Static analysis of simply supported functionally graded ... · In this article, static analysis of...

International Journal of Solids and Structures 43 (2006) 3230–3253

www.elsevier.com/locate/ijsolstr

Static analysis of simply supported functionally gradedand layered magneto-electro-elastic plates

Rajesh K. Bhangale 1, N. Ganesan *

Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Received 15 February 2005Available online 18 July 2005

Abstract

In this article, static analysis of functionally graded, anisotropic and linear magneto-electro-elastic plates have beencarried out by semi-analytical finite element method. A series solution is assumed in the plane of the plate and finiteelement procedure is adopted across the thickness of the plate such a way that the three-dimensional character ofthe solution is preserved. The finite element model is derived based on constitutive equation of piezomagnetic materialaccounting for coupling between elasticity, electric and magnetic effect. The present finite element is modeled with dis-placement components, electric potential and magnetic potential as nodal degree of freedom. The other fields are cal-culated by post-computation through constitutive equation. The functionally graded material is assumed to beexponential in the thickness direction. The numerical results obtained by the present model are in good agreement withavailable functionally graded three-dimensional exact benchmark solutions given by Pan and Han [Pan, E., Han, F., inpress. Green�s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J. Sol-ids Struct.]. Numerical study includes the influence of the different exponential factor, magneto-electro-elastic proper-ties and effect of mechanical and electric type of loading on induced magneto-electro-elastic fields. In addition furtherstudy has been carried out on non-homogeneous transversely isotropic FGM magneto-electro-elastic plate available inthe literature [Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropicmagneto-electro-elastic plates].� 2005 Published by Elsevier Ltd.

Keywords: FGM; Magneto-electro-elastic; Static; Finite element

0020-7683/$ - see front matter � 2005 Published by Elsevier Ltd.doi:10.1016/j.ijsolstr.2005.05.030

* Corresponding author. Tel.: +91 44 22578174/5171; fax: +91 44 22578502.E-mail addresses: [email protected] (R.K. Bhangale), [email protected] (N. Ganesan).

1 Ph.D. Scholar, IIT Madras.

mailto:[email protected]

mailto:[email protected]

R.K. Bhangale, N. Ganesan / International Journal of Solids and Structures 43 (2006) 3230–3253 3231

1. Introduction

Literature dealing with research on the behaviour of magneto-electro-elastic structures has gained moreimportance recently as these smart or intelligent materials have ability of converting energy from one formto the other (among magnetic, electric and mechanical energy) (Nan, 1994; Harshe et al., 1993). Such mate-rials can exhibits field coupling that is not present in any of the monolithic constituent material. With appli-cation in ultrasonic imaging devices, sensors, actuators, transducers and many other emerging components,there is a strong need for theories or techniques that can predict the coupled response of theses so calledsmart materials, as well as structure composed of them. Studies on static and dynamic behaviour on platesas well as infinite cylinder has been dealt in the literature. Pan (2001) derived an exact closed-form solutionfor the simply supported and multilayered plate made of anisotropic piezoelectric and piezomagnetic mate-rials under a static mechanical load. Pan and Heyliger (2002) solved the corresponding vibration problem.Piezoelectric and piezomagnetic composites exhibit coupling effect of electric and magnetic fields. In mostof the studies these composite materials has been used as layers or as multiphase. The behaviour of finitelylong cylindrical shells under uniform internal pressure has been studied by Wang and Zhong (2003) andthey conclude that piezoelectric and Piezomagnetic composites in general have the coupling effect whichis two orders higher than that of single-phase magnetoelectric constituent materials. Micro-mechanicalanalysis of fully coupled electro-magneto-thermo-elastic composites has been carried out by Aboudi(2001) for the prediction of the effective moduli of magneto-electro-elastic composites. Li (2000) studiedthe multi-inclusion and inhomogeneity problems in a magneto-electro-elastic solid. From the literature sur-vey, it is found that only few studies have been reported on magneto-electro-elastic structures by finite ele-ment analysis. Free vibration behaviour of infinitely long magneto-electro-elastic cylindrical shell has beenstudied by Buchanan (2003) by using semi-analytical finite element method. Buchanan (2004) has studiedthe behaviour of layered versus multiphase magneto-electro-elastic infinite long plate composites by finiteelement method. Wang et al. (2003) derived the state vector approach to analysis of multilayered magneto-electro-elastic plates for mechanical and electrical loading. Lage et al. (2004) developed a layerwise partialmixed finite element model for magneto-electro-elastic plate. The studies on non-homogeneous magneto-electro-elastic structure are less in the literature. Chen and Lee (2003) adopted state space formulationto derive equations for non-homogeneous transversely isotropic magneto-electro-elastic plates. Accordingto Chen et al. (2005), like other advanced material the functionally graded magneto-electro-elastic struc-tures will be appear soon and they carried out free vibration analysis of non-homogeneous transversely iso-tropic magneto-electro-elastic plates.

The special feature of graded spatial compositions (non-homogeneous) associated with FGM providesfreedom in the design and manufacturing of novel structures. Still it is great challenges in the numericalmodeling and simulation of the FGM structure (Liew et al., 2003; Pan and Han, in press).

Recently Pan and Han (2005) presented an exact solution for functionally graded and layered magneto-electro-elastic plates by Pseudo-Stroh formalism and they mentioned that their results can serve as abenchmarks for finite element and boundary element solutions of the problem. Hence in the presentstudy, static analysis of functionally graded and layered magneto-electro-elastic plates has been carriedout by using series solution in conjunction with finite element approach. In addition the main aim ofthe study is to evaluate the influence of functionally grading on the mechanical, electrical and magneticbehaviour.

2. Constitutive equations

The coupled constitutive equations for anisotropic and linearly magneto-electro-elastic solids can bewritten as

3232 R.K. Bhangale, N. Ganesan / International Journal of Solids and Structures 43 (2006) 3230–3253

rk ¼ CjkSk � ekjEk � qkjH k ð1ÞDk ¼ ejkSk þ ejkEk þ mjkHk ð2ÞBj ¼ qjkSk þ mjkEk þ ljkH k ð3Þ

where rk denotes stress, Dj is the electric displacement and Bj is the magnetic induction. Cjk, ejk and ljk arethe elastic, dielectric and magnetic permeability coefficients. ekj, qkj and mjk are piezoelectric, piezomagneticand magnetoelectric material coefficients. Apparently, various uncoupled cases can be reduced from Eqs.(1)–(3). A completely coupled magneto-electro-elastic material matrix, assuming a hexagonal crystal class,for above constitutive equations given below by Buchanan (2003):

r1

r2

r3

r4

r5

r6

D1

D2

D3

B1

B2

B3

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;

¼

C11 C12 C13 0 0 0 0 0 e31 0 0 q31

C12 C11 C13 0 0 0 0 0 e31 0 0 q31

C13 C13 C33 0 0 0 0 0 e33 0 0 q33

0 0 0 C44 0 0 0 e15 0 0 q15 0

0 0 0 0 C44 0 e15 0 0 q15 0 0

0 0 0 0 0 C66 0 0 0 0 0 0

0 0 0 0 e15 0 e11 0 0 m11 0 0

0 0 0 e15 0 0 0 e11 0 0 m11 0

e31 e31 e33 0 0 0 0 0 e33 0 0 m33

0 0 0 0 q15 0 m11 0 0 l11 0 0

0 0 0 q15 0 0 0 m11 0 0 l11 0

q31 q31 q33 0 0 0 0 0 m33 0 0 l33

2666666666666666666666664

3777777777777777777777775

S1

S2

S3

S4

S5

S6

E1

E2

E3

H 1

H 2

H 3

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;

ð4Þ

The strain–displacement relations are

rxx ¼ r1 ¼ouox

; ryy ¼ r2 ¼ovoy

; rzz ¼ r3 ¼owoz

ryz ¼ r4 ¼ovozþ ow

oy; rxz ¼ r5 ¼

ouozþ ow

ox; rxy ¼ r6 ¼

ouoyþ ov

ox

ð5Þ

where u, v and w are mechanical displacements in coordinate directions x, y and z.The electric field vector Ei is related to the electric potential U as shown below:

Ex ¼ E1 ¼ �o/ox

; Eh ¼ E2 ¼ �o/oy

; Ez ¼ E3 ¼ �o/oz

ð6Þ

The magnetic field Hi is related to magnetic potential w as shown below:

H x ¼ H 1 ¼ �owox

; H h ¼ H 2 ¼ �owoy

; H z ¼ H 3 ¼ �owoz

ð7Þ

3. Finite element formulation

Recently Buchanan (2004) has analyzed free vibration behaviour for infinitely long plate. In the presentwork finite series solution has been assumed satisfying boundary conditions for simply supported plates hasbeen adopted. The finite element model has been used in the thickness direction. For a general loading theshape functions are as follows:


uðx; y; zÞ ¼XN

n¼1

XM

m¼1

U nmðzÞ cosnpLx

� �x sin

mpLy

� �y

vðx; y; zÞ ¼XN

n¼1

XM

m¼1

V nmðzÞ sinnpLx

� �x cos

mpLy

� �y

wðx; y; zÞ ¼XN

n¼1

XM

m¼1

W nmðzÞ sinnpLx

� �x sin

mpLy

� �y

uðx; y; zÞ ¼XN

n¼1

XM

m¼1

UnmðzÞ sinnpLx

� �x sin

mpLy

� �y

wðx; y; zÞ ¼XN

n¼1

XM

m¼1

WnmðrÞ sinnpLx

� �x sin

mpLy

� �y

ð8Þ

where n and m being two positive integers and N and M are the number of terms in the series to be ac-counted for the general loading. In the present study analysis has been carried out similar to the reportedby Pan and Han (2005) for m = n = 1. In the end the analysis has been reduced for finite element in thick-ness direction still retaining the three-dimensional dependence the solution based on the choice of n and m.The analysis is carried out by two noded finite element and the assumed shape functions are

Ui ¼ ½N u�fUg; U ¼ ½Nu�fUg; W ¼ ½Nu�fWg ð9Þ
where
N 1 ¼ 1� zi

ziþ1 � zi

� �; N 2 ¼

zi

ziþ1 � zi

� �

For a coupled filed problem finite element equations are as follows:

½Kuu�fUg þ ½Ku/�f/g þ ½Kuw�fwg ¼ 0

½Ku/�TfUg � ½K//�f/g � ½K/w�fwg ¼ 0

½Kuw�TfUg � ½K/w�Tf/g � ½Kww�fwg ¼ 0

ð10Þ

Various stiffness matrices are defined as shown below:

½Kuu� ¼ cZ½Bu�T½C�½Bu�dz

½Ku/� ¼ cZ½Bu�T½e�½B/�dz

½Kuw� ¼ cZ½Bu�T½q�½Bw�dz

½K//� ¼ cZ½B/�T½e�½B/�dz

½Kww� ¼ cZ½Bw�T½l�½Bw�dz

½K/w� ¼ cZ½B/�T½m�½Bw�dz

ð11Þ

where C = 0.25LxLy.[Bu], [B/], [Bw] represents the strain–displacement, electric field–electric potential and magnetic field–

magnetic potential relations respectively. The component matrices for Eq. (11) are


½Bu� ¼ ½Lu�½Nu� ¼

o

ox0 0

0o

oy0

0 0o

oz

0o

ozo

oyo

oz0

o

oxo

oyo

ox0

266666666666666666664

377777777777777777775

�N 1 0 0 N 2 0 0

0 N 1 0 0 N 2 0

0 0 N 1 0 0 N 2

264

375 ð12Þ

½Bu� ¼

� npLx

� �N 1 0 0 � � �

0 � mpLx

� �N 1 0 � � �

0 0oN 1

oz� � �

0oN 1

ozmpLx

� �N 1 � � �

oN 1

oz0

npLx

� �N 1 � � �

mpLx

� �N 1

npLx

� �N 1 0 � � �

26666666666666666666664

37777777777777777777775

ð13Þ

where there are three additional columns for N2. The matrix [B/] is developed using Eq. (6) and is asfollows:

B/

� �¼ L/

� �N/

� �¼

o

oxo

oyo

oz

26666664

37777775

N 1 N 2½ � ¼

� npLx

� �N 1 � np

Lx

� �N 2

� mpLy

� �N 1 � mp

Ly

� �N 2

oN 1

ozoN 2

oz

266666664

377777775

ð14Þ

Similarly for Eq. (7) gives

½Bw� ¼ ½Lw�½Nw� ¼

o

oxo

oyo

oz

26666664

37777775

N 1 N 2½ � ¼

� npLx

� �N 1 � np

Lx

� �N 2

� mpLy

� �N 1 � mp

Ly

� �N 2

oN 1

ozoN 2

oz

266666664

377777775

ð15Þ

In the present study the Gaussian integration scheme has been implemented to evaluate integrals in-volved in different matrices. The functionally graded material properties based on the assumptions madeas described in the next section are accounted by evaluating the material properties at Gaussian points.


Problem has been solved by decoupled way. In the case of mechanical load the conventional elastic stiffnessmatrix of the system has been inverted to obtain the mechanical displacements. Neglecting the couplingbetween magnetic and piezoelectric the system of equation pertaining to be solved to obtain the potentialin the plate, based on the mechanical displacements. Similarly magnetic potential has been evaluated byconsidering the coupling between displacements and magnetic.

4. Analytical model of FGM materials properties

4.1. Functionally and layered magneto-electro-elastic plate

First to start with validation of the present formulation the layered FGM plate (Model-I) made up ofthree layers is considered reported by Pan and Han (2005). The three layers have equal thickness of0.1 m and the horizontal dimensions of the plate are Lx · Ly = 1 m · 1 m. Two functionally graded and lay-ered sandwich plates with stacking sequence BaTiO3/COFe2O4/BaTiO3 (B/F/B) and CoFe2O4/BaTiO3/COFe2O4 (F/B/F) are shown in Fig. 1(a). Both top and bottom layers are functionally graded with thesymmetric exponential variation as shown in Fig. 1(b). Five different exponential factors, i.e.,g = �10, �5, 0, 5, 10 (1/m), were studied.

The functionally graded material with exponential variation in the thickness direction (z-direction), thematerial coefficients are given by

Homogeneous CoFe2O4 (BaTiO3)

Functionally Graded BaTiO3 (CoFe2O4)

Functionally Graded BaTiO3 (CoFe2O4) 0.1

0.1

0.1

X, u

Y, v Z, w

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.05

0.10

0.15

0.20

0.25

0.30

Thi

ckne

ss (

m)

Proportional Factor in Material Properties

η=-10

η=-5.0

η= 0.0

η= 5.0

η=10.0

(b)

(a)

Fig. 1. (a) Coordinate and configuration of functionally graded and layered magneto-electro-elastic plate under study (Model-I).(b) Variation of FGM proportional coefficient (egz) across the thickness of the FGM and layered plate.

-0

-0

-0

0

0

0

0

Thi

ckne

ss (

m)

(c)

Fig. 2.distribmagneperme


CikðzÞ ¼ C0ikegz; eikðzÞ ¼ e0

ikegz; qikðzÞ ¼ q0ikegz

eikðzÞ ¼ e0ikegz; likðzÞ ¼ l0

ikegz; dikðzÞ ¼ d0ikegz

ð16Þ

where g is the exponential factor governing the degree of the material gradient in the z-direction, and thesuperscript 0 is attached to indicate the z-independent factors in the material coefficients. For g = 0 corre-sponds to the homogeneous material case.

4.2. Functionally graded (non-homogeneous) magneto-electro-elastic plate

Further study has been carried out for non-homogeneous transversely isotropic FGM magneto-electro-elastic plate reported in the literature (Chen et al., 2005). The present study considers functionally gradedmaterial composed of piezoelectric and magnetostrictive material. The grading is accounted across thethickness of the FGM magneto-electro-elastic plate as shown in Fig. 2(a). An advantage of a plate madeof a FGM over a laminated plate is that material properties vary continuously in a FGM but are discon-tinuous across the adjoining layers in a laminated plate. This has been achieved by grading the volume frac-tion of particular material governed by power law index. Fig. 2(b) depicts the through-the-thicknessdistribution of the volume fraction for different power law indexes n. Consider an FGM magneto-

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

n=25.0n=1000.0

n=10.0

n=1.0

n=5.0

n=2.0

n=0.02

n=0.5

n=0.1

n=1.0

Z/h

Volume fraction

n=0.0

(b)

CoFe2O4 rich

BaTiO3 rich

Graded region

+h/2

-h/2

X, u

Y, v Z, w

(a)

11 10 9 8 7 6 5 4 3 2 1.15

.10

.05

.00

.05

.10

.15

Dielectric coeffcients ε11

n=0.0 n=0.2 n=0.1 n=0.5 n=1.0 n=2.0 n=5.0 n=10.0 n=15.0 n=1000.0

0 100 200 300 400 500 600-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

n=0.0 n=0.2 n=0.1

n=0.5 n=1.0

n=2.0 n=5.0 n=10.0 n=15.0 n=1000.0

Thi

ckne

ss (

m)

Magnetic permeability (μ11)(d)

(a) Coordinate and configuration of functionally graded magneto-electro-elastic plate (Model-II). (b) Through-the-thicknessution of the volume fraction for different values of the power law index n. (c) Variations of effective dielectric constants oftostrictive material along the thickness direction of functionally graded plate (Model-II). (d) Variations of effective magneticability constants of magnetostrictive material along the thickness direction of Functionally graded plate (Model-II).


electro-elastic plate having horizontal dimensions Lx and Ly = 1 m · 1 m and thickness h = 0.3. In the pres-ent analysis it is assumed that the composition is varied from the bottom surfaces to top surface, i.e., thetop surface of the plate is piezoelectric-rich, whereas the bottom surface is magnetostrictive-rich. In addi-tion material properties are graded throughout the thickness direction according to volume fraction powerlaw distribution. Present study considers smooth and continuous variation of the volume fraction of eitherpiezoelectric or magnetostrictive material governed by the power law index.

A simple power law type definition for the volume fraction of the metal across the thickness direction ofthe FGM plate is assumed as given in Eq. (17):

V B ¼2zþ h

2h

� �n

ð17Þ

where h is the thickness of the plate, z is the thickness coordinates (0 6 z 6 h), and n is the power law index.The bottom surface of the plate (z = �h/2) is CoFe2O4-rich whereas the top surface (z = h/2) of the plate isBaTiO3-rich. The sum total volume of the constituent materials, BaTiO3(B) and CoFe2O4(F) should be

V B þ V C ¼ 1 ð18Þ

Based on the volume fraction definition and law of mixtures, the effective material property definitionfollows:

ðMPÞeff ¼ ðMPÞtopV B þ ðMPÞbottomV C ð19Þ

�MP� is general notation for material property. Making use of Eqs. (17)–(19) the following effective elas-tic, piezoelectric, piezomagnetic, dielectric and magnetic permeability, and thermal properties definitionscan be written as

Ceff ¼ ðCB � CF Þ2zþ h

2h

� �n

þ CF ð20aÞ

eeff ¼ ðeB � eF Þ2zþ h

2h

� �n

þ eF ð20bÞ

qeff ¼ ðqB � qF Þ2zþ h

2h

� �n

þ qF ð20cÞ

eeff ¼ ðeB � eF Þ2zþ h

2h

� �n

þ eF ð20dÞ

leff ¼ lB � lFð Þ 2zþ h2h

� �n

þ lF ð20eÞ

In Eqs. (20a)–(20e), �eff� stands for effective material properties obtained by above equations for particularpower law index n. Material coefficients of the piezoelectric BaTiO3 and magnetostrictive CoFe2O4 is givenin Appendix. In addition variation of effective dielectric coefficients and effective magnetic permeabilitywith respect to power law index n across the thickness direction is shown in Fig. 2(c) and (d). It is seen thatfor power law index n = 1.0 the variation of effective material property is linear.

5. Result and discussion

In this section, we present some numerical results by using the present finite element formulation used inthis paper. First a benchmark problem (Pan and Han, 2005) is considered for validating the present finite


element method. Second another functionally graded model available in the literature (Chen et al., 2005) isconsidered subjected to both mechanical and electric loading.

5.1. Mechanical load on the top surface of the functionally graded and layered plate

A simply supported FGM and layered square plate of thickness 0.3 m is subjected to a distributed loadon a top surface with the sinusoidal distribution sinðp=LxÞ sinðp=LyÞ. Consistent load vector approach isused in finite element sense to evaluate the response of the system. Responses are calculated for fixed hor-izontal coordinates (x, y) = (0.75Lx, 0.25Ly).

Fig. 3(a) and (b) shows the variation of the in plane elastic displacement ux (=�uy) in the FGM B/F/Band F/B/F respectively. It is seen that as exponential factor �g� is decreases magnitude of horizontal dis-placement ux increases. This behaviour is similar on the top and bottom surfaces. In addition it is noticedthat variation of elastic displacements is similar for both stacking sequence with slight difference in magni-tude. Magnitude of the elastic displacements in B/F/B is larger than that of F/B/F. Fig. 3(c) and (d) showsthe elastic displacement uz for B/F/B and F/B/F respectively. Vertical displacement uz increases withdecreasing of exponential factor in a plate. Here g = 0 pertains to homogeneous B/F/B and F/B/F platesreported by Pan (2001), Wang et al. (2003) and Lage et al. (2004) and it is found that present results are ingood correlations observed between them. Fig. 4(a) and (b) shows the variation of the / electric potential

-4.00E-012 -2.00E-012 0.00E+000 2.00E-012 4.00E-0120.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

B

Thi

ckne

ss (

m)

Ux

-4.00E-012 -2.00E-012 0.00E+000 2.00E-012 4.00E-0120.00

0.05

0.10

0.15

0.20

0.25

0.30

F

F

Thi

ckne

ss (

m)

Ux

B

0.00E+000 2.00E-012 4.00E-012 6.00E-012 8.00E-012 1.00E-0110.00

0.05

0.10

0.15

0.20

0.25

0.30

B

B

Thi

ckne

ss (

m)

UZ

F

0.00E+000 2.00E-012 4.00E-012 6.00E-012 8.00E-012 1.00E-0110.00

0.05

0.10

0.15

0.20

0.25

0.30

F

F

Thi

ckne

ss (

m)

UZ

B

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

(a) (b)

(c) (d)

Fig. 3. Elastic displacement components uxand uz (m) variation across the thickness direction in FGM magneto-electro-elastic plate.

0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 0.0024 0.00280.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10η=-5.0η=0.0η=5.0η=10.0

B

F

B

Thi

ckne

ss (

m)

φ0.0020 0.0022 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034

0.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

Thi

ckne

ss (

m)

φ

F

-3.0x10-6 -2.8x10-6 -2.6x10-6 -2.4x10-6 -2.2x10-6 -2.0x10-6 -1.8x10-60.00

0.05

0.10

0.15

0.20

0.25

0.30

B

B

F

Thi

ckne

ss (

m)

ψ-3.0x10-6 -2.5x10-6 -2.0x10-6 -1.5x10-6 -1.0x10-6 -5.0x10-7

0.00

0.05

0.10

0.15

0.20

0.25

0.30

F

F

B

Thi

ckne

ss (

m)

ψ

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

(a) (b)

(c) (d)

Fig. 4. Electric potential / (V) and magnetic potential w (C/s) distribution across the thickness in FGM magneto-electro-elastic platefor different exponential factor caused by a mechanical load acts on the top surface: / in (a) and w in (c) for B/F/B, and / in (b) and win (d) for F/B/F.


for B/F/B and F/B/F cases across the thickness direction. While Fig. 4(c) and (d) shows the variation ofmagnetic potential w across the thickness for the B/F/B and F/B/F cases. As exponential factor �g� increasesthe magnitude of the electric and magnetic potential decreases. In addition it is noticed that slope of thepotentials are discontinuous across the thickness even though potentials are continuous.

Fig. 5(a) and (b) shows the distribution of stress component rxx (=ryy) across the thickness direction forB/F/B and F/B/F stacking sequence respectively. It is observed that the horizontal stress is discontinuousacross the interfaces and it is very sensitive to the exponential factor �g�.

Fig. 5(c) and (d) shows the variation of rzz for the B/F/B and F/B/F. It is apparent that both stackingsequences produce nearly the same stress distribution pattern, even though elastic constants for the twomaterials are considerably different. Please see Appendix. Fig. 6(a) and (b) shows the shear stress variationacross the thickness. It is seen that both stacking sequence produce nearly the similar pattern and magni-tude of B/F/B is slightly larger.

Fig. 7(a–b) and (c–d) show the electric displacements Dx (=�Dy) and Dz, and the magnetic induction Bx

(=�By) and Bz are shown in Fig. 8(a–b) and (c–d) for B/F/B and F/B/F cases. It is observed that electricdisplacement and magnetic inductions are very much sensitive for different exponential factor �g�. Further itis noticed that because of material property discontinues in different layer the horizontal electric displace-ment and magnetic induction are discontinuous across the interface.

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

B

Thi

ckne

ss (

m)

σxx

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

F

Thi

ckne

ss (

m)

σxx

σzz σzz

0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

B

Thi

ckne

ss (

m)

0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

F

Thi

ckne

ss (

m)

(a) (b)

(c) (d)

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

η=-10η=-5.0η=0.0η=5.0η=10.0

Fig. 5. Stress component distribution rxx and rzz (N/m2) across the thickness direction: rxx in (a) and rzz in (c) for the B/F/B, and rxx

in (b) and rzz in (d) for F/B/F case.

-0.5 -0.4 -0.3 -0.2 -0.1 0.00.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10η=-5.0η=0.0η=5.0η=10.0F

B

B

τxz

Thi

ckne

ss (

m)

-0.5 -0.4 -0.3 -0.2 -0.1 0.00.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10η=-5.0η=0.0η=5.0η=10.0B

F

F

τxz

Thi

ckne

ss (

m)

(a) (b)

Fig. 6. Variation of the shear stress across the thickness direction: sxz in (a) for B/F/B and sxz in (b) for F/B/F.


-8.00E-011 -4.00E-011 0.00E+000 4.00E-011 8.00E-011 1.20E-0100.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10η=-5.0η= 0.0η= 5.0η=10.0

B

F

B

Thi

ckne

ss (

m)

Dx

-8.00E-011 -4.00E-011 0.00E+000 4.00E-011 8.00E-011 1.20E-010

Dx

0.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

F

Thi

ckne

ss (

m)

-1.20E-011 -8.00E-012 -4.00E-012 0.00E+000 4.00E-012 8.00E-012 1.20E-0110.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10.0η=-5.0η=0.0η=5.0η=10.0

B

F

B

Thi

ckne

ss (

m)

Dz Dz

-8.00E-013 -4.00E-013 0.00E+000 4.00E-013 8.00E-013 1.20E-0120.00

0.05

0.10

0.15

0.20

0.25

0.30η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

F

B

F

Thi

ckne

ss (

m)

(a) (b)

(c) (d)

Fig. 7. Electric displacements components Dx and Dz (C/m2) along the thickness direction in FGM magneto-electro-elastic plate fordifferent exponential factor caused by a mechanical load acts on the top surface: Dx in (a) and Dz in (c) for the B/F/B case, and Dx in(b) and Dz in (d) for the F/B/F case.


5.2. Electrical potential (load) on the top surface of the functionally graded and layered plate

The FGM magneto-electro-elastic plate is loaded electrically on the top surface of sinusoidal distribution/ ¼ /0 sinðp=LxÞ sinðp=LyÞ by an amount of /0 = 1 V. The response has been calculated at same locationi.e. horizontal coordinate fixed at (x, y) = (0.75Lx, 0.25Ly).

Fig. 9(a) and (b) shows the distribution of the elastic displacement ux (=�uy) in the FGM B/F/B and F/B/F cases in plate. Fig. 9(c) and (d) shows the elastic displacement uz for B/F/B and F/B/F respectively. It isnoticed that elastic displacements in the plate are completely dissimilar as compared to caused by mechan-ical load. In addition influence of stacking sequence has been felt.

The variation of the magnetic potential w across the thickness of the FGM B/F/B and F/B/F plate hasbeen shown in Fig. 10(a) and (b). It is noticed that magnitude of the magnetic potential in B/F/B is higherthan that of F/B/F. In addition it is felt that for actuation purpose the large magnitude potential can beinduced in bottom layer under the influence of electric potential on top surface.

Fig. 11(a) and (b) shows the distribution of the stress components rxx (=�ryy) across the thickness of theplate for B/F/B and F/B/F cases respectively. The effect of stacking sequence has been felt on the stresscomponents both in terms of curved shape and magnitude as compared to mechanical load reported inFig. 5. Effect of exponential factor has been felt in graded layers (top and bottom) for both stacking

-4.00E-010 0.00E+000 4.00E-010 8.00E-010 1.20E-0090.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

B

Thi

ckne

ss (

m)

-4.00E-009 -2.00E-009 0.00E+000 2.00E-009 4.00E-0090.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

B

Thi

ckne

ss (

m)

-8.00E-011 -4.00E-011 0.00E+000 4.00E-0110.00

0.05

0.10

0.15

0.20

0.25

0.30η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

B

F

B

Thi

ckne

ss (

m)

Bz

Bx Bx

-4.00E-010 0.00E+000 4.00E-0100.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

FT

hick

ness

(m

)

Bz

(a) (b)

(c) (d)

η=-10.0η=-5.0η=0.0η=5.0η=10.0

Fig. 8. Magnetic induction components Bx and Bz (Wb/m2) along the thickness direction in FGM magneto-electro-elastic plate fordifferent exponential factor caused by a mechanical load acts on the top surface: Bx in (a) and Bz in (c) for the B/F/B case, and Bx in (b)and Bz in (d) for F/B/F case.


sequence. Fig. 11(c) and (d) shows the variation of shear stress across the thickness. It is seen that bothstacking sequence produce different pattern and magnitude of F/B/F is larger than B/F/B.

Fig. 12(a–b) and (c–d) show the electric displacements Dx (=�Dy) and Dz, and the magnetic inductionBx (=�By) and Bz are shown in Fig. 13(a–b) and (c–d) for B/F/B and F/B/F cases. Here also stacking se-quence plays an important role in terms of shape and magnitude of the electric displacement and magneticinduction. In addition it is seen that electric displacement effect is felt more in top layers as compared toother under the application of electric potential. This is hold good for both stacking sequences. In additionit is observed the discontinuity at the interfaces of the layers. Regarding the magnetic induction in F/B/Fcase a relatively large magnetic field can be still induced in the magnetostrictive layer even if an electric loadis applied in the case of F/B/F case. Similar observations have been reported by Pan and Han (2005).

5.3. Mechanical loading on the top surface of the FGM plate (Model-II)

The FGM plate is subjected to mechanical load on the top surface with the following sinusoidaldistribution. sinðp=LxÞ sinðp=LyÞ; r0 = 1 N/m2 at fixed horizontal coordinates (x, y) = (0.75Lx, 0.25Ly).Consistent load vector approach has been used. Fig. 14(a) and (b) shows the variation of the inplane elasticdisplacement ux (=�uy) and uz in the FGM magneto-electro-elastic plate for different power law index �n�.

-1.00E-011 0.00E+000 1.00E-011 2.00E-011 3.00E-0110.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

B

Ux

Thi

ckne

ss (

m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Thi

ckne

ss (

m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Thi

ckne

ss (

m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Thi

ckne

ss (

m)

-8.00E-013 -4.00E-013 0.00E+000 4.00E-013 8.00E-013

=10.0

F

B

Ux

F

-4.00E-011 -3.50E-011 -3.00E-011 -2.50E-011 -2.00E-011 -1.50E-011 -1.00E-011

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

B

F

B

Uz

-4.00E-012 -3.00E-012 -2.00E-012 -1.00E-012

B

F

F

Uz

(a) (b)

(c) (d)

Fig. 9. Elastic displacement components ux and uz (m) distribution across the thickness direction in FGM magneto-electro-elastic platefor different exponential factor caused by an electric potential on the top surface: ux in (a) and uz in (b) for B/F/B, and ux in (c) and uz in(d) for F/B/F.

0.0 1.0x10-6 2.0x10-6 3.0x10-6 4.0x10-60.00

0.05

0.10

0.15

0.20

0.25

0.30η=-10.0η=-5.0η= 0.0η= 5.0η=10.0

η=-10.0η=-5.0η= 0.0η= 5.0η=10.0

B

F

B

Thi

ckne

ss (

m)

Magnetic Potential ψ

-1.0x10-7 -5.0x10-8 0.0 5.0x10-8 1.0x10-7 1.5x10-7 2.0x10-70.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

F

Thi

ckne

ss (

m)

Magnetic Potential ψ(a) (b)

Fig. 10. Magnetic potential w (C/s) distribution across the thickness in FGM magneto-electro-elastic plate: w in (a) for B/F/B and w in(b) for F/B/F.


-0.8 -0.4 0.0 0.4 0.80.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

F

σxx

Thi

ckne

ss (

m)

-8 -4 0 4 8 12 160.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10.0η=-5.0

η=5.0η=10.0 F

B

B

σxx

Thi

ckne

ss (

m)

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.080.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

B

τxz

Thi

ckne

ss (

m)

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.20.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

F

τxz

Thi

ckne

ss (

m)

η=0.0

η=-10.0η=-5.0

η=5.0η=10.0

η=0.0

η=-10.0η=-5.0

η=5.0η=10.0

η=0.0

η=-10.0η=-5.0

η=5.0η=10.0

η=0.0

(a) (b)

(c) (d)

Fig. 11. Stress component distribution rxx and sxz across the thickness direction of FGM plate: rxx in (a) for the B/F/B, and rxx in(b) for F/B/F case; sxz in (c) for the B/F/B and (d) for F/B/F case.


Here n = 0.0 corresponds to homogeneous piezoelectric material (BaTiO3) while n = 1000.0 corresponds tomagnetostrictive materials (CoFe2O4). The power law index value n other than two extreme values governsthe distribution of properties of BaTiO3 and CoFe2O4 mixture in FGM magneto-electro-elastic plate. Var-iation of the composition of piezoelectric and magnetostrictive is linear for power law index n = 1.0. It isseen that as power law index n decreases the magnitude of horizontal displacement ux and uz increases, as itis approaching towards the homogeneous BaTiO3. This phenomenon can be explained by looking at elasticproperties of both structures, as elastic modulus of the BaTiO3 is smaller than the CoFe2O4. This behaviouris similar on the top and bottom surfaces. The other power law index behaviour is lies between the twoextreme homogeneous materials.

Fig. 15(a) and (b) shows the variation of the / electric potential and magnetic potential w across thethickness direction for FGM plate. It is seen that as electric potential / increases as decrease in powerlaw index. It is interesting to note that the magnitude of electric potential is very small for power law indexn = 1000.0. This is due to the fact that for pure magnetostrictive materials, piezoelectric (eij) coefficients arezero. In case of magnetic potential distribution as power law index increases the magnitude of w increasesfrom n = 0.0 and attains maximum at n = 1000.0. In case of magnetostrictive material the piezoelectriccoefficients (qij) are zero. Fig. 16(a) and (b) shows the distribution of stress component rxx (=ryy) andrz across the thickness direction in Fig. 17 for FGM plate. It is seen that magnitude of rz is around 2.5times of the inplane stresses. Inplane stresses are predominant as compared to normal stress. Further itis seen that all power law index will produce nearly the same magnitude for rz. Fig. 18(a) and (b) shows

0.00E+000 2.00E-008 4.00E-008 6.00E-0080.00

0.05

0.10

0.15

0.20

0.25

0.30

η=-10.0η=-5.0η=0.0η=5.0η=10.0

Dx

F

B

B

Thi

ckne

ss (

m)

0.00E+000 5.00E-010 1.00E-009 1.50E-0090.00

0.05

0.10

0.15

0.20

0.25

0.30

Dx

B

F

F

Thi

ckne

ss (

m)

-3.00E-008 -2.00E-008 -1.00E-008 0.00E+0000.00

0.05

0.10

0.15

0.20

0.25

0.30

B

B

F

Thi

ckne

ss (

m)

Dz

-1.00E-009 -8.00E-010 -6.00E-010 -4.00E-010 -2.00E-010 0.00E+0000.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

F

Thi

ckne

ss (

m)

Dz

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

(a) (b)

(d)(c)

Fig. 12. Electric displacements components Dx and Dz (C/m2) along the thickness direction in FGM magneto-electro-elastic plate fordifferent exponential factor caused by a mechanical load acts on the top surface: Dx in (a) and Dz in (b) for the B/F/B case, and Dx in(c) and Dz in (d) for the F/B/F case.


the electric displacements Dx (=�Dy) and Dz, and the magnetic induction Bx (=�By) and Bz are shown inFig. 19(a) and (b) for FGM magneto-electro-elastic plate. It is observed that electric displacement and mag-netic inductions are very much sensitive for different power law index. For n = 0.0 the magnitude of electricdisplacement is higher and then it reduces slowly as increase in n and finally reduces to zero for n = 1000.0.The magnitude is higher at top surface. While magnetic induction is higher for n = 1000.0 and then reducesslowly as decrease in power law index and finally becomes zero for n = 0.0 (piezoelectric material). In addi-tion it is observed that there is a dramatic variation of the electric displacement and magnetic inductionacross the thickness of the FGM plate.

In the literature mainly the study deals with the influence of mechanical load on the displacement, stres-ses, electric potential, magnetic potential, electric displacement and magnetic induction. The maximumabsolute values of different quantities for different power law index n are listed in Table 1 caused bymechanical load. Generally use of these materials as sensing devices is to be attempted. It is felt that fora sensor it is preferable to have maximum electric potentials if electric quantities is used to be a sensor.As expected for n = 0.0 electric potential is maximum. In contrast it is preferable to use n = 1000.0 if mag-netic field is used as a sensor. Intermediate value of n will better if both the magnetic and electric potentialused for the sensing. From this study it may possible to choose appropriate for grading governed by powerlaw index depending on applications.

-1.20E-008 -8.00E-009 -4.00E-009 0.00E+000 4.00E-009 8.00E-0090.00

0.05

0.10

0.15

0.20

0.25

0.30η=-10.0η=-5.0η=0.0η=5.0η=10.0

B

F

B

Thi

ckne

ss (

m)

Bx

-8.0x10-10 -6.0x10-10 -4.0x10-10 -2.0x10-10 0.0 2.0x10-10 4.0x10-100.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

F

Thi

ckne

ss (

m)

Bx

-1.60E-009 -1.20E-009 -8.00E-010 -4.00E-010 0.00E+000 4.00E-0100.00

0.05

0.10

0.15

0.20

0.25

0.30

F

B

B

Y A

xis

Titl

e

Bz

-2.0x10-11 0.0 2.0x10-11 4.0x10-11 6.0x10-11 8.0x10-11 1.0x10-100.00

0.05

0.10

0.15

0.20

0.25

0.30

B

F

F

Thi

ckne

ss (

m)

Bz

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

η=-10.0η=-5.0η=0.0η=5.0η=10.0

(a) (b)

(c) (d)

Fig. 13. Magnetic induction components Bx and Bz (Wb/m2) along the thickness direction in FGM magneto-electro-elastic plate fordifferent exponential factor caused by a mechanical load acts on the top surface: Bx in (a) and Bz in (b) for the B/F/B case, and Bx in (c)and Bz in (d) for F/B/F case.

-2.00E-012 -1.00E-012 0.00E+000 1.00E-012 2.00E-012-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Thi

ckne

ss (

m)

Ux

Power law index (n)0.00.20.51.05.01000.0

4.00E-012 5.00E-012 6.00E-012 7.00E-012-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Thi

ckne

ss (

m)

Uz

Power law index (n)0.00.20.51.05.01000.0

(a) (b)

Fig. 14. Elastic displacement variation across the thickness: (a) Ux and (b) Uz.


0.0 4.0x10-4 8.0x10-4 1.2x10-3 1.6x10-3 2.0x10-3 2.4x10-3-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Electric Potential (a)

-3.0x10-6 -2.0x10-6 -1.0x10-6 0.0 1.0x10-6-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Magnetic Potential (b)φ ψ

Fig. 15. Potential distribution across the thickness: (a) electric and (b) magnetic.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

σxx

0.0 0.1 0.2 0.3 0.4 0.5-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15Power law index (n)

0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

σzz(a) (b)

Fig. 16. Stress variation across the FGM plate thickness: (a) rxx and (b) rzz.

0.45 0.30 0.15 0.00 0.15-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

τxz

Fig. 17. Variation of shear stress across the thickness of FGM plate.


-4.00E011 0.00E+000 4.00E-011 8.00E-011-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.01000.0T

hick

ness

(m

)

Dx(a)-8.0x10-12 -4.0x10-12 0.0 4.0x10-12 8.0x10-12 1.2x10-11 1.6x10-11

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Dz(b)

Fig. 18. Variation of electric displacement across the thickness of plate: (a) Dx and (b) Dz.

-3.00x10-9 -1.50x10-9 0.00 1.50x10-9 3.00x10-9 4.50x10-9 6.00x10-9-0.15

-0.10

-0.05

0.00

0.05

0.10


0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Bx(a)-2.00E-009 -1.50E-009 -1.00E-009 -5.00E-010 0.00E+000

-0.15

-0.10

-0.05

0.00

0.05

0.10


0.00.20.51.05.01000.0

Bz

Thi

ckne

ss (

m)

(b)

Fig. 19. Variation of magnetic induction across the thickness of plate: (a) Bx and (b) Bz.


5.4. Electrical potential (load) on the top surface of the FGM plate (Model-II)

The FGM magneto-electro-elastic plate is loaded electrically on the top surface of sinusoidal distribution/ ¼ /0 sinðp=LxÞ sinðp=LyÞ by an amount of /0 = 1V. The response has been calculated at same location i.e.horizontal coordinate fixed at (x, y) = (0.75Lx, 0.25Ly). Consistent load vector approach is used in finiteelement sense to evaluate the response of the system.

Fig. 20(a) and (b) shows the distribution of the elastic displacement ux (=�uy) in the FGM plate. It isnoticed that elastic displacements pattern in the plate are completely dissimilar as compared to caused bymechanical load. For power law index n = 0.0 the maximum inplane displacements has been noticed andsubsequently there is decrease in displacements as increase in the power law index, n = 1000.0 there is min-imum elastic displacement. This fact can be explained by looking at the properties of magnetostrictivematerial where piezoelectric coefficients are zero and the present study magneto electric coupling has beenneglected.

Fig. 21(a) shows the variation of the / electric potential for FGM plate across the thickness direction. Asn increases the magnitude of electric potential is increases and attains maximum for n = 5.0. Further it isnote that magnitude of all power law indexes is 0.5 at the top surface of the plate. Fig. 21(b) shows the

Table 1Results for simply supported FGM plate (Model-II) with applied mechanical sinusoidal traction

x = 0.75, y = 0.25 n = 0.0 (BaTiO3) n = 0.2 n = 1.0 n = 5.0 n = 1000.0 (CoFe2O4)

ux [m] �2.18 · 10�12 �1.97 · 10�12 �1.75 · 10�12 �1.66 · 10�12 �1.37 · 10�12

uy [m] 2.18 · 10�12 1.97 · 10�12 1.75 · 10�12 1.66 · 10�12 1.37 · 10�12

w [m] 6.55 · 10�12 6.19 · 10�12 5.74 · 10�12 5.42 · 10�12 5.16 · 10�12

/ [V] 0.00235 0.0023 0.0021 0.00139 1.27 · 10�12

w [C/s] 0.0 �3.93 · 10�7 �2.30 · 10�6 �2.61 · 10�6 �2.84 · 10�6

rxx [Pa] 1.26 1.21 1.15 1.10 1.35ryy [Pa] �1.26 �1.21 �1.15 �1.10 �1.35rzz [Pa] 0.49 0.49 0.49 0.49 0.49rzx [Pa] �0.396 �0.389 �0.391 �0.395 �0.389Dx [C/m2] �2.56 · 10�11 �2.21 · 10�11 �1.48 · 10�11 �6.14 · 10�12 0.0Dx [C/m2] 2.56 · 10�11 2.21 · 10�11 1.48 · 10�11 6.14 · 10�12 0.0Dz [C/m2] 1.40 · 10�11 1.35 · 10�11 1.04 · 10�11 2.89 · 10�12 0.0Bx [Wb/m2] 0.0 1.51 · 10�9 3.64 · 10�9 4.42 · 10�9 4.88 · 10�9

Bx [Wb/m2] 0.0 �1.51 · 10�9 �3.64 · 10�9 �4.42 · 10�9 �4.88 · 10�9

Bz [Wb/m2] 0.0 �2.31 · 10�9 �8.10 · 10�9 �1.29 · 10�9 �1.59 · 10�9

Maximum (in absolute value) across the thickness computed at x = 0.75, y = 0.25.

0.00E+000 1.00E-011 2.00E-011 3.00E-011-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Thi

ckne

ss (

m)

Ux

Power law index (n)0.00.20.51.05.01000.0

(a)-1.0x10-10 -8.0x10-11 -6.0x10-11 -4.0x10-11 -2.0x10-11 0.0

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Thi

ckne

ss (

m)

Uz

Power law index (n)0.00.20.51.05.01000.0

(b)

Fig. 20. Elastic displacement variation across the thickness: (a) Ux and (b) Uz.


variation of the magnetic potential across the thickness direction. It is seen that w vary dramatically acrossthe thickness of the FGM plate.

Fig. 22(a) and (b) shows the distribution of the stress components rxx (=�ryy) and rzz across the thick-ness of the FGM plate respectively. It is observed that for n = 0.0 the magnitude of rxx is higher. This phe-nomenon has been expected as the elastic displacements are higher for n = 0.0. In case of rzz distribution itis seen that magnitude of stress distribution is higher for power law index n = 1.0 where the compositionvariation is linear for functionally graded magneto-electro-elastic material. In contrast to mechanical loadhere completely different magnitude and pattern of stresses has been observed.

Fig. 23(a) and (b) shows the electric displacements Dx (=�Dy) and Dz, and the magnetic induction Bx

(=�By) and Bz are shown in Fig. 24(a) and (b) for FGM magneto-electro-elastic plate. It is observed thatelectric displacement and magnetic inductions are very much sensitive for different power law index. Forn = 0.0 the magnitude of electric displacement is higher and then it reduces slowly as increase in n and

2.5x10-1 3.0x10-1 3.5x10-1 4.0x10-1 4.5x10-1 5.0x10-1-0.15

-0.10

-0.05

0.00

0.05

0.10


0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Electric Potential (a)-4.0x10-5 -2.0x10-5 0.0 2.0x10-5 4.0x10-5 6.0x10-5 8.0x10-5

-0.15

-0.10

-0.05

0.00

0.05

0.10


0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Magnetic Potential (b) ψφ

Fig. 21. Potential distribution across the thickness: (a) electric and (b) magnetic.

8 6 4 2 0 2 4-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.0

0.2 0.5 1.0 5.0 1000.0

Thi

ckne

ss (

m)

σxx

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0-0.15

-0.10

-0.05

0.00

0.05

0.10


0.0

0.2

0.5

1.0

5.0

1000.0

Thi

ckne

ss (

m)

σzz(a) (b)

Fig. 22. Stress variation across the thickness of FGM plate: (a) rxx and (b) rzz.

0.0 4.0x10-9 8.0x10-9 1.2x10-8 1.6x10-8 2.0x10-8 2.4x10-8-0.15

-0.10

-0.05

0.10

Power law index (n)0.00.20.51.05.01000.0

Thi

ckne

ss (

m)

Thi

ckne

ss (

m)

Dx

Dz

-3.0x10-8

-2.5x10-8

-2.0x10-8

-1.5x10-8

-1.0x10-8

-5.0x10-9 0.0

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Power law index (n)0.00.20.51.05.0

0.00

0.15

0.05

(a) (b)

1000.0

Fig. 23. Variation of electric displacement across the thickness of plate: (a) Dx and (b) Dz.


-1.20E-007 -8.00E-008 -4.00E-008 0.00E+000 4.00E-008 8.00E-008-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Thi

ckne

ss (

m)

Bx

Power law index (n)0.0 0.2 0.5 1.0 5.0 1000.0

0.0 1.0x10-8 2.0x10-8 3.0x10-8 4.0x10-8 5.0x10-8 6.0x10-8-0.15

-0.10

-0.05

0.00

0.05

0.10


0.0 0.2 0.5 1.0 5.0 1000.0

Thi

ckne

ss (

m)

Bz(a) (b)

Fig. 24. Variation of magnetic induction across the thickness of plate: (a) Bx and (b) Bz.

Table 2Material coefficients of the magneto-electro-elastic plate

CoFe2O4 BaTiO3

C11 286 166C12 173 77C13 170 78C33 269.5 162C44 45.3 43

e15 0 11.6e31 0 �4.4e33 0 18.6

e31 0.08 11.2e33 0.093 12.6

l11 �5.9 0.05l33 1.57 0.1

q15 560 0q31 580 0q33 700 0

m11 0 0m33 0 0

Cij in 109 N/m2, eij in C/m2, eij in 10�9 C/V m, qij in N/(A m), lij in 10�4 N s2/C2, mij in 10�9 N s/V C.


finally reduces to zero for n = 1000.0. Magnetic induction is zero for the n = 0.0 and then magnitude in-creases and attains maximum at n = 5.0 and finally reduces to zero for n = 1000.0. This is due to the cou-pling between piezoelectric and magnetostrictive material has been neglected in the present study. Magneticinduction changes dramatically across the thickness of the FGM plate.

6. Conclusion

A series solution is assumed in the plane of the plate and finite element procedure is adopted across thethickness of the plate to solve the magneto-electro-elastic plate under mechanical and electrical loading.


The model is derived based on constitutive equation of piezomagnetic material. Coupling between elastic-ity, electric and magnetic effects are included in the analysis. The magneto electric coupling is neglected. Theresults of the present model are in good agreement with the exact benchmark solution for functionallygraded and layered magneto-electro-elastic plate given by Pan and Han (2005). The FGM plate is gradedin the thickness direction and a simple power law index will govern the magneto-electric constituents profileacross the thickness. It is found that different exponential factors in Functionally graded and layered plate(Model-I) and different power law index in functionally graded plate (Model-II) produce different magni-tudes of the response curve. As compare to mechanical loading, the electric potential produce completelydifferent normal stress components distribution for different exponential factor and different power lawindex. In addition stacking sequences have substantial effect on the induced magnetic, electric, and elasticfields. The advantage of functionally graded material model over the layered model is that there is no dis-continuity between the electric, magnetic, electric displacement and magnetic induction between the layers.It is felt that the present numerical study is highly useful for characterising FGM magneto-electro-elasticsystem for use as a sensors or actuators.

Acknowledgement

Authors would like to thank Professor E. Pan and F. Han, Department of Civil Engineering, Universityof Akron, OH, USA for giving valuable suggestion and providing literature.

Appendix

See Table 2.

References

Buchanan, G.R., 2003. Free vibration of an infinite magneto-electro-elastic cylinder. J. Sound Vib. 268, 413–426.Buchanan, G.R., 2004. Layered verses multiphase magneto-electro-elastic composites. Composites B (Engg) 35, 413–420.Chen, W.Q., Lee, K.Y., 2003. Alternative state space formulations for magneto electric thermo elasticity with transverse isotropy and

the application to bending analysis of nonhomogeneous plate. Int. J. Solids Struct. 33 (7), 977–990.Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates.

J. Sound Vib. 279, 237–251.Harshe, G., Dougherty, J.P., Newnham, R.E., 1993. Theoretical modeling of multilayered magnetoelectric composites. Int. J. Appl.

Electromag. Mater. 4, 145–159.Lage, G.R., Soares, C.M., Soares, C.A., Reddy, J.N., 2004. Layer wise partial mixed finite element analysis of magneto-electro-elastic

plates. Comput. Struct. 76, 299–317.Li, J.Y., 2000. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. Int. J.

Eng. Sci. 38, 1993–2011.Liew, K.M., Yang, J., Kitipronchai, S., 2003. Postbuckling of piezoelectric FGM plates subjected to thermo-electro-mechanical

loading. Int. J. Solids Struct. 40, 3869–3892.Nan, C.W., 1994. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B 50, 6082–

6088.Pan, E., 2001. Exact solution for simply supported and multilayered magneto-electro-elastic plates. J. Appl. Mech., ASME 68, 608–

618.Pan, E., Han, F., in press. Green�s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J.

Solids Struct., doi:10.1016/j.ijsolstr.2004.11.003.Pan, E., Han, F., 2005. Exact solution for functionally graded and layered magneto-electro-elastic plates. Int. J. Eng. Sci. 43, 321–

339.

http://dx.doi.org/10.1016/j.ijsolstr.2004.11.003


Pan, E., Heyliger, P.R., 2002. Free vibrations of simply supported and multilayered magneto-electro-elastic plates. J. Sound Vib. 252,429–442.

Wang, X., Zhong, Z., 2003. A finitely long circular cylindrical shell of piezoelectric/piezomagnetic composite under pressuring andtemperature change. Int. J. Eng. Sci. 41, 2429–2445.

Wang, J., Chen, L., Fang, S., 2003. State vector approach to analysis of multilayered magneto-electro-elastic plates. Int. J. SolidsStruct. 40, 1669–1680.

Static analysis of simply supported functionally graded ... · In this article, static analysis of...

Documents

Transcript of Static analysis of simply supported functionally graded ... · In this article, static analysis of...