tboiestricr th. Soli., int stn sll ate se t10.

s, ntio

1

tsbfi

IsPfficurvature changes 2.

Stoneys formula involves the following assumptions:was derived by relaxing the assumption v of curvature equibi-axiality 2. Related analyses treating discontinuous films in theform of bare periodic lines 3 or composite films with periodic

Js

1

Downloai Both the film thickness hf and substrate thickness hs areuniform, the film and substrate have the same radius R,and hfhsR;

ii The strains and rotations of the plate system are infinitesi-mal;

iii Both the film and substrate are homogeneous, isotropic,and linearly elastic;

iv The film stress states are in-plane isotropic or equibiaxialtwo equal stress components in any two, mutually or-thogonal in-plane directions while the out-of-plane directstress and all shear stresses vanish;

v The systems curvature components are equibiaxial twoequal direct curvatures while the twist curvature vanishesin all directions; and

vi All surviving stress and curvature components are spa-tially constant over the plate systems surface, a situationwhich is often violated in practice.

Despite the explicitly stated assumptions, the Stoney formula is

line structures e.g., bare or encapsulated periodic lines have alsobeen derived 46. These latter analyses have removed assump-tions iv and v of equibiaxiality and have allowed the existenceof three independent curvature and stress components in the formof two, nonequal, direct components and one shear or twist com-ponent. However, the uniformity assumption vi of all of thesequantities over the entire plate system was retained. In addition tothe above, single, multiple and graded films and substrates havebeen treated in various large deformation analyses 710.These analyses have removed both the restrictions of an equibi-axial curvature state as well as the assumption ii of infinitesimaldeformations. They have allowed for the prediction of kinemati-cally nonlinear behavior and bifurcations in curvature states thathave also been observed experimentally 11,12. These bifurca-tions are transformations from an initially equibiaxial to a subse-quently biaxial curvature state that may be induced by an increasein film stress beyond a critical level. This critical level is inti-mately related to the systems aspect ratio, i.e., the ratio of in-planeto thickness dimension and the elastic stiffness. These analysesalso retain the assumption vi of spatial curvature and stress uni-formity across the system. However, they allow for deformationsto evolve from an initially spherical shape to an energetically

Contributed by the Applied Mechanics Division of ASME for publication in theOURNAL OF APPLIED MECHANICS. Manuscript received October 18, 2006; final manu-cript received January 14, 2007. Review conducted by Robert M. McMeeking.

276 / Vol. 74, NOVEMBER 2007 Copyright 2007 by ASME Transactions of the ASMEX. Feng

Y. Huang

Department of Mechanical and IndustrialEngineering,

University of Illinois,Urbana, IL 61801

A. J. RosakisGraduate Aeronautical Laboratory,California Institute of Technology,

Pasadena, CA 91125

On the SFilm/SuNonunifCurrent methodologmeasurements are sremain uniform oveworkers [Acta Mech2500 (2005); Thin SMech. Mater. Structto nonuniform misfidepend nonlocally omethods, however, athe thin film/substranonuniform substratmisfit strain. DOI:Keywords: thin filmstress-curvature rela

IntroductionStoney 1 used a plate system composed of a stress bearing

hin film, of uniform thickness hf, deposited on a relatively thickubstrate, of uniform thickness hs, and derived a simple relationetween the curvature, , of the system and the stress, f, of thelm as follows:

f =Eshs

2

6hf1 s1

n the above the subscripts f and s denote the thin film and sub-trate, respectively, and E and are the Youngs modulus andoissons ratio, respectively. Equation 1 is called the Stoneyormula, and it has been extensively used in the literature to inferlm stress changes from experimental measurement of systemded 22 Apr 2008 to 129.105.86.142. Redistribution subject to ASMoney Formula for a Thinstrate System Withrm Substrate Thicknessused for the inference of thin film stress through system curvaturetly restricted to stress and curvature states which are assumed toe entire film/substrate system. Recently Huang, Rosakis, and co-inica, 21, pp. 362370 (2005); J. Mech. Phys. Solids, 53, 2483d Films, 515, pp. 22202229 (2006); J. Appl. Mech., in press; J.

press] established methods for the film/substrate system subjectrain and temperature changes. The film stresses were found toystem curvatures (i.e., depend on the full-field curvatures). Thesessume uniform substrate thickness, which is sometimes violated inystem. Using the perturbation analysis, we extend the methods tohickness for the thin film/substrate system subject to nonuniform1115/1.2745392

onuniform misfit strain, nonuniform substrate thickness, nonlocalns, interfacial shears

often arbitrarily applied to cases of practical interest where theseassumptions are violated. This is typically done by applyingStoneys formula pointwise and thus extracting a local value ofstress from a local measurement of the system curvature. Thisapproach of inferring film stress clearly violates the uniformityassumptions of the analysis and, as such, its accuracy as an ap-proximation is expected to deteriorate as the levels of curvaturenonuniformity become more severe.

Following the initial formulation by Stoney, a number of exten-sions have been derived to relax some assumptions. Such exten-sions of the initial formulation include relaxation of the assump-tion of equibiaxiality as well as the assumption of smalldeformations/deflections. A biaxial form of Stoney formula withdifferent direct stress values and nonzero in-plane shear stressE license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

ffn

nacaoRssbsfmtecaf

tstfiissmatces

2

osm

dmm

simplicity. The substrate is modeled as a plate since hsR. TheYoungs modulus and Poissons ratio of the film and substrate are

Fw

J

Downloaavored shape e.g., ellipsoidal, cylindrical or saddle shapes thateatures three different, still spatially constant, curvature compo-ents 11,12.

The above-discussed extensions of Stoneys methodology haveot relaxed the most restrictive of Stoneys original assumptionvi of spatial uniformity which does not allow either film stressnd curvature components to vary across the plate surface. Thisrucial assumption is often violated in practice since film stressesnd the associated system curvatures are nonuniformly distributedver the plate area. Recently Huang et al. 13 and Huang andosakis 14 relaxed the assumption vi and also iv and v to

tudy the thin film/substrate system subject to non-uniform, axi-ymmetric misfit strain in thin film and temperature change inoth thin film and substrate, respectively, while Ngo et al. 15tudied the thin film/substrate system subject to arbitrarily nonuni-orm e.g., nonaxisymmetric misfit strain and temperature. Theost important result is that the film stresses depend nonlocally on

he substrate curvatures, i.e., they depend on curvatures of thentire substrate. The relations between film stresses and substrateurvatures are established for arbitrarily nonuniform misfit strainnd temperature change, and such relations degenerate to Stoneysormula for uniform, equibiaxial stresses and curvatures.

Feng et al. 16 relaxed part of the assumption i to study thehin film and substrate of different radii, i.e., the thin film has amaller radius than the substrate. Ngo et al. 15 further relaxedhe assumption i for arbitrarily nonuniform thickness of the thinlm. The main purpose of the present paper is to relax the remain-

ng portion in assumption i, i.e., the uniform thickness of theubstrate. To do so we consider the case of thin film/substrateystem with nonuniform substrate thickness subject to nonuniformisfit strain field in the thin film. Our goal is to relate film stresses

nd system curvatures to the misfit strain distribution, and to ul-imately derive a relation between the film stresses and the systemurvatures that would allow for the accurate experimental infer-nce of film stress from full-field and real-time curvature mea-urements.

Governing Equations and Boundary ConditionsConsider a thin film of uniform thickness hf which is deposited

n a circular substrate of thickness hs and radius R Fig. 1. Theubstrate thickness is nonuniform, but is assumed to be axisym-etric hs=hsr for simplicity, where r and are the polar coor-

inates. The film is very thin, hfhs, such that it is modeled as aembrane, and is subject to nonuniform misfit strain m. Here theisfit strain is also assumed to be axisymmetric m=mr for

ig. 1 A schematic diagram of a thin film/substrate systemith the cylindrical coordinates r , ,z

ournal of Applied Mechanicsded 22 Apr 2008 to 129.105.86.142. Redistribution subject to ASMdenoted by Ef, f, Es, and s, respectively.Let uf and us denote the displacements in the radial direction in

the thin film and substrate, respectively. The in-plane membranestrains are obtained from = u,+u, /2 for infinitesimal de-formation and rotation, where ,=r,. The linear elastic consti-tutive model, together with the vanishing out-of-plane stress zz=0, give the in-plane stresses as

=E

1 21 + 1 + m ,

where E ,=Ef, f in the thin film and Es,s in the substrate, andthe misfit strain m is only in the thin film. The nonvanishing axialforces in the thin film and substrate are

Nr =Eh

1 2durdr + urr 1 + m2

N =Eh

1 2durdr + urr 1 + mwhere h=hf in the thin film and hsr in the substrate, and onceagain the misfit strain m is only in the thin film.

Let w denote the lateral displacement in the normal z direc-tion. The curvatures are given by =w,. The bending mo-ments in the substrates are

Mr =Eshs

3

121 s2d2wdr2 + s1r dwdr

3

M =Eshs

3

121 s2sd2wdr2 + 1r dwdr

For nonuniform misfit strain distribution m=mr, the shearstress along the radial direction at the film/substrate interface doesnot vanish, and is denoted by . The in-plane force equilibriumequations for the thin film and substrate, accounting for the effectof interface shear stress , becomes

dNrdr

+Nr N

r

= 0 4

where the minus sign in front of the interface shear stress is forthe thin film, and the plus sign is for the substrate. The momentand out-of-plane force equilibrium equations for the substrate are

dMrdr

+Mr M

r+ Q hs

2 = 0 5

dQdr

+Qr

= 0 6

where Q is the shear force normal to the neutral axis. Equation6, together with the requirement of finite Q at r0, gives Q0.

The substitution of Eq. 2 into 4 yields the governing equa-tions for u and ,

d2ufdr2

+1r

dufdr

ufr2

=

1 f2

Efhf + 1 + f

dmdr

7

ddrhsdusdr + usr 1 sdhsdr usr = 1 s2Es 8

Equations 3, 5, and 6 give the governing equation for w and,

ddrhs3d2wdr2 + 1r dwdr 1 s1r dhs3dr dwdr = 61 s2Es hs 9

NOVEMBER 2007, Vol. 74 / 1277E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

The continuity of displacements across the film/substrate inter-face requires

Ec

n

ws

3s

n

w

s

ff

u

t

Ts

Ttis

shear stress vanishes even for nonuniform substrate thickness.We use the perturbation method to write us as

1

Downloauf = us hs2

dwdr

10

quations 710 constitute four ordinary differential equationsODEs for uf, us, w, and . The ODEs are linear, but have non-onstant coefficients.

The boundary conditions at the free edge r=R require that theet forces and net moments vanish,

Nrf + Nr

s= 0 11

Mr hs2

Nrf

= 0 12

here the superscripts f and s denote the film and substrate, re-pectively.

Perturbation Method for Small Variation of Sub-trate Thickness

In the following we assume small variation of substrate thick-ess

hs = hs0 + hs = hs0 + hs1 13here hs0 constant is the average substrate thickness, andhsr is the substrate thickness variation which satisfieshs hs0; hsr is also written as hs1 in 13, where 01 is a small, positive constant, and hs1=hs1r is on the

ame order as hs0.We use the perturbation method to solve the ODEs analytically

or 1. Two possible scenarios are considered separately in theollowing:

i The substrate thickness variation hs is on the same orderas the thin film thickness hf, i.e., hshf. This is repre-sented by =hf /hs0 1. For this case the film stressesand system curvatures are identical to their counterpartsfor a constant substrate thickness hs0. This is because theStoney formula 1, as well as all its extensions, holds onlyfor thin films, hfhs. As compared to unity one, termsthat are on the order of Ohf /hs are always neglected. Inthis case the difference between the film stresses or sys-tem curvatures, for nonuniform substrate thickness hsand those for constant thickness hs0 is on the order ofOhs /hs0 as compared to unity, which is the same asOhf /hs since hshf, and is therefore negligible.

ii The substrate thickness variation hs is much larger thanthe thin film thickness hf, i.e., hs hf. This is repre-sented by hf /hs0 1. In the following we focus onthis case and use the perturbation method for 1 toobtain the analytical solution.

Elimination of from 7 and 8 yields an equation for uf ands. For hf /hs01, uf disappears in this equation, which becomeshe governing equation for us,

ddrhsdusdr + usr 1 sdhsdr usr = Efhf1 f 1 s

2

Es

dmdr

14

he above equation, together with 8, gives the interface sheartress

= Efhf

1 fdmdr

15

his is a remarkable result that holds regardless of the substratehickness and boundary conditions at the edge r=R. Therefore, thenterface shear stress is proportional to the gradient of misfittrain. For uniform misfit strain mr=constant, the interface

278 / Vol. 74, NOVEMBER 2007ded 22 Apr 2008 to 129.105.86.142. Redistribution subject to ASMus = us0 + us1 16

where 1, us0 is the solution for a constant substrate thicknesshs0, and is given by Huang et al. 13

us0 =Efhf

1 f

1 s2

Eshs0 1r0r

md +1 s1 + s

m

2r 17

and

m =2R20

R

md

is the average misfit strain in the thin film; us1 in 16 is on thesame order as us0. In the following we use u to denote du /dr.The substitution of 16 and 17 into 14 and the neglect ofO2 terms give the following linear ODE with constant coeffi-cients for us1,

us1 + us1r = 1 shs1hs0 us0r hs1hs0us0 + us0r 18

Its general solution is

us1r = hs1hs0

us0 +12r0

r

1 + s + 1 s r22hs1 hs0 us0d+

A2

r 19

where the constant A is to be determined. The total substrate dis-placement is then given by

usr = 2 hshs0us0 + 12r0r

1 + s + 1 s r22

hshs0

us0d +A2

r 20

The substitution of 15 into 9 yields the governing equationfor the displacement w,

hs3w + wr 1 shs3w

r=

6Efhf1 f

1 s2

Eshsm

21

Its perturbation solution can be written as

w = w0 + w1 22

where w0 is the solution for a constant substrate thickness hs0, andis given by Huang et al. 13

w0 = 6Efhf

1 f

1 s2

Eshs02 1r0

r

md +1 s1 + s

m

2r 23

and once again

m =2R20

R

md

is the average misfit strain in the thin film; w1 in 22 is on thesame order as w0. Equations 2123 give the following linearODE with constant coefficients for w1

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w1 + w1r = 6Efhf1 f 1 s

2

Eshs02

hs1hs0

m 3hs1hs0w0 + w0r

I

wfor w is obtained from 22 as

The displacement uf in the thin film is then obtained from us in20 and w in 26 via 10.

28

w

i

= 6Efhf 1 s + hs 1m 2 hs 1md

1 +2

30

t

a

wOs

J

Downloa1 f Eshs02 hs0 r2 0

1r2

0

r

2m + 3

The thin film stresses are obtained from the constitutive rela-ions

rrf

=

Ef1 f

2uf + f ufr 1 + fmnd

f

=

Ef1 f

2 fuf + ufr 1 + fmhere uf is given in 10. The sum of thin film stresses, up to the2 accuracy as compared to unity, is related to the misfit

train by

ournal of Applied Mechanicsded 22 Apr 2008 to 129.105.86.142. Redistribution subject to ASMhs0

s0

md +31 s

2mhshs0 d

rrf +

f=

Ef1 f

2m 31

The difference of thin film stresses rr

f

f is on the order ofOEf

2 /Esmhf /hs0, which is very small as compared to rrf

+f

. Therefore only its leading term is presented

rrf

f

= 4EfEfhf

1 f2

1 s2

Eshs0 m 2r20r

md 324.1 Special Case: Uniform Misfit Strain. For uniform misfit

strain distribution m=constant and nonuniform substrate thick-

NOVEMBER 2007, Vol. 74 / 1279w = 4 3 hshs0w0 + 32r0r

1 + s + 1 s r22hshs0 w0d+B2

r + 3Efhf

1 f

1 s2

Eshs02

1r

0

r ddr2 2hshs0 1

md 26

4 Thin-Film Stresses and System CurvaturesThe system curvatures rr=d2w /dr2 and = 1/rdw /dr are

obtained from 26. Their sum rr+ is given in terms ofthe misfit strain by

= 6Efhf

1 f

1 s2

Eshs02 3 2

hshs0m + 4 3 hshs0 + 31 s2 hs hs0hs0 1 s1 + sm

+0

r 31 s2 0

md mhshs0 d 29here

m =2R20

R

md

s the average misfit strain in the thin film. The difference of system curvatures rr is given by

2

4 3 hshs0m 2r20r

mdr+ 31 shs1hs0

w0

r24

ts general solution is

w1 = 3hs1hs0

w0 +32r0

r

1 + s + 1 s r22hs1 hs0 w0d+

B2

r + 3Efhf

1 f

1 s2

Eshs02

1r

0

r ddr2 2hs1hs0 md

25here the constant B is to be determined. The complete solutionThe constants A and B, or equivalently, A and B, are deter-mined from the boundary conditions 11 and 12 as

A = 1 s

R2 0R R2 2

hshs0

us0d 27

B = 31 s

R2 0R R2 2

hshs0

w0d 6Efhf

1 f1 sEshs0

21R2

0

R dd1 + sR2 + 1 s2hshs0 1mdE license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

ness, the interface shear stress in 15 vanishes. The thin filmstresses become constant and equibiaxial, and are given by

T

wt

u

dT

w

Tt

31 and 32 depend only on the misfit strain and are independent of substrate thickness. The system curvatures in 29 and 30 involvet for a discontinuous h . However, it appears only in the integrations

uni

5 Stifor

e sub

w

0

R

i .

1

Downloahe derivative of substrate thickness hs, which is not well defineduch that 29 and 30 still hold.

In the following, we extend the Stoney formula for arbitrary non

Extension of Stoney Formula for Nonuniform MisfitIn this section we extend the Stoney formula for arbitrary nonun

stablishing the direct relation between the thin-film stresses and

m = 1 f6Efhf

Es1 s

2hs

2 1 s

2hs

2

+12

r

R

1 3s

1 sR2 0

R

2

here

hs2 =

2R2

s the average of hs2, and we have used 30 in establishing 37

280 / Vol. 74, NOVEMBER 2007rrf

= f

=

Ef1 f

m 33

he curvatures in 29 and 30 become

= 12Efhf

1 f1 sEshs0

2 1 5 s2 hshs0 1+ 1 2s

hs hs0hs0

m34

= 18Efhf

1 f

1 s2

Eshs02 hshs0 2r20r hshs0 dm

hich are neither constant nor equibiaxial for varying substratehickness.

Figure 2 shows a substrate with a step change in thickness; aniform thickness h in the outer region rRin and a slightlyifferent value hh in the inner region rRin, whereh h. The average thickness becomes hs0=hhRin2 R2 .he curvature in the circumferential direction is

= 6Efhf

1 f1 sEsh2

m1 +h2h 5 s 1 sRin2R2 for r Rin

1 +h2h 5 s 1 sRin2R2 31 + s1 Rin2r2 for r Rin 35

hich is a constant in the inner region, and is continuous across r=Rin. The curvature in the radial direction rr is the same constant as in the inner region; however, it is discontinuous across r=Rin, and is given by

rr = 6Efhf

1 f1 sEsh2

m1 +h2h 5 s 1 sRin2R2 for r Rin

1 +h2h 5 s 1 stRin2R2 31 + s1 + Rin2r2 for r Rin 36

he continuous and discontinuous rr are illustrated in Fig. 2. Similar discontinuity in rr has been observed for varying thin filmhickness 17,18.

It should be pointed out that the results in this section hold for discontinuous substrate thickness. This is because the film stresses in

Fig. 2 a A schematic diagram of a thin film/substrate systemwith a step change in substrate thickness. b The normalizedsystem curvatures rr=rr /0 and = /0, where 0=6Efhf /1f / 1s /Esh2m, h /2h=0.1, s=0.27, and Rin=R /3.ded 22 Apr 2008 to 129.105.86.142. Redistribution subject to ASMs

form misfit strain distribution and nonuniform substrate thickness.

rain Distribution and Nonuniform Substrate Thicknessm misfit strain distribution and nonuniform substrate thickness bystrate curvatures. We invert the misfit strain from 29 as

31 shs2hs

hs0d

hs2hs

hs0d 37

hs2d

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The thin film stresses are obtained by substituting 37 into 31 and 32 as

rrf +

f=

Es31 s

2hfhs

2 1 s

2hs

2

+12

r

R

1 3s 31 shs2

hshs0

d

1 sR2 0

R

2 hs2

hshs0

d 38rr

f

f=

2Efhs031 + f

39

Equations 38 and 39 provide direct relations between filmsw

htdlue

t

Escsdpg

tm

R

Thermal Stresses in Passivated Metal Interconnects, J. Appl. Phys., 86, pp.60886095.

5 Shen, Y. L., Suresh, S., and Blech, I. A., 1996, Stresses, Curvatures, andShape Changes Arising From Patterned Lines on Silicon Wafers, J. Appl.Phys., 80, pp. 13881398.

J

Downloaournal of Applied Mechanics NOVEMBER 2007, Vol. 74 / 1281tresses and system curvatures. The system curvatures in 38 al-ays appear together with the square of substrate thickness, i.e.,

s2 and hs

2. It is important to note that stresses at a point in thehin film depend not only on curvatures at the same point localependence, but also on curvatures in the entire substrate non-ocal dependence via the term hs

2 and the integrals in 38. Forniform substrate thickness, 38 and 39 degenerate to Huangt al. 13

The interface shear stress can also be directly related to sys-em curvatures via 15 and 37

=Es

61 s2 ddr hs2 12 1 3shs2 31 shs2 hshs0

40quation 40 provides a way to determine the interface sheartresses from the gradients of system curvatures once the full-fieldurvature information is available. Since the interfacial sheartress is responsible for promoting system failures throughelamination of the thin film from the substrate, Eq. 40 has aarticular significance. It shows that such stress is related to theradient of rr+, as well as to the magnitude of rr+ andrr for nonuniform substrate thickness.In summary, 3840 provide a simple way to determine the

hin film stresses and interface shear stress from the nonuniformisfit strain in the thin film and nonuniform substrate thickness.

eferences1 Stoney, G. G., 1909, The Tension of Metallic Films Deposited by Electroly-

sis, Proc. R. Soc. London, Ser. A, 82, pp. 172175.2 Freund, L. B., and Suresh, S., 2004, Thin Film Materials; Stress, Defect For-

mation and Surface Evolution, Cambridge University Press, Cambridge, U.K.3 Wikstrom, A., Gudmundson, P., and Suresh, S., 1999, Thermoelastic Analysis

of Periodic Thin Lines Deposited on a Substrate, J. Mech. Phys. Solids, 47,pp. 11131130.

4 Wikstrom, A., Gudmundson, P., and Suresh, S., 1999, Analysis of Averageded 22 Apr 2008 to 129.105.86.142. Redistribution subject to ASM6 Park, T. S., and Suresh, S., 2000, Effects of Line and Passivation Geometryon Curvature Evolution During Processing and Thermal Cycling in CopperInterconnect Lines, Acta Mater., 48, pp. 31693175.

7 Masters, C. B., and Salamon, N. J., 1993, Geometrically Nonlinear StressDeflection Relations for Thin Film/Substrate Systems, Int. J. Eng. Sci., 31,pp. 915925.

8 Salamon, N. J., and Masters, C. B., 1995, Bifurcation in Isotropic Thin Film/Substrate Plates, Int. J. Solids Struct., 32, pp. 473481.

9 Finot, M., Blech, I. A., Suresh, S., and Fijimoto, H., 1997, Large Deformationand Geometric Instability of Substrates With Thin-Film Deposits, J. Appl.Phys., 81, pp. 34573464.

10 Freund, L. B., 2000, Substrate Curvature Due to Thin Film Mismatch Strainin the Nonlinear Deformation Range, J. Mech. Phys. Solids, 48, p. 1159.

11 Lee, H., Rosakis, A. J., and Freund, L. B., 2001, Full Field Optical Measure-ment of Curvatures in Ultra-Thin Film/Substrate Systems in the Range ofGeometrically Nonlinear Deformations, J. Appl. Phys., 89, pp. 61166129.

12 Park, T. S., and Suresh, S., 2000, Effects of Line and Passivation Geometryon Curvature Evolution During Processing and Thermal Cycling in CopperInterconnect Lines, Acta Mater., 48, pp. 31693175.

13 Huang, Y., Ngo, D., and Rosakis, A. J., 2005, Non-Uniform, AxisymmetricMisfit Strain: In Thin Films Bonded on Plate Substrates/Substrate Systems:The Relation Between Non-Uniform Film Stresses and System Curvatures,Acta Mech. Sin., 21, pp. 362370.

14 Huang, Y., and Rosakis, A. J., 2005, Extension of Stoneys Formula to Non-Uniform Temperature Distributions in Thin Film/Substrate Systems. The Caseof Radial Symmetry, J. Mech. Phys. Solids, 53, pp. 24832500.

15 Ngo, D., Huang, Y., Rosakis, A. J., and Feng, X., 2006, Spatially Non-Uniform, Isotropic Misfit Strain in Thin Films Bonded on Plate Substrates:The Relation Between Non-Uniform Film Stresses and System Curvatures,Thin Solid Films, 515, pp. 22202229.

16 Feng, X., Huang, Y., Jiang, H., Ngo, D., and Rosakis, A. J., 2006, The Effectof Thin Film/Substrate Radii on the Stoney Formula for Thin Film/SubstrateSubjected to Non-Uniform Axisymmetric Misfit Strain and Temperature, J.Mech. Mater. Struct., in press.

17 Brown, M., Rosakis, A. J., Feng, X., Feng, X., Huang, Y., and stndag, E.,2006, Thin Film/Substrate Systems Featuring Arbitrary Film Thickness andMisfit Strain Distributions. Part II: Experimental Validation of the Non-LocalStress/Curvature Relations, Int. J. Solids Struct., 44, pp. 17551767.

18 Brown, M. A., Park, T. S., Rosakis, A. J., Ustundag, E., Huang, Y., Tamura,N., and Valek, B., 2006, A Comparison of X-Ray Microdiffraction and Co-herent Gradient Sensing in Measuring Discontinuous Curvatures in Thin Film:Substrate Systems, ASME J. Appl. Mech., 73, pp. 723729.E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm