Research Article | Open Access
F. Schöpfer, F. Binder, A. Wöstehoff, T. Schuster, "A Mathematical Analysis of the Strip-Element Method for the Computation of Dispersion Curves of Guided Waves in Anisotropic Layered Media", Mathematical Problems in Engineering, vol. 2010, Article ID 924504, 17 pages, 2010. https://doi.org/10.1155/2010/924504
A Mathematical Analysis of the Strip-Element Method for the Computation of Dispersion Curves of Guided Waves in Anisotropic Layered Media
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.
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  or Rose . 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 . 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  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  and Aalami , or Kausel , Galán and Abascal , the excellent book of Liu and Xi , and the many references therein. Also the works of Gavrić , Bartoli et al. , Marzani et al. , and Treyssède  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 , 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 Section 3. 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 , 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  and Conway .
Let be real or complex Hilbert spaces with scalar products . By we denote the space of continuous linear operators from to and set . For we denote by the adjoint operator defined by Range and nullspace are denoted by and , respectively. We say that is self-adjoint if , and positive if
Lemma 2.1. Let be self-adjoint. If is injective and has closed range, then it is bijective and hence continuously invertible.
Proof. For , let be the orthogonal complement of . The assertion follows from which means that is also surjective.
Lemma 2.2. An operator is injective and has closed range if and only if there exists some constant such that
Proof. See, for example, the study by Schröder in .
We obtain the following.
Corollary 2.3. An operator is injective and has closed range if there exists some constant such that
For , consider the generalized eigenvalue problem That is, “for which do nontrivial solutions to (2.7) exist?” In case is bijective, (2.7) is equivalent to the standard eigenvalue problem and we readily obtain the following proposition which we state for the case . It applies accordingly to the finite-dimensional case with finite sequences , .
Proposition 2.4. Let be self-adjoint and positive. Assume that is compact and injective and that is bijective. Then there exists a decreasing sequence (counted with multiplicity) and a sequence such that and each can be uniquely represented by an -convergent series with an -sequence . Especially, defines a scalar product on which induces an equivalent norm on , and is an orthonormal basis with respect to this scalar product.
Proof. Since is self-adjoint, bijective, and positive, indeed defines a scalar product on which induces an equivalent norm on . With respect to this scalar product, the injective and compact operator is self-adjoint and positive. Therefore (2.9) is equivalent to the standard eigenvalue problem for the injective, compact, self-adjoint, and positive operator .
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 . For a profound study of the mathematical theory of elasticity in tensor notation, we refer to the book of Marsden and Hughes in . (i)Differential operator matrix is given as, with the constant matrices (ii)Displacement vector is given as (iii)Strain vector is given as (iv)Strain-displacement relation is given as (v)Stress vector is given as (vi)And the generalized Hooke Law is given as with being the matrix of material constants. In the following, is supposed to be real, symmetric, and positive definite (spd). For an isotropic material we have, for example, with Lamé's constants , .
Now consider a laminated plate of thickness in direction of the -axis (middle , top , bottom ) and infinite in the - plane; see Figure 1. The plate consists of layers. Layer has thickness with and is supposed to consist of homogenous, anisotropic, elastic material with density and material constants .
Let 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 The traction-free boundary conditions on the top and bottom surfaces are Besides continuity of the displacement vector, the following interface conditions concerning continuity of the stresses are supposed to hold: As ansatz for a wave mode, we take a plane harmonic wave travelling in the - plane with -dependent amplitude vector , real circular frequency , and real wave vector . For such a wave mode, the wave equation (3.9), boundary (3.10) and interface conditions (3.11) reduce to with the differential operator matrix given as Obviously for complex conjugation we have The question is “for which combinations of circular frequencies and wave vectors do nontrivial solutions of (3.13), (3.14), and (3.15) exist?” The answer leads to the dispersion relations and is given in the next section. For better readability we will write instead of in the following.
4. Weak Form of the Reduced Wave Equation and Related Eigenvalue Problem
We define the piecewise constant functions of density and matrix of material constants as For suitable virtual displacements , which we define later, the weak form of (3.13), (3.14), and (3.15) can then be written as
The next proposition shows that strong and weak solutions coincide for smooth enough functions.
Proof. Note that we shortly write meaning that there is a function such that . By the smoothness assumptions on , relation (3.17), and layerwise partial integration on the right-hand side of (4.2), we see that for all (4.2) is equivalent to
Let fulfill this equation for all . At first we fix a layer and choose arbitrary functions that have compact support in the interior of the layer. For such functions, (4.3) reduces to
from which we infer that 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
Now by successively choosing functions that equal in a vicinity of one of the points and that are everywhere else, we see that 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 be the Sobolev space of all complex-valued square integrable functions on whose distributional first derivative can also be identified with a square integrable function; see, the study by Adams in . For simplicity in the following we use the same symbol also for and likewise for other spaces like the spaces of square integrable functions and continuously differentiable functions since it will become clear from the context how many components the vector-valued functions have. Endowed with the scalar product the space is a complex Hilbert space. The space is continuously embedded and dense in and the inclusions and are compact. See, for example, the study by in . Multiplications with the piecewise constant function and the piecewise constant spd-matrix , respectively, define bijective, positive, self-adjoint, continuous linear operators on the respective -spaces. Furthermore, the differential operator matrix (3.16) defines a continuous linear operator . For functions , which we assume in the following, the left-hand side of (4.2) can then be written as with the compact, injective, positive, and self-adjoint operator , and the right-hand side can be written as with the positive and self-adjoint operator . Thus, a function fulfills (4.2) for all if and only if which is equivalent to the generalized eigenvalue problem with and the positive and self-adjoint operator for an arbitrary . We will need this disturbance with to prove that the operator is also bijective and hence can apply Proposition 2.4 to guarantee nontrivial solutions. By Lemma 2.1 and Corollary 2.3, bijectivity of follows by showing that there exists some constant such that To show this, we write Since layerwise is an spd-matrix and , there is a constant such that pointwise a.e. and thus we have with and the Hermitian matrix where . For all real , the matrix is positive definite since for all eigenvalues are strictly positive. Hence there is a constant such that pointwise a.e. and by the definition of the scalar product (4.6) we arrive at Now we can apply Proposition 2.4 to the pair , and get the following.
Proposition 4.2. There exists an increasing sequence and a sequence such that and each can be uniquely represented by an -convergent series with an -sequence .
Proof. Let the assertions of Proposition 2.4 appropriately hold for the pair , . At first we observe that by (2.10) we have and hence and increasingly. By (2.10), on one hand, we then trivially have and, on the other hand, we further conclude that Finally each can be uniquely represented by an -convergent series with an -sequence . Evaluating the scalar product of both sides of the above equation with , we get
The next proposition together with Proposition 4.1 finally shows that all these weak -solutions are indeed strong solutions.
Proposition 4.3. If fulfills (4.2) for all , then is continuous and layerwise .
Proof. We inductively show that the distributional derivatives of can be identified with smooth functions. Expanding the operator according to its definition and rearranging (4.2) give We fix a layer and take an arbitrary -function with compact support in . As is symmetric, for those the left-hand side can be written as We integrate by parts on the right-hand side to get The operator is invertible since is positive definite and is injective; hence the previous equation for all is equivalent to where Equation (4.31) states that the second distributional derivative of restricted to can be identified with . Since , we have . Consequently, we may assume that and hence . From this we in turn infer that , and (4.31) ensures that ; that is, . Repeating this argument proves the assertion.
We conclude this section by deriving a mode decomposition of general waves travelling in the plate. For a function , we denote by its Fourier transform in the wavenumber domain Let again 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 to these equations, we see that then fulfills and the boundary and interface conditions (3.14) and (3.15). Suppose that for all and (or at least for the ones of interest) the functions and are in . Analogously to the previous considerations, the weak form of (4.34), (3.14), and (3.15) is then equivalent to According to Proposition 4.2 and (2.11), we can represent by a series with the eigenvectors as basis functions We remark that for different we have different basis functions , hence the dependence ; but for fixed we fix this basis for all times , hence the dependence . Furthermore we assume that can indeed be represented with ; that is, summation and time derivation commute. Applying the orthogonality relations in (4.21) to (4.35), we get for fixed the decoupled system of ordinary differential equations in :
with general solutions Assuming that the following operations are valid, we get the mode decomposition
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 and stiffness matrix ; 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 . 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 .
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 with -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  and Altenbach et al. in . We used as Young's moduli GPa, , as Poisson’s ratios , , and as shear moduli , where the ratio determines the degree of anisotropy. The material behaves isotropic for while for it becomes transversely isotropic with rotational symmetry around the -axis. In this case the matrix of material constants can be written as with
The plate is assumed to have a thickness of and a mass density of . We computed the circular frequency for a fixed circular wavenumber in dependence of the propagation direction and for different ratios . Some results for the first four modes are displayed in Figure 2 which illustrate the increasingly anisotropic behaviour. In the strictly transversely isotropic case , we labeled the modes according to their isotropic counterparts, but it becomes clear from Figure 3 that the distinction between pure - and -modes is growing less obvious as both kinds of modes get components in all directions.
The work of the authors is being supported by Deutsche Forschungsgemeinschaft (DFG) under Schu 1978/4-1.
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