Interphase effect on the controlled frequency response of three-phase smart magneto-electro-elastic plates embedded with active constrained layer damping: FE study

In this article, the controlled frequency response of three-phase smart magneto-electro-elastic (TPS-MEE) composite plates treated with active constrained layer damping (ACLD) is studied using finite element (FE) method. The composite consists of Cobalt Ferrite piezomagnetic matrix embedded with carbon fibers in the piezoelectric shell. A three-dimensional FEformulation is developed through the principle of virtual work. The ACLD treatment comprises of the constraining layer of 1–3 piezoelectric composite as well as the constrained layer of viscoelastic material. The coupling characteristics and hence the stiffness of the smart composite significantly change with the thickness of the piezoelectric interphase. Therefore, emphasis is made on the effect of ACLD treatment on the vibration attenuation of the TPS-MEE plates with different interphase thickness and material. In addition, special attention is paid to the influence of geometrical skewness associated with different interphase thickness on the vibration amplitude and the control voltage.


Introduction
Among the many available smart materials, magneto-electro-elastic (MEE) materials have proved to be a potential candidate for many engineering applications due to its inherent coupling capabilities. This material is comprised of piezoelectric and piezomagnetic phases and exhibits a superior magnetoelectric (ME) coupling which is not observed in the individual phases [1,2]. Boomgard and Born [3] for the first time attempted to fabricate the ME composites using Barium Titanate (BaTiO 3 ) and Nickel Ferrite (NiFe 2 O 4 ) through sintering technique. Boomgard et al [2] found that the ME properties of MEE materials are significantly affected by the combined phases of Barium Titanate (BaTiO 3 ) -Cobalt Ferrite (CoFe 2 O 4 ) present in the composite. Many researchers have attempted to study the material properties of MEE composites incorporating numerous computational methods such as micromechanical approaches [4][5][6] (e.g. asymptotic homogenization method [7][8][9], self-consistent method [10,11], and Mori-Tanaka method [12][13][14]). Shen and Li [15], Sevastionav and Kachanov [16] and Aboudi [17] adopted the homogenization technique to the estimation of the coupled effective material coefficients of magneto-electro-thermo-elastic (METE)materials. That the inhomogeneity associated with the microstructure of METE material affects the effective material constants was thoroughly studied by Koutsawa et al [18][19][20]. The influence of multi-inclusions and arbitrary lamination on the coupled characteristics of MEE material was investigated by Li [21] and Giordano et al [22], respectively. Lee et al [23] incorporated finite element (FE) methods to determine the MEE material properties. Tang and Yu [24][25][26] and Yu and Tang [27,28] adopted variational asymptotic methods for unit cell homogenisation (VAMUCH) to predict the local fields of METE materials. Similarly, asymptotic homogenization scheme [29][30][31][32] is also found to be effective in predicting the characteristics properties of METE materials.
A structure in practical operation usually is subjected to hazardous vibrations which deteriorate its performance. In addition, it may also reduce the service life of the structure due to resonance. One of the solutions that the researchers have effectively found out to attenuate the unwanted vibrations is through the active constrained layer damping (ACLD) treatment. In this technique the constraining layer of 1-3 piezoelectric composite (PZC) patch and the constrained layer low stiff viscoelastic material are both used. Previous studies have highlighted that in contrast to the monolithic piezoelectric patch, superior damping properties are exhibited by the piezoelectric fiber reinforced epoxy matrix (i.e. the 1-3 PZC) [62] as a constrained layer. The credibility of adopting ACLD treatment is demonstrated in various research works. Ray and Pradhan [63] probed the damped frequency characteristics of composite plates through FE methods. In addition, Ray and coworkers assessed the attenuation response of functionally graded [64][65][66], laminated [67][68][69] and sandwich [70,71] plates with ACLD treatment. The effect of ACLD patch on the damping behaviour of skew plates was  Table 1. Material properties of TPS-MEE material with different interphase thickness and volume fraction [82][83][84].

Material property
Material  [73][74][75] and shell [76]structures. In addition, Vinyas and co-researchers [77][78][79] probed the influence of geometrical skewness on the controlled frequency response of layered, multiphase and functionally graded MEE plates, respectively. On the other hand, the intensification of MEEcoupling phenomenon is significantly affected by the interphase region. The interphase region is a result of a chemical reaction between the matrix material and the reinforcement phase or the reinforcements may be protectively coated [80]. This interphase region can be treated as the third phase in the MEE composite and it exhibits different properties from the reinforcements and matrix. Notable consequences of the interphases linking the matrix and reinforcement have been observed on the inclusive characterization of ME smart material [81]. However, systematic analyses on the frequency response of MEE plates considering the effect of interphase are limited [82][83][84].
Here, we make the first endeavor to investigate the influence of ACLD treatment of the damped vibrations of three phase smart magneto-electro-elastic (TPS-MEE) plates made of carbon fiber coated with piezoelectric material and embedded in piezomagnetic matrix. Also, emphasis has been made to evaluate the effects of interphase on the controlled linear response of TPS-MEE plate for the first time. To this end, a threedimensional FE model is developed using the principle of virtual work. The method of condensation is applied to obtain governing equations of motion which takes the coupling factors into consideration. The viscoelastic layer of ACLD is modeled through complex modulus approach [85]. In addition, a special emphasis has been made on evaluating the effect of coupling fields associated with interphase, volume fraction and geometrical skewness on the attenuated frequencies of TPS-MEE plates.   thickness is incorporated in any of the smart structures, a drastic variation in its mechanical responses is noticed. Hence, the present article aims at demonstrating the controlled frequency response of TPS-MEE plates ( figure 1(b)) embedded with ACLD treatment. The material properties corresponding to the different volume fraction and interphase thickness of TPS-MEE material is tabulated in table 1.

Material properties, constitutive relations and governing equations
The constitutive equations of TPS-MEE material with various interphase thickness can be expressed as follows: in which, 'Vf' and 'in' denotes the volume fraction and interphase thickness of TPS-MEE material. Also, [¯] relates to the elastic stiffness coefficients. The piezoelectric coefficient, the magnetostrictive coefficient where, f and ψ are the electric and magnetic potential functions that can be represented as, The viscoelastic constrained layer of ACLD is modelled through the complex modulus approach (CMA) as follows [85]: where, ¢ G , n and h corresponds to the storage modulus, Poisson's ratio and loss factor of the viscoelastic layer, respectively.
The axial displacement components u and v of the TPS-MEE plate is assumed to vary along x-andy-, respectively according to the relation as follows [77]: Meanwhile, a higher order transverse displacement component w is adopted to enhance the vertical attenuation of ACLD patch and it can be shown as follows: The mid-surface displacement components along x-, y-and z-axes are denoted by u 0 , v 0 and w 0 respectively. Also, θ x (θ y ), κ x (κ y )and γ x , (γ y ) represents generalised rotations of the normal to mid-plane of the substrate, viscoelastic layer and piezoelectric patch about the y-axis (x-axis), respectively. Further, the terms θ z andκ z are the gradients of transverse displacements with respect to z-axis associated with the TPS-MEE plate and viscoelastic layer, respectively.
The governing equations of motion are derived incorporating the FE methods. A FE mesh with size 10×10 and eight noded isoparametric quadrilateral element is employed in this study. Invoking the principle of virtual work and skew transformations, the equations of motion of TPS-MEE plate embedded with ACLD treatment can be represented as [77]:

Problem description
The linear controlled frequency response of simply supported TPS-MEE plate with ACLD patch ( figure 1(c)) is studied. To this end, the geometrical dimensions of TPS-MEE plate chosen for this study are: length a=0.5 m, breadth b=0.5 m, thickness H=3 mm. The reference plane of the TPS-MEE plate is chosen at z=0. The constraints employed in this study can be expressed as follows:

( )
The ACLD patch is attached to the TPS-MEE substrate at its centre ( figure 1(c)). Further, it is designed in such a manner that the constraining layer of 1-3 PZC patch is of thickness h p =250 μm and the constrained viscoelastic layer is of thickness h v =50.8 μm. The parameters λ and α represents the skew angle of the TPS-MEE plate and piezoelectric fiber orientation angle of the 1-3 PZC patch. The TPS-MEE plate, is excited by force of 2 N at the position (a/2, b/4 cosλ, H/2).

Results and discussions
The influence of interphase on the damped frequency of ACLD treated TPS-MEE plate with various skew angles is demonstrated. The credibility of the FE formulation is justified in prior assuming the inactive state of ACLD. The results presented in tables 2 and 3 affirm that the proposed formulation properly accommodates the coupling and geometrical skewness effects and yield reliable results. Therefore, it can be extended to evaluate the damped frequency response of TPS-MEE plate. In this analysis, unless mentioned otherwise, the interphase piezoelectric material PZT-7A, piezoelectric fiber orientation of 1-3 PZC patch α=0°oriented in yz-plane, control gain K d =600 and a center ACLD patch is considered.

Effect of interphase thickness
The influence of interphase thickness (δ) on the damped frequency of ACLD treated rectangular TPS-MEE plate (λ=0°) with a volume fraction V f =0.2 to 0.5 is probed. Four different interphase thickness of PZT-7A viz, δ = 0.36, 0.25, 0.13 and 0.01 is considered for this study. From figures 2(a)-(d) it can be witnessed that the maximum linear frequency response functions for deflection is noticed for δ=0.01. The reason can be due to the minimum values of material constants, the TPS-MEE plate with δ=0.01 experiences minimal stiffness in contrast to the other interphase thickness. Therefore, the required control voltage to attenuate the vibrations are also higher for TPS-MEE plate with δ=0.01 as depicted in figure 3 for all the volume fractions.

Effect of skew angle
The degree of coupling associated with the TPS-MEE plate drastically changes with the geometrical skewness of the plate. Figures 4(a)-(d) encapsulate the influence of skew angle of TPS-MEE plate with V f =0.2 to 0.5 and δ=0.36. It can be inferred from these plots that as the skew angle improve the linear frequency response functions for deflection and control voltage reduces. This can be attributed to the fact that the degree of coupling and hence the stiffness of the TPS-MEE plate improveswith the higher skew angle. The study is extended to evaluate the influence of interphase thickness associated with skew angle. From figures 5 and 6, it can be witnessed that with the improvement in the value of δ and skew angle λ, the attenuation capabilities of the plate significantly improves. Therefore it can be justified that both δ and skew angle λ are the prominent parameters contributing to the damping characteristics of TPS-MEE plate.

Effect of volume fraction
The effect of volume fraction of TPS-MEE material associated with different skew angle and interphase thickness on the linear frequency response functions of ACLD treated TPS-MEE plate is shown in tables 4 and 5 in terms of the maximum deflection and control voltage, respectively. It can be assessed from these results that as the volume fraction of TPS-MEE material increases from 0.2 to 0.5, the deflection and control voltage increases. This can be attributed to the fact that higher volume fraction of TPS-MEE material possess decreased percentage of CoFe 2 O 4 phase which has superior elastic stiffness coefficient than the other phases of TPS-MEE material. In addition, the conclusions drawn in the previous section with respect to interphase thickness and skew angles can be observed here also.

Effect of interphase material
The piezoelectric material has a significant influence on the overall coupled characteristics of TPS-MEE material. An attempt has been made to assess the influence of prominent piezoelectric interphase material such as PZT-5A and PZT-7A on the linear frequency response functions of deflections and control voltage. From figure 7, it is evident that in contrast to PZT-5A a predominant effect of PZT-7A is witnessed on the enhanced coupled frequency of the TPS-MEE plate. In other words, due to minimal stiffness, a higher deflection of the TPS-MEE plate with PZT-5A as the interphase material is noticed. The same trend is observed for the control voltage as well as shown in figure 8.

Effect of coupling
The coupling between magnetic, electric and elastic fields results in an effective vibration control of the TPS-MEE plate. However, when coupling between these fields are neglected, the structural behaviour of TPS-MEE plate marginally varies. Therefore, it is necessary to compare the linear frequency response of TPS-MEE plate with and without coupling effects. In addition, the contribution of interphase thickness and skew angles on the coupling effect needs to be determined. From figures 9 and 10, it can be seen that in contrast to lesser value of interphase thickness (δ=0.01), a higher value of interphase thickness (δ=0.36) results in a higher discrepancy of deflection and control voltage. In other words, the influence of coupling fields increases as the interphase thickness improves. In addition, magneto-elastic field tends to increase the deflection marginally more than the elastic fields. However, it is witnessed that when complete coupling is established between magnetic, electric and elastic fields, the maximum damping can be achieved. In addition, from tables 6-9 it can be seen that a higher percentage of contribution on the coupled behaviour of TPS-MEE plate is noticed for δ=0.36 which improves  with higher skew angle and volume fraction. The schematic representation of discrepancies existing between coupled and uncoupled values of deflection and control voltage corresponding to skewed TPS-MEE plate with V f =0.2 and δ=0.36 for both PZT-5A and PZT-7A is shown in figures 10(a) and (b), respectively.

Conclusions
In this article the damped linear frequency response of active constrained layer damping (ACLD) treated three phase smart magneto electro elastic (TPS-MEE) plate with skewed edges is investigated through finite element methods. The TPS-MEE plate is composed of carbon fiber/piezoelectric interphase/piezomagnetic material. The equations of motion are derived using the principle of virtual work. The numerical results suggest that the attenuation capability of TPS-MEE plate improves with the increase in the interphase thickness. An improvement in the volume fraction results in increased damping phenomenon. In addition, PZT-7A is found to be a better interphase material when compared to PZT-5A. The linear frequency response functions for deflection and control voltage increases with the reduction in the skew angle of the plate. Finally, the results reveal that higher couplings between magnetic, electric and elastic fields are noticed for TPS-MEE plate with higher interphase thicknessδ=0.36, skew angle λ=45°and V f =0.2. It is firmly believed that the results of this article will serve as benchmark for upcoming researches on TPS-MEE structural analysis.