanicalstiffener-substrate interaction. Melan analyzed the contact prob-lem of an infinite bar loaded by a longitudinal force applied at itsmidpoint and bonded to an infinite or semi-infinite plate. He ob-tained a closed form solution for the interfacial shear stress andthe axial force acting in the bar, founding a singularity of the inter-facial stress of logarithmic kind in the neighborhood of the point ofload application. The problem of an infinite flat sheet axially loadedat the boundary and stiffened by a finite cover was considered alsoby Reissner (1940). Buell (1948) studied a semi-infinite stiffenerbonded to a semi-infinite plate through a proper series expansionof the Airy stress function for the cover. Koiter (1955) solved theproblem of the load diffusion in an infinite elastic layer stiffenedby a semi-infinite sheet axially loaded at one end by using theGreen function of the infinite or semi-infinite elastic space. Hesolved the singular integral equation of Prandtl type governingthe problem by using Mellin transformation and Laplace trans-forms, founding an interfacial stress characterized by a square-rootsingularity in the neighborhood of the sheet edge. Brown (1957)studied the same problem considering various load conditions byusing complex potentials. Rybakov and Cherepanov (1977) ana-lyzed the frictionless contact problem of an infinite elastic rod peri-odically riveted in some points to a semi-infinite plate by adoptinga complex potentials representation for the displacement field andthe internal forces in the stiffener. In this reference, the forces act-ing in the rivets are determined by solving a system of linear alge-braic equations. Later, the same author solved the problem by

http://dx.doi.org/10.1016/j.ijsolstr.2011.03.001

mailto:[email protected]

http://dx.doi.org/10.1016/j.ijsolstr.2011.03.001

http://www.sciencedirect.com/science/journal/00207683

http://www.elsevier.com/locate/ijsolstr

1916 L. Lanzoni / International Journal of Solids and Structures 48 (2011) 1915–1926

taking into account a fracture occurring in the stiffener (Rybakov,1982).

The study of the coating-substrate system was extended tobonded structures of finite dimension also. For instance, Benscoter(1949) studied the contact problem of a finite strip welded to aninfinite shell by using the stress field of an elastic half plane loadedby a horizontal force acting at its free surface. He evaluated themagnitude of the axial force in the cover by solving numericallya singular integral equation. Arutiunian (1968) solved the problemadopting a series representation for the unknown interfacial stress,solving the integral equation by means of contour integrals in com-plex domain. Later, a simpler method was proposed by Morar andPopov (1971) and Erdogan and Gupta (1971), who solved the prob-lem by using series of Chebyshev polynomials to approximate theinterfacial shear stress.

Then, various methods have been proposed to evaluate thestress and strain fields in single or multi-coated systems. For exam-ple, Hu (1979) adopted a finite difference technique to evaluate thestress transferred to the surface of an elastic half plane by a bondedfilm. A proper refinement of the grid elements was used by theauthor to correctly estimate the interfacial stress near the filmedge. Conventional finite element (FE) programs were used byDjabella and Arnell (1993) to evaluate the contact stress in multi-layered systems subjected to a specified pressure distribution act-ing on a portion of the boundary. Jain et al. (1995) investigated thestress field in an elastic layer induced by a bonded strip-like filmvia FE models, founding the distribution of the shear and normalstresses in the substrate and the axial stress in the overlying film.A numerical analysis based on a proper boundary element (BE)method has been used by Takahashi and Shibuya (1997) to inves-tigate the singular nature of the interfacial stress in film-substratesystems subjected to thermoelastic strain. The authors analyzedthe interfacial stress distribution of a coating made of differentsuperconducting ceramics bonded to a ceramic or metallicsubstrate.

By means of the elementary theory of bent plates, Zhang et al.(2005) evaluated the stress distribution in multilayered systemsunder thermal strain, comparing the results to the numerical solu-tion provided by FE simulations. However, the authors recognizedthat their model can not be used to estimate the strength of inter-facial stress singularities properly.

Here, some analytical models based on the use of orthogonalChebyshev polynomials are proposed to study the contact problemof thin structures bonded to a 2D elastic substrate under severalloading conditions. Perfect adhesion between the bonded structureand the underlying half plane is supposed. This condition leads to asingular integral equation which can be reduced to an infinite sys-tem of linear algebraic equations. The solution of the algebraic sys-tem allows the evaluation of the displacement and stress fields ofthe system, with particular reference to the strength of stress sin-gularities in the neighborhood of the points of load application andat the edges of the coating. The mechanical interaction betweenthe bonded structure and the substrate is governed by a rigidityparameter involving the axial stiffness of the cover and themechanical parameters of the underlying substrate.

a b

Fig. 2.1. Film bonded to a half plane and subjected to (a) u

The paper is organized as follows. The problem of a coatingbonded to the substrate and subjected to thermal load or, similarly,two opposite axial forces acting at the coating ends is considered inSection 2. The film-substrate interaction is discussed varying therigidity parameter of the system. The comparison with the Koitersolution in terms of strength of stress singularity is given also. Acoating with variable thickness under thermal load is solved inSection 3. Interfacial stress, axial displacement and axial force inthe coating are reported for a given geometric configuration. Thecontact problem of a coating subjected to a single axial force ap-plied at one edge is solved and discussed in Section 4 for severalvalues of the rigidity parameter, founding a large agreement withrespect to the strength of stress singularities predicted by theKoiter solution. Section 5 deals with the study of a film partiallydetached from the underlying half space. For a certain location ofthe interior detachment, a comparison with the results obtainedby solving the contact problem of a perfectly bonded film is re-ported. The interfacial stress singularities in the neighborhood ofthe ends of the contact region have been investigated. Finally,the problem of a coating subjected to an interior axial force is trea-ted in Section 6. A comparison between the obtained results andthe Melan solution is given and discussed.

2. Film subjected to two opposite axial forces or a uniformthermal variation

In the present section, the problem of an elastic film of totallength 2a and thickness d welded to an elastic half-plane and sub-jected to a uniform thermal variation �DT or, similarly, loaded bytwo opposite axial forces F applied at its ends, as reported inFig. 2.1, is considered. A Cartesian coordinate system (O,x,y) placedat the interface between the film and the underlying half plane isadopted, with the origin located at the midpoint of the film. Linearelastic and isotropic behavior is assumed for both the half planeand coating. Generalized plane stress or plane strain conditioncan be considered; nevertheless, in the present study, a planestrain regime is assumed. In that follows, E0, m0 and a0 representthe Young modulus, the Poisson ratio and the coefficient of thermalexpansion of the film, respectively, whereas E and m are the Youngmodulus and Poisson ratio of the half plane.

The contact region [�a,a] equals the length of the film. Thethickness d of the coating is assumed very thin, making it possibleto neglect its flexural stiffness and, in turn, to ignore the effectsproduced by the vertical component of stress. Thus, only interfacialshear stress s(x) arises across the contact region.

The assumption of perfect adhesion between the coating andthe half space leads to the following singular integral equationwith Cauchy kernel for the unknown interfacial shear stress:

1� m20

E0d

Z a

xsðnÞdnþ

2 1� m2� �

Ep

Z a

�a

sðnÞn� x

dn ¼ De; for jxj

6 a; ð2:1Þ

where De = (1 + m0) a0DT or De ¼ �Fð1� m20Þ=ðE0 dÞ represents the

constant component of horizontal strain of the coating due to

niform thermal load or (b) two opposite axial forces.

L. Lanzoni / International Journal of Solids and Structures 48 (2011) 1915–1926 1917

thermal load or two opposite axial forces, respectively, and theGreen function (Cerruti solution) for a homogeneous half spaceloaded by a tangential force acting on its free surface has been used(Kachanov et al., 2003).

In Arutiunian (1968), Eq. (2.1) is solved through a techniquebased upon contour integrals in complex domain, as reported inMuskhelishvili (1953), by assuming the unknown function s(n) inpower series.

In the following, a simpler approach is adopted by approximat-ing the contact shear stress through series of orthogonal Cheby-shev polynomials displaying a square root singularity at the filmedges. The same procedure was recently proposed by Villaggio(2003) to investigate the brittle detachment of a stiffener bondedto an elastic half plane, founding the critical load that causes thecrack propagation by adopting the Griffith criterion for fracture.

Indeed, the use of Chebyshev polynomials in this kind of prob-lems leads to some important advantages. In fact, the exact solu-tion of Eq. (2.1) is not known in closed form, and the use ofChebyshev polynomials appears the most efficient method to solveEq. (2.1) since its high rate of convergence. In addition, the singu-larity of the unknown function is assumed explicitly; thus, the sin-gular nature of the shear stress is preserved independently to thenumber of terms used in the series expansions. Therefore, theChebyshev polynomials allow to remove the singularity in the inte-gral equation, obtaining an algebraic system which can be rapidlysolved without any supplementary manipulation. The regularityconditions of the algebraic system and the rate of convergence ofthe method are known (e.g. Erdogan, 1969).

Thus, the following expression for the interfacial shear stress isadopted:

sðxÞ ¼ Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðx=aÞ2

q X1n¼1;3;5

CnTnðx=aÞ; for jxj 6 a; ð2:2Þ

Tn(x/a) being the orthogonal Chebyshev polynomials of first kind oforder n.

Then, by introducing expression (2.2) in the Eq. (2.1), and apply-ing the Bubnov–Galerkin method, the following infinite system ofalgebraic equations for the unknown coefficients Cn is found:

X1n¼1;3;

Cn4p

1� m2� �Bnm þ

cm

dnm

� �¼ 2De

pAm; m ¼ 1;3;5; . . . ;1;

ð2:3Þ

dnm being the Kronecker delta, the expressions of coefficients Bnm

and Am have been reported in Appendix (see Eqs. (A1) and (A2),

a b

Fig. 2.2. Film subjected to thermal variation �DT: dimensionless (a) interfacial shear strefor negative x coordinate.

respectively), and c ¼ a Eð1�m20Þ

E0d . The dimensionless parameter c canbe interpreted as the compliance of the coating with respect thatof the half plane and, as showed in the following, it governs themechanical response of the film-substrate system.

The series in Eq. (2.3) is truncated to N terms, ad the system issolved finding the coefficients Cn (n = 1,3, . . . ,N). The surface dis-placements for the half plane can be evaluated by superposition,integrating the Green functions for a distribution of tangentialforces:

uðtÞ ¼ �2a 1� m2� � XN

n¼1;3;

Cn

nTnðtÞ;

vðtÞ ¼ að1� 2mÞð1þ mÞXN

n¼1;3;

Cn

nUn�1ðtÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� t2

p; ð2:4Þ

where relation (A3) has been used (see Appendix).The longitudinal stress in the coating r0(x) is obtained by inte-

grating the contact shear stress in the region [x,a]. The boundarycondition of vanishing tractions at the half plane surface alongthe y direction requires ryy(x,0) = 0. By using the constitutive rela-tions it follows that the horizontal stress on the half plane surfacerxx(x,0) can be expressed as a function of the horizontal compo-nent of the elastic strain only, namely rxxðx;0Þ ¼ ½E=ð1� m2Þ�ee

0ðxÞ.So (jtj 6 1):

r0ðtÞ ¼ E0cXN

n¼1;3

Cn

nUn�1ðtÞ


p; rxxðt;0Þ ¼ �2E

XN

n¼1;3

CnUn�1ðtÞ:

ð2:5ÞAs predicted by Eq. (2.2), the shear stress exhibits a square root sin-gularity at the edges of the film. The singular behavior of the inter-facial stress at the film ends (x = ±a) can be properly investigated bymeans of the shear stress singularity factor KII:

K IIð�aÞ ¼ limx!�a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pða� xÞ

psðxÞ ¼ �E

ffiffiffiffiffiffiapp XN

n¼1;3

Cn: ð2:6Þ

2.1. Results

In the present section, the mechanical response of a coating-substrate system subjected to a thermal load �DT is discussed. Anumber of terms equals to 40 has been considered in the seriesexpansion, leading to an accurate evaluation of the mechanicalresponse of the system. The dimensionless shear stress s(x)/(EDe)and axial displacement u(x)/(a De) are shown in Fig. 2.2(a) and(b), respectively, for different values of the parameter c.

ss and (b) axial displacement of the coating for different values of c. Symmetry holds

Fig. 2.4. Film subjected to a thermal variation �DT: normalized shear stresssingularity factors KII in the neighborhood of the film ends varying c andcomparison with K�II factors provided by the Koiter solution.


As c increases, both interfacial shear stress and horizontal dis-placement (in modulus) decrease, due to the diminishing of thecoating stiffness and, in turn, the effects transmitted to the halfplane. In fact, c is proportional to the ratio E/E0, and it can be inter-preted as a measure of compliance of the cover with respect to thatof the half space. Moreover, for high values of c, the effects of thecoating are significant at the edges of the coating only; conversely,for small values of c, the contact shear stress and the axial dis-placement become considerable near the middle of the cover also.It should be noted that the relative displacement among the edgesof the coating depends on coefficient C1 only, namely:

uðaÞ � uð�aÞ½ �=a ¼ cpC1=2� 2De: ð2:7Þ

Moreover, it is worth noting that in the limiting case of c ? 0(very stiff cover), the horizontal displacement tends to display analmost linear trend, and the coating behaves like a truss subjectedto thermal load only, since in this case the coating is much stifferthan the substrate which has no sensible constraint effects onthe film. This is confirmed by the asymptotic value of the unknowncoefficients of system (2.3):

C1 ¼ De=2; and C3;C5; . . . ;CN ¼ 0; for c! 0: ð2:8Þ

Thus, from Eq. (2.4)1, the total axial displacement of the coatingu(a) � u(�a) turns out to be 2a(1 � m2)De, as expected for a rigidfilm subjected to an axial uniform strain De, whereas the interfacialshear stress can be approximate as:

sðxÞ= EDeð Þ ¼ 1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðx=aÞ2

q ; for c! 0: ð2:9Þ

This behavior is reflected in Fig. 2.3(a)–(b) also, where the dimen-sionless normal stress in the coating r0(t)/(E0De) and the half planesurface rxx(t,0)/(EDe) are reported. It should be noted that, differ-ently to the shear stress, the horizontal component of normal stressrxx(x, 0) takes finite values in each point of the contact region.

The normalized shear stress singularity factors KII(a)/(E De a0.5)are reported in Fig. 2.4(a) varying the parameter c. As c increases,the normalized KII factors decrease due to the fact that the magni-tude of the shear stress decreases, as showed by Fig. 2.2(a). Con-versely, for stiff coatings, the KII factor increases, and for thelimiting case of an inextensible film one finds:

K IIð�aÞ ¼ �Effiffiffiffiffiffiapp De

2; for c! 0; ð2:10Þ

or, in dimensionless form, KII(±a)/(E a0.5De) = p0.5/2 ffi 0.886227.

a

Fig. 2.3. Film subjected to thermal variation �DT: dimensionless normal stress (a) in thenegative x coordinate.

As reported previously, the loading case of a thermal variation issimilar to that of two opposite axial forces applied at the film ends.This load condition resembles the situation of the problem of Koit-er, who solved the contact problem of a semi-infinite rod bondedto a semi-infinite plate and loaded at the edge by an axial forceF, founding the following expression for the axial stress N(x) inthe coating (Grigolyuk and Tolkachev, 1987, p. 150):

NðxÞF¼ 1�

ffiffiffiffiffiffiffiffi2cxp

r1� cx 0:25425� 0:1061 LogðcxÞð Þð

"

þ c2x2 0:008627� � � �ð Þ�#; ð2:11Þ

where x denotes the longitudinal coordinate of the coating startingform the point of load application and the width of the rod-substrate system is taken unitary. By differentiating Eq. (2.11) withrespect x, the expression for the shear stress is obtained, fromwhich, accordingly to Eq. (2.6), one finds:

K�IIð�aÞ ¼ E Deffiffiffiffiffiffiffiffia=c

p= 1� m2

0

� �: ð2:12Þ

The strength of interfacial stress predicted by Koiter, in dimen-sionless form, i.e. K�IIð�aÞ=ðEDea0:5=ð1� m2ÞÞ, is reported in

b

coating and (b) at the half plane surface for different values of c. Symmetry holds for


Fig. 2.4(a). As predicted by Eq. (2.12), the singularity strength ofthe interfacial stress is monotonic decreasing with c. As shown,a large agreement between the stress singularity factors KII(±a)and K�IIð�aÞ is found, confirming the square root singular behaviorof the Koiter solution. Thus, the Koiter solution can be used as avalid approximation to assess interfacial stress concentrations incoating-substrate systems provided that, approximately, c > 10.Moreover, it is worth noting that, differently to the stress fieldin the neighborhood of an interfacial crack tip predicted by linearelastic fracture mechanics, here no oscillations of the shear andnormal stresses are found.

3. Film with variable thickness under thermal load

In the present section, a coating of length 2a having variablethickness and bonded to a homogeneous isotropic half plane, as re-ported in Fig. 3.1, is considered. It is assumed that a uniform ther-mal variation �DT acts on the system; however, coatings subjectedto other loading conditions and having more general thicknessvariations can be easily treated through the same approach.

By following the method reported in the previous section, thecompatibility condition between the strain of the half plane sur-face and that of the bonded coating leads to the following algebraicsystem:

X1n¼1;2;

Cn 2 1�m2� �

Bnmþcl

nDl nmþ

cr

nDr nm

n o¼DeAm; m¼ 1;2;3; . . . ;

ð3:1Þ

where cl ¼Eað1�m2

0ÞE0dl

; cr ¼Eað1�m2

0ÞE0dr

; b ¼ al�aralþar

and expressions of Dlnm,Dr mn have been reported in equations (A4) of the Appendix. Notethat, differently from Eq. (2.3), expression (3.1) contains even terms

Fig. 3.1. Coating having variable thickness under thermal load.

a b

Fig. 3.2. Coating having variable thickness subjected to thermal load �DT: dimensionlessvarying cr.

also, due to the fact that symmetry with respect the y axis is lost be-cause the variation of coating thickness.

The algebraic system (3.1) is solved for the unknown coeffi-cients Cn, leading to the evaluation of the interfacial stress and,in turn, the stress and displacement fields of the system.

The horizontal components of displacement and stress of thecoating and the half plane surface can be evaluated through Eqs.(2.4), (2.5), provided that also the even terms are included in theseries.

Similarly to expression (2.6), the KII factors are evaluated asfollows:

K IIðaÞ ¼ Effiffiffiffiffiffiapp XN

n¼1;2;

Cn; K IIð�aÞ ¼ Effiffiffiffiffiffiapp XN

n¼1;2;

ð�1ÞnCn: ð3:2Þ

It should be noted that, for b = ±1, the solution of the contact prob-lem for a coating of uniform thickness with c = cl or c = cr, respec-tively, is exactly retrieved.

3.1. Results

The results reported in the present section are related to acoating having variable thickness for which b = 0 and cl = 10.The dimensionless shear stress and the axial displacement are re-ported in Fig. 3.2(a)–(b), respectively, for several values of cr. Asreported in the figure, the shear stress drops in correspondenceof thickness variation, except for cr = 10, since that for this valueof cr a coating with uniform thickness is retrieved. The jumpexhibited by the shear stress is due to the fact that the horizontalstrain and, in turn, the normal stress in the film is discontinuousin the point of thickness variation, as showed by Fig. 3.3(a). Thepeak of the shear stress assumes finite values, and its magnitudeincreases (in modulus) increasing the thickness variation of thefilm, as showed by Fig. 3.4. As expected, for large values of cr,the axial displacement is nearly linear in the stiffer part of thecoating, whereas it assumes a non monotonic trend in the leftside of the system, as showed in Fig. 3.2(b). Note that the pointof zero displacement moves toward the midpoint of the stifferpart of the film. The KII factors at the edges of the film are re-ported in Fig. 3.3(b) varying cr. Similarly to the case of a coatinghaving uniform thickness, a monotonic decreasing trend of theKII(±a) factors with cr is found. As expected, the KII factors atthe left edge of the film are not sensibly affected by variationsof cr; in particular, for cr > 20 an almost constant value of KII(�a)is obtained. Note that KII(a) = KII(�a) for cr = 10.

(a) interfacial shear stress and (b) axial displacement of the coating for b = 0, cl = 10

a b

Fig. 3.3. Coating having variable thickness subjected to thermal load �DT: dimensionless (a) normal stress in the coating for some values of cr and (b) KII factors at the filmedges for b = 0, cl = 10 varying cr.

Fig. 3.4. Dimensionless shear stress in the coating across the thickness variation(x = 0) for b = 0 and cl = 10 varying cr.


4. Film subjected to an axial force applied to a film end

In the present section, the contact problem of an elastic filmbonded to a half-plane and loaded by an axial force F applied atone edge, as reported in Fig. 4.1, is considered.

The interfacial shear stress is represented in the form:

sðxÞ ¼ Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðx=aÞ2

q X1n¼0;1;2;

CnTnðx=aÞ; for jxj 6 a: ð4:1Þ

The equilibrium condition of the coating along the axial directioncan be written as:Z a

�asðxÞ dx ¼ F; ð4:2Þ

Fig. 4.1. Film bonded to a half plane and subjected to an axial force at one end.

leading to the evaluation of coefficient C0:

C0 ¼F

Eap¼ F

cpE0d: ð4:3Þ

The compatibility condition among the horizontal strain of the halfplane and that of the coating inside the contact region jxj 6 a leadsto the following algebraic system:

XN

n¼1;2;

Cn 2 1�m2� �Bnmþ

cm

p2

dnm

n o¼� p

mð�1Þmþ1

; m¼ 1;2;3; . . . ;N;

ð4:4Þ

where the series has been truncated to the Nth term.The horizontal and vertical displacements of the half plane sur-

face take the form:

uðtÞ ¼ �a2ð1� m2Þ C0 Logð2Þ þXN

n¼1;2;

Cn

nTnðtÞ

" #; for jtj 6 1;

ð4:5Þ

vðtÞ ¼ að1� 2mÞð1þ mÞ C0 arcosðtÞ þXN

n¼1;2;

Cn

nUn�1ðtÞ


p" #;

for jtj 6 1; ð4:6Þ

whereas the horizontal component of stress in the coating is evalu-ated through the following expression (jtj 6 1):

r0ðtÞ ¼ E0c C0 arcosðtÞ þXN

n¼1;2;

Cn

nUn�1ðtÞ


p" #; ð4:7Þ

and the horizontal stress in the half plane surface is given byexpression (2.5)2 with n = 1,2,3,. . .,N.

The KII factors are evaluated through expression (2.6), wherethe series starts from n = 0, taking into account the coefficient C0

also.For the limiting case of very stiff coatings one finds:

C1;C2; . . . ;CN;! 0; for c! 0: ð4:8Þ

It follows that:

K IIð�aÞ ¼ �Effiffiffiffiffiffiapp

C0; for c! 0; ð4:9Þ

or, in dimensionless form, KII(±a)/(F/a0.5) = p�0.5 ffi 0.564190.


4.1. Results

In the present section, some results related to the contact prob-lem of a coating subjected to an axial force applied at its left edgeare reported. The interfacial shear stress is showed in Fig. 4.2(a) indimensionless form, namely s(x)/(F/a), for different values of c. Forvery stiff coatings, the shear tractions are reasonably approximatedby the symmetric law:

sðxÞ=ðF=aÞ ¼ 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðx=aÞ2

q ; for c! 0: ð4:10Þ

It is worth noting that, in the neighborhood of the point of loadapplication, the shear stress increases as c increases. Conversely, atthe opposite edge of the coating, the shear stress increases as c de-creases. This behavior is reflected in Fig. 4.3(b), where the KII fac-tors at the film edges are reported varying c. As shown in thefigure, the strength of interfacial stress singularity at the loadededge is monotonic increasing with c, and its trend largely agreeswith that predicted by Koiter also for small values of c. Note that,in order to plot the K�II factors provided by the Koiter solution, aproper normalization of the stress singularity factors has beenadopted in Fig. 4.3(b) (cf. Tullini et al., submitted for publication):

K�IIð�aÞ=ðF=ffiffiffiapÞ ¼ ffiffiffi

cp

: ð4:11Þ

a b

Fig. 4.2. Dimensionless (a) interfacial shear stress and (b) ax

a

Fig. 4.3. Dimensionless (a) normal stress in the coating for differe

As shown, the free edge of the film is characterized by KII factorsmonotonic decreasing with c. The dimensionless axial displace-ment, i.e. u(x)/(aF/(E0d)), is reported in Fig. 4.2(b) and it is quitesimilar to that of a coating subjected to thermal load (see Sec-tion 2.1). The axial displacement between the edges of the coatingcan be evaluated as:

uðaÞ � uð�aÞ½ � ¼ acp C0 � 0:5C1ð Þ: ð4:12Þ

Thus, it follows that [u(a) � u(�a)]/a tends to unity for c ? 0, asfound for a bar axially loaded by an end force of magnitude F andsubjected to an uniform distribution of tractions along its lengthequilibrating the force F.

The dimensionless normal stress in the coating r0(x)/(F/d) is re-ported in Fig. 4.3(a). As expected, an almost linear trend is foundfor a rigid film. Conversely, as c increases, the stress tends to local-ize near the point of load application, and it rapidly diminishesgoing toward the free edge of the film.

5. Detached film under thermal load

An elastic thin film having total length 2a and constant thick-ness d, subjected to a uniform thermal variation �DT (or, similarly,two opposite axial forces, see Section 2) and partially detached

ial displacement of the coating for different values of c.

b

nt values of c and (b) KII factors at the film edges varying c.


from the underlying half plane, is studied in the present section(see Fig. 5.1).

The system can be analyzed by considering two separate coat-ings, named ‘‘l’’ and ‘‘r’’ in Fig. 5.2, having length 2al and 2ar,respectively, and subjected to thermal load and an unknown axialforce F, acting at its ends, transmitted by the detached portion ofthe coating (cf. Wang and Meguid, 2000). Two unknown distribu-tions of interfacial shear stress, sl(n) and sr(n), occur across the con-tact region. The y axis of the reference system is placed at themidpoint of the internal debonding whose length equals 2ad.

The condition of perfect adhesion between the coatings and thehalf plane, together with the compatibility condition of the longi-tudinal displacement of the coating and the half plane in the de-tached region leads to the following equations:

u;xðxÞ ¼ u0l;xðxÞ; for � 2al � ad 6 x 6 �ad; ð5:1aÞu;xðxÞ ¼ u0r;xðxÞ; for ad 6 x < ad þ 2ar; ð5:1bÞuðadÞ � uð�adÞ ¼ u0rðadÞ � u0lð�adÞ; ð5:1cÞ

where u,x(x) denotes the axial strain of the half plain, u0l,x(x) andu0r,x(x) represent the axial strain of the coatings ‘‘l’’ and ‘‘r’’, u0r(ad)and u0l(�ad) are the horizontal displacements at the right and leftend of the coatings and u(ad) � u(�ad) is the relative displacementof the corresponding points of the half plane.

As usual, the interfacial stresses are assumed as:

slðxÞ ¼Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� xþalþadal

� �2r X1

n¼0;1;2;

ClnTnxþ al þ ad

al

;

for � 2al � ad 6 x 6 �ad; ð5:2aÞ

srðxÞ ¼Effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� x�ad�arar

� �2r X1

n¼0;1;2;

CrnTnx� ad � ar

ar

;

for ad 6 x < ad þ 2ar: ð5:2bÞ

The equilibrium condition of the two bonded portion of the coatingalong the x axis leads to the determination of the coefficients Cl0 andCr0:

Fig. 5.1. Film partially detached from the underlying half plane and subjected tothermal load �DT.

Fig. 5.2. Sketch for studying the contact problem of a partially detached coatingunder thermal load �DT.

Cl0 ¼F

Ealp; Cr0 ¼

FEarp

: ð5:3Þ

By introducing expressions Eq. (5.2) in Eqs. (5.1a)–(5.1c), the fol-lowing equations are found:

X1j¼1;2;

Clj 2 1� m2� �þcl

j

ffiffiffiffiffiffiffiffiffiffiffiffi1� t2

l

q� �Uj�1ðtlÞþ2 1� m2� �

X1

j¼0;1;

Crj

tr þffiffiffiffiffiffiffiffiffiffiffiffit2

r �1q j

ffiffiffiffiffiffiffiffiffiffiffiffit2

r �1q ¼ clCl0 p�arcosðtlÞð Þ�De; for jtlj6 1;

ð5:4aÞ

�X1

j¼1;2;

Crj 2 1� m2� �þ cr

j


r

q� �Uj�1 trð Þ � 2 1� m2� �

X1

j¼0;1;

Clj

tl �ffiffiffiffiffiffiffiffiffiffiffiffiffit2

l � 1q j

ffiffiffiffiffiffiffiffiffiffiffiffiffit2

l � 1q ¼ crCr0 arcos trð Þ � De; for jtrj 6 1;

ð5:4bÞ

2 1� m2� �Cl0Al0 þ

X1j¼1;2;

CljAlj � Cr0Ar0 �X1

j¼1;2;

CrjArj

!¼

¼ 2 ad 1� m20

� �F= E0dð Þ � 2 ad De; for jtlj 6 1; ð5:4cÞ

where the result (A5) has been used, and tl ¼ xþalþadal

; tr ¼x�ad�ar

ar; cl ¼

Ealð1�m20Þ

E0d ; cr ¼Ear ð1�m2

0ÞE0d . The coefficients Al0, Alj, Ar0 and

Arj are reported in expressions (A6).If the series expansions are truncated to the Nth term and Eqs.

(5.4a), (5.4b) are imposed in N collocation points inside the regions�2al � ad 6 x 6 �ad and ad 6 x < ad + 2ar, respectively, an algebraicsystem of 2N + 1 equations is obtained, which can be solved for theunknown coefficients Clj, Crj (j = 1,2, . . . ,N) and the force F acting atthe ends of the detached portion of the coating. As known, the opti-mal collocation points tlk, trk are the roots of the Chebyshev polyno-mial of first kind of order N, namely:

tlk; trk ¼ cosð2k� 1Þ

2Np

� �; for k ¼ 1;2; . . . ;N: ð5:5Þ

5.1. Results

Fig. 5.3(a)–(b) show the dimensionless interfacial stress s(x)/(EDe) and the axial displacement u(x)/(a De) of a coating havingc = 20, detached between 0.3a and 0.4a, i.e. for al = 1.6 a andar = ad = 0.2 a. The interfacial stress is similar to that of a perfectlybonded film, except in the neighborhood of the detached zone,where it is unbounded. This similarity occurs also for the axial dis-placement, as showed in Fig. 5.3(b). It should be noted that a smalldifference with respect to a perfectly bonded film occurs in corre-spondence of the detached region, where the axial displacement ofthe detached coating exhibits a linear trend. The axial displace-ment of the coating exhibits a continuous slope, since the axialforce and the axial strain are continuous despite the presence ofa detached zone. The dimensionless axial stress r0(x)/(E0De) is al-most coincident with that of a bonded film in correspondence ofthe bonded region, as reported in Fig. 5.4(a), whereas it equalsthe constant value F/d in the detached portion of the coating. Thedimensionless stress singularity factors KII/(E

paDe) at the edges

of the film and at the ends of the detached region are reported inFig. 5.4(b) varying c. Similarly to the case of a perfectly bonded filmloaded by thermal variation, the KII factors at the film edges aremonotonic decreasing with c, and are well approximated by the

a b

Fig. 5.3. Detached film subjected to thermal load �DT: dimensionless (a) interfacial shear stress and (b) axial displacement of the coating for c = 20.

a b

Fig. 5.4. Detached film loaded subjected to thermal load �DT: dimensionless (a) normal stress in the coating for c = 20 and (b) KII factors at the film edges and at the ends ofthe detached region varying c.

Fig. 6.1. Film bonded to a half plane and subjected to an internal axial force.


Koiter solution. The KII factors in the neighborhood of the film ends,namely x = �2al � ad and x = ad + 2ar are almost coincident. At theends of the internal debonding the dimensionless KII factors as-sume lower values, and for x = ad the strength of interfacial stresssingularity is greater than that evaluated at x = �ad. This is dueto the fact that the portion of the coating at the right side of the de-tached region is stiffer than the other portion, thus exhibiting lar-ger values of KII, according to the trend of KII factors showed inFig. 2.4 for a perfectly bonded film.

Fig. 6.2. Sketch for studying the contact problem of a film bonded to a half planeand subjected to an interior axial force.

6. Film subjected to an interior axial force

In the present section, a coating subjected to an interior axialforce is considered, as reported in Fig. 6.1.

Similarly to the procedure used to study a partially detachedfilm, the coating can be divided in two portions having length 2al

and 2ar, respectively, subjected to the axial forces F1 and F � F1 ap-plied at its ends, as shown in Fig. 6.2.

Instead of condition (5.1c), here the adjunctive compatibilitycondition consists in imposing the same relative displacements


at the limits of the contact region between the half plane and thefilm, namely:

uð2arÞ � uð�2alÞ ¼ u0lð0Þ � u0lð�2alÞ þ u0rð2arÞ � u0rð0Þ; ð6:1Þ

with obvious significance of symbols u(x), u0l(x), u0r(x).The procedure is analogous to the one reported in the previous

section to solve the contact problem of a partially detached film;for this reason, it has been omitted here.

In the neighborhood of the point of load application the ob-tained results have been compared to the Melan solution, here re-ported for unitary width of the film-substrate system (Grigolyukand Tolkachev, 1987, p. 142):

NðxÞF¼ 1

psi

cx2

� �cos

cx2

� �� ci

cx2

� �sin

cx2

� �h i; ð6:2Þ

sðxÞF¼ � c

2ci

cx2

� �cos

cx2

� �þ si

cx2

� �sin

cx2

� �h i; ð6:3Þ

where N(x) and s(x) denote the axial force in the film and the inter-facial shear stress, and ci(x) and si(x) are the sine and cosine integralfunctions, respectively.

a b

Fig. 6.3. Film subjected to an internal axial force at the midpoint: dimensionless (a) intethe coating for some values of c. Symmetry holds for negative x coordinate.

a b

Fig. 6.4. (a) Film subjected to an internal axial force at the midpoint: dimensionless axialholds for negative x coordinate), (b) KII factors at x = a of a film loaded by an axial force

6.1. Results

The results reported in the present section refer to a coating axi-ally loaded at the midpoint, i.e. for al = ar = 0.5 a. The interfacialshear traction s(x)/(F/a) and the axial displacement u(x)/(aF/(E0d))of the film are reported in dimensionless form in Fig. 6.3(a)–(b)for c = 1, 10, 100. As expected, for low values of c, the Melan solu-tion agrees with the obtained results in the region close to thepoint of load application, whereas for c P 100 the Melan solutionpractically holds for all points of the coating.

As depicted in Fig. 6.3(a), as c increases, the shear stress tends toconcentrate in the neighborhood of the point of load application,whereas it tends to diminish at the edges of the film, similarly tothe case of a coating subjected to an axial force acting at one end(see Fig. 4.2(a)). Moreover, for a very rigid film, an almost bilineartrend is assumed by the axial displacement, which decreasesincreasing c. The dimensionless axial stress in the film r(x)/(F/d)is reported in Fig. 6.4(a) and, similarly to Fig. 4.3(a), it rapidly van-ishes going toward the film edges. The KII factors in the neighbor-hood of the film edge (x = a) are reported in Fig. 6.4(b) varying c fortwo load conditions: one curve refers to an axial force applied atthe midpoint (x = 0), whereas the other is related to an axial force

rfacial shear stress and comparison to the Melan solution, (b) axial displacement of

force in the film compared with the Melan solution for some values of c (symmetryacting at x = 0 and x = �a varying c.


acting at the edge (x = �a). Accordingly to Eq. (4.9), both curvesstart from 0.564190. As expected, the curve related to the pointof load application x = �a stays below the other, confirming thatan axial force acting at the midpoint of the film produces larger ef-fects at the film edge than the same force applied to the oppositeedge of the coating. As expected, this difference diminishesincreasing c, as showed by Fig. 6.4(b), and for large values of c, al-most the same KII(a) factors are expected for these two loadingconditions.

7. Conclusion

In the present paper, the contact problem of thin coating struc-tures bonded to a homogeneous elastic half plane has been consid-ered. The singular interfacial stress arising across the contactregion has been approximated via series expansion of orthogonalChebyshev polynomials. This allows to reduce the compatibilitycondition of strains among the coating structure and the underly-ing substrate to a linear algebraic system. The rigidity parameter cinvolving the geometric dimension of the coating and mechanicalparameters of the half plane and the bonded structure governsthe film-substrate mechanical interaction. The contact problemof thin structures having several geometric configurations underdifferent loading conditions has been examined. A coating sub-jected to a uniform thermal variation has been studied first, found-ing analogies with the load case of two opposite axial forces actingat the film ends. Then, a coating having non uniform thickness sub-jected to thermal load has been considered. The load case of a hor-izontal force applied at one edge of the film has been studied also.The obtained results in terms of stress singularity factors havebeen compared with those predicted by the Koiter solution, found-ing a large agreement between them. In particular, the Koiter solu-tion appears adequate to predict stress concentrations in theneighborhood of the film edges approximately for c > 10. A coatingpartially detached from the half plane under thermal load has beenconsidered, founding at the coating edges almost the same valuesof stress singularities of a bonded coating. Conversely, in the neigh-borhood of the limits of the detached region, lower values of thestress intensity factors have been found. The contact problem ofa coating loaded by an internal axial force has been solved also,and the obtained results have been compared with those predictedby the Melan solution. In particular, for this load condition, theMelan solution is retrieved for almost the entire length of the coat-ing provided that the parameter c is large enough to resemble aninfinite strip, namely for c P 100. Actually, the analytical resultsagree well with respect to the Melan solution also for lower valuesof c in the neighborhood of the point of load application.

Appendix A

Some formulas and results used in the main text are reported inthe following.

The coefficients Bnm and Am in Eq. (2.3) assume the form:

Bnm ¼ Bmn ¼Xminfn;mg

k¼1;2;

1þ cos½ðn�mÞp�n�mþ 2k� 1

; for m ¼ 1;3;5; . . . ;1;

ðA1Þ

Am ¼1� cosðmpÞ

m; for m ¼ 1;3;5; . . . ;1: ðA2Þ

The expression (2.4)1 for the axial displacement has been obtainedby using the result:Z 1

�1

TnðtÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� t2p ln jx� tj dt ¼

�p lnð2Þ; for n ¼ 0;� p

n TnðtÞ; for n > 0:

(ðA3Þ

The expressions of coefficients Dl nm, Dr nm used in Eq. (3.1) are:

Dl nm¼Dl mn¼Z b

�1sinðn#Þsinðm#Þd#; Dr mn¼

�Dl mn; if n–m;p2�Dl mn; if n¼m:

(:

ðA4Þ

Eq. (5.4) have been obtained by using the following identities:

Z 1

�1

TnðtÞðt�xÞ

ffiffiffiffiffiffiffiffiffiffiffiffi1�t2p dt¼

0; for n¼0; jxj<1;pUn�1ðxÞ; for n–0; jxj<1;

�p x�signðxÞffiffiffiffiffiffiffiffiffiffiffiffix2�1p� �n

= signðxÞffiffiffiffiffiffiffiffiffiffiffiffix2�1p� �

;

for nP0; jxj>1:

8>>>><>>>>:

ðA5Þ

The coefficients Al0, Alj, Ar0 and Arj used in Eq. (5.4c) are:

Al0 ¼ al LogðalÞ � 2 Logffiffiffiffiffiadpþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiad þ alp� ��

;

Alj ¼ al �1þ 2ad þ al

al�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ 2ad þ al

al

2s0

@1A

j264

375,

j;

Ar0 ¼ �ar LogðarÞ � 2 Logffiffiffiffiffiadpþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiad þ arp� ��

;

Arj ¼ ar ð�1Þj � �2ad þ ar

arþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ 2ad þ ar

ar

2s0

@1A

j264

375,

j:

ðA6Þ

References

Arutiunian, N.Kh., 1968. Contact problem for a half-plane with elasticreinforcement. J. Appl. Math. Mech. (PMM) 32 (4), 632–646.

Benscoter, S., 1949. Analysis of a single stiffener on an infinite sheet. ASME J. Appl.Mech. 16, 242–246.

Brown, E.H., 1957. The diffusion of load from a stiffener into an infinite elastic sheet.Proc. R. Soc. Lond. A 239, 296–310.

Buell, E.L., 1948. On the distribution of plane stress in a semi-infinite plate withpartially stiffened edge. J. Math. Phys. 26 (4), 223–233.

Djabella, H., Arnell, R.D., 1993. Finite element comparative study of elastic stressesin single, double layer and multilayered coated systems. Thin Solid Films 235(1–2), 156–162.

Erdogan, F., 1969. Approximate solutions of systems of singular integral equations.SIAM J. Appl. Math. 17 (6), 1041–1059.

Erdogan, F., Gupta, G.D., 1971. The problem of an elastic stiffener bonded to a halfplane. ASME J. Appl. Mech. 38 (4), 937–941.

Freund, L.B., 2000. Substrate curvature due to thin film mismatch strain in thenonlinear deformation range. J. Mech. Phys. Solids 48, 1159–1174.

Freund, L.B., Suresh, S., 2003. Thin film materials. Stress, Defect formation andSurface evolution. Cambridge University Press.

Gevorgian, S., 2009. Ferroelectrics in Microwave Devices, Circuits and Systems.Physics, Modelling, Fabrication and Measurements. Springer, London.

Grigolyuk, EI., Tolkachev, V.M., 1987. Contact Problems in the Theory of Plates andShells. Mir Publishers, Moscow.

Hsueh, C.H., 2002. Thermal stresses in elastic multilayer systems. Thin Solid Films418, 182–188.

Hu, S.M., 1979. Film-edge-induced stress in substrates. J. Appl. Phys. 50 (7), 4661–4666.

Jain, S.C., Harker, A.H., Atkinson, A., Pinardi, K., 1995. Edge-induced stress and strainin stripe films and substrates: a two-dimensional finite element calculation. J.Appl. Phys. 78 (3), 1630–1637.

Kachanov, M.L., Shafiro, B., Tsukrov, I., 2003. Handbook of Elasticity Solutions.Kluwer Academic Publishers, Dordrecht.

Koiter, W.T., 1955. On the diffusion of load from a stiffener into a sheet. Q. J. Mech.Appl. Math. 8 (2), 164–178.

Melan, E., 1932. Der Spannungszustand der durch eine Einzelkraft im Innernbeanspruchten Halbscheibe. ZAMM – J. Appl. Math. Mech./Z. Angew. Math.Mech. 12 (6), 343–346.

Morar, G.A., Popov, G.Ia, 1971. On a contact problem for a half-plane with elasticcoverings. J. Appl. Math. Mech. (PMM) 35 (1), 172–178.

Muskhelishvili, N.I., 1953. Singular Integral Equation. Noordhoff, Groningen.Reissner, E., 1940. Note on the problem of the distribution of stress in a thin elastic

sheet. Proc. Natl. Acad. Sci. USA 26, 300–305.Rybakov, L.S., 1982. On discrete interaction of a plate and a damaged stringer. J.

Appl. Math. Mech. (PMM) 45, 127–133.Rybakov, L.S., Cherepanov, G.P., 1977. Discrete interaction of a plate with a semi-

infinite stiffener. J. Appl. Math. Mech. (PMM) 41, 322–328.Shen, Y.L., 2010. Constrained Deformation of Materials. In: Devices, Heterogeneous

Structures and Thermo-Mechanical Modeling. Springer, New York.


Takahashi, M., Shibuya, Y., 1997. Numerical analysis of interfacial stress and stresssingularity between thin films and substrates. Mech. Res. Commun. 24 (6), 597–602.

Villaggio, P., 2003. Brittle detachment of a stiffener bonded to an elastic plate. J. Eng.Math. 46, 409–416.

Wang, X.D., Meguid, S.A., 2000. On the electroelastic behaviour of a thinpiezoelectric actuator attached to an infinite host structure. Int. J. Solids.Struct. 37, 3231–3251.

Zhang, X.C., Xu, B.S., Wang, H.D., Wu, Y.X., 2005. Analytical modelling of edge effectson the residual stresses within the film/substrate systems. I: interfacial stresses.Thin Solid Films 488, 274–282.

Tullini, N., Tralli, A., Lanzoni, L., submitted for publication. Interfacial shear stressanalysis of bar and thin film bonded to 2D elastic substrate using a coupled FE-BIE method.

analysis of stress singularities in thin coatings bonded to a semi-infinite elastic substrate

Documents