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.

REFLECTION OF AN ACOUSTICAL PRESSURE PULSE FROM A LIQUID‐SOLID PLANE BOUNDARY

Geophysics ◽  
1956 ◽  
Vol 21 (1) ◽  
pp. 71-87 ◽  
Author(s):  
T. W. Spencer
Keyword(s):  
Surface Wave ◽  
Poisson’S Ratio ◽  
Velocity Ratio ◽  
Elastic Solid ◽  
Vertical Axis ◽  
Vertical Displacement ◽  
Poisson's Ratio ◽  
Plane Boundary ◽  
The Laplace Transform ◽  
Compressional Velocity

The problem treated is concerned with predicting the transient response of a system composed of a liquid layer, bounded above by a vacuum and below by a perfectly elastic solid, when excited by an arbitrary pressure applied uniformly over the surface of a spherical cavity located in the fluid. The Laplace transform of the displacement response is expressed in terms of an integral which is expanded in such a way that each term describes the contribution from one of the image sources. Each term may be evaluated exactly at points located on a vertical axis passing through the source. The final expression for the vertical displacement at axial points is composed of the acoustic, after‐flow, and correction terms. In solids for which Poisson’s ratio is greater than one third the initial variation of the correction is toward positive values (corresponding to motion directed toward the interface). For Poisson’s ratio less than one third the initial variation may be either positive or negative depending on the magnitude of the compressional velocity ratio. A surface wave is shown to exist regardless of the choice of parameters. The surface wave velocity is always less than it would be in the absence of the liquid.

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.

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.

Nonlinear singular one dimensional thermo-elasticity coupled system and double Laplace decomposition methods

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6269-6280
Author(s):  
Hassan Gadain
Keyword(s):  
Laplace Transform ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Coupled System ◽  
Decomposition Methods ◽  
One Dimensional ◽  
Convergence Proof ◽  
Double Laplace Transform ◽  
Adomian Decomposition

In this work, combined double Laplace transform and Adomian decomposition method is presented to solve nonlinear singular one dimensional thermo-elasticity coupled system. Moreover, the convergence proof of the double Laplace transform decomposition method applied to our problem. By using one example, our proposed method is illustrated and the obtained results are confirmed.

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

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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