Analysis of wave attenuation in unidirectional viscoelastic composites by a differential scheme

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Abstract

Wave attenuation characteristics of unidirectional fiber-reinforced polymer composites are analyzed theoretically by a micromechanical differential (incremental) scheme for their macroscopic acoustic properties. In this approach, the effect of neighboring fibers on the wave scattering by a single fiber is accounted for in an approximate and averaged sense. The analysis also takes into account the viscoelastic nature of the matrix, so it yields the attenuation of the composite due to both wave scattering loss and the viscoelastic absorption loss. As an example, frequency-dependent attenuation coefficients of the longitudinal and transverse waves are computed for unidirectional carbon/epoxy composites. The computed results are favorably compared to the experimental data regarding their dependence on the frequency and the fiber volume fraction. The analysis shows that the attenuation in these composites is principally governed by the wave absorption of the epoxy matrix for a practical frequency range.

Introduction

Evaluation of anisotropic stiffness of composites based on ultrasonic velocity measurements has been in wide practice for a long time [1]. In addition, ultrasonic attenuation measurements attract increasing attention regarding nondestructive characterization of composites. In particular, ultrasonic attenuation characteristics of fiber-reinforced plastics have been correlated to their porosity and defect concentrations, and further to their strength performance [2], [3], [4]. Lhermitte et al. [5] reported measurements of frequency-dependent attenuation behavior of unidirectional carbon-fiber-reinforced plastics for different propagation and polarization directions. In a series of detailed study, Hosten and co-workers [6], [7] have investigated experimental characterization principles for anisotropic viscoelastic properties of composites by ultrasonic methods. Other studies have been performed regarding the effect of moisture absorption on the ultrasonic attenuation behavior [8]. In order to enhance composites characterization by attenuation measurements, however, there is an obvious need for sound theoretical interpretation of wave attenuation processes in composites.

One of the dominant sources of wave attenuation in composite materials is wave scattering by the composite microstructure. Microscopic wave scattering by the embedded fibers and the resulting wave propagation behavior in fiber-reinforced composites have been the subject of extensive theoretical investigation, c.f. [9], [10], [11], [12], [13], [14], [15], [16] to name but a few. In polymer-based composites, the wave attenuation not only arises from scattering but also from energy absorption due to viscoelastic dissipation. Such effects can be in principle accounted for in most of the foregoing scattering theories by replacing the elastic moduli of the constituents with their complex moduli. However, neither the influence of viscoelastic properties of the polymeric constituents nor correspondence between the theoretical predictions and actual measurements has been fully explored in the situation where viscoelastic nature dominates.

In this context, theoretical works by Beltzer and Brauner [17], [18], [19] are worth attention who analyzed wave scattering and the resulting average wave propagation behavior in two-phase media accounting explicitly for the lossy nature of the matrix. In contrast to conventional multiple scattering theories that require configurational averaging of the scattered wave field, their theory rests on the evaluation of the energy loss of the plane wave in terms of the single-inclusion scattering characteristics and the attenuation property of the matrix. Their model has been recently re-examined, and after certain modifications applied to ultrasonic attenuation in unidirectional fiber-reinforced plastics, particulate polymer composites and polymer blends [20], [21], [22], where the model has proved to give a reasonable description of the attenuation spectra obtained by actual measurements. These studies have highlighted the importance of properly accounting for the matrix viscoelasticity in modeling wave scattering and attenuation in polymer composites.

It is to be noted, however, that the analysis in [20], [21], [22] was based on an assumption of single and independent scattering: the modeling neglected the influence of neighboring inclusions on the scattering response of a single generic inclusion and will be valid for dilute concentrations of inclusions. As fiber-reinforced plastics in practical use usually have high fiber volume fractions, it remained as a task of further study to deal with composites with such high fiber fractions. To this end, the above analysis was extended by incorporating a so-called differential scheme to approximately account for the effects of neighboring fibers on the single fiber scattering [23]. The idea of differential (incremental) scheme originally stems from a micromechanics model for effective properties of two-phase composites [24], [25], and has been employed already in [17] in the context of wave scattering. As a preliminary step, the analysis in [23] treated a simple case of shear waves (SH waves), where the wave polarization was parallel to the fibers and no mode conversion occurred so that the problem to be analyzed possessed a scalar nature. The wave mode analyzed in [17] was also SH wave in unidirectional composites.

The objective of this paper is to consider more general cases of longitudinal and transverse waves propagating in unidirectional viscoelastic composites, and also to discuss the analytical results in the light of actual measurements of ultrasonic attenuation in fiber-reinforced plastics. In what follows, first the independent scattering model examined in [20], [21], [22] is outlined briefly in chapter 2, and combined with a differential scheme in chapter 3 to construct a set of differential relations for the analysis of ultrasonic attenuation in unidirectional composites. The single fiber scattering problems are described in chapter 4 for longitudinal as well as transverse incident waves. Finally in chapter 5, numerical results are presented for the dependence of attenuation coefficients in unidirectional carbon-fiber-reinforced epoxy composites on the frequency and the fiber volume fraction. In particular, it is shown that for a practical range of frequency, the matrix viscoelastic effect is a dominant factor of the composite attenuation. As a result, the attenuation coefficient of the composite is found to be a decreasing function of the fiber volume fraction. The results are also discussed in comparison to actual measurements [26].

Section snippets

Independent scattering model for viscoelastic composites

Fig. 1 shows the schematic of a unidirectional composite consisting of a viscoelastic matrix (indicated by the subscript 1) and unidirectionally arranged elastic fibers of radius a and volume fraction φ (indicated by the subscript 2). The number density of fibers ns is related to φ by φ=πa2ns. As is often the case for practical fiber-reinforced plastics, the matrix is assumed to be isotropic and the fibers transversely isotropic, so the resulting unidirectional composite also possesses

Differential scheme for wave attenuation in unidirectional composites

In order to apply the above formalism to composites with non-dilute fiber concentrations, the above model is incorporated into a framework of differential scheme, which has been well known in micromechanics [24], [25]. Beltzer and Brauner [17] and later Biwa et al. [23] adopted this concept for the scalar-wave scattering analysis. According to the differential scheme, the effective properties of composites with non-dilute fiber distribution are described in differential (incremental) forms. The

Material parameters of the matrix and fiber

Based on the theoretical model described above, the ultrasonic attenuation spectra in unidirectional carbon-fiber-reinforced epoxy composites are computed for the three wave modes illustrated in Fig. 2.

To characterize isotropic linear viscoelastic properties of the matrix, the phase velocities and attenuation coefficients of epoxy resin (#340, Mitsubishi Rayon Co. Ltd.) are identified from ultrasonic measurements for the frequency range of 2–8 MHz in the case of longitudinal wave, and 2–4 MHz

Results and discussion

Fig. 5 shows the computed frequency dependence of the attenuation coefficients of the unidirectional CFRP with different fiber volume fractions, φ=0.2, 0.4, 0.6, for (a) longitudinal wave, (b) transversely polarized transverse wave, and (c) axially polarized transverse wave. The corresponding attenuation coefficients of the matrix are illustrated together, which have been obtained by the aforementioned fitting procedure to the measured data. In every case, the attenuation coefficient of the

Conclusion

In this paper, wave scattering and attenuation properties of unidirectional fiber-reinforced polymer composites have been analyzed theoretically by a micromechanical differential scheme. The viscoelastic property of the polymer matrix has been taken into account explicitly. The set of differential relations has been constructed for macroscopic acoustic properties of the composite for an infinitesimal change of the fiber volume fraction, starting from a micromechanical stiffness model and the

Acknowledgements

Financial support from the Japan Society for the Promotion of Science under the Grant-in-Aid for Encouragement of Young Scientists (No. 12750070) to S. B. is gratefully acknowledged. The courtesy of Mitsubishi Rayon Co., Ltd., Products Development Laboratories is also appreciated for supplying CFRP and epoxy samples for the attenuation measurement discussed here.

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