Effect of Rotation on the Propagation of Waves in Hollow Poroelastic Circular Cylinder with Magnetic Field

A.M. Farhan

Effect of Rotation on the Propagation of Waves in Hollow Poroelastic Circular Cylinder with Magnetic Field

A.M. Farhan1,2

. Employing Biot’s theory of wave propagation in liquid saturated porous media, the effect of rotation and magnetic field on wave propagation in a hollow poroelastic circular of infinite extent are investigated. An exact closed form solution is presented. General frequency equations for propagation of poroelastic cylinder are obtained when the boundaries are stress free. The frequencies are calculated for poroelastic cylinder for different values of magnetic field and rotation. Numerical results are given and illustrated graphically. The results indicate that the effect of rotation, and magnetic field are very pronounced. Such a model would be useful in large-scale parametric studies of mechanical response.

Wave propagation, Rotation, Magnetic field, Poroelastic medium, Natural frequency.

1 Introduction

The study of wave propagation over a continuous media is of practical importance in the field of engineering, medicine and bio-engineering. [Abd-Alla, et al. (2016)]investigated the reflection of Plane Waves from studied the electro-magneto-thermoelastic Half-space with a Dual-Phase-Lag Model. [Ahmed and Abd-Alla (2002)] studied the electromechanical wave propagation in a cylindrical poroelastic bone with cavity. [Abd-Alla, et al.(2011)] investigated the wave propagation modeling in cylindrical human long wet bones with cavity. [Abd-Alla and Abo-Dahab (2013)] discussed the effect of magnetic field on poroelastic bone model

for internal remodeling. [Abo-Dahab, et al.(2014)] investigated the effect of rotation on wave propagation in hollow poroelastic circular cylinder. [Abd-Alla and Yahya(2013)] studied the wave propagation in a cylindrical human long wet bone.[Biot(1955)] studied the theory of elasticity and consolidation for a porous anisotropic solid.[Biot(1956)] studied the theory of propagation of elastic waves in a fluid-saturated porous solid. [Brynk, et al. (2011)] investigated the experimental poromechanics of trabecular bone strength: role of Terzaghi's effective stress and of tissue level stress fluctuations. [Cardoso and Cowin (2012)] discussed the role of structural anisotropy of biological tissues in poroelastic wave propagation. [Cui, et al. (1997)] studied the poroelastic solutions of an inclined borehole. Transactions. [Cowin (1999)] studied the bone poroelasticity. [El-Naggar, et al. (2001)] investigated the analytical solution of electro-mechanical wave propagation in long bones. [Gilbert, et al. (2012)]investigated a quantitative ultrasound model of the bone with blood as the interstitial fluid. [Love(1944)] studied a theoretical on the mathematical theorey of elasticity.[Matuszyk and Demkowicz (2014)] found the solution of coupled poroelastic/acoustic/elastic wave propagation problems using automatic-adaptivity . [Misra and Samanta (1984)] studied the wave propagation in tubular bones. [Mathieu, et al. (2012)] investigated the influence of healing time on the ultrasonic response of the bone-implant interface. [Marin, et al. (2015)] discussed the structural continuous dependence in micropolar porous bodies. [Marin (2010)] studied the harmonic vibrations in thermoelasticity of microstretch materials. [Marin, M.(1997)] found the weak solutions in elasticity of dipolar bodies with voids. [Morin and Hellmich (2014)] investigated a multiscale poro-micromechanical approach to wave propagation and attenuation in bone. [Nguyen, et al. (2010)] studied the poroelastic behaviour of cortical bone under harmonic axial loading: A finite element study at the osteonal scale. [Papathanasopoulou, et al. (2002)] investigated a poroelastic bone model for internal remodeling. [Potsika, et al. (2014)] discussed the application of an effective medium theory for modeling ultrasound wave propagation in healing long bones. [Qin, et al. (2005)] studied the thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads. [Shah (2011)]investigated the flexural wave propagation in coated poroelastic cylinders with reference to fretting fatigue. [SHARMA and Marin. M. (2013)] investigated the effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space [Yoon and Katz (1976)] studied the ultrasonic wave propagation in human cortical bone—II. Measurements of elastic properties and microhardness. [Wen (2010)] studied the Meshless local Petrov–Galerkin (MLPG)method for wave propagation in 3D poroelastic solids.

In the present, the wave propagation in a cylindrical poroelastic medium with cavity is studied. The frequency equation for poroelastic medium is obtained. From measurements of the density, angular velocity, and bone thickness, the coefficients of the poroelastic medium may be evaluated. The frequencies are calculated for poroelastic medium is obtained for various values of rotation and magnetic field are given in graphs. The propagation of flexural waves in an infinite cylindrical element which is porous in nature is considered and numerical results are carried out. The results indicate that the effect of magnetic field and rotation are very pronounced.[Parnell, et al. (2012)] studied the analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone.

2 Formulation of the problem

(2) The magnetic stress is

where τijis the average stress of solid, τ is the average stress of fluid per unit of mass, andis the magnetic stress with elastic constants cij,M,Q,R and

The equation of the flow [Papathanasopoulou, et al. (2002)] is

The strains are expressed as

and dilation of the phases asand ε=vi,i.

In general, the stress-strain relation for a piezoelectric body can be written in the following way in matrix notation:

where emkand Ekare, respectively, the piezoelectric strain constants and the component of the electrical field.

The last term in Eq. (5) is ignored in Eq. (2) for simplifying the calculation. But this step can be justified by the results of [Yoon and Katz (1976)], who showed that the piezoelectric stiffening in bones in the ultrasonic wave propagation is negligibly small.

The equations of motion are

where, ρ is the density of the bone,is the rotation vector,is the magnetic field acts normal on the planeand t is the time.

Substituting from equations (1) into equations (6), we obtain

3 Solution of the problem

Let

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

We can write the Eq. (10) in the determinant form:

Evaluating the determinant form, the following equations are obtained:

The general solutions of equation (14) can be obtained by using Mathematica program in terms of the Bessel functions of the first and second kind J and Y respectively as

where αi2are the non-zero roots of the equation

The roots of the equation (16) by using Mathematica program are

Solving equations (17) we obtain diand ei

(17) Solving Eq. (11) we have

4 Frequency equation

The boundary conditions for traction free inner and outer surfaces of the hollow poroelastic cylinder are

Equation (20) is called the characteristic frequency equation. The elementis analytically expressed in terms of the elastic constants of the material. Eq. (20) is a transcendental equation of the frequency and wave number. The roots of Eq. (20)provide the dispersion curves of the guided modes. i.e. the wave number as a function of frequency

Figure 1: Variations of the roots with respect to the rotation

Figure 2: Variations of with respect to the rotation with the variation of

Figure 3: Variations of with respect to the rotation with the variation of

Figure 4: Variations of the determinant with respect to the rotation with the variation of

5 Numerical results and discussion

The numerical results for the frequency equation are computed for the wet bone. Since the frequency equation is transcendental in nature, there are an infinite number of roots for the frequency equation. The results are evaluated in the rangewith the ratio ofand the thickness. The values of the elastic constant of the bone are taken from [5] and the poroelastic constant is evaluated from the expression given by

where c is taken to be zero for the incompressibility for the fluid.

Table 1: The approximate geometry of the femur and the material constants which are used in the computations.

6 Conclusion

The investigation of propagation of wave in hollow poroelastic circular cylinder of infinite extent has led to the following conclusion:

(i) The frequency equation of free vibrations is independent of the nature of surface,rotation, magnetic field and presence of fluid in poroelastic media.

(ii) By comparing figures 1–4, it was found that the frequency equation, wave velocity,and attenuation coefficient have the same behavior in both media; but, with the passage of rotation, magnetic field, density, frequency and thickness, numerical values of frequency in the poroelastic cylinder are large in comparison due to the influences of rotation and magnetic field.

(iii) The frequency equation is obtained by considering the material as transversely isotropic in nature.

(iv)The results presented in this paper should prove to be useful for researchers in material science and designers of new materials and bones.

Appendix A:

Abd-Alla, A.M.; Othman, M.I.A.; Abo-Dahab,S.M.(2016): Reflection of Plane Waves from Electro-magneto-thermoelastic Half-space with a Dual-Phase-Lag Model,CMC: Computers, Materials & Continua, vol.51 (2),pp. 63-79

Ahmed, S. M.; Abd-Alla, A. M.(2002): Electromechanical wave propagation in a cylindrical poroelastic bone with cavity,Applied Mathematics and Computation, vol.pp.133, 257–286.

Abd-Alla, A. M.; Abo-Dahab, S. M.; Mahmoud,S. R.(2011): Wave propagation modeling in cylindrical human long wet bones with cavity,Meccanica,vol. 46,pp.1413–1428.

Abd-Alla, A. M. and Abo-Dahab, S.M. (2013): Effect of magnetic field on poroelastic bone model for internal remodeling,Applied Mathematics and Mechanics,vol. 34, pp.889–906.

Abo-Dahab, S. M.; Abd-Alla, A. M. and Alqosami, S.(2014): Effect of Rotation on Wave Propagation in Hollow Poroelastic Circular Cylinder,Mathematical Problems in Engineering, vol. 2014 pp.16.

Abd-Alla, A. M.; Yahya, G. A.(2013): Wave Propagation in a Cylindrical Human Long Wet Bone,J. Comput. Theor. Nanosci., vol. 10, 750-755, (2013).

Biot, M. A.(1955): Theory of elasticity and consolidation for a porous anisotropic solid,”Journal of Applied Physics,vol. 26, no. 2, 182–185, (1955).

Biot, M. A.(1956): Theory of propagation of elastic waves in a fluid-saturated porous solid. I: low-frequency range,Acoustical Society of America,Vol. 28, 168–178.

Brynk, T.; Hellmich, C.; Fritsch, A.; Zysset, P.; Eberhardsteiner, J.(2011):Experimental poromechanics of trabecular bone strength: Role of Terzaghi's effective stress and of tissue level stress fluctuations,Journal of Biomechanics, Vol. 44, no. 3,pp.501-508.

Cardoso, L.; Cowin, S. C. (2012): Role of structural anisotropy of biological tissues in poroelastic wave propagation,Mechanics of Materials, vol. 44, pp.174-188.Cui, L. A.; Cheng, H. D. and Abousleiman,Y.(1997): Poroelastic solutions of an inclined borehole. Transactions of ASME,Journal of Applied Mechanics,vol. 64,pp.32–38.

Cowin, S. C.(1999): Bone poroelasticity.Journal of Biomechanics, vol. 32,pp.217–238.

El-Naggar, A. M.; Abd-Alla, A. M.; Mahmoud, S. R.(2001): Analytical solution of electro-mechanical wave propagation in long bones,Applied Mathematics and Computation, vol. 119, pp.77–98.

Gilbert, R. P.; Guyenne, P.; Ou, M. Y.(2012): A quantitative ultrasound model of the bone with blood as the interstitial fluid,Mathematical and Computer Modelling,vol. 55, pp.2029–2039.

Love, A. E.H.(1944): A theoretical on the mathematical theorey of elasticity.Dover,New York.

Matuszyk, P. J.; Demkowicz, L. F.(2014): Solution of coupled poroelastic/acoustic/elastic wave propagation problems using automatic-adaptivity,Computer Methods in Applied Mechanics and Engineering, vol. pp.281,54-80.

Misra, J. C.; Samanta, S. C.(1984): Wave propagation in tubular bones,International Journal of Solids and Structures, vol. 20, no. 1, pp.55-62.

Mathieu, V.; Vayron, R.; Soffer, E.; Anagnostou, F.(2012): Influence of healing time on the ultrasonic response of the bone-implant interface,Ultrasound in Medicine& Biology, vol. 38, no. 4, pp.611-618.

Marin, M.; Abd-Alla, A.M., Raducanu, D.; Abo-Dahab, S.M. (2015): Structural continuous dependence in micropolar porous bodies,CMC: Computers, Materials &Continua,45 (2), pp.107-125

Marin, M.(1997): On weak solutions in elasticity of dipolar bodies with voids,J.Comp.

Appl. Math.,vol. 82 (1-2), PP.291-297.

Morin, C.; Hellmich, C.(2014): A multiscale poromicromechanical approach to wave propagation and attenuation in bone,Ultrasonics,vol. 54, pp.1251-1269.

Nguyen, Vu-Hieu; Lemaire, T.; Naili, S.(2010): Poroelastic behaviour of cortical bone under harmonic axial loading: A finite element study at the osteonal scale,Medical Engineering & Physics,vol. 32, no. 4, pp.384-390.

Papathanasopoulou, V. A.; Fotiadis, D.; Foutsitzi, I. G.; Massalas, C. V. (2002):A poroelastic bone model for internal remodeling,International Journal of Engineering Science, vol. 40, no. 5, pp.511-530.

Potsika, V. T.; Grivas, K. N. ; Protopappas, V. C. ; Vavva, M. G. ; Raum, K.;

Rohrbach, D. ; Polyzos, D.; Fotiadis, D. I.(2014): Application of an effective medium theory for modeling ultrasound wave propagation in healing long bones,Ultrasonics, Vol. 54, no. 5, pp.1219-1230.

Parnell, W.J.; Vu, M. B.; Grimal, Q.; Naili, S.(2012): Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone,Biomechanics and Modeling in Mechanobiology,vol. 11, no. 6, pp.883-901.

Qin, Qing-Hua; Qu, C.; Ye, J.(2005): Thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads,Biomaterials,vol. 26, no. 33,pp.6798-6810.

Sharma, K.; Marin, M.(2013): Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space, U.P.B.Sci. Bull., Series A-Appl. Math. Phys., vol.75(2), pp. 121-132.

SHARMA, A., Marin. M.(2013): Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space, U.P.B.Sci.Bull., Series A-Appl. Math. Phys.,vol.75(2), PP.121-132.

Shah, A.(2011): Flexural wave propagation in coated poroelastic cylinders with reference to fretting fatigue.Journal of Vibration and Control, vol. 17, pp.1049–1064.

Yoon, H. S.; Katz, J. L.(1976): Ultrasonic wave propagation in human cortical bone—II. Measurements of elastic properties and microhardness,Journal of Biomechanics, vol. 9, pp.459-462,9, pp.463-464.

Wen, P. H.(2010): Meshless local Petrov–Galerkin (MLPG) method for wave propagation in 3D poroelastic solids,Engineering Analysis with Boundary Elements,vol. 34, pp.315-323.

1Physics Department, Faculty of Science Jazan University-K.S.A.

2Physics Department, Faculty of Science, Zagazig University, Zagazig,