Flexural vibration of pyrocomposite solid cylinder

Flexural vibration of an infinite Pyrocomposite circular cylinder made of inner solid and outer hollow pyroelectric layer belonging to 6mm-class bonded together by a Linear Elastic Material with Voids (LEMV) is studied. The exact frequency equation is obtained for the traction free outer surface with continuity conditions at the interfaces. Numerical results in the form of data and dispersion curves for the first and second mode of the flexural vibration of the cylinder ceramic 1 / Adhesive / ceramic 2 by taking the adherents as BaTio3 and the adhesive layer as an existing Carbon Fibre Reinforced Polymer (CFRP) or as a hypothetical LEMV layer with and without voids are compared with a pyroelectric solid cylinder. The damping is analyzed through the imaginary part of the complex frequencies. | Flexural vibration | pyrocomposite | solid cylinders | LEMV | CFRP |


INTRODUCTION
The applications of pyroelectric materials and pyroelectric ceramic/polymer composite materials are many and [1] - [10] are a few in particular.Dispersion characteristics of wave propagation in pyoelectric plate and cylinder have been studied by Paul and Raman [11]- [13].Paul and Nelson [14] have extended the study of Vasudeva and Govinda Rao [15]- [16] on the influence of distributed voids in the interfacial LEMV adhesive zones of the isotropic Sandwich plate to the flexural vibration of Piezo composite hollow cylinder.A continuum theory of LEMV with distinct properties has been developed by Cowin and Nunziato [17].In layered composites pores or voids are found in the interface region and it is known to affect the estimation of physical and mechanical properties of the composites [18].Voorhees and Green [19] have studied the mechanical behavior of sandwich composites made of thin porous core and denser face materials.Damage detection and vibration control of a new smart board designed by mounting piezoelectric fibers with metal cores on the surface of a CFRP composite was studied by Takagikiyoshi [20].
In the present analysis flexural vibration of pyrocomposite circular solid cylinder of crystal class 6mm with LEMV/CFRP as a bonding layer is considered.The frequency equation for flexural vibration of solid cylinder has been derived for traction free shorted outer surface with interface continuity conditions on both sides of the LEMV layer.Numerical work is carried out and the dispersion curves for the flexural vibration of the pyrocomposite solid cylinder with middle core LEMV/CFRP are compared with that of pyroelectric solid cylinder.

GOVERNING EQUATIONS
The equations governing elastic, electric and thermal behavior are given by Mindlin [21] - [22]  are elastic, piezoelectric, dielectric, thermal stress coefficients and pyroelectric constants respectively.The comma followed by an independent variable denotes partial differentiation of that coefficient with respect to that independent variable and is to denote the constants and variables of inner and outer pyroelectric materials of hexagonal (class 6 mm).For crystal class 6 mm, the material constants are The equations of flexural motion, Gauss's equation and the entropy equation in cylindrical polar coordinates r,θ, z for class 6 are Where, k is the wave number, p is the angular frequency and 1 − = i . We introduce the non-dimensional quantities x and ε such that , and h = h 3 -h 0 (h 0 , h 3 are inner and outer radius of the cylinders) thickness of the composite solid cylinder.Using the above solution, Eqn (3) can be rewritten as   ( ) The solutions of the Eqn (5) are taken as In the context of the theory of LEMV, the equations of motion and balance of equilibrated force are given by [23] Nelson V. K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 The solution for Eqn. ( 9) is taken as Substituting Eqn.(10) in Eqn ( 9) and using the dimensionless variables x and ε, the Eqn.( 10) becomes
The solutions of the Eqn.(11) are taken as The interface condition 0 , = r ψ on the void volume fraction field ψ is suggested by Atkin et al [24].(When a material with voids comes into contact with another material without voids).The frequency equation is obtained as 23 x 23 determinantal equation, on substituting the solutions in the boundary-interface conditions.It is written in symbolic form as ( ) The non-zero elements at ) and the other nonzero elements at the interfaces x = x 1 can be obtained on replacing J 0 by J 1 and Y 0 by Y 1 in the above elements.They are ( )( ) . The non -zero elements at x = x 2 by varying j from 6 to 10 are,

NUMERICAL RESULTS
Zeros of the frequency equations are evaluated using Muller's method [25].The elastic, piezoelectric, dielectric and pyroelectric constants for BaTio 3 are taken from Ref. [26]- [27].The material constants of LEMV bonding layer are taken as the hypothetical material no.2, in Table III of Puri and Cowin [28].The value of dimensionless number N, which is void volume measure factor, defined in eq.(3.4) of Ref [28], and the value of N is found to be 0 ≤ N ≤ 0.66.The material constants of CFRP bonding layer are taken from [29].For all the numerical calculations, the inner/outer and middle/outer radius of the cylinders are taken as a/c = 0.6666 and b/c = 0.7333 respectively.The complex frequencies for the flexural waves in the first and second modes are given in Tables 1  and 2. The imaginary parts of the frequencies representing the attenuation of the flexural vibration of Pyro laminated-LEMV (with and without voids) and Pyro laminated -CFRP cylinders are compared with pyroelectric solid cylinder.The dispersion curves for the real part of frequency against the dimensionless wave number for the interfacial layers LEMV (N=0 and N=0.33)/CFRP and pyroelectric solid cylinder are plotted for the first and second flexural mode in Figs. 1 and 2 respectively.

CONCLUSION
The frequency equation for free flexural vibration of pyrocomposite solid cylinder with LEMV as core material is derived.From the numerical data, an increase in imaginary part of the frequencies which is a measure of attenuation of the composite vibration is observed with voids/pores in the core material than the vibration of the pyroelectric solid cylinder.The present model with CFRP core may have a similar practical application discussed in [5]- [6].
the other nonzero element at the interface x = x 3 can be obtained on replacing J 0 by J 1 and Y 0 by Y 1 in the above elements.They are ( ) stresses, strains, electric displacements, electric fields, entropy and temperature.Here,l vC is the specific heat capacity, 0 θ is the reference temperature, and l ρ is the density.

D
and the entropy σ satisfy the following equations for flexural vibration of hexagonal symmetry Nelson V. K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 v and w are the displacements along r, θ, z direction, φ is the electric potential, ρ is the mass density and t is the time.The solutions of Eqn.(3) is considered in the form Nelson V. K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 and

h
can be evaluated using the following relations: ) are material constants characterizing the core of LEMV, ρ is the density and µ λ , are the Lame's constants and ψ is the new kinematical variable associated with a material with voids comes into contact with another material without voids.

(
The governing equation for CFRP core material can be deduced from Eqn. (9) by taking the void volume fraction 0 = ψ , and the Lame's constants asThe frequency equation has been derived by using the following boundary and interface conditions (i) Since the outer surface are traction free and coated with electrodes which is shorted, the boundary conditions become On the interfaces (inner and middle, outer and middle), the continuity conditions are At non-pyroelectric core material) and 0 , = r ψ due to void volume fraction field with 2 , 1 = l Nelson V. K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 Nelson V. K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286 Nelson V. K. and Karthikeyan S. / Journal of Fundamental Sciences 4 (2008) 269-286

.
In the case of without voids in the interface region, the frequency equation is obtained by taking 0 = ψ in Eqn.(14) which reduces to a 21 x 21 determinantal equation.The frequency equations derived above are valid for different inner and outer materials of 6mm class and arbitrary thickness of layers.

Table 1
Complex frequencies for different values of real wave numbers in the first flexural mode of the pyrocomposite solid cylinder

Table 2
Complex frequencies for different values of real wave numbers in the first flexural mode of the pyrocomposite solid cylinder.