Computers, Materials & Continua DOI:10.32604/cmc.2021.012586 | |

Article |

Wave Propagation Model in a Human Long Poroelastic Bone under Effect of Magnetic Field and Rotation

1Department of Mathematics, Sohag University, Sohag, Egypt

2Department of Statistics, University of Jeddah, College of Science, Jeddah, Saudi Arabia

3Department of Mathematics, South Valley University, Qena, 83523, Egypt

4Department of Engineering Physics and Instrumentation,National Institute of Applied Sciences and Technology, Carthage University, Tunisia

5Department of Physics, Taif University, Taif, Saudi Arabia

6Deanship of Scientific Research, King Abdulaziz University, Jeddah, Saudi Arabia

*Correspondence: A. M. Abd-Alla. Email: mohmrr@yahoo.com

Received: 10 June 2020; Accepted: 20 August 2020

Abstract: This article is aimed at describing the way rotation and magnetic field affect the propagation of waves in an infinite poroelastic cylindrical bone. It offers a solution with an exact closed form. The authors got and examined numerically the general frequency equation for poroelastic bone. Moreover, they calculated the frequencies of poroelastic bone for different values of the magnetic field and rotation. Unlike the results of previous studies, the authors noticed little frequency dispersion in the wet bone. The proposed model will be applicable to wide-range parametric projects of bone mechanical response. Examining the vibration of surface waves in rotating cylindrical, long human bones under the magnetic field can have an impact. The findings of the study are offered in graphs. Then, a comparison with the results of the literature is conducted to show the effect of rotation and magnetic field on the wave propagation phenomenon. It is worth noting that the results of the study highly match those of the literature.

Keywords: Propagation of waves; rotation; magnetic field; poroelastic; wet bone; natural frequency; magnetic field

One of the highly considerable clinical methods for identifying the integrity of bones in vivo is radiographic examination, though, X-ray cannot detect when the loss of a bone decreases less than 30%. By the same token, periodic X-rays can always be utilized in monitoring the healing of fractures although evaluating the healing degree is subjective and often inaccurate. Natall et al. [1] examined bones as a material from a biomechanical perspective. The authors of [2–7] explored various issues related to the propagation of waves within poroelastic cylinders. In regard to a porous anisotropic solid, Biot [8] introduced the theory of elasticity and consolidation. In another study, Biot [9] discussed the theory of elastic wave propagation in a solid that is fluid-saturated and porous. Cardoso et al. [10] investigated the role of the biological tissue structural anisotropy in the poroelastic propagation of waves. The authors of [11] solved issues related to the propagation of coupled poroelastic/acoustic/elastic waves through automatic hp-adaptively. In 3D poroelastic solids, Wen [12] used the meshless local Petrov–Galerkin method for the propagation of waves. Morin et al. [13] investigated the arduous multiscale poromicrodynamics method that is effective in the diverse bone tissues. Employing an iterative active medium approximation, Potsika et al. [14] introduced the model ultrasound propagation of waves in the healing of long bones. The authors of [15] analyzed theoretically the process of internal bone restoration motivated by a medullary pin. Nguyen et al. [16] investigated the performance of the flows of interstitial fluid in cortical bones controlled by axial cyclic harmonic loads that mimic the behavior of in vivo bones while doing daily activities, such as going for a walk. Misra et al. [17] derived the relation of dispersion for axisymmetric acoustic wave propagation along a long composite bone. Qin et al. [18] predicted theoretically the remodeling of the surface bone in the diaphysis of the long bone under different external loads controlled by the theory of adaptive elasticity. Mathieu et al. [19] studied biomechanically the performance of the bone-dental implant interface as an environmental task by taking into account the in silico, in vivo, and ex vivo projects on animal models. Brynk et al. [20] evaluated relevant experimental findings within a microporomechanic theoretical framework. Parnell et al. [21] compared the theoretical estimates of the active elastic moduli of cortical bone at the meso- and macroscales. Shah [22] studied the near-surface condition of stress established under the oscillatory contact between the artificial components that have a considerable role in defining fretting severity. Gilbert et al. [23] investigated the viscous interstitial fluid that plays a role in the ultrasound insonification of non-defatted cancellous bone. The authors of [24] solved analytically the noticeably long borehole in the isotropic and poroelastic medium inclined to the far-field principal stresses. Cowin [25] developed the interaction model of fluid and solid stages of a fluid-saturated porous medium. The effectiveness of bone healing in the ultrasonic reaction of the titanium implants that take the shape of coins and inserted in rabbit tibiae was discussed by Mathieu et al [26]. Singhal et al. [27] investigated the interior restoration of bone by defining the process that enables the bones to have the histological structure to modify within areas of long mechanical load. Kumha [28] investigated the shear wave in a primarily stressed poroelastic medium that has corrugated boundary surfaces inserted between a higher material strengthened with fiber and isotropic inhomogeneous half-space.

Abo-Dahab et al. [29] investigated the analytical solution for surface waves’ remodeling in the long bones under the magnetic field and rotating. Farhan [30] discussed the effect of rotation on the propagation of waves in a hollow poroelastic circular cylinder with a magnetic field. Marin et al. [31] investigated the structural continuous dependence in micropolar porous bodies. Abo-Dahab et al. [32] discussed the effect of rotation on the propagation of waves model in a human long poroelastic bone.

In this paper, the way rotation and magnetic field affect the propagation of waves in an infinite poroelastic cylindrical bone is discussed. The paper provides a solution with an exact closed form. The authors got and examined numerically the general frequency equation of the poroelastic bone. Moreover, they calculated the frequencies of the poroelastic bone for different values of the magnetic field and rotation. Unlike the results of the previous studies, the authors noticed little frequency dispersion in the wet bone. The proposed model will be applicable to wide-range parametric projects of bone mechanical response. Examining the vibration of surface waves in rotating cylindrical, long human bones under the magnetic field can have an impact. The findings of the study are offered in graphs. Then, a comparison with the results of the literature is conducted. It is worth noting that the results of the study highly match those of the literature.

Take into account a hollow cylinder in the form of a geometric approximation to a long bone that is well-defined in the cylindrical coordinates

where

The equation of the fluid is

where

The dilation

The motion equations are

where

Replacing from Eq. (1) into Eq. (5), the result becomes

To obtain a solution to Eq. (7), use the following solution in the field equations

where

Replacing from Eq. (1) into Eqs. (3) and (5) and using Eqs. (6) and (7), the following equations are obtained:

Introducing the parameter as

where

Because the fluid flow through the bone boundaries does not happen while exploring the wave propagation,

where

Estimating the determinant form, we have these equations:

where

The solution of Eq. (10) are

where

where

The solution of Eq. (11) is

where

To have the boundary conditions that are free of traction, stress must disappear on the internal and external surfaces of the hollow cylinder, as follows:

where

Eqs. (8), (15) and (18) together with Eq. (19) and combining

where the coefficients of

The roots of Eq. (20) afford the curves of dispersion of the guided modes, namely the wavenumber as a frequency function.

5 Frequency Equation: Special Cases

The frequency Eq. (20) degenerates into the product of two determinants

where

The terms

If the motion becomes independent of the angular coordinate

The terms

Now, Eq. (23) is satisfied if

5.3 Motion Independent of

If the wavenumbers

The terms

6 Numerical Results and Discussion

The numerical results of the equation of frequency are calculated for the wet bone. The roots are obtained for

where

where

The human bone porosity within the age group 35–40 years is estimated as 0.24 [1]. To evaluate one more poroelastic constant, the following equation is defined

Fig. 1 shows a considerable modification of the absolute value of

Fig. 2 displays various coefficients of

Fig. 3 graphically portrays the variations of the absolute of the coefficients for the poroelastic bone of

Fig. 4 shows the variations of the scalar equation

Fig. 5 graphically irradiates the effect of the variations of the scalar equation

Fig. 6 illustrates the variations of the scalar equation

Fig. 7 displays the variations of the scalar equation