# A Mathematical Analysis of the Strip-Element Method for the Computation of Dispersion Curves of Guided Waves in Anisotropic Layered Media

Authors: F. Schöpfer; F. Binder; A. Wöstehoff; T. Schuster
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

Dispersion curves of elastic guided waves in plates can be efficiently
computed by the Strip-Element Method. This method is based on a finite-element
discretization in the thickness direction of the plate and leads to an eigenvalue
problem relating frequencies to wavenumbers of the wave modes. In this paper
we present a rigorous mathematical background of the Strip-Element Method
for anisotropic media including a thorough analysis of the corresponding infinite-dimensional
eigenvalue problem as well as a proof of the existence of eigenvalues.


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## Body

## 1. Introduction

In recent years there has been considerable interest in the study of the behaviour of elastic guided waves in plates due to their potential use in Nondestructive Evaluation (NDE) and Structural Health Monitoring (SHM); see, for example, the comprehensive books of Giurgiutiu [1] or Rose [2]. Already in isotropic plates Lamb waves are dispersive and the dispersion relations expressed by the Rayleigh-Lamb equations must be computed numerically; see, the study by Achenbach in [3]. Elastic wave propagation in layered and anisotropic media is an even more complex problem, and efficient numerical methods are required to obtain dispersion curves. Among those methods are the Transfer Matrix Method and the Global Matrix Method and we refer to the study by Lowe in [4] for an overview. One of the most efficient and flexible methods is based on a finite-element discretization in the thickness direction of the plate and leads to a generalized eigenvalue problem relating frequencies to wavenumbers of the Lamb wave modes. This method is known, amongst others, as the strip-element method (SEM) or layer-element method or semianalytic finite-element-method; see, for example, the early works of Dong and Nelson [5] and Aalami [6], or Kausel [7], Galán and Abascal [8], the excellent book of Liu and Xi [9], and the many references therein. Also the works of Gavrić [10], Bartoli et al. [11], Marzani et al. [12], and Treyssède [13] could be of interest.So far, there seems to be no strict mathematical analysis of the underlying infinite-dimensional eigenvalue problem in the anisotropic case. For the isotropic case we recommend reading the paper by Bouhennache in [14], who also uses a more abstract setting. Since the SEM has been successfully applied in practice for several years, we think that it is worthwhile to start such an analysis and give here a mathematical proof of the existence of eigenvalues.In the next section we recall some basic facts about generalized eigenvalue problems in Hilbert spaces. The differential equation governing the wave propagation in laminated plates is formulated in Section3. In the main Section 4 we analyse the weak form of those equations for the elementary Lamb wave modes, show that weak and strong solutions coincide and are layerwise 𝒞∞, and prove existence of weak solutions and eigenvalues of the related eigenvalue problem. Finally we present some numerical results illustrating the increasingly direction-dependent behaviour of transversely isotropic material with an increasing degree of anisotropy.


## 2. Generalized Eigenvalue Problems in Hilbert Spaces

Although the results of this section are wellknown, they might not always be explicitly found in the given form and therefore we prove some of them for the convenience of the reader. For further reading we suggest the books of Lax [15] and Conway [16].Letℋ,ℋ1,ℋ2 be real or complex Hilbert spaces with scalar products 〈·∣·〉. By ℒ(ℋ1,ℋ2) we denote the space of continuous linear operators from ℋ1 to ℋ2 and set ℒ(ℋ):=ℒ(ℋ,ℋ). For T∈ℒ(ℋ1,ℋ2) we denote by T*∈ℒ(ℋ2,ℋ1) the adjoint operator defined by
(2.1)〈Tv∣u〉ℋ2=〈v∣T*u〉ℋ1,∀v∈ℋ1,u∈ℋ2.
Range and nullspace are denoted by ℛ(T) and 𝒩(T), respectively. We say that T∈ℒ(ℋ) is self-adjoint if T*=T, and positive if
(2.2)〈u∣Tu〉ℋ≥0,∀u∈ℋ.Lemma 2.1.
LetT∈ℒ(ℋ) be self-adjoint. If T is injective and has closed range, then it is bijective and hence continuously invertible.Proof.
For𝒳⊂ℋ, let 𝒳⊥={x∈ℋ∣〈x∣y〉=0,for ally∈ℋ} be the orthogonal complement of 𝒳. The assertion follows from
(2.3)ℛ(T)=ℛ(T)¯=𝒩(T*)⊥=𝒩(T)⊥={0}⊥=ℋ,
which means that T is also surjective.Lemma 2.2.
An operatorT∈ℒ(ℋ1,ℋ2) is injective and has closed range if and only if there exists some constant c>0 such that
(2.4)∥Tu∥ℋ2≥c∥u∥ℋ1,∀u∈ℋ1.Proof.
See, for example, the study by Schröder in [17].We obtain the following.Corollary 2.3.
An operatorT∈ℒ(ℋ) is injective and has closed range if there exists some constant c>0 such that
(2.5)〈u∣Tu〉ℋ≥c∥u∥ℋ2,∀u∈ℋ.Proof.
Together with the Cauchy-Schwartz inequality, we obtain from (2.5)
(2.6)∥u∥ℋ∥Tu∥ℋ≥〈u∣Tu〉ℋ≥c∥u∥ℋ2,∀u∈ℋ.
With Lemma 2.2, we can conclude the assertion.ForM,S∈ℒ(ℋ), consider the generalized eigenvalue problem
(2.7)Mu=λSu.
That is, “for which λ∈ℝ,ℂ do nontrivial solutions u∈ℋ to (2.7) exist?” In case S is bijective, (2.7) is equivalent to the standard eigenvalue problem
(2.8)S-1Mu=λu,
and we readily obtain the following proposition which we state for the case dimℋ=∞. It applies accordingly to the finite-dimensional case with finite sequences λn, un.Proposition 2.4.
LetM,S∈ℒ(ℋ) be self-adjoint and positive. Assume that M is compact and injective and that S is bijective. Then there exists a decreasing sequence 0<λn→0 (counted with multiplicity) and a sequence un∈ℋ such that
(2.9)Mun=λnSun,(2.10)1λn〈un∣Mum〉ℋ=δn,m=〈un∣Sum〉ℋ,
and each u∈ℋ can be uniquely represented by an ℋ-convergent series
(2.11)u=∑n=1∞αnun,
with an ℓ2-sequence αn=〈u∣Sun〉ℋ. Especially, 〈v∣u〉S:=〈v∣Su〉ℋ defines a scalar product on ℋ which induces an equivalent norm on ℋ, and un is an orthonormal basis with respect to this scalar product.Proof.
SinceS is self-adjoint, bijective, and positive, 〈v∣u〉S:=〈v∣Su〉ℋ indeed defines a scalar product on ℋ which induces an equivalent norm on ℋ. With respect to this scalar product, the injective and compact operator M̃:=S-1M is self-adjoint and positive. Therefore (2.9) is equivalent to the standard eigenvalue problem M̃un=λnun for the injective, compact, self-adjoint, and positive operator M̃∈ℒ{ℋ,〈·∣·〉S}.


## 3. Differential Equations in Matrix Notation

We recall some relations of elasticity theory in matrix notation which is especially suited for the finite-element formulation [9]. For a profound study of the mathematical theory of elasticity in tensor notation, we refer to the book of Marsden and Hughes in [18]. (i)
Differential operator matrixL is given as,
(3.1)L=Lx∂x+Ly∂y+Lz∂z,
with the constant matrices
(3.2)Lx=(100000000000001010),Ly=(000010000001000100),Lz=(000000001010100000).(ii)
Displacement vectoru is given as
(3.3)u=u(x,y,z,t)=(ux,uy,uz)T.(iii)
Strain vectorϵ is given as
(3.4)ϵ=(ϵx,ϵy,ϵz,γyz,γxz,γxy)T.(iv)
Strain-displacement relation is given as(3.5)ϵ=Lu.(v)
Stress vectorσ is given as
(3.6)σ=(σx,σy,σz,τyz,τxz,τxy)T.(vi)
And the generalized Hooke Law is given as(3.7)σ=Cϵ,
with C=(Cij)i,j=1,…,6 being the matrix of material constants. In the following, C is supposed to be real, symmetric, and positive definite (spd). For an isotropic material we have, for example,
(3.8)Cisotropic=(λ+2μλλ000λλ+2μλ000λλλ+2μ000000μ000000μ000000μ),
with Lamé's constants λ, μ.Now consider a laminated plate of thickness2H in direction of the z-axis (middle z=0, top z=+H, bottom z=-H) and infinite in the x-y plane; see Figure 1. The plate consists of N layers. Layer l=1,…,N has thickness hl=Hl-Hl-1 with -H=H0<H1<⋯<HN=+H and is supposed to consist of homogenous, anisotropic, elastic material with density ρl and material constants Cl.Figure 1
Laminate and coordinate systems.Letu=(ux,uy,uz)T=u(x,y,z,t) be the displacement vector of a wave travelling in the plate in the absence of external forces. In each layer the elastic wave equation in matrix notation is
(3.9)ρlü=LTClLu,∀x,y,tandallz∈(Hl-1,Hl).
The traction-free boundary conditions on the top and bottom surfaces are
(3.10)LzTCNLu|z=+H=LzTC1Lu|z=-H=0.
Besides continuity of the displacement vector, the following interface conditions concerning continuity of the stresses are supposed to hold:
(3.11)LzTClLu|z=Hl=LzTCl+1Lu|z=Hl,∀l=1,…,N-1.
As ansatz for a wave mode, we take a plane harmonic wave travelling in the x-y plane
(3.12)u(x,y,z,t)=û(z)ei(kxx+kyy±ωt),
with z-dependent amplitude vector û(z)=(ûx(z),ûy(z),ûz(z))T, real circular frequency ω, and real wave vector k=(kx,ky)T. For such a wave mode, the wave equation (3.9), boundary (3.10) and interface conditions (3.11) reduce to
(3.13)-ρlω2û=LkTClLkû,∀z∈(Hl-1,Hl),l=1,…,N,(3.14)LzTCNLkû|z=+H=LzTC1Lkû|z=-H=0,(3.15)LzTClLkû|z=Hl=LzTCl+1Lkû|z=Hl,∀l=1,…,N-1,
with the differential operator matrix Lk given as
(3.16)Lk=ikxLx+ikyLy+Lz∂z.
Obviously for complex conjugation we have
(3.17)L¯k=-ikxLx-ikyLy+Lz∂z.
The question is “for which combinations of circular frequencies ω and wave vectors k do nontrivial solutions û of (3.13), (3.14), and (3.15) exist?” The answer leads to the dispersion relations ω(k) and is given in the next section. For better readability we will write ω instead of ω(k) in the following.


## 4. Weak Form of the Reduced Wave Equation and  Related Eigenvalue Problem

We define the piecewise constant functions of densityρ and matrix of material constants C as
(4.1)ρ(z)=ρl,C(z)=Cl,∀z∈(Hl-1,Hl),l=1,…,N.
For suitable virtual displacements v̂=v̂(z), which we define later, the weak form of (3.13), (3.14), and (3.15) can then be written as
(4.2)ω2∫-H+Hv̂¯Tρûdz=∫-H+H(Lkv̂¯)TC(Lkû)dz.The next proposition shows that strong and weak solutions coincide for smooth enough functions.Proposition 4.1.
Suppose thatû=û(z) is continuous and layerwise 𝒞2; that is, û∈𝒞([-H,+H]) and û|(Hl-1,Hl)∈𝒞2([Hl-1,Hl]), l=1,…,N. Then û fulfills (4.2) for all v̂∈𝒞1([-H,+H]) if and only if  û is a solution of (3.13), (3.14), and (3.15).Proof.
Note that we shortly writeû|(Hl-1,Hl)∈𝒞2([Hl-1,Hl]) meaning that there is a function ũ∈𝒞2([Hl-1,Hl]) such that û|(Hl-1,Hl)=ũ|(Hl-1,Hl). By the smoothness assumptions on û, relation (3.17), and layerwise partial integration on the right-hand side of (4.2), we see that for all v̂∈𝒞1([-H,+H]) (4.2) is equivalent to
(4.3)ω2∫-H+Hv̂¯Tρûdz=-∫-H+Hv̂¯TLkTCLkûdz+∑l=1N[v̂¯TLzTClLkû]Hl-1Hl.
Let û fulfill this equation for all v̂∈𝒞1([-H,+H]). At first we fix a layer l=1,…,N and choose arbitrary functions v̂ that have compact support in the interior (Hl-1,Hl) of the layer. For such functions, (4.3) reduces to
(4.4)ω2∫Hl-1Hlv̂¯Tρlûdz=-∫Hl-1Hlv̂¯TLkTClLkûdz
from which we infer that û fulfills (3.13). Repeating this for all layers, we conclude that the integrals on the left- and right-hand side of (4.3) coincide and hence (4.3) reduces to
(4.5)0=∑l=1N[v̂¯TLzTClLkû]Hl-1Hl.
Now by successively choosing functions v̂ that equal 1 in a vicinity of one of the points -H=H0,H1,…,HN=+H and that are 0 everywhere else, we see that û also fulfills (3.14) and (3.15).
The converse assertion is obvious.To prove existence of nontrivial weak solutions, we will transform (4.2) into a generalized eigenvalue problem in a suitable Hilbert space. Let H1=H1(-H,+H) be the Sobolev space of all complex-valued square integrable functions on (-H,+H) whose distributional first derivative can also be identified with a square integrable function; see, the study by Adams in [19]. For simplicity in the following we use the same symbol H1 also for (H1)3=H1×H1×H1,(H1)6,…, and likewise for other spaces like the spaces of square integrable functions L2=L2(-H,+H) and continuously differentiable functions 𝒞1=𝒞1([-H,+H]),…, since it will become clear from the context how many components the vector-valued functions have. Endowed with the scalar product
(4.6)〈v̂∣û〉H1=〈v̂∣û〉L2+〈∂zv̂∣∂zû〉L2=∫-H+Hv̂(z)¯Tû(z)dz+∫-H+H∂zv̂(z)¯T∂zû(z)dz,
the space H1 is a complex Hilbert space. The space 𝒞1 is continuously embedded and dense in H1 and the inclusions H1↪𝒞 and J:H1↪L2 are compact. See, for example, the study by Maz'ja in [20]. Multiplications with the piecewise constant function ρ>0 and the piecewise constant spd-matrix C, respectively, define bijective, positive, self-adjoint, continuous linear operators on the respective L2-spaces. Furthermore, the differential operator matrix Lk (3.16) defines a continuous linear operator Lk:H1→L2. For functions û,v̂∈H1, which we assume in the following, the left-hand side of (4.2) can then be written as
(4.7)ω2∫-H+Hv̂¯Tρûdz=ω2〈Jv̂∣ρJû〉L2=ω2〈v̂∣J*ρJû〉H1=ω2〈v̂∣Mû〉H1
with the compact, injective, positive, and self-adjoint operator M:=J*ρJ, and the right-hand side can be written as
(4.8)∫-H+H(Lkv̂¯)TC(Lkû)dz=〈Lkv̂∣CLkû〉L2=〈v̂∣Lk*CLkû〉H1=〈v̂∣Skû〉H1
with the positive and self-adjoint operator Sk:=Lk*CLk. Thus, a function û∈H1 fulfills (4.2) for all v̂∈H1 if and only if
(4.9)ω2〈v̂∣Mû〉H1=〈v̂∣Skû〉H1,∀v̂∈H1,
which is equivalent to the generalized eigenvalue problem
(4.10)ω2Mû=Skû⇔(γ+ω2)Mû=(γM+Sk)û⇔λMû=Sγû
with λ:=γ+ω2 and the positive and self-adjoint operator Sγ:=γM+Sk for an arbitrary γ>0. We will need this disturbance with γM to prove that the operator Sγ is also bijective and hence can apply Proposition 2.4 to guarantee nontrivial solutions. By Lemma 2.1 and Corollary 2.3, bijectivity of Sγ follows by showing that there exists some constant c>0 such that
(4.11)〈û∣Sγû〉H1≥c∥û∥H12,∀û∈H1.
To show this, we write
(4.12)〈û∣Sγû〉H1=γ〈û∣Mû〉H1+〈û∣Skû〉H1=∫-H+Hγû¯Tρû+(Lkû¯)TC(Lkû)dz.
Since layerwise C is an spd-matrix and ρ>0, there is a constant c1>0 such that pointwise a.e. 						(4.13)γû¯Tρû+(Lkû¯)TC(Lkû)≥c1(γû¯Tû+(Lkû¯)T(Lkû)),
and thus we have
(4.14)〈û∣Sγû〉H1≥c1∫-H+Hγû¯Tû+(Lkû¯)T(Lkû)dz=c1∫-H+Hũ¯TQkũdz,
with
(4.15)ũ=(ûx,ûy,ûz,∂zûx,∂zûy,∂zûz)T
and the Hermitian matrix
(4.16)Qk:=(|k|2+γkxky0000kxky|k|2+γ000000|k|2+γ-ikx-iky000ikx10000iky010000001),
where |k|2=kx2+ky2≥0. For all real k=(kx,ky), the matrix Qk is positive definite since for γ>0 all eigenvalues
(4.17)λ1,2=1,λ3,4=|k|2+γ±kxky,λ5,6=12(1+|k|2+γ±(1+|k|2+γ)2-4γ)
are strictly positive. Hence there is a constant c2>0 such that pointwise a.e. 						(4.18)ũ¯TQkũ≥c2ũ¯Tũ,
and by the definition of the scalar product (4.6) we arrive at
(4.19)〈û∣Sγû〉H1≥c1c2∫-H+Hũ¯Tũdz=c1c2∥û∥H12.
Now we can apply Proposition 2.4 to the pair M, Sγ and get the following.Proposition 4.2.
There exists an increasing sequence0≤ωn2→∞ and a sequence ûn∈H1 such that
(4.20)ωn2Mûn=Skûn,(4.21)(γ+ωn2)〈ûn∣Mûm〉H1=δn,m=γ+ωn2ωn2〈ûn∣Skûm〉H1,
and each û∈H1 can be uniquely represented by an H1-convergent series
(4.22)û=∑n=0∞αnûn,
with an ℓ2-sequence αn=((γ+ωn2)/ωn2)〈û∣Skûn〉H1.Proof.
Let the assertions of Proposition2.4 appropriately hold for the pair M, Sγ. At first we observe that by (2.10) we have
(4.23)λ1=〈û1∣Sγû1〉H1〈û1∣Mû1〉H1=γ+〈û1∣Skû1〉H1〈û1∣Mû1〉H1≥γ,
and hence ωn2:=λn-γ≥0 and ωn2→∞ increasingly. By (2.10), on one hand, we then trivially have
(4.24)(γ+ωn2)〈ûn∣Mûm〉H1=λn〈ûn∣Mûm〉H1=δn,m,
and, on the other hand, we further conclude that
(4.25)δn,m=〈ûn∣Sγûm〉H1=γ〈ûn∣Mûm〉H1+〈ûn∣Skûm〉H1=γγ+ωn2δn,m+〈ûn∣Skûm〉H1.
Finally each û∈H1 can be uniquely represented by an H1-convergent series
(4.26)û=∑n=0∞αnûn,
with an ℓ2-sequence αn=〈û∣Sγûn〉H1. Evaluating the scalar product of both sides of the above equation with Skûm, we get
(4.27)〈û∣Skûm〉H1=∑n=0∞αn〈ûn∣Skûm〉H1=αmωm2γ+ωm2.The next proposition together with Proposition4.1 finally shows that all these weak H1-solutions are indeed strong solutions.Proposition 4.3.
Ifû∈H1 fulfills (4.2) for all v̂∈H1, then û is continuous and layerwise 𝒞∞.Proof.
We inductively show that the distributional derivatives ofû can be identified with smooth functions. Expanding the operator L according to its definition and rearranging (4.2) give
(4.28)∫-H+H∂zv̂¯TLzTCLz∂zûdz=∫-H+Hv̂¯T(ω2ρ+(ikxLx+ikyLy)TCLk)ûdz-∫-H+H∂zv̂¯TLzTC(ikxLx+ikyLy)ûdz.
We fix a layer l and take an arbitrary 𝒞∞-function v̂ with compact support in (Hl-1,Hl). As Cl is symmetric, for those v̂ the left-hand side can be written as
(4.29)∫Hl-1Hl∂z(LzTClLz)v̂¯T∂zûdz.
We integrate by parts on the right-hand side to get
(4.30)∫Hl-1Hl∂z(LzTClLz)v̂¯T∂zûdz=∫Hl-1Hlv̂¯T(ω2ρ+(ikxLx+ikyLy)TClLk)ûdz+∫Hl-1Hlv̂¯TLzTCl(ikxLx+ikyLy)∂zûdz.
The operator LzTClLz is invertible since Cl is positive definite and Lz is injective; hence the previous equation for all v̂ is equivalent to
(4.31)∫Hl-1Hl∂zv̂¯T∂zûdz=-∫Hl-1Hlv̂¯TAlûdz,
where
(4.32)Alû:=-(LzTClLz)-1(ω2ρ+(ikxLx+ikyLy)TClLk+LzTCl(ikxLx+ikyLy)∂z)û.
Equation (4.31) states that the second distributional derivative ∂z2û|(Hl-1,Hl) of û restricted to (Hl-1,Hl) can be identified with Alû|(Hl-1,Hl). Since û∈H1, we have Alû|(Hl-1,Hl)∈L2(Hl-1,Hl). Consequently, we may assume that û|(Hl-1,Hl)∈H2(Hl-1,Hl) and hence û|(Hl-1,Hl)∈𝒞1([Hl-1,Hl]). From this we in turn infer that Alû|(Hl-1,Hl)∈𝒞([Hl-1,Hl]), and (4.31) ensures that ∂z2û|(Hl-1,Hl)∈𝒞([Hl-1,Hl]); that is, û|(Hl-1,Hl)∈𝒞2([Hl-1,Hl]). Repeating this argument proves the assertion.We conclude this section by deriving a mode decomposition of general waves travelling in the plate. For a functionu=u(x,y), we denote by û=û(kx,ky) its Fourier transform in the wavenumber domain
(4.33)û(kx,ky)=1(2π)2∬ℝ2u(x,y)e-i(kxx+kyy)dxdy.
Let again u=(ux,uy,uz)T=u(x,y,z,t) be the displacement vector of a wave travelling in the plate in the absence of external forces (e.g., as a result of an initial excitation which now has stopped) such that it fulfills (3.9), (3.10), and (3.11). By formally applying the Fourier transform with respect to (x,y) to these equations, we see that û=û(k,z,t)=û(kx,ky,z,t) then fulfills
(4.34)ρlû̈=LkTClLkû,∀z∈(Hl-1,Hl),l=1,…,N
and the boundary and interface conditions (3.14) and (3.15). Suppose that for all k and t (or at least for the ones of interest) the functions z↦û(k,z,t) and z↦û̈(k,z,t) are in H1. Analogously to the previous considerations, the weak form of (4.34), (3.14), and (3.15) is then equivalent to
(4.35)Mû̈=-Skû.
According to Proposition 4.2 and (2.11), we can represent û(k,z,t) by a series with the eigenvectors ûn(k,z) as basis functions
(4.36)û(k,z,t)=∑n=0∞αn(k,t)ûn(k,z),û̈(k,z,t)=∑n=0∞α̈n(k,t)ûn(k,z).
We remark that for different k we have different basis functions ûn=ûn(k), hence the dependence αn=αn(k); but for fixed k we fix this basis for all times t, hence the dependence αn=αn(k,t). Furthermore we assume that û̈ can indeed be represented with α̈n; that is, summation and time derivation commute. Applying the orthogonality relations in (4.21) to (4.35), we get for fixed k the decoupled system of ordinary differential equations in t:
(4.37)α̈n(k,t)1γ+ωn(k)2=-αn(k,t)ωn(k)2γ+ωn(k)2⇔α̈n(k,t)+ωn(k)2αn(k,t)=0,with general solutions(4.38)αn(k,t)=an(k)eiωn(k)t+bn(k)e-iωn(k)t.
Assuming that the following operations are valid, we get the mode decomposition
(4.39)u(x,y,z,t)=∬ℝ2û(kx,ky,z,t)ei(kxx+kyy)dkxdky=∑n=0∞∬ℝ2(an(k)ûn(k,z)eiωn(k)t+bn(k)ûn(k,z)e-iωn(k)t)×ei(kxx+kyy)dkxdky.


## 5. Numerical Results

The generalized eigenvalue problem can efficiently be solved by a suitable finite-element discretization, leading to a finite-dimensional generalized eigenvalue problem of the same form of (4.10) with mass matrix M and stiffness matrix Sk; see [7–9]. A comprehensive treatment of the mathematical theory and practice of FEM can be found in the study by Ern and Guermond in [21]. For theoretical results concerning order-preserving convergence of the eigenvalues of the discretized problem to the eigenvalues of the infinite-dimensional problem, we refer to the studies by Yang and Chen in [22].As an application we want to examine how dispersion curves vary with an increasing degree of anisotropy. To do so, we consider a transversely isotropic material withy-axis as the axis of rotational symmetry. We note that unidirectional fibre-reinforced composites can often be approximately modelled by transversely isotropic material so that symmetry-axis and fibre direction coincide. For a description of the elastic material constants in this case we refer to the studies by Ambartsumyan in [23] and Altenbach et al. in [24]. We used as Young's moduli E1=E3=70 GPa, E2=θE1, as Poisson’s ratios ν32=ν31=ν13=ν12=0.3, ν23=ν21=ν32/θ, and as shear moduli G23=G12=G13=E1/(2(1+ν23)), where the ratio θ=E2/E1 determines the degree of anisotropy. The material behaves isotropic for θ=1 while for θ>1 it becomes transversely isotropic with rotational symmetry around the y-axis. In this case the matrix C of material constants can be written as
(5.1)C=((1-ν322/θ)E1Δ(ν32+ν322)E1Δ(ν32+ν322/θ)E1Δ000(1-ν322)θE1Δ(ν32+ν322)E1Δ000(1-ν322/θ)E1Δ000E12+2ν3200symmetricE12+2ν320E12+2ν32),
with
(5.2)Δ:=1-(2θ+1)ν322-2θν323.The plate is assumed to have a thickness ofd=1mm and a mass density of ρ=2700kg·m-3. We computed the circular frequency ω(k,ϕ) for a fixed circular wavenumber in dependence of the propagation direction ϕ and for different ratios θ=E2/E1. Some results for the first four modes are displayed in Figure 2 which illustrate the increasingly anisotropic behaviour. In the strictly transversely isotropic case θ>1, we labeled the modes according to their isotropic counterparts, but it becomes clear from Figure 3 that the distinction between pure SH- and S-modes is growing less obvious as both kinds of modes get components in all directions.Direction dependence of circular frequencyω(k,ϕ)[s-1] at circular wavenumber k=1000 m-1 for different ratios E2/E1. As one can see, each mode has its own individual anisotropic behaviour; even the sequence of the wave modes ordered by their frequencies is angle dependant.Components of the displacement vectoru=(ux,uy,uz) over the thickness of the plate at ϕ=70∘ of S0- and SH1-modes for different ratios E2/E1. Here ux and uy are the in-plane components in longitudinal and transversal directions, respectively. Obviously the distinction between shear-horizontal and in-plane Lamb-waves becomes less clear in an anisotropic material. This means especially that common measuring techniques quantifying the z component will detect also the SH-modes.



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*Source: 924504-2010-07-27.xml*