Research Article  Open Access
S_{1}ZGV Modes of a Linear and Nonlinear Profile for Functionally Graded Material Using Power Series Technique
Abstract
The present work deals with functionally graded materials (FGM) isotropic plates in the neighborhood of the firstorder 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 problem are 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 does not affect the linear dependence of ZGV frequency as function of cutoff frequency ; only the slope is slightly varied.
1. 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 firstorder 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 highsensitivity imaging applications. With aircoupled 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 nanometerscale 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 [8–10] and analytical methods [11–16] 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 for 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 23. The last paragraph is devoted to the discussion and the main results.
2. 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 () plane and the Lamb waves propagate in the positive direction of the axis. The material properties can be expressed as [18–21] 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 is the thickness coordinate and “” is a gradient coefficient. According to this distribution, the bottom surface () of the functionally graded plate is pure metal and the top surface () 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 . 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 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 . The considered FGM materials are isotropic so their elastic constants are expressed in terms of Lame’s coefficients and ; this leads to , , and .
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 , that is, Equations (6) are relative to the motion along and ; they reveal coupling between both displacement amplitudes 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 and 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 to zero we can obtain two recursive equations. At this level any couple () can be expressed as a function of the quadruplet ; this is true for the displacement components. Accordingly, any physical magnitude will have a fourdimensional vector form. For , 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 , , () are deduced from the properties of the FGM basic materials (see (9)). and are equal to zero if .
The next step consists of putting boundary conditions in a suitable matrix form. Stress components written with respect to on both sides of the plate give rise to a square matrix () dependent on and . For a given frequency, the secular equation leads to the corresponding wave number and obviously to the phase velocity. Then, the dispersion curves of symmetric and antisymmetric propagative Lamb modes are represented by a set of branches in the plane .
From the recursive relationships one can deduce the convergence criteria: The convergence condition of the solution is satisfied when and . That has been checked for the selected basic materials.
4. 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 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 and 2. When the gradient coefficient is equal to one, the left side is metalrich and the right side is ceramicrich. The parameter gives the mass rate of metal in the FGM plate. Anywhere in the plate the mass rate of metal is increased when comparatively with . 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 cutoff frequencies are also reported for both cases linear () and nonlinear ().

4.1. 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 firstorder symmetric (S_{1}) mode [3–5]. 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 (Crceramic) 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.
4.2. 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 (). 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 .
(a)
(b)
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 (). 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.
4.3. Effect of FGM Proprieties on the Shape Factor β
For a homogenous plate, it has been shown that the S_{1}ZGV frequency varies linearly with the cutoff frequency () according to the relation , where is the shape factor introduced by Sansalone et al. [23, 24].
These ZGV and cutoff 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 cutoff 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, is smaller than whereas, for the second couple, is greater than . 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].
4.4. 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.
(a)
(b)
Using the mechanical displacements and 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 and are comparatively high in the neighborhood of the free surfaces. The obtained profiles for the longitudinal and the transverse components, respectively, and are coherent with the symmetrical character of the ZGV mode.
Besides we verify on Figure 6(b) that stress components and 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.
5. 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 cutoff 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2014 M. Zagrouba et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.