The effect of initial stress and magnetic field on wave propagation in human dry bones
© Mahmoud et al.; licensee Springer. 2014
Received: 1 February 2014
Accepted: 11 April 2014
Published: 27 May 2014
The aim of the present paper is to study the influence of initial stress and magnetic field on the propagation of harmonic waves in a human long dry bone as transversely isotropic material, subject to the boundary conditions that the outer and inner surfaces are traction free. The equations of elastodynamics are solved in terms of displacements. The natural frequency of plane vibrations in the case of harmonic vibrations has been obtained. The frequencies and phase velocity are calculated numerically, the effects of initial stress and magnetic field are discussed. Comparisons are made with the result in the absence of initial stress and magnetic field.
The investigation of wave propagation over a continuous medium has very important application in the fields of engineering, medicine and in bioengineering. Application of the poroelastic materials in medicinal fields such as cardiovascular, dental and orthopedics is well known. The dry bone is piezoelectric in the classical sense [1, 2], i.e., mechanical stress results in electric polarization (the indirect effect); and an applied electric field causes strain (the converse effect). Since that time, many others have confirmed the capacity of bones to produce piezoelectric potentials . Electrical properties of bone are relevant not only as a hypothesized feedback mechanism for bone adaptation and remodeling, but also in the context of external electrical stimulation of bone in order to aid its healing and repair . In orthopedics, the propagation of wave over bone is used in monitoring the rate of fracture healing. There are two types of osseous tissue such as trabecular or cancellous and cortical or compact bone, which are of different materials with respect to their mechanical behavior. In macroscopic terms, the porosity percentage in the compact bone is 3-5%, whereas in the cancellous or trabecular the porosity percentage is up to 90% .
Mahmoud [1, 5, 6] investigated the wave propagation under the effects of initial stress, rotation and magnetic field in cylindrical poroelastic bones, a granular medium and a porous medium. Theoretical analyses of bone piezoelectricity may be relevant to the issue of bone remodeling. Recent thorough studies have explored electromechanical effects in wet and dry bone [7, 8]. They suggest that two different mechanisms are responsible for these effects: classical piezoelectricity mainly due to the molecular asymmetry of collagen in dry bone and streaming potentials found in moist or living bone and generated by the flow of a liquid across charged surfaces. The second mechanism was argued by dielectric measurements, and it was suggested that the electromechanical effect in wet (fluid saturated) bone is not due to a piezoelectric effect . Abd-Alla and Mahmoud [10, 11] solved a magneto-thermoelastic problem in a rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model and investigated analytical solution of wave propagation in non-homogeneous orthotropic rotating elastic media. Abd-Alla et al.  studied the propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under the influence of gravity field. Honarvarla et al. , Ding et al.  studied the elasticity of transversely isotropic materials. Chen et al. [15, 16] investigated the free vibration and general solution of non-homogeneous transversely isotropic magneto-electroelastic hollow cylinders. Abd-Alla et al. [17, 18] studied the problem of transient coupled thermoelasticity of an annular fin and the problem of radial vibrations in a non-homogeneous isotropic cylinder under the influence of initial stress and magnetic field. Mofakhami et al.  studied the finite cylinder vibrations with different end boundary conditions. Abd-Alla et al. [20, 21] studied the effect of rotation, magnetic field and initial stress on peristaltic motion of micropolar fluid and investigated the effect of rotation on a non-homogeneous infinite cylinder of orthotropic material.
In this paper, the equations of elastodynamics for transversely isotropic material under the effect of initial stress and magnetic field are solved in terms of displacement potentials. Also, this paper is concerned with the determination of phase velocity and the eigenvalues of natural frequency of plane vibrations of bones under the effect of initial stress and magnetic field for different boundary conditions in the cases of harmonic vibrations. The numerical results of the frequency equation are discussed in detail for transversely isotropic material and the effect of initial stress and magnetic field for different cases is indicated by figures.
2 Formulation of the problem
Consider a homogeneous and transversely isotropic long bone as a hollow cylinder of inner radius a and outer radius b taking the cylindrical polar coordinates such that the z-axis points vertically upward along the bone axis.
3 Solution of the problem
where and are the Bessel functions of the first order. In the following section, solutions of hollow circular cylinders with three different boundary conditions are performed.
4 Boundary conditions and frequency equation
5 Numerical results and discussion
For the numerical calculation of dimensionless frequency and phase velocity under the effect of initial stress and magnetic field , one shall investigate the frequency equations given by (21) numerically for a particular model. Since these equations are an implicit function relation of dimensionless frequency, one can proceed with finding the variation of frequency with ratio h. Once the frequency has been computed, the corresponding effect of initial stress and magnetic field on the frequency equations for dimensionless frequency (the eigenvalues) can be studied by taking values of ratio h. As an illustrative purpose of the foregoing solutions, the cylinder has the following geometric and material constants which are used in the computations given by [1, 2, 14]: , , , , , .
This study has presented the effect of initial stress and magnetic field on surface wave dispersion in bone. The phase velocity and the dimensionless frequency for this problem are obtained from the dimensionless frequency equation. A numerical method has been presented for obtaining the estimates of phase velocity and dimensionless frequencies of vibration of transversely isotropic bone using the half-interval method. The eigenvalues are calculated for different cases and compared with those reported in the absence of initial stress and magnetic field . The effects of initial stress and magnetic field on the dimensionless frequencies and the phase velocity were indicated by figures.
This article (project) was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-075-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.
- Mahmoud SR: Wave propagation in cylindrical poroelastic dry bones. Appl. Math. Inf. Sci. 2010, 4(2):209-226.MathSciNetGoogle Scholar
- Abd-Alla AM, Mahmoud SR, Abo-Dahab SM: Wave propagation modeling in cylindrical human long wet bones with cavity. Meccanica 2011, 46(6):1413-1428. 10.1007/s11012-010-9398-5MathSciNetView ArticleGoogle Scholar
- Ramtani S: Electro-mechanics of bone remodeling. Int. J. Eng. Sci. 2008, 46: 1173-1182. 10.1016/j.ijengsci.2008.06.001MathSciNetView ArticleGoogle Scholar
- Ramtani S, Zidi M: A theoretical model of the effect of continuum damage on a bone adaptation model. J. Biomech. 2001, 34: 471-479. 10.1016/S0021-9290(00)00215-3View ArticleGoogle Scholar
- Mahmoud SR: Influence of rotation and generalized magneto-thermoelastic on Rayleigh waves in a granular medium under effect of initial stress and gravity field. Meccanica 2012, 47(7):1561-1579. 10.1007/s11012-011-9535-9MathSciNetView ArticleGoogle Scholar
- Mahmoud SR: Effect of rotation and magnetic field through porous medium on peristaltic transport of a Jeffrey fluid in tube. Math. Probl. Eng. 2011., 2011: Article ID 971456Google Scholar
- Nickerson DP, Smith NP, Hunter PJ: A model of cardiac cellular electromechanics. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2001, 359(1783):1159-1172. 10.1098/rsta.2001.0823View ArticleGoogle Scholar
- Kohl P, Sachs F: Mechanoelectric feedback in cardiac cells. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2001, 359(783):1173-1185.View ArticleGoogle Scholar
- Eringen AC: Electromagnetic theory of microstretch elasticity and bone modeling. Int. J. Eng. Sci. 2004, 42: 231-242. 10.1016/S0020-7225(03)00288-XMathSciNetView ArticleGoogle Scholar
- Abd-Alla AM, Mahmoud SR: Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model. Meccanica 2010, 45(4):451-462. 10.1007/s11012-009-9261-8MathSciNetView ArticleGoogle Scholar
- Abd-Alla AM, Mahmoud SR: Analytical solution of wave propagation in non-homogeneous orthotropic rotating elastic media. J. Mech. Sci. Technol. 2012, 26(3):917-926. 10.1007/s12206-011-1241-yView ArticleGoogle Scholar
- Abd-Alla AM, Mahmoud SR, Abo-Dahab SM, Helmi MIR: Propagation of S -wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field. Appl. Math. Comput. 2011, 217(9):4321-4332. 10.1016/j.amc.2010.10.029MathSciNetView ArticleGoogle Scholar
- Honarvarla F, Enjilela E, Sinclair A, Mirnezami S: Wave propagation in transversely isotropic cylinders. Int. J. Solids Struct. 2007, 44: 5236-5246. 10.1016/j.ijsolstr.2006.12.029View ArticleGoogle Scholar
- Ding HJ, Chen WQ, Zhang L: Elasticity of Transversely Isotropic Materials. Springer, Berlin; 2006.Google Scholar
- Chen WQ, Lee KY, Ding HJ: On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic hollow cylinders. J. Sound Vib. 2005, 279: 237-251. 10.1016/j.jsv.2003.10.033View ArticleGoogle Scholar
- Chen WQ, Lee KY, Ding HJ: General solution for transversely isotropic magneto-electro-thermo-elasticity and potential theory method. Int. J. Eng. Sci. 2004, 42: 1361-1379. 10.1016/j.ijengsci.2004.04.002View ArticleGoogle Scholar
- Abd-Alla AM, Mahmoud SR, Abo-Dahab SM: On problem of transient coupled thermoelasticity of an annular fin. Meccanica 2012, 47(5):1295-1306. 10.1007/s11012-011-9513-2MathSciNetView ArticleGoogle Scholar
- Abd-Alla AM, Mahmoud SR: On problem of radial vibrations in non-homogeneity isotropic cylinder under influence of initial stress and magnetic field. J. Vib. Control 2013, 77(12):269-276.MathSciNetGoogle Scholar
- Mofakhami MR, Toudeshky HH, Hashemi SH: Finite cylinder vibrations with different end boundary conditions. J. Sound Vib. 2006, 297: 293-314. 10.1016/j.jsv.2006.03.041View ArticleGoogle Scholar
- Abd-Alla AM, Yahya GA, Mahmoud SR, Alosaimi HS: Effect of the rotation, magnetic field and initial stress on peristaltic motion of micropolar fluid. Meccanica 2012, 47(6):1455-1465. 10.1007/s11012-011-9528-8MathSciNetView ArticleGoogle Scholar
- Abd-Alla AM, Mahmoud SR, AL-Shehri NA: Effect of the rotation on a non-homogeneous infinite cylinder of orthotropic material. Appl. Math. Comput. 2011, 217(22):8914-8922. 10.1016/j.amc.2011.03.077View ArticleGoogle Scholar
- Mahmoud SR, Abd-Alla AM, AL-Shehri NA: Effect of the rotation on plane vibrations in a transversely isotropic infinite hollow cylinder. Int. J. Mod. Phys. B 2011, 25(26):3513-3528. 10.1142/S0217979211100928View ArticleGoogle Scholar
- El-Naggar AM, Abd-Alla AM: On a generalized thermo-elastic problems in an infinite cylinder under initial stress. Earth Moon Planets 1987, 37: 213-223. 10.1007/BF00116637View ArticleGoogle Scholar
- Marin M, Agarwal RP, Mahmoud SR: Nonsimple material problems addressed by the Lagrange’s identity. Bound. Value Probl. 2013., 2013: Article ID 135Google Scholar
- Marin M, Agarwal RP, Mahmoud SR: Modeling a microstretch thermoelastic body with two temperatures. Abstr. Appl. Anal. 2013., 2013: Article ID 583464 10.1155/2013/583464Google Scholar
- Lu H, Sun L, Sun J: Existence of positive solutions to a non-positive elastic beam equation with both ends fixed. Bound. Value Probl. 2012., 2012: Article ID 56Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.