nondestructive testing

MECHANICAL ENGINEERING THEORY AND APPLICATIONS

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MECHANICAL ENGINEERING THEORY AND APPLICATIONS

NONDESTRUCTIVE TESTING: METHODS, ANALYSES AND APPLICATIONS

EARL N. MALLORY EDITOR

Nova

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

vii Nondestructive Materials Characterization by Magnetic Sensing Hector Carreon Experimental and Numerical Method for Nondestructive Ultrasonic Defect Detection D. Cerniglia and A. Pantano Investigation of Thermal Properties of Steel Undergoing Heat Treatment by the Photothermal Deflection Technique: Correlation with Mechanical Properties Taher Ghrib, Imen Gaied and Noureddine Yacoubi

1

63

95

Chapter 4

Machine Thermal Diagnostics Latest Advances L. Burstein

147

Chapter 5

Scanning Acoustic Correlation Microscopy M.Saint-Paul

185

Index

199

PREFACE The authors of this book present and review data on nondestructive testing discussing such topics as: a novel noncontacting thermoelectric method for nondestructive detection of material imperfections in metals by magnetic sensing; laser generated ultrasound as a tool for defect detection combined with air-coupled receivers; the photothermal deflection technique; thermal diagnostic testing; and acoustic microscopy. Chapter 1- This research work presents a novel noncontacting thermoelectric method for nondestructive detection of material imperfections in metals. The method is based on magnetic sensing of local thermoelectric currents around imperfections when a temperature gradient is established throughout a conducting specimen by external heating and cooling. The surrounding intact material serves as the reference electrode therefore the detection sensitivity could be very high if a sufficiently sensitive magnetometer is used in the measurements. This self-referencing, noncontacting, nondestructive inspection technique offers the following distinct advantages over conventional methods: high sensitivity to subtle variations in material properties, unique insensitivity to the size, shape, and other geometrical features of the specimen, noncontacting nature with a substantial stand-off distance, and the ability to probe relatively deep into the material. The potential applications of this method cover a very wide range from detection metallic inclusions and segregations, inhomogeneities, and tight cracks to characterization of hardening, embrittlement, fatigue, texture, and residual stresses. During this research work was laid down the groundwork of this new field of nondestructive materials characterization and with the result of recent technological advances in the development of highsensitivity magnetic sensors, such as Giant Magneto-Resistive (GMR)

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Earl N. Mallory

detectors, Spin Dependent Tunneling (SPD) detectors, fluxgates and, especially, high-temperature Superconductive QUantum Interference Device (SQUID) magnetometers, it has become feasible to adapt the noncontacting magnetic thermoelectric inspection to many applications of great interest in NDT. We have successfully adapted the noncontacting thermoelectric method to a series of nondestructive materials characterization applications that are currently not accessible by any other known inspection method. In particular, we verified experimentally a series of analytical models capable of predicting the thermoelectric signatures produced by surface-breaking and subsurface inclusions under external thermal excitation for different lift-off distances between the sensor and the surface of the specimen. In addition, we studied the feasibility of nondestructive detection and characterization of cracks and voids in textured polycrystalline materials, phase anomalies and anisotropic effects of the microstructure, and thermally induced residual stress relief in surface treated components. Chapter 2- Ultrasonic methods are well known as powerful and reliable tools for defect detection. Conventional ultrasonic techniques rely generally on piezoelectric transducers where transmission of energy to the material is achieved with contact. In the last decades, focus and interest have been directed to non-contact sensors and methods, showing many advantages over contact techniques where inspection depends on contact conditions (pressure, coupling medium, contact area). The growing interest is also due to the further development of air-coupled probes, thanks to new materials for acoustic devices and manufacturing technologies. The use of the laser as a tool for ultrasonic defect detection is also an emerging approach in the industry and holds substantial promise as inspection is remote, feasible in a hostile environment, can be automated and also performed with the test object in motion. Moreover, thanks to its ability to produce frequencies in the MHz range, laser-generated ultrasound enables fine spatial resolution of defects. The non-contact hybrid ultrasonic method described here is of interest for many applications, requiring periodic in-service inspection or after manufacturing. Despite the potential impact of laser-generated ultrasound in many areas of industry, robust tools for studying the phenomenon are lacking and thus limit the design and optimization of non-destructive testing and evaluation techniques. Ultrasonic waves propagate through the structure interacting with defects, corners and curved surfaces, causing reflection and mode conversion. Moreover, interference between waves can produce a more complex pattern. This makes the laser-generated ultrasound propagation in complex structures

Preface

ix

an intricate phenomenon extremely hard to analyze. Only simple geometries can be studied analytically. Numerical techniques found in literature have proved to be limited in their applicability by the frequencies in the MHz range and very short wavelengths. The acoustic field in complex structures should be well understood for each application to optimize sensitivity toward a particular type of defect. A specific numerical method is presented in this chapter to efficiently and accurately solve ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies and through air. Tests simulated with numerical analysis are replicated experimentally for validation. The numerical technique provides a valuable tool for studying the lasergenerated ultrasound propagation and for designing and optimizing nondestructive testing and evaluation techniques. The information that can be acquired can be very valuable for choosing the right setup and configuration when performing non-contact hybrid ultrasonic inspection. Chapter 3- In this work we present a new method, based on the photothermal deflection technique, which permits the simultaneous determination of thermal conductivity and thermal diffusivity of steel undergoing a heat treatment (carburizing, nutriding, electroerosion, Jominy test). This method consists of the deposition of a thin graphite layer on the treated surface steel which will absorb the totality of the incident light and will play the role of a heat source. The local thermal properties of hardened steel are determined by drawing the experimental amplitude and phase curves of the photothermal signal versus square root modulation frequency and to compare them to the corresponding theoretical ones. The best coincidence between these curves is obtained for a unique and known thermal diffusivity and thermal conductivity. The main interest of this method is that such obtained thermal properties are correlated to hardness. In some cases, the thermal properties are related to hardness through an empiric mathematical law which permits us to deduce the hardness of steel without measuring it. Chapter 4- The chapter presents a systematic description of the advantages of a novel theory and a friction heat-based method for evaluation of the technical condition and residual service life of frictional machinery. The diagnostic parameter considered is the peak point of the temperature change rate observed on the real machine during the starting stage, and is applicable for mechanisms in which friction of the rubbing parts leads to wear and to increased heat release. The relevant relationships for the diagnostic parameter and its time dependence ("reference dependence") are derived for two time approaches - short-term (starting time) and long-term (service time) - on the

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Earl N. Mallory

basis of the machine’s heat balance. The theoretical values of the residual service life (RSL) obtained by these parameters are compared with their counterparts by the traditional empirical method. It is shown that the theory is valid and applicable in real machine diagnostics. The reliability of the predicted values of the reference dependence and RSL is assessed by a Monte-Carlo simulation procedure described in detail. Measured temperatures were varied repeatedly within the temperature error range, yielding repeated values of the parameters. It is shown that in 95% of the cases the thermal method entails a two-sided error of at most 1.7% and the discrepancy between the deterministic and simulated averages does not exceed 0.04 %. The assessment indicates the possibility of determining the RSL for the small machine populations often involved in practical exploitation. The tests were carried out on the SAWA power reduction gears of KOCKS heavy portal transtainer cranes; the results for automotive brakes are also given. Chapter 5- Acoustic microscopy has proven to be a powerful tool for the non-destructive evaluation of defects, delaminations, cracks and porosity in microlectronic packages. The through transmission mode is used to detect the existence of defects while the reflected signal provide information about the depth of the defects. Detection and image formation are generally based on signal amplitude. However, for advanced packaging formats that contain thin layers of sub-wavelength thickness, the reflected ultrasonic echoes are overlapped. In the case of multilayered composite plates having thickness comparable to the acoustic pulse spatial length, serious problems are found. One of the major problems of ultrasonic testing is the difficulty of interpretation of the recorded echoes. Lack of adequate time resolution makes the identification of the individual echoes impossible. Conventional rules for interpretation fail to explain the acoustic images. An approach for detection and imaging based on the correlation procedures is presented in this chapter. The shape change between two signals is quantified by forming a normalized short time cross correlation between the two signals. The cross correlation values of the acoustic waves reflected by the sample allow to image internal features. The cross correlation value is a quantitative measure of the severity of structural damage, it can be used to discriminate between an undamaged and a damaged structure.

In: Nondestructive Testing: Methods, Analyses… ISBN: 978-1-60876-157-9 Editors: Earl N. Mallory, pp. 1-62 © 2010 Nova Science Publishers, Inc.

Chapter 1

NONDESTRUCTIVE MATERIALS CHARACTERIZATION BY MAGNETIC SENSING Hector Carreon Instituto de Investigaciones Metalúrgicas, Edif.”U” Ciudad Universitaria, Morelia, Mich. México 58000-888

ABSTRACT This research work presents a novel noncontacting thermoelectric method for nondestructive detection of material imperfections in metals. The method is based on magnetic sensing of local thermoelectric currents around imperfections when a temperature gradient is established throughout a conducting specimen by external heating and cooling. The surrounding intact material serves as the reference electrode therefore the detection sensitivity could be very high if a sufficiently sensitive magnetometer is used in the measurements. This self-referencing, noncontacting, nondestructive inspection technique offers the following distinct advantages over conventional methods: high sensitivity to subtle variations in material properties, unique insensitivity to the size, shape, and other geometrical features of the specimen, noncontacting nature with a substantial stand-off distance, and the ability to probe relatively deep into the material. The potential applications of this method cover a very wide range from detection metallic inclusions and segregations, inhomogeneities, and tight cracks to characterization of hardening,

2

Hector Carreon embrittlement, fatigue, texture, and residual stresses. During this research work was laid down the groundwork of this new field of nondestructive materials characterization and with the result of recent technological advances in the development of high-sensitivity magnetic sensors, such as Giant Magneto-Resistive (GMR) detectors, Spin Dependent Tunneling (SPD) detectors, fluxgates and, especially, high-temperature Superconductive QUantum Interference Device (SQUID) magnetometers, it has become feasible to adapt the noncontacting magnetic thermoelectric inspection to many applications of great interest in NDT. We have successfully adapted the noncontacting thermoelectric method to a series of nondestructive materials characterization applications that are currently not accessible by any other known inspection method. In particular, we verified experimentally a series of analytical models capable of predicting the thermoelectric signatures produced by surfacebreaking and subsurface inclusions under external thermal excitation for different lift-off distances between the sensor and the surface of the specimen. In addition, we studied the feasibility of nondestructive detection and characterization of cracks and voids in textured polycrystalline materials, phase anomalies and anisotropic effects of the microstructure, and thermally induced residual stress relief in surface treated components.

1. INTRODUCTION A variety of different physical principles have been exploited for nondestructive detection, localization, and characterization of material imperfections in metals. The most popular techniques rely on ultrasonic, eddy current, x-ray radiographic, magnetic, thermal, and microwave principles. A common feature of these conventional methods is that they are sensitive to both intrinsic material (e.g., electrical and thermal conductivity, permeability, elastic stiffness, density, etc.) and spurious geometrical (e.g., size, shape, surface roughness, etc.) parameters. Unfortunately, these two classes of properties are often very difficult to separate, which sets the ultimate limit for the detectable smallest and/or weakest material imperfection. Conventional thermoelectric techniques, that have been used in nondestructive materials characterization for several decades, are essentially free from these geometrical limitations, i.e., they are sensitive to intrinsic material variations only regardless of the size, shape, and surface quality of the specimen to be tested. Essentially all existing thermoelectric NDT techniques are based on the well-known Seebeck effect that is commonly used in thermocouples to

Nondestructive Materials Characterization by Magnetic Sensing

3

measure temperature at the junction of two different conductors. Ideally, regardless of the temperature difference between the junctions, only thermocouples made of different materials, or more precisely, materials of different thermoelectric power, will generate a thermoelectric signal. This unique feature makes the simple thermoelectric tester one of the most sensitive material discriminators used in nondestructive inspection. The thermoelectric power of metals is sensitive to a variety of material properties that can affect the measurement. Clearly, chemical composition exerts the strongest effect on the thermoelectric properties and accordingly the basic application of conventional thermoelectric materials characterization is metal sorting [1]. However, it is well known that under special conditions materials of identical chemical composition can also produce an efficient thermocouple as a result of different heat treatment, hardening, texture, residual stress, fatigue, etc., which can be further exploited for nondestructive testing of materials [2-7]. In spite of its obvious advantages over other methods, thermoelectric testing is rarely used in nondestructive testing (NDT) because of the requirement that a metallic contact be established between the specimen and the reference electrode. The resulting thermoelectric offset can be reduced, but not entirely eliminated, by decreasing the thermal and electrical resistance between the specimen and the reference electrode, e.g., via better cleaning or imposing higher contact pressure. Ultimately, the presence of this imperfect contact limits the detectability of small variations in material properties by the conventional thermoelectric technique [4]. On the other hand, the new noncontacting thermoelectric method uses the surrounding intact material as the reference probe; thus provides perfect interface between the region to be tested and the surrounding material. In the self-referencing thermoelectric method the material imperfections naturally form thermocouples in the specimen itself and, in the presence of an externally induced temperature gradient, these innate thermocouples produce thermoelectric currents around the imperfections that can be detected bias the magnetic flux density B by magnetic sensors from a significated lift-off distance between the tip of the sensor and the material surface imperfection. Even when the material imperfections are rather deep below the surface. It is also well known that the noncontacting thermoelectric technique is very sensitive to the presence of foreign body inclusions, when the thermoelectric power of the affected region is significantly different from that of the surrounding medium [810]. Figure 1 shows a schematic diagram of the noncontacting thermoelectric measurements process in the presence of material imperfections as most often used in nondestructive materials characterization.

4

Hector Carreon

Figure 1. A schematic diagram of noncontacting thermoelectric detection of material imperfections by magnetic monitoring

In this research work, we demonstrated that the noncontacting thermoelectric technique can be used to detect various imperfections in metals, including foreign body inclusions and more subtle local property variations caused by service or manufacturing related effects such as cold work, localized texture, residual stress, fretting damage, etc. [11-20]. Like most other methods used in NDE, the detection sensitivity of the noncontacting thermoelectric method is ultimately limited by temporally coherent material noise rather than temporally incoherent electrical noise, which could be easily eliminated by simple time averaging. Strictly speaking, material “noise” is really a spurious background “signature” that is often called noise only because it interferes with, and potentially conceals, the flaw signals to be detected. The main sources of this adverse background signature in thermoelectric NDE are macrostructural features such as case hardening, cold work, texture induced anisotropy, residual stress, etc., while the inherent microstructural inhomogeneity caused by, e.g., the grain structure in polycrystalline metals, is less important because of the lack of sufficient spatial resolution. The main goal of this research work was to lay down the groundwork necessary to develop this new field of nondestructive testing and materials characterization based on no contacting magnetic detection of thermoelectric currents. In order to achieve this ambitious goal we had to study not only the flaw signals produced by different types of material imperfections, but also the thermoelectric noise or background signature produced by inherent material variations, that together determine the probability of detection (POD) of a certain type of imperfection and thereby the ultimate sensitivity threshold of the method. This research work involved closely related theoretical and experimental efforts that led to a better understanding of the underlying


5

physical phenomena, the development of new, predictive analytical models, more sensitive experimental procedures, and, ultimately, increased POD for small inclusions and weak material imperfections.

2. MODELING OF THE SIGNAL FROM ISOTROPIC SPHERICAL INCLUSIONS In this section we presented a summary of the analytical model developed by Nayfeh and Nagy [12]. First, it was considered an infinite homogeneous medium containing a spherical inclusion of a different material when the system is subjected to a uniform temperature gradient. After deriving the governing equations for the coupled thermal and electrical fields and satisfying all boundary conditions, first the electrical current densities in the host and the inclusion can be calculated, then the magnetic field of the thermoelectric current can be determined by integration using the Biot-Savart law. Thermoelectricity is a result of intrinsically coupled transport of electricity and heat in metals. The electrical current density j and thermal flux h produced by a given combination of electrochemical potential Φ and temperature T distributions are given by [21] ⎡ j ⎤ ⎡σ ε ⎤⎡−∇Φ ⎤ , ⎢h⎥ = ⎢ ε κ⎥⎢ −∇ T ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(1)

where σ denotes the electrical conductivity measured at uniform temperature, κ is the thermal conductivity for zero electrical field, and ε and ε are thermoelectric coupling coefficients that can be expressed by the absolute thermoelectric power S of the material as ε = σ S and ε = σ S T . The thermal conductivity for zero electrical field κ can be easily expressed by the thermal conductivity of the material for zero electrical current, k , which is often easier to determine experimentally, as κ = k + σ S 2 T . The difference between these two thermal conductivities is due to the thermoelectric coupling in the material. It can write that κ = k (1 + η) , where η =σ S 2T / k is a dimensionless factor that provides a measure of the strength of coupling between thermal and electrical transports. For typical metals, the coupling −3 −2 factor is relatively small somewhere between 10 and 10 , an important fact that will be exploited in the following calculations.

6

Hector Carreon The total energy flux h + Φ j includes the thermal flux plus an additional

term representing the changing electric potential of the electrons. The rate at which heat is evolved, per unit volume, at any point in the material is ∇ ⋅ h + ∇Φ ⋅ j , where it was exploited Maxwell's law that ∇ ⋅ j = 0 . If it now assumes the coupling coefficients ε and ε to be small such that their individual squares and products can be neglected, then for thermal loading we can neglect ∇Φ ⋅ j and, for the steady state, it can concluded that ∇ ⋅ h ≈ 0 . Imposing the conditions that the divergences of both the electrical current density and the thermal flux vanish and noting that σ κ − ε ε ≠ 0 , Eq. (1) requires that the Laplacians of T and Φ vanish individually, i.e.,

∇ 2 T = 0 and ∇ 2 Φ = 0.

(2)

For a homogeneous isotropic medium σ, κ, ε and ε are scalar quantities that do not depend on the spatial coordinates, though generally they do depend on temperature, especially ε and ε . In the first-order approximation of Eq. (1), the temperature dependence of σ can be neglected and the curl of the thermoelectric current ∇ × j = − σ ∇ × ∇Φ − ε ∇ × ∇T − ( ∂ε / ∂T )∇T × ∇T is zero. Actually, this outcome does not change even if the temperature σ is accounted for by an additional dependence of term − ( ∂σ / ∂T )∇T × ∇Φ , as ∇T will be parallel with ∇Φ . Since the divergency of the thermoelectric current is inherently zero, in the absence of an external electric source, the current density itself must be identically zero everywhere in the medium. This means that, regardless of the size, shape, and material properties of a homogeneous isotropic specimen, no thermoelectric currents will be generated by any type of heating or cooling. In other words, the presence of any magnetically or otherwise detected thermoelectric current will positively identify the specimen as either inhomoge-neous or anisotropic.

2.1. Infinite Homogeneous Medium Containing a Spherical Inclusion First, it was considered an infinite homogeneous medium (host) containing a spherical inclusion of a different material having radius a . To differentiate between the properties of the two media, it was designated those


7

of the inclusion by a prime. The system is subjected to a thermal flux h0 far away from the inclusion and directed along the x3 -axis of the Cartesian coordinate system ( x1, x2 , x3) as illustrated in Figure 2. In the absence of the inclusion, solutions to the above presented coupled field equations are given by,

h ε h0 x3 T = T0 = − 0 x3 and Φ = Φ 0 = κ κσ

(3)

It is advantageous to introduce a spherical polar coordinate system ( r, θ, ϕ ) so that θ is the polar angle measured from the x3 direction and ϕ is the azimuthal angle measured from the x1 direction. In these spherical coordinates, Eq. (3) can be expressed as

h ε h0 r cos θ . T0 = − 0 r cos θ and Φ0 = κ κσ x 1

(4)

x3 H P θ

r

φ j

x2

inclusion h 0

Figure 2. The coordinate system used to study the magnetic field produced by the thermoelectric currents around a spherical inclusion embedded in an otherwise homogeneous specimen

With these solutions as a guide, in the presence of the spherical inclusion it seeks solutions of the form

T = f (r ) cosθ and Φ = g (r ) cosθ .

(5)

8

Hector Carreon

These solutions have to satisfy Eq. (2), respectively. For the axisymmetric situation under consideration, the Laplacian operator in spherical coordinates is widely available and is given as

∇2 =

∂2

2 ∂ 1 ∂2 ∂ + + ( + cot ) θ ∂θ ∂r 2 r ∂r r 2 ∂θ2

(6)

Combinations of Eqs. (5) and (6) lead to the following formal solutions for the temperature and electric potential for both the host and the inclusion

B

*

B*

T = ( Ar + 2 ) cos θ , Φ = ( A r + 2 ) cos θ , r r * T ' = D r cos θ , Φ' = D r cos θ

(7)

where bondedness of the solutions at r = 0 , i.e., at the center of the inclusion, is satisfied and A, A* , B , B* , D, and D* are currently unknown constants to be determined from the appropriate boundary conditions. At the interface between the host and the inclusion ( r = a ), the interface continuity conditions require that both the temperature and the electrical potential be continuous, namely T = T ' , Φ = Φ ' . Furthermore, the normal (radial) components of both the electrical current density and thermal flux are also continuous at r = a , jr = j 'r , and hr = h'r . It should be mentioned that the continuity of the thermal flux is an approximation based on ∇ ⋅ h = 0 , which is used instead of the continuity of the total energy flux of ∇ ⋅ h + ∇Φ ⋅ j in the weak thermoelectric coupling approximation. As such, it clearly neglects the so-called Peltier heat generated at the interface between the host and the inclusion as a result of the weak * * * thermoelectric currents. The unknown constants A, A , B , B , D, and D can be determined by imposing the above boundary on the formal solutions of Eq. (7) and requiring that in the limit of r → ∞ the solutions reduce to Eq. (4). After some algebraic reductions while exploiting small coupling [12,22], it can be obtained for j and j ' as


j =

9

a3 h0

Gs (2 cos θ er + sin θ eθ ) r3 and j' = h0 Gs (2 cos θ er − 2 sin θ eθ ) ,

(8)

where

Gs = 3

ε' σ − ε σ' . (σ' + 2 σ )(κ' + 2 κ )

(9)

Of course similar expressions can be obtained for h and h' , but need not be reported here. It now remains to obtain expressions for the magnetic field components Hϕ and H'ϕ . We recall that the electrical current density j is related to the magnetic field H by ∇ × H = j . In spherical coordinates, we have

∇×H =

∂Hϕ Hϕ 1 ∂Hϕ ( + cot θ Hϕ ) er − ( + ) eθ , r ∂θ ∂r r

(10)

where it can be exploited the fact that, due to the axial symmetry of the problem, both Hr and Hθ identically vanish everywhere. From Eqs. (10) we solve for Hϕ as

Hϕ = a 3 h0 Gs

sin θ

r2

and H' ϕ = h0 Gs r sin θ .

(11)

2.2. Numerical Results It was assumed that the host is infinite so that the analysis could be simplified as a result of the perfect axial symmetry. In particular, due to this axial symmetry, T, Φ, h, j , and H were all independent of ϕ , and hϕ , jϕ , Hr , and Hθ all vanished. Needless to say that this assumption renders the problem somewhat hypothetical since the magnetic field cannot be practically measured inside the specimen. In order to realistically model a

10

Hector Carreon

large but finite-size specimen, it can be either truncated or sectioned the infinite specimen so that the measurement of the magnetic field could be done on the outside. Generally, this requirement renders the calculations very difficult since the spatial distribution of the thermoelectric current density 3 becomes distorted as all the current that would otherwise spread out as 1 / r is squeezed inside the contour of the finite specimen. In order to show how dramatically this will affect the magnetic field, let us consider a cylindrical specimen of radius b with its axis along the x3 direction. Unless b >> a , the thermoelectric current density distribution will be significantly different from that in an infinite medium, but it will be still axisymmetric. In this case, the resulting magnetic field can be readily calculated from Stokes' theorem

∫ H ds = ∫∫ jdA ,

(12)

where the integrations on the left and right sides are carried out over the circumference and the area of the same continuous surface, respectively. Outside the specimen ( r > b ), j = 0 so that the total electrical current on the right side of Eq. (12) is identically zero, therefore the magnetic field completely vanishes. It is clear that, by virtue of Stokes' theorem, any other axisymmetric arrangement would also result in the complete disappearance of the external magnetic field. In other words, the magnetic field to be measured in the case of an embedded inclusion is caused by asymmetric distortion of the thermoelectric current distribution and it will be inevitably rather weak. This means that noncontacting magnetic detection of a spherical inclusion is possible only if it is close enough to the surface of the specimen so that the thermoelectric currents are deflected by the surface contour. Otherwise, the thermoelectric currents form a toroid that produces no magnetic field on the outside. To simplify the following calculations, all spatial coordinates will be normalized to the radius of the spherical inclusion as ξ = x / a . The magnetic field can be also written in a normalized form as H = H 0 F (ξ ) , where H0 = a h0 Gs combines the size of the inclusion a , the thermoelectric contrast Gs and the externally induced heat flux h0 into a single scalar constant characterizing the strength of the magnetic field while F(ξ ) is a universal spatial distribution function for all spherical inclusions. In order to better facilitate the estimation of the absolute strength of the magnetic


11

field in a given experimental arrangement, we can re-write H0 with the temperature gradient ∇T that would prevail in the vicinity of the inclusion if it were not there. From Eq. (3), h0 = − κ ∇T and

H0 = − a ∇T σ SSR Γs ,

(13)

where SSR = S ' − S is the relative thermoelectric power of the inclusion with respect to the host and Γs is the normalized contrast coefficient

Γs =

3 κ' σ (1 + 2 ) (2 + ) σ' κ

.

(14)

It should be noted that, for weak material inhomogeneity ( κ ' ≈ κ and σ' ≈ σ ), the normalized contrast coefficients of spherical and cylindrical inclusions approach 1/3 and 1/2, respectively.

2.3. Half-Space with a Surface-Breaking Spherical Inclusion The closer the inclusion to the surface, the stronger the outside magnetic field. Maximum detectability is reached when the inclusion is cut halfway through by the surface as it is shown in Figure 3. In this case, the coupled thermoelectric problem in the remaining half-space is exactly the same as it was before, though the resulting magnetic field is not axisymmetric anymore. In particular, T, Φ , h and j will be still independent of ϕ and hϕ and jϕ both vanish in both the host and the inclusion, but H will now depend on ϕ , and neither Hr nor Hθ will automatically vanish. Actually, it is advantageous to describe the magnetic field in Cartesian coordinates since in that way at least one of the components, namely H3, does identically vanish. This can be easily illustrated by recasting Eq. (11) as follows

F = for the host and

− ξ 2 i + ξ1 j (ξ12 + ξ 22 + ξ 32 )3 / 2

(15)

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Hector Carreon

F = − ξ 2 i + ξ1 j

(16)

for the inclusion, where i and j are the Cartesian unit vectors in the x1 and x2 directions, respectively. Since the magnetic field of the thermoelectric current is no longer axisymmetric, the sought magnetic field cannot be simply calculated from Stokes' theorem and we have to rely on the Biot-Savart law H (x)= ∫∫∫ V

j ×(x− X) 4π x− X

3

dX 1 dX 2 dX 3 ,

(17)

where the integration is carried out over the entire half-space, V , and inside the inclusion j should be substituted by j' . x1

free surface h0

x3

inclusion j

Figure 3. Schematic diagram of a surface-breaking spherical inclusion and the resulting symmetric thermoelectric current distribution

Nondestructive Materials Characterization by Magnetic Sensing (a) ξ1 = 0.5, F1 = 0.5

13

(b) ξ1 = 1, F1 = 0.171

∇T

∇T

Figure 4. Two-dimensional distributions of the normal component of the magnetic field parallel to the free surface at two different lift-off distances for a semi-spherical inclusion with its center lying on the surface ( the peak normalized flux density F1 is indicated for comparison)

Peak Magnetic Field and Half-Width

10 Half-Width (w ) 1

0.1 Normal Peak Magnetic Field ( F1 ) 0.01

0.001 0.01

0.1

1

10

Normalized Lift-Off Distance ( ξ1 )

Figure 5. The two principal parameters of the field distribution, namely the peak of the normalized magnetic field and the half-distance between the peaks, as functions of the normalized lift-off distance

Figure 4 shows the two-dimensional distributions of the normal component of the magnetic field F1 ( ξ 2 , ξ 3 ) taken in planes parallel to the free surface at two different lift-off distances ( ξ1 = 0.5 and 1) for a semi-

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Hector Carreon

spherical inclusion with its center lying on the surface. The characteristic shape produced by the positive and negative peaks in the magnetic field distribution are very typical in noncontacting thermoelectric detection of inclusions and can be easily exploited by digital image processing and feature extraction techniques to increase the probability of detection for inclusions. The larger the lift-off distance, the lower the peak magnetic field and the wider the field distribution. The latter can be quantitatively characterized by the lateral distance of the peaks from the center of the inclusion, w, which we are going to call the half-width of the bi-polar signature. Since the field distribution remains very similar regardless of the lift-off distance, it is reasonable to choose these two parameters, namely the peak magnetic field and the half-distance between the peaks, to characterize the whole field distribution. Figure 5 shows these two parameters as functions of the normalized lift-off distance. The solid lines are numerical results while the dashed lines are far-field asymptotes. At small lift-off distances, the peaks are located directly above the circumference of the inclusion, i.e., w = 1, and their magnitude approaches 0.5 [12]. At large distances, the field distribution o spreads out so that the peaks occur at φ ≈ 45 , i.e., w ≈ ξ1 / 2 , and their magnitude is inversely proportional to the square of the normalized lift-off distance.

2.4. Half-Space with a Subsurface Spherical Inclusion In order to simulate fully embedded hidden inclusions, it can be assumed that the center of a spherical inclusion lies below the surface at a depth d , that is deeper than its radius, i.e., it is not breaking the surface at all. In this case, symmetry to the sectioning plane can be retained by considering a pair of spherical inclusions as shown in Figure 6. By assuming that the inclusions are only slightly different from the host or the separation 2 d between them is large with respect to their radius a , the interaction between the two inclusions can be neglected and the resulting thermoelectric current distribution can be approximated simply by superimposing the currents produced by the individual inclusions. Because of its simplicity, Nagy and Nayfeh [12] adapt this approximation to spherical inclusions embedded below the surface at shallow depths, only when the thermal and electrical conductivities of the inclusion are similar to those of the host, i.e., in the weak contrast limit. In this case, it is replaced ξ1


15

by ξ1 + δ in Eqs. (15) and (1), where δ = d / a denotes the normalized depth, so that

− ξ 2 i + (ξ1 + δ ) j [(ξ1 + δ )12 + ξ 22 + ξ 32 ]3 / 2

F =

(18)

for the host and

F = − ξ 2 i + (ξ1 + δ) j

(19)

for the inclusion. Directly on the surface ( ξ1 = 0 ), by virtue of symmetry, the magnetic field exhibits only normal component that can be readily obtained from Eq. (18)

F1 = −

2

(δ +

ξ2

ξ 22

+

ξ 32 )3 / 2

and F2 =

δ 2

(δ +

ξ 22

+ ξ 32 )3 / 2

(20)

Figure 7 shows the surface scan ( ξ1 = 0, ξ 2 , ξ 3 ) of the normal ( F1) component of the normalized magnetic field for a subsurface spherical inclusion at a depth of d = a and d = 2a respectively based on Eq. (20). As it could be expected qualitatively from Figure 1, the thermoelectric currents flow in opposite directions along two loops on the opposite sides of the inclusions relative to the direction of the heat flux, therefore the magnetic field is asymmetric with respect to this direction. One of the most crucial questions to be answered is how deeply inclusions can be buried and still detected. Figure 8 shows the maximum normal F1( ξ1 = 0, ξ 2 = − δ / 2 , ξ 3 = 0) and tangential F2 (ξ1 = 0, ξ 2 = 0, ξ 3 = 0) components of the normalized magnetic field on the surface for a subsurface spherical inclusion as a function of the normalized depth δ . The magnetic field to be detected rapidly decreases with the depth of the inclusion. The normal component drops from F1 ≈ 0.4 at d = a to as low as F1 ≈ 0.024 at d = 3 a . This sharp decay bodes ill for detection of inclusions buried deeply in the specimen, although our previous example indicates that, thanks to the exceptional sensitivity of SQUID −3 magnetometers, we might tolerate F1 values as low as 10 ÷ 10 −4 .

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x1 mirror image of the inclusion h0

free surface

x3

h0

d

j inclusion

Figure 6. A Schematic diagram of the inclusion buried below the surface and its its mirror image above it.

∇T

a)

∇T

b)

Figure 7. Two-dimensional distributions of the normal component of the normalized magnetic field for a subsurface spherical inclusion at two different depths a) d = a and b) d = 2a.


17

Normalized Magnetic Field

1 Maximum Normal Component ( F1) Maximum Lateral Component ( F2)

0.8 0.6 0.4 0.2 0 1

1.5

2

2.5

3

Normalized Depth ( δ )

Figure 8. The maximum normal F1( ξ1 = 0, ξ 2 = − δ / 2 , ξ 3 = 0) and tangential F2 ( ξ1 = 0, ξ 2 = 0, ξ 3 = 0) components of the normalized magnetic field on the surface for a subsurface spherical inclusion as a function of the normalized depth

δ.

3. EXPERIMENTAL INVESTIGATION OF THE SIGNAL FROM ISOTROPIC SPHERICAL INCLUSIONS One of the primary goals of this research work was to experimentally verify that the theoretical models can accurately predict the thermoelectric signature of surface-breaking, subsurface, and deeply embedded inclusions. The experimental results on spherical tin inclusions in a copper host provided excellent verification of the analytical predictions and also highlighted the crucial areas where significant future improvements are needed. In this section, it will briefly discuss the experimental setup and procedure used to verify that the above described analytical model truthfully captures the main features of the thermoelectrically generated magnetic field and accurately predicts its magnitude over a wide range of inclusion sizes and lift-off distances.

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3.1. Thermoelectric Detection of Surface-Breaking Spherical Tin Inclusions in Copper 3.1.1. Experimental method In the following, it will describe the experimental setup and procedure used to verify that the analytical model of Ref. [12,22], which it was briefly summarized in the previous section, truthfully captures the main features of the thermoelectrically generated magnetic field and accurately predicts its magnitude over a wide range of inclusion sizes and lift-off distances. Figure 9 shows a schematic diagram of the experimental arrangement. It was prepared a series of semi-spherical pure tin inclusions embedded in two pure copper bars of 12.7mm × 38.1mm × 500mm dimensions. First, it was prepared the semispherical holes by milling, then it was heated the specimens to approximately 300°C and filled the holes with molten tin, and finally milled the surface flat after the specimen has cooled down. The diameter of the inclusions varied from 2.38mm to 12.7mm and the center of each inclusion was at the level of the specimen's surface. The distance between inclusions was approximately 75mm to avoid interference between their individual magnetic fields. Both ends of the cooper bar were perforated by a series of holes and equipped with sealed heat exchangers to facilitate efficient heating and cooling and then mounted on a non-magnetic translation table for scanning. d fluxgate gradiometer heat flux

ec b g

hot (cold) water

cold (hot) water

copper specimen tin inclusions translation table

scanning

Figure 9. A schematic diagram of the experimental arrangement.


19

a) bar stock, ~ 4 nT

b) from plate, ~ 1 nT

c) bar stock after annealing, < 0.5 nT

T

T

Figure 10. Case hardening induced background signature in copper rods (∇T ≈ 0.5 o C/cm, 2 mm lift-off distance, 3"x3" scanning dimension)

The relevant physical properties of pure copper and tin were taken from standard references with the exception of the absolute thermoelectric power of

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Hector Carreon

tin at room temperature, which was measured by a Koslow TE-3000 thermoelectric instrument on the largest inclusion itself since it was found to be significantly affected by melting and subsequent recrystallization during sample preparation [13]. The TE-3000 is a conventional thermoelectric alloy tester used in NDT, that provides relative readings only, therefore it was first absolute calibrated by materials of known absolute thermoelectric power such as standard thermocouple alloys like Alumel and Chromel and pure copper. In order to better separate the sought magnetic signals of truly thermoelectric origin from potentially much stronger spurious artifacts, it was necessary to adopt a series of protective measures. Since the earth’s static magnetic field is about 100 μT, i.e., orders of magnitude stronger than the fields generated by thermoelectric currents around metallic inclusions, dc measurements are rendered completely useless. In the experiments, it was used ac coupling with a high-pass filter of very low cut-off frequency at 0.01 Hz. Since sufficiently fast alternating heating could not be implemented because of the inherently sluggish thermal response of the specimen, all measurements were done under steady-state thermal condition that was achieved in a few minutes after starting the heating and cooling. The pseudo-dynamic magnetic signals required for ac detection were produced by laterally (normal to the heat flux) scanning the specimen at a speed of 20 mm/s. As it was previously mentioned, the magnetic field is asymmetric to the principal direction of heating, therefore the signal to be detected does not exhibit a significant dc component. In addition to the relatively fast lateral scanning (“line” direction), it was also scanned the specimens at a much lower rate in the axial direction (“frame” direction). In this way, a 76.2mm × 76.2mm scan of 200 × 200 pixels took about 13 minutes. It should be mentioned that it is absolutely necessary to move the specimen with respect to the magnetometer rather than the other way around. As a result of the distortions caused by the presence of ferromagnetic objects in the surrounding, the earth’s magnetic field is inhomogeneous enough that any motion of the magnetometer would produce unacceptable levels of spurious signals. Of course all the moving parts of the translation table have to be made of nonmagnetic materials and even nonmagnetic materials of high susceptibility should be avoided. The signals to be detected are generated in the close vicinity of the magnetometer therefore they significantly vary from point to point, i.e., they exhibit strong gradients. Extraneous signals typically originate at larger distances from the magnetometer, therefore they smoothly vary from point to point, i.e., they exhibit relatively small gradients. In order to exploit this difference, we used a pair of detectors in a gradiometric arrangement. The


21

primary sensor closer to the specimen picks up a much stronger signal from the inclusion than the secondary sensor further away, while the two sensors exhibit essentially the same sensitivity for sources at large distances. The baseline distance (b in Figure 9) was chosen to be 28.6 mm in our case (generally, baseline optimization depends on the spatial distribution of the magnetic field to be measured). Further reduction of the baseline distance would improve the rejection of extraneous signals but would also reduce the sensitivity to the thermoelectric signals to be detected. The ends of the copper bar were simultaneously heated and cooled by running water to temperatures of ≈+10 °C and ≈+40 °C, respectively. The actual temperature difference between the ends of the bar was monitored during the measurements by thermocouple thermometers and the temperature gradient was kept at 0.7 °C/cm, which is more than sufficient to produce detectable magnetic signals in high-conductivity materials like copper and tin. It should be mentioned that most structural metals exhibit much lower electrical conductivity than copper. For example, in Ti-6Al-4V, the most popular aerospace titanium alloy, the electrical conductivity is only

σ ≈ 5.8 × 10 5 A / Vm , i.e., two orders of magnitude lower than that of copper, therefore the expected magnetic signals are also proportionally lower Like most other methods used in NDT, the noncontacting thermoelectric method is ultimately limited by temporally coherent material noise rather than temporally incoherent electrical noise that could be easily eliminated by simple time-averaging. Strictly speaking, material “noise” is really unwanted background “signal” that is often called noise only because it interferes with, and often conceals, the flaw signals to be detected. The main sources of such adverse background signals in thermoelectric NDT are macrostructural features such as case hardening, cold work, texture-induced anisotropy, residual stress, etc., while small-scale microstructural features such as grains are less important because of the lack of sufficient spatial resolution. The peak magnetic flux density of the background signature in pure copper bar stock was ≈ 8 nT, which is actually larger than the signals produced by the smallest tin inclusions used in the experiments. This background signature is due to case hardening and axial texture caused by cold rolling during manufacturing of the bar stock. These effects are much smaller in a bar cut from a larger plate (≈ 2 nT), but can be more or less eliminated (< 0.5 nT) only by appropriate annealing (30 minutes at 700 °C in a vacuum furnace)as shown in Figure10. Since this signature is essentially the same everywhere along the length of the bar, its adverse effect on flaw detection can be significantly reduced by

22

Hector Carreon

subtraction. The copper specimens used in the following experiments were all cut from a 12.7-mm-thick plate, but instead of the rather troublesome annealing process we chose to simply subtract the reference material signature whenever it reached above 10 % of the signal from the inclusion.

3.1.2. Experimental results Figure 11 shows examples of the magnetic images obtained from 6.35and 9.53-mm-diameter surface-breaking semi-spherical tin inclusions embedded in copper. These pictures were taken at g = 2 mm distance above the surface (see Figure 9). This apparent lift-off distance is the actual gap between the tip of the magnetometer probe and the specimen. However, the sensing element of the fluxgate is an e = 15 mm long ferromagnetic rod of d = 2 mm diameter centered in a c = 25 mm long case. It should be emphasized that, besides its much lower sensitivity with respect to a SQUID magnetometer, a fluxgate detector also suffers from its larger size that adversely affects the absolute accuracy of measurements for fields that vary over the sensor volume. The geometric center of the fluxgate is approximately 12.5 mm below the tip of the case, i.e., the 2-mm apparent lift-off corresponds to a much larger 14.5-mm actual lift-off distance. This crude approximation, however, will not be sufficient for the purposes of quantitative comparison to our analytical predictions, therefore later we are going to use a more accurate method. The goal is only to establish an empirical relation between the peak value and the half-width of the bi-polar magnetic signature on one side and the inclusion diameter and lift-off distance on the other side. The measured magnetic field distributions are similar in shape to the analytical predictions previously shown in Figure 4. As expected, the characteristic bi-polar lobes change sign when the direction of the temperature gradient in the specimen is reversed. These lobes get larger and the magnitude of the magnetic flux decreases when the lift-off distance is increased. In Figure 12a, compares the experimentally measured and theoretically predicted magnetic flux densities for all the different diameters and lift-off distances. Another quantitative parameter that can be readily used to compare the analytical predictions and experimental observations is the half-width of the bi-polar magnetic signature, which was defined as half of the lateral separation between the positive and negative peaks. In the amplitude measurements, it was simply subtracted the predicted magnetic field at the location of the secondary sensor from that of the primary sensor to account for the weak, but not entirely negligible secondary signal. Clearly, simple subtraction of the half-widths predicted at the levels of the primary and


23

secondary sensors would not give the right answer, therefore it was compared the experimental results simply to the predicted values at the level of the primary sensor. Figure 12b shows our theoretical predictions (solid line) and experimental results (symbols) for the half-width of the magnetic signature as a function of lift-off. There is a good agreement between the theoretical and experimental data except when the normalized lift-off distance exceeds 10. Since the peaks are not only smaller but also less sharp in the case of large normalized lift-off distances, it is not surprising that the accuracy of the measured half-width also declines.

a) 6.35-mm-diameter inclusion on the top, B ≈ 31 nT

T

T=0

T

b) 9.53-mm-diameter inclusion on the top, B ≈ 95 nT Figure 11. Magnetic images of surface-breaking semi-spherical tin inclusions on the top of a copper host (∇T ≈ 0.7 °C/cm, 2 mm lift-off distance, 76.2 mm × 76.2 mm scanning dimension, the peak magnetic flux density is indicated for comparison)

24

Hector Carreon 15

1000

100

Normalized Half-Width

Experimental Flux Density [nT]

12.7 mm 9.53 mm 6.35 mm 4.76 mm

10

3.18 mm 2.38 mm

1

10 2.38 mm

3.18 mm

5 4.76 mm 6.35 mm 9.53 mm 12.7 mm

0.1

0 0.1

1

10

100

Theoretical Flux Density [nT]

(a)

1000

0

5

10

15

20

Normalized Lift-Off

(b)

Figure 12. Comparison between theoretical predictions (solid line) and experimental results (symbols) for (a) the peak magnetic flux density and (b) the half-width of the magnetic signature for surface-breaking tin inclusions in copper.

3.2. Thermoelectric Detection of Subsurface Tin Inclusions In Copper 3.2.1. Experimental method In order to compare the magnitude, over a range of inclusion sizes and liftoff distances, of the analytical magnetic field produced by subsurface semispherical inclusions embedded in a homogeneous host material under external thermal excitation with respect to the experimental magnetic field, a plate of cooper was selected as the host medium and tin as the embedded inclusion. The copper bar (specimen) was cut from a copper plate with 12.7mm × 38.1mm × 500mm dimensions. The copper specimen was milled to obtain different semispherical holes ranging from 12.7 to 3.18mm at the specimen surface with a distance of 75mm between them in order to avoid interference between their individual magnetic fields. The depth of each hole was the same as its radius. Then the specimen was heated to approximately ≈ +300 °C. The holes of the specimen were filled with molten pure tin in order to obtain surface semi-spherical inclusions embedded in the copper specimen. Finally, the specimen was cooled down and milled the surface flat. Both ends of the specimen were drilled with several holes through the thickness and equipped


25

with sealed heat exchangers to heat and cool it by running water in each side generating a temperature gradient into the specimen, particularly in the boundary between the host medium and the inclusion. The copper specimen with different embedded surface breaking tin inclusions was turned upside down so that the surface-breaking inclusions were at the bottom simulating subsurface semi-spherical inclusions. The copper specimen was mounted and equipped by the heater exchangers to heat and cool it simultaneously by running water at temperature of ≈ +10°C and ≈ + 40°C, respectively. All the assembly was mounted on a translation table for scanning. Since the temperature of the cold and hot water in the laboratory inevitably fluctuated by a couple degrees, the actual temperature difference between the ends of the bar was monitored during the magnetic flux density measurements. The temperature gradient was between 0.5°C/cm and 0.7° C/cm, which is more than enough to produce detectable magnetic signals in high-conductivity materials such copper and tin. The magnetic flux density measurements of the different diameter subsurface semi-spherical tin inclusions were detected by a fluxgate (magnetometer) sensor in both temperature gradient directions as shown in Figure 13. The reported experimental data were always obtained as the difference between measurements taken at opposite temperature gradient directions.

Figure 13. Schematic diagram of the experimental set up

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Hector Carreon

The physical properties of pure copper σ = 59.7 ×106 A/Vm, κ =399 ×VA/m°C, S =1.72 ×10-6 V/°C and tin σ = 8.31 ×106 A/Vm, κ = 62.4 ×VA/m °C, S = -1.73 ×10-6 V/°C were used to make a quantitative comparison to the analytical predictions. These physical properties were taken from standard book references with the exception of the absolute thermoelectric power of tin inclusion at room temperature, which was measured by a Koslow TE-3000 thermoelectric instrument on the largest inclusion itself since it was found to be significantly affected by the sample preparation process (melting and recrystallization). The Koslow is a thermoelectric alloy tester used in NDT that provides relative readings with arbitrary units; therefore it was necessary to calibrate the equipment with materials of known absolute thermoelectric power. Figure 14 shows the relative readings between 25°C and 125° C measured by a Koslow TE-3000 thermoelectric instrument for the case of alumel and chromel alloys, pure copper and the tin inclusion. The copper specimen was scanned with a 3 axis magnetic field sensor (Mag-03 Bartington) that has a sensitivity of 10μT/V. This fluxgate (magnetometer) sensor has three sensing elements individually potted. The dimension of the sensing element of each sensor (fluxgate) is a cylinder of 15mm length × 1mm radius independently encapsulated with reinforced epoxy with dimensions of 8 × 8 × 25mm. In the experiment, only a pair of fluxgate sensors configurated in a gradiometric arrangement were used to detect the magnetic flux density produced by subsurface tin inclusions embedded in a copper specimen as shown in Figure 13. The primary sensor close to the specimen detects a much stronger signal from the inclusion than the secondary sensor further away, the two sensors display the same sensitivity for sources at large distances [13,23]. To calculate a quantitative comparison to the analytical predictions, the geometric center of the fluxgate was located approximately 12.5 mm below the tip of the epoxy-encapsulated case, i.e., the g = 2 mm apparent lift-off corresponding to a much larger 14.5-mm actual liftoff distance. The baseline distance b and the inclusion depth distance d were chosen to be 28.6 mm and 11.5 mm respectively. The baseline distance and inclusion depth optimization depend on the spatial distribution of the magnetic field to be measured [23].

3.2.2. Experimental results Figure 15 shows how the peak-to-peak magnetic flux density changes with the lift-off distance between 1 and 8 mm for five inclusions of different diameters between 12.7 and 3.18 mm. The solid lines represent the analytical


27

predictions while the solid points represent the experimental results based on the material properties listed . The lift-off distance was corrected for the depth of the sensing element below the surface of the probe and also for the inclusion depth distance, but no other adjustments were made. The results plotted in Figure 15 compare the experimentally measured and theoretically predicted magnetic flux densities for different diameters and lift-off distances. Considering the crude approximations used in the theoretical model, the large number of independent material parameters involved in the phenomenon and their inherent uncertainties, as well as the potential experimental errors associated with the measurements, the agreement between experimental results and analytical predictions over a range of more than two orders of magnitude is satisfactory. These results unequivocally prove that the theoretical approach adapted for modeling purposes and the experimental method chosen to map the thermoelectric magnetic field are both fundamentally correct and reliable. Furthermore, the small changes in the slope sign are quiet different than that expected from the simple theoretical model which assumes that the specimen is infinitely wide beside the strong effect of the intrinsic material background magnetic signature which limits the detectability of small, weak or subtle imperfections by the thermoelectric method.

400 chromel

Reading [a. u.]

300 200 100

tin copper

0 alumel

-100 -200 -300 -30

-20

-10

0

10

20

30

Absolute Thermoelectric Power [µV/C] Figure 14.- Relative readings measured by a Koslow TE-3000 thermoelectric instrument for the alumel and chromel alloys, pure copper and the tin inclusion.

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Hector Carreon

Magnetic Flux Density [nT]

1000 diameter [mm]

100 10

12.7 9.53

1

6.35 4.76 3.18

0.1 0

1

2

3

4

5

6

7

8

9

10

Lift-Off [mm] Figure 15. Comparison between the experimentally measured and theoretically predicted peak-to-peak magnetic flux densities plotted as functions of the lift-off distance for subsurface semi-spherical tin inclusions in copper.

Figure 16 shows the magnetic images recorded from two subsurface semispherical tin inclusions. These measurements were taken from the copper specimen of 12.7 mm-thick copper bar turned upside down so that the surfacebreaking inclusions were at the bottom simulating subsurface inclusions with a depth distance given by d. These pictures were taken at 2 mm distance above the copper specimen surface. This apparent lift-off distance is the distance between the tip of the fluxgate magnetometer and the copper specimen surface. However, the sensing element of the fluxgate magnetometer is a 15mm-long ferromagnetic rod buried at an average distance of 12.5 mm below the surface of the probe. Therefore, for the purposes of comparison with the analytical predictions of the magnetic flux density, this experimental configuration corresponds to a much larger 14.5 mm lift-off distance beside the case of fully embedded subsurface inclusions. The measured magnetic field distributions are very similar in shape to the analytical predictions shown in Figure 7. As expected, the characteristic main bi-polar lobes change sign when the direction of the temperature gradient in the specimen is reversed. These main lobes get wider and the magnitude of the magnetic flux decreases when the lift-off distance is increased (obviously the inclusion depth distance d is also affected). However the magnetic field became significantly weaker in the case of subsurface inclusions and the spatial distribution of the field also revealed some distortions. In addition to the previously observed two main


29

lobes, two weaker secondary lobes could also be observed. This particular feature is not predicted by the simple analytical model and is most probably associated with the finite width (12.7 mm) of the copper bar, which is expected to affect the measurements in a greater extent when the inclusion is rather deep below the surface due to the fact that the intrinsic material background signature affected the POD of subtle material flaws in the noncontacting thermoelectric technique for QNDE material characterization.

∇T

∇T = 0

∇T

a) 6.35 mm-diameter subsurface semi-spherical inclusion, B ≈ 2.3 nT

∇T

∇T = 0

∇T

b) 9.53 mm-diameter subsurface semi-spherical inclusion, B ≈ 8.6 nT Figure 16. Magnetic images of subsurface semi-spherical tin inclusions ( ∇T ≈ 0.7 o C/cm, 2 mm lift-off distance, 76.2 mm × 76.2 mm scanning dimension, the peak magnetic flux density B is indicated for comparison).

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Hector Carreon

4. THERMOELECTRIC DETECTION OF HARD ALPHA INCLUSION IN TI-6AL-4V The experimental results reported by author and et al. [13,23] clearly verify the feasibility of nondestructive evaluation of tin inclusions embedded in a copper specimen by thermoelectric means. The obvious next question to be addressed is whether the non-contacting thermoelectric method is applicable to other engineering materials of special interest to the aerospace industry. In particular, it is very important to develop NDE techniques for high-strength, high-temperature engine materials such as titanium-base alloys. So, we conducted thermoelectric measurements on uncracked hard-alpha inclusions with different nitrogen content in Ti-6Al-4V. Like most engine materials, titanium-base alloys exhibit much lower thermal and electrical conductivity and more significant microstructural inhomogeneity and anisotropic texture than single-phase pure copper.

Figure 17. Cracking in the hard-alpha (HA) inclusion and through the diffusion zone in a high nitrogen, large surface defect specimen


31

4.1. State of Art Initiation sites for LCF failure of titanium-base alloys in the components of interest can occur at nitrogen-rich inclusions.These inclusions generally have a core of TiN, which is surrounded by a layer of α-titanium, which in turn is surrounded by a layer of β-titanium. These inclusions are frequently referred to as a “hard alpha” inclusions. Hard-alpha inclusions are high interstitial defects and are regions of much higher hardness than the surrounding material. The increased brittleness of alpha inclusions is the result of excess nitrogen and oxygen that increase the beta transus of the material. These inclusions may appear during the manufacturing process if the starting material is already contaminated, if the proper temperature is not maintained to break up the alpha regions, or the material becomes contaminated during the melt process [24]. The hard-alpha titanium with the added nitrogen is much more brittle than the base metal,and a study was done to experimentally showed the transition from plastic deformation to brittle fracture. Currently, hard-alpha inclusions can be detected only if they contain cracks or voids as shown in Figure 17. In general the hard alpha inclusion core tended to exhibit single continuous cracks with more numerous sites of damage typically apparent in the diffusion zone. This means that uncracked and non-voided hard-alpha inclusions go undetected and represent a major hidden problem. Several candidate methods for nondestructive evaluation were identified including mainly ultrasonics and Photon Induced Positron Annihilation. Present technology for ultrasonic detection of this type of defects includes a multizone system that uses multiple channels employing focused transducers and analog electronics. It uses four to eight transducers of 5 MHz each focused at a different depth in the material, thus inspecting the material uniformly [25,26]. Another current method for detection is Photon Induced Positron Annihilation (PIPA) developed by Positron Systems. This method promises to detect smaller, buried inclusions that other nondestructive methods cannot detect. The principle of PIPA involves penetrating the material with a photon beam to create positrons. The positrons are attracted to the defects and when they eventually collide with electrons in the material and are destroyed, gamma ray energy is released. This energy is distinct and readable and allows for characterization of the defect.With this method, defects can be detected in their earliest stage before any failure can occur [27,28]. The purpose of this investigation was to suggest an alternative nondestructive method of characterizing nitrogen-enriched specimens that are free from cracks and voids. We presents experimental results for the particular

32

Hector Carreon

case of uncracked hard-alpha inclusions in Ti-6Al-4V under external thermal excitation which truthfully captures the main features of the thermoelectrically generated magnetic field and predicts its nitrogen content ranging from 1.6% to 5.9%. And it also shows experimental evidence of the strong effect produced by highly anisotropic materials such as titanium-base alloys in thermoelectric NDE.

4.2. Experimental Method In this section, the experimental setup will be described and procedure used to characterize the TiN inclusions with different nitrogen content embbeded in a Ti-6AL-4V specimen. Also, it will be investigated how the magnetic signal to be detected depends on the physical properties of the host and the inclusion. We conducted thermoelectric measurements on four titanium specimens with uncracked hard-alpha inclusions with different nitrogen content namely 1.6%, 2.6%, 3.5% and 5.9% supplied by MLLP of AFRL. The TiN samples are representative of Ti-6Al-4V engine material since when nitrogen enters Ti-6Al-4V, the aluminum and vanadium is mostly replaced with nitrogen, thus leaving a mostly TiN inclusion like the specimens used in this investigation. Analysis of the nitrogen content in weight percent, for the uncraked hard-alpha inclusions, is shown in Table I. All compositions reported in this article are in weight percent. The four specimens with TiN inclusions with different nitrogen content were cut from Ti-6Al-4V block with the same dimensions 15.5 mm × 15.5 mm × 25 mm. The Ti-N inclusions were cylindrical shapes with a diameter of ∼ 0.19 mm and ∼ 0.19 mm of depth and with the center of each Ti-N inclusion at the level of the specimen surface (top side). Table I. Nitrogen content in weight percent for the uncraked hard alpha inclusions Specimen 1N 2N 3N 5N

Nitrogent Content % Wt 1.6 2.6 3.5 5.9

Nondestructive Materials Characterization by Magnetic Sensing a)

33

fluxgate gradiometer y

heat sink compound

hot (cold) water

heat flux copper support

specimen x

cold (hot) water

translation table scanning

Figure 18. Set up experimental configuration for the characterization of the TiN inclusion.

Figure 18 shows a schematic diagram for the experimental arrangement used to study the different thermoelectric signatures produced by the uncracked hard-alpha inclusions embedded in Ti-6Al-4V. Each specimen was mounted into two copper supporters which were perforated by a series of holes and equipped with sealed heat exchangers to facilitate efficient heating and cooling and then mounted on a nonmagnetic translation table for scanning. In order to get a better heat transfer between the specimen and the copper supporters, it was put a layer of heat sink compound silicone. The ends of the copper supporters were simultaneously heated and cooled by running water to temperatures of 85°C and 10°C respectively. However, the actual temperature difference between the ends of the Ti-6Al-4V specimen was monitored during the measurements by thermocouple thermometers and the temperature gradient was kept at 12 °C/cm, which is more than sufficient to produce detectable magnetic signals in high-strength, high-temperature engine materials such as titanium-base alloys [15] . A pair of fluxgate sensors configured in a differential arrangement was used to detect the thermoelectric signals from the specimen. The primary sensor, which is closer to the specimen, measures a stronger signal than the secondary sensor, while the two sensors exhibit essentially the same sensitivity for external sources at larger distances, which are rejected accordingly. In each case the so called lift-off distance, i.e., the gap between the primary sensor and the surface of the specimen was 2 mm. The specimen was scanned in normal

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Hector Carreon

sensor polarization. The magnetic signals produced by the TiN inclusions were detected by horizontally scanning the specimen at the center of the bar in a direction normal to the heat flux. Figure 18 also shows how the normal magnetic signatures can be recorded when scanning in the “width-direction” of the specimen. In order to separate the magnetic signature of the TiN inclusion from the adverse intrinsic material background signature. First, the scans were taken by rotating the specimen 90° around its axis to record the magnetic signature at four specimen orientations (0°, 90°,180°,270°) and secondly the specimen was flipped at top and bottom sides respectively. Altogether, eight independent signatures can be measured using four specimen orientations and two specimen sides. However, all magnetic signatures were recorded on four titanium specimens with uncracked hard-alpha inclusions with different nitrogen content namely 1N, 2N, 3N and 5N. Therefore, the experimental data to be presented later represents thirty-two measurements. Because of the weak magnetic flux densities B (≈10 pT and 300 nT) created by the thermoelectric currents around the material imperfection zone with temperature gradients of ≈ 0.1-15 °C/cm. Numerous additional measures were taken to assure that the magnetic signatures were recorded with minimal distortion and these measures are described in detail in Reference [13].

4.3. Experimental Results In the next step, all specimens were inspected by the noncontacting thermoelectric technique described above. Figure 19 shows the average of six magnetic signatures recorded from the bottom side (right) and top side (left) of the Ti-6Al-4V specimen with different nitrogen content of hard alpha inclusions a) 1N, b) 2N, c) 3N and d) 5N as a function of the four specimen orientations (0°, 90°,180°,270°) at 2-mm lift-off distance and 12°C/cm temperature gradient. In general, on the specimens at the top side (left) the peak-to-peak value of the measured magnetic flux density increased from 5 nT to 25 nT as nitrogent content increased from 1.6% to 5.9% and the variation within the six series of repeated magnetic measurements was found to be at an acceptable level. In comparison, on the specimens at the bottom side (right) the peak-to-peak value of the magnetic flux density was approximately 65 % lower. However, by comparing the magnetic signatures recorded after rotating the specimen around its principal (0°, 90°,180°,270°) axes. It can not be


35

established whether the actual magentic signature is dominated by the intrinsic thermoelectric anisotropy or the TiN inclusions. The measured magnetic signature of highly textured materials such as Ti-6Al-4V often exhibit a significative intrinsic thermoelectric background due to macrostructural features.The strong effect of the intrinsic material background magnetic signature limits the detectability of small, weak or subtle imperfections by the thermoelectric method [13,15,23,29]. Therefore the measured magnetic signature reported in Figure 19 is due to a combination of the intrinsic material thermoelectric anisotropy and the TiN inclusion signature. In order to separate them, Figure 20 shows the average of the magnetic signatures for the four specimen orientations (0°,90°,80°,270°) of the top and bottom sides of all four specimens with different nitrogen content 1N, 2N, 3N and 5N (the forward and backward scanning parts of the signatures were averaged for better accuracy) and Figure 21 shows the averaged magnetic signatures for the four specimen orientations (0°,90°,80°,270°) of the bottom and top sides of all four specimens with different nitrogen content 1N, 2N, 3N and 5N after baseline subtraction (the baseline was calculated as the average of the four bottom signatures). As we can see the magnetic magnetic signature due to the different TiN inclusion nitrogent content is much higher comparing to the instrinsic material background anisotropy after baseline subtraction and averaged by the four specimen orientations. A number of important conclusions can be drawn from these results. First, the magnetic signature produced by TiN inclusions is proportional to the nitrogent content , which is very favorable from the point of view of quantitative detection assessment. Second, even the lowest nitrogent content TiN inclusion (1.6%) is well above the detection threshold of the noncontacting thermoelectric measurement, therefore the absolute sensitivity of the technique seems to be sufficient for most practical applications. Figure 22 shows the peak-to-peak magnetic flux density as a function of the nitrogent content of hard alpha inclusions embedded in a Ti-6Al-4V specimen. The right set of data represent the experimental results of the magnetic flux density at the top side of the specimen specimen where the surface-breaking TiN inclusions are located while the left set of data represent the experimental results of the magnetic flux density at the bottom side of the specimen with different nitrogent content. In both cases the magnetic signature for all the specimens were corrected by the baseline subtraction and averaged by the four specimen orientations.

36

Hector Carreon 14

14 1N

10

1N

10 6

6

2

2

-2

-2

-6

0°

90°

0°

90°

180°

270°

-6 180°

-10

270°

-10

-14 0

5

10

15

20

25

30

35

40

-14

45

0

5

10

15

Position [mm]

20

25

30

35

40

45

Position [mm]

(a) 14

14

Flux Density [nT]

Flux Density [nT]

6 2 -2 0°

-6

90°

180° 270°

-10

2N

10

2N

10

6 2 -2 0°

-6

90°

180° 270°

-10 -14

-14 0

5

0

10 15 20 25 30 35 40 45

5

10 15 20 25 30 35 40 45

Position [mm]

Position [mm]

(b) 14

14 3N

Flux Density [nT]

10

3N

10 6

6

2

2

-2

-2

-6

-6

0°

-10

0°

90°

180° 270°

90°

180° 270°

-10 -14

-14 0

0

5 10 15 20 25 30 35 40 45

10 15 20 25 30 35 40 45 Position[mm]

Position [mm]

(c) Figure 19. (Continued)

5

Nondestructive Materials Characterization by Magnetic Sensing 14

14

5N

10 6 2 -2 0°

-6

90°

180° 270°

-10

5N

10 Flux Density [nT]

Flux Density [nT]

37

6 2 -2 -6 -10

0°

90°

180° 270°

-14

-14 0

0 5 10 15 20 25 30 35 40 45

5 10 15 20 25 30 35 40 45

Position [mm]

Position [mm]

(d) Figure 19. Averaged magnetic signatures from the bottom and top sides of a) 1N, b) 2N, c) 3N and d) 5N specimen.

Clearly, the measured magnetic flux density significatly increased with the nitrogent content of hard alpha inclusions at the top side of the Ti-6Al-4V specimen. While a relatively strong baseline signature is presented along the Ti-6Al-4Vspecimen at the bottom side. Since this baseline is essentially the same everywhere along the length of the specimen, its adverse effect on flaw detection is reduced by subtraction. However, the efficiency of baseline subtraction is badly reduced by inevitable local variations and experimental uncertainties, therefore, like other similar image processing tricks, it has its limitations in improving POD. Figure 23 shows the magnetic images recorded from the four Ti-N inclusions in Ti-6Al-4V at top side with different nitrogen content. The measured peak to peak magnetic flux density is also listed for the experimental results. These magnetic pictures were taken at 2 mm distance above the specimen surface. This apparent lift-off distance is the distance between the tip of the fluxgate magnetometer and the specimen surface.

38

Hector Carreon 2.5

2.5

1N

Top

Top

Bottom

1.5 Flux Density [nT]

Flux Density [nT]

1.5

0.5

-0.5

-1.5

2N

Bottom

0.5

-0.5

-1.5

-2.5

-2.5

0

5

10 15 20 25 30 35 40 45

0

5

10 15 20 25 30 35 40 45

Position [mm]

Position [mm]

2.5

2.5 Top

1.5

0.5

-0.5

5N

Top

3N

Bottom

Flux Density [nT]

Flux Density [nT]

1.5

Bottom

0.5

-0.5

-1.5

-1.5

-2.5

-2.5 0

5

10 15 20 25 30 35 40 45 Position [mm]

0

5

10 15 20 25 30 35 40 45 Position [mm]

Figure 20. Averaged magnetic signatures for four directions of the bottom and top sides of the four specimens.

The measured magnetic field distributions are very similar in shape to the results published by author et al. in the particular case of surface-breaking tin inclusions embbeded in a copper specimen [13]. In this respect, the surfacebreaking TiN inclusions measurements are representative of a real application since the base metal is a not necessarily homogeneous or texture-free such as copper specimen. As we expected, the main lobes get wider and the magnitude of the magnetic flux decreases when the nitrogent content is decreased and the spatial distribution of the field revealed some distortions. This particular feature is most probably associated with the intrinsic material background signature affected the POD of subtle material flaws in the noncontacting thermoelectric technique. This intrinsic material background signature is due to the manufacturing process used to fabricate stock materials (bar, billet, plate ect.) that tend to induce a preferred crystallographic orientation due to the restricted nature of mechanical slip (dislocations), leaving the material with a remarkable macroscopic anisotropy [30].

Nondestructive Materials Characterization by Magnetic Sensing 3

3 1N

Top 2

2

1 0 -1

2N

Top

Bottom

Flux Density [nT]

Flux Density [nT]

39

Bottom

1 0 -1

-2

-2 -3 0

5

-3

10 15 20 25 30 35 40 45

0

Position [mm]

5

10 15 20 25 30 35 40 45 Position [mm]

3

3

Top Bottom

1 0 -1 -2

5N

Top 2 Flux Density [nT]

Flux Density [nT]

2

3N

Bottom

1 0 -1 -2

-3

-3

0

5

10 15 20 25 30 35 40 45 Position [mm]

0

5

10 15 20 25 30 35 40 45 Position [mm]

Figure 21. Averaged magnetic signatures for four directions of the bottom and top sides of all four specimens after baseline subtraction.

5. THERMOELECTRIC SIGNATURE PRODUCED BY RESIDUAL STRESS The noncontacting thermoelectric method can be used to characterize the prevailing residual stress in conducting specimens, which is usually produced by plastic deformation during either manufacturing or service. One particular application of great interest is the nondestructive assessment of residual stresses in surface-treated metals. It is well known that surface properties play a major role in determining the overall performance and, in particular, the fatigue resistance of structural components. Shot peening, one of the most popular surface improvement methods, induces compressive residual stresses in the surface layers of metallic parts via bombarding the them with a stream

40

Hector Carreon

of high-velocity shots as it is schematically shown in Figure 24. As the plastically deformed surface layer tries to expand relative to the intact interior of the specimen, compressive residual stress develops parallel to the surface at shallow depths, while beneath this layer a reaction-induced tensile stress results. Generally, the compressive stress at the surface is several times greater than the subsurface tensile stress. This near-surface compressive stress offsets any service-imposed tensile stress, retards fatigue crack nucleation and growth, and ultimately extends the fatigue life of the part as shown in Figure 25. In addition to the primary residual stress effect, shot peening also causes an adverse geometrical side effect by roughening the surface and certain relatively subtle variations in material properties, such as increased hardness and texture, that are consequences of the significant plastic deformation through cold work.

Flux Density [nT]

6

Inclusion at top

Inclusion at bottom

5 4 3 2 1 0 1.6

2.6

3.5

5.9

1.6

2.6

3.5

5.9

TiN Nitrogen Content [wt%] Figure 22. Peak-to-peak magnetic signatures for all four specimens after baseline subtraction and averaged by the four specimen orientations (10 nT/V detection sensitivity).


1N (0.90 nT)

2N (3.30 nT)

3N (4.2 nT)

5N (5.2 nT)

∇T

∇T

41

Figure 23. Magnetic images of TiN inclusions with different nitrogent content ( ∇T ≈ 12oC/cm, 2 mm lift-off distance, 76.2 mm × 76.2 mm scanning dimension). The measured peak magnetic flux density is also listed for the experimental results

42

Hector Carreon

surface roughness

residual stress

cold work

Figure 24. A schematic diagram of shot peening and its three major effects.

Figure 25. Effect of shot peening on fatigue life in a 7075 Aluminum

5.1. State of Art Residual stress analysis by nondestructive methods is a highly developed field using a great variety of different physical principles ranging from radiography, ultrasonics, electromagnetism, and ferromagnetism to assess the absolute level and relative distribution of elastic stresses prevailing in the material [31]. The most advanced technique for measuring residual stress in


43

crystalline materials is based on X-ray diffraction (XRD), which measures changes in atomic inter-planar spacing to determine the magnitude of the prevailing elastic strain (stress). Neutrons can penetrate many millimeters into most engineering materials, while X-rays are typically absorbed within a surface layer of 5-20 µm. In comparison, the crucial compressive part of typical residual stress profiles ranges from 50 µm to 500 µm. Therefore, residual stress assessment by XRD is nondestructive only within a very shallow surface layer. To probe the residual stress below the surface, successive layers must be removed, usually through etching or electropolishing, i.e., in a destructive manner. The removal of material also alters the stress field, and thus requires theoretical corrections of the measured values. Furthermore, since the method probes only the surface, the results can be easily skewed by spurious effects in the extremely shallow top layer. In spite of the troublesome and destructive sectioning required by the low penetration depth, XRD is probably the most accurate and reliable method for residual stress assessment in surface-treated metals. One of the main reasons for this is that XRD methods are not significantly influenced by additional variations in material properties such as hardness, plastic strain, or texture [3234]. In recent years, several candidates were identified for subsurface nondestructive stress evaluation including ultrasonic, eddy current, and other methods. For a long time, the characteristic dependence of ultrasonic surface wave velocity on stress has been thought to be very promising for residual stress measurements in surface- treated metals, though these expectations have remained largely unfulfilled as far as shot-peened specimens are concerned [35-37]. One of the main problems is that the most adverse side effect of shot peening, namely the rough surface topography, is also the most persistent. It is unaffected at best, or even exacerbated, by extended service during which the primary residual stress gradually relaxes. Eddy current conductivity measurements suffer from essentially the same limitation [38,39] except in the case of nickel-base superalloys [40-42], which is not surprising since eddy currents exhibit similar “surface hugging” behavior as Rayleigh-type surface waves. It was found that in surface wave velocity and electrical conductivity measurements the difference between lowstress- ground and shot-peened parts of the same specimen is essentially unaffected by annealing, which clearly indicates that the observed phenomenon is mainly due to surface roughness that increases the path length of surface-hugging Rayleigh waves and eddy currents. Because of the diminishing contribution of the slowly fading residual stress effect during long-term service, any NDE method that is expected to

44

Hector Carreon

reliably characterize the thermal relaxation process in shot-peened metals must be entirely insensitive to the persistent surface roughness effect. We have already demonstrated that the noncontacting thermoelectric technique is very sensitive to the presence of foreign body inclusions, when the thermoelectric power of the affected region is significantly different from that of the surrounding medium [13,19,23]. The question arises whether mere plastic deformation of the material can produce a perceivable thermoelectric contrast with respect to the surrounding intact host. To answer this question, Figure 26 compares the magnetic scans of two apparently similar 9.53-mm-diameter surface holes in copper. Figure 26a corresponds to a semi-spherical hole produced by low-stress milling which is expected to generate only negligible hardening and residual stress below the machined surface. In comparison, Figure 26b shows the results from an otherwise similar semi-spherical indentation produced by pressing a stainless steel ball into the material in a manner that simulates a single impact during shot-peening. As a result of plastic deformation, the surrounding material below and around the indentation is substantially hardened and supports significant residual stresses. In order to demonstrate that the proposed thermoelectric method could readily detect thermally induced stress release, Figures 26c and 26d show the magnetic images of the same two specimens after annealing in a vacuum furnace for 30 minutes at 700 °C. All the effects of plastic deformation during indentation, as well as the much weaker manufacturing texture found in the original bar stock, are gone. As a result, the thermoelectric currents are also eliminated and the measured magnetic field is essentially zero.

5.2. Monitoring Residual Stress Relaxation in Copper 5.2.1. Thermal stress release In this research work, it was conducted a detailed experimental investigation of all the material properties that significantly change during shot peening in order to establish how they individually and collectively affect the recorded magnetic signatures and to verify that the residual stress effect dominates the outcome of the measurement. It was postulated in this study that the penetration depth and the particular depth-profile of the residual stress are primarily determined by material and manufacturing process variables and will change accordingly during thermally activated stress release. Different surface treatments produce not only different residual stress amplitudes and penetration depths, but, even more importantly, different cold work amplitudes


45

and penetration depths, therefore exhibit very different thermal relaxation behaviors. Generally, the rate of thermally induced relaxation towards equilibrium is proportional to the existing deviation from equilibrium at that particular location and at that particular instance of time. In some cases of thermo-mechanical stress relaxation, the penetration depth and the particular depth-profile of the residual stress will not change significantly during thermally activated stress release [43]. a) milled

b) pressed

c) milled, annealed

d) pressed, annealed

T

T

plastic zone

Figure 26. Comparison between (a) a semi-spherical hole produced by low-stress milling and (b) an otherwise similar semi-spherical indentation produced by pressing a stainless steel ball into the material. The scans on the right (c and d) were taken after annealing (VT = 0.5 ºC/cm, 2 mm lift-off distance, 76.2 mm × 76.2 mm)

46

Hector Carreon

Residual Stress [a. u.]

0% 25% 50%

proportional

75%

enhanced

100%

Depth [a. u.] Figure 27. A schematic illustration of proportional stress relief that does not affect the normalized depth distribution of the residual stress and enhanced stress relief due to near-surface cold work

Figure 27 shows a schematic illustration of such proportional stress relief (solid lines) that does not affect the normalized depth distribution of the residual stress, though it significantly reduces its magnitude. However, this model neglects that certain types of surface treatments, especially shot peening, departs a very strong cold work to the material, that substantially reduces the stress relaxation temperature at shallow depth below the surface. Therefore, in many cases, the relaxation directly at and below the surface is much stronger than at larger depths and the depth-profile of the residual stress significantly distorts during the relaxation. Since the thermoelectric method measures a weighted average of the material variation below the surface, absolute calibration of the technique is very difficult if not impossible. Instead, like in almost all other cases of nondestructive materials characterization, we have to rely on empirical calibration curves that are obtained separately for different materials, different surface treatments, and possibly even different levels of surface treatment. On actual shot-peened specimens the residual stress and cold work effects can be best modified by appropriately chosen heat treatment that simulates thermally activated stress release during service. Hardness increases with the degree of cold work [44]. Figure 28 shows the measured Rockwell F hardness


47

of cold-rolled C11000 copper as a function of annealing temperature in a vacuum furnace for 30 minutes. There is a sharp drop at around 440°C where recrystallization occurs. Based on this observation, it was selected a heat treatment annealing at 315°C for 30 minutes to induce stress release without recrystallization, i.e., approximately 125°C below the measured transition temperature. This difference is necessary to account for the additional drop in recrystallization temperature that is anticipated due to the higher degree of cold work directly below the shot-peened surface. These results are intended only as demonstrations of the much more rigorous procedure to be followed in the next stage, when stress release will be accomplished in several steps and in each step destructive X-ray diffraction measurements will be carried out to determine the remaining residual stress level as well as the local anisotropic texture as characterized by pole figures.

5.2.2. Experimental results In the next step, all the specimens were inspected by the noncontacting thermoelectric technique. Figure 29 shows the schematic diagram of the noncontacting thermoelectric method as used for the characterization of shotpeened specimens. Since the generated magnetic field is perpendicular to both the heat flux in the specimen (parallel to the surface) and the gradient of the material property (normal to the surface), the magnetometer was polarized in the tangential direction Characteristic magnetic profiles were obtained from the shot-peened copper specimens along with that of an unpeened bar at ∇T ≈ 2.5 °C/cm temperature gradient and 2-mm lift-off distance. Because of the relatively low level of the magnetic signatures to be measured, special measures were taken to eliminate spurious effects of the Earth magnetic field, magnetic interference from the electrical power system and surrounding instruments, and other extraneous magnetic sources. These protective measures were discussed in great detail in reference [13]. As an example, Figure 30 shows the magnetic scans on C11000 copper specimens before and after shot peening taken in horizontal sensor polarization at ∇T≈2.5 °C/cm temperature gradient for two opposite heating directions. The peak to peak magnetic flux is indicated for comparison. Figure 30b shows the magnetic images of the copper specimen with a 14 Almen peenig intensity. While, Fig 30a shows the magnetic images of the unpeened copper specimen. The specimen without shot-peened exhibits a background signature of ≈ 2 nT peak-to-peak magnetic flux density that is caused by intrinsic anisotropic texture of the material [15,20]. After shot peening, the magnetic signature

48

Hector Carreon

increased to ≈ 20 nT peak to peak amplitude. The main advantage of the thermoelectric method is that it is sensitive only to the “material” effects of the damage, namely residual stress, local texture, and increased dislocation density, but it is entirely insensitive to its “geometrical” by-product, i.e., the rough surface topography [45]. However, the detectability of weak imperfections is obviously adversely affected by the presence of the rather strange looking background signature in the intact material, that can be reduced only by annealing above the recrystallization temperature, which would also completely change the existing microstructure.

Figure 28. Rockwell F hardness versus annealing temperature in cold-rolled C11000 copper (in a vacuum furnace after 30 minutes) fluxgate gradiometer

shot-peened surface

T

thermoelectric current

Figure 29. A modified schematic diagram of the noncontacting thermoelectric method as used for the characterization of shot-peened specimens


49

Figure 31 shows the measured magnetic signatures on three series of C11000 copper specimens before and those of two series after stress release for 30 minutes at 315 °C as function of the shot peening intensity at 2-mm liftoff distance and 2.5 °C/cm temperature gradient. On the intact specimens (i.e., before stress release) the peak-to-peak value of the measured magnetic flux density increased from ≈5 nT to ≈20 nT as the peening intensity increased from Almen 2 to Almen 16 and the variation within the three series was found to be at an acceptable level. In comparison, on the stress released specimens the magnetic flux density was approximately 75% lower.

(a) before shot-peened, B ≈ 2nT

(b) after shot-peened, B ≈ 20nT

∇T

∇T

Figure 30. Magnetic signatures produced by copper specimens before and after shotpeened (∇T ≈2.5 °C/cm temperature gradient, 2 mm lift-off distance, 76.2 mm × 76.2 mm scanning dimension)

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Hector Carreon

The next task was to repeat all these measurements after different degrees of partial stress release to separate the individual effects of residual stress, cold work induced texture and hardening, and surface topography on the measured thermoelectric signature. However, one series of shot-peened specimens were sent for X-ray diffraction analysis. X-ray diffraction residual stress measurements were performed using a two-angle sine-squared-psi technique, in accordance with SAE J784a, employing the diffraction of Mn K-alpha radiation from the (311) planes of the FCC structure of the C11000 series copper. The fully corrected residual stress distributions measured as function of depth before stress release and after the first partial stress release are presented in Figure 32 for nine specimens. The residual stress data before stress release show that the maximum compressive layer was found close below the surface on all but the unpeened specimen (Almen 0). Generally, the results indicate increasing depth of compression with increasing intensity. The surface residual stress slightly decreases in magnitude as the peening intensity increases, which is typical of work hardening materials. In comparison, the residual stress data after the first partial stress release show a substantial reduction in residual stress by approximately a factor of two at all Almen intensities (the weighed average reduces even more due to the almost complete relaxation at and directly below the surface). As a by-product of the above described residual stress measurement, the obtained X-ray diffraction data can be also used to quantitatively assess the degree of cold work below the shot-peened surface. Figure 33 shows the subsurface cold work distribution for nine C11000 copper specimens of different shot peening intensity before and after the first partial stress release as measured by the (311) diffraction peak width, which is a sensitive function of the hardness and the degree to which the material has been cold worked. In work hardening materials, the diffraction peak width increases significantly as a result of an increase in the average microstrain and the reduced crystallite size produced by cold working [43]. The (311) diffraction peak width can be indicative of how the material may have been processed and the degree to which it has been plastically deformed. The results before stress release indicate that maximum cold working or hardness occurs at the surface of the shot-peened samples. The surface cold working does not differ significantly with peening intensity, however the depth of the plastically deformed layer appears to increase slightly with increased peening intensity. In comparison, after the first partial stress release the cold work is essentially gone at all Almen intensities.

Almen 12

Flux Density [5 nT/div]

Position [20 mm/div]




Almen 8


before stress release

Almen 14 Position [20 mm/div]

51



Almen 6





Almen 2



Almen 0





after stress release (30 min, 315 °C)

Figure 31. Magnetic signatures on three series of C11000 copper specimens before and those of two after stress release as functions of the shot peening intensity.

-25 -75

Almen 0

-125 0

-100 -150 -200

Almen 2

-250

0.2 0.4 0.6

0

0 -50 -100 -150 -200

Almen 6

-250 0

-50 -100 -150 -200

Almen 8

-250 0

-100 -150 -200

Almen 12

-250 0

-100 -150 -200

Almen 4

-250 0

0 -50 -100 -150 -200

Almen 10

-250

0.2 0.4 0.6

0 -50 -100 -150 -200

Almen 14

-250

0.2 0.4 0.6 Depth [mm]


0

0.2 0.4 0.6 Depth [mm]

0.2 0.4 0.6 Depth [mm]

0

Depth [mm]

Residual Stress [MPa]


Depth [mm]

-50

-50

0.2 0.4 0.6

0

0.2 0.4 0.6

0

0

Depth [mm]



Depth [mm]


25

-50


75

0

0.2 0.4 0.6 Depth [mm]


125


Hector Carreon


52

0 -50 -100 -150 -200

Almen 16

-250 0

0.2 0.4 0.6 Depth [mm]


Figure 32. Subsurface residual stress distribution for nine C11000 copper specimens of different shot peening intensity as measured by X-ray diffraction.


1 Almen 0

0 0

0.2

0.4

2

1 Almen 2

0

0.6

0

Depth [mm]

0.6

0

Almen 6 0

0.2

0.4

Almen 8

0 0

Depth [mm]

0.2

0.4

0

0.2

0.4

0.6

0

0.6

Depth [mm]


0.2

0.4

0.6

Depth [mm] 2

1 Almen 14

0 0

0.2

0.4

Depth [mm]

0.6

Peak Width [deg]

Peak Width [deg]

Almen 12

0

Almen 10

0

2

1

0.6

1

Depth [mm]

2

0.4

2

1

0.6

0.2

Depth [mm]

Peak Width [deg]

1

0

Almen 4

0

2 Peak Width [deg]

Peak Width [deg]

0.4

1

Depth [mm]

2

Peak Width [deg]

0.2

Peak Width [deg]

2 Peak Width [deg]

Peak Width [deg]

2

53

1 Almen 16

0 0

0.2

0.4

0.6

Depth [mm]


Figure 33. Subsurface cold work distribution for nine C11000 copper specimens of different shot peening intensity as measured by the (311) diffraction peak width.

Another effect of cold work in the material is the development of a localized sub-surface texture, which can be also studied by X-ray diffraction. The data necessary to construct the (200) pole figures were obtained using copper K-alpha radiation and a Schulz back-reflection pole figure device mounted on an automated Bragg-Brentano X-ray diffractometer. After subtraction of the background, the corrected pole figure was constructed in

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Hector Carreon

spherical coordinates by linear interpolation of the data collected on the equal area spherical net. An analytical solution was used to correct for defocusing intensity losses, which occur as the specimen is tilted in the X-ray beam. Figure 34 shows the plane (200) back-reflection pole figures for four C11000 copper specimens of different shot peening intensity (light areas - 50-100 % relative to average, dark areas 100-200 % relative to average, the vertical axis corresponds to the length of the bar). Generally, the pole figures of all samples appear to have a nearly random azimuthal orientation. The unpeened specimen exhibits a “spotty” appearance, which decreases with increasing peening intensity. The lack of preferred orientation in the azimuthal direction indicates that the initial annealing successfully removed the original texture of the bar stock as intended. After shot peening, the plane of the surface still remains essentially isotropic, but the emergence of a perceivable polar texture clearly indicates the effect of substantial plastic deformation at the surface. Figure 35 shows that the measured magnetic signature is more or less a linear function of the shot peening intensity and that this signature gradually decreases during relaxation to essentially zero in fully recrystallized specimens. These trends are very promising for the feasibility of nondestructive monitoring of thermal relaxation in shot-peened copper specimens, but they do not provide unequivocal evidence whether the magnetic signature is caused mainly by the presence of residual stresses, the presence of cold work, or a certain combination of both. In order to establish the relative role of the competing residual stress and cold work contributions in the measured thermoelectric signature, beside measuring the thermoelectric signature at different stages of thermal relaxation, it was also monitored the decay of all relevant material properties such as residual stress, hardness, texture, and dislocation density using destructive micro-indentation and X-ray diffraction measurements. Figure 36 shows the thermal relaxation of the integrated residual stress, integrated cold work, and peak- to-peak magnetic signature in Copper C11000 for different Almen intensities. A statistical comparison of these integrated residual stress and cold work results to the magnitude of the decaying magnetic signature revealed that in shot-peened C11000 copper, on the average, approximately 64% of the thermoelectric signal is due to residual stresses and the remainder of the signal is caused by coldwork while the contribution of surface roughness is negligible [18].


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5.3. Monitoring Residual Stress Relaxation in Nickel-Base Superalloys The experimental results obtained clearly verify the feasibility of nondestructive evaluation of thermal relaxation in shot-peened C11000 copper. Although the development of a quantitative residual stress measurement method might require additional more accurate and more detailed tests, the obvious next question to be addressed is whether the thermoelectric method is applicable to other engineering materials of special importance to the aerospace industry. In particular, it is very important to develop NDE techniques for high-strength, high-temperature engine materials such as nickel-base superalloys. Therefore, as a final experiment, we conducted a series of thermoelectric measurements on shot-peened nickel-base superalloy specimens IN100 both before and after partial stress relaxation. Figure 37 shows the average peak-to-peak amplitudes of the magnetic signatures recorded on IN100 nickel-base superalloy specimens as functions of the shot peening intensity before and after partial stress release (≈ 25 °C/cm temperature gradient, vertical polarization).

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Figure 34. Plane (200) back-reflection pole figures for four C11000 copper specimens of different shot peening intensity (light areas - 50-100 % relative to average, dark areas 100-200 % relative to average)

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The most obvious difference between these IN100 results and the corresponding data on copper (see Figure 35) is that the magnetic signature does not completely vanish in the untreated specimens. This background signature is caused by the anisotropic texture and inhomogeneity of the specimen and is the subject of a separate study [15,29,46]. It should be mentioned that a similar, though somewhat weaker signature was also observed in cold rolled copper specimens, therefore the material was annealed before shot peening to eliminate this adverse feature. In this respect, the IN100 measurements are more representative of a real application when the base metal is not necessarily homogeneous or texture-free even before surface treatment. Currently, efforts are under way to establish the most effective methods of baseline compensation based on the specific features of the baseline signature [19,20]. 25 before relaxation relaxation at 235 ºC relaxation at 275 ºC relaxation at 315 °C 2nd relaxation at 315 °C 3rd relaxation at 460 °C recrystallization at 600 °C

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These experimental results illustrate the potential for a new noncontacting thermoelectric NDE technique to detect plastic deformation and the presence


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of residual stresses, which are very difficult to characterize by other, more conventional NDE methods. Beside the primary residual stress effect, the thermoelectric method is also sensitive to the secondary “material” effects of shot peening (local texture, increased dislocation density, hardening), but it is entirely insensitive to its “geometrical” by-product, i.e., the rough surface topography. The initial experimental results indicate that unequivocal separation of residual stress relaxation from the parallel decay of secondary cold work effects is not feasible at this point and further research is needed to better understand the relationship of these two contrast mechanisms, especially in the case of low-conductivity materials like titanium alloys and nickel-base superalloys [47,48]. Despite the considerable complexity of the problem, our results indicate that nondestructive evaluation of near surface material variations in surface-treated metals is feasible, though additional research is needed to further optimize this promising NDE method. It is expected that the noncontacting thermoelectric technique could be also used for the characterization of different surface-treated metals, including laser shock peened and low plasticity burnished components. This nondestructive inspection technique is aimed primarily at quantitatively assessing the degree of stress release during long- term service at elevated temperatures. This goal can be achieved by measuring the thermoelectric signature that is related to the weighted average of the compressive residual stress below the surface. This limited single parameter characterization is acceptable since the residual stress profile is primarily determined by material and manufacturing process variables and will not change significantly its shape or average depth during thermally activated stress release except for the well-known accelerated nearsurface relaxation due to the presence of cold work. Further work is needed to verify the feasibility of this technique by destructive profiling of the residual stress in specimens at different stages of thermal stress release using X-ray diffraction measurements. Although it is expected that the thermoelectric method will be found to be sensitive to secondary cold work effects as well as to primary residual stress effects, it still can be a very useful NDE tool since quantitative assessment of the level and distribution of cold work in surfacetreated metals is of primary importance from the point of view of thermomechanical stability of the beneficial residual stresses.

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6. CONCLUSION This research work was aimed at developing a new noncontacting thermoelectric method for nondestructive detection of material imperfections in metals. The experimental results indicate that spatially incoherent material noise produced by microstructural features (e.g., grains, second phases, or precipitations) exert a negligible effect on noncontacting thermoelectric inspection because of the low spatial resolution of the measurement. Therefore, the efforts were focused on the investigation of the spatially coherent background signature produced by macrostructural features (e.g., heat-affected, work-hardened, textured, or stressed regions). This spurious background signature is called material noise only because it interferes with, and often conceals, the flaw signal to be detected and ultimately determines the achievable probability of detection (POD) of a given flaw, although in certain applications it can be used as a useful thermoelectric signal to characterize the intact material itself. Many manufacturing processes produce hardened, textured, recrystallized, or otherwise modified surface layers that often sustain significant levels of residual stresses. As a result, the material properties in the affected skin region are perceivably different from those of the intact material in the interior of the specimen. We have established that the noncontacting thermoelectric method is uniquely sensitive to subtle variations in material properties and that it is essentially insensitive to geometrical features such as surface curvature and roughness. The potential applications of this method cover a very wide range from detection metallic inclusions and segregations, inhomogeneities, and tight cracks to characterization of hardening, embrittlement, fatigue, texture, and residual stresses. In this research work, we laid down the foundations of a new field of nondestructive materials characterization and made substantial advance towards all of the originally proposed research goals. We have successfully adapted the noncontacting thermoelectric method to a series of nondestructive materials characterization applications that are currently not accessible by any other known inspection method. In particular, we developed and experimentally verified a series of analytical models capable of predicting the thermoelectric signatures produced by inclusions of different material properties and geometrical features. In addition, we studied the feasibility of nondestructive detection and characterization of cracks and voids in textured polycrystalline materials, phase anomalies and anisotropic effects of the microstructure, and thermally induced residual stress relief in surface treated components. Building on the extensive and very promising results of this research work, it

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will be possible to develop numerous new NDT techniques that will find application in energy production (nuclear, oil, and gas industries, power generators, etc.), material manufacturing (aluminum, steel, titanium, nickelbase super-alloys, etc.), and in the transportation industry (aerospace, automobile, etc.).

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

Henry, EB; Stuart, CM; Tomasulo, W. Nondestructive Testing Handbook, ASNT, Columbus, 1995, 9, 363. Stuart, CM. J. Testing Eval., 1987, 15, 224. Morgner W. Mat. Eval., 1991, 9, 1081. Hu, J; Nagy, P. B. Appl. Phys. Lett. 1998, 73,467. Nagy, PB; Hu, J. Rev. Prog. Quant. NDE, Plenum, New York, 1998, 17, 1573. Carreon, H; Medina, A. Nondestruct. Test. Eval. , 2007, 22/4, 299. Carreon, H. NDT. & E International, 2006, 39/6,433. Hinken, JH; Tavrin Y. Rev. Prog. Quant. NDE, AIP, Melville, 2000, 19, 2085. Maslov, K; Kinra, VK. Mat. Eval., 2001, 59, 1081. Tavrin, Y; Krivoy, GS; Hinken, JH; Kallmeyer, JP. Rev. Prog. Quant. NDE, AIP, Melville, 2001, 20,1710. Carreon, H; Nagy, PB. Proc. 7th NDE Topical Conf, ASME, New York, 2001, 20, 209. Nagy, PB; Nayfeh, AH. J. Appl. Phys., 2000, 87,7481. Carreon, H; Nagy, PB; Nayfeh, AH. J. Appl. Phys., 2000, 88, 6495. Nayfeh, AH; Faidi, WI. Eur. Phys. J. Appl. Phys., 2002, 19, 153. Nayfeh, AH; Carreon, H; Nagy, PB. J. Appl. Phys., 2002, 91, 225. Nayfeh, AH; Faidi, WI; Jaghoub, MI. Eur. Phys. J. Appl. Phys., 2003, 22, 103. Carreon, H; Lakshminarayan, B; Faidi, WI; Nayfeh, AH; Nagy, PB. NDT. & E International, 2003, 36, 339. Carreon, H; Nagy, PB; Blodgett, MP. Res. Nondestr. Eval., 2002, 14,59. Carreon, H. J. Alloys & Compounds, 2007, 427/1-2, 183. Carreon, H. Wear, 2008, 465/1-2,255. Nye, JF. Physical Properties of Crystals, Their Representation by Tensors and Matrices, Clarendon Press, Oxford, 1985, 96.


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[22] Nayfeh, AH; Carreon, H; Nagy, PB. Rev. Prog. Quant. NDE, AIP, Melville, 2001, 20, 1686. [23] Carreon, H. NDT. & E International, 2006, 39/1:22. [24] Dillard, AB; Clark, KR; Denda, T; Hendrix, BC; Tien, JK. Proc. Conf. on Electron Beam Melting and Refining - State of the Art, 1992, 277. [25] Gigliotti, MF; Perocchi, LC; Nieters, EJ; Gilmore, RS. Rev. Prog. Quant. NDE, AIP, Melville, 1995, 14, 2089. [26] Bartos, JL; Copley, DC; Gilmore, RS; Howard, PJ. Titanium: Science and Technology, 1996, 2, 1513. [27] Leverant GR; Littlefield, DL; McClung, RC; Millwater, HR; Wu, JY; ASME International Gas Turbine & Aeroengine Congress, 1997, 97-GT22. [28] McKeighan, PC; Nicholls, AE; Perocchi, LC; McClung, RC. Nontraditional Methods of Sensing Stress, Strain, and Damage in Materials and Structures, ASTM, West Conshohocken, PA, 2000, 2, 1323. [29] Carreon, H; Lakshminarayan, B; Nagy, PB. Rev. Prog. Quant. NDE, AIP, Melville, 2004, 23, 445. [30] Blodgett, M; Eylon, D. J. Nondestructive Eval., 2001, 20, 1. [31] Hauk, V. Structural and Residual Stress Analysis by Nondestructive Methods, Elsevier, Amsterdam, 1997, 56. [32] Prevéy, PS. IITT International, Gournay-Sur-Marne, France 1990, 81. [33] Eguiluz, AG; Maradudin, AA. Phys. Rev., 1983, 28,728. [34] Krylov, VV; Smirnova, ZA. Sov. Phys. Acoust., 1990, 36, 583. [35] Thompson, RB; Lu, WY; Clark, AV. Jr. Handbook of Measurement of Residual Stress Fairmont Press, Lilbum, 1996, 149. [36] Lavrentyev, AI; Stucky, PA; Veronesi, WA. Rev. Prog. Quant. NDE, AIP, Melville, 2000, 19, 1621. [37] Glorieux, C; Gao, W; Kruger, SE; Rostyne, KV; Lauriks, W; Thoen, J. J. Appl. Phys., 2000, 88,4394. [38] Schoenig, FC; Soules, JA; Chang, H; DiCillo, JJ. Mat. Eval., 1995, 53, 22. [39] Chang, H; Schoenig, FC; Soules, JA. Mat. Eval., 1999, 57, 1257. [40] Blodgett, MP; Nagy, PB. J. Nondestr. Eval., 2004, 23, 107. [41] Yu, F; Nagy, PB. J. Appl. Phys., 2004, 96, 1257. [42] Carreon, H. Nondestruct. Test. Eval., 2009, in press. [43] Prevéy, PS; Hornbach, D; Mason, P. Proc. 17th Heat Treating Society Conference, Eds. D.L. Milam et.al; ASM, Materials Park, OH, 1998, 3. [44] Carter, GF; Paul, DE. Materials Science & Engineering, ASM,

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International, 1991, 50. [45] Carreon, H; Nagy, PB. NDE & Char. Eng. Mat. Reliability & Durability Predict. AMD-NDE ASME, New York, 2000, 18,31. [46] Carreon, H; Nayfeh, AH; Nagy, PB. Rev. Prog. Quant. NDE, AIP, Melville, 2002, 21,1455. [47] Lakshminarayan, B; Carreon, H; Nagy, PB. Rev. Prog. Quant. NDE, AIP, Melville, 2003, 22, 1523. [48] Carreon, H. Proc. 136th TMS Annual Meeting, Characterization of Minerals, Metals, and Materials, TMS, 2007, 25.

In: Nondestructive Testing: Methods, Analyses… ISBN: 978-1-60876-157-9 Editor: Earl N. Mallory, pp. 63-94 © 2010 Nova Science Publishers, Inc.

Chapter 2

EXPERIMENTAL AND NUMERICAL METHOD FOR NONDESTRUCTIVE ULTRASONIC DEFECT DETECTION D. Cerniglia and A. Pantano* Dipartimento di Meccanica, University of Palermo, viale delle Scienze, 90128, Palermo, Italy

ABSTRACT Ultrasonic methods are well known as powerful and reliable tools for defect detection. Conventional ultrasonic techniques rely generally on piezoelectric transducers where transmission of energy to the material is achieved with contact. In the last decades, focus and interest have been directed to non-contact sensors and methods, showing many advantages over contact techniques where inspection depends on contact conditions (pressure, coupling medium, contact area). The growing interest is also due to the further development of air-coupled probes, thanks to new materials for acoustic devices and manufacturing technologies. The use of the laser as a tool for ultrasonic defect detection is also an emerging approach in the industry and holds substantial promise as inspection is remote, feasible in a hostile environment, can be automated and also performed with the test object in motion. Moreover, thanks to its ability *

Corresponding author: Fax: +39-091484334, E-mail: [email protected], apantano@dima. unipa.it,

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D. Cerniglia and A. Pantano to produce frequencies in the MHz range, laser-generated ultrasound enables fine spatial resolution of defects. The non-contact hybrid ultrasonic method described here is of interest for many applications, requiring periodic in-service inspection or after manufacturing. Despite the potential impact of laser-generated ultrasound in many areas of industry, robust tools for studying the phenomenon are lacking and thus limit the design and optimization of non-destructive testing and evaluation techniques. Ultrasonic waves propagate through the structure interacting with defects, corners and curved surfaces, causing reflection and mode conversion. Moreover, interference between waves can produce a more complex pattern. This makes the laser-generated ultrasound propagation in complex structures an intricate phenomenon extremely hard to analyze. Only simple geometries can be studied analytically. Numerical techniques found in literature have proved to be limited in their applicability by the frequencies in the MHz range and very short wavelengths. The acoustic field in complex structures should be well understood for each application to optimize sensitivity toward a particular type of defect. A specific numerical method is presented in this chapter to efficiently and accurately solve ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies and through air. Tests simulated with numerical analysis are replicated experimentally for validation. The numerical technique provides a valuable tool for studying the laser-generated ultrasound propagation and for designing and optimizing non-destructive testing and evaluation techniques. The information that can be acquired can be very valuable for choosing the right setup and configuration when performing non-contact hybrid ultrasonic inspection.

1. INTRODUCTION In the last two decades, there has been a growing use of laser-generated ultrasound as a tool for defect detection, combined with air-coupled receivers [1-2]. One of the main advantages of this method is the capability of using ultrasound frequencies in the MHz range, enabling fine spatial resolution for the detection of defects. Another fundamental benefit in using a laser to generate acoustic waves consists in making possible remote inspection, which does not depend on contact conditions (pressure, coupling medium, contact area) and allows development of automated systems and operation in a hostile environment, by guiding the laser beam through proper delivery optics. Ultrasound waves propagate through the structure interacting with defects,

Experimental and Numerical Method for Nondestructive Ultrasonic… 65 corners and curved surfaces, causing reflection and mode conversion. This makes the laser-generated ultrasound propagation in complex structures an intricate phenomenon extremely hard to analyze. Ultrasound wave propagation can be studied analytically by solving the governing equations of motion together with the related boundary condition. One of the initial works on the generation of thermoelastic waves in a body induced by a surface heat source was produced by White [3] in the early 1960s. His effort can be considered the first-generation model which failed to predict the shear and surface waves the laser pulse generates. Better insight was later obtained in the second generation point source models initiated by Scruby et al. [4] who identified that in the thermoelastic regime the laser heated region acts as a surface center of expansion. Later, Rose [5] gave a more rigorous mathematical basis. However, point source theory had some limits; in particular it didn’t consider thermal diffusion. McDonald [6] developed the third generation models, which take into account thermal diffusion and finite shape of the laser pulse, leading to an excellent agreement between theory and experiment. However, analytical models become intractable for complicated geometries or for a specimen with defects. Another approach to obtaining a solution for problems involving laser-generated ultrasound waves is a numerical method. There are basically two numerical methods which can be used for this problem: the finite element (FE) method or the boundary element (BE) method. The BE method has the advantage that just the surface of the specimen needs to be discretized, but it has some limitations that made the scientific community choose the FE method as the preferred tool in most studies on ultrasound wave propagation [7-19]. Finite element modeling allows studying of the interaction of waves with defects but requires respecting strict rules for spatial and temporal discretization, as a result when high frequencies and great dimensions are involved the number of freedom degrees can become rapidly huge, making it often impossible to solve real wave propagation problems. Laser-generated ultrasound has frequencies in the MHz range requiring extremely small time increments between the solutions. Moreover these high frequencies have very short wavelengths, and since for a reasonable spatial resolution of the propagating waves the size of the finite elements must be at least 1/20 of the shortest wavelength to be analyzed [e.g., 8], the elements must be very small. This explains why we couldn’t find in the vast literature on the subject any work involving ultrasound frequencies in the MHz range traveling in bodies larger than a few centimeters in the three directions. This can lead the scientific and industrial

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community to think that this class of problems cannot be studied with modern workstations by use of the FE method. The objective of this research is to prove that using an explicit integration rule together with diagonal (“lumped”) element mass matrices, instead of the almost universally adopted implicit integration rule, to integrate the equations of motion in a dynamic analysis, it is possible to efficiently and accurately solve ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies. Since laser generation must be coupled to air-receivers for a truly remote inspection system, the propagation of acoustic waves through air is also modeled to simulate the complete experimental set-up.

2. LASER-BASED ULTRASOUND Ultrasonic methods represent a powerful and reliable tool for nondestructive inspection (NDI) of materials and components. Traditional ultrasonic techniques rely generally on piezoelectric transducers where transmission of energy to the material is achieved with direct contact by a liquid or grease couplant, with full immersion of transducer and sample in a water tank, or flooding by a water-jet the gap between transducer and material. The main drawback of conventional methods is the need to maintain consistent coupling; thus rapid and automatic scanning of contact transducers is not easy, especially over awkward geometries, and inspection of large areas is time consuming. In the last decades, focus and interest have been directed to non-contact ultrasonic sensors and methods, that are couplant free hence showing many advantages over contact techniques where inspection depends on contact conditions (pressure, coupling medium, contact area). The growing interest is also due to the further development of air-coupled probes, thanks to new materials for acoustic devices and manufacturing technologies [20-21]. The use of lasers as a tool for defect detection is also an emerging approach in the industry and holds substantial promise as the laser is a remote source, usable in severe conditions and on parts with complex geometries or in motion; inspection can be automated and performed in a vibratory environment. Although lasers are less compact and portable than ultrasonic transducers, the beam can be easily steered by means of mirrors or optical fibers into surfaces difficult to reach and where access is limited.

Experimental and Numerical Method for Nondestructive Ultrasonic… 67 The use of a laser to generate acoustic waves in solids, the effects produced by high-power beams at opaque surfaces and mechanisms involved in laser generation of acoustic waves have been well characterized [22-23]. The beam of coherent radiation emitted by the laser, incident on the surface, is partially absorbed and some reflected or scattered by the surface; generation of ultrasound happens according to two different mechanisms, depending upon the power density. At low power density, the radiation absorbed from the laser is converted into heat, localized to a layer of few microns in the surface. The resulting temperature rise produces transient thermoelastic expansions, consequent stresses and then elastic wave perturbation. Other physical processes are involved with negligible effects if compared to the dominant mechanism of thermoelastic expansion. At higher optical power density, depending on the material, the surface temperature of the material rapidly rises to its vaporization temperature and ultrasound generation occurs in the ablative regime. In this case, few micrometers of material are ablated at the surface. A problem associated with the ablative regime is the visible damage caused at the surface. As a result, a thermoelastic regime is preferred for NDI applications. The limiting factor usually is the threshold for damage in the irradiated material. When required, generation efficiency (i.e., the amplitude of the acoustic signals) can be enhanced by constraining the surface or by applying a surface coating [1]. Roughness, impurities, and oxide layers on the surface contribute to a supplementary absorption of laser radiation. The characteristics of the laser that affect efficiency in generating acoustic waves are wavelength, pulse energy, beam profile and pulse duration. Optical to acoustic energy conversion efficiency in the material depends on its optical, thermal and elastic properties. For a given pulse energy, ultrasonic generation is more efficient if the energy is compressed in a very short laser pulse. Pulse length controls also ultrasonic frequency range, in an inversely proportional manner; if the pulse duration is short (order of nanosecond) the expansion is in the frequency of ultrasound and frequency spectrum is broadband. Typically, for pulses of tens of nanoseconds, most of the spectrum energy is delivered at frequencies below 50 MHz. Formed beam sources, created by modifying the shape of the illuminated area, can be used to control wave-fronts and frequency of laser-generated acoustic signals in a way to optimize its directivity, to enhance interaction and sensitivity to a specific type of defect [24-25]. Spatial frequency of sources allows us to control the wavelength λ of the wave, that is expected to have a center frequency at f=c/λ (c is the velocity of the wave). Thus, for instance,

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since surface waves propagate at the solid surface with their motion confined to a region whose depth is about the wavelength, by controlling λ is possible to control the extent of the region being inspected. Formed laser light pulses can be created through the use of lenticular arrays, shadow masks, optical diffraction gratings, multiple lasers, etc. Wavefront and frequency should be chosen to match the frequency sensing capabilities of the receiver. While beam directivity is fixed for transducers, directivity of ultrasonic beam in the material depends on the regime (conical lobe for thermoelastic, omnidirectional for ablation and constrained) [22]. More directional ultrasonic paths can be obtained with line source or multi-line source; thus total laser energy is directed toward the generation of a particular mode of ultrasound, increasing efficiency of generation. Figures 1 and 2 show the acoustic map (normalized amplitude) of guided waves, starting at 78 mm away from the source, where total energy is focused to one line source or to multi-lines (six lines at 1 mm x 14 mm). Directionality of the acoustic field is very strong in both cases but energy is better used with the multi-line source, enhancing the amplitude of the selected mode. 118

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Angle of incidence of the laser beam on the material does not influence propagation of ultrasound, if the shape of the irradiated area is equal. Thus beam can be directed to the surface with high angles off axis, allowing inspection of parts where access is limited [1]. Laser-generated ultrasound propagation in complex structures is an intricate phenomenon hard to analyze. The complex acoustic field comes from the full complement of ultrasonic waves generated by the laser (longitudinal, shear, head and Rayleigh modes and, in thin plate, guided waves). Interference between waves, resulting from a certain number of coherent sources (i.e. diffracted waves), can produce a more complex pattern. Acoustic field and its directivity in the structure, dynamic of wave propagation and interference with echoes reflected at the boundaries should be well understood for each application to optimize sensitivity toward a particular type of defect. It is also important to discern wave interactions with microstructure (grain boundaries, inclusions and discontinuities) and defects to improve repeatability and probability of detection. A thorough understanding of directivity of lasergenerated ultrasound in complex structures for the different regimes (thermoelastic, ablative, constrained) or formed beam sources (point, line, multi-line) is essential for mapping the acoustic field and to visualize perturbations in the wave-propagation.

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For a completely remote inspection system, laser-generation unit should be coupled to laser receivers. However, since laser interferometers as receivers have lower sensitivity than conventional transducers, particularly on rough or poorly reflecting surfaces, usually alternatives are searched. Air-coupled piezoelectric transducers represent a good choice, when NDI testing is performed below 5 MHz, for their capability to receive efficiently ultrasound through air. If higher frequencies are required (e.g. for material characterization), then the use of laser receivers is forced. The main limit associated with air-coupling is the large acoustic-impedance mismatch at the solid/air interface that causes reflection of most of the incident energy (close to 100%). Besides, attenuation of ultrasound in air is significant at frequencies above 1 MHz, as attenuation depends mainly on frequency of the wave and propagating medium. It increases with the square of frequency and the square root of temperature, is proportional to the atmospheric pressure and humidity is an additional factor. Literature provides an approximate value for the coefficient of attenuation when medium is air at standard condition (293.15 K and 1.01325·105 Pa), α= 1.83 ·10-11f 2. Even though high frequency components attenuate severely in air, sensitivity to the defects of interest can be still high. The fairly broadband signal produced by a short pulse laser is thus narrowed by both the frequency response of the air-coupled transducer and the air-gap acting like a low pass filter. Figure 3 shows the scheme of the laser/air-sensor system that includes, in its simplest configuration, optics (mirrors and focussing lenses) for the laser beam, signal receiver (with amplifiers and filters) and acquisition boards.

3. MODELING PROCEDURES 3.1. Explicit Dynamic Analysis for Wave Propagation The literature on FE modeling of ultrasound wave propagation problems shows an extensive use of the implicit integration rule to integrate the equations of motion. However, while there are several classes of problems where the implicit integration is more computationally efficient than explicit, there are technical reasons that make explicit dynamic analysis far superior in simulating the ultrasound wave propagation problems.

Experimental and Numerical Method for Nondestructive Ultrasonic… 71

Figure 3. Scheme of the laser/air-coupling setup.

Wave propagation problems are characterized by high frequencies. The time frames of interest are short; usually what needs to be observed is the time that the wave takes to go through the entire structure. Thus during the transient, the stress wave propagation and the behavior at the stress wave front can be analyzed. In an explicit dynamic analysis the size of the time step is small, because the central-difference operator is only conditionally stable and requires that within one time increment the information does not propagate across more than one element [e.g., 26]. For wave propagation this means that time step must be smaller than the time required for a stress wave to cross the smallest element in the model, therefore if the elements of the mesh are extremely small or the velocity of the wave in the material is high the time steps need to be very short. Here this severe requirement is not a serious inconvenience because for wave propagation studies a small time increment is always necessary for accuracy. The main advantage of the implicit operator over the explicit one is that being unconditionally stable there is no a limit on the size of the time increment due to stability reasons, however the short time increments that wave propagation problems requires for accuracy makes this benefit not useful making the implicit dynamic analysis much more expensive than the explicit one. The implicit method is economically attractive only when the time step can be much larger than what would be used in an explicit method, but it is not case of the wave propagation studies. There are other factors in favor of the explicit integration. In explicit methods displacements are calculated in terms of information consisting of displacements and time derivatives of displacements known at the beginning of an increment. Consequently the global mass and stiffness matrices need not be formed and inverted, with great computational saving with respect to

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implicit dynamic analysis. Also because the element stiffness matrix doesn’t need to be formed or stored, explicit methods can deal with large 3D models requiring much less disk space and memory than the implicit for the same simulation. Because of its formulation the explicit method is also much easier to implement than the implicit one, and can efficiently deal with material nonlinearities while implicit method in presence of severe nonlinearities may have convergence difficulties [e.g., 26]. In summary an explicit dynamic analysis is far more computationally efficient for the analysis of large models with relatively short dynamic response times, as it is the case for ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies. In addition the computational efficiency of the explicit procedure with respect to the implicit one for wave propagation problems can be further enhanced by use of diagonal element mass matrices. Simulations of laser generated ultrasound waves require a coupled thermal-stress analysis. Here we used a fully coupled thermal-stress analysis where the mechanical solution response is obtained using an explicit centraldifference integration rule, and the heat transfer equations are integrated using an explicit forward-difference time integration rule. The mechanical solution response is based upon the implementation of an explicit integration rule together with the use of diagonal element mass matrices. The equation of motion at a specific instant of time is:

[ M ]{d&&}n + [C ]{d&}n + [ K ]{d }n = {F }n

(1)

where the subscript n marks the time increment number, [ M ] is the mass matrix, [C ] is the damping matrix, [ K ] is the stiffness matrix, {F } is the external load vector, and

{d }

is the displacements vector. In the explicit

method:

{d }n+1 = f ({d }n ,{d&}n ,{d&&}n ,{d }n−1 ,...)

(2)

The equations of motion for the body are integrated using the explicit central difference integration rule:

Experimental and Numerical Method for Nondestructive Ultrasonic… 73 1 ({d }n+1 − {d }n −1 ) 2Δt

(3)

1 ({d }n+1 − 2 {d }n + {d }n −1 ) Δt 2

(4)

{d&}

n

{d&&}

n

{}

=

=

{}

where d& is the velocity and d&& is the acceleration. Substituting equations (3) and (4) in (1): 1 ⎤ 1 1 ⎡ 1 (5) ⎢⎣ Δt 2 M + 2Δt C ⎥⎦ {d }n +1 = { F }n − [ K ]{d }n + Δt 2 [ M ] ( 2 {d }n − {d }n −1 ) + 2Δt [C ]{d }n −1

If [ M ] and [C ] are diagonal then the equations become uncoupled and the displacements at time n+1 can be obtained without solving simultaneous equations. The “lumped” mass matrix formulation determines a mass matrix that is diagonal, satisfying the requirement for uncoupling the equations. The “lumped” mass matrix is obtained by placing particle masses at the node of an element such that the sum of the masses gives the total element mass. The use of diagonal element mass matrices is extremely important for the computational efficiency of the explicit procedure. The internal force vector

[ K ]{d }n

can be computed by summation of elements contributions such that

a global stiffness matrix need not be formed. Moreover, in addition to uncoupling the equations, lumped mass matrices determine larger stable time increments with respect to consistent mass matrices. The heat transfer equations are integrated using the explicit forwarddifference time integration rule:

{T }n+1 = {T }n + Δtn+1 {T& }n

(6)

where T is the temperature. If a lumped capacitance matrix is used the integration is explicit. Since both the mechanical solution response and the heat transfer equations are explicit, the heat transfer and mechanical solutions are obtained simultaneously by an explicit coupling. Therefore, no iterations or tangent stiffness matrices are required.

74


In explicit dynamics procedures temporal and spatial resolution of the FE simulations are of fundamental importance for the stability and the accuracy of the solution. An approximation to the stability limit is often written as the smallest transit time of a dilatational wave across any of the elements in the mesh Δt ≈

Lmin CL

(7)

where Lmin is the smallest element dimension in the mesh and CL is the dilatational wave speed. Laser-generated ultrasound wave requires very small time step in order to accurately resolve their high frequencies components. The works of several researches [e.g., 8, 16] have established that accurate solution can be obtained if: Δt =

1 20 f max

(8)

where f max is the highest frequency of interest. A reasonable spatial resolution of the propagating waves can be obtained [i.e. 8, 16] when the size of the finite elements must be at least 1/10 of the shortest wavelength to be analyzed. le =

where le is the element length and

λmin 10

(9)

λmin is the shortest wavelength of interest.

3.2. Propagation of Sound Waves through Air As it has been underlined in the introduction a fundamental benefit in using laser to generate acoustic waves consists in making possible remote inspection, which does not depend on contact conditions and allows development of automated systems and dynamic operations. However for an

Experimental and Numerical Method for Nondestructive Ultrasonic… 75 inspection that operates fully in remote also the receiver must work without needing contact. If a non-contact transducer is used in reception then in order to simulate the complete experimental setup it is required to model the propagation of the acoustic waves through air. The degree of freedom of the finite elements used to model air is the acoustic pressure, which is the local pressure deviation from the ambient pressure caused by a sound wave. The FE modeling of ultrasound wave propagation through air requires a significant computational effort because the speed of sound in air is 343 m/s, which is a value much lower than the speed of the longitudinal wave in solids, for example for steel it takes the value of about 5990 m/s. Since the frequency of the ultrasound waves generated in a solid remain unchanged when they are transmitted from the vibrating surface of the solid to the air, the wavelength must be significantly reduced. This follows from the relation: Css = λair f

(10)

where Css is the speed of sound in air 343 m/s,

λair is the wavelength in air,

and f is the frequency that for laser-generated ultrasound wave is in the MHz range. For example if a wave has been generated in a solid made of steel, the ratio between the wavelength in steel and the wavelength air will be: CL λsteel f = Css λair f

⇒

λsteel CL 5990 = = = 17.463 λair Css 343

⇒ λair =

λsteel 17.463

(11)

According to equation (9), a reasonable spatial resolution of the waves propagating through air will now require a size of the finite elements that is about 17.5 times smaller than the size of the finite elements used to model steel. In case of wave with frequency f = 1MHz equation (9) would require

leair =

air

where le

analyzed.

λair 10

=

343 m Css s = 10 ⋅ f 10 ⋅ 1 ⋅ 106 1

(

s

)

= 34.3 ⋅ 10−6 m = 0.0343mm

(12)

is the required element length for the FE model of the portion of air

76


This result clearly states that it is very important to limit the amount of air to be considered in the FE analyses; otherwise the computational cost would often exceed the resources available. A way to solve the problem is to implement non-reflecting boundary conditions, which are used for exterior problems such as a structure vibrating in an acoustic medium of infinite extent. A non-reflecting boundary condition allows all outgoing waves to exit the domain at the boundary where they have been imposed without reflection. Thus, unless in the proximity of the transducer there are other solids where the acoustic waves could be reflected, non-reflecting boundary conditions allow modeling only the portion of air between the non-contact transducer and the solid under testing. The non-reflecting boundary conditions that have been implemented in our numerical model are able to avoid any reflections for every possible angle of incidence.

4. RESULTS The use of laser-generated ultrasound as tool for defect detection has been recently extended to rails. In this study, the attention was focused on assessing the capability of the presented numerical tool in simulating laser-generated ultrasound waves propagation with frequencies in the MHz range. Both numerical and experimental analyses were conducted on a 136 lb AREMA rail. Several cases have been considered; here we present results reproducing NDE testing of the rail head, with and without defects, and of the rail web. First the numerical approach is validated by comparison with analytical results for transient guided waves propagating in a circular annulus.

4.1. Comparison with Analytical Solution - Circular Annulus In order to validate the proposed approach, the accuracy of the numerical simulations is tested against an analytical solution [8, 27]. Liu and Qu developed a general method of solution for obtaining the transient guided waves propagating in the circumferential direction of a circular annulus subjected to time-dependent surface traction [27]. They assumed that the inner surface of the annulus is traction free and the outer surface of the annulus is subjected to a time-dependent transient excitation. The guided circumferential waves induced by this transient excitation were studied to understand the

Experimental and Numerical Method for Nondestructive Ultrasonic… 77 propagation characteristics of various wave modes. The method of eigenfunction expansion was used to solve the transient wave propagation problem. The time-dependent response of the annulus is then obtained by superimposing all eigenfunctions over all possible frequencies. Moser et al. [8] used the approach developed by Liu and Qu [27] to verify their results computed using a finite element approach based on the implicit integration. The ring structure under consideration is shown in [8, Figure 5], and its material properties (steel) are summarized in [8, Table 2] as Material II. The ring is loaded on its outer surface with a point force, f(t), that acts perpendicular to the surface. The time function of the input load, f(t), and its frequency content are given in [8, Figure 6]. The radial displacements, ur, from the analytical solution computed at a 90° angle between the source and the receiver are plotted in Figure 4 and compared with the FE results from the presented approach. The agreement in the solutions is extremely good, proving that the explicit method can reach the same level of accuracy as the implicit one in modeling wave propagation problems.

4.2. Testing of the Rail Head without Defects The first experimental setup is shown in Figures 5 and 6. The laser beam was focused to a spot on the head of the 136 lb AREMA rail that was 1 mm in diameter and located in the middle of the side of the head. The laser had a maximum energy of 0.75 J and the pulse duration was 6⋅10-9 seconds. The laser spot area was heated by the optical energy which generated the ultrasound wave via the thermoelastic coupling effect. In the numerical simulation, the rail model was meshed to match the experimental configuration and measure the displacements and stresses generated by the ultrasound wave. In the experiment a piezoelectric transducer, with a sensitive plate diameter of 12.7mm and a central frequency of 1 MHz, was used to record a voltage signal which is proportional to the surface displacements on the top of the head, Figure 6 a). The 136 lb AREMA rail is made of alloy steel where small percentage of the following elements are added to carbon Mn, P, S, and Si; other elements that can be added to meet the mechanical property requirements are Ni, Cr, Mo, V [28]. In the numerical study the following material properties are assumed for the rail: a density of 7800 Kg/m3, a Young’s modulus of 207⋅109 Pa, a Poisson’s ratio of 0.3, a thermal expansion

78


Radial Displacements [mm]

coefficient of 1.3⋅10-5 K-1, a specific heat capacity of 490 J⋅Kg-1⋅K-1, and a thermal conductivity of 46 W⋅m-1⋅K-1. Based on the numerical formulation described in the previous paragraphs, the thermoelastic generated ultrasound waves are calculated in a finite element model that reproduces in detail the geometry of the rail, Figure 6 b). Figure 7 shows a map of the displacements, as computed by the FE tool, which do not exceed the value of 1⋅10-4 mm, allowing for a clear detection of the wave front after 5⋅10-6 seconds from the laser pulse. The limit in the displacements map is needed because the displacements are very high at the location where the laser beam hits the rail, thus without a limit all the grey scale variations would be located at the laser spot and all the other areas would fall into the last displacement range of the map which is represented by the lighter grey. 3.5E-08

Analytical Solution

3.0E-08

Finite Element Solution 2.5E-08 2.0E-08 1.5E-08 1.0E-08 5.0E-09 0.0E+00 -5.0E-09 -1.0E-08 -1.5E-08 0.E+00

Time [s] 1.E-05

2.E-05

3.E-05

4.E-05

5.E-05

Figure 4. Comparison of the analytical solution and the FE solution.

Figure 5. Photo of the experimental setup for the study of the rail head.

6.E-05


(a)

(b)

Figure 6. (a) Schematic of the experimental setup and b) finite element model of the rail head without defects.

Figure 7. Displacements map from the FE approach. The wave front location is pointed. Black regions have displacements exceeding the value of 1⋅10-4 mm.

In Figure 8 a) the graph shows the output of the piezoelectric transducer located at the top of the rail head as function of time. Figure 8 b) presents the surface normal displacements measured by the numerical approach at the same location. Examining the arrival of the different waves shown in Figures 8 a) and b), we can distinguish that the first wave front corresponds to the theoretical arrival time of the longitudinal wave, C1, propagating directly from the source to the receiver. The distance from the laser spot to the receiver is

80


about 43 mm; the theoretical arrival time, which corresponds to both the experimental and the numerical value, is t1=43/5.99=7.2 μs. The theoretical arrival time of the shear wave that travels directly from the source to the receiver, C2, can be computed to be t2=43/3.2=13.4 μs, knowing that the shear wave velocity in steel is 3.2 mm⋅μs-1. Finally the Rayleigh surface wave, CR, traveling along the surface of the rail a distance of about 53.5 mm at a speed 2.97 mm⋅μs-1 arrives to the receiver after about 18 μs. The propagation of the Rayleigh surface wave can be clearly seen in the numerical simulation; here it is not shown for brevity. Experiments and simulations match rather well, proving the efficiency of the method in simulating laser generated ultrasound wave propagation with frequencies in the MHz range.

4.3. Testing of the Rail Web The second experimental setup is shown in Figure 9 a). The laser beam was focused to a spot on the web of the 136 lb AREMA rail whose center is located 141mm under the top of the rail. The laser beam did hit the web of the rail with an angle of 50° with respect to the vertical, determining an elliptical laser spot area whose minor axis is 1 mm. The laser had a maximum energy of 0.75 J and the pulse duration was 6⋅10-9 seconds. In the numerical simulation, the rail model was meshed to match the experimental configuration and measure the displacements and stresses generated by the ultrasound wave. In the experiment a piezoelectric transducer, with a sensitive plate diameter of 12.7 mm and a central frequency of 1 MHz, was used to record a voltage signal which is proportional to the surface displacements on the top of the head. The physical properties of the 136 lb AREMA rail have been reported in the previous paragraph. Figure 10 shows a map of the displacements, as computed by the FE tool, which do not exceed the value of 1⋅10-6 mm, allowing for a clear detection of the wave front after 20⋅10-6 seconds from the laser pulse. The limit in the displacements map is needed because the displacements are very high at the location where the laser beam hits the rail, thus without a limit all the grey scale variations would be located at the laser spot and all the other areas would fall into the last displacement range of the map which is represented by the lighter grey.


(a)

(b) Figure 8. (a) Voltage, which is proportional to the surface displacements, as function of time recorded by the transducer located at the top of the rail head; b) Surface normal displacements measured by the numerical approach at the same location.

82


(a)

(b)

Figure 9. (a) Schematic of the experimental setup and b) finite element model for the study of the rail web.

Figure 10. Displacements map at the central section of the rail from the FE approach for the study of the web. The wave front location is pointed. Black regions have displacements exceeding the value of 1⋅10-6 mm.


(a)

(b) Figure 11. Study of the rail web: a) Voltage, which is proportional to the surface displacements, as function of time recorded by the piezoelectric transducer located at the top of the rail head; b) Surface normal displacements measured by the numerical approach at the same location.

In Figure 11 a) the graph shows the output of the piezoelectric transducer located at the top of the rail head as function of time. The transducer output is voltage which is proportional to the surface displacements. Figure 11 b) presents the surface normal displacements measured by the numerical approach at the same location. Examining the arrival of the different waves shown in figure 11 a) and b), we can distinguish that the first wave front

84


corresponds to the theoretical arrival time of a longitudinal wave, C1, propagating directly from the source to the receiver. The distance from the laser spot to the receiver is about 146 mm and the longitudinal wave speed for the steel is 5.99 mm⋅μs-1; the theoretical arrival time, which corresponds to both the experimental and the numerical value, is t1=146/5.99=24.4 μs. The following three arrivals, indicated as W1, W2 and W3 in Figure 11, are originated from longitudinal waves that are reflected a few times from the sides of the web before arriving at the top of the rail head where the receiver is located. Results show a very good agreement between experiments and numerical solutions.

4.4. Testing of the Rail Head with Defect The third experiment setup is shown in Figure 12 a). Also in this case the laser beam was focused to a spot on the head of the 136 lb AREMA rail, but in this case the rail head had a defect of the type known as vertical split head (VSH), which is shown in Figure 13. The vertical split head defect grows vertically through the rail head. The defect runs through the entire longitudinal length of the rail. The laser beam was 1 mm in diameter and located in the middle of the side of the head. The laser had a maximum energy of 0.75 J and the pulse duration was 6⋅10-9 seconds. In the numerical simulation the rail model tries to match the experimental configuration, but the geometry of the vertical split head defect is not clearly known. Thus in the FE model a defect has been introduced which approximately reproduce the defect visible on the final section of the rail, which is however far from the location where the laser beam hits the rail. Not knowing the exact defect geometry and location we do not expect a perfect match among experimental and the numerical results. Here we study how the presence of the defect in the FE model can affect the ultrasound waves that arrive to the point where the transducer is located. The results should qualitatively match the experiment where it was found that the presence of the defect visibly attenuates the initial signal. In Figure 14 results for the rail head with and without defect are compared. In the rail without defect the first wave arrives after about 11.4 μs and the displacements values in the time frame 12 μs to 15 μs are in the 10-6 mm range. In the rail with defect the first wave arrives after about 12.5 μs and the displacements values in the time frame 12.5 μs to 15 μs are in the 10-7 mm range, much attenuated with respect to the rail

Experimental and Numerical Method for Nondestructive Ultrasonic… 85 without defect. This result is qualitatively in good agreement with the experiments.

(a)

(b) Figure 12. (a) Schematic of the experimental setup and b) finite element model of the rail head with defect.

86


Figure 13. Photo of a rail head defect of the type known as vertical split head (VSH).

Figure 14. Surface normal displacements measured by the numerical approach on the opposite side of the rail head with respect to laser spot. Results for the rail head with and without defect are compared.


Figure 15. Displacement maps at four different times from the FE model for the study of the head with defect. Black regions have displacements exceeding the value of 1⋅109 mm in order to clearly identify the wave front.

Figure 15 shows the displacements maps of the head with defect at four different times, from 2 μs to 8 μs, computed by the numerical tool. Black regions have displacements exceeding the value of 1⋅10-9 mm in order to clearly identify the wave front. The waves cannot move through the defect, so when the wave front reaches the discontinuity in the material propagates around it. This causes a delay in the arrival time and an attenuation of the initial displacements field at the opposite side of the rail head where the displacements are detected. In Figure 16 the limit in the displacements map has been raised to 1⋅10-5 mm in order to visualize the main features of the wave propagation in the rail head with defect. Numerical results are shown for the rail central cross section, where the laser beam hits the rail, in a 2D view and in a 3D view, where half of the FE model has been removed so that the central section could be exposed. The displacement maps shown in Figure 16 have been computed after 18µs from the laser shot; at this time the Rayleigh surface wave, which is not affected by the presence of the defect, have traveled along the surface of the rail the distance of about 53.5 mm from the laser spot to the top of the head at a speed of 2.97 mm/μs. That's why we find the Rayleigh surface wave in the location indicated in Figure 16.

88


Figure 16. Displacement maps at time=18µs from the FE model for the rail cross section. a) 2D, b) 3D image where half of the FE model has been removed. Black regions have displacements exceeding the value of 1⋅10-5 mm.

4.5. Testing of the Rail Head without Defects Using a NonContact Transducer The fourth experimental setup is shown in Figure 17. As in section 4.4, the laser beam was focused to a spot on the head of the 136 lb AREMA rail that was 1 mm in diameter and located in the middle of the side of the head. The only difference between this test and the one in section 4.4 is that here a noncontact transducer is used to receive the ultrasonic waves traveling through the air gap. The distance between the side surface of the rail head and the noncontact transducer is 10 mm. The transducer has a central frequency of 1 MHz. In the numerical model the properties of air has been taken as follow: adiabatic bulk modulus 1.42⋅105 Pa, and density 1.2 kg/m3. The thermoelastic generated ultrasound waves travel through the rail head and arrive to the side surface where the non-contact transducer is located. The ultrasound waves then move through the air for 10 mm before reaching the transducer that will record the pressure variations on its surface. Figures 18 from a) to i) plot the pressure maps in the section of the portion of air between the non-contact transducer and the side surface of the rail head at nine different times, the section is normal to the rail axis and passing through the center of mass of the portion of air. As it was seen in Figure 14 for the rail without defect, the ultrasound waves travel through the rail head in about t1=11.4 μs, then it will travel through the air with the well known speed of

Experimental and Numerical Method for Nondestructive Ultrasonic… 89 sound 343 m/s. Since the distance covered by the wave through the air is 10 mm, it will need t2=10/343000=29.1 μs to reach the transducer; thus the total time from the laser pulse is about t1+ t2 = 40 μs. Figures 18 from a) to f) show the maps of the acoustic pressure, which is the local pressure deviation from the ambient pressure caused by a sound wave. Figure 18 a) shows the ultrasound waves traveling through the air after 20⋅10-6 seconds since the laser pulse. As computed after about 40 μs the wave has reached the transducer, this is shown in Figure 18 c) and also in Figure 18 f) where a limit of 1⋅10-8 N/mm2 in the maximum acoustic pressure map is imposed to highlight that the wave front arrives at the theoretical time since it is not perfectly clear in Figure 18 c). Several waves with a high acoustic pressure arriving to the transducer can be noted in Figures 18 d) to f), this will be confirmed by the analysis of the experimental results obtained in the time frame 55⋅10-6 to 70⋅10-6 seconds. Figures 18 g) to i) show the acoustic pressure map for the final part of the analysis.

Transducer Laser beam Air

Air gap 10 mm

(a)

(b)

Figure 17. (a) Schematic of the experimental setup and b) finite element model of the rail and of the portion of air between the non-contact transducer and the side surface of the rail head.

90


Time = 20 µs

a)

Time = 70 µs

f)

Time = 30 µs

b)

Time = 80 µs

g)

Time = 40 µs

c)

Time = 90 µs

h)

Time = 50 µs

d)

Time = 100 µs

i)

Time = 60 µs

e)

Time = 40 µs Press. limit 10-8 N/mm2

f)

Figure 18. Acoustic pressure maps in the central section of the portion of air between the non-contact transducer and the side surface of the rail head at nine different times.

In Figure 19 a) the graph shows the output of the piezoelectric non-contact transducer as function of time. Figure 19 b) presents the acoustic pressure measured by the numerical approach at a point in the air located at the center of the receiving surface of the non-contact transducer. The arrival time of the first wave, marked with the arrow number 1 in Figures 19 a) and b), is about 40 μs as computed previously as sum of the time needed to the longitudinal wave to go through the rail head plus the time traveled through the air by the acoustic wave. After the arrival of the first group of waves, a group of stronger waves is recorded both in the experimental and the numerical results, as marked with the arrow number 2 in Figures 19 a) and b). We have already seen these waves with a high acoustic pressure arriving to the transducer in Figures 18 d) to f) in the time frame 55⋅10-6 to 70⋅10-6 seconds. Experiments and simulations match rather well even if a non-contact transducer is used, proving the efficiency of the method in simulating laser generated ultrasound wave propagation with frequencies in the MHz range both in steel and in air.


(a)

(b) Figure 19. (a) Voltage, which is proportional to the air pressure, as function of time recorded by the transducer located at the side of the rail head; b) air pressure measured by the numerical approach at the same location.

In the paragraph on the modeling procedures we have discussed the need to impose non-reflecting boundary conditions on the surfaces of the air that are not in contact with the rail head. In Figures 20 a) to e) the acoustic pressure maps in absence of non-reflecting boundary conditions are reported for the

92


time frame 50⋅10-6 to 90⋅10-6 seconds. As it can be noted a large amount of reflections are generated; unwanted artificial reflections of waves at boundaries that are not present in the reality but are needed to limit the computational expense, can strongly affect the accuracy of the simulation.

5. CONCLUSION Despite the potential impact of laser-generated ultrasound in many areas of science and industry, robust tools for studying the phenomenon are lacking. The laser-generated ultrasound propagation in complex structures is an intricate phenomenon extremely hard to analyze. Only simple geometries can be studied analytically. Numerical techniques found in literature have proved to be limited in their applicability by the frequencies in the MHz range and very short wavelengths. In summary, here we have established that an explicit dynamic analysis together with the use of diagonal element mass matrices is far more computationally efficient than the implicit integration rule for the analysis of large models with relatively short dynamic response times, as it is the case for ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies and through air. Thus the proposed numerical technique provides a valuable tool to simulate completely the noncontact ultrasonic system and for designing and optimizing inspection configurations.

Time = 50 µs

a)

Time = 60 µs

b)

Time = 70 µs

c)

Time = 80 µs

d)

Time = 90 µs

e)

Figure 20. Pressure maps in the central section of the portion of air between the noncontact transducer and the side surface of the rail head for the model where the air has reflecting boundary conditions in the time range 50⋅10-6 –90⋅10-6 seconds.

Experimental and Numerical Method for Nondestructive Ultrasonic… 93 The numerical approach has been validated by comparison with analytical results for transient guided waves propagating in a circular annulus. Then the attention was focused on assessing the capability of the presented numerical method as a tool for defect detection in rails by laser generated ultrasound waves propagation. The information that can be acquired can be very valuable for choosing the right setup when performing NDE tests. Both numerical and experimental analysis were conducted on a 136 lb AREMA rail, and a good agreement among the solutions was found.

REFERENCES American Railway Engineering and Maintenance-of-Way Association: Manual for Railway Engineering AREMA: Lanham MD USA, 2000. Bartoli, I; Marzani, A; Lanza di Scalea, F; Viola, EJ. Sound Vibration, 2006, vol. 295, 685-707. Bhardwaj, MC. Non-contact ultrasound: the last frontier in non-destructive testing and evaluation; Encyclopedia of Smart Materials; John Wiley & Sons: New York, 2001. Blomme E; Bulcaen D; Declercq F. Ultrasonics, 2002, vol. 40, 153-157. Cook, R; Malkus, D; Plesha, M. Concepts and Applications of Finite Element Analysis; John Wiley & Sons: New York, 1989. Cosenza, C; Kenderian, S; Djordjevic, BB; Green, R. Jr; Pasta, A. IEEE Trans. Ultrason. Ferroelec. Freq. Control., 2007, vol. 54, 147-156. Gavric, L. J. Sound Vibration, 1995, vol. 185, 531- 543. Glushkov, E; Glushkova, N; Ekhlakov, A; Shapar, E. Wave Motion, 2006, vol. 43, 458- 473. Hassan, W; Veronesi, W. Ultrasonics, 2003, vol. 41, 41-52. Jeong, H; Park, MC. Res. Nondestr. Eval., 2005, vol. 16, 1-14. Kenderian, S; Cerniglia, D; Djordjevic, BB; Green, RE. Jr. Res. Nondestr. Eval., 2005, vol. 16, 195- 207. Kenderian, S; Djordjevic, BB; Cerniglia, D; Garcia, G. Insight, 2006, vol. 48, 336-341. Kenderian, S; Djordjevic, BB; Green, R. Jr; IEEE Trans. Ultrason. Ferroelec. Freq. Control., 2003, vol. 50, 1057-1064. Liu, G; Qu, J. J. Acoust. Soc. Am., 1998, vol. 104, 1210-1220. McDonald, FA. Appl. Phys. Lett., 1990, vol. 56, 230- 232. Moser, F; Jacobs, LJ; Qu, J. NDT&E International 1999, vol. 32, 225- 234.

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Rose, LRF. J. Acoust. Soc. Am., 1984, vol. 75, 723-732. Sanderson, R; Smith, S. Insight 2002, vol. 44, 359-363. Scruby, CB; Dewhurst, RJ; Hutchins, DA; Palmer, SB. J. Appl. Phys., 1980, vol. 51, 6210- 6216. Scruby, CB; Drain, LE. Laser Ultrasonics: Techniques and Applications Adam Hilger: Bristol, 1990. Telschow, KL; Conant RJ. J. Acoust. Soc. Am.1990, vol. 88, 1494-1502. Terrien, N; Royer, D; Lepoutre, F; Deom A. Ultrasonics 2007, vol. 46, 251265 Wang, J; Shen, Z; Xu, B; Ni, X; Guan, J; Lu, J. Appl. Phys., A 2006, vol. 84, 301-307. White, R. J. Appl. Phys., 1963, vol. 34, 3559- 3567. Xu, B; Shen, Z; Ni, X; Lu, J. J. Appl. Phys., 2004, vol. 95, 2116- 2122. Xu, B; Shen, Z; Wang, J; Ni, X; Guan, J; Lu, J. Appl. Phys., 2006, vol. 99, 33508-1 to 33508-7. Zerwer, A; Polak, MA; Santamarina, JC. J. Nondestr. Eval., 2003, vol. 22, 3952. Zhou, S; Reynolds, P; Krause, R; Buma, T; Donnell, M; Hossack, J. A. IEEE Trans. Ultrason. Ferroelec. Freq. Control., 2004, vol. 51, 11781186.

In: Nondestructive Testing: Methods, Analyses… ISBN: 978-1-60876-157-9 Editor: Earl N. Mallory, pp. 95-146 © 2010 Nova Science Publishers, Inc.

Chapter 3

INVESTIGATION OF THERMAL PROPERTIES OF STEEL UNDERGOING HEAT TREATMENT BY THE PHOTOTHERMAL DEFLECTION TECHNIQUE: CORRELATION WITH MECHANICAL PROPERTIES Taher Ghrib*, Imen Gaied and Noureddine Yacoubi Equipe photothermique de Nabeul. IPEIN, 8000 Merazka, Nabeul, Tunisia

ABSTRACT In this work we present a new method, based on the photothermal deflection technique, which permits the simultaneous determination of thermal conductivity and thermal diffusivity of steel undergoing a heat treatment (carburizing, nutriding, electroerosion, Jominy test). This method consists of the deposition of a thin graphite layer on the treated surface steel which will absorb the totality of the incident light and will play the role of a heat source. The local thermal properties of hardened steel are determined by drawing the experimental amplitude and phase curves of the photothermal signal versus square root modulation *

Corresponding author: [email protected], Tel: +216 97 465 933

96

Taher Ghrib, Imen Gaied and Noureddine Yacoubi frequency and to compare them to the corresponding theoretical ones. The best coincidence between these curves is obtained for a unique and known thermal diffusivity and thermal conductivity. The main interest of this method is that such obtained thermal properties are correlated to hardness. In some cases, the thermal properties are related to hardness through an empiric mathematical law which permits us to deduce the hardness of steel without measuring it.

Keywords: Thermal conductivity, thermal diffusivity, hardness, Photothermal Deflection Technique, steel, carburizing, nutriding, electroerosion, Jominy test, semiconductor.

1. INTRODUCTION There are many methods based on the Photothermal Deflection (PTD) Technique [1] which may be used to determine thermal properties of materials. In this work we compare three methods. The first one consists of heating the sample placed in air by a uniform modulate light and to study the variation of the photothermal deflection signal versus square root modulation frequency. The second method has the same experimental conditions as the first method except that the sample is covered by a thin graphite layer. The third method, which is a spectroscopic one and where the sample is immerged in CCl4, consists of drawing the experimental phase variations versus wavelength at a fixed modulation frequency. The best method will be used to determine the thermal properties of steel which has undergone a heat treatment. The use of the heat treatments to improve hardness and the resistance of alloys goes back to moved back times [2, 3]. As the determination of the mechanical properties uses generally destructive methods such as (Hardness, Elasticity, stress, Tenacity, …), it is preferable to relate the mechanical properties to other kinds of physical properties which determination is easy by using nondestructive techniques such as the law of Petch [4] which relates the grain size to the elasticity constant Re = Re 0 + β / d

where Re0 is edge elasticity, β is a

constant that depends on materials and d is grain diameter. Photothermal methods [6-14], which are non-destructive techniques, are widely used for carrying out the thermal and optical properties of materials. In this work we have used the photothermal deflection techniques (PTD) [1,6,7,8,12,14] in order to determine in a first step the local thermal conductivity and thermal diffusivity of steel having undergone a heat treatment

Investigation of Thermal Properties of Steel Undergoing Heat…

97

in surface as carburized, nitriding [15,16] and electoerosion[17] and in volume as a Jominy test [18]. In a second step, we will try to correlate the thermal properties to the mechanical one. In some cases one can deduce an empirical formula relating the thermal properties to hardness.

2. PRINCIPLE OF THE PTD TECHNIQUE The PTD technique consists in heating the absorbing sample using a modulated light pump beam. The optical absorption of the sample will generate a thermal wave that will propagate into the sample and in the surrounding fluid medium, inducing a temperature gradient then a refractive index gradient in the fluid. A Laser probe beam skimming the sample surface and crossing the region with inhomogeneous refractive index gradient is deflected. Its deflection ψ may be related to the thermal properties of the sample.

3. THEORY 3.1. Heat Transfer by Conduction Mode Contrary to the transfer by convection, the transfer by conduction is done without a macroscopic matter movement, it occurs in a medium out of thermal equilibrium (the temperature is not uniform). An isotropic and homogeneous medium, of density ρ, specific heat capacity c, into which can exist an internal energy source of power density Pth characterized by a temperature gradient which is the origin of a heat flow defined by the Fourier law [19] :

Φ = − K grad

T

(1)

Where K is a constant characteristic of the medium called thermal conductivity expressed in W.m-1.K-1, the sign (-) indicates that the flow is in the direction of the decreasing temperatures. This flow obeys to the general heat equation which in the presence of a heat source is written as:

98

Taher Ghrib, Imen Gaied and Noureddine Yacoubi

Pump beam

x x0 Probe beam

Deflection ψ

y

Fluid (f)

Sample (s) Backing (b) Figure 1. Schematic representation of the PTD principle.

div Φ + ρ c

∂T = Pth ∂t

(2)

In the case of a heat source lack the equations (1) and (2) give:

div ( grad T ) = One poses D = 1

ρ c ∂T K

∂t

i.e

ΔT =

ρ c ∂T K

∂t

.

K called thermal diffusivity of the medium expressed in m2.sρc

. One obtains the equation (3): [19-21]

Δ T

1 D

=

∂ T ∂ t

(3)

If the thermal diffusion process is unidirectional along the x axis:

∂ 2T ∂x2

=

1 D

∂T ∂t

(4)


99

If the medium contains a heat source which can be due to an optical absorption of the incidental light modulated at the frequency f =

ω , with an 2π

absorption coefficient α, equation4 may be written as:

∂ 2T 1 ∂T = − A e α x (1 + e 2 ∂x D ∂t

jω t

)

(5)

A e α x ( 1 + e j ω t ) represent the heat source term. With A = α η I 0 and η is the conversion rate of the light intensity into heat 2K

Where

by de-energizing nonradiative take equal to 1 thereafter. In this work we take into account only the energy transfer by conduction because the periodic temperature elevation on the sample surface is very small so one can neglect the heat transfer by radiation and by convection.

3.2. Calculation of the Laser Probe Beam Deflection Ψ The laser probe beam which passes parallel to the sample surface at a z0 distance undergoes a periodic deflection (figure 1) due to the periodic variations of the refractive index of the fluid in the vicinity of the sample; its trajectory obeys to the general luminous ray equation: [22]

d dr (n ) = − grad n ds ds

(7)

Where n is the medium refractive index and s the curvilinear X-coordinate of the luminous ray along its trajectory; and r is the vector position of a point M of this trajectory. As the angle Ψ is very small, we have:

ds =

(dx )2

+ (dy

d dx dn . (n ) =− dy dy dx

Either

)2

≈ dy then

its

expression

is

simplified

as:

100


n

l / 2 dn dx =−∫ dy − l / 2 dx dy

(8)

Where l is the width of the pump beam. While introducing the fluid temperature Tf we obtain:

dx 1 l / 2 dn dTf =− ∫ dy dy n −l / 2 dTf dx

(9)

dn is the represent the variation of the fluid refractive index with dT f temperature. In our experiment, the temperature interval in the fluid is very weak, hence

dT f dx

dn dTf

can be taken as a constant and removed from the integral as

is independent of y the Eq. (9) can be rewritten as :

ψ=

dx l dn dT f =− dy n dT f dx

(10)

if one considers T0 the temperature rises at the sample surface it may be written as:

T 0 = T 0 e iθ

Where T0 and θ are respectively the amplitude

and argument of T0. For any point M of the fluid, distant of x from the sample surface the temperature is given by:

T f =T0 e

−σ f x

e jω t

Where

σ f = (1+ j)

πf Df

Thus one determines the complex deflection of the probe beam [1,6, 7]:

Investigation of Thermal Properties of Steel Undergoing Heat… x

j (θ + − dx 2 L dn μ ψ = = T0 e f e dy n μ f dT f

=ψ e

5π x − ) 4 μf

101

e jω t (11)

j ( ωt + ϕ )

Where μ = D f is the thermal diffusion length in the fluid, ψ f π f

and ϕ

are

respectively the amplitude and the argument of the laser pump beam deflexion given by: x

− 2 L dn μ ψ = T0 e f n μ f dT f

(12)

And

ϕ=−

x

μf

+θ +

5π 4

(13)

One can note that these expressions are function of thermal properties of all media and also of the modulation frequency. A comparison between the theoretical and experimental curves one will allow us to determine the thermal properties of all studied materials.

3.3. Calculation of the Periodic Elevation Temperature T0 at the Sample Surface 3.3.1. Case of bulk sample If the studied sample is composed of only one layer as shown on figure 2, of which one notes by ls its thickness, Ks its thermal conductivity and Ds its thermal diffusivity. The sample is fixed at a support of lb thickness, of thermal conductivity Kb and thermal diffusivity Db all this medias are in a fluid of thermal conductivity Kf, thermal diffusivity Df and lf thickness (figure 2).

102

Taher Ghrib, Imen Gaied and Noureddine Yacoubi Probe laser beam

Sample

Backing -ls

-lb-ls

x

0

Fluid lf

Figure 2. Different regions crossed by the heat in the case of bulk sample.

In the case of uniform heating where only the sample is supposed to absorb the incidental light, the heat equations in the various media are written as follow:

∂ 2T f ∂x

2

∂ 2Ts ∂x 2 ∂ 2Tb ∂x

2

=

=

1 ∂T f D f ∂t

=

1 ∂Ts − Aeα x (1 + e j ω t ) si − l s ≤ x ≤ 0 Ds ∂t

1 ∂Tb Db ∂t

si 0 ≤ x ≤ l f

(14)

(15)

si − l s − lb ≤ x ≤ −l s (16)

As the system of detection which we use is only sensitive to the periodic signal the periodic solution of these equations is as follow:

e jωt

si 0 ≤ x ≤ l f

(17)

Ts ( x, t ) = (U eσ s x + V e −σ s x − Eeα x ) e jω t

si − l s ≤ x ≤ 0

(18)

si − lb − l s ≤ x ≤ −l s

(19)

T f ( x, t ) = T0 e

−σ f x

Tb ( x, t ) = W eσ b ( x+ ls ) e j ω t

Investigation of Thermal Properties of Steel Undergoing Heat… Where T0, U, V, E and W are complex constant and σi = (1+ j) π f Di

103

, Di is the

thermal diffusivity of medium i. The heat flow in each medium is written:

φ f = −K f

φs = − K s φb = − K b

∂T f

si 0 ≤ x ≤ l f

∂x

∂Ts ∂x

(20)

si − ls ≤ x ≤ 0

∂Tb ∂x

(21)

si − ls − lb ≤ x ≤ −ls

(22)

Then after derivation of the previously found expressions one obtains:

φ f = K f σ f T0e

−σ f x

e j ωt

si 0 ≤ x ≤ l f

φs (x) = −Ksσ s (U eσ x − V e−σ x − s

φb (x) = −Kb σ b W eσ

b

s

( x+l )

e jωt

α α x jωt Ee )e σs

(23)

si − ls ≤ x ≤ 0 (24)

si − ls − lb ≤ x ≤ −ls

(25)

Constant U, V, W, E and T0 are given by writing the conditions of temperature and heat flow of continuity at the various interfaces.

in x = 0 T f ( x = 0) = Ts ( x = 0) and Φ f ( x = 0) = Φ s ( x = 0). in x = −l s Ts ( x = −l s ) = Tb ( x = −l s ) and Φ s ( x = −l s ) = Φ b ( x = −l s ). Then one obtains:

104


T0 = U + V − E K f σ f T0 = K s σ s (U − V +

α E) σs

U e −σ s ls + V eσ s ls − E e −α ls = W K s σ s (U e −σ s ls − V eσ s ls +

If one poses: b =

K bσ b K sσ s

,g=

α E e −α l ) = K b σ bW σs

K fσ f K sσ s

s

et r =

α σs

One obtains

T0 = U + V − E

(26)

g T0 = U − V + rE

(27)

U e −σ s l s + V e σ s l s − E e −α l s = W

(28)

U e −σ s ls − V eσ s ls + rE e −α ls = bW (26) + (27) give U =

(29)

1 (( 1 + g ) T 0 + (1 − r ) E ) 2

And (26) - (27) give V =

1 ((1 − g ) T0 + (1 + r ) E ) 2

By multiplying the equation (28) by b and identification with the equation (29) and by replacing U and V by their expressions one obtains:

T0 = − E [(1 − r )(1 + b) eσ s ls − (1 + r )(1 − b) e −σ s ls + 2(r − b) e −α ls ] / [(1 + g ) (1 + b) eσ s ls − (1 − g )(1 − b) e −σ s ls ]

(30)


105

E is obtained by identification by replacing Ts by its expression (equation 18) in the equation (15) one obtains:

E=

α I0 A = 2 α − σ s 2 K s (α 2 − σ s2 ) 2

One notes according to equation 30 that the temperature T0 on the surface of the sample is a function of the optical and thermal properties of the sample as well as thermal properties of the fluid and the backing and the modulation frequency.

3.3.2. Sample composed of a layer deposed on a substrate Fernelius [23] was the first who developed a theoretical model of a sample composed of a layer deposed on a substrate .Only the layer is supposed to absorb the incident light. The heat equations in each medium (figure 3) may be written as follows: ∂ 2T f

=

1 ∂T f D f ∂t

si 0 ≤ x ≤ l f

( 31)

∂ 2Tc 1 ∂Tc = − Aeα x (1 + e j ω t ) si − lc ≤ x ≤ 0 Dc ∂t ∂x 2

( 32)

∂ 2Ts 1 ∂Ts = Ds ∂t ∂x 2

( 33)

∂x

2

si − lc − ls ≤ x ≤ −ls

∂ 2Tb 1 ∂Tb = ∂x 2 Db ∂t

si − lc − ls − lb ≤ x ≤ − lc − ls

( 34)

The periodic solutions of the temperature in each medium are: −σ f x jωt

Tf (x) = T0e

si 0 ≤ x ≤ l f

(35)

Tc (x) = (U eσs x +V e−σs x − Eeα x )e jωt

si −lc ≤ x ≤ 0

( 36)

Ts (x) = (X eσs (x+lcs) +Y e−σs (x+lc ) )e jωt

si −lc −ls ≤ x ≤ −lc

( 37)

si −lb −ls −lc ≤ x ≤ −lc −ls

(38)

e

Tb (x) =W eσb ( x+ lc + ls ) e jωt

106


Backing

Layer

Substrate -lc

-ls-lc

-lb-ls-lc

x

Probe laser beam Fluid

0

lf

Figure 3. Different regions crossed by the heat in the case of a layer deposed on a substrate.

And the heat flows in the various media are: −σ f x jωt

Φf = Kf σf T0e

si 0 ≤ x ≤ lf

e

α σc

(39)

Φc(x) = − Kcσc(Ueσc x −V e−σc x − Eeαx) ejωt si −lc ≤ x ≤ 0

(40)

Φs (x) = − Ksσs (X eσs (x+lc ) −Y e−σs (x+lc ) ) e jωt si −ls −lc ≤ x ≤ −ls

(41)

σb (x+ls −lc ) jωt

Φb(x) =−Kb σb We

e

Also we pose b =

si −lb −ls −lc ≤ x ≤ −ls −lc

(42)

K fσ f K bσ b K σ α ,c= c c ,g = et r = K sσ s K sσ s K cσ c σc

By applying the continuity conditions of temperature and heat flow at the different interfaces, one obtains:

at x = 0 U + V = T0 + E U − V = g T0 − rE at x = −l c X + Y = U e − σ c l c + V e σ c l c − E e −α l c X − Y = c (U e −σ c lc − V e σ c lc − r E e −α lc ) at x = −l c − l s X e −σ s ls + Y e σ s ls = W X e −σ s ls − Y e σ s ls =b W


107

Then

U = 12 [(1 + g )T0 + (1 − r ) E ] V = 12 [(1 − g )T0 + (1 + r ) E ]. X = 12 [(1 + c ) e − σ c l c U + (1 − c ) e σ c l c V − (1 + c r ) E e − α l c ] Y = 12 [(1 − c ) e − σ c l c U + (1 + c ) e σ c l c V − (1 − c r ) E e − α l c ] et (1 − b ) X e − σ s l s = (1 + b ) Y e σ s l s These equations may be expressed in a matrix form:

⎛U ⎞ ⎜ ⎟ ⎜V ⎟ = ⎜E⎟ ⎝ ⎠ ⎛X ⎜ ⎜Y ⎜E ⎝

⎛1 + g 1 − r 0 ⎞⎛ T0 ⎞ ⎟⎜ ⎟ 1⎜ ⎜1 − g 1 + r 0 ⎟⎜ E ⎟ 2⎜ 0 2 ⎟⎠⎜⎝ E ⎟⎠ ⎝ 0

⎛ (1 + c ) e − σ c l c ⎞ ⎟ 1⎜ −σ l ⎟ = ⎜ (1 − c ) e c c 2 ⎜ ⎟ 0 ⎠ ⎝

(1 − c ) e σ c l c (1 − c ) e σ c l c 0

− (1 + c r ) e − α l c ⎞ ⎛ U ⎞ ⎟⎜ ⎟ − (1 − c r ) e − α l c ⎟ ⎜ V ⎟ ⎟⎜ ⎟ 2 ⎠⎝ E ⎠

Finally one can obtain the expression of T0

T0 = E [(1 − b ) e −σ sls [(1 − r )(1 − c ) e σ clc + (1 + r )(1 + c ) e − σ c lc − 2 (1 + r c ) e −α lc ] − (1 + b ) e σ sls [(1 − r )(1 + c ) e σ clc + (1 + r )(1 − c ) e −σ c lc − 2 (1 − r c ) e −α lc ]] /[(1 + b ) e σ sls [(1 + g )(1 + c ) e σ clc + (1 − g )(1 − c ) e −σ clc ] − (1 − b ) e −σ s ls [(1 + g )(1 − c ) e σ clc + (1 − g )(1 + c ) e −σ c lc ]]

( 43 )

3.3.3. Case of n layers deposed on a substrate If we consider now the general case of a sample formed by n layers on a substrate (figure 4) where Ki , Di and li are respectively the thermal conductivity, the thermal Diffusivity and the thickness of the layer i.

108

Taher Ghrib, Imen Gaied and Noureddine Yacoubi Baking

Layer (n-1) Layer (n) Fluid

Layer (1)

x

−ln....−l2

−ln....−l1

−ln....−ln−1

−ln

x=0

Figure 4. different regions crossed by the heat in the case of n layers deposed on a substrate. The temperatures inside each layer are written as: −σ x Tf (x,t) =T0e f e jω t if 0≤ x ≤ l f σ x

Tn (x,t) =(Xn e n + Yn e−σn x −En eαn x ) e jω t

if − ln ≤ x ≤ 0

Tn−1(x, t) =(Xn−1 eσn−1 (x+ln ) + Yn−1e−σn−1 (x+ln ) −En−1eαn−1 (x+ln ) ) e jω t

if − ln − ln−1 ≤ x ≤ −ln

.......... .......... .......... .......... .......... .......... .......... ........ T1(x, t) =(X1eσ1 ( x+ln +ln−1 +...+l2 ) + Y1e−σ1 ( x+ln + ln−1 +...+l2 ) −E1eα1 ( x+ ln +ln−1 +...+l2 ) )ejω t if − ln − ln−1...− l1 ≤ x ≤ −ln − ln−1...− l2 σb ( x+ln +ln −1 +...+l2 +l1 ) jω t

Tb (x, t) =W e

e

if − ln − ln−1...− l1 − lb ≤ x ≤ − ln − ln−1...− l1 By writing the heat flow expressions in each medium one obtains: −σ f x jω t

φf (x,t) =Kf σf T0e

if 0≤ x ≤ l f

e

α σ x φn(x,t) =−Knσn (Xn e −Yn e−σ x − n Eneα x) ejω t σn n

n

φn−1(x,t) =−Kn−1σn−1(Xn−1 eσ

if −ln ≤ x ≤ 0

n

n−1 (x+ln )

αn−1 En−1eα σn−1

−Yn−1e−σn−1 (x+ln ) −

n−1 (x+ln )

) ejω t

if −ln −ln−1 ≤ x ≤ −ln

.......... .......... .......... .......... .......... .......... .......... .......... ......

φ1(x,t) =−K1σ1(X1eσ (x+l +l 1

n

n −1 +...+l2 )

α σ1

−Y1e−σ1 (x+ln +ln−1 +...+l2 ) − 1 E1eα1 (x+ln +ln−1 +...+l2 ) )ejωt if −ln −ln−1...−l1 ≤ x ≤ −ln −ln−1...−l2

σb (x+ln +ln−1 +...+l2 +l1 ) jω t

φb(x,t) =−Kbσb We

e

if −ln −ln−1...−l1 −lb ≤ x ≤ −ln −ln−1...−l1

The temperature and heat flow continuity at the interfaces x=-ln , -ln –ln-1, -lnln-1..-l2 permit to obtain:


− 1 ⎞⎛ X n −1 ⎞ ⎛ e −σ n ln ⎛1 1 ⎜ ⎟⎜ ⎟ ⎜ − r2 ⎟⎜ Yn −1 ⎟ = ⎜ c n e −σ nln ⎜1 − 1 ⎜ 0 0 E / E ⎟⎜ E ⎟ ⎜ 0 n n −1 ⎠⎝ n −1 ⎠ ⎝ ⎝

Gn

⎛ X n −1 ⎞ ⎟ ⎜ ⎜ Y n −1 ⎟ = D n ⎜E ⎟ ⎝ n −1 ⎠

⎛Xn⎞ ⎟ ⎜ ⎜ Yn ⎟ ⎜E ⎟ ⎝ n⎠

e σ nln −c n e

− e −α n l n

σ nln

− c n rn e

0

−α n l n

1

⎛ X n −1 ⎞ ⎟ ⎜ i.e. ⎜ Y n −1 ⎟ = G n−1 .D n ⎜E ⎟ ⎝ n −1 ⎠

⎞⎛Xn ⎞ ⎟⎜ ⎟ ⎟ ⎜ Yn ⎟ ⎟⎜ ⎟ ⎠ ⎝ En ⎠

⎛Xn⎞ ⎟ ⎜ ⎜ Yn ⎟ = M n ⎜E ⎟ ⎝ n⎠

⎛Xn⎞ ⎟ ⎜ ⎜ Yn ⎟ ⎜E ⎟ ⎝ n⎠

−1 ⎞ ⎛1 1 ⎟ ⎜ Where Gn = ⎜ 1 − 1 − rn −1 ⎟ , ⎜0 0 E / E ⎟ n n −1 ⎠ ⎝ ⎛ e −σ n l n ⎜ Dn = ⎜ c n e −σ nln ⎜ ⎝ 0

e σ n ln −cn e

− e −α n l n

σ nln

0

− c n rn e 1

−α n ln

⎞ ⎟ ⎟ and ⎟ ⎠

M n =Gn−1 .Dn .

In the same way for i from n to 2 we can write ⎛ X i −1 ⎞ ⎟ ⎜ −1 ⎜ Y i −1 ⎟ = G i . D i ⎜E ⎟ ⎝ i −1 ⎠

⎛Xi⎞ ⎟ ⎜ ⎜ Yi ⎟ = M i ⎜E ⎟ ⎝ i⎠

⎛1 ⎜ Where G i = ⎜ 1 ⎜0 ⎝

1 −1

c1 =

0

109

⎛Xi⎞ ⎟ ⎜ ⎜ Yi ⎟ ⎜E ⎟ ⎝ i⎠

⎛ e−σili eσili −e−αi li ⎞ ⎞ ⎟ ⎜ −σ l ⎟ − ri ⎟ , Di =⎜cie i i −cieσili − ci ri e−αi li ⎟ , ⎟ ⎜ E i +1 / E i ⎟⎠ 0 1 ⎠ ⎝ 0

−1

αi I0 K σ K 1σ 1 K σ . , c2 = 2 2 ,......, cn = n n and Ei = 2 K i (α i2 − σ i2 ) K2 σ 2 K 3σ 3 Kf σf

110

Taher Ghrib, Imen Gaied and Noureddine Yacoubi ⎛ X1⎞ ⎜ ⎟ ⎜ Y1 ⎟ = M 2 ..... M ⎜E ⎟ ⎝ 1⎠

Then

⎛Xn ⎜ n ⎜ Yn ⎜E ⎝ n

⎞ ⎟ ⎟= ⎟ ⎠

⎛ m 11 ⎜ ⎜ m 21 ⎜m ⎝ 31

m 12 m 22 m 32

m 13 m 23 m 33

⎞⎛Xn ⎟⎜ ⎟ ⎜ Yn ⎟⎜E ⎠⎝ n

⎞ ⎟ ⎟ ⎟ ⎠

⎧ X 1 = m11 X n + m12Yn + m13 E n ⎩ Y1 = m21 X n + m22Yn + m23 E n

In this case we write ⎨

The writing of the heat flow and temperature continuity at the interfaces x=0 and x= -ln –ln-1-...-l1 give respectively:

E E 1 1 X n = (1 − g )T0 + (1+ rn ) n , Yn = (1 + g )T0 + (1− rn ) n . 2 2 2 2 And (1 − b) e

−σ 1 l1

X 1 −(1 + b) eσ1 l1 Y1 − (r1 −b) e −α1 l1 E1 = 0

(44) .

Then X1 =

m 11 m ((1 − g )T 0 + (1 + rn ) E n ) + 12 ((1 + g )T 0 + (1 − rn ) E n ) + m 13 E n 2 2

and Y1 =

m 21 m ((1 − g )T 0 + (1 + rn ) E n ) + 22 ((1 + g )T 0 + (1 − rn ) E n ) + m 23 E n 2 2

Then X 1 = ( m 11 (1 − g ) + m 12 (1 + g )) T 0 + ( m 11 (1 + rn ) + m12 (1 − rn ) + 2 m13 ) E n 2 2 and

Y1 =(m21 (1 − g)+ m22 (1 + g))

T0 E + (m21 (1 + rn ) + m22 (1 − rn )+ 2 m23 ) n 2 2

That is to say X 1 =η1T0 +η 2 E n and Y1 =η 3T0 +η 4 E n .


111

By replacing X 1 and Y1 by its expressions in equation (44) one obtains:

(1− b) e−σ1l1 (η1T0 +η2 En )−(1+ b) eσ1l1 (η3T0 +η4 En )−(r1 −b)e−α1l1 E1 =0 This gives

((1−b)η1 e−σ1l1 −(1+b)η3eσ1l1 )T0 = ((1+b)η4eσ1l1 −(1−b)η2e−σ1l1 )En +(r1−b)e−α1l1 E1 Finally

[

][

T0 = ((1+ b)η4eσ1l1 −(1− b)η2e−σ1l1 )En +(r1 −b) e−α1l1 E1 (1− b)η1 e−σ1 l1 −(1+ b)η3eσ1l1

]

3.4. Optimization of Experimental Conditions for Determining the Thermal Properties of the Graphite Layer and the Sample As the studied samples are in general metallic then they are highly reflective so the detected photothermal signal will be very small and it become necessary to cover them with a graphite layer which will absorb the totality of the incidental light and will play the role of a heat source. To determine the thermal properties of these materials it is initially necessary to know the thermal properties of the graphite layer.

3.4.1. Study of the thermal properties of the graphite layer The graphite layer that we propose to determine it’s thermal properties is deposed on a sample with known thermal properties such as copper of which thermal conductivity Ks=400W.m-1.K-1and thermal diffusivity Ds=0,99m2.s-1.

112


3.4.1.1. Case where the graphite layer is thermally thick: Determination of its thermal diffusivity A layer is known as thermally thick if it’s thickness lc is higher than it’s

Dc . πf

thermal diffusion length μc ( lc > μc ) where μc =

In this case the periodic temperature elevation T0 at the sample surface (Eq. 43) is simplified and can be written as follow:

T0 =

e −α lc ≈ 0 et e −σ

Since

c

1− r E 1+ g lc

(45)

≈0

By replacing E by its expression in equation 45 one obtains:

1 + (1 − i ) T0 =

α 2 a

c

α

1 + g

2

A − 2 ia

2

T0

πf

where ac =

2 c

Dc

2

⎛ ⎛ α ⎞ α ⎞ ⎜⎜ 1 + ⎟⎟ + ⎜⎜ ⎟⎟ 2 ac ⎠ ⎝ ⎝ 2 ac ⎠ = ⎛⎛ ⎞ ⎞ ⎜ ⎜ 1 + α ⎟ + i α ⎟ (1 + g ⎟ ⎜⎜ 2 ac ⎠ 2 a c ⎟⎠ ⎝⎝

(

A α

2

α

4

)

+ i 2 a c2 + 4 a c4

)

Consequently the amplitude and phase of T0 may be written as: 2

T0 = A

And

⎛ ⎛ α α ⎞ ⎜⎜ 1 + ⎟⎟ + ⎜⎜ a 2 c ⎝ ⎠ ⎝ 2 ac 4 (1 + g ) α + 2 a c4

θ = − arctg(

(

α α + 2a c

)

)+ arctg(

2 ac2

α2

⎞ ⎟⎟ ⎠

)

2

( 46 )

(47)


113

One note according to the equations (46) and (47) that both amplitude and phase are independent of the thermal conductivity of the graphite layer but depend on its thermal diffusivity. The curves of figure 5, represents the theoretical variation of the normalized amplitude and phase of the photothermal signal according to the square root modulation frequency in the case of a graphite layer of thickness lc = 50 μm deposited on a copper sample of thickness 1mm for different values of thermal diffusivity Dc at a fixed thermal conductivity value equal to 0,1Wm-1K-1.


0,9 0,8 0,7 0,6

100 -6 2 -1 3×10 m .s -6 2 -1 2.5×10 m .s -6 2 -1 2×10 m .s -6 2 -1 1.5×10 m .s -6 2 -1 1×10 m .s

80 Phase(degree)

-6 2 -1 3×10 m .s -6 2 -1 2.5×10 m .s -6 2 -1 2×10 m .s -6 2 -1 1.5×10 m .s -6 2 -1 1×10 m .s

1,0

60 40 20

0,5

0

0,4 2

4

6

8

10

12

14

1/2 Square root modulation frequency (Hz )

2

16

4

6

8

10

12

14

16


Figure 5. Variation of the normalized amplitude and phase of Photothermal signal versus square root modulation frequency for various values of thermal diffusivity of the graphite layer whose thickness lc=50µm and thermal conductivity Kc=0,1W.m-1 .


0,9 0,8 0,7

80 Phase(degree)

-1 -1 Kc= 0.1 W.m .K -1 -1 Kc= 0.5 W.m .K -1 -1 Kc= 1 W.m .K -1 -1 Kc= 5 W.m .K

1,0

-1 -1 Kc= 0.1 W.m .K -1 -1 Kc= 0.5 W.m .K -1 -1 Kc= 1 W.m .K -1 -1 Kc= 5 W.m .K

60

40

20

0,6 6

8

10

12

14

16

18


20

6

8

10

12

14

16

18

20


Figure 6. Variation of the normalized amplitude and phase of Photothermal signal versus square root modulation frequency for various values of thermal conductivity of the graphite layer whose thickness lc=50µm and thermal diffusivity Dc=1×10-6m2/s.

114


One notes according to these curves, that both amplitude and phase are very sensitive to the layer thermal diffusivity. Now if we fixes the value of the thermal diffusivity and varies the thermal conductivity one notes according to the curves of figures 6 and 7 obtained respectively for graphite layer thicknesses of 50μm and 100 μm for a thermal diffusivity Dc=1×10-6m2.s-1 that both amplitude and phase of the photothermal signal are insensitive to the thermal conductivity. Thus one can conclude that to determine thermal diffusivity it is necessary to deposit a graphite layer of a thickness higher than 50 μm. 1,0

-1 -1 Ks=0.1W.m .K -1 -1 Ks=10W.m .K -1 -1 Ks=100W.m .K

0,8 -1 -1 Ks=0.1W.m .K -1 -1 Ks=10W.m .K -1 -1 Ks=100W.m .K

0,6

-1 -1 Ks=400W.m .K

Phase(degree)


110

100

-1 -1 Ks=400W.m .K

90

80

0,4 4

6

8

10

12

14


4

6

8

10

12

14


Figure 7. Variation of the normalized amplitude and phase of Photothermal signal versus square root modulation frequency for various values of thermal conductivity of the graphite layer whose thickness lc=100µm and thermal diffusivity Dc=1×10-6m2/s.

3.4.1.2. Case of thermally thin graphite layer: Determination of its thermal conductivity A graphite layer is known as thermally thin if its thickness lc is less than the thermal diffusion length. In this case −σc lc

e

σc lc

≈e

-α l

= 1 and e

= 0 since α ≈ 10 7 m −1 .


115

So the expression of the periodic temperature elevation T0 may be simplified as follow:

T0 = − E

[(1 + b )(1 − rc )e σ s ls − (1 − b )(1 + r c ) e −σ s l s ]

120

-1 -1 0.05 W.m .K -1 -1 0.5 W.m .K

0,9 0,8

100

-1 -1 0.05 W.m .K -1 -1 0.25 W.m .K

0,7

Phase(degree)


1,0

0,6 0,5

-1 -1 0.5 W.m .K -1 -1 1 W.m .K -1 -1 3 W.m .K

0,4 0,3 2

( 46 )

(1 + b )(1 + c g ) eσ sls − (1 − b ) (1 − c g ) e −σ s ls

4

6

8

10

12

14


-1 -1 0.25 W.m .K -1 -1 1 W.m .K -1 -1 3 W.m .K

80 60 40

16

2

4

6

8

10

12

14


16

Figure 8. Amplitude and phase of the signal versus the square root frequency for different values of thermal conductivity of the graphite layer and for a thermal diffusivity Dc = 1,5×10-6m2.s-1

One can notice that T0 is independent on the thermal diffusivity of the graphite layer and is sensitive only to its thermal conductivity. This is confirmed by the theoretical curves of figures 8 and 9. By choosing a thickness of graphite lc= 2μm, One notes according to the theoretical curves of figure 8 which gives the variations of the normalized amplitude and phase according to the square root modulation frequency for various values of thermal conductivity with a constant thermal diffusivity Dc= 1,5×10-6m2.s-1 that the signal is very sensitive to the thermal conductivity of the layer on the other hand the curves of figure 8 which gives the variations of the normalized amplitude and phase for a thickness of the layer lc=2μm and for various values

116


of thermal diffusivity in the interval of 0,5×10-6m2.s-1 to 1×10-4m2.s-1 with a graphite layer thermal conductivity Kc=0,35 W.m-1.K-1 that the signal is insensitive to the thermal diffusivity. In conclusion to determine the thermal diffusivity of the graphite layer it must be thermally thick (lc>μc) while to determine its thermal conductivity it must be thermally thin (lc