Geometrically-induced singularities in functionally graded magneto-electro-elastic wedges

International Journal of Mechanical Sciences 89 (2014) 242–255

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Geometrically-induced singularities in functionally graded magneto-electro-elastic wedges C.S. Huang n, C.N. Hu Department of Civil Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30050, Taiwan

art ic l e i nf o

a b s t r a c t

Article history: Received 13 November 2013 Received in revised form 10 April 2014 Accepted 14 September 2014 Available online 19 September 2014

Asymptotic solutions are proposed for geometrically-induced magneto-electro-elastic (MEE) singularities at the vertex of a rectilinearly polarized wedge that is made of functionally graded magneto-electro-elastic (FGMEE) materials. The material properties are assumed to vary along the thickness of the wedge, and the direction of polarization may not be parallel or perpendicular to the thickness direction. An eigenfunction expansion approach with the power series solution technique and domain decomposition is adopted to establish the asymptotic solutions by directly solving the three-dimensional equations of motion and Maxwell's equations in terms of mechanical displacement components and electric and magnetic potentials. Since the direction of polarization can be arbitrary in space, the in-plane components of displacement, electric and magnetic fields are generally coupled with the out-of-plane components. The solutions are presented here for the first time. The correctness of the proposed solutions is confirmed by comparing the orders of MEE singularities with the published results for wedges under anti-plane deformation and in-plane electric and magnetic fields. The proposed solutions are further employed to investigate the effects of the direction of polarization, vertex angle, boundary conditions and materialproperty gradient index on the MEE singularities in BaTiO3–CoFe2O4 wedges. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Singularities Magnet-electro-elastic wedge Functional graded material Three-dimensional asymptotic solution

1. Introduction Magneto-electro-elastic (MEE) materials exhibit mechanicalelectric-magnetic coupling effects and have a wide range of applications in engineering structures, while functionally graded materials (FGMs) have the special feature of graded spatial compositions, which affords freedom in the design and manufacturing of novel structures. It is interesting and important to investigate geometrically-induced magneto-electro-elastic singularities in wedges made of functionally graded magneto-electro-elastic (FGMEE) materials because such singularities normally initiate cracking and worsen the functioning of structures. Numerous studies of geometrically-induced stress singularities in elastic wedges have been conducted, and most of them were reviewed by Sinclair [1]. Williams [2,3] was at the forefront of investigating stress singularities at the vertex of an elastic wedge under extension or bending. Following his work, various investigations have been carried out on elastic wedges that are made of single and multiple materials, drawing on plane elasticity theory [4–12], plate theories [13–16], and three-dimensional elasticity

n

Corresponding author. E-mail address: [email protected] (C.S. Huang).

http://dx.doi.org/10.1016/j.ijmecsci.2014.09.009 0020-7403/& 2014 Elsevier Ltd. All rights reserved.

theory [17,18]. Huang and Chang [19,20] examined geometricallyinduced stress singularities in FGM plates using classical plate theory and third-order shear deformation theory. A few studies of geometrically-induced electroelastic singularities in piezoelectric wedges have been performed, and most of them make the plane strain assumption or the generalized plane deformation assumption. Using the plane strain assumption, Xu and Rajapakse [21] employed Lekhnitskii's complex potential functions to examine the in-plane electroelastic singularities at the vertex of a piezoelectric wedge, while Shang and Kitamura [22] utilized a modified version of the general solution of Wang and Zheng [23]. Hwu and Ikeda [24] made the generalized plane strain assumption and proposed an extended Stroh formulation to investigate the in-plane and out-of-plane electroelastic singularities in wedges. Chu et al. [25–27] adopted the generalized plane deformation assumption and conducted a series of analytical studies of geometrically-induced electroelastic singularities. Scherzer and Kuna [28], Chen et al. [29] and Chen and Ping [30] presented finite element solutions for in-plane singular electroelastic states at the vertex of a wedge. Based on the three-dimensional piezoelasticity theory, Huang and Hu [31] proposed asymptotic solutions at the vertex of a wedge. Several studies have addressed magneto-electro-elastic singularities in MEE wedges and all of them considered wedges under

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

anti-plane deformation and in-plane electric and magnetic fields and assumed that all of the physical quantities are independent of the coordinates along the thickness. Liu and Chue [32] applied the Mellin transform to analyze the singularities at the vertex of a bimaterial MEE wedge, while Sue et al. [33] employed the complex potential function with an eigenfunction expansion method. Liu [34] extended the solution of Liu and Chue [32] to examine the magneto-electro-elastic singularities at the apex of an MEE wedge-junction structure, accounting for the air effect. A review of the literature reveals a complete lack of investigation of geometrically-induced magneto-electro-elastic singularities in MEE wedges and FGMEE wedges based on the three-dimensional magneto-electro-elasticity theory. Accordingly, this paper aims to derive a three-dimensional asymptotic solution for the MEE singularities at the vertex of an FGMEE wedge with an arbitrary direction of polarization by extending the solutions of Huang and Hu [31] for a piezoelectric wedge. The eigenfunction expansion method is combined with the power series technique and a domain decomposition scheme to solve the three-dimensional equations of motion and Maxwell's equations in terms of mechanical displacement components, electric potential and magnetic potential in a cylindrical coordinate system. The correctness of the proposed solution is confirmed by comparison with the published results of Liu and Chue [32] for MEE wedges under anti-plane deformation and in-plane electric and magnetic fields. The proposed solutions are further applied to determine the orders of the MEE singularities in BaTiO3–CoFe2O4 wedges whose material properties may vary along the thickness. The results demonstrate the influence of the orientation of polarization, vertex angle, material-property gradient index and boundary conditions on the orders of MEE singularities in a wedge.

Consider an FGMEE wedge with vertex angle β, shown in Fig. 1, where the x–y–z coordinate system is used to describe the material anisotropy, and the x–y–z coordinate system is used to describe the geometry of the wedge. The FGMEE material is z

assumed to be a transversely isotropic material with the polarization direction along the z axis. The material properties are assumed to vary along the z axis. The x–y–z coordinate system is not generally coincident with the x–y–z coordinate system, so the in-plane and out-of-plane physical quantities (mechanical displacement, electric and magnetic fields) are generally coupled. The linear constitutive equations for the FGMEE material in the x–y–z coordinate are given by

 fσ g ¼ ½cfεg  ½eT E  ½dT H ;

ð1aÞ




 D ¼ ½efεg þ ½η E þ ½g  H ;

ð1bÞ




 B ¼ ½efεg þ ½g  E þ ½μ H ;

ð1cÞ

f εxx

y

εyy

y

x

x y

O

x

Fig. 1. Coordinate systems for a wedge.

σ zx

σ xy gT

and

fεg ¼

T

εzz

the electric and magnetic fields, respectively. 2

c11 6c 6 12 6 6 c13 6 ½c ¼6 6 0 6 6 0 4 0 0

6 ½e ¼ 4 0 e31 2

0 6 ½d ¼ 4 0 d31 η11

6 ½η ¼ 4 0 0

g 11 6 ½g ¼ 4 0 0

O

σ yz

the electric displacement and magnetic flux vectors, respectively;

 T T E ¼ f Ex Ey Ez g and H ¼ f H x H y H z g are the vectors of

2

h

σ zz

2εyz 2εzx 2εxy g are the stress and strain vectors,

 T T respectively; D ¼ f Dx Dy Dz g and B ¼ f Bx By Bz g are

2

z

σ yy

fσg ¼ f σ xx

where

2

2. Basic equations of 3D magneto-electro-elasticity

243

c12

c13

0

0

0

c11

c13

0

0

0

c13

c33

0

0

0

0

0

c44

0

0

3

0

0

0

c44

0

0

0

0

0

c11  c12 2

3

0

0

0

e15

0

0

e15

0

07 5

e31

e33

0

0

0

0

0

0

d15

0

0

d15

0

7 0 5;

d31

d33

0

0

0

0

0

3

0

0 7 5; η33

0

0

η11

g 11 0

2

0

0

μ11

6 ½μ ¼ 4 0 0

7 7 7 7 7 7; 7 7 7 5

3

0 μ11 0

0

3

0 7 5; μ33

3

0 7 5: g 33

ð2Þ

where ½ c , ½ e , ½ d , ½ η , ½ μ  and ½ g  are the elastic stiffness constant, piezoelectric coefficient, piezomagnetic coefficient, dielectric constant, magnetic permeability, and magnetoelectric coefficient matrices, respectively, and all of these are functions of z. The cylindrical coordinate system ðr; θ; zÞ (Fig. 1) is used to investigate the MEE singularities as r approaches zero in the wedge. Hence, Eqs. (1) and (2) have to be transformed to ðr; θ; zÞ from ðx; y; zÞ and rewritten as  T fσ g ¼ ½cfεg  ½eT fEg  d fH g;

ð3aÞ

fDg ¼ ½efεg þ ½ηfEg þ ½g fH g;

ð3bÞ

  fBg ¼ d fεg þ ½g fEg þ ½μfH g;

ð3cÞ

244

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

where 2

c11

6 6 c12 6 6 c13 6 ½c  ¼ 6 6 c14 6 6 c15 4 2

c16

c12

c13

c14

c15

c22 c23

c23 c33

c24 c34

c25 c35

c24

c34

c44

c45

c25

c35

c45

c55

c26

c36

c46

c56

e11 e12 e13 e14 e15 6 ½e ¼ 4 e21 e22 e23 e24 e25 e31 e32 e33 e34 e35 2 d11 d12 d13 d14 d15   6 d ¼ 4 d21 d22 d23 d24 d25 2

d31 η11

d32 η12

d33 d34 3

η13

6 7 ½η ¼ 4 η12 η22 η23 5; η13 η23 η33 2 3 g 11 g 12 g 13 6 7 ½g  ¼ 4 g 12 g 22 g 23 5: g 13 g 23 g 33

c16

  ∂c66 ∂c56 ∂ ∂c56 ∂c55 ∂ þr þr þ c þ 15 ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z  2 ∂c26 ∂c25 1 ∂ þ  c22 þ þr  ρn 2 unr ∂θ ∂z r 2 ∂t

∂2 c26  ∂2 ∂2 ∂2 þ c16 2 þ 2 þ c45 2 þ ðc12 þ c66 Þ r∂r ∂θ ∂r r ∂θ2 ∂z þ

3

7 c26 7 7 c36 7 7 7; c46 7 7 c56 7 5 c66 3 e16 e26 7 5 e36 3 d16 d26 7 5; d36

d35 2 μ11 6 ½μ ¼ 4 μ12 μ13

μ12 μ22 μ23

μ13

3

μ23 7 5; μ33

  Now the material properties matrices ½c, ½e, d , ½η, ½μ and ½g  are related to ½c, ½e, ½d, ½η, ½μ and ½g , respectively, and are the functions of θ, z and the angles between the axes of x–y–z and x–y–z. These relations are specified in Appendix A. To develop the asymptotic solutions to examine the geometricallyinduced magneto-electro-elastic singularities as r approaches zero in a wedge, the body forces, free electric charges and magnetic charges are ignored, and the equations of motion and Maxwell's equations are ∂σ rr 1 ∂σ rθ ∂σ rz ðσ rr  σ θθ Þ ∂ 2 un ¼ ρn 2r ; þ þ þ r ∂θ r ∂r ∂z ∂t

ð4aÞ

∂σ rθ 1 ∂σ θθ ∂σ θz σ rθ ∂ 2 un þ þ þ2 ¼ ρn 2θ ; r ∂θ ∂r ∂z r ∂t

ð4bÞ

∂σ rz 1 ∂σ θz ∂σ zz σ rz ∂ 2 un þ þ þ ¼ ρn 2z ; r ∂θ ∂r ∂z r ∂t

ð4cÞ

1 ∂ðrDr Þ 1 ∂Dθ ∂Dz þ þ ¼ 0; r ∂r r ∂θ ∂z

ð4dÞ

1 ∂ðrBr Þ 1 ∂Bθ ∂Bz þ þ ¼ 0; r ∂r r ∂θ ∂z

ð4eÞ

where ρn is the density of the material, and uni ði ¼ r; θ; zÞ is the displacement component in the i direction. Using electric field-electric potential and magnetic fieldmagnetic potential relations, ∂ϕn 1 ∂ϕn ∂ϕn ; Eθ ¼  ; Ez ¼  ; r ∂θ ∂r ∂z n n ∂ψ 1 ∂ψ ∂ψ n ; Hθ ¼  and H z ¼  ; Hr ¼  r ∂θ ∂r ∂z Er ¼ 

ð5Þ

as well as the strain–displacement relations, Eqs. (4a)–(4e) can be expressed in terms of mechanical displacement components (unr , unθ and unz ), electric potential (ϕn ) and magnetic potential (ψ n ) as follows:

 ∂2 c66  ∂2 ∂2 2c16 ∂2 þ c þ c11 2 þ 2 55 r ∂r ∂θ ∂r r ∂θ2 ∂z2  2  2 ∂ 2c56 ∂ ∂c16 ∂c15 ∂ þ þ c11 þ þr þ 2c15 ∂r ∂z r ∂θ ∂z ∂θ ∂z r∂r

∂2 ∂2 þ ðc14 þ c56 Þ þ ðc25 þ c46 Þ ∂r ∂z r∂θ ∂z  ∂c66 ∂c56 ∂ þr þ  c26 þ ∂θ ∂z r∂r  ∂c26 ∂c25 ∂ þr þ  c22 c66 þ ∂θ ∂z r 2 ∂θ  ∂c46 ∂c45 þr þ c14  c24  c56 þ ∂θ ∂z   ∂ ∂c66 ∂c56 1 n þ c26   u  r∂z ∂θ ∂z r 2 θ

∂2 c46  ∂2 ∂2 ∂2 þ ðc13 þ c55 Þ þ c15 2 þ 2 þ c35 2 þ ðc14 þ c56 Þ r∂r ∂θ ∂r r ∂θ2 ∂z  ∂2 ∂2 ∂c56 ∂c55 ∂ þ ðc36 þ c45 Þ þ c15  c25 þ þr  ∂r ∂z r∂θ ∂z ∂θ ∂z r∂r  ∂c46 ∂c45 ∂ þr þ  c24 þ ∂θ ∂z r 2 ∂θ   ∂c36 ∂c35 ∂ un þr þ c13  c23 þ ∂θ ∂z r∂z z

∂2 e26  ∂2 ∂2 ∂2 þ e11 2 þ 2 þ e35 2 þ ðe16 þ e21 Þ r∂r ∂θ ∂r r ∂θ2 ∂z ∂2 ∂2 þ ðe15 þ e31 Þ þ ðe25 þ e36 Þ ∂r ∂z r∂θ ∂z   ∂e16 ∂e15 ∂ ∂e26 ∂e25 ∂ þ e22 þ þr þr þ e11  e12 þ ∂θ ∂z r∂r ∂θ ∂z r 2 ∂θ   ∂e36 ∂e35 ∂ ϕn þ e31  e32 þ þr ∂θ ∂z r∂z

 ∂2 d26 ∂2 ∂2 ∂2 þ d11 2 þ þd35 2 þ ðd16 þd21 Þ 2 2 r∂r ∂θ ∂r r ∂z ∂θ ∂2 ∂2 þ ðd25 þ d36 Þ þ ðd15 þ d31 Þ ∂r ∂z r∂θ ∂z   ∂d16 ∂d15 ∂ ∂d26 ∂d16 þ  d22 þ þr þr þ d11  d12 þ ∂θ ∂z r∂r ∂θ ∂z   ∂ ∂d36 ∂d36 ∂ ψ n ¼ 0; þr  2 þ d31  d32 þ ∂θ ∂z r∂z r ∂θ

ð6aÞ

∂2 c26  ∂2 ∂2 ∂2 c16 2 þ 2 þc45 2 þ ðc12 þ c66 Þ 2 r∂r ∂θ ∂r r ∂θ ∂z  ∂2 ∂2 ∂c12 ∂c14 ∂ þ ðc25 þ c46 Þ þ 2c16 þc26 þ þr þ ðc14 þ c56 Þ ∂r ∂z r∂θ ∂z ∂θ ∂z r∂r   ∂c26 ∂c46 ∂ ∂c25 ∂c45 þr þ r þ c þ 2c þ þ c22 þ c66 þ 24 56 ∂θ ∂z r 2 ∂θ ∂θ ∂z  

∂ ∂c22 ∂c24 1 n ∂2 c22  ∂2 ∂2 þ c26 þ þr  þ c44 2 ur þ c66 2 þ 2 2 2 r∂z ∂θ ∂z r ∂r r ∂θ ∂z  2c26 ∂2 ∂2 ∂2 þ 2c46 þ 2c24 þ r ∂r ∂θ ∂r ∂z r∂θ ∂z   ∂c26 ∂c46 ∂ ∂c22 ∂c24 ∂ þ þr þr þ c66 þ ∂θ ∂z r∂r ∂θ ∂z r 2 ∂θ   2 ∂c24 ∂c44 ∂ ∂c26 ∂c46 1 n ∂ þ  c66  þr r  ρ unθ þ c46 þ ∂θ ∂z r∂z ∂θ ∂z r 2 ∂t 2

∂2 c24  ∂2 ∂2 ∂2 þ c56 2 þ 2 þ c34 2 þ ðc25 þ c46 Þ 2 r∂r ∂θ ∂r r ∂θ ∂z  ∂2 ∂2 ∂c25 ∂c45 ∂ þ ðc23 þ c44 Þ þ 2c56 þ þr þ ðc36 þ c45 Þ ∂r ∂z r∂θ ∂z ∂θ ∂z r∂r    ∂c24 ∂c44 ∂ ∂c23 ∂c34 ∂ un þr þr þ 2c36 þ þ c46 þ ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z z

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

∂2 e22  ∂2 ∂2 ∂2 þ ðe14 þ e36 Þ þ e16 2 þ 2 þ e34 2 þ ðe12 þ e26 Þ 2 r∂r ∂θ ∂r r ∂θ ∂z  ∂2 ∂2 ∂e12 ∂e14 ∂ þ ðe24 þ e32 Þ þ 2e16 þ þr  ∂r ∂z r∂θ ∂z ∂θ ∂z r∂r   ∂e22 ∂e24 ∂ ∂e32 þr þ 2e36 þ þ e26 þ ∂θ ∂z r 2 ∂θ ∂θ 

 2 2 ∂e34 ∂ ∂ d ∂ ∂2 ∂2 22 ϕn þ d16 2 þ þr þd34 2 þ ðd12 þd26 Þ 2 2 ∂z r∂z r∂r ∂θ ∂r r ∂z ∂θ  2 2 ∂ ∂ ∂d12 ∂d14 ∂ þ ðd14 þ d36 Þ þ ðd24 þ d32 Þ þ 2d16 þ þr ∂r ∂z r∂θ ∂z ∂θ ∂z r∂r    ∂d22 ∂d24 ∂ ∂d32 ∂d34 ∂ ψ n ¼ 0; þr þr þ 2d36 þ þ d26 þ ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z ð6bÞ

∂2 c46  ∂2 ∂2 ∂2 ∂2 þ ðc13 þ c55 Þ þ c35 2 þ ðc14 þ c56 Þ c15 2 þ 2 r∂r ∂θ ∂r ∂z ∂r r ∂θ2 ∂z  ∂2 ∂c14 ∂c13 ∂ þ c15 þ c25 þ þr þ ðc36 þ c45 Þ r∂θ ∂z ∂θ ∂z r∂r   ∂c46 ∂c36 ∂ ∂c45 ∂c35 ∂ þr þ r þ c þ c þ þ c24 þ 55 23 ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z  

∂c24 ∂c23 1 n ∂2 c24  ∂2 þ þr u þ c56 2 þ 2 ∂θ ∂z r 2 r ∂r r ∂θ2 2 ∂ ∂2 ∂2 þ ðc36 þ c45 Þ þ c34 2 þ ðc25 þ c46 Þ r∂r ∂θ ∂r ∂z ∂z  ∂2 ∂c46 ∂c36 ∂ þ þr þ ðc23 þ c44 Þ r∂θ ∂z ∂θ ∂z r∂r   ∂c24 ∂c23 ∂ ∂c44 þ  c46 þ þr þ  c36 þ c45 þ 2 ∂θ ∂z r ∂θ ∂θ   ∂c34 ∂ ∂c46 ∂c36 1 n þ  þr r u ∂θ r∂z ∂θ ∂z r 2 θ 

∂2 c44  ∂2 ∂2 2c45 ∂2 þ 2c35 þ þ c55 2 þ 2 þ c 33 r ∂r ∂θ ∂r r ∂θ2 ∂z2 

  ∂ ∂2 ∂2 e24 ∂2  ρn 2 unz þ e15 2 þ 2  r∂z ∂r r ∂θ2 ∂t 2 2 ∂ ∂ ∂2 þ ðe13 þe35 Þ þ ðe23 þ e33 2 þ ðe14 þ e25 Þ r∂r ∂θ ∂r ∂z ∂z   ∂2 2c34 ∂2 ∂c45 ∂c35 ∂ þ þ c55 þ þr  ∂r ∂z r ∂θ ∂z ∂θ ∂z r∂r   ∂c44 ∂c34 ∂ ∂c34 ∂c33 þr þr þ c35 þ þ 2 ∂θ ∂z r ∂θ ∂θ ∂z  ∂2 ∂e14 ∂e13 ∂ þ e34 Þ þ e15 þ þr r∂θ ∂z ∂θ ∂z r∂r    ∂e24 ∂e23 ∂ ∂e34 ∂e33 ∂ ϕn þr þ r þ e þ þ 35 ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z

 2 ∂2 d24 ∂ ∂2 ∂2 þ d15 2 þ þ d33 2 þ ðd14 þ d25 Þ 2 2 r∂r ∂θ ∂r r ∂z ∂θ ∂2 ∂2 þ ðd23 þ d34 Þ þ ðd13 þ d35 Þ ∂r ∂z  r∂θ ∂z ∂d14 ∂d13 ∂ ∂d24 ∂d23 ∂ þ þr þr þ d15 þ ∂θ ∂z r∂r ∂θ ∂z r 2 ∂θ   ∂d34 ∂d33 ∂ ψ n ¼ 0; þr þ d35 þ ∂θ ∂z r∂z

∂2 e26  ∂2 ∂2 ∂2 ∂2 þ ðe15 þ e31 Þ e11 2 þ 2 þ e35 2 þ ðe16 þ e21 Þ r∂r ∂θ ∂r ∂z ∂r r ∂θ2 ∂z  ∂2 ∂e21 ∂e31 ∂ þ e11 þ e12 þ þr þ ðe25 þe36 Þ r∂θ ∂z ∂θ ∂z r∂r   ∂e26 ∂e36 ∂ ∂e25 ∂e35 ∂ þ e22 þ þr þ r þ e þe þ 15 32 ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z  

  2 2 ∂e22 ∂e32 1 ∂ e22 ∂ þ þr  2 unr þ e16 2 þ 2 ∂θ ∂z r ∂r r ∂θ2

245

∂2 ∂2 ∂2 ∂2 þ ðe14 þ e36 Þ þ ðe24 þ e32 Þ þ e34 2 þ ðe12 þe26 Þ r∂r ∂r ∂z r∂θ ∂z  ∂z  ∂θ ∂e26 ∂e36 ∂ ∂e22 ∂e32 ∂ þ e26 þ þr þr þ ∂θ ∂z r∂r ∂θ ∂z r 2 ∂θ    ∂e24 ∂e34 ∂ ∂e26 ∂e36 1 n þ  þr r þ e14  e36 þ u ∂θ ∂z r∂z ∂θ ∂z r 2 θ

  ∂2 e24 ∂2 ∂2 ∂2 þ e15 2 þ 2 þ e33 2 þ ðe14 þ e25 Þ r∂r ∂θ ∂r r ∂θ2 ∂z ∂2 þ ðe13 þ e35 Þ ∂r ∂z  ∂2 ∂2 ∂e25 ∂e35 ∂ þ ðe23 þ e34 Þ þ e15 þ þr  ∂r ∂z r∂θ ∂z ∂θ ∂z r∂r    ∂e24 ∂e25 ∂ ∂e23 ∂e33 ∂ un þr þr þ þ e13 þ ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z z 

∂ 2 η  ∂ 2 ∂2 2η12 ∂2 ∂2 þ 2η13  η11 2 þ 22 þ η33 2 þ 2 2 r ∂r ∂θ ∂r ∂z ∂r r ∂θ ∂z   2η23 ∂2 ∂η12 ∂η13 ∂ þ η11 þ þr þ r ∂θ ∂z ∂θ ∂z r∂r    ∂η22 ∂η23 ∂ ∂η23 ∂η ∂ ϕn þ þr þ r 33 þ η13 þ 2 ∂θ ∂z r ∂θ ∂θ ∂z r∂z 

∂ 2 g  ∂ 2 ∂2 2g 12 ∂2  g 11 2 þ 22 þ g 33 2 þ 2 2 r ∂r ∂θ ∂r r ∂θ ∂z   ∂2 2g 23 ∂2 ∂g 12 ∂g ∂ þ þ g 11 þ þ r 13 þ 2g 13 ∂r ∂z r ∂θ ∂z ∂θ ∂z r∂r    ∂g 22 ∂g ∂ ∂g ∂g ∂ ψ n ¼ 0; þr 23 2 þ g 13 þ 23 þ r 33 þ ∂θ ∂z r ∂θ ∂θ ∂z r∂z þ ðe13 þ e35 Þ

ð6dÞ

 2 ∂2 d26 ∂ ∂2 ∂2 ∂2 þ ðd15 þ d31 Þ d11 2 þ þd35 2 þ ðd16 þ d21 Þ 2 2 r∂r ∂θ ∂r ∂z ∂r r ∂z ∂θ  ∂2 ∂d21 ∂d31 ∂ þ d11 þd12 þ þr þ ðd25 þ d36 Þ r∂θ ∂z ∂θ ∂z r∂r   ∂d26 ∂d36 ∂ ∂d25 ∂d35 ∂ þr þr þ d15 þ d32 þ þ d22 þ 2 ∂θ ∂z r ∂θ ∂θ ∂z r∂z  

 2 ∂d22 ∂d32 1 n ∂2 d22 ∂ ∂2 þr þ þd34 2 ur þ d16 2 þ 2 2 2 ∂θ ∂z r ∂r r ∂z ∂θ

ð6cÞ

∂2 ∂2 þ ðd14 þ d36 Þ þ ðd12 þ d26 Þ r∂r ∂θ ∂r ∂z  ∂2 ∂d26 ∂d36 ∂ þ þr þ ðd24 þ d32 Þ r∂θ ∂z ∂θ ∂z r∂r   ∂d22 ∂d32 ∂ ∂d24 ∂d34 ∂ þr þ r þ d  d þ þ  d26 þ 14 36 ∂θ ∂z r 2 ∂θ ∂θ ∂z r∂z  

 2 2 ∂d26 ∂d36 1 n ∂ d24 ∂ ∂2 þ  r u þ d15 2 þ þ d33 2 ∂θ ∂z r 2 θ ∂r r 2 ∂θ2 ∂z 2  ∂ ∂2 ∂2 þ ðd13 þ d35 Þ þ ðd23 þ d34 Þ þ d14 þ d25 r∂r ∂θ ∂r ∂z r∂θ ∂z   ∂d25 ∂d35 ∂ ∂d24 ∂d34 ∂ þ þr þr þ d15 þ ∂θ ∂z r∂r ∂θ ∂z r 2 ∂θ 

 ∂d23 ∂d33 ∂ ∂ 2 g  ∂ 2 ∂2 unz  g 11 2 þ 22 þr þ d13 þ þ g 33 2 2 2 ∂θ ∂z r∂z ∂r r ∂θ ∂z   2g 12 ∂2 ∂2 2g 23 ∂2 þ 2g 13 þ  þ r ∂r ∂θ ∂r ∂z r ∂θ ∂z   ∂g 12 ∂g 13 ∂ ∂g 22 ∂g 23 ∂ þ þ g 11 þ þr þr ∂θ ∂z r∂r ∂θ ∂z r 2 ∂θ   ∂g ∂g ∂ ϕn þ g 13 þ 23 þ r 33 ∂θ ∂z r∂z 

∂ 2 μ  ∂ 2 ∂2 2μ12 ∂2 ∂2 þ2μ13 þ μ  μ11 2 þ 22 þ 33 r ∂r ∂θ ∂r ∂z ∂r r 2 ∂θ2 ∂z2    2μ23 ∂2 ∂μ ∂μ ∂ ∂μ22 ∂μ ∂ þ þ μ11 þ 12 þ r 13 þr 23 2 þ r ∂θ ∂z ∂θ ∂z r∂r ∂θ ∂z r ∂θ   ∂μ ∂μ ∂ ψ n ¼ 0; þ μ13 þ 23 þ r 33 ð6eÞ ∂θ ∂z r∂z

246

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

3. Construction of asymptotic solution The eigenfunction expansion approach, which Hartranft and Sih [7] had used for 3D elastic wedges, is adopted herein. The mechanical displacements, electric potential and magnetic potential are expressed as

1 1  ðmÞ unr ðr; θ; z; t Þ ¼ ∑ ∑ r λm þ n U^ n ðθ; zÞ eiωt ; ð7aÞ m¼1n¼0

unθ ðr; θ; z; t Þ ¼

1

∑ 1

∑ 1

∑

ð7cÞ

 1 ^ ðnmÞ ðθ; zÞ eiωt ; ∑ r λm þ n Φ

ð7dÞ

 1 ðmÞ ∑ r λm þ n Ψ^ n ðθ; zÞ eiωt ;

ð7eÞ

m¼1n¼0

ψ n ðr; θ; z; t Þ ¼

 ^ ðmÞ ðθ; zÞ eiωt ; ∑ r λm þ n W n 1

m¼1n¼0

ϕn ðr; θ; z; t Þ ¼

ð7bÞ

m¼1n¼0

unz ðr; θ; z; t Þ ¼

 1 ðmÞ ∑ r λm þ n V^ n ðθ; zÞ eiωt ;

1

∑

m¼1n¼0

The characteristic values λm may be real or complex for MEE wedges. For FGMEE wedges, λm is expected to depend on z because the material properties are functions of z. To ensure the finite displacements, electric field and magnetic field at r ¼ 0, the real part of λm is required to be positive. Notably, when the real part of λm (Re[λm ]) is less than unity, the order of the singularities of stress, electric displacement and magnetic flux is Re[λm ]  1. Substituting Eqs. (7a)–(7e) into Eqs. (6a)–(6e), carefully rearranging and considering the terms with the lowest order of r yields, ðmÞ ðmÞ 2 ^ ðmÞ ^ ðmÞ ∂2 U^ 0 ∂U^ 0 ^ ðmÞ þ p ∂ V 0 þ p ∂V 0 þ p þ p U 1 2 0 3 4 ∂θ ∂θ ∂θ2 ∂θ2 ðmÞ 2 ^ ðmÞ ^ ðmÞ ∂ W0 ∂W 0 ^ ðmÞ þ p8 W þp7 þ p5 V^ 0 þ p6 0 ∂θ ∂θ2 ðmÞ 2 ^ ðmÞ 2 ^ ðmÞ ^ ∂ Φ0 ∂Φ ^ ð0mÞ þ p12 ∂ Ψ 0 þ p9 þ p10 0 þ p11 Φ 2 ∂θ ∂θ ∂θ2 ðmÞ ∂Ψ^ ðmÞ þ p13 0 þ p14 Ψ^ 0 ¼ 0; ∂θ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ∂2 V^ 0 ∂V^ ∂2 U^ 0 ∂U^ þ q1 0 þ q2 V^ 0 þ q3 þ q4 0 2 2 ∂θ ∂θ ∂θ ∂θ 2 ^ ðmÞ ^ ðmÞ ðmÞ ð m ∂ ∂ W W 0 0 ^ Þ þ q8 W þ q7 þ q5 U^ 0 þ q6 0 ∂θ ∂θ2 ð m Þ ð m Þ 2 ^ ðmÞ ^ ^0 ∂2 Φ ∂Φ ^ ð0mÞ þq12 ∂ Ψ 0 þ q9 þ q10 0 þ q11 Φ 2 ∂θ ∂θ ∂θ2 ðmÞ ∂Ψ^ ðmÞ þ q13 0 þ q14 Ψ^ 0 ¼ 0; ∂θ 2 ^ ðmÞ ^ ðmÞ ^ ðmÞ ^ ðmÞ ∂2 W ∂W 0 0 ^ ðmÞ þ r 3 ∂ U 0 þ r 4 ∂U 0 þr þ r W 1 2 0 ∂θ ∂θ ∂θ2 ∂θ2 2 ^ ðmÞ ^ ðmÞ ðmÞ ð m Þ ∂ ∂ V V 0 þ r 7 0 þr 8 V^ 0 þ r 5 U^ 0 þ r 6 ∂θ ∂θ2 ðmÞ 2 ^ ðmÞ 2 ^ ðmÞ ^ ∂ Φ0 ∂Φ0 ^ ð0mÞ þ r 12 ∂ Ψ 0 þr þ r9 þ r Φ 10 11 ∂θ ∂θ2 ∂θ2 ðmÞ ^ ∂Ψ ðmÞ þ r 13 0 þ r 14 Ψ^ 0 ¼ 0; ∂θ 2 ^ ðmÞ ^ ðmÞ ^ ð0mÞ ∂Φ ^ ð0mÞ þ s3 ∂ U 0 þ s4 ∂U 0 þs Φ 2 ∂θ ∂θ ∂θ2 ∂θ2 ðmÞ ðmÞ 2 ^ ^ ðmÞ ðmÞ ∂ V0 ∂V þ s5 U^ 0 þ s6 þ s7 0 þ s8 V^ 0 ∂θ ∂θ2

2 ^ ðmÞ ^ ðmÞ ^ ðmÞ ∂2 W ∂W 0 0 ^ ðmÞ þ s12 ∂ Ψ 0 þ s þ s W 10 11 0 ∂θ ∂θ2 ∂θ2 ðmÞ ^ ∂Ψ ðmÞ þ s13 0 þ s14 Ψ^ 0 ¼ 0; ∂θ

þ s9

ð8aÞ

ðmÞ ðmÞ 2 ^ ðmÞ ^ ðmÞ ∂2 Ψ^ 0 ∂Ψ^ 0 ^ ðmÞ þ t 3 ∂ U 0 þ t 4 ∂U 0 þt 5 U^ ðmÞ Ψ þ t þt 1 2 0 0 ∂θ ∂θ ∂θ2 ∂θ2 ðmÞ ðmÞ 2 ^ ðmÞ 2 ^ ðmÞ ^ ^ ðmÞ ∂ V0 ∂V 0 ∂ W0 ∂W 0 ^ V þ t þ t6 þ t þ t þ t 7 8 0 9 10 ∂θ ∂θ ∂θ2 ∂θ2 ðmÞ 2 ^ ðmÞ ^ ðmÞ ∂ ∂ Φ Φ ð m Þ 0 ^ ^ 0 ¼ 0; þ t 11 W þ t 13 0 þt 14 Φ 0 þ t 12 ∂θ ∂θ2

ð8eÞ

where pi , qi , r i , si and t i (i ¼1–14) are functions of θ and z and are defined in Appendix B. Careful examination of Eqs. (8a)–(8e) reveals that when λm at z ¼ zl is sought, the material properties at z ¼ zl must be used to determine pi , qi , r i , si and t i . Eq. (8) represent a set of ordinary differential equations with variable coefficients. The three displacement components, electric potential and magnetic potential are generally coupled in Eqs. (8a)–(8e). The exact closed-form solutions to Eqs. (8a)–(8e) are intractable, if they exist. The power series method can be directly applied to establish a general solution to these ordinary differential equations. Very high-order terms are commonly needed to obtain an accurate solution and they typically cause numerical difficulties. To overcome these difficulties, a domain decomposition technique is applied in conjunction with the power series method to find a general solution to Eqs. (8a)–(8e). The whole domain of θ is divided into numerous sub-domains (Fig. 2). A series solution for each sub-domain is constructed directly using the traditional power series method. Then, a general solution over the whole θ domain is obtained by assembling the solutions in all sub-domains and imposing continuity conditions between each pair of adjacent sub-domains. Notably, this solution procedure is also valid for a wedge that has different materials in different sub-domains of θ. To develop the series solution for sub-domain i where θi  1 r θ r θi , the variable coefficients at z ¼ zl in Eqs. (8a)–(8e) are expressed by Taylor's series with respect to the middle point of the sub-domain θi , K

k ðiÞ pj ðθ; zl Þ ¼ ∑ χ j k θ  θi ; k¼0

K

k ðiÞ qj ðθ; zl Þ ¼ ∑ κ j k θ  θi ; k¼0

ð8bÞ

ð8cÞ

ðmÞ

^0 ∂2 Φ

ð8dÞ

þs1

Fig. 2. Sub-domains for θ A ½0; β

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255 K

k ðiÞ r j ðθ; zl Þ ¼ ∑ ϑj k θ  θi ;

K

k ðiÞ sj ðθ; zl Þ ¼ ∑ ξj k θ  θi ;

k¼0

k¼0

ðiÞ

k t j ðθ; zl Þ ¼ ∑ ζ j k θ  θi : K

ð9Þ

k¼0

Similarly, the solutions to Eqs. (8a)–(8e) in the sub-domain are expressed as

j ðiÞ ðmÞ U^ 0i ¼ ∑ A^ j θ θi ; J

j¼0

^ ðmÞ W 0i ðmÞ Ψ^ 0i

J

¼ ∑

j¼0

¼

j ðiÞ C^ j θ  θi ;

J

j ðmÞ ðiÞ V^ 0i ¼ ∑ B^ j θ  θi ; J

^ ðiÞ θ θi j ¼ ∑D j

ð10Þ

Substituting Eqs. (9) and (10) into Eqs. (8a)–(8e) yields the following recurrence relations among the coefficients in Eq. (10): ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ^ ^ A^ j þ 2 þ χ 3 0 B^ j þ 2 þ χ 6 0 C^ j þ 2 þ χ 9 0 D j þ 2 þ χ 12 0 E j þ 2 (

 j1 ðiÞ ðiÞ ðiÞ ðiÞ 1 ∑ ðk þ 2Þðk þ 1Þ χ 3 j  k B^ k þ 2 þ χ 6 j  k C^ k þ 2 ¼ ðj þ2Þðjþ 1Þ k ¼ 0  ðiÞ ðiÞ ðiÞ ðiÞ  ^ ^ þ χ þ χ9 j  kD E 12 kþ2 jk kþ2 j

þ ∑

 ðk þ1Þ

k¼0

þ

ðiÞ

^ ðiÞ þ χ ðiÞ E^ ðiÞ χ 11 j  k D 14 j  k k k

ðiÞ

;

ð11aÞ

ðiÞ

ðiÞ

ðiÞ ðiÞ ^ ðiÞ þ ðϑ13 ÞðiÞ E^ ðiÞ A^ k þ 1 þ ðϑ7 Þðj iÞ k B^ k þ 1 þ ðϑ10 Þðj iÞ k D kþ1 jk kþ1

ðiÞ

ðiÞ

) ;

ðiÞ ðiÞ ðiÞ ^ ðiÞ and E^ ðiÞ with j Z 2 are determined from cients A^ j , B^ j , C^ j , D j j

ðiÞ ðmÞ ^ ðiÞ U^ ðmÞ ðθ; zÞ þ D ^ ðiÞ U^ ðmÞ ðθ; zÞ þ C^ 1 U^ 0i5 ðθ; zÞ þ D 0 0i6 1 0i7



ðiÞ ðmÞ ðiÞ ðmÞ þ E^ 0 U^ 0i8 ðθ; zÞ þ E^ 1 U^ 0i9 ðθ; zÞ;

#) ;

ð12aÞ

ðiÞ ðmÞ ðiÞ ðmÞ ðmÞ ðiÞ ðmÞ V^ 0i ðθ; zÞ ¼ A^ 0 V^ 0i0 ðθ; zÞ þ A^ 1 V^ 0i1 ðθ; zÞ þ B^ 0 V^ 0i2 ðθ; zÞ ðiÞ ðmÞ ðiÞ ðmÞ þ B^ 1 V^ 0i3 ðθ; zÞ þ C^ 0 V^ 0i4 ðθ; zÞ ðiÞ ðmÞ ^ ðiÞ V^ ðmÞ ðθ; zÞ þ D ^ ðiÞ V^ ðmÞ ðθ; zÞ þ C^ 1 V^ 0i5 ðθ; zÞ þ D 0 0i6 1 0i7 ðiÞ ðmÞ ðiÞ ðmÞ þ E^ 0 V^ 0i8 ðθ; zÞ þ E^ 1 V^ 0i9 ðθ; zÞ;

ð12bÞ

^ ðmÞ ðθ; zÞ þ A^ ðiÞ W ^ ðmÞ ðθ; zÞ þ B^ ðiÞ W ^ ðmÞ ðθ; zÞ ^ ðmÞ ðθ; zÞ ¼ A^ ðiÞ W W 0i 0i0 0i1 0 0i2 0 1 ðiÞ ^ ðmÞ ðθ; zÞ þ C^ ðiÞ W ^ ðmÞ ðθ; zÞ þ C^ ðiÞ W ^ ðmÞ ðθ; zÞ þ B^ 1 W 0 1 0i3 0i4 0i5

^ ðiÞ W ^ ðmÞ ðθ; zÞ þ D ^ ðmÞ ðθ; zÞ þ E^ ðiÞ W ^ ðmÞ ðθ; zÞ ^ ðiÞ W þD 0 0i6 1 0i7 0 0i8 ðiÞ ^ ðmÞ ðθ; zÞ; þ E^ 1 W 0i9



ðiÞ

ð12cÞ ðiÞ

ðiÞ

mÞ mÞ mÞ ^ ð0imÞ ðθ; zÞ ¼ A^ 0 Φ ^ ð0i0 ^ ð0i1 ^ ð0i2 Φ ðθ; zÞ þ A^ 1 Φ ðθ; zÞ þ B^ 0 Φ ðθ; zÞ

ðiÞ ðiÞ ðiÞ ðϑ2 Þðj iÞ k C^ k þ ðϑ5 Þðj iÞ k A^ k þ ðϑ8 Þðj iÞ k B^ k ðiÞ

ð11eÞ

ðiÞ ðmÞ ðiÞ ðmÞ þ B^ 1 U^ 0i3 ðθ; zÞ þ C^ 0 U^ 0i4 ðθ; zÞ

 ðiÞ ðk þ 1Þ ðϑ1 Þðj iÞ k C^ k þ 1 þ ðϑ4 Þðj iÞ k

^ þ ðϑ14 ÞðiÞ E^ þ ðϑ11 Þðj iÞ k D k jk k

) ;

ðiÞ ðmÞ ðiÞ ðmÞ ðmÞ ðiÞ ðmÞ U^ 0i ðθ; zÞ ¼ A^ 0 U^ 0i0 ðθ; zÞ þ A^ 1 U^ 0i1 ðθ; zÞ þ B^ 0 U^ 0i2 ðθ; zÞ

ðiÞ ðiÞ ðiÞ ^ ðiÞ þ ðϑ12 ÞðiÞ E^ ðiÞ C^ j þ 2 þ ðϑ3 Þð0iÞ A^ j þ 2 þ ðϑ6 Þð0iÞ B^ j þ 2 þ ðϑ9 Þð0iÞ D jþ2 0 jþ2 (

j1 1 ¼ ∑ ðk þ 2Þðk þ 1Þ ðj þ2Þðjþ 1Þ k ¼ 0   ðiÞ ðiÞ ^ ðiÞ þ ðϑ12 ÞðiÞ E^ ðiÞ  ðϑ3 Þðj iÞ k A^ k þ 2 þ ðϑ6 Þðj iÞ k B^ k þ 2 þ ðϑ9 Þðj iÞ k D kþ2 jk kþ2

þ

ðiÞ



ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ^ ðiÞ , D ^ ðiÞ , E^ ðiÞ and E^ ðiÞ are Eqs. (11a)–(11e) if A^ 0 , A^ 1 , B^ 0 , B^ 1 , C^ 0 , C^ 1 , D 0 1 0 1 known. As a result, the solutions of Eqs. (8a)–(8e) in sub-domain i can be expressed as

ð11bÞ



ð11dÞ

Careful examination of Eqs. (11a)–(11e) reveals that the coeffi-

#)

 ðiÞ ðiÞ ðiÞ ^ ðiÞ þ ðκ14 ÞðiÞ E^ ðiÞ þ ðκ2 Þðj iÞ k B^ k þ ðκ 5 Þðj iÞ k A^ k þ ðκ 8 Þðj iÞ k C^ k þ ðκ 11 Þðj iÞ k D k jk k

k¼0

) ;

ðiÞ ðiÞ ðiÞ ðiÞ ^ ðiÞ E^ j þ 2 þ ðξ3 Þð0iÞ A^ j þ 2 þ ðξ6 Þð0iÞ B^ j þ 2 þ ðξ9 Þð0iÞ C^ j þ 2 þ ðξ12 Þð0iÞ D jþ2 (

j  1 1 ¼ ∑ ðk þ 2Þðk þ 1Þ ðj þ2Þðjþ 1Þ k ¼ 0   ðiÞ ðiÞ ðiÞ ^ ðiÞ  ðξ3 Þðj iÞ k A^ k þ 2 þ ðξ6 Þðj iÞ k B^ k þ 2 þ ðξ9 Þðj iÞ k C^ k þ 2 þ ðξ12 Þðj iÞ k D kþ2

^ þ ðξ11 Þðj iÞ k C^ k þ ðξ14 Þðj iÞ k D k

ðiÞ ^ ^ þ ðκ 4 Þðj iÞ k A^ k þ 1 þ ðκ 7 Þðj iÞ k C^ k þ 1 þ ðκ10 Þðj iÞ k D k þ 1 þ ðκ 13 Þj  k E k þ 1

j

ðiÞ

k¼0



k¼0

þ ∑

ðiÞ

þ ðξ11 Þðj iÞ k C^ k þ ðξ14 Þðj iÞ k E^ k



ðiÞ ðiÞ ðiÞ ^ ðiÞ A^ k þ 1 þ ðξ7 Þðj iÞ k B^ k þ 1 þ ðξ10 Þðj iÞ k C^ k þ 1 þ ðξ13 Þðj iÞ k D kþ1  ð i Þ ð i Þ ð i Þ þ ðξ2 Þðj iÞ k E^ k þ ðξ5 Þðj iÞ k A^ k þ ðξ8 Þðj iÞ k B^ k

ðiÞ ðiÞ ðiÞ ^ ðiÞ þ ðκ 12 ÞðiÞ E^ ðiÞ B^ j þ 2 þ ðκ 3 Þð0iÞ A^ j þ 2 þ ðκ6 Þð0iÞ C^ j þ 2 þ ðκ9 Þð0iÞ D jþ2 0 jþ2 (

 j1 ðiÞ 1 ðiÞ ^ ðiÞ ∑ ðk þ 2Þðk þ 1Þ ðκ3 Þj  k Ak þ 2 þ ðκ6 Þðj iÞ k C^ k þ 2 ¼ ðj þ2Þðjþ 1Þ k ¼ 0 "   j ðiÞ ^ ðiÞ þ ðκ 12 ÞðiÞ E^ ðiÞ þ ∑ þ ðκ 9 Þðj iÞ k D ðk þ 1Þ ðκ 1 Þðj iÞ k B^ k þ 1 kþ2 jk kþ2 ðiÞ

k¼0

 j ðiÞ þ ∑ ðk þ 1Þ ðξ1 Þðj iÞ k E^ k þ 1 þ ðξ4 Þðj iÞ k

ðiÞ ^ ðiÞ ðiÞ χ 1 j  k Ak þ 1 þ χ 4 j  k

ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ^ ^ B^ k þ 1 þ χ 7 j  k C^ k þ 1 þ χ 10 j  k D k þ 1 þ χ 13 j  k E k þ 1  ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ þ χ 2 j  k A^ k þ χ 5 j  k B^ k þ χ 8 j  k C^ k

 j ^ ðiÞ þ ðξ ÞðiÞ : þ ∑ ðk þ 1Þ ðξ1 Þðj iÞ k D 4 jk kþ1

j¼0

j ðiÞ ∑ E^ j θ  θi : j¼0 J

"

^ ðiÞ þ ðξ ÞðiÞ A^ ðiÞ þ ðξ ÞðiÞ B^ ðiÞ þ ðξ ÞðiÞ C^ ðiÞ þ ðξ ÞðiÞ E^ ðiÞ D 3 0 jþ2 6 0 jþ2 9 0 jþ2 12 0 j þ 2 jþ2 (

j1 1 ¼ ∑ ðk þ 2Þðk þ 1Þ ðjþ 2Þðj þ 1Þ k ¼ 0   ðiÞ ðiÞ ðiÞ ðiÞ  ðξ3 Þðj iÞ k A^ k þ 2 þ ðξ6 Þðj iÞ k B^ k þ 2 þ ðξ9 Þðj iÞ k C^ k þ 2 þ ðξ12 Þðj iÞ k E^ k þ 2

ðiÞ ðiÞ ðiÞ ðiÞ A^ k þ 1 þ ðξ7 Þðj iÞ k B^ k þ 1 þ ðξ10 Þðj iÞ k C^ k þ 1 þ ðξ13 Þðj iÞ k E^ k þ 1  ^ ðiÞ þ ðξ ÞðiÞ A^ ðiÞ þ ðξ ÞðiÞ B^ ðiÞ þ ðξ2 Þðj iÞ k D 5 jk k 8 jk k k

j¼0

^ ð0imÞ Φ

247

ðiÞ ðmÞ ðiÞ ðmÞ ^ 0i3 ðθ; zÞ þ C^ 0 Φ ^ 0i4 ðθ; zÞ þ B^ 1 Φ ðiÞ

ð11cÞ

ðiÞ

ðiÞ

mÞ mÞ mÞ ^ Φ ^ Φ ^ ð0i5 ^ ð0i6 ^ ð0i7 ðθ; zÞ þ D ðθ; zÞ þ D ðθ; zÞ þ C^ 1 Φ 0 1 ðiÞ

ðiÞ

mÞ mÞ ^ ð0i9 ^ ð0i9 ðθ; zÞ þ E^ 1 Φ ðθ; zÞ; þ E^ 0 Φ

ð12dÞ

248

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

ðiÞ ðmÞ ðiÞ ðmÞ ðiÞ ðmÞ ðmÞ Ψ^ 0i ðθ; zÞ ¼ A^ 0 Ψ^ 0i0 ðθ; zÞ þ A^ 1 Ψ^ 0i1 ðθ; zÞ þ B^ 0 Ψ^ 0i2 ðθ; zÞ ðiÞ ðmÞ ðiÞ ðmÞ þ B^ 1 Ψ^ 0i3 ðθ; zÞ þ C^ 0 Ψ^ 0i4 ðθ; zÞ ðiÞ ðmÞ ^ ðiÞ Ψ^ ðmÞ ðθ; zÞ þ D ^ ðiÞ Ψ^ ðmÞ ðθ; zÞ þ C^ 1 Ψ^ 0i5 ðθ; zÞ þ D 0i6 0i7 0 1 ðiÞ ðmÞ ðiÞ ðmÞ þ E^ 0 Ψ^ 0i9 ðθ; zÞ þ E^ 1 Ψ^ 0i9 ðθ; zÞ;

ð12eÞ

Dividing the overall domain of θ into n sub-domains yields 10n ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ^ ðiÞ , D ^ ðiÞ , E^ ðiÞ and E^ ðiÞ for i¼1, 2, coefficients A^ 0 , A^ 1 , B^ 0 , B^ 1 , C^ 0 , C^ 1 , D 0 1 0 1 …n, which are to be determined. The continuity conditions between pairs of adjacent sub-domains provide 10(n  1) equations, which are, at θ ¼ θi (i¼1, 2, …,n-1)

unrðiÞ ¼ unrði þ 1Þ ;

unθðiÞ ¼ unθði þ 1Þ ;

unzðiÞ ¼ unzði þ 1Þ ;

σ nrθðiÞ ¼ σ nrθði þ 1Þ ;

σ nθθðiÞ ¼ σ nθθði þ 1Þ ;

DnθðiÞ ¼ Dnθði þ 1Þ ;

ϕnðiÞ ¼ ϕnði þ 1Þ ;

ð13aÞ

σ nzθðiÞ ¼ σ nzθði þ 1Þ ;

BnθðiÞ ¼ Bnθði þ 1Þ ;

ð13bÞ

ψ nðiÞ ¼ ψ nði þ 1Þ :

ð13cÞ

The homogenous boundary conditions at θ ¼ 0 and θ ¼ β give another 10 equations. The homogeneous boundary conditions can be specified as follows: n

n

n

n

n

Consequently, 10n homogeneous algebraic equations are established for the 10n coefficients to be determined. To have nontrivial solutions for the coefficients requires zero determinant of the 10n  10n matrix that is formed from the coefficients in the established 10n homogeneous algebraic equations. The values of λm are the roots of the zero determinant and are determined by the numerical approach of Müller [35]; they are ordered as Re[λi ] rRe[λi þ 1 ] (i¼1, 2, 3,…).

4. Verification and comparison The BaTiO3–CoFe2O4 particulate composite is a typical MEE material and is considered herein. According to the macroscopic mixture rule, the composite material properties of this MEE material are related to the material properties of BaTiO3 and CoFe2O4 by [36] κij ¼ κ Bij V I þ κFij ð1  V I Þ;

ð15Þ

where κ Bij and κ Fij denote the material properties of BaTiO3 and CoFe2O4 (Table 1), respectively, and V I is the volume fraction of the

n

σ rθ ¼ σ θθ ¼ σ zθ ¼ 0 ðtractionfreeÞ or ur ¼ uθ ¼ uz ¼ 0 ðclampedÞ; ð14aÞ Dnθ ¼ Bnθ ¼ 0 ðmagneto  electrically openÞ

or

ϕn ¼ ψ n ¼ 0 ðmagneto  electrically closedÞ;

ð14bÞ

Table 1 Material properties. Parameters

BaTiO3 [37]

CoFe2O4 [37]

c11 c12 c13 c33 c44 e15 e31 e33

d15 (N/Am)

166 77 78 162 43 11.6  4.4 18.6 0

286.0 173.0 170.5 269.5 45.3 0 0 0 550.0

d31 (N/Am)

0

580.3

d33 (N/Am) η11 (C2/Nm2) η33 (C2/Nm2) μ11 (Ns2/C2) μ33 (Ns2/C2) g 11 (Ns/VC) g 33 (Ns/VC)

0

699.7

112.0  10  10 126.0  10  10 5.0  10  6 10.0  10  6 0 0

0.8  10  10 0.93  10  10 590.0  10  6 157.0  10  6 0 0

(GPa) (GPa) (GPa) (GPa) (GPa) (C/m2) (C/m2) (C/m2)

Fig. 3. Variation of Re[λ1 ] with vertex angle for MEE wedges.

Table 2 Convergence of Re[λ1] for MEE wedges. Vertex angle

β ¼ 360 3

Material

Boundary conditions

BaTiO3–CoFe2O4VI ¼50%

FOCO

BaTiO3–CoFe2O4V ðI1Þ =V ðI2Þ ¼ 50%/20%

FOFO

BaTiO3–CoFe2O4V ðI1Þ =V ðI2Þ ¼ 90%/10%

Number of sub-domains

3 4 6 8 3 4 6 8 3 4 6 8

Terms

Published results

6

7

8

9

10

12

14

15

0.2482 0.2504 0.2500 0.2500 0.4978 0.4963 0.4993 0.4999 0.4962 0.4977 0.4993 0.4999

0.2492 0.2501 0.2499 0.2500 0.4916 0.4993 0.4999 0.4999 0.4933 0.4972 0.4996 0.5000

0.2462 0.2500 0.2500 0.2500 0.4417 0.4999 0.4999 0.4999 0.4823 0.4986 0.4999 0.4998

0.2489 0.2498 0.2500 0.2500 0.4980 0.4999 0.5000 0.4999 0.4887 0.4998 0.5000 0.5000

0.2492 0.2495 0.2500 0.2500 0.4998 0.4984 0.5000 0.5000 0.4978 0.4989 0.5000 0.5000

0.2495 0.2500 0.2500 0.2500 0.4750 0.4999 0.4999 0.5000 0.4983 0.4998 0.4999 0.5000

0.2499 0.2500 0.2500 0.2500 0.5000 0.4999 0.5000 0.4999 0.4999 0.4999 0.5000 0.5000

0.2499 0.2500 0.2500 0.2500 0.4999 0.4999 0.5000 0.5000 0.4999 0.4999 0.5000 0.5000

0.2500 [32]

0.5000 [32]

0.5000 [32]

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

249

inclusion BaTiO3. The material properties in Table 1 were taken from Wang et al. [37]. For simplicity, four letters are utilized to specify the boundary conditions along the two radial side faces of a wedge with vertex angle β. The first and third letters indicate mechanical boundary conditions at θ ¼ 0 and θ ¼ β, respectively; C and F specify clamped and traction-free, respectively. Similarly, the second and fourth letters represent the magneto-electric boundary conditions at θ ¼ 0 and θ ¼ β, respectively; C and O denote magneto-electrically closed and open boundary conditions, respectively. Since MEE singularities in an MEE wedge that is subjected to the out-of-plane deformation and in-plane electric and magnetic fields have been studied, wedges with the direction of polarization along the

z axis are analyzed herein to verify the proposed solutions. For such wedges, the out-of-plane deformation and in-plane electric and magnetic fields are independent of in-plane deformation and out-ofplane electric and magnetic fields according to Eqs. (6a)–(6e). Notably, Eqs. (8a)–(8e) are independent of z for an MEE wedge. Consequently, the results of Liu and Chue [32], who assumed that all physical quantities are independent of z, are used for comparison. Table 2 shows values of Re[λ1 ] that were obtained by the present approach and by that of Liu and Chue [32] for three wedges with a vertex angle β ¼ 3601. One wedge is made of BaTiO3 and CoFe2O4 with V I ¼ 50% under FOCO boundary conditions. The other two are under FOFO boundary conditions and made of BaTiO3 and CoFe2O4 with V I ¼ 50% and 90% for 01r θ o 1801 and with V I ¼ 20% and 10% for

Fig. 4. Variation of Re[λ1 ] with polarization direction for MEE wedges with β ¼ 2701.

Fig. 6. Variation of Re[λ1 ] with polarization direction for bi-material MEE wedges with β ¼ 2701.

z

h x

Fig. 5. Variation of Re[λ1 ] with vertex angle for bi-material MEE wedges.

Fig. 7. Distribution of λ1 along the thicknesses of FGMEE wedges with γ ¼ 01 and β ¼ 3601.

250

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1801 o θ r 3601, respectively. Notably, for comparison, μ11 ¼ 100  10  6 Ns2 =C2 for CoFe2O4, as in the work of Liu and Chue [32] was used for the results in Table 2. The present results were obtained using different numbers of sub-domains and polynomial terms for each subdomain. The convergent results were obtained by increasing the number of sub-domains or the number of polynomial terms for each sub-domain. The convergent results agree excellently with the results of Liu and Chue [32].

5. Numerical results and discussion Following the confirmation of the correctness of the proposed solution, the asymptotic solution is further employed to

investigate the MEE singularities in the MEE and FGMEE wedges, taking into account the effects of the polarization direction, vertex angle, material-property gradient index and boundary conditions. In the following, only Re[λ1 ] is shown in a graphic form because the dominant singularity order at the vertex of a wedge depends on Re[λ1 ]. Notably, the vertical axis of a figure is labeled λ1 when all λ1 are real; otherwise, the vertical axis is labeled Re[λ1 ]. The numerical results were obtained by dividing a wedge into eight sub-domains in θ with 10 polynomial terms per sub-domain. The x–y–z coordinate system is formed by rotating the x–y–z coordinate system counter-clockwise about the y-axis through an angle γ, so the angle between the polarization direction of an MEE or FGMEE wedge and its thickness direction is γ.

z

x

Fig. 8. Distribution of Re[λ1 ] along the thicknesses of FGMEE wedges with γ ¼ 451 and β ¼ 3601: (a) FOFO boundary conditions, and (b) FCFC boundary conditions.

Fig. 9. Distribution of Re[λ1 ] along the thicknesses of FGMEE wedges with γ ¼ 901 and β ¼ 3601: (a) FOFO boundary conditions, and (b) FCFC boundary conditions.

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

5.1. MEE wedges Figs. 3 and 4 show the results concerning Re[λ1 ] for BaTiO3– CoFe2O4 wedges with V I ¼ 50% and different polarization directions and boundary conditions. Fig. 3 plots the variation of Re[λ1 ] with the vertex angle β, whereas Fig. 4 depicts the variation of Re [λ1 ] with the polarization direction for wedges with β ¼ 2701. Notably, some values of λ1 in Figs. 3 and 4 are complex. For example, all λ1 values in Fig. 3 for wedges with γ ¼ 451 and COCO boundary conditions are complex, and λ1 values in Fig. 4 for wedges with γ between 281 and 1521 under FCFC boundary conditions are complex. These figures demonstrate that the strength of MEE singularities in a wedge generally increases with β, and that free–free mechanical boundary conditions lead to stronger singularities than clamped–clamped boundary conditions. Fig. 4 indicates that changing the direction of polarization can change Re[λ1 ] by up to approximately 10%. The arrangements considered in Figs. 5 and 6 are the same as those in Figs. 3 and 4, respectively, except that bi-material wedges are considered in Figs. 5 and 6. The bi-material wedges in Fig. 5 are made of BaTiO3–CoFe2O4 with V I ¼ 50% in 01 rθ o 1801 and BaTiO3–CoFe2O4 with V I ¼ 20% in 1801 o θ r β. Fig. 5 also reveals that Re[λ1 ] generally decreases as β increases. Notably, not all values of λ1 are complex for wedges with COCO boundary conditions and γ ¼ 451, and λ1 is real when β is between 2121 and 3361. Fig. 6 considers bi-material wedges with β=2701, made of BaTiO3CoFe2O4 with VI ¼20% in 01 rθo 1801 and BaTiO3-CoFe2O4 with VI ¼ 50% in 1801 oθ r2701, and indicates that indicates that changing the polarization direction can change Re[λ1 ] by up to around 15%.

251

FGMEE wedges under consideration herein consist of BaTiO3– CoFe2O4 with V I varying along the thickness direction according to a simple power law VI ¼

z ~ n; h

ð16Þ

where n~ is the material-property gradient index, which indicates the degree of variation of the material properties along its thickness h, and 0 r z r h. The wedges consist of CoFe2O4 only at their bottom surfaces (z¼0), and BaTiO3 at their top surfaces (z¼ h). Figs. 7, 8, and 9 display the variations of Re[λ1 ] or λ1 along the thickness of the FGMEE wedges with γ¼01, 451 and 901, respectively. The wedges have vertex angle β ¼ 3601, free–free mechanical

z

h

x

5.2. FGMEE wedges As mentioned in Section 1, no investigation of geometricallyinduced singularities in an FGMEE wedge has been published. The special feature of the graded spatial compositions of FGMEE materials enables these materials to be used in a very wide range of applications. The distribution of the singularities at the vertex of an FGMEE wedge along its thickness is of particular interest.

z

h x

Fig. 10. Distribution of λ1 along the thicknesses of FGMEE wedges with γ ¼ 01 and β ¼ 2701.

Fig. 11. Distribution of Re[λ1 ] along the thicknesses of FGMEE wedges with γ ¼ 451 and β ¼ 2701: (a) FOFO boundary conditions, and (b) FCFC boundary conditions.

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C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

boundary conditions and n~ ¼ 0:5; 1; 4 and 50. Fig. 7 demonstrates that for wedges with γ ¼ 01 under FOFO and FCFC boundary conditions, the material-property gradient index does not affect the strength of the singularities at the vertex (λ1 ¼ 0:5), which remains constant along the thickness direction. Boundary conditions FOFO and FCFC yield identical results. Figs. 8 and 9 reveal that when γ ¼ 451 and 901, the values of Re[λ1 ] vary with z/h and n~ by less than 7%. The values of Re[λ1 ] for wedges with FOFO boundary conditions generally increase ~ but these trends are not with an increase of z/h and a decrease in n, observed under FCFC boundary conditions. The arrangements considered in Figs. 10, 11 and 12 are the same as those in Figs. 7, 8, and 9, respectively, except that wedges with β ¼ 2701 are considered in Figs. 10–12. Fig. 10 reveals that Re[λ1 ] does not depend on z/h or n~ for wedges with γ ¼ 01 under FOFO or FCFC

boundary conditions, and boundary conditions FOFO and FCFC yield identical results. Figs. 11 and 12 discover that Re[λ1 ] may significantly ~ and Re[λ1 ] may vary by about 4% and 12% for depend on z/h and n, FOFO and FCFC wedges, respectively. The values of Re[λ1 ] for wedges

Fig. 12. Distribution of Re[λ1 ] along the thicknesses of FGMEE wedges with γ ¼ 901 and β ¼ 2701: (a) FOFO boundary conditions, and (b) FCFC boundary conditions.

Fig. 13. Distribution of Re[λ1 ] with along the thicknesses of FGMEE wedges with β ¼ 2701 and COCO boundary conditions: (a) γ ¼ 01, (b) γ ¼ 451, and (c) γ ¼ 901.

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

~ with γ ¼ 901 generally increase with increasing z/h and decreasing n, but wedges with γ ¼ 451 do not exhibit these trends. To investigate the effects of mechanical boundary conditions on Re[λ1 ], Fig. 13 depicts the variation of Re[λ1 ] or λ1 along the thickness direction of COCO wedges with β ¼ 2701 and various ~ n~ ¼ 0:5; 1; 4 and 50). The values of γ (γ ¼ 01; 451 and 901) and n( ~ Re[λ1 ] generally increase with an increase in z/h or a decrease in n. The variations of λ1 with z/h and n~ are less than 2% for wedges with γ ¼ 01, and Re[λ1 ] may vary by up to 7% and 13% for γ ¼ 451 and 901, respectively.

6. Concluding remarks Based on three-dimensional magneto-electro-elasticity theory, this study established asymptotic solutions for the geometricallyinduced magneto-electro-elastic singularities at the vertex of a rectilinearly polarized wedge that is made of functionally graded magneto-electro-elastic materials. The asymptotic solutions were developed using an eigenfunction expansion approach with the power series solution technique and a domain decomposition technique to solve the three-dimensional equations of motion and Maxwell's equations in terms of mechanical displacement components and electric and magnetic potentials in a cylindrical coordinate system. An arbitrary polarization direction of material yields coupling among in-plane and out-of-plane physical quantities (mechanical displacement, electric and magnetic fields), considerably complicates the asymptotic solutions. The correctness of the solutions was confirmed by performing convergence studies of Re[λ1 ] for BaTiO3–CoFe2O4 wedges with polarization in the thickness direction, and comparing the convergent Re[λ1 ] with the results published for wedges under anti-plane deformation and in-plane electric and magnetic fields. The proposed solutions were further employed to determine the variations of Re[λ1 ] with the vertex angle (β) and the position along the thickness (z/h) for MEE and FGMEE wedges made of BaTiO3–CoFe2O4 particulate composite under various boundary conditions and with various polarization directions (γ ¼ 01; 451 and 901). For FGMEE wedges, the volume fraction of the inclusion BaTiO3 was assumed to vary in the thickness direction in a manner governed by a simple power law with the material-property gradient index, and the effects of the materialproperty gradient index (n~ ¼ 0:5; 1; 4; and 50) on Re[λ1 ] were examined. Numerical results reveal that a larger vertex angle yields stronger MEE singularities at the vertex of a wedge, and free–free mechanical boundary conditions lead to stronger singularities than do clamped– clamped boundary conditions. The polarization direction may significantly affect Re[λ1 ]. When FGMEE wedges are polarized along the thickness direction (γ ¼ 01), n~ and z/h have no effect on the orders of singularities in the wedges with β ¼ 2701 and 3601 under FOFO or FCFC boundary conditions. When the thickness direction is not the direction of polarization, the orders of singularities in an FGMEE wedge may depend significantly on n~ and z/h. For example, Re[λ1 ] may vary with n~ and z/h by up to about 12% for a wedge with β ¼ 2701, γ ¼ 901 under FCFC boundary conditions. Notably, the variation of orders of singularities with z/h substantially complicates the determination of stress intensity factors for an FGMEE plate with a V-notch or crack.

Acknowledgment The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract no. NSC 102-2221-E-009-071.

253

Appendix A

h i   ½c ¼ ½Tσ ½K½c½KT ½Tε 1 ; ½e ¼ ½TD ½L½e½KT ½Tε 1 ; d ¼ ½TD ½L d ½KT ½Tε 1 ; ½η ¼ ½TD ½L½η½LT ½TE 1 ; ½μ ¼ ½TB ½L½μ½LT ½TH 1 ; ½g  ¼ ½TE ½L½g ½LT ½TB 1

where 2

cos 2 θ 6 2 6 sin θ 6 6 0 6 ½Tσ ¼ 6 6  cos θ sin θ 6 6 0 4 0 2

cos 2 θ 6 2 6 sin θ 6 6 0 6 ½Tε ¼ 6 6 2 cos θ sin θ 6 6 0 4 0

2

sin θ cos 2 θ 0

0 0 1

2 cos θ sin θ  2 cos θ sin θ 0

cos θ sin θ 0 0

0 0 0

cos 2 θ  sin θ 0 0

2

2

0 cos θ  sin θ

0 0

cos θ sin θ  cos θ sin θ

0 0

0 2 cos θ sin θ 0 0

1 0 0 0

0 2 cos 2 θ  sin θ 0 0

0 0 cos θ  sin θ

cos θ

sin θ

0

3

6 7 ½TE ¼ ½TD ¼ ½TB ¼ ½TH ¼ 4  sin θ cos θ 0 5; 0 0 1 2 3 2 cos ðx; xÞ cos ðx; yÞ cos ðx; zÞ l11 l12 6 7 6 ½L ¼ 4 cos ðy; xÞ cos ðy; yÞ cos ðy; zÞ 5 ¼ 4 l21 l22 cos ðz; xÞ cos ðz; yÞ cos ðz; zÞ l31 l32 2 2 2 2 3 l11 l12 l13 " # K1 2K2 6 2 2 2 7 7 l ½K ¼ ; K1 ¼ 6 4 21 l22 l23 5; K 3 K4 2 2 2 l31 l32 l33 2 3 l12 l13 l13 l11 l11 l12 6l l 7 K2 ¼ 4 22 23 l23 l21 l21 l22 5; l32 l33

l33 l31

l31 l32

l21 l31

l22 l32

l23 l33

6 K3 ¼ 4 l31 l11 l32 l12 l11 l21 l12 l22 2 l22 l33 þ l23 l32 6 K4 ¼ 4 l32 l13 þ l33 l12 l12 l23 þ l13 l22

3

0 0 0

sin θ cos 2 θ

2

2

0 0 0

l13

7 7 7 7 7 7; 0 7 7 7 sin θ 5 cos θ 0 0 0 0 sin cos

3

l23 7 5; l33

3

l33 l13 7 5; l13 l23 l23 l31 þ l21 l33

l21 l32 þ l22 l31

3

l33 l11 þ l31 l13

l31 l12 þ l32 l11 7 5:

l13 l21 þ l11 l23

l11 l22 þ l12 l21

Appendix B

  1 ∂c66 1 ∂c16 ∂c26 ; p2 ¼  c22 þ ; 2λm c16 þ λ2m c11 þ λm c66 c66 ∂θ ∂θ ∂θ

 c26 1 ∂c26 ; p3 ¼ ; p4 ¼ ðλm  1Þc66 þ λm c12  c22 þ c66 c66 ∂θ  1 ∂c66 c46 ; p6 ¼ p5 ¼ ðλm  1Þ λm c16  c26 þ ; c66 ∂θ c66

 1 ∂c46 ; λm ðc14 þ c56 Þ  c24 þ p7 ¼ c66 ∂θ  1 ∂c56 e26 ; p9 ¼ p8 ¼ λm λm c15  c25 þ ; c66 ∂θ c66

 1 ∂e26 ; λm ðe16 þe21 Þ  e22 þ p10 ¼ c66 ∂θ  1 ∂e16 d26 p11 ¼ ; p12 ¼ λm λm e11  e12 þ ; c66 ∂θ c66

 1 ∂d26 ; p13 ¼ λm ðd16 þ d21 Þ  d22 þ c66 ∂θ p1 ¼

3 7 7 7 7 7 7; 7 7 7 θ5 θ

254

p14 ¼

C.S. Huang, C.N. Hu / International Journal of Mechanical Sciences 89 (2014) 242–255

 1 ∂d16 : λm λm d11  d12 þ c66 ∂θ

  1 ∂d21 ∂d22 þ ; λm λm d11 þ d12 þ μ22 ∂θ ∂θ

 d22 1 ∂d22 ; ; t7 ¼  λm ðd12 þ d26 Þ  d26 þ t6 ¼  μ μ22 ∂θ  22 1 ∂d26 ; t8 ¼  ðλm  1Þ λm d16 þ μ22 ∂θ

 d24 1 ∂d24 ; t9 ¼  ; t 10 ¼  λm ðd14 þ d25 Þ þ μ22 μ22 ∂θ  1 ∂d25 ; λm λm d15 þ t 11 ¼  μ22 ∂θ

 g 1 ∂g 2λm g 12 þ 22 ; t 12 ¼ 22 ; t 13 ¼ μ22 μ22 ∂θ  1 ∂g 12 t 14 ¼ : λm λm g 11 þ μ22 ∂θ t5 ¼ 



 1 ∂c22 1 ∂c26 ; q2 ¼ ; 2λm c26 þ ðλm 1Þ ðλm þ 1Þc66 þ c22 c22 ∂θ ∂θ

 c26 1 ∂c26 q3 ¼ ; ; q4 ¼ λm ðc12 þ c66 Þ þc66 þ c22 þ c22 c22 ∂θ

 1 ∂c12 ∂c22 q5 ¼ þ ; ðλm þ 1Þðλm c16 þc26 Þ þ λm c22 ∂θ ∂θ

 c24 1 ∂c24 q6 ¼ ; ; q7 ¼ λm ðc25 þ c46 Þ þc46 þ c22 c22 ∂θ

 ∂c25 q8 ¼ λm ðλm þ1Þc56 þ ; ∂θ

 e22 1 ∂e22 q9 ¼ ; ; q10 ¼ λm ðe12 þ e26 Þ þ e26 þ c22 c22 ∂θ

 1 ∂e12 ; λm ðλm þ 1Þe16 þ q11 ¼ c22 ∂θ

 d22 1 ∂d22 ; ; q13 ¼ λm ðd12 þ d26 Þ þ d26 þ q12 ¼ c22 c22 ∂θ

 1 ∂d12 : q14 ¼ λm ðλm þ 1Þd16 þ c22 ∂θ q1 ¼

References

  1 ∂c44 1 ∂c45 c46 ; r2 ¼ ; r3 ¼ 2λm c45 þ λm λm c55 þ c44 c44 ∂θ ∂θ c44

 1 ∂c46 ; r4 ¼ λm ðc14 þ c56 Þ þ c24 þ c44 ∂θ

  1 ∂c14 ∂c24 c24 r5 ¼ þ ; r6 ¼ λm λm c15 þc25 þ ; c44 ∂θ ∂θ c44

 1 ∂c24 r7 ¼ ; λm ðc25 þ c46 Þ  c46 þ c44 ∂θ  1 ∂c46 e24 r8 ¼ ; r9 ¼ ðλm  1Þ λm c56 þ ; c44 ∂θ c44

  1 ∂e24 1 ∂e14 ; r 11 ¼ ; r 10 ¼ λm ðe14 þe25 Þ þ λm λm e15 þ c44 c44 ∂θ ∂θ

 d24 1 ∂d24 ; ; r 13 ¼ λm ðd14 þ d25 Þ þ r 12 ¼ c44 c44 ∂θ  1 ∂d14 : λm λm d15 þ r 14 ¼ c44 ∂θ r1 ¼

  1 ∂η 1 ∂η 2λm η12 þ 22 ; s2 ¼ λm λm η11 þ 12 ; η22 η22 ∂θ ∂θ

 1 ∂e26 ; s4 ¼  λm ðe16 þ e21 Þ þ e22 þ η22 ∂θ

  1 ∂e21 ∂e22 s5 ¼  þ λm λm e11 þ e12 þ η22 ∂θ ∂θ

 e22 1 ∂e22 s6 ¼  ; s7 ¼  ; λm ðe12 þ e26 Þ  e26 þ η22 η22 ∂θ  1 ∂e26 s8 ¼  ðλm 1Þ λm e16 þ ; η22 ∂θ

 e24 1 ∂e24 s9 ¼  ; s10 ¼  ; λm ðe14 þe25 Þ þ η22 η22 ∂θ  1 ∂e25 s11 ¼  λm λm e15 þ ; η22 ∂θ

 g 1 ∂g s12 ¼ 22 ; s13 ¼ 2λm g 12 þ 22 ; η22 η22 ∂θ  1 ∂g 12 s14 ¼ : λm λm g 11 þ η22 ∂θ s1 ¼

  1 ∂μ 1 ∂μ 2λm μ12 þ 22 ; t 2 ¼ λm λm μ11 þ 12 ; μ22 μ22 ∂θ ∂θ

 1 ∂d26 ; λm ðd16 þ d21 Þ þ d22 þ t4 ¼  μ22 ∂θ t1 ¼

s3 ¼ 

e26 ; η22

t3 ¼ 

d26 ; μ22

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