PHASE VELOCITY AND ATTENUATION OF SH WAVES IN A FIBER- REINFORCED COMPOSITE Ruey-Bin Yang and Ajit K. Mal Department of Mechanical, Aerospace and Nuclear Engineering University of California, Los Angeles, CA90024 INTRODUCTION We consider SH wave propagation in a fiber-reinforced composite which consists of a homogeneous, isotropic matrix, containing long, parallel, randomly distributed circular fibers of identical properties. The scattering of waves in the elastically inhomogeneous medium results in a frequency dependent velocity and attenuation of the coherent wave. The overall dynamic response of the medium may be conveniently expressed by means of the complex wave number, (k), describing the coherent wave propagation: (k) +ia(w) Yew) (1) where V(w) and a(w) denote the shear wave velocity and attenuation, respectively. The problem of the propagation of multiple-scattered waves from a random distribution of objects has been studied in the literature for over three decades [1-10]. Foldy [1], Waterman and Truell [2], Twersky [3] and Mal and Knopoff [4] developed statistical procedures for specifying (k) in terms of the micro-structure of the medium. Sayers [5] and Datta et al [6] have used similar methods for particle-reinforced composites. Bose and Mal [7, 8] have studied analytically the problem of the scattering of a plane wave by a large number of cylindrical inclusions, arbitrarily distributed in an infinite matrix material. Considering the positions of the fibers to be random, they used a statistical approach through the introduction of a "pair correlation function" and the so called quasicrystalline approximation to obtain the propagation characteristics of the average wave. The theory leads to the well known Hashin and Rosen's formulas [11] for the static moduli. Review of Progress in Quantitative Nondestructive Evaluation, Vol. 12 Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New York, 1993 155
However, the effective dynamic constants may be quite sensitive to the choice of the pair correlation function at higher concentrations. In general, the available experimental data show a reasonable agreement with the theory only at low volume concentrations (c < 0.1) of the scatterers. An experimentally observed fact in the scattering of electromagnetic waves by a dense distribution of discrete scatterers is that the assumption of independent scattering leads to overestimation of the scattering effects [10]. To account for this problem, the Generalized Self Consistent Method [12] is applied to the study of multiple scattering of elastic waves by a random distribution of fibers. In this paper, a modified Waterman/Truell [2] method is used to calculate the scattering coefficients that are needed for the determination of the overall wavenumber (k). The effective phase velocity and attenuation are then determined for a wide range of frequencies and concentrations. SCATTERING BY CYLINDRICAL INCLUSIONS USING GENERALIZED SELF CONSISTENT MODEL The Generalized Self Consistent Model is shown in Fig. 1. The cylindrical inclusion of radius a is embedded in a concentric annulus of the matrix material of radius b, which in turn is embedded in an infinite medium possessing the unknown effective properties (JJ.) and (p). The ratio of radii ajb is related to volume fraction c by (2) (p) = The effective density (p) is defined as cp 1 + (l-c)p2 (3) y ikx e 1\)-!-L' P x Fig. 1. The Generalized Self Consistent Model. 156
Suppose that a plane time harmonic plane SH wave is generated at infinity propagating in a direction perpendicular to the fibers. Then, suppressing the time factor exp( -iwt) and dropping the angle brackets of the effective properties, i.e., replacing (k) by k, the total displacement w (parallel to the fibers) can be written as w_e ikx + L A"Hl)(kr)ei"e, r>b (4) n.. - eo - L [B"Hl)(k{) + CnH2)(klr)]ei"e, a<r<b (5) n--oo n--oo (6) where Hnw and Hn{2} are the Hankel functions of the first and second kind of order n respectively. The coefficients An, En> Cn, Dn are in general complex and may be obtained from the boundary conditions. The conditions of continuity of the shear stresses and displacements at the interfaces r = a, b yield (7) i "J (kb) + A H(l)(kb) - B H(1)(k b) + C H(2)(k b) n nn nn 1 nn 1 (8) (9) D J (k a) - B H(l)(k a) + C H(2)(k a) "a " 2 na nina n 1 i-l 1 a a a (10) The constants An, En, Cn, Dn can be obtained from Eqs. (7) - (10). THE EFFECTIVE SHEAR MODULUS In the static case, the average strain in the inclusion is defined by -I 1 f2nfa 1 e. - - e.rdrd6 lj Tta 2 0 0 lj (11) 157
and the average strain in the matrix is given by - 1 i2nfb e.. = e.. rdrd6 'I 1t(b2 _ a2) 0 a 'J (12) If we use the same averaging procedure in the dynamic case, then it can be shown that (13) (14) The effective shear modulus can be calculated from the formula (15) THE MODIFIED WATERMANjTRUELL METHOD The general approach for solving multiple scattering problem involves a configurational averaging technique using the joint probability distribution for the occurrence of a given configuration of scatterers. If the effects of the correlation in position can be neglected, the effective complex wave number (k) of the scattering medium can be obtained in terms of the properties of inclusion and matrix, the volume fraction of the inclusion and frequency. The approach described in Waterman and Truell [2] for a distribution of spherical inclusions can be used for cylindrical inclusions to yield the following formula for the overall complex wavenumber (k): (16) where no is the number of scatterer per unit area (c = no 7ra 2 ) and f(o) and f(7r) are the forward and backward scattering amplitudes of a single fiber embedded in the matrix. However, it can be easily shown that the formula does not reduce to the Hashin and Rosen's result in the static limit. In order to obtain an estimate which reduces to the proper static limit and gives reasonable results at high concentrations, the Generalized Self Consistent Model (GSCM) and an iterative numerical scheme are employed 158
here. It is assumed that each fiber is surrounded by a concentric cylinder as in Fig. 1 and the composite is embedded in an effective medium with properties (p; and (k;. Then the scattering formula (16) is modified to nof _(lt_)]2 1-[1-2inof(O)]2 _[_2i --=c (k)2 (k)2 (17) where fro) and {(Tr) are calculated from (18) n-- oo (19) n--oo and An is given by Eqs. (7) - (10). It should be noted that An is a transcendental function of the unknown (k; and an explicit solution for (k; is difficult to obtain. An iterative procedure is used to solve for (k;. We start by assuming that the effective medium has the wave number kl. Using Eqs. (18) and (19), we calculate the forward and backward scattered amplitudes. The homogenization is carried out by employing Eq. (17) and a new value of (k) is obtained. Next, we substitute the corrected effective wave number (k; for kl in Eq. (17) and this procedure is repeated until convergence is achieved. The Rayleigh limit of the modified Waterman/Truell method can be obtained by using the asymptotic expansions of the Bessel and Hankel functions in A.. for ka < < 1. The first two coefficients in GSCM are given by (20) Al = {l- c_ m -I} _{I + c_ m -I} lti 2 III m + 1 m + 1 -(kb)--'---------- 4 {l _c_ m -I} + {I + c_ m -I} III m + 1 m + 1 (21) where m = ij.2/ ij.l. Substituting Eqs. (18) - (21) into Eq. (17) the effective elastic modulus (ij.; / ij.l can be obtained. The static limit is = l+c(m-l)/(m+l) ILl 1-c(m -1)/(m + 1) Eq. (22) is the same as that given in Hashin and Rosen [11]. (22) 159
NUMERICAL RESULTS Computations were carried out for graphite-epoxy composite with the properties of the constituents P2/Pl = 1.79/1.26, fj.2/fj.l = 27.58/1.595. The results obtained by the application of the various theories are presented in Fig. 2. The averaging procedure used in calculating the static moduli using Generalized Self Consistent Model (GSCM) is given in Eq. (15). However, this formula gives no attenuation, i.e., the imaginary part of effective wave number is zero and the modulus becomes negative at higher frequencies. However, in the static limit this formula reduces to Hashin and Rosen's result. Thus, we can conclude that the averaging procedure using the direct GSCM is valid only at low frequencies. The phase velocity obtained by the -- - - GSCM -.---- WATERMAN/TRUELL MODEL - MODIFIED WATERMAN/TRUELL MODEL 1.25 -,-------------------, 1.20 '-" Q) 1.10,,,, I I I I, 1.05 1. 00 +nttttttrrrrtt"lttttttrrrrtt"lttttttt"ttrtt"l"tttttt-rn 0.0 0.2 0.4 0.6 0.8 1.0 k 1a Fig. 2. Phase velocity of SH waves in graphite-epoxy composites for volume fraction c = 0.1. modified Waterman/Truell method can be seen to be almost constant at low frequencies and then decreases slightly with increasing frequency. This model gives results which converge to the Hashin and Rosen's static limit. The attenuation curves calculated from this model are shown in Fig. 3, indicating that they increase rapidly at higher frequencies. Figs. 4 shows the the static and dynamic moduli versus volume fraction c. The dynamic effect can be seen to be very small. The results show that the modified Waterman/Truell method permits the limiting case of full packing of the fibers. The attenuation curves shown in Fig. 5 indicate that at any frequency the attenuation increases initially with concentration, attains a maximum and then declines to zero as the concentration approaches 1. 160
0.040,----------------, 0.035 0.030,-.. 0.025 ::::::;0.020 S... 0.015 C = 0.3 0.010 0.005 C = 0.5 0.2 0.4 0.6 O.B 1.0 kla Fig. 3. Attenuation of SH waves for various concentrations. -- static limit 3.5 r-._.._.._..-'k::<.c:: a _=--=I.::.2'----J 3.0.- 1.5.....-/... 1.0 'W;m=======TTT==-rrr=,..,-ri 0.0 0.2 0.4 0.6 0.8 1.0 Volume Fraction (c) Fig. 4. Phase velocity of SH waves as a function of volume fraction. -- kle = 1.0 - - - - k,a = 0.6 ------- kab = 0.5 0.05,..------------'-----, 0.04,;; 0.03 ""'- J 0.02,,,,,, 0.01 0.2 0.4 0.6 O.B Volume Fraction (e) Fig. 5. Attenuation of SH waves for various frequencies. 1.0 161
CONCLUDING REMARKS In the past, theoretical predictions of the overall elastodynamic constants have been shown to be in good agreement with experimental results at relatively low frequencies (when the wavelengths are large compared to fiber diameter) and at low volume concentrations of the fibers. At high volume concentration, the effect of multiple scattering by fibers becomes significant, and the theoretical estimates are not satisfactory. To account for this problem at high concentrations, the Generalized Self Consistent Method (GSCM) has been employed to modify the multiple scattering formulation proposed by Waterman and Truell. The results for the SH case appear to work well at higher concentrations, and in a large frequency range including the limiting static case (zero frequency). ACKNOWLEDGEMENT This research was supported by the Mechanics Division of the Office of Naval Research under Contract NOO0l4-90-J-1867. REFERENCES 1. L. L. Foldy, 1945, "The Multiple Scattering of Waves," Phys. Rev. Vol. 67, pp. 107-119. 2. P. C. Waterman and R. Truell, 1961, "Multiple Scattering of Elastic Waves," J Math. Phys., Vol. 2, pp. 512-537. 3. V. Twersky, 1962, "On Scattering of Waves by Random Distributions. I. Free-Space Scatter Formalism," J. Math. Phys., Vol. 3, pp. 700-715. 4. A. K. Mal and 1. Knopoff, 1967 "Elastic Wave Velocities in two component systems", J. Inst Math. Appl., Vol 3, pp. 376-387. 5. C. M. Sayers, 1980, "On the Propagation of Ultrasound in Highly Concentrated Mixtures and Suspensions," J Phys. D: Appl. Phys., Vol. 13, pp. 179-184. 6. S. K. Datta, H. M. Ledbetter, Y. Shindo and A. H. Shah, 1988, "Phase Velocity and Attenuation of Plane Elastic Waves in a Particle-reinforced Composite Medium," Wave Motion, Vol. 10, pp. 171-182. 7. S. K. Bose and A. K. Mal, 1973, "Longitudinal Shear Waves in a Fiber Reinforced Composite," Int. J. Solids Structures, Vol. 9, pp. 1075-1085. 8. S. K. Bose and A. K. Mal, 1974, "Elastic Waves in a Fiber-reinforced Composite,"J Meeh. Phys. Solids, Vol. 22, pp. 217-229. 9. A. I. Beltzer and N. Brauner, 1987, "The Dynamic Response of Random Composites by a Causal Differential Method," Meeh. Materials Vol. 6, pp. 337-345. 10. 1. Tsang, J. A. Kong and H. Habashy, 1982, "Multiple Scattering of Acoustic Waves by Random Distribution of Discrete Spherical Scatterers with the Quasicrystalline and Percus-Yevick Approximation," J. Aeoust.Soe. Am. Vol. 71, pp. 552-558. 11. Z. Hashin and R. W. Rosen, 1964, "The Elastic Moduli of Fiber Reinforced Materials," J. Appl. Meeh., Vol. 31, pp. 223-232. 12. R. M. Christensen, 1990, "A critical Evaluation for a Class of Micromechanic Models," J. Meeh. Phys. Solids, Vol. 27, pp. 379-404. 162