scholarly journals On generalized thermoelastic disturbances in an elastic solid with a spherical cavity

1991 ◽  
Vol 4 (3) ◽  
pp. 225-240 ◽  
Author(s):  
Basudeb Mukhopadhyay ◽  
Rasajit Bera ◽  
Lokenath Debnath
Keyword(s):  
Laplace Transform ◽  
Classical Theory ◽  
Spherical Cavity ◽  
Dynamic Pressure ◽  
Elastic Solid ◽  
Dynamical Theory ◽  
The Laplace Transform ◽  
Generalized Thermoelastic ◽  
Short Time ◽  
The Waves

In this paper, a generalized dynamical theory of thermoelasticity is employed to study disturbances in an infinite elastic solid containing a spherical cavity which is subjected to step rise in temperature in its inner boundary and an impulsive dynamic pressure on its surface. The problem is solved by the use of the Laplace transform on time. The short time approximations for the stress, displacement and temperature are obtained to examine their discontinuities at the respective wavefronts. It is shown that the instantaneous change in pressure and temperature at the cavity wall gives rise to elastic and thermal disturbances which travel with finite velocities v1 and v2(>v1) respectively. The stress, displacement and temperature are found to experience discontinuities at the respective wavefronts. One of the significant findings of the present analysis is that there is no diffusive nature of the waves as found in classical theory.

scholarly journals Thermoelastic waves without energy dissipation in an unbounded body with a spherical cavity

2000 ◽  
Vol 23 (8) ◽  
pp. 555-562 ◽  
Author(s):  
D. S. Chandrasekharaiah ◽  
K. S. Srinath
Keyword(s):  
Energy Dissipation ◽  
Spherical Cavity ◽  
Small Time ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Thermoelastic Body ◽  
Without Energy Dissipation ◽  
The Waves ◽  
Thermoelastic Waves ◽  
Stress Free Boundary

The linear theory of thermoelasticity without energy dissipation is employed to study waves emanating from the boundary of a spherical cavity in a homogeneous and isotropic unbounded thermoelastic body. The waves are supposed to be spherically symmetric and caused by a constant step in temperature applied to the stress-free boundary of the cavity. Small-time solutions for the displacement, temperature, and stress fields are obtained by using the Laplace transform technique. It is found that there exist two coupled waves, of which one is predominantly elastic and the other is predominantly thermal, both propagating with finite speeds but with no exponential attenuation. Exact expressions for discontinuities in the field functions that occur at the wavefronts are computed and analysed. The results are compared with those obtained earlier in the contexts of some other models of thermoelasticity.

The impact of a rigid circular cylinder on an elastic solid

1962 ◽  
Vol 255 (1053) ◽  
pp. 153-191 ◽  
Keyword(s):  
Laplace Transform ◽  
Elastic Body ◽  
Contact Area ◽  
Elastic Solid ◽  
Coupled System ◽  
Poisson's Ratio ◽  
Normal Displacement ◽  
Plane Boundary ◽  
The Laplace Transform ◽  
Perfect Adherence

A method is described for approximating to any degree of accuracy the solution of the following problem: An elastic body which is bounded by a plane on one side, but extends to infinity otherwise, is hit by a circular disk of given mass, radius, and initial speed perpendicular to the plane boundary. The whole surface of the disk enters into contact with the elastic body at the same time and stays in contact at all its points from then on. The disk is assumed to be rigid, i.e. it does not allow the particles of the elastic body in the contact area to move relative to each other in a direction perpendicular to the plane boundary. For the relative motion of these particles parallel to the face of the disk several conditions are considered, representing perfect lubrication, various degrees of viscous friction and perfect adherence. With the help of various Mellin transformations a method is indicated which leads to an expansion of the motion in powers of the Laplace transform variable. The case of perfect adherence needs some special consideration, and a simple approximation for the static problem is found. The case of perfect lubrication is then treated in more detail by a different method which replaces the condition of constant normal displacement in the contact area by an equivalent number of requirements for certain averages over the normal displacement in the contact area. The condition of rigidity for the disk is not exactly satisfied, but it is possible to judge the accuracy of the approximation (with the help of asymptotic expansions in the Laplace transform variable) at the initial time, when discrepancies are largest. The concept of ‘mode of vibration’ is introduced. Any transient in the coupled system of elastic body and rigid disk can be described as superposition of modes, each of which is an exponentially damped harmonic oscillation of the coupled system with a frequency and damping constant independent of the particular transient. The motion of the impinging disk is then seen to be dominated mostly by the lowest mode, provided the mass of the disk is not too small. The displacement perpendicular to the boundary outside of the contact area has been calculated. This calculation is not more difficult than the corresponding one in the case of a point-like source at the plane boundary of an elastic solid. Numerical computations were carried out for the case of perfect lubrication with the help of the Elecom digital computer in order to determine the first two modes and their contributions to the motion of the disk. As long as Poisson’s ratio for the elastic solid exceeds 1/4, the results do not depend strongly on the value of Poisson’s ratio. The ratio of the areal mass densities of the disk to the elastic solid is the main parameter of the theory. The shear wave velocity of the elastic solid determines the time scale of the motion.

Laplace wavelet transform theory and applications

2017 ◽  
Vol 24 (9) ◽  
pp. 1600-1620 ◽  
Author(s):  
Tariq Abuhamdia ◽  
Saied Taheri ◽  
John Burns
Keyword(s):  
Wavelet Transform ◽  
Differential Equations ◽  
Laplace Transform ◽  
Linear Time ◽  
Time Frequency ◽  
Linear Time Invariant ◽  
The Laplace Transform ◽  
Invariant Differential ◽  
Short Time ◽  
Time Invariant

This study introduces the theory of the Laplace wavelet transform (LWT). The Laplace wavelets are a generalization of the second-order under damped linear time-invariant (SOULTI) wavelets to the complex domain. This generalization produces the mother wavelet function that has been used as the Laplace pseudo wavelet or the Laplace wavelet dictionary. The study shows that the Laplace wavelet can be used to transform signals to the time-scale or time-frequency domain and can be retrieved back. The properties of the new generalization are outlined, and the characteristics of the companion wavelet transform are defined. Moreover, some similarities between the Laplace wavelet transform and the Laplace transform arise, where a relation between the Laplace wavelet transform and the Laplace transform is derived. This relation can be beneficial in evaluating the wavelet transform. The new wavelet transform has phase and magnitude, and can also be evaluated for most elementary signals. The Laplace wavelets inherit many properties from the SOULTI wavelets, and the Laplace wavelet transform inherits many properties from both the SOULTI wavelet transform and the Laplace transform. In addition, the investigation shows that both the LWT and the SOULTI wavelet transform give the particular solutions of specific related differential equations, and the particular solution of these linear time-invariant differential equations can in general be written in terms of a wavelet transform. Finally, the properties of the Laplace wavelet are verified by applications to frequency varying signals and to vibrations of mechanical systems for modes decoupling, and the results are compared with the generalized Morse and Morlet wavelets in addition to the short time Fourier transform’s results.

scholarly journals Propagation of non-stationary antisymmetric kinematic perturbations from a spherical cavity in Cosserat medium

2020 ◽  
pp. 201-210
Author(s):  
D. V Tarlakovskii ◽  
Van Lam Nguyen
Keyword(s):  
Laplace Transform ◽  
Spherical Cavity ◽  
Bessel Functions ◽  
Initial Conditions ◽  
Displacement Vector ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Spherical Coordinate ◽  
Cosserat Medium ◽  
The Laplace Transform

We consider a space filled with a linearly elastic Cosserat medium with a spherical cavity under given nonstationary antisymmetric surface perturbations, which are understood as the corresponding analogue of classical antiplane deformations. The motion of a medium is described by a system of three equations with respect to nonzero components of the displacement vector and potentials of the rotation field, written in a spherical coordinate system with the origin at its center of the cavity. The initial conditions are assumed to be zero. To solve the problem, we use decomposition of functions to Legendre and Gegenbauer polynomials, as well as the Laplace transform in time. As a result, the problem is reduced to independent systems of ordinary differential equations with the Laplace operator for the coefficients of the series. A statement about the structure of the general solution of this system is formulated. Images of the series coefficients are presented in the form of linear combinations of boundary conditions with coefficients - transformants of surface influence functions, the explicit formulas for which include the Bessel functions of a half-integer index. Due to the complexity of these expressions, to determine the originals in the linear approximation, the method of a small parameter is used, which is taken as a coefficient characterizing the relationship between the displacement and rotation fields. Then, taking into account the connection between the Bessel functions and elementary functions, the images are written in the form of linear combinations of exponentials with coefficients - rational functions of the transformation parameter. The further procedure for inverting the Laplace transform is carried out using residues. It is shown that there are three wave fronts corresponding to a shear wave modified with allowance for free rotation and two rotation waves. Examples of calculations for a granular composite of aluminum shot in an epoxy matrix are presented.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

scholarly journals Algebraic inversion of the Laplace transform

2005 ◽  
Vol 50 (1-2) ◽  
pp. 179-185 ◽  
Author(s):  
P.G. Massouros ◽  
G.M. Genin
Keyword(s):  
Laplace Transform ◽  
The Laplace Transform
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