Finite Element Simulation and Parameter Optimization
of a Flexible Tactile Pressure Sensor Array
by
Shira M. Lee
Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the
Requirements for the Degree of
Rachelnrnf qcience
at the
Massachusetts Institute of Technology
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MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
June 2005
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0 8 2005
LIBRARIES
©)2005 Shira M. Lee. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole or in part.
A
Signature of Author:
Department of Mechanical Engineering
X~~f
)May
)~
Certified by:
6,2005
-
Mandayam Srinivasan
Senior Research Scientist
Thesis Supervisor
)
Accepted by:
Ernest Cravalho
Chairman, Undergraduate Thesis Committee
.ARCHIVES
Finite Element Simulation and Parameter Optimization
of a Flexible Tactile Pressure Sensor Array
by
Shira M. Lee
Submitted to the Department of Mechanical Engineering on May 6, 2005 in Partial Fulfillment
of the Requirements for the Degree of Master of Science in Mechanical Engineering
ABSTRACT
A finite element model was developed to optimize design of a flexible tactile sensor. The sensor
consists of layers of thin-film copper and PDMS, and the model can be used to determine the
effects on sensor sensitivity and durability of variations in material properties and geometry.
The model was used to study the effect of variations in copper thickness. Four copper
thicknesses, 0.3gm, 0.5pm, 3[im, and 9pm, were analyzed under a range of pressure loads.
The thickness of the copper affected both the stress in the material and the displacement of the
copper when a pressure load was applied to the sensor model. The stress in the sensor was
highest for 3pm copper, potentially causing decreased durability in this sensor. The separation
between the copper strips beneath the pressure load was highest for 9plm copper, so this sensor
may have lower accuracy for small loads. Thin copper strips are challenging to manufacture, so
the largest but most accurate and durable copper strip thickness, 0.5ptm, is recommended
this analysis.
Thesis Supervisor: Mandayam Srinivasan
Title: Senior Research Scientist
2
from
Acknowledgements
I am grateful to Dr. Gang Liu for his guidance, teaching, and assistance throughout this
project. I also thank Mandayam Srinivasan for supporting the project and Prof. Ian Hunter and
Dr. Andrew Taberner for generously allowing me to use their ANSYS software.
3
Table of Contents
Abstract..........................................................................................
Acknowledgements
.............................................................................
2
3
1.0 Introduction and Project Background...................................................
1.1 Use of Finite Element Analysis for the Sensor................................
5
6
1.2 Goal for Current Analysis ........................................................
7
1.3 Sensor Structure and Parameters ................................................
1.3.1 Material Properties .....................................................
2.0 Finite Element Analysis ..................................................................
2.1 Primary Assumptions ............................................................
3.0 Results ......................................................................................
3.1 Displacement .....................................................................
3.2 Stress ..............................................................................
3.3 Electrostatic Analysis ............................................................
4.0 Discussion and Conclusions .............................................................
4.1 Finite Element Capabilities ......................................................
4.2 Interpretation of Results and Recommendations .............................
4.3 Future Use of the Model.........................................................
Appendix A: System of Units and Material Property Data..............................
Appendix B: Model development procedure ..............................................
Appendix C: Contour plots of FEA solutions...........................................
4
8
9
1
15
16
16
20
23
27
27
30
30
32
34
37
1.0 Introduction and Project Background
Sensors that mimic the sense of touch by providing feedback about forces on their
surfaces have an extensive range of applications. If the sensor is thin, flexible and robust, it can
be used as a coating for robotic hands so robots can take feedback on the objects they touch.
Such sensors can also cover artificial organs used to train surgeons, so the output from the
sensors can indicate whether the surgeons should apply more or less force to the organs. One of
these tactile sensors is being developed in the MIT Laboratory for Human and Machine Haptics.
This sensor is designed to be thin, flexible and robust and to produce an electric signal that
indicates the location and magnitude of any forces on the surface.
The sensor is composed of thin-film copper and a silicone-based polymer called
polydimethylsiloxane (PDMS). The thin-film copper is deposited in two layers using an
evaporation method in a vacuum chamber. There are two different compositions of PDMS used
in the sensor, and the copper strips are deposited with polymer A between them and polymer B
along the outside (Figure 1). Each copper layer is not continuous, but is instead composed of
rows of uniformly spaced parallel strips. The copper strips in one copper layer are laid
perpendicular to the copper strips in the second layer, so that each copper strip crosses every
copper strip of the opposite layer, at a perpendicular intersection (Figure 2). When a pressure is
applied to the surface of the sensor, the copper strips at all nearby intersections (nodes) are
brought closer to each other, and the capacitance of the sensor changes. A voltage is applied to
one end of each copper strip, and the opposite end is connected to a computer-based
signal
processing circuit that processes variations in the sensor's electrical signal to indicate the
magnitude and location of surface forces.
Copper
Polymer A
Figure 1: The sensor has five layers, an inner layer of polymer A, two copper
strips, and outer layers of polymer B.
5
q
1
~k
46.5mm
l.
11 j1k 1 IIII
5
__
-Y1
i
1.5mm
-I
-t
-t
4
mm
Wm"
now
mm
am
-0
mm
NW
/Ai'11111114r1
I\
Bottom electrode
copper strap
Ternl
lead Bording Point
s
COpprsp
Top electrode
copper
copstrap
Figure 2: Sensor array with two layers of copper strips. The strips in each layer
are parallel but the two layers are perpendicular. The sensor has 32X32 sensor
elements and is about 0.3tm thick. Image courtesy of Dr. Gang Liu.
To perform as accurately as possible, the sensor must be sensitive to variations in
pressure near the range OMPato 0.1MPa. The sensor is designed to be used to mimic human
skin, and this range of pressures is the range to which human skin is most sensitive.' Both the
geometry and material properties of the sensor materials can be adjusted to improve the sensor
performance. The geometry of the thin-film copper affects its structural and electromagnetic
properties. The width and thickness of the copper strips can be varied to adjust the stiffness of
the sensor and the sensitivity of its electrical output. The PDMS is formed by blending a base
and a catalyst, and varying the ratio of base to catalyst changes the material properties of the
polymer. The base-to-catalyst ratio can be varied separately for polymer A and polymer B to
adjust their stiffness. The stiffness of these individual polymers and the thickness of each
polymer layer can be adjusted, along with the copper geometry, to produce a sensor with desired
sensitivity to deformation.
1.1 Use of Finite Element Analysis for the Sensor
Finite element analysis (FEA) is a powerful tool for investigating the effects of variations
in physical parameters on the behavior of a structure. The analysis is most useful when
parameter variations are difficult, expensive, or too numerous to test efficiently by
experimentation. FEA involves building a software model of the geometry and loading of a
Data provided by Dr. Gang Liu
6
structure. The structure is then divided into finite elements (meshed), and the state of any
mechanical, electrical, or thermal parameters of interest are determined across each element. If
the elements are small enough, the finite solution is an accurate but efficient method of
approximating the solution for an infinite number of points across the structure.
Optimization of sensor design involves variation of numerous parameters, including the
geometry of each layer and the properties of each material, under a range of expected loads.
Optimization through experimentation alone would be extremely time-consuming, as numerous
combinations of small variations in geometry, material type, and load would have to be carefully
manufactured and then tested. Finite element analysis can be used to test many combinations
quickly, by varying the parameters in a model and comparing the analysis results. Once an
optimized set of parameters is determined with FEA, these parameters can be tested
experimentally to confirm the accuracy of the model optimization and to make slight adjustments
in the sensor.
A variety of finite element analysis software packages are commercially available. All
analysis for this project was done using ANSYS 9.0 University Research Edition.
1.2 Goal for Current Analysis
The copper strips in the sensor are thin-film copper several hundred nanometers thick.
Changes in the thickness of the copper affect the stiffness of the strips and the overall stiffness of
the sensor. The stiffness of the sensor changes the deformation between copper strips resulting
from pressure loads. The deformation, in turn, determines the electrical output of the capacitive
sensor. Thicker copper strips may increase the stiffness of the sensor and decrease its sensitivity,
but sensors with thick strips are much easier to fabricate. Ease of fabrication must be balanced
with sensitivity, and finite element analysis can be used to determine the range of copper
thicknesses that can produce the desired sensor sensitivity.
Finite element models of a segment of the sensor with four different thicknesses of
copper were built and meshed. The models were analyzed under a range of pressure loads to
determine their sensitivity. The deformation in each model was analyzed quantitatively to
determine which copper thicknesses would produce distinct deformations according to the
magnitude of the load applied. The capacitance of the sensor is directly related to the separation
between the copper strips, so the separation was determined as a function of pressure magnitude
for each copper thickness. The thickness of the copper also affects the stress in the copper strips.
If the axial stress reaches the yield stress of the material, the durability of the sensor decreases.
The axial stress along the copper strips was determined as a function of pressure magnitude and
copper strip thickness.
The models built for analysis of copper strip thickness can be easily modified to test the
effect of changes in material properties of any of the three materials by varying the material
property data and re-solving the model. The models can also be modified to test the effect of
changes in geometric parameters besides copper thickness, such as copper width and polymer
thickness.
The geometric and material parameters can be varied individually, to isolate the effect
of one parameter, or can be varied in combination to optimize the complete sensor design. The
models built for this analysis form a foundation for extensive future analysis.
7
1.3 Sensor Structure and Parameters
All geometric and material properties except the thickness of the copper were set
according to sensor prototypes currently being developed at MIT. The model is a twodimensional cross-section of the sensor. The cross-section is cut along one copper strip in the
bottom layer so that it bisects the perpendicular strips in the top layer. The cross-section was
selected to be long enough to bisect three top strips, so the model includes three copper nodes
and the space between each node.
The dimensions of the model are shown in Figures 3 and 4. Each copper strip is I mm
long, and the space between each strip is 0.Smm. The bottom layer of PDMS, polymer B, is
200pm thick. The bottom layer of copper strips is deposited on top of this outer polymer, and
then a layer of polymer A 20tm thick is laid on top of the copper. The top layer of copper strips
is then laid on top of this middle polymer layer, and a top layer of polymer B 1 0im thick is
used to cover the top of the copper. Four thicknesses (th(ou)of the copper strips were tested:
0.3pm, 0.5pm, 3tm, 9jtm.
thcu
Figure 3: Vertical dimensions of the sensor model. The thickness of the copper
layers is varied in each model between 0.3pm and 9pm.
1.0mm
1.0mm
0.5mm
0.5mm
1.0mm
1.0mm
0.5mm
0.5mm
1.0mm
1.0mm
Figure 4: Horizontal dimensions of the sensor model. Each copper node is 1.0mm
wide and the space between each node is 0.5mm.
8
1.3.1 Material Properties
The material properties of each material in the sensor are listed in Table 1. The
dimensions are all in the gMKSv (microMeter-Kilogram-Second-Volt) system of units, a system
developed to be used in nano-scale electromechanical devices. The conversion from MKS units
to pMKSv is included in Appendix A.
Table 1: Material Properties used in the sensor modell
Mechanical Parameter
Young's Modulus
FMKSv Unit and Value
MPa=kg/(im)(s) 2
Copper
1.1 OE+05
PDMS, middle
PDMS, outer
Poisson's ratio
Copper
PDMS
Thermal Parameter
Conductivity
0.0682
0.2029
(constant)
0.343
0.499
pW/([am)K=(kg)( gm)/(K)(s)3
Copper
3.85E+08
PDMS
Specific Heat
Copper
PDMS
1.50E+05
2
pJ/(kg)K=(im) 2/(K)(s)
3.85E+14
1.46E+15
Electrical Parameter
Conductivity
pS/[m=(pA)^2(s)
Copper
PDMS
3
/(kg)( gm) 3
58823529.41
2.50E-14
Resistivity
T0,um=(kg)(
Copper
PDMS
Relative Permittivity
Copper
PDMS
(constant)
Permittivity of free space
pF/gim
2
(S) 3
0.017
4.00E+19
9.99E+59
2.75
~8.85E-06
____________________
Permeability of free space
am) 3/(pA)
TH/grm
Amrani, Liu & Aluru. "Re-configurable Fluid Circuits by PDMS Elastomer Micromachining;"
12th International Conference on MEMS, MEMS 99, pp.222-227, Orlando, FL, 1998.
"Copper, Cu; Annealed," MatWeb.com, The Online Materials Database; Automation Creations,
1996.
CRC Handbook of Chemistry and Physics, CRC Press, 2004.
Polymer Data Handbook, Oxford University Press, 1999.
9
1.26E-24
The stiffness of each polymer was determined experimentally from polymers used in
sensor prototypes. Experiments were conducted to measure increments in stress and strain for
uniaxial compression and then release of the compression for each polymer sample. The stressstrain curves are all convex up, but there is some hysteresis between the curves generated in push
and release. The polymers were approximated as linear materials to simplify the model, and the
slope of the linear approximation of the push curve was used as the estimate of Young's
Modulus. The stress-strain curves for each polymer are shown in Figures 5 and 6. Both
polymers are from the Dow Coring Corporation; the middle polymer is brand name HS IV, and
the outer polymer is brand name Sylgard 184.
u.U0300
0.0250
0.0200
0
* push
A release
10
(n
om.o15o
-
Linear (push)
0.0100
0.0050
0 000
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
Strain
Figure 5: Experimental uniaxial compression data for the middle layer of PDMS
(polymer A). A linear approximation of the push data was used to determine a
value of 0.0682MPa for Young's Modulus of polymer A. Data supplied by Dr.
Liu.
10
* push
A release
-Linear (push)
IL
£
uX
i
0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600 0.0700 0.0800 00900 0.1000
Strain
Figure 6: Experimental uniaxial compression data for the outer layers of PDMS
(polymer B). A linear approximation of the push data was used to determine a
value of 0.2029MPa for Young's Modulus of polymer B. Data supplied by Dr.
Liu.
For all structural analysis, the bottom line of the sensor was fixed for all degrees of
freedom, because in practice the sensor is fixed along the bottom surface. A uniformly
distributed pressure load was applied along the top line of the top polymer layer over the center
node of copper, to simulate distributed pressures on the top surface of the sensor.
The mechanical parameters of greatest interest were the vertical (y) component of
displacement, especially the minimum distance between the copper strips, and the horizontal (x)
component of stress. The distance between the copper strips determines the capacitance of the
sensor, so the vertical displacement of each strip is directly related to the electrical output of the
sensor. The stress in the copper strips affects their durability, especially their likelihood to fail or
fatigue during use.
2.0 Finite Element Analysis
Finite element modeling consists of four primary steps. First, a solid model of the
geometry is constructed. For the sensor, a two-dimensional cross-section was drawn across three
nodes (intersections of the upper and lower copper strips). The dimensions of this solid model
were set according to Figures 3 and 4. Four models were built, one for each thickness of copper.
The computing power and time required for FEA increases with the number of elements
in a model. An effective method of minimizing the number of elements is to cut the model along
any plane of symmetry, apply symmetry boundary conditions to all planes of symmetry, and
analyze the smallest unique portion of the model. The sensor model is symmetric about a
vertical line down the middle of the center node, so only the left half of the sensor model was
11
meshed. A symmetry boundary condition was enforced along the middle vertical line of the
center node.
The second step in building a finite element model is to divide the model into finite
elements. This is called meshing. The type and shape of element affects the accuracy and
complexity of the solution. For the sensor model, planar, rectangular elements with eight nodes
were chosen. The division of each area into elements was controlled manually to produce the
desired number of elements in each layer of the model. To analyze variation in parameters
across the copper and the middle layer of polymer, for example, there had to be multiple
elements across the thickness of each of these thin layers. The horizontal divisions for these thin
layers were also controlled, to produce elements with moderate aspect ratios. Elements with
moderate aspect ratios produce more accurate solutions than long thin elements. At the same
time, the number of elements needed to be minimized to allow the computer to process the
analysis. The mesh of the sensor model is shown in Figures 7-10.
Outer node
f
Outer
node:CentenodeI
Center node
,
~,,
.
s
,
,
~~I ~~s
Plane of
I
I
I
II
II
';
I
3I
Figure 7: Mesh of a segment of the sensor, with copper 0.5[tm thick.
12
Concentrated
mesh
Figure 8: Mesh of the center copper node with O.5gmthick copper; the elements
closest to the copper and middle polymer layer are smallest, to accurately model
stress and displacement distributions near these thin areas.
Polymer B
Polymer A
Copper
strips
Polymer B
Figure 9: Mesh of the sensor with 0.51amcopper strips. The magnification shows
the middle layer of polymer with the thin copper strips above and below it and the
outer polymer surrounding the copper.
13
I
I
I
I
I
I
V
I
I
I
I
I
I
}Copper
elements
Figure 10: Mesh of the copper model with 0.5pm copper strips. The
magnification shows the mesh within the copper strips, which has multiple layers
to allow variation in stress and displacement across the thickness of the copper.
In the third step of the analysis, mechanical or electric loads were applied to the finite
element model. The mechanical loads used on the sensor included a constraint on the bottom
line for all degrees of freedom. The second constraint of interest was a pressure on the top line
of the model representing a pressure on the upper surface of the sensor. A distributed pressure
load was applied to the top of the top layer of polymer over the center node of the model. All
mechanical forces are shown in Figure 11. Five distinct analyses were conducted to solve each
model for five different pressure magnitudes, 0.02 MPa, 0.04MPa, 0.06 MPa, and 0.08 MPa.
Distributed
pressure
Plane of
Fixed surface (all degrees of freedom)
.-
symmetry
Figure 11:Mechanical loads on the sensor model, including pressure on the top
surface of the center node, symmetry at the middle of the center node, and zero
displacement along the bottom surface.
14
The final step in FEA is solving the model and processing results. Each model was
solved to determine the magnitude of vertical (y) displacement and horizontal (x) stress
throughout the sensor after application of each load magnitude. The analysis was static because
the desired information corresponded to the final state of the sensor after application of the load.
For each solution, contour plots of the vertical displacement and horizontal stress were produced
throughout the sensor, with magnified plots of specific areas of interest. The magnified plots
included the point of maximum horizontal stress, the vertical displacement of the copper strips at
the center of the pressure load, and the displacement and stress at the edge of the node under
pressure.
A more detailed list of analysis steps for this model, including specific ANSYS
commands, is included in Appendix B.
2.1 Primary Assumptions
A primary goal of this project was to develop a finite element model that was functional,
efficient, and could be modified to analyze the effect of a large variety of parameters. Several
assumptions were made to simplify the model to allow it to be processed efficiently in ANSYS.
Some of these assumptions can be modified in future analyses to reflect the more complex (often
nonlinear) behavior of the materials in the sensor.
For this analysis, the material properties of copper were assumed to be the material
properties of bulk copper. Nano-scale copper has material properties that depend on its
geometry, but these properties are not well understood. It is necessary to do experimental
materials testing of specific copper geometries to determine the nano-scale properties. The
initial FEA was used to narrow the range of copper geometries to test experimentally. Once such
testing is done to determine the material properties of copper for the thicknesses most likely to be
used in the sensor, this more accurate material data can be input to the model.
For electrical analysis, the copper was assumed to be a perfect conductor with a relative
permittivity of infinity. However, the maximum value of relative permittivity allowed in
ANSYS is le60, so this value was used for copper's relative permittivity instead of infinity.
The PDMS has a Poisson's ratio of approximately 0.5, so the maximum value allowed in
ANSYS, 0.499, was used for both PDMS variations. The stiffness of both PDMS variations was
calculated from a linear approximation of nonlinear experimental data for uniaxial compression.
Poisson's ratio and all electrical properties were assumed to be standard for PDMS and to not
differ between the two PDMS variations, polymers A and B, used in the sensor. Experimental
testing of these properties for each PDMS variation is necessary if greater accuracy is desired.
Before conducting these experimental tests for numerous PDMS variations, the stiffness of the
PDMS for an optimized sensor can be determined using the model. Experimental testing can
then be conducted to determine the base-to-catalyst ratios that produce the desired stiffness.
PD)MScompositions that most closely produce the desired stiffnesses can be experimentally
tested to determine their actual stress-strain curves, these curves can be used in new FEA, and
optimized polymers can be developed through iterations of this cycle. Analysis of PDMS
behavior with cyclical loads should also be analyzed and tested because of the hysteresis
apparent in the stress-strain curves between push and release. This hysteresis should be
incorporated into material models for simulations of sensor loading and unloading so that the
sensor behavior can be optimized for the repetitive loading it will encounter in applications.
15
3.0 Results
The most important parameters investigated using FEA were the vertical displacement of
the copper strips relative to each other and the magnitude of horizontal stress in the copper.
Contour plots were generated using ANSYS to display these parameters for each model.
The model consists of three copper nodes with a pressure applied across the center one,
but the model is symmetric about a vertical line down the middle of the center node. All results
are shown with the left half of the sensor only; the right half is a mirror image of the contours in
the figures. All figures are drawn to scale and all units are [MKSv (microMeter, Kilogram,
Second, Volt).
3.1 Displacement
The vertical displacement of the sensor with 0.5tm thick copper under a load of 0.06MPa
is shown in Figure 12. Similar contour plots of vertical displacement were produced for loads of
0.02MPa, 0.04MPa, 0.06MPa, and 0.08MPa for copper strips of thicknesses 0.3pm, .5pm, 3pm,
and 9tm. All results are shown in Appendix C. As shown in Figure 12, the highest magnitude
of displacement occurs directly under the pressure load. The highest magnitude of displacement
is negative because of the downward pressure force, so ANSYS designates this maximum
(negative) displacement
as MN. The greatest upward displacement is designated as MX in the
figures because it is the greatest displacement in the positive y-direction.
NODAL
AN
SOLUTION
MAY
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=O-DM4X =80.788
SMN =-80.765
SMX =1.213
1
005
15:12:39
Distributed
pressure
Maximum
displacement
Plane of
symmetry
4
Max upward
displacement
Figure 12: Vertical displacement (m) of the sensor with 0.51m copper under a
load of 0.06MPa.
The capacitance of the sensor is determined by the distance between the copper strips, so
the primary locations of interest for vertical displacement were at the center of pressure and at
the minimum distance between the copper strips. For some of the models, the minimum
16
separation of the copper strips occurred at the edge of the copper node, while for others it
occurred at the center of pressure. A magnified image of the vertical displacement at the center
of pressure for the sensor with 0.5pm thick copper under a load of O.6MPa is shown in Figure
13. The position of the copper strips is known before application of the load, so the vertical
displacement can be used to determine the distance between the strips after deformation.
Copper
strips
Plane of
symmetry
4-
Figure 13: Vertical displacement (m) of the 0.5tm thick copper strips under a
load of 0.06MPa.
The vertical line on the far right is the plane of symmetry and
the center of the pressure load.
The O.5pmthick copper has smaller strip separation at the center of pressure than at the
node edge for every load magnitude. The vertical displacement at the edge of the copper node
for 0.51m thick copper under a load of 0.6MPa is shown in Figure 14. For the sensor with 9Im
thick copper, however, the smallest strip separation is located at the node edge for all load
magnitudes, rather than at the center of pressure. This does not occur because the 9ptm strips
deform more, but because pressure on the strips deforms the polymer between the strips at the
edge of the copper node. The vertical displacement at the edge of the copper node for 9pamthick
copper under a load of 0.06MPa is shown in Figure 15.
17
AN
NODAL SOLUTION
MAY 1 200g
3TEP=1
SUB =1
15:14:30
UY
RSYS=O
DMX =80. 788
SMN =-80.785
SMX =1. 213
dge of top
-
opper strip
Copper
strips
1..1Zi
-19. Zu6
-43. 886
-68. 486
Figure 14: Vertical displacement (pm) of 0.5gm-thick copper under a weight of
.06MPa, at the edge of the center copper node.
AN
NODAL SOLUTION
MAY 1
005
12:59:00
STEP=1
SUB =1
TIME=1
(AVG)
UY
RSYS=0
DMX =70.53
, ,_
OM =-70.53
81M2 =1.293
ls\-
is.
Edge of top
copper strip
./
g?
Copper
strips
-70.53
-59. 756
-48.983
-38.21
-27.436
-16.663
-5.889
1.293
Figure 15: Vertical displacement (pm) of 9pm thick copper under a weight of
0.06MPa, at the edge of the center copper node. The edge of the copper strip is
the location of the shortest distance between the copper strips.
18
The separation between the copper strips at the center of pressure (middle of the center
node) and edge of the center node are listed in Table 2, along with the location of the minimum
distance between copper strips.
Table 2: Location of minimum distance between copper strips
Copper
thickness (pm)
Pressure
(MPa)
Separation at
pressure center (pm)
Separation at
node edge (m)
Location of minimum separation
9
0.02
17.61
15.21
Edge of node
9
9
3
3
3
3
0.5
0.5
0.5
0.5
0.3
0.3
0.3
0.3
0.04
0.06
0.02
0.04
0.06
0.08
0.02
0.04
0.06
0.08
0.02
0.04
0.06
0.08
12.82
9.23
15.80
11.60
7.40
3.20
15.90
11.80
7.70
3.60
15.90
11.79
7.69
3.58
10.42
5.64
15.80
11.60
7.40
3.20
17.27
14.53
11.80
9.07
17.26
14.53
11.79
9.06
Edge of node
Edge of node
Equal at edge and center
Equal at edge and center
Equal at edge and center
Equal at edge and center
Center of pressure
Center of pressure
Center of pressure
Center of pressure
Centerof pressure
Center of pressure
Center of pressure
Center of pressure
The capacitance of the sensor depends on the average distance between the copper strips
at each node. This separation distance is affected by the separation at the center of pressure and
at the edge of the node, but the electric field spreads out at the edge of the node so the separation
distance at the edge has less influence on capacitance than the separation at the node's center.
The separation at the center of the node was used as a standard of comparison for vertical
displacement. This center separation is plotted in Figure 16 as a function of the thickness of the
copper strips and the magnitude of the pressure load. The distance between the copper strips has
low dependence on the thickness of the strips for the three thinnest strips, though the dependence
increases with pressure magnitude. The differences between the three thinnest copper
thicknesses are less than 1% for the 0.02MPa load, rising to only 11% for the 0.08MPa load.
The difference between the copper separation for the 9m thick copper and the 3m thick copper
for the 0.06MPa load is much higher, about 24%. (Accurate results were not achieved for the
0.08MPa load on the 9pm copper.)
19
20
,14
....I
18
16
0
1.
1
0.5um
* *CU
( 12
*9u
CU
*3um CU
to 0
0
0.3umrn
CU
8-
6
(U
(D4
a4
a
.R
:> 2
0
0.00
0.02
0.06
0.04
0.08
0.10
Pressure (MPa)
Figure 16: Distance (m) between the two layers of copper strips at the center of
pressure, as a function of the magnitude of the pressure load and the thickness of
the copper strips.
3.2 Stress
The horizontal component of stress was plotted throughout the sensor, and the image
generated for the 0.5gm copper under a load of 0.06MPa is shown in Figures 17 and 18. The
stress state throughout the polymer layers is approximately uniform, but the magnitude of stress
in the copper increases where the copper is deformed. The maximum magnitude of horizontal
stress is indicated by MX for the maximum positive stress and MN for the maximum negative
stress. Figure 17 shows the location of the maximum magnitude of horizontal stress, marked by
MX.
20
ne of
imetry
Figure 17: Horizontal component of stress (MPa) in sensor with 0.5pm thick
copper strips under a weight of 0.06MPa. The location of the maximum stress
magnitude is marked by MX and is located near the edge of the center node.
AN
NODAi SOLUTON
t"=1
-- _ (AVG)
..--
Wa47.q*8
~~~~~~~~~~~~~~~~1:
13:Z
MAY1 2005
-
'
:"'
::.
=..
__--__
_,_,,_
Max
magnitude
of x-stress
-'
- -
> Copper
strips
4 -27. 98
-27.98
-1-4604
228
-1.228
.4.
25.524 .
--
._..
Figure 18: Horizontal component of stress (MPa) in sensor with 0.5pm thick
21
copper strips under a weight of 0.06MPa, magnified at the point of maximum
stress magnitude.
The maximum and minimum of stress values in each model were located in the sensor
and recorded. These values are plotted as a function of copper thickness and pressure magnitude
in Figure 19. The increase in stress as a function of pressure load is perfectly linear for each
thickness of copper, as is expected with perfectly linear materials. The percent difference
between stresses in different copper geometries under a given pressure load is the same for every
load magnitude.
150
100
a
50
* 9um max
A
*
A
*
0
0A
e.
0
0
0.01
0X2
0.03
0.04
3umrn
min
0.05
0.06
0.07
~~~A
a
9umrn
min
* 3um max
A ,
o
0.08
o
A
-50
0.
A 0.5um max
A 0.5ummin
* 0.3ummax
o 0.3um min
A
0~~~~~
1-
^~~~~~~~~~~~~~~~~~~~~~
-100
-150
Pressure(MPa)
Figure 19: Horizontal stress in the sensor plotted as a function of pressure
magnitude for each thickness of copper. For each geometry, the increase in stress
with pressure magnitude is linear (zero measurable error).
The location of maximum and minimum displacement and stress varied with different
thicknesses of copper. The deformed sensor is divided into regions in Figure 20. The location of
the displacement and stress extremes are listed in Table 3 according to the letter in Figure 20 to
which their location most closely corresponded. The maximum and minimum stress are always
located in close proximity to each other, so they are listed as one entity.
22
Center node
Outer node
A-
t -IA
I
I
I
I
I
I
B
C
D
E
F
G
H
I
Figure 20: Division of the model into regions for description of location of
extreme values of displacement and stress. This model is the vertical
displacement of the sensor with 0.S5imthick copper strip under 0.06MPa. The
max upward displacement is located at D and the max magnitude of displacement
is located at H, as designated in Table 3.
Table 3: Location of Extreme Displacement and Stress Values for each Copper Thickness.
Copper
Max and Min XMax upward YMax magnitude YThickness (prm) Stress Location
Displacement
Displacement Location
Location
(downward)
9
3
0.5
0.3
_
H, bottom CU strip
E, bottom CU strip
G, top CU strip
G, top CU strip
A
C
D
D
I
I
H
H
The maximum magnitude of displacement occurs near the pressure center in all cases.
The maximum upward displacement value is close to zero; it is where the sensor moves in the
opposite direction of the pressure force. This occurs because the pressure bends the sensor down
at the center node and the copper strips tilt so the sections far from the pressure are lifted. The
location of this maximum upward displacement is farther from the pressure sensor for thicker
copper because the thicker copper bends less easily. There is less bend in the thicker copper
strips, so they reach their maximum lift farther away from the pressure center.
The extreme values of stress are located where the copper is deformed and under
pressure, often near bends in the strips. The extreme values of stress are always in the copper,
not the polymer. The maximum and minimum values are located close to each other, on the top
and bottom of the same copper strip. For the thinner copper strips they are located in the top
copper strip, and for the thicker copper strips they are located in the bottom copper strip.
3.3 Electrostatic Analysis
The finite element model was designed to be used for either mechanical or electrostatic
analysis, or a coupled electromechanical analysis. Although electrostatic behavior was not the
23
subject of this project, the model was solved for an electrostatic solution to make sure it was
capable of producing such a solution. The same geometry and mesh density were used for the
electrostatic and mechanical models, but the element type was changed to an electrostatic
element compatible with the mechanical element type. The mechanical loads were removed for
the electrostatic analysis, and electric loads were applied to the model. Electric loads included a
voltage of 20V applied to the top layer of copper strips and a voltage of OV(ground) applied to
the bottom layer of copper. The copper was idealized as a perfect conductor, with uniform
voltage throughout.
The undeformed model was analyzed to determine the electric potential and electric field
throughout the undeformed sensor. The electric potential in the sensor is shown in Figures 21
and 22.
NODAL
AN
SOLUTION
MAY
STEP=1
SUB =1
TIME=1
VOLT
RSYS=0
SMX =20
2 2005
14:36:38
(AVG)
Edge of top
copper strip
X-
I
I
I
l
0
6
3
Copper
strips
12
18
9
15
20
Figure 21: Electric potential (V) in the undeformed sensor (with 0.5pm thick
copper strips). Loads of 20V were applied to each copper strip in the top layer
and a load of OVwas applied to the bottom copper strip.
24
NODAL SOLUTION
STEP=i1
SUB =1
TIME=I
VOLT
RSYS=0
SMX =20
Edge of top
copper strip
(AVG)
\
11\
Copper
strips
Figure 22: Electric potential (V) in the undeformed sensor (with O.51amthick
copper strips) at the edge of a top copper strip.
The electric field resulting from the same load condition is shown in Figure 23. The
electric field is concentrated where the copper strips are in close proximity and reaches a
maximum on the top copper strip at the edges of each copper node. The area at the edge of one
node is shown in Figure 24.
25
NODAL
AN
SOLUTION
STEP=1
SUB =1
TIME=1
EFSUM
RSYS=0
SMX =3.27
MAY
2 2005
14:34:44
(AVG)
Edge of top
copper strip
I
i
I:
Copper
strips
!
-,
0
.490559
tttl'
=
.981118
1.472
1.962
2.453
2.943
3.27
Figure 23: Electric field (V/pm) in the undeformed sensor (with 0.5pm thick
copper strips) with a load of 20V on each top copper strip and a load of OV on the
bottom copper strip.
, Edge of top
copper strip
Copper
strips
Figure 24: Electric field (V/jm) in the undeformed sensor (with 0.51amthick
copper strips) with a load of 20V on each top copper strip and a load of OVon the
bottom copper strip, magnified at the edge of one copper node.
26
The electrical and mechanical models can be coupled to conduct an analysis that accounts
for both electrical and mechanical behavior. The loads can either be applied and analyzed
simultaneously, or they can be sequentially coupled, with a solution produced for an applied
pressure and then for a voltage applied to the deformed structure. This coupled analysis can be
used in the future to determine how mechanical and electric behavior influence each other and to
achieve solutions that more accurately reflect experimental behavior.
4.0 Discussion and Conclusions
Each solution was tested to determine whether the finite element model's behavior
corresponded to the theories used to run the analysis, and comparisons of the specific solutions
were used to develop recommendations for sensor optimization.
4.1 Finite Element Capabilities
The finite element model produced results that corresponded to theoretical predictions of
actual behavior under the model's simplified assumptions. Magnitude of stress and
displacements had approximately a linear relationship to load magnitude, for one geometric
condition, as is expected with linear material stiffness. For loads between 0.02MPa and
0.06MPa, the elements deformed according to conservation of mass, so they were thinned and
elongated under pressure, with deformations applied from each element to its neighbors. The
same was true for loads of 0.08MPa on all copper thicknesses except 9pm. An accurate mesh
deformation is shown at the center of pressure in Figure 25 and at the edge of the center node in
Figure 26. In these figures, the number of elements in the deformed and undeformed mesh are
equal, but the elements in the deformed mesh have changed shape under pressure.
Copper 1.11-
-
l- ...
... 1- 111 I''..''....
Copper
Polymer A
I= = - =:
= =d := =.__=
I==:_ = :I= := -
Polymer A
Copper.
1-1.1 .... 1111.11.111-11-....... I ---.
Coppeir
t
l
I--1
-
-ii iii
l
Ii
I I
T
Plane of -- ,
symmetry
':
Plan e of
I
symi rnetry
I
Figure 25: Mesh near the center of pressure for 3m thick copper strips under
0.08MPa pressure. The figure on the left is the undeformed mesh (no load) and
the figure on the right is the deformed mesh. There are an equal number of
elements showing in the deformed mesh and the undeformed mesh, but the
elements in the deformed mesh are thinned and elongated under pressure.
27
Edge
of top
opperstrip PolymerA
Polymer
dgeof top
)pperstrip
Copper
strips
> Copper
strips
I
lrUi
,
/.U.
IVII1
1
s
taal LIB
UF
U
f'+I,- '11tl6+- -Atl!
IIUJ
1UVI JttIll
+lk.-I,
lilN
1
UIJA
q+-.'3LI
plJa
under 0.08MPa pressure. There are an equal number of elements showing in the
undeformed mesh (left) and the undeformed mesh (right), but the elements in the
deformed mesh are bent under pressure.
However, for loads of O. MPa on all geometries and 0.08MPa on the 9gm copper
thickness, the size of the elements was not sufficient to capture the large deformations. A small
gradient in deformations from one element to the next was not possible, so elements put under
greatest pressure were displaced past the neighboring elements, and the elements overlapped
each other. The overlapped deformation occurred at the center of pressure or at the edge of the
center node, at the location of minimum strip separation.
A deformed mesh for 3pgmand 0.5 gm thick copper strips under loads of 0.1MPa are
shown in Figure 27 for the section near the center of pressure. In the 3gm model, the elements of
the copper strips have not only overlapped each other, but have completely overlapped the
polymer between them. In the 0.5tm model, the middle layer of polymer is visible between the
copper strips, but thin sections of overlapped polymer elements are visible near the top and
bottom edge of the polymer layer.
'4-Pro
5
1
i
Figure 27: Deformed mesh of the copper strips and middle layer of polymer near
the center of pressure in the 3tm model (left) and 0.5pm model (right) under a
load ofO. IMPa. In the 3tm model, the copper strips completely overlap the
middle polymer layer. In the 0.5gtmmodel small sections of the overlapped
polymer elements are visible at the top and bottom of the layer.
28
In some models, the overlapped mesh occurred at the edge of the center copper node
instead of at the center of pressure. One such case, for 9tm thick copper strips under a load of
0.1 MPa, is shown in Figure 28. Here, the copper strips overlap each other and the layer of
polymer between them.
- Edge of top
copper strip
Polymer A
I
Element
overlap
Figure 28: Deformed mesh in the 9m model near the edge of the center node
under 0.1MPa. The elements of the top copper layer are displaced past the middle
polymer layer and part of the bottom copper layer, and the elements are too large
to capture this displacement.
The mesh was too large to accurately solve the model for a load of 0.1 MPa on all copper
thicknesses and for a load of 0.8MPa on the 0.9gm thick copper. To improve the mesh, the
number of horizontal rows of elements in the copper and middle polymer layers must be
increased. The height of each element decreases as the number of rows within each material
layer increases. In order to form rows of elements, the horizontal width of each element cannot
be much larger than the height of each element. As a result, the number of elements in the
copper and middle polymer layers must be increased drastically for small increases in the
number of element rows. The mesh density in these material layers could not be increased high
enough to produce accurate results without exceeding the element limit imposed by the available
version of ANSYS.
The second limitation was the thickness of the copper strips that could be tested. In order
to accurately capture the stress variation across the copper strips, there had to be several layers of
elements across the thickness of each strip. For thinner copper strips, the maximum thickness of
each element layer was reduced. To mesh the model with rows of elements in these layers, the
spacing of the elements along the length of each strip had to be increased for thinner copper
strips, and so the number of elements in the model increased. For copper strips of 0.1I[m, the
total number of elements required to mesh the model exceeded the number of elements in the
available version of ANSYS. The thinnest strips accurately analyzed were 0.3Rm thick.
29
4.2 Interpretation of Results and Recommendations
The finite element models analyzed for this project were used to determine how copper
strip thickness affects the sensitivity and durability of the sensor. This information can be used
to choose a limited range of copper strip thickness to be experimentally tested for verification of
the model's predictions and further optimization.
The analysis shows that the separation between copper strips 9pm thick at the center of
pressure is significantly (24%) more than thinner strips. Sensors produced with this thick copper
are easier to manufacture but produce different electrical signals than sensors with thinner strips.
These 9gm sensors may be less sensitive to changes in pressure load, especially at small load
magnitudes. Also, the 9gm copper strips are brought so close together at the edge of the copper
node that the sensor may resist deformation or may be damaged at high loads. It is necessary to
test the variations in capacitance carefully through experimentation to determine whether the
9pm strips produce an accurate sensor.
The stress in the copper is 28.8% higher in the 0.5gm thick strips than in the 0.3gm thick
strips, 22.8% higher in the 91m thick strips than in the 0.3gm thick strips, and 10.2% higher in
the 3gm thick strips than in the 9tpm thick strips. The higher the stress in the copper strips, the
more likely the copper is to decrease in performance quality through use. The highest stress is in
the 3gm thick copper strips, so the sensor made with these strips has the least durability. The
copper should be tested to determine whether the stress level in the 3tm thick copper strips
would significantly affect their behavior under the expected loads and through the expected
lifetime of the sensor. If the stress level significantly decreases the durability of the copper, a
different strip thickness should be used to ensure accuracy of the sensor throughout its lifetime.
Loads of 0.1MPa are expected in the applications of the sensor, but these loads could not
be tested using this finite element model. Sensors should be studied carefully to make sure loads
above 0.08MPa are accurately converted to electrical signals, because it is possible that the
deformation at these load levels is so great that the material does not deform according to the
same patterns observed for smaller loads.
According to the analysis, the 0.5gm and 0.3gm thick copper strips combine high
durability and high sensitivity to produce sensors that are expected to have the highest quality
performance of the geometries tested. The thinner copper strips are more challenging to
manufacture, so the 0.5pm copper strips are optimal for use in the sensor.
4.3 Future Use of the Model
Finite element analysis is a powerful tool for optimizing the design of the sensor, but it
should not be used without experimental verification and iterations with improved modeling
accuracy. The results of this analysis should be used to choose copper strip thicknesses near
0.5gm to be tested experimentally. The actual material properties and durability of nano-scale
strips of this thickness should be determined and input to the model to improve the model's
accuracy. The model should also be used to vary the stiffnesses and thicknesses of polymers A
and B to determine which stiffnesses and thicknesses are optimal for each layer and to vary the
width of the copper strips and spacing between them to determine the optimal geometry for the
sensor. Once these and any additional parameters of interest are optimized through FEA,
materials with the optimized characteristics should be produced and tested experimentally to
verify the model and to further optimize the design. Nonlinear material properties from
30
experimental data should be input to the model and analyzed. Finally, sensors with the
optimized parameters should be produced and tested experimentally to verify the whole model
and to fine-tune the sensor design. Finite element modeling significantly reduces the number of
combinations of parameters that must be tested to optimize the sensor design, but it cannot
replace experimentation. This model forms a foundation for extensive future analysis and
experimental testing.
31
Appendix A
System of Units and Material Property Data
Table Al: Conversion from MKS units to gMKSv (microMeter-Kilogram-Second-volt) units I
Multiply
by this
Number
To
Obtain
pMKSv
Unit
pm
pN
Dimension
pm
(kg)(pm)/(s)2
Mechanical
Parameter
Length
Force
MKS
Unit
m
N
Dimension
m
(kg)(m)/(s)2
Time
s
s
1
s
s
Mass
kg
kg
1
kg
kg
Pressure
Pa
kg/(m)(s)
2
01 E-06
MPa
kg/(pm)(s)
2
kg/(m)(s)
3
01 E-06
MPa
kg/(pm)(s)
2
Pa
kg/(m)(s)
4
01 E-06
MPa
kg/(pm)(s)
2
Thermal
Parameter
Conductivity
Specific Heat
Electrical
Parameter
W/mK
J/(kg)K
(kg)(m)/(K)(s)A^
(m)2/(K)(s)2
Voltage
V
(kg)(m)
Stress
Young's
Pa
Modulus
2
01E+06
1.OOE+06 pW/(pm)K
1.OOE+12 pJ/(kg)K
(kg)(pm)/(K)(s) ~
(m)2/(K)(S) 2
/(A)(s)~
1
V
(kg)(pm) 2/(pA)(s)3
Conductivity
S/m
(A) (s)3/(kg)(m)3
1
pS/pm
(pA)A2(s)3/(kg)(pm)3
Resistivity
Qm
(kg)(m)3/(A)2' (s)3
Permittivity
F/m
(A)'(s)
2
4
4
/(kg)(m ~)
4
Capacitance
F
(A)2(s) /(kg)(m )
Electric Field
V/m
V/m
1.OOE+06 Topm
(kg)(pm)3/(pA)^2(s)3
1.OOE+06
pF/pm
(pA) 2 (s) /(kg)(pm)3
1.OOE+12
pF
(pA) 2 (s) 4 /(kg)(pm) 2 '
V/pm
(kg)(pm)/(s)3(pA)
1.OOE-06
System of Units, ANSYS Coupled-Field Analysis Guide, ANSYS Release 8.1 Documentation Preview
32
Table A2: Material properties conversion from MKS to pMKSv units
Mechanical
Parameter
Young's
Modulus
MKS
Unit
Dimension
Pa
kg/(m)(s)4
Copper
1.10OE+ 11
1.10E+11
01E-06
PDMS, middle
PDMS, outer
Poisson's ratio
Copper
46614
202854
constant
0.343
4.66E+04
2.03E+05
01E-06
01E-06
PDMS
Thermal
Parameter
Conductivity
Copper
PDMS
Specific Heat
Copper
PDMS
Electrical
Parameter
Conductivity
Copper
PDMS
Resistivity
Copper
PDMS
Relative
Permittivity
Copper
01E-06
To Obtain
pMKSv
Unit
Dimension
MPa
kg/(pm)(s)2
1.10OE+05
0.0682
0.20285
in ANSYS
1.OOE+00
0.343
must be
below 0.5
0.499
constant
3.43E-01
0.5
W/mK
Multiply by
this Number
5.00E-01
(kg)(m)/(K)(s)3
1.1E+05
0.0682
0.20285
in ANSYS
0.343
0.499
385
1460
1.00E+06 pW/(pm)K
1.OOE+06 3.85E+08
1.OOE+06 1.50E+05
1.OOE+12 pJ/(kg)K
1.OOE+12 3.85E+14
1.OOE+12 1.46E+15
(kg)(pm)/(K)(s)3
3.85E+08
1.5E+05
(pm) 2/(K)(s)2
3.85E+14
1.46E+15
S/m
5.88E+07
2.50E-14
Dm
1.70E-08
4.00E+13
(A)2(s)3/(kg)(m)3
58823529.4
2.5E-14
(kg)(m)3/(A)2(s)3
1.7E-08
4E+13
1 pS/pm
1
5.88E+07
1
2.50E-14
1.00E+06 Topm
1.OOE+06
0.017
1.OOE+06 4.00E+19
(pA)f(s)~/(kg)(pm)3
58823529.412
03E-14
(kg)(pm)3/(pA)2 (s)3
0.017
4.00E+19
constant
constant
385
0.15
J/(kg)K
385
1.46E+03
(m)2/(K)(s)2
infinite
385
0.15
infinite
in ANSYS
must be
below 1e60
33
9.99E+59
in ANSYS
9.99E+59
Appendix B
Model Development Procedure
These are instructions according to my preferred modeling and analysis methods. There are
alternate approaches, but these instructions can be used to replicate the model that produced the
results included in this paper. These instructions used the GUI because I am more experienced
with this interface; however all actions have equivalent commands for the command-line
interface.
Designate a structural discipline for any mechanical analysis and an electromagnetic discipline
for an analysis that includes electricity. This is done in Preferences.
Enter the preprocessor.
Set the element type. Use Plane82 for structural analysis and Plane 121 for electric field analysis.
These element type were chosen because they are planar eight-node elements and are compatible
for coupled electromechanical coupled analysis.
Menu Path: Preprocessor>Element Type>Add
Input material parameters. There are three materials used in this model, copper and two
polymers.
The material properties for all these materials are listed in Table 1. Input the material
properties into the material models. All properties were inputted in gMKSv (microMeter,
Kilogram, Second, Volt) units, which are listed in Appendix A. These properties can be written
to a material data file to be used in future models.
Menu Path: Preprocessor>Material Props>Material Models
Menu Path: Preprocessor>Material Props>Write to/Read from File
Build the geometry.
The geometry consists of rectangular areas, which were built according to the dimensions shown
in Figures # and #. The geometry was constructed by creating one rectangle and dividing it into
the components of the sensor. After divisions, a merge command must be used to ensure there
are no duplicate entities. The model is symmetric, so, to produce an efficient analysis, only half
the geometry was modeled and symmetry boundary conditions were used along the vertical
middle line of the center node (see Loads).
Menu Path: Preprocessor>Modeling>Create>Areas>Rectangle>By 2 Corners
Menu Path: Preprocessor>Modeling>Operate>Booleans>Divide
Menu Path: Preprocessor>Numbering Controls>Merge Items
Menu Path: Preprocessor>Numbering Controls>Compress Numbers
Mesh the model
The model must be meshed very carefully to ensure that the mesh is fine enough in all areas,
especially those with concentrated loads or extreme dimensions, but large enough to be
processed with the available computing power. First, element and material types must be
assigned to each area.
Menu Path: Preprocessor>Meshing>Mesh Attributes>Picked Areas
34
Size Controls
The size of the elements was set manually. The number of divisions required for each line varied
with copper thickness and increased with pressure load. These size controls were chosen to
make sure the variations in displacement and stress across each material layer were accurate, the
mesh was small in concentrated areas but large along the edges of the model (to reduce the
number of required elements), and the aspect ratios of individual elements were not extreme.
In order to use create a mesh that is concentrated along one side of each area and expands across
the areas, the number of elements on the sides of each area must be equal and the difference
between the number of elements on the top and bottom of each model must be an even number.
Each area must be meshed with free meshing, so the elements can conform to the changing size
controls across each area.
Menu Path: Preprocessor>Meshing>Size Contrls>ManualSize>Lines>Picked Lines
Menu Path: Preprocessor>Meshing>Mesh>Areas>Free>Pick
All
Element Connectivity
Merge and renumber all items to ensure that there are no extra nodes or elements, so all entities
at the same location are considered one entity.
Menu Path: Preprocessor>Numbering Controls>Merge Items
Menu Path: Preprocessor>Numbering Controls>Compress Numbers
Loads
For mechanical models, set the displacement of all bottom lines of the model to be zero, for all
degrees of freedom. This edge of the model is fixed. Add a pressure load to the line on top of
the center node of copper, indicated by the arrow in Figure #. The pressure loads used in this
analysis ranged from 0.02-0. MPa, corresponding to intervals in the Og-lOg range intended for
experimentation. A symmetry boundary condition must be placed on the plane of symmetry in
the middle of the center copper node.
Menu Path: Preprocessor>Loads>Define Loads>Apply>Structural>Displacement>On Lines
Menu Path: Preprocessor>Loads>Define Loads>Apply>Structural>Pressure>On Lines
Menu Path: Preprocessor>Loads>Define Loads>Apply>Structural>Displacement>Symmetry
B.C.>On Lines
For electrical models, all applied loads were voltages on boundaries. A voltage of 20V was
applied to the areas corresponding to copper segments in the top row, and a voltage of 0V was
applied to the areas corresponding to the copper in the bottom row.
Menu Path: Preprocessor>Loads>Define Loads>Apply>Electric>Boundary>Voltage>On Areas
Coupled Analysis
For coupled analysis, it is necessary to save the electric physics environment and the structural
physics environment. The physics environment includes the geometry, mesh, and loads in a
model. Physics environments can be written when the model is ready to be solved and then read
whenever the environment should be applied in the coupled analysis.
Menu Path: Preprocessor>Physics>Environment>Write
35
Solution
All analysis conducted were static analysis, which is the default in ANSYS. Once the loads and
mesh are created, the model can be solved.
Menu Path: Solution>Analysis Type>New Analysis
Menu Path: Solution>Solve>Current
LS
Post processing
Once the model is solved, the last set of results should be read and then plotted. Plots can
include the deformed mesh of the model and contours of the vertical (y) displacement or
horizontal (x) stress. The number of contours can be increased by setting the plot controls, and
hardcopies of the images can be written to image files.
Menu Path: General Postprocessor>Read Results>Last Set
Menu Path: General Postprocessor>Plot Results>Deformed Shape
Menu Path: General Postprocessor>Plot Results>Contour Plot>Nodal Solu
Menu Path: General Postprocessor>Plot Results>Contour Plot>Element Solu
Utility Bar: PlotCtrls>Style>Contours>Uniform Contours>Number of contours
Utility Bar: PlotCtrls>Device Options>Use extra colors for>Contours
Utility Bar: PlotCtrls>Hard Copy>To File
36
Appendix C
Contour plots for all load cases for all copper thicknesses.
NODAL
AN
SOLUTION
MAY
STEP=1
SUB
1 2005
12:41:06
=1
TIME=1
UY
(AVG)
RSYS=O
DMX
=23.51
SMN
=-23.51
SMX
=.430989
-16.328
-23.51
-19.919
-9.145
-1.963
-12 .737
Figure C 1: Y-dclisplacement (m)
-5.554
.430989
of copper 9[tm thick under a pressure of O.O2MPa.
305
:17
-so
-23.51
-16.328
-19.919
-^
.,
-M
-9.145
-12.737
-5.554
W
-1.96
.4308
.·430989
Figure C2: Y-displacement (pm) of copper 9pm thick under a weight of 0.02MPa. The vertical
line farther right is the center of pressure.
37
A
NODAL SOLUTION
MAY 1 2005
12: 41:55
STEP=1
SU =1
TIME=1
SX
(AVG)
RSYS=0
DMX =Z3.51
SMN =-20.423
SMX =20.633
.~-~~ ..B____
.
______________~~
-20.423
-8.106
-14.264
4. 211
-1.
16.527
948
10.369
20.633
Figure C3: X-component of stress (MPa) of copper 91m thick under a weight of O.O2MPa.
AN
NODAL SoLUrXOS
MAY
TIMI--1
8x,W }
2005
.. -
R8Y8=0
SMa =20.
1
1Z:;42:15
af =1
.'<
633
4.211
-8.106
14264.-
-i4~64-16.527
-1.948
10.
369
20.633
Figure C4: X-component of stress (MPa) of copper 9prmthick under a weight of 0.02MPaat the
point of maximum and minimum stress.
38
AN
NODAL SOLUTION
MAY 3. 2005
12:45:01
STEP=1
SUB =1
TIME--1
UY
(AVG)
RSYS=0
DMX =23.51
BMN =-23.51
SMX =.430989
------------
-23 .51
-16.328
-19.919
-
- -----
-9. 145
-1.963
-12.737
-5.554
.430989
Figure C5: Y-displacement (m) of copper 9pm thick under a weight of O.O2MPa,at the edge of
the center copper node.
Figure C6: X-component of stress (MPa) of copper 9jim thick under a weight of O.O2MPa,at the
edge of the center copper node, mesh showing.
39
AN
NODAL SOLUTION
MAY 1 2005
12: 4 9:59
STEP=1
SUB =1
TIME=--i1
UY
(AVG)
RSYS=0
DMX =47.02
SMN =-47.02
SMX =. 861978
-47. 02
-39. 838
-32. 655
-25. 473
-18.291
-11. 108
-3. 926
.861978
Figure C7: Y-displacement (m) of copper 9m thick under a pressure of 0.04MPa.
AN
XW0AZ EsmunWxO
M z 1 2005
12:53:15
UY
(AVG)
RSYB=O
DM.X=47.02
8m=-=47.02
,M
-47.02
-39. $38
-32105
- -
-&473
-1e,291
-11.106
-3.!
266
861978
Figure C8: Y-displacement (pm) of copper 9m thick under a weight of 0.04MPa. The vertical
line farther right is the center of pressure.
40
AN
NODAL SOLUTION
MAY 1 2005
12:50:43
STEP=l
SUB =1
TIME=1
SX
(AVG)
RSYS=0
DMX =47.02
SMN =-40.846
SMX =41.266
---------
-40.846
-28.529
-16.212
-3.895
8.421
20.738
33.055
41.266
Figure C9: X-component of stress (MPa) of copper 9um thick under a weight of O.O4MPa.
AN
NODAL SOLUTION
MAY 1 2005
STEP=I
SUB
12:51:01
=1
TIM
(AVG)
ex
SMX =41.266
t4o
.. ..
.
WV
-40.846
-28. 529
-16.212
..
-
-3.895
,
,
8.421
,
20.738
33.05S
41. 266
Figure CIO: X-component of stress (MPa) of copper 9pim thick under a weight of O.O4MPaat the
point of maximum and minimum stress.
41
AN
ANY
1 2005
NODALSOLUTION
MAY
STEP1i
STEP-SUB =1
TIME=1
UY
(AVG)
RSOYS=0
DMX =47.02
1 2005
1Z252:26
SMM -47.02
SMX .861978
Mx
-47. 02
_39, 838 -32. 655
-39.
883
~
-25. 473
-18. 291
108
-11. 108
-3. 926 86197
.861978
Figure C 1: Y-displacement (rm) of copper 9m thick under a weight of O.O4MPa,at the edge
of the center copper node.
Figure C12: X-component of stress (MPa) of copper 9im thick under a weight of O.O4MPa,at
the edge of the center copper node, mesh showing.
42
NODAL
AN
SOLUTION
NAY
STEP=1
SUB =1
TIME-i
UY
(AVG)
RSYSO=0
DMX =70.53
SMN =-70.53
SMX =1.293
.,
1 2005
12:56:55
-
11
i
-70.53
-59.756
-48.983
-38.21
-27. 436
-16.663
-5. 889
1.293
Figure C13: Y-displacement ([tm)of copper 9tm thick under a pressure of 0.06a
AN
io4vaL SoT. UXO,
MAY I 2005
enal
UY
Iat
s5027
(AVG)
RSYS=0
DMX=70.53
SMN =-70.53
8m -1. 93
-59,756
-48, 983
-38.21
-27. 436
-16.663
-5. 889
1.293
Figure C 14: Y-displacement (im) of copper 9rm thick under a weight of 0.06MPa. The vertical
line farther right is the center of pressure.
43
NODAL
AN
SOLUTION
MAY
STEP=i
SUB =1
TIME=1
(AVGS)
SX
RSYS=0
DMX =70.53
SMN =-61.269
SMX =61.899
1 2005
12:57:19
--------
,J
- - ------
-61.
269
-42. 793
-24.
318
-5. 843
12. 632
49
31.107
582 .
61.899
Figure C 15: X-component of stress (MPa) of copper 9tpmthick under a weight of O.06MPa.
(AVG)
-61.Z2
57'41
--
RSY=O
70. 53
DMX
SW
x zo00$
"..
'-.-
s' =1
'S!IMt=i M
W
-12
9S!~~~~~~1!~~~BPW1.
SX
~~AN
*
NODAL SOLU'IAON-
,
,
9
H i
·
-61. 269
-42. 793
-24.31U
I
,,.', .'
-5. 843
5.
.~~~~~
.
,.
--fi.563Z
:-
tB.OUC
31-107
61. 8E99
Figure C 16: X-component of stress (MPa) of copper 9pm thick under a weight of O.06MPa at the
point of maximum and minimum stress.
44
NODAL
AN
SOLUTION
MAY 1 2005
STEP1SUB =1
TIME-1
UY
(AVG)
RSYS=O
DMX =70.53
SMN =-70.53
SMX =1.293
-70.53
12:59:00
-59.756
-48.983
-38.21
-27.436
-16.663
-5. 889
1.293
Figure C17: Y-displacement (tmn)of copper 9ptm thick under a weight of 0.06MPa, at the edge
of the center copper node.
Figure C 18: X-component of stress (MPa) of copper 9pm thick under a weight of O.06MPa, at
the edge of the center copper node, mesh showing.
45
AN
NODAL SOLUTION
MAY 1 2005
13:46:53
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =27.481
SMN =-27.481
SMX =.51271
-27.48
1
-19.083
-10.685
-2.287
-14.884
-23.282
-6.486
.51271
Figure C19: Y-displacement (tm) of copper 3tm thick under a pressure of 0.02MPa.
,AN
S8cDA$ 5LVT0r
W
1 2005
13:49:
su ;
SAW)
111 7 ,, - '.
-
37
I
l
8KR -2.481
88I 5i73.
_______________________________
I
-
-. 60$
8-AAs84
-6.486
-z2,zI
II
1/
.51271
Figure C20: Y-displacement (tm) of copper 3ptm thick under a weight of 0.02MPa. The vertical
line farther right is the center of pressure.
46
NODAL
AN
SOLUTION
MAY
STEP=1
1 2005
13:47:32
=1
SUB
TIME=1
SX
(AVG)
RSYS=O
DMX
=27.481
SMN
=-22.85
SMX
=22.967
……---
- -- - -
ia
--
- ------ - -- ----…
…
--
-
-
-
-
-
-
-
-
, L ,5_
-22.85
-9.105
4. 64
-15.978
18.385
-2.233
11.513
22.967
Figure C21: X-component of stress (MPa) of copper 3 pm thick under a weight of O.O2MPa.
AN
NODAL SOLUTION
-AY 1-n
~~~~~~~STES=1~~~~fl~~
~13:48:44
TIME=
,
sx
(AVG)
RSY8=0
DMX =27.481
SMN -22.85
MX =22.967
-22.85
-15.178
- -5
'
-2.
-2. 233
4.64
11.513
18.385
22.967
Figure C22: X-component of stress (MPa) of copper 3m thick under a weight of O.O2MPaat the
point of maximum and minimum stress.
47
AN2
NODAL SOLUTION
STEP51
SUB
SIUB =1
=1
MAY I 2005
13:50:12
TIME=1
rUY
(AVG)
RSYS=0
DMX =27.481
SMN =-27.481
SMX =. 51271
.l__ _ _
-27.481
_
_
-19.083
-23
282
_
_
_
-10.
_
-_
_
_
685
_
_
-2.287
-14.884
-6.486
.51271
Figure C23: Y-displacement (pm) of copper 3m thick under a weight of O.O2MPa,at the edge of
the center copper node.
Figure C24: X-component of stress (MPa) of copper 3pm thick under a weight of O.O2MPa,at
the edge of the center copper node, mesh showing.
48
NODAL
AN
SOLUTION
MAY
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =54.961
SMN =-54.961
SMX =1.025
1 2005
13:54:47
37,
IMMMINS
-54. 961
--
-38. 165
-46. 563
-29. 767
-21. 369
-4. 573
-12.971
1.025
Figure C25: Y-displacement (plm)of copper 3im thick under a pressure of 0.04MPa.
MO6AL
AN
TXV
MA
WI!RP1
0G.1.
tI
I
(AV&)
JsSo
~
mW=~:.c
. ~:-'.';¥,
'.f £:.r
.
U4Z
-. - . , I
,,
-
--l',OZ$
.'
·
St
, . i"
<-
'
'
1 2005
13:56:29
,
.
~~~~~~ ~.: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
.. ~~ ~ ~ ..~ ~ ~ ~ .
~[-
-
~
.
.f
. i-
)
.
-46563
38.165
-29.767:
-21.369
-12.971
-4.5
U
'3
1. 025
Figure C26: Y-displacement (m) of copper 3pm thick under a weight of 0.04MPa. The vertical
line farther right is the center of pressure.
49
A
NODAL SOLUTION
MAY 1 2005
13:55:14
STEP=l
SUB =1
TIME=l
SX
(AVS)
RSYS=0
DMX =54.961
SMN =-45.701
MX =45.934
S~~~~~~~
-18.211
-45.701
-31.956
36.77
9.28
-4.465
23.025
45.934
Figure C27: X-component of stress (MPa) of copper 3im thick under a weight of 0.04MPa.
a
- -AN
SoLuToN
NODAb
STEP1~~~~~~~~~~~~~~~~~~~~~~jv:,
sVBli
TIM
sX
1 S.S
AYI
AY3. 25
"1·
(Ae)'
<w
:
>'5~~~~~~~~~~~~~~~~~~~~~~,
9MN
=4.i1
,'"',.''
i
:_
34 =5.934
,77 ,
9.28
. 0
-18.211
'-31956
-4.465
2, 1
'
-
I
''~
..
-'.
45
4
Figure C28: X-component of stress (MPa) of copper 3gm thick under a weight of 0.04MPa at the
point of maximum and minimum stress.
50
AN
NODAL SOLUTION
MAY 1 2005
13:56:57
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =54.961
SMN =-54.961
SMX =1.025
----------
Em~~~~I l. - - -54.961
-46. 563
-e11-1
-38. 165
l -
-29.767
- f----21. 369
-12.971
-4. 573
1. 025
Figure C29: Y-displacement (pm) of copper 3tm thick under a weight of O.O4MPa,at the edge
of the center copper node.
Figure C30: X-component of stress (MPa) of copper 3tm thick under a weight of O.O4MPa,at
the edge of the center copper node, mesh showing.
51
NODAL
AN
SOLUTION
MAY
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =8Z. 442
SMN =-8Z.442
SMX =1.538
,
-82. 442
-69. 845
1 2005
14:01:21
,
-57.248
-32. 054
-44. 651
-6. 86
-19. 457
1.538
Figure C3 1: Y-displacement (tm) of copper 3pm thick under a pressure of 0.06MPa.
AN
NODAL SOLUTION
y
STP-1
UB =1
W
mm
1 2005
14:02:52
(Av-)
Ac
4- ,
e,"
-82.442,
ff)X
=1.538
~
... ~"
..
:"-...
'
/~~~~~~~~~~~~~~~
'\'
.:
.
I
.....
.
-
.
. .
">t>.
'\,,,-', , .,; ', ' .. -,' ~~~~~~~~~~~~~~~~~~~~:
:!ijh';
-82. 442
'-~ ~sx
, .054. :~
" ~ -I,,-69.-9
45 457.Z48
'
5 - .:
-is,,,
>
8
6
1. 538
Figure C32: Y-displacement (tm) of copper 31tm thick under a weight of 0.06MPa. The vertical
line farther right is the center of pressure.
52
NODAL
AN
OLUTION
MAY 1 2005
14:01:48
STEP=1
SUB =1
TIME=1
8X
RSYS=0
(AV(O)
DMX =82.44Z
SMN =-68.551
SMX =68.9
. _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n=
.
::
,
.__
....
.
......
...
_
__¢
I
-68.551
-47.934
-27.316
-6.698
13.92
34.538
55.155
68.9
Figure C33: X-component of stress (MPa) of copper 3pm thick under a weight of 0.06MPa.
AN
NODAL SOLUTION
STEP:1
SUB =1
TIME=i
Sx
RSY2=03
MAY
1 2005
12:57- 41
(AVG)
DM =70.53
SMI
- 61. 269
qR II
-
--
1-
42. 793
-42. 793
--- - --- -1
-Z4. 318
-I-,--
----
-5. 843
12-.,3Z
3,
31.
bC7
4982'
1.899
61.
899
Figure C34: X-component of stress (MPa) of copper 3m thick under a weight of 0.06MPa at the
point of maximum and minimum stress.
53
AN
NODAL SOLUTION
MAY 1 2005
14:03:22
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =82 .442
SMN=-82.442
SMX =1. 538
---
-82.442
____
---
--------
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~_
-57.248
-32. 054
-6.86
-69.845
-44. 651
-19. 457
1.538
Figure C35: Y-displacement (pm) of copper 3pm thick under a weight of 0.06MPa, at the edge
of the center copper node.
Figure C36: X-component of stress (MPa) of copper 3pm thick under a weight of 0.06MPa, at
the edge of the center copper node, mesh showing.
54
AN
NODAL SOLUTION
MAY 1 2005
14:10:18
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =109.922
SMN =-109.922
SMX =2.051
-76.33
-109.922
-93.
-42.738
-9.146
-59.534
126
2.051
-25.942
Figure C37: Y-displacement (im) of copper 3pm thick under a pressure of 0.08MPa.
NDA
'I;Eu
8TIX
t
AN
.-
-
- ,'j
1 2005
14:12:10
-9.1
6
4-z
X.
=1
TiXL-i'-]
I -VG)
S
DMX =109.
922
M= -109.922
X =2.051
'-, -l
-109.922
-76.33
-93.126
-59.534
-42.738
-25.942
U
2 .051
Figure C38: Y-displacement (um) of copper 3pm thick under a weight of 0.08MPa. The vertical
line farther right is the center of pressure.
55
iN
NODAL SOLUTION
MAY
STEP=1
SUB =1
TIME=1
SX
(AVG)
RSY8=0
DMX =109.922
SMN =-91.402
SMX =91.867
-36. 421
91.402
-63.
911
73. 54
18.56
-8.
1 2005
14:10:55
931
46.05
91.
867
Figure C39: X-component of stress (MPa) of copper 3m thick under a weight of O.08MPa.
AN
NODAL SOL1UTION
-
RS~~~s~~~o
oX
=.10 -
-
.:".' -'
'"'
(AVG),
DX =1O9.922
XMN =-91. 4OZ
=MX
91.867
..
-,/Y
.
. X
-
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..
-..
..
.
.
'6..
,;.' '.
:'s~
-.~3
-91.40Z
46.05
-36..
.6
Figure C40: X-component of stress (MPa) of copper 3tm thick under a weight of O.O8MPaat the
point of maximum and minimum stress.
56
NODAL
AN
SOLUTION
~
S3~~~~~TSE~~P=~
SUB
SUB
MAY 1 2005
14:13:56
=1
=1
TIME=1
UY
(AVG)
RSYS=O
DMX =109.922
SMN =-109.922
SMX =2.051
-109.922
-42.738
-76.33
-93.126
-9.146
-59.534
-25.942
2.051
Figure C41: Y-displacement (m) of copper 31amthick under a weight of O.O8MPa,at the edge
of the center copper node.
Figure C42: X-component of stress (MPa) of copper 31imthick under a weight of O.O8MPa,at
the edge of the center copper node, mesh showing.
57
AN
NODAL SOLUTION
MAY 1 2005
14:59:39
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=O
DMX =26.929
SMN =-26. 928
SMX =.404461
I
-26.928
-10.529
-18.729
-22.829
-2.329
-6.429
-14.629
.4044 61
Figure C43: Y-displacement (lm) of copper 0.5pm thick under a pressure of O.O2MPa.
1kN
NODAL SOLUTION
MA
STEP-1
BUB =1
TIME=1
UY
1 2005
15:01:32
(AVG)
SMXN=.40446- 928
SMX =.409461
-2G.928
-18.729
-22.829
.
-10.529
-. 429
-14.629
:2 3
9
S
.404461
Figure C44: Y-displacement (m) of copper 0.5pm thick under a weight of 0.02MPa. The
vertical line farther right is the center of pressure.
58
AN
NODAL SOLUTION
MAY 1 2005
15:00:15
STEP=1
SUB =1
TIME=1
SX
(AVG)
RSYS0
DMX =26.929
SMN =-13.785
SMX =15.939
I-
-
-------
l~~~~t
, -13.785
-1--'
-4.868
-9.327
4.049
12.967
-. 409217
8.508
15.939
Figure C45: X-component of stress (MPa) of copper 0.5[tm thick under a weight of O.O2MPa.
AN
NODAL SOLUTION
STEP=1
AY
sUA =l
0
RISYS=0
1
005
is:.00:52
(AVG)
(
--
---
DM1 =26. 92 9
SMN =-13.785
5M =15. 939
:
-a.,o
-9.327
-.4517
-.409217
8
8.508i
12.967
Figure C46: X-component of stress (MPa) of copper 0.5tm thick under a weight of 0.O2MPa at
the point of maximum and minimum stress.
59
NODAL
AN
SOLUTION
MAY
STEP-1
TIM8=tJY
1 2005
15:02:01
(AVG)-_
DMX =26.929
SMN =-26.928
Sm4X =.404461
-26.928
-18.729
-22.8Z9
-10.529
-2.329
-14.629
-6.429
.404461
Figure C47: Y-displacement (Rm) of copper 0.5pm thick under a weight of 0.02MPa, at the edge
of the center copper node.
Figure C48: X-component of stress (MPa) of copper 0.5 im thick under a weight of O.02MPa, at
the edge of the center copper node, mesh showing.
60
NODAL
SOLUTION
MAY
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =53.859
SMN =-53.857
SMX =.808922
-53. 857
-45. 657
-37. 457
-29. 257
-21. 057
-12. 858
1 2005
15:-05: 46
-4. 658
.808922
Figure C49: Y-displacement (m) of copper 0.5[tm thick under a pressure of O.O4MPa.
AN
OLUTION
NODAL
MAkY 1 2005
15: 07 18
STEP=1
SUB
=1
TIME=1
AVG)
_[
DMX 53.859
SMN =- 53.857
SMX =.808922
-53.857
-45. 657
-37.457
-29. 257
-2 i
...
:
-12. 8S
-t.6
. 808922
Figure C50: Y-displacement (m) of copper 0.5gimthick under a weight of 0.04MPa. The
vertical line farther right is the center of pressure.
61
NODAL
SOLUTION
M
I
MAY 1 2005
15:06:13
STEP=1
SUB =1
TIME=1
(AVG)
SX
RSYS0
DMX =53.859
SMN =-27.57
SMX =31.878
i
AK
I
-
--------
I
25.933
8.099
-9.736
-27.57
31.878
17.016
-. 818434
-18.653
Figure C5 1: X-component of stress (MPa) of copper 0.5tpmthick under a weight of O.04MPa.
N
NODAL SOLUTION
1 2005
15
,_
~
ex
(AVG)
:
~
~V
'1f
'
.
"f
i'
-
...
-18.6.
3'
'
'.
.
-
'-
'
.
=2
sm
Sam =31.878
'
31,
-
87
Figure C52: X-component of stress (MPa) of copper 0.5prmthick under a weight of 0.04MPa at
the point of maximum and minimum stress.
62
AN
NODAL SOLUTION
MAY 1
005
15:07:46
STEP=-i
SUB =1
UY
(Av
RSYBD0
DMX =53.859
SMN =-53.857
SMX =. 808922
-
-53 857
~~
~
-45. 657
~
-37.457
~
-29.257
~
~
-21.057
~
12. 858
-
-
-4.658
.
808922
Figure C53: Y-displacement (tm) of copper 0.5pamthick under a weight of 0.04MPa, at the edge
of the center coppernode.
Figure C54: X-component of stress (MPa) of copper 0.5grmthick under a weight of 0.04MPa, at
the edge of the center copper node, mesh showing.
63
NODAL
STEP=1
SUB =1
TIME=1
UY
RSYS=0
AN
SOLUTION
MAY 1 2005
15:12:39
(AVS)
DMX =80.788
SMN =-80.785
SMX =1.213
-56.186
-80.785
-68.486
-31.586
-6. 986
-43.886
-19.286
1.213
Figure C55: Y-displacement (pm) of copper 0.5pm thick under a pressure of 0.06MPa.
AN
NODAL SOLUTION
MAY
STEP=1
SUB =1
TIME-1
uEY
(AVG)
RSYS=
DMX =80.788
SMN -80.785
SMX =1.213
-80.785
-68. 486
-56.186
-43.886
-31.586
2005
1-1.-S
-6. 986
1.213
Figure C56: Y-displacement (um) of copper 0.5grmthick under a weight of 0.06MPa. The
vertical line farther right is the center of pressure.
64
AN
NODAL SOLUTION
MAY
STEP=1
SUB =1
TIME=1
X
1 2005
15:13:06
(AVG)
RSYS=O
DMX =80.788
SMN =-41.356
SMX =47.818
I-
- --------
- ----------- -
-
--41.356
-:
-14.604
-27.98
12.148
:
l~~~~~~~~~~~~~~~~~~~~~
38.9
-1.228
25.524
47.818
Figure C57: X-component of stress (MPa) of copper 0.5ytmthick under a weight of 0.06MPa.
AN
NODAL SOLUTION
MAY 1 2005
15 13:23
8T9*~~~~~~~~~E~=Iw~~~~~~
9tl
=1
TIM0--I
W"-
=47(AVO
-lwg~ w
-41.3$6
-27.98
-14604
-
-i.
I.';:'
.
25.524
Figure C58: X-component of stress (MPa) of copper 0.5pm thick under a weight of 0.06MPa at
the point of maximum and minimum stress.
65
AN
NODAL SOLUTION
.. Y
STEP=1
i
2005
15:14:30
SUB =1
UY
RSYS=O
DMX=80.788
SMN=-80.785
SMX =1.213
-80.785
-68.486
-56.186
-43. 886
-31.586
-19.286
-6. 986
1.213
Figure C59: Y-displacement (pim)of copper 0.5pamthick under a weight of O.06MPa, at the edge
of the center copper node.
Figure C60: X-component of stress (MPa) of copper 0.5prmthick under a weight of 0.06MPa, at
the edge of the center copper node, mesh showing.
66
NODAl
AN
SOLUTION
MAY 1 2005
15:32:01
8TEP=1
SUB
=1
TIME=1
UY
(AVG)
RSYS=O
DMX =107.717
SMN
=-107.714
SMX
=.
618
J11111111111111=
-107.714
-91.314
-74.914
-42.115
-9.315
-58.515
-25.715
1.6 18
Figure C61: Y.-displacement (m) of copper 0.5pm thick under a pressure of 0.08MPa.
NODAL
AN
9SOLUTION
2 2005
0:28:42
MAY
STE1
SUB =1
TIMEi
ty
(AVG)
DMX =107.717
SM
=-107.714
X =1. 618
I
1
....
-107.714
-74.914
-91
.314
-42.115
-9.31S5
-58 .515
-25.715
1.618
Figure C62: Y-displacement (pm) of copper 0.5pm thick under a weight of 0.08MPa. The
vertical line farther right is the center of pressure.
67
NODAL
N
SOLUTION
MAY
STEP=1
SUB =1
TIME=I
(AVG)
SX
RSYS=O
DMX =107.717
SMN =-55.141
SMX =63.757
1 2005
15:32:19
__.
--
10-
-55. 14 1
37 36
-37.306
- 19.471
16.198
637
-1.637
I
34 0
34.032
51.
673.
757
63.757
Figure C63: X-component of stress (MPa) of copper 0.5tpmthick under a weight of O.08MPa.
(AVG)
DMX=107.717
$MN=-55.141
gm =63 .757
44---_,w,
n,;
_,~.,,7
.
JUp
19.471
9559
-l_ All)34.032
*
.
VW
51 867
16 198
63.757
f
Figure C64: X-component of stress (MPa) of copper 0.5pm thick under a weight of 0.08MPa at
the point of maximum and minimum stress.
68
AN
NOCDALSOLUTION
MAY
STEPI
1 2005
US In S9
UY
(A~t
RSYS=0
DMX =107.717
SMN =-107. 714
SMX =1.
18
-107
-74.914
.714
-91.
314
-42.
-9.315
115
-58.515
-25.715
1.618
Figure C65: Y-displacement (rm) of copper 0.5pm thick under a weight of O.O8MPa,at the edge
of the center copper node.
Figure C66: X-component of stress (MPa) of copper 0.5pm thick under a weight of 0.08MPa, at
the edge of the center copper node, mesh showing.
69
Figure C67: Y-displacement (,tm) of copper 0.3pm thick under a pressure of 0.02MPa.
AN
NODAL SOLUTION
MAY
STB P1
SUB 21
TIME-1
UY
2 2005
12:08:40
-
(AVG)
-
RSYS=0
SMN
: ...
-
VM4 =268..............
-26.958
SMX =. 401349
,
,:
Wz
.11 ,
-26.958
i7----l-
-22.854
--18.75
: II
:,
11
-1,
Britoil
" ---::!:
-1044z.
-14.40,
-
-6.438
-2. 3.35
10134
.40134!
Figure C68: Y-displacement (ptm)of copper 0.3pm thick under a weight of 0.02MPa. The
vertical line farther right is the center of pressure.
70
-
NODAL
AN
SOLUTION
MAY 2 2005
12: 07:35
STEP=1
SUB
=1
TIME=1
SX
RSYS=0
DMX =26.959
SMN
=-7.
SMX
=11.353
(AVG)
937
=
_ ______.
________
____S.__ __
-7 937
-5.043
______ .
._____B
_______
-2.15
.743688
.
:
7
3.______ ________6.__
9___________
3.637
9.424
6.531
_
_J
11.353
Figure C69: X-component of stress (MPa) of copper 0.31tmthick under a weight of O.O2MPa.
NODAL SOLUTION
MAY
MM
SUB
1~
TIME=1
SX
AN
2 2005
==12:07:55
(AVG)
RSYS=0
DMX =26. 959
SMN* =7, 937
SMX =11. 353
MRSOMMMU
- .
Z-I ,---.043
-5 .043
-111-----
11- --
-1.1-
.743668
"
3.637
6.531
9.4Z4
11.353
Figure C70: X-component of stress (MIPa)of copper 0.3pm thick under a weight of 0.02MPa at
the point of maximum and minimum stress.
71
-
SUB =1
TIME=
UY
ju~~.O
fl~~aTION
MAY
~~~~~~~~~~~~~~STEP=1
=
2 205
~~12:
10:22
(AVG)
RSYS=0
DMX =26.959
SMN =-26. 958
SMX =. 401349
-26.958
-18.75
-22.854
-10.542
-2.335
-14.646
-6.438
.401349
Figure C71: Y-displacement ([tm) of copper 0.3pm thick under a weight of O.02MPa, at the edge
of the center copper node.
Figure C72: X-component of stress (MPa) of copper 0.3pm thick under a weight of O.O2MPa,at
the edge of the center copper node, mesh showing.
72
NODAL
AN
SOLUTION
MAY
STEP=1
SUB =1
TIME=1
UY
(A.VG)
RSYS=0
DMX =53.918
SMN =-53.916
SMX =.802697
2 2005
11:57:39
-. 1
k,-
-53. 916
-45.708
-37 .5
-29.293
I
i
i
1.
-21.085
i
-12.877
-4.669
.802697
Figure C73: Y-displacement (m) of copper 0.3pm thick under a pressure of 0.04MPa.
AN
NODAL SOLUTION
MAY 2 2005
11: 59:30
STEP=
SUB 1
TIM2=1
UY
RSYS=0
rAVa}
DMX -.
53r91S
DP.'
? !-
SMN -53. 916
SMX'. 802697
-53. 916
-45. 708
-37.5
-29.293
n-
-21.085
-12. 877
-4.669
.802697
Figure C74: Y-displacement (pm) of copper 0.3pm thick under a weight of 0.04MPa. The
vertical line farther right is the center of pressure.
73
NODAL
SOLUTION
MAY
2 2005
11 :58:13
STEP=1
SUB =1
TIME=l
SX
(AVG)
RSYS=0
DMX =53.918
SMN =-15. 873
SMX =22.706
-15.873
-4.299
7.274
-10.086
18.848
1.487
13.061
22.706
~~~~~~
Figure C75: X-component of stress (MPa) of copper 0.3 tmthick under a weight of O.04MPa.
sus ~~ S
=StEP1
== =
9 IME=il
>-
I1~~~~~~~~~~~AY
284-1
btQO
,
-.
.~~~~~~~~~~~~~~~~S
',S 'zoo
.
. .,
-!.i...
RSYS¢O~~
-
DMX =53. 918
=-1.. 873
S.X=22.706
SMN
\,'
'q-
5,.
-15
-10.086
3-.4
2
7.274
~99 , .....-. ;.,:t7
.
-
1.487
13.01
'-
22.%70
Figure C76: X-component of stress (MPa) of copper 0.3tm thick under a weight of 0.04MPa at
the point of maximum and minimum stress.
74
NODAL
M
MAY
SOLUTION
AN
2
2005
12:00:41
SUB
=1
TIME=
UY
(AVG)
RSYS=0
DMX
SMN
=53.918
=-53. 916
SMX
=. 802697
-37.5
-53.916
-45.708
-21.085 0
-24.669
-29.293
-12.877
--
.802697
Figure C77: Y-displacement (pm) of copper 0.3[tm thick under a weight of O.O4MPa,at the edge
of the center copper node.
Figure C78: X-component of stress (MPa) of copper 0.3[tm thick under a weight of O.04MPa, at
the edge of the center copper node, mesh showing.
75
-80. 874
-68. 562
-56.251
-43. 939
-31. 627
-19. 315
-7. 004
1.204
Figure C79: Y-displacement (pm) of copper 0.3pm thick under a pressure of 0.06MPa
AN
NODAL SOLUTION
STEP=1
SUB =1
TIME=1
UY
RSYS=0
DMX =U.
MAY 2 2005
11:50:24
(AVIG)
876
2198
-1. Z-,
,j\
'-%,.
,, .
-80.874
-68. 562
-56.251
1
i
.
-
-7.004
1.204
Figure C80: Y-displacement (Lm) of copper 0.3pm thick under a weight of 0.06MPa. The
vertical line farther right is the center of pressure.
76
NODAL
SOLUTION
MAY
3TEP=1
SUB =1
TIME=1
2 2005
11: 49:11
I
SX
(AVG)
RSYS=0
DMX =80.876
SMN =-23.81
SMX =34.059
_
_ __
_ -__
_ -_
_
_ _-----------
---23.81
<
-6.449
-15.129
r
10.911
28.272
2.231
19.592
34.059
Figure C81: X-component of stress (MPa) of copper 0.3ptmthick under a weight of O.O6MPa.
MAY 2 2005
11:t;49;38
STgP=}
TIME51
SX
(AVG)
SMN -23.81
SM =34059
-23.81
-IS.J29
-15.129
-6.449
10.911
19.52
I.231
-2821
o
34,059
Figure C82: X-component of stress (MPa) of copper 0.3pm thick under a weight of O.O6MPaat
the point of maximum and minimum stress.
77
NODAL
A
SOLUTION
MAY
STEP~l
~sa=1
TIMR
UY
(AVGm
RSYS=0
DMX =80. 876
SMN =-80.874
SMX =1.204
2 2005
-11:51:34
-
-
-80. 874
-68.562
-56.251
-43.939
-31 .627
-19.315
-7.004
1.204
Figure C83: Y-displacement (pm) of copper 0.3pm thick under a weight of O.O6MPa,at the edge
of the center copper node.
Figure C84: X-component of stress (MPa) of copper 0.3pm thick under a weight of 0.O6MPa, at
the edge of the center copper node, mesh showing.
78
NODAL
AN
OLUTION
MAY
STEP=1
SUB =1
TIME=1
UY
(AVG)
RSYS=0
DMX =107.835
SMN =-107.832
sMX =1.605
-107.832
-42.17
-75.001
-91.416
2 2005
11:31:37
-9.338
-58.585
-25.754
1.605
Figure C85: Y-displacement ([tm) of copper 0.3,um thick under a pressure of 0.08MPa
NODAL
AN
SOLUTION
NAY
STBP=I
SUB =1
TIME=ZI
UY
(AVG)
RSYS0
DMX =107.835
SMN =-107.83Z
SM =1. 605
-----
-75.001
-107.832
-91.416
005
2
11:3( :51
-
-9. 338
-42. 17
-58.585
-25.754
Figure C86: Y-displacement (m) of copper 0.3pm thick under a weight of 0.08MPa. The
vertical line farther right is the center of pressure.
79
AN
NODAL SOLUTION
MAY Z Z005
11:31:57
STEP=1
SUB =1
TIME=I
SX
(AVG)
RSYS=0
DMX =107.835
SMN =-31.746
SMX =45.412
i
j
II
-31 746
-8.599
-20.173
14.548
37.696
2.975
26.122
45.412
Figure C87: X-component of stress (MPa) of copper 0.3pm thick under a weight of 0.08MPa.
.
SUB =1
81
$X
R8YS50
AN
.
_
(AVG)
DX =1-i07.835
MW
=-31. 746
SM =45.412
_
-31.746
-20.173
14.540
-8.599
26.122
2.975
37.69
-
_
45.412
Figure C88: X-component of stress (MPa) of copper 0.3gm thick under a weight of 0.08MPa at
the point of maximum and minimum stress.
80
AN
NO
MAY
ST
Su
TI
UY
RZ
DM
SN
SN
91.416
-58.585
4
005
15:47:14
-25.754
Figure C89: Y-displacement (rm) of copper 0.3ptmthick under a weight of 0.08MPa, at the edge
of the center copper node.
Figure C90: X-component of stress (MPa) of copper 0.3pm thick under a weight of 0.08MPa, at
the edge of the center copper node, mesh showing.
81