SUPERVISOR DECLARATION “I hereby declare that I have read...
Transcript of SUPERVISOR DECLARATION “I hereby declare that I have read...
SUPERVISOR DECLARATION
“I hereby declare that I have read this thesis and in my opinion this report is sufficient in
terms of scope and quality for the award of the degree of
Bachelor of Mechanical Engineering (structure and material)”
Signature : ...................................................
Supervisor : …………………………………
Date : …………………………………
Signature : ...................................................
2nd Supervisor : …………………………………
Date : …………………………………
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DECLARATION
“I hereby declare that the work in this report is my own except for summaries and
quotations which have been duly acknowledged.”
Signature : ..............................................
Author : ………………………………..
Date : ………………………………..
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FOR MY BELOVED PARENTS,
SIBLINGS AND MY FREINDS
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ACKNOWLEDGEMENTS
Assalamualaikum w.b.t
On this occasion, I wish to express, and thanks to my supervisor, Dr. Ruztamreen
b. Jenal, He has served me for help in completing the first undergraduate. Not forgetting
my parents and friends who helped me a lot in the completion of a Bachelor Degree
Project report. Without them I might not be fully complete these papers. Their service is
really huge to me.
Not forgetting the other parties involved, either directly or indirectly. Only God
alone can give back to them. With this, once again I extend my thank-you very much.
Wassalam.
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ABSTRACT
Non linear acoustics is one of the analytical methods used for the purpose of
reviewing whether a specimen has a disability or not. Apart from studying the specimen,
it also aims to analyze the defects that occur in the specimen. These methods include the
use or transfer of a function called "transfer function". Transfer function is used to
generate a graph and the graph, we can make comparisons when there is a defect in
material or not.
This relationship involves the material elasticity and nonlinear acoustic effects.
Elasticity of materials depends on the properties of young modulus of plate such as
aluminum which has elasticity 72400 MPa. Aluminum is easily established, cheap and
readily available. In addition, aluminum is a medium elasticity causes aluminum are
choose as a specimens selected experiments.
However, in this investigation, we used ABAQUS software is a software technology
without involving the destruction of the plate and analysis Finite Element Method
(FEM). These methods include the investigation of the properties of materials
(Aluminum). As such we can see the properties of aluminum that has a density of 2780
kg / m³ and the Poisson ratio of 0.33. From these characteristics, we can see the changes
that occur in the specimens, especially in the frequency. This is because the main focus
is the frequency.
Plates that do not have a disability will produce the same output frequency to input
frequency. This will happen because of reflection back to the plate when no valid target
out. Therefore, this method is very good because it can be reviewed in detail in the event
of a defect in the plate. However, this method requires high expertise because it requires
analysts with knowledge to make the analysis of published data output. Of the results
obtained showed that contrary to graph theory. Graph obtained is a not straight lined.
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ABSTRAK
Non linear akustik adalah salah satu kaedah analisis yang digunapakai untuk tujuan
mengkaji spesimen samada mempunyai kecacatan ataupun tidak. Selain daripada mengkaji
spesimen, ia juga bertujuan menganalisis tahap kecacatan yang berlaku pada specimen. Kaedah
ini meliputi penggunaan persamaan fungsi pemindahan atau dikenali “transfer function”. Fungsi
pemindahan digunakan untuk menghasilkan graf dan daripada graf tersebut , kita akan dapat
membuat perbandingan apabila wujudnya kecacatan pada bahan ataupun tidak.
Hubungan ini melibatkan kekenyalan bahan dengan kesan non linear akustik. Kekenyalan
bahan bergantung kepada sifat modulus young bahan tersebut contohnya aluminium yang
mempunyai kekenyalan 72400 Mpa. Aluminium bersifat mudah dibentuk, murah dan mudah
didapati. Selain itu kerana sifat kekenyalannya yang sederhana menyebabkan alluminium dipilih
sabagai spesimen ujikaji.
Walaubagaimanapun, dalam penyiasatan ini, kita menggunakan perisian ABAQUS iaitu satu
perisian menggunakan teknologi tanpa melibatkan kemusnahan plat dan melibatkan analisis
unsur tidak terhingga. Kaedah ini meliputi penyiasatan melalui sifat-sifat bahan (alluminium)
tersebut. Sepertimana kita dapat melihat sifat aluminum yang mempunyai ketumpatan 2780
kg/m³ dan nisbah poisson 0.33. Daripada ciri-ciri tersebut, maka kita boleh melihat perubahan
yang berlaku pada spesimen terutama melalui frekuensinya. Ini kerana fokus utama adalah
kepada frekuensi.
Plat yang tidak mempunyai kecacatan akan menghasilkan frekuensi output yang sama
dengan frekuensi input. Ini kerana akan berlaku pantulan balik pada plat apabila tiada sasaran
keluar berlaku. Oleh itu, kaedah ini adalah sangat bagus kerana ia dapat mengkaji secara
terperinci apabila berlaku kecacatan pada plat. Walaubagaimanapun, kaedah ini memerlukan
kepakaran yang tinggi kerana ia memerlukan penganalisis yang berilmu untuk membuat analisis
semula terhadap data output yang diterbitkan. Daripada keputusan yang diperolehi menunjukkan
graph yang diperolehi bercanggah dengan teori. Graph yang diperolehi tidak bergaris lurus.
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TABLE OF CONTENTS
CHAPTER CONTENTS PAGES
ACKNOWLEDGEMENT iv
ABSTRACT v
ABSTRAK vi
CONTENTS vii
LIST OF FIGURE ix
LIST OF TABLE xii
NOMENCLATURE xii
APPENDIXS
1.1 Gantt Chart PSM 1
1.2 Gantt Chart PSM II
1.3 Result Frequency uncracked (1-6)
1.4 Result Frequency uncracked (932-939)
1.5 Result Frequency cracked (1-6)
1.6 Transfer Function Matrix
1.7 R Value Analysis
CHAPTER 1 INTRODUCTION
1.1 Background 1
1.2 Objective 3
1.3 Outline of the Thesis 4
1.4 Problem Statement 4
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CHAPTER 2 DAMAGE DETECTION BY USING NONLINEAR
ACCOUSTIC EFFECT
2.1 Introduction 5
2.2 Crack Detection Method 7
2.2.1 Analytical Modeling Method 7
2.2.2 Finite Element Method (Fem) 9
2.3 Non-Linear Acoustics 11
2.3.1 Surface Acoustic Waves 11
2.3.2 Nonlinear Acoustic Mechanisms 13
2.3.3 Non Linear Elasticity 13
2.4 Harmonics Generation 15
2.5 Sidebands Generation And Amplitude
Modulation
15
2.6 Summary 16
CHAPTER 3 FINITE ELEMENT ANALYSIS
3.1 Flow Analysis 17
3.2 Modal Analysis 19
3.3 Modal Analysis Using Finiti Element
Method (FEM) 19
3.4 Uncracked Plate 20
3.5 Cracked Plate 23
3.6 Frequency Response Matrix *H(ώ)+ 26
3.7 Model Diagram 27
3.8 Modulus Intensity ( R Value) 28
CHAPTER 4 MODAL ANALYSIS OF ALUMINUM
4.1 Finite Element Analysis (FEA) Result 29
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4.2 Mode Shape 29
4.2.1 Mode Shape Uncracked Results 30
4.2.2 Mode Shape Cracked Stiffness 34
4.3 Natural Frequency Response Function
For Plates With Several Of Stiffness Value
At Center Area
37
4.4 Transfer Function Result 39
CHAPTER 5 RESULT ANALYSIS
5.1 Analysis Results 41
5.2 Optimizing Of Frequency Range 41
5.3 Natural Frequency Shifting 43
5.4 Modulation Intensity (R Value) Against
Stiffness
45
5.5 Conclusion 47
CHAPTER 6 CONCLUSION AND RECOMMENDATIONS
6.1 SUMMARY OF RESEARCH 48
6.2 CONCLUSIONS 49
6.3 RECOMMENDATIONS 50
CHAPTER 7 REFERENCES
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LIST OF FIGURE
NO TITLE PAGES
Figure 2.1 : specimen with impact damage area 6
Figure 2.2 : step simulation to analysis of specimen 7
Figure 2.3 receptance analysis 9
Figure2.4 : A cracked beam is decomposed into three plain beams
where the cracked section is represented by a short reduced
cross section beam
10
Figure 2.5 : (a) Finite Element Method 10
(b) Average of R value against crack size with various
damping factors
11
Figure 2.6 : Particles move in ellipses in a surface acoustic wave. The
amplitudes decrease exponentially with depth
12
Figure 2.7 : stress-strain curve for sandstone of nonlinear and linear
elasticity
14
Figure 2.8 : stress-strain curve for the glass characteristics of nonlinear
and linear elasticity
14
Figure 3.1 : procedure of analysis 18
Figure 3.2 : Method to create and analysis part using Finite Element
Analysis 0f ABAQUS software
20
Figure 3.3 : Model Plate for uncracked (150 mm x 400 mm x 2 mm ) 22
Figure 3.4 : Material properties of the aluminum plate model for FE
analysis
23
Figure 3.5 : model aluminum plate 24
Figure 3.6 : selected stiffness first and second plate 25
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Figure 3.7 : dimension of sensor 27
Figure 4.1 : picture uncracked with stiffness, E = 69 Gpa 33
Figure 4.3 : Plate response 38
Figure 5.2 : Graph optimizing frequency range and total quantity 1700 42
Figure 5.4 : result of natural frequency shifting versus stiffness for the
aluminum plate at various vibration modes excitation from
FE method
44
Figure 5.6 : Graph of material stiffness against the R value 46
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LIST OF TABLE
NO TITLE PAGES
Table 4.2 : natural frequency result 37
Table 4.4 : natural frequency from z-direction 39
Table 5.1 : Total number of Frequency range
42
Table 5.3 : Percentage of natural frequency shifting of uncracked and
cracked plate
43
Table 5.5 : Diagram shows the stiffness reduction with the average R
value
45
NOMENCLATURE
Bo Mean value of the modulation signal
Bı Peak-to-peak variation around the modulation
E Modulus of elasticity
fο Fundamental frequency
H i j (w) Transfer function value at frequency between points i and j
wn Natural frequency at mode n
U i/j Mode shape value at location i/j for mode n
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CHAPTER 1
INTRODUCTION
1.1 BACKGROUND
In today's world, engineering is an important area to meet modern times.
Development time is the time of the sophisticated and people can move fast without
any restrictions. As now, researchers may be studying equipment or materials
without destroying the material. In this analysis involves the simulation using
ABAQUS software. ABAQUS is a suite of software applications for Finite
Element Analysis (FEA) and computer-aided engineering.
ABAQUS is used in the automotive, aerospace, and industries products. The
software is popular with academic and research institutions due to the wide material
modeling capability, and the program's ability to be customized. Software ABAQUS
also provides a good collection of multiphysics capabilities, such as coupled
acoustic-structural, piezoelectric, and structural-pore capabilities, making it attractive
for production-level simulations where multiple fields need to be coupled. However,
this research is to use the software to investigate the relationship between material
stiffness with non linear acoustics effect using plates.
Stiffness is a property of a solid body. Stiffness can to classify as the
resistance of an elastic body to deformation by an applied force along a given degree
of freedom (DOF) when a set of loading points and boundary conditions are
prescribed on the elastic body. It is an extensive material property. The stiffness of a
structure is of principal importance in many engineering applications, so the modulus
of elasticity is often one of the primary properties considered when selecting a
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material. A high modulus of elasticity is sought when deflections are undesirable,
while a low modulus of elasticity is required when flexibility is needed.
While, non linear acoustic is to review, evaluate and analyze the relationship
between material stiffness with nonlinear acoustic. In this situation, two different
properties of structure will be placed elasticity properties investigated and different
properties of a material considered as a defect (damage). It is an essential element
either qualitatively or quantitatively to determine the presence of damage to prevent
any consequences that could lead to catastrophe. Nonlinear are used in the method
for crack detection in metallic structures. This method involved frequency
modulation, side bands, mode shape and changes in frequencies. Although all these
effects are crucial for incipient damage detection, their physical explanation is still
not well understood [4]. However, it is generally agreed that the interaction between
the high-frequency acoustical wave and the low-frequency modal excitation is
important for crack detection. The application of this method in real engineering
fields is very limited.
Damage in material can happen by many mechanisms. Damage is one that is
common in structural materials. As such, damage modeling has been a remarkably
active trend in the Engineering community since the 50s, so that it is largely beyond
our scope even to try to review the huge existing literature on this subject. This paper
covers methods of detecting damage (different properties) in the structure and
frequency response. Damage in structures can be defined as changes of material or
geometric properties in a structure that could affect the structure’s performance. In
solid material like metal, Jean Lemaitre (2005) [2] defined damage as the creation
and growth of micro voids or micro cracks that create discontinuities in a
homogeneous material.
Damage detection in structural materials is an important for public safety. It
is required that the neighborhood of the damage is known a priori and part of the
structure under inspection is readily accessible. Different techniques and methods for
damage detection are available in the literature which can be used and classified.
Example methods are visual, acoustic, magnetic field and eddy current techniques.
While for Nondestructive Test/Evaluation (NDT/NDE) methods, widely used NDT
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techniques are dye penetration, magnetic particle, eddy-current, radiography,
ultrasonic and acoustic emission, as summarized by Staszewski et. al. (2004) [3] and
Gdoutus [5].
In this technique the condition of a structure is determined and quantified by
inspecting changes in its global structural characteristics. Simulation is one of the
methods are categorized as Non-Destructive Testing (NDT) which is a method used
to detect or measure defects of a material or system without damaging the material
being tested. This method plays an important role in medical technology, quality
control and shelf life determination of a structure. It is widely used in the
manufacture of petrochemicals, power sources, transportation and civil engineering.
There are a variety of NDT techniques and these techniques should be used together,
depending on the material being tested. Almost all metal or non-metallic materials
tested by NDT techniques. For the sake of safety, reliability and operational life, it is
essential to monitor the health status of structural systems.
1.2 OBJECTIVE
Nonlinear acoustics will be used for damage detection in a structure. The main
focus will be in simulation of Finite Element Analysis (FEA) using ABAQUS
software. The ultimate objective of the research work presented to analyze the
Modulus Intensity (R value) behind these nonlinear acoustic modulations in the
presence of the crack. In order to achieve these objectives that project aims to:
i. To evaluate the effect of stiffness changes in small area that representing
fatigue crack on plate.
ii. To determine frequencies of stiffness versus natural frequency shifting
iii. To study relative and relation of material stiffness versus R value.
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1.3 OUTLINE OF THE THESIS
A brief summary of the thesis is presented in this section. The research work
undertaken consists of two major parts. Firstly, finite element modeling is used to
establish excitation frequencies for nonlinear acoustics. Secondly, analyze result to
perform and to investigate Modulus Intensity (R value) for nonlinear acoustic
modulations in the presence of the stiffness.
Damage detection by using nonlinear acoustic effects are reviewed in a chapter 2.
This chapter indicates the explanation of crack detection method and analytical
modeling method.
Finite Element Analysis (FEA) is described in chapter 3. Finite Element Analysis
focused to the uncrack and crack of the aluminum plate. Analysis of structure used
the ABAQUS software.
Next chapter discussed the modal analysis of the aluminum plate that conducted
to know the mode shape of the structure. This chapter consists of the result uncrack
and cracked plate structure of the mode shape and frequency response function.
Result analysis can to get after modal analysis is being described. Relation
between material stiffness against modulation intensity (R Value) can to look. Last
chapter is a conclusion and recommendation of the thesis. Scope of this study is only
limited the analytical analysis using Finite Element Method with use of ABAQUS
software.
1.4 PROBLEM STATEMENT
The mechanism of the nonlinear acoustic effect is not quite established R.B
Jenal (2010) had assumed that one of the mechanisms is due to the behavior of the
fatigue crack surfaces interaction. From the above hypothesis this study is aimed to
relate the relation of the material stiffness surfaces interaction with the high
frequency vibration.
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CHAPTER 2
DAMAGE DETECTION BY USING NONLINEAR ACOUSTIC EFFECT
2.1 INTRODUCTION
There are many ways that we can use to analyze whether the product can be
safely used or whether it is in good condition or not. Non-destructive testing is the
best method. This is because, it not only saves cost but also the products used for the
analysis can also be used again. An examples of the ways that we commonly as
magnetic particle, dye penetration, eddy current, radiography, ultrasound and others
against summarized by Staszewski et. al. (2004) and Gdoutus. However, the overall
methodology used to test the different deformities.
Magnetic particle is a method of using the principle that a flaw in magnetic
material produces distortion in an induced magnetic field. The method is easy, fast
and economical to apply, but similarly to the dye penetration method it can only be
used to detect cracks or damage near the applied surface.
Dye penetration is a method of applying colour or fluorescent dye onto the
cleaned surface of a component to detect any surface flaws. After applying the dye
onto any surface and applying a post-penetrant material such as chalk, flaws will
appear as coloured lines. It is a fast method for damage detection and can detect
small cracks, but it is only capable of detecting surface flaws.
Eddy-current is a method of using the change of impedance in a coil caused
by the eddy current from a tested conductor surface. A coil with alternating current is
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placed near the conductor surface to induce an eddy current. The sensitivity of the
method to the defect is dependent on the penetration depth of the eddy current into
the conductor and is influenced by the frequency of the alternating current, the
magnetic permeability and electrical conductivity of the conductor, and the geometry
of the coil and conductor. Therefore the method is highly sensitive for defects near
the conductor surface but it is difficult to relate the defect size to the impedance
change and the impedance change is also affected by others factors.
Radiography is the oldest NDT method and uses X- or γ-ray to detect a
defect. The X-ray is transmitted to a tested material and the emerging radiation is
measured. If the material contains defects or variations in its thickness or density, the
emerged radiation intensity will not be uniform. This method is suitable for detecting
volumetric defects. However it is important to know the orientation of the defect
beforehand to get the best effect.
Ultrasound is a method of transmitting ultrasonic waves into a test material.
Any defects and boundaries in the specimen will reflect a pulse wave and the
reflection waves are measured. By using the reflection wave data, the defect size and
location can be estimated. It is a very effective method for detecting defects in most
positions, gives a quick response, is economical, is applicable to thick material, and
is highly portable in-situ. However, it is very difficult to distinguish between cracks
and other types of defect and it has limited application to certain specimen geometry.
We can see an example as follows:
Figure 2-1 : specimen with impact damage area (2)
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The above diagram (figure 2-1) shows an example of the damage that we can
analyze occurred in a surface structure. Damage occurred in the middle. It may be
due to various factors.
2.2 CRACK DETECTION METHOD
Detection of damage in a structure can be done in various ways. Each of the
methods used would have its own advantages and disadvantages. However, damage
detection can to divide two main categories: experimental and analysis. This part will
present analytical modeling examples by previous researchers using nonlinear
acoustics methods for defects like stiffness material.
2.2.1 ANALYTICAL MODELLING METHOD
In the analysis of the structure, a mathematical formula can also be used for
the detection of non-linear effects caused by wave distortion when interacted with
nonlinearity or non-uniformity of its passage medium. Mathematical modeling
method can to use to show non linear wave effect graph.
Figure 2-2: step simulation to analysis of specimen
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That graph can to show via of non linear phenomena of sideband. To make a
structural analysis of ABAQUS software should be used first to obtain the response
frequency and nonlinear wave effect. Figure 2-2 show step simulation of the
specimen. So we can see the relationship between the stiffness of the (R value). From
this, they derived the sideband amplitude ratio over the fundamental frequency f0
amplitude, R value, as:
2 │Ho ( f 0 ) – Hc ( f 0 )│ 2 B1 R = – ––––––––––––––––– = – – ……………....... Eq. 2.1
│Ho ( f 0 ) + Hc ( f 0 )│ B0
Where HO and HC are transfer function values when the crack is fully open and
closed
HO C BO = ––––––– is mean value of the modulation signal 2
HO - HC BO = ––––––– is the peak-to-peak variation around the modulation 2
The transfer functions were computed by using a standard analytical formula written
as
U inU jn
H i j (w) = ∑ ––––––––––––––––––––––––– ………………………………………………….. Eq. 2.2
n wn2 + i wnw / Qn - w2
Where H i j (w) denotes transfer function value at frequency w between points i and j.
U i/j n is mode shape value at location i/j for mode n
wn is natural frequency at mode n
Qn is damping factor at mode n
They then used Equations 2-1 and 2-2 in two modeling methods to quantify the
nonlinear effects. The method is reacceptance finite element method.
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2.2.2 FINITE ELEMENT METHOD (FEM)
The finite element method (FEM) originated from the need for solving
complex elasticity and structural analysis problems. The finite element method
(FEM) (its practical application often known as finite element analysis (FEA)) is a
numerical technique for finding approximate solutions specializations of the
mechanical engineering such as automotive industries commonly use integrated FEM
in design and development of the structure. In a structural simulation, FEM helps
tremendously in producing stiffness and strength visualizations and also in
minimizing weight, materials, and costs.
FEM is also popular method for analyzing structural vibration responses.
With rapid development in computing and software technology, this method has
become more reliable and problem solving has become faster. The results from this
analysis also showed clearly the dependence of R value on the fundamental
frequency and damping factor may also affect sensitivity of the nonlinear acoustics
method. The fundamental frequency resulted from the FEM. It also shows the
sensitivity of the R value.
Figure 2-3: Ratio of first sideband amplitude over fundamental frequency
amplitude, R value, against the ultrasound frequency resulted (R.B Jenal- fatigue
crack detection using nonlinear acoustic – analysis of vibro-acoustic modulations) :
Figure 2-3 : (a) receptance analysis (Duffour)
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Receptance analysis is an analytical method of decomposing a system into a
few sub-systems to analyze a system’s vibration responses. As presented by Duffour
et al., this method was used to model a beam schematically described in Figure 2-4.
The beam was divided into three sections, where the cracked section was represented
by a very short beam (no.2) with a reduced cross-section area compared to the other
two sections (nos 1 and 3). The cross-section area was dependent on the crack depth.
Figure 2-4: A cracked beam is decomposed into three plain beams where the cracked
section is represented by a short reduced cross section beam (Duffour)
First they calculated the natural frequencies and mode shape coefficients of
each beam section by using standard theory for the longitudinal vibration of uniform
bars. The results were substituted into Equation 2-2 and then the results from each
section were assembled by using a coupling formula for linear system in series.
Finally the R values were determined by using Equation 2-1 within a frequency
range.
The results from this analysis showed that the sensitivity of using the
nonlinear acoustic effect for damage detection is extremely dependent on the
fundamental frequency and it should be close to the resonances of the specimen.
Damping factors for the specimen also may affect this method sensitivity.
Figure 2-5 : (a) Finite Element Method (Duffour)
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Figure 2-5 (b) and (c): Ratio of first sideband amplitude over fundamental frequency
amplitude, R value, against the ultrasound frequency resulted ((R.B Jenal- fatigue
crack detection using nonlinear acoustic – analysis of vibro-acoustic modulations) :
Figure 2-5 : (b) Average of R value against crack size with various damping factors
(Duffour)
2.3 NON-LINEAR ACOUSTICS
Non-linear acoustics is a branch of physics dealing with sound waves being
distorted. The amplitude dependence is due to the nonlinear response of the medium
in which the frequency propagates, and not to the nonlinear behavior of the sound
source. According to the linear theory of acoustics, increasing the level of a source
by 10 dB results in precisely the same sound field as before, just 10 dB more intense.
Linear theory also predicts that only frequency components radiated directly by the
source can be present in the sound field.
2.3.1 SURFACE ACOUSTIC WAVES
Surface waves take many forms in nature, science, and technology. They
include ultrasonic surface waves at the interface between a solid on one hand and
vacuum, gas, liquid or another solid on the other. A common feature of all kinds of
surface waves is that most of energy is localized near the surface, within a depth of
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about one wavelength. Instead of propagating thought-out the whole three-
dimensional medium, the energy remains localized at the surface and spreads out
primarily in the two-dimensional (2D) interface region.
Waves illustrated the principal features of elastic surface waves. The particle
motion can be easily visualized when a waves passes a leaf floating on the waves
surface. The leaf moves to and fro, but also up and down around its original position.
The radius of the particle orbit is equal to the wave amplitude (H.Peter - Surface
acoustics waves materials science).
Elastic surface waves, usually called surface acoustic waves (SAWs) were
discovered in 1885 by Lord Rayleigh. SAWs depend on the elastic forces acting
between the constituent atoms. The internal forces of medium, or stress, are assumed
to depend only on the deformation of the material, or strain , measured relative to the
undisturbed state. In the bulk of an elastic material, the longitudinal and transverse
waves modes are independent and propagate with different velocities, but in surface
waves the two modes are coupled. Due to the asymmetry of the elastic forces at the
surface, the motion normal to the surface may be different from that in the direction
of wave propagation along the surface. Consequently, in the elastic medium, the
particle motion is elliptical polarized. The depth dependence of the particle
displacement and polarization are illustrated in figure 2.6.
Figure 2-6: particles move in ellipses in a surface acoustic wave. The amplitudes
decrease exponentially with depth (H.Peter )