NEAR FIELD TRANSIENT AXISYMMETRIC WAVES IN LAYERED STRUCTURES: EFFECTS OF WEAK COUPLING

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, (1997) NEAR FIELD TRANSIENT AXISYMMETRIC WAVES IN LAYERED STRUCTURES: EFFECTS OF WEAK COUPLING C. CETINKAYA*, J. BROWN, A. A. F. MOHAMMED AND A. F. VAKAKIS Department of Mechanical and Industrial Engineering, 1206 ¼. Green, ºniversity of Illinois at ºrbana, Champaign, ºrbana, I 61801, º.S.A. SUMMARY Axisymmetric stress wave transmission through the leading layers of layered structures of infinite radial but finite axial extent is numerically studied by employing two different computational approaches: a technique based on the numerical inversion of Double Integral Transformations (DIT), and a Finite Element (FE) analysis. Considering the first approach, careful selections of the limits of the numerical inversions and the sampling rates are required in order to overcome inherent numerical instabilities associated with exponential dichotomy. This type of numerical instability is more evident in layered media with weak coupling. In such systems, direct multiplications of layer transfer matrices are avoided by employing a global scheme to assemble well-conditioned global transfer matrices. Moreover, the specific structure of the propagation and attenuation zones of the structure are taken into account for increasing the efficiency and effectiveness of the transfer matrix manipulations. Satisfactory agreement between the DIT and FE numerical results is observed, at least for early times. Close to the region of application of the external pressure, the FE simulations suffer from the discretization of the applied load, node-to-node oscillations and reflections from infinite elements ( silent boundaries ). Using the aforementioned numerical techniques, transient wave transmission in two-layered systems (one with weak and one with strong interlayer coupling) is considered, and the effects of weak coupling on the wave transmission is studied. We show that at early times, weak coupling results in stress localization in the region close to the applied pressure, a result which can have potential application in the use of layered media as shock isolators by John Wiley & Sons, Ltd. KEY WORDS: layered media; stress wave propagation; integral transform; propagation zone; attenuation zone 1. INTRODUCTION Layered media are commonly used in various engineering applications, such as, in the manufacturing of composite materials and in the study of reinforced composites flexible systems like plates and shells. Additional applications can be found in the areas of seismology and non-destructive evaluation. Accurate numerical computation of stress wave propagation in layered media leads to a better physical understanding of the stress transmission properties of such systems, to the development of reliable techniques for the detection of defects in layered materials by * Adjunct Assistant Professor, and Member of Research and Development Staff, Wolfram Research Inc. Graduate Research Assistant, Member of Technical Staff, Hughes Aircraft Co. Graduate Research Assistant Associate Professor CCC /97/ $17.50 Received 5 December by John Wiley & Sons, Ltd. Revised 9 September 1996

2 1640 C. CETINKAYA E¹ A. non-destructive testing, and to a refined interpretation of ultrasonic signals used for determining the elastic properties of composite layered structures. Elastic wave propagation in layered media has been studied extensively in the literature. Homogenization techniques, such as the effective modulus theory, have been developed which approximate 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. More exact methods take into account the variation of the elastic properties of the medium. Rizzi and Doyle developed a spectral method based on the fast Fourier transform to study transient waves in layered media. The computation of transient elastic waves in layered media is a challenging numerical task. This is due to numerical instabilities associated, for example, with, (a) exponential dichotomy (i.e., the presence of very large or very small eigenvalues in the corresponding transfer matrices, (b) poles of transfer functions on the real axis of the complex plane resulting in numerical transform inversions that are difficult to perform; (c) large numbers of eigenfunctions needed in modal expansions due to complicated geometries; and, (d) node-to-node oscillations and imperfectly absorbing boundaries in finite element formulations. An extensive discussion of some of the aforementioned numerical instabilities will be given in the following sections. Numerical instabilities in finite-difference schemes used in modelling wave propagation in layered or general heterogeneous media (such as those associated with dispersion of Fourier components) are discussed in References In Reference 21 axisymmetric transient waves in a circular disk of finite radius and thickness were studied using finite-difference and modal superposition schemes; a stability criterion for the numerical accuracy of the finite-difference algorithm was derived in the same work. In Reference 22, exact solutions for scattering of harmonic waves from layered interfaces of planar, cylindrical and spherical geometries were computed and dynamic stress concentrations were studied. General analytical and numerical techniques based on Green s function formulations for analysing transient waves in layered systems were examined in Reference 23. In Reference 24, the transient dispersion of stress pulses in the leading layers of a one-dimensional, weakly coupled bilayered system was asymptotically studied by evaluating approximately the inverse Fourier transform integrals, thereby expressing the dispersed pulse as a summation of convolutions of the applied load with low- and high-frequency kernels. Additional asymptotic techniques for studying transient waves in homogeneous or heterogeneous elastic media are included in the monographs by Achenbach and Miklowitz. In the monograph by Nayfeh transient waves in anisotropic layered media are studied using the Cagniardde Hoop transformation method, and elegant asymptotic approximations for axisymmetric travelling waves in layered half-spaces are derived. The structures considered in this work are composed of periodic sets of stiff and soft homogeneous and isotropic layers of infinite radial extent; layered media composed or alternating ceramic and polymeric layers are examples. When the soft layers are of small thicknesses, weak coupling between the stiff layers results, and the composite structure can be shown to possess densely packed natural frequencies, along with large attenuation zones in the frequency radial wave number domain. Two numerical techniques are employed to study near field transient wave propagation in the leading layers, the first based on Double Integral Transforms (DIT), and the second on Finite Elements (FE). Due to the complexity of the problem, extensive symbolic algebra and parallel large-scale computations computations are employed, and special formulations are developed to overcome the numerical instabilities. After presenting the essentials of the two techniques, we study near field transient waves propagating through the layers of two structures, and examine the effects of weak inter-layer coupling on the transmission and reflection patterns of waves at layer interfaces by John Wiley & Sons, Ltd.

3 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES THE DOUBLE INTEGRAL TRANSFORM (DIT) TECHNIQUE The layered structure considered is depicted in Figure 1. It consists of five biperiodic sets of linearly elastic, isotropic layers with perfect bonding between them. The bottom surface is traction-free, whereas an external distributed force is applied on the top surface. The applied force is assumed to be applied in the axial direction and to be axisymmetric; the footprint where the pressure field is applied is assumed to be circular of radius 0 25 in, and its temporal dependence is trapezoidal with a total duration of 25 μs. The loading condition at the top surface can be expressed mathematically as σ "f (r) f (t) (1) where the functions f (t) and f (t) are depicted in Figure 2. As in the previous works by Cetinkaya and Vakakis two different layered configurations are considered, labelled as Systems I and II with elastic and geometric properties listed in Appendix I. The difference between Systems I and II is the thickness of the soft layer B (cf. Figure 1); indeed, the stiff layers A of System I are coupled by thin soft layers B, whereas in System II there is stronger coupling between the stiff layers. The difference between the mechanical impedances of layers A and B is characterized by the non-dimensional parameters τ and τ for shear and pressure (longitudinal) waves, respectively (cf. Appendix I). As discussed by Cetinkaya and co-workers in 1994, τ and τ determine the level of coupling and the strength of mode conversion between the shear and pressure waves the more they deviate from unity the weaker the coupling is. Figure 1. The bilayered periodic system: (a) general configuration, (b) a single periodic set, (c) fictitious layer for computing transformed stresses and displacements inside a layer 1997 by John Wiley & Sons, Ltd.

4 1642 C. CETINKAYA E¹ A. Figure 2. Applied external pressure σ "f (r) f (t): (a) time dependence f (t), (b) radial dependence f (r), (c) Fourier transform of f (t) versus normalized frequency ω, (d) Hankel transforms of orders zero ( ) and one (--- --) of f (r) versus normalized wavenumber k A transfer matrix formulation for axisymmetric wave propagation was derived in References 7 and 8 by Fourier-transforming the elasticity equations with respect to time and Hankel-transforming with respect to the radial variable. This approach is similar to that employed by Miklowitz in his study of the axisymmetric response of an infinite elastic plate forced by two symmetrically placed point-loads on its free surfaces. We denote the Fourier and Hankel transform variables by ω and k, respectively. At this point we note that ω and k represent normalized frequency and radial wave number variables, respectively; ω is the normalized frequency corresponding to the scaled time τ"(c T /H )t, where t is physical time, c T the phase velocity of shear waves in layer A, and H the thickness of layer A; similarly, k is the normalized radial wave number corresponding to the scaled radial variable ρ"r/h. Considering the ith periodic set, where i"1,..., 5, we define the (2 1) vectors d and d to denote the transformed displacements on the upper and lower surfaces of the periodic set, respectively, and the (2 1) vectors f and f to denote the corresponding transformed stresses: d "(un, un ), f "(σn, σn ) (2) where superscript T denotes the transpose of a vector, un (r, k, ω), un (r, k, ω) the transformed radial and axial displacement components at the mth interface between periodic sets (cf. Figure 1), σn (k, ω), σn (k, ω) the corresponding transformed axial and shear components of the stress tensor, overbars denote Fourier transform, and superscripts denote the order of Hankel transform. Then the following matrix relation for a biperiodic set holds (d f ) "[¹ ](d f ) (3) where [¹ ] is the (4 4) transfer matrix. The components of the transfer matrix depend on the Laplace and Hankel transform variables ω and k, and on the elastic and geometric properties of the layers. Considering now the layered structure of Figure 1, the stresses and displacements at 1997 by John Wiley & Sons, Ltd.

5 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1643 the top surface can be related to the corresponding quantities at the bottom by the following expression: (d f ) "[¹ ] (d f ) (4) where f denotes the (2 1) applied (external) stress vector and f "(0 0) due to the traction free condition at the bottom surface. In principle, the resulting equations in (4) can be solved for d and d, which enable the computation of the displacement and stress at the mth interface using the formula: (d f ) "[¹ ] (d f ), m"2,..., 5 (5) Expressions (4) and (5), although conceptually simple, are difficult to numerically implement due to the problem of exponential dichotomy, i.e., of very large and very small eigenvalues of the matrix [¹ ] inside Attenuation Zones (AZs) of the layered system. To eliminate the potential numerical instabilities, Cetinkaya and Vakakis employed an alternative methodology based on the Direct Global Matrix Approach. In this approach, instead of solving successively for the state vectors at different interfaces, a 20-tuple global state vector S is formed and solved in a single step. In addition, the applied external force is embedded in a similar (20 1) global forcing vector F: S"(d L d L f L d L f L d L ) (6) F"(f L f L (00)(00)) The vectors S and F are then related by a (20 20) global transfer matrix C which is predominantly diagonal and relatively well-conditioned; for a complete listing of the entries of C the reader is referred to References 7 and 8. The transformed displacements and stresses at all interfaces of the layered system can then be computed simultaneously by solving the following numerical matrix equation: CS"F N S"C F (7) Since the condition number for matrix C is relatively low, the numerical operations associated with (7) are free of arithmetic operations over very large and very small numbers, which is the primary reason for the loss of precision in computing [¹ ]. As a result, the inversion (7) is free of numerical instabilities in the integration domain of interest. In essence, successive multiplications of the ill-conditioned matrix [¹ ] are avoided by expanding the dimension of the solution space. The principal objective of this work is to compute near-field elastic waves in the leading layers of the layered structure of Figure 1, in order to study the effect of the layering on the axisymmetric wave propagation. Considering the ith bilayered set, i"1,..., 5, the transformed components un (k, ω), un (k, ω) and σn (k, ω), un (k, ω) at a distance h inside layer A are computed as follows. A fictitious layer C is assumed to exist (cf. Figure 1(c)), with the state at the top of this layer, (d, f ), being already computed by inversion (7). Layer C has identical elastic properties with layer A and its thickness is equal to the distance h. Hence, a transfer matrix formulation can be employed to compute the transformed stress components: (d, f ) "[¹ ](d, f ) (8) where the (4 4) transfer matrix [¹ ] is defined in explicit form in Appendix B. The transient elastic waves inside the layers of Systems I and II are then obtained by numerically inverting the double-transformed components un, un and σn, σn by John Wiley & Sons, Ltd.

6 1644 C. CETINKAYA E¹ A. In what follows, we will consider transient waves inside the layer A of the first periodic set of the layered system (setting i"1 in relation (8)); however, a similar analysis can be performed to compute transient responses inside any other layer of the structure of Figure 1. The following integrals must be evaluated in order to compute the transient responses at radial distance r and depth h from the top surface of the layered structure: R(r, h, t)" f (k)kj (rk) fm (ω)rm (h, k, ω)e dωdk (9) S(r, h, t)" f (k)kj (rk) fm (ω)sm (h, k, ω)e dωdk where RM (h, k, ω) denotes the transformed components un and σ, and SM (h, k, ω) the components un and σn. In (9), f (k) denotes the zeroth-order Hankel transform of the radial component of the applied load f (r), and fm (ω) the Fourier transform of the transient component of the external load f (t). To evaluate the double integral (9) the following computational procedure is followed. For fixed radial wave number k, the inverse Fourier transform is employed to eliminate the frequency ω from RM (h, k, ω) and SM (h, k, ω); the resulting transient responses corresponding to the constant radial wave number k are termed transient wavemodes. For fixed k the transient wavemodes of inversions (9) are denoted by R (h, k, t) and S (h, k, t), and are defined by the following expressions: R (h, k, t)" S (h, k, t)" fm (ω) RM (h, k, ω)e dω (10a) fm (ω) SM (h, k, ω)e dω Note that the transient wavemodes corresponding to k"0 correspond to one-dimensional (pressure or shear) wave propagation, with no radial dependence. In the second and final step of the computation, the transient wavemodes are multiplied by appropriate Bessel functions and are superposed over k in the Hankel transform inversion in order to obtain the transient responses R(r, h, t) and S(r, h, t): R(r, h, t)" S(r, h, t)" f (k)kj (rk)r (h, k, t)dk f (k) kj (rk)s (h, k, t)dk (10b) The frequency inversions (10a) are numerically computed using the inverse fast Fourier transform (FFT). The finite upper- and lower-frequency limits of the numerical integration are chosen based on the frequency content of the transient dependence of the applied excitation. Denoting by ω "2 57 the Nyquist frequency of f (t) (as mentioned previously, ω represents a normalized frequency), the limits for the inverse FFT are chosen as [!ω, ω ], a frequency range which assures convergence and accuracy of the numerical FFT inversions. The inverse FFTs are performed using the routine ZFFT1D from the Library SgiMath on the Silicon Graphics Power Challenge Supercomputer System of the National Center for Supercomputing Applications (NCSA) at the University of Illinois by John Wiley & Sons, Ltd.

7 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1645 In the second step of the computation, the Hankel inversions (10b) are performed. For computational purposes, the upper limit of the integrations, k*;r, is taken as the wave number above which the kernels of the integrals (10b) become sufficiently small. This assures accuracy of the numerics, and the inverse Hankel transforms are approximated as follows: R(r, h, t)+ S(r, h, t)+ p k f (k) kr (h, k, t)j (rk)dk (p!1) k p k f (k) ks (h, k, t)j (rk)dk (p!1) k where, N is the total number of sampling points in the radial wave number direction, and the step size for k is defined as, k"k*/(n!1). The quantities f (k)kr (h, k, t) and f (k)ks (h, k, t) in the above integrands are curve-fitted by splines to increase the efficiency of the numerical computations since the kernel is a smooth function. This computational approximation was first used in References 7 and 8 to compute stresses at interfaces between layers; more details of the use of spline approximations to approximate integrals of the general form of equations (11) can be found in these references. For near-field computations, the upper limit for the wave number, k*, must be selected with care since at high wave numbers exponential dichotomy hampers the accuracy of the integrals in equations (11). Considering the integrand of the second of expressions (11), the wave number at which f (k) becomes small is approximately equal to k +40 (cf. Figure 2). Hence, at least in principle, one could choose k**k to assure the accuracy of the approximations (11); in actuality, however, the value of k is so high that it may become impractical to evaluate the corresponding transient wavemodes S (h, k, t) within the limits of a floating point arithmetic precision. This becomes evident by considering the dependence of R (h, k, t) and S (h, k, t) onk. Both of these quantities contain entries of the tansfer matrices [¹ ] and [¹ ], and these entries, in turn, contain terms of the form ξ(cosh, sinh ), "η(k, H, h) where H denotes the ratio of the thicknesses of layers A and B within one periodic set (cf. Figure 1), and h the axial position of the transient calculation. Assuming that h is sufficiently large (i.e., that the transient calculation is far from the point of application of the external excitation), for high wave numbers k the arguments become exponentially large, causing severe numerical inaccuracies; this is one of the main sources of exponential dichotomy and can only be eliminated by restricting k to small values. Denoting by k the value of the wave number above which the integrands R (h, k, t) and S (h, k, t) become sufficiently small, k* must necessarily satisfy k**k ; in this case one refers to the transient wavemodes instead of the wave number content of the forcing function in order to fix the upper limit of the integration (11). However, when h is small (i.e. for near field computations), it can be shown that the arguments remain relatively small even for high wave numbers and no numerical instabilities occur in the computations; in this case k should be used as the upper limit for the Hankel inversions. From the above discussion, one formulates the following criterion for choosing k*: k**min(k, k ) Fortunately, the knowledge of the structure of the propagation and attenuation zones (PZs and AZs) of Systems I and II offers considerable help in further improving the selection of k*. The concept of PZs and AZs in a periodic system of infinite spatial extent is discussed in the works by Mead. For the layered systems under consideration, PZs are defined as regions in the frequency-wave number plane for which waves with fixed frequencies and wave numbers propagate unattenuated through the layered medium. On the contrary, waves with frequencies and (11) (12a) 1997 by John Wiley & Sons, Ltd.

8 1646 C. CETINKAYA E¹ A. Figure 3. Propagation and attenuation zones of Systems I and II wave numbers in AZs represent near-field solutions and possess exponentially decaying envelopes. The PZ AZ diagrams are obtained by studying the eigenvalues of the transfer matrix [¹ ] (cf. expression (3)) [7]. Although the PZs and AZs have strict meaning only for layered systems with an infinite number of layers, resonance (Lamb) curves of layered structures with finite number of layers are predominantly spaced inside PZs of the corresponding infinite systems, and, as a result, their dynamic response is significantly affected by the spacing of the PZs and AZs. In Figure 3 the PZs and AZs of Systems I and II with an infinite number of layers are presented. As shown in References 7 and 8, the slope of the first boundary curve in the (ω, k) plane approaches the phase velocity of Rayleigh waves in layer A as ω, kpr; this Rayleigh asymptote provides an alternative way of selecting k*. Referring to Figure 3, and considering the Rayleigh asymptote, one defines as k +5 0 the wave number corresponding to the Nyquist frequency of the excitation ω. By choosing k**k, one eliminates attenuating waves (near-field solutions) with large wave numbers from the numerical calculation (11), a step which eliminates the problem of exponential dichotomy. However, such an approximation only works sufficiently far from the top layer of the system, since only then near-field waves contribute minimally to the system response. In near-field transient responses, i.e., close to the region of application of the external pressure field, waves in AZs are of major importance and must be included in the numerical computations. Taking into account (12a) and the above discussion, it is concluded that the following criterion for k* assures the accurate numerical computation of the inverse Hankel transforms (11): k**min(k, k, k ) (12b) In Figure 4 the upper limit k* used in the transient computations inside the layers of Systems I and II is depicted as a function of h. The selection of k* was based on criterion (12b). Note that k* increases for small values of h, a feature which reflects the importance of waves in AZs on the computations of near-field transient responses. Having fixed the upper limit of the integration (11), the transient response of the layered systems are computed using a high-performance integration routine; in the simulations a sevenpoint quadrature integration algorithm was utilized. The code used to perform the aforementioned numerical computations is written in FORTRAN and double-precision complex variables 1997 by John Wiley & Sons, Ltd.

9 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1647 Figure 4. Upper limits of the inverse Hankel transforms for Systems I and II versus depth z are used throughout. The code was parallelized and executed on a 12-processor vector machine. Parallel computing well suits this problem, due to the fact that the computations which require many Floating-point Operations Per Second (FLOPS) are neither iterative nor recursive. In the simulations the matrix [¹ ] was sampled at 10,000 frequencies in the range [0, ω ], and at 400 wave numbers in the range [0, k*]. At each sample, the inversion of the (20 20) global transfer matrix C is required, an operation which requires 1200 M multiplications for M inversions of the [¹ ] matrix. The method of direct multiplication of [¹ ] would only require 16 M multiplications, but numerical instabilities due to exponential dichotomy would be encountered in such an operation. The computation of the displacement components posed some additional problems associated with singularities close to (k, ω)"(0, 0), and some special remarks are in order at this point. To compute the transient displacement components u (r, t), u (r, t) at depth h from the top interface of the layered structure, one needs to perform the following double inversions: u (r, t)" f (k)kj (rk) fm (ω)un (r, k, ω)e dwdk (13) u (r, t)" f (k)kj (rk) fm (ω)un (r, k, ω)e dwdk Computations of transient displacements, velocities and accelerations in elastic media is a more challenging technical task than the computation of the stress components. In addition to the previously mentioned numerical difficulties which are encountered in the numerical evaluation of expressions (13), the integrands un (k, ω) and un (k, ω) possess an ω singularity at (k, ω)"(0, 0) which introduces additional numerical instabilities in the computations. To circumvent this problem, we considered accelerations instead of displacement components in the numerical inversions. In the transformed domain, the acceleration components are computed as an (k, ω)"!ω un (k, ω) an (k, ω)"!ω un (k, ω) (14) 1997 by John Wiley & Sons, Ltd.

10 1648 C. CETINKAYA E¹ A. which remove the singularity at the origin of the transform domain. Moreover, to further improve the numerical accuracy of the results the transient wavemodes for the accelerations were computed by numerical convolutions instead of using the inverse FFT routine: where an (k, t)"f (t) * A (k, t) an (k, t)"f (t) * A (k, t) A (r, t)"!ω un (k, ω)e dω (15) A (r, t)"!ω un (k, ω)e dω and asterisk denotes the convolution operation. This computational procedure assures the elimination of numerical offsets of the response at time t"0. After computing the transient wavemodes for the accelerations, the transient responses are evaluated by numerically inverting the Hankel transforms as discussed in the previous section. A final note concerns the limitations of the DIT technique. A first limitation concerns the relative inaccuracy of the methodology at higher times. The reason is, that at higher times the transient wavemodes (corresponding to fixed k) are highly oscillatory due to multiple wave reflections at interfaces between layers. If the sampling in k is not fine, these high oscillations result in numerical inaccuracies. This problem can only be surpassed by finer wave number sampling, a procedure which is computationally involved. The second limitation is that components of the stress tensor other than σ and σ cannot be computed using the previous transfer matrix methodology. Additional stress components can only be computed by using gradients of displacement components in the spatial domain, and for this one needs fine sampling and an efficient interpolating scheme. In the following sections the Finite Element (FE) technique for computing the stresses and displacements inside the layers is discussed, and a comparison of DIT and FE results is presented. 3. FINITE ELEMENT (FE) APPROACH COMPARISON OF RESULTS By using finite element analysis, we mainly verified the implementation of the DIT technique and the associated tedious mathematical formulation (see Reference 7 for details) as well as the numerical results (e.g. the arrival times of propagating waves and the magnitudes of wave fronts) to a lesser degree. The layered structure was modelled using ABAQUS axisymmetric four-node elements CAX4R. These solid elements employ a single-point reduced integration scheme with hourglass control. Reduced integration was chosen because it lowers the cost of forming an element and decreases significantly the output file size without incurring a significant loss in accuracy. Because the layered structure of Figure 1 is infinite in the radial direction, the FE model incorporates energy-absorbing infinite elements which minimize reflections in the radial direction from the boundary of the near field. These silent boundaries were constructed using the four-node axisymmetric elements CINAX4 of ABAQUS; they produce accurate results for low-frequency wave propagation, but their performance is affected by node-to-node resonances, and thus, it is relatively poor in handling high-frequency components. In addition, surface waves such as Rayleigh waves are not absorbed to a reasonable degree by such elements. In 1997 by John Wiley & Sons, Ltd.

11 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1649 Figure 5. The finite element mesh used for the transient computations constructing the mesh for the numerical computations, the silent boundaries were placed far from the z-axis (cf. Figure 1) in order to minimize the effects of spurious wave reflections at early times. In addition, orienting the silent boundary parallel to the z-axis optimizes the capacity of the infinite elements to absorb energy (except for energy associated with Rayleigh waves) from the finite mesh, assuming that the incident wave fronts are predominantly in the radial direction. The mesh used for the transient computations consists of 17,375 total elements and 17,640 nodal points, and is depicted in Figure 5. More details concerning the FE model can be found in the thesis by Brown. Of the three types of direct integration approaches available in ABAQUS, the implicit integration method (Hilber Hughes Taylor operator) is chosen, which solve iteratively the dynamic equilibrium equations at each time increment using Newton s method. This implicit integration method is unconditionally stable, i.e., there is no mathematical limit of the time increment that can be used, and, since the problem is linear, there are no concerns about cost and reliability issues. For the FE computations presented herein, a coefficient of artificial damping of α"!0 16 was used, as suggested by the ABAQUS documentation. An automatic time incrementation schemes is employed with the implicit dynamic integration operator and the total interval of the FE simulations is 100 μs. The simulations were carried out in NCSA s Silicon Graphics Power Challenge Supercomputer System. In Figures 6 9 we present a comparison between the DIT and FE simulations for the stresses σ, σ and the accelerations a, a, at axial distances h"0 31 and 0 76 in inside layer A of the first periodic set of System I, and radial distance equal to r"0, 0 25, 0 50 and 0 75 in. A general conclusion is that the simulations agree better for earlier times (before 50 μs) than for later times; this result is to be expected from the previous discussion on the limitations of the DIT and FE methods. The results for both stress components agree well. In early response times the FE simulations for the axial acceleration, a, agree well with the corresponding DIT results away from the region where the external pressure is applied. Even the near-field solutions the convergence of the two methods is reasonable considering the numerical error introduced in the FE computations of a, which utilize a differentiation process. As for the radial acceleration a, the results of the two methods differ throughout the layered medium for the entire duration of the analysis by John Wiley & Sons, Ltd.

12 1650 C. CETINKAYA E¹ A. Figure 6. Comparison between DIT ( ) and finite element (---- -) computations for the acceleration component a at various locations inside the first layer A of System I The FE results are, in general, more oscillatory than the DIT ones, a fact which may be attributed to node-to-node resonances occurring in the course of the FE analysis. This is because the integration points (where σ is computed) are not coinciding to the nodal points. After 50 μs considerable differences between the FE and DIT results are observed, as can be seen from 1997 by John Wiley & Sons, Ltd.

13 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1651 Figure 7. Comparison between DIT ( ) and finite element ( ) computations for the stress component σ at various locations inside the first layer A of System I Figure 9, where the shear stresses predicted by the DIT technique tend to diminish at later times, whereas the FE results exhibit non-decreasing oscillations. This behaviour of the FE results must be attributed to wave reflections from the silent boundary. Finally, it is difficult to determine the cause of the differences in the radial acceleration results, which may be attributed to a combination of the aforementioned factors by John Wiley & Sons, Ltd.

14 1652 C. CETINKAYA E¹ A. Figure 8. Comparison between DIT ( ) and finite element (--- --) computations for the stress component σ at various locations inside the first layer A of System I Despite the errors that arise in the transient simulations after 50 μs, two general trends emerge. First, that the DIT and FE simulations agree between farther away from the symmetry axis r"0; as illustrated in Figures 6 9, the differences between the wave profiles tend to disappear with increasing radial direction. A second pattern is noticeable in the axial direction: as the depth from 1997 by John Wiley & Sons, Ltd.

15 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1653 Figure 9. Comparison between DIT ( ) and finite element (- ----) computations for the acceleration component a at various locations inside the first layer A of System I the top surface increases the agreement between the two simulations improves. Both of these trends point to the same conclusion, namely, that the FE results improve in accuracy away from the region of application of the external pressure. One explanation for this behaviour is that, in the FE formulation the pressure on the top surface is approximated by a set of point loads applied to the surface nodes within a 0 25 in radius from the z-axis. This approximation introduces errors 1997 by John Wiley & Sons, Ltd.

16 1654 C. CETINKAYA E¹ A. on the near-field simulations, which are more prominent in the region directly below the applied pressure field. Away from this region, however, the effect of the point loads diminishes. Despite the aforementioned limitations of the FE technique, its significant advantage over the DIT method is that it permits the computation of stress components which cannot be directly computed using the double integral transfer matrix formulation. Hence, using the FE method one can compute equivalent (effective) stresses (using, for example, the Von Mises criterion) which enable the study of stress concentration in the layered structure before and after the application of the external pressure. As an application of the computational techniques developed, in the next section we study the different patterns of wave transmission in Systems I and II, in order to determine the effects of weak coupling between layers on the transmission and reflection of waves at the layer interfaces. 4. EFFECTS OF WEAK LAYER COUPLING ON THE WAVE TRANSMISSION Upon examination of the structure of the PZs and AZs of Systems I and II (cf. Figure 3), we note that the weakly coupled System I (corresponding to large values of ratios of mechanical impedances τ and τ (cf. Appendix I)) possesses narrower ranges of wave transmission in the frequency-wave number domain than the strongly coupled System II (for a detailed asymptotic analysis of the PZs and AZs see Reference 37). Since energy transmission in the axial direction can only be carried out by propagating waves (and not by near field attenuating motions, the different level of interlayer coupling is expected to affect the wave transmission in the two systems. A first indication that this is indeed the case was reported in References 7 and 8, where it was shown that in the weakly coupled System I there is greater axial attenuation of stresses at layer interfaces compared to System II. In this section we utilize stress wave computations inside layers to study the effects of layer coupling on the near field wave transmission. External pressure fields identical to the one depicted in Figure 2 are applied to each of the Systems I and II and stress waves inside layers are computed by employing the DIT and FE techniques. When weak coupling between layers A exists, waves propagating in the positive axial direction (downwards) possess high-reflection coefficients at layer interfaces. In addition, reflected waves propagating in the negative axial direction (upwards) possess higher transmission coefficients than waves travelling downwards. As a result, in System I stresses can be localized in the leading layers, i.e., can be confined close to the point of application of the external load, and axial wave transmission is more impaired compared to System II. This localization of stresses can be observed in the FE computations of the Von Mises equivalent stresses depicted in Figures 10 and 11. By making use of the axisymmetry of the problem we only depict the stresses in the semi-infinite strip r*0, 0)z)5(H #H ). The contour plots depicted in these figures depict contours of equal magnitudes of Von Mises stresses in Systems I and II at time instants t"11 01, 11 76, and μs, and were constructed by interpolating the FE results. Such contour plots are especially useful in the wave propagation problem under consideration, since they provide the spatial distribution of stresses and displacements throughout the entire layered medium at a specific time instant. Note the strong axial penetration of stresses in the strongly coupled System II, instead of the more radial stress distributions in System I. Moreover, note the higher stress intensity in the two leading layers of System I, indicating stress localization. In Figures 12 and 13 we present total energy distributions in the three leading layers of the two systems at certain time instants. These simulations were generated by FE computations, and the data were interpolated using Mathematica; the thickness of layer B is exaggerated in order to depict the energy distribution inside that layer. We note that at early times the total energy of System I is confined in the leading layer A, indicating strong reflection of waves at the interface 1997 by John Wiley & Sons, Ltd.

17 Figure 10. Contour plots of Von Mises equivalent stresses for System I at time instants: (a) t"11 01, (b) 11 76, (c) 12 51, (d) ks NEAR FIELD TRANSIENT AXISYMMETRIC WAVES ( 1997 by John Wiley & Sons, Ltd. 1655

18 Figure 11. Contour plots of Von Mises equivalent stresses for System II at time instants: (a) t"11 01, (b) 11 76, (c) 12 51, (d) ks 1656 C. CETINKAYA E¹ A. ( 1997 by John Wiley & Sons, Ltd.

19 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1657 Figure 12. Contour plots of the total energy in the three leading layers of System I at various time instants (external load is removed at t"25 00 μs with layer B; clearly, the weak coupling between layers delays the transmission of the wave fronts through layer B. At the same time instant, there is no such energy localization in System II, indicating that energy has transmitted through the layer interfaces with comparatively little obstruction. After the removal of the external pressure field (after 25 μs) the total energy levels drop significantly and in both systems energy is dispersed mainly in the radial direction. To examine the transmission of stresses through the soft layer, the stress components σ and σ were computed at two sensor locations above and below, respectively, the first layer B of Systems I and II, at a radial distance r"0 125 in from the axis of symmetry. These simulations were performed using the DIT technique, and the results are depicted in Figures 14 and 15. Generally, the magnitudes of both stress components are higher in System I, above and below the layer B. In addition, in System I note the high attenuations of the stress components as they are transmitted through layer B, and the high-amplitude, low-frequency tails in σ following the transmission of the main pulse. Similar tails can be observed in the σ plots of System II, albeit of higher frequencies and of lower amplitudes. The dispersed stress waves of Figures 14 are 1997 by John Wiley & Sons, Ltd.

20 1658 C. CETINKAYA E¹ A. Figure 13. Contour plots of the total energy in the three leading layers of System II at various time instants similar in form to those studied in Reference 32, where asymptotic approximations of dispersed stress pulses propagating in semi-infinite, one-dimensional, weakly coupled layered media were developed, by analytically evaluating the corresponding inverse Fourier integrals; the tail accompanying the propagation of the main transmitted pulse was analytically studied using lowand high-frequency asymptotic approximations. It is suggested that a similar asymptotic analysis could be performed to analyse the radial and axial dispersion of stress waves in the semi-infinite, weakly coupled System I. This analysis would involve, however, analytic approximations of Fourier, as well as, Hankel inversions, and, thus, would be more demanding. Extensive numerical simulations of wave transmission through the layers of Systems I and II have been carried out in the thesis by Mohammed and Brown, but these are not presented here due to space limitations. Generally, it was found that weak coupling between layers significantly impairs wave transmission in the axial direction, whereas at the same time, facilitates wave propagation in the radial direction. Moreover, FE simulations indicate that the coupling between layers affects the displacement magnitudes in the two systems under consideration: the weaker the coupling, the softer the system, and the greater the downward displacement the top interface under identical loading conditions. This is analogous to a single-degree-of-freedom 1997 by John Wiley & Sons, Ltd.

21 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1659 Figure 14. Stress σ at radius r"0 125 in above and below the layer B of the first periodic set of : (a) System I, (b) System II Figure 15. Stress σ at radius r"0 125 in above and below the layer B of the first periodic set of : (a) System I, (b) System II 1997 by John Wiley & Sons, Ltd.

22 1660 C. CETINKAYA E¹ A. spring mass system under impulsive excitation: a mass grounded with a weaker stiffness possesses a larger amplitude than does one with a stiffer stiffness. 5. CONCLUDING REMARKS Two computational approaches were used in this work to study axisymmetric wave transmission inside the layers of two-layered structures: a DIT technique based on numerical inversions of Fourier and Hankel transformations, and a Finite Element (FE) analysis. To overcome numerical instabilities associated with exponential dichotomy in the DIT method a global diagonalization scheme was employed instead of performing direct transfer matrix multiplications. It was shown that in order to compute near field waves close to the point of application of the external load, a careful selection of the upper wave number limit for the numerical Hankel inversion is required. Generally, the near field DIT results are more accurate than the corresponding FE simulations, which suffer from inherent numerical instabilities due to the FE discretization of the distributed external load, node-to-node oscillations, or reflections from infinite elements. However, away from the near field of the applied external pressure both DIT and FE results are in satisfactory agreement, at least at early times. As discussed in the thesis by Mohammed, a direct transfer matrix multiplication can still be used to compute the transfer functions in the DIT method, provided that preliminary filtering of the propagation constants in the propagation zones of the layered medium is performed. This filtering effectively poses an upper bound for the real parts of the propagation constants which are the main source of exponential dichotomy, while it preserves the imaginary parts. Preliminary computations of transfer functions using filtered propagation constants indicate that this method could be of potential use, at least in the computations of transient stresses and displacements away from the near field of the applied pressure field. A new feature of the presented DIT computational technique is that it takes into account the specific structure of the propagation and attenuation zones (PZs and AZs) of the layered medium in accurately determining integration domains for the transform inversions. This greatly contributes in eliminating numerical instabilities due to exponential dichotomy, and in incorporating the underlying physics in the computations: indeed, it is critical to select the sampling rates for the transfer functions based on the widths and number of the PZs. This ensures that the dominant excited modes for wave transmission are taken into account in the numerical simulations. As an application of the numerical techniques, wave propagation in two-layered structures was studied. The two structures are composed of biperiodic sets of stiff-soft layers, and they differ in the amount of coupling between their stiff layers. For the weakly coupled layered system, it was shown that wave transmission is impaired in the axial direction and enhanced in the radial one. Such a result can have potential application in the use of layered media as shock isolators of sensitive surfaces. APPENDIX I Elastic and geometric properties of systems I and II The layered Systems I and II are each composed of five biperiodic sets, with each set composed of two layers labeled A and B. The layers have the following properties by John Wiley & Sons, Ltd.

23 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1661 System I Layer A ( stiff layer): E "45 10 lb in " N m ν"0 25 ρ " lb kg s" in m H "0 5 in"12 7 mm λ " lb in " N m μ " lb in " N m c T " in s " m s c L " in s " m s Layer B ( soft layer): E "2 10 lb in " N m ν"0 48 ρ " lb kg s" in m H "0 01 in"0 254 mm λ " lb in " N m μ " lb in " N m c T "8219 in s " m s c L "41913 in s " m s where for each layer, E denotes the modulus of elasticity, ν the Poisson s ratio, ρ the material density, H the layer thickness, λ, μ the Lamé constants, and c, c the phase velocities for shear and longitudinal waves, respectively. The impedance parameters τ and τ are defined as τ "(ρ c T )/(ρ c T ), and τ "(ρ c )/(ρ c L ); for System I these impedance parameters assume the values τ "89 9 and τ " by John Wiley & Sons, Ltd.

24 1662 C. CETINKAYA E¹ A. System II Layer A ( stiff layer): E "45 10 lb in " N m ν"0 25 ρ " lb kg " in m H "0 5 in"12 7 mm λ " lb in " N m μ " lb in " N m c T " in s " m s c L " in s " m s Layer B ( soft layer): E "2 10 lb in " N m ν"0 48 ρ " lb kg s" in m H "0 01 in"0 254 mm λ " lb in " N m μ " lb in " N m c T "8219 in s " m s c L "41913 in s " m s τ "8 99 and τ " by John Wiley & Sons, Ltd.

25 NEAR FIELD TRANSIENT AXISYMMETRIC WAVES 1663 APPENDIX II Explicit form of transfer matrix [T ] The ijth entry of the (4 4) matrix [T ] is denoted by ¹ (i, j ), and is defined as follows: ¹ (1, 1)" 1 p!2k cosh g H h #(hm #k ) cosh hm h H ¹ (1, 2)" 1 p!k g (hm #k ) sinh g H h #2hM k sinh hm h H ¹ (1, 3)" H p!k μ cosh g H h #k μ cosh hm h H ¹ (1, 4)" H p!k gμ sinh g H h #hm μ sinh hm h H ¹ (2, 1)" 1 p 2gk sinh g H h!k h (hm #k ) sinh hm H hm ¹ (2, 2)" 1 p (hm #k ) cosh g H h!2k sinh hm h H ¹ (2, 3)" H p g μ sinh g H h!k h sinh μhm H hm ¹ (2, 4)" H p k μ cosh g H h!k μ cosh hm h H ¹ (3, 1)" 1 Hp 2k(hM #k )μ cosh g H h!2k(hm #k )μ cosh hm h H ¹ (3, 2)" 1 Hp k μ (hm #k ) sinh g H h!4hm k μ sinh hm h H ¹ (3, 3)" 1 p (hm #k ) cosh g H h #2k cosh hm h H ¹ (3, 4)" 1 p k g (hm #k )sinh g H h!2khm sinh hm h H ¹ (4, 1)" 1 Hp!4gk μ sinh g H h #μ h (hm #k ) sinh hm H hm ¹ (4, 2)" 1 Hp!2k(hM #k )μ cosh g H h #2k(hM #k )μ cosh hm h H ¹ (4, 3)" 1 p!2gk sinh g H h #μ h (hm #k ) sinh hm H hm ¹ (4, 4)" 1 p!2k cosh g H h #(hm #k ) cosh hm h H 1997 by John Wiley & Sons, Ltd.

26 1664 C. CETINKAYA E¹ A. Since we are computing transient waves inside a layer A of the structure, the following definitions refer to layer A: C " λ#2μ, C " μ ρ ρ, a"c, C g "k # p a, hm "k #p where λ, μ are the Lamé constants, ρ the layer density, k the normalized Hankel transform variable, H is the layer thickness, h is the depth inside the layer where the computation is performed (i.e., the thickness of the fictitious layer, cf. Figure 1(c)), and p"jωa, where ω is the normalized Fourier (frequency) variable. ACKNOWLEDGEMENTS This work was partially supported by NSF Young Investigator Award CMS (A.F.V.); Dr. Devendra Garg is the Grant Monitor. Additional support was provided by the National Center for Supercomputer Applications (NCSA) of the University of Illinois in the form of Supercomputer time allocation. This support is gratefully acknowledged. The authors would like to thank Professor J. Ghaboussi of the Department of Civil Engineering of the University of Illinois for sharing with them his expertise in Finite Element analysis. REFERENCES 1. J. D. Achenbach (ed.), Evaluation of Materials and Structures by Quantitative ºltrasonics, CISM Courses and Lectures No. 330, International Centre for Mechanical Sciences, Springer, Wien, New York, F. Santosa and W. W. Symes, A dispersive effective medium for wave propagation in periodic composite, SIAM J. Appl. Math., 51, (1991). 3. E. Sanchez-Palencia, Nonhomogeneous Media and»ibration ¹heory, Lecture Notes in Physics, Vol. 127, Springer, Berlin and New York, T. J. Delph, G. Herrmann and R. K. Kaul, Harmonic wave propagation in a periodically layered infinite elastic body: antiplane strain, J. Appl. Mech., 45, (1978a). 5. T. J. Delph, G. Herrmann and R. K. Kaul, Harmonic wave propagation in a periodically layered infinite elastic body: plane strain, analytical results, J. Appl. Mech., 46, (1979b). 6. T. J. Delph, G. Herrmann and R. K. Kaul, Harmonic wave propagation in a periodically layered infinite elastic body: plane strain, numerical results, J. Appl. Mech., 47, (1979). 7. C. Cetinkaya, Axisymmetric elastic wave propagation in weakly coupled layered media: analytical and computational studies, Ph.D. ¹hesis, University of Illinois, Urbana, IL, C. Cetinkaya and A. F. Vakakis, Transient axisymmeric stress wave propagation in weakly coupled layered structures, J. Sound»ib., 182, (1995). 9. S. A. Rizzi and J. F. Doyle, A spectral element approach to wave motion in layered solids, J.»ib. Acoust., 114, (1992). 10. T. Kundu and A. K. Mal, Elastic waves in a multilayered solid due to a dislocation source, ¼ave Motion, 7, (1985). 11. A. K. Mal, Wave propagation in layered composite laminates under periodic surface loads, ¼ave Motion, 10, (1988). 12. R. L. Weaver, W. Sachse and K. Y. Kim, Transient elastic waves in a transversely isotropic plate, ¹AM Report No. 740, ºI º-ENG , University of Illinois at Urbana, Champaign, Urbana, IL, M. El-Raheb and P. Wagner, Coupled transient response of tiles bonded elastically to a finite flexible plate, J. Acoust. Soc. Am., 77, (1985). 14. M. El-Raheb and P. Wagner, Transient waves in a thick disk, J. Acoust. Soc. Am., 90, (1993). 15. T. Hughes and W. K. Liu, Implicit explicit finite elements in transient analysis: stability theory, J. Appl. Mech., 45, (1978). 16. T. Hughes and W. K. Liu, Implicit explicit finite elements in transient analysis: implementation and numerical examples, J. Appl. Mech., 45, (1978b). 17. J. Bielak, L. F. Kallivokas, J. Xu and R. Monopoli, Finite element absorbing boundary for the wave equation in a halfplane with an application to engineering seismology, Proc. 3rd Int. Conf. on Mathematical and Numerical Aspects of ¼ave Propagation, Mandelieu, La Napoule, April 24 28, SIAM, Philadephia, PA, by John Wiley & Sons, Ltd.