Abstract

The objective of this paper is to study the effect of rotation on the wave propagation in an infinite poroelastic hollow circular cylinder. The frequency equation for poroelastic hollow circular cylinder is obtained when the boundaries are stress free and is examined numerically. The frequency, phase velocity, and attenuation coefficient are calculated for a pervious surface for various values of rotation, wave number, and thickness of the cylinder which are presented for nonaxial symmetric vibrations for a pervious surface. The dispersion curves are plotted for the poroelastic elastic behavior of the poroelastic material. Results are discussed for poroelastic material. The results indicate that the effect of rotation, wave number, and thickness on the wave propagation in the hollow poroelastic circular cylinder is very pronounced.

1. Introduction

The study of wave propagation over a continuous medium is of practical importance in the fields of engineering, medicine, and bioengineering. Application of the poroelastic materials in medical fields such as orthopedic, dental, and cardiovascular is well known. In orthopedics, wave propagation over bone is used in monitoring the rate of fracture healing. There are two types of osseous tissue such as cancellous or trabecular and compact or cortical bone, which are of different materials, with respect to their mechanical behavior. In macroscopic terms, the percentage of porosity in the cortical bone is 3–5%, whereas, in the trabecular or cancellous, the percentage of porosity is up to 90% [1]. In fact, in a recent article, Ahmed and Abd-Alla [1] investigated the electromechanical wave propagation in a cylindrical poroelastic bone with cavity. Analytical solution of electromechanical wave propagation in long bones has been obtained by El-Naggar et al. [2]. Abd-Alla et al. [3] studied the wave propagation modeling in cylindrical human long wet bones with cavity. Hart [4] investigated the theoretical study of the influence of bone maturation rate on surface remodeling predictions. Qin et al. [5] studied the thermoelectroelastic solutions for surface bone remodeling under axial and transverse loads. Martínez et al. [6] discussed the external bone remodeling through boundary elements and damage mechanics. Computational simulation of simultaneous cortical and trabecular bone change in human proximal femur during bone remodeling has been investigated by Jang and Kim [7]. Tsili [8] studied the theoretical solutions for internal bone remodeling of diaphyseal shafts using adaptive elasticity theory. Cowin and Firoozbakhsh [9] investigated the bone remodeling of diaphyseal surfaces under constant load: theoretical predictions, a contribution to the mechanics and thermodynamics of surface growth. Application to bone external remodeling has been studied by Ganghoffer [10]. Sims and Gooi [11] investigated bone remodeling: multiple cellular interactions required for coupling of bone formation and resorption. Zumsande et al. [12] discussed the general analysis of mathematical models for bone remodeling. Malachanne et al. [13] studied the numerical model of bone remodeling sensitive to loading frequency through a poroelastic behavior and internal fluid movements. A model for mechanical adaptation of trabecular bone incorporating cellular accommodation and effects of microdamage and disuse has been studied by Vahdati and Rouhi [14]. Hazelwood et al. [15] investigated the mechanistic model for internal bone remodeling exhibiting different dynamic responses in disuse and overload. Qu et al. [16] studied the hypothetical mechanism of bone remodeling and modeling under electromagnetic loads. Papathanasopoulou et al. [17] investigated the poroelastic bone model for internal remodeling. Isotropic continuum damage/repair model for alveolar bone remodeling has been studied by Mengoni and Ponthot [18]. Boyle and Kim [19] investigated the three-dimensional microlevel computational study of Wolff’s law via trabecular bone remodeling in the human proximal femur using design space topology optimization. Wang et al. [20] studied the theoretical analysis of alendronate and risedronate effects on canine vertebral remodeling and microdamage. A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling has been discussed by Peterson and Riggs [21]. Qin and Ye [22] studied the thermoelectroelastic solutions for internal bone remodeling investigated under axial and transverse loads. Boyle and Kim [23] studied the comparison of different hip prosthesis shapes considering microlevel bone remodeling and stress-shielding criteria using three-dimensional design space topology optimization. Cowin and Van Buskirk [24] investigated surface bone remodeling induced by a medullary pin. Biot [25] explained the general theory of three-dimensional consolidation. Hegedus and Cowin [26] studied bone remodeling II, small strain adaptive elasticity. The extensive literature on the topic is now available and we can only mention a few recent interesting investigations in [27–33]. Recently, Abd-Alla and Abo-Dahab [34] investigated magnetic field effect on poroelastic bone model for internal remodeling.

In the present analysis, the free vibrations of an infinite hollow poroelastic circular cylinder are studied employing general displacement components in cylindrical polar coordinates, following Biot’s [35] theory. The general frequency hollow cylinder is homogeneous and isotropic. Degenerate cases of the general frequency equation of pervious surfaces, when the longitudinal wave number is considered. The numerical result displayed by figures and the physical meaning are explained. The results and discussions presented in this study may be helpful to further understand wave propagation in hollow poroelastic circular cylinder.

2. Governing Equations

Let us consider that the equations of motion of a homogeneous, isotropic poroelastic solid in the presence of dissipation are where is the Laplace operator, , are displacements of solid and liquid, respectively, and are the dilatations of solid and liquid, , , , and are all poroelastic constants, and are the mass coefficients following Biot [35].

The stress-strain equations for an isotropic poroelastic material of solid and liquid are given as

3. Solution of the Problem

Let us consider to be the cylindrical polar coordinates. Consider a homogeneous, isotropic, poroelastic cylinder with inner and outer radii being and , respectively, so that the thickness of poroelastic cylinder is , whose axis is in the direction of -axis; let where , are displacements of solid and liquid, respectively, , are scalar potentials, and , are vector potentials.

Assuming that from (1) and (3), we get where .

Assuming that where is the axial wave number, is an integer number of waves around the circumference or also known as angular wave number, and is circular frequency. From the second and third equations of (5) with (4), when the first two equations of (5) remain the same, they are reduced to From (7) and (6), we obtain where The general solution of (8) can be obtained in terms of the Bessel function of the first and second kind and depending on its arguments , , .

From (8), we obtain where is Bessel function of the first kind and order and is the Bessel function of the second kind and order . Consider that are positive or negative and where , are the dilatational wave velocities of first and second kind, respectively, and is shear wave velocity.

The gauge invariance property, following the analysis of [36], is used to eliminate two integration constants from (10). Any one of the potential functions , , or can be set equal to zero, without loss of generality of the solution. Setting , we can obtain The displacement vector of solid with the help of (3) and (4) is given by Substituting from (6) into (14), the displacement components of solid are Substituting from (13) into (15), the solid displacement is where “prime” over a quantity denotes differentiation with respect to being the absolute values of , respectively.

The dilatations of solid and liquid media are Substituting from (16) into strain displacement relations and then using (2), the stresses and the liquid pressures are where the coefficients are given as In (23), where we considered