1. Introduction
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.
1.1. Lamb Waves in a Homogeneous Isotropic Plate
The first works [1, 2] on waves propagating in an infinite isotropic homogeneous plate with the traction-free boundary surfaces were done at the assumption that the wavelength is much longer than the plate thickness.
The complete theory of harmonic Lamb waves free from the long wavelength limit assumption was presented in [3]. The starting point of the Lamb theory is considering the equation of motion in the form
(1)cP∇div u-cSrot rot u=∂2u∂t2,
where u is the displacement field and cP and cS are velocities of the longitudinal and transverse bulk waves, respectively:
(2)cP=λ+2μρ, cS=μρ.
In (2), λ and μ are Lamé constants and ρ is the material density. Then, the displacement field was represented in terms of scalar (Φ) and vector (Ψ) potentials
(3)u=∇Φ+rotΨ.
The potentials were assumed to be harmonic in time:
(4)Φ(x,t)=Φ′(x)eiωt, Ψ(x,t)=Ψ′(x)eiωt.
Substituting representation (4) into (1) yields two independent Helmholtz equations:
(5)(Δ+ω2cP2)Φ′=0, (Δ+ω2cS2)Ψ′=0.
To define the spatial periodicity and to simplify the analysis, the splitting spatial argument is needed:
(6)x=(x·n)n+(x·ν)ν+(x·w)w,
where n is the unit wave vector, ν is the unit normal to the median plane of the plate, and w=n×ν.
Remark 1.
For the considered waves, it was further assumed that the displacement field does not depend upon argument x·w. That allowed Lamb [3] to consider scalar potentials Ψ and Ψ′ in (4) instead of vector ones (actually, Lamb considered the vector potential composed of one nonvanishing component (Ψ·w)w).
The further assumption relates to the periodicity of the potentials in the direction of propagation
(7)Φ′(x)=φ(x′′)ex′, Ψ′(x)=ψ(x′′)ex′,
where the dimensionless complex coordinates x′ and x′′ are
(8)x′=irx·n, x′′=irx·ν.
In (8), i=-1 and r is the wave number related to the wavelength l by
(9)r=2πl.
Substituting representations (7) into (5) results in the decoupled ordinary differential equations
(10)d2φdx′′2+(1-c2cP2)φ=0, d2ψdx′′2 +(1-c2cS2)ψ=0,
where the phase speed c relates to the frequency and the wave number by the following relation:
(11)c=ωr.
The general solution of (10) can be written in the form
(12)φ(x′′)=C1sinh(γ1x′′)+C2cosh(γ1x′′),ψ(x′′)=C3sinh(γ2x′′)+C4cosh(γ2x′′),
where
(13)γ1=(1-c2cP2)1/2, γ2=(1-c2cS2)1/2.
The unknown coefficients in (12) are defined (up to a multiplier) from the following boundary conditions on the free surfaces:
(14)tν≡(λtr(∇u)I+μ(∇u+∇ut))·ν=0,x·ν=±h,
where 2h is the depth of the plate. Substitution representation (3) into boundary conditions (14) yields boundary conditions written in terms of potentials φ′′ and ψ′′:
(15)(λΔΦ′I+2μ(∇∇Φ′+12(+(∇ rot Ψ′)t∇rotΨ′ +(∇ rot Ψ′)t)(∇∇Φ′+12)(∇∇Φ′+12)·ν=0,x·ν=±h.
Substituting solutions (12) into (15), in view of Remark 1, yields the desired dispersion equation, found in [1, 3]:
(16)tanh(γ2rh)tanh(γ1rh)-(4γ1γ2(1+γ22)2)±1=0.
The sign “+” refers to symmetrical and “−” to antisymmetrical modes. In view of expressions (11) and (13), the obtained dispersion equation implicitly determines the phase velocity c as a function of frequency. Equations for velocities related to the long wave and short wave limits were found in [3].
Taking the short wave limit at rh→∞ in (16) yields
(17)1=(4γ1γ2(1+γ22)2).
The latter expression coincides with the secular equation for the speed of Rayleigh waves derived in [4], and hence the first limiting speed coincides with the speed of Rayleigh wave
(18)c1,lim=cR.
Analysis of (16) at rh→0 (the long wave limit) yields the following equation [5]:
(19)γ2γ1=(4γ1γ2(1+γ22)2),
from where the values for two limiting velocities can be obtained:
(20)c2,lim=2cS1-cS2cp2.(21)c3,lim=cS.
Expression (20) coincides with the limiting velocity value found in [1, 2] and differs from the anticipated value for the long wave limiting velocity of the sound waves in rods.
For the antisymmetric fundamental mode, (16) was analytically solved in [6] using the perturbation technique. Earlier studies of the dispersion curves for Lamb waves in a layer contacting with a half-space were mainly associated with the geophysical applications [7–10]. The analysis of the dispersion curves at different Poisson’s ratios (including negative values) was done in [11–13]. The points of intersection of the dispersion curves were studied in [14].
1.2. Group Velocity
Consider now the notion of the group velocity introduced by Stokes [15] for description of the wave package propagation in hydrodynamics and later extrapolated to acoustic waves in the theory of elasticity; see [16–18]. Formally, the group velocity can be defined by
(22)cgroup=dωdr.
Numerical studies [19–25] of the group velocity dispersion, mainly at Poisson’s condition λ=μ, confirmed Rayleigh’s anticipation [16, 17] that the negative values of the group velocity can appear at the very small wave numbers. Later on, numerical computations [25] revealed the existence of a broader range of negative group velocities at Poisson’s ratios belonging to the interval 0.31<ν<0.45.
Several additional equations for computing dispersion of the group velocity can be constructed from (16). For example, substituting the phase speed defined by (11) into (13), denoting the left-hand side of (16) by F±(r,ω), and assuming that ω is a function of r, the derivative of (16) with respect to r takes the form
(23)∂F±(r,ω)∂r+∂F±(r,ω)∂ωdωdr=0,
from where, in view of (22), the secular equation for the group speed takes the form
(24)cgroup=-(∂F±(r,ω))/∂r(∂F±(r,ω))/∂ω.
In (24), r is considered as a function of ω. Theoretical studies of the phase, group, and ray velocities were done in [26].
1.3. Homogeneous Anisotropic Plates: Three-Dimensional Formalism
In the first works on Lamb waves propagating in anisotropic plates, a three-dimensional formalism was used. Initially, that formalism was developed in [27] for analysis of Rayleigh waves in an anisotropic half-space; later on, this method was applied to half-spaces with different groups of elastic symmetry [28–32]. With necessary modification, the approach [27] was used for analyzing Lamb waves in anisotropic plates [33–43].
All these publications, except [33] where a more complicated case of a cylindrically anisotropic plate was considered, actually exploited the following representation for the displacement field:
(25)u(x,t)=(∑k=16Ckmkeγkx′′︸uk(x′′))ei(rn·x-ωt),
where Ck are arbitrary complex coefficients determined up to a multiplier by satisfying conditions at the plate boundaries, uk is the displacement field of the kth partial wave, mk is a vectorial, generally, complex amplitude, determined by the Christoffel equation (this equation will be introduced later on), and γk is a root of the Christoffel equation. Note that, according to (8), the coordinate x′′ is imaginary. Six partial waves in (25) correspond to six (not necessary aliquant) roots of the Christoffel equation.
Substituting representation (25) into equation of motion
(26)A(∂x,∂t)u≡divxC··∇xu-ρu¨=0,
where C is the fourth-order elasticity tensor assumed to be positive definite, yields the Christoffel equation
(27)[(γkν+n)·C·(n+γkν) -ρc2I]·mk=0,
where I is the unit diagonal matrix. Equation (27) admits the equivalent form
(28)det[(γkν+n)·C·(n+γkν)-ρc2I]=0.
The left-hand side of (28) represents a polynomial of degree six with respect to the Christoffel parameter γk. Equations (27), (28) show that roots γk and the corresponding eigenvectors mk can be considered as functions of the phase speed c.
Remark 2.
(a) For Rayleigh waves, the roots γk in representation (25) should be complex with Im(γk)<0; this ensures attenuation of Rayleigh wave in the “lower” half-space (ν·x)<0. The condition Im(γk)<0 confines the admissible speed interval and decreases the number of summation terms in (25). If Re(γk)=0 for all partial waves composing Rayleigh wave, then such a wave is called the genuine Rayleigh wave; if Re(γk)≠0 for some k, then such a wave is called the generalized Rayleigh wave [28]. For Lamb waves, the cases Re(γk)=0 and Re(γk)≠0 are usually not distinguished.
(b) Within the discussed formalism, the case of appearing multiple roots γk and the coincident kernel eigenvectors mk was considered in [44] with application to Rayleigh waves and in [45] for obtaining the dispersion equation of subsonic Lamb waves.
(c) For anisotropic plates, the limiting velocities (20), (21) were computed in [46, 47].
To obtain the dispersion equation, consider the traction-free boundary conditions
(29)tν|x·ν=±h≡ ± ν·C ··∇xu|x·ν=±h=0,
where 2h denotes the overall thickness of the plate. Substituting representation (25) into the boundary conditions (29) yields the dispersion equation in the form
(30)∑k=16Ck(γkν·C·ν+ν·C·n)·mke±irγkh=0.
Finally, (30) can be rewritten in a form better suited for numerical computations:(31)det((γ1ν·C·ν+ν·C·n)·m1e+irγ1h⋯(γ6ν·C·ν+ν·C·n)·m6e+irγ6h(γ1ν·C·ν+ν·C·n)·m1e-irγ1h⋯(γ6ν·C·ν+ν·C·n)·m6e-irγ6h)=0.The dispersion equation (31) defines the phase speed as a function of the wave number or, in view of (11), as a function of the circular frequency.
1.4. Homogeneous Anisotropic Plates: Stroh Six-Dimensional Formalism
Initially, Stroh formalism [48] was applied to analysis of Rayleigh waves propagating on a free boundary of an anisotropic half-space [49–55]. The case of the nonsemisimple degeneracy of the fundamental matrix was considered in [53] (that case is associated with appearing multiple roots and the coincident kernel eigenvectors in the Christoffel equation). In [56, 57], the Stroh formalism was applied to the description of Lamb waves propagating in the homogeneous anisotropic plates.
Following [49], the displacement field for Rayleigh or Lamb wave is searched in the form
(32)u(x,t)=f(x′′)ei(rn·x-ωt)
with the unknown amplitude f regarded as a function of the imaginary coordinate x′′ defined by (8). Substituting representation (32) into equation of motion (26) yields
(33)(A1∂x′′2+(A2+A2t)∂x′′+A3)·f(x′′)=0,
where
(34)A1=ν·C·ν, A2=ν·C·n, A3=n·C·n-ρc2I.
The matrix A1 is nondegenerate due to the assumption of positive definiteness of the elasticity tensor.
Remark 3.
For an isotropic material, matrices (34) take the following form:
(35)A1=(λ+2μ000μ000μ), A2=(0λ0μ00000),A3=(μ-ρc2000λ+2μ-ρc2000μ-ρc2).
Up to the harmonic multiplier ei(rn·x-ωt), the surface tractions on the planes parallel to the free boundary (or the median surface of a plate) can be written in terms of matrices (34):
(36)tν(x′′)≡(A1∂x′′+A2)·f(x′′).
The main idea of Stroh formalism is in rewriting the equation of motion (26) in terms of the displacements and surface tractions. Multiplying both sides of (36) by matrix A1-1 yields the following expression for the derivative ∂x′′f(x′′):
(37)∂x′′f(x′′)=A1-1·tν(x′′)-A1-1·A2·f(x′′).
Combining now (33)–(37) produces the desired equation of motion written in terms of vectors f and tν:
(38)∂x′′(ftν)=N·(ftν),
where N is the fundamental matrix [58]
(39)N=(-A1-1·A2A1-1A2t·A1-1·A2-A3-A2t·A1-1).
Now, the general solution of (38) can be represented in the form
(40)(ftν)=exp(x′′N)·C→6,
where C→6 is the six-dimensional generally complex vector of the unknown coefficients; that vector can be defined (up to a multiplier) by the boundary conditions. Substituting solution (40) into boundary conditions (29) yields(41a)(f(+irh)0)=exp(+irhN)·C→6,(41b)(f(-irh)0)=exp(-irhN)·C→6.Excluding vector C→6 from (41a) and substituting the resultant expression into (41b) gives the following dispersion equation for an anisotropic plate:
(42)det((0,I)·exp(-2irhN))·(I0)=0.
Despite the obvious simplicity of deriving (42), the relevant works on Lamb waves in anisotropic plates [56, 57] use a more complicated procedure.
1.5. Multilayered Plates: The Transfer Matrix Method
The transfer matrix method suggested for analyzing propagation of Lamb and Rayleigh waves multilayered isotropic media was introduced in [59, 60]. Up to now, the transfer matrix method was combined with the three-dimensional formalism. The main idea of the method is to exclude the unknown coefficients Ck in representation (25) expressing them in terms of the only partly known surface tractions and displacements on one of the outer surfaces of the plate; such a procedure is similar to (40a), (41b). If the plate contains several layers, the interface conditions can also be expressed in terms of these surface tractions and displacements via specially constructed transfer matrices. Finally, the boundary conditions on the other outer surface are expressed in terms of the surface tractions and displacements from the first outer surface.
Since its introduction, the transfer matrix method was applied to finding the dispersion relations for Lamb waves propagating in both isotropic [60–62] and monoclinic (or with higher symmetries) [63, 64] multilayered plates. Despite the abundance and simplicity, the original variant of the transfer matrix method revealed some numerical instability, especially when high frequencies and large depths of the layers were considered. To overcome this problem, the numerically stable algorithms were suggested [65–70]. Differing in details, these algorithms have the common principle idea of eliminating the terms containing large exponentials. Such a procedure, developed in [65], is known as the δ-matrix method [66, 67]. The δ-matrix method can also be applied for analyzing dispersion of Lamb waves in a single-layered plate; see [65].
In the following sections, the transfer matrix method will be coupled with the Cauchy six-dimensional formalism, and a numerically stable approach resembling [65–70] will be worked out.
1.6. Multilayered Plates: The Global Matrix Method
The global matrix method introduced in [71] appeared more numerically stable than the original version of the transfer matrix method; see [62, 65] for discussions. In [72–77] applications of the global matrix method to analyses of Lamb wave dispersion are presented.
There are several variants of constructing the global matrix. For a traction-free plate containing n-homogeneous anisotropic layers, the matrix equation related to the global matrix method can be written in the following form:
(43)0→6n=((0,I)·D1+D1--D2+D2--D3+⋯⋯(0,I)·Dn-)·(C→1C→2C→3⋮C→n),
where C→k∈ℂ6, k=1,…,n, are six-dimensional generally complex vectors containing the unknown coefficients in representation (25); Dk± are 6×6-matrices defining displacements by (25) and surface tractions by (30) on both surfaces of a layer: (44)Dk±=(mk1e±irγk1hk⋯mk6e±irγk6hk(γk1ν·Ck·ν+ν·Ck·n)·mk1e±irγk1hk⋯(γk6ν·Ck·ν+ν·Ck·n)·mk6e±irγk6hk).
Despite the reported numerical stability [72–74], the comparative study of the original transfer matrix method [58, 59] and the global matrix method [71] revealed that actually both methods exhibit some intrinsic instability resulting in loss of accuracy at high frequencies and large depths of the layers [78]. To overcome the persistent instability, in [78] a more refined rearrangement of the exponential terms, appearing in the global matrix method, was suggested.