Nonlinear static and dynamic analysis of mulltilayer nanocomposite structures in solar cell

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VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

VU MINH ANH

NONLINEAR STATIC AND DYNAMIC

ANALYSIS OF MULTILAYER

NANOCOMPOSITES STRUCTURES IN

SOLAR CELL

MASTER’S THESIS

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VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

VU MINH ANH

NONLINEAR STATIC AND DYNAMIC

ANALYSIS OF MULTILAYER

NANOCOMPOSITES STRUCTURES IN

SOLAR CELL

MAJOR: INFRASTRUCTURE ENGINEERING

RESEARCH SUPERVISORS: 
Prof. Dr. Sci. NGUYEN DINH DUC

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ACKNOWLEDGEMENT

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TABLE OF CONTENTS

ACKNOWLEDGEMENT ... I

LIST OF FIGURES ... IV

LIST OF TABLES ... VI

NOMENCLATURES AND ABBREVIATIONS ... VII

ABSTRACT ... VIII

CHAPTER 1: INTRODUCTION ... 1

1.1. Background ... 1

1.2. Research objectives ... 7

1.3. The layout of the thesis ... 7

CHAPTER 2: LITERATURE REVIEW ... 9

2.1. Literature review in Outside Vietnam ... 9

2.2. Literature review in Vietnam ... 13

CHAPTER 3: METHODOLOGY ... 15

3.1. Modelling of SC ... 15

3.2. Methodology ... 16

3.3. Basic Equation ... 17

3.4. Boundary Conditions ... 22

3.5. Nonlinear Dynamic Analysis ... 23

3.6. Nonlinear Static Stability ... 25

CHAPTER 4: RESULTS AND DISCUSSION ... 27

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III

4.2. Natural frequency ... 27

4.3. Dynamic response ... 28

4.4. Frequency – amplitude relation ... 32

4.5. Nonlinear Static ... 33

4.6. Critical buckling load ... 35

CHAPTER 5: CONCLUSIONS AND FURTHER WORKS ... 37

5.1. Conclusions ... 37

5.2. Future works ... 37

APPENDIX ... 39

PUBLICATIONS ... 40

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LIST OF FIGURES

Figure 1.1 Modelling of surface transitions organic solar cell. ... 4
Figure 1.2 Modelling of a solar cell using perovskite as a light-sensitive substance and 
structure of the energy zone of the solar cell. ... 6
Figure 3.1 Geometry and coordinate system of nanocomposite multilayer SC. ... 16

Figure 4.1 Influence of ratio a b/ on the SC’s nonlinear dynamic response

(

Px =0,Py =0 .

)

... 29
Figure 4.2 Influence of ratio a/h on the SC’s nonlinear dynamic response

(

Px =0,Py =0 .

)

... 29

Figure 4.3 Effect of the exciting force amplitude Q on the dynamic response of SC

(

Px =0,Py =0 .

)

... 30

Figure 4.4 Influence of the pre-loaded axial compression

P

x on SC’ dynamic response.

... 30

Figure 4.5 Effect of the pre-loaded axial compression Py on the dynamic response of
SC. ... 31

Figure 4.6 Effect of initial imperfection

W

0 on the dynamic response of the SC. ... 31

Figure 4.7 Influence of external force Fon SC’ frequency – amplitude curves. ... 32

Figure 4.8 The influence of initial geometrical imperfection on the SC’ stability with
uniaxial compressive load. ... 33

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Figure 4.10 The effect of elastic foundations on the SC’ the load – deflection amplitude
curve. ... 34

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LIST OF TABLES

Table 4. 1 Initial thickness and properties materials of layers of SC. ... 27

Table 4. 2 Effects of the thickness of layers and modes on natural frequencies of the 
nanocomposite multilayer organic solar cell. ... 28

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VII

NOMENCLATURES AND ABBREVIATIONS

SC Solar Cell

a The length of SC

b The width of SC

h The thickness of SC

Oxyz The space coordinates system

En The elastic modulus and Poisson ratio
0, ,0 0

u v w The displacements in the x, y and z directions

k The Winker foundation

k The Pasternak foundation

W =µh The initial imperfection

IM Immovable

FM Freely movable

GPa GygaPascal =109 Pascal 
,

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VIII

ABSTRACT

This thesis focuses on the study of the mechanical behavior of Solar Cell (SC). As we
know, SC is a sustainable solution for energy supply and greenhouse gas reduction.
Therefore, SC is currently hot topic that attracted a lot of interest from scientists
worldwide. Currently, there are many research about SC such as Replace component
materials in OS or how to improve SC performance … However, there is very little
research on mechanical properties, mechanical behavior of SC. Therefore, this thesis
will focus on investigation of mechanical behavior of SC. SC are made by 5 layers
under mechanical load. Besides, SC are supported by elastic foundations: winker
foundation and Pasternak foundation. In details, nonlinear static and dynamic analysis
of multilayer nanocomposite structure in solar cell will be investigated. By using
classical plate theory, Glarkin method along with Runge - Kutta method, the effect of
geometrical parameter, elastic foundation, load, and imperfection on the nonlinear
static and dynamic analysis of multilayer nanocomposite structure in solar cell will be
described in detail. Besides, some numerical results as critical buckling load and
natural frequency also will be shown.

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CHAPTER 1: INTRODUCTION

1.1. Background

It is predicted that all the oil of the world will be depleted by 2050. Finding a
renewable energy sources to replace the exhausted fossil fuel power has become an
urgent issue.

Since 1953 when D. Chapin, C. Fuller and G. Pearson Silicon introduced the 2 cm2 Si
solar cell with power conversion efficiency 4%, this solar cell has been developed
through many generations. Now, it is commercial with power conversion efficiency
40%. This attracts numerous researchers, which is represented through an increasing
number of researches about solar cell in recent years. Therefore, it is necessary to have
better understand the operational methods and physics behaviors of solar cell to
increase its power conversion efficiency.

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because it acts as a greenhouse gas. The cause of the greenhouse effect is that the
atmosphere reflects the infrared (heat radiation) back to Earth. This effect is essential
for life on Earth because radiation balances the sun, the atmosphere and surface of the
Earth. It leads to an average temperature of 14 0C on the earth's surface. Without this
effect, the surface temperature of the earth would be -15 0C (Würfel, 2009). The impact
of global warming is very serious and the potential consequence is the rise of sea level.
In addition, desertification and dehydration are likely to collapse the entire ecosystem,
change in ocean currents; which lead to the imbalance of natural life. Because of these
risks, scientists have been looking for solutions to reduce greenhouse gas emissions.
The solution of using renewable energy sources has attracted large numbers of
scientists and the field of solar cells has boomed (S. Chu, 2017). The contribution of
this energy source is increasing in total world energy demand. At present, developed
countries have used solar power plants to contribute to national energy such as the
United States, Germany, Japan and China. In Vietnam, we also have two solar power

plants under construction in Quang Ngai and Binh Thuan province.

Although the development of solar cells has gone through many development cycles,
we can divide it into three main generations:

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high stability to the environment. But the silicon wafer is not a good light absorber, so
it is usually thick and hard. These solar panels are very complex to produce and
relatively expensive. But their efficiency is up to 25% - the highest level for
commercial applications. In addition, scientists have developed a multilayer
configuration, different semiconductors with different band widths. Their efficiency
has reached 65% because it can absorb a wider range of solar spectrum.

§ The second generation of solar cells consists of thin film solar cells which
are primarily made of Cadmium Telluride (CdTe) and Copper Selium Indium Gallium
(CIGS). Both of them are rare and toxic metals. This type of solar cell is manufactured
by depositing one or more thin film photovoltaic materials onto glass, plastic or metal.
They absorb the light from 10 to 100 times as much as silicon, so the thickness of the
photoelectric material is just a few micrometers (the thickness of human hair is 90 µm).
The efficiency of these solar cells has reached more than 20% but they have the
potential to achieve the same efficiency as the first generation of solar cells according
to scientists. However, they are made of heavy metals, which have an adverse effect on
the environment.

§ The main goals of the third generation of solar cells are to improve the
efficiency while keeping low costs. The third generation includes thin film solar cells
that use light-sensitive pigments, organic solar cells, quantum dot solar cells and
peroskite solar cells. The advantages of these solar cells are cheap, easy manufacturing.
Their efficiency has recently reached 20.1% and they have the potential to reach 31%.

This makes them to be the most promising photovoltaic technology today. This shows
the importance of the mechanical and physical behaviours study of the third generation
of solar cells.

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and OTFS due to their benefits such as abundant materials, large-scale, low-energy
fabrication methods. At present, organic solar cell (OSC) technology is considered as
one of the most promising cost-effective alternative and environmentally friendly
electric generation. Nanocomposite materials will be novel materials in the near future
because of their outstanding properties. These are combined the advantages of both
organic and inorganic materials as well as surmounted the disadvantages of them. In
term of mechanics, the composite materials are more stability than organic or inorganic
materials. They also give distinct properties in comparison with photovoltaic devices.
By the ways, we would observe interesting effects and therefore having ability to open
new application in the field of nanotechnology.

The model of surface transitions organic solar cell

Figure 1.1 Modelling of surface transitions organic solar cell (google).

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separated at the interface between Acceptor and Donnor. Solar cell transitions face is
the simplest structure of organic solar cells. The advantage of transient solar cells is
that it reduces the recombinant exciton by reducing the travel distance of the exciton.
In contrast, the downside of solar cell transitions is that the surface is small, which
leads to the reduction of efficiency in exciton separation.

The model of the solar cell uses perovskite as a light-sensitive substance

Perovskite material is used in solar cells for the first time using a light-sensitive
substance. In particular, perovskite nanocrystals are used as optical absorbers instead of
light sensitive ones. Perovskite nanoparticles will be adsorbed onto the surface of the
capillary oxide layer as TiO2 ... and absorb light. The electron transporter in this case is

TiO2 and the hole transporter is a liquid electrolyte. Perovskite in this case only takes

on a role of absorbing light. The transfer of charge in the battery will be done by ETL
and HTM. The efficiency of this battery is about 3.8% and 3.1% by using
CH3NH3PbI3 and CH3NH3PbBr3 as optical absorbers. The efficiency of this solar
cell is low and it goes down quickly within minutes. The reason of this phenomenon is
the rapid decomposition of perovskite in liquid electrolytes. Therefore, liquid
electrolytes have been replaced by solid-state carriers to improve the efficiency of the
solar cell (Figure 1.2). The efficiency and the stability of the cell have been
significantly improved by continuously improving the efficiency of the transportation
and collecting holes in the HTM layer. It has reached 9.7% (H.S. Kim, 2012)by using
Spiro-MeOTAD and 12% by replacing PTAA for HTM (J. H. Heo, 2013)

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Perovskite material, instead of absorbing on the capillary material, penetrates
completely into the hollow spaces between the TiO2 nanoparticles. efficiency of this

structure was recorded up to 15% (J. Burschka, 2013)

Figure 1.2 Modelling of a solar cell using perovskite as a light-sensitive substance and 
structure of the energy zone of the solar cell (google).

Advantages of solar cells

Main advantages of organic solar:

• Cost of production is low because it can be made using roll to roll or low
molecular weight technology (N. N. Dinh, 2017) (Nam, 2014)

• High flexibility and high performance.

• Nontoxicity, rich materials, light weight (few grams per m2).
• Applications in mobile devices

• No refinement in fabrication.

The main advantages of perovskite solar cells:

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7
1.2. Research objectives

The research objective of this thesis is to study nonlinear static stability and dynamic
response of organic solar subjected to mechanical load. Therefore, in order to have
remarkable results, this Master thesis will set goals that need to be achieved as below:

v Studies on nonlinear static stability of structure in solar cell subjected to
mechanical load to determine the critical loads and the load – deflection curves.
The effects of geometrical parameters, material properties, imperfections, loads
on the nonlinear static stability of next-generation solar cells will be also
discussed.

v Investigations on nonlinear dynamic analysis on the structure in solar cell
subjected to mechanical load. The natural frequency of free and forced vibration,

the deflection – time, frequency – amplitude curves and dynamic critical
buckling loads of organic solar structures are determined. In numerical results,
the effects of the material properties, geometrical parameters, imperfections and
loads on the nonlinear dynamic analysis on the structure in solar cell structures
will be analyzed.

1.3. The layout of the thesis

This thesis focuses on the investigation of nonlinear static and dynamic analysis of
multilayer nanocomposites structure solar cell under mechanical load. Classical plate
theory and boundary condition are proposed to obtained the numerical results and
figure results. In order to understand the problem as well as to get the best results, this
dissertation has taken steps according to the structure below:

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The research background, the necessity of this thesis along with overview of research
situation will mentioned. Formation, development, types as well as advantages and
disadvantages of OS will also be introduced.

Ø Chapter 2: Literature review

Chapter 2 will show some research papers are related to this thesis’s topic. In those
research papers, they also pointed out the outstanding results obtained from their
research as well as those research’s limitation. Since then, the objective of the thesis
will be more clearly defined. This chapter also explains why an investigation of
nonlinear static and dynamic analysis of OS is important.

Ø Chapter 3: Methodology

Chapter 3 will introduce the method used to approach and solve problems. In details,
classical plate theory along with some basic equation such as Hook’Law, the nonlinear
equilibrium equations… will be used. Besides, boundary conditions also will be
described. Additionally, the results are helped by some software.

Ø Chapter 4: Numerical results and discussion

In this chapter, the numerical results such as critical buckling load and natural
frequency will be shown. Furthermore, the effect of geometrical parameters, material
properties, imperfections, loads also will considered in the form of figures. The results
shown will also include discussion.

Ø Chapter 5: Conclusions

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CHAPTER 2: LITERATURE REVIEW

2.1. Literature review in Outside Vietnam

In 1953, the first sillic solar cell with a performance of about 4% was fabricated at Bell
laboratories after six years of p-n junctions’ discovery by William B. Shockley,
Walther H. Brattain and John Bardeen. The first module of solar cell was built as a
power source for the spacecraft five years later. In 1960, commercial modules were
produced with power conversion efficiency 14%. These modules are mainly used as
power supplies for telecommunication systems. In the early years of development, this
source of energy was very expensive with an estimate of 100 EUR/W. However, the
price of this energy source has declined in recent years. Therefore, it can be widely
spread all over the world. For example, the price of a module of solar cell dropped
from 3 USD/W in 2008 (the price of a module of solar cell to generate 1W of energy

under 1 sun sunlight density) to about 0.5 USD/W in 2017 (pvXchange, 2017)

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second breakthrough was the invention of transient block heterostructure by
simultaneously depositing two materials of different electrical properties. After these
achievements, the number of publications has grown exponentially over the past
decade. Photoelectric conversion efficiency was above 10% (Green et al, 2017), (He et
al, 2015), (Hou, Inganäs, Friend, & Gao, 2018), (Pelzer & Darling, 2016), (Yusoff et al,
2015), (Zhao et al, 2016), (Zheng et al, 2018), (Zheng et al, 2017), (Zimmermann et al,
2014). The reason for these successes is the enormous potential applications of organic
semiconductor materials (Dinh, 2016), (Lu et al, 2015), (Mazzio & Luscombe, 2015).
Generally, Perovskite is an oxide layer with the chemical formula ABX3. This material
has the well-known and widely studied physical properties as magnetic, ferroelectric,
and two-dimensional conductive material. Recently, halide perovskite has attracted
considerable interest in the fields of materials research as well as chemistry and physics.
This is explained by the high performance of solid-state solar cells based on perovskite
halide, reaching 17.9% in 2014 after reaching 9.7% for the first time in 2012. In 2009,
Miyakasa and his colleagues used a perovskite CH3NH3PbI3 metal-organic hybrid in a
solar cell using a light-sensitive colorant with an efficiency of 3.8%. By using
surface-active CH3NH3PbI3 and TiO2 nanoparticles, research group of Park gained a 6.5%

performance in 2011. In 2012, Park and Gratzel (Kim et al, 2012) replaced hole
transport layer based on liquid by solid Spiro-MeOTAD due to corrosion problems
related to liquid electrolytes. Unexpectedly, this increased the efficiency up to 9.7%.
Lee and Snaith (Lee, Teuscher, Miyasaka, Murakami, & Snaith, 2012) gained an
efficiency of 7.6% by using a similar structure. They also found that the efficiency
could be as high as 10.9% by replacing the conductive layer TiO2 with insulating oxide

Al2O3. Although there is still debate over the efficiency of solar cells between the use

of Al2O3 and TiO2, this finding also indicates that perovskite can transport electrons

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cell gained a 15.4% efficiency (M. Liu, 2013). Recently, Seok and his colleagues used
the CH3NH3Pb(I1-xBrx) composite material and their efficiency hit a record of 16.2% to

17.9% by adjusting the thickness ratio of the layers and chemical composition
( Other developmental
directions of perovskite include adjusting the properties of perovskite by controlling
the chemical composition, developing effective manufacturing methods and optimizing
the hole transport layer and properties at the interface surfaces.

Unlike p-n junctions in semiconductor such as p-n (Si) or n-p (GaP), heterojunctions
are a junction formed between two dissimilar crystalline semiconductors. Solar
nanocomposite materials include heterojunctions of inorganic semiconductors and
organic semiconductor. Different types of nanocomposite materials are increasingly
being studied because they are widely applied in many types of components with
specific properties. Some new photovoltaic materials and components using layered
structure are gradually replace traditional inorganic electric components, forming the
field of “organic electrics”. Typical solar cells consist of inorganic solar cell, organic
solar cell, perovskite solar cell, etc. (Burlakov, Kawata, Assender, Briggs, & Samuel,
2005). There are processes occurring in solar cell: excitons created by light absorption,
they diffuse and separate into free electrons and free holes, electrons and holes move to
the corresponding electrodes to generate a photovoltaic (for organic solar cell).

Apart from the ability to raise optical conversion efficiency, multilayer structures also
help to raise the mechanical stability and the life of components. Thickness
optimization of each layer and thermal stress of this structure in solar cell has been

investigated. The charge separation process is improved by implanting layers in
nanoscale thickness which include materials such as C60 (Kawata, et al, 2005),

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layered structure give higher conversion efficiency than monolithic structures.
Environmentally friendly materials and lower costs than traditional structures
(Haugeneder et al, 1999), (Dittmer et al, 2000). In the world, the research on
monolayer and layered nanomaterials has attracted interest from many research groups,
for example in USA, UK, France, Germany, Italy, Canada, Japan, Singapore, Korea,
etc. Based on nanocomposite thin films, high-quality and environmentally friendly
solar cells such as organic solar cell and perovskite solar cell are being researched,
developed and applied in practice.

Experimentally, authors at Stanford University (Kline & McGehee, 2006) showed that
in conjugated polymers (CP), the surface morphology and thickness strongly affect the
capacity for carrier transport. Low-order CP thin films give higher carrier mobility.
The change in surface structure at the boundary between two CP layers change the
mobility of electrons and holes moving through contact boundary. CP is used to
fabricate organic solar cells and perovskite solar cells. The carrier trapping distribution
was studied using the technique “Thermally stimulated current- TSC" (Wurzburg
University, Germany) (Schafferhans, Baumann, Deibel, & Dyakonov, 2008). The
research on carrier transport through contact boundary of the P3HT polymer /
electrodes demonstrated that the charge separation process is strongly depend on
electron and hole mobility (National Institute of Standards and Technology,
Gaithersburg, Maryland, USA) (Germack et al, 2009). These processes are strongly
influenced by mechanical and thermal loading.

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graphene reinforced nanocomposite plates under different load is investigated
(Shingare & Kundalwal, July 2019). In order to indicate the effect of the geometrical
on the mechanical behavior of sandwich wide panels, the Extended High-order
Sandwich Panel Theory is used (Yuan & Kardomateas, September 2018). By using
Petrov – Glerkin method, the mechanical of functionally graded viscoelastic hollow
cylineder under effect of thermo – mechanical load (Akbari, Bagri, & Natarajan, 1
October 2018). By using finite elements model with eight degrees and three nodes of
freedom per node along with high oder shell deformation theory, the effect of
temperature environment combined with mechanical load on the functionally graded
plates (Moita et al, 15 October 2018). Nonlinear dynamic response of sandwich
S-FGM are supported by elastic foundation subjected to thermal environment by using
galerkin method and classical plate theory (Singh & Harsha, July–August 2019).
2.2. Literature review in Vietnam

For organic solar cells, recently, there are some domestic research groups such as Dr.
Dinh Van Chau’s group at VNU-University of Engineering and Technology, Prof. Le
Van Hieu’s group at Ho Chi Minh National University, Prof. Pham Thu Nga, Prof.
Pham Duy Long’s group at Institute of Material Science. They have been interested in
this field during two last decades. For perovskite solar cells, there are some research
groups such as Nguyen Duc Cuong et al. from VNU-University of Engineering and
Technology, and Dr. Nguyen Tran Thuat at VNU-Hanoi University of Science.

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In term of situation, the research projects in Vietnam have the same research direction
compared with foreign countries. The difference in between Vietnam and other
countries is that researchers in foreign countries were furnished modern devices with
high qualification. As a result, the quality of scientific research from aboard is more
dominant than Vietnam. However, it is able to see that domestic research are also
gradually integrating into the research of advanced groups in the world. For example,

there are a lot of articles were published on international respected papers such as
Composite structure, Journal of Sandwich Structures and Materials, Journal of
Vibration and Control, Thin Solid Films, Solar Energy Materials & Solar Cells, J.
Nanomaterials, J. Nanotechnology, v.v... Beside that Vietnamese researchers were also
invited to take international meetings and workshops in specific fields. Based on initial
results, we can apply to stabilize layer structure during fabrication process, although
PCE and active area were still limited (ex: OSCs were fabricated in Faculty of
Engineering Physics and Nanotechnology, VNU-University of Engineering and
Technology).

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CHAPTER 3: METHODOLOGY

3.1. Modelling of SC

The OS basically consists of at least five transparent substrate layers as shown in
Figure 3.1. The substrate may include polyester or many other transparent materials,
sometimes a type of stainless steel is used. But, in this case, the substrate used is glass
and designed on the back of the cell. Superstrate materials can be coated with a
transparent conductive oxide (TCO), such as indium tin oxide (ITO); Poly
(3,4-ethy-lenedioxythiophene) poly (styrenesulfonate) (PEDOT: PSS) is considered the best
option to prevent diffusion into active layers caused by anode and bias factors due to
formation. electrostatic trap centers. This protection layer is placed between the active
layer and the anode. To enhance the performance of organic solar cells, a layer of
material appears with the mixing of regioregular polyi (3-hexylthiophene) (P3HT) and
phenyl-C61-butyric methyl acid (PCBM) with the condition Heat annealing and
mixing ingredients and finally the Al layer covers the upper surface of the cell.

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Figure 3.1 Geometry and coordinate system of nanocomposite multilayer SC.

a) 2D model b) 3D model
3.2. Methodology

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such as Matlab, Maple, etc. Some numerical results are given and compared with the
one of other authors to verify the accuracy of the study.

3.3. Basic Equation

In order to investigate nonlinear static and dynamic analysis of multilayer
nanocomposite structure in SC subjected to mechanical load, this thesis used classic
plate theory along with Galerkin method. Noting that SC rested on elastic foundations.
The strains - displacements along with the Von Karman nonlinear terms as (DD & BO,
1975), (Reddy, 2004), (Duc, 2014)

2 0
2

0
2 0
0
2
0
2 0
;
2
x x
y y
xy xy
w
x
w
z
y
w
x y

e

g

ổ ả ử
ỗ ữ
ả
ỗ ữ
ổ ử
ổ ử
ỗ ữ
ỗ ữ ả
ỗ ữ=

- ỗ ữ
ỗ ữ
ỗ ữ ả
ỗ ữ
ỗ ữ ỗ ữ
ố ứ ố ứ ỗ ả ữ
ỗ ữ
ỗ ả ả ữ
ố ứ
(1)
with
2
0 0
2 0
2
0
2

0 0 2 0

2
0

2 0

0 0 0 0

2
1
, ;
2
2
x x
y y
xy xy

u w w

x x

x
k

v w w

k

y y y

k

w

u v w w

x y

y x x y

e
e
g
ổ ả ổả ử ử
ổ ả ử
ỗ + ỗ ữ ữ 
-ỗ ữ
ả ả
ỗ ố ứ ữ ỗ ả ữ
ổ ử ỗ ữ ổ ử
ỗ ữ
ổ ử
ỗ ữ=ỗ ả + ả ữ ỗ ữ= -ả
ỗ ữ
ỗ ữ
ỗ ữ ỗ ả ố ả ứ ữ ỗ ữ ả
ỗ ữ
ỗ ữ ỗ ữ ỗ ữ
ố ứ ỗả ả ả ả ữ ố ứ ỗ ả ữ
ỗ- ữ
+ +
ỗ ả ả ả ả ữ ỗố ả ả ữứ
ỗ ữ
ố ứ
(2)

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' ' '

12 22 26

' ' '

11 12 16

' ' '

26 16 66

y x

y
x

xy k k xy

k

Q Q Q

s

e

s

g

ổ ử
ổ ử ổ ử
ỗ ữ
ỗ ữ =ỗ ữ
ỗ ữ
ỗ ữ ỗ ữ
ỗ ữ
ỗ ữ
ỗ ữ ố ứ
ố ø è ø
(3)
in which

(

)

' ' '

12 2 11 2 22 2

' ' '

66 16 26

, ,

1 1 1

, 0, 0.

2 1

vE E E

Q Q Q

v v v

E

Q Q Q

v
= = =
- -
-= = =
+
(4)

The moment resultants (M ) and force (i N ) of the SC are determined by i

[ ]

1
1
1
1
, , , .
, , , ,
k

k
k
k
h
n

i i k

k h
h
n

i i k

k h

M z dz i y x xy

N dz i y x xy

s
s

-=
=
= =
= =

å ò

(5)

Replace equation (1) and equation (3) into equation (5) and equation (5) are obtained
as

0 0

11 12 11 12

0 0

12 22 12 22

0
66 66

0 0

11 12 11 12

0 0

12 22 12 22

0
66 66

x x y x y
y x y x y
xy xy xy

N A A B k B k

N A B k

M B B D k D k

N B D k

e e
e e
g
e e
e e
g
= + + +
= + + +
= +
= + + +
= + + +
= +
(6)

</div>
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The nonlinear equilibrium equations of SC rested on elastic foundation according to
the classical theory is given:

Nonlinear dynamic analysis of OS:

, ,

, , , , ,

2 2 2

, 2 2 2 1 1 2

0,
0,

2 2

w
.

xy x y y
x x xy y

x xx xy xy y yy x xx xy xy

y yy

N N

M M M N w N w

w w

N w k k w

x y r t

+ =
+ =
+ + + +
ỉ¶ ¶ ư ¶
+ - ỗ + ữ + =
ả ả ả
ố ứ

(7a)

Nonlinear static stability of OS:

, ,

, , , , ,

2 2

, 2 2 2 1

0,
0,

2 2

xy x y y
x x xy y

x xx xy xy y yy x xx xy xy
y yy

N N

M M M N w N w

w w

N w k k w

x y

+ =
+ =
+ + + +
ỉ¶ ¶ ư
+ - ỗ + ữ+ =
ả ả
ố ứ

(7b)

The geometrical compatibility equation for an imperfect OS is written as:

2 0 2 0 2 0 2 2 2

2 2 2 2

2 2 * 2 2 * 2 2 *

2 2 2 2 2 .

y x xy xy x y

w w w

x y x y x y y x

w w w w w w

x y x y x y y x

e

g

¶ +¶ -¶ =¶ -¶ ¶
¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶
¶ ¶ ¶ ¶ ¶ ¶
- +
-¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

(8)

Airy’ function f(x,y) is defined by

, , , , , .

y xx x yy xy xy

N = f N = f N = -f (9)

</div>
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0 * * * *

24 23 12 22

0 * * * *

14 13 11 12

0 * *

32 31

,
,
.

y y x x y

x y x x y

xy xy xy

C k C k C N C N

C k C N

e
e
g
= + + +
= + + +
= +

(10)

The linear parameters *

ij

C are given in Appendix.

Replace equation (10) and equation (6) along with the support of equation (8) into
equation (7a) and equation (7b), the equation (7a) and equation (7b) is rewritten as
follows

Nonlinear dynamic analysis of OS:

4 4 4 4 4 4

11 4 12 4 13 2 2 14 4 15 4 16 2 2

2 2 2 2 2 2 2

2 2 2 2 2

2 .

f f f w w w

S S S S S S

x y x y x y x y

f w f w f w

q

y x y x x y x y

r

t

¶ + ¶ + ¶ + ¶ + ¶ + ¶
¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶
¶ ¶ ¶ ¶ ¶ ¶ ¶
+ - + + =
¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶
(11a)

Nonlinear static stability of OS:

4 4 4 4 4 4

11 4 12 4 13 2 2 14 4 15 4 16 2 2

2 2 2 2 2 2

2 2 2 2 2 0.

f f f w w w

S S S S S S

x y x y x y x y

f w f w f w

q

y x y x x y x y

</div>
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(

)

(

)

(

)

(

)

(

)

* *

11 12 22 11 12

* *

12 22 12 12 11

* * * *

13 11 11 22 22 12 12 66 31

* *

14 12 23 11 13 11

* *

15 22 24 12 14 22

* * * *

22 23 66 32 11 14 12 24

16 *

12 13 66 12 12

,
S
2 2
,
S
4
4

S B C B C

B C B C

S B C B C B C B C

S B C B C D

B C B C D

B C B C B C B C

S

B C D D D

= +
= +
= + +
-= - + +
= - + +
ổ + + + ử
= -ỗỗ ữữ
+ + + +
è ø

Equation (12) is equation showing appearance of geometrical imperfection. equation
(12) is written from equation (11).

Nonlinear dynamic analysis of OS:

4 4 4 4

11 4 12 4 13 2 2 14 4

4 4 2 2 2 *

15 4 16 2 2 2 2 2

2 2 2 * 2 2 2 * 2

2 2 2 2

2 .

f f f w

S S S S

x y x y x

w w f w w

S S

y x y y x x

f w w f w w

y x x y x y x y y r t

¶ + ¶ + ¶ + ¶
¶ ¶ ¶ ¶ ¶
ỉ ư
¶ ¶ ¶ ¶ ¶
+ + + ỗ + ữ
ả ả ả ả ố ả ¶ ø
ỉ ư ỉ ư

¶ ¶ ¶ ¶ ¶ ¶ ả
- ỗ + ữ+ ỗ + ữ=
ả ả ả ảố ¶ ¶ ø ¶ è ¶ ¶ ø ¶

(12a)

Nonlinear static stability of OS:

4 4 4 4

11 4 12 4 13 2 2 14 4

4 4 2 2 2 *

15 4 16 2 2 2 2 2

2 2 2 * 2 2 2 *

2 2 2

2 0.

f f f w

S S S S

x y x y x

w w f w w

S S

y x y y x x

f w w f w w

y x x y x y x y y

¶ + ¶ + ¶ + ¶
¶ ¶ ¶ ¶ ¶
ỉ ư
¶ ¶ ả ả ả
+ + + ỗ + ữ
ả ả ả ¶ è ¶ ¶ ø
ỉ ư ỉ ư
¶ ¶ ¶ ả ả ả
- ỗ + ữ+ ỗ + ữ=
ả ả ¶ ¶è ¶ ¶ ø ¶ è ¶ ¶ ø
(12b)

</div>
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2 2 2 2

0 * * * *

11 2 12 2 13 2 14 2

2 2 2 2

0 * * * *

22 2 12 2 23 2 24 2

2 2

0 * *

31 32
,
,
2 .
x
y
xy

f f w w

C C C C

y x x y

f f w w

C C C C

x y x y

f w

C C

x y y x

e
e
g
¶ ¶ ¶ ¶
= + -
-¶ ¶ ¶ ¶
¶ ¶ ¶ ¶
= + -
-¶ ¶ ¶ ¶
¶ ¶
= -
-¶ -¶ ¶ ¶
(13)

The linear parameters are given in Appendix.

In order to obtain an imperfect plate’s compatibility equation, inserting equation (13)
into equation (8):

4 4 4 4

* * * *

11 4 12 2 2 13 2 2 14 4

4 4 4 4

* * * *

22 4 12 2 2 23 4 24 2 2

2 2 2

4 4

* *

31 2 2 32 2 2 2 2

2 2 * 2 2 * 2 2

2 2 2

2

xy x y

y x xy xy x

f

w

C

y

x y

y

f

w

C

x

x y

x

x y

w

f

w

C

x y

y

x

w

x

y

x y x y

y

¶

+

¶

-

¶

-

¶

¶ ¶

¶

+

¶

-

¶

-

¶

¶ ¶

¶

¶ ¶

¶

¶

¶

+

=

-¶ -¶

¶ ¶

¶

¶

¶

¶

¶

-

+

-¶

¶

¶ ¶ ¶ ¶

¶

*
2

..

y

x

¶

(14)

3.4. Boundary Conditions

In this section, boundary conditions will be introduced. Assuming that the SC’ edges
are simply supported (SS) that have two different cases:

Case 1: the freely movable (FM) plate
0

xy x

N = =w M = , = 0

x x

N N at

x

=

0, ,

a

xy y

N = =w M = , = 0

y y

N N at

y

=

0,

b

(15)

</div>
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0

x

M

= = =

u w

, = 0

x x

N N at

x

=

0, ,

a

y

M = = =v w , = 0

y y

N N at

y

=

0,

b

. (16)

The approximate solutions of

w

and f will be determined based on above boundary 
conditions. It is shown as:

sin sin ,

sin sin .

n m

w W y x

w h y x

d l

µ d l

=
=

(17a)

2 4

2 2

3 1 0 0

cos 2 os os

1 1

sin sin cos 2 .

2 2

n m n

m n m y x

f B y B c xc y

B x y B x N x N y

d l d

l d l

= + +

+ + + +

(17b)

Therein, the amplitude deflection is called W and W0 =const is a known initial
amplitude.

a

=m

p

/ ,a

b

=n

p

/ , ,b m n=1, 2,… are numbers of half waves in ,x y

direction, respectively and W is amplitude of deflection.

(

1 4

)

i

B i= ÷ are determined by substituted equations (17a) and (17b) into the
compatibility equation (14), obtained

(

)

(

)

2 * 2 1 * 2

11 22
3 4
2 2
, ,
32 32
0.
m n
n m

h W W h W W

B B

A A

B B

µ a µ d

d a

+ +

= =

= = (18)

3.5. Nonlinear Dynamic Analysis

</div>
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(

)

(

)(

)

(

)

(

)

(

)

(

)

1 2 3 4

2 0 2 0

0 0 0 0

0 0 6 1 2

2 2

x y

W W W W W

w

W W T q

TW W T W T W W T W W

T W N N W

t

b

r

a

+ + + + + + +
+ + - + ¶
¶

+ + = (19)

where linear parameter

S i

i

( 1,6)

=

are mentioned in Appendix.

Consider a OS with freely movable edges only subjected to uniform external pressure
sin

q Q= Wt ( Q is the amplitude of uniformly excited load, W is the frequency of the 
load) and uniform compressive forces Px and Py (Pascal) on the edges x=0,a and

y= b. In this case, 0 = - , 0 =

-x x y y

N P h N P h and Eq. (19) is reduced to

(

)

(

)(

)

(

)

(

)

(

)

(

)

0 0 0 0

0 0 6 1

1 2 3 4

2 2
5 2
2 2
w
sin .
x y

TW W W TW TW W T W W

T W P P h W

W W W W

W W T Q t

t

a

b

r

+ + +
+ + +
+ + + +
¶
+ W
¶

- + = (20)

By using Eq. (20), three aspects are taken into consideration: fundamental frequencies
of natural vibration of the SC, frequency – amplitude relation of nonlinear free
vibration and nonlinear dynamic response of SC. The nonlinear dynamic responses of
the SC can be obtained by solving this equation combined with initial conditions to be
assumed as (0) 0,W dW (0) 0

dt

= = by using the fourth – order Runge – Kutta method.

In other hand, from equation (20) as well as using explicit expression, the fundamental
frequencies of a perfect SC be determined approximately as

(

)

2 5
1
2
.

w

a

r

b

+
+ +

= - x y

mn

P P

S S h (21)

</div>
(35)<div class='page_container' data-page=35>

25
load, equation (20) has of the form

(

)

2 3

w

sin( ) 0,

¶ + + + - W =

¶ mn

W

W MW NW F t

t

(22)

with

(

)

(

)

2 2

1 4

2 2 2 2

5 5

, , .

a b a b

= = =

+ + m x + n y + + m x + n y

S

S S

S P P

S

h S P P

M N F Q

S S h

(23)

In order to determined amplitude – frequency relation, W =AsinWt is chosen along
with applying Galerkin method for equation (22). From that, the amplitude – frequency 
relation of nonlinear forced vibration is obtained

2 2
2
8 3
1 0.
3 4

c

p

w

ổ ử
- -ỗ + ữ+ =

ố ứ mn

F

MA NA

A (24)

where

c

= W

w

mn (25)

If F = , i.e. no excitation acting on the SC, equation (22) can be written as form 0

2 2 1 8 3 2 .

3 4
w w

p
ổ ử
= ỗ - + ÷
è ø

NL mn MA NA (26)

3.6. Nonlinear Static Stability

</div>
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(

)

(

)(

)

(

)

(

)

(

)

(

)

1 2 3 4

2 0 2 0

2 2

m x n y

T T W h

TW W h W W W h T W W h

N N W h T W h

µ µ µ

a b µ µ

+ + +

+ + + +

- + + + + =

(27)

The linear parametersSi are mentioned in Appendix.

According to the case the freely movable (FM) plate, the plate is uniformly compressed
by forces Px and Py at the edges x=0,a and y=0,b.

0 = - , 0 = - .

x x y y

N P h N P h (28)

In order to analyze the static post-buckling and buckling behaviors, the nonlinear
equation is determined by replacing equation (28) in equation (27)

(

)

(

)

(

)

1 1

2 2 2

2 .

x

m m m

W

W H h

H W H

P

h W W W

a µ a µ a µ

= - -

-+ + (29)

For SC perfection µ=0, Eq (29) leads to

1 1

2 2 2

2
.

x

m m m

W
H h

H H

P W

h

a

= - - - (30)

The upper compression load makes OS perfect branching in a branched manner that
can be obtained by taking the limit of Fx function when W- >0

1
2 .
xupper
m
H
F
h

a

</div>
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CHAPTER 4: RESULTS AND DISCUSSION

4.1. Introduction

This chapter shows and discuss the obtained results that will show the effect of factors
such as geometrical parameter, elastic foundations and initial imperfection on nonlinear
static and dynamic analysis of SC with the properties such as initial thickness, poison
parameter, modulus Young given as in table 4.1. In order to has those results, Eq (20)
and Eq (29) are used along with a b= =0.03 ;m m n

(

) ( )

= 1,1 .

Table 4.1Initial thickness and properties materials of layers of SC.

(

)

E GPa r

(

kg m/ 3

)

v h

Al 70 2601 2701- 0.35 100nm

ITO 116 7120 0.35 120nm

PEDOT: PSS 2.3 1000 0.4 50nm

P3HT: PCBM

6

1200 1500- 0.23 170nm

Glass Substrate 69 2400 0.23 550 mµ

4.2. Natural frequency

</div>
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value of the two frequencies is the same, it will lead to a sudden and significant change
in deflection amplitude.

Table 4.2 Influenceof modes( , )m n and ratioh a/ on the SC’s natural oscillation 
frequency.

h a

( )

m n,

( )

1,1

( )

1,3

( )

3,3

( )

3,5

0.05 19001 71192 171100 286770
0.1 19032 71243 171290 287090
0.15 19040 71261 171360 287200
0.2 19733 72940 177600 297540
0.25 19808 73121 178270 298660
4.3. Dynamic response

With Px =0,Py =0, Figure 4.1 illustrates the influence of geometrical parameter a b/
on SC’ nonlinear dynamic response. It easy see that increasing the ratio a b/ leads to
the amplitude of the SC increases. In details, the amplitude of SC has highest value at

a b = 2 and at a b/ = 1 the amplitude of SC has the smallest values.

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Figure 4.1 Influence of ratio a b/ on the SC’s nonlinear dynamic response

(

Px =0,Py =0 .

)

Figure 4.2 Influence of ratio a/h on the SC’s nonlinear dynamic response

(

Px =0,Py =0 .

)

0 1 2 3 4 5 6 7

x 10-3
-1.5

-1
-0.5
0
0.5
1
1.5x 10

-5

(m,n)=(1,1), q=300sin(3500t)
t(s)

W(m)

a/b = 1
a/b = 1.5
a/b = 2

0 1 2 3 4 5 6 7

t(s) 10-3

-8
-6

-4
-2
0
2
4
6
8

W(m)

10-6

(m,n)=(1,1), =0.1, T=0 , a/b=1, N=1, q=1800sin(1600t)

</div>
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Figure 4.3 Influence of the exciting force amplitude Q on the SC’s dynamic response

(

Px =0,Py =0 .

)

Figure 4.3 can explain how much influence harmonic uniform exciting force has on the
SC’ dynamic response. In figure 4.3, amplitudes Q=100 N m/ 2Q=250 N m/ 2 and

300 /
=

Q N m are considered. As expect, the reduction of excitation force amplitude

Q decreases the SC’ nonlinear dynamic amplitude while the vibaration period still the

same.

Figure 4.4 Influence of the pre-loaded axial compression

P

x on SC’ dynamic response.

0 1 2 3 4 5 6 7

x 10-3
-3

-2
-1
0
1
2
3x 10

-6

(m,n)=(1,1), q=Qsin(3500t), Px = 0, Py = 0.

t(s)

W(m)

Q = 100 N/m2
Q = 250 N/m2
Q = 300 N/m2

0 1 2 3 4 5 6 7

x 10-3
-4

-3
-2
-1
0
1
2
3
4x 10

-6

(m,n)=(1,1), Py = 0, q=300sin(3500t)
t(s)

W(m)

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Figure 4.5 Effect of the pre-loaded axial compression Py on the SC’ dynamic response.

The pre-loaded axial compression P Px, y are considered in figure 4.4 and figure 4.5
with various values. It can be noted that the higher value of the pre-loaded axial
compression is, the higher nonlinear dynamic amplitude of the organic solar cell is.

Figure 4.6 Effect of initial imperfection

W

0 on the dynamic response of the SC.

0 1 2 3 4 5 6 7

x 10-3
-4

-3
-2
-1
0
1
2
3
4x 10

-6

(m,n)=(1,1), Px = 0 , q=300sin(3500t)
t(s)

W(m)

Py = 0
Py = 10 MPa
Py = 17 MPa

0 1 2 3 4 5 6 7

x 10-3
-12

-10
-8
-6
-4
-2
0
2
4x 10

-6

(m,n)=(1,1), Px = 0, Py = 0, q=300sin(3500t)

t(s)

W(m)

W0 = 0

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Figure 4.6 indicate that the SC’ dynamic response is impacted by initial imperfection

W

. Clearly, the amplitudes of SC’ nonlinear vibration will change much when the
amplitude of initial imperfection rise.

4.4. Frequency – amplitude relation

Figure 4.7 shows the influence of external force F on the frequency – amplitude
relations of SC’ frequency – amplitude curves. As can be seen, when the excitation
force decreases, the curves of forced vibration are closer to the curve of free vibration.

Figure 4.7 Influence of external force Fon SC’ frequency – amplitude curves.

0 10 20 30 40 50 60 70 80

0
0.02
0.04
0.06
0.08
0.1

(m,n)=(1,1), Px = 0, Py = 0, W0 = 0,q=300sin(3500t)

Frequency ratio

Ampl

itu

F = 0 N

F = 2 x 107 N

</div>
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33
4.5. Nonlinear Static

Figure 4.8 The influence of initial geometrical imperfection on the SC’ stability with 
uniaxial compressive load.

The effect of imperfection about geometrical on the SC’ load – deflection amplitude
curve is described in figure 4.8. It can be noted that there is always 1 point where SC's
load carrying capacity will be changed under the influence of initial gemetrical
imperfection. Specifically, the SC’s static stability will be negatively affected by
imperfection parameter along with condition is small defection’value. The SC's load
carrying capacity will reduce when the rising the initial imperfection. However, when
passing the point that is mentioned above, the SC's load carrying capacity will rise with
the increasing of initial imperfection.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

W/h

0
0.05
0.1
0.15
0.2
0.25

(GPa)

a=b; m=n=1; k1 = 0.1 GPa/m; k2 = 0.02 GPa.m
= 0

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Figure 4.9 The influence of a/b ratio the SC’ load – deflection amplitude curve.

The effect of a b/ ratio on the SC’ load – deflection amplitude curve is demonstrated in
figure 4.9. From figure 4.9, there is 1 point like in the discussion of the figure 4.8.
Besides, the static stability will reduce when incresing ratio a b/ . On contrary, the load
rise of ratio a/b inrease static stability of SC.

Figure 4.10 The effect of elastic foundations on the SC’ the load – deflection amplitude 
curve.

0 0.5 1 1.5 2 2.5 3

W/h

0
0.1
0.2
0.3
0.4
0.5
0.6

(GPa)

(1): k1 = 0.1 GPa/m; k2 = 0.02 GPa.m
(2): k1 = 0.2 GPa/m; k2 = 0.04 GPa.m
(3): k1 = 0.4 GPa/m; k2 = 0.05 GPa.m (3)

(2)
(1)

a=b; m=n=1.
= 0

</div>
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Figure 4.10 shows the influnce of elastic foundation on the SC’ load – deflection
ampiltude curved. It can be seen that the rise of the modulus k k will improve the SC’ 1, 2
load carring capacity while initial imperfection still the same. It demonstrates that
elastic foundations have significant influence on the SC’s static stability.

4.6. Critical buckling load

Table 4.3 Effects of the elastic foundations and ratio a/b on the SC’ critical buckling 
load (unit: GPa).

1; 2

k k

(

GPa m GPa m / ; .

)

a b

1 1.5 2 2.5

0.1; 0.02 0.29368 0.71574 1.4681 2.6607

0.1; 0.04 0.29371 0.71580 1.4682 2.6610

0.2; 0.02 0.29373 0.71585 1.4684 2.6614

0.2; 0.04 0.29376 0.71593 1.4685 2.6615

0.3; 0.02 0.29383 0.71611 1.4696 2.6623

0.3; 0.04 0.29387 0.71617 1.4704 2.6633

</div>
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determined that will very helpful. If there are the same values elastic foundation and
/

</div>
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CHAPTER 5: CONCLUSIONS AND FURTHER WORKS

5.1. Conclusions

This thesis investigates the nonlinear static and dynamic analysis of multilayer

nanocomposite structure in solar cell. In order to evaluate the role of geometrical
parameter, initial imperfection, elastic foundation and load on the nonlinear static and
dynamic analysis of multilayer nanocomposite structure in solar cell; classical theory,
Galerkin method as well as Runge – Kutta method are used. This thesis has some
remarkable conclusions as: The geometrical parameter, elastic foundation has positive
influence on the nonlinear static and dynamic analysis of solar cell. The increasing of
excitation force amplitude Q and Px rise nonlinear dynamic amplitude of solar cell.

The amplitudes of SC’ nonlinear vibration will change much when the amplitude of
initial imperfection rise. Natural frequency of OS will significant changes according to
geometrical parameter and mode m, n. Changing the value of elastic foundations and 
geometrical parameter will lead to changing of critical buckling load.

This thesis has some remarkable results. These results are the scientific basis to
improve currently durability of solar panels. Based on the obtained results, designers,
scientists and manufacturers will select geometric parameters to ensure load capacity of
solar panels. Besides, it ensures the electrical performance of solar panels.

5.2. Future works

</div>
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</div>
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APPENDIX

(

)

(

)

11 2 12 2 22 11

1 1

16 26 66

11 2 12 2 22 11

1 1

16 26 66

1
3
2

11 2 12

1
; ; ;
1 1
0; ;
2 1
; ; ;
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</div>

Nonlinear static and dynamic analysis of mulltilayer nanocomposite structures in solar cell

<b>NONLINEAR STATIC AND DYNAMIC </b>

<b>ANALYSIS OF MULTILAYER </b>

<b>NANOCOMPOSITES STRUCTURES IN </b>

<b>SOLAR CELL </b>

<b>MASTER’S THESIS </b>

<b>NONLINEAR STATIC AND DYNAMIC </b>

<b>ANALYSIS OF MULTILAYER </b>

<b>NANOCOMPOSITES STRUCTURES IN </b>

<b>SOLAR CELL </b>

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