Elastic Solutions for a Saturated Isotropic Half Space Subjected to a Fluid Line Sink

2013 ◽  
Vol 405-408 ◽  
pp. 275-284 ◽  
Author(s):  
John C.C. Lu
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Mass Conservation ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Cylindrical Coordinate ◽  
Line Sink ◽  
The Mathematical Model

The study derives the closed-form solutions of the long-term elastic consolidation subjected to the fluid line sink in a homogeneous isotropic elastic half space aquifer. The Hankel transform in a cylindrical coordinate system is employed to develop the analytical elastic solutions. Derivations of governing equations are based on the mathematical model of Biots theory of poro-mechanics, and the half space aquifer is modelled as a saturated porous stratum which is bounded by a horizontal surface. The total stresses of the aquifer obey Newtons second law and Hookes law. Besides, the mass conservation and Darcys law are introduced to formulate the governing equations of pore fluid flow. The software Mathematica is used to complete the symbolic integrations and obtain the closed-form solutions. The solutions can be applied in dewatering operations of compressible aquifer.

Closed-Form Solutions of the Elastic Half Space due to a Line Heat Source

2014 ◽  
Vol 638-640 ◽  
pp. 2082-2091
Author(s):  
John C.C. Lu ◽  
Feng Tsai Lin
Keyword(s):  
Heat Source ◽  
Closed Form ◽  
Half Space ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Line Heat Source ◽  
Closed Form Solutions ◽  
Line Sink ◽  
Vertical Displacements ◽  
Thermoelastic Response

Thermoelastic response due to a line heat source is analog to poroelastic reaction caused by a fluid line sink. In this study, the strata are modeled as a thermoelastic or poroelastic half space bounded by horizontal surface in the mathematical model. Thermomechanics and poromechanics are applied on the formulation of basic governing equations, and an analogy is drawn to show the similarity. Using Hankel transform technique and approaching symbolic integral through Mathematica, the closed-form solutions of the horizontal and vertical displacements due to a fluid line sink are obtained. The displacements produced by the line heat source are described through analog quantities between thermoelasticity and poroelasticity. The solutions can be applied to dewater operations and build waste repository.

Approximate Closed-Form Solutions for the Shift Mechanics of Rubber Belt Variators

2009 ◽  
Author(s):  
Marco Cammalleri ◽  
Francesco Sorge
Keyword(s):  
Closed Form ◽  
Mass Conservation ◽  
Experimental Tests ◽  
Speed Ratio ◽  
Equilibrium Equations ◽  
Closed Form Solutions ◽  
Rubber Belt ◽  
Linear Differential ◽  
The Mathematical Model ◽  
Transport Theorem

The mechanical behavior of V-belt variators during the speed ratio shift is different from the steady operation as a gross radial motion of the belt is superimposed to the circumferential motion. The theoretical analysis involves equilibrium equations similar to the steady case, but requires a re-formulation of the mass conservation condition making use of the Reynolds transport theorem. The mathematical model of the belt-pulley coupling implies the repeated numerical solution of a strongly non-linear differential system. Nevertheless, an attentive observation of the numerical diagrams suggests simple and useful closed-form approximations for the four possible working modes of any pulley, opening/closing, driver/driven, whose validity ranges over most practical cases. The present analysis focuses on the development of such simplified solutions, succeeding in an excellent matching with the numerical plots, and on the comparison of the theory with some experimental tests on a motorcycle variator, revealing a very good agreement.

Contact Stress Analysis of a Layered Transversely Isotropic Half-Space

1992 ◽  
Vol 114 (2) ◽  
pp. 253-261 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Region ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Stress Components ◽  
Contact Failure

The three-dimensional problem of contact between a spherical indenter and a multi-layered structure bonded to an elastic half-space is investigated. The layers and half-space are assumed to be composed of transversely isotropic materials. By the use of Hankel transforms, the mixed boundary value problem is reduced to an integral equation, which is solved numerically to determine the contact stresses and contact region. The interior displacement and stress fields in both the layer and half-space can be calculated from the inverse Hankel transform used with the solved contact stresses prescribed over the contact region. The stress components, which may be related to the contact failure of coatings, are discussed for various coating thicknesses.

scholarly journals Surface Displacements due to a Steadily Moving Point Force

1996 ◽  
Vol 63 (2) ◽  
pp. 245-251 ◽  
Author(s):  
J. R. Barber
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Elastic Half Space ◽  
Point Force ◽  
Two Dimensional ◽  
Normal Surface ◽  
Displacement Fields ◽  
Linear Superposition ◽  
Surface Displacements ◽  
Stress And Displacement Fields

Closed-form expressions are obtained for the normal surface displacements due to a normal point force moving at constant speed over the surface of an elastic half-space. The Smirnov-Sobolev technique is used to reduce the problem to a linear superposition of two-dimensional stress and displacement fields.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

Generalized Brinkman Type Dusty Fluid Model for Blood Flow

2020 ◽  
pp. 154-170
Author(s):  
Muhammad Saqib ◽  
Sharidan Shafie ◽  
Ilyas Khan
Keyword(s):  
Blood Flow ◽  
Magnetic Parameter ◽  
Fluid Model ◽  
Cylindrical Tube ◽  
Hankel Transform ◽  
Governing Equations ◽  
Two Phase ◽  
Cylindrical Coordinate ◽  
Long Time ◽  
Fractional Parameter

This chapter is dedicated to studying the magnetic blood flow with uniformly distributed magnetite dusty particles (MDP) in a cylindrical tube. For this purpose, the two-phase fractional Brinkman type fluid model is considered. The fractional governing equations are modeled in the cylindrical coordinate system taking into consideration the magnetization of the fluid due to the applied magnetic field. The fractional governing equations are subjected to physical initial and boundary conditions. The joint Laplace and Hankel transform is employed to develop exact analytical solutions. The obtained solutions are computed numerically and plotted in different graphs. It is noticed that for a long time the blood and MDP velocities increase with increasing values of the fractional parameter. In contrast, this effect reverses for a shorter time. In the case of the magnetic parameter, both velocities are decreased with increasing values of the magnetic parameter.

The In-Plane Loading of a Rigid Disk Inclusion Embedded in an Elastic Half-Space

1991 ◽  
Vol 58 (2) ◽  
pp. 362-369 ◽  
Author(s):  
A. P. S. Selvadurai ◽  
B. M. Singh ◽  
M. C. Au
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Space Region ◽  
Rigid Disk ◽  
Lateral Translation

The paper examines the problem of the in-plane loading of a rigid disk inclusion which is embedded in bonded contact with an isotropic elastic half-space region. The governing coupled integral equations, derived via a Hankel transform technique, are evaluated numerically to generate results for the in-plane stiffness of the rigid disk inclusion and the rotation which accompanies the lateral translation.

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