Dynamic Responses of Magneto-Thermo-Electro-Elastic Shell Structures with Closed-Circuit Surface Condition

INTRODUCTION

Structronics is concept of “Structures+Electronics”, which are synergistic integration of smart, adaptive or responsive materials, that contains the main structure and the distributed functional materials (e.g., piezoelectric, piezomagnetic, electrostrictive, magnetostrictive and alike materials). Furthermore, structronic refer to a class of structures had the capability of simultaneously sensing/actuating, mechanical, electrical, magnetic and even thermal effects, as well as simultaneously generating control forces to eliminate the undesirable effects or to enhance the desirable one. Whereas, Structronics are largely improves the working performance and lifetime of devices that construct from it (Badri and Al-Kayiem, 2012a; Bassiouny, 2006). Several accurate solutions of structronics shell have been presented using 3-D and 2-D theories or the discrete layer approaches. The exact closed-form solutions for multilayered piezo-electric-magnetic and purely elastic plates have been proved for special cases of Pan’s analysis. Heyliger and Pan (2004) demonstrated the free vibration analysis of the simply supported and multilayered Magneto-electro-elastic (MEE) plates under cylindrical bending.

Then, Heyliger et al. (2004) studied two cases of the MEE plates subjected to static fields, one under cylindrical bending and the other of completely traction-free under surface potentials. Following up the previous Stroh formulation, Pan and Han (2005) presented the 3-D solutions of multilayered Functionally Graded (FG) and MEE plates. Wang et al. (2003) proposed a modified state vector approach to obtain 3-D solutions for MEE laminates, based on the mixed formulation of solid mechanics.

By an asymptotic approach, Tsai et al. (2008) studied 3-D static and dynamic behavior of doubly curved FG-MEE shells under the mechanical load, electric displacement and magnetic flux by considered the edge boundary conditions as full simple supports.

In comparison with the recently development of smart shell it could be said that the literature dealing with theoretical work in piezolaminated shell concerning coupled field phenomena in general and in magneto-thermo-electro-elastic (MTEE) in particular, is rather scarce, especially for shear deformation studies.

In this study, a fundamental theory of piezolaminates shell/plates based on the First-order Transversely Shear Deformations Theory (FSDT) will be developed. New issues elicited by the structural lamination, such as the distributions of center deflection over the thickness of shell are addressed.

The results supplied herein are expected to provide a foundation for the investigation of the interactive effects among the thermal, magnetic, electric and elastic fields in thin-walled structures and of the possibility to apply the MTEE adapting.

FOUNDATIONS THEORY

In order to be reasonably self-contained, in what follows, here will summarize the fundamental physical laws that govern the conservation law of electro-magnetic field and they are:

THEORY OF VARIATIONAL PRINCIPLE

The energy functional are important for their use in approximate methods as well as deriving a consistent set of equations of motion coupled with free charge equations and its boundary conditions (Reddy, 1984; Bao, 1996; Tzou et al., 2004; Badri and Al-Kayiem, 2012b). In summary, the total energy of a shell element is defined as:

(1)

where, P is total potential energy:

(2)

where, Q (δ_i, ε_i, g₁, T), t (δ_i, ε_i, g₁) and W (δ_i, ε_i, g₁) are the thermodynamic potential “Gibbs free energy”, tractions and the work done by body force, electrical and magnetic charge, respectively. Moreover, the kinetic energy is:

(3)

Substituting Eq. 2 and 3 into Eq. 1 yields:

The kinetic energy of the shell can be expressed as:

(4)

Based on the conservation laws of electro-magnetic field, the linear thermodynamic potential energy Q for quasi-static infinitesimal reversible system, subject to mechanical, electric, magnetic and thermal influences from its surroundings, can be approximated by:

Means that and are the dependent variables of Q, while , , x₁ and are the natural independent variables. In order to obtain the thermodynamic potential for which these variables are natural, is performed (Perez-Fernandez et al., 2009), that is:

where, Q is commonly known as Gibbs free energy, the superscripts indicate that the magnitudes must be kept constant when measuring them in the laboratory frame. The constitutive relations can be expressed formally by differentiation of Q corresponding to each dependent variable as:

(5)

Then the total thermodynamic potential is given by:

(6)

While the tractions are:

(7)

Moreover, the external study is:

(8)

where, F^S_α, F^S_R and F^S_n, are the distributed forces in α, β and ζ directions, respectively and C^s_α and C^S_R are the distributed couples about the middle surface of the shell. In addition F^ε, C^ε, F^/ and C^/ are the distributed forces and couples due to electrical and magnetic charge.

Substituting Eq. 6-8 in Eq. 2 and equating the resulted equation with Eq. 1, yields the equations of motion of piezolaminated shell as shown in Eq. 9 below.

Note that, the kinetic relations (i.e., the force and moment resultants per unit length at the boundary Ω) are obtained by integrating the stresses over the shell thickness as in Eq. 10.

(9)

Not that, the temperature τ is a known function of position. Thus, temperature field enter the formulation only through constitutive equations. While I₁, I₂ and I₃ are, the inertia terms and they define as:

And:

where, (I^k) is the mass density of the kth layer of the shell per unit midsurface area. While the energy expressions described above are used to derive the equations of motion:

(10)

Also, can rewrite Eq. 10 in term of constitutive relations Eq. 5 directly as that expressed below in Eq. 11.

Thus, the constitutive terms in Eq. 9 could be replaced by the kinetic relations Eq. 11 for a reason of casting the equation of motion to be dependent of forces and moment resultant as well as to reduce the volume integral to double integral.

By recasting Eq. 9 to put in the familiar form, the governing equations of motion and the equation charge equilibrium for first-order shearable deformation case could be derived based on the fundamental Lemma of calculus of variations; e.g., by integrating the field gradients by parts to relieve the virtual fields and setting its coefficients to zero individually.

EQUATIONS OF MOTION

In order to solve the resulted equation of motion, we introduce the following assumptions to cast the equation of motion in thick (or shear deformation) shell theories. Furthermore, the deepness (or shallowness) of the shell, is also one criterion used in developing shell equations (Badri and Al-Kayiem, 2012c).

Thus, shell is referred to as a shallow, when it has infinity R_αβ and the term (1+ζ/R_i) = 1: Where R_i is either of the curvature parameter R_α, R_β, or R_αβ (Qatu, 2004). If it is represented by the plane coordinate systems for the case of rectangular orthotropy, this leads to constant Lame parameters (i.e., A, B = 1). In additional, the radii of curvature are assumed very large compared to the in-plane displacements. i.e., u_i/R_i = 0, where i = α,β and α, u_i = u_o, or v_o.

Hence, the procedure outlined above, is valid irrespective of using the Navier solution. The Navier-type solution can be applied to obtain exact solution as (K_ii + λ² M_ii) {Δ} = {F}, which is an eigenvalue problem. For nontrivial solution, the determinant of the matrix in the parenthesis is set to zero. Then the configuration of K_ii terms for SS-1, cross-ply and rectangular plane form is listed in the Appendix.