Study of wave propagation in elastic and thermoelastic solids

Transcription

1 Study of wave propagation in elasti and thermoelasti solids Thesis submitted in partial fulfillments of the requirements For the award of the degree of Master of Siene In Mathematis and Computing Submitted by Gurpreet Singh under the guidane of Dr. Satish Kumar Sharma to the Shool of Mathematis and Computer Appliations Thapar University Patiala (Punjab) India July 013 i

2 DEDICATED TO MY FATHER LATE S. KARAM SINGH AND MY TEACHERS ii

3 iii

4 iv

5 ABSTRACT Lamb wave (1917), whih propagate in solid plates with free boundaries, are elasti perturbations for whih displaement ours both in the diretion of wave propagation and perpendiular to the plane of plate. These waves are ommonly used in ultrasoni nondestrutive testing appliations. Rayleigh surfae waves have been well reognized in the study of seismology, geophysis and geodynamis. These types of surfae waves propagate in half spae. In the present work, a mathematial analysis has been done to study Rayleigh and Lamb waves in Thermoelasti solids in the ontext of Generalized theories of Thermoelastiity. The thesis has been divided into four hapters. Chapter 1 inludes the disussion of lassial theory of elastiity and Generalized Hooke s law. This hapter also inludes a brief theory of Thermoelastiity and Generalized Thermoelastiity. In Chapter, we have studied Rayleigh waves in homogeneous isotropi elasti half spae and Lamb waves in homogeneous isotropi elasti plate. Formulation of the problem is done and equations are solved by assuming the boundary onditions. Chapter 3 inludes the disussion of Generalized Thermoelasti waves in homogeneous isotropi plates. The seular equations are derived and disussed in this hapter. The variations of phase veloity with wave number are shown graphially. Chapter 4 inludes the disussion of Rayleigh waves in Generalized Thermoelasti half-spae. The seular equations are derived and disussed in this hapter. The variations of phase veloity with wave number are also shown graphially. v

6 Contents List of symbols Chapter 1 Introdution 1.1 Theory of elastiity 1 1. Generalized Hooke s law 1.3 Wave propagation in elasti solids Waves in infinite media Waves in semi infinite media Waves in plate Thermoelastiity Generalized theory of Thermoelastiity. 7 Chapter Rayleigh waves in a homogeneous isotropi half spae and Rayleigh lamb wave propagation in homogeneous isotropi elasti plate..1 Introdution 9. Rayleigh waves 9.3 Lamb waves. 13 Chapter 3 Propagation of Thermoelasti waves in isotropi plates. 3.1 Introdution Formulation of the problem and its solution Seular equations Numerial results and Disussion. 3 vi

7 Chapter 4 Propagation of Rayleigh wave in homogenous isotropi solids. 4.1 Introdution 5 4. Formulation of the problem and its solution Seular equations Disussion. 8 Conlusions 30 Referenes 31 vii

8 List of symbols, Elasti onstants Density ij Stress tensor ij Strain tensor q, Potential funtion u Displaement vetor Thermal expansion C e Speifi heat K e T Thermal ondutivity Dilatation Temperature hange ij Kroneker s delta t0, t 1 Relaxation time viii

9 Chapter 1 Introdution 1.1 Theory of elastiity Solid bodies an be divided into namely two ategories; elasti body and plasti body. A body is alled elasti if it returns to its original shape upon the removal of applied fores. All bodies exhibit elasti behavior under suffiiently small loads. The mathematial analysis of elasti behavior of solid body is alled theory of elastiity. A body that does not return to its original shape or size, upon the removal of deforming fore is alled plasti body. When an elasti material is deformed due to an external fore, it experienes internal fores that oppose the deformation and restore it to its original state, if the external fore is no longer applied. There are various elasti moduli, suh as Young s modulus, the shear modulus, and the bulk modulus, all of whih are measures of the inherent stiffness of a material as a resistane to deformation under an applied load. The various moduli apply to different kinds of deformation. For instane, Young's modulus applies to uniform extension, whereas the shear modulus applies to shearing. The elastiity of materials is desribed by a stress-strain urve, whih shows the relation between stress (the average restorative internal fore per unit area) and strain (the relative deformation). For most metals or rystaline materials, the urve is linear for small deformations, and hene the stress-strain relationship an adequately be desribed by Hooke s law and higherorder terms an be ignored. However, for larger stresses beyond the elasti limit the relation is no longer linear. For even higher stresses, materials exhibit plasti behavior, that is, they deform irreversibly and do not return to their original shape after stress is no longer applied. For rubberlike materials suh as elastomers, the gradient of the stress-strain urve inreases with stress, meaning that rubbers progressively beome more diffiult to streth, while for most metals, the gradient dereases at very high stresses, meaning that they progressively beome easier to streth. 1

10 The theory of elastiity is onerned with the displaements and angular movements of the volume element with respet to their neighborhood. If the volume elements were small enough, their movements an be desribed ompletely in terms of three positional oordinates of eah element. Likewise, when the volume elements are small enough, the interations between eah element and its neighborhood an be expressed in terms of trative fore alone. Now from these ideas the state of strain in solid at any given point an be expressed by resolving the displaement of elementary volume, originally loated at suh points, along three mutually perpendiular diretions and differentiating three omponent of displaement again along eah of three axis in turn. Thus, we obtain nine omponent of strain tensor. We onsider the trative fore ating on an infinitesimal area drawn normal to three oordinate planes in turn at the position and again resolving these trative fore along eah of the three oordinate axes. We thus obtain the nine omponents of stress tensor. 1. Generalized Hooke s law In stress strain relationship Hooke s law states that the stress is a quantity that is proportional to the fore ausing deformation and strain is a measure of the degree of deformation. Stress an be divided into two omponents; normal and shearing stress. In 3 dimensional axes of the ubi body, stress an be resolved into three parts, one normal stress and shearing stress whih itself an be resolved into two omponents parallel to the diretion of two oordinates. The Generalized Hooke s law states that nine omponent of stress ats on the fae of the ube. These are ( xx, xy, xz, yx, yy, yz, zx, zy, zz ) where the first suffix refers to the normal to the plane on whih shear stress ats and seond suffix refers to the diretion of the shear on the plane. Similarly strain an also be resolved into two omponents; longitudinal and shearing strain. There are nine omponent of strain. These are (,,,,,,,, ). xx xy xz yx yy yz zx zy zz

11 In general ij and will have nine omponents and ijkl will have 81 omponents. Considering kl the symmetry of stress and strain omponents, the number of independent omponent redues from 81 to 36. We write Hooke s law in the form Here, where I, j,.k, l=1,,3. ij ijkl kl For other rystal the number of independent onstants is diminished by reason of their symmetry property. For monolini materials there are only 13 independent elasti onstants. In ase of orthotropi materials the number of elasti onstant is further redued by 4 from thirteen to nine. For transversely isotropi material there are only five independent onstants. Finally, there are only two independent elasti onstants known as Lame s parameter for isotropi elasti material. The onstitutive equation for lassial isotropi elastiity is u ( u u ) where and are the Kroneker s delta and stress tensor ij r, r ij i, j j, i ij ij respetively. 1.3 Wave propagation in elasti solids Depending upon the nature of medium and the boundary onditions the phenomenon of wave propagation an be divided into following ategories. 1. Waves in infinite media. Waves in semi infinite media 3. Waves in plate. 3

12 1.4 Waves in infinite media given by The equation of motion for a homogeneous isotropi medium without body fore is u u u ( ) (. ) t Where u ( u1, u, u3) is the displaement vetor. are, Lami s onstant and is density. We take u ( u1, u, u3) and (0,,0) for two- dimensional problems. Using Helmholtz deomposition, this vetor an be deomposed in terms of salars and vetor potentials, u q q and ψ are the salars and vetor potentials respetively. The resolution of a vetor field into the gradient of a salar and the url of a zero- divergene vetor is due to a theorem by Helmholtz. The ondition. 0 provides the neessary additional ondition to uniquely determine the three omponents of u. Now Inserting u in basi equation and interhanging the order of some operations, we obtain t q t we take 1 and. q Therefore waves may propagate in the interior of elasti solid at two different veloities. 1 is the wave veloity of longitudinal and is the veloity of shear waves. In longitudinal waves, the osillations our in the diretion of wave propagation. Sine ompressional and dilatational fores are ative in these waves. They are also alled density waves beause their partile density flutuates as they move. In the transverse or shear wave, the partiles osillates at right angle. Shear waves require an aoustially solid material for effetive propagation and therefore, are 4

13 not effetively propagated in materials suh as liquids or gasses. Shear waves are weaker than longitudinal waves. Shear waves are usually generated in materials using some of the energy from longitudinal waves. A variety of terminology exists for the two wave- types. Longitudinal waves are also alled irrotational, Dilatational and primary waves(p). The Shear waves are also alled equi-voluminal, distortional, rotational and seondary waves (S). The P and S waves designations have arisen in seismology, where they are also oasionally pituresquely designated as the push and shake waves. 1.5 Waves in semi infinite media There are different types of waves in semi infinite media. But in the present work we are onsidering only the types of waves whose effets are losely onfined to the surfae. These types of waves were first investigated by Lord Rayleigh who showed that there is a wave type that ould propagate along surfaes, suh that the motion assoiated with the wave deayed exponentially with the distane into material from the surfae. This type of surfae wave is alled Rayleigh wave. Longitudinal and transverse both motions may be found in solids as Rayleigh surfae wave. When Rayleigh surfae wave passes through a solid, the partiles in a solid moves in elliptial paths, with the major axis of the ellipse perpendiular to the surfae of solid. The simplest medium in whih Rayleigh wave propagate is homogeneous isotropi half spae. The transverse veloity is slightly greater than the veloity of Rayleigh waves. 1.6 Waves in plates Here we onsider the propagation of waves in a plate having tration- free boundary. This is the ase of greatest pratial interest and is the lassial ase first studied by Rayleigh and Lamb (Graff 1991) and in general are known as Lamb waves. These are the elasti waves whose partile motion lies in the plane that ontains the diretion of wave propagation. These type of waves an travel long distanes with little attenuation. Rayleigh Lamb theory applies to the propagation of ontinuous, straight rested waves in an infinite plate with free surfae. The Rayleigh-Lamb frequeny equation gives the relationship between phase veloity and wave number. The displaement and stress distribution funtions an be obtained after the Rayleigh-Lamb frequeny are solved. As in Rayleigh waves whih propagates along single free surfaes, the partile motion in Lamb wave is elliptial with its x 5

14 and z omponents depending on the depth within the plate. In one family of modes, the motion is symmetrial about the mid thikness plate. In the other family it is antisymmetri. When Lamb waves are transmitted, partiles move in one of two different ways. If the partile motion is symmetri to the mid surfae, it is alled symmetri Lamb wave. If the partile motion is antisymmetri with respet to the mid surfae, it is alled anti symmetri Lamb waves. Furthermore, a number of modes exist for eah type of the Lamb waves. Lamb wave an be generated in a plate with free boundaries with an infinite number of modes for both symmetri and antisymmetri displaements within the layer. The symmetri modes are also alled longitudinal modes beause the average displaement over the thikness of the plate or layer in the longitudinal diretion. The anti-symmetri modes are observed to exhibit average displaement in the transverse diretion and these modes are also alled flexural modes. Symmetri and anti-symmetri Lamb waves have different phase and group veloity, as well as distribution of partile displaement and stress through the plate thikness. 1.7 Thermoelastiity Thermoelastiity deals with the study of thermodynamial system of bodies in equilibrium, whose interations with surrounding are limited to mehanial work, heat exhange and external work. We know from experiment that deformation of a body is assoiated with a hange of heat. The time varying loading of a body auses in it not only displaements but also temperature distribution hanging in time. Conversely, the heating of a body produes in it deformation and temperature hange. The motion of a body is haraterized by mutual interation between deformation and temperature fields. The domain of siene dealing with the mutual interation of these fields is alled Thermoelastiity. The theory of oupling of thermal and the strain fields give rise to oupled theory of Thermoelastiity and was first postulated by Duhamel [1837] shortly after the formulation of elastiity. He derived the equations for the distributions of strains in an elasti medium subjeted to temperature gradient and introdued the dilatation term in heat ondution equation. However, this equation was not well grounded in the thermo dynamial sense. Next, an attempt at thermo dynamial Justifiation of this equation was undertaken by Voigt and Jefreys [1930]. Biot[1956] gave the full justifiation of the thermal ondutivity equation on the basis of 6

15 thermodynamis of irreversible proesses. Biot also presented the fundamental methods for solving the thermo elastiity equation as well also variational theorem. Thermoelastiity desribes a broad range of phenomena; it is the generalization of the lassial theory of elastiity and of the theory of thermal ondutivity. It is known, that researh in the field of thermo elastiity was preeded by broad-sale investigation within the framework of what is alled the theory of thermal stresses. By this term we mean the investigation of strains and stresses produed by heating a body, with the simplifying assumption that the deformation of an elasti body does not affet the thermal ondutivity. Coupled theory of Thermoelastiity The oupling between thermal and strain fields give rise to oupled theory of Thermoelastiity. For stati problems this oupling vanishes and two fields beomes independent of eah other.the oupling effets were onsidered by Weiner [1957] and Lesson[1957] in their works. Nowaki [196] disussed the propagation of longitudinal waves in an unbounded Thermoelasti medium. 1.8 Generalized theories of Thermoelastiity The basi governing equations of Thermoelastiity in the usual framework of linear oupled Thermoelastiity onsist of wave type (hyperboli) equations of motion and diffusion type (paraboli) equation of heat ondution. It is observed that a part of solution of the energy tends to infinity. This implies that if an isotropi homogeneous elasti medium is subjeted to thermal or mehanial disturbanes, the effets in the temperature and displaement fields are felt at an infinite distane from the soure of disturbane instantaneously. This implies that a part of solution has infinite veloity of Researhers suh as kaliski[1965], Lord and Shulman[1967]. Fox[1969], Gurtin and Pipkkin[1969] have tried to modify the Fourier law of heat ondution. These works inlude the time needed for aeleration of the hest flow, in the heat ondution equation along with the oupling between temperature and stain fields. This paradox in existing oupled theory of Thermoelastiity has also been disussed by Boley[1956]. Thermoelastiity overs a wide range of extensions of lassial dynamial oupled Thermoelastiity. This theory eliminates the paradox of an infinite veloity of propagation and is based upon the more general linear funtional relationship between the heat flow and 7

16 temperature gradients. The term generalized Thermoelastiity stands for Hyperboli Thermoelastiity in whih thermo mehanial load applied to the body is transmitted in a wave like manner throughout the body. In the present work we have applied the following two models of Thermoelastiity namely, Lord-Shulman(LS) and Green-Lindsay(GL) Lord Shulman theory Lord and Shulman[1967] have formulated a generalized dynamial theory of Thermoelastiity by using Maxwell-Cattano law that generalizes the Fourier s heat ondution equation by introduing a single relaxation time needed for aeleration of heat flow Green Lindsay (GL) Green and Lindsay[197] have also obtained a generalization of oupled theory of Thermoelastiity whih proved that the seond sound effets are short lived. Their analysis is based on modified form of entropy prodution inequality. In this theory the onstitutive relations for stress and entropy are generalized by introduing two different relaxation times into onsideration onstrained by inequality t 1 t 0 0. Basi differenes between GL and LS theory (1) GL theory modifies both onstitutive equation and energy equation. But LS theory modifies only the energy equation. () GL involves two relaxation times while LS theory involves one relaxation time. (3) The energy equation of LS theory depends on strain veloity and strain aeleration but orresponding equation of GL theory depends only on the strain onstraints. (4) In GL theory the heat annot propagate with a finite speed unless the stresses depend on temperature veloity. But aording to LS theory the heat an propagate with a finite speed even though the stresses are independent of temperature veloity. 8

17 Chapter - Rayleigh wave in a homogeneous isotropi elasti half spae and Rayleigh Lamb wave Propagation in homogeneous isotropi elasti plate..1 Introdution Rayleigh or surfae waves travel along the surfae of a relative thik solid material penetrating to a depth of one wavelength. Rayleigh waves are useful beause they are very sensitive to surfae defets and sine they will follow the surfae around, so these an be used to assess the surfae, the other waves might have diffiulty in reahing. Lamb wave an propagates in thin metals. Lamb wave are omplex vibrational waves that travel through the entire thikness of a material. These provide a means for inspetion of very thin materials. The propagation of lamb waves depend on density, elasti, and material properties of a omponent. With Lamb waves, a number of modes of a partiles vibrations are possible, but most two ommon are symmetrial and asymmetrial. In this hapter, we have derived the seular equations for Rayleigh and Rayleigh Lamb type wave propagation.. Rayleigh wave Consider a homogeneous, isotropi elasti half-spae. We introdue a Cartesian oordinate system whose (x-plane oinides with the surfae of the medium, and z- axis is positive downwards). the z axis is pointing vertially downward into the semi spae. The x-axis is taken along the diretion of wave propagation so that all partiles on a line parallel to y- axis are equally displaed and hene all the quantities are independent of y- oordinate. The surfaes are assumed to be stress free. 9

18 Diretion of propagation x-axis z-axis Fig.1 (geo.mff.uni.z/vyuka/novotony-seismisurfaewaves) The basi governing equations of generalized elastiity in the absene of body fores and heat soures are u. u The onstitutive relation is given by u t (..1) u ( u u ) ij r, r ij i, j j, i (..) Where u ( u, u, u ) is the displaement vetor, is the density. 1 3 We define the quantities x z, x, z, t t, u u, * * * ' ' ' * ' 1 x z 1 1 T0 w w,,,,. * ' 1 ' ij ij 1 T0 1 T0 (..3) 1and * are longitudinal and shear veloities. w is harateristi frequeny of medium. For two dimensional problem, we take 10

19 u u u 1, 0, 3 The equations in the non dimensional form for Rayleigh wave an be written as u u u u 1 x xz x t u u u u 1 z xz z t (..4) (..5) In order to solve above equations, we introdue potential funtion q and in the solid through the relations q u1 x z (..6) u 3 q z x (..7) Substituting (..6) and (..7) in equations (..4) and (..5) we obtained t q 0 (..8) 1 0 t Now on Substituting (..6) and (..7) in onstitutive relation, we obtain (..9) zz q ( q, xx, xz ) (..10) ( q ) zx, xz, xx, xz (..11) Boundary onditions The surfae of the solid is assumed to be free of stresses, ouples so that, zz 0, 0 zx zz 0, 0 zx 11

20 The boundary ondition is to be satisfied at the surfae (z=0) of the solid half spae...1 Solution of the problem We assume the solution of the form, q q() z e ( x t ) (..1) () ( x t ) ze (..13) Where is the non dimensional phase veloity, is the irular frequeny, is wave number Upon using solutions in (..8) and (..9), we get * * ( ) z z 1 1 q ( Ae B e ) e i x t (..14) * * ( xt ) z ( A e B e ) e z (..15) Where (1 ) * and (1 ) * Sine we are primarily interested in surfae wave propagation along the free surfae of half spae. So hoose the partial wave that satisfied the radial ondition. That is all field variables should be bounded for loation deep within the solid. Therefore, we hoose the solution for q and as 1 * i ( x t) z q B e e (..16) B e * z ( x t) e (..17) Now applying q and in and zz zx zz B ( ) B * * 1 * * ZX B( ) B1 1

21 Now invoking boundary ondition, zz xz 0 B ( ) B 0 (..18) * * 1 B( ) B 0 * * 1 (..19) This is a homogeneous system of equation for the unknown amplitude B and B 1 The system of equations (..15) and (..16) has a non trivial solution if orresponding determinant equals to zero. Now solving determinant, we get * * i * * i 0 * * * 4 0 (..0) This is the Rayleigh equation whih is same as is obtained and disussed in by Graff (1991).3 Rayleigh lamb wave We onsider an infinite homogeneous isotropi elasti plate of thikness d. We take the origin of the oordinate system (x, y, z) on the middle surfae of the plate. The x-y plane is hosen to oinide with the middle surfae and the z- axis normal to it along the thikness. The surfae are assumed to be stress free.the basi governing equation, onstitutive relation and partial differential equations for potential funtion of Rayleigh-lamb wave are same as that of Rayleigh wave equations (..1 to..9). Boundary onditions The boundaries of the plate are assumed to be stress free. Therefore, the non dimensional mehanial boundary onditions are given by 13

22 zz 0 at z= d xz For Rayleigh lamb type waves, solution for potential funtions are obtained as q ( Asin z B os z) e ' ' i ( xt) ' where ( 1) and E sin z F os z e ' ' i ( xt ) where ' ( 1) On using (..6) and (..7) in onstitutive relation, we get zz A z B z i E z F z ' ' ' ' ' ' ( ) ( sin os ) ( os sin ) ' ' ' ' ' ' xz i Aos z Bsin z E sin z F os z The displaement are obtained from equation (..6) and (..7) as (..1) (..) ' ' ' ' ' i u1 i ( Asin z Bos z) ( E os z F sin z) e ( xt) ' ' ' ' ' ' u3 A z B z i E z F z e ( ) (( os sin ) ( sin os )) i x t.3.1 Derivation of Seular Equations Invoking the boundary onditions at the surfae of q and, we obtain a system of four simultaneous linear equation as below: z d of the plate and using equations p( As B ) q( E Fs ) p( As B ) q( E Fs ) f ( A Bs ) r( Es F ) f ( A Bs ) r( Es F )

23 Where ' ' ' ' ' ' ' ' p ( ), q i, 1 os z, os z, s1 sin z, s sin z, f i, r ( ) For symmetri modes, the redued system of equations an be written in the matrix form as ' ' p os d q os d ' ' f sin d r sin d B 0 E = 0 Sine the system of equation is homogeneous, the determinant of oeffiient has to vanish for non trivial solution, whih results in the frequeny equation. pr d d qf d d ' ' ' ' os sin os sin 0 it an be written as ' ' ' tan d 4 tan d ( ) ' ' ' ' where ( 1), ( 1) (..3) Similarly for anti-symmetri modes, the redued system of equations an be written in the matrix form as ' ' psin d qsin d A 0 ' ' f os d r os d F 0 whih gives the Rayleigh Lamb frequeny equation for the propagation of anti symmetri waves in a plate ' ' ' tan d 4. (..4) tan ' d ' Combining (..3) and (..4), ' 1 ' ' tan d 4 ' tan d ' Here +1 is for anti symmetri mode and 1 for symmetri mode. 15

24 The equations are same as obtained and disussed by Graff(1991). 16

25 CHAPTER 3 Propagation of Thermoelasti waves in isotropi plates. 3.1 Introdution The onept of generalized Thermoelastiity overs a wide range of extensions of lassial dynamial oupled Thermoelastiity. Thermoelastiity proposed by Lord and Shulman in1967 (L S model) in whih, in omparison to the lassial theory, the Fourier law of heat ondution is modified by taking into onsideration a single relaxation time. To explain and remove the paradox of infinite veloity of heat propagation, some researhers derived and formulated generalized theories of thermo elastiity. The propagation of plane harmoni waves in homogeneous transversely isotropi materials has also been studied in generalized theories of thermo elastiity. In the present work we have onsidered the propagation of plane waves in an infinite homogeneous, isotropi plate of thikness d. In the ontext of generalized theories of Thermoelastiity. Here we have reviewed the work arried out by Sharma etal. [000]. (Generalized Thermoelasti waves in homogeneous isotropi plates, J. Aoust. So. Am. 108(), August 000) 3. Formulation of the problem and its solution We onsider an infinite homogeneous isotropi thermally onduting elasti plate of thikness d initially at uniform temperature T 0. We take origin of the oordinate system (x,y,z) on the middle surfae of the plate. The x y plane is hosen to oinide with the middle surfae and the z axis normal to it along the thikness. The surfaes z= d are assumed to be (i) stress free insulated and (ii) rigid fixed insulated. z = -d z = +d x-axis z-axis 17

26 The basi governing equations of generalized Thermoelastiity in the absene of heat soures and body fores are ( ) (. ) ( k 1 ) u u T t T u (3.1) K T Ce( T t0t ) T0 ( 1 kt0 ). u t t (3.) where u=,, u u u is the displaement vetor, 1 3 T(x,y,z,t) is the temperature hange and are Lame s parameters; K is thermal ondutivity; and e are the density and speifi heat at onstant strain respetively. (3 ), is the linear thermal expansion and e is the dilatation. t t The dot notation denotes time differentiation and ij is the Kroneker s delta. Here K=1 for Lord Shulman (LS) theory and K= for Green- Lindsay (GL) theory. The thermal relaxation times t0 and t1satisfies the inequalities t 0 t 1 0 for GL theory only. We take the x z plane as the plane of inidene and we assume that the solutions are expliitly Independent of y, but impliit dependene is there so that the transverse omponent u of displaement is non vanishing. The governing equations in non-dimensional form an be written as: u1 u3 u1 u1 (1 ) ( T kt1t ) x xz z x t (3.3) u 3 u1 3 3 (1 ) u ( T kt1t ) u z xz x z t (3.4) T T u1 u3 u1 u3 ( T t 0T ) 1 kt0 x z t x t z t x t z (3.5) Here we have defined the quantities, 18

27 x x /, t t ' * ' * i i i u u / T ' * i 1 i 0 T T / T, t t ' ' * t t, ( ) / k ' * * 0 0 T0 / e ( ) /( ) 1 ( ) / / e (3.6) The onstitutive relation is given as : ij ur, rij ( ui, j uj, j ) ( T t T 1 k ) ij t (3.7) Boundary onditions (a) Mehanial boundary onditions The non-dimensional mehanial boundary ondition at z (1)For stress free boundary onditions- d are given by: k u 1 u T t T ,3 1,1 1 (3.8) 13 u1,3 u3,1 0 (3.9) ()For rigidly fixed boundary onditions- u1 u u3 0 (b)thermal ondition The non dimensional thermal boundary onditions at z T ht 0 z d are given by: 19

28 Where h is the surfae heat transfer oeffiient h 0 Corresponds to Thermally Insulated Boundary Using Helmholtz deomposition, Now, u q q t T q t (3.10) 1 1 k and 1 0 (3.11) and T T T t t q t t t t (3.1) 0 1k 0 Now from equation (3.10), we get i ( x t ) Esin z F os z e (3.13) And from equation (3.10) and (3.1) we get sin os sin os q A m z B m z e C m z D m z e i ( x t ) i ( xt ) 1 1 T i m Ar B m Cr D e 1 1 i ( xt ) (3.14) (3.15) Displaement u1, u3 are given by, q u1 x z, 0

29 u 3 q z x Putting the values of q and, we get i ( xt ) u1 i Ar1 B1 Cr D ( E Fr) e, (3.16) i ( xt ) u m Br A m Dr C i F Cr e (3.17) Here, t i, 1 0` t0 1k i, 1 1` t1 k i, r sin z, os z, ri sin mi z, i os mi z, i 1, m m 1 1 1, 1, a a 1, 1, a a i i i 1 1 1, 1 0 0` ` 1 4 0` Seular equations Invoking the boundary onditions at the surfaes z of six simultaneous linear equations for stress free onditions as p( As B Cs D ) q( E Fr) f ( A Bs ) g( C Ds ) h( Er F) 0 l( A Bs ) n( C Ds ) p( As B Cs D ) q( E Fr) f ( A Bs ) g( C Ds ) h( Er F) l( A Bs ) n( C Ds ) 0 Where d of the plate. We obtain a system

30 p (1 ), q=i, f = m i, g = m i, h=( ) 1 l ( m ) m, n=( m ) m 1 1 r sin z, os z, s sin m z, os m z, i 1, i i i i Equations possess nontrivial solution if the determinant of the oeffiients of amplitudes [A,B,C,D,E,F] T vanishes. For skew symmetri mode, under stress free onditions we obtained the matrix as (( 1 ) ) T (( 1 ) ) T it m1i mi m1 m1 m m ( ) ( ) 0 =0 (3.18) Equations for the symmetri mode an also be obtained in the similar way. After applying lengthy algebrai redutions and manipulations, we obtain the equations for a plate with thermally insulated boundaries as: 1 1 tanm1d m 1( m 1 ) tanmd 4 m 1( m m 1 ) tand m ( m ) tand m Where +1 orresponds to skew symmetri and -1 orresponds to symmetri modes. Similarly, Invoking the boundary onditions at the surfaes z system of six simultaneous linear equations for rigidly fixed ase as i( As B Cs D ) ( r) m ( A Bs ) m ( C Ds ) i ( Er F) l( A Bs ) n( C Ds ) i( As B Cs D ) ( r) m ( A Bs ) m ( C Ds ) i ( Er F) l( A Bs ) n( C Ds ) d of the plate. We obtain a whih has nontrivial solution if the determinant of the oeffiients of amplitudes [A,B,C,D,E,F] T vanishes. For skew symmetri mode, under rigidly fixed onditions we obtained the matrix as

31 PHASE VELOCITY 0 it it T 1 4 m m i 1 m1 m1 m m ( ) ( ) 0 =0 (3.19) Similarly, after obtaining the orresponding equations for symmetri modes we an write the frequeny equation for both symmetri and skew symmetri modes as: 1 1 tanm1d m 1( m 1 ) tanmd m 1( m m 1 ) tand m ( m ) tand m Here the subsript +1 orresponds to skew symmetri and -1 refers to symmetri modes 3.4 Numerial results and disussion To illustrate the theoretial results obtained in the preeding setions, we now present some numerial results. The material for this purpose is aluminum-epoxy omposite, the physial data for whih is given as , = Nm, = Nm, =.1910 kg m, K=.508 J/ms C, C Jkg / C, e T 3 C, t s, t s, d= The phase veloity of symmetri and skew symmetri modes of wave propagation have been omputed for various value of wave number from the dispersion relations for stress free insulated boundary onditions. The orresponding lassial and modified dispersion urves for Rayleigh Lamb types are presented in Fig. 1 in the ontext of LS theories of Thermoelastiity N=0(A) N=1(A) N=(A) N=3(A) N=0(S) N=1(S) N=(S) N=3(S)

32 The phase veloity of the lowest skew symmetri mode is observed to inrease from zero value at vanishing wave number to beome loser the Rayleigh wave veloity at higher values of the wave number, where as in the ase of lowest symmetri mode it derease from a value greater than unity towards the Rayleigh veloity asymptotially with an inrease in wave number. The phase veloity of higher modes of propagation, symmetri and skew symmetri, attain quite large values at vanishing wave number, whih sharply slash down to beome steady with inreasing values of wave number. 4

33 Chapter 4 Propagation of Rayleigh wave in Thermoelasti solid. 4.1 Introdution In this hapter the work arried out in hapter 3 is being extended to study the Rayleigh wave propagation in a Thermoelasti half spae. Here I have reviewed the work done by Sharma etal. in the absene of mirostreth and miropolarity (Effet of miropolarity, mirostreth and relaxation times on Rayleigh Surfae waves in Thermoelasti Solids, Int J. of Appl. Math and Meh. 5():17-38,009.) 4. Formulation of the problem and its solution Consider a homogeneous, isotropi and elasti half spae for Rayleigh wave. We introdue a Cartesian oordinate system whose (x-plane oinides with the surfae of the medium, and z- axis is positive downwards). the z axis is pointing vertially downward into the half-spae. The x- axis is taken along the diretion of wave propagation so that all partiles on a line parallel to y- axis are equally displaed and hene all the quantities are independent of y- oordinate. The surfaes are assumed to be stress free. Equations 3.1 to 3.13 are same as obtained and disussed in hapter 3. Boundary onditions (a)mehanial boundary ondition k u 1 u T t T ,3 1,1 1 (4.1) 13 u1,3 u3,1 0 (4.) (b)thermal boundary ondition 5

34 T ht 0 z Where h is the surfae heat transfer oeffiient h 0 Corresponds to Thermally Insulated Boundaries. Here, we obtain the solutions A e B e e (4.3) m4z m4 z i ( xt) 4 4 ( ) m1 z m1 z mz mz 1 1 q A e B e e A e B e e and i ( x t ) i( xt ) 1 1 T i m A e B e m A e B e e 1 m z m z m z m z i ( xt ) (4.4) Where m m (1 a ) 1 1 (1 a ) Sine displaement u1, u3an be obtained by using the relations q u1 x z, u 3 q z x Sine we are primarily interested in surfae wave propagation along the free surfae of half spae. So hoose the partial wave that satisfied the radial ondition. That is all field variables should be bounded for loation deep within the solid. Therefore, we hoose the solution as B e mz 4 ( ) 4 ( e i x t ) 6

35 m1z mz q B e e B e e and i ( x t ) i( xt ) 1 1 T i m B e m B e e 1 m z m z i ( xt) (4.5) 4.3 Seular equations Invoking the boundary onditions at the surfaes z 0. We obtain a system of six simultaneous linear equations p( B B ) qb m i B m i B ( m ) B m m B m m B e mz p( A A ) qa m i A m i A ( m ) A Where m m A m m A p 1, q= i m 4 In eah ase whih has nontrivial solution if the determinant of the oeffiients of amplitudes A, B, A, B, A, B vanishes T Under stress free onditions, we obtained the matrix as ((1 ) ) ((1 ) ) i m4 m1i mi m4 m1 m1 m m 0 ( ) ( ) 0 And solving determinant for stress free onditions, we get m1 m m1m m1m m4 m1 m 4 ( ) 0 Now put the value of m1, m, m 4 (4.6) 7

36 Phase veloity. if we take 1a 1 1 and 1a and m 4 4 we get, This equation is the seular equations for Thermoelasti Rayleigh waves and is same as disussed in detail by many authors suh as Sharma etal[000], Lokett[1958], Chadwik and Windle[1964], Atkin and Chadwik[1981], Nayfeh and Nasser[1971]. 4.4 Disussion Variations of phase veloity with respet to wave number in the ase of elastiity have been studied in Figure CT LS GL Wave number FIG : Variation of phase veloity with wave number It is observed that the phase veloity of Rayleigh waves in half spae dereases sharply from its peak values in the wave number range 0 to 1, inreases moderately in the wave number range 1 8

37 to and beomes dispersion less for wave number range greater than. The phase veloity profiles are signifiantly dispersive at small values of wave number and beome dispersion less with inreasing wave number. 9

38 Conlusion A Thermoelasti model for studying Rayleigh and Rayleigh Lamb type waves has been presented for insulated boundary onditions. Seular equations are obtained for both Rayleigh and Rayleigh Lamb waves and variations of phase veloity with the wave number have been explained graphially also. Rayleigh waves are widely used for material haraterization, to disover the mehanial and strutural properties of the objets. Guided waves like Rayleigh and Lamb wave have great potential for non destrutive evaluation. In the present work, modified seular equations have been derived onsidering Thermoelasti onditions, appliation of our model may be useful for further appliations of Rayleigh and Lamb waves. 30

39 Referenes Atkin, R.J. and Chadwik. P., Surfae waves in heat onduting elasti bodies. Journal of Thermal Stresses, 4, Biot, M.A., Mehanis of Inremental Deformations. Wiley, New York. Boley, B.A., High temperature strutures and materials. Paragamon Press Oxford, Chadwik, P. and Windle, D.W., Propagation of Rayleigh waves along isothermal and insulated boundaries. Pro. Roy. So. Am., 80, Chanderasekharaiah, D.S., Thermoelastiity with seond sound. Applied mehanis, 39, Duhamel, J.M.C., Seond memoire Sur les phenomes thermo-meaniques. J. Eole Polytehn.,15,1-57. Fox, N., Generalized thermoelastiity. International journal of Engineering Siene, 7, Graff, K.F., Wave Motion Elasti Solids. (Oxford engineering siene series), Dover Publiations; New ed. Green, A.E. and Lindsay, K.A., 197. Thermoelastiity. journal of elastiity.,, 1-7. Gurtin. M.E. and Pipkin, A.C., A general theory of heat ondution with finite wave speed. Arhive for Rational Mehanis and Analysis, 31, Jeffreys, H., Thermodynamis of an elasti solid. Mathematial proeedings of Cambridge philosophial Soiety, 6, Lesson, M., The motion of thermoelasti solids. Quarterly journal of Applied Mathematis, 15, Lokett, F.G., Effet of thermal properties of solid on the veloity of Rayleigh waves. Jounal of mehanis and physis of solids., 7, Lord, H.W. and Shulman, Y., A generalized dynamial theory of thermoelastiity. Journal of mehanis and physis of solids, 15, Love, A.E.H., A Treatise on the Mathematial Theory of Elastiity. (4th ed.) Dover Publiations, New York. Nayfeh, A.H. and Nasser, S.N., Thermoelasti waves in solids with relaxation. Ata Mehania, 1,pp

40 Nowaki, W., 196. Thermoelastiity. Pergamon Press Oxford. Rayleigh, L On progressive waves. Pro. Lond. Math. So., S1-9, 1-6. Rayleigh, L On waves propagation on the plane surfae of an elasti solid. Pro. Lond. Math. So., 17, Rayleigh, L On the propagation of waves through a stratified medium, with speial referene to the question of refletion. Pro. Roy. So. Lond. A, 86(586), Sharma, J.N., Kumar, S. and Sharma, Y.D Effet of miropolarity, mirostreth and relaxation times on Rayleigh surfae waves in thermoelasti solids. Int. J.of Appl. Math and Meh. 5(), Sharma, J.N., Singh, D. and Kumar,R.,000 Generalized thermoelasti waves in homogeneous isotropi plates J. Aoust. So. Am.108(). Voigt, W., Theoretishe Studin uber die Elastiitatsverhaltnisse der. Krystalle. Abh.Ges. Wiss. Gottingen, 34. Weiner, J., A uniqueness theorem for oupled thermoelasti problem. Quarterly Journal of Applied Mathematis, 15,