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Physics Procedia 70 (2015) 811 - 814

2015 International Congress on Ultrasonics, 2015 ICU Metz

Multiple scattering of elastic waves in unidirectional composites

with coated fibers

Shiro Biwa a*, Takuto Sumiya a

a Graduate School of Engineering, Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8540, Japan

Abstract

A computational method for analyzing the two-dimensional multiple scattering of elastic waves in unidirectional fiber-reinforced media, consisting of elastic cylindrical fibers arranged in an infinitely extended elastic matrix, is presented. The method is based on the eigenfunction expansion of the displacement potentials and the numerical collocation technique to obtain the expansion coefficients. In the present study, the formulation is extended to incorporate the presence of a coating layer between the fibers and the matrix. As an example, the transmission characteristics of longitudinal as well as transverse waves in a unidirectional composite with square array of fibers are analyzed, and the phase velocities in the composite are evaluated. The effect of the coating layer on the wave transmission and stop-band formation behavior is also demonstrated. ©2015TheAuthors.Publishedby Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015

Keywords: Composite material; Elastic wave; Multiple scattering; Eigenfunction expansion; Collocation method; Stop band

1. Introduction

Multiple scattering of elastic waves in fiber-reinforced composite materials is an important subject regarding the design for dynamic loading as well as the ultrasonic nondestructive testing. From a theoretical point of view, many versions of the multiple scattering theory and micromechanical models have been applied to analyze the elastic wave propagation characteristics in fiber-reinforced composites, e.g. [1, 2]. Direct computational approaches were also employed to analyze the multiple scattering and the resulting wave propagation characteristics of SH (shear horizontal) waves in fiber-reinforced composites [3-8]. Recently, these approaches were extended to two-dimensional elastic waves in fiber-reinforced composites [9]. In many of the previous works, however, the fibers were assumed to be directly bonded to the matrix. In some composites, the fibers are often multilayered, for example

* Corresponding author. Tel.: +81-75-383-3796, E-mail address: biwa@kuaero.kyoto-u.ac.jp

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Scientific Committee of ICU 2015

doi:10.1016/j.phpro.2015.08.165

having a coating layer [10]. In this study, the formulation of the previous work [9] is extended to incorporate the presence of a coating layer between the fibers and the matrix. As a numerical example, the effect of the coating layer on the propagation behavior of longitudinal as well as transverse waves in a fiber-reinforced composite is analyzed.

2. Formulation

The problem to be considered here is the time-harmonic, two-dimensional motion of an infinite isotropic medium (Lamé constants X\, / and density p\) containing unidirectionally aligned N circular cylindrical isotropic elastic fibers (radius a, Lamé constants A2, /2 and density p) with isotropic elastic coating layer (outer radius b, Lamé constants A3, / and density p3) as shown in Fig. \. The wave speeds in each medium are given by cLa = {(Aa + 2/a)/pa}\/2 and cTa = (Lapa)12 , and the wave numbers kLa = cdcLaand kTa = (dcTa (a= \, 2, 3), where c = 2nf is the angular frequency f is the frequency). The elastodynamic multiple scattering problem is formulated in terms of the displacement potentials O(r) and ¥(r) which satisfy the two-dimensional Helmholtz equation. First the exciting fields at the ith fiber are given by

O!',exo (r) = ®inc (r) + ^ ®j,sca (r) , r ,exc (r) = Yinc (r) + ^ Yj,sca (r) ,

j=\ j *'

where Oinc(r) and ¥inc(r) are the incident fields propagating in the x1 direction, and 0'''sca(r) and ¥j,sca(r) are the scattered fields by the jth fiber. The exciting and the scattered fields can be expressed as

®!,exc(r) = XE„'J„(kLi | r- r |)exp(i«0;) , r,exc(r) = ^F„'J„(kn | r -r |)exp(in0,-) , (2a)

®!,sca(r) = X A'Hn (kLi | r — r |) exp(mdj) , r,sca(r) = £ B*Hn (kn | r — r |)exp(in^.) , (2b)

where En', F„', An', Bn (n = 0, ± 1, ±2, ...; i = 1, 2,..., N ) are unknown coefficients, Jn and Hn are the nth-order Bessel and Hankel functions of the first kind, respectively, and $ is the polar angle of the position r viewed from the position r (center of the th fiber). The fields refracted in the th fiber and in the coating layer are expressed as

©i,ref (r) = £cnjn(kL2 | r — r |)exp(in^i) , Yi,ref(r) = £D^Jnfe | r — r |)exp(in0.) , (3a)

®i,coat(r) = ¿RJ(kL31 r — r |) + bn%(kL31 r — r |)}exp(in^i), (3b)

^,coat(r) = Yj{c^Jn(kT3 |r-r |) + dn'Yn(kT3 |r-r |)}exp(in^) :

where Cn', Dn, aj, bj, cj, dj ( n = 0, ± \, ± 2,... ; i = \, 2,..., N ) are unknown coefficients, and Yn is the nth-order Bessel function of the second kind. The relations between the expansion coefficients in Eqs. (2) and (3) are obtained

Fig. 1 A unidirectional composite with coated fibers, subjected to the incidence of plane longitudinal or transverse wave.

by applying the boundary conditions at the fiber-coating as well as the coating-matrix interfaces, and expressed in the following matrix form.

(( Bn en D\ an K cn d'„ ) = [M ]

By rewriting the components of the 8X2 matrix [M] as Sn = Mn, Tn = M12, Un = M21, Vn = M22, the scattered and the exciting field coefficients are connected as

An' = SnEn + TnFn', Bn' = UnEn' + VnFn'. (5)

Substituting Eqs. (2) and (5) in Eq. (1) leads to the following equations which hold at arbitrary r in the matrix.

£En'Jn(*L1 |r-r I)exp(m0î) = Olnc(r) + £ £(SmEmJ + TmFj)Hm(kL1 | r-r, |)exp(im0,), (6a)

j=1 m=-^

2 Fn'Jn (*T1 |r - r I) exp(ln0¿ ) = r1» + 2 X (UmEj + VmFj )Hm (kri |r - r, I) exp(lm0j ). (6b)

In the numerical analysis, the infinite series in Eq. (6) are truncated at a finite but sufficiently large number, and the above equations are evaluated at collocation points, which are taken at the interface between the matrix and each coating layer. Then, the unknown coefficients can be determined as a solution of a linear system of equations.

3. Numerical example and results

As a numerical example, the elastic wave propagation in a unidirectional composite with Ti-alloy matrix (Ai = 103 GPa, jUi = 44.8 GPa, p = 5400 kg/m3), SiC fibers (a = 67 |im, A = 92.1 GPa, ^ = 177 GPa, p = 3200 kg/m3) and carbon-rich coating layers (b = 70 jm, A3 = 21.8 GPa, j = 4.6 GPa, p3 = 2100 kg/m3) is analyzed. The coating layers are assumed to be perfectly bonded to the fibers and the matrix. For comparison, the composite without the coating layers is also considered, with the same material parameters for the matrix and fibers but the fiber radius 67 jm replaced by 70 jm. Although the method can be applied to arbitrary arrangement of coated fibers, in this study they are arranged in a square array with the volume fraction of coated fibers 25 %. By utilizing the periodicity of the fiber arrangement, only the fibers in a fundamental block [9] are explicitly accounted for, and the fiber arrangement in the matrix is assumed to consist of the repetition of this block in the x2 direction. As shown in Fig. 2, the present analysis employs the fundamental blocks containing two fiber rows in the x2 direction.

For the square arrangement of 80 X 2 fibers in the fundamental block as shown in Fig. 2 (a), the wave fields are computed for the incidence of either plane longitudinal (P) or plane transverse (SV) waves. From the obtained wave fields, the wavelengths in the fiber-reinforced region can be evaluated, which then gives the phase velocities in the

Fig. 2 Fundamental blocks with (a) Í array of fibers.

x2 and (b) 5x2 square

Longitudinal wave

a—"-A-—-a-—-h-

Transverse wave

8—--i—-8—-8-

—■— With coating layer

-O— Without coating layer

0.02 0.04 0.06 0.08 Normalized frequency bfcT1

Fig. 3 Phase velocities of longitudinal and transverse waves in the composite.

. ¡2 -(a) Longitudinal wave incidence

a i 1 -

With coating layer Without coating layer _i_i_i_1__i_

0.1 0.2 0.3 0.4 Normalized frequency bfcT1

2 ■ (b) Transverse wave incidence

2 t 1 -

-With coating layer !

----Without coating

0.1 0.2 0.3 0.4 Normalized frequency bflcT1

Fig. 4 Energy transmission spectra of (a) longitudinal and (b) transverse waves when the fibers are with or without coating.

composite. The normalized phase velocities of P and SV waves in the composite (cL/cT1 and cT/cT1) in a relatively low frequency range are shown in Fig. 3 with and without the presence of coating layers. As expected, the presence of coating layer reduces the phase velocities to a certain extent.

The square arrangement of 5X2 fibers in the fundamental block as shown in Fig. 2 (b) is also analyzed, and the energy transmission coefficient of P and SV waves are determined. The frequency dependence of the energy transmission coefficients is shown in Fig. 4 for P and SV waves, with and without the coating layers. It is noted that these results are in good agreement with the previous finite element simulations [11]. The square array of fibers causes the reduction of energy transmission at certain frequencies due to the Bragg reflection. Stop-band formation is observed in the numerical results with only five layers of fibers in the x1 direction. In Fig. 4 (a), a stop band of the P wave at around bfcT1 = 0.38 in the absence of coating shifts to lower frequency in the presence of coating, which is likely due to the lower wave speed of the coated-fiber composite. In Fig. 4 (b), while a clear stop band appears for the SV wave at around bfcT1 = 0.16 in the absence of coating layer, it becomes very shallow in the presence of coating layer. This can be explained by the relatively low stiffness of the coating, which moderates the magnitude of wave scattering by each fiber.

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