research article s -zgv modes of a linear and...

Research ArticleS1-ZGV Modes of a Linear and Nonlinear Profile forFunctionally Graded Material Using Power Series Technique

M Zagrouba1 M S Bouhdima12 and M H Ben Ghozlen1

1 Laboratory of Physics Materials Faculty of Sciences University of Sfax BP 1171 3000 Sfax Tunisia2 Superior Institute of Applied Science and Technology University of Gabes 6072 Zrig Gabes Tunisia

Correspondence should be addressed to M S Bouhdima mbouhdimayahoofr

Received 14 May 2014 Revised 24 July 2014 Accepted 5 August 2014 Published 21 September 2014

Academic Editor Rama B Bhat

Copyright copy 2014 M Zagrouba et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The present work deals with functionally gradedmaterials (FGM) isotropic plates in the neighborhood of the first-order symmetriczero group velocity (S

1-ZGV) point The mechanical properties of functionally graded material (FGM) are assumed to vary

continuously through the thickness of the plate and obey a power law of the volume fraction of the constituents Governingequations for the problemare derived and the power series technique (PST) is employed to solve the recursive equationsThe impactof the FGM basic materials properties on S

1-ZGV frequency of FGM plate is investigated Numerical results show that S

1-ZGV

frequency is comparatively more sensitive to the shear modulusThe gradient coefficient p does not affect the linear dependence ofZGV frequency 119891

119900as function of cut-off frequency 119891

119888 only the slope is slightly varied

1 Introduction

A functionally graded material (FGM) is a kind of aninhomogeneousmaterialThe characterization ofmechanicalproperties of materials is important for testing their struc-tural integrity Lamb waves are frequently employed in theultrasonic characterization of thin plates [1] As an importantproperty of Lamb waves the zero group velocity (ZGV) atthe frequency minimum 119891

119900of the first-order symmetric (S

1)

continues to be of an interest for the scientific community [23] Tolstoy and Usdin pointed out that for the S

1Lambmode

group velocity vanishes at a particular point of the dispersioncurve and predicted that this zero group velocity pointmust be associated with a sharp continuous wave resonanceand ringing effect [4] Holland and Chimenti demonstratedthe exploitation of this mode for high-sensitivity imagingapplicationsWith air-coupled transducers they observed thetransparency of a plate due to the S

1mode ZGV resonance

[5]The S1-ZGV frequency is obviously sensitive to mechan-

ical properties and to any change in the plate thickness Toexploit this phenomenon recent works evoke the idea thatit may be suitable for the measurement of nanometer-scale

thickness variations in homogeneous plates [6 7] Due to theresulting differential equations of variable coefficients associ-ated with the spatial variation of the material properties thewave propagation in FGM remains difficult to analyze Somenumerical [8ndash10] and analytical methods [11ndash16] have beenapplied in order to study the wave propagation behavior inan inhomogeneousmediumwithmaterial properties varyingcontinuously along the depth direction In an effort to showthe interest of ZGV in the study of FGMmaterials Bouhdima[15] first discussed the effect of the linear variation ofmechan-ical properties along the thickness plate on the S

1-ZGV using

the power series technique (PST) To our knowledge noreports have been published on the relationship between theS1-ZGV frequency and material properties in an inhomo-

geneous free standing plate Previous investigations on S1-

ZGV phenomenon are limited to inspection experiments onhomogenous plates and mainly focused on measuring thethickness of a coating on a relatively thin plate [3 7 17] Thepresent investigation includes different kinds of FGM plateswith various basicmaterials to extract the effects produced bymechanical parameters variation All the selected materialsfor the illustration are in agreement with the convergence cri-terion The PST has been used and the recursive relationship

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2014 Article ID 401042 7 pageshttpdxdoiorg1011552014401042

2 Advances in Acoustics and Vibration

for 119901 = 2 is derivedThe performed developments permit theevaluation of the impact of the nonlinear profile ondispersioncurves of LambwavesThe effect of the shearmodulus on thisfrequency is highlighted Furthermore the relative variationof the stress and the mechanical displacement is investigatedat the S

1-ZGV frequency

The FGM plate data and the basic theory of PST arereported in Sections 2-3 The last paragraph is devoted to thediscussion and the main results

2 Statement of the Problem andTheoretical Study

A functionally graded plate with thickness ldquo119889rdquo is consideredhere It is assumed that the mechanical properties of FGMvary continuously through the thickness of plateThemotionis restricted in the (119909

1 1199093) plane and the Lamb waves

propagate in the positive direction of the1199091-axisThematerial

properties 120572 can be expressed as [18ndash21]

120572 (1199093) = 120572119888times 119865119888+ 120572119898times 119865119898 (1)

where 119865119898and 119865

119888are the volume fractions and the subscripts

119898 and 119888 denote the metallic and ceramic constituentsrespectively 119865

119888and 119865

119898follow a simple power law as

119865119888= (

1199093

119889+

1

2)119901

119865119898

= 1 minus 119865119888 (2)

where 1199093is the thickness coordinate and ldquo119901rdquo is a gradient

coefficient According to this distribution the bottom surface(1199093= minus1198892) of the functionally graded plate is puremetal and

the top surface (1199093= 1198892) is pure ceramic and for different

values of ldquo119901rdquo one can obtain different volume fractions ofmetal

The constitutive equations can be expressed as follows

119879119894119895= 119862119894119895119896119897

119878119896119897

119878119894119895=

1

2(120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)

120597119879119894119895

120597119909119895

= 120588 times 119894

(3)

In (3) 119879119894119895and 119878119896119897are the stress and strain tensors119862

119894119895119896119897are the

elastic coefficients 120588 is the density and 119906119894is the component

of the mechanical displacement in the 119894th directionOn the basis of the previous assumption of plane strain

the displacement components can be described as

1199061(1199091 1199093 119905) = 119880

1(1199093) exp [119894 (119896119909

1minus 120596119905)]

1199062= 0

1199063(1199091 1199093 119905) = 119894119880

3(1199093) exp [119894 (119896119909

1minus 120596119905)]

(4)

where 119896 is the wave number 120596 denotes the frequency and119894 = radicminus1 Note that for convenient description 119894 is introducedto make the first and third displacement components in

phase quadrature so that the polarization locus becomeselliptical Additionally the recursive process inherent to thePST method will have a suitable form On the other handand for brevity the complex exponential exp[119894(119896119909

1minus 120596119905)] is

omitted below From (3)-(4) the governing equations in aninhomogeneous FGM plate are rewritten as follows

11988844119880101584010158401+ 11988810158404411988010158401+ (1205881205962 minus 119888

111198962)1198801minus (11988813

+ 11988844) 11989611988010158403

minus 1198961198881015840441198803= 0

11988811119880101584010158403+ 11988810158401111988010158403+ (1205881205962 minus 119888

441198962)1198803+ (11988813

+ 11988844) 11989611988010158401

+ 1198881015840131198961198801= 0

(5)

The symbols (1015840) and (10158401015840) represent the first and second dif-ferentials with respect to 119909

3 The considered FGM materials

are isotropic so their elastic constants are expressed in termsof Lamersquos coefficients 120582 and 120583 this leads to 119888

11= 120582 + 2 sdot 120583

11988813

= 120582 and 11988844

= 120583Then (5) can be transformed into the following forms

120583119880101584010158401+ 12058310158401198801015840

1+ (1205881198882 minus 120582 minus 2120583) 1198962119880

1minus (120582 + 120583) 1198961198801015840

3

minus 11989612058310158401198803= 0

(120582 + 2120583)119880101584010158403+ (120582 + 2120583)

1015840

11988010158403+ (1205881198882 minus 120583) 1198962119880

3+ (120582 + 120583) 1198961198801015840

1

+ 12058210158401198961198801= 0

(6)

For Lambwaves that propagate in the FGMplate the tractionfree boundary condition should be satisfied at the top andbottom surfaces (119909

3= plusmn1198892) that is

11987913

(1199093= plusmn

119889

2) = 0 119879

33(1199093= plusmn

119889

2) = 0 (7)

Equations (6) are relative to the motion along 1199091and 119909

3 they

reveal coupling between both displacement amplitudes 1198801

and 1198803

3 Used Method

To solve the differential equation with variable coefficientswe use the PST method [15 16] Regarding the longitudinaland the shear wave amplitudes for Lamb guided waves thePST method specifies that 119880

1and 119880

3can take the following

forms

1198801= sum119899

119860119899(1199093

119889)119899

1198803= sum119899

119861119899(1199093

119889)119899

(8)

Advances in Acoustics and Vibration 3

It is assumed that the parameters of the FGM possess thefollowing form

120582 (1199093) =

119901

sum119894=0

119886119894(1199093

119889)119894

120583 (1199093) =

119901

sum119894=0

119887119894(1199093

119889)119894

120588 (1199093) =

119901

sum119894=0

119888119894(1199093

119889)119894

(9)

Substituting (8) and (9) into (6) and by equating the coef-ficients of (119909

3119889)119899 to zero we can obtain two recursive

equations At this level any couple (119860119899 119861119899) can be expressed

as a function of the quadruplet 1198600 1198601 1198610 1198611 this is true

for the displacement components Accordingly any physicalmagnitude will have a four-dimensional vector form For 119901 =2 the corresponding recursive relationships involving119860

119899and

119861119899are written below

(119899 + 2) (119899 + 1) 119887119900119860119899+2

+ (119899 + 1)21198871119860119899+1

minus 119896119889 (119899 + 1) (119886119900+ 119887119900) 119861119899+1

+ [11989621198892 (1198881199001198622 minus 119886

119900119900) + 119899 (119899 + 1) 119887

2] 119860119899

minus 119896119889 [1198991198861+ (119899 + 1) 119887

1] 119861119899

+ 11989621198892 (11988811198622 minus 119886

11)119860119899minus1

minus 119896119889 [(119899 minus 1) 1198862+ (119899 + 1) 119887

2] 119861119899minus1

+ 11989621198892 (11988821198622 minus 119886

22)119860119899minus2

= 0

(119899 + 2) (119899 + 1) 119886119900119900119861119899+2

+ (119899 + 1)211988611119861119899+1

+ 119896119889 (119899 + 1) (119886119900+ 119887119900) 119860119899+1

+ [11989621198892 (1198881199001198622 minus 119887

119900) + 119899 (119899 + 1) 119886

22] 119861119899

+ 119896119889 [1198991198871+ (119899 + 1) 119886

1] 119860119899+ 11989621198892 (119888

11198622 minus 119887

1) 119861119899minus1

+ 119896119889 [(119899 minus 1) 1198872+ (119899 + 1) 119886

2] 119860119899minus1

+ 11989621198892 (11988821198622 minus 119887

2) 119861119899minus2

= 0

(10)

with 119886119900119900

= 119886119900+ 2119887119900 11988611

= 1198861+ 21198871 11988622

= 1198862+ 21198872 119896 is the

wave number and 119862 is the phase velocity Some explorationsrelated to coefficients denoted 119886

119894 119887119894 119888119894(119894 = 0 1 2) are deduced

from the properties of the FGM basic materials (see (9)) 119860119895

and 119861119895are equal to zero if 119895 lt 0

The next step consists of putting boundary conditions ina suitable matrix form Stress components 119879

1198943written with

respect to 1198600 1198601 1198610 1198611 on both sides of the plate give rise

to a square matrix (4 times 4) dependent on 120596 and 119896 For a givenfrequency the secular equation leads to the correspondingwave number and obviously to the phase velocity Then the

Table 1 The physical properties of basic materials

Material 120582 (Gpa) 120583 (Gpa) 120588 (Kgm3) 119891119900(MHz)

Cu 79291 37313 8930 117Ni 13327 81679 8907 166Cr 742 1025 7190 185Si 861 795 2329 302Ceramic 138 11811 3900 287

dispersion curves of symmetric and antisymmetric propaga-tive Lamb modes are represented by a set of branches in theplane (120596 119888)

From the recursive relationships one can deduce theconvergence criteria

lim119899rarrinfin

119860119899+1

119860119899

=1198871

119887119900

lim119899rarrinfin

119861119899+1

119861119899

=1198861+ 21198871

119886119900+ 2119887119900

(11)

The convergence condition of the solution is satisfied when|1198871119887119900| lt 1 and |(119886

1+ 21198871)(119886119900+ 2119887119900)| lt 1 That has been

checked for the selected basic materials

4 Results and Discussion

An artificial FGM is composed of two different kinds ofmaterial and the volume fraction of each material variesalong the thickness [22] As it is mentioned above andaccording to (1) and (2) both density and elastic constantsof FGMmaterial are functions of 119909

3coordinate The physical

properties of basic materials used in this study are shown inTable 1

In the present work metals are associated either withsilicon or ceramic The linear and nonlinear graded variationof volume fraction of metallic phase through the platethickness are investigated below on the basis of (1)

Figure 1 shows the variations of volume fraction of metal-lic phase through the plate thickness for 119901 = 1 and 2 Whenthe gradient coefficient 119901 is equal to one the left side is metal-rich and the right side is ceramic-rich The 119865

119898parameter

gives the mass rate of metal in the FGM plate Anywhere inthe plate the mass rate of metal is increased when 119901 = 2comparatively with 119901 = 1 for high values of 119901 the changetrend of properties is more pronounced

To study the S1-ZGV modes of the FGM plate different

kinds of FGM are considered The investigation includes dif-ferent basic materials accordingly the nature of the ceramicandor the nature of metal is changed The FGMs consideredin this study are reminded in Table 2 Similarly their S

1-ZGV

frequency 119891119900and cut-off frequencies 119891

119888are also reported for

both cases linear (119901 = 1) and nonlinear (119901 = 2)

41 Effect of Graded Variation on the S1-ZGV Mode The

dispersion curves provide information on the properties ofmaterials Some branches of the dispersion curves exhibitminima for nonzero wave numbers Such phenomenon hasbeen observed very early for the first-order symmetric (S

1)

mode [3ndash5]The dispersion curves of Lambwaves in an FGM


Table 2 S1-ZGV and cut-off frequencies of the different FGMs for119901 = 1 and 119901 = 2

FGM plate 119901 = 1 119901 = 2

119891119900(MHz) 119891

119888(MHz) 119891

119900(MHz) 119891

119888(MHz)

Cr-ceramic 229 240 215 225Cr-Si 230 240 218 228Ni-ceramic 215 225 200 212Ni-Si 214 228 201 214Cu-ceramic 186 194 159 169Cu-Si 180 187 162 171

0

01

02

03

04

05

06

07

08

09

1

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

x3d

Fm

() p = 1 p = 2

Figure 1 Variations of the volume fraction of metallic phasethrough the dimensionless thickness of the functionally grated plate

plate are located between those for the two correspondinghomogeneous plates [15 16] (Figure 2)

The dispersion curves and the S1-ZGV frequency are

influenced not only by the gradient functions but also by thegradient coefficients 119901 From Figure 3 one can see how theS1-ZGV frequency is sensitive to the FGM profiles When 119901

evolves from the linear to nonlinear case an appreciable shifttowards lower frequency is observed In fact to elucidate ourperception of S

1-ZGV modes for linear and nonlinear FGM

profile different basic materials are consideredThe observed shift towards low frequency in the above

plot (Cr-ceramic) has been checked for the other couplesof basic materials That shift is expected since nonlinear 119901corresponds to a FGM plate closer to the metallic phase

42 Influence of Shear Modulus on the S1-ZGV Frequency

From the investigation of FGM plates where the ceramicis kept unchanged one can see from Figure 4(a) that thelayout of different dispersion curves for the FGMs plates iscoherent with their metals shear modulus (120583Cu lt 120583Ni lt120583Cr) Accordingly the S1-ZGV frequency value seems to besensitive to the nature of the metallic component In thenumerical analysis the variation of metals shear modulus(from 817 Gpa to 1025 Gpa) corresponds to a S

1- ZGV fre-

quency shift about 240KHz Conversely the effect of ceramic

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

Ni-ceramicCr-ceramicCu-ceramic

S1

Figure 2 The dispersion curves of Lamb waves in the plates of CrFGM and ceramic

1500 2000 2500 3000 3500 4000 4500 5000 5500Frequency (MHz)

Phas

e velo

city

(ms

)

0

05

1

15

2

25

3

35

4

45

5times104

p = 1

p = 2

S1

Figure 3 S1-ZGV frequency shift when 119901 evolves from the linear to

nonlinear case

shear modulus is not so significant (Figure 4(b)) In factdespite a large deviation of ceramic shear modulus (795GPafor ceramic and 1181 GPa for Si) the S

1-ZGV frequency shift

does not exceed 10KHz Such shift increases according to thegradient coefficient 119901

Additionally the numerical investigation includes theeffect associated with the plate thickness variation from 119889 to2119889 As it was reported previously in literature [7 16] the ZGVfrequency exhibits a linear behavior with respect to (119889minus12)


Dispersion

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

Ni-ceramicCr-ceramic

Cu-ceramic

S1

(a)

Cr-SiCr-ceramic

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

S1A1

(b)

Figure 4 Influence of the shear modulus on the S1-ZGV frequency (a) metals shear modulus (b) ceramic shear modulus

15

2

25

3

35

4

15 2 25 3 35 4

fo

(MH

z)

fc (MHz)p = 1

p = 2

1205731 = 0953

1205732 = 0957

Figure 5 Variation of the 1198781-ZGV frequency 119891

119900versus the cut-off

frequency 119891119888for different values of 119901

That is still true for a nonlinear profile Accordingly the S1-

ZGV frequency reveals a high sensitivity either to the metalshear modulus or to the plate thickness That result can beexploited in the study of the microsystems

43 Effect of FGM Proprieties on the Shape Factor 120573 Fora homogenous plate it has been shown that the S

1-ZGV

frequency 119891119900varies linearly with the cut-off frequency (119891

119888)

according to the relation 119891119900= 120573 lowast 119891

119888 where 120573 is the shape

factor introduced by Sansalone et al [23 24]These ZGV and cut-off frequencies are studied when the

plate thickness of FGM plate is increased from 119889 to 2119889 Forthe nonlinear profile of the FGM plate the obtained results

reveal that the ZGV frequency119891119900presents the same behavior

a linear variation in terms of the cut-off frequency is reportedin Figure 5 The obtained linear variation in the case of thenonlinear FGM plate seems to be in agreement with theliterature [14] The gradient coefficient 119901 does not affect thelinear dependence of119891

119900as function of119891

119888 but 120573 varies slightly

(see Figure 5)Moreover 120573 undergoes slight change when 119901 varies from

1 to 2 That is mainly produced by a small change of thePoissonrsquos ratio 120592 due to its local character To illustrate how 119891

119900

depends on 119891119888 two kinds of FGM plates have been selected

Crceramic andNiceramic For the first couple ]Cr is smallerthan ]Ceramic whereas for the second couple ]Ni is greaterthan ]Ceramic The reported shift in Figure 5 is coherent withthe corresponding Poissonrsquos ratios The obtained result isconsistent with the relationship of 120573 according to ] given byClorennec [4]

44 Mechanical Displacement and Distribution of Stress Atthe dispersive region of the S

1mode the phase velocity

decreases rapidly when the frequency passes from 119891119888to 119891119900

and the guided wave leaves its steady character At the cut-off frequency (119891

119888) the whole surface is vibrating in phase

Conversely with a finite wave number ZGV modes give riseto local resonances So we focus on this resonance frequency119891119900to study the vibratory structure of the S

1Lamb modes in

the FGM plate In Figure 6 the variation of the stress and themechanical displacement are plotted as function of the depthat the 119891

119900frequency

Using the mechanical displacements 1198801(1199093) and 119880

3(1199093)

plotted in Figure 6(a) we can describe the wave power pene-tration through the thickness of FGM plate Because of theasymmetric properties of the FGM plate the displacement


minus05 0 05minus04

minus02

0

02

04

06

08

1

x3d

Disp

lace

men

t

U1

U2

(a)

minus05 0 05minus20

minus15

minus10

minus5

0

5times1013

Stre

ss

T13T33

x3d

(b)

Figure 6 Profiles of displacement (a) and stress (b) in the nonlinear ceramic-chromium FGM plate

amplitudes do not reveal a symmetric character as obtainedfor the homogeneous plate [3 20] The amplitudes of 119880

1and

1198803are comparatively high in the neighborhood of the free

surfaces The obtained profiles for the longitudinal and thetransverse components respectively 119880

1and 119880

3are coherent

with the symmetrical character of the ZGV modeBesides we verify on Figure 6(b) that stress components

11987913

and 11987933

vanish on free sides of the FGM plate Thispermits to be ensured about the computation process Thesame plots performed for the linear FGM profile [16] notincluded here show that displacement components are moresensitive than stress components to 119901 coefficient

5 Conclusion

Using the power series technique we have analytically solvedthe propagation of Lamb waves As an originating phe-nomenon the S

1-ZGV Lamb mode in a functionally graded

plate is studied Based on the PST governing equations forthe problem of Lamb waves that propagate in an FGM plateare derived For the kinds of FGM discussed in this paperthe S1-ZGV frequency in an FGM plate is between those for

the two corresponding homogeneous plates Moreover theS1-ZGV value depends on metal shear modulus and gradient

coefficient 119901Hence in both cases (linear and nonlinear) the metal

shear modulus influences enormously the S1-ZGV

On the other hand the linear dependence between S1-

ZGV frequency 119891119900and cut-off frequency 119891

119888is still observed

even for nonlinear profile The ZGV frequency provides alocal measurement of Poissonrsquos ratio At the point corre-sponding to ZGV frequency the displacement componentsare more sensitive than stress components

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press New York NY USA 1999

[2] D Clorennec C Prada and D Royer ldquoLocal and noncontactmeasurements of bulk acoustic wave velocities in thin isotropicplates and shells using zero group velocity Lamb modesrdquoJournal of Applied Physics vol 101 Article ID 034908 2007

[3] M Ces D Clorennec D Royer and C Prada ldquoThin layerthickness measurements by zero group velocity Lamb moderesonancesrdquo Review of Scientific Instruments vol 82 no 11Article ID 114902 2011

[4] I Tolstoy and E Usdin ldquoWave propagation in elastic plates lowand highmode dispersionrdquoThe Journal of the Acoustical Societyof America vol 29 pp 37ndash42 1957

[5] S D Holland and D E Chimenti ldquoAir-coupled acoustic imag-ing with zero-group-velocity Lamb modesrdquo Applied PhysicsLetters vol 83 no 13 pp 2704ndash2706 2003

[6] C Prada D Clorennec and D Royer ldquoLocal vibration of anelastic plate and zero-group velocity Lamb modesrdquo Journal ofthe Acoustical Society of America vol 124 no 1 pp 203ndash2122008

[7] D Clorennec C Prada and D Royer ldquoLaser ultrasonicinspection of plates using zero-group velocity lamb modesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 57 no 5 pp 1125ndash1132 2010

[8] G R Liu X Han and K Y Lam ldquoAn integration technique forevaluating confluent hypergeometric functions and its applica-tion to functionally graded materialsrdquo Computers amp Structuresvol 79 no 10 pp 1039ndash1047 2001


[9] X Han G R Liu K Y Lam and T Ohyoshi ldquoQuadratic layerelement for analyzing stress waves in FGMS and its applicationin material characterizationrdquo Journal of Sound and Vibrationvol 236 no 2 pp 307ndash321 2000

[10] X Han and G R Liu ldquoElastic waves in a functionally gradedpiezoelectric cylinderrdquo Smart Materials and Structures vol 12no 6 pp 962ndash971 2003

[11] Z H Qian F Jin Z K Wang and K Kishimoto ldquoTransversesurfacewaves on a piezoelectricmaterial carrying a functionallygraded layer of finite thicknessrdquo International Journal of Engi-neering Science vol 45 pp 455ndash466 2007

[12] X Y Li Z K Wang and S H Huang ldquoLove waves in func-tionally graded piezoelectric materialsrdquo International Journal ofSolids and Structures vol 41 no 26 pp 7309ndash7328 2004

[13] V Vlasie and M Rousseau ldquoGuided modes in a plane elasticlayer with gradually continuous acoustic propertiesrdquo NDT andE International vol 37 no 8 pp 633ndash644 2004

[14] J Liu and Z Wang ldquoStudy of the propagation of Rayleighsurface waves in a graded half-spacerdquoChinese Journal of AppliedMechanics vol 21 pp 106ndash109 2004

[15] XCao F Jin and I Jeon ldquoCalculation of propagation propertiesof Lambwaves in a functionally gradedmaterial (FGM) plate bypower series techniquerdquo NDT and E International vol 44 no1 pp 84ndash92 2011

[16] M S Bouhdima M Zagrouba and M H Ben Ghozlen ldquoThepower series technique and detection of zero-group velocityLamb waves in a functionally graded material platerdquo CanadianJournal of Physics vol 90 no 2 pp 159ndash164 2012

[17] C PradaO Balogun andTWMurray ldquoLaser-based ultrasonicgeneration and detection of zero-group velocity Lamb waves inthin platesrdquo Applied Physics Letters vol 87 no 19 Article ID194109 2005

[18] J N Reddy and C D Chin ldquoThermomechanical analysis offunctionally graded cylinders and platesrdquo Journal of ThermalStresses vol 21 no 6 pp 593ndash626 1998

[19] J N Reddy C N Wang and S Kitipornchai ldquoAxisymetricbending of functionally graded circular and annular platerdquoEuropean Journal of MechanicsmdashASolids vol 18 pp 185ndash1991999

[20] Z-Q Cheng andR C Batra ldquoThree-dimensional thermoelasticdeformations of a functionally graded elliptic platerdquoCompositesB Engineering vol 31 no 2 pp 97ndash106 2000

[21] J Woo and S A Maguid ldquoNonlinear analysis of functionallygraded plates and shallow shellsrdquo International Journal of Solidsand Structures vol 38 pp 7409ndash7421 2001

[22] L S Ma and T J Wang ldquoNonlinear bending and post-bucklingof a functionally graded circular plate under mechanical andthermal loadingsrdquo International Journal of Solids and Structuresvol 40 no 13-14 pp 3311ndash3330 2003

[23] M Sansalone and N J Carino ldquoNational Bureau of StandardsrdquoReport NBSIR 86-3452 NBSIR Gaithersburg Md USA 1986

[24] A Gibson and J S Popovics ldquoLambwave basis for impact-echomethod analysisrdquo Journal of Engineering Mechanics vol 131 no4 pp 438ndash443 2005

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




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DistributedSensor Networks



for 119901 = 2 is derivedThe performed developments permit theevaluation of the impact of the nonlinear profile ondispersioncurves of LambwavesThe effect of the shearmodulus on thisfrequency is highlighted Furthermore the relative variationof the stress and the mechanical displacement is investigatedat the S

1-ZGV frequency

The FGM plate data and the basic theory of PST arereported in Sections 2-3 The last paragraph is devoted to thediscussion and the main results

2 Statement of the Problem andTheoretical Study

A functionally graded plate with thickness ldquo119889rdquo is consideredhere It is assumed that the mechanical properties of FGMvary continuously through the thickness of plateThemotionis restricted in the (119909

1 1199093) plane and the Lamb waves

propagate in the positive direction of the1199091-axisThematerial

properties 120572 can be expressed as [18ndash21]

120572 (1199093) = 120572119888times 119865119888+ 120572119898times 119865119898 (1)

where 119865119898and 119865

119888are the volume fractions and the subscripts

119898 and 119888 denote the metallic and ceramic constituentsrespectively 119865

119888and 119865

119898follow a simple power law as

119865119888= (

1199093

119889+

1

2)119901

119865119898

= 1 minus 119865119888 (2)

where 1199093is the thickness coordinate and ldquo119901rdquo is a gradient

coefficient According to this distribution the bottom surface(1199093= minus1198892) of the functionally graded plate is puremetal and

the top surface (1199093= 1198892) is pure ceramic and for different

values of ldquo119901rdquo one can obtain different volume fractions ofmetal

The constitutive equations can be expressed as follows

119879119894119895= 119862119894119895119896119897

119878119896119897

119878119894119895=

1

2(120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)

120597119879119894119895

120597119909119895

= 120588 times 119894

(3)

In (3) 119879119894119895and 119878119896119897are the stress and strain tensors119862

119894119895119896119897are the

elastic coefficients 120588 is the density and 119906119894is the component

of the mechanical displacement in the 119894th directionOn the basis of the previous assumption of plane strain

the displacement components can be described as

1199061(1199091 1199093 119905) = 119880

1(1199093) exp [119894 (119896119909

1minus 120596119905)]

1199062= 0

1199063(1199091 1199093 119905) = 119894119880

3(1199093) exp [119894 (119896119909

1minus 120596119905)]

(4)

where 119896 is the wave number 120596 denotes the frequency and119894 = radicminus1 Note that for convenient description 119894 is introducedto make the first and third displacement components in

phase quadrature so that the polarization locus becomeselliptical Additionally the recursive process inherent to thePST method will have a suitable form On the other handand for brevity the complex exponential exp[119894(119896119909

1minus 120596119905)] is

omitted below From (3)-(4) the governing equations in aninhomogeneous FGM plate are rewritten as follows

11988844119880101584010158401+ 11988810158404411988010158401+ (1205881205962 minus 119888

111198962)1198801minus (11988813

+ 11988844) 11989611988010158403

minus 1198961198881015840441198803= 0

11988811119880101584010158403+ 11988810158401111988010158403+ (1205881205962 minus 119888

441198962)1198803+ (11988813

+ 11988844) 11989611988010158401

+ 1198881015840131198961198801= 0

(5)

The symbols (1015840) and (10158401015840) represent the first and second dif-ferentials with respect to 119909

3 The considered FGM materials

are isotropic so their elastic constants are expressed in termsof Lamersquos coefficients 120582 and 120583 this leads to 119888

11= 120582 + 2 sdot 120583

11988813

= 120582 and 11988844

= 120583Then (5) can be transformed into the following forms

120583119880101584010158401+ 12058310158401198801015840

1+ (1205881198882 minus 120582 minus 2120583) 1198962119880

1minus (120582 + 120583) 1198961198801015840

3

minus 11989612058310158401198803= 0

(120582 + 2120583)119880101584010158403+ (120582 + 2120583)

1015840

11988010158403+ (1205881198882 minus 120583) 1198962119880

3+ (120582 + 120583) 1198961198801015840

1

+ 12058210158401198961198801= 0

(6)

For Lambwaves that propagate in the FGMplate the tractionfree boundary condition should be satisfied at the top andbottom surfaces (119909

3= plusmn1198892) that is

11987913

(1199093= plusmn

119889

2) = 0 119879

33(1199093= plusmn

119889

2) = 0 (7)

Equations (6) are relative to the motion along 1199091and 119909

3 they

reveal coupling between both displacement amplitudes 1198801

and 1198803

3 Used Method

To solve the differential equation with variable coefficientswe use the PST method [15 16] Regarding the longitudinaland the shear wave amplitudes for Lamb guided waves thePST method specifies that 119880

1and 119880

3can take the following

forms

1198801= sum119899

119860119899(1199093

119889)119899

1198803= sum119899

119861119899(1199093

119889)119899

(8)



120582 (1199093) =

119901

sum119894=0

119886119894(1199093

119889)119894

120583 (1199093) =

119901

sum119894=0

119887119894(1199093

119889)119894

120588 (1199093) =

119901

sum119894=0

119888119894(1199093

119889)119894

(9)






119899and


(119899 + 2) (119899 + 1) 119887119900119860119899+2

+ (119899 + 1)21198871119860119899+1

minus 119896119889 (119899 + 1) (119886119900+ 119887119900) 119861119899+1

+ [11989621198892 (1198881199001198622 minus 119886

119900119900) + 119899 (119899 + 1) 119887

2] 119860119899

minus 119896119889 [1198991198861+ (119899 + 1) 119887

1] 119861119899

+ 11989621198892 (11988811198622 minus 119886

11)119860119899minus1

minus 119896119889 [(119899 minus 1) 1198862+ (119899 + 1) 119887

2] 119861119899minus1

+ 11989621198892 (11988821198622 minus 119886

22)119860119899minus2

= 0

(119899 + 2) (119899 + 1) 119886119900119900119861119899+2

+ (119899 + 1)211988611119861119899+1

+ 119896119889 (119899 + 1) (119886119900+ 119887119900) 119860119899+1

+ [11989621198892 (1198881199001198622 minus 119887

119900) + 119899 (119899 + 1) 119886

22] 119861119899

+ 119896119889 [1198991198871+ (119899 + 1) 119886

1] 119860119899+ 11989621198892 (119888

11198622 minus 119887

1) 119861119899minus1

+ 119896119889 [(119899 minus 1) 1198872+ (119899 + 1) 119886

2] 119860119899minus1

+ 11989621198892 (11988821198622 minus 119887

2) 119861119899minus2

= 0

(10)

with 119886119900119900

= 119886119900+ 2119887119900 11988611

= 1198861+ 21198871 11988622

= 1198862+ 21198872 119896 is the


119894 119887119894 119888119894(119894 = 0 1 2) are deduced




1198943written with








lim119899rarrinfin

119860119899+1

119860119899

=1198871

119887119900

lim119899rarrinfin

119861119899+1

119861119899

=1198861+ 21198871

119886119900+ 2119887119900

(11)


1+ 21198871)(119886119900+ 2119887119900)| lt 1 That has been








119898parameter




1-ZGV






1)




FGM plate 119901 = 1 119901 = 2

119891119900(MHz) 119891

119888(MHz) 119891

119900(MHz) 119891

119888(MHz)


0

01

02

03

04

05

06

07

08

09

1


x3d

Fm

() p = 1 p = 2











1- ZGV fre-


1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


S1


1500 2000 2500 3000 3500 4000 4500 5000 5500Frequency (MHz)

Phas

e velo

city

(ms

)

0

05

1

15

2

25

3

35

4

45

5times104

p = 1

p = 2

S1


nonlinear case






Dispersion

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


Cu-ceramic

S1

(a)

Cr-SiCr-ceramic

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

S1A1

(b)


15

2

25

3

35

4

15 2 25 3 35 4

fo

(MH

z)

fc (MHz)p = 1

p = 2

1205731 = 0953

1205732 = 0957







1-ZGV


119888)











119900









1Lamb modes in


119900frequency


3(1199093)



minus05 0 05minus04

minus02

0

02

04

06

08

1

x3d

Disp

lace

men

t

U1

U2

(a)

minus05 0 05minus20

minus15

minus10

minus5

0

5times1013

Stre

ss

T13T33

x3d

(b)



1and



1and 119880

3are coherent


11987913

and 11987933


5 Conclusion













References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120582 (1199093) =

119901

sum119894=0

119886119894(1199093

119889)119894

120583 (1199093) =

119901

sum119894=0

119887119894(1199093

119889)119894

120588 (1199093) =

119901

sum119894=0

119888119894(1199093

119889)119894

(9)






119899and


(119899 + 2) (119899 + 1) 119887119900119860119899+2

+ (119899 + 1)21198871119860119899+1

minus 119896119889 (119899 + 1) (119886119900+ 119887119900) 119861119899+1

+ [11989621198892 (1198881199001198622 minus 119886

119900119900) + 119899 (119899 + 1) 119887

2] 119860119899

minus 119896119889 [1198991198861+ (119899 + 1) 119887

1] 119861119899

+ 11989621198892 (11988811198622 minus 119886

11)119860119899minus1

minus 119896119889 [(119899 minus 1) 1198862+ (119899 + 1) 119887

2] 119861119899minus1

+ 11989621198892 (11988821198622 minus 119886

22)119860119899minus2

= 0

(119899 + 2) (119899 + 1) 119886119900119900119861119899+2

+ (119899 + 1)211988611119861119899+1

+ 119896119889 (119899 + 1) (119886119900+ 119887119900) 119860119899+1

+ [11989621198892 (1198881199001198622 minus 119887

119900) + 119899 (119899 + 1) 119886

22] 119861119899

+ 119896119889 [1198991198871+ (119899 + 1) 119886

1] 119860119899+ 11989621198892 (119888

11198622 minus 119887

1) 119861119899minus1

+ 119896119889 [(119899 minus 1) 1198872+ (119899 + 1) 119886

2] 119860119899minus1

+ 11989621198892 (11988821198622 minus 119887

2) 119861119899minus2

= 0

(10)

with 119886119900119900

= 119886119900+ 2119887119900 11988611

= 1198861+ 21198871 11988622

= 1198862+ 21198872 119896 is the


119894 119887119894 119888119894(119894 = 0 1 2) are deduced




1198943written with








lim119899rarrinfin

119860119899+1

119860119899

=1198871

119887119900

lim119899rarrinfin

119861119899+1

119861119899

=1198861+ 21198871

119886119900+ 2119887119900

(11)


1+ 21198871)(119886119900+ 2119887119900)| lt 1 That has been








119898parameter




1-ZGV






1)




FGM plate 119901 = 1 119901 = 2

119891119900(MHz) 119891

119888(MHz) 119891

119900(MHz) 119891

119888(MHz)


0

01

02

03

04

05

06

07

08

09

1


x3d

Fm

() p = 1 p = 2











1- ZGV fre-


1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


S1


1500 2000 2500 3000 3500 4000 4500 5000 5500Frequency (MHz)

Phas

e velo

city

(ms

)

0

05

1

15

2

25

3

35

4

45

5times104

p = 1

p = 2

S1


nonlinear case






Dispersion

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


Cu-ceramic

S1

(a)

Cr-SiCr-ceramic

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

S1A1

(b)


15

2

25

3

35

4

15 2 25 3 35 4

fo

(MH

z)

fc (MHz)p = 1

p = 2

1205731 = 0953

1205732 = 0957







1-ZGV


119888)











119900









1Lamb modes in


119900frequency


3(1199093)



minus05 0 05minus04

minus02

0

02

04

06

08

1

x3d

Disp

lace

men

t

U1

U2

(a)

minus05 0 05minus20

minus15

minus10

minus5

0

5times1013

Stre

ss

T13T33

x3d

(b)



1and



1and 119880

3are coherent


11987913

and 11987933


5 Conclusion













References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











FGM plate 119901 = 1 119901 = 2

119891119900(MHz) 119891

119888(MHz) 119891

119900(MHz) 119891

119888(MHz)


0

01

02

03

04

05

06

07

08

09

1


x3d

Fm

() p = 1 p = 2











1- ZGV fre-


1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


S1


1500 2000 2500 3000 3500 4000 4500 5000 5500Frequency (MHz)

Phas

e velo

city

(ms

)

0

05

1

15

2

25

3

35

4

45

5times104

p = 1

p = 2

S1


nonlinear case






Dispersion

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


Cu-ceramic

S1

(a)

Cr-SiCr-ceramic

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

S1A1

(b)


15

2

25

3

35

4

15 2 25 3 35 4

fo

(MH

z)

fc (MHz)p = 1

p = 2

1205731 = 0953

1205732 = 0957







1-ZGV


119888)











119900









1Lamb modes in


119900frequency


3(1199093)



minus05 0 05minus04

minus02

0

02

04

06

08

1

x3d

Disp

lace

men

t

U1

U2

(a)

minus05 0 05minus20

minus15

minus10

minus5

0

5times1013

Stre

ss

T13T33

x3d

(b)



1and



1and 119880

3are coherent


11987913

and 11987933


5 Conclusion













References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Dispersion

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104


Cu-ceramic

S1

(a)

Cr-SiCr-ceramic

Frequency (MHz)

Phas

e velo

city

(ms

)

1 15 2 25 3 35

times106

0

05

1

15

2

25

3

35

4

45

5times104

S1A1

(b)


15

2

25

3

35

4

15 2 25 3 35 4

fo

(MH

z)

fc (MHz)p = 1

p = 2

1205731 = 0953

1205732 = 0957







1-ZGV


119888)











119900









1Lamb modes in


119900frequency


3(1199093)



minus05 0 05minus04

minus02

0

02

04

06

08

1

x3d

Disp

lace

men

t

U1

U2

(a)

minus05 0 05minus20

minus15

minus10

minus5

0

5times1013

Stre

ss

T13T33

x3d

(b)



1and



1and 119880

3are coherent


11987913

and 11987933


5 Conclusion













References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus05 0 05minus04

minus02

0

02

04

06

08

1

x3d

Disp

lace

men

t

U1

U2

(a)

minus05 0 05minus20

minus15

minus10

minus5

0

5times1013

Stre

ss

T13T33

x3d

(b)



1and



1and 119880

3are coherent


11987913

and 11987933


5 Conclusion













References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article s -zgv modes of a linear and...

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