AXISYMMETRIC ELASTIC WAVES IN WEAKLY COUPLED LAYERED MEDIA OF INFINITE RADIAL EXTENT

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1 Journal of Sound and Vibration (1995) 182(2), AXISYMMETRIC ELASTIC WAVES IN WEAKLY COUPLED LAYERED MEDIA OF INFINITE RADIAL EXTENT C. CETINKAYA Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A. A. F. VAKAKIS Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A. AND M. EL-RAHEB Central Research, Engineering Laboratory, The Dow Chemical Company, Midland, Michigan, 48640, U.S.A. (Received 8 September 1993, and in final form 24 February 1994) The propagation of axisymmetric waves in layered periodic elastic media of infinite radial extent is investigated. Hankel and Laplace transforms are employed, to convert radial and time dependence of displacement and stress within a layer to frequency and radial wavenumber. Continuity of stress and displacement is imposed at the interface between layers yielding transfer matrices. The structure of propagation and attenuation zones (PZ s and AZ s) of the system with infinite number of layers is studied. When the ratios of shear and longitudinal mechanical impedances ( T, L) of the two layers are large, the PZ s of the layered system become narrow, approaching certain limiting curves in the frequency wavenumber plane. For large ( T, L) an asymptotic analysis is employed analytically to approximate the width of the PZ s. Numerical computations of PZ s are given, which are in good agreement with the analytical predictions. The spacing of the resonance points of a layered system with finite number of layers is found to depend mainly on the structure of the PZ s of the corresponding system with an infinite number of layers. 1. INTRODUCTION Elastic wave propagation in layered media has been studied extensively in the literature. One-dimensional wave propagation was studied in reference [1], adopting the effective modulus theory, which approximates the layered medium as a homogeneous but transversely isotropic continuum, when the thicknesses of the layers are small compared to the wavelengths of the propagating waves. Certain inaccuracies of the effective modulus theory are discussed in references [2, 3], and alternative methods, such as the effective stiffness theory, are employed which approximately take into account the wave dispersion. More exact methods take into account the compatibility of stress and displacement at the interface between layers. Robinson [4] studied SH-waves propagating at arbitrary angles in layered media, and showed that the dispersion relation of plane waves defines Currently at Wolfram Research, Inc., Champaign, Illinois 61820, U.S.A X/95/ $08.00/ Academic Press Limited

2 284 C. CETINKAYA ET AL. a surface in the frequency wavenumber space. In references [5 8], Floquet theory was employed to study the properties of the dispersion surface of propagating SH-, P- and SV-waves. Herrmann and Hemami [9] considered plane strain (P- and SV-) wave propagation in a periodically layered half-space with layers parallel to the free surface. They found that Rayleigh waves propagating along the traction-free boundary are highly dispersive, and that some higher dispersion branches of the surface waves may be discontinuous. Rousseau [10] computed the propagation and attenuation zones in the frequency wavenumber plane for oblique Floquet waves in a periodic medium of fluid and elastic layers. Rizzi and Doyle [11] developed a spectral method based on the Fast Fourier Transform to study transient waves in layered media. In reference [12], the structure of propagation and attenuation zones (PZ s and AZ s) for one-dimensional waves in layered media was examined. Parameters controlling the width of the PZ s were identified. The concept of fixed points relates the behavior of the propagation zones to the ratio of impedances of the layers in the limit when this ratio is large. Analytic approximations defining the PZ s were derived. Layered systems with weak disorder were also addressed in that work. Numerical computations of the transient responses of layered systems were performed in additional works. Thomson [13] adopted a matrix formulation in treating plane waves in layered media. Haskell [14] extended Thomson s formulation, and derived the dispersion relation for Rayleigh and Love surface waves in bi-periodic and tri-periodic media. Transfer matrix methods suffer from possible numerical instabilities arising from evanescent waves (exponential dichotomy). Kundu and Mal [15] proposed a pole removal method to eliminate this numerical instability, and numerically computed the transient responses of a layered system forced by harmonic and impulsive point loads. This method was modified by Mal [16], and applied to transient waves in anisotropic layered systems. Recently, Tenenbaum and Zindeluk [17] derived an exact algebraic solution for computing the transient response of one-dimensional layered media excited by arbitrary input pulses. In the field of anisotropic layered media, Anderson [18] studied harmonic waves in an elastic composite structure with transversely isotropic layers in the axial direction. Shah and Datta [19] used a stiffness method and Hamilton s princple to treat harmonic waves in periodically laminated infinite media. Braga [20] applied Floquet theory to waves in anisotropic layered media of infinite, semi-infinite and finite axial extent. In these studies, the governing sextic matrix equation excluded source and body forces. An exact solution for the dispersion relation of Floquet waves was obtained by applying Stroh s formalism [21]. This work examines axisymmetric waves in a bi-periodic medium of infinite extent in the radial direction. The formulation followed herein is similar to that employed by Miklowitz [22], who studied the axisymmetric response of an infinite elastic plate forced by two symmetrically placed point loads on its free surfaces. The main goal of the present study is to understand the structure of propagation and attenuation zones (PZ s and AZ s) for axisymmetric wave propagation in the system with infinite layers, and to derive asymptotic relations for the boundaries of the PZ s in the limit when the ratios of the shear and longitudinal mechanical impedances ( T, L ) of the two layers become large. Furthermore, the concept of fixed points introduced in reference [12] is extended to the two-dimensional axisymmetric problem, to study the PZ s and AZ s of layered systems with large T and L. A secondary goal of the work is to investigate the resonances of a free free system with a finite number of periodic sets, and to determine the design parameters which control the spacing of the resonance curves in the frequency wavenumber plane.

3 ELASTIC WAVES TRANSFER MATRIX OF A PERIODIC SET The layered system investigated in this section consists of an infinite number of periodic sets. Each set is composed of two homogeneous, isotropic, and linearly elastic layers of constant thickness and of infinite radial extent (cf., Figure 1(a)). A local cylindrical co-ordinate system, (r,, z), is introduced for each layer (cf., Figure 1(b)). Assuming axisymmetry, the displacement field in a single layer, u(r, z, t) = (u r (r, z, t), u z (r, z, t), u = 0), can be expressed in terms of the Helmholtz decomposition as u = +, (1) where (r, z, t) and (r, z, t) = ( r (r, z, t), (r, z, t), z (r, z, t)) = (0, (r, z, t), 0) are scalar and vector potentials respectively. The differential equations governing the potentials are decoupled [23], and assume the form 2 = , 2 = , (2a, b) where, = c L /c T, c 2 L = ( + 2 )/d, c 2 T = /d, and are Lame s constants, d is the density of the material, = r/h and = z/h are scaled cylindrical co-ordinates, = s t, s = c T /H, H is the layer thickness, and the Laplacian operator in equations (2) is defined as 2 = (1/H 2 )[ 2 / 2 + (1/ ) / + 2 / 2 ]. Figure 1. (a) The layered system under consideration; (b) the cylindrical co-ordinate system for a single layer.

4 286 C. CETINKAYA ET AL. The scaled radial, and axial components of the displacement vector u are defined by u = Hu r, and u = Hu z, respectively, and are computed as u (,, ) =, u (,, ) = + ( ), u (,, ) = 0, (3) where = /H and = /H are scaled potentials. Scaled components for the stress tensor in the layer are defined as = H 2 zz, = H 2 zr,.... Along the two boundaries of a layer, = 0, 1, the only non-zero stresses are and. The quantities q, q r, 1 and 1 r at the boundaries are now defined as (, 0, ) = q (, ), (, 0, ) = q (, ), (, 1, ) = 1 (, ), (, 1, ) = 1 (, ), (4) Similarly, along the boundaries the displacements components are functions of and. As in equation (4), define s, s r, w and w r by u (, 0, ) = s (, ), u (, 0, ) = s (, ), u (, 1, ) = w (, ), u (, 1, ) = w (, ). (5) In addition, zero initial conditions are prescribed to the scaled potentials, (,, 0) = (,, 0) = (,, 0)/ = (,, 0)/ = 0. Following Miklowitz [22], Hankel transforms of order zero and one in are applied to equations (2a) and (2b), respectively. The equations are then Laplace transformed in, yielding the ordinary differential equations k2 + p2 2 0 = (k 2 + p 2 ) 1 = 0, (6a, b) where overbars denote Laplace transform, superscripts denote the order of Hankel transform, and p and k are Laplace and Hankel transform variables, respectively. Equations (6) yield solutions in the form: 0 (p, k, ) = A 1 (p, k) e g + A 2 (p, k) e +gt, 1 (p, k, ) = B 1 (p, k) e h + B 2 (p, k) e +h, (7a, b) where A i and B i, i = 1, 2, are undetermined coefficients depending on p and k, g 2 = k 2 + p 2 / 2 and h 2 = k 2 + p 2. Expression the displacements u and u, and stresses and in terms of the potential and, and applying Hankel and Laplace transforms on the resulting expressions, the following equations are obtained: ū 1 (p, k, ) = k 2 0 1, u 0 (p, k, ) = k 1 + 0, 0 (p, k, ) = k ( + 2 ) k 1 2, 1 0 (p, k, ) = 2k k 2. (8) 1 2 Substituting equations (7) into equations (8), and casting the resulting algebraic expressions into matrix form, provides a relation between transformed stresses and displacements and the coefficients A i and B i, i = 1, 2: [T( )]{A 1 A 2 B 1 B 2 } T = {u 1 u } T, (9)

5 where the matrix [T( )] is given by: ELASTIC WAVES 287 k e g k e g h e h h e h [T( )] = g e g g e g k e h k e h. e g e g 2 kh e h 2 k h e h 2 k g e g 2 kg e g (h 2 + k 2 ) e h (h 2 + k 2 ) e h and = k 2 + ( + 2 )g 2. At this point, one applies Laplace and Hankel transforms on the boundary displacements (5) and stresses (4) in order to eliminate the time and radial variables from the problem. Making use of the matrix relation (9) at the boundaries = 0 and = 1, and eliminating the coefficients A i and B i, one obtains a matrix equation relating stresses and displacements at the two boundaries: {s 1 s 0 q 0 q 1 } T = [T(0)][T(1)] 1 {w 1 w } T, (10) where, consistent with the previous notation, an overbar denotes a Laplace transform, and a superscript the order of the applied Hankel transform. At this point, in order to obtain a compact description of the displacement and stresses, one defines the 2 1 vectors d L and d R to denote the transformed displacements on the upper and lower surfaces of the layer, respectively, while the 2 1 vectors f L and f R denote the corresponding transformed stresses. These vectors are related to the scaled transformed displacements and stresses at the boundaries of the layer by f L = H {q q } T, f R = H 2 { } T, d L = H {s s } T and d R = H {w w } T. Equation (10) can then be expressed in terms of these vectors as {Hd R H 2 f R } T = [R layer ]{Hd L H 2 f L } T {d R f R } T = [R layer ]{d L f L } T, (11) where [R layer ] = diag (1, 1, 1, 1)[T(1)][T(0)] 1, diag ( ) stands for diagonal matrix, and the 2 2 submatrices [R layerij ] of the 4 4 matrix [R layer ] are related to those of [R layer ] by [R layer12 ] = H[R layer12 ], [R layer21 ] = H 1 [R layer21 ] and [R layerii ] = [R layerii ], i = 1, 2. Equation (11) relates the state vectors at the two boundaries of a single layer, and can be used to obtain the transfer matrix of the periodic set of the layered system. Now consider a periodic set of two perfectly bonded layers, j and j + 1 (cf., Figure 1(a)). Imposing continuity of displacement and stress at the interface between layers, {d R f R } T j = {d L f L } T j + 1, and employing equations (11) for each of the two layers, one obtains a relation of the form: {d L f L } T j + 2 = [T set ]{d L f L } T j = t 11 t 21 t 12 t 22 {d L f L } T j, (12) where {d L f L } T i, i = j, j + 2, is the state vector of displacements and stresses on the upper boundary of layer i, and t pq, p, q = 1, 2, are 2 2 submatrices of the 4 4 matrix [T set ]. Matrix [T set ] is the transfer matrix of the periodic set for axisymmetric wave propagation. This matrix provides the relation between the displacement and stress vectors at the two free surfaces of a periodic set. In computing [T set ], a local set of cylindrical co-ordinates is defined for each layer, and the radial and axial co-ordinates are normalized with respect to the layer thickness, in accordance to the previously outlined formulation. The evaluation in compact form of the entries of [T set ] was performed using the computer algebra system MATHEMATICA TM. It was shown that [T set ] can be placed in the form [T set ] = [T b ][T a ], (13)

6 288 C. CETINKAYA ET AL. where subscripts a and b in equation (13) are employed to distinguish between the elastic properties and dimensions of layers j and j + 1, respectively. The element of the matrices [T a ] and [T b ] were explicitly computed as where 11 t [T a,b ] = F a,b (p) 12 t, (14) t 21 t 22 a,b [t 11 ] a = (k 2 + h 2 a ) cosh h a 2k 2 cosh g a (k(k 2 + h 2 a )/h a ) sinh h a + 2g a k sinh g a [t 12 ] a = G a k cosh h a k cosh g a (k 2 /h a ) sinh h a g a sinh g a 2h a k sinh h a (k(k 2 + ha 2 )/g a ) sinh g a 2k 2 cosh h a + (k 2 + ha) 2 cosh g a, h a sinh h a (k 2 /g a ) sinh g a k cosh g a k cosh h a, [t 21 ] a = G 1 a 2k(k2 + ha 2 )(cosh h a cosh g a ) 4h a k 2 sinh h a + ((k 2 + ha 2 ) 2 /g a ) sinh g a ((k 2 + ha 2 ) 2 /h a ) sinh h a 4g a k 2 sinh g a 2k(k 2 + ha 2 )(cosh g a cosh h a ), [t 22 ] a = (k 2 + h 2 a ) cosh g a 2k 2 cosh h a (k(k 2 + h 2 a )/h a ) sinh h a 2g a k sinh g a 2h a k sinh h a + (k(k 2 + ha 2 )/g a ) sinh g a 2k 2 cosh g a + (k 2 + ha 2 ) cosh h a, F a (p) = p 2, G a = (H a u 1 a ). (15a) Expressions for [t ij ] b in [T b ] follow from equations (15a), by replacing subscript a with b. The coefficients F b (p) and G b are given by F b (p) = (p T ) 2, G b = (H a T T 1 a ). (15b) In deriving the above expressions, it was assumed that H a = H b, where H a,b denote the thicknesses of layers a and b; moreover, in equations (15), a denotes the Lame constant of layer a. The quantities T, T, h a,b and g a,b are defined by h 2 a = k a, g 2 a = k 2 2, h 2 b = k a 2 T, g 2 b = k L, T = (H b /H a )(c Ta /c Tb ), L = (H b /H a )(c La /c Lb ) = T ( a / b ), T = (d a /d b )(c Ta /c Tb ), L = (d a /d b )(c La /c Lb ) = T ( a / b ), (16a) where the Laplace variable p was expressed as p = j a, j = ( 1) 1/2 ; a,b = c La,b /c Ta,b, (c La,b, c Ta,b ) are phase velocities of longitudinal and shear waves in layers a and b, and d a,b are the densities of layers a and b. In physical terms, T and L are the ratios of mechanical impedances of the layers for shear and longitudinal wave propagation, whereas T and L are the ratios of travel times of waves propagating at shear and longitudinal phase velocities, respectively, through the thicknesses of the two layers. Note that L = (H b /H a )(c La /c Lb ) = T ( a / b ) (16b) and that L does not appear explicitly in the following analytical derivations. Of particular interest is the case in which T and L are of O(1), and T and L are small (of O( ), 1) or large (of O(1/ )) quantities. These values correspond to layered systems consisting of periodic sets composed of stiff and soft layers. Vakakis et al. [12] examined one-dimensional waves in layered media consisting of weakly coupled periodic sets and showed that unattenuated Floquet waves can only propagate in very narrow frequency ranges (PZ s). This greatly affected the propagation of transient waves generated by externally applied impulses.

7 ELASTIC WAVES 289 The derivation of the transfer matrix of the single two-layered set is a prerequisite for the study of free and forced axisymmetric wave propagation in the periodic system with an infinite number of layers. In the following section, explicit expressions for propagation constants of Floquet waves are derived, and PZ s and AZ s of the infinite layered system are studied in the frequency radial wavenumber plane. 3. SYSTEM WITH INFINITE NUMBER OF LAYERS: PROPAGATION AND ATTENUATION ZONES Axisymmetric harmonic waves propagating through the layered system of infinite axial and radial extent can be studied using the notion of a propagation constant,, which relates the state vectors at the two boundaries of a set by {d L f L } T j + 1 = e {d L f L } T j. (17) Combining equations (17) and (12) yields a characteristic determinant in : det [t 12 ] 1 [t 11 ] [t 22 ][t 12 ] 1 + [t 12 ] 1 e ([t 21 ] [t 22 ][t 12 ] 1 [t 11 ]) e = 0. (18) The solutions for exist in two positive and negative pairs, corresponding to waves propagating in opposite directions of the layered system. Depending on the material and geometric parameters of the system, and on the frequency and radial wavenumber k, the propagation constants can assumed imaginary, real or even complex values. When is purely imaginary, (waves within a PZ), cosh is real and in the range cosh [ 1, 1]. In this case waves represented by equation (17) travel unattenuated, and Im ( ) is the constant phase difference between motions at the two boundaries of the periodic set. When is real (waves within an AZ), cosh is real in the range cosh [ 1, 1], and the corresponding waves decay exponentially. When is complex, cosh is also complex (waves within an AZ), and the corresponding waves propagate with exponentially decaying envelopes. These waves are termed complex modes of the layered system. Thus, depending on the character of, axisymmetric waves propagate freely within PZ s, decay exponentially or possess exponentially decaying envelopes. As shown by Mead [24, 25], only waves inside PZ s can propagate energy through the layered system. The boundaries computed by equation (18) define PZ s and AZ s in the (k, ) plane, which will be termed bounding curves (BC s). Exact solutions for the propagation constant can be determined by recognizing that, since the solutions of occur in positive negative pairs, relation (18) yields an equation with a symmetric structure of the form: e 4 + R 1 e 3 + R 2 e 2 + R 1 e + 1 = 0. (19) Equation (19) can be solved in closed form, leading to the following analytical expression for cosh : cosh 1,2 = (R 1 /4) (1/2)[(R 2 1/4) R 2 + 2]. (20) Equation (20) provides two of the propagation constants ; the remaining two are equal to their negatives. Coefficients R 1 and R 2 in equation (20) are functions of k,, T, T and a, and are evaluated by the following expressions: R 1 = tr [T set ], R 2 = [(x 2 1 2x 2 x 3 + x 4 4) + det [t 12 ] 1 [t 11 ] [t 22 ][t 12 ] 1 ]/det [t 12 ] 1, (21)

8 290 C. CETINKAYA ET AL. where tr [ ] denotes the trace of a matrix, and x i, i = 1, 2, 3, 4 are the four elements of matrix [t 12 ] 1 : [t 12 ] 1 = x 1 x 3 x 2 x 4. (22) Equations (20) (22) define the propagation constants for axisymmetric wave propagation in the layered system, and are used numerically to compute the boundaries between PZ s and AZ s of a layered system of infinite axial extent. It is of interest to examine the propagation constants corresponding to layered media with finite values of T, L, a and b, and large values of T and L. As mentioned in the previous section, this choice of parameters corresponds to systems composed of alternating stiff and soft layers. Introducing the small parameter, 1, the previous assumptions are summarized as T = ˆ T/, L = ˆ L/, ˆ T, ˆ L, T, L, a, b of O(1). (23) Substituting equations (23) into equation (20), one seeks an asymptotic solution for cosh in the following power series form: cosh 1,2 = ( ˆ T/ )C 1 (k, ; T, L, a ) + C 2 (k, ; T, L, a ) + O( ), (24) where the coefficients C 1 and C 2 in equation (24) are O(1) quantities independent of T. Their analytic expressions are derived by substituting equation (24) into equation (20) and matching respective powers of. These calculations were carried out using MATHEMATICA TM, and closed form expressions for C 1 and C 2 are listed in Appendix A. In equation (24), propagation constants 1 and 2 correspond to the (+) and ( ) signs, respectively. Expressions (24) can be used analytically to study the structure of the PZ s of the layered system for large values of the parameter T. Depending on, two types of bounding curves (BC s) separating PZ s and AZ s are possible. Bounding curves of the first type separate regions with imaginary from regions with real, whereas bounding curves of the second type separate regions with imaginary from those with complex. Bounding curves of the second type are points of generation of complex modes. BC s of the first type are computed by the imposing relation cosh = 1 (BC s of the first type). (25) In one-dimensional propagation, only BC s of the first type are possible since no complex modes exist. In Figure 2, a typical plot of the real part of cosh versus frequency is Figure 2. The real part of cosh versus scaled frequency for fixed k., Real modes (Im (cosh )=0);, complex modes (Im (cosh ) 0).

9 ELASTIC WAVES 291 Figure 3. The limiting curves of a layered system with T = 0 720, L = 0 304, a = and T. depicted, for fixed k. Points a, b, c, and d are boundary points of the first type satisfying relation (25). All branches shown represent real modes (with vanishing imaginary part of cosh ), with the exception of branch ef which represents a complex mode (with a non-zero imaginary part of cosh ). The extensions of branches ab and cd coalesce into point e, which is the point of generation of the complex branch. Combining equations (24) and (25) leads to the following analytic expressions for the BC s of the first type. These expressions are valid for sufficiently large values of parameters T and L : C + 1 (k, ; T, L, a ) + ( / ˆ T)[C + 2 (k, ; T, L, a ) + ] + O( 2 ) = 0, (26a) C 1 (k, ; T, L, a ) + ( / ˆ T)[C 2 (k, ; T, L, a ) + ] + O( 2 ) = 0, (26b) where 1. In the limit = 0 (i.e., infinite values of T and L ), equations (26) degenerate to: C + 1 (k, ; T, L, a ) = 0, C 1 (k, ; T, L, a ) = 0. (27a, b) For fixed values of T, L and a, and for real and positive k and, equations (27) define lines in the (k, ) plane, which will be termed limiting curves. It can be shown that the roots of equations (27a, b) are obtained by solving the algebraic equation det [t 21 ] a det [t 12 ] b = 0 8k 2 (k 2 + h 2 a ) 2 (cosh h a cosh g a ) cosh h a + [(16g 2 ah 2 ak 4 (k 2 + h 2 a ) 4 )/g a h a ] sinh g a sinh h a = 0, (28a) k 2 [(cosh h b cosh g b ) 2 sinh 2 h b + sinh 2 g b ] + [(g 2 bh 2 b + k 4 )/g b h b ] sinh g b sinh h b = 0, (28b) where the various variables in equations (28) are defined by relations (16). Equation (28a) represents the Rayleigh Lamb dispersion equation for a layer (a) with traction-free boundaries, whereas equation (28b) represents the dispersion relation for a layer (b) with clamped boundaries. The asymptotic behavior of the limiting curves defined by equation (28a) have been studied in reference [23]. Restricting the analysis to real and positive values of k and, it can be shown that for large k and, limiting curves in the (k, ) plane within the region k a 0 asymptotically approach the Rayleigh wave dispersion relation for layer a, whereas curves within the region a k 0 asymptotically approach the line = k/ a. Similarly, limiting curves defined by equation (28b) within the region T a k L 0 asymptotically approach the line = k/( T a ) for large values of k and.

10 292 C. CETINKAYA ET AL. In Figure 3, limiting curves of a layered system with T = 0 720, L = and a = are depicted. The pair of limiting curves emanating from (k, ) = (0, 0) are the only curves which approach the Rayleigh wave dispersion relation. The limiting lines emanating from the points (k, ) = (0, i / L ), (0, i /( a T )), (0, i ), (0, i / a ), i = 0, 1, 2, 3,..., correspond to natural frequencies of a free free layer a and a clamped clamped layer b when no radial dependence of the motion exists (k = 0). In the limit as k 0, two uncoupled, one-dimensional, shear and longitudinal plane waves result. Considering equations (26), for small values of, BC s of the first type can be regarded as perturbations of the limiting lines (28). Hence, for large values of T the limiting curves can be considered as backbone curves of the PZ s. A perturbation analysis can then be employed to compute the BC s of the layered system. The perturbation methodology will be demonstrated by computing the BC s close to the mth limiting curve = 0 (m) (k). In what follows, only the BC s which are solutions of equation (26a) will be considered; however, a similar perturbation analysis can be carried out to study PZ s generated by relations (26b). The BC s are expressed in the series form = (m) (k; ) = (m) 0 (k) + n = 1 n (m) n (k), (29) where = (m) (k; ) denotes the BC in the (k, ) plane. Substituting equation (29) into equation (26a), expanding the function C + 1 in Taylor series about = 0 (m) (k), and matching coefficients of equal powers of, yields the following solution for the first order approximation, 1 (m) (k): 1 (m) (k) = (1/ ˆ T)[C + 2 (k, 0 (m) 1 (k, 0 (k)) + ] (m) (k)) C+ 1 + O( 2 ) = 0. (30) Solution (30) is valid only when C + 1 (k, 0 (m) (k))/ 0. Setting = 1 gives the analytic approximations of two BC s, which define a PZ. Higher order corrections to the BC s can be computed considering coefficients of higher powers of in equation (26a). BC s computed from equations (29) and (30) define PZ s with widths of O( ). Note that if C + 1 (k*, *)/ 0 at a point (k*, *) on a limiting curve, the Implicit Function Theorem (IFT) guarantees that close to (k*, *) the solution of C + 1 (k, ) = 0 can be represented by a unique function = 0 (m) (k), satisfying the expressions C + 1 (k, 0 (m) (k)) = 0 and * = 0 (m) (k*). For points (k*, *) on limiting curves where C + 1 (k*, *)/ = 0, the conditions of the IFT are not satisfied, and the solutions of C + 1 (k, ) = 0 cannot be represented locally by functions. These degenerate points are intersections of two or more solutions, and represent singularities of the limiting curves. Consider such a degenerate point (k*, *) on the intersection of two limiting curves. At the neighbourhood of this point, the solutions for the limiting curves cannot be represented by functional relations of the form = (k), or k = k( ). This is due to the fact that degenerate points represent the intersection of two or more branches of solutions of equations (27a) or (27b), and, hence, are points in the (k, ) plane where the IFT fails. Assuming that the degenerate point (k*, *) is the intersection of two solutions of equation (27a), the IFT fails at (k*, *), provided that the following conditions are satisfied (degenerate points of solutions of equation (27b) are treated similarly): C + 1 (k*, *) = 0, C + 1 (k*, *)/ = 0, C + 1 (k*, *)/ k = 0. (31) Since no functional relations for the limiting curves exist close to (k*, *), the boundaries of PZ s in the vicinity of (k*, *) cannot be represented by the perturbation series (29) and (30), and an alternative analytical methodology must be employed for their computation. For sufficiently small values of, the boundaries of PZ s are approximately

11 ELASTIC WAVES 293 computed by relations (26). Expanding equation (26a) in Taylor series close to the degenerate point (k*, *), and taking into account conditions (31), one obtains the following relation between k and on the bounding curves: (1/2) 2 C + 1 (k*, *) ( *) C + 1 (k*, *) ( *)(k k*) 2 k + (1/2) 2 C + 1 (k*, *) (k k*) k 2 + ( / ˆ T)[C (k*, *) + ] + ( / ˆ T) C+ 2 (k*, *) + ( / ˆ T) C+ 2 (k*, *) k ( *) (k k*) + O( * p k k* q ) + O( * m k k* n ) = 0, (32) where p, q, m and n are non-negative integers satisfying p + q = 3, m + n = 2 and = 1. Equation (32) can be regarded as a quadratic equation in ( *). Assuming that (k k*) = O( 1/2 ) and using the notation (k k*) = 1/2 k, the solution of equation (32) for ( *) is computed as where ( *) = 1/2 { (L 2 /2L 1 ) k [{(L 2 /2L 1 ) 2 (L 3 /L 1 )} k 2 ( ˆ T) 1 (L 4 /L 1 )] 1/2 } + O( 3/2 ), (33) L 1 = 2 C + 1 (k*, *)/ 2, L 2 = 2 C + 1 (k*, *)/ k, L 3 = 2 C + 1 (k*, *)/ k 2, L 4 = 2[C + 2 (k*, *) + ]. (34) Solution (33) computes the boundaries of PZ s close to a degenerate point of limiting curves, and indicates that the widths of PZ s close to degenerate points ((k k*) = O( 1/2 )) are of O( 1/2 ). A similar analytic result can be derived if one assumes that (k k*) = k. Hence, it is concluded that sufficiently close to the non-degenerate point (k*, *), the widths of the PZ s of the layered system are of O( 1/2 ). It was previously found that PZ s in neighborhoods of non-degenerate points of limiting curves are merely of O( ) (cf., relations (29) and (30)). Thus, PZ s in neighborhoods of degenerate points are wider than PZ s close to nondegenerate ones. This interesting analytical conclusion will be verified by the numerical calculations that follow. It must be stated that analogous results were obtained for one-dimensional wave propagation in layered media [12]. The previous analysis showed that, as the impedance ratio T of the layered system increases (i.e., as 0), the widths of all PZ s of the first type diminish. For large values of T (of O(1/ )), the widths of the PZ s become of O( ), except for regions close to degenerate points of limiting curves, in which their widths increase, becoming of O( 1/2 ). At the limit T = ( = 0) all PZ s collapse into their neighboring limiting curves. Hence, the widths of the PZ s of the layered system can be made arbitrarily small by increasing the impedance ratio T of the two layers of the periodic set. As shown in a later section, the structure of the PZ s of the layered system with infinite number of layers greatly affects the spacing of the natural frequency curves of the corresponding system with finite axial extent. BC s of the second type separate propagating waves from complex modes. Numerical simulations indicate that BC s of this type are limited in number and small in extent. Moreover, the analytical study of BC s of the second type requires elaborate computations,

12 294 C. CETINKAYA ET AL. since these BC s are lines of coalescence of two branches of propagation constants in the (k, ) plane. At BC s of the second type, the partial derivatives (cosh 1,2 )/ of the two coalescing branches of propagation constants tend to ( ). Considering expression (20) (or, the approximate expression (24) at the limit of large T ). BC s of the second type are computed by the relations cosh 1 = cosh 2 = p, p real, p 1, / cosh 1 0, / cosh 2 0 BC s of the second type. (35) A bounding point defined by equations (35) is point f in Figure 2. Considering the same figure, it is noted, that although point e is also a point of generation of complex modes, it is not a boundary point of a second type, since there cosh 1 = cosh 2 = p 1. For large values of T, a perturbation analysis similar to the previous one can be employed to derive analytic approximations for BC s of the second type. It can then be shown that the widths of the resulting PZ s decrease with increasing T, vanishing at the limit T = ( = 0). The previous analysis was carried out under the assumption of large T. When T is a quantity of O(1), the boundaries between PZ s and AZ s can only be computed numerically by employing expression (20). For the numerical applications reported in this work, only real and positive values for k and were considered. The BC s in the (k, ) plane were computed using equation (20) while fixing k to determine the ranges of for which is imaginary. These ranges of define the PZ s of the layered system. In Figure 4, the leading PZ s of a layered system with T = 0 304, L = , a = and T = are depicted. Dashed lines denote limiting curves and solid lines denote BC s. Consistent with the theoretical prediction, the limiting curves form backbone curves for the PZ s; each limiting curve is close to a PZ. In Figure 4, two distinct set of PZ s are superimposed, corresponding to the two propagation constants 1 and 2 determined by equation (20). The crossings between limiting curves give rise to a complicated structure for the PZ s. Figure 4. The PZ s of a layered system with T = 0 304, L = , a = and T = , Limiting curves;, PZ s of the first type; PZ s of the second type.

13 ELASTIC WAVES 295 Figure 5. The PZ s of a layered system with T = 0 304, L = , a = and T = , Limiting curves;, PZ s of the first type; PZ s of the second type. Branches ab and cd are BC s of the second type, whereas all other BC s are of the first type. Regions in Figure 4 outside PZ s are AZ s of the layered system. As discussed in references [9] and [20], in a layered system with traction-free surfaces Rayleigh waves exist in AZ s of the corresponding infinite system. To study the structure of the PZ s when T is increased, a layered system with T = was considered, with all other parameters remaining constant. In Figure 5, the BC s of this system are depicted. Note that all PZ s are narrow, and approach the limiting curves. This is in accordance with previous analytical predictions. The branches of complex modes diminish in size, and a new complex branch ef is generated close to = 4. Points I and II are degenerate points on limiting lines representing intersections of solutions of equations (27a) or (27b). In accordance with theoretical predictions, the widths of the PZ s of the layered system are very small (of O(1/ T )), except in the neighborhoods of the two degenerate points I and II, where the widths of the PZ s attain relatively larger magntiudes (of O(1/ T 1/2 )). To test the validity of the asymptotic analysis, a comparison between the predictions of the asymptotic theory and numerical computations is carried out in Figure 6, where the third PZ of the layered system with T = is considered. A good agreement between the asymptotic and numerical solutions is observed for this value of T. This result shows that the asymptotic expressions developed in this work can be employed to analytically study the structure of the PZ s of layered systems with large values of the impedance parameter T. It must be pointed out that the dispersion curves for axisymmetric elastic wave propagation are identical in structure to those for wave propagation in plain strain. However, the integral transformations used for deriving the dispersion curves differ in the two cases and, in addition, there is a different physical interpretation for the wavenumber k. All of the previous results were derived for layered media of infinite axial extent. In the next section, the free dynamics of a layered system with finite number of layers is examined. For this system, it will be shown that the spacing of the resonance curves in the (k, ) plane is mainly determined by the structure of the PZ s of the corresponding system with an infinite number of layers.

14 296 C. CETINKAYA ET AL. Figure 6. The BC s of the third PZ of the layered system with T = 0 304, L = , a = and T = , Numerical solutions;, limiting curves;, asymptotic solutions. 4. SYSTEM WITH FINITE NUMBER OF LAYERS: RESONANCE CURVES When a system with a finite number of layers is considered, boundary relations for the stresses and displacements must be imposed in order to satisfy the state of the system at its end boundaries. Consider the axisymmetric motion of a layered system composed of n periodic sets (2n layers). Assuming that layers 1 and 2n possess traction-free upper and lower surfaces, respectively, the following stress relations must be satisfied: (f L ) 1 = (f L ) 2n + 1 = 0, (36) where (f L ) i denotes the vector of transformed stresses acting on the upper boundary of layer i, and the notation of sections 2 and 3 is employed. Using the transfer matrix relation (12), and taking into account equation (36), the state vectors at the upper and lower surfaces of the layered system can be related by {d L 0} T 2n + 1 = [T set ] n {d L 0} T 1 n t 11 n t 21 n t 12 n t 22 {d L 0} T 1, (37) where n t ij, i, j = 1, 2, denote the 2 2 submatrices of the overall transfer matrix [T set ] n. The resonance curves of the free free layered system can then be computed by the expression det n t 21 = 0. (38) The solutions of equation (38) are one-dimensional curves in the frequency radial wavenumber, (, k), plane. When k = 0, equation (38) provides the natural frequencies of the one-dimensional bi-periodic layered system with no radial dependence. For non-zero values of k, one can use equation (38) to determine the effects of axisymmetric radial dispersion on the natural frequencies of the layered system. The corresponding eigenfunctions of the layered system are evaluated in employing the transfer matrices developed in the previous sections. The resonance curves of a system with 2n = 10 layers and parameters T = 0 304, L = , a = and T = , are depicted in Figures 7 and 8. In the same figures, the boundary curves (BC s) separating PZ s and AZ s of the respective systems with infinite number of layers, n =, are also shown (these superimposed BC s are identical to those depicted in Figures 4 and 5). It is observed that inside each of the

15 ELASTIC WAVES 297 Figure 7. The resonance curves of a system with n = 5, T = 0 304, L = , a = and T = Numbered resonance points correspond to the eigenfunctions of Figure 9., Resonance curves;, boundaries of PZ s. PZ s there exist (n 1) = 4 resonance curves of the free free systems. The only resonance curves inside AZ s are those corresponding to a single free free periodic set. Hence, the resonance curves are mainly clustered inside the PZ s of the corresponding systems with infinite numbers of layers. As T increases (cf., Figure 8), the clusters of resonance curves inside PZ s become increasingly dense, and each cluster is separated from its adjacent ones by AZ s of increasingly large widths. Indeed, as T, the widths of the resonance clusters become of O( 1 T ) (close to non-degenerate points of limiting curves), or of O( 1/2 T ) (in the neighborhoods of degenerate points). This conclusion can be verified by the numerical computations depicted in Figure 8 ( T = ), were the clusters of resonance curves Figure 8. The resonance curves of a system with n = 5, T = 0 304, L = , a = and T = , Resonance curves;, boundaries of PZ s.

16 298 C. CETINKAYA ET AL. Figure 9. The resonance curves of a system with n = 5, T = 0 304, L = , a = and T = , corresponding to resonance points (a) 1, (b) 2, (c) 3, and (d) 4 of Figure 7. are everywhere dense, except in the neighborhoods of the degenerate points I and II. Similar results are obtained when the number of periodic sets, n, is increased: each PZ contains clusters of n 1 resonance curves, and the only resonances in AZ s are those of the free free single periodic set. The eigenfunctions corresponding to resonance points 1, 2, 3 and 4 of Figure 7 are shown in Figure 9. Each eigenfunction is depicted in terms of axial and radial displacements at interfaces between layers. Point 1 (k = 0 25, = ) is a resonance point slightly outside the first PZ, and the corresponding eigenfunctions have attenuating (or expanding) spatial distributions. Points 2 (k = 0 50, = ) and 3 (k = 0 50, = ) are resonances inside PZ s and their eigenfunctions are spatially extended through all the layers of the system. Point 4 (k = 0 50, = ) is located in an AZ and the corresponding eigenfunctions are spatially localized with attenuating (or expand-

17 ELASTIC WAVES 299 ing) spatial distributions. Hence, depending on the positions of the resonances in the (k, ) plane (in AZ s or in PZ s), the corresponding eigenfunctions are spatially localized or spatially extended. 5. DISCUSSION Axisymmetric wave propagation in a bi-periodic system with infinite radial extent was studied. Propagating and evanescent waves were analyzed in the scaled frequency ( ) scaled radial wavenumber (k) plane. Using computer algebra, an exact analytical expression for the propagation constant was derived in terms of the elements of the transfer matrix of a single periodic set. The structure of the PZ s of the layered structure was analyzed. In the limit of infinite impedance ratios, T, the PZ s degenerate into certain limiting curves. These curves act as backbone curves for the PZ s of the layered system, a feature which allows an approximate analytic study of PZ s at the limit of large values of T. It was found that, for increasing values of T, the widths of the PZ s decrease as ( 1 T ) close to non-degenerate points of limiting curves, or as ( 1/2 T ) close to degenerate points. Analytical approximations for the bounding curves between PZ s and AZ s were derived. Numerical computations of PZ s agree closely with theoretical predictions. The resonance curves in the (k, ) plane and the corresponding eigenfunctions of a free free system with n = 5 layers were numerically computed. The positions of the resonance curves correlate with the positions of the PZ s of the layered system with infinite number of layers. Each PZ contains a cluster of n 1 resonance curves, and the only resonances inside AZ s are those of the single free free periodic set. It is noted that similar results were found by Mead [24] for one-dimensional wave propagation in periodic systems. As the impedance parameter T increases, the clusters of resonance curves become increasingly dense, in accordance to the diminishing of the respective PZ s where they lie. The diminishing of the widths of the PZ s were increasing T for axisymmetric wave propagation in layered systems is of considerable practical interest as far as the forced transient response of these systems is concerned. Forced transient waves in AZ s attenuate as they propagate through the layers. When a layered system with thin PZ s and dense clusters of resonance curves is excited by a external impulse, most of the generated waves are expected spatially to localize close to the point of application of the external force, resulting in spatial confinement of the induced vibrational energy. ACKNOWLEDGMENTS This work was supported by a Grant from Central Research of the Dow Chemical Company, Midland, Michigan. REFERENCES 1. S. M. RYTOV 1956 Soviet Physics Acoustics 2, Acoustical properties of a thinly laminated medium. 2. C. T. SUN, J. D. ACHENBACH and G. HERRMANN 1968 Journal of Applied Mechanics 35, Time-harmonic waves in a stratified medium, propagating in the direction of the layering. 3. E. H. LEE and W. H. YANG 1973 SIAM Journal of Applied Mathematics 25, On waves in composite materials with periodic structure. 4. C. W. ROBINSON 1972 Sandia Laboratories Report SCL-RR Shear waves in layered composites. 5. T. J. DELPH, G. HERRMANN and R. K. KAUL 1978 Journal of Applied Mechanics 45, Harmonic wave propagation in a periodically layered infinite elastic body: antiplane strain.

18 300 C. CETINKAYA ET AL. 6. T. J. DELPH, G. HERRMANN and R. K. KAUL 1979 Journal of Applied Mechanics 46, Harmonic wave propagation in a periodically layered infinite elastic body: plane strain, analytical results. 7. T. J. DELPH, G. HERRMANN and R. K. KAUL 1979 Journal of Applied Mechanics 47, Harmonic wave propagation in a periodically layered infinite elastic body: plane strain, numerical results. 8. A. A. GOLEBIEWSKA 1980 Journal of Applied Mechanics 47, On dispersion of periodically layered composites in plane strain. 9. G. HERRMANN and M. HEMAMI 1982 Journal of Applied Mechanics 49, Plane-strain surface waves in a laminated composite. 10. M. ROUSSEAU 1989 Journal of the Acoustical Society of America 86, Floquet wave properties in a periodically layered medium. 11. S. A. RIZZI and J. F. DOYLE 1992 Journal of Vibration and Acoustics 114, A spectral element approach to wave motion in layered solids. 12. A. F. VAKAKIS, M. EL-RAHEB and C. CETINKAYA 1994 Journal of Sound and Vibration 172, Free and forced dynamics of a class of periodic elastic systems. 13. W. THOMSON 1950 Journal of Applied Physics 21, Transmission of elastic waves through a stratified medium. 14. N. A. HASKEL 1953 Bulletin of the Seismic Society of America 43, The dispersion of surfaced waves in multi-layered media. 15. T. KUNDU and A. K. MAL 1985 Wave Motion 7, Elastic waves in a multilayered solid due to a dislocation source. 16. A. K. MAL 1988 Wave Motion 10, Wave propagation in layered composite laminates under periodic surface loads. 17. R. A. TENENBAUM and M. ZINDELUK 1992 Journal of the Acoustical Society of America 92, An exact solution for the one-dimensional elastic wave equation in layered media. 18. D. L. ANDERSON 1961 Journal of Geophysical Research 66, Elastic wave propagation in layered anisotropic media. 19. A. H. SHAH and S. K. DATTA 1982 International Journal of Solids and Structures 18, Harmonic waves in a periodically laminated medium. 20. A. M. B. BRAGA 1990 Ph.D. dissertation, Stanford University, Palo Alto, California. Wave propagation in anisotropic layered composites. 21. A. N. STROH 1962 Journal of Mathematical Physics 41, Steady-state problems in anisotropic elasticity. 22. J. MIKLOWITZ 1962 Journal of Applied Mechanics 29, Transient compressional waves in an infinite elastic plate or elastic layer overlying a rigid half-space. 23. J. D. ACHENBACH 1987 Wave Propagation in Elastic Solids. New York: North-Holland. 24. D. J. MEAD 1975 Journal of Sound and Vibration 40, Wave propagation and natural modes in periodic systems, I: mono-coupled systems. 25. D. J. MEAD 1975 Journal of Sound and Vibration 40, Wave propagation and natural modes in periodic systems, II: multi-coupled systems with and without damping. APPENDIX A: VARIOUS QUANTITIES Expressions for the coefficients C 1 and C 2 of the asymptotic expression for cosh (equation (24)), were computed using MATHEMATICA TM, as follows: where C 1 = 1 /4 1 2( 2 1/4 Q 1 /Q 2 ) 1/2, C 2 = 2 /4 1 4( 2 1/4 Q 1 /Q 2 ) 1/2 [ 1 2 /2 + (Q 1 Q 3 Q 4 Q 2 )/Q 2 2], (A1) 1 = (z 11 + z 41 ), 2 = (q 10 + q 40 + z 10 + z 40 ), Q 1 = y 2 12 y 22 y 32, Q 2 = v 11 v 41 v 21 v 31, Q 3 = v 21 v 30 v 20 v 31 + v 11 v 40 + v 10 v 41, Q 4 = 2y 11 y 12 y 22 y 31 y 21 y 32, y 11 = q 20 v 31 q 10 v 41 v 41 z 10 v 40 z 11 + v 21 z 30 + v 20 z 31,

19 ELASTIC WAVES 301 y 12 = v 41 z 11 + v 21 z 31, y 22 = v 41 z 21 + v 21 z 41, y 31 = q 40 v 31 q 30 v 41 + v 31 z 10 + v 30 z 11 v 11 z 30 v 10 z 31, y 21 = q 20 v 11 + q 10 v 21 v 41 z 20 v 40 z 21 + v 21 z 40 + v 20 z 41, y 32 = v 31 z 11 v 11 z 31. (A2) The quantities q ij, v ij and z ij, i = 1, 2, 3, 4, j = 0, 1, depend on the variables k,, T, L and a and are computed by the following relations: q i = q i0 + (1/ T )q i1, z i = z i0 + T z i1, v i = v i0 + T v i1 (i = 1, 2, 3, 4), (A3) where q i, z i and v i are related to the 2 2 submatrices of the transfer matrix of the periodic set, [T set ] (cf. equations (12) (15)): [t 22 ] = q 1 q 3 q 2 q 4, [t 11] = z 1 z 3 z 2 z 4, [t 12] = v 1 v 3 v 2 v 4. (A4 A6) Based on equations (A3) and (A6), the explicit expressions for v ij are v 10 = ( cosh g a + cosh h a )( 2k 2 cosh g b + (k 2 + h 2 b ) cosh h b ) + g a sinh g a k2 sinh h a h a 2h b sinh h b (k2 + hb 2 ) sinh g a g a v 11 = (cosh h b cosh g b )(2k 2 cosh h a + (k 2 + h 2 a ) cosh g a ) k/p4 2 T, + h b sinh h b k2 sinh g b g b 2g a sinh g a + (k2 + ha 2 ) sinh h a h a v 20 = k2 sin g a g a + h a sinh h a ( 2k2 cosh g b + (k 2 + h 2 b ) cosh h b ) + (cosh g a cosh h a ) 2k2 h b sinh h b k2 (k 2 + h 2 b ) sinh g b g b k/p4 T, 1/p4 2 T, v 21 = (cosh h b cosh g b ) h a sinh h a + k2 (k 2 + ha 2 ) sinh g a g 2k2 a + h b sinh h b k2 sinh g b g b ( 2k2 cosh g a + (k 2 + ha 2 ) cosh h a ) T, 1/p4 v 30 = k2 sinh h a h a + (g a sinh g a )( 2k 2 cosh h b + (k 2 + h 2 b ) cosh g b ) + ( cosh g a + cosh h a ) 2k2 g b sinh g b k2 (k 2 + h 2 b ) sinh h b h b 1/p4 2 T, v 31 = (cosh g b cosh h b ) g a sinh g a + k2 (k 2 + ha 2 ) sinh h a h 2k2 a + g b sinh g b k2 sinh h b h b ( 2k2 cosh h a + (k 2 + ha 2 ) cosh g a ) T, 1/p4

20 302 C. CETINKAYA ET AL. v 40 = (cosh g a cosh h a )( 2k 2 cosh h b + (k 2 + h 2 b ) cosh g b ) + h a sinh h a k2 sinh g a g a 2g b sinh g b (k2 + hb 2 ) sin h b h b k/p4 2 T, v 41 = (cosh g b cosh h b )( 2k 2 cosh g a + (k 2 + h 2 a ) cosh h a ) + g b sinh g b k2 sinh h b h b 2h a sinh h a + (k2 + ha 2 ) sinh g a g a where p = j a. Similar expressions can be derived for q ij and z ij. k/p4 T, (A7)