The Application of Continued Fractions to Wave Propagation in a Semi-Infinite Elastic Cylindrical Membrane

1969 ◽  
Vol 36 (3) ◽  
pp. 420-424 ◽  
Author(s):  
J. E. Akin ◽  
J. Counts
Keyword(s):  
Wave Propagation ◽  
Laplace Transform ◽  
Exponential Function ◽  
Continued Fractions ◽  
Wave Speed ◽  
Axial Stress ◽  
Rational Approximations ◽  
Stress Resultant ◽  
The Laplace Transform ◽  
Standard Techniques

The Laplace transform of the axial stress resultant in an impacting semi-infinite elastic cylindrical membrane is generated in the form of an exponential function involving the wave speed, and a series in powers of the reciprocal of the transform parameter. Continued fractions are used to obtain rational approximations for the transform represented by the series. The rational approximations, which are in the form of the ratio of two polynomials, are inverted by standard techniques. Results are presented for the axial stress resultant for small and large values of time and are compared with previously published results.

Sensitivity of the Shear Wave Speed-Stress Relationship to Soft Tissue Material Properties and Fiber Alignment

2021 ◽  
Author(s):  
Jonathon Blank ◽  
Darryl Thelen ◽  
Matthew S. Allen ◽  
Joshua Roth
Keyword(s):  
Wave Propagation ◽  
Shear Modulus ◽  
Shear Wave ◽  
Wave Speed ◽  
Axial Stress ◽  
Beam Model ◽  
Fiber Alignment ◽  
Wave Speeds ◽  
Shear Wave Speed ◽  
Constitutive Properties

The use of shear wave propagation to noninvasively gauge material properties and loading in tendons and ligaments is a growing area of interest in biomechanics. Prior models and experiments suggest that shear wave speed primarily depends on the apparent shear modulus (i.e., shear modulus accounting for contributions from all constituents) at low loads, and then increases with axial stress when axially loaded. However, differences in the magnitudes of shear wave speeds between ligaments and tendons, which have different substructures, suggest that the tissue’s composition and fiber alignment may also affect shear wave propagation. Accordingly, the objectives of this study were to (1) characterize changes in the apparent shear modulus induced by variations in constitutive properties and fiber alignment, and (2) determine the sensitivity of the shear wave speed-stress relationship to variations in constitutive properties and fiber alignment. To enable systematic variations of both constitutive properties and fiber alignment, we developed a finite element model that represented an isotropic ground matrix with an embedded fiber distribution. Using this model, we performed dynamic simulations of shear wave propagation at axial strains from 0% to 10%. We characterized the shear wave speed-stress relationship using a simple linear regression between shear wave speed squared and axial stress, which is based on an analytical relationship derived from a tensioned beam model. We found that predicted shear wave speeds were both in-range with shear wave speeds in previous in vivo and ex vivo studies, and strongly correlated with the axial stress (R2 = 0.99). The slope of the squared shear wave speed-axial stress relationship was highly sensitive to changes in tissue density. Both the intercept of this relationship and the apparent shear modulus were sensitive to both the shear modulus of the ground matrix and the stiffness of the fibers’ toe-region when the fibers were less well-aligned to the loading direction. We also determined that the tensioned beam model overpredicted the axial tissue stress with increasing load when the model had less well-aligned fibers. This indicates that the shear wave speed increases likely in response to a load-dependent increase in the apparent shear modulus. Our findings suggest that researchers may need to consider both the material and structural properties (i.e., fiber alignment) of tendon and ligament when measuring shear wave speeds in pathological tissues or tissues with less well-aligned fibers.

Two-Dimensional Wave Propagation in Realistic Viscoelastic Materials

1965 ◽  
Vol 32 (2) ◽  
pp. 303-314 ◽  
Author(s):  
R. J. Arenz
Keyword(s):  
Wave Propagation ◽  
Wave Speed ◽  
Broad Band ◽  
Series Representation ◽  
Steady Motion ◽  
Two Dimensional ◽  
Stress Distributions ◽  
Response Characteristics ◽  
The Laplace Transform ◽  
Inversion Procedure

A solution method using realistic (broad-band) viscoelastic response characteristics covering approximately ten decades of logarithmic time is presented for wave propagation in two-dimensional geometries. A Dirichlet series representation of the viscoelastic mechanical properties and a numerical collocation inversion procedure overcome many of the computational difficulties associated with the Laplace-transform approach to dynamic linear viscoelasticity and also afford a complete time solution. Stress distributions are given for two complementary cases, (a) the viscoelastic half-space subjected on the upper surface to the steady motion of a step pressure load moving supersonically relative to any wave speed in the material, and (b) the semi-infinite thin plate similarly loaded on its edge.

scholarly journals BEM ANALYSIS OF WAVE PROPAGATION IN POROVISCOELASTIC LAYERED HALFSPACE AND HALFSPACE WITH CAVITY

2020 ◽  
Vol 82 (3) ◽  
pp. 364-376
Author(s):  
A.A. Ipatov
Keyword(s):  
Wave Propagation ◽  
Boundary Value Problem ◽  
Laplace Transform ◽  
Boundary Element ◽  
Half Space ◽  
Viscoelastic Material ◽  
Boundary Value ◽  
Heaviside Function ◽  
Biot Model ◽  
The Laplace Transform

The paper is dedicated to the wave propagation a porous-viscoelastic material. As a mathematical model of a fully saturated poroelastic medium, we consider the Biot model with four basic functions – pore pressure and skeleton movements. The Biot model is supplemented by the principle of elastic and viscoelastic reaction correspondence. The skeleton of a porous material is assumed to be viscoelastic material. A model of a standard viscoelastic solid is spplied to describe the viscoelastic properties of a skeleton. The initial boundary-value problem is reduced to a boundary-value problem by formal application of the Laplace transform. To solve boundary integral equations, the boundary element method is performed. Quadrangular eight-node biquadratic elements are used for boundary element discretization. Numerical integration is carried out according to Gaussian quadrature formulas using algorithms for lowering the order and eliminating features. To obtain a solution in explicit time, numerical inversion of the Laplace transform is applied based on the Durbin algorithm with a variable frequency step. This study is a development of the existing boundary-element technique for solving problems on layered porous-elastic half-spaces. This will allow you to take into account the heterogeneity of the soil in depth. The problem of the action of a vertical force in the form of the Heaviside function on the surface of a layered porous-elastic half-space and a half-space with a cavity is considered. Variants of a homogeneous and heterogeneous half-space are considered. Under the model of heterogeneity we understand the piecewise homogeneous solid. The responses of the boundary displacements on the surface of the half-space are presented. The effect of the viscoelastic material model parameter on the dynamic response of displacements is demonstrated. It is established that the viscosity parameters have a significant effect on the nature of the distribution of parameters of wave processes.

The Laplace transform induced by the deformed exponential function of two variables

2018 ◽  
Vol 21 (3) ◽  
pp. 775-785
Author(s):  
Predrag M. Rajković ◽  
Miomir S. Stanković ◽  
Sladjana D. Marinković
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Kernel Function ◽  
Exponential Function ◽  
Liouville Type ◽  
Type Integral ◽  
The Laplace Transform ◽  
Function Of Two Variables ◽  
Differential And Integral Equations

Abstract Based on the easy computation of the direct transform and its inversion, the Laplace transform was used as an effective method for solving differential and integral equations. Its various generalizations appeared in order to be used for treating some new problems. They were based on the generalizations and deformations of the kernel function and of the notion of integral. Here, we expose our generalization of the Laplace transform based on the so-called deformed exponential function of two variables. We point out on some of its properties which hold on in the same or similar manner as in the case of the classical Laplace transform. Relations to a generalized Mittag-Leffler function and to a kind of fractional Riemann-Liouville type integral and derivative are exhibited.

scholarly journals The Solutions of Some Riemann–Liouville Fractional Integral Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 154
Author(s):  
Karuna Kaewnimit ◽  
Fongchan Wannalookkhee ◽  
Kamsing Nonlaopon ◽  
Somsak Orankitjaroen
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Exponential Function ◽  
Fractional Integral ◽  
Laplace Transform Technique ◽  
The Laplace Transform

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form I0+3σy(t)+a·I0+2σy(t)+b·I0+σy(t)+c·y(t)=f(t), where I0+σ is the Riemann–Liouville fractional integral of order σ=1/3,1,f(t)=tn,tnet,n∈N∪{0},t∈R+, and a,b,c are constants, by using the Laplace transform technique. We obtain solutions in the form of Mellin–Ross function and of exponential function. To illustrate our findings, some examples are exhibited.

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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