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
AlKayiem, 2012a; Bassiouny, 2006). Several accurate
solutions of structronics shell have been presented using 3D and 2D theories
or the discrete layer approaches. The exact closedform solutions for multilayered
piezoelectricmagnetic 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
Magnetoelectroelastic (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 tractionfree under surface potentials. Following
up the previous Stroh formulation, Pan and Han (2005)
presented the 3D solutions of multilayered Functionally Graded (FG) and MEE
plates. Wang et al. (2003) proposed a modified
state vector approach to obtain 3D solutions for MEE laminates, based on the
mixed formulation of solid mechanics.
By an asymptotic approach, Tsai et al. (2008)
studied 3D static and dynamic behavior of doubly curved FGMEE 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 magnetothermoelectroelastic (MTEE)
in particular, is rather scarce, especially for shear deformation studies.
In this study, a fundamental theory of piezolaminates shell/plates based on
the Firstorder 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 thinwalled structures and of the possibility to apply the MTEE adapting.
FOUNDATIONS THEORY
In order to be reasonably selfcontained, in what follows, here will summarize
the fundamental physical laws that govern the conservation law of electromagnetic
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 AlKayiem, 2012b). In summary, the total energy of a shell element is
defined as:
where, P is total potential energy:
where, Q (δ_{i}, ε_{i}, g_{1}, T), t (δ_{i},
ε_{i}, g_{1}) and W (δ_{i}, ε_{i},
g_{1}) 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:
Based on the conservation laws of electromagnetic field, the linear thermodynamic
potential energy Q for quasistatic 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 ,
,
x_{1} and
are the natural independent variables. In order to obtain the thermodynamic
potential for which these variables are natural, is performed (PerezFernandez
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 study is:
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. 68 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.
Not that, the temperature τ is a known function of position. Thus, temperature
field enter the formulation only through constitutive equations. While I_{1},
I_{2} and I_{3} 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:
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 firstorder 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
AlKayiem, 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 inplane 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 Naviertype solution can be applied to obtain exact solution as
(K_{ii} + λ^{2} 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 SS1, crossply and rectangular plane form is listed in the Appendix.
RESULTS AND DISCUSSION
To prove the validity of the developed theory, laminated composite square plate
(a/b = 1) with both the upper and lower surfaces embedded smart materials is
considered. The plate structure considered here is made of BaTiO_{3}
and CoFe_{2}O_{4} composite material. The material properties
are given in several papers like (Badri and AlKayiem, 2011ac)
and it will not be included here.
First, the example of sandwich piezoelectric and magnetostrictive plate that
studied and analyzed exactly by various researchers e.g., Pan
and Heyliger (2001) and Chen et al. (2005)
is considered here for validation and comparison. Table 1
gives the lowest five frequency parameters:
of the fundamental vibrational mode (m = n = 1) which is of practical importance
(Pan and Heyliger, 2002), whereas, δ_{max}
being the maximum of the δ_{ii} in the whole sandwich plate and
ρ_{max} = 1, which was defined by Pan and
Heyliger (2002) and adopted by Chen et al. (2005).
While in Table 1 it clearly seen that the frequency results
obtained by the present model are in close agreement with those obtained by
Chen et al. (2005) using alternative state space
formulations:
Note that Table 1 shows the frequencies of the first class
of vibration only. It is worth to highlight that the 1st mode of vibration shows
100% agreement with literature, while the discrepancy in other higher modes
are negligible in practical sense. Further results and conclusions about the
classes of vibration can be found by Pan and Heyliger (2001)
and Chen et al. (2005) for BaTiO_{2}/CoFe_{2}O_{4}
sandwich plate. In fact, those results have been successfully reproduced and
discrepancy around 5% is observed.
It should be mentioned here that present model has been verified for results
available in literature for pure elastic shell by letting Q_{ii} and/or
κ_{ii} equal to zero and rigorous agreement was found.
While the bonded error for plate results were predicted and explained as due
to the assumption of specialization of shell theory to plate by letting R_{α}
= R_{β} = R_{α,β} = ∞. In essence the plate
can be regarded as a special case of the present analysis, but in fact it has
a purpose of verification with literature only. In the other hand, Fig.
1 shows the center deflections ,
angle of twist ψ_{α} and ψ_{β}, inplane
displacement u and v, electrical potential φ and magnetic potential θ
sensory responses for sandwich shell formed from three smart layers. It is perceived
that the elastic deflections, electrical potential and magnetic potential have
similar occurrence.
Table 1: 
Comparison of recent results of the lowest 5 frequency parameters
of the sandwich plate with results of Pan and Heyliger
(2002) 

† Note that the present results are for shell of (R_{α},
R_{β}, R_{α,β} = ∞) and the shear correction
factor used in FSDT is (κ^{2} = 5/6), *(P/M/P) is denoting
Piezoelectric [inner]/Magnetostrictive [middle]/Piezoelectric [outer] 

Fig. 1: 
The uncontrolled responses of laminated composite spherical
shell of at which the right side is representing the P/M/P scheme when the
left side is being for M/P/M 
It is interesting to note, that the sensory responses have simple discriminate
behavior against the variation in the shell dimensions.
CONCLUSION
In this study a model is developed for dynamic analysis of piezolaminated shell
structure and embedded smart material lamina and influenced by MTEE load. The
fundamental theory is derived based on FSDT involving CodazziGauss geometrical
discretion. The theory is casted in version of piezolaminated plate of rectangular
planeform (for purpose of validation and verification only). At which the generic
forcedsolution procedures for the response were derived and its frequency parameters
were evaluated in simply supported boundary condition.
Results have shown a close agreement with those obtained by Chen
et al. (2005). Furthermore, the present model has been verified for
results available in literature for pure elastic shell by assuming Q_{ii}
and/or κ_{ii} equal to zero and rigorous agreement was also found.
The present results may serve as a reference in developing the piezolaminated
shell theories and to improve the benchmark solutions for judging the existence
of imprecise theories and other numerical approaches.
ACKNOWLEDGMENTS
The authors would like to acknowledge Universiti Teknologi PETRONAS for sponsoring
the research work under the GA scheme.
APPENDIX
NOMENCLATURE
Latin symbols:
a, b 
= 
Length and width of the shell in (m) 
A_{i}, A,B 
= 
Lame’ parameter 
ε 
= 
Electric displacement 

= 
Electrical charge density 

= 
Electric current density or magnetic charge density 

= 
Thermal forces resultant 

= 
Elastic body forces in (N) 

= 
Magnetic inductions 
K 
= 
Kinetic energy 

= 
Unit outward normal 

= 
Edge forces, shear forces and its free traction resultant 

= 
Edge moment, higher shear terms and its free traction resultant 

= 
Gibbs free energy in (Joule) 

= 
Midsurface displacements of the shell 

= 
Radius of curvature 

= 
Stress field and Free elastic tractions, respectively in (Pa) 
t 
= 
Surface tractions 

= 
Thermal Gain (Entropy) 
W 
= 
Work (body forces) in (N m) 
Greek symbols
α, β, ζ 
= 
Curvilinear coordinates, α and β for the reference
surface and _{ζ} for the normal axis 

= 
Thermomagnetic 
ε_{i, j} 
= 
Elastic strain 

= 
Magnetoelectric 

= 
Thermal properties (Pa/°C^{2}) 

= 
Magnetoelastic 

= 
Thermoelastic 

= 
Magnetic (Ns^{2}/C^{2}) 
ζ 
= 
Electric field 

= 
Thermoelectric 
ρ_{o} 
= 
Density 

= 
Elastic properties in (pa) 
τ 
= 
Thermal Field 

= 
Electric properties in (C^{2}/Nm^{2}) 

= 
Electroelastic 
φ 
= 
Electric potential 
x 
= 
Magnetic field vector 

= 
Magnetic potential 
ψ_{α}, ψ_{β} 
= 
Midsurface rotations 