Effect of Inter-Fibre Distance on Energy Transfer in Unidirectional Composites Containing Ultrasonic Waves

Abstract

Interaction of time harmonic monochromatic anti-plane (SH) waves with a pair of parallel unidirected elastic cylindrical fibres embedded in an unbounded elastic matrix is considered. The classical method of separation of variables along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for cylindrical wave functions are employed to obtain a series solution in terms of Fourier coefficients. The presented development is illustrated with numerical examples in which the fibres are insonified at end-on/broadside incidence for a relatively wide range of nondimensional frequencies. Particular attention has been focused on the effects of lateral fibre spacing on energy transfer between the fibres and matrix. The numerical results reveal that, while inter-fibre scattering (interference) has a small influence on the normalized fibre energies for broadside incidence at all frequencies, it has a considerable impact on the energy distribution for end-on incidence (fibres positioned in the eclipse configuration) at intermediate and high frequencies. The proposed model can complement acousto-ultrasonic (AU) techniques in non-destructive evaluation of unidirectional fibre-reinforced composite materials which is generally hampered by the highly dense concentration of the fibres. Limiting case is considered and fair agreement with a well-known solution is established.

Ultrasonic Energy Transfer and Stress Concentrations in a Single-Fiber Composite with Absorbing Interface Layer

“…In this subsection, the pertinent integrals for the dynamic mechanical energy induced by the elastic waves refracted along the fiber [66,70] will be derived. First, the potential energy U F per unit length of fiber in the z-direction is obtained by integrating the strain energy density over volume [66,70]:…”

“…Next, in order to obtain a single numerical measurement of energy, it is common to average the total energy (E F ¼ U F þ T F ) over one time period of the harmonic oscillation. Therefore, the time-averaged energy hEi F may be defined by [66,70]…”

“…. , 18Þ are provided in Appendix B. Consequently, we may advantageously consider a dimensionless fiber energy parameter, hei P, SV, SH F ¼ hEi F =hEi P, SV, SH M , which gives a measure of the fiber energy relative to the energy that the matrix would contain if there were no fiber present [66,70]. This completes the necessary background required for the exact analysis of the problem.…”

“…If the fiber response could be measured, the local maxima would appear in a frequency spectrum of the measured response. Thus, it is sensible to expect that certain frequency components in a measured response may contain information about the fiber [66,70]. Figures 4-6 indicate that the fiber energy curves associated with the composites with relatively low fiber/matrix density ratio (i.e., Carbon and Aramid fibers) exhibit the least number of local maxima.…”

“…It can serve as the basis for analysis of problems involving a spatial distribution of viscoelastically coated fibers in an elastic matrix [62][63][64]. The calculated energy transfer between the fiber and matrix can be interpreted in the context of phenomena observed in acousto-ultrasonic (AU) experiments on fiber reinforced composite materials [65][66][67][68][69][70], which may be strongly influenced by state the interphase region. It can particularly be of practical value in quantitative characterization of the adhesive (dissipative) fiber-matrix bond by correlating certain parameters in the frequency spectrum of the detected waveform to mechanical characteristics of the interface layer [9,42,45,71,72].…”