Sensitivity of Dynamic Response of a Sim ply Supported Functionally Graded Magneto-electro-elastic Plate to its Elastic Parameters

G.Q.Xie,2 and M.X.Chi

1　Introduction

Many scholars have increasingly focused on the properties of magneto-electroelastic structures employed as these smart or intelligent materials have ability of converting energy from one form to the other(among magnetic,electric and me-

chanical energy)[Nan(1994);Harshe et al.(1993)].Applied in ultrasonic imaging devices,sensors,actuators,transducers and many other emerging components,it is a strong necessity for these smart or intelligent materials and structure to study dynamic and static characteristics of theses by theories or techniques.Static and dynamic behavior of plates as well as in finite cylinder has been studied in the literature.Pan(2001)obtained an exact closed-form solution for the simply supported and multilayered plate composed of anisotropic piezoelectric and piezomagnetic materials under a static mechanical load.Pan and Heyliger(2002)solved the corresponding dynamic problem.Static behavior of a functionally graded magnetoelectro-elastic hollow sphere subjected to hydrothermal loading in the spherically symmetric state was studied by M.Saadatfar and M.Aghaie-Khafri(2014).The dynamic response of a rotating radically polarized functionally graded piezoelectric hollow cylinder was investigated by A.H.Akbarzadeh and Z.T.Chen(2012).Coupling effect of electric and magnetic fields was found in piezoelectric and piezomagnetic composites.Jianguo Wang and Xuefeng Li(2008)derived analytical solutions for the magneto-electric effect of multilayered magneto-electro-elastic media by using the transfer matrix method.M R Sedighi and M.Shakeri(2009)obtained a three-dimensional elasticity solution of functionally graded piezoelectric cylindrical panels.Bishay,Sladek,Sladek and Atluri(2012)used hybrid/mixed finite elements and node-wise material properties to analyze functionally graded magneto-electro-elastic composites Buchanan(2004)used finite element method to study the behavior of layered versus multiphase magneto-electro-elastic in finite long plate composites Wang et al.(2003)had carried out analysis of multilayered magnetoelectro-elastic plates for mechanical and electrical loading by the state vector approach.Free vibration of a magneto-electro-elastic plate resting on a Pasternak foundation was investigated by Yansong Li and Jingjun Zhang(2014)based on Mindlin theory Rajesh K.Bhangale and N.Ganesan(2006)derived semi-analytical finite element method to static analysis of simply supported functionally graded and layered magneto-electro-elastic plates In the present study,a series solution in conjunction with finite element approach is extended to dynamic response sensitivity analysis of a functionally graded magneto-electro-elastic plates The studies on dynamic sensitivity analysis of the functionally graded magnetoelectro-elastic structure to material parameters is less in the literature,Dynamic response sensitivity analysis is essential to optimization design and inverse technique of smart material.

2　Basic equations

The coupled physic equations for anisotropic and linear magneto-electro-elastic solids are given by:

vector andBis the magnetic induction vectorC,gandμare the elastic,dielectric and magnetic permeability coefficient matricesq,eandαare piezoelectric,piezomagnetic and magnetoelectric material coefficient matrices.

The strain-displacement relations are

Where

The operatorLdis

In Eq.(3),u,v,ware elastic displacement component in coordinate directionsx,yandz,respectively

The electric field vectorEis related to the electric potentialφas follows:

The relations between magnetic field and magnetic potential are given by:

In Eq.(5)and Eq.(6),Lφis an operator as shown below:

3　Dynamics differential equation

The functionally graded material parameters are assumed to obey exponential law in the thickness direction.The plate is divided into 3 nodal surfaces namely upper,middle and lower nodal surfaces in the thickness z direction.The following shape functions are adopted

The displacementUand the electrostatic potentialφand the static magnetic potentialψare approximated by the shape functions and the degrees of freedom of 3 nodal surfaces.

Where

Wherei=l,m,u,ldenotes lower nodal surface,mmiddle nodal surface anduupper nodal surface.

In the present work a set of finite series solution agreed with the boundary conditions for the simply supported rectangular plates(a×b)has been adopted.The generalized displacement functions are as follows:

Wherenandmare two positive integers,NandMare the term number of the series to be accounted for the general excitation.

Coupled equations of a finite layer element are given by:

Where

In Eq.(13)

In Eq.(14)

WhereFis the external traction vector applied on the nodal surface.

In Eq.(15)

In Eq.(16)

Where

Where

Assembling matrices of all the elements,we obtain the entire dynamic differential equation of the functionally graded magneto-electro-elastic plate:

In the present work we only consider sensitivity of dynamic response to material elastic parameters,and take no account of sensitivity of dynamic response to material density mass and dimension sizes.

If a harmonic mechanical traction vectorsin(ωt)is applied on the top surface of the plate,and in Eq.(11)m=n=1 is adopted,the solution corresponding to Eq.(28)has the form:

Substituting Eq.(29)into Eq.(28),we have

Where

If material elastic parameters are taken as the design variables ‘θ’,applying a partial differential on Eq.(30)with respect toθ,we have

4　Numerical exam ple

For all the subsequent numerical examples,the functionally graded magnetoelectro-elastic plate are taken with the material parameters are given inAppendix AConsider a FGM magneto-electro-elastic plate having horizontal dimensionsa=0.5m andb=0.5m and thicknessh=0.04m,The mechanic model of the FGM magneto-electro-elastic plate is shown in Fig.1.

The follow ing dimensional parameters are adopted:

Where10-6Vs/Am2.For the mechanical loads,q0=1N/m2and for the electrode excitation,is a constant expressing the value of the electrostatic potential,and for the magnetic pole excitation,is a constant expressing the value of the static magnetic potential.A normal harmonic mechanical excitationis applied on the top surface of plate.

Figure 1:Mechanic model of FGM magneto-electro-elastic plate.

Eq.(32)can be written as:

andcan be obtained from the last two equations of Eq.(32)is defined as sensitivity of a variable ‘w’to a parameterhas the physical meaning as shown below:

We first obtain{?}tfrom the solution of Eq.(30),and then obtainby substituting{?}tinto Eq.(33)and solving Eq.(33).The numerical results are obtained by MATLAB program,and the figures are plotted by Origin 8.

Fig.2 shows the sensitivity distributions of the dimensionless deflection amplitude of the plate to elastic parametersCl(i,j)across the thickness forη=1.It is observed from Fig.2 that dimensionless deflection amplitudes of both the surfacesandis hardly sensitive to all the elastic parameters The dimensionless deflection amplitude is the most sensitive to the two parametersCl(1,2)andCl(2,2),however,Cl(1,2)andCl(2,2)have opposite effects on vibration control.In other words,the parameterCl(2,2),when increase,can suppress the vibration,butCl(1,2)when increase,can cause the more strong vibration The dimensionless deflection amplitude is insensitive to elastic parameterCl(5,5).Sensitivities of the dimensionless deflection amplitudes betweenandtoCl(1,2)andCl(1,3)are positive;toandCl(4,4)are negative.A positive sensitivity means that the amplitude increases with the increase of parameters,while a negative sensitivity instead.

Fig.3 shows the sensitivity distributions of the dimensionless deflection amplitude of the plate to elastic parametersCl(i,j)across the thickness forη=3 andη=10,respectively.It is seen from Fig.3 that deflection amplitude is the most sensitive to elastic parameterCl(6,6),the dimensionless de flection amplitudes of both the surfacesforη=3 andforη=10 are hardly sensitive to all the elastic parameters.In Fig,2 it is medium,but in Fig.3 the absolute value of sensitivity of the dimensionless de flection amplitude of the plate toCl(6,6)is the largest namely,the relative sensitivity of the dimensionless deflection amplitude of the plate toCl(6,6)will increase with the increase of exponent factor‘η’.

In Fig.4 the first two digital inCijklnamelyijdenotesCl(i,j),and the nextkorkldenotes the value ofη,such asC1110 denotes the elastic constantCl(1,1)andη=10.It is seen from Fig.4 that as the exponent factor‘η’increases the magnitude of sensitivity of the dimensionless deflection amplitude of the plate to the elastic parameters decreases.

Figure 2:Sensitivity distributions of the dimensionless de flection amplitude of the plate to elastic parameters Cl(i,j)across the thickness(η=1).

Figure 3:Sensitivity distributions of the dimensionless de flection amplitude of the plate to elastic parameters Cl(i,j)across the thickness(η =3 and η =10).

Fig.5 shows the sensitivity distributions of the dimensionless electric potential amplitudes of the plate to elastic parameterCl(i,j)across the thickness(η=1).It is seen from the Fig.5 that the dimensionless electric potential amplitudes of the plate are the most sensitive to the two elastic parametersCl(1,2)andCl(1,1).However,Cl(1,2)andCl(1,1)have opposite effects on vibration control namely,Cl(1,1),when increase,can suppress the vibration of electric potential,butCl(1,2),when increase,can lead to the more strong vibration of electric potential The dimensionless electric potential amplitudes is hardly sensitive to these elastic parametersCl(3,3),Cl(4,4)andCl(5,5).Sensitivity curves of the electric potential amplitude to these elastic parametersCl(1,1),Cl(1,2)andCl(2,2)are parabola The dimensionless electric potential amplitude of the top surface of the plate is hardly sensitive to all the elastic parameters.Sensitivity of the dimensionless electric potential amplitude of the plate to the elastic parametersCl(1,2),Cl(1,3)andCl(6,6)are positive;to the elastic parametersCl(1,1),Cl(2,2)andCl(2,3)are negative.

Figure 4:Sensitivity distributions of dimensionless deflection amplitude of the plate to elastic parameters Cl(i,j)across the thickness(η =1η =3,η =10).

Figure5:Sensitivity distributions of the dimensionless electric potential amplitudes of the plate to elastic parameter Cl(i,j)across the thickness(η=1).

Figure6:Sensitivity distributions of the dimensionless deflection amplitude of the plate to elastic parameter Cl(i,j)across the thickness(η=3).

Fig.7 shows the sensitivity distributions of the dimensionless electric potential amplitudes of the plate to elastic parameterCl(i,j)across the thickness forη=1,η=3,η=10,respectively.It is also seen from the comparison of the sensitive curves forη=1,η=3 andη=10 in Fig.7 that as the exponent‘η’increases the magnitudes of sensitivity of the dimensionless electric potential amplitudes of the plate to the elastic parameters decrease.

Fig.8 shows the sensitivity distribution of the dimensionless magnetic potential amplitude of the plate to the elastic parameterCl(i,j)across the thickness(η=1).It has been noticed that sensitivity curves of the magnetic potential amplitude to these elastic parametersCl(1,1),Cl(1,2),Cl(1,3),Cl(2,2),Cl(2,3)are parabola.Further it is seen from Fig.8 that the dimensionless magnetic potential amplitudes of the plate are the most sensitive to the two elastic parametersCl(1,2)andCl(1,1).However,Cl(1,2)andCl(1,1)have opposite effects on vibration control.Cl(1,1),when increase,can suppress vibration of magnetic potential,butCl(1,2),when increase,can cause the more strong vibration of magnetic potential The values of sensitivity of magnetic potential to these elastic parametersCl(1,2),Cl(1,3),Cl(2,3)andCl(6,6)are positive;to these elastic parametersCl(1,1)andCl(2,2)are negative.The dimensionless magnetic potential amplitude of the plate is hardly sensitive toCl(3,3),Cl(4,4)andCl(5,5).

The sensitivity distributions of the dimensionless magnetic potential amplitudes of the plate to elastic parameterCl(i,j)across the thickness(η=3)is shown in Fig.9.It can be seen from Fig.9 that magnitude of sensitivity to elastic parameterCl(i,j)gradually reduce from the bottom surface(ˉz=0)to the top surface(ˉz=1)of the plate.The dimensionless magnetic potential amplitude of the plate is hardly sensitive to the elastic parameterCl(4,4).In Fig.8 it is very small,but in Fig.9 magnitude of sensitivity of magnetic potential amplitude to the elastic parameterCl(6,6)is larger than that to the other parameters except forCl(1,2).In other words,as the exponent factor‘η’increases the relative sensitivity of the dimensionless magnetic potential amplitude of the plate to the elastic parameterCl(6,6)increases.

Figure7:Sensitivity distributions of the dimensionless deflection amplitude of the plate to elastic parameter Cl(i,j)across the thickness(η =1η =3?η =10).

Figure8:Sensitivity distribution of the dimensionless magnetic potentialamplitude of the plate to the elastic parameter Cl(i,j)across the thickness(η=1).

Figure9:Sensitivity distribution of the dimensionless magnetic potential amplitude of the plate to the elastic parameter Cl(i,j)across the thickness(η=3).

Figure 10:Sensitivity distribution of the dimensionless magnetic potential amplitude of the plate to the elastic parameter Cl(i,j)across the thickness(η =1η =3?η =10).

Fig.10 shows the sensitivity distribution of the dimensionless magnetic potential amplitude of the plate to the elastic parameterCl(i,j)across the thickness,forη=1η=3,η=10,respectively.It is seen from the comparison of the sensitive curves forη=1,η=3,η=10 that magnitude of sensitivity of the dimensionless magnetic potential amplitude of the plate to the elastic parameters decreases as the exponent factor‘η’increases.

5　Conclusion

We can obtain the following conclusions from the numerical example:

1.Forη=1 the sensitivity curves of the dimensionless amplitude of the electric and magnetic potentials of the plate to these elastic parametersCl(1,1),Cl(1,2)andCl(2,2)are parabola.But forη=3 the parabola disappears as the exponent factor‘η’increases magnitude of the sensitivity to elastic parameterCl(i,j)reduce from the bottom(ˉz=0)to the top surface(ˉz=1)of the plate.

2.There is a surface within the plate.De flection amplitude of the surface is hardly sensitive to all the elastic parameters.

3.The dimensionless electric and magnetic potential amplitudes of the top surface of the plate is hardly sensitive for all the elastic parameters no matter how much the exponent factor‘η’is.

4.The dimensionless amplitudes of electric potential,magnetic potential and elastic de flection are all insensitive to the elastic parameterCl(4,4).

5.The relative sensitivities of the dimensionless de flection,electric and magnetic potentials amplitude of the plate toCl(6,6)increase as the exponent factor‘η’increases.

6.As the exponent factor‘η’increases the sensitivity of dynamic response of the FGM plate to its elastic parameters decreases.

Acknowledgement:This work is supported by Foundation of China under the Grant Number 11372109.

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