Sagittal Elastic Waves Research Articles

Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method

The finite element (FE) method offers an efficient framework to investigate the evolution of phononic crystals which possess materials or geometric nonlinearity subject to external loading. Despite its superior efficiency, the FE method suffers from spectral distortions in the dispersion analysis of waves perpendicular to the layers in infinitely periodic multilayered composites. In this study, the analytical dispersion relation for sagittal elastic waves is reformulated in a substantially concise form, and it is employed to reproduce spatial aliasing-induced spectral distortions in FE dispersion relations. Furthermore, through an anti-aliasing condition and the effective elastic modulus theory, an FE modeling general guideline is provided to overcome the observed spectral distortions in FE dispersion relations of infinitely periodic multilayered composites, and its validity is also demonstrated.

Comparative study of the sagittal elastic waves in metallic and semiconductor multilayer systems between periodic and Fibonacci superlattices

We study the sagittal elastic waves in periodic and Fibonacci superlattices by means of the frequency dispersion relation and the density of states. We study also the intermediate multilayer structures between both systems, which can be seen as periodic or Fibonacci systems with planar defects. Applications are made to specific metallic and semiconductor systems as representative of the wide gap and small gap systems, respectively, in order to see the influence of the different elastic properties on the gaps and acoustic properties of these multilayer systems.

Elastic waves in quasiperiodic structures

We study the transverse and sagittal elastic waves in different quasiperiodic structures by means of the full transfer-matrix technique and surface Green-function matching method. The quasiperiodic structures follow Fibonacci, Thue-Morse and Rudin-Shapiro sequences, respectively. We consider finite structures having stress-free bounding surfaces and different generation orders, including up to more than 1000 interfaces. We obtain the dispersion relations for elastic waves and spatial localization of the different modes. The fragmentation of the spectrum for different sequences is evident for intermediate generation orders, in the case of transverse elastic waves, whereas, for sagittal elastic waves, higher generation orders are needed to show clearly the spectrum fragmentation. The results of Fibonacci and Thue-Morse sequences exhibit similarities not present in the results of Rudin-Shapiro sequences.

Elastic waves in graded-composition systems

We study the transverse and sagittal elastic waves in graded-composition systems, having a parabolic or linear grading, by means of the full transfer matrix technique and the surface Green function matching method. We obtain the velocities of the different waves as a function of the thickness of the structure, and we compare the variation of the velocities of these waves for the different profiles employed in our calculations. The spatial localization of the different modes is also considered.

Sagittal Elastic Waves at the Interface between a Superlattice and a Substrate: Theoretical Analysis of the Density of States and Reflection Coefficientsrik

We investigate the existence and behavior of localized and resonant acoustic modes of sagittal polarization associated with the interface between a superlattice (SL) and a substrate. These modes appear as well-defined peaks in the density of states, either inside the minigaps or inside the bulk bands of the SL and the substrate. It is shown that the interface modes strongly depend on the acoustic impedances of the substrate. Furthermore, a phonon incident on the superlattice from a substrate is partially reflected in the substrate, therefore the amplitudes and the phases of the reflection coefficients enable us to deduce the frequency ranges of the bulk bands and the minigaps of the superlattice as well as the frequency levels of the interface resonant modes. The densities of states and reflection coefficients, which are calculated as function of the frequency ω and the wave vector k∥ (parallel to the interfaces), are obtained from an analytical determination of the Green's function for the superlattice/substrate interface. Specific application of our analytical results are given for a W/Al isotropic superlattice in contact with a substrate with different acoustic impedances.

Sagittal elastic waves at the interface between a superlattice and a substrate

We investigate the existence and behavior of localized and resonant acoustic modes of sagittal polarization associated with the interface between an isotropic medium and a superlattice, consisting of alternating isotropic layers of two materials. These modes appear as well-defined peaks in the density of states. The local and total densities of states are obtained from an analytical determination of the Green function for the superlattice/substrate system. The reflection coefficients in the substrate are also derived and compared to the density of states. We show that the existence and dispersion of interface modes are dependent upon the relative parameters of the substrate and the superlattice, but also upon the nature of the film in the superlattice, which is in contact with the substrate.

Sagittal elastic waves in Fibonacci superlattices

We study the sagittal elastic waves propagating along the [100] direction in Fibonacci superlattices, formed by the repetition of GaAs and AlAs slabs in $A$ and $B$ constituent blocks following the Fibonacci sequence.We consider infinite repetition of a given Fibonacci generation and the finite Fibonacci generation with free surfaces, as a comparison. We have considered in our study systems ranging from the second to the eighth generation. We employ the surface Green-function matching for $N$ nonequivalent interfaces to obtain the dispersion relation and the density of states of the different systems. In this way it is possible to obtain the spatial localization of the different modes. We have studied the influence of the increasing Fibonacci generations on the dispersion relation of the sagittal elastic modes.

Elastic waves in polytype superlattices

We study the shear horizontal and sagittal elastic waves in polytype superlattices, formed by the periodic repetition of three different constituent materials. We use the surface Green function matching method for N non-equivalent interfaces in order to obtain the dispersion relations and the density of states. In this way it is possible to obtain the spatial localization of the different modes. The influence of the variation of the thickness of the constituent materials on the dispersion relation of the elastic modes is studied.

Sagittal elastic waves in cubic superlattices

Using a recent extension of the surface Green function matching (SGFM) method, we calculate bulk bands for sagittal waves in superlattices made of alternating layers of two different elastic cubic media. The secular equation, involving a surface projection of the superlattice Green function, is written in terms of the bulk Green functions for each medium. This is applied to the calculation of dispersion relations for GaAs-AlAs and Al-W superlattices, two systems exhibiting appreciable differences in their gap patterns. The analysis can be readily used to construct the complete Green function of the system, hence to calculate any desired spectral function. Application is made to study the spatial dependence of the spectral strength (local density of states) corresponding to various representative modes in Al-W superlattices, separating out the longitudinal from the total strength.

Sagittal elastic waves in cubic superlattices

Sagittal elastic waves in cubic superlattices

Sagittal elastic waves in infinite and semi-infinite superlattices

Elastic waves, polarized in the sagittal plane, are investigated in infinite and semi-infinite superlattices made from alternate layers of two isotropic media. By the use of a transfer-matrix method, we obtain explicit equations for the dispersion of the bulk modes as well as of the surface modes of a semi-infinite superlattice. We present a few illustrations of the theory for Al-W superlattices in which the relative thicknesses of the two media or the thickness of the layer at the surface are varied.