Anisotropic, non-uniform misf it strain in a thin film ... Anisotropic, non... · Anisotropic,...

Interaction and Multiscale Mechanics,, Vol. 1, No. 1 (2007) 123-142 123

Anisotropic, non-uniform misfit strain in a thin film bonded on a plate substrate

Y. Huang†

Department of Civil and Environmental Engineering and Department of Mechanical Engineering,

Northwestern University, Evanston, IL 60208, USA

D. Ngo

Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, USA

X. Feng

Department of Engineering Mechanics, Tsinghua University, Beijing, China

A.J. Rosakis

Graduate Aeronautical Laboratory, California Institute of Technology, Pasadena, CA 91125, USA

(Received July 13, 2007, Accepted November 5, 2007)

Abstract. Current methodologies used for the inference of thin film stresses through curvature measurementsare strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. These methodologies have recently been extended to non-uniform stress and curvature statesfor the thin film subject to non-uniform, isotropic misfit strains. In this paper we study the same thin film/substrate system but subject to non-uniform, anisotropic misfit strains. The film stresses and system curvaturesare both obtained in terms of the non-uniform, anisotropic misfit strains. For arbitrarily non-uniform, anisotropicmisfit strains, it is shown that a direct relation between film stresses and system curvatures cannot beestablished. However, such a relation exists for uniform or linear anisotropic misfit strains, or for the averagefilm stresses and average system curvatures when the anisotropic misfit strains are arbitrarily non-uniform.

Keywords: anisotropic film misfit strains and stresses; non-uniform film stresses and system curva-tures; stress-curvature relations; non-local effects; interfacial shear.

1. Introduction

Stoney (1909) used a plate system composed of a thin film, of thickness hf, deposited on a

relatively thick substrate, of thickness hs, and derived a simple relation between the curvature, κ, of

the system and the stress, σ ( f ), of the film as follows:

† Professor, Corresponding Author, E-mail: [email protected]

124 Y. Huang, D. Ngo, X. Feng and A.J. Rosakis

(1)

In the above the subscripts “f ” and “s” denote the thin film and substrate, respectively, and E and

v are the Young’s modulus and Poisson’s ratio. Eq. (1) is called the Stoney formula, and it has been

extensively used in the literature to infer film stress changes from experimental measurement of

system curvature changes (Freund and Suresh 2004).

The Stoney formula was derived for an isotropic “thin” solid film of uniform thickness deposited

on a much “thicker” plate substrate based on a number of assumptions. The assumptions include the

following: (1) Both the film thickness hf and the substrate thickness hs are uniform and hf << hs << R,

where R represents the characteristic length in the lateral direction (e.g. system radius R shown in

Fig. 1); (2) The strains and rotations of the plate system are infinitesimal; (3) Both the film and

substrate are homogeneous, isotropic, and linearly elastic; (4) The film stress states are in-plane

isotropic or equi-biaxial (two equal stress components in any two, mutually orthogonal in-plane

directions) while the out-of-plane direct stress and all shear stresses vanish; (5) The system’s

curvature components are equi-biaxial (two equal direct curvatures) while the twist curvature

vanishes in all directions; and (6) All surviving stress and curvature components are spatially

constant over the plate system’s surface, a situation which is often violated in practice.

The assumption of equi-biaxial (κxx = κyy = κ, κxy = κyx = 0) and spatially constant curvature (κ

independent of position) is equivalent to assuming that the plate system would deform spherically

under the action of the film stresses. If this assumption were to be true, a rigorous application of the

Stoney formula would indeed furnish a single film stress value. This value represents the common

magnitude of each of the two direct stresses in any two, mutually orthogonal directions (i.e.

σxx = σyy = σ ( f ), σxy = σyx =0, σ ( f ) independent of position). This is the uniform stress for the entire

film and it is derived from measurement of a single uniform curvature value which fully

characterizes the system provided the deformation is indeed spherical.

Despite the explicitly stated assumptions of spatial stress and curvature uniformity, the Stoney

formula is often, arbitrarily, applied to cases of practical interest where these assumptions are violated.

This is typically done by applying the Stoney formula pointwise and thus extracting a local value of

stress from a local measurement of the curvature of the system. This approach of inferring film stresses

clearly violates the uniformity assumptions of the analysis and, as such, its accuracy as an

σ f( ) Eshs

2κ

6hf 1 vs–( )-------------------------=

Fig. 1 A schematic diagram of the thin film/substrate system, showing the cylindrical coordinates (r, θ, z)

Anisotropic, non-uniform misfit strain in a thin film bonded on a plate substrate 125

approximation is expected to deteriorate as the levels of curvature non-uniformity become more severe.

Following the initial formulation by Stoney, a number of extensions have been derived by various

researchers who have relaxed some of the other assumptions (other than the assumption of uniformity)

made by his analysis. Such extensions of the initial formulation include relaxation of the assumption of

equi-biaxiality as well as the assumption of small deformations/deflections. A biaxial form of Stoney

formula, appropriate for anisotropic film stresses, including different stress values at two different

directions and non-zero, in-plane shear stresses, was derived by relaxing the assumption of curvature

equi-biaxiality (Freund and Suresh 2004). Related analyses treating discontinuous films in the form of

bare periodic lines (Wikstrom et al. 1999a) or composite films with periodic line structures (e.g. bare or

encapsulated periodic lines) have also been derived (Shen et al. 1996, Wikstrom et al. 1999b, Park and

Suresh 2000). These latter analyses have also removed the assumption of equi-biaxiality and have

allowed the existence of three independent curvature and stress components in the form of two, non-

equal, direct components and one shear or twist component. However, the uniformity assumption of all

of these quantities over the entire plate system was retained. In addition to the above, single, multiple

and graded films and substrates have been treated in various “large” deformation analyses (Masters and

Salamon 1993, Salamon and Masters 1995 Finot et al. 1997, Freund 2000). These analyses have

removed both the restrictions of an equi-biaxial curvature state as well as the assumption of infinitesimal

deformations. They have allowed for the prediction of kinematically nonlinear behavior and bifurcations

in curvature states. These bifurcations are transformations from an initially equi-biaxial to a

subsequently biaxial curvature state that may be induced by an increase in film stresses beyond a critical

level. This critical level is intimately related to the system’s aspect ratio, i.e., the ratio of in-plane to

thickness dimension and the elastic stiffness. These analyses also retain the assumption of spatial

curvature and stress uniformity across the system. However, they allow for deformations to evolve from

an initially spherical shape to an energetically favored shape (e.g. ellipsoidal, cylindrical or saddle

shapes) which features three different, still spatially constant, curvature components (Lee et al. 2001).

None of the above-discussed extensions of the Stoney methodology have relaxed the most

restrictive of Stoney’s original assumption of spatial uniformity which does not allow either film

stress or curvature components to vary across the plate surface. This crucial assumption is often

violated in practice since film stresses and the associated system curvatures are non-uniformly

distributed over the plate area. Huang and Rosakis (2005) and Huang et al. (2005) have recently

made progress to remove the two restrictive assumptions of the Stoney analysis relating to spatial

uniformity and equi-biaxiality. They have studied the cases of thin film/substrate systems subject to

non-uniform but axisymmetric temperature distribution T(r) and misfit strain εm(r), respectively. Their

results show that the relations between film stresses and system curvatures feature not only a “local

part which involves a direct dependence of stresses on curvatures at the same point, but also a “non-

local part which reflects of the effect of curvatures at other points on the location of scrutiny. The

“non-local effect comes into play in the axisymmetric analysis via the average curvature in the thin

film. The “non-local” analysis has been extended to general non-uniform temperature (Huang and

Rosakis 2007) and misfit strains (Ngo et al. 2006), thin film with non-uniform thickness (Ngo et al.

2007) or different radius from the substrate radius (Feng et al. 2006). The X-ray diffraction and

coherent gradient sensing experiments have verified the non-local analysis (Brown et al. 2006, 2007).

The main purpose of the present paper is to extend the nonl-local analysis for the general case of a thin

film/substrate system subject to arbitrary anisotropic misfit strain distribution . Our goal is to

relate film stresses and system curvatures to the misfit strain distribution, and explore a relation between

the film stresses and the system curvatures for general anisotropic misfit strain distributions.

εij

mr θ,( )


2. Governing equations

A thin film of radius R and thickness hf is deposited on a substrate of the same radius and

thickness hs, and hf << hs << R. The Young’s modulus and Poisson’s ratio of the film and substrate

are denoted by Ef, vf, Es and vs, respectively. The thin film is subject to arbitrary anisotropic and

non-uniform misfit strains in the film plane, where r and θ are polar coordinates (Fig. 1).

For convenience we use cos2θ + ,

and , where x and y are the Cartesian coordinates. For

uniform misfit strains , and = constants in the Cartesian coordinates (Freund and Suresh

2004), is also uniform, but and become linear combinations of cos2θ and sin2θ.

The thin film is modeled as a membrane that has no resistance against bending due to its small

thickness hf << hs. Let and denote the displacements in the radial (r) and circumferential (θ)

directions. The strains in the thin film are , and + −

.

The stresses in the thin film can be obtained from the linear elastic constitutive model as

(2)

The membrane forces in the thin film are , and .

For non-uniform misfit strains distribution, the normal stress traction still vanishes, but the

shear stresses and at the interface do not vanish anymore, and are denoted by τr and τθ,

respectively. The equilibrium equations for the thin film, accounting for the effect of interface shear

stresses τr and τθ, become

(3)

The substitution of Eq. (2) into (3) yields the following governing equations for , , τr and τθ

(4a)

εij

mr θ,( )

εΣm 1

2--- εrr

mεθθm

+( ) 1

2--- εxx

mεyy

m+( ) ε∆

m 1

2--- εrr

mεθθm

–( ) 1

2--- εxx

mεyy

m–( )==,= = εxy

m2θsin

γm

2εrθ

m2εxy

m2θcos εxx

mεyy

m–( ) 2θsin–= =

εxx

mεyy

m, εxy

m

εΣm

ε∆m

γm

ur

f( )uθ

f( )

εrr

∂ur

f ( )

∂r----------- εθθ

ur

f ( )

r--------=

1

r---

∂uθ

f ( )

∂θ-----------+,= γrθ

1

r---

∂ur

f ( )

∂θ-----------=

∂uθ

f ( )

∂r-----------

uθ

f ( )

r--------

σrr σθθ+Ef

1 vf–-----------

∂ur

f ( )

∂r-----------

ur

f ( )

r-------- 1

r---

∂uθ

f ( )

∂θ----------- 2εΣ

m–+ +⎝ ⎠

⎛ ⎞=

σrr σθθ–Ef

1 vf+------------

∂ur

f ( )

∂r-----------

ur

f ( )

r--------–

1

r---

∂uθ

f ( )

∂θ----------- 2ε∆

m––⎝ ⎠

⎛ ⎞=

σrθ

Ef

2 1 vf+( )--------------------

1

r---

∂ur

f( )

∂θ----------

∂uθ

f( )

∂r----------

uθ

f( )

r------- γm

––+⎝ ⎠⎛ ⎞=

Nr

f ( )hf σrr Nθ

f ( )hf σθθ=,= Nrθ

f ( )hf σrθ=

σzz

σrz σθz

∂Nr

f ( )

∂r------------

Nr

f ( )Nθ

f ( )–

r----------------------- 1

r---

∂Nrθ

f ( )

∂θ------------ τr–+ + 0=

∂Nrθ

f ( )

∂r------------ 2

r---Nrθ

f( ) 1

r---

∂Nθ

f ( )

∂θ------------ τθ–+ + 0=

ur

f ( )uθ

f ( )

∂2ur

f ( )

∂r2

------------- 1

r---

∂ur

f ( )

∂r-----------

ur

f ( )

r2

--------1 vf–

2-----------

1

r2

----∂2

ur

f ( )

∂θ2

-------------1 vf+

2------------

1

r---

∂2uθ

f ( )

∂r∂θ-------------

3 vf–

2-----------

1

r2

----∂uθ

f ( )

∂θ-----------–+ +–+

1 vf

2–

Ef hf

------------τr 1 vf+( )∂εΣ

m

∂r-------- 1 vf–( )

∂ε∆m

∂r-------- 1 vf–( )2

r---ε∆

m 1 vf–

2-----------

1

r---

∂γm

∂θ--------+ + ++=


(4b)

Let and denote the displacements in the radial (r) and circumferential (θ) directions at

the neutral axis (z = 0) of the substrate, and w the displacement in the normal (z) direction. It is

important to consider w since the substrate can be subject to bending and is modeled as a plate. The

strains in the substrate are given by , and

The stresses in the substrate can then be obtained from the linear

elastic constitutive model as

(5)

The forces in the substrate are obtained by averaging the stresses over the thickness as

, , and .

The moments in the substrate are obtained from as + ,

and .

The shear stresses τr and τθ at the thin film/substrate interface are equivalent to the distributed

forces τr in the radial direction and τθ in the circumferential direction, and bending moments hs / 2τr

and hs / 2τθ applied at the neutral axis (z = 0) of the substrate. The in-plane force equilibrium

equations of the substrate then become

(6)

1 vf+

2------------

1

r---

∂2ur

f ( )

∂r∂θ-------------

3 vf–

2-----------

1

r2

----∂ur

f ( )

∂θ-----------

1 vf–

2-----------

∂2uθ

f ( )

∂r2

------------- 1

r---

∂uθ

f ( )

∂r-----------

uθ

f ( )

r2

--------–+⎝ ⎠⎜ ⎟⎛ ⎞ 1

r2

----∂2

uθ

f ( )

∂θ2

-------------+ + +

1 vf

2–

Ef hf

------------τθ 1 vf+( )1

r---

∂εΣm

∂θ-------- 1 vf–( )1

r---

∂ε∆m

∂θ--------

1 vf–

2-----------

∂γm

∂r---------- 1 vf–( )γm

r-----+ +–+=

ur

s( )uθ

s( )

εrr

∂ur

s( )

∂r---------- z

∂2w

∂r2

--------- εθθur

s( )

r------- 1

r---

∂uθ

s( )

∂θ---------- z

1

r---

∂w

∂r-------

1

r2

----∂2

w

∂θ2

---------+⎝ ⎠⎛ ⎞–+=,–= γrθ =

1

r---

∂ur

s( )

∂θ----------

∂uθ

s( )

∂r----------

uθ

s( )

r------- 2z–

∂∂r-----

1

r---

∂w

∂θ-------⎝ ⎠

⎛ ⎞+–+

σrr

Es

1 vs

2–

------------∂ur

s( )

∂r---------- vs

ur

s( )

r------- 1

r---

∂uθ

s( )

∂θ----------+⎝ ⎠

⎛ ⎞ z∂2

w

∂r2

--------- vs1

r---

∂w

∂r-------

1

r2

----∂2

w

∂θ2

---------+⎝ ⎠⎛ ⎞+–+

⎩ ⎭⎨ ⎬⎧ ⎫

=

σθθ

Es

1 vs

2–

------------ vs

∂ur

s( )

∂r----------

ur

s( )

r------- 1

r---

∂uθ

s( )

∂θ----------+ + z vs

∂2w

∂r2

---------∂w

∂r-------

∂w

∂r-------

1

r2

----∂2

w

∂θ2

---------++⎝ ⎠⎛ ⎞–=

σrθ

Es

2 1 vs+( )--------------------

1

r---

∂ur

s( )

∂θ----------

∂uθ

s( )

∂r----------

uθ

s( )

r------- 2z

∂∂r-----

1

r---

∂w

∂θ-------⎝ ⎠

⎛ ⎞––+=

Nr

s( ) Es hs

1 vs

2–

------------=

∂ur

s( )

∂r---------- vs

ur

s( )

r------- 1

r---

∂uθ

s( )

∂θ----------+⎝ ⎠

⎛ ⎞+ Nθ

s( ) Es hs

1 vs

2–

------------= vs

∂ur

s( )

∂r----------

ur

s( )

r------- 1

r---

∂uθ

s( )

∂θ----------++ Nrθ

s( ) Es hs

2 1 vs+( )-------------------=

1

r---

∂ur

s( )

∂θ----------

∂uθ

s( )

∂r----------

uθ

s( )

r-------– ⎠

⎞+⎝⎛

zσij zd

hs/2

hs/2

∫– Mr

Es hs

3

12 1 vs

2–( )

----------------------=∂2

w

∂r2

--------- vs1

r---

∂w

∂r-------

1

r2

----∂2

w

∂θ2

---------⎠⎞

⎝⎛

Mθ

Es hs

3

12 1 vs

2–( )

---------------------- vs∂2

w

∂r2

---------1

r---

∂w

∂r-------

1

r2

----∂2

w

∂θ2

---------+ +⎝ ⎠⎛ ⎞= = Mrθ

Es hs

3

12 1 vs+( )-----------------------=

∂∂r-----

1

r---

∂w

∂θ-------⎝ ⎠

⎛ ⎞

∂Nr

s( )

∂r-----------

Nr

s( )Nθ

s( )–

r----------------------

1

r---

∂Nrθ

s( )

∂θ----------- τr+ + + + 0=


The substitution of , and in terms of the displacements into the above equation yields

the following governing equations for , and τθ

(7a)

(7b)

The out-of-plane moment and force equilibrium equations are given by

(8)

(9)

where Qr and Qθ are the shear forces normal to the neutral axis. Elimination of Qr and Qθ from the

above two equations in conjunction with the moments-displacement relation, give the following

governing equations for w, τr and τθ

(10)

where

The continuity of displacements across the thin film/substrate interface requires

(11)

Eqs. (4), (7), (10) and (11) constitute seven ordinary differential equations for , , , ,

w, τr and τθ. Under the limit hf << hs these seven equations can be decoupled to solve , first,

followed by w, then and , and finally τr and τθ, as discussed in the following.

(i) Elimination of τr and τθ from Eqs. (4) and (7) yields two equations for , , , and .

For hf << hs, and disappear in these two equations which give the following governing

equations for and only,

(12a)

∂Nrθ

s( )

∂r----------- 2

r---Nrθ

s( ) 1

r---

∂Nθ

s( )

∂θ----------- τθ+ + + 0=

Nr

s( )Nθ

s( ), Nrθ

s( )

ur

s( )uθ

s( )τr, ,

∂2ur

s( )

∂r2

------------ 1

r---

∂ur

s( )

∂r----------

ur

s( )

r2

-------1 vs–

2------------

1

r2

----∂2

ur

s( )

∂θ2

------------1 vs+

2------------

1

r---

∂2uθ

s( )

∂r∂θ------------

3 vs–

2------------

1

r2

----∂uθ

s( )

∂θ----------–+ +–+

1 vs

2–

Es hs

------------τr–=

1 vs+

2------------

1

r---

∂2ur

s( )

∂r∂θ------------

3 vs–

2------------

1

r2

----∂ur

s( )

∂θ----------

1 vs–

2------------

∂2uθ

s( )

∂r2

------------ 1

r---

∂uθ

s( )

∂r----------

uθ

s( )

r2

-------–+⎝ ⎠⎜ ⎟⎛ ⎞ 1

r2

----∂2

uθ

s( )

∂θ2

------------+ + +1 vs

2–

Es hs

------------τθ–=

∂Mr

∂r---------

Mr Mθ–

r------------------ 1

r---

∂Mrθ

∂θ------------ Qr

hs

2----τr–+ + + 0=

∂Mrθ

∂r------------ 2

r---Mrθ

1

r---

∂Mθ

∂θ---------- Qθ

hs

2----τθ–+ + + 0=

∂Qr

∂r---------

Qr

r----- 1

r---

∂Qθ

∂θ---------+ + 0=

∇2 ∇2w( )

6 1 vs

2–( )

Es hs

2--------------------

∂τr

∂r-------

τr

r---- 1

r---

∂τθ∂θ--------+ +⎝ ⎠

⎛ ⎞=

∇2 ∂2

∂r2

------- 1

r---

∂∂r-----

1

r2

----∂2

∂θ2

--------+ +=

ur

f ( )ur

s( ) hs

2----

∂w

∂r-------–= uθ

f( )uθ

s( ) hs

2----

1

r---

∂w

∂θ-------–=,

ur

f ( )uθ

f ( )ur

s( )uθ

s( )

ur

s( )uθ

s( )

ur

f ( )uθ

f ( )

ur

f ( )uθ

f ( )ur

s( )uθ

s( )

ur

f ( )uθ

f ( )

ur

s( )uθ

s( )

∂2ur

s( )

∂r2

------------ 1

r---

∂ur

s( )

∂r----------

ur

s( )

r2

-------1 vs–

2------------

1

r2

----∂2

ur

s( )

∂θ2

------------1 vs+

2------------

1

r---

∂2uθ

s( )

∂r∂θ------------

3 vs–

2------------

1

r2

----∂uθ

s( )

∂θ----------–+ +–+

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

------------ 1 vf+( )∂εΣ

m

∂r-------- 1 vf–( )

∂ε∆m

∂r-------- 1 vf–( )2

r---ε∆

m 1 vf–

2-----------

1

r---

∂γs

∂θ-------+ + +=

1 vs+

2------------

1

r---

∂2ur

s( )

∂r∂θ------------

3 vs–

2------------

1

r2

----∂ur

s( )

∂θ----------

1 vs–

2------------

∂2uθ

s( )

∂r2

------------ 1

r---

∂uθ

s( )

∂r----------

uθ

s( )

r2

-------–+⎝ ⎠⎜ ⎟⎛ ⎞ 1

r2

----∂2

uθ

s( )

∂θ2

------------+ + +


(12b)

(ii) Elimination of and from Eqs. (4) and (11) gives τr and τθ in terms of , and w

(and ).

(iii) The substitution of τr and τθ in (ii) into Eq. (10) yields the following governing equation for the

normal displacement w. For hf << hs, the governing equation becomes

(13)

This biharmonic equation can be solved analytically, which gives the substrate displacement w.

(iv) The displacements and are obtained from Eq. (11). The leading terms of the interface

shear stresses τr and τθ are then obtained from Eq. (4) as

(14)

These are remarkable results that hold regardless of boundary conditions at the edge r = R. Therefore

the interface shear stresses are proportional to the gradients of misfit strains. For uniform misfit strain

, and =constants in the Cartesian coordinates (Freund and Suresh 2004), the interface shear

stresses do NOT vanish unless = =constant and =0 (i.e. the isotropic Stoney formula).

We expand the arbitrary non-uniform misfit strain distributions , and γ m (r, θ) to

the Fourier series in order to solve the above partial differential equations. analytically

(15)

where , , ,

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

------------ 1 vf+( )1

r---

∂εΣm

∂θ-------- 1 vf–( )1

r---

∂ε∆m

∂θ--------

1 vf–

2-----------

∂γm

∂r-------- 1 vf–( )γ

m

r-----+ +–=

ur

f ( )uθ

f ( )ur

s( )uθ

s( )

εΣm

ε∆m

γm, ,

∇2 ∇2w( ) 6

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------–

1 vf+( ) ∂2εΣm

∂r2

---------- 1

r---

∂εΣm

∂r--------

1

r2

----∂2

εΣm

∂θ2

----------+ +⎝ ⎠⎜ ⎟⎛ ⎞

1 vf–( )+∂2

ε∆m

∂r2

---------- 3

r---

∂ε∆m

∂r--------

1

r2

----∂2

ε∆m

∂θ2

----------–+⎝ ⎠⎜ ⎟⎛ ⎞

1 vf–( )+1

r---

∂2γm

∂r∂θ------------

1

r2

----∂γ

m

∂θ--------+

⎝ ⎠⎜ ⎟⎛ ⎞

=

ur

f ( )uθ

f ( )

τr

Ef hf

1 vf

2–

------------– 1 vf+( )∂εΣ

m

∂r-------- 1 vf–( )

∂ε∆m

∂r-------- 1 vf–( )2

r---ε∆

m 1 vf–

2-----------

1

r---

∂γm

∂θ--------+ + +=

τθEf hf

1 vf

2–

------------– 1 vf+( )1

r---

∂εΣm

∂θ-------- 1 vf–( )1

r---

∂ε∆m

∂θ--------

1 vf–

2-----------

∂γm

∂θ-------- 1 vf–( )γ

m

r-----+ +–=

εxx

mεyy

m, εxy

m

εxx

mεyy

mεxy

m

εΣm

r θ,( ) ε∆m

r θ,( )

εΣm

r θ,( ) εΣcm n( )

r( ) nθ εΣsm n( )

n 1=

∞

∑ r( ) nθsin+cosn 0=

∞

∑=

ε∆m

r θ,( ) ε∆cm n( )

r( ) nθ ε∆s

m n( )

n 1=

∞


∞

∑=

γmr θ,( ) γc

m n( )r( ) nθ γs

m n( )

n 1=

∞


∞

∑=

εΣcm 0( )

r( ) 1

2π------ εΣ

mr θ,( ) θd

0

2π

∫= ε∆c

m 0( )r( ) 1

2π------ ε∆

mr θ,( ) θd

0

2π

∫= γc

m 0( )r( ) 1

2π------ γm

r θ,( ) θd0

2π

∫= εΣcm n( )

r( ) =


, (n ≥1),

, and

(n ≥ 1). Without losing generality, we focus on the cos nθ term in and and

sin nθ term in in the following. The corresponding displacements and interface shear

stresses can be expressed as

, , (16)

Eq. (12) then gives two ordinary differential equations for and , which have the general

solution

(17a)

(17b)

1

π--- εΣ

mr θ,( ) nθ θ, ε∆c

m n( )r( ) 1

π--- ε∆

mr θ,( ) nθ θdcos

0

2π

∫=dcos0

2π

∫ γc

m n( )r( ) 1

π---= γm

r θ,( )0

2π

∫ ncos θ θd

εΣsm n( )

r( )1

π---= εΣ

mr θ,( ) nsin θ θd

0

2π

∫ ε∆sm n( )

r( ) 1

π--- ε∆

mr θ,( ) nsin θ θd

0

2π

∫= γs

m n( )r( ) 1

π--- γm

r θ,( ) nsin θ θd0

2π

∫=

εΣm

r θ,( ) ε∆m

r θ,( )γm

r θ,( )

ur

s( )ur

sn( )r( ) nθcos= uθ

s( )uθ

sn( )r( ) nsin θ= w w

n( )r( ) nθcos=

ur

sn( )uθ

sn( )

ur

sn( ) 1

8---

Ef hf

1 vf

2–

-------------1 vs+

Es hs

-------------

4 1 vf+( ) 1 vs–( ) r1 n+( )–

ηn 1+

0

r

∫ εΣcm n( )

dη rn 1–

+ η1 n–

R

r

∫ εΣcm n( )

dη

1 vf–( )+ 2 1 vs–( ) 1 vs+( )n–[ ]r1 n+

ηn 1+( )–

2ε∆cm n( ) γs

m n( )+( )dη

R

r

∫

1 vf–( )+ 2 1 vs–( ) 1 vs+( )n+[ ]r1 n–η

n 1–2ε∆c

m n( ) γsm n( )

–( )dη0

r

∫

1 vf–( )– 1 vs+( )nr

1 n+( )–η

n 1+2ε∆c

m n( ) γsm n( )

–( )dη0

r

∫

r–n 1–

η1 n–

2ε∆cm n( ) γs

m n( )+( )dη

R

r

∫⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

1 vs–1 vs+

2------------n–⎝ ⎠

⎛ ⎞A0r1 n+

D0rn 1–

–+

uθ

sn( ) 1

8---

Ef hf

1 vf

2–

-------------1 vs+

Es hs

-------------

4 1 vf+( ) 1 vs–( ) r1 n+( )–

ηn 1+

0

r

∫ εΣcm n( )

dη rn 1–

– η1 n–

R

r

∫ εΣcm n( )

dη

1 vf– 1 vs+( )n 4+[ ]r1 n+η

n 1+( )–2ε∆c

m n( ) γsm n( )

+( )dηR

r

∫

1 vf–( )+ 1 vs+( )n 4–[ ]r1 n–η

n 1–2ε∆c

m n( ) γsm n( )

+( )dη0

r

∫

1 vf–( )– 1 vs+( )nr

1 n+( )–η

n 1+2ε∆c

m n( ) γsm n( )

–( )dη0

r

∫

r+n 1–

η1 n–

2ε∆cm n( ) γs

m n( )+( )dη

R

r

∫⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

1 vs+

2------------n 2+⎝ ⎠

⎛ ⎞ A0r1 n+

D0rn 1–

+( )+


where A0 and D0 are constants to be determined, and the condition of finite displacements at the

center r = 0 has been used.

The normal displacement is obtained from the biharmonic Eq. (13) as

(18)

where A1 and B1 are constants to be determined, and the condition of finite w at the center r = 0 has been

used. The displacements in the thin film are obtained from the interface continuity condition (11).

It is important to point out that Eqs. (17) and (18) hold for n > 0. For n = 0 the displacements are

given and discussed in details in Section 5.

3. Boundary conditions

The first two boundary conditions at the free edge r = R require that the net forces vanish,

and at r = R (19)

which give A0 and D0 as

(20a)

(20b)

under the limit hf << hs. The other two boundary conditions at the free edge r = R are the vanishing

of net moments, i.e.,

wn( ) 3

4---–

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

1 vf+( )4n--- r

nη

1 n–

R

r

∫ εΣcm n( )

dη rn–

– ηn 1+

0

r

∫ εΣcm n( )

dη

1 vf–( )+ rn 2+

ηn 1+( )–

R

r

∫ 2ε∆cm n( ) γs

m n( )+( )dη r

2 n–η

n 1–2ε∆c

m n( ) γs

m n( )–( )dη

0

r

∫+

1 vf–( )– rn

η1 n–

R

r


m n( )+( )dη r

n–η

n 1+2ε∆c

m n( ) γs

m n( )–( )dη

0

r

∫+

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

A1rn 2+

B1rn

+ +

Nr

f ( )Nr

s( )+ 0= Nrθ

f ( )Nrθ

s( )+ 0=

A0

1

4---

Ef hf

1 vf

2–

------------1 vs+

Es hs

------------

4 1 vf+( )1 vs–

1 vs+------------R

2– n 1+( )η

1 n+

0

R

∫ εΣcm n( )

dη

1 vf–( )– nR2– n 1+( )

ηn 1+

0

R


m n( )–( )dη

1 vf–( )+ n 1–( )R 2– nη

n 1–

0

R


m n( )–( )dη

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

D0

1

8---–

Ef hf

1 vf

2–

------------1 vs+

Es hs

------------

4 1 vf+( ) 1 vs–( ) 1 n+( )R 2– nη

1 n+

0

R

∫ εΣcm n( )

dη

1 vf–( )+ 1 vs+( )n2R

2– n 1–( )η

n 1–

0

R


m n( )–( )dη

1 vf–( )– 1 vs+( )n n 1+( )R 2– nη

1 n+

0

R

∫ 2ε∆c

m n( ) γs

m n( )–( )dη

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=


and at r = R, (21)

which give A1 and B1 as

(22a)

(22b)

4. Thin-film stresses and system curvatures

We provide the general solution that includes both cosine and sine terms in this section. The

system curvatures are

(23)

The sum of system curvatures is related to the misfit strains by

Mr

hs

2----Nr

f ( )– 0= Qr

1

r---

∂∂θ------ Mrθ

hs

2----– Nrθ

f ( )

⎝ ⎠⎛ ⎞– 0=

A1

3

4---–

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

1 vs–

3 vs+------------

4 1 vf+( )R 2– n 1+( )η

1 n+

0

R

∫ εΣcm n( )

dη

1 vf–( )– n 1–( )R 2– nη

n 1–

0

R


m n( )–( )dη

1 vf–( )+ nR2– n 1+( )

ηn 1+

0

R


m n( )–( )dη

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

B1

3

4---

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

4 1 vf+( )1 vs–( )3 vs+( )

----------------n 1+

n----------R

2n–η

n 1+

0

R

∫ εΣcm n( )

dη

1 vf–( )+1 n

2–

n------------

1 vs–

3 vs+------------

1

n---

3 vs+

1 vs–------------–⎝ ⎠

⎛ ⎞R2– n 1–( )

ηn 1–

0

R


m n( )–( )dη

1 vf–( )+1 vs–( )3 vs+( )

---------------- n 1+( )R2– n

ηn 1+

0

R


m n( )–( )dη

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

=

κrr

∂2w

∂r2

---------= κθθ1

r---

∂w

∂r-------

1

r2

----∂2

w

∂θ2

---------+= κrθ∂∂r-----

1

r---

∂w

∂θ-------⎝ ⎠

⎛ ⎞=, ,

κrr κθθ+ 3–Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------=

∗

2 1 vf+( )εΣm

2 1 vf–( )ε∆m

4 1 vf–( ) η1–ε∆cm 0( )

R

r

∫ dη 1 vf+( )1 vs–

1 vs+------------

4

R2

----- ηεΣcm 0( )

0

R

∫ dη+ + +

1 vf–( ) n 1+( )rnnθcos η

n 1+( )–2ε∆c

m n( )γs

m n( )+( )dη

R

r

∫ nθ ηn 1+( )–

2ε∆sm n( )

γc

m n( )–( )dη

R

r

∫sin+n 1=

∞

∑+

1 vf–( )– n 1–( )r n–nθcos η

n 1–2ε∆c

m n( )γs

m n( )–( )dη

0

r

∫ nθ ηn 1–

2ε∆sm n( )

γc

m n( )+( )dη

0

r

∫sin+n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫


(24)

The average curvature sum over the entire thin film is then

obtained as

(25)

where is the average misfit strain sum. The subtraction of the average curvature

sum from Eq. (24) gives

3–Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

1 vs–

3 vs+------------

∗

4 1 vf+( ) n 1+( )rn rn

R2 n 1+( )

---------------- nθcos ηn 1+

εΣcm n( )

dη0

R

∫ nθ ηn 1+

εΣsm n( )

dη0

R

∫sin+n 1=

∞

∑

1 vf–( )– n2

1–( ) rn

R2n

------- nθcos ηn 1–

2ε∆cm n( )

γs

m n( )–( )dη

0

R

∫ nθ ηn 1–

2ε∆sm n( )

γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑

1 vf–( ) n n 1+( ) rn

R2 n 1+( )

---------------- nθcos ηn 1+

2ε∆cm n( )

γs

m n( )–( )dη

0

R

∫ nθ ηn 1+

2ε∆s

m n( )γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑+

⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

κrr κθθ+1

πR2

--------- θ η κrr κθθ+( ) ηd0

R

∫d0

2π

∫=

κrr κθθ+ 12Ef hf

1 vf–-----------

1 vs–

Es hs

2------------εΣ

m–=

εΣm 1

πR2

--------- θ ηεΣm

ηd0

R

∫d0

2π

∫=

κrr κθθ κrr κθθ+–+ 3–Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------=

∗

2 1 vf+( ) εΣm

εΣm

–( ) 2 1 vf–( )ε∆m

4 1 vf–( ) η1–ε∆c

m 0( )

R

r

∫ dη+ +

1 vf–( ) n 1+( )rnnθcos η

n 1+( )–2ε∆c

m n( )γs

m n( )+( )dη

R

r

∫ nθ ηn 1+( )–

2ε∆s

m n( )γc

m n( )–( )dη

R

r

∫sin+n 1=

∞

∑+

1 vf–( )– n 1–( )r n–nθcos η

n 1–2ε∆c

m n( )γs

m n( )–( )dη

0

r

∫ nθ ηn 1–

2ε∆s

m n( )γc

m n( )+( )dη

0

r

∫sin+n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

3–Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

1 vs–

3 vs+------------


(26)

The difference between two curvatures, , and the twist are given by

(27)

∗

4 1 vf+( ) n 1+( ) rn

R2 n 1+( )

---------------- nθcos ηn 1+

εΣcm n( )

dη0

R

∫ nθ ηn 1+

εΣsm n( )

dη0

R

∫sin+n 1=

∞

∑

1 vf–( )– n2

1–( ) rn

R2n

------- nθcos ηn 1–

2ε∆c

m n( )γs

m n( )–( )dη

0

R

∫ nθ ηn 1–

2ε∆s

m n( )γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑

1 vf–( )+ n n 1+( ) rn

R2 n 1+( )

---------------- nθcos ηn 1+

2ε∆c

m n( )γs

m n( )–( )dη

0

R

∫ nθ ηn 1+

2ε∆sm n( )

γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

κrr κθθ– κrθ

κrr κθθ–3

2---–

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------=

∗

4 1 vf+( )εΣm

4 1 vf–( )ε∆m

1 vf+( )–8

r2

---- η0

r

∫ εΣcm 0( )

dη+

4+ 1 vf+( ) n 1–( )rn 2–nθcos η

1 n–εΣcm n( )

dηR

r

∫ nθ η1 n–

εΣsm n( )

dηR

r

∫sin+n 1=

∞

∑

4– 1 vf+( ) n 1+( )r n 2+( )–nθcos η

n 1+εΣcm n( )

dη0

r

∫ nθ ηn 1+

εΣsm n( )

dη0

r

∫sin+n 1=

∞

∑

1 vf–( )+ n n 1+( )rnnθcos η

n 1+( )–2ε∆c

m n( )γs

m n( )+( )dη

R

r

∫ nθ ηn 1+( )–

2ε∆s

m n( )γc

m n( )–( )dη

R

r

∫sin+n 1=

∞

∑

1 vf–( )+ n n 1–( )r n–nθcos η

n 1–2ε∆c

m n( )γs

m n( )–( )dη

0

r

∫ nθ ηn 1–

2ε∆sm n( )

γc

m n( )+( )dη

0

r

∫sin+n 1=

∞

∑

1 vf–( )– n n 1–( )rn 2–nθcos η

1 n–2ε∆c

m n( )γs

m n( )+( )dη

R

r

∫ nθ η1 n–

2ε∆sm n( )

γc

m n( )–( )dη

R

r

∫sin+n 1=

∞

∑

1 vf–( )– n n 1+( )r n 2+( )–nθcos η

n 1+2ε∆c

m n( )γs

m n( )–( )dη

0

r

∫ nθ ηn 1+

2ε∆sm n( )

γc

m n( )+( )dη

0

r

∫sin+n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

3

2---–

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

1 vs–

3 vs+------------


(27)

∗

4 1 vf+( ) n n 1+( ) rn

R2 n 1+( )

---------------- nθcos ηn 1+

εΣcm n( )

dη0

R

∫ nθ ηn 1+

εΣsm n( )

dη0

R

∫sin+n 1=

∞

∑

1 vf–( )– n n2

1–( ) rn

R2n

------- nθcos ηn 1–

2ε∆cm n( )

γs

m n( )–( )dη

0

R

∫ nθ ηn 1–

2ε∆s

m n( )γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑

1 vf–( )+ n2

n 1+( ) rn

R2 n 1+( )

---------------- nθcos ηn 1+

2ε∆c

m n( )γs

m n( )–( )dη

0

R

∫ nθ ηn 1+

2ε∆sm n( )

γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑

4– 1 vf+( ) n2

1–( )rn 2–

R2n

--------- nθcos ηn 1+

εΣcm n( )

dη0

R

∫ nθ ηn 1+

εΣsm n( )

dη0

R

∫sin+n 1=

∞

∑

1 vf–( )+ n 1–( ) n2

1–( )3 vs+

1 vs–------------⎝ ⎠

⎛ ⎞2

+rn 2–

R2 n 1–( )

---------------

nθcos ηn 1–

2ε∆c

m n( )γs

m n( )–( )dη

0

R

∫

nθ ηn 1–

2ε∆sm n( )

γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑

1 vf–( )– n n2

1–( )rn 2–

R2n

--------- nθcos ηn 1+

2ε∆c

m n( )γs

m n( )–( )dη

0

R

∫ nθ ηn 1+

2ε∆s

m n( )γc

m n( )+( )dη

0

R

∫sin+n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

κrθ3

4---

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------=

∗

4 1 vf+( ) n 1–( )rn 2–nθ η

1 n–εΣcm n( )

dη nθcos η1 n–

εΣsm n( )

dηR

r

∫–R

r

∫sinn 1=

∞

∑

4 1 vf+( ) n 1+( )r n 2+( )–nθ η

n 1+εΣcm n( )

dη nθcos ηn 1+

εΣsm n( )

dη0

r

∫–0

r

∫sinn 1=

∞

∑+

1 vf–( )+ n n 1+( )rnnθ η

n 1+( )–2ε∆c

m n( )γs

m n( )+( )dη

R

r

∫sin nθcos– ηn 1+( )–

2ε∆s

m n( )γc

m n( )–( )dη

R

r

∫n 1=

∞

∑

1 vf–( )– n n 1–( )r n–nθ η

n 1–2ε∆c

m n( )γs

m n( )–( )dη

0

r

∫sin nθcos– ηn 1–

2ε∆sm n( )

γc

m n( )+( )dη

0

r

∫n 1=

∞

∑

1 vf–( )– n n 1–( )rn 2–nθ η

n 1–2ε∆c

m n( )γs

m n( )+( )dη

R

r


2ε∆sm n( )

γc

m n( )–( )dη

R

r

∫n 1=

∞

∑

1 vf–( )+ n n 1+( )r n 2+( )–nθ η

n 1+2ε∆c

m n( )γs

m n( )–( )dη

0

r

∫sin nθcos– ηn 1+

2ε∆s

m n( )γc

m n( )+( )dη

0

r

∫n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫


(28)

The stresses in the thin film are obtained from Eq. (2). Specifically, the sum of stresses +

is related to the misfit strains by

(29)

The difference between stresses, , and shear stress are given by

(30)

(31)

5. Limiting cases

We present a few limit cases to further illustrate the thin film stresses and system curvatures in

Section 4.

3

4---+

Ef hf

1 vf

2–

------------1 vs

2–

Eshs

2------------

1 vs–

3 vs+------------

∗

4 1 vf+( ) n n 1+( ) rn

R2 n 1+( )

---------------- nθ ηn 1+

εΣcm n( )

dη nθcos ηn 1+

εΣsm n( )

dη0

R

∫–0

R

∫sinn 1=

∞

∑

1 vf–( )– n n2

1–( ) rn

R2n

------- nθ ηn 1–

2ε∆c

m n( )γs

m n( )–( )dη

0

R


2ε∆sm n( )

γc

m n( )+( )dη

0

R

∫n 1=

∞

∑

1 vf–( )+ n2

n 1+( ) rn

R2 n 1+( )

---------------- nθ ηn 1+

2ε∆cm n( )

γs

m n( )–( )dη

0

R


2ε∆s

m n( )γc

m n( )+( )dη

0

R

∫n 1=

∞

∑

4– 1 vf+( ) n2

1–( )rn 2–

R2n

--------- nθ ηn 1+

εΣcm n( )

dη nθcos ηn 1+

εΣsm n( )

dη0

R

∫–0

R

∫sinn 1=

∞

∑

1 vf–( )+ n 1–( ) n2

1–( )3 vs+

1 vs–------------⎝ ⎠

⎛ ⎞2

+rn 2–

R2 n 1–( )

---------------

nθ ηn 1–

2ε∆cm n( )

γs

m n( )–( )dη

0

R

∫sin

nθcos– ηn 1–

2ε∆s

m n( )γc

m n( )+( )dη

0

R

∫⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

n 1=

∞

∑

– 1 vf–( ) n n2

1–( )rn 2–

R2n

--------- nθ ηn 1+

2ε∆cm n( )

γs

m n( )–( )dη

0

R


2ε∆sm n( )

γc

m n( )+( )dη

0

R

∫n 1=

∞

∑⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫

σrr

f( )

σθθ

f( )

σrr

f( )σθθ

f( )+

Ef

1 vf–----------- 2εΣ

m–( )=

σrr

f( )σθθ

f( )– σrθ

f( )

σrr

f( )σθθ

f( )–

Ef

1 vf+------------ 2ε∆

m–( )=

σrθ

f( ) Ef

2 1 vf+( )-------------------- γm

–( )=


5.1 Uniform misfit strains in the Cartesian coordinates

Freund and Suresh (2004) obtained the solution for arbitrarily anisotropic but uniform misfit

strains in the Cartesian coordinates, , and = constants. For this case

is a constant, but sin 2θ and sin 2θ depend

on θ. These give the non-vanishing coefficients of the Fourier series of the misfit strains as

, and . Eqs. (24)-(28) give the system

curvatures, which can be transformed to curvatures in the Cartesian coordinates as

(32)

which are also constant curvatures. The thin-film stresses in the Cartesian coordinates can be

obtained from Eqs. (29)-(31) as

(33)

which are constant stresses in the thin film. Elimination of misfit strains from Eqs. (32) and (33)

gives the relation between thin film stresses and system curvatures

(34)

which is identical to Freund and Suresh (2004).

εxx

mεyy

m γxy

m2εxy

mεΣm

= 1

2--- εxx

mεyy

m+( )

ε∆m 1

2--- εxx

mεyy

m–( ) 2θ

1

2---γxy

m+cos= γm γxy

m2θcos εxx

mεyy

m–( )–=

εΣcm 0( ) 1

2---=

εxx

mεyy

m+( ) ε∆c

m 2( ) 1

2---γs

m 2( )–

1

2--- εxx

mεyy

m–( )= = ε∆c

m 2( ) 1

2---γc

m 2( ) 1

2---γxy

m= =

κxx κyy+ 6Efhf

1 vf–-----------

1 vs–

Eshs

2------------ εxx

mεyy

m+( )–=

κxx κyy–

κxy⎩ ⎭⎨ ⎬⎧ ⎫

6Efhf

1 vf+------------

1 vs+

Eshs

2------------

εxx

mεyy

m–

γxy

m

2------

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

–=

σxx

f( )σyy

f( )+

Ef

1 vf–----------- εxx

mεyy

m+( )–=

σxx

f( )σyy

f( )–

σxy

f( )⎩ ⎭⎨ ⎬⎧ ⎫ Ef

1 vf+------------–

εxx

mεyy

m–

γxy

m

2------

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=

σxx

f( )σyy

f( )+

Ef hf

2

6 1 vs–( )hf

------------------------ κxx κyy+( )–=

σxx

f( )σyy

f( )–

σxy

f( )⎩ ⎭⎨ ⎬⎧ ⎫ Es hs

2

6 1 vs+( )hf

-------------------------κxx κyy–

κxy⎩ ⎭⎨ ⎬⎧ ⎫

=


5.2 Axisymmetric normal misfit strains

We consider the axisymmetric normal misfit strains and , which give

and . The non-vanishing displacement in the substrate

is

(35)

The normal displacement is given by

(36)

which gives the non-vanishing system curvatures as

(37)

The non-zero stresses in the thin film are and .

Eq. (37) seems to provide two equations to determine and (and therefore the thin-film

stresses) in terms of curvatures. However, these two equations are NOT independent, as to be

shown in the following.

The average curvature sum over the entire thin film can be

obtained from Eq. (37) as

(38)

where and are the average of and , respectively. It is clear that

and satisfy the Stoney formula. The subtraction of Eq. (38) from Eq. (37) yields

εrr

mεrr

mr( )= εθθ

mεθθm

r( )=, γrθ

m0=

εΣm 1

2--- εrr

mr( ) εθθ

mr( )+[ ]= ε∆

m 1

2--- εrr

mr( ) εθθ

m– r( )[ ]=

ur

s( ) Ef hf

1 vf

2–

------------1 vs

2–

Es hs

------------1 vf+

r------------ ηεΣ

mηd

0

r

∫ 1 vf–( )rε∆m

η----- ηd

R

r

∫ 1 vf+( )1 vs–

1 vs+------------

r

R2

----- ηεΣm

ηd

0

R

∫+ +=

dw

dr------- 6

Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------

1 vf+

r------------ ηεΣ

mηd

0

r

∫ 1 vf–( )rε∆m

η----- ηd

R

r

∫ 1 vf+( )1 vs–

1 vs+------------

r

R2

----- ηεΣm

ηd

0

R

∫+ +⎩ ⎭⎨ ⎬⎧ ⎫

=

κrr κθθ+ 6Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------–

1 vf+( )εΣm

1 vf–( )ε∆m

2 1 vf–( )ε∆m

η----- ηd

R

r

∫+ +

1 vf+( )1 vs–

1 vs+------------

2

R2

----- ηεΣm

ηd

0

R

∫+⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

κrr κθθ– 6Ef hf

1 vf

2–

------------1 vs

2–

Es hs

2------------– 1 vf+( )εΣ

m1 vf–( )ε∆

m 2 1 vf–( )

r2

-------------------– ηεΣm

ηd

0

r

∫+⎩ ⎭⎨ ⎬⎧ ⎫

=

σrr

f( )σθθ

f( )+

Ef

1 vf–----------- 2εΣ

m–( )= σrr

f( )σθθ

f( )–

Ef

1 vf+------------ 2ε∆

m–( )=

εΣm

ε∆m

κrr κθθ+1

R2

----- η κrr κθθ+( ) ηd

0

R

∫=

κrr κθθ+ 12Ef hf

1 vf–-----------

1 vs–

Eshs

2------------εΣ

m–

6 1 vs–( )hf

Eshs

2------------------------σrr

f( ) σθθ

f( )+= =

εΣm σrr

f( ) σθθ

f( )+ εΣ

m σrr

f( ) σθθ

f( )+ σrr

f( ) σθθ

f( )+

κrr κθθ+


(39)

It can be shown that, if the misfit strains satisfy , the right

sides of both Eq. (39) vanish. The curvatures then become uniform and equi-biaxial, κrr =

but the stresses are still non-uniform and non-equibiaxial given by

and . Therefore, for axisymmetric misfit strains, the thin-film

stresses may not be expressed in terms of the system curvatures. This point will become clearer in

the next section 35.

5.3 Axisymmetric shear misfit strain

We consider the axisymmetric shear misfit strain and . The substrate

displacement in the radial direction vanishes, , and that in the circumferential direction is

given by

(40)

The normal displacement also vanishes w = 0, which gives vanishing system curvatures

(41)

The normal stresses in the thin film are also zero, but the shear stress does not vanish

(42)

It is clear that, for axisymmetric shear misfit strain, the non-vanishing thin-film stresses cannot be

expressed in terms of the vanishing curvatures.

6. Extension of Stoney formula for nonuniform anisotrpic misfit strains

Freund and Suresh (2004) obtained the anisotropic relation between thin film stresses and system

κrr κθθ κrr κθθ+–+ 6Ef hf

1 vf

2–

------------1 vs

2–

Eshs

2------------–

1 vf+( )εΣm

1 vf–( )ε∆m

2 1 vf–( )ε∆m

η----- ηd

R

r

∫+ +

1 vf+( )–2

R2

----- ηεΣm

ηd

0

R

∫⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

=

κrr κθθ– 6Ef hf

1 vf

2–

------------1 vs

2–

Eshs

2------------– 1 vf+( )εΣ

m1 vf–( )ε∆

m 2 1 vf+( )

r2

--------------------– ηεΣm

ηd

0

r

∫+⎩ ⎭⎨ ⎬⎧ ⎫

=

1 vf–( )ε∆m

2 1 vf+( )

r2

--------------------= ηεΣm

η 1 vf+( )εΣm

–d0

r

∫

κθθ =

6–Ef hf

1 vf–-----------

1 vs–

Eshs

2------------εΣ

mσrr

f ( )=

Ef

1 vf–-----------

2

r2

----–⎝⎛

η

0

r

∫ εΣm

ηd ) σθθ

f( ) Ef

1 vf–----------- 2εΣ

m 2

r2

----+– ηεΣm

ηd

0

r

∫⎝ ⎠⎜ ⎟⎛ ⎞

=

εrr

mεθθm

0= = γrθ

m γmr( )=

ur

s( )0=

ur

s( ) Ef hf

1 vf–-----------

1 vs+

Eshs

2------------r

γm

η----- ηd

R

r

∫=

κrr κθθ κrθ 0= = =

σrr

f ( )σθθ

f ( )0= = σrθ

f ( ) Ef

2 1 vf+( )-------------------- γm

–( )=,


curvatures for uniform misfit strains. In this section we extend it to nonuniform, linearly distributed

misfit strains, i.e.,

(43)

where a and b are constants, and are the average misfit strains, which can be related to the

average system curvatures by

(44)

The constants a and b in Eq. (43) can be obtained by averaging and over

the entire thin film as

(45)

where and are the average of and , respectively.

The thin-film stresses in the Cartesian coordinates can be obtained from Eqs. (29)-(31) as

(46)

Elimination of misfit strains from Eqs. (44) and (46) gives the relation between thin film stresses

and system curvatures

εxx

m

εyy

m

εxy

m⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫ εxx

m

εyy

m

εxy

m

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

1 ax by+ +( )=

εij

m

κxx κyy+ 6Ef hf

1 vf–-----------

1 vs–

Eshs

2------------– εxx

mεyy

m+=

κxx κyy–

κxy⎩ ⎭⎨ ⎬⎧ ⎫

6Ef hf

1 vf+------------

1 vs+

Eshs

2------------–

εxx

mεyy

m–

εxy

m

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=

x κxx κyy+( ) y κxx κyy+( )

a2 3 vs+( )

R2

--------------------

1 vs+( ) κxx κyy+( )1 vs–( )

2----------------- κxx κyy–( )– x κxx κyy+( ) 1 vs–( )κxyy κxx κyy+( )–

1 vs+( ) κxx κyy+( )[ ]2 1 vs–( )

2----------------- κxx κyy–( )

2

– 1 vs–( )κxy[ ]2

–

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

b2 3 vs+( )

R2

--------------------

1 vs+( ) κxx κyy+( )1 vs–( )

2----------------- κxx κyy–( )+ y κxx κyy+( ) 1 vs–( )κxyx κxx κyy+( )–

1 vs+( ) κxx κyy+( )[ ]2 1 vs–( )

2----------------- κxx κyy–( )

2

– 1 vs–( )κxy[ ]2

–

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

x κxx κyy+( ) y κxx κyy+( ) x κxx κyy+( ) y κxx κyy+( )

σxx

f ( )σyy

f ( )+

Ef

1 vf–-----------– εxx

mεyy

m+( )=

σxx

f ( )σyy

f ( )–

σxy

f ( )⎩ ⎭⎨ ⎬⎧ ⎫ Ef

1 vf+------------–

εxx

mεyy

m–

εxy

m⎩ ⎭⎨ ⎬⎧ ⎫

=


(47)

7. Concluding remarks and discussion

The stresses and curvatures are given in terms of anisotropic misfit strains in Section 4. For

uniform misfit strains in Cartesian coordinates, the direct relation (34) between the thin-film stresses

and system curvatures is established, and it is identical to Freund and Suresh (2004). However, for

axisymmetric normal and shear misfit strains in Sections 34 and 35, such a film stress-curvature

relation cannot be established because some components of anisotropic misfit strains give vanishing

system curvatures but non-vanishing film stresses. This observation of no direct relation between

film stresses and system curvatures also holds for non-uniform, anisotropic misfit strains. It is

somewhat puzzling why a direction relation can be established for uniform, anisotropic misfit

strains (in Cartesian coordinates) as in Eq. (34) but not for non-uniform misfit strains.

The average curvatures in Cartesian coordinates provide an explanation. The average curvature

sum over the entire thin film in Eq. (38) can be rewritten in terms of the Cartesian components as

(48)

The curvature components and in Cartesian coordinates can be obtained from

and in Eqs. (27) and (28), and their average over the entire thin film gives

(49)

Eqs. (43) and (44) suggest that the average misfit strains (and average film stresses) can be linked

directly to the average curvatures. In fact, they become identical to Eq. (32) if the average misfit

strains are replaced by uniform misfit strains.

The subtraction of curvatures by their averages gives , and

in terms of , and . However, these relations

cannot be inverted to express the misfit strain deviation in terms of the curvature deviation

. This is because all curvatures are related to the same displacement w such that their

derivatives are not independent. For example, for axisymmetric misfit strains in Section 34, the

derivatives of curvatures satisfy . This relation becomes trivial

for uniform curvatures. For non-uniform curvatures, however, it indicates that the derivatives of

curvatures, or equivalently the curvature deviation , are not independent. This is the reason

that the misfit strain deviation cannot be solved from the curvature deviation .

σxx

f ( )σyy

f ( )+

σxx

f ( )σyy

f ( )–

σxy

f ( )⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

Eshs

2

6 1 vs

2–( )hf

-------------------------

1 vs+( ) κxx κyy+( )

1 vs–( ) κxx κyy–( )

1 vs–( )κxy⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

1 ax by+ +( )=

κxx κyy+( ) 6–Ef hf

1 vf–-----------

1 vs–

Eshs

2------------εxx

mεyy

m+=

κxx κyy– κxy

κrr κθθ– κrθ

κxx κyy–

κxy⎩ ⎭⎨ ⎬⎧ ⎫

6Ef hf

1 vf+------------

1 vs+

Eshs

2------------–

εxx

mεyy

m–

εxy

m

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=

κxx κyy κxx κyy+–+ κxx κyy– κxx κyy––

κxy κxy– εxx

mεyy

mεxx

mεyy

m+–+ εxx

mεyy

m– εxx

mεyy

m–– εxy

mεxy

m–

εαβm

εαβm

–

καβ καβ–

d

dr----- r

2κrr κθθ–( )[ ] r

2 d

dr----- κrr κθθ+( )=

καβ καβ–

εαβm

εαβm

– καβ καβ–


However, for linear misfit strain distributions, the direct relation between the thin film stresses and

system curvatures can be established.

The interface shear stresses are related to the gradient of misfit strains via Eq. (14), and cannot be

given in terms of curvatures directly.

References

Brown, M.A., Park, T.S., Rosakis, A.J., Ustundag, E., Huang, Y., Tamura, N. and Valek, B. (2006), “Acomparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures inthin film: substrate systems”, J. Appl. Mech., 73, 723-729.

Brown, M.A., Rosakis, A.J., Feng, X., Huang, Y. and Ustundag, E. (2007), “Thin film/substrate systemsfeaturing arbitrary film thickness and misfit strain distributions: Part II. Experimental validation of the non-local stress-curvature relations”, Int. J. Solids Struct., 44, 1755-1767.

Feng, X., Huang, Y., Jiang, H., Ngo, D. and Rosakis, A.J. (2006), “The effect of thin film/substrate radii on theStoney formula for thin film/substrate subjected to non-uniform axisymmetric misfit strain and temperature”.J. Mech. Mater. Struct., 1, 1041-1054.

Finot, M., Blech, I.A., Suresh, S. and Fijimoto, H. (1997), “Large deformation and geometric instability ofsubstrates with thin-film deposits” J. Appl. Phys., 81, 3457-3464.

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Huang, Y., Ngo, D. and Rosakis, A.J. (2005), “Non-uniform, axisymmetric misfit strain in thin films bonded onplate substrates/substrate systems: The relation between non-uniform film stresses and system curvatures”,Acta Mechanica Sinica., 21, 362-370.

Huang, Y. and Rosakis, A.J. (2007), “Extension of Stoney's formula to arbitrary temperature distributions in thinfilm/substrate systems”, J. Appl. Mech., 74, 1225-1233.

Lee, H., Rosakis, A.J. and Freund, L.B. (2001), “Full field optical measurement of curvatures in ultra-thin film/substrate systems in the range of geometrically nonlinear deformations”, J. Appl. Phys., 89, 6116-6129

Master, C.B. and Salamon, N.J. (1993), “Geometrically nonlinear stress-deflection relations for thin film/substratesystems”, Int. J. Engrg. Sci., 31, 915-925.

Ngo, D., Feng, X., Huang, Y., Rosakis, A.J. and Brown, M.A. (2007), “Thin film/substrate systems featuringarbitrary film thickness and misfit strain distributions: Part I. Analysis for obtaining film stress from nonlocalcurvature information”, Int. J. Solids Struct., 44, 1745-1754.

Ngo, D., Huang, Y., Rosakis, A.J. and Feng, X. (2006), “Spatially non-uniform, isotropic misfit strain in thinfilms bonded on plate substrates: the relation between non-uniform stresses and system curvatures”, Thin SolidFilms., 515, 2220-2229.

Park, T.S. and Suresh, S. (2000), “Effects of line and passivation geometry on curvature evolution duringprocessing and thermal cycling in copper interconnect lines”, Acta Materialia., 48, 3169-3175.

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