Polarization of the Longitudinal Pochhammer–Chree Waves

2020 ◽  
Vol 22 (4) ◽  
pp. 1329-1336
Author(s):  
Alla V. Ilyashenko ◽  
Sergey V. Kuznetsov
Keyword(s):  
Wave Speed ◽  
Phase Speed ◽  
Harmonic Waves ◽  
Wave Polarization ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
Shear Wave Speed ◽  
The Matrix ◽  
Circular Frequency ◽  
Analytical Expressions

AbstractThe exact solutions of the linear Pochhammer – Chree equation for propagating harmonic waves in a cylindrical rod, are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear wave speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer – Chree wave can propagate at this phase speed.

Abnormality of the longitudinal Pochhammer–Chree waves in the vicinity of C2 phase speed

2018 ◽  
Vol 24 (23) ◽  
pp. 5642-5649 ◽  
Author(s):  
Sergey V Kuznetsov
Keyword(s):  
Spectral Analysis ◽  
Phase Speed ◽  
Harmonic Waves ◽  
Wave Polarization ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
The Matrix ◽  
Circular Frequency ◽  
Shear Speed ◽  
Analytical Expressions

The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed and analytical expressions for displacement fields are obtained. The variation of wave polarization on the free surface due to the variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree waves can propagate at this phase speed.

Longitudinal Pochhammer — Chree Waves in Mild Auxetics and Non-Auxetics

2018 ◽  
Vol 35 (3) ◽  
pp. 327-334 ◽  
Author(s):  
A. V. Ilyashenko ◽  
S. V. Kuznetsov
Keyword(s):  
Poisson’S Ratio ◽  
Wave Speed ◽  
Poisson's Ratio ◽  
Abnormal Behavior ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
Shear Wave Speed ◽  
The Matrix ◽  
The Waves ◽  
Analytical Expressions

ABSTRACTThe exact solutions of Pochhammer — Chree equation for propagating harmonic waves in isotropic elastic cylindrical rods, are analyzed. Spectral analysis of the matrix dispersion equation for the longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of the wave polarization due to variation of Poisson’s ratio for mild auxetics (Poisson’s ratio is greater than -0.5) is analyzed and compared with the non-auxetics. It is observed that polarization of the waves for both considered cases (auxetics and non-auxetics) exhibits abnormal behavior in the vicinity of the bulk shear wave speed.

Image Contrast of Dislocation Loops Near a Free Surface

1977 ◽  
Vol 35 ◽  
pp. 52-53
Author(s):  
S. M. Ohr
Keyword(s):  
Free Surface ◽  
Dislocation Loop ◽  
Image Contrast ◽  
Stress System ◽  
Shear Stresses ◽  
Dislocation Loops ◽  
Axially Symmetric ◽  
Displacement Fields ◽  
Elastic Fields ◽  
Analytical Expressions

The image contrast of dislocation loops computed in the past has made use of the displacement fields which do not take into account the presence of stress- free foil surfaces. The free surface modifies the elastic fields around a dislocation loop and hence can influence the image contrast observed in the electron microscope. The effect can be significant particularly when the loops lie close to one of the foil surfaces. In general, the elasticity problem of dislocation loops that takes the free surface into account is difficult to handle mathematically. In the present paper, the method of Bastecka1 was extended to obtain explicitly the analytical expressions for the displacement fields around a pure edge circular dislocation loop lying parallel to the foil surface. In this method, the stress fields of an image dislocation loop and another axially symmetric stress system were added in order to eliminate the normal as well as shear stresses at the surface.

Effect of Impermeable Boundaries on the Propagation of Rayleigh Waves in an Unsaturated Poroelastic Half-Space

2010 ◽  
Vol 26 (4) ◽  
pp. 501-511 ◽  
Author(s):  
Y.-S. Chen ◽  
W.-C. Lo ◽  
J.-M. Leu ◽  
Alexander H.-D. Cheng
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Wave Speed ◽  
Water Saturation ◽  
Excitation Frequency ◽  
Phase Speed ◽  
High Water ◽  
Attenuation Coefficients ◽  
Shear Wave Speed ◽  
Dilatational Wave

ABSTRACTThis study presents an analytical model for describing propagation of Rayleigh waves along the impermeable surface of an unsaturated poroelastic half-space. This model is based on the existence of the three modes of dilatational waves which employ the poroelastic equations developed for a porous medium containing two immiscible viscous compressible fluids (Lo, Sposito and Majer, [13]). In a two-fluid saturated medium, the three Rayleigh waves induced by the three dilatational waves can be expressed as R1, R2, and R3 waves in descending order of phase speed magnitude. As the excitation frequency and water saturation are given, the dispersion equation of a cubic polynomial can be solved numerically to obtain the phase speeds and attenuation coefficients of the R1, R2, and R3 waves. The numerical results show the phase speed of the R1 wave is frequency independent (non-dispersive). Comparatively, the R1 wave speed is 93 ∼ 95% of the shear wave speed, and 28% to 49% of the first dilatational wave speed for selected frequencies between 50Hz & 200Hz and relative water saturation ranging from 0.01 to 0.99. However, the R2 and R3 waves are dispersive at the frequencies examined. The ratios of R2 and R3 wave phase speeds to the second and third dilatational wave speeds fall between 56% and 90%. The R1 wave attenuates the least while the R3 wave has the highest attenuation coefficient. Furthermore, the phase speed of the R1 wave under an impermeable surface approaches 1.01 ∼ 1.37 times of the R1 wave under a permeable boundary. Impermeability has significant influence on the phase speeds and attenuation coefficients of the R1 and R2 waves at high water saturation due to the existence of confined fluids.

Dynamic fibre sliding along debonded, frictional interfaces

2006 ◽  
Vol 462 (2068) ◽  
pp. 1081-1106 ◽  
Author(s):  
Q.D Yang ◽  
A Rosakis ◽  
B.N Cox
Keyword(s):  
Wave Speed ◽  
Constitutive Law ◽  
Inertial Effects ◽  
Matrix Interface ◽  
Complete Understanding ◽  
Displacement Fields ◽  
Loading Rates ◽  
The Matrix ◽  
Loading Problem ◽  
Fibre Matrix

The problem is considered of a fibre that is driven dynamically, by compression at one end, into a matrix. The fibre is not initially bonded to the matrix, so that its motion is resisted solely by friction. Prior work based on simplified models has shown that the combination of inertial effects and friction acting over long domains of the fibre–matrix interface gives rise to behaviour that is far more complex than in the well-known static loading problem. The front velocity may depart significantly from the bar wave speed and regimes of slip, slip/stick and reverse slip can exist for different material choices and loading rates. Here more realistic numerical simulations and detailed observations of dynamic displacement fields in a model push-in experiment are used to seek more complete understanding of the problem. The prior results are at least partly confirmed, especially the ability of simple shear-lag theory to predict front velocities and gross features of the deformation. Some other fundamental aspects are newly revealed, including oscillations in the interface stresses during loading; and suggestions of unstable, possibly chaotic interface conditions during unloading. Consideration of the experiments and two different orders of model suggest that the tentatively characterized chaotic phenomena may arise because of the essential nonlinearity of friction, that the shear traction changes discontinuously with the sense of the motion, rather than being associated with the details of the constitutive law that is assumed for the friction. This contrasts with recent understanding of instability and ill-posedness at interfaces loaded uniformly in time, where the nature of the assumed friction law dominates the outcome.

Toroidal Vibrations of Anisotropic Spheres With Spherical Isotropy

1998 ◽  
Vol 65 (1) ◽  
pp. 59-65 ◽  
Author(s):  
K. T. Chau
Keyword(s):  
Wave Speed ◽  
Transversely Isotropic ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Shear Wave Speed ◽  
Spherical Surfaces ◽  
Spherical Isotropy ◽  
Anisotropic Spheres ◽  
Circular Frequency ◽  
Isotropic Solids

This paper derives the exact frequency equation for the toroidal mode of vibrations for a spherically isotropic elastic sphere. The vibrations of spherically isotropic solids are solved by introducing two wave potentials (Φ and Ψ) such that the general solutions for free vibrations can be classified into two independent modes of vibrations, namely the “toroidal” and “spheroidal” modes. Both of these vibration modes can be written in terms of spherical harmonics of degree n. The frequency equation for the toroidal modes is obtained analytically, and it depends on both n and β[=(C11-C12)/(2C44)], where C11C12, and C44 have the usual meaning of moduli and are defined in Eqs. (2)–(3); and, as expected, Lamb’s (1882) classical frequency equation is recovered as the isotropic limit. Numerical results show that the normalized frequency ωa/Cs increases with both n and β, where ω is the circular frequency of vibration, a the radius of the sphere, and Cs is the shear wave speed on the spherical surfaces. The natural frequencies for spheres of transversely isotropic minerals and crystals, with β ranging from 0.3719 to 1.8897, are also tabulated. However, two coupled differential equations are obtained for the spheroidal modes, which remain to be solved.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

scholarly journals Spatial variations in Achilles tendon shear wave speed

2014 ◽  
Vol 47 (11) ◽  
pp. 2685-2692 ◽  
Author(s):  
Ryan J. DeWall ◽  
Laura C. Slane ◽  
Kenneth S. Lee ◽  
Darryl G. Thelen
Keyword(s):  
Achilles Tendon ◽  
Shear Wave ◽  
Wave Speed ◽  
Spatial Variations ◽  
Shear Wave Speed

Measured temperature and pressure dependence of Vp and Vs in compacted, polycrystalline sI methane and sII methane–ethane hydrate

2003 ◽  
Vol 81 (1-2) ◽  
pp. 47-53 ◽  
Author(s):  
M B Helgerud ◽  
W F Waite ◽  
S H Kirby ◽  
A Nur
Keyword(s):  
High Pressure ◽  
Shear Wave ◽  
Pressure Dependence ◽  
Gas Hydrate ◽  
Pressure Vessel ◽  
Wave Speed ◽  
Measured Temperature ◽  
Shear Wave Speed ◽  
Temperature And Pressure ◽  
Fixed End

We report on compressional- and shear-wave-speed measurements made on compacted polycrystalline sI methane and sII methane–ethane hydrate. The gas hydrate samples are synthesized directly in the measurement apparatus by warming granulated ice to 17°C in the presence of a clathrate-forming gas at high pressure (methane for sI, 90.2% methane, 9.8% ethane for sII). Porosity is eliminated after hydrate synthesis by compacting the sample in the synthesis pressure vessel between a hydraulic ram and a fixed end-plug, both containing shear-wave transducers. Wave-speed measurements are made between –20 and 15°C and 0 to 105 MPa applied piston pressure. PACS No.: 61.60Lj

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