Dynamic stability of nanocomposite viscoelastic cylindrical shells coating with a piezomagnetic layer conveying pulsating fluid flow

1 Introduction

Composites offer advantageous characteristics of different materials with qualities that none of the constituents possess. Smart composite addressed in this paper has attracted more attention amongst researchers due to the provision of new properties and exploiting unique synergism between materials. Piezoelectric materials produce an electric field when deformed, and undergo deformation when subjected to an electric field. The coupling nature of piezoelectric materials has attracted wide applications in electro-mechanical and electrical devices, such as actuators, sensors and transducers [1].

Since this paper studies the dynamic response of a composite cylinder conveying pulsating fluid, the introduction is divided into three parts including magneto-electro-elastic shells, pulsating fluid and composite structures.

Three-dimensional (3D) solutions for the static analysis of doubly curved functionally graded (FG) magneto-electro-elastic shells were presented by Tsai et al. [2]. They used an asymptotic approach to solve 29 basic equations that reduced to 10 differential equations under elastic, electric and magnetic fields. The governing equations were derived by considering the first-order approximation to the 3D magneto-electro-elasticity. Their results could be used as a reference to develop an advanced approximate shell theory and as the benchmark solutions. Wu and Tsai [3] analyzed a 3D free vibration of doubly curved FG magneto-electro-elastic shells. They used closed-circuit surface conditions and a perturbation method. They introduced a method of multiple scales to eliminate the secular terms in various order problems of their formulation. The coupled classical shell theory was derived in this work as a first-order approximation to the 3D magneto-electro-elasticity. Also some benchmark solutions for the free vibration analysis of FG elastic and piezoelectric plates were used to validate the present asymptotic formulation.

Torsional buckling of a piezoelectric polymeric cylindrical shell made from polyvinylidene fluoride (PVDF) subjected to combined electro-thermo-mechanical loads was analyzed by Mosallaie Barzoki et al. [4]. The core was modeled as an elastic medium by Winkler and Pasternak modules and the shell was reinforced by armchair double-walled boron nitride nanotubes. Mechanical, electrical and thermal characteristics of the composite shell were determined based on micromechanical modeling. Their results indicated that buckling strength increases substantially as harder foam cores are employed. Nonlinear dynamics of pipes conveying pulsating fluid was investigated considering the effect of motion constraints modeled as cubic springs by Wang [5]. He derived the partial differential equation with a higher mean flow velocity and using the Galerkin method and fourth-order Runge-Kutta scheme. He found an analytical model to exhibit rich and variegated dynamical behaviors that include quasi-periodic and chaotic motions. Ariaratnam and Namachchivaya [6] assumed that the flow velocity is harmonically perturbed about a constant mean value and analyzed the dynamic stability of supported cylindrical pipes conveying fluid. They used a method of averaging to explicit stability conditions for perturbations and the Floquet theory to extend the stability boundaries. Their results have shown the effects of the mean flow velocity, dissipative forces, boundary conditions and virtual mass on the extent of parametric instability regions.

Also, nonlinear dynamics of pipes conveying pulsatile fluid was investigated by Panda and Kar [7] where the pipe was subjected to combination and principal parametric resonance. In this research, the fluid velocity harmonically varied to a constant mean velocity. They utilized the method of multiple scales to solve first-order ordinary differential governing equations considering the associated boundary conditions. Their results have shown the response in directly excited and indirectly excited modes due to modal interaction. Liang and Su [8] studied the stability of a single-walled carbon nanotube conveying fluid where the internal flow was pulsating and viscous. Based on nonlocal elasticity theory, partial differential equations were obtained using the Galerkin method as well as averaging equations for the first two modes. They discussed the stability regions in the frequency-amplitude plane and the influences of the nonlocal effect, viscosity and some system parameters. Tan and Tong [9] presented the closed-form formulas for the effective constants of composite materials under various loading assumptions using the linear piezoelectric theory and iso-field assumptions. They found that the predicted electro-elastic constants under various loading assumptions are generally bounded by those obtained under single- and multiple-loading assumptions. In fact, they showed that the effective constants along the fiber direction obtained under the single-loading assumption are the same as those evaluated using the rule of mixtures. They also developed in another work [10] two micromechanics models, referred to as “XY PEMFRC model” and “YX PEMFRC model”, to investigate the electro-magneto-thermo-elastic properties for piezoelectric-magnetic fiber reinforced composite (PEMFRC) materials operating in the linear regime. The theoretical and experimental findings in piezoelectric nanostructures were presented by Fang et al. [11], including piezoelectric nanowires, nanoplates, nanobeams, nanofilms, nanoparticles and piezoelectric heterogeneous materials containing piezoelectric nano-inhomogeneities. First, they delineated the types of piezoelectric nanostructured materials and their wide applications. Then, they illustrated the theoretical foundations including the definition of surface stress, electric displacement, surface constitutive relations, surface equilibrium equations and nonlocal piezoelectricity.

This paper aims to study the dynamic stability of a viscoelastic nanocomposite cylindrical shell conveying pulsating fluid flow made of PVDF and reinforced by boron nitride nanotubes (BNNTs) with a viscoelastic piezomagnetic coating layer of iron oxide (CoFe₂O₄). In order to control the stability of the system, the cylindrical shell is subjected to applied electric and magnetic fields in the longitudinal direction. The nanocomposite cylindrical shell is embedded in a viscoelastic medium which is simulated by a visco-pasternak model. The governing equations are derived using Hamilton’s principle, which are then solved by the differential quadrature method (DQM) for clamped boundary conditions. The influences of fluid velocity, geometrical parameters of the shell, viscoelastic foundation, orientation angle and percentage of BNNTs in polymer on the resonance frequency of the shell are investigated.

2 Structural definition

A schematic diagram of a double-layered shell embedded in a viscoelastic foundation is illustrated in Figure 1, in which geometrical parameters such as length L, average radius R and thickness h₁+h₂ are also indicated.

Figure 1:

A schematic diagram of a double-layered shell modeled embedded in a viscoelastic foundation.

As can be seen from Figure 1, the shell consists of two layers:

• Viscoelastic nanocomposite cylindrical shell

• Viscoelastic piezomagnetic coating layer.

From the strain-displacement relations, the stress and moment resultants are obtained separately for each layer. Then the total energy which includes the energies of the viscoelastic nanocomposite cylindrical shell and viscoelastic piezomagnetic coating layer is obtained using Hamilton’s principle in three orthogonal directions of the cylindrical coordinate system.

3 Motion equations

4 Solution procedures

The DQM is used in this study to solve the equations of motion. In this method, differential equations change into first algebraic equations with first order and weighting coefficients. In other words, the partial derivatives of a function (F) are approximated by a specific variable, at discontinuous points in the domain as a set of weighting series and its amount represented by the function itself at that point and other points throughout the domain. Let F be a function representing u̅, v̅, w̅ and Φ with respect to variables ξ and θ in the following domain of (0<ξ<L, 0<θ<2π) having N_ξ ×N_θ grid points along these variables. The nth-order partial derivative of F(ξ, θ) with respect to ξ, the mth-order partial derivative of F(ξ, θ) with respect to θ and the (n+m)th-order partial derivative of F(ξ, θ) with respect to both ξ and θ may be expressed discretely [18] at the point (ξ_i , θ_i) as follows:

where Aik(n) and Bjl(m) are the weighting coefficients associated with the nth-order partial derivative of F(ξ, θ) with respect to ξ at the discrete point ξ_i and the mth-order derivative with respect to θ at θ_i , respectively, whose recursive formulae can be found in Ref. [19]. A more superior choice for the positions of the grid points is Chebyshev polynomials as expressed in [19].

According to the DQM, mechanical clamped and free electrical boundary conditions at both ends of the shell may be written as follows:

Applying weighting coefficients and using Equations (23)–(27), a set of algebraic governing equations are obtained:

where [K], [M] and [C] are the stiffness, mass and damping matrix, respectively. It is assumed that the flow velocity is harmonically fluctuating and has the following form [19]:

where v₀ is the average flow velocity, α is the amplitude harmonic and ω is the frequency.

Substituting the pulsatile fluid velocity in Equation (33), the stiffness matrix coefficients and damping coefficient matrix are separated as follows:

Finally, by substituting Equation (34) in Equation (35):

where [K]^f, [K]^ff and [C]^f are stiffness and damping coefficient matrix of pulsating fluid. The bulletin helper method is used to solve Equation (36). In this way, the displacement vector {d} can be considered as follows [20]: