International Journal of Solids and Structures 60–61 (2015) 66–74

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsolst r

Static deformation of a spherically anisotropic and multilayeredmagneto-electro-elastic hollow sphere

http://dx.doi.org/10.1016/j.ijsolstr.2015.02.0040020-7683/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 330 972 6739 (O); fax: +1 330 972 6020.E-mail address: pan2@uakron.edu (E. Pan).

J.Y. Chen a, E. Pan b,⇑, P.R. Heyliger c

a School of Mechanical Engineering, Zhengzhou University, Zhengzhou 450001, PR Chinab Department of Civil Engineering, University of Akron, OH 44325, USAc Department of Civil and Environmental Engineering, Colorado State University, CO 80523, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 December 2014Received in revised form 3 February 2015Available online 13 February 2015

Keywords:Spherical system of vector functionsPropagation matrix methodMultiferroics compositeMultilayered MEE sphere

In this paper, we present an analytical solution on the general static deformation of a sphericallyanisotropic and multilayered magneto-electro-elastic (MEE) hollow sphere. We first express the generalsolution in each layer in terms of the spherical system of vector functions where two transformations ofvariables are also proposed to achieve the analytical results. The spherical system of vector functions canbe applied to expand any vector as well as scalar function, and it further automatically separates the stat-ic deformation into two independent sub-problems: The LM-type and N-type. The LM-type is associatedwith the spheroidal deformation and is coupled further with the electric and magnetic fields. The N-typeis associated with the torsional deformation and is purely elastic and independent of the electric andmagnetic fields. To solve the multilayered spherical problem, the propagation matrix method is intro-duced with the propagation matrix being simply the exponential matrix for each layer. By assumingthe continuity conditions on the interface between the adjacent spherical shells, the solution can be sim-ply propagated from the inner surface to the outer surface of the layered and hollow MEE sphere so thatspecific boundary value problems can be solved. As numerical examples, a three-layered sandwich hol-low sphere with different stacking sequences under different boundary conditions is studied. Our resultsillustrate the influence of the stacking sequences while showing the effectiveness of the proposedmethod.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Magneto-electro-elastic (MEE) materials and structures attractmany researchers because of their excellent ability of convertingenergy among the electric, mechanical, and magnetic fields. Assuch, they have huge potential applications in smart sensors,actuators, filters among others, as reviewed in material communityas multiferroics and multiferroic composites (Eerenstein et al.,2006; Nan et al., 2008).

In the past two decades, studies of these solids were concentrat-ed primarily on MEE layered plates under both static and dynamicdeformation. Pan (2001) and Wang and co-workers (2003) ana-lyzed multilayered MEE rectangular plates under static loadingswith the propagation matrix method. In constructing thepropagation matrix, they employed the Stroh formalism and thestate-space method. Pan and Heyliger (2002) and Chen andco-workers (2007a) investigated the corresponding free vibration

problem in the layered rectangular and multilayered MEE plates.The dispersion relation of wave propagation in multilayered MEEplates with infinite dimensions in the horizontal plane was derivedand analyzed by Chen and co-workers (2007b) and Yu and co-workers (2012). More recent studies include guided waves in lay-ered MEE bars with rectangular cross-sections (Yu et al., 2014) andfree vibrations in layered plates under more complicated lateralboundary conditions (Chen et al., 2014).

Compared to the horizontally layered structure, a layeredsphere poses more of a challenge. Ding and Chen (1996) calculatedthe natural frequencies of an elastic spherically isotropic hollowsphere submerged in a compressible fluid medium. Based on thestate-space method and variables separation techniques, Chenand Ding (2001) and Chen and co-workers (2001) investigated,respectively, the free vibration of hollow elastic sphere and thestatic deformation of multilayered piezoelectric hollow spheres.Heyliger and Wu (1999) derived an analytical solution for the radi-al deformation of a layered piezoelectric sphere. However, to thebest of the authors’ knowledge, the general static deformation ofa multilayered MEE hollow sphere with spherical anisotropy has

J.Y. Chen et al. / International Journal of Solids and Structures 60–61 (2015) 66–74 67

not been analyzed so far. Furthermore, our recent studies indicatethat a spherical and layered shell structure could be more efficientand powerful than a cylindrical and layered shell in terms of theproduced coupling effect between the magnetic and electric fields(under preparation). This absence motivates the present study.

This paper is organized as follows: After the introduction, inSection 2, we describe the problem to be solved and present thebasic equations in spherical coordinates for the sphericallyanisotropic MEE material. In Section 3, we introduce the sphericalsystem of vector functions along with its properties. In Section 4,the general solutions in the analytical form for each layer andthe propagation matrix in terms of the exponential matrix arederived. Numerical examples are presented in Section 5 and con-clusions are drawn in Section 6.

2. Problem description and basic equations

We assume, as shown in Fig. 1, a spherically anisotropic and p-layered MEE hollow sphere. The ith layer has a radius of ri on itsinterior interface and ri+1 on its exterior interface. Thus, the innerand outer surface of the layered sphere has, respectively, a radiusof r1 and rp+1, and is under suitable boundary conditions which willbe presented later. The interface between the adjacent layers isassumed to be perfectly connected. In other words the extendeddisplacements and tractions in r-direction are continuous. Spheri-cal anisotropy means that in the spherical coordinates (r, h, /), thecenter of the spherical isotropy is coincident with the origin andthe symmetry axis of the material is along r-direction. We furthermention that only under this assumption of spherical anisotropy,we can derive the analytical solutions as presented in this paper;the general anisotropic case has to be solved numerically.

For this general problem, the following equations must be sat-isfied at every point inside each of the individual layers in termsof the spherical coordinates. The general constitutive relations ofthe MEE structure with spherical anisotropy are given as follows:

rhh ¼ c11chhþ c12cuuþ c13crr � e31Er �q31Hr ; rrh ¼ 2c44crh� e15Eh�q15Hh

ruu ¼ c12chhþ c11cuuþ c13crr � e31Er �q31Hr ; rru ¼ 2c44cru� e15Eu�q15Hu

rrr ¼ c13chhþ c13cuuþ c33crr � e33Er �q33Hr ; rhu ¼ 2c66chu

Dh ¼ 2e15chr þ e11Ehþa11Hh; Du ¼ 2e15cur þ e11Euþa11Hu

Dr ¼ e31ðchhþcuuÞþ e33crr þ e33Er þa33Hr

Bh ¼ 2q15chr þa11Ehþl11Hh; Bu ¼ 2q15cur þa11Euþl11Hu

Br ¼ q31ðchhþcuuÞþq33crr þa33Er þl33Hr

ð1Þ

Fig. 1. A hollow MEE sphere made of p layers with its inner surface at r1 and outersurface at rp+1. Both surfaces are subjected to suitable extended displacement andtraction boundary conditions.

where rij, Di and Bi are the stress, electric displacement and mag-netic induction, respectively; cij, Ei and Hi are the strain, electricfield and magnetic field, respectively; cik, eik and lik are the elastic,dielectric, and magnetic permeability coefficients, respectively; eik

and qik are the piezoelectric and piezomagnetic coefficients, respec-tively; aij are the electromagnetic coefficients. It is noted that anadditional relationship c11 = c12 + 2c66 holds for the sphericallyanisotropic material with the r-axis being the axis of symmetry.

In Eq. (1), the strains, electric and magnetic fields are related tothe elastic displacement and electric and magnetic potentials by

crr ¼ @ur@r ; chh ¼ 1

r@uh@h þ

urr ; cuu ¼ 1

r sin h@uu@u þ

uhr cot hþ ur

r

2crh ¼ @uh@r �

uhr þ 1

r@ur@h ; 2cru ¼ 1

r sin h@ur@u þ

@uu@r �

uur

2chu ¼ 1r@uu@h �

uur cot hþ 1

r sin h@uh@u

Er ¼ � @/@r ; Eh ¼ � 1

r@/@h ; Eu ¼ � 1

r sin h@/@u

Hr ¼ � @w@r ; Hh ¼ � 1

r@w@h ; Hu ¼ � 1

r sin h@w@u

ð2Þ

where ui (i = r, h, u) are components of elastic displacement, / andw are electric and magnetic potentials.

Under static deformation and in the absence of external forces,the equations of equilibrium are written in the following form:

@rrr@r þ 1

r@rrh@h þ 1

r sin h@rru@u þ 1

r ð2rrr � rhh � ruu þ rrh cot hÞ ¼ 0@rrh@r þ 1

r@rhh@h þ 1

r sin h@rhu@u þ 1

r ½ðrhh � ruuÞ cot hþ 3rrh� ¼ 0@rru@r þ 1

r@rhu@h þ 1

r sin h@ruu@u þ 1

r ð3rru þ 2rhu cot hÞ ¼ 0

ð3Þ

Similarly, the static Maxwell equations of the electric and magneticfields without electric and magnetic sources are given as

@Dr

@rþ 1

r@Dh

@hþ 1

r sin h@Du

@uþ 1

rð2Dr þ Dh cot hÞ ¼ 0

@Br

@rþ 1

r@Bh

@hþ 1

r sin h@Bu

@uþ 1

rð2Br þ Bh cot hÞ ¼ 0

ð4Þ

3. Spherical system of vector functions

To solve the problem described in Section 2, we introduce thefollowing spherical system of vector functions (Ulitko, 1979):

Lðh;u; n;mÞ ¼ erSðh;u; n;mÞMðh;u; n;mÞ ¼ rrS ¼ eh@h þ eu

@usin h

� �Sðh;u; n;mÞ

Nðh;u; n;mÞ ¼ rr� ðerSÞ ¼ ðeh@u

sin h� eu@hÞSðh;u; n;mÞ

ð5Þ

where er, eh, and eu are the unit vectors, respectively, along r-, h-and u-directions, and S is a scalar function defined by

Sðh;u; n;mÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2nþ 1Þðn�mÞ!

4pðnþmÞ!

sPm

n ðcos hÞeimu;

jmj 6 n and n ¼ 0;1;2; . . . ð6Þ

with Pmn being the associated Legendre function (Abramovitz and

Stegun, 1972). It is noted that the scalar function S satisfies the fol-lowing Helmholtz equation

1sin h

@

@hsin h

@S@h

� �þ 1

sin2 h

@2S@u2 þ k2S ¼ 0 ð7aÞ

or

@S2

@h2 þ cot h@S@hþ 1

sin2 h

@2S@u2 þ k2S ¼ 0 ð7bÞ

where k2 = n(n + 1).The following identity for any f(h,u) is also very useful when

deriving our solutions

68 J.Y. Chen et al. / International Journal of Solids and Structures 60–61 (2015) 66–74

@

@h1

sin h@f@u

� �þ cot h

sin h@f@u� 1

sin h@2f@h@u

¼ 0 ð8Þ

It is easy to show that the spherical system of vector functions (5) iscomplete and orthogonal in the following sense.R 2p

0 duR p

0 Lðh;u; n;mÞ � �Lðh;u; n0;m0Þ sin hdh ¼ dnn0dmm0R 2p0 du

R p0 Mðh;u; n;mÞ � �Mðh;u; n0;m0Þ sin hdh ¼ k2dnn0dmm0R 2p

0 duR p

0 Nðh;u; n;mÞ � �Nðh;u; n0;m0Þ sin hdh ¼ k2dnn0dmm0

ð9Þ

where dij are the components of the Kronecker delta and an overbarindicates complex conjugate.

4. Solutions in terms of spherical system of vector functions

4.1. Expansions of physical quantities in each spherical layer

Owing to the orthogonal properties (9), we can expand any vec-tor, such as the elastic displacement u, electric potential / andmagnetic potential w, as

uðr;h;uÞ¼X1n¼0

Xn

m¼�n

½ULðrÞLðh;uÞþUMðrÞMðh;uÞþUNðrÞNðh;uÞ�

¼X1n¼0

Xn

m¼�n

eh UMðrÞ@hþUNðrÞ@u

sinh

� �þeu

(UMðrÞ

@u

sinh�UNðrÞ@h

)þerULðrÞ

#Sðh;uÞ

/ðr;h;uÞ¼X1n¼0

Xn

m¼�n

UðrÞSðh;uÞ

wðr;h;uÞ¼X1n¼0

Xn

m¼�n

WðrÞSðh;uÞ:

ð10Þ

For easy presentation, the scalar function S(h, u; n, m) is written asS(h, u) or S later on. For the same reason, the dependence of theexpansion coefficients UL, UM, UN, U and W on (n,m) is also omitted.

Similarly, the traction vector t, the electric displacement vectorD and magnetic induction vector B can be expressed as

tðr; h;uÞ � rrrer þrrheh þrrueu

¼X1n¼0

Xn

m¼�n

erTLSþ eh TM@S@hþ TN

sin h@S@u

� �þ eu

TM

sin h@S@u� TN

@S@h

� �

Dðr; h;uÞ ¼X1n¼0

Xn

m¼�n

erDLSþ eh DM@S@hþ DN

sin h@S@u

� �þ eu

DM

sin h@S@u�DN

@S@h

� �

Bðr;h;uÞ ¼X1n¼0

Xn

m¼�n

erBLSþ eh BM@S@hþ BN

sin h@S@u

� �þ eu

BM

sin h@S@u� BN

@S@h

� �

ð11Þ

Also in terms of the spherical system of vector functions, the strains,the electric and magnetic fields can be expressed by the coefficientsof the elastic displacements and the electric and magnetic potentials

crr ¼ U0LS; chh ¼UL

rSþ 1

rUM

@2S

@h2 þ UN@

@h@S

sin h@u

!ð12aÞ

cuu ¼UL

rSþ 1

r sin hUM

sin h@2S@u2 � UN

@S2

@h@u

!þ cot h

rUM

@S@hþ UN

sin h@S@u

� �

ð12bÞ

2crh ¼UL

r@S@hþ U0M

@S@hþ U0N

sin h@S@u

� �� UM

1r@S@hþ UN

r sin h@S@u

� �ð12cÞ

2cru ¼1

r sin hUL

@S@uþ U0M

sin h@S@u� U0N

@S@h

� �� UM

r sin h@S@u� UN

1r@S@h

� �ð12dÞ

2chu ¼1r

UM@

@h1

sin h@S@u

� UN

@2S

@h2

!� cot h

rUM

sin h@S@u� UN

@S@h

� �

þ 1r sin h

UM@2S@h@u

þ UN

sin h@2S@u2

!ð12eÞ

Er ¼ �U0S; Eh ¼ �Ur@S@h

; Eu ¼ �U

r sin h@S@u

ð12fÞ

Hr ¼ �W0S; Hh ¼ �Wr@S@h

; Hu ¼ �W

r sin h@S@u

ð12gÞ

in which the superscript prime ‘‘0’’ indicates the derivative withrespect to the radial coordinate r.

Substituting Eqs. (11) and (12) into Eq. (1) and comparing thecoefficients on both sides of the resulting equations, we find

TL ¼ 2c13UL

r� k2c13

UM

rþ c33U0L þ e33U

0 þ q33W0

TM ¼ c44UL

rþ U0M �

UM

r

� �þ e15

Urþ q15

Wr

TN ¼ c44 U0N �UN

r

� �

DL ¼ 2e31UL

r� k2e31

UM

rþ e33U0L � e33U

0 � a33W0

BL ¼ 2q31UL

r� k2q31

UM

rþ q33U0L � a33U

0 � l33W0

ð13Þ

Furthermore, making use of Eqs. (1)–(4) and (11), we obtain

T 0L�k2 TM

rþ2TL

r�1

rð2c13U0L�2e31U

0 �2q31W0Þ�2UL

r2 Cþ 1r2 Ck2UM ¼0

T 0Mþ3r

TMþ1rðc13U0Lþe31U

0 þq31W0Þþ 1

r2 CULþUM

r2 2c66ð1�k2Þ�k2c12� �

¼0

T 0Nþ3r

TNþ1r2

UNc66ð2�k2Þ¼ 0

D0L�k2 e15

r1r

ULþU0M�UM

r� e11

e15

1rU�a11

e15

1rW

� �þ2

rDL ¼0

B0L�k2 q15

r1r

ULþU0M�UM

r�a11

q15

1rU�l11

q15

1rW

� �þ2

rBL ¼0

ð14Þwhere C = c11 + c12.

It is obvious from Eqs. (13) and (14) that TN and UN are uncou-pled form other variables and that they are purely elastic, meaningthat they are independent of the electric and magnetic fields. Wecall this the N-type problem, which is also associated with the tor-sional deformation only. The N-type problem satisfies the follow-ing set of first-order differential equations

U0NT 0N

" #¼

1=r 1=c44

� c66r2 ð2� k2Þ �3=r

" #UN

TN

ð15Þ

The remaining part is called LM-type problem, which couples theelastic, electric and magnetic fields together. The deformation inthis case is associated with the spheroidal deformation. It is gov-erned by the following set of first-order differential equations

R11 0R21 rI

U 0

T 0

¼

H11=r I

H21=r H22

UT

ð16Þ

in which each sub-matrix is given by

U ¼ ½UL UM U W �t ; T ¼ ½ TL TM DL BL �t ð17aÞ

R11 ¼

c33 0 e33 q33

0 c44 0 0e33 0 �e33 �a33

q33 0 �a33 �l33

26664

37775; R21 ¼

�2c13 0 2e31 2q31

c13 0 e31 q31

0 �k2e15 0 00 �k2q15 0 0

26664

37775

ð17bÞ

J.Y. Chen et al. / International Journal of Solids and Structures 60–61 (2015) 66–74 69

H11 ¼

�2c13 k2c13 0 0�c44 c44 �e15 �q15

�2e31 k2e31 0 0�2q31 k2q31 0 0

26664

37775 ð17cÞ

H21 ¼

2C �k2C 0 0

�C �2c66½1� k2� þ k2c12 0 0

k2e15 �k2e15 �k2e11 �k2a11

k2q15 �k2q15 �k2a11 �k2l11

2666664

3777775;

H22 ¼

�2 k2 0 0

0 �3 0 0

0 0 �2 0

0 0 0 �2

266664

377775 ð17dÞ

Both Eq. (15) for the N-type and (16) for the LM-type are first-orderdifferential equations with variable coefficients. In other words,their coefficients are functions of the variable r. To find the generalsolutions to these equations, we first recast Eqs. (15) and (16) into

rU0Nr2T 0N

" #¼

1 1=c44

�c66ð2� k2Þ �3

UN

rTN

R11 0R21 I

rU 0

r2T 0

¼

H11 I

H21 H22

UrT

ð18Þ

To solve Eq. (18), we let r ¼ rien, with 0 6 n 6 ni and ni ¼ lnðriþ1=riÞto change Eq. (18) to (derivative is now with respect to n)

U0NrT 0N

" #¼

1 1=c44

�c66ð2� k2Þ �3

UN

rTN

R11 0R21 I

U 0

rT 0

¼

H11 I

H21 H22

UrT

ð19Þ

Introducing

�TN ¼ rTN; �T ¼ rT ð20Þ

we obtain the following first-order differential equations with con-stant coefficients

U0N�T 0N

" #¼ BN UN

�TN

U 0

�T 0

¼ B

U�T

ð21Þ

In Eq. (21),

BN ¼1 1=c44

�c66ð2� k2Þ �2

B ¼R�1

11 0

�R21R�111 I

" #H11 IH21 H22 þ I

ð22Þ

Fig. 2. Variation of the dimensionless elastic displacement component urc44=ðqr2Þin (a) and the dimensionless stress component rrr/(qc44) in (b) along the radialdirection (r/r2) in a spherically anisotropic piezoelectric hollow sphere made ofBaTiO3 under a uniform external pressure q on its outer surface r2. The electricshorted circuit is applied on both the inner (r = r1) and outer (r = r2) surfaces.

4.2. Solution and propagation matrices of N- and LM-typedeformations

4.2.1. Spherically symmetric deformationBefore we present the solutions to Eq. (21) and the correspond-

ing propagation matrices, we first discuss the special and simpledeformation corresponding to n = 0 and m = 0 in our spherical sys-tem of vector functions. It is obvious that for this case we do nothave the N-type solution, and that the LM-type solution is reducedto the solution associated with L component only. More specifical-ly, when n = 0, we find that Eq. (16) needs to be reorganized byintroducing the extended displacements and tractions as

U ¼ ½UL;U;W�t; T ¼ ½TL;DL;BL�t ð23Þ

This is equivalent to removing the second row and second columnin the involved matrices while also letting n = 0 in other elements.Thus, the size of the matrices defined in Eq. (17) will all be reducedto 3 � 3, with the following new elements

R11 ¼c33 e33 q33

e33 �e33 �a33

q33 �a33 �l33

264

375; R21 ¼

�2c13 2e31 2q31

0 0 00 0 0

264

375 ð24aÞ

H11 ¼�2c13 0 0�2e31 0 0�2q31 0 0

264

375; H21 ¼

2C 0 00 0 00 0 0

264

375;

H22 ¼�2 0 00 �2 00 0 �2

264

375 ð24bÞ

Therefore, instead of a 4 � 4 system, for the special case of n = 0,all the involved intermediate matrices are 3 � 3. We also point outthat since n = 1 involves rigid-body motion (Watson and Singh,1972), its solution will not be discussed in this paper.

4.2.2. Solution matrices and propagation matricesWe present the solutions and propagation matrices for both the

N-type and LM-type. For the LM-type the solution correspondingto n = 0 will be formally the same as for the case of n P 2, but withthe involved vector and matrix sizes being reduced, and with theintermediate matrices for n = 0 being those given by Eq. (24).

For each given layer with constant material properties, the solu-tions of Eq. (21) can be assumed as

UNðnÞ�TNðnÞ

¼ expðBNnÞ

UNð0Þ�TNð0Þ

UðnÞ�TðnÞ

¼ expðBnÞ

Uð0Þ�Tð0Þ

ð25Þ

Table 1Material properties of piezoelectric BaTiO3, magnetic CoFe2O4, and MEE materials MEE which is made of 50% BaTiO3 and 50% CoFe2O4 (cij in 109 N/m2, eij in C/m2, qij in N/Am, eij in10�9 C2/(Nm2), lij in 10�6 Ns2/C2, aij in 10�12 Ns/VC).

Properties BaTiO3 CoFe2O4 MEE Properties BaTiO3 CoFe2O4 MEE

c11 166 286 213 e31 = e32 �4.4 0 �2.71c22 166 286 213 e33 18.6 0 8.86c12 77 173 113 e24 = e15 11.6 0 0.15c13 78 170.5 113 q31 = q32 0 580.3 222c23 78 170.5 113 q33 0 699.7 292c33 162 269.5 207 q24 = q15 0 550 185c44 43 45.3 49.9 e11 = e22 11.2 0.08 0.24c55 43 45.3 49.9 e33 12.6 0.093 6.37c66 44.5 56.5 50 l11 = l22 5 590 201a11 0 0 �5.23 l33 10 157 83.9a33 0 0 2750 – – – –

70 J.Y. Chen et al. / International Journal of Solids and Structures 60–61 (2015) 66–74

For ith layer, with its inner and outer interfaces at r = ri and ri+1,which corresponds 0 and ni, the extended displacement and trac-tion components at its interfaces are connected by

UNðniÞ�TNðniÞ

¼ expðBN

i niÞUNð0Þ�TNð0Þ

¼ expðBN

i niÞUNðni�1Þ�TNðni�1Þ

UðniÞ�TðniÞ

¼ expðBiniÞ

Uð0Þ�Tð0Þ

¼ expðBiniÞ

Uðni�1Þ�Tðni�1Þ

ð26Þ

We point out that there is no summation over the repeatedindex i on the right-hand side of Eq. (26). With the relation (26),we can propagate the solution from the inner surface (r1, n = 0)to the outer surface (rp+1, np ¼ lnðrpþ1=rpÞ) of the layered MEE hol-low sphere to arrive at

UNðnpÞ�TNðnpÞ

" #¼ pN UNð0Þ

�TNð0Þ

UðnpÞ�TðnpÞ

" #¼ p

Uð0Þ�Tð0Þ

ð27Þ

where

pN ¼ expðBNp npÞ expðBN

p�1np�1Þ . . . expðBN1 n1Þ

p ¼ expðBpnpÞ expðBp�1np�1Þ . . . expðB1n1Þð28Þ

Eq. (27) is a simple relation and, for given boundary conditions onboth the inner and outer surfaces, can be solved for the involvedunknowns. We present the following example for the LM-type toillustrate.

We assume that, at the outer surface r = rp+1, the radial tractionin its dimensionless form is applied, as

rrr ¼ r0Pnðcos hÞ=cmax ð29Þ

where cmax is the maximum value of all elastic coefficients amongall layers and Pn represents the n-order Legendre function. In addi-tion, we assume that all other elastic traction components on boththe inner and outer surfaces are zero and the electric and magneticfields are short circuited at those locations (i.e., U(r1) = U(rp+1) =W(r1) = W(rp+1) = 0).

Then, Eq. (27) for the LM-type can be written as

U1ðnpÞ0

rpþ1 �rrr

�T2ðnpÞ

26664

37775 ¼

p11 p12 p13 p14

p21 p22 p23 p24

p31 p32 p33 p34

p41 p42 p43 p44

26664

37775

U1ð0Þ00

�T2ð0Þ

26664

37775 ð30Þ

in which �rrr ¼ r0=cmax, U1 ¼ ½UL UM �t , �T2 ¼ ½ rDL rBL �t . From Eq.(30), one can determine the unknowns at the inner surface r = r1 as

U1ð0Þ�T2ð0Þ

¼

p21 p24

p31 p34

�1 0rpþ1 �rrr

ð31Þ

With the solved coefficients at the inner surface, a propagatingrelation similar to Eq. (26) can be applied to find the expansioncoefficients at any r-level in any given layer. For instance, to obtainthe coefficients of the extended displacement and traction vectorsat r, with rj�1 < r < rj in layer j, we propagate the solution from theinner surface to find

UðnÞ�TðnÞ

¼ expðBjnÞ expðBj�1nj�1Þ . . . expðB1n1Þ

Uð0Þ�Tð0Þ

ð32Þ

Substituting the expansion coefficients from Eq. (32) into Eqs. (10)and (11), we can then obtain the extended displacements and trac-tions at any r-level in any layer as functions of spherical coordinates(r, h, u). Thus, the boundary value problem is finally solved.

5. Numerical examples

5.1. Verification of the analytical solution

Before presenting the numerical results, we have first comparedour solution to existing solutions for the reduced cases. We applyour formulation to the reduced piezoelectric case where a piezo-electric hollow sphere is made of BaTiO3 with its inner radius, r1,being half of its outer radius r2. It is assumed that the short-circuit-ed electric boundary condition is applied to both the inner andouter surfaces and that a uniform external pressure q (i.e., rrr = �q)is applied on the outer surface of the sphere while other elastictraction components are zero on both the inner and outer surfaces.The material coefficients of the piezoelectric BaTiO3 are the sameas those in Chen and co-workers (2001). Fig. 2a and b show the dis-tribution of the dimensionless stress rrr/(qc44) and the dimension-less displacement urc44=ðqr2Þ. These distributions are identical tothose of Chen and co-workers (2001).

5.2. A hollow sandwich sphere under uniform external pressure

Having validated our solutions, we now apply them to a three-layered sandwich hollow sphere made of magneto-electro-elasticmaterials. We first consider the spherically symmetric deformationcorresponding to n = 0. The following three different stackingsequences are studied (from inner layer to the outer layer): (1)B/F/B, (2) F/B/F, (3) MEE/MEE/MEE (simply MEE), where ‘‘B’’denotes BaTiO3, ‘‘F’’ denotes CoFe2O4, and ‘‘MEE’’ denotes theMEE material made of 50% BaTiO3 and 50% CoFe2O4. The third caseactually corresponds to a homogeneous spherical shell made of thecoupled MEE material. We denote the outer radius of the layeredhollow sphere by r4 = R, and let the inner radius of the hollowsphere be located at r1 = 0.4R with each of the three layers havingequal thickness of 0.2R. The material constants are listed in Table.1and are taken from Chen and co-workers (2007a,b) and Xue andPan (2013) with the MEE material properties being predicted based

Fig. 3. Variation of the extended displacements and stresses along the radial direction of the hollow sandwich sphere made of magneto-electro-elastic materials with itsouter surface being under uniform pressure p = r0/cmax. The dimensionless elastic displacement component urcmax/(Rr0) in (a), electric potential /emax/r0 in (b), magneticpotential wqmax/r0 in (c), radial stress rrr/r0 in (d), circumferential stress rhh/r0 in (e), radial electric displacement Drcmax/(r0emax) in (f), and radial magnetic induction Brcmax/(r0qmax) in (g). The dashed thin vertical lines denote the two interfaces in the sandwich sphere.

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on the micromechanics of Kuo and Pan (2011). As for boundaryconditions, we assume that the outer surface of the sphere is undera uniform external pressure, i.e., rrr = �r0 and that the other elastictraction components are zero on both the inner and outer surfacesof the layered sphere. Furthermore, electric and magnetic short-circuit is assumed on both the inner and outer surfaces. For easypresentation, all quantities are normalized by following the sameapproach used in Chen and co-workers (2007a,b).

Fig. 3 show the variation of the induced extended displace-ments and stresses in dimensionless form with respect to the

normalized radius r/R. The elastic displacement component urcmax/(Rr0) is presented in Fig. 3a, the electric potential /emax/r0 inFig. 3b, the magnetic potential wqmax/r0 in Fig. 3c, the radial trac-tion rrr/r0 in Fig. 3d, the circumferential traction rhh/r0 in Fig. 3e,the radial electric displacement Drcmax/(r0emax) in Fig. 3f, and theradial magnetic induction Brcmax/(r0qmax) in Fig. 3g. While cmax isthe maximum elastic coefficients among all materials, emax andqmax are, respectively, the maximum absolute values of the piezo-electric and magnetostrictive coefficients of the given materials.From Fig. 3a–c, by comparing to the homogeneous MEE results,

Fig. 4. Variation of the extended displacements and stresses along the radial direction at fixed h = 0� of the hollow sandwich sphere made of magneto-electro-elasticmaterials with its outer surface being under an axisymmetric external pressure rrr = r0P2(cosh)/cmax. The dimensionless elastic displacement component urcmax/(Rr0) in (a),electric potential /emax/r0 in (b), magnetic potential wqmax/r0 in (c), radial stress rrr/r0 in (d), circumferential stress rhh/r0 in (e), radial electric displacement Drcmax/(r0emax)in (f), and radial magnetic induction Brcmax/(r0qmax) in (g). The dashed thin vertical lines denote the two interfaces in the sandwich sphere.

72 J.Y. Chen et al. / International Journal of Solids and Structures 60–61 (2015) 66–74

the elastic displacement and electric and magnetic potentials areclearly affected by the layering. From Fig. 3d, we can observe thatthe radial stress is only slightly influenced by the layering and thatthe traction boundary condition is satisfied which verifies againour solution. The circumferential stress, however, shows obviousdependence on layering as can be observed from Fig. 3e, along withits expected jump across the interfaces. It is particularly interestingthat, while the magnetostrictive layer F corresponds to a largeincrease in the stress magnitude in the middle layer of the sand-wich B/F/B, the piezoelectric layer B helps to reduce it substantiallyin the middle layer of the sandwich F/B/F. From Fig. 3f and g, weobserve that, compared to the semi-coupled B/F/B or F/B/F cases,

the truly coupled MEE layer would induce much large magnitudeof the electric displacement and magnetic induction, which is par-ticular true in the inner layer.

5.3. A hollowed sandwich sphere under an axisymmetric externalpressure

Here we assume the same sandwich structure as in Section 5.2,but under an axisymmetric external pressure rrr = r0P2(cosh)/cmax

on its outer surface. The response of the layered sphere is plottedalong the radial direction for fixed angle h = 0�. Fig. 4 shows thatdistribution of the extended displacements and stresses along

Fig. 5. Distribution of dimensionless displacement component uhcmax/(R/r0) in (a)and stress component rrh/r0 in (b) as functions of (r,h) in the homogeneous MEEhollow sphere under an axisymmetric external pressure rrr = r0P2(cosh)/cmax

applied on its outer surface r = R.

J.Y. Chen et al. / International Journal of Solids and Structures 60–61 (2015) 66–74 73

radial direction in the same sandwich hollow spheres as in Fig. 3discussed in Section 5.2. Similar to Fig. 3, the extended displace-ments (i.e., radial elastic displacement in Fig. 4a, electric potentialin Fig. 4b, and magnetic potential in Fig. 4c) depend significantlyupon layering of the sphere, but the radial stress component isnearly independent of layering and MEE coupling. Furthermore,for the order 2 case, material layering has only slight effect onthe circumferential stress, even though one can still observe its dis-continuities across the interface of different layers (Fig. 4e). Com-paring the electric displacement and magnetic induction fororders 0 and 2 (i.e., Figs. 3f and g vs. 4f and g), one can clearlynotice the difference. While for the case of order 0, the largest mag-nitude of these electric and magnetic quantities is reached on theinner surface (r/R = 0.4) by the fully coupled homogeneous MEEhollow sphere (Fig. 3f and g), for the case of order 2, the B/F/B sand-wich sphere has the largest magnitude of the electric displacementon the inner surface (Fig. 4f) and the F/B/F sandwich sphere has thelargest magnitude of the magnetic induction on the inner surface(Fig. 4g).

Shown in Fig. 5a and b are the variations of the elastic displace-ment component uh and the stress component rrh in both the radialand circumferential directions when the upper surface of thehomogeneous hollow MEE sphere is under the same axisymmetricexternal pressure of order 2 as in Fig. 4. These two figures illustratehow the elastic displacement and stress vary as functions of (r, h).While these 3D contour plots are smooth, there are minima andmaxima within the (r, h)-plane one may need to pay attention to,as can be clearly observed from Fig. 5b for the shear stressdistribution.

6. Conclusions

In this paper, we have derived the analytical solutions for a lay-ered hollow sphere made of spherically anisotropic magneto-elec-tro-elastic materials. The spherical system of vector functions isintroduced to express the solutions and variable transformationis carried out twice in order to reduce the system of differentialequations to a standard one with constant coefficients. For themultilayered case, the propagator matrix method is employed withthe propagating matrix in each layer being simply the exponentialmatrix. We further point out that under the spherical system ofvector functions, the spheroidal and torsional deformations canbe easily separated with the multiphase coupling being involvedin the spheroidal deformation only. The special case of uniformdeformation corresponding to order 0 is also discussed. As numer-ical examples, we have presented the results for both orders 0 and2 for three layered hollow spheres made of piezoelectric BaTiO3,magnetostrictive CoFe2O4, and the coupled MEE material underexternal pressures of orders 0 and 2. The stacking sequences stud-ied are B/F/B, F/B/F, and the homogenous MEE spheres. The influ-ence of layering, multiphase coupling, as well as different ordersof loading, on the elastic, electric and magnetic quantities is illus-trated clearly. Specifically, we find that:

1. The presence of the magnetostrictive layer causes relativelylarge stress in the B/F/B sphere while the piezoelectric layerreduces the corresponding stress.

2. The fully coupled MEE sphere induces large values of electricdisplacement and magnetic induction in the innermost layer.

3. Unlike the majority of the MEE field variables, the componentsof radial stress have little dependence on stacking sequence.

These features would be valuable references when designinglayered spherical structures made of MEE materials.

Acknowledgments

This work was supported by the National Natural Science Foun-dation of China (No. 11172273) and Bairen Project of Henan Pro-vince (China).

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