Propagation of Longitudinal Waves in Circularly Cylindrical Bone Elements

1971 ◽  
Vol 38 (3) ◽  
pp. 578-584 ◽  
Author(s):  
J. L. Nowinski ◽  
C. F. Davis
Keyword(s):  
Linear Equations ◽  
Elastic Solid ◽  
Variable Coefficients ◽  
External Loading ◽  
Regular Singular Point ◽  
Two Phase ◽  
Longitudinal Waves ◽  
Living Bone ◽  
Poroelastic Material ◽  
Biot’S Equations

Two-phase poroelastic material is taken as a model of the living bone in the sense that the osseous tissue is treated as a linear isotropic perfectly elastic solid, and the fluid substances filling the pores as a perfect fluid. Using Biot’s equations, derived in his consolidation theory, four coupled governing differential equations for the propagation of harmonic longitudinal waves in circularly cylindrical bars of poroelastic material are derived. A longer manipulation reduces the task of solution to a single ordinary differential equation with variable coefficients and a regular singular point. The equation is solved by Frobenius’ method. Three boundary conditions on the curved surface of the bar, expressing the absence of external loading and the permeability of the surface, supply a system of three linear equations in three unknown coefficients. A nontrivial solution of the system gives two phase velocities of propagation of longitudinal waves in agreement with the finding of Biot for an infinite medium. A simplification to the purely elastic case yields the elementary classical result for the longitudinal waves.

A Two-Phase Algorithm for the Chebyshev Solution of Complex Linear Equations

1994 ◽  
Vol 15 (6) ◽  
pp. 1440-1451
Author(s):  
Dirk P. Laurie ◽  
Lucas M. Venter
Keyword(s):  
Linear Equations ◽  
Two Phase ◽  
Complex Linear

On the Natural Lubrication of Synovial Joints: Normal and Degenerate

1977 ◽  
Vol 99 (2) ◽  
pp. 163-172 ◽  
Author(s):  
Joseph M. Mansour ◽  
Van C. Mow
Keyword(s):  
Articular Cartilage ◽  
Solid Phase ◽  
Articular Surface ◽  
Moving Load ◽  
Frictional Resistance ◽  
Elastic Solid ◽  
Surface Load ◽  
Two Phase ◽  
Tissue Properties ◽  
Synovial Joints

Fluid flow and mass transport mechanisms associated with articular cartilage function are important biomechanical processes of normal and pathological synovial joints. A three-layer permeable, two-phase medium of an incompressible fluid and a linear elastic solid are used to model the flow and deformational behavior of articular cartilage. The frictional resistance of the relative motion of the fluid phase with respect to the solid phase is given by a linear diffusive dissipation term. The subchondral bony substrate is represented by an elastic solid. The three-layer model of articular cartilage is chosen because of the known histological, ultrastructural, and biomechanical variations of the tissue properties. The calculated flow field shows that for material properties of normal healthy articular cartilage the tissue creates a naturally lubricated surface. The movement of the interstitial fluid at the surface is circulatory in manner, being exuded in front and near the leading half of the moving surface load and imbibed behind and near the trailing half of the moving load. The flow fields of healthy tissues are capable of sustaining a film of fluid at the articular surface whereas pathological tissues cannot.

On the correctness of a phenomenological model of equilibrium phase transitions in a deformable elastic medium

1992 ◽  
Vol 3 (2) ◽  
pp. 181-191
Author(s):  
A. M. Meirmanov ◽  
N. V. Shemetov
Keyword(s):  
Mathematical Model ◽  
Continuous Medium ◽  
Equilibrium Phase ◽  
Elastic Solid ◽  
Complementary Energy ◽  
Structure Of Solutions ◽  
Two Phase ◽  
Pure Phase ◽  
The Mathematical Model ◽  
Uniqueness Of A Solution

In this paper we investigate the mathematical model of the equilibrium of a finite volume in ℝn (n = 1,2, 3) of a two-phase continuous medium, under the assumption that each pure phase is an isotropic elastic solid. The main results in this paper are:(i) the existence and uniqueness of a solution of this mathematical model;(ii) a discussion of the stress-strain law associated with the free energy of this two-phase continuous medium, which is multiple-valued due to the non-smoothness of the Gibbs potential (complementary energy);(iii) a description of the structure of solutions in plane strain.

Plane Deformations of an Inhomogeneity–Matrix System Incorporating a Compressible Liquid Inhomogeneity and Complete Gurtin–Murdoch Interface Model

2018 ◽  
Vol 85 (12) ◽  
Author(s):  
Ming Dai ◽  
Min Li ◽  
Peter Schiavone
Keyword(s):  
Analytic Solution ◽  
Elastic Solid ◽  
Compressible Liquid ◽  
External Loading ◽  
Interface Model ◽  
Interface Tension ◽  
Soft Solids ◽  
Remote Loading ◽  
Interface Elasticity ◽  
Plane Deformations

We consider the plane deformations of an infinite elastic solid containing an arbitrarily shaped compressible liquid inhomogeneity in the presence of uniform remote in-plane loading. The effects of residual interface tension and interface elasticity are incorporated into the model of deformation via the complete Gurtin–Murdoch (G–M) interface model. The corresponding boundary value problem is reformulated and analyzed in the complex plane. A concise analytical solution describing the entire stress field in the surrounding solid is found in the particular case involving a circular inhomogeneity. Numerical examples are presented to illustrate the analytic solution when the uniform remote loading takes the form of a uniaxial compression. It is shown that using the simplified G–M interface model instead of the complete version may lead to significant errors in predicting the external loading-induced stress concentration in gel-like soft solids containing submicro- (or smaller) liquid inhomogeneities.

Wavy Motion of Viscous Bubbly Liquid in Tubes of Orthotropic Material

2012 ◽  
Vol 2 (1) ◽  
pp. 39-47
Author(s):  
Rafael Yusif Amenzadeh ◽  
Akperli Reyhan Sayyad ◽  
Faig Bakhman Ogli Naghiyev
Keyword(s):  
Orthotropic Material ◽  
Volume Fraction ◽  
Linear Equations ◽  
Bubbly Liquid ◽  
Air Bubbles ◽  
Two Phase ◽  
One Dimensional ◽  
Wave Characteristics ◽  
Infinite Tube ◽  
Limited Process

This article investigates the pulsating flow of a compressible two-phase bubble of viscous fluid contained in an elastic orthotropicle direct axis tube. In this work, one-dimensional linear equations have been used. It is assumed that the tube is rigidly attached to the certain environment. In the case of finite length the pressure is applied at the end of its faces. In the limited process, relations obtained for a very long tube. Such a description, in a sense generalizes and strengthens the work of this type. In the numerical experiment a semi-infinite tube with flowing water containing small amount of air bubbles is considered. The influence of volume fraction of bubbles on wave characteristics is determined.

scholarly journals A general existence result for isothermal two-phase flows with phase transition

2019 ◽  
Vol 16 (04) ◽  
pp. 595-637
Author(s):  
Maren Hantke ◽  
Ferdinand Thein
Keyword(s):  
Phase Transition ◽  
Euler Equations ◽  
Equations Of State ◽  
Linear Equations ◽  
Two Phase Flows ◽  
Kinetic Relation ◽  
Two Phase ◽  
Uniqueness Results ◽  
Wide Range ◽  
Isothermal Euler Equations

Liquid–vapor flows with phase transitions have a wide range of applications. Isothermal two-phase flows described by a single set of isothermal Euler equations, where the mass transfer is modeled by a kinetic relation, have been investigated analytically in [M. Hantke, W. Dreyer and G. Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71(3) (2013) 509–540]. This work was restricted to liquid water and its vapor modeled by linear equations of state. The focus of this work lies on the generalization of the primary results to arbitrary substances, arbitrary equations of state and thus a more general kinetic relation. We prove existence and uniqueness results for Riemann problems. In particular, nucleation and cavitation are discussed.

Seismic Response of Dam-Sediment-Water on Fluid-Solid Coupling

2013 ◽  
Vol 353-356 ◽  
pp. 2645-2651
Author(s):  
Yi Zhi Yan ◽  
Chang Xin Xiong ◽  
Zhi Min Su
Keyword(s):  
Porous Medium ◽  
Seismic Response ◽  
Compressible Fluid ◽  
Water Reservoir ◽  
Elastic Solid ◽  
Hydrodynamic Pressure ◽  
Calculation Model ◽  
Sediment Thickness ◽  
Two Phase ◽  
Model Based

Studying the important effects of sediments on the seismic response of dams ,This paper established the calculation model based on regarding the water reservoir as compressible fluid ,the dam and the foundation as an elastic solid, the sediment as Liquid-Solid two-phase porous medium. The results showed that the sediment thickness and properties have important effects on the dam seismic. Increasing the thickness of sediment ,the seismic response of acceleration significantly decreased, the hydrodynamic pressure significantly reduced , which is benefited to the safety of the dam.

scholarly journals Propagation of Plane Waves in a Thermally Conducting Mixture

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Baljeet Singh
Keyword(s):  
Free Surface ◽  
Longitudinal Wave ◽  
Plane Waves ◽  
Generalized Thermoelasticity ◽  
Elastic Solid ◽  
Reflection Coefficients ◽  
Governing Equations ◽  
Transverse Waves ◽  
Longitudinal Waves ◽  
Thermally Conducting

The governing equations for generalized thermoelasticity of a mixture of an elastic solid and a Newtonian fluid are formulated in the context of Lord-Shulman and Green-Lindsay theories of generalized thermoelasticity. These equations are solved to show the existence of three coupled longitudinal waves and two coupled transverse waves, which are dispersive in nature. Reflection from a thermally insulated stress-free surface is considered for incidence of coupled longitudinal wave. The speeds and reflection coefficients of plane waves are computed numerically for a particular model.

scholarly journals Influence of Poroelasticity on the 3D Seismic Response of Complex Geological Media

2017 ◽  
Vol 47 (2) ◽  
pp. 34-60 ◽  
Author(s):  
Frank Wuttke ◽  
Petia Dineva ◽  
Ioanna-Kleoniki Fontara
Keyword(s):  
Elastic Inclusion ◽  
Low Frequency ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Computationally Efficient ◽  
Two Phase ◽  
Computational Environment ◽  
Local Site ◽  
Biot’S Equations ◽  
Water Saturated

AbstractElastic wave propagation in 3D poroelastic geological media with localized heterogeneities, such as an elastic inclusion and a canyon is investigated to visualize the modification of local site responses under consideration of water saturated geomaterial. The extended computational environment herein developed is a direct Boundary Integral Equation Method (BIEM), based on the frequency-dependent fundamental solution of the governing equation in poro-visco elastodynamics. Bardet’s model is introduced in the analysis as the computationally efficient viscoelastic isomorphism to Biot’s equations of dynamic poroelasticity, thus replacing the two-phase material by a complex valued single-phase one. The potential of Bardet’s analogue is illustrated for low frequency vibrations and all simulation results demonstrate the dependency of wave field developed along the free surface on the properties of the soil material.

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