1. 1
M20100376
Development of an Analytical tool for Multilayer Stack
Assemblies
Rammohan B , Sanjay Singh Chauhan, Arvind Krishna
Delphi Automotive Systems
Copyright © 2011 SAE INDIA
ABSTRACT
The development of an Analytical model for
multilayer stack subjected to temperature
change is demonstrated here. Thin continuous
layers of materials bonded together deform as
a plate due to their differing coefficients of
thermal expansion on subjecting the bonded
materials to the change in temperature.
Applications of such structures can be found in
the electronic industry for the study of
warpage issues in printed circuit boards or in
the aerospace industry as laminated thin
sheets used as skin structures for load bearing
members such as wings and fuselage. In
automotive electronics, critical high-power
packages (IGBT, Power FETs) include several
layers of widely differing materials (Aluminum,
Solder, Copper, Ceramics) subjected to wide
temperature cyclic ranges. Modeling of such
structures by the application of three
dimensional finite element methods is usually
time consuming and may not exactly predict
the inter-laminar strains. An attempt has been
made here to obtain the closed form solution
for such a multilayered stack using a set of
recursive polynomial equations on subjecting
the stack to temperature change under steady
state conditions. Based on the closed form
solution technique, a simple excel based tool
has been developed to predict the radius of
curvature, bending strains at both the bottom
and the top layer of stack and compared with
the Numerical standard codes. Results show a
close comparison for both the analytical and
the finite element simulation. Analytical
solutions are further extended to predict the
interlayer displacements.
INTRODUCTION
Co-efficient of Thermal expansion (CTE) and
Fatigue induced due to thermal loads on
electronic devices are a major concern for the
failure of the electronic devices. Inter-laminar
strains that are developed due to the bending
of the stack results in de-bonding of such
multilayered assemblies. In this study, a
relatively simple analytical excel based tool is
developed based on linear first order
polynomial equations to obtain the closed form
solution to predict the radius of curvature,
warpage, bending strains at both top and
bottom layers of the stack. Commercially
available software - Ansys has been used to
verify the VBA- Excel analytical tool.
ANALYTICAL MODEL
An analytical model based on conventional thin
plate theory has been used here. The
assumptions are such that the dimensions of
the stack in the thickness direction are less
than the other two directions. The stack tends
to deform symmetrically on the application of
temperature gradient. Bending moment, radius
of curvature, in-plane forces, and the strains at
both top and bottom layer of the stack are
deduced using the set of equations in [1].

2. 2
Fig 1a: Illustrative example: undeformed stack
Fig 1b: Illustrative example: Deformed stack
The stack shown in the Fig 1a & 1b above has
the same dimensions even in the out of plane
direction. Radius of curvature of the stack in
the in-plane direction is larger than thickness of
the stack.
r
tE
M
i
ii
i
1
)1(12
3

 = 2
2
dx
wd
From the basic static equilibrium at the neutral
axis, the summation of all forces and moments
about the neutral axis is zero.
(In the Equation 3 above the index k ranges
from 1 to j).
Since the interlayer displacements (u) between
the two consecutive layers are the same, and
then re-writing equations we get,
Equations 1 to 4 are a set of recursive
polynomial equations which are solved
numerically using FORTRAN code[1]
. In this
paper a closed form solution is developed
using the following algorithm with F1 and 1/r
being the primary unknowns.
 iF and in the flow chart
above leads to a set of recursive equations
which can be reduced to 2x2 matrix of the form
[A]{B}={C}. The coefficients A11,A12,A21 and
A22 obtained as a function of Young’s
Modulus, thickness and Poisson’s Ratio is
shown below.
0 iF
0)
2
(   j
kji
t
tFM
   
r
tt
T
tE
F
tE
F ii
ii
ii
ii
ii
ii
2
)1()1( 1
1
11
11 


 






0)
2
(   j
kji
t
tFM
 
 
 
 
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
 
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i
j
i
j
i
ii
N
i
i
N
k k
kk
n
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j
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ii
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j
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ii
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j
j
N
i i
ii
t
tT
tE
T
tE
t
t
tE
ttt
A
t
t
tE
tE
t
A
T
tE
T
tE
ttt
A
tE
tE
A
T
T
r
F
AA
AA
12
1
1
3
1 1
1
1
22
1211
11
21
2
1
2
1
1
12
11
1
2
11
1
2221
1211
2)1(
)1(122
21
2
)
2
(
1
1
2
1
12
)(2
1
)
1
(1
1













(1)
(2)
(3)
(4)
(5)
r

3. 3
Thus a closed form equation is obtained as a
function of only two variables which can be
solved using simple matrix inversions. By
knowing the force in the first layer and the
radius of curvature of the stack , bending
moments and forces in all other layers can be
computed. This can be further extended to
obtain the forces and moments in all layers of
stack, using the recursive functions explained
above.
COMPARISON USING ANSYS
A 3-D Finite Element model has been created
in Ansys to exactly simulate the Multilayered
stack when subjected to a temperature
change under steady state condition. 2-D and
3-D structural solid elements [2,3]
have been
used for creating the three dimensional FE
Model. Each layer has a different co-efficient of
thermal expansion, thickness and also has
different Modulus of Elasticity.
A close correlation is observed between VBA
excel tool outputs and FE results shown in
Table-1 and 2. Similarly analytical tool gives a
very close comparison for prediction of
warpage shown in Fig (4) in the out of plane
direction by providing the inputs of the stack
assembly assumed in Pan et al[1]
.
Fig 2: Displacements for three layered stack
Table 1: Comparison of results for a 3 layered stack
Comparison between the Multi-stack tool and
the Ansys results are tabulated in the Example
1. Similar comparison for a five layered stack is
shown below in Table 2.
Fig 3: Displacements for five layered stack
Table 2: Comparison of results for a 5 layered stack
Fig 4: Warpage from the excel tool for a 5 layered
stack – A close correlation with Pan et al [1]
.
Fig 5: Intermediate displacements from the excel
tool for a 5 layered stack –A close correlation with
Pan et al [1].

4. 4
CONCLUSION
First order polynomial linear equations have
been used for developing closed form solutions
using thin plate theory. This correlates well with
3-D finite Element method (FEM) and can be
used for quicker study for linear models without
actually modeling the 3-D structure in FEM.
The results obtained from the tool are the
warpage, bending strains both at the top and
bottom layer of stack, forces and moments on
each layer of the stack.
REFERENCES
1. Tsung-Yu Pan and Yi-Hsin Pao,
“Deformation in Multilayer stack
assemblies”, ASME Journal of Electronic
Packaging, Vol.30, pp. 30-34, 1980.
2. Cook, R. D., Concepts and Applications of
Finite Element Analysis, 2nd ed. Wiley, New
York, 1981
3. Theory Reference for ANSYS and ANSYS
Workbench: - Release 11.0 Documentation
for ANSYS
Symbols used
D = flexural rigidity, Nmm2
E = Young's modulus, N/mm2
F= Force, N
L= length and width of each layer in the stack,
mm
M = Bending moment, N mm
R = Radius of curvature, mm
ΔT= change in temperature
t = thickness of each layer
u= Displacement in x-direction, mm
V = Displacement in y-direction, mm
w= Displacement in z-direction, mm
Subscripts
1,2,..., j ,...
α = thermal expansion coefficient, mm/mm/°C
v = Poisson's ratio
j= layer indicator