X. Feng
Y. Huang
Department of Mechanical and Industrial
Engineering,
University of Illinois,
Urbana, IL 61801
A. J. Rosakis
Graduate Aeronautical Laboratory,
California Institute of Technology,
Pasadena, CA 91125
On the Stoney Formula for a Thin
Film/Substrate System With
Nonuniform Substrate Thickness
Current methodologies used for the inference of thin film stress through system curvature
measurements are strictly restricted to stress and curvature states which are assumed to
remain uniform over the entire film/substrate system. Recently Huang, Rosakis, and coworkers [Acta Mech. Sinica, 21, pp. 362–370 (2005); J. Mech. Phys. Solids, 53, 2483–
2500 (2005); Thin Solid Films, 515, pp. 2220–2229 (2006); J. Appl. Mech., in press; J.
Mech. Mater. Struct., in press] established methods for the film/substrate system subject
to nonuniform misfit strain and temperature changes. The film stresses were found to
depend nonlocally on system curvatures (i.e., depend on the full-field curvatures). These
methods, however, all assume uniform substrate thickness, which is sometimes violated in
the thin film/substrate system. Using the perturbation analysis, we extend the methods to
nonuniform substrate thickness for the thin film/substrate system subject to nonuniform
misfit strain. 关DOI: 10.1115/1.2745392兴
Keywords: thin films, nonuniform misfit strain, nonuniform substrate thickness, nonlocal
stress-curvature relations, interfacial shears
1
Introduction
Stoney 关1兴 used a plate system composed of a stress bearing
thin film, of uniform thickness h f , deposited on a relatively thick
substrate, of uniform thickness hs, and derived a simple relation
between the curvature, ␬, of the system and the stress, ␴共f兲, of the
film as follows:
␴共 f 兲 =
Eshs2␬
6h f 共1 − ␯s兲
共1兲
In the above the subscripts f and s denote the thin film and substrate, respectively, and E and ␯ are the Young’s modulus and
Poisson’s ratio, respectively. Equation 共1兲 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 关2兴.
Stoney’s formula involves the following assumptions:
共i兲
共ii兲
共iii兲
共iv兲
共v兲
共vi兲
Both the film thickness h f and substrate thickness hs are
uniform, the film and substrate have the same radius R,
and h f hs R;
The strains and rotations of the plate system are infinitesimal;
Both the film and substrate are homogeneous, isotropic,
and linearly elastic;
The film stress states are in-plane isotropic or equibiaxial
共two equal stress components in any two, mutually orthogonal in-plane directions兲 while the out-of-plane direct
stress and all shear stresses vanish;
The system’s curvature components are equibiaxial 共two
equal direct curvatures兲 while the twist curvature vanishes
in all directions; and
All surviving stress and curvature components are spatially constant over the plate system’s surface, a situation
which is often violated in practice.
Despite the explicitly stated assumptions, the Stoney formula is
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received October 18, 2006; final manuscript received January 14, 2007. Review conducted by Robert M. McMeeking.
1276 / Vol. 74, NOVEMBER 2007
often arbitrarily applied to cases of practical interest where these
assumptions are violated. This is typically done by applying
Stoney’s formula pointwise and thus extracting a local value of
stress from a local measurement of the system curvature. This
approach of inferring film stress clearly violates the uniformity
assumptions of the analysis and, as such, its accuracy as an approximation is expected to deteriorate as the levels of curvature
nonuniformity become more severe.
Following the initial formulation by Stoney, a number of extensions have been derived to relax some assumptions. Such extensions of the initial formulation include relaxation of the assumption of equibiaxiality as well as the assumption of small
deformations/deflections. A biaxial form of Stoney formula 共with
different direct stress values and nonzero in-plane shear stress兲
was derived by relaxing the assumption 共v兲 of curvature equibiaxiality 关2兴. Related analyses treating discontinuous films in the
form of bare periodic lines 关3兴 or composite films with periodic
line structures 共e.g., bare or encapsulated periodic lines兲 have also
been derived 关4–6兴. These latter analyses have removed assumptions 共iv兲 and 共v兲 of equibiaxiality and have allowed the existence
of three independent curvature and stress components in the form
of two, nonequal, direct components and one shear or twist component. However, the uniformity assumption 共vi兲 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 关7–10兴.
These analyses have removed both the restrictions of an equibiaxial curvature state as well as the assumption 共ii兲 of infinitesimal
deformations. They have allowed for the prediction of kinematically nonlinear behavior and bifurcations in curvature states that
have also been observed experimentally 关11,12兴. These bifurcations are transformations from an initially equibiaxial to a subsequently biaxial curvature state that may be induced by an increase
in film stress beyond a critical level. This critical level is intimately related to the systems aspect ratio, i.e., the ratio of in-plane
to thickness dimension and the elastic stiffness. These analyses
also retain the assumption 共vi兲 of spatial curvature and stress uniformity across the system. However, they allow for deformations
to evolve from an initially spherical shape to an energetically
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simplicity. The substrate is modeled as a plate since hs R. The
Young’s modulus and Poisson’s ratio of the film and substrate are
denoted by E f , ␯ f , Es, and ␯s, respectively.
Let u f and us denote the displacements in the radial direction in
the thin film and substrate, respectively. The in-plane membrane
strains are obtained from ␧␣␤ = 共u␣,␤ + u␤,␣兲 / 2 for infinitesimal deformation and rotation, where ␣ , ␤ = r,␪. The linear elastic constitutive model, together with the vanishing out-of-plane stress ␴zz
= 0, give the in-plane stresses as
␴␣␤ =
E
关共1 − ␯兲␧␣␤ + ␯␧␬␬␦␣␤ − 共1 + ␯兲␧m␦␣␤兴,
1 − ␯2
where E , ␯ = E f ,␯ f in the thin film and Es,␯s in the substrate, and
the misfit strain ␧m is only in the thin film. The nonvanishing axial
forces in the thin film and substrate are
Nr =
Fig. 1 A schematic diagram of a thin film/substrate system
with the cylindrical coordinates „r , ␪ , z…
favored shape 共e.g., ellipsoidal, cylindrical or saddle shapes兲 that
features three different, still spatially constant, curvature components 关11,12兴.
The above-discussed extensions of Stoney’s methodology have
not relaxed the most restrictive of Stoney’s original assumption
共vi兲 of spatial uniformity which does not allow either film stress
and 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 nonuniformly distributed
over the plate area. Recently Huang et al. 关13兴 and Huang and
Rosakis 关14兴 relaxed the assumption 共vi兲 关and also 共iv兲 and 共v兲兴 to
study the thin film/substrate system subject to non-uniform, axisymmetric misfit strain 共in thin film兲 and temperature change 共in
both thin film and substrate兲, respectively, while Ngo et al. 关15兴
studied the thin film/substrate system subject to arbitrarily nonuniform 共e.g., nonaxisymmetric兲 misfit strain and temperature. The
most important result is that the film stresses depend nonlocally on
the substrate curvatures, i.e., they depend on curvatures of the
entire substrate. The relations between film stresses and substrate
curvatures are established for arbitrarily nonuniform misfit strain
and temperature change, and such relations degenerate to Stoney’s
formula for uniform, equibiaxial stresses and curvatures.
Feng et al. 关16兴 relaxed part of the assumption 共i兲 to study the
thin film and substrate of different radii, i.e., the thin film has a
smaller radius than the substrate. Ngo et al. 关15兴 further relaxed
the assumption 共i兲 for arbitrarily nonuniform thickness of the thin
film. The main purpose of the present paper is to relax the remaining portion in assumption 共i兲, i.e., the uniform thickness of the
substrate. To do so we consider the case of thin film/substrate
system with nonuniform substrate thickness subject to nonuniform
misfit strain field in the thin film. Our goal is to relate film stresses
and system curvatures to the misfit strain distribution, and to ultimately derive a relation between the film stresses and the system
curvatures that would allow for the accurate experimental inference of film stress from full-field and real-time curvature measurements.
2
Governing Equations and Boundary Conditions
Consider a thin film of uniform thickness h f which is deposited
on a circular substrate of thickness hs and radius R 共Fig. 1兲. The
substrate thickness is nonuniform, but is assumed to be axisymmetric hs = hs共r兲 for simplicity, where r and ␪ are the polar coordinates. The film is very thin, h f hs, such that it is modeled as a
membrane, and is subject to nonuniform misfit strain ␧m. Here the
misfit strain is also assumed to be axisymmetric ␧m = ␧m共r兲 for
Journal of Applied Mechanics
冋
冋
ur
Eh dur
+ ␯ − 共1 + ␯兲␧m
2
r
1 − ␯ dr
册
册
dur ur
Eh
N␪ =
+ − 共1 + ␯兲␧m
␯
dr
1 − ␯2
r
共2兲
where h = h f in the thin film and hs共r兲 in the substrate, and once
again the misfit strain ␧m is only in the thin film.
Let w denote the lateral displacement in the normal 共z兲 direction. The curvatures are given by ␬␣␤ = w,␣␤. The bending moments in the substrates are
Mr =
Eshs3
12共1 −
␯s2兲
Eshs3
冉
冉
1 dw
d 2w
+ ␯s
r dr
dr2
d2w 1 dw
+
M␪ =
2 ␯s
dr2 r dr
12共1 − ␯s 兲
冊
冊
共3兲
For nonuniform misfit strain distribution ␧m = ␧m共r兲, the shear
stress along the radial direction at the film/substrate interface does
not vanish, and is denoted by ␶. The in-plane force equilibrium
equations for the thin film and substrate, accounting for the effect
of interface shear stress ␶, becomes
dNr Nr − N␪
+
⫿␶=0
dr
r
共4兲
where the minus sign in front of the interface shear stress is for
the thin film, and the plus sign is for the substrate. The moment
and out-of-plane force equilibrium equations for the substrate are
hs
dM r M r − M ␪
+
+Q− ␶=0
dr
r
2
共5兲
dQ Q
+ =0
dr
r
共6兲
where Q is the shear force normal to the neutral axis. Equation
共6兲, together with the requirement of finite Q at r⫽0, gives Q⫽0.
The substitution of Eq. 共2兲 into 共4兲 yields the governing equations for u and ␶,
d␧m
d2u f 1 du f u f 1 − ␯2f
− =
+
␶ + 共1 + ␯ f 兲
dr2 r dr r2
Efhf
dr
冋冉
dus us
d
hs
+
dr
dr
r
冊册
− 共1 − ␯s兲
1 − ␯s2
dhs us
=−
␶
dr r
Es
共7兲
共8兲
Equations 共3兲, 共5兲, and 共6兲 give the governing equation for w and
␶,
冋冉
d 3 d2w 1 dw
h
+
dr s dr2 r dr
冊册
− 共1 − ␯s兲
1 dhs3 dw 6共1 − ␯s2兲
=
hs␶ 共9兲
r dr dr
Es
NOVEMBER 2007, Vol. 74 / 1277
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The continuity of displacements across the film/substrate interface requires
u f = us −
hs dw
2 dr
Mr −
hs 共 f 兲
N =0
2 r
共11兲
3 Perturbation Method for Small Variation of Substrate Thickness
In the following we assume small variation of substrate thickness
共13兲
where hs0 共⫽constant兲 is the average substrate thickness, and
⌬hs共r兲 is the substrate thickness variation which satisfies
兩⌬hs 兩 hs0; ⌬hs共r兲 is also written as ␤hs1 in 共13兲, where 0
⬍ ␤ 1 is a small, positive constant, and hs1 = hs1共r兲 is on the
same order as hs0.
We use the perturbation method to solve the ODEs analytically
for ␤ 1. Two possible scenarios are considered separately in the
following:
共i兲
The substrate thickness variation ⌬hs is on the same order
as the thin film thickness h f , i.e., ⌬hs ⬃ h f . This is represented by ␤ = h f / hs0 共1兲. For this case the film stresses
and system curvatures are identical to their counterparts
for a constant substrate thickness hs0. This is because the
Stoney formula 共1兲, as well as all its extensions, holds only
for thin films, h f hs. As compared to unity 共one兲, terms
that are on the order of O共h f / hs兲 are always neglected. In
this case the difference between the film stresses 共or system curvatures, ¯兲 for nonuniform substrate thickness hs
and those for constant thickness hs0 is on the order of
O共⌬hs / hs0兲 共as compared to unity兲, which is the same as
O共h f / hs兲 since ⌬hs ⬃ h f , and is therefore negligible.
共ii兲 The substrate thickness variation ⌬hs is much larger than
the thin film thickness h f , i.e., 兩⌬hs 兩 h f . This is represented by h f / hs0 ␤ 共1兲. In the following we focus on
this case and use the perturbation method 共for ␤ 1兲 to
obtain the analytical solution.
Elimination of ␶ from 共7兲 and 共8兲 yields an equation for u f and
us. For h f / hs0 1, u f disappears in this equation, which becomes
the governing equation for us,
冋冉
冊册
冋冕
E f h f 1 − ␯s2 1
us0 =
1 − ␯ f Eshs0 r
␧m =
冉
共15兲
This is a remarkable result that holds regardless of the substrate
thickness and boundary conditions at the edge r = R. Therefore, the
interface shear stress is proportional to the gradient of misfit
strain. For uniform misfit strain ␧m共r兲 = constant, the interface
1278 / Vol. 74, NOVEMBER 2007
2
␲R2
册
共17兲
冕
R
␩ ␧ md ␩
0
冋 冉
冊
冊册
⬘
共18兲
Its general solution is
us1共r兲 = −
+
1
hs1
us0 +
hs0
2r
冕冋
r
␩ 1 + ␯s + 共1 − ␯s兲
0
册
⬘ 共␩兲
r2 hs1
u 共␩兲d␩
␩2 hs0 s0
A
r
2
共19兲
where the constant A is to be determined. The total substrate displacement is then given by
冉
us共r兲 = 2 −
⫻
冊
1
hs
us0 +
hs0
2r
冕冋
r
␩ 1 + ␯s + 共1 − ␯s兲
0
r2
␩2
册
hs⬘共␩兲
␤A
us0共␩兲d␩ +
r
hs0
2
共20兲
The substitution of 共15兲 into 共9兲 yields the governing equation
for the displacement w⬘,
冋冉
hs3 w⬙ +
w⬘
r
冊册
⬘
− 共1 − ␯s兲共hs3兲⬘
6E f h f 1 − ␯s2
w⬘
⬘
h s␧ m
=−
r
1 − ␯ f Es
共21兲
Its perturbation solution can be written as
w⬘ = w0⬘ + ␤w1⬘
共22兲
where w⬘0 is the solution for a constant substrate thickness hs0, and
is given by Huang et al. 关13兴
冋冕
E f h f 1 − ␯s2 1
2
1 − ␯ f Eshs0
r
and once again
E f h f d␧m
1 − ␯ f dr
0
1 − ␯s ␧m
r
1 + ␯s 2
⬘ us0
hs1
hs1
us1 ⬘
us0
= 共1 − ␯s兲
u⬘ +
−
r
hs0 r
hs0 s0 r
⬘ +
us1
The above equation, together with 共8兲, gives the interface shear
stress
␶=−
␩␧m共␩兲d␩ +
is the average misfit strain in the thin film; us1 in 共16兲 is on the
same order as us0. In the following we use u⬘ to denote du / dr.
The substitution of 共16兲 and 共17兲 into 共14兲 and the neglect of
O共␤2兲 terms give the following linear ODE with constant coefficients for us1,
w0⬘ = − 6
− 共1 − ␯s兲
r
and
dhs us E f h f 1 − ␯s2 d␧m
=
共14兲
dr r 1 − ␯ f Es dr
dus us
d
hs
+
dr
dr
r
共16兲
where ␤ 1, us0 is the solution for a constant substrate thickness
hs0, and is given by Huang et al. 关13兴
共12兲
where the superscripts f and s denote the film and substrate, respectively.
hs = hs0 + ⌬hs = hs0 + ␤hs1
us = us0 + ␤us1
共10兲
Equations 共7兲–共10兲 constitute four ordinary differential equations
共ODEs兲 for u f , us, w, and ␶. The ODEs are linear, but have nonconstant coefficients.
The boundary conditions at the free edge r = R require that the
net forces and net moments vanish,
Nr共 f 兲 + Nr共s兲 = 0
shear stress vanishes 共even for nonuniform substrate thickness兲.
We use the perturbation method to write us as
2
␧m =
␲R2
r
␩␧m共␩兲d␩ +
0
冕
1 − ␯s ␧m
r
1 + ␯s 2
册
共23兲
R
␩ ␧ md ␩
0
is the average misfit strain in the thin film; w⬘1 in 共22兲 is on the
same order as w⬘0. Equations 共21兲–共23兲 give the following linear
ODE with constant coefficients for w1⬘
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冉
w1⬙ +
冋 冉
冊
w1⬘ ⬘
w0⬘
6E f h f 1 − ␯s2 hs1
hs1
=−
␧⬘ − 3
w⬙ +
2
r
1 − ␯ f Eshs0
hs0 m
hs0 0 r
+ 3共1 − ␯s兲
+
冕冋
r
0
⬘
⬘ w0⬘
hs1
hs0 r
The displacement u f in the thin film is then obtained from us in
共20兲 and w⬘ in 共26兲 via 共10兲.
The constants A and B, or equivalently, ␤A and ␤B, are determined from the boundary conditions 共11兲 and 共12兲 as
共24兲
Its general solution is
3
hs1
w1⬘ = − 3 w0⬘ +
hs0
2r
冊册
册
r2 h⬘ 共␩兲
␩ 1 + ␯s + 共1 − ␯s兲 2 s1 w0⬘共␩兲d␩
␩
hs0
B
E f h f 1 − ␯s2 1
r+3
2
2
1 − ␯ f Eshs0
r
冕 冋
r
0
册
d
hs1共␩兲
共r2 − ␩2兲
␧m共␩兲d␩
d␩
hs0
⫻
冉
冊
+
冕冋
冕 再
0
册
冕
冕 再关
R
where the constant B is to be determined. The complete solution
for w⬘ is obtained from 共22兲 as
r
3共1 − ␯s兲
R2
␤B = −
共25兲
3
hs
w⬘ +
w⬘ = 4 − 3
hs0 0 2r
1 − ␯s
R2
␤A = −
0
d
d␩
R
0
冕
R
0
R2 − ␩2 hs⬘共␩兲
us0共␩兲d␩
␩
hs0
R2 − ␩2 hs⬘共␩兲
E f h f 1 − ␯s 1
w⬘共␩兲d␩ − 6
2
␩
hs0 0
1 − ␯ f Eshs0
R2
共1 + ␯s兲R2 + 共1 − ␯s兲␩2兴
冋
r
0
冋
d
h s共 ␩ 兲
共r2 − ␩2兲
−1
d␩
hs0
4
␬⌺ = −
E f h f 1 − ␯s2
6
2
1 − ␯ f Eshs0
冉
冦 冕冋
3−2
where
Thin-Film Stresses and System Curvatures
冊 冋
册
hs
hs 3共1 − ␯s兲 hs − hs共0兲 1 − ␯s
␧m + 4 − 3
+
␧m
2
hs0
hs0
hs0
1 + ␯s
r
+
0
冕
3共1 − ␯s兲
␩2
␧m =
␩
␳␧m共␳兲d␳ − ␧m共␩兲
0
2
␲R2
冕
册
hs⬘共␩兲
d␩
hs0
冧
␬⌬ = − 6
冦
冉
␩ ␧ md ␩
0
冊冉 冕
冉 冊 冕冋
4−3
hs
hs0
␧m −
2
hs
E f h f 1 − ␯s2
+
− 1 ␧m − 2
2
hs0
r
1 − ␯ f Eshs0
−
1
r2
冕
r
0
冋
r
2
r2
␩ 2 ␧ m共 ␩ 兲 +
␩␧m共␩兲d␩
0
r
␩
0
冋
u⬘f + ␯ f
uf
− 共1 + ␯ f 兲␧m
r
册
冋
␯ f u⬘f +
uf
− 共1 + ␯ f 兲␧m
r
册
and
共f兲
␴␪␪
=
Ef
1 − ␯2f
where u f is given in 共10兲. The sum of thin film stresses, up to the
O共␤2兲 accuracy 共as compared to unity兲, is related to the misfit
strain by
Journal of Applied Mechanics
␩
冊
册
h s共 ␩ 兲
− 1 ␧m共␩兲d␩
hs0
3共1 + ␯s兲
The thin film stresses are obtained from the constitutive relations
共29兲
R
is the average misfit strain in the thin film. The difference of system curvatures ␬⌬ ⬅ ␬rr − ␬␪␪ is given by
1−
␧m共␩兲d␩
The system curvatures ␬rr = d2w / dr2 and ␬␪␪ = 共1 / r兲共dw / dr兲 are
obtained from 共26兲. Their sum ␬⌺ ⬅ ␬rr + ␬␪␪ is given in terms of
the misfit strain by
共26兲
␯2f
册冎
共28兲
册冎
⫻␧m共␩兲d␩
Ef
h s共 ␩ 兲
−1
hs0
r 2 h ⬘共 ␩ 兲
␩ 1 + ␯s + 共1 − ␯s兲 2 s w0⬘共␩兲d␩
␩
hs0
␤B
E f h f 1 − ␯s2 1
r+3
2
2
1 − ␯ f Eshs0
r
共f兲
␴rr
=
共27兲
2
冕
␩
0
␳␧m共␳兲d␳ +
册
hs⬘共␩兲
3共1 − ␯s兲
d␩
␧m
2
hs0
共f兲
共f兲
␴rr
+ ␴␪␪
=
冧
共30兲
Ef
共− 2␧m兲
1 − ␯f
共f兲
共31兲
共f兲
The difference of thin film stresses ␴rr − ␴␪␪ is on the order of
共f兲
O共共E2f / Es兲␧m共h f / hs0兲兲, which is very small as compared to ␴rr
共f兲
+ ␴␪␪ . Therefore only its leading term is presented
共f兲
共f兲
␴rr
− ␴␪␪
= 4E f
冋 冕
2
E f h f 1 − ␯s2
␧m − 2
r
1 − ␯2f Eshs0
r
0
␩␧m共␩兲d␩
册
共32兲
4.1 Special Case: Uniform Misfit Strain. For uniform misfit
strain distribution ␧m = constant 共and nonuniform substrate thickNOVEMBER 2007, Vol. 74 / 1279
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ness兲, the interface shear stress in 共15兲 vanishes. The thin film
stresses become constant and equibiaxial, and are given by
共f兲
共f兲
␴rr
= ␴␪␪
=
Ef
共− ␧m兲
1 − ␯f
共33兲
The curvatures in 共29兲 and 共30兲 become
␬⌺ = − 12
再
冉
5 − ␯s hs
E f h f 1 − ␯s
1−
−1
2
1 − ␯ f Eshs0
2
hs0
+ 共1 − 2␯s兲
冎
hs − hs共0兲
␧m
hs0
E f h f 1 − ␯s2
␬⌬ = 18
2
1 − ␯ f Eshs0
再
hs 2
−
hs0 r2
冕
r
0
冊
冎
共34兲
h 共␩兲
␩ s d␩ ␧m
hs0
which are neither constant nor equibiaxial for varying substrate
thickness.
Figure 2 shows a substrate with a step change in thickness; a
uniform thickness h in the outer region 共r ⬎ Rin兲 and a slightly
different value h − ⌬h in the inner region 共r ⬍ Rin兲, where
Fig. 2 „a… A schematic diagram of a thin film/substrate system
with a step change in substrate thickness. „b… The normalized
system curvatures ␬ˆ rr = ␬rr / ␬0 and ␬ˆ ␪␪ = ␬␪␪ / ␬0, where ␬0
= 6„Efhf / 1 − ␯f… / „1 − ␯s / Esh2…␧m, ⌬h / 2h = 0.1, ␯s = 0.27, and Rin
= R / 3.
Ⲑ
兩⌬h 兩 h. The average thickness becomes hs0 = h − ⌬h共 R2in R2 兲.
The curvature in the circumferential direction is
E f h f 1 − ␯s
␧m
␬␪␪ = − 6
1 − ␯ f E sh 2
冦
冋
冋
册
1+
2
Rin
⌬h
5 − ␯s − 共1 − ␯s兲 2
2h
R
1+
2
2
Rin
Rin
⌬h
5 − ␯s − 共1 − ␯s兲 2 − 3共1 + ␯s兲 1 − 2
2h
R
r
for r ⬍ Rin
冉 冊册
for r ⬎ Rin
冣
共35兲
冣
共36兲
which 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 = − 6
E f h f 1 − ␯s
␧m
1 − ␯ f E sh 2
冦
冋
冋
册
1+
2
Rin
⌬h
5 − ␯s − 共1 − ␯s兲 2
2h
R
1+
2
2
Rin
Rin
⌬h
5 − ␯s − 共1 − ␯st兲 2 − 3共1 + ␯s兲 1 + 2
2h
R
r
for r ⬍ Rin
冉 冊册
for r ⬎ Rin
The continuous ␬␪␪ and discontinuous ␬rr are illustrated in Fig. 2. Similar discontinuity in ␬rr has been observed for varying thin film
thickness 关17,18兴.
It should be pointed out that the results in this section hold for discontinuous substrate thickness. This is because the film stresses in
共31兲 and 共32兲 depend only on the misfit strain and are independent of substrate thickness. The system curvatures in 共29兲 and 共30兲 involve
the derivative of substrate thickness hs⬘, which is not well defined for a discontinuous hs. However, it appears only in the integration
such that 共29兲 and 共30兲 still hold.
In the following, we extend the Stoney formula for arbitrary nonuniform misfit strain distribution and nonuniform substrate thickness.
5
Extension of Stoney Formula for Nonuniform Misfit Strain Distribution and Nonuniform Substrate Thickness
In this section we extend the Stoney formula for arbitrary nonuniform misfit strain distribution and nonuniform substrate thickness by
establishing the direct relation between the thin-film stresses and substrate curvatures. We invert the misfit strain from 共29兲 as
1 − ␯ f Es
␧m = −
6E f h f 1 − ␯s2
where
冦
1 − ␯s 2
hs ␬⌺
2
hs2␬⌺ −
+
−
1
2
冕关
冕
R
共1 − 3␯s兲␬⌺共␩兲 − 3共1 − ␯s兲␬⌬共␩兲兴hs2共␩兲
r
1 − ␯s
R2
R
␩2关␬⌺共␩兲 − ␬⌬共␩兲兴hs2共␩兲
0
hs2␬⌺ =
2
R2
冕
hs⬘共␩兲
d␩
hs0
hs⬘共␩兲
d␩
hs0
冧
共37兲
R
␩hs2␬⌺d␩
0
is the average of hs2␬⌺, and we have used 共30兲 in establishing 共37兲.
1280 / Vol. 74, NOVEMBER 2007
Transactions of the ASME
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The thin film stresses are obtained by substituting 共37兲 into 共31兲 and 共32兲 as
共f兲
␴rr
+
共f兲
␴␪␪
共f兲
共f兲
␴rr
− ␴␪␪
=−
=
Es
3共1 −
␯s2兲h f
2E f hs0
␬⌬
3共1 + ␯ f 兲
冦
hs2␬⌺ −
+
−
1
2
冕
1 − ␯s 2
hs ␬⌺
2
R
关共1 − 3␯s兲␬⌺共␩兲 − 3共1 − ␯s兲␬⌬共␩兲兴hs2共␩兲
r
1 − ␯s
R2
冕
R
␩2关␬⌺共␩兲 − ␬⌬共␩兲兴hs2共␩兲
0
共39兲
Equations 共38兲 and 共39兲 provide direct relations between film
stresses and system curvatures. The system curvatures in 共38兲 always appear together with the square of substrate thickness, i.e.,
hs2␬⌺ and hs2␬⌬. It is important to note that stresses at a point in the
thin film depend not only on curvatures at the same point 共local
dependence兲, but also on curvatures in the entire substrate 共nonlocal dependence兲 via the term hs2␬⌺ and the integrals in 共38兲. For
uniform substrate thickness, 共38兲 and 共39兲 degenerate to Huang
et al. 关13兴
The interface shear stress ␶ can also be directly related to system curvatures via 共15兲 and 共37兲
␶=
Es
6共1 −
␯s2兲
再
hs⬘
1
d 2
共hs ␬⌺兲 − 关共1 − 3␯s兲hs2␬⌺ − 3共1 − ␯s兲hs2␬⌬兴
2
dr
hs0
冎
共40兲
Equation 共40兲 provides a way to determine the interface shear
stresses from the gradients of system curvatures once the full-field
curvature information is available. Since the interfacial shear
stress is responsible for promoting system failures through
delamination of the thin film from the substrate, Eq. 共40兲 has a
particular significance. It shows that such stress is related to the
gradient of ␬rr + ␬␪␪, as well as to the magnitude of ␬rr + ␬␪␪ and
␬rr − ␬␪␪ for nonuniform substrate thickness.
In summary, 共38兲–共40兲 provide a simple way to determine the
thin film stresses and interface shear stress from the nonuniform
misfit strain in the thin film and nonuniform substrate thickness.
关5兴
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
References
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关4兴 Wikstrom, A., Gudmundson, P., and Suresh, S., 1999, “Analysis of Average
Journal of Applied Mechanics
关17兴
关18兴
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d␩
hs0
hs⬘共␩兲
d␩
hs0
冧
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NOVEMBER 2007, Vol. 74 / 1281
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