B. K. Henderson

The transmission of a longitudinal wave through a layer of spherical inclusions with a random or periodic arrangement

Abstract The transmission of a normally incident plane longitudinal wave through a single layer of spherical inclusions embedded in a polyester matrix is measured. Area fraction of inclusions is varied from 0.05 (a very dilute suspension) to 0.86 (densely packed). Two specimens are manufactured for each area fraction: in the first, the spheres are distributed in a random manner; in the second, they are arranged in a periodic (square) array; otherwise, the two are identical. The transmission spectra are measured at wavelengths that are large, equal or small compared to the two characteristic lengths of the composites, namely, the particle radius and, in the case of the periodic composites, the inter-particle distance. The transmission spectra are characterized by several resonances; the first of these corresponds to the excitation of the rigid-body translation resonance. It also happens to be the most dominant resonance for both the periodic and the random composites. In the case of the periodic composites, the transmission spectra are characterized by at least one additional resonance that can be unambiguously attributed to the periodicity of the lattice.

Lattice resonances of a planar array of spherical inclusions: An experimental study

This paper is concerned with an experimental investigation of the excitation of lattice resonances in a single periodic (square) array of glass inclusions embedded in an elastic matrix. A normally-incident plane longitudinal wave is used to excite the resonances. The excitation of the lattice resonances is identified through a comparison of the transmission and reflection spectra of the periodic layer with those of an equivalent random layer. At the lattice-resonant frequencies, standing shear waves are excited within the plane of the layer along certain directions of lattice symmetry. These waves are measured along five different symmetry directions by the use of shear wave transducers. The transmission and reflection spectra reveal at least three lattice resonances; these are corroborated by the shear wave spectra.

Elastodynamic response of a coplanar periodic layer of elastic spherical inclusions

Reflection and transmission spectra of a plane longitudinal wave normally incident on a coplanar periodic array of identical spherical elastic inclusions in a solid (polyester) matrix are measured at wavelengths that are comparable to the inter-particle distance. These spectra exhibit pronounced Woods anomalies i.e., a sharp variations in amplitude at the onset of a shear wave diffraction order as well as a drop in the transmission and a corresponding maximum in the reflection coefficients due to an enhancement of the resonance of the particles (for the case reported herein, this is the rigid body translational “dipole” resonance). An approximate low frequency (ka<1) self-consistent model is developed in which multiple scattering is explicitly taken into account by adding appropriate terms to the well-known solution for the scattering of a plane longitudinal wave by a single spherical particle in an unbounded matrix. The results of the numerical calculations show excellent agreement with the experimental data.

Elastodynamic response of layers of spherical particles in hexagonal and square periodic arrangements

The effect of particle arrangement on the transmission and reflection of ultrasonic longitudinal waves through a single layer of spherical inclusions is studied. Specimens were manufactured, each consisting of a layer of glass spheres embedded in a polyester matrix arranged in either a square periodic, hexagonal periodic, or random array, with projected layer area fractions ranging from 0.14 to near close-packed. The transmission and reflection coefficient spectra of a normally-incident plane longitudinal wave were measured over a range of wavelengths that are large, equal, and small compared to the two characteristic lengths of the composites, namely, the particle radius and, in the case of the periodic composites, the unit cell dimension. Periodic spectra are characterized by extrema, which are attributed to lattice resonance. Spectra from specimens with random layers of equal area fraction do not exhibit any extrema. Measured resonance frequencies, which may be accurately predicted for low to midrange area factions using the concept of the reciprocal lattice, are different for square and hexagonal lattice structures of similar area fraction and identical unit cell dimension.

Ultrasonic diffraction by a square periodic array of spheres

Shear waves propagated as a result of the diffraction of a broadband normally incident plane ultrasonic longitudinal pulse by a square periodic array of spherical elastic inclusions are measured at specific angles from the normal. As expected, peaks in the spectra were observed at frequencies given by the condition of constructive interference.

Acoustic Response of a Layer of Spherical Inclusions with a Random or Periodic Arrangement

Starting with the classic work of Ying and Truell [1], the scattering of a plane elastic wave by an isolated elastic sphere embedded in an unbounded medium has been studied in great detail. Similarly, the propagation of an effective elastic wave in an elastic matrix containing a random or periodic distribution of inclusions has received considerable attention. By comparison, an intermediate level of microstructure — a single layer of inclusions in an elastic matrix — has received very little attention. Apart from the fact that this problem is worth studying in its own right because of its inherent value as a canonical problem in elastodynamics of materials with a microstructure, it has applications in geophysics and quantitative nondestructive evaluation.

Elasto-optic technique for measurement of elastic wave propagation in solids

The modulation of the optical path of the beam of a laser vibrometer in a specimen under acoustic excitation is measured at two planes, separated by a precisely known distance. The phase shift and the decrease in magnitude are used to calculate the phase velocity and attenuation, respectively. The method is demonstrated for a homogeneous specimen, and the results compare favorably with those obtained by a conventional ultrasonic technique. The method is then applied to measure specular and first diffraction-order reflection from a coplanar periodic array of particles in an elastic matrix and phase velocity spectra in a tetragonal periodic particulate composite. As expected, in a periodic composite the establishment of dispersive Floquet-type waves is observed throughout the entire periodic particulate composite.

Acoustic Response of a Layer of Spherical Inclusions with a Hexagonal or Square Periodic Arrangement

Starting with the work of Wolf [1], the scattering of a plane elastic wave by an isolated sphere embedded in an unbounded (or bounded) medium has been studied in great detail [2–3]. Similarly, the propagation of an effective elastic wave in an elastic matrix containing a random or periodic distribution of inclusions (particles, voids, cracks, etc.) has received considerable attention. The literature concerning these two problems is so extensive that its review is beyond the scope of this paper. By comparison, an intermediate level of microstructure, an elastic matrix containing a single layer of a random or periodic distribution of inclusions, has received very little attention. Although this problem is worth examining in its own right because of its inherent value as a canonical problem of elastodynamics of materials with microstructure, it also has applications in geophysics, quantitative nondestructive evaluation, and the design of ultrasound absorptive materials.

Resonant Scattering of an Elastic Wave by a Layer Containing a Random or Periodic Distribution of Inclusions

Wave propagation through particulate composites has received considerable attention in recent years. The dispersive wave propagation through particulate composites with both random and periodic distributions has been studied theoretically [1] and experimentally [2–7]. The response of a layered composite with a finite number of layers can be predicted using the acoustic complex-valued transfer functions for a single layer [8]. The effect of the in-plane structure of an inclusion layer and resonance of individual particles on the wave propagation phenomena have been studied [9]. For a single layer of inclusions, it was shown that the arrangement of the inclusions has a significant effect on a wave propagating normal to the layer. The objective of this work is to study further the effect of the in-plane structure of an inclusion layer and acoustical properties of individual particles on the wave propagation phenomena.

Elastodynamic Response of a Periodic Layer of Spherical Particles Containing Vacancies

Abstract This paper is concerned with the influence of vacancies on the elastodynamic response of a periodic (square) array of identical spherical elastic inclusions embedded in an unbounded elastic matrix. A response function of the array is defined as H*(ω) ≡ R*(ω)/T*(ω), where R*(ω) and T*(ω) are, respectively, the reflection and transmission spectra of the lattice. H*(ω) was measured for a “perfect lattice,” i.e., one without any vacancies, and was found to be characterized by lattice resonances. H*(ω) was also measured for lattices containing one and three vacancies within the 72 lattice site area insonified by the ultrasonic beam. A counter-intuitive observation is that the presence of even one vacancy significantly reduces the amplitude of the fundamental lattice resonance. Furthermore, in the case of the specimen containing three closely spaced vacancies, the reduction is not three times the reduction due to one vacancy; it is significantly less than that.