S1-ZGV Modes of a Linear and Nonlinear Profile for Functionally Graded Material Using Power Series Technique

The present work deals with functionally gradedmaterials (FGM) isotropic plates in the neighborhood of the first-order symmetric zero 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. Governing equations for the problemare derived, and the power series technique (PST) is employed to solve the recursive equations.The impact of 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 modulus.The gradient coefficient p does not affect the linear dependence of ZGV frequency f o as function of cut-off frequency f c ; only the slope is slightly varied.


Introduction
A functionally graded material (FGM) is a kind of an inhomogeneous material.The characterization of mechanical properties of materials is important for testing their structural integrity.Lamb waves are frequently employed in the ultrasonic characterization of thin plates [1].As an important property of Lamb waves, the zero group velocity (ZGV) at the frequency minimum   of the first-order symmetric (S 1 ) continues to be of an interest for the scientific community [2,3].Tolstoy and Usdin pointed out that for the S 1 Lamb mode, group velocity vanishes at a particular point of the dispersion curve and predicted that this zero group velocity point must be associated with a sharp continuous wave resonance and ringing effect [4].Holland and Chimenti demonstrated the exploitation of this mode for high-sensitivity imaging applications.With air-coupled transducers, they observed the transparency of a plate due to the S 1 mode ZGV resonance [5].
The S 1 -ZGV frequency is obviously sensitive to mechanical properties and to any change in the plate thickness.To exploit this phenomenon recent works evoke the idea that it may be suitable for the measurement of nanometer-scale thickness variations in homogeneous plates [6,7].Due to the resulting differential equations of variable coefficients associated with the spatial variation of the material properties, the wave propagation in FGM remains difficult to analyze.Some numerical [8][9][10] and analytical methods [11][12][13][14][15][16] have been applied in order to study the wave propagation behavior in an inhomogeneous medium with material properties varying continuously along the depth direction.In an effort to show the interest of ZGV in the study of FGM materials, Bouhdima [15] first discussed the effect of the linear variation of mechanical properties along the thickness plate on the S 1 -ZGV using the power series technique (PST).To our knowledge, no reports have been published on the relationship between the S 1 -ZGV frequency and material properties in an inhomogeneous free standing plate.Previous investigations on S 1 -ZGV phenomenon are limited to inspection experiments on homogenous plates and mainly focused on measuring the thickness of a coating on a relatively thin plate [3,7,17].The present investigation includes different kinds of FGM plates with various basic materials, to extract the effects produced by mechanical parameters variation.All the selected materials for the illustration are in agreement with the convergence criterion.The PST has been used and the recursive relationship 2 Advances in Acoustics and Vibration for  = 2 is derived.The performed developments permit the evaluation of the impact of the nonlinear profile on dispersion curves of Lamb waves.The effect of the shear modulus on this frequency is highlighted.Furthermore, the relative variation of the stress and the mechanical displacement is investigated at the S 1 -ZGV frequency.
The FGM plate data and the basic theory of PST are reported in Sections 2-3.The last paragraph is devoted to the discussion and the main results.

Statement of the Problem and Theoretical Study
A functionally graded plate with thickness "" is considered here.It is assumed that the mechanical properties of FGM vary continuously through the thickness of plate.The motion is restricted in the ( 1 ,  3 ) plane and the Lamb waves propagate in the positive direction of the  1 -axis.The material properties  can be expressed as [18][19][20][21] where   and   are the volume fractions and the subscripts  and  denote the metallic and ceramic constituents, respectively.  and   follow a simple power law as where  3 is the thickness coordinate and "" is a gradient coefficient.According to this distribution, the bottom surface ( 3 = −/2) of the functionally graded plate is pure metal and the top surface ( 3 = /2) is pure ceramic, and for different values of "" one can obtain different volume fractions of metal.
The constitutive equations can be expressed as follows: In (3),   and   are the stress and strain tensors,   are the elastic coefficients,  is the density, and   is the component of the mechanical displacement in the th direction.
On the basis of the previous assumption of plane strain, the displacement components can be described as where  is the wave number,  denotes the frequency, and  = √ −1.Note that for convenient description  is introduced to make the first and third displacement components in phase quadrature so that the polarization locus becomes elliptical.Additionally the recursive process inherent to the PST method will have a suitable form.On the other hand and for brevity, the complex exponential exp[( 1 − )] is omitted below.From ( 3)-( 4), the governing equations in an inhomogeneous FGM plate are rewritten as follows: The symbols (  ) and (  ) represent the first and second differentials with respect to  3 .The considered FGM materials are isotropic so their elastic constants are expressed in terms of Lame's coefficients  and ; this leads to  11 =  + 2 ⋅ ,  13 = , and  44 = .Then ( 5) can be transformed into the following forms: For Lamb waves that propagate in the FGM plate, the traction free boundary condition should be satisfied at the top and bottom surfaces ( 3 = ±/2), that is, Equations ( 6) are relative to the motion along  1 and  3 ; they reveal coupling between both displacement amplitudes  1 and  3 .

Used Method
To solve the differential equation with variable coefficients, we use the PST method [15,16].Regarding the longitudinal and the shear wave amplitudes for Lamb guided waves, the PST method specifies that  1 and  3 can take the following forms: It is assumed that the parameters of the FGM possess the following form: Substituting ( 8) and ( 9) into ( 6) and by equating the coefficients of ( 3 /)  to zero we can obtain two recursive equations.At this level any couple (  ,   ) can be expressed as a function of the quadruplet { 0 ,  1 ,  0 ,  1 }; this is true for the displacement components.Accordingly, any physical magnitude will have a four-dimensional vector form.For  = 2, the corresponding recursive relationships involving   and   are written below: with is the wave number, and  is the phase velocity.Some explorations related to coefficients denoted   ,   ,   ( = 0, 1, 2) are deduced from the properties of the FGM basic materials (see (9)).  and   are equal to zero if  < 0. The next step consists of putting boundary conditions in a suitable matrix form.Stress components  3 written with respect to { 0 ,  1 ,  0 ,  1 } on both sides of the plate give rise to a square matrix (4 × 4) dependent on  and .For a given frequency, the secular equation leads to the corresponding wave number and obviously to the phase velocity.Then, the From the recursive relationships one can deduce the convergence criteria: The convergence condition of the solution is satisfied when That has been checked for the selected basic materials.

Results and Discussion
An artificial FGM is composed of two different kinds of material and the volume fraction of each material varies along the thickness [22].As it is mentioned above and according to (1) and ( 2), both density and elastic constants of FGM material are functions of  3 coordinate.The physical properties of basic materials used in this study are shown in Table 1.
In the present work metals are associated either with silicon or ceramic.The linear and nonlinear graded variation of volume fraction of metallic phase through the plate thickness are investigated below on the basis of (1).
Figure 1 shows the variations of volume fraction of metallic phase through the plate thickness for  = 1 and 2. When the gradient coefficient  is equal to one, the left side is metalrich and the right side is ceramic-rich.The   parameter gives the mass rate of metal in the FGM plate.Anywhere in the plate the mass rate of metal is increased when  = 2 comparatively with  = 1.for high values of , the change trend of properties is more pronounced.
To study the S 1 -ZGV modes of the FGM plate, different kinds of FGM are considered.The investigation includes different basic materials; accordingly the nature of the ceramic and/or the nature of metal is changed.The FGMs considered in this study are reminded in Table 2. Similarly their S 1 -ZGV frequency   and cut-off frequencies   are also reported for both cases linear ( = 1) and nonlinear ( = 2).

Effect of Graded Variation on the S 1 -ZGV Mode.
The dispersion curves provide information on the properties of materials.Some branches of the dispersion curves exhibit minima for nonzero wave numbers.Such phenomenon has been observed very early for the first-order symmetric (S 1 ) mode [3][4][5].The dispersion curves of Lamb waves in an FGM  plate are located between those for the two corresponding homogeneous plates [15,16] (Figure 2).The dispersion curves and the S 1 -ZGV frequency are influenced not only by the gradient functions but also by the gradient coefficients .From Figure 3, one can see how the S 1 -ZGV frequency is sensitive to the FGM profiles.When  evolves from the linear to nonlinear case an appreciable shift towards lower frequency is observed.In fact, to elucidate our perception of S 1 -ZGV modes for linear and nonlinear FGM profile, different basic materials are considered.
The observed shift towards low frequency in the above plot (Cr-ceramic) has been checked for the other couples of basic materials.That shift is expected since nonlinear  corresponds to a FGM plate closer to the metallic phase.

Influence of Shear Modulus on the S 1 -ZGV Frequency.
From the investigation of FGM plates, where the ceramic is kept unchanged, one can see from Figure 4(a) that the layout of different dispersion curves for the FGMs plates is coherent with their metals shear modulus ( Cu <  Ni <  Cr ).Accordingly, the S 1 -ZGV frequency value seems to be sensitive to the nature of the metallic component.In the numerical analysis, the variation of metals shear modulus (from 81.7 Gpa to 102.5 Gpa) corresponds to a S 1 -ZGV frequency shift about 240 KHz.Conversely the effect of ceramic shear modulus is not so significant (Figure 4(b)).In fact, despite a large deviation of ceramic shear modulus (79.5 GPa for ceramic and 118.1 GPa for Si) the S 1 -ZGV frequency shift does not exceed 10 KHz.Such shift increases according to the gradient coefficient .
Additionally the numerical investigation includes the effect associated with the plate thickness variation from  to 2.As it was reported previously in literature [7,16], the ZGV frequency exhibits a linear behavior with respect to ( −1/2 ).That is still true for a nonlinear profile.Accordingly the S 1 -ZGV frequency reveals a high sensitivity either to the metal shear modulus or to the plate thickness.That result can be exploited in the study of the microsystems.

Effect of FGM Proprieties on the Shape
Factor .For a homogenous plate, it has been shown that the S 1 -ZGV frequency   varies linearly with the cut-off frequency (  ) according to the relation   =  *   , where  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  to 2.For the nonlinear profile of the FGM plate, the obtained results reveal that the ZGV frequency   presents the same behavior; a linear variation in terms of the cut-off frequency is reported in Figure 5.The obtained linear variation in the case of the nonlinear FGM plate seems to be in agreement with the literature [14].The gradient coefficient  does not affect the linear dependence of   as function of   , but  varies slightly (see Figure 5).
Moreover,  undergoes slight change when  varies from 1 to 2. That is mainly produced by a small change of the Poisson's ratio  due to its local character.To illustrate how   depends on   , two kinds of FGM plates have been selected Cr/ceramic and Ni/ceramic.For the first couple, ] Cr is smaller than ] Ceramic whereas, for the second couple, ] Ni is greater than ] Ceramic .The reported shift in Figure 5 is coherent with the corresponding Poisson's ratios.The obtained result is consistent with the relationship of  according to ], given by Clorennec [4].

Mechanical Displacement and Distribution of Stress.
At the dispersive region of the S 1 mode, the phase velocity decreases rapidly when the frequency passes from   to   and the guided wave leaves its steady character.At the cutoff frequency (  ), the whole surface is vibrating in phase.Conversely, with a finite wave number, ZGV modes give rise to local resonances.So, we focus on this resonance frequency   to study the vibratory structure of the S 1 Lamb modes in the FGM plate.In Figure 6, the variation of the stress and the mechanical displacement are plotted as function of the depth at the   frequency.
Using the mechanical displacements  1 ( 3 ) and  3 ( 3 ) plotted in Figure 6(a), we can describe the wave power penetration through the thickness of FGM plate.Because of the asymmetric properties of the FGM plate, the displacement amplitudes do not reveal a symmetric character as obtained for the homogeneous plate [3,20].The amplitudes of  1 and  3 are comparatively high in the neighborhood of the free surfaces.The obtained profiles for the longitudinal and the transverse components, respectively,  1 and  3 are coherent with the symmetrical character of the ZGV mode.
Besides we verify on Figure 6(b) that stress components  13 and  33 vanish on free sides of the FGM plate.This permits to be ensured about the computation process.The same plots performed for the linear FGM profile [16], not included here, show that displacement components are more sensitive than stress components to  coefficient.

Conclusion
Using the power series technique, we have analytically solved the propagation of Lamb waves.As an originating phenomenon, the S 1 -ZGV Lamb mode in a functionally graded plate is studied.Based on the PST, governing equations for the problem of Lamb waves that propagate in an FGM plate are derived.For the kinds of FGM discussed in this paper, the S 1 -ZGV frequency in an FGM plate is between those for the two corresponding homogeneous plates.Moreover, the S 1 -ZGV value depends on metal shear modulus and gradient coefficient .
Hence, in both cases (linear and nonlinear), the metal shear modulus influences enormously the S 1 -ZGV.
On the other hand, the linear dependence between S 1 -ZGV frequency   and cut-off frequency   is still observed even for nonlinear profile.The ZGV frequency provides a local measurement of Poisson's ratio.At the point corresponding to ZGV frequency, the displacement components are more sensitive than stress components.

Figure 1 :
Figure 1: Variations of the volume fraction of metallic phase through the dimensionless thickness of the functionally grated plate.

1 Figure 2 : 1 Figure 3 : S 1 -
Figure 2: The dispersion curves of Lamb waves in the plates of Cr, FGM and ceramic.

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

Table 1 :
The physical properties of basic materials.