Lamb Modes for an Isotropic Incompressible Plate

Abstract

Lamb modes for an incompressible isotropic plate behave in a manner different from those for a compressible plate. The plateau region disappears and anomalous behavior of modes does not exist.

1. Introduction

For an infinite isotropic plate of thickness 2ℎ, the dispersion relation for symmetric Lamb modes is given by [1] tan𝑞ℎtan𝑝ℎ=−4𝑘²𝑝𝑞𝑞²−𝑘²², (1) where 𝑝=𝜔²𝑐²_𝐿−𝑘²,𝑞=𝜔²𝑐²_𝑇−𝑘². (2)

In (1), 𝑐_𝑇 and 𝑐_𝐿, respectively, denote the phase speeds of the transverse and longitudinal bulk waves in the material. Also 𝜔 and 𝑘, respectively, denote the frequency and the wave number of the mode. The phase velocity, 𝑐, of a mode is given by 𝜔𝑐=𝑘. (3) Differentiation with respect to 𝜔 gives 𝑑𝑐=1𝑑𝜔𝑘−𝜔𝑘²𝑑𝑘,𝑑𝜔𝑑𝑐=1𝑑𝜔𝑘𝑐1−𝑐_𝑔, (4) where 𝑐_𝑔=𝑑𝜔/𝑑𝑘 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 ℎ𝑘, the phase speeds of all modes except the lowest 𝑆₀ mode, asymptotically approaches 𝑐_𝑇 as the frequency 𝜔 tends to infinity, whereas that of the 𝑆₀ approaches 𝑐_𝑅, the speed of the Rayleigh wave in the material. 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.

Figure 1 

Spectrum of Lamb modes for a steel plate having plateau region.

If the normalized phase velocity is plotted as a function of frequency, by using (1), the 𝑆₁ 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 𝑆₁ mode for an isotropic material with 𝜈=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 𝜈=1/3, each of the mode 𝑆_3𝑛+1, 𝑛=0,1,2,… is found to be anomalous. Theoretical explanation of this peculiar behavior of the anomalous modes has been presented in Werby and Überall [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 𝜔_𝑛(𝑘)−𝜔_𝑛(0) for small values of the wave number 𝑘. 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, 𝑥₁𝑥₂𝑥₃, is chosen in such a way that 𝑥₂-axis is normal to the plate surface. With the (𝑥₁,𝑥₂) as plane of motion, the displacement components (𝑢₁, 𝑢₂, 𝑢₃) are such that 𝑢_𝑖=𝑢_𝑖𝑥₁,𝑥₂,𝑡,𝑖=1,2,𝑢₃≡0, (5) and the incompressibility condition, ∇⋅𝐮=0, reduces to 𝑢_1,1+𝑢_2,2=0. (6) The constitutive relation for an incompressible elastic material is given by 𝜎_𝑖𝑗=−𝑝𝛿_𝑖𝑗+𝑐_{𝑖𝑗𝑘𝑙}𝜖_𝑘𝑙, (7) where 𝜎_𝑖𝑗 is the stress tensor, 𝑝 is an arbitrary hydrostatic pressure associated with the incompressibility constraint, 𝑐_{𝑖𝑗𝑘𝑙} are the elastic constants, and 𝜖_𝑘𝑙 is the strain tensor. Traction free boundary conditions lead to the following dispersion relation for symmetric modes: tan𝑠₁𝑘ℎtan𝑠₂=𝑠𝑘ℎ¹1−𝑠²₂²𝑠₂1−𝑠²₁², (8) [8], where 𝑠²₁ and 𝑠²₂ are roots of the following quadratic equation: 𝑠⁴+2−𝜌𝑐²𝜇𝑠²+1−𝜌𝑐²𝜇=0, (9) hence, 𝑠²₁=−1, 𝑠²₂=𝜌𝑐²/𝜇−1, so (8) becomes tan𝑦²−1𝑥(𝑥)=4tanh𝑦²−12−𝑦²², (10) where we have defined the dimensionless wave number ℎ𝑘 by 𝑥 and normalized velocity (𝑐/𝑐_𝑇)(𝑐_𝑇=𝜇/𝜌) by 𝑦.

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 𝑆₁ 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 𝜈=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.

2. Dispersion Curves

Shape of the dispersion curves, in 𝑘−𝑐 plane, for a compressible plate, was discussed by Ahmad [12]. The plateau region occurs because, as can be easily shown, the slope of the 𝑛th mode, when 𝑦=𝜅, is given by 𝑑𝑦𝑑𝑥_𝑦=𝜅𝜅=−²−1^3/2𝜅²−2²𝑛𝜅⁴𝜋. (11) It is clear that for large 𝑛, the slope, while remaining negative approaches zero.

The dispersion curves for an incompressible isotropic plate, in the 𝑘−𝑐 plane, are shown in Figure 2.

Figure 2 

Symmetric Lamb modes for an incompressible isotropic plate in 𝑘−𝑐 plane.

All curves, except the lowest 𝑆₀ mode, approach the line 𝑦=1 corresponding to 𝑐=𝑐_𝑇. Also the phase speed of 𝑆₀ mode approaches the speed of the Rayleigh wave which is the unique root of the equation as follow: 4𝑐1−²𝑐²_𝑇=𝑐2−²𝑐²_𝑇². (12) Explanation of the above features follows on the same lines as for a compressible plate [12].

Since (10) does not depend on 𝑐_𝐿, so 𝑦=𝜅 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 [2–6] that the Lamb modes of a compressible isotropic plate in 𝜔−𝑐 plane exhibit a phenomenon known as “anomalous dispersion.” The 𝑆₁ mode in steel plate, as in most other materials, undergoes a turning point when 𝑘_𝑇=2.686, and group velocity is negative in the interval [2.686,2.873]. This is highlighted in Figure 3.

Figure 3 

The 𝑆1 Lamb mode for a steel plate in 𝜔−𝑐 plane.

A question naturally arises whether the anomaly persists even when the incompressibility constraint is enforced. To verify this define 𝑢=𝜔ℎ𝑐_𝑇=𝑐𝑘ℎ𝑐_𝑇=𝑦𝑥. (13)

Equation (10) in terms of 𝑢 and 𝑦 becomes tan𝑦²−1𝑢/𝑦=4tanh𝑢/𝑦𝑦²−12−𝑦²². (14)

Dispersion curves in the 𝑢−𝑦 plane, which is the same as 𝜔−𝑐 plane are shown in Figure 4.

Figure 4 

Symmetric Lamb modes for an incompressible isotropic plate in 𝑢−𝑦 plane.

The anomalous behavior of the 𝑆₁ mode disappears. Analytically this can be seen by examining slope of the modes given by (14) first when 𝑦≫1, secondly when 𝑦→1⁺.

We rewrite (14) in the form 𝑓𝑢,𝑦=tan𝑦²−1𝑢/𝑦(2−𝑦²)²𝑢−tanh𝑦4𝑦²−1=0. (15)

For 𝑦≫1, (14) becomes 𝑢4tanh𝑦≃tan(𝑢)𝑦³. (16) Since for large 𝑦𝑢tanh𝑦≃0, (17) hence, (16) leads to tan(𝑢)≃0, (18) or 𝑢_𝑛=𝑛𝜋+𝜖,𝑛=0,1,2,3,…, (19) where 𝜖 is an infinitesimally small positive number.

Now we will calculate 𝑑𝑦/𝑑𝑢 by the following formula: 𝑑𝑦𝑑𝑢=−𝜕𝑓/𝜕𝑢.𝜕𝑓/𝜕𝑦 (20) We will show that 𝑑𝑦/𝑑𝑢<0 for large value of 𝑦 for all modes.

For large value of 𝑦, the partial derivatives can be approximated as follows: 𝜕𝑓≃𝜕𝑢4−𝑦⁴cos𝑢,𝜕𝑓𝜕𝑦≃−𝑢𝑦cos𝑢+4𝑦³sin𝑢. (21) Then 𝑑𝑦𝑑𝑢_{𝑢=𝑢𝑛}𝑦≃−³𝑢_𝑛+4𝑦²tan𝑢_𝑛. (22)

Since 𝑢_𝑛+4𝑦²tan𝑢_𝑛>0, therefore 𝑑𝑦𝑑𝑢<0 (23) for all modes.

Now let 𝑦²=1+𝜖². (24) Dispersion relation (14) can be approximated in the following way: tan(𝜖𝑢)≃tanh𝑢4𝜖1−𝜖²²≃4𝜖1+2𝜖²()≃4𝜖,tan𝜖𝑢≃4𝜖tanh𝑢. (25) The derivative 𝑑𝑦/𝑑𝑢 can be easily shown to be, for large 𝑢_𝑛, as 𝑑𝑦≃𝜖𝑑𝑢−𝑢²_𝑛.+4 (26)

Thus, the modes 𝑆_𝑛(𝑛≠0) start off with negative slopes and the slope retains its sign till the end. Therefore, no anomalous behavior is expected.

Acknowledgments

This work has been fully supported by the University Research Fund of the National University of Sciences and Technology. Faiz Ahmad is grateful to the Higher Education Commission of Pakistan for financial support. Helpful comments of reviewers led to improvement of the paper.