Propagation of Lamb waves in multilayered elastic anisotropic plates is studied in the framework of combination of the six-dimensional Cauchy formalism and the transfer matrix method. The closed form secular equations for dispersion curves of Lamb waves propagating in multilayered plates with arbitrary elastic anisotropy are obtained.

Herein, a brief introduction to the theory of Lamb waves and a review of some of the most important works on this matter are presented.

The first works [

The complete theory of harmonic Lamb waves free from the long wavelength limit assumption was presented in [

In (

The potentials were assumed to be harmonic in time:

Substituting representation (

To define the spatial periodicity and to simplify the analysis, the splitting spatial argument is needed:

For the considered waves, it was further assumed that the displacement field does not depend upon argument

The further assumption relates to the periodicity of the potentials in the direction of propagation

In (

Substituting representations (

The unknown coefficients in (

The sign “+” refers to symmetrical and “−” to antisymmetrical modes. In view of expressions (

Taking the short wave limit at

Expression (

For the antisymmetric fundamental mode, (

Consider now the notion of the group velocity introduced by Stokes [

Numerical studies [

Several additional equations for computing dispersion of the group velocity can be constructed from (

In the first works on Lamb waves propagating in anisotropic plates, a three-dimensional formalism was used. Initially, that formalism was developed in [

All these publications, except [

Substituting representation (

(a) For Rayleigh waves, the roots

(b) Within the discussed formalism, the case of appearing multiple roots

(c) For anisotropic plates, the limiting velocities (

To obtain the dispersion equation, consider the traction-free boundary conditions

Finally, (

Initially, Stroh formalism [

Following [

For an isotropic material, matrices (

Up to the harmonic multiplier

The main idea of Stroh formalism is in rewriting the equation of motion (

Combining now (

The transfer matrix method suggested for analyzing propagation of Lamb and Rayleigh waves multilayered

Since its introduction, the transfer matrix method was applied to finding the dispersion relations for Lamb waves propagating in both isotropic [

In the following sections, the transfer matrix method will be coupled with the Cauchy six-dimensional formalism, and a numerically stable approach resembling [

The global matrix method introduced in [

There are several variants of constructing the global matrix. For a traction-free plate containing

Despite the reported numerical stability [

In the following, another variant of a six-dimensional formalism for analyzing dispersion of Lamb waves in plates with arbitrary anisotropy is presented. Since the developed formalism is based on reduction of the second-order equation (

For the considered formalism, the representation for harmonic Lamb wave is searched in the form (

Similar to (

The surface traction vector is defined by (

Equation (

Equation (

If both sides of a plate are traction free, then

Matrices

The condition (

Taking in (

Substituting matrices (

If both sides of a plate are clamped, then

Herein, two types of mixed boundary conditions at a boundary surface are considered: (i) vanishing normal surface-traction and vanishing tangential displacements and (ii) vanishing tangential surface-tractions and vanishing normal displacements. Boundary conditions (i) can be written in the form

Consider an

Assuming that propagation of the Lamb wave in each layer is determined by (

Now, multiplying both sides of (

As it was with (

While the global matrix equation (

Performing numerical computations with the exponential matrices leads to numerical instability associated with either loss of precision or overflow due to the presence of the exponential terms with large positive powers.

This problem can be relatively easily solved by a suitable normalization. Let the fundamental matrices

With decomposition (

Let

As reported in [

While being more time consuming, the first measure allows one to avoid possible overlook of a root. For example, consider a typical case with the following diagonal matrix:

Necessity to perform multiprecision computations can be demonstrated by considering the following matrix:

In numerical examples considered below mantissas from ~25 decimal digits (quadruple precision) to ~1000 decimal digits were used, depending upon the analyzed problem.

Three plates made of materials with cubic symmetry are considered, all with unit material density. To estimate deviation of elastic moduli of a cubic crystal from the isotropic elastic constants, the following measure can be introduced:

In (

The first considered crystal has elastic parameters that only slightly differ from the isotropic ones

The second cubic crystal has the following elastic parameters:

The third considered crystal belongs to a set of auxetic materials characterized by negative Poisson’s ratios [

For these crystals, vectors

Variation of limiting velocity

The plots in Figure

In the NDT, the limiting Lamb wave propagating with the velocity

Herein, two multilayered traction-free plates with alternating anisotropic layers are considered; the depth of each layer is

The alternating layers are made of hexagonal crystal SiC with _{3}N_{4} (

It should be noted that the SiC crystal is piezoelectric, so in a more rigorous approach [

The computed bulk wave velocities (in m/s) for the plane waves with normal

Dispersion curves for 3-layerd (a) and 11-layered (b) traction-free plates with alternating SiC-Si_{3}N_{4} layers.

These plots were constructed by the described transfer matrix method in combination with the quadruple precision computations (mantissas with ~34 decimal digits).

One of the most interesting observations concerns almost vertical parts of the dispersion curves at low-frequency limits; these are clearly seen in plots for both of the plates. The corresponding relative limiting velocity increases with increase of number of layers; however, the relative limiting velocity does not exceed unity, meaning that

The Cauchy six-dimensional formalism is developed for analyzing propagation of Lamb waves in anisotropic multilayered plates. The presented numerical examples reveal its ability to obtain dispersion curves for plates with large number of anisotropic layers.