Lamb Modes for an Isotropic Incompressible Plate

c T as the frequency ω tends to infinity, whereas that of the S 0 approaches c R , the speed of the Rayleigh wave in thematerial. Near their crossing points of the line c = c L , the curves flatten out before embarking once again on the downward journey. This flat region is called the plateau region [2] as shown in Figure 1. If the normalized phase velocity is plotted as a function of frequency, by using (1), the S 1 mode in most materials, exhibits a region in which phase velocity is directed opposite to the group velocity [2]. A mode possessing such a region is called an anomalous mode. Tolstoy and Üsdin [3] were the first to observe this phenomenon for S 1 mode for an isotropic material with ] = 1/4. Negishi [4] observed this phenomenon in aluminum. Prada et al. [5] observed the anomalous behavior of Lamb modes in ω − k plane for Duralumin plate. For an isotropic material with Poisson’s ratio ] = 1/3, each of the mode


Introduction
For an infinite isotropic plate of thickness 2ℎ, the dispersion relation for symmetric Lamb modes is given by [1] tan ( ℎ) tan ( ℎ) = − 4 2 In (1), and , respectively, denote the phase speeds of the transverse and longitudinal bulk waves in the material. Also and , respectively, denote the frequency and the wave number of the mode. The phase velocity, , of a mode is given by = . ( Differentiation with respect to gives where = / denotes the group velocity. When the dispersion curves corresponding to (1) are plotted, depicting the normalized phase velocity as a function of dimensionless wave number ℎ , the phase speeds of all modes except the lowest 0 mode, asymptotically approaches as the frequency tends to infinity, whereas that of the 0 approaches , the speed of the Rayleigh wave in the material. Near their crossing points of the line = , the curves flatten out before embarking once again on the downward journey. This flat region is called the plateau region [2] as shown in Figure 1.
If the normalized phase velocity is plotted as a function of frequency, by using (1), the 1 mode in most materials, exhibits a region in which phase velocity is directed opposite to the group velocity [2]. A mode possessing such a region is called an anomalous mode.
Tolstoy andÜsdin [3] were the first to observe this phenomenon for 1 mode for an isotropic material with ] = 1/4. Negishi [4] observed this phenomenon in aluminum. Prada et al. [5] observed the anomalous behavior of Lamb modes in − plane for Duralumin plate. For an isotropic material with Poisson's ratio ] = 1/3, each of the mode 3 +1 , = 0, 1, 2, . . . is found to be anomalous. Theoretical explanation of this peculiar behavior of the anomalous modes has been presented in Werby andÜberall [2]. Shuvalov and Poncelet [6] have investigated the dispersion relation of a plate of unrestricted anisotropy. They identify anomalous modes by looking at the sign of the coefficient of ( )− (0) for small values of the wave number . Recently Hussain and Ahmad [7] considered zero-group velocity (ZGV) points in the spectrum of Lamb modes in compressible orthotropic plate. It was found that, in addition to modes with a single ZGV point, a large number of modes exist with multiple points.
In this paper we have studied the Lamb modes for an incompressible isotropic material for anomalous behavior of the modes. In [8,9], details of governing equations for incompressible solids have been given. A cartesian coordinate system, 1 2 3 , is chosen in such a way that 2 -axis is normal to the plate surface. With the ( 1 , 2 ) as plane of motion, the displacement components ( 1 , 2 , 3 ) are such that and the incompressibility condition, ∇ ⋅ u = 0, reduces to The constitutive relation for an incompressible elastic material is given by where is the stress tensor, is an arbitrary hydrostatic pressure associated with the incompressibility constraint, are the elastic constants, and is the strain tensor. Traction free boundary conditions lead to the following dispersion relation for symmetric modes: [8], where 2 1 and 2 2 are roots of the following quadratic equation: hence, 2 1 = −1, 2 2 = 2 / − 1, so (8) becomes  where we have defined the dimensionless wave number ℎ by and normalized velocity ( / ) ( = √ / ) by . The dispersion curves are plotted using a recent technique of Honarvar et al. [10]. It is found that there exists no mode with anomalous behavior. This result is derived analytically.
Rogerson [8] and Ogden and Roxburgh [9] discussed the dispersion relation asymptotically. The main result of the present work is to keep focus on the fact that incompressibility constraint in an isotropic material suppresses the ZGV phenomenon which exists in 1 and several other modes of a compressible material [2]. It appears that absence of ZGV Lamb modes in an incompressible isotropic plate, at least for materials with ] = 0 or 1, was known in Russia in the 1980s [11]. However, to the best of our knowledge, there does not exist any analytical proof of this fact in the literature.

Dispersion Curves
Shape of the dispersion curves, in − plane, for a compressible plate, was discussed by Ahmad [12]. The plateau region occurs because, as can be easily shown, the slope of the th mode, when = , is given by It is clear that for large , the slope, while remaining negative approaches zero. The dispersion curves for an incompressible isotropic plate, in the − plane, are shown in Figure 2.
All curves, except the lowest 0 mode, approach the line = 1 corresponding to = . Also the phase speed of 0 mode approaches the speed of the Rayleigh wave which is the unique root of the equation as follow: Mathematical Problems in Engineering Explanation of the above features follows on the same lines as for a compressible plate [12].
Since (10) does not depend on , so = has no significance, and there is no plateau region for curves in Figure 2. The equation does not involve any material parameter; hence, Figure 2 represents the dispersion curves for all incompressible isotropic materials.
It is well known [2][3][4][5][6] that the Lamb modes of a compressible isotropic plate in − plane exhibit a phenomenon known as "anomalous dispersion. " The 1 mode in steel plate, as in most other materials, undergoes a turning point when = 2.686, and group velocity is negative in the interval [2.686, 2.873]. This is highlighted in Figure 3.
A question naturally arises whether the anomaly persists even when the incompressibility constraint is enforced. To verify this define Equation (10) in terms of and becomes Dispersion curves in the − plane, which is the same as − plane are shown in Figure 4.
The anomalous behavior of the 1 mode disappears. Analytically this can be seen by examining slope of the modes given by (14) first when ≫ 1, secondly when → 1 + .
We rewrite (14) in the form For ≫ 1, (14) becomes Since for large hence, (16) leads to tan ( ) ≃ 0, where is an infinitesimally small positive number. Now we will calculate / by the following formula: We will show that / < 0 for large value of for all modes. For large value of , the partial derivatives can be approximated as follows: Since + 4 2 tan > 0, therefore for all modes.
The derivative / can be easily shown to be, for large , as Thus, the modes ( ̸ = 0) start off with negative slopes and the slope retains its sign till the end. Therefore, no anomalous behavior is expected.