This paper presents an elastic analysis of a multilayered system due to the gradient residual stresses. Given the residual strain distribution, the elastic deformation of a multilayered structure can be determined using the derived general solution. Bimaterial cantilevers are widely used as sensing and actuation components in MEMS devices. Assuming the residual strain in each layer of such a bimaterial cantilever is linearly distributed, the error induced by evaluating its curvature without the gradient item is studied. The distributions of normalized deformation strain and real strain in a bilayered cantilever is also given. The positions of deformation plane and neutral plane in that cantilever as well in three more cases are summarized. The presented results in this paper shows that missing the gradient residual stress in multilayered systems may lead to serious misunderstanding of the deformation, and therefore, the model built in this paper may help in the development of multilayered structures in MEMS devices.

1.
Lavrik
N V
,
Sepaniak
M J
, and
Datskos
P G
,
2004
,
Cantilever transducers as a platform for chemical and biological sensors
,
Rev. Sci. Instrum.
,
75
,
1
25
.
2.
Lai
T
,
Shi
Z
,
Perazzo
T
, and
Majumdar
A
,
1997
,
Optimization and performance of high-resolution micro-optomechanical thermal sensors
,
Sensors Actuators A
,
58
,
113
9
.
3.
Senesac
L R
,
Corbeil
J L
,
Rajic
S
, and
Datskos
P G
,
2003
,
IR imaging using uncooled microcantilever detectors
,
Ultramicroscopy
,
97
,
451
8
.
4.
Binnig
G
,
Despont
M
,
Drechsler
U
,
Haberle
W
,
Lutwyche
M
,
Vettiger
P
,
Mamin
H J
,
Chui
B W
, and
Kenny
T W
,
1999
,
Ultrahigh-density atomic force microscopy data storage with erase capability
,
Appl. Phys. Lett.
,
74
,
1329
31
.
5.
Chu
W-H
,
Mehregany
M
, and
Mullen
R L
,
1993
,
Analysis of tip deflection and force of a bimetallic cantilever microactuator
,
J. Micromech. Microeng.
,
3
,
4
7
.
6.
Nix
W D
,
1989
,
Metall. Trans. A
,
20A
,
2217
40
.
7.
Zhang
X
,
Chen
K-S
,
Ghodssi
R
,
Ayvon
A A
, and
Spearing
S M
,
2001
,
Residual stress and fracture in thick tetraethylorthosilicate (TEOS) and silane-based PECVD xxide films
,
Sensors Actuators A
,
91
,
373
80
.
8.
Marcus
P M
,
1996
,
Curvature of multilayer epitaxial films
,
J. Appl. Phys.
,
79
,
8364
6
.
9.
Yang
E H
,
Yang
S S
, and
Yoo
S H
,
1995
,
A technique for quantitative determination of the profile of the residual stress along the depth of p+ silicon films
,
Appl. Phys. Lett.
,
67
,
912
4
.
10.
Kamiya
S
,
Sato
M
,
Saka
M
, and
Abe
H
,
1999
,
Residual stress distribution in the direction of the film normal in thin diamond films
,
J. Appl. Phys.
,
86
,
224
9
.
11.
Thompson
C V
and
Carel
R
,
1996
,
Stress and grain growth in thin films
,
J. Mech. Phys. Solids
,
44
,
657
73
.
12.
Fang
W
and
Wickert
J A
,
1995
,
Comments on measuring thin-film stresses using bi-layer micromachined beams
,
J. Micromech. Microeng.
,
5
,
276
81
.
13.
Fang
W
and
Wickert
J A
,
1996
,
Determining mean and gradient residual stresses in thin films using micromachined cantilevers
,
J. Micromech. Microeng.
,
6
,
301
9
.
14.
Greek
S
and
Chitica
N
,
1999
,
Deflection of surface-micromachined devices due to internal homogeneous or gradient stresses
,
Sensors Actuators
,
78
,
1
1
15.
Hubbard
T
and
Wylde
J
,
2000
,
Residual strain and resultant postrelease deflection of surface micromachined structures
,
J. Vac. Sci. Technol. A
,
18
,
734
7
.
16.
Stoney
G G
,
1909
,
The tension of metallic films deposited by electrolysis
,
Proc. R. Soc. London, Ser. A
,
82
,
172
5
.
17.
Brenner
A
and
Senderoff
S
,
1949
,
Calculation of stress in elctrodeposits from the curvature of a plated strip
,
J. Res. Natl. Bur. Stand.
,
42
,
105
23
.
18.
Atkinson
A
,
1995
,
Macro- and microstress analysis in solgel derived Pb(ZrxTi1-x)O3 thin films
,
Br. Ceram. Proc.
,
54
,
1
7
.
19.
Freund
L B
,
Floro
J A
, and
Chason
E
,
1999
,
Extension of the Stoney formula for substrate curvature to configurations with thin substrate or large deformations
,
Appl. Phys. Lett.
,
74
,
1987
9
.
20.
Klein
C A
,
2000
,
How accurate are Stoney’s equation and recent modification
,
J. Appl. Phys.
,
88
,
5487
9
.
21.
Hsuch
C-H
,
2002
,
Modelling of elastic deformation of multilayers due to residual stresses and external bending
,
J. Appl. Phys.
,
91
,
9652
6
.
22.
Schweitz
J-A
,
1991
,
A new and simple micromechanical approach to the stress-strain characterization of thin coatings
,
J. Micromech. Microeng.
,
1
,
10
5
.
23.
Min
Y-H
and
Kim
Y-K
,
2000
,
In situ measurement of residual stress in micromachined thin films using a specimen with composite-layered cantilevers
,
J. Micromech. Microeng.
,
10
,
314
321
.
24.
Hou
M T-K
and
Chen
R-S
,
2004
,
A new residual stress measurement method using ultra-wide micromachined bilayer cantilevers
,
J. Micromech. Microeng.
,
14
,
490
6
.
25.
Vinson J R and Chou T W, 1975, Composite Materials and Their Use in Structures (London: Elsevier-Applied Science Publishers)
26.
Vinson J R and Sierakowski R L, 1986, The Behavior of Structures Composed of Composite Materials (Kluwer Acadmeic Publishers).
27.
Freund
L B
,
1996
,
Some elementary connections between curvature and mismatch strain in compositionally graded thin films
,
J. Mech. Phys. Solids
,
44
,
723
736
.
28.
Finot
M
and
Suresh
S
,
1996
,
Small and large deformation of thick and thin-film multi-layers: effects of layer geometry, plasticity and compositional gradients
,
J. Mech. Phys. Solids
,
44
,
683
721
.
29.
Pulskamp
J S
,
Wickenden
A
,
Polcawich
R
,
Piekarski
B
, and
Smith
G
,
2003
,
Mitigation of residual film stress deformation in multilayer microelectromechanical systems cantilever devices
,
J. Vac. Sci. Technol. B
,
21
,
2482
6
.
30.
Malzbender
J
,
2004
,
Mechanical and thermal stresses in multilayered materials
,
J. Appl. Phys.
,
95
,
1780
2
.
31.
Townsend
P H
,
Barnett
D M
, and
Brunner
T A
,
1987
,
Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate
,
J. Appl. Phys.
,
62
,
4438
44
.
This content is only available via PDF.
You do not currently have access to this content.