Multiple Scattering Simulation of Ultrasonic Shear Wave in Unidirectional Carbon/Epoxy Composites
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1 Materials Transactions, Vol. 48, No. 6 (27) pp. 96 to 2 Special Issue on Advances in Non-Destructive Inspection and Materials Evaluation #27 The Japanese Society for Non-Destructive Inspection Multiple Scattering Simulation of Ultrasonic Shear Wave in Unidirectional Carbon/Epoxy Composites Shiro Biwa, Takushi Kamiya 2; * and Nobutada Ohno 3 Department of Energy Conversion Science, Graduate School of Energy Science, Kyoto University, Kyoto 66-85, Japan 2 Department of Micro System Engineering, Nagoya University, Nagoya , Japan 3 Department of Computational Science and Engineering, Nagoya University, Nagoya , Japan A computational multiple scattering simulation method was applied to analyze the characteristics of the ultrasonic shear wave that propagates in unidirectional carbon-fiber-reinforced epoxy composites with its polarization direction parallel to the fibers. The numerical simulations were carried out for regular as well as random fiber arrangements and for different fiber volume fractions. The results were combined with the one-dimensional theory describing the macroscopic propagation behavior, in order to identify the phase velocity and the attenuation coefficient of the composite. The phase velocity and the attenuation coefficient were found to depend significantly on the fiber volume fraction, but less so on the fiber arrangement in the frequency range examined here. Furthermore, the present analysis showed a good agreement with the experimental data. [doi:.232/matertrans.i-mra27846] (Received August 28, 26; Accepted February 27, 27; Published May 25, 27) Keywords: multiple scattering, unidirectional composite, shear wave, polymer-matrix composite, phase velocity, attenuation. Introduction *Graduate Student, Nagoya University. Present Address: DENSO Corporation, Aichi , Japan (a) (c) (b) x 3 x 2 x Polarization direction Fiber direction Fig. Three through-thickness wave modes in unidirectional fiberreinforced composites, (a) longitudinal wave, (b) shear wave polarized perpendicular to the fibers, and (c) shear wave polarized parallel to the fibers. Ultrasonic nondestructive characterization of fiber-reinforced composite materials is becoming increasingly important 6) as these materials find yet wider applications owing to their high stiffness-to-weight and strength-to-weight ratios. In fiber-reinforced polymer-matrix composites, ultrasonic waves attenuate significantly due to the viscoelastic nature of the polymer matrix as well as the scattering by the reinforcing fibers. The frequency-dependent attenuation brings about the waveform change of an ultrasonic pulse as it propagates through a composite component. It is highly desirable to understand these features appropriately when interpreting the waveform information obtained in ultrasonic measurements. Recently, the ultrasonic attenuation in unidirectional carbon-fiber-reinforced epoxy composite laminates was analyzed 7,8) using a micromechanical model called the differential (incremental) scheme, 9) and the theoretical predictions were compared to the experimental data. ) Among the three wave modes propagating in the thickness direction (c.f. Fig. ), the longitudinal wave (Fig. (a)) and the shear wave polarized perpendicular to the fibers (Fig. (b)) showed good agreement. 8) On the other hand, for the shear wave polarized parallel to the fibers (Fig. (c)), some discrepancy was found between the theory and the experiment. The discrepancy observed for the third mode can be partly due to the inaccuracy of the adopted theoretical model. Namely, the approximation involved in the model may become less valid because of the dense fiber concentration in the composite and the particularly high mismatch of the acoustic impedance between the fiber and the matrix for this mode due to the high anisotropy of the carbon fiber. For this reason, it is intriguing to perform a more direct numerical simulation which does not involve such theoretical approximations, in order to clarify the propagation characteristics and to interpret the experimental data. In the present study, the characteristics of the above third mode (Fig. (c)), referred to as the SH (shear horizontal) mode in analogy with seismology, are analyzed by a computational multiple scattering simulation which has been recently formulated based on the eigenfunction expansion of the time-harmonic multiple scattering field and a numerical collocation technique. ) This technique does not rely on various simplifying assumptions as employed in existing micromechanical models, and can handle arbitrary spatial fiber arrangements. Although this method was originally used for composites with elastic matrix and elastic fibers, 3) it is applied here to viscoelastic matrix and elastic fibers in order to model polymer-matrix composites. Below, the wave propagation characteristics are analyzed by this simulation, and from the results, the phase velocity and the attenuation coefficient of the composite are identified. The obtained results are discussed for different fiber volume fractions and fiber arrangements, and compared to the experimental data.
2 Multiple Scattering Simulation of Ultrasonic Shear Wave in Unidirectional Carbon/Epoxy Composites 97 Incident wave x 2 r i+hi 2 H r i r Fundamental block r i -Hi 2 2. Multiple Scattering Simulation Procedure L Semi-infinite Reinforced region Semi-infinite matrix (matrix + fibers) matrix Fig. 2 Computational model of a unidirectional composite. 2. Formulation The computational procedure for the time-harmonic SH wave propagation in unidirectional fiber-reinforced composites is outlined. ) The infinitely extended viscoelastic matrix (complex shear modulus and density ) and the elastic fibers (shear modulus 2, density 2 and radius a) are assumed to be homogeneous and isotropic in the plane normal to the fibers. Time-harmonic wave motions are considered, and the temporal factor expð i!tþ is omitted in all relevant quantities, where! is the angular frequency. The displacement field uðx ; x 2 Þ that is polarized parallel to the x fibers (c.f. Fig. (c)) obeys the two-dimensional scalar Helmholtz 2 u þ k 2 u ¼ ; ðþ 2 where k ( ¼ ; 2) are the wave numbers in the matrix ( ¼ ) and the fiber ( ¼ 2), respectively. The wave number in each phase is given by! k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼! þ i ð ¼ ; 2Þ; ð2þ = c where c and are the phase velocity and the attenuation coefficient, respectively, for each phase. It is noted that for a viscoelastic material, the wave number is a complex quantity since the attenuation coefficient is nonzero. As in the previous treatments, 3) the fiber arrangement is specified in a fundamental block with length L and height H as shown in Fig. 2, which contains N f fibers. The blocks with the same fiber arrangement are repeated infinitely in the x 2 direction to build up the whole fiber arrangement. Therefore, the computational model consists of a fiber-reinforced region of the length L, < x < L, embedded between the semiinfinitely extended matrix media occupying x < and x > L. When the plane harmonic shear wave, defined to be of unit amplitude at x ¼, i.e., expðik x Þ, is incident in the positive x direction, each fiber is insonified by the so-called exciting field, which is the sum of the incident wave and the scattered waves from other fibers. Expressing the exciting as well as scattered fields by the eigenfunction expansion, 4) the governing system of equations is given by X n¼ a i n J nðk jr r i jþ expðin i Þ¼expðik x Þþ X þ X X p¼ m¼ p6¼ p¼ X N f j¼ j6¼i X m¼ T m a j m H mðk jr ðr j þ phi 2 ÞjÞ expðim jp Þ T m a i m H mðk jr ðr i þ phi 2 ÞjÞ expðim ip Þ; ð3þ where J n ðþ and H n ðþ are the nth order Bessel and Hankel functions of the first kind, a i n (n ¼ ; ; 2;...; i ¼ ; 2;...; N f ) are the unknown expansion coefficients, and T n (n ¼ ; ; 2;...) are the components of Waterman s transfer matrix, 5) namely, T n ¼ k Jn ðk aþj n ðk 2 aþ 2 k 2 J n ðk aþjn ðk 2aÞ 2 k 2 H n ðk aþjn ðk 2aÞ k Hn ðk aþj n ðk 2 aþ ; ð4þ where a prime denotes the derivative of the Bessel or Hankel function with respect to its argument. Furthermore, the angle variables i are the polar angle from the ith fiber position r i to a generic position r, and ip are the angle from the position of the ith fiber position in other blocks r i þ phi 2 to r, where p is an integer and i 2 is the unit vector in the x 2 direction. Equations (3) hold at arbitrary points in the matrix. In the numerical analysis, the eigenfunction expansions are truncated at a finite level, i.e. up to n ¼n max, and N f ð2n max þ Þ unknown coefficients a i n are determined by a collocation method. Namely, eq. (3) is evaluated at (2n max þ ) collocation points on the boundary for each of N f fibers, and the resulting system of linear equations is solved numerically. The infinite sums with respect to the parameter p are also truncated at a finite value, p max, in the numerical procedure. The values of n max and p max are determined by checking the satisfactory convergence of the solution at reasonable computational costs. Once the coefficients a i n are determined, the displacement can be calculated by uðx ; x 2 Þ¼expðik x Þþ XN f X n max Xpmax i¼ n¼ n max p¼ p max T n a i n H nðk jr ðr i þ phi 2 ÞjÞ expðin ip Þ for an arbitrary position r ¼ðx ; x 2 Þ in the matrix. In particular, eq. (5) is used to compute the transmitted wave for x > L, from which the transmission coefficient is extracted in order to identify the macroscopic property of the composite as explained later. ð5þ
3 98 S. Biwa, T. Kamiya and N. Ohno (a) (b) exp(ik x ) Aexp(ik eff x ) Texp(ik x ) Rexp(-ik x ) Bexp(-ik eff x ) (c) (d) L x Matrix Reinforced region Matrix ρ, µ ρeff, µ eff ρ, µ Fig. 4 Schematic illustration of the macroscopic one-dimensional model with the wave field representations, where the coefficients R, T, A and B are determined by the continuity conditions. Fig. 3 Spatial fiber arrangements assumed in the fundamental block, (a) square arrangement, (b) hexagonal arrangement (type-), (c) hexagonal arrangement (type-2) and (d) random arrangement. 2.2 Numerical models Based on the formulation outlined above, the SH wave propagation in a unidirectional composite made of the epoxy matrix and carbon fibers was analyzed. The epoxy matrix and the carbon fibers were assumed to be linear viscoelastic and linear elastic, respectively, and their property data were set according to the previous analysis, 8) as ¼ 23 [kg/m 3 ], c ¼ 97 [m/s], ¼ 2:2 þ 226 6!=ð2Þ [m ] (! is measured in rad/s), 2 ¼ 67 [kg/m 3 ], c 2 ¼ 379 [m/s] and a ¼ 3:5 6 [m]. Four different types of fiber arrangements were considered in this analysis, namely, a square arrangement, two hexagonal arrangements with different orientations (type- and type-2), and random arrangements which were generated using the RSA (random sequential adsorption) technique, 6) to examine the influence of the fiber arrangement on the propagation behavior. Examples of these fiber arrangements are illustrated in Fig. 3, where the fundamental blocks are enclosed by dashed lines (for (a) to (c), three blocks are shown to ease the visualization of the arrangement). The number of the fibers arranged in the fundamental block and its sizes were varied in order to see the effect of the fiber volume fraction as well as the length of the reinforced region L. As a result, the simulations were performed for the fiber volume fraction in the range of to 7%, except for the random arrangements which were constructed up to 5% only. Also, these simulations were made for two different frequencies of practical interest, namely, 2 MHz and 3 MHz for which the experimental data are available for comparison. In the present simulations, the ratios of the fiber radius or the fiber-to-fiber distance to the wavelength were always small enough, so that the truncation in the series of eq. (3) was satisfactorily made with n max ¼ 2 and p max ¼ 5 after preliminary calculations. 3. Identification of Phase Velocity and Attenuation Coefficient Based on the results of the multiple scattering simulation, the macroscopic ultrasonic property of the composite is identified. To this purpose, the composite region < x < L is replaced by a homogeneous medium with the equivalent macroscopic property, i.e. phase velocity and attenuation coefficient. For this homogenized medium, the transmission/ reflection of the plane wave, expðik x Þ, can be analyzed as a one-dimensional problem as shown in Fig. 4. An elementary calculation accounting for the continuity conditions at x ¼ and x ¼ L leads to the following expression for the complex coefficient of the transmitted wave; T ¼ 4 k eff k eff expð ik LÞ ð k þ eff k eff Þ 2 expð ik eff LÞ ð k eff k eff Þ 2 expðik eff LÞ ; ð6þ where! 2 eff ¼ eff ; k eff ¼! þ i eff ; ð7þ k eff c eff using the density eff, the complex shear modulus eff, the phase velocity c eff and the attenuation coefficient eff of the effective medium. Furthermore, the effective density is given by eff ¼ð Þ þ 2 ; ð8þ where is the fiber volume fraction. The transmitted wave is written by u ¼ T expðik x Þ ¼jTjexpð x Þ exp i! x þ ; ð9þ c using eq. (2), where jtj ¼ðTT Þ =2 (asterisk denotes the complex conjugate) and ¼ tan ðim½tš= Re½TŠÞ characterize the effect of the fiber reinforcement on the amplitude and the phase, respectively, of the transmitted wave. These quantities are given by the phase velocity c eff and the attenuation coefficient eff of the effective medium from eqs. (6) and (7). Eqs. (6) and (7) can be used to determine c eff and eff when the complex quantity T is given, provided that the parameters
4 Multiple Scattering Simulation of Ultrasonic Shear Wave in Unidirectional Carbon/Epoxy Composites 99 Displacement field, Re[u] 2 (a) - -2 Displacement field, Re[u] Reinforced region - 2 Propagation distance, x / mm 2 (b) - -2 Reinforced region - 2 Propagation distance, x / mm Displacement field, Re[u] - -2 of the matrix (with subscript ) are known. Namely, eqs. (6) and (7) yield a set of nonlinear equations for c eff and eff, which can be solved numerically. In the present study, the complex transmission coefficient T was obtained from each multiple scattering simulation. In this way, the phase velocity c eff and the attenuation coefficient eff of the composite were determined for each volume fraction and for each fiber arrangement. 4. Results and Discussion 2 (c) Reinforced region Propagation distance, x / mm Fig. 5 Displacement fields for the 2 8 square arrangement (3 MHz), for the fiber volume fraction (a) 3%, (b) 5% and (c) 7%. 4. Results of multiple scattering simulation As typical results from the multiple scattering simulations, the wave displacement fields (distribution of Re½uŠ) along the horizontal line x 2 ¼ H=2, are shown in Fig. 5 for the fiber volume fraction of (a) 3%, (b) 5% and (c) 7%. These results correspond to the 3 MHz frequency and the square fiber arrangement (c.f. Fig. 3(a)) consisting of 8 fibers in the propagation direction and 2 fibers in the perpendicular direction in the fundamental block (denoted simply as 2 8 hereafter). The fiber-reinforced regions in Fig. 5(a) (c) are delineated with dashed lines, which lengths are (a) L ¼ :96 [mm], (b) L ¼ :72 [mm] and (c) L ¼ :593 [mm], respectively. It is seen that the wave decays as it propagates, but to different degrees in the matrix and fiber-reinforced regions. Also, the presence of the fibers makes the wavelength elongated in < x < L, and influences the amplitude as well as the phase of the transmitted wave in the region x > L. In each multiple scattering simulation, the quantities and lnðtt Þ were extracted from the obtained displacement field. For the square fiber arrangements with the fiber volume fraction 25% and the frequency 3 MHz, these quantities, in the form of =L and lnðtt Þ=L, are plotted as open circles in ϕ/l / rad mm One-dimensional theory Multiple scattering simulation Length of reinforced region, L / mm Fig. 6 Variation of =L with the length of the reinforced region for the square arrangement, the frequency 3 MHz, and the fiber volume fraction 25%. ln(tt*)/l / mm One-dimensional theory Multiple scattering simulation. 2 Length of reinforced region, L / mm Fig. 7 Variation of lnðtt Þ=L with the length of the reinforced region for the square arrangement, the frequency 3 MHz, and the fiber volume fraction 25%. Figs. 6 and 7, respectively, for different L obtained by changing the number of the fibers in the fundamental block from 2 2 (L ¼ :248 [mm]) to 2 5 (L ¼ :86 [mm]) in the propagation direction. From these data, the phase velocity c eff and the attenuation coefficient eff of the carbon/epoxy composite were determined as the properties of the equivalent medium, as explained above in connection to the one-dimensional problem. The obtained values for c eff and eff were very close to each other when different fiber numbers, from 2 2 to 2 5, are used in the simulation and the identification. Using the properties c eff and eff determined for the 2 8 arrangement, the theoretical results using eq. (6) are drawn in Figs. 6 and 7 as solid curves. It is seen that the theoretical curves oscillate with the length of the fiber reinforced region, and approach certain asymptotic values. The agreement between the theoretical curves, determined by the results of the 2 8 arrangement, and the plots for other fiber numbers gives confidence to the present identification procedure. Such identification tasks were performed for different fiber volume fractions and different fiber arrangements. Although an arbitrary length of the fiber-reinforced region can be employed in the multiple scattering simulation as confirmed in the above results, it is advantageous to use a greater L since the aforementioned oscillation tends to be smaller as L becomes longer. With longer L, however, a greater number of
5 2 S. Biwa, T. Kamiya and N. Ohno Phase velocity, ceff / m s / Square (2MHz / 3MHz) / Hexagonal- (2MHz / 3MHz) / Hexagonal-2 (2MHz / 3MHz) / Random (2MHz / 3MHz) / Experimental data (2MHz / 3MHz) Fiber volume fraction, φ (%) Fig. 8 Relation between the phase velocity of the composite and the fiber volume fraction, for different fiber arrangements in comparison to the experimental data (2 MHz and 3 MHz). the fibers need to be dealt with in the simulation. For the trade-off with the computational cost, the following fiber numbers were assumed for the identification procedure; (a) square: N f ¼ 6, (b) hexagonal type-: N f ¼ 74, (c) hexagonal type-2: N f ¼ 42, (d) random: N f ¼ 32, which gave similar lengths L for the same. For the cases with high fiber volume fraction, however, increased numbers as (b) N f ¼ 48, (c) N f ¼ 86 were also used for the hexagonal arrangements. 4.2 Phase velocity and attenuation of the composite The phase velocity of the composite is plotted in Fig. 8 as a function of the fiber volume fraction for 2 and 3 MHz and for different fiber arrangements. The results for three different random arrangements are plotted for each fiber volume fraction, which eventually appear to overlap to each other. As seen in Fig. 8, the phase velocity is an increasing function of the fiber volume fraction, while the influences of the frequency and the fiber arrangement are almost negligible. The experimental data 8,) for this wave mode, obtained for different unidirectional carbon/epoxy composite samples with random fiber distributions and fiber volume fractions ranging between 5 and 6%, are also shown in Fig. 8, where good agreement can be observed between the present analysis and the measured data. The corresponding results for the attenuation coefficient are shown in Fig. 9, for (a) 2 MHz and (b) 3 MHz. First, it is observed that the attenuation coefficient is higher for higher frequency, which is a common feature of polymer-based materials. Second, the attenuation coefficient is found to be a decreasing function of the fiber volume fraction. In particular, the decrease in the attenuation coefficient is almost proportional to the fiber volume fraction. Third, like the phase velocity, the influence of the fiber arrangement is found to be insignificant, although the results of three random arrangements appear to scatter slightly as the fiber volume fraction becomes large (up to 5%). Experimental data 8,) are also included in Fig. 9. Considering the experimental error and the sample-to-sample variations which are inherent in the attenuation measurements, the agreement between the present analysis and the experiment is considered to be quite satisfactory. From these results, the attenuation in this type of composites is seen to be essentially governed by the Attenuation coefficient, α eff / m - Attenuation coefficient, α eff / m - 5 (a) 2 MHz 4 3 viscoelastic property of the epoxy matrix in the frequency range examined here. In the results shown here, the fiber arrangement has been found to have negligible influence on the macroscopic wave propagation behavior. This is in fact an expected feature, since the fiber radius and the fiber spacing are smaller than the wavelengths by about two orders of magnitude in the examples considered here, which means that the wave characteristics are more or less governed by the macroscopic property of the composite. When the frequency is much higher, the situation can be different and significant effects of the local fiber arrangement can occur, 3) though it is considered to be difficult to observe such effects in polymermatrix composites, since the attenuation is quite substantial at such high frequencies. 5. Conclusion 2 Square Hexagonal- Hexagonal-2 Random Experimental data Fiber volume fraction, φ (%) (b) 3 MHz 6 4 Square 2 Hexagonal- Hexagonal-2 Random Experimental data Fiber volume fraction, φ (%) Fig. 9 Relation between the attenuation coefficient of the composite and the fiber volume fraction, for different fiber arrangements in comparison to the experimental data, (a) 2 MHz and (b) 3 MHz. The propagation characteristics of the ultrasonic shear wave in unidirectional carbon-fiber-reinforced epoxy composites, with its polarization parallel to the fibers, have been analyzed with the aid of the computational multiple scattering simulation. The multiple scattering simulation results have been combined with the macroscopic one-dimensional theory in order to identify the phase velocity and the attenuation coefficient of the composite. These characteristics have been demonstrated for different fiber volume fractions and different fiber arrangements. As a result, the following features have been clarified. () The phase velocity and the attenuation coefficient are not very sensitive to the fiber arrangement in a practical frequency range examined here, due to the fact that the
6 Multiple Scattering Simulation of Ultrasonic Shear Wave in Unidirectional Carbon/Epoxy Composites 2 wavelength is sufficiently large compared to the microstructural scales such as fiber radius and fiber spacing. (2) The phase velocity and the attenuation coefficient are influenced significantly by the fiber volume fraction. The phase velocity increases while the attenuation coefficient decreases with the increase of the fiber volume fraction. (3) The results of the present analysis are in good agreement with the experimental data. REFERENCES ) R. A. Kline: Nondestructive Characterization of Composite Media, (Technomic, Lancaster, 992), pp ) B. Hosten, M. Deschamps and B. R. Tittmann: J. Acoust. Soc. Am. 82 (987) ) D. S. Forsyth, S. O. Kasap, I. Wacker and S. Yannacopoulos: Trans. ASME J. Eng. Mater. Tech. 6 (994) ) S. Boudouin and B. Hosten: Ultrasonics 34 (996) ) Y. Mizutani and M. Takemoto: Jpn. J. Appl. Phys. 37 (998) ) Y. Okabe and N. Takeda: J. Soc. Mater. Sci. Jpn. 47 (998) (in Japanese). 7) S. Biwa, Y. Watanabe and N. Ohno: Mater. Sci. Res. Int. STP 2 (2) ) S. Biwa, Y. Watanabe and N. Ohno: Compos. Sci. Tech. 63 (23) ) A. I. Beltzer and N. Brauner: Mech. Mater. 6 (987) ) Y. Watanabe, S. Biwa and N. Ohno: J. Soc. Mater. Sci. Jpn. 5 (22) (in Japanese). ) S. Biwa, S. Yamamoto, F. Kobayashi and N. Ohno: Int. J. Solids Struct. 4 (24) ) F. Kobayashi, S. Biwa and N. Ohno: Int. J. Solids Struct. 4 (24) ) S. Biwa, F. Kobayashi and N. Ohno: Mech. Mater. 39 (27). 4) Y.-H. Pao and C.-C. Mow: Diffraction of Elastic Waves and Dynamic Stress Concentrations, (Crane Russak, New York, 97), pp. 3. 5) P. C. Waterman: J. Acoust. Soc. Am. 45 (969) ) J. Feder: J. Theo. Biol. 87 (98)
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