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]
(1)tan(qh)tan(ph)=-4k2pq(q2-k2)2,
where
(2)p=ω2cL2-k2,q=ω2cT2-k2.

In (1), cT and cL, 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
(3)c=ωk.
Differentiation with respect to ω gives
(4)dcdω=1k-ωk2dkdω,dcdω=1k(1-ccg),
where cg=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 cT as the frequency ω tends to infinity, whereas that of the S0 approaches cR, the speed of the Rayleigh wave in the material. Near their crossing points of the line c=cL, 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.

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 S3n+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 ωn(k)-ωn(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
(5)ui=ui(x1,x2,t),i=1,2,u3≡0,
and the incompressibility condition, ∇·u=0, reduces to
(6)u1,1+u2,2=0.
The constitutive relation for an incompressible elastic material is given by
(7)σij=-pδij+cijklϵkl,
where σij is the stress tensor, p is an arbitrary hydrostatic pressure associated with the incompressibility constraint, cijkl are the elastic constants, and ϵkl is the strain tensor. Traction free boundary conditions lead to the following dispersion relation for symmetric modes:
(8)tan(s1kh)tan(s2kh)=s1(1-s22)2s2(1-s12)2,
[8], where s12 and s22 are roots of the following quadratic equation:
(9)s4+(2-ρc2μ)s2+(1-ρc2μ)=0,
hence, s12=-1, s22=ρc2/μ-1, so (8) becomes
(10)tan(y2-1x)tanh(x)=4y2-1(2-y2)2,
where we have defined the dimensionless wave number hk by x and normalized velocity (c/cT)(cT=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
(11)dydx|y=κ=-(κ2-1)3/2(κ2-2)2nκ4π.
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.

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=cT. Also the phase speed of S0 mode approaches the speed of the Rayleigh wave which is the unique root of the equation as follow:
(12)41-c2cT2=(2-c2cT2)2.
Explanation of the above features follows on the same lines as for a compressible plate [12].

Since (10) does not depend on cL, 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 kT=2.686, and group velocity is negative in the interval [2.686,2.873]. This is highlighted in 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
(13)u=ωhcT=ckhcT=yx.

Equation (10) in terms of u and y becomes
(14)tan(y2-1u/y)tanh(u/y)=4y2-1(2-y2)2.

Dispersion curves in the u-y plane, which is the same as ω-c plane are shown in 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
(15)f(u,y)=tan(y2-1u/y)(2-y2)2-tanh(uy)4y2-1=0.

For y≫1, (14) becomes
(16)4tanh(uy)≃tan(u)y3.
Since for large y(17)tanh(uy)≃0,
hence, (16) leads to
(18)tan(u)≃0,
or
(19)un=nπ+ϵ,n=0,1,2,3,…,
where ϵ is an infinitesimally small positive number.

Now we will calculate dy/du by the following formula:
(20)dydu=-∂f/∂u∂f/∂y.
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:
(21)∂f∂u≃(4-y4)cosu,∂f∂y≃-(uycosu+4y3sinu).
Then
(22)dydu|u=un≃-y3un+4y2tanun.

Since un+4y2tanun>0, therefore
(23)dydu<0
for all modes.

Now let
(24)y2=1+ϵ2.
Dispersion relation (14) can be approximated in the following way:
(25)tan(ϵu)tanhu≃4ϵ(1-ϵ2)2≃4ϵ(1+2ϵ2)≃4ϵ,tan(ϵu)tanhu≃4ϵ(1-ϵ2)2tan(ϵu)≃4ϵtanhu.
The derivative dy/du can be easily shown to be, for large un, as
(26)dydu≃ϵ2-un+4.

Thus, the modes Sn(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.

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