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.

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.

scholarly journals Rotational Effects on Propagation of Rayleigh Wave in a Micropolar Piezoelectric Medium

2018 ◽  
Vol 48 (2) ◽  
pp. 93-105 ◽  
Author(s):  
Baljeet Singh ◽  
Ritu Sindhu
Keyword(s):  
Rayleigh Wave ◽  
Half Space ◽  
Wave Speed ◽  
Propagation Speed ◽  
Transversely Isotropic ◽  
Frequency Equation ◽  
Piezoelectric Medium ◽  
Effects Of Rotation ◽  
Radiation Conditions ◽  
Relevant Material

Abstract In this paper, the governing equations of a linear, homogeneous and transversely isotropic rotating micropolar piezoelectric medium are solved for surface wave solutions. The appropriate solutions satisfying the radiation conditions are obtained in a half-space. These solutions are applied to suitable boundary conditions at the free surface of the half-space. A frequency equation for Rayleigh wave is obtained for both charge free and electrically shorted cases. Using iteration method, the non-dimensional wave speed of Rayleigh wave is computed for relevant material constants modelling the medium. The effects of rotation, piezoelectricity, frequency and material parameters are observed graphically on the propagation speed.

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

Near-Tip Fields for Penny-Shaped Cracks in Magnetoelectroelastic Media

2006 ◽  
Vol 312 ◽  
pp. 41-46 ◽  
Author(s):  
Bao Lin Wang ◽  
Yiu Wing Mai
Keyword(s):  
Closed Form ◽  
Transversely Isotropic ◽  
Crack Plane ◽  
Axially Symmetric ◽  
Crack Configuration ◽  
Penny Shaped Crack ◽  
Electric And Magnetic Fields ◽  
Shaped Crack ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

This paper solves the penny-shaped crack configuration in transversely isotropic solids with coupled magneto-electro-elastic properties. The crack plane is coincident with the plane of symmetry such that the resulting elastic, electric and magnetic fields are axially symmetric. The mechanical, electrical and magnetical loads are considered separately. Closed-form expressions for the stresses, electric displacements, and magnetic inductions near the crack frontier are given.

scholarly journals A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

2021 ◽  
Author(s):  
Chennakesava Kadapa
Keyword(s):  
Wave Propagation ◽  
Wave Speed ◽  
Critical Time ◽  
Incompressible Material ◽  
Implicit Scheme ◽  
Bulk Wave ◽  
Lumped Mass ◽  
Time Step ◽  
Material Models ◽  
Shear Wave Speed

AbstractThis paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

Diagnosis of liver fibrosis based on quantification of factors associated with shear wave speed

Choonpa Igaku ◽  
2021 ◽  
Author(s):  
Hiroko IIJIMA ◽  
Toshifumi TADA ◽  
Hiroyuki HACHIYA ◽  
Takashi NISHIMURA ◽  
Junko NISHIMURA ◽  
...  
Keyword(s):  
Liver Fibrosis ◽  
Shear Wave ◽  
Wave Speed ◽  
Shear Wave Speed ◽  
Factors Associated

A Source Physics Interpretation of Nonself-Similar Double-Corner-Frequency Source Spectral Model JA19_2S

2022 ◽  
Author(s):  
Chen Ji ◽  
Ralph J. Archuleta
Keyword(s):  
Stress Drop ◽  
Wave Speed ◽  
Dynamic Stress ◽  
Corner Frequency ◽  
Rupture Velocity ◽  
Scaling Relation ◽  
Shear Wave Speed ◽  
Dynamic Stress Drop ◽  
Static Stress Drop ◽  
Frequency Source

Abstract We investigate the relation between the kinematic double-corner-frequency source spectral model JA19_2S (Ji and Archuleta, 2020) and static fault geometry scaling relations proposed by Leonard (2010). We find that the nonself-similar low-corner-frequency scaling relation of JA19_2S model can be explained using the fault length scaling relation of Leonard’s model combined with an average rupture velocity ∼70% of shear-wave speed for earthquakes 5.3 < M< 6.9. Earthquakes consistent with both models have magnitude-independent average static stress drop and average dynamic stress drop around 3 MPa. Their scaled energy e˜ is not a constant. The decrease of e˜ with magnitude can be fully explained by the magnitude dependence of the fault aspect ratio. The high-frequency source radiation is generally controlled by seismic moment, static stress drop, and dynamic stress drop but is further modulated by the fault aspect ratio and the relative location of the hypocenter. Based on these two models, the commonly quoted average rupture velocity of 70%–80% of shear-wave speed implies predominantly unilateral rupture.

scholarly journals A soft cervix, categorized by shear-wave elastography, in women with short or with normal cervical length at 18–24 weeks is associated with a higher prevalence of spontaneous preterm delivery

2018 ◽  
Vol 46 (5) ◽  
pp. 489-501 ◽  
Author(s):  
Edgar Hernandez-Andrade ◽  
Eli Maymon ◽  
Suchaya Luewan ◽  
Gaurav Bhatti ◽  
Mohammad Mehrmohammadi ◽  
...  
Keyword(s):  
Cohort Study ◽  
Preterm Delivery ◽  
Shear Wave ◽  
Wave Speed ◽  
Shear Wave Elastography ◽  
Cervical Length ◽  
Singleton Pregnancy ◽  
Short Cervix ◽  
Shear Wave Speed ◽  
Spontaneous Preterm Delivery

AbstractObjective:To determine whether a soft cervix identified by shear-wave elastography between 18 and 24 weeks of gestation is associated with increased frequency of spontaneous preterm delivery (sPTD).Materials and methods:This prospective cohort study included 628 consecutive women with a singleton pregnancy. Cervical length (mm) and softness [shear-wave speed: (SWS) meters per second (m/s)] of the internal cervical os were measured at 18–24 weeks of gestation. Frequency of sPTD <37 (sPTD<37) and <34 (sPTD<34) weeks of gestation was compared among women with and without a short (≤25 mm) and/or a soft cervix (SWS <25thpercentile).Results:There were 31/628 (4.9%) sPTD<37 and 12/628 (1.9%) sPTD<34 deliveries. The combination of a soft and a short cervix increased the risk of sPTD<37 by 18-fold [relative risk (RR) 18.0 (95% confidence interval [CI], 7.7–43.9); P<0.0001] and the risk of sPTD<34 by 120-fold [RR 120.0 (95% CI 12.3–1009.9); P<0.0001] compared to women with normal cervical length. A soft-only cervix increased the risk of sPTD<37 by 4.5-fold [RR 4.5 (95% CI 2.1–9.8); P=0.0002] and of sPTD<34 by 21-fold [RR 21.0 (95% CI 2.6–169.3); P=0.0003] compared to a non-soft cervix.Conclusions:A soft cervix at 18–24 weeks of gestation increases the risk of sPTD <37 and <34 weeks of gestation independently of cervical length.

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