INTRODUCTION
Magneto-thermo-electro-elastic (MTEE) concept is a 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). Which refer to a class
of structures that had the capability of simultaneously sensing/actuating; mechanical,
electrical, magnetic and even thermal effects, as well as simultaneously generating
a control forces to eliminate the undesirable effects or to enhance the desirable
one. Whereas, structronics are largely improving the working performance and
lifetime of devices that construct from it (Bassiouny, 2006;
Badri and Al-Kayiem, 2011a-c).
Several accurate solutions of MTEE plate have been presented using 3-D and 2-D
theories or the discrete layer approaches. The exact closed-form solutions for
multilayered piezoelectric-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 MEE plates under cylindrical bending. Then, 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 and FG 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 functionally graded
MEE shells under the mechanical load, electric displacement and magnetic flux
by consideration the edge boundary conditions as full simple supports. In comparison
with the recent development of 3D solutions of smart plate, we found that the
literature dealing with theoretical work in smart composites plate concerning
coupled field phenomena in general and in MTEE in particular, is rather scarce,
especially for shear deformation studies. In addition, the distribution of sensors
and actuators in the plate structure are not well understood.
In this study, a theory of laminated composite MTEE plates based on the First-order
Transversely Shearable (FSDT) model will be developed. New issues elicited by
the structural lamination, such as the distributions of center deflection over
the thickness of plate 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.
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
equation and the boundary conditions (Reddy, 1984; Bao,
1996; Tzou et al., 2004; Badri
and Al-Kayiem, 2012a-c). In summary, the total energy
of a shell element is defined as:
where, p is total potential energy:
where, Q (si, εj, gl, t), t (si,
εj, gl) and W (si, εj,
gl) 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:
Substituting Eq. 2 and 3 into Eq.
1 yields:
The kinetic energy of the shell can be expressed as:
It is known that, for quasi-static infinitesimal reversible processes, the
linear thermodynamic potential energy Q of a system subject to mechanical, electric,
magnetic and thermal influences from its surroundings, can be approximated by:
where, sij, εk, gl and t are the dependent
variables of Q, while εij, ξk, xl
and τ are the natural independent variables. In order to obtain the thermodynamic
potential for which these variables are natural, is performed by 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:
Then the total thermodynamic potential is given by:
While the tractions are:
Moreover, the external work is:
where, 
,
and 
are the distributed forces in α, β and ζ directions, respectively
and 
and 
are the distributed couples about the middle surface of the shell. In addition
fε, cε, fg and cg 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 after expanding the
terms:
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:
where, Ik is the mass density of the kth layer of the shell per
unit mid-surface area. While the energy expressions described above are used
to derive the equations of motion. 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 plate thickness as in Eq.
10. Or we can rewrite Eq. 10 in term of constitutive
relations Eq. 5a and b 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.
Through, a recast of Eq. 9 to put in the familiar form, the
governing equations of motion and the equation charge equilibrium for first-order
shearable deformation case can be derived based on the fundamental Lemma of
calculus of variations. By integrating the displacement gradients by parts to
relieve the virtual displacements and setting its coefficients to zero individually:
EQUATIONS OF MOTION
In order to solve the equation of motion, we introduce the following assumptions
to cast the equation of motion in thick (or shear deformation) plate theories.
Where the deepness (or shallowness) of the shell, is One criterion used in developing
plate equations. Thus, shell is referred to as a plate, when it has zero curvature
or infinity radius of curvature (i.e., the term 1+ι/R1: where,
R1 is either of the curvature parameter Rα, Rβ,
or Rαβ (Qatu, 2004; Badri
and Al-Kayiem, 2011a, b). 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 addition, the radii of curvature
are assumed to be very large compared to the in-plane displacements (i.e., ui/Ri
= 0, where I = α, β and α, β and αβ, u0
or v0).
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 (kij+λ2Mij) {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 kij
terms for SS-1, cross-ply and rectangular plane form is given by Badri
and Al-Kayiem (2012b).
ILLUSTRATED EXAMPLE
In the present examine, laminated composite square plate (a/b-1) with both
the upper and lower surfaces embedded smart materials is considered. The plate
structures considered here are made of Terfonal-D smart composite material.
The material properties are given in several papers and books like (Reddy,
2004; Badri and Al-Kayiem, 2011c) and it will not
repeat here. The adhesive used to bond the structural layers or smart-material
layers are neglected in the analysis. The laminated composite structures are
composed of N layers and all the layers are assumed to be of the same thickness.
The side-to-thickness ratios stack range (a/h = 10 to a/h = 100) are considered
to represent the thick and thin laminated composites. Four different laminations
schemes (i.e., symmetric cross-ply, symmetric angle-ply, symmetric general angle-ply
and asymmetric general angle-ply laminates) under SS-1 boundary condition are
considered in this study.
| Table 1: |
Static analysis of nondimensionalized center deflection as
and load parameter 
of laminated composite plate (a/b = 1, CFRP and Terfonal-D, 10-layer and
SS-1) |
 |
As a baseline of computer simulation, unless otherwise specified, symmetric
cross-ply laminates with (SS-1) boundary condition are mainly used. The HSDT
that developed by Lee (2004), are used here in the verifications.
The shear correction factor used in FSDT is (K2 = 5/6). Numerical
values of nondimensional center deflection as function of the load parameter
are tabulated in Table 1 and the effects of two kind of plate
thickness are studied. As stated earlier by Tsai et al.
(2008), that the distribution of displacements through the thickness by
kinematics field in classical plate theories may lead to unexpected error.
Consequently, the Higher-order Shears Deformation Theory (HSDT) that allows
the transversal displacement w and its corresponding strain εςς,
to vary nonlinearly through the cross-thickness, should be more accurate. Thus,
a correspondence has been observed between the results of the presented theory
with those obtained by Lee (2004) that use an exact model
based on a HSDT and satisfactory agreement is found.
Even though, shear deformation theory is relevant in the stress calculations
but still not essentially for electric and magnetic potentials as well electric
displacement and magnetic induction. Whereas only including of nonlinear constitutive
relations of smart materials in the structural analysis could justify the discrepancies
found in the predictions with shear deformation theories. A similar conclusion
was also reported by Lee (2004).
In the other hand, Fig. 1-3 show the magnetic
potential 
,
electrical potential φ, center deflections w, angle of twist ψα
and ψβ the in-plane displacement u and v responses for
sandwich plate formed from three smart layers.
|
|
| Fig. 1(a-b): |
The uncontrolled magnetic responses of laminated composite
plate of (a/b = 1 and m = n = 5) (a) P/M/P and (b) M/P/M scheme, P: BaTiO2,
M: CoFe2O4 |
|
| Fig. 2(a-b): |
The uncontrolled electrical responses of laminated composite
plate of (a/b = 1 and m = n = 5) (a) P/M/P and (b) M/P/M scheme, P: BaTiO2,
M: CoFe2O4 |
|
| Fig. 3(a-b): |
The uncontrolled elastic responses of laminated composite
plate of (a/b = 1 and m = n = 5) (a) P/M/P and (b) M/P/M scheme, P: BaTiO2,
M: CoFe2O4 |
It is perceived that the elastic deflections, electrical potential and magnetic
potential have similar occurrence. It is interesting to note that the sensory
responses have simple discriminate behavior against the variation in the plate
dimensions.
CONCLUSION
In this study, a model is developed for static and dynamic analysis of MTEE
and multilayered plate structure and/or plate embedded a smart material lamina
and influenced by MTEE load. The fundamental theory is derived based on FSDT
involving Codazzi-Gauss geometrical discretion. The theory is casted in version
of general laminated composite plate of rectangular plane-form, in which the
generic forced-solution procedures for the response were derived and its mode
shapes were evaluated in simply supported boundary condition. Thus the center
deflection was selected among the primary variable for validation and verification
purpose. Whereas, result have been shown a close agreement with those of HSDT
that obtained by previous researchers. The present results may serve as a reference
in developing the MTEE plate theories and to improve the benchmark solutions
for judging the existence of imprecise theories and other numerical approaches.
ACKNOWLEDGMENT
The authors would like to acknowledge Universiti Teknologi PETRONAS for sponsoring
the research work under the GA scheme.
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