Analytical Solutions for Magneto Electro Elastic Beams Under Uniform or Linear Electric and Magnetic Potential

7/25/2019 Analytical Solutions for Magneto Electro Elastic Beams Under Uniform or Linear Electric and Magnetic Potential

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Aanalytical Solutions for Magneto-electro-

elastic Beams under Uniform or Linear

Electric and Magnetic Potential

Jiang Ai-min, Wu Yi-li

West Branch of Zhejiang University of Technology

Quzhou,324000, P. R.China

E-mail: [email protected]

Abstract-For the orthotropic magneto-electro-elastic

plane problem, a series of magneto-electro-elastic

beams is solved and the corresponding exact or

analytical solutions are obtained with the trial-

and-error method on the basis of the general solution

in the case of four distinct eigenvalues, in which all

physical quantities are expressed by four

displacement functions in terms of harmonic

polynomials. These are magneto-electro-elastic

rectangular beam with rigid body displacements,

identical electric and magnetic potential, beams of

free ends under uniform electric potential and

magnetic potential, and cantilever beam under linear

electric potential and magnetic potential. The exact

and analytical solutions obtained in this paper are

also useful for study of other problems relating to

more complicated loads and boundary conditions by

the superposition principle.

Key words-analytical solution; magneto

-electro-elastic beam; harmonic function

INTRODUCTION

Magneto-electro-elastic materials possess

simultaneously magneto-electro-elastic, piezo-

magnetic and magnetoelectric effects. Liu,et

al.[1] obtained Greens functions for an infinite

two-dimensional anistropic magneto-electro-

elastic medium containing an elliptical cavity

based on the extended Stroh Formalism. Pan[2]

derived three-dimensional Greens functions in

anisotropic magneto-electro-elastic full space,

half space, and bi-materials based on the

extended Stroh formalism by applying the

two-dimensional Fourier transforms. Wang and

Shen[3] obtained the general solution expressed

by five harmonic functions and applied the

derived general solution to find the fundamental

solution for a generalized dislocation and also to

derive Greens functions for a semi-infinite

magneto-electro-elastic solid. Hou,et al.[4]

analyzed the elliptical Hertizan contact of

transversely isotropic magneto-electro-elastic

bodies with the general solutions in terms of

harmonic functions. Chen,et al[5] obtained

analytical solutions of simply supported

magneto-electro-elastic circular plate under

uniform loads with a general solution in forms of

harmonic functions. Ding and Jiang[6] obtained

the fundamental solution of infinite magneto-electro-elastic solid with the method of trial-and-

error on the basis of the general solution in the

case of distinct eigenvalues , which is expressed

in five harmonic functions and the same as in

Wang and Shen[3], and derived the boundary

integral formulation. Jiang and Ding[7] derived

the general solution in terms of four harmonic

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displacement functions for magneto-

electro-elastic plane problem and solved a series

of problems by the trial-and-error method.

In this paper, we continue to obtain the

analytical solutions for the magneto-electro-elastic

plane problems on the basis of Jiang and Ding[7].

With the trial-and-error method, the exact solution

for beam of free ends under uniform electrical

potential and magnetic potential are derived.

Furthermore, we also give out the analytical

solutions in harmonic polynomials to cantilever

beam under linear electric potential and magnetic

potential on the upper and bottom surfaces. All

these solutions can serve as benchmarks for

numerical methods such as the finite element

method, the boundary element method, etc.

. GENERAL SOLUTION TO THE PLANE

PROBLEM OF

MAGNETO-ELECTRO-ELASTIC SOLID

For the magneto-electro-elastic plane-strain

problems, the general solution in the case of

distinct eigenvalues has been derived and

expressed in four harmonic functions in Jiangand Ding[7] as follows:

=

=

4

1j

j

xu

,

= =

4

1j j

j

mjjm zksw

,

=

=

4

12

2

4

j j

j

jxz

,

=

=

4

12

2

j j

j

mjmz

,

=

=

4

1

2

j j

j

mjjmzx

s

, )3,2,1( =m (1)

where the generalized displacements and stresses

are defined as follows:

ww =1 , w2 = , =3w ,

z =

1 ,

zD=2 , zB=3 ,

xz =

1 ,

xD=2 , xB=3 (2)

),(),( wuxzz )( zx DD and )( zx BB are the

components of stress, displacement, electric

displacement and magnetic induction,

respectively; and are the electric

potential and magnetic potential, respectively.

The functions j in (1) satisfy the

following equations :

02

2

2

2

=

+

j

jzx , )4~1( =j (3)

where zsz jj = , js are the four roots of the

following equations (We take Re ( js )0) :

052

4

4

3

6

2

8

1 =++ asasasasa (4)

where the coefficients )3~1(, =mk mjmj

in (1) and )5~1( =nan in (4) are the same as

those listed in Hou, et al.[4].

.EXACT SOLUTIONS FOR TWO

SIMPLE PROBLEMS

A. Rigid Body Displacements Identical Electric

Potential and Magnetic Potential

In the paper of Jiang and Ding[7], we have

obtained the exact solution as follows :

zuu 00 += , xww 00 = ,

0= , 0= (5)

0===xzzx , 0== zx DD ,

0==zx BB (6)

where 0000 ,,, wu and 0 are constants

denoting rigid body displacements, rigid body

rotation, identical electric potential and magnetic

potential .

Beam of Two Free Ends under Uniform

Electrical Potential and Magnetic Potential

we constitute the displacement function as

)( 222 jjj zxA = , )4~1( =j 7

where jA2 are unknown constants to be

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determined.

Substituting (7) into (1), we will have

==

==4

1

2

4

1

22,2

j

jjmjjm

j

j zAkswAxu

,2,04

1

24=

==j

jjxm A

=

=4

1

22

j

jmjm A , )3~1( =m (8)

The boundary conditions are

2/hz = 0=z 0=xz

2/0

= , 2/0

= (9a)

2/Lx = 0=x 0=xz

0=x

D , 0=x

B 9b

Substituting the expressions of x , z ,

and into (9) leads to :

==

==

==

==

,2,0

,2,0

0

4

1

23

24

1

24

0

4

1

22

24

1

21

j

jjj

j

jj

j

jjj

j

jj

AkshA

AkshA

10

The four unknown constants jA2 can be

calculated from (10). Superposing the solutionabove on the exact solution (5, 6) for a

rectangular beam with identical electrical

potential, and taking 000 ==wu and

00 = , we will have the following solution

,2,24

1

21

24

1

2 ==

==j

jjj

j

j AkszwAxu

=

=

4

1

22

2

0 ,2j

jjj Aksz

,2

4

123

2

0

== jjjj Aksz

,0===xzzx

,24

1

22=

=j

jjz AD

,0,0 ==xx BD

=

=4

1

23 ,2j

jjz AB (11)

Because 0 and 0 are two arbitrary

constants, one can appoint arbitrary uniform

electrical potentials on the two surfaces

( 2/hz = ). For the same reason, one can also

appoint arbitrary uniform magnetic potentials as

0 and 0 are arbitrary constants. The

solution (11) is the exact one for beam of two

free ends under uniform electrical potential.

. CANTILEVER BEAM WITH CROSS

FORCE, POINT CHARGE AND POINT

CURRENT AT FREE END

The boundary conditions of a cantilever

beam with cross force P in zdirection, pointcharge Q and point current Jat free end, are

2/hz = 0=z 0=xz

0=z

D 0=z

B (12a)

0=x 0=x 12/

2/Qdz

h

h xz =

+

2

2/

2/QdzD

h

h x =

+

3

2/

2/QdzD

h

h x =

+

(12b)

Lx= , 0=z 0=u 0=w

0/ = xw (12c)

where QQPQ == 21 , and JQ =3 .

In the paper of Jiang and Ding[7], the

corresponding analytical solution has been

obtained to constitute the displacement function

and superposing the solution for magneto-

electro-elastic beam with rigid body

displacements, identical electric potential and

magnetic potential.

. CANTILEVER BEAM UNDER

LINEARELECTRICAL AND MAGNETICPOTENTIAL ON THE SURFACES

we constitute the displacement function:

)3( 2332 jjjjj xzxAxzB +=

)4~1( =j 13

where jB2 and jA3 are unknown constants to

be determined.

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Substituting (13) into (1), we have

=

+=4

1

3

22

2 ])33([j

jjjj AzxBzu

=

=

4

1

32 )6(j

jjjmjjm AxzxBksw ,

== ==4

13

4

134 6,6

jjmjm

jjjx AxAx

=

=4

1

32 )6(j

jjjmjjm AzBs

)3~1( =m 14The boundary conditions are:

2/hz += 0=z xh

x 1)2

,( =+

0=xz xhx 2)2/,( =+ (15a)

2/hz = 0=z xh

x 1)

2

,( = ,

0=xz xhx 2)2/,( = (15b)

0=x +

=

2/

2/0

h

h xdz

+

=

2/

2/0

h

h xzdz

+

==

2/

2/)3~1(,0

h

h m mdz (15c)

0, == zLx 0=u 0=w

0/ = xw 15dSubstituting (14) into (15a, 15b) and the

second equation of (15d), yields

,01

1 31==

j jj

A ,03

1 31

2

==

j jjj

As

,2/)(34

1

1132

2=

=j

jjj Aksh

2/)(34

1

2233

2=

=j

jjj Aksh 16

,04

1

21=

=j

jjj Bs ,0

4

1

21=

=j

jjj Bks

,2/)(4

1

1122=

+=j

jjj Bks

2/)(4

1

2223=

+=j

jjj Bks 17

Then,jA3 can be determined from (16) and jB2

from (17). Furthermore, in order to satisfy the

third equation of (15c), one should superpose the

solution above on the corresponding analytical

solution for a cantilever magneto-electro-elastic

beam under loads at free end discussed in

Section, and let

+

=

==2/

2/

4

1

2

h

hj

jmjjmm BshdzQ ,

)3~1( =m 18Furthermore, the first equation of (15d) can be

satisfied by superposing on the rigid body

displacement determined as follows

=

==4

1

3

2

0 3j

jALuu (19)

ACKNOWLEDGMENT

The work was supported by Zhejiang Natural

Science Foundation (No.Y605040, Y608118).

REFERENCES

[1] J. X. Liu, X. L. Liu, and Y. B. Zhao,Greens functions

for anisotropic magneto-electro-elastic solids with an

elliptical cavity or a crack, Int. J. Eng. Sci. vol.39,

pp.1405-1418, 2001.

[2] E. Pan, Three-dimensional Greens function in

anisotropic magneto-electro-elastic bimaterials , Z.

angew. Math.Phys, vol.53, pp.815-838, 2002.

[3] X. Wang, Y. P. Shen, The general solution of

three-dimensional problems in magneto-electro-elastic

media, Int. J. Eng. Sci. vol.40, pp.1069-1080, 2002.

[4] P. F. Hou, Y. T. Leung Andrew, and H. J. Ding, The

elliptical Hertizan contact of transversely isotropic

magneto-electro-elastic bodies, Int. J. Solids Struct. ,

vol.40, pp.2833~2850, 2003.

[5] J. Y. Chen, H. J. Ding, and P. F. Hou, Analytical

solutions of simply supported magneto-electro-elastic

circular plate under uniform loads, Journal of Zhejiang

University SCIENCE, vol.4(5), pp.560-564, 2003.

[6] H. J. Ding, A. M. Jiang, Fundamental solutions for

transversely isotropic magneto-electro-elastic media

and boundary integral formulation , Science in

China(Series E), vol.46(6), pp.607-619, 2003.

[7] A. M. Jiang, H. J. Ding, Analytical solutions to

magneto-electro-elastic beams, Structural Engineering

and Mechanics, vol.18(2), pp.195-209, 2004

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Analytical Solutions for Magneto Electro Elastic Beams Under Uniform or Linear Electric and Magnetic Potential