Nonlinear static and dynamic analysis of mulltilayer nanocomposite structures 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 nanocomposit[r]

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

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

I

ACKNOWLEDGEMENT

II

TABLE OF CONTENTS

ACKNOWLEDGEMENT I

LIST OF FIGURES IV

LIST OF TABLES VI

NOMENCLATURES AND ABBREVIATIONS VII

ABSTRACT VIII

CHAPTER 1: INTRODUCTION

1.1. Background

1.2. Research objectives

1.3. The layout of the thesis

CHAPTER 2: LITERATURE REVIEW

2.1. Literature review in Outside Vietnam

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

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

IV

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 Px 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 W0 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

V

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

VI

LIST OF TABLES

Table Initial thickness and properties materials of layers of SC 27

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

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

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

1

k The Winker foundation

2

k The Pasternak foundation

0

W =µh The initial imperfection

IM Immovable

FM Freely movable

GPa GygaPascal =109 Pascal ,

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 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

1

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 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

2

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:

3

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

4

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)

5

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)

6

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:

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:

8

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 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 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

9

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 USD/W in 2008 (the price of a module of solar cell to generate 1W of energy under sun sunlight density) to about 0.5 USD/W in 2017 (pvXchange, 2017)

10

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

11

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 (http://www.nrel.gov/ncpv/images/efficiency_chart.jpg) 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),

12

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

13

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, 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

14

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)

15

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

16

a)

b)

Figure 3.1 Geometry and coordinate system of nanocomposite multilayer SC

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

17

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 2 0 2 ; x x y y xy xy w x w z y w x y e e e e g g ổ ả ỗ ữ ả ỗ ữ ổ ổ 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 0 2

0

0

2

2

0 0

1 , ; 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 ỉ ¶ ỉ¶ ư ỉ ¶ ỗ + ỗ ữ ữ -ỗ ữ ả ả ỗ ố ứ ữ ỗ ả ữ ổ ç ÷ ỉ ç ÷ ỉ ç ÷=ç 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)

18

' ' '

12 22 26

' ' '

11 12 16

' ' '

26 16 66

,

y x

y x

xy k k xy

k

Q Q Q

Q Q Q

Q Q Q

s e e s s g ỉ ỉ ỉ ỗ ữ ỗ ữ =ỗ ữ ỗ ữ ỗ ữ ỗ ữ ỗ ữ ỗ ữ ỗ ữ ố ø è ø è ø (3) in which ( ) ' ' '

12 11 22

' ' '

66 16 26

, ,

1 1

, 0,

2

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 , , , , , , , 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 å ò (5)

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

0

11 12 11 12

0

12 22 12 22

0 66 66

0

11 12 11 12

0

12 22 12 22

0 66 66

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

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

N A A B k B k

N A A B k B k

N A B k

M B B D k D k

M B B D k D k

N B D k

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

19

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 1

0, 0,

2

w

xy x y y x x xy y

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

N N

N N

M M M N w N w

w w

N w k k w

x y r t

+ = + = + + + + ổả ả ả + - ỗ + ữ + = ả ả ả ố ứ (7a)

Nonlinear static stability of OS:

, ,

, ,

, , , , ,

2

, 2

0, 0,

2

0

xy x y y x x xy y

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

N N

N N

M M M N w N w

w w

N w k k w

x y + = + = + + + + ổả ả + - ỗ + ữ+ = ¶ ¶ è ø (7b)

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

2 2 2

2 2

2 2 * 2 * 2 *

2 2 2

y x xy xy x y

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 e g

¶ +¶ -¶ =¶ -¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ - + -¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ (8)

Airy’ function f(x,y) is defined by

, , , , ,

y xx x yy xy xy

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

20

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 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

11 12 13 2 14 15 16 2

2 2 2 2

1

2 2 2

w

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

11 12 13 2 14 15 16 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

21 ( ) ( ) ( ) ( ) ( ) * *

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 , S 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 + + + = -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

11 12 13 2 14

4 2 *

15 16 2 2

2 2 * 2 *

1

2 2

w

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 ¶ ¶ ả ả ả + + + ỗ + ữ ả ¶ ¶ ¶ è ¶ ¶ ø ỉ ỉ ả ả ả ả ả ả ả - ỗ + ữ+ ỗ + ữ= ả ả ả ảố ả ¶ ø ¶ è ¶ ¶ ø ¶ (12a)

Nonlinear static stability of OS:

4 4

11 12 13 2 14

4 2 *

15 16 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

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

22

2 2

0 * * * *

11 12 13 14

2 2

0 * * * *

22 12 23 24

2

0 * *

31 32 , , 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

* * * *

11 12 2 13 2 14

4 4

* * * *

22 12 2 23 24 2

2 2

4

* *

31 2 32 2 2

2 * 2 * 2

2 2

2

2

xy x y

y x xy xy x

f f w w

C C C C

y x y x y y

f f w w

C C C C

x x y x x y

w w w

f w

C C

x y x y x y y x

w w w w w w

x y x y x y y

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

xy x

N = =w M = , =

x x

N N at x=0, ,a

xy y

N = =w M = , =

y y

N N at y=0,b

(15)

23

0

x

M = = =u w , =

x x

N N at x=0, ,a

y

M = = =v w , =

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

n m

w W y x

w h y x

d l

µ d l

= =

(17a)

2

2

3 0

cos os os

1

sin sin cos

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

direction, respectively and W is amplitude of deflection

( 4)

i

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

( ) ( )

2 * *

11 22 2 , , 32 32 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

24

( ) ( )( ) ( )

( ) ( )( )

1

2

5

0 0

2

0

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 ii( 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

0,

y= b In this case, = - , =

-x x y y

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

( ) ( )( ) ( )

( ) ( ) ( )

0 0

2

0

1

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)

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 w a r b + + +

= - x y

mn

P P

S S h (21)

25 load, equation (20) has of the form

( )

2

2

2 w sin( ) 0,

¶ + + + - W =

¶ mn

W

W MW NW F t

t

(22)

with

( ) ( )

2

1

2 2

1

5

, ,

r

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 3 c p w ỉ - -ỗ + ữ+ =

ố ứ mn

F

MA NA

A (24)

where

/

c = W wmn (25)

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

2 1 .

3 w w p ổ = ỗ - + ữ è ø

NL mn MA NA (26)

3.6 Nonlinear Static Stability

26

( ) ( )( ) ( )

( )( ) ( )

1

2

5

2

0

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

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 = - , = - .

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

2 2

2

2

x

m m m

W

W

W H h

H W H

P

h W W W

µ

a µ a µ a µ

+

+

= - -

-+ + (29)

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

3

1

2 2

2

x

m m m

W H h

H H

P W

h

a a 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 xupper m H F h a

27

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

28

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

29

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

x 10-3 -1.5

-1 -0.5 0.5 1.5x 10

-5

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

W(m)

a/b = a/b = 1.5 a/b =

0

t(s) 10-3

-8 -6 -4 -2

W(m)

10-6

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

30

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

2

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 Px on SC’ dynamic response

0

x 10-3 -3

-2 -1 3x 10

-6

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

t(s)

W(m)

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

0

x 10-3 -4

-3 -2 -1 4x 10

-6

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

W(m)

31

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 W0 on the dynamic response of the SC

0

x 10-3 -4

-3 -2 -1 4x 10

-6

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

W(m)

Py = Py = 10 MPa Py = 17 MPa

0

x 10-3 -12

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

-6

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

t(s)

W(m)

W0 =

32

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

0

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

de

A)

F = N

F = x 107 N

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 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 1.2 1.4 1.6 1.8

W/h

0 0.05 0.1 0.15 0.2 0.25

Fx

(GPa)

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

34

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 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 1.5 2.5

W/h

0 0.1 0.2 0.3 0.4 0.5 0.6

Fx

(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 =

35

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;

k k (GPa m GPa m / ; )

/

a b

1 1.5 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

36

determined that will very helpful If there are the same values elastic foundation and /

37

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

38

39

APPENDIX

( )

( )

11 12 22 11

1

16 26 66

1

11 12 22 11

1

16 26 66

1

11 12

1 ; ; ; 1 0; ; ; ; ; 1

0; B ;

2

;

1 12

n n

k k k k k

k k k k

n

k k

k k

n n

k k k k k k k

k k k k

n

k k k

k k

n

k k k

k k k k

E h v E h

A A A A

v v

E h

A A A

v E h z v E h z

B B B B

v v

E h z

B B

v

E h v

D h z D

v = = = = = = = = = = - -= = = + = = = - -= = = + ổ = ỗ + ữ = - ố ứ å å å å å å å ( ) 22 11

16 26 66

1

; ;

1

0; D ;

2 12

n

k k k

k k

n

k k

k k

k k

E h z

D D

v

E h

D D h z

v = = = -ỉ = = = ỗ + ữ + ố ứ ồ ồ

2 * * 66

12 22 11 31 32

66 66

* 22 * 12 * 12 12 22 11 * 12 22 22 12

11 12 13 14

* 11 * 12 11 11 12 * 12 12 11 22

22 23 24

1

,C ,C

,C ,C ,C

,C ,C

B

A A A

A A

A A A B A B A B A B

C

A A B A B A B A B

C

D = - + = =

-= = - = =

D D D D

-

-= = =

D D D

( )

4 2 2

1 14 15 16

11 12

2 * *

22 11

4

3 * *

22 11

4

2

3

16 16

m n m n m

n m n m

n m

H S S S k k

ab S S H A A H A A

a b a b a b

b a b a

40

( ) (( ))

11 12

1 * *

22 11

* * * * * 2

14 23 32 13 24

4 2

2 11 12 13 * 4 * 4 * * * 2 2

11 22 12 12 31

4

3 *

22

* *

14 23 32

4 , , , 32

9

n m m n

n m m n

S S

T

A A ab

A A A A A

T S S S

A A A A A

T

A

A A A

T

ab

a b

b a a b

a b a b

b a a b

b b a b a ổ ửổ- =ỗ + ữỗ ữ è ø è ø é éë + - - - ùûù ê ú = + + é + + + + ù ê ë û ú ë û = -+ -ổ = ỗ - ữ ố ứ ( ) ( ) ( )( )

* * * 2

13 24

* * * * * 2

11 22 12 12 31

4 2

5 14 15 16

6 , , A A

A A A A A

T S S S W W

T ab

a b

b a a b

a b a b

-

-+ + + +

= + + +

=

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