Available online at www.sciencedirect.com

SciericeDirect

Physics Procedia 2 (2009) 1103-1111

www.elsevier.com/locate/procedia

Proceedings of the JMSM 2111 conference

Comparison of Two Numerical Method for Determination of Floquet Waves in Periodic Multilayer

M. Ben Amor*, M. H. Ben Ghozlen

Laboratoire de Physique des Matériaux, Faculté des Sciences de Sfax, BP 1171, 3000 Sfax,Tunisia. Received 1 January 2009; received in revised form 31 July 2009; apcepted h1 August 2009

Abstract

The present work deals with Floquet waves determination in multilayered composites using two methods: Floquet theory and spectral analysis of displacement field (SADF). Based on a recently-developed stiffness matrix, a recursive process gives interfacial displacement anywhere in the multilayer as function of deepness. A fast-Fourier transform (FFT) procedure has been used in order to extract the wave numbers propagating in the multilayer. The Floquet waves parameters as function of frequency and incident angle are calculated separately and a comparative study has been achieved for these two methods. For different incidence angle, the upper frequency bound of this homogenization domain is estimated. A consistent algorithm is adopted to develop an inverse procedure for the determination of the materials constants of multidirectional composite. Computational example for quasi-isotropic composite is given and compared with Floquet Homogenization method (FHM). © 2009 Elsevier B.V. All rights reserved

PACS: 43.20.Bi, 43.35.Cg

Keyswords: Layered anisotrpic media, Waves propagation,Spectral analysis ; Homogenization; Stiffness matrix.

1. Introduction

Due to the importance of the composite material for structural applications, significant research has been devoted to elastic wave propagation in such materials. A convenient method for analyzing wave propagation in composite media is the use of Floquet theory. This approach has been frequently used by several authors [1,2,3] to study steady state wave propagation in periodic elastic composites. It is well known that Floquet response is closely related to the periodic character of the multilayer along deepness axis. Nayfeh [4] developed a transfer matrix method for general multilayered anisotropic media, obtained the Floquet wave equation for the periodic case and considered several numerical examples. Experimental and theoretical examples for multidirectional composites are given by Chimenti and Nayfeh [4,5]. Braga and Herrmann [6] also addressed periodic arbitrarily anisotropic media with a transfer matrix method using the Stroh formalism. They performed a very detailed analysis of the Floquet wave characteristic equation and studied stop and pass bands (Brillouin zones). Potel and Bellevall [7,8] applied the

* Corresponding author. Tel.: +216 74676617; fax: +216 74676617. E-mail address: morched_benamor@yahoo.fr.

doi:10.1016/j.phpro.2009.11.069

Floquet wave solution to finite periodic media and specifically addressed composite media homogenization using Floquet wave theory. They obtained results equivalent to low frequency homogenization used in mechanics of composite materials. To overcome the computational instabilities of the transfer matrix, a different approach, was developed to investigate the dynamics for wave propagation in periodic structure. The formalism of stiffness matrix introduced by Wang and Rajapkse [9] into the symmetry planes of an orthorhombic medium and by Wang and Rokhlin [10] for a composite medium made up of monoclinic layer, represents an effective solution for the layers great thickness and the incidental waves in high frequency range. Indeed, this method ensures a better homogenization of the square matrix of sixth rank that connects the parameters relating to two successive interfaces. The Floquet wave spectrums of multilayered composites stop and pass bands and slowness surfaces have been extensively investigated in Refs. [3, 11]. To obtain the Floquet wave characteristic equation one should apply the periodicity condition [1,3,12]. It is well-known that the stiffness matrix method remains the numerical stable procedure especially when the cell number or the frequency becomes large.

In this work, we are interested in the stop-pass diagram where the multilayer could be compared to an homogenous material. In this paper a novel approach is proposed to compute the Floquet waves spectrum. In this method, a combination of stiffness matrix and spectral analysis of displacement field (SADF) is developed to solve the wave propagation in the periodic structure. In Section 2, a mathematical statement of the stiffness matrix method necessary for further developments is briefly reviewed. In section 3, we describe a novel approach, recently published by us [13], for computing the Floquet waves number based on spectral analysis. In this method, the displacement field and the fast Fourier transform techniques (FFT) are combined to compute the Floquet wave numbers propagating in the multilayer. Next in section 4, a homogenization method is developed for the multilayer to determine the material constants of multi-ply composite employing an inversion scheme. Finally, section 5 a conclusion must be drawn.

2. Stiffness matrix

In the jth layer, the displacement vector may be written as the summation of six partial waves [10]:

uj = X (ad„P„deia'<z-Zj-1' + a"Pueia'<z-Zj) )j ei(o>x-m) (1)

where u ' = (u"'" ,u"'" ,u"'" ) j, n = (1,2,3) denotes n partial wave. The superscripts d and u represent the

j x y z J

downward (+z) and the upward (-z) travelling plane wave modes respectively; P'¡j'u = (P^" ,Pyd" Pf'" fn are the unit displacement polarization vectors corresponding to waves with (odz )n and (ouz )n wave vector respectively.

The displacement polarization vector Pd'u and third component wavenumber(odz'u )n are determined by solving the Christoffel equation (2) and applying Snell's law cr{ = , where j =1, 2,...N and 0"° is the x projection of the wavenumber for the incident wave:

(CjUojal -pa2Sik)Pk = 0, (2)

where Cjjkl represent the layer elastic constants and p the density.

The stress component vector tj = (13j'tl2 )T on the x-y plane parallel to the layer surface can be related to each of the plane wave displacement field using Hook's law:

tj = X (aidieia'(z~zj-1) + au„d u„ei0z(z-Zj) )j ei(°*x-«) (3)

Where the components (d± )j of vector d j are related to the polarization vector

(Pf'u) by (dd'u )j = (Ci3UoiPtd'u )j, (Wang, 2001).

The displacement u j and stresses t j on the upper and lower interfaces bounding layer j can be represented in matrix form:

p,!h„

Pd PjHd

Dj DdHd

Where Pju (3x3) matrix gathers together polarization vectors, Ad u the amplitude vectors of the waves going down (d) and up (u) and Hdu a diagonal matrix which contain the corresponding wave phase shift.

The cell stiffness matrix Kc connecting separately displacements and stresses, on the level of the two interfaces of

the jth layer, is obtained by substituting into Eq. (4) the amplitude vector Aju fromEq. (5) [11].

Kc( 6x6 )--

PjH r 11

Due to the periodicity of the composite structure a free wave is propagating in the periodic medium called Floquet wave. The theory of the periodic mediums known as Floquet theory, establishes a bound between displacement and stresses at the entry and the exit of the cell, this bound represents the difference of phase which accompanies the crossing by the cell for thickness. To find the Floquet wave properties one applies the periodicity conditions [1,3,6].

" t- u

= e zh c .

t + u +

where hc represents the thickness of the unit cell and az the Floquet wave z-component. Eq. (7) relates the stresses and displacements on the top (t+ ,u+) and bottom (t- ,u- ) surfaces of the cell (Fig.1).

The Floquet wave spectrum of multilayered composites, pass-stop bands and slowness surfaces have been extensively investigated in [1,3,11]. To obtain the Floquet wave characteristic equation one should apply the periodicity condition. Different methods can be used to obtain the Floquet wave characteristic; among them the transfer matrix d and Stiffness matrix methods are often used [1,9,10]. The Floquet wave numbers a are found by searching for the roots of the determinant of the linear system (6) which has six solutions for eia zhc .

3. Spectral analysis of displacement field

In order to investigate the dynamics for wave propagation in an anisotropic periodic system, the displacement field through the multilayer is calculated using the stiffness matrix method. A novel approach for the determination of the Floquet wave is proposed.

The model utilized the mechanical characteristics of the carbon-epoxy summarized in table1.

In the present work, for Floquet wave computation we utilized the (SADF) method developed for layered composites. In order to investigate the dynamics for wave propagation in an anisotropic periodic system, the displacement field through the multilayer is calculated using the stiffness matrix method. A novel approach for the determination of the Floquet wave is proposed. An illustration is performed on carbon-epoxy multilayered medium where period stacking sequences is [0/45/90/-45].

Fig.2. Displacement field along x of [0/45/90/-45] carbon-epoxy versus depth, with the incidence of 12° at 1 MHz.

Fig.3. Displacement field along x of [0/45/90/-45] carbon-epoxy versus depth, with the incidence of 12° at 1 MHz.

When the frequency of the incident wave is sufficiently low and the wavelength is much larger then the cell-thickness, the behaviour of the propagating waves extends this periodicity to both displacements and stresses fields. The Fourier analysis of the displacement or stress fields shows the excited Floquet modes, three picks are expected when the obtained wave number is real. For instance the displacement field calculated at an incidence of 12° and a frequency fixed at 1MHz is given by Fig.2 and Fig.3. The corresponding Fourier analysis spectral is reported in Fig.4. The plot displays three peaks which can be related to three propagating Floquet waves i.e. the quasi-longitudinal wave and the two quasi-shears waves. The obtained Floquet wave number are respectively aL = 0.250 mm-1, aS1 0.435 mm-1 and gS2 = 0.516 mm-1.The relation between the so-obtained wave numbers and the frequency is plotted in Fig.5. The dispersion curves are associated to the quasi-longitudinal wave and the quasi-shear wave. This plot show that, for a limiting frequency bounded to 1.7 MHz, we may consider the multilayer as an effective homogeneous anisotropic medium supporting at most three plane waves. It is noted that for the wave number varies

linearly as for a homogenous material. Above this frequency domain, the dispersion curves become fairly complex, which indicates that the material becomes dispersive and has a nonhomogeneous character.

1400 ! i ....... !!!!!!! i 0.435 i i i i i ____i.......:.......L......:......j......j.......L......

i i .......!......Ii i .......:......J. i i i i i i ......!.......f......!......1......1.......i...... i i i i i : ......:.......L......;......J......J.............

□ 250i ....... ! i i ...j. i i i i i i ......I.......[......!......!......1.......1...... i i i i i : ......i.......i......;......;......j.......i......

; i .......

i i i ...j. i i i i i i 0516 ; ; ; ;

0 .......r iJ n I .......:......"......:......:.......:...... :::::: • -J____i_^_i__j_

□ 0 2 0 4 0 6 0.8 1 12 1.4 IE wave number (1/mm)

Fig.4. Spectrum of total displacement field of [0/45/90/45] carbon-epoxy with the incidence of 12°, at 0.6 MHz frequency.

OÔJ |OOOOf

dence=12° ........... ® ®

............

Fig.5. Comparison between dispersion curve of [0/45/90/-45] carbon-epoxy at an incidence of 12°obtained by FHM: (o) and SADF methods: (• ).

-100 -80 -60 -40 -20 0 20 40 60 80 100 ange dincidence

Fig.6. Spectrum of pass and stop bands for the three Floquet waves for [0/45/90/-45] composite as a function of incident angle G and incident plane orientation angle a, frequency is 2.25 MHz.

Frequency (MHz)

Fig.7. Pass and stop bands of the three Floquet waves as function of incident angle and frequency for [0/45/90/-45] periodic composite. The propagating plane is oriented along 0° fiber direction.

To check the accuracy of the numerical calculation, we reported in the same plot the slowness surface and dispersion curves obtained by SDAF method and Floquet theory. As can be seen in Fig.5, the results obtained by Stiffness matrix method are very close to those calculated by SADF method. Ideally, inside the homogenization

domain both methods provide identical results. The plot of the pass-stop bands diagram associated to Floquet waves is given by Fig.6 and Fig.7.

The region located at the low frequency range look similar to the diagram of an homogenous material, only critical angles are slightly dependant on frequency. The criterion adopted for homogenization, is expressed by X>2hc where X is the weakest wavelength, and hc the cell thickness [3]. That means the Floquet wavelength along z axis must be at least two times hc the spatial period of the multilayer. The curve bounding the homogenization zone is drawn on the plot by discontinuous line. The spectrum of pass and stop bands for the three Floquet waves for [0/45/90/45] composite is represented as a function of incident angle 9 and incident plane orientation angle a at frequency 2.25 MHz (Fig.6) and as a function of incident angle 9 and frequency with the propagating plane oriented at a = 0° (Fig.7). White domains correspond to all three propagating waves, light gray to two propagating waves and darker gray to one propagating wave, black corresponds to the stop band for all three waves (no propagating waves permitted). The frequency indicated by the arrow 'A' corresponds to the polar diagram shown in Fig. 6. The dashed line indicated by the arrow 'B' shown in (Fig.7) corresponds to the homogenization domain.

4. Effective properties of multi-ply composites using SADF

It is well-known that for an arbitrary frequency and propagation direction in a homogeneous anisotropic medium, the Floquet wave numbers (velocities) are equal to the wave numbers of plane waves (velocities) in this medium calculated from the spectrum associated the displacement field. Thus, to replace the actual multilayered we define the effective medium as an anisotropic homogeneous media from which the Floquet wave number are equal to the wave number of plane wa ves in original medium. This replacement does not change the Floquet wave numbers or other properties due the equality between displacements and stresses.

At low frequencies we homogenize the composite using the SA method. More details of the method can be found in a Ref. [3] transforming the composite to a dispersive homogeneous anisotropic medium.

The plot of the dispersion curves show that, for a limiting frequency, we may consider the multilayer as an effective homogeneous anisotropic medium supporting at the most three plane waves. The displacement field has been computed in the z-direction, and a similar FFT procedure has been employed in order to extract the Floquet wave number and phase velocity in the composite. These Floquet wave numbers are equal to the wave numbers of plane waves in original medium.

The set of effective elastic constants is obtained from the obtained Floquet wave numbers. To do this, we select an arbitrary set of elastic constants of the effective medium as initial guesses and calculate the wave numbers of the propagated plane waves for the homogeneous medium using SADF. The initial guesses are very important to save a calculation time and to obtain a quickly convergent solution. Since the initial guesses for the effective elastic constants are arbitrary, the wave numbers in the effective homogeneous medium are different from the Floquet wave numbers in the periodic medium. The determined elastic constants are then used again in the SADF to calculate the spectrum corresponding to the displacement field. The final determined elastic constants are guaranteed to produce the effective spectrum associated to the displacement filed that are very close to the calculated ones when fed into the SA DF. The unknown effective elastic constants are obtained by a least-squares nonlinear minimization of function (Y).

Y = "2 X(of(9i) -af (9i))2, (8)

where aF(9i)is the set of calculated Floquet wave numbers propagating at angles 9i and aff(0i) are the calculated wave numbers for an effective medium, n is the number of independent parameters to be determined which depends on the cell structure, m is the number of wave numbers values for different propagation directions 9i used.

Figure 8 outlines the algorithm to obtain the effective elastic constants of a homogenized periodic medium from the Floquet wave numbers. As shown in Fig.4, the Floquet wave numbers are calculated, using the FFT procedure of the displacement filed, at different wave propagation planes rotated by angle a varying from 0° to 90° with a step of 10° around the symmetry axis z (Fig.1). In each propagation plane, 50 different propagation angles 9i are used. The extracted wave numbers are used to calculate the wave numbers propagating in the effective homogeneous medium in the same propagation directions. The (Y) function consisting of the summation of the square difference

of calculated Floquet waves number and the calculated waves number for an effective medium for different propagation direction. An optimization algorithm based on the Simplex method [14,15] predefined in Matlab environment was used.

As an example, we will use [0/45/90/-45] composites to calculate the effective elastic constants using the algorithm described above.

For the quasi-isotropic composite [0/45/90/-45], in the homogenization frequency domain, the x-y plane (the plane of cell boundaries) is a plane of isotropy. Thus the material can be modelled as a transversely isotropic medium with five independent elastic constants.

To validate the SADF approach, we compare the effective elastic constants at very low frequency with static calculations of composite elastic constants using the Floquet wave homogeneous method (FHM). The cell [0/90] is used as an example. The frequency used to calculate effective elastic constants using the FHM is 100 Hz. As can be seen in Table 2, these results are in agreement with those obtained by Wang et al. [3] who used the FHM.

Tablel. Lamina properties

Elastic constants (GPa)

C11 143.2

C22 15.8

C12 7.5

C23 8.2

C55 7.0

Density (gcm-3) 1.6

Thickness (mm) 0.194

Table2. Results calculated by SAFD method and Floquet wave homogenization method (FHM) (unit: GPa) Cii C33 C12 C23 C44 C66

FHM 79.427 15.800 7.507 7.853 4.927 7.000 SAFD 79.499 15.769 7.500 7.850 4.943 6.999

Fig.8: Elastic constants determination by using optimization algorithm of calculated: Wav number as function of direction of propagation.

5. Conclusion

This work deals with Floquet waves determination in periodic multilayered composites using two numerical methods: the spectral analysis of displacement field and Floquet wave theory. Based on displacement field analysis of the displacement field, a fast-Fourier transform (FFT) processing is used to compute the Floquet wave characteristic propagating in the multilayer. From the dispersion curves, we identify the homogenization frequency domain, within which the periodic medium is equivalent to an effective homogeneous anisotropic medium. The extracted wave numbers are used as the inputs for the SAFD model. The outputs are the elastic constants of multidirectional composite. Since the initial guesses for the effective elastic constants are arbitrary, the wave numbers in the effective homogeneous medium are different from the Floquet wave numbers in the periodic

medium. Computational examples for quasi-isotropic composite is given and founded in agreement with the results obtained by FHM method.

The spectral analysis may have some advantages and it also provides an alternative computational approach. Firstly it gives a best description of the displacement distribution any where in the multi-ply composite. Secondly, it is physically more significant. Since the spectrum give not only the actual Floquet waves (only the exited modes) propagating in the effective medium but also information about the amplitude and the energy repartition of each mode.

The technique presented in this paper can be extended to the homogenization of multi-ply-composite containing anomalies, a case of missing layer, which affect slightly the periodicity of the stratified multilayer.

References:

[1] C. Potel, J.F de Belleval, Y. Gargouri, J. Acoust. Soc. Am. 97 (1995) 2815

[2] A.H. Nayfeh, wave propagation in layered anisotropic media (North-Holland, Amsterdam, 1995)

[3] L.Wang, S.I.Rokhlin, J. Acoust. Soc. Am. 112 (2002)38

[4] A.H. Nayfeh, J. Acoust. Soc. Am. 89 (1991)1521

[5] D.E. Chimenti, A.H. Nayfeh, J. Acoust. Soc. Am. 87 (1990)1409

[6] A.M. Braga, G. Hermann, J. Acoust. Soc. Am. 91 (1992)1211

[7] C. Potel, J.F de Belleval, J. Acoust. Soc. Am. 93 (1993)2669

[8] C.Potel, J.F de Belleval, J. Appl. Phys. 77 (1995) 6152

[9] Y.Wang, R.K.N.D Rajapakse, J. Appl. Mech. 61 (1994) 339

[10]S.I. Rokhlin, L.Wang, J. Acoust. Soc. Am 112 (2002) 822

[11]S.I. Rokhlin, L.Wang, International Journal of Solids and Structures 39 ( 2002) 5529

[12] L. Wang., S.I. Rokhlin, Analysis of ultrasonic wave propagation in multiply composites: homogenization and effective anisotropic media. In Tompson, D.O., Chimenti, D.E. (Eds), Review of Progress in QNDE, vol.20B. Plenum, New York, (2000)1015

[13] M. Ben Amor, M.H. Ben Ghozlen, Mechanics Research Communications 34 (2007) 488

[14] J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wrigh, SIAM J. OPTIM. 9 (1998) 112

[15] J.A. Nelder, R .Mead, Comput. J. 7 (1965) 308