Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space

1962 ◽  
Vol 29 (1) ◽  
pp. 53-60 ◽  
Author(s):  
Julius Miklowitz
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Integral Transform ◽  
Formal Solution ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Frequency Equation ◽  
Point Load ◽  
Phase Method ◽  
Stationary Phase Method

The problem treated is that of an infinite free plate excited symmetrically by two equal and normally opposed step point-loads on its faces. The problem is equivalent to that of the surface normal point-load excitation of an infinite elastic layer, half the thickness of the plate, overlying a rigid half-space with lubricated contact. The formal solution is obtained from the equations of motion in linear elasticity with the aid of a double integral transform technique and residue theory. The stationary phase method, and known characteristics of the governing Rayleigh-Lamb frequency equation, are used to analyze and evaluate numerically the far field displacements. It is shown that the head of the disturbance is composed predominantly of the low-frequency long waves from the lowest mode of wave transmission.

Transient Compressional Waves in an Infinite Elastic Plate With a Circular Cylindrical Cavity

1964 ◽  
Vol 31 (4) ◽  
pp. 627-634 ◽  
Author(s):  
R. A. Scott ◽  
Julius Miklowitz
Keyword(s):  
Stationary Phase ◽  
Cylindrical Cavity ◽  
Integral Transform ◽  
Formal Solution ◽  
Linear Equations ◽  
Normal Displacement ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Hole Radius ◽  
Circular Cylindrical Cavity

The problem treated is that of an infinite free plate with a circular cylindrical cavity subjected to a step normal displacement. The linear equations of elasticity are employed and the formal solution is obtained using a multi-integral transform technique, necessitating the introduction of extended Hankel transforms, and residue theory. Some properties of the Rayleigh-Lamb frequency equation, pertinent to the inversion process, are derived. Numerical information for the far field, showing the effect of the hole radius on the displacements, is obtained using the stationary phase method and, in the case of the radial displacement, the solution is compared with the corresponding slab solution. The results show that at a given station the plate-cavity solution approaches that for the slab, as the hole radius decreases. The head of the pulse and stationary-phase approximations to the corresponding horizontal slab displacement are also compared, and some discrepancies between the two are found in the vicinity of the wave-front arrival time.

scholarly journals The near-resonant regimes of a moving load in a three-dimensional problem for a coated elastic half-space

2016 ◽  
Vol 22 (1) ◽  
pp. 89-100 ◽  
Author(s):  
Bariş Erbaş ◽  
Julius Kaplunov ◽  
Danila A Prikazchikov ◽  
Onur Şahin
Keyword(s):  
Half Space ◽  
Moving Load ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Point Load ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Dispersive Effect ◽  
Mach Cones ◽  
Integral Solutions

This paper deals with the three-dimensional analysis of the near-resonant regimes of a point load, moving steadily along the surface of a coated elastic half-space. The approach developed relies on a specialized hyperbolic–elliptic formulation for the wave field, established earlier by the authors. Straightforward integral solutions of the two-dimensional perturbed wave equation describing wave propagation along the surface are derived along with their far-field asymptotic expansions obtained using the uniform stationary phase method. Both sub-Rayleigh and super-Rayleigh cases are studied. It is shown that the singularities arising at the contour of the Mach cones typical of the super-Rayleigh case, are smoothed due to the dispersive effect of the coating.

Dynamic Response of a Rigid Foundation Subjected to a Distance Blast

2012 ◽  
Author(s):  
Deji Ojetola ◽  
Hamid R. Hamidzadeh
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Integral Transform ◽  
Numerical Technique ◽  
Formal Solution ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Rigid Foundation ◽  
Transform Method

Blasts and explosions occur in many activities that are either man-made or nature induced. The effect of the blasts could have a residual or devastating effect on the buildings at some distance within the vicinity of the explosion. In this investigation, an analytical solution for the time response of a rigid foundation subjected to a distant blast is considered. The medium is considered to be an elastic half space. A formal solution to the wave propagations on the medium is obtained by the integral transform method. To achieve numerical results for this case, an effective numerical technique has been developed for calculation of the integrals represented in the inversion of the transformed relations. Time functions for the vertical and radial displacements of the surface of the elastic half space due to a distant blast load are determined. Mathematical procedures for determination of the dynamic response of the surface of an elastic half-space subjected to the blast along with numerical results for displacements of a rigid foundation are provided.

Dispersive Pulse Propagation Parallel to the Interfaces of a Laminated Composite

1969 ◽  
Vol 36 (3) ◽  
pp. 479-484 ◽  
Author(s):  
J. C. Peck ◽  
G. A. Gurtman
Keyword(s):  
Wave Propagation ◽  
Half Space ◽  
Fourier Transforms ◽  
Formal Solution ◽  
Low Frequency ◽  
Laminated Composite ◽  
Propagation Distance ◽  
Step Function ◽  
Algebraic Expression ◽  
Mathematical Treatment

This paper presents a theoretical analysis of the geometric dispersion of transient stress waves in a linearly elastic laminated composite. The loading is a uniform pressure of step-function time-dependence, applied to a half space. The laminates are perpendicular to the half-space boundary. The mathematical treatment is borrowed from the theory of wave propagation in rods. Fourier transforms are applied to time and the coordinate in the propagation direction. Inversion of the spatial transform by residues yields a formal solution in the form of an infinite series of integrals. Each of these integrals is the contribution to the transient response from a mode of sinusoidal wave propagation. Application of the saddle-point technique for long-time asymptotic approximation indicates that the low-frequency portion of the integral from the first mode gives the dominant contribution, called the head-of-the-pulse approximation. The form of the expression for the head-of-the-pulse approximation leads to the definition of a characteristic dispersion time τ. Since τ is a single quantity which describes the dispersion of the wave, it simplifies parametric studies. A closed-form algebraic expression for τ is presented, which has a simple dependence on the propagation distance and spacing of the laminates. Numerical examples for boron-epoxy and glass-epoxy laminates are given.

Acoustic Radiation From Stiffened Cylindrical Shells With Constrained Layer Damping

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
Xiongtao Cao ◽  
Hongxing Hua ◽  
Zhenguo Zhang
Keyword(s):  
Cylindrical Shell ◽  
Sound Pressure ◽  
Acoustic Radiation ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Constrained Layer Damping ◽  
Acoustic Characteristics ◽  
Stiffened Cylindrical Shell

Acoustic radiation from cylindrical shells stiffened by two sets of rings, with constrained layer damping (CLD), is investigated theoretically. The governing equations of motion for the cylindrical shell with CLD are described on the basis of Sanders thin shell theory. Two sets of rings interact with the host cylindrical shell only through the normal line forces. The solutions are derived in the wavenumber domain and the stationary phase method is used to find an analytical expression of the far-field sound pressure. The effects of the viscoelastic material core, constrained layer and multiple loadings on sound pressure are illustrated. The helical wave spectra of sound pressure and the radial displacement clearly show the vibrational and acoustic characteristics of the stiffened cylindrical shell with CLD. It is shown that CLD can effectively suppress the radial vibration and reduce acoustic radiation.

Solution of contact problems for a transversely isotropic elastic layer bonded to an elastic half-space

2009 ◽  
Vol 223 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
V I Fabrikant
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Integral Transform ◽  
Transversely Isotropic ◽  
Contact Problems ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Electrostatic Problem ◽  
Crack Problems ◽  
New Interpretation

The idea, first used by the author for the case of crack problems, is applied here to solve a contact problem for a transversely isotropic elastic layer bonded to an elastic halfspace, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the layer's free surface. The governing integral equation is derived; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged discs in the shape of the domain of contact. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.

Transient Wave Propagation in a Symmetrically Layered Elastic Plate

1974 ◽  
Vol 41 (3) ◽  
pp. 684-690 ◽  
Author(s):  
D. C. Viano ◽  
J. Miklowitz
Keyword(s):  
Near Field ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Short Wave ◽  
Frequency Equation ◽  
Layered Plate ◽  
Transient Wave ◽  
Long Time ◽  
Transient Wave Propagation ◽  
Front Running

The response of a symmetrically layered plate subjected to symmetric step normal line face loads is determined. The solution, which is based on the equations of motion from linear elasticity, is obtained in the form of an infinite series of integrals with each term being the contribution from a branch of the underlying frequency equation. Through numerical analysis several of the lower branches of the frequency equation, which depend on layered plate ratios (four material and one thickness) are evaluated and used to write the transient response in the near field. Integrations based on the first five branches show the dominance of the lowest mode in the solution. Applying arguments of stationary phase to the spectrum, yields low frequency-long wave and high frequency-short wave approximations for use in obtaining the far field-long time disturbance. The former governs a front running “head of the pulse” for the layered plate, and the latter, later arriving Rayleigh and Stoneley waves.

SH-Wave Scattering From the Interface Defect

2017 ◽  
Vol 5 (1) ◽  
pp. 45-50
Author(s):  
Myron Voytko ◽  
◽  
Yaroslav Kulynych ◽  
Dozyslav Kuryliak
Keyword(s):  
Half Space ◽  
Wave Diffraction ◽  
Scattered Field ◽  
Wave Scattering ◽  
Elastic Layer ◽  
Analytical Form ◽  
Structure Parameters ◽  
Sh Wave ◽  
Interface Defect ◽  
Hopf Method

The problem of the elastic SH-wave diffraction from the semi-infinite interface defect in the rigid junction of the elastic layer and the half-space is solved. The defect is modeled by the impedance surface. The solution is obtained by the Wiener- Hopf method. The dependences of the scattered field on the structure parameters are presented in analytical form. Verifica¬tion of the obtained solution is presented.

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