Wave motion in multi-layered transversely isotropic porous media by the method of potential functions

2019 ◽  
Vol 25 (3) ◽  
pp. 547-572 ◽  
Author(s):  
Hamid Teymouri ◽  
Ali Khojasteh ◽  
Mohammad Rahimian ◽  
Ronald Y S Pak
Keyword(s):  
Differential Equations ◽  
Half Space ◽  
Riemann Surfaces ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Transformation Method ◽  
Harmonic Excitation ◽  
Form Solution ◽  
Potential Functions ◽  
Special Cases

Wave propagation in a multi-layered transversely isotropic porous medium has been considered in this paper, which consists of n parallel layers overlying on a half-space. Potential functions are used to solve elastodynamic differential equations of the poroelastic medium. Time-harmonic excitation is assumed and the procedure of solution is performed in the frequency domain. Generalized reflection and transmission matrices are generated for compressional and shear waves separately. By means of the Hankel transformation method, coupled differential equations are altered to ordinary ones and Riemann surfaces are used to establish the path of integrations. A closed-form solution is described to reach Green’s functions of displacements and stresses. Some special cases of excitations are discussed and verification of the solution is presented. The numerical results of a three-layered medium on a porous half-space are determined and discussed.

Solution of the Contact Problem of a Rigid Conical Frustum Indenting a Transversely Isotropic Elastic Half-Space

2013 ◽  
Vol 81 (4) ◽  
Author(s):  
X.-L. Gao ◽  
C. L. Mao
Keyword(s):  
Contact Problem ◽  
Closed Form ◽  
Half Space ◽  
Cone Angle ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Elastic Half Space ◽  
Form Solution ◽  
Potential Functions ◽  
Displacement Method

The contact problem of a rigid conical frustum indenting a transversely isotropic elastic half-space is analytically solved using a displacement method and a stress method, respectively. The displacement method makes use of two potential functions, while the stress method employs one potential function. In both the methods, Hankel's transforms are applied to construct potential functions, and the associated dual integral equations of Titchmarsh's type are analytically solved. The solution obtained using each method gives analytical expressions of the stress and displacement components on the surface of the half-space. These two sets of expressions are seen to be equivalent, thereby confirming the uniqueness of the elasticity solution. The newly derived solution is reduced to the closed-form solution for the contact problem of a conical punch indenting a transversely isotropic elastic half-space. In addition, the closed-form solution for the problem of a flat-end cylindrical indenter punching a transversely isotropic elastic half-space is obtained as a special case. To illustrate the new solution, numerical results are provided for different half-space materials and punch parameters and are compared to those based on the two specific solutions for the conical and cylindrical indentation problems. It is found that the indentation deformation increases with the decrease of the cone angle of the frustum indenter. Moreover, the largest deformation in the half-space is seen to be induced by a conical indenter, followed by a cylindrical indenter and then by a frustum indenter. In addition, the axial force–indentation depth relation is shown to be linear for the frustum indentation, which is similar to that exhibited by both the conical and cylindrical indentations—two limiting cases of the former.

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

Postbuckling Analysis of a Nonlocal Nanorod Under Self-Weight

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Chinnawut Juntarasaid ◽  
Tawich Pulngern ◽  
Somchai Chucheepsakul
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Nonlocal Elasticity ◽  
Nonlinear Differential Equations ◽  
Closed Form Solution ◽  
Constraint Equation ◽  
Form Solution ◽  
Postbuckling Behavior ◽  
Euler Beam ◽  
Softening Behavior

This paper presents the postbuckled configurations of simply supported and clamped-pinned nanorods under self-weight based on elastica theory. Numerical solution is considered in this work since closed-form solution of postbuckling analysis under self-weight cannot be obtained. The set of nonlinear differential equations of a nanorod including the effect of nonlocal elasticity are investigated. The constraint equation at boundary condition technique is introduced for the solution of postbuckling analysis. In order to solve the set of nonlinear differential equations, the shooting method is utilized, where the set of these equations along with boundary conditions are integrated by the fourth-order Runge-Kutta algorithm. Numerical results are obtained and the highlighting influences of the nonlocal elasticity on postbuckling behavior of nanorods are discussed. The obtained results indicate that the rotation angle and the postbuckled configurations of nanorods are varied by changing the nonlocal elasticity parameter. The effect of nonlocal elasticity shows the softening behavior in comparison with the Euler beam. The present formulation together with constraint boundary condition technique is an effective solution for postbuckling analysis of a nanorod under self-weight including the effect of nonlocal elasticity.

A note on the summation of divergent power series

1969 ◽  
Vol 66 (1) ◽  
pp. 109-114 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Severe Restriction ◽  
Mathematical Problems ◽  
Divergent Power Series ◽  
Euler Transformation ◽  
Direct Use

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

Axisymmetric Boussinesq problem of a transversely isotropic half space with surface effects

2018 ◽  
Vol 24 (5) ◽  
pp. 1425-1437 ◽  
Author(s):  
Jing Jin Shen
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Stiffness ◽  
Surface Elasticity ◽  
Transversely Isotropic ◽  
Surface Effects ◽  
Building Blocks ◽  
Material Anisotropy ◽  
Residual Surface Stress ◽  
Special Cases

A transversely isotropic half space with surface effects subjected to axisymmetric loadings is investigated in terms of the Lekhnitskii formulism. Surface effects including residual surface stress and surface elasticity are introduced by using the Gurtin–Murdoch continuum model. With the aid of the Hankel transforms, solutions corresponding to several different axisymmetic loadings are derived and used to determine the influence of surface effects on contact stiffness in nanoindentations. Numerical results are provided to show the influence of surface effects and material anisotropy on the material behaviours. Meanwhile, the obtained analytical Green’s functions for two special cases can be used as building blocks for further mixed boundary value problems.

Diffraction by a half-plane perpendicular to the distinguished axis of a general gyrotropic medium

1977 ◽  
Vol 55 (4) ◽  
pp. 305-324 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Diffraction Problem ◽  
Plane Waves ◽  
Closed Form Solution ◽  
Basic Feature ◽  
Form Solution ◽  
Electromagnetic Plane Wave ◽  
Edge Diffraction ◽  
Special Cases

An exact, closed-form solution is found for the following half-plane diffraction problem: (I) The medium surrounding the half-plane is both electrically and magnetically gyrotropic. (II) The scattering half-plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The incident electromagnetic plane wave propagates in a direction normal to the edge of the half-plane.The formulation of the problem leads to a coupled pair of Wiener–Hopf equations. These had previously been thought insoluble by quadratures, but yield to a newly discovered technique : the Wiener–Hopf–Hilbert method. A basic feature of the problem is its two-mode character i.e. plane waves of both modes are necessary for the spectral representation of the solution. The coupling of these modes is purely due to edge diffraction, there being no reflection coupling. The solution obtained is simple in that the Fourier transforms of the field components are just algebraic functions. Properties of the solution are investigated in some special cases.

Fundamental Solution via Invariant Approach for a Brain Tumor Model and its Extensions

2014 ◽  
Vol 69 (12) ◽  
pp. 725-732 ◽  
Author(s):  
Andrew G. Johnpillai ◽  
Fazal M. Mahomed ◽  
Saeid Abbasbandy
Keyword(s):  
Brain Tumor ◽  
Partial Differential Equations ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Tumor Models ◽  
Partial Differential ◽  
Equivalence Transformations ◽  
Extended Class

AbstractWe firstly show how one can use the invariant criteria for a scalar linear (1+1) parabolic partial differential equations to perform reduction under equivalence transformations to the first Lie canonical form for a class of brain tumor models. Fundamental solution for the underlying class of models via these transformations is thereby found by making use of the well-known fundamental solution of the classical heat equation. The closed-form solution of the Cauchy initial value problem of the model equations is then obtained as well. We also demonstrate the utility of the invariant method for the extended form of the class of brain tumor models and find in a simple and elegant way the possible forms of the arbitrary functions appearing in the extended class of partial differential equations. We also derive the equivalence transformations which completely classify the underlying extended class of partial differential equations into the Lie canonical forms. Examples are provided as illustration of the results.

scholarly journals The Sustainable Characteristic of Bio-Bi-Phase Flow of Peristaltic Transport of MHD Jeffrey Fluid in the Human Body

2018 ◽  
Vol 10 (8) ◽  
pp. 2671 ◽  
Author(s):  
Ahmed Zeeshan ◽  
Nouman Ijaz ◽  
Tehseen Abbas ◽  
Rahmat Ellahi
Keyword(s):  
Volume Fraction ◽  
Closed Form Solution ◽  
Pressure Rise ◽  
Expansion Method ◽  
Form Solution ◽  
Peristaltic Transport ◽  
Jeffrey Fluid ◽  
Physical Parameters ◽  
Phase Flow ◽  
Special Cases

This study deals with the peristaltic transport of non-Newtonian Jeffrey fluid with uniformly distributed identical rigid particles in a rectangular duct. The effects of a magnetohydrodynamics bio-bi-phase flow are taken into account. The governing equations for mass and momentum are simplified using the fact that wavelength is much greater than the amplitude and small Reynolds number. A closed-form solution for velocity is obtained by means of the eigenfunction expansion method whereby pressure rise is numerically calculated. The results are graphically presented to observe the effects of different physical parameters and the suitability of the method. The results for hydrodynamic, Newtonian fluid, and single-phase problems can be respectively obtained by taking the Hartmann number (M = 0), relaxation time (λ1=0), and volume fraction (C = 0) as special cases of this problem.

Rayleigh waves in transversely isotropic thermoelastic diffusive half-space

2008 ◽  
Vol 86 (9) ◽  
pp. 1133-1143 ◽  
Author(s):  
R Kumar ◽  
T Kansal
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Chemical Potential ◽  
Transversely Isotropic ◽  
Surface Wave Propagation ◽  
Surface Displacements ◽  
Special Cases ◽  
Secular Equations ◽  
The Media ◽  
Straight Lines

The present investigation is devoted to the study of the propagation of Rayleigh waves in a homogeneous, transversely isotropic, thermoelastic diffusive half-space subjected to stress-free, thermally insulated and (or) isothermal, and chemical potential boundary conditions, in the context of the theory of coupled thermoelastic diffusion. Secular equations for surface-wave propagation in the media being considered are derived. The surface-particle paths during the motion are found to be elliptical, but degenerate into straight lines in case where there is no phase difference between the horizontal and vertical components of the surface displacements. The phase velocity; attenuation coefficient; specific loss of energy; and the amplitudes of surface displacements, temperature change, and concentration are computed numerically and presented graphically to depict the anisotropy and diffusion effects. Some special cases of frequency equations are also deduced from the present investigation. PACS Nos.: 62.20.–x, 62.20.D–, 62.20.de, 62.20.dj, 62.20.dq, 62.30.+d, 66.10.C–, 66.10.cd, 66.10.cg, 66.30.–h

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