Transient elastic waves in a transversely isotropic laminate impacted by axisymmetric load

A method of reverberation-ray matrix has been extended to the investigation of the field of wave propagation in a transversely isotropic laminate. By using the decomposition in a local coordinate system, any complicated waves can be separated into a departing part and an arriving part, which are expressed in the local scattering matrix at structural interface. Together with the local phase matrix, we obtain a certain wave transmitting from a layer to the neighboring one. Thus, the wave propagation in the whole laminate can be described when assembling the local information with global phase and global permutation matrices. This method is perfectly suitable for evaluating the transient waves involving a large number of generalized-rays. In this paper, the method is applied to laminate made of transversely isotropic material. Numerical results show the influence of the change of thickness and elastic constants of the layers on the wave propagation in laminate.     

Elastic constants of natural quartz

The elastic constants of a natural-quartz sphere using resonance-ultrasound spectroscopy (RUS) are measured. The measurements of the near-traction-free vibrational frequencies of the sphere are matched with the predicted frequencies from the dynamic theory of elasticity, with optimized estimates for the elastic constants driving the differences between these sets of frequencies to a minimal value. The present computational model, although based on earlier approaches, is the first application of RUS to trigonal-symmetry spheres. Quartz shows six independent elastic constants, and our estimates of these constants are close to those computed by other means. Except for C14, after a 1% mass-density correction, natural quartz and cultured quartz show the same elastic constants. Natural quartz shows higher internal frictions.     

Theories of surface elasticity for nanoscale objects

The emergence of nanotechnology has driven recent interest in systems having surface atoms as a significant fraction of all atoms present, in particular nano-sheets (ultra-thin slabs), nano-wires, and nano-particles. In these systems, the bulk (i.e. non-surface region or interior) is typically strained in response to the stress of the surface. This elastic strain of the bulk in turn changes the surface lattice constants. Since the bulk and the surface are coupled, the problem must be solved self-consistently. Solving this problem requires a quantitative model of the surface elastic properties which are different from the bulk. In this paper we consider various models that have been proposed for surface elasticity. Our goal is to elucidate the relationship between two contrasting approaches: (1) the Shuttleworth equation which defines a surface stress based on the strain derivative of the surface energy and (2) the Gurtin-Murdoch (GM) theory which considers the surface layer as a membrane with residual strain and with elastic constants different from the bulk. The GM theory is analogous to the 2-D Frenkel-Kontorova (FK) model and can be used to obtain quantitative parameters for the FK model. We present an embedded atom method calculation of the surface elastic constants of Cu(1 1 1) using the GM theory with the surface represented by a membrane one atomic layer thick. This quantitative approach describes the elastic properties of surfaces in a physically appealing way. Just as the bulk elastic constants provide direct information regarding the stress/strain relationship in a bulk material, the surface elastic constants provide similar information for a surface monolayer. This theory will allow elasticity analysis and atomistic calculations of properties of nano-scale objects.     

First-principles investigation of the phonon spectrum of β-GaS

This paper presents the results of the first-principles density functional theory calculation of the phonon spectrum of the ??-GaS semiconductor with a layered structure. The elastic constants and velocities of sound along and across the layers of the ??-GaS semiconductor have been determined. Investigation of the equilibrium structure and the phonon spectrum of the (0001) surface of the ??-GaS crystals has demonstrated that the bulk and surface structural and dynamical properties of these crystals differ only slightly. The calculated frequencies and symmetries of phonon modes at the center of the Brillouin zone of the semiconductor are in satisfactory agreement with the experimental data obtained from the Raman and infrared spectra.     

Third-order elastic constants of ZnS and ZnSe

The complete set of non-vanishing third-order elastic constants of the semiconductors ZnS and ZnSe is obtained theoretically. The strain energy density is estimated using finite strain elasticity theory by considering the interactions up to two nearest neighbours of each atom in the unit cell of these compounds. This energy density is compared with the strain dependent lattice energy density from the continuum model approximation. The second-order parameter of the potential function φ is obtained from the measured principal axis C_ij. The third-order potential parameter is estimated by assuming a Lennard-Jones type of interatomic potential. The interlattice displacements as well as the second-order elastic constants are evaluated along with the six third-order elastic constants of ZnS and ZnSe. Using these second- and third-order elastic constants of ZnS, the pressure derivatives of second-order elastic constants are evaluated. The second- and third-order elastic constants of ZnSe are compared with the available experimental values. The third-order elastic constants show anisotropy in different directions.     

Third-order elastic constants of the alloy Fe72Pt28

The complete sets of second- and third-order elastic constants of the cubic Fe₇₂Pt₂₈ have been obtained using the strain energy density derived from interactions up to three nearest neighbours of each atom in the unit cell. The finite strain elasticity theory has been used to get the strain energy density of Fe₇₂Pt₂₈. The strain energy density is compared with the strain-dependent lattice energy density obtained from the continuum model approximation and the expressions for the second- and third-order elastic constants of Fe₇₂Pt₂₈ are given. The second-order potential parameter is deduced from the measured second-order elastic constants of Fe₇₂Pt₂₈ and the third-order potential parameter is estimated from the Lennard-Jones inter-atomic potential for Fe₇₂Pt₂₈. The inter-lattice displacements; the three independent second-order elastic constants and the six independent third-order elastic constants of Fe₇₂Pt₂₈ are also determined. The second-order elastic constants are compared with the experimental elastic constants of Fe₇₂Pt₂₈. We also study the effect of pressure on the second-order elastic constants of Fe₇₂Pt₂₈.     

The molecular-statistical calculation of the elastic constants of uniaxial nematics

Using the molecular-statistical theory of the thermodynamic properties of nematics and perturbation theory, the expressions for the elastic constants of uniaxial nematics are derived in terms of the mutual interactions between their constituent molecules. Next the elastic constants of the Maier-Saupe model are calculated and compared with the original result. The appearing difference is discussed briefly. Finally, in view of existing discrepancies, the elastic constants of the Onsager model are recalculated. For that purpose the hard rod interaction is conceived as the limit of a non-singular interaction. The elastic constants of hard core models are shown to increase with increasing temperature at constant density. In view of this behaviour the usefulness of hard core models for an understanding of the nematic state seems rather limited.     

Prediction of dislocation formation in epitaxial multilayers subject to in-plane loading

A theoretical model is described to predict equilibrium distributions of misfit dislocations in one or more anisotropic epitaxial layers of a multilayered system deposited on a thick substrate. Each layer is regarded as having differing elastic and lattice constants, and the system is subject to biaxial in-plane mechanical loading. A stress transfer methodology is developed enabling both the stress and displacement distributions in the system to be estimated for cases where the interacting dislocations are of a pure edge configuration. Energy methods are used to determine equilibrium distributions of the dislocations for given external applied stress states. It is shown that the new model accurately reproduces known exact analytical solutions for the special case of just one isotropic epitaxial layer applied to an isotropic semi-infinite substrate having the elastic constants of the substrate but differing lattice constants. The model is used to consider equilibrium dislocation distributions in capped epitaxial systems with misfit dislocations. It is shown that the simplifying assumptions often made in the literature, regarding the uniformity of elastic properties and the neglect of anisotropy, can lead to critical thicknesses being underestimated by 15–18%. The application of uniaxial tensile stresses increases the value of critical thicknesses. The model can be used to analyse dislocations in various non-neighbouring layers provided the dislocation density has the same value in all layers in which dislocations have formed. This type of analysis enables the prediction of the deformation of metallic multilayers subject to mechanical and thermal loading.     

First-principles study of the phonon spectrum of ɛ-GaSe

The phonon spectrum and phonon density of states of ?-GaSe layered semiconductor have been studied from the first principles in the linear-response approximation. The elastic constants and acoustic velocities along and across layers have been determined. The study of the equilibrium structure and phonon spectrum of the (0001) surface of ?-GaSe has demonstrated that the volume and surface structural dynamic properties of these crystals differ insignificantly. The calculated frequencies and symmetries of the phonon modes in the center of the Brillouin zone are in good agreement with the experimental data obtained from the Raman scattering and infrared spectra.     

Effect of diffusion layers and defect layer on acoustic phonons transport through the structure consisting of different films

By using the continuum elastic approximation model and the transfer matrix method, we investigate the effect of diffusion layers and defect layer on acoustic phonons transport through the structure consisting of different films. Our work show that most acoustic phonons can easily pass the structure, but some only have much less transmission probabilities and form corresponding dips in the transmission spectrum. With the change of the structure parameters such as the width of diffusion layers and defect layer, the number of unit cell and the density of containing Al in diffusion layers and defect layer, the magnitude of the frequencies of acoustic phonons corresponding to the dips almost remain unchanged, but the transmission coefficients corresponding to the dips change at different degree, and the transmission probabilities of some frequencies are very sensitive to the variation of the above-mentioned structure parameters. These results can provide some references in controlling the transmission coefficients of acoustic phonons, devising parts of acoustic apparatus and theoretical investigation related.     

Effective optical constants and effective optical properties of ultrathin trilayer structures

This work presents an extension of the characteristic effective medium approximation (CEMA) to ultrathin trilayer systems. The extension has been carried out analytically and is supported by corresponding calculations of the effective optical constants of Cu-Au-Cu and Ag-SiO-Ag trilayer systems using the CEMA approximation. This work is in essence a generalization of the characteristic effective medium approximation introduced earlier for ultrathin bilayer structures. This method is used to derive the effective optical constants of a trilayer system, consisting of three thin layers with each constituent layer of thickness much less than the wavelength of the incident radiation. Within this regime a trilayer system is viewed as one effective layer referred to as an effective stack (ES) with well defined effective optical constants, which can be used to calculate the optical properties of the trilayer stack within a specified wavelength range. The CEMA based calculations of the effective optical constants are applied to two trilayer systems with a total of five stacks. Three are Cu-Au-Cu and two are Ag-SiO-Ag stacks. The thicknesses of the parent layers in the Cu-Au-Cu stack range from 3 to 30 nm for Cu and 4 to 40 nm for Au; in the Ag-SiO-Ag stack the constituent layers are 6 nm for Ag, but range from 5 to 10 nm for SiO. This study is for normal or near normal incidence spectroscopy in a wavelength range that extends from visible to near infrared. The agreement between CEMA based ES stack results and those of the standard CMT technique is very satisfactory.     

Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.     

Discrete and continuum models for calculating the phonon spectra of carbon nanotubes

The vibration spectrum of perfect carbon nanotubes is studied using a two-parametric potential which includes pairwise and three-particle interatomic interactions. This potential was proposed by Keating and allows one to take into account the elasticity of pairwise interatomic bonds and the elasticity associated with a change in the angle between directional interatomic bonds in covalent crystals. Using the Keating potential, the vibration spectrum of a graphite monolayer is calculated and fitted to the vibration spectrum of crystalline graphite, thereby determining the parameters of the potential. With these parameters, the phonon spectra of perfect monolayer graphite nanotubes are calculated. A continuum model, in which a monolayer nanotube is represented as an elastic cylindrical shell of a finite thickness, is also discussed. Within this model, the vibration spectrum of a nanotube is calculated numerically in the long-wavelength limit as a function of the radius and thickness of the nanotube.     

Scale-dependent effects on wave propagation in magnetically affected single/double-layered compositionally graded nanosize beams

This article deals with the wave propagation analysis of single/double layered functionally graded (FG) size-dependent nanobeams in elastic medium and subjected to a longitudinal magnetic field employing nonlocal elasticity theory. Material properties of nanobeam change gradually according to the sigmoid function. Applying an analytical solution, the acoustical and optical dispersion relations are explored for various wave number, nonlocality parameter, material composition, elastic foundation constants, and magnetic field intensity. It is found that frequency and phase velocity of waves propagating in S-FGM nanobeam are significantly affected by these parameters. Also, presence of cut-off and escape frequencies in wave propagation analysis of embedded S-FGM nanobeams is investigated.     

Simulation and fabrication study of porous silicon photonic crystal

The present work reports design and fabrication of porous silicon based one-dimensional (1D) photonic crystal. Distributed Bragg reflector (DBR) is a 1D photonic crystal composed of multilayer stack of high and low refractive index layers. Design of porous silicon DBR is a complex one and requires appropriate control in optical parameters of its constituent layers. In order to design DBR, two porous silicon single layer samples were fabricated using current density of 10 and 50 mA/cm². Optical characterization of single layer samples showed series of interference fringes. Reflective interferometric Fourier transform spectroscopy (RIFTS) method was employed to determine optical constants of porous silicon single layers. DBR simulation was carried out based on transfer matrix method. DBR was then fabricated using optical parameters obtained from RIFTS method. Reflection bandwidth of prepared DBR was found to be 216 nm, which is comparable to the simulated value of 203 nm.     

Lattice instability and martensitic transformation in LaAg predicted from first-principles theory

The electronic structure, elastic constants and lattice dynamics of the B(2) type intermetallic compound LaAg are studied by means of density functional theory calculations with the generalized gradient approximation for exchange and correlation. The calculated equilibrium properties and elastic constants agree well with available experimental data. From the ratio between the bulk and shear moduli, LaAg is found to be ductile, which is unusual for B(2) type intermetallics. The computed band structure shows a dominant contribution from La 5d states near the Fermi level. The phonon dispersion relations, calculated using density functional perturbation theory, are in good agreement with available inelastic neutron scattering data. Under pressure, the phonon dispersions develop imaginary frequencies, starting at around 2.3 GPa, in good accordance with the martensitic instability observed above 3.4 GPa. By structural optimization the high pressure phase is identified as orthorhombic B(19).     

Raman scattering and the low-frequency vibrational spectrum of glasses

We present a theory of low-frequency Raman scattering in glasses, based on the concept that light couples to the elastic strains via spatially fluctuating elasto-optic (Pockels) constants. We show that the Raman intensity is not proportional to the vibrational density of states (as was widely believed), but to a convolution of Pockels constant correlation functions with the dynamic strain susceptibilities of the glass. Using the dynamic susceptibilities of a system with fluctuating elastic constants we are able for the first time to describe the Raman intensity and the anomalous vibration spectrum of a glass on the same footing. Good agreement between the theory and experiment for the Raman spectrum, the density of states, and the specific heat is demonstrated at the example of glassy As(2)S(3).     

Characterization of a cylindrical rod by inversion of acoustic scattering data

In this paper, a new approach is proposed for nondestructive characterization of immersed and embedded isotropic rod-shaped samples by inversion of acoustic scattering data. The normal mode expansion technique is used for modelling the scattered field and the compression incident and compression scattered waves are considered. Genetic algorithm is the inversion technique used for estimating the elastic wave velocities and density of the rods from their measured backscattered pressure spectrum. The inversion technique is capable of computing the parameter values that best fit a particular set of data. A perturbation study is conducted on the sensitivity of the resonance frequencies to changes in elastic properties and density of the rods. The numerical results indicate that proper selection of resonance frequencies leads to accurate measurement of elastic constants and density. The proposed approach showed very good convergence and the results obtained were found to agree very well with available data.     

Acoustic response from a bubble pulsating near a fluid layer of finite density and thickness

A theory is developed that allows one to consider the dynamics of an acoustically induced bubble near a fluid layer of finite density and thickness. The theory reveals that, as far as the scattered field of a bubble in the far-field zone is concerned, the layer thickness is a very important factor because the behavior of the scattered field in the cases of infinite and finite layers is qualitatively different even if both layers are of the same density. The amplitude of the scattered pressure from a bubble pulsating in the vicinity of an infinite layer is larger than that for the same bubble in an unbounded fluid, while in the case of a finite layer, on the contrary, the amplitude of the scattered pressure for a bubble near the layer is smaller than that in an unbounded fluid. It is also shown that the higher the layer density, the greater the difference between the scattered pressure amplitudes for infinite and finite layers.     

Heterogeneous flame propagation in the self-propagating high-temperature synthesis (SHS) process in multi-layer foils for three components system: Theory and experimental comparisons

A theory for the heterogeneous flame propagation in the self-propagating high-temperature synthesis (SHS) process that proceeds in multi-layer foils, consisting of alternating layers of constituents, has been extended to three-component systems, by describing a premixed mode of bulk flame propagation supported by a non-premixed reaction that proceeds at the layer surface of a constituent with higher melting-point. The formulation allows for finite rate Arrhenius reaction at the layer surface, temperature-sensitive Arrhenius mass diffusion in the liquid phase, and existence of intermixed region between constituent layers. It is confirmed that thickness ratio of the constituent layers is related to mixture ratio and that thickness ratio of the intermixed region and the constituent layer is related to degree of dilution. Results show that the burning velocity first increases, reaches the maximum, and then decreases rapidly, with decreasing alloy layer thickness, that the decrease in the burning velocity can be attributed to the increase in the substance in the intermixed region between constituent layers, playing a role of diluent, and that the increase in the activity and/or amount of the less-active component in the alloy suppresses the burning velocity. A preheating effect for the burning velocity is also predicted. In experimental comparisons, it has been demonstrated that the analytical results agree with available experimental data in the literature, indicating that the present formulation has captured the essential features of the adiabatic, heterogeneous SHS process.