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Applied Mathematical Modelling 52 (2017) 197–214

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Applied Mathematical Modelling

journal homepage: www.elsevier.com/locate/apm

Magneto–electro interaction of two offset indenters in

frictionless contact with magnetoelectroelastic materials

Yue-Ting Zhou

a , ∗, Sheng-Jie Pang

a , Yong Hoon Jang

b

a School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 20 0 092, PR China b School of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea

a r t i c l e i n f o

Article history:

Received 12 April 2017

Revised 1 July 2017

Accepted 20 July 2017

Available online 27 July 2017

Keywords:

Magneto-electro interaction

Magnetoelectroelastic materials

Electrically-conducting and

magnetically-conducting

Closed forms

Singularity

Multi-field coupling

a b s t r a c t

Within the theory of linear full-field magneto–electro–elasticity, magneto–electro interac-

tion of two electrically-conducting and magnetically-conducting indenters acting over the

surface of magnetoelectroelastic materials widely used in practical industries is examined.

The operation theory, Fourier transform technique and integral equation technique are em-

ployed to address the two-dimensional, mixed boundary-value problem explicitly. The sur-

face stresses, electric displacement and magnetic induction and their respective intensity

factors are obtained in closed forms for two perfectly conducting semi-cylindrical inden-

ters. Degradation from two perfectly conducting semi-cylindrical indenters to one single

perfectly conducting cylindrical indenter is discussed. Numerical analyses are detailed to

reveal the effects of the interactions between two semi-cylindrical indenters on contact

behaviors subjected to multi-field loadings.

© 2017 Published by Elsevier Inc.

1. Introduction

The indentation technique involves the application of a well-defined indenter to deform the testing materials and charac-

terize their mechanical performances [1] . For example, since residual stresses exert pronounced influences on materials’ me-

chanical behaviors including fatigue, fracture, wear and friction, one can use the instrumented indentation technique [2] to

measure the residual stress field easily with comparison of other methods, such as the hole-drilling and layer-removing

techniques, curvature measurement, ultrasonic methods, X-ray and neutron diffraction [3] .

The indentation problems of magnetoelectroelastic materials (MEE) exhibiting a magneto–electric effect motivated a

number of experimental fabrications and theoretical predictions for gaining a better understanding of the interaction of

microstructures and coupling effects [4] . Indenters with several specific geometries were concerned. Hou et al. [5] studied

the elliptical Hertzian contact problem of magnetoelectroelastic materials based on the fundamental solutions [6] . Chen et

al. [7] established a general theory of the indentation for the flat, conical and spherical indenter acting on the surface of

magnetoelectroelastic materials. Zhou and Lee [8] and Zhou and Kim [9] developed a basic theory of the sliding contact of

magnetoelectroelastic materials subject to a rigid indenter with flat profile, parabolic profile, triangular profile or cylindrical

profile illustrating that the indenter profile greatly affects the contact behavior. Recently, Li et al. [10] presented fundamental

solutions for the contact problem of a magnetoelectroelastic half-space punched by a smooth and rigid half-infinite inden-

ter. In the above-mentioned papers, the media were only subject to one indenter. Multiple indenters should be considered

to reveal how the indenter spacing affects the contact behaviors. For example, two or more indenters ( Fig. 1 ) are used to

∗ Corresponding author.

E-mail address: [email protected] (Y.-T. Zhou).

http://dx.doi.org/10.1016/j.apm.2017.07.041

0307-904X/© 2017 Published by Elsevier Inc.


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198 Y.-T. Zhou et al. / Applied Mathematical Modelling 52 (2017) 197–214

Fig. 1. Two indenters acting on Solid-phase-sintered Silicon Carbide Ceramic (SSiC) [11] .

Fig. 2. Schematic figure of magnetoelectroelastic materials under two symmetrical, perfectly conducting indenters – two semi-cylindrical indenters as

example.

detect the fracture toughness of Solid-phase-sintered Silicon Carbide Ceramic (SSiC) by the Vickers and Knoop indentation

method [11] . The Vickers indenters or Knoop indenters are loaded on the SSiC sample surfaces with different loads. Beyond

the critical loads, surface cracks make the sample break. Thus, the surface damage mechanism under double indenters needs

to be explored. On the other hand, the indenters occupy various profiles [12] , and different profile has different effect on

the contact behavior. Addressing two collinear indenters acting on piezoelectric materials, one kind of single-phase magne-

toelectroelastic materials, Wang et al. [13] found that the indenter tip fields were greatly affected by the relative distance

between two collinear indenters whose decrease leads to weakening of the quantities near the inner tips of the indenters.

Due to multiple fields coupling, the quantities in magnetoelectroelastic materials must be disturbed by two indenters, which

makes the interaction effect between two indenters deserve to be studied.

This article conducts an exact contact analysis of magnetoelectroelastic materials under two electrically-conducting and

magnetically-conducting indenters. Singular integral equations with kernel like 1 / ( χ2 − x 2 ) because of the interaction of two

indenters are obtained and solved analytically, which makes the solutions have a quite different form. The fields disturbed

by the two perfectly conducting semi-cylindrical indenters are given explicitly. Figures are drawn to show that the quantities

around the tips can be adjusted by selecting proper groups of the multi-field loadings due to the coupling properties.

2. Formulation of the problem

There are two electrically-conducting and magnetically-conducting indenters acting symmetrically about z -axis on the

surface of magnetoelectroelastic materials ( Fig. 2 ). Each of the two rigid indenters is pressed by an external loading P , an

accumulated electric charge Q and an accumulated magnetic influx M .

2.1. Basic equations

The constitutive equations are [14]

σ = cS − e E − h H , (1)

D = e T S + εE + dH , (2)

B = h

T S + dE + μH , (3)

Y.-T. Zhou et al. / Applied Mathematical Modelling 52 (2017) 197–214 199

where the superscript T denotes the transposition, and σ , S , D , E , B and H are, respectively, the vectors of the stress, strain,

electric displacement, electric field, magnetic induction, and magnetic field, which are given as follows:

σ =

(σxx σzz σxz

)T , S =

(u ,x w ,z

1 2 ( u ,z + w ,x )

)T , (4)

D =

(D x D z

)T , E =

(−ϕ ,x −ϕ ,z

)T , (5)

B =

(B x B z

)T , H =

(−ψ ,x −ψ ,z

)T , (6)

where the comma stands for the differentiation with respect to the corresponding coordinate variables, u and w are the

mechanical displacement components, and ϕ and ψ represent the electric potential and the magnetic potential.

In Eqs. (1 –3 ), c , e , h , ε, d and μ are the matrices of the elastic coefficients, electromechanical coupling coefficients, magne-

tomechanical coupling coefficients, dielectric permeability coefficients, magneto–electro coupling coefficients, and magnetic

permeability coefficients, which take the following forms for linearly, transversely isotropic magnetoelectroelastic materials:

c =

[

c 11 c 13 0

c 13 c 33 0

0 0 2 c 44

]

, e =

[

0 e 31

0 e 33

2 e 15 0

]

, (7)

h =

[

0 h 31

0 h 33

2 h 15 0

]

, ε =

[ε 11 0

0 ε 33

], (8)

d =

[d 11 0

0 d 33

], μ =

[μ11 0

0 μ33

]. (9)

The equilibrium equations free of any generalized body source are

σxx,x + σxz,z = 0 , σxz,x + σzz,z = 0 , (10)

D x,x + D z,z = 0 , (11)

B x,x + B z,z = 0 . (12)

2.2. Modeling of contact problem of two perfectly conducting indenters

It is noted that for the fracture problem, there is a gap between the upper and lower crack surfaces, and air may enter

crack gap. Thus, the flux of an electric field through the crack gap is not always zero. Hao and Shen [15] proposed the

semi-permeable crack face condition to describe this situation with the impermeable and permeable cases as limiting cases.

For the present contact problem, the indenter and magnetoelectroelastic materials contact well in the contact region. Thus,

it is assumed that there is no air inside the contact region, and perfectly conducting boundary conditions are chosen.

The x > 0 part of the system is considered since the stated problem is symmetric with respect to x = 0. Denote the contact

area as 0 < l 1 < x < l 2 in x > 0 part ( Fig. 2 ).

Since each indenter is electrically-conducting and magnetically-conducting, one has

ϕ(x, 0) = ϕ 1 (x ) , 0 < l 1 < x < l 2 , (13)

ψ(x, 0) = ψ 1 (x ) , 0 < l 1 < x < l 2 , (14)

with φ1 ( x ) and ψ 1 ( x ) denoting the electric potential and the magnetic potential inside the contact area 0 < l 1 < x < l 2 .

The perfectly conducting property makes the normal components of the electric displacement and the magnetic influx

non-zero inside the contact area, though which are free outside the contact area. The surface normal stress is also not zero

inside the contact area while keeps free outside the contact area.

D z (x, 0) =

{−q (x ) , 0 < l 1 < x < l 2

0 , 0 < x < l 1 , x > l 2 , (15)

B z (x, 0) =

{−m (x ) , 0 < l 1 < x < l 2

0 , 0 < x < l 1 , x > l 2 , (16)

σzz (x, 0) =

{−p(x ) , 0 < l 1 < x < l 2

0 , 0 < x < l 1 , x > l 2 , (17)


where q ( x ), m ( x ), and p ( x ) are unknown functions. The accumulated electric charge Q , the accumulated magnetic induction

M and the total indentation force P can be found by integrating q ( x ), m ( x ), and p ( x )

∫ −l 1

−l 2

q (x ) dx =

∫ l 2

l 1

q (x ) dx = Q , (18)

∫ −l 1

−l 2

m (x ) dx =

∫ l 2

l 1

m (x ) dx = M , (19)

∫ −l 1

−l 2

p(x ) dx =

∫ l 2

l 1

p(x ) dx = P . (20)

The surface shear stress keeps zero

σxz (x, 0) = 0 . (21)

The indenter profile is described as

w (x, 0) = w 1 (x ) , 0 < l 1 < x < l 2 , (22)

with w 1 ( x ) being a known function.

The following regularity conditions should be satisfied at infinity:

u (x, z) , w (x, z) , ϕ(x, z) , ψ(x, z) → 0 ,

√

x 2 + z 2 → ∞ . (23)

3. General solutions

Considering Eqs. (1 –12 ), one may discover the general solutions as [16] (u w φ ψ

)T =

(h̄ k 1 h̄ k 2 h̄ k 3 h̄ k 4

)T λ̄(x, z) , (24)

where λ̄(x, z) is an auxiliary function, which satisfies Eq. (26) as will be seen later, and � kn ( k , n = 1, 2, 3, 4) are the cofactors

of the operator matrix K given in the Appendix . In the following, � 2 n ( n = 1, 2, 3, 4) will be used, which takes the form

h̄ 21 = 11 ∂ 6

∂ x 5 ∂z + 12

∂ 6

∂ x 3 ∂ z 3 + 13

∂ 6

∂ x∂ z 5 ( n = 1 ) ,

h̄ 2 n = n 1 ∂ 6

∂ x 6 + n 2

∂ 6

∂ x 4 ∂ z 2 + n 3

∂ 6

∂ x 2 ∂ z 4 + n 4

∂ 6

∂ z 6 ( n = 2 , 3 , 4 ) ,

(25)

where kn ( k , n = 1, 2, 3, 4) are given in the Appendix .

In Eq. (24) , the following relationship holds for the function λ̄(x, z) :

det ( K ) λ̄(x, z) = 0 . (26)

Applying the Fourier cosine transform with respect to x to Eq. (26) arrives at

� 1 ∂ 8 ˜ λ̄

∂ z 8 + � 2 ω

2 ∂ 6 ˜ λ̄

∂ z 6 + � 3 ω

4 ∂ 4 ˜ λ̄

∂ z 4 + � 4 ω

6 ∂ 2 ˜ λ̄

∂ z 2 + � 5 ω

8 ˜ λ̄ = 0 , (27)

where � j ( j = 1 , . . . , 5) are given in the Appendix , and

˜ λ̄( ω, z ) is defined as

˜ λ̄( ω, z ) =

∫ ∞

0

λ̄( x, z ) cos ( ωx ) dx . (28)

Considering ˜ λ̄( ω, z ) = e ωυz , one can obtain the characteristic equation associated with Eq. (27) as

� 1 υ8 + � 2 υ

6 + � 3 υ4 + � 4 υ

2 + � 5 = 0 . (29)

where υ represents the root.


According to the eigenvalue properties of Eq. (29) and using the Almansi’s theorem [17] , one may express ˜ λ̄( ω, z ) as

follows:

˜ λ̄( ω, z ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

4 ∑

k =1

O k ̃ λ̄k ( ω, z ) , υ1 � = υ2 � = υ3 � = υ4

2 ∑

k =1

O k ̃ λ̄k ( ω, z ) +

4 ∑

k =3

O k ( ωz ) k −3 e υ3 ωz , υ1 � = υ2 � = υ3 = υ4

2 ∑

k =1

( O 2 k −1 e υk ωz + O 2 k ωz e υk ωz ) , υ1 = υ2 � = υ3 = υ4 > 0

O 1 e υ1 ωz +

4 ∑

k =2

O k ( ωz ) k −2 e υ2 ωz , υ1 � = υ2 = υ3 = υ4 > 0

4 ∑

k =1

O k ( ωz ) k −1 e υ1 ωz , υ1 = υ2 = υ3 = υ4 > 0

, (30)

where O k ( k = 1, 2, 3, 4) are the unknown functions to be determined from the boundary conditions, and

˜ λ̄k ( ω, z ) depend

on whether the eigenvalues are real or complex. For example, in the case of υ1 � = υ2 � = υ3 � = υ4 , if there are K 1 positive real

roots expressed as υm

= ζm

( m = 1, …, K 1 ) and K 2 pairs of conjugate complex roots with positive real parts expressed as

υn = ξn = αn + i βn ( n = 1, …, K 2 ) with i 2 = −1, ˜ λ̄m

( ω, z ) can be given as

˜ λ̄m

( ω, z ) = e ζm ωz ( m = 1 , . . . , K 1 ) , (31)

˜ λ̄K 1 +2 n −1 ( ω, z ) = e αn ωz cos ( βn ωz ) ( n = 1 , . . . , K 2 ) , (32)

˜ λ̄K 1 +2 n ( ω, z ) = e αn ωz sin ( βn ωz ) ( n = 1 , . . . , K 2 ) . (33)

Note that when υ1 � = υ2 � = υ3 � = υ4 , one may obtain K 1 = 0 and K 2 = 2, or K 1 = 1 and K 2 = 1, or K 1 = 4 and K 2 = 0 since

K 1 + 2 K 2 = 4 is required.

In the υ1 � = υ2 � = υ3 = υ4 case, ˜ λ̄1 ( ω, z ) and

˜ λ̄2 ( ω, z ) can be given in the same way as shown in Eq. (31) .

Various field quantities can be given on the basis of ˜ λ̄( ω, z ) with the consideration of the constitutive Eqs. (1 –3 )

(u σxz D x B x

)T =

2

π

∫ ∞

0

4 ∑

k =1

O k ω

6 sin ( ωx ) ×(ϒ(k )

u ωϒ(k ) xz ωϒ(k )

dx ωϒ(k )

bx

)T dω, (34)

(w ϕ ψ σxx σzz D z B z

)T =

2

π

∫ ∞

0

4 ∑

k =1

O k ω

6 cos ( ωx )

×(ϒ(k )

w

ϒ(k ) ϕ ϒ(k )

ψ

ωϒ(k ) xx ωϒ(k )

zz ωϒ(k ) dz

ωϒ(k ) bz

)T

dω, (35)

where known functions ϒ(k ) l

( ω, z ) ( k = 1, …, 4, l = u , w , φ, ψ , xx , zz , xz , dz , dx , bz , bx ) are given as

ϒ(m ) l

( ω, z ) = γl ( ζm

) e ζm ωz ( m = 1 , . . . , K 1 ) , (36)

ϒ( K 1 +2 n −1) l

( ω, z ) =

[γ (C)

l ( αn , βn ) cos ( βn ωz ) − γ (S)

l ( αn , βn ) sin ( βn ωz )

]e αn ωz

( n = 1 , . . . , K 2 ) , (37)

ϒ( K 1 +2 n ) l

( ω, z ) =

[γ (S)

l ( αn , βn ) cos ( βn ωz ) + γ (C)

l ( αn , βn ) sin ( βn ωz )

]e αn ωz

( n = 1 , . . . , K 2 ) , (38)

where γ l ( ζ m

) ( m = 1, …, K 1 , l = u , w , φ, ψ , xx , zz , xz , dx , dz , bx , bz ), γ (C) l

( αn , βn ) and γ (S) l

( αn , βn ) ( n = 1, …, K 2 , l = u , w , φ,

ψ , xx , zz , xz , dx , dz , bx , bz ) are given in the Appendix (results for distinctive eigenvalues case are presented in what follows

since the commercially available MEE, which are transversely isotropic, generate distinctive eigenvalues).

4. Singular integral equations for two perfectly conducting indenters

Considering boundary conditions Eqs. (22) , ( 13 ) and ( 14 ) results in singular integral equations as follows: ∫ l 2

l

�11 p(χ ) + �12 q (χ ) + �13 m (χ )

χ2 − x 2 dχ = − π

2 x

∂ w 1 ( x )

∂x , l 1 < x < l 2 , (39)

1


∫ l 2

l 1

�21 p(χ ) + �22 q (χ ) + �23 m (χ )

χ2 − x 2 dχ = − π

2 x

∂ ϕ 1 ( x )

∂x , l 1 < x < l 2 , (40)

∫ l 2

l 1

�31 p(χ ) + �32 q (χ ) + �33 m (χ )

χ2 − x 2 dχ = − π

2 x

∂ ψ 1 ( x )

∂x , l 1 < x < l 2 , (41)

where �mn ( m , n = 1, 2, 3) are given as

�m 1 =

4 ∑

k =1

(−1) k +1 C (1 k )

B

det ( C B ) ×

⎧ ⎪ ⎨

⎪ ⎩

ϒ(k ) w

(ω, 0) , m = 1

ϒ(k ) ϕ (ω, 0) , m = 2

ϒ(k ) ψ

(ω, 0) , m = 3

, (42)

�m 2 =

4 ∑

k =1

(−1) k +1 C (3 k )

B

det ( C B ) ×

⎧ ⎪ ⎨

⎪ ⎩

ϒ(k ) w

(ω, 0) , m = 1

ϒ(k ) ϕ (ω, 0) , m = 2

ϒ(k ) ψ

(ω, 0) , m = 3

, (43)

�m 3 =

4 ∑

k =1

(−1) k C (4 k )

B

det ( C B ) ×

⎧ ⎪ ⎨

⎪ ⎩

ϒ(k ) w

(ω, 0) , m = 1

ϒ(k ) ϕ (ω, 0) , m = 2

ϒ(k ) ψ

(ω, 0) , m = 3

, (44)

where C (lk ) B

(l, k = 1 , . . . , 4) are the complement minors of the matrix C B = [ C lk ], ( l , k = 1, …, 4) taking the following forms:

C 1 k = ϒ(k ) zz ( ω, 0 ) , C 2 k = ϒ(k )

xz ( ω, 0 ) , C 3 k = ϒ(k ) dz

( ω, 0 ) ,

C 4 k = ϒ(k ) bz

( ω, 0 ) (k = 1 , . . . , 4) . (45)

The singular integral equations, Eqs. (39) and ( 40 ), and Eqs. (18 –20 ) can be normalized as ∫ +1

−1

� j1 p (0) (ς ) + � j2 q

(0) (ς ) + � j3 m

(0) (ς )

ς − τdς = −πZ j (τ ) , −1 < τ < +1 ,

j = 1 , 2 , 3 , (46)

∫ 1

−1

p (0) (ς ) dς =

2 P

d 0 , (47)

∫ 1

−1

q (0) (ς ) dς =

2 Q

d 0 , (48)

∫ 1

−1

m

(0) (ς ) dς =

2 M

d 0 , (49)

where

p (0) (ς ) =

p(χ )

χ, q (0) (ς ) =

q (χ )

χ, m

(0) (ς ) =

m (χ )

χ, (50)

ς =

χ2 − l 0 d 0

, τ =

x 2 − l 0 d 0

, d 0 =

( l 2 ) 2 − ( l 1 )

2

2

, l 0 =

( l 2 ) 2 + ( l 1 )

2

2

, (51)

Z 1 (τ ) =

1

x

∂ w 1 (x )

∂x , Z 2 (τ ) =

1

x

∂ ϕ 1 (x )

∂x , Z 3 (τ ) =

1

x

∂ ψ 1 (x )

∂x . (52)

5. Exact contact analysis

5.1. Two perfectly conducting semi-cylindrical indenters

For two perfectly conducting semi-cylindrical indenters acting on the surface of MEE with a constant electric potential

and a constant magnetic potential, one may obtain

w 1 (x ) = w 0 +

x 2 , l 1 < x < l 2 , (53)

2 R


ϕ 1 (x ) = ϕ 0 , (54)

ψ 1 (x ) = ψ 0 , (55)

where w 0 , φ0 and ψ 0 are constants, and R is the radius of the semi-cylindrical indenter.

Note that the first contact point ( x = l 1 ) is not on the apex of the semi-cylindrical shape of the indenter, and the maxi-

mum penetration depths vary with different values of l 1 . Single semi-cylindrical indenter has been studied by Hwu and Fan

[18] and Guler and Erdogan [19] . Two semi-cylindrical indenters also enable one to give analytical solutions.

Considering Eqs. (39 –41 ), ( 46 ), ( 53–55 ), one has

Z 1 (τ ) =

1

R

, (56)

Z 2 (τ ) = Z 3 (τ ) = 0 . (57)

Since q ( x ) = q 1 ( x ) + q 2 ( x ) and m ( x ) = m 1 ( x ) + m 2 ( x ) for the conducting semi-cylindrical indenter, in which q 1 ( x ) and m 1 ( x )

are caused by the electric potential and the magnetic potential and q 2 ( x ) and m 2 ( x ) caused by the normal mechanical load

P , one has

q (0) (ς ) = q (0) 1

(ς ) + q (0) 2

(ς ) , (58)

m

(0) (ς ) = m

(0) 1

(ς ) + m

(0) 2

(ς ) . (59)

Thus, Eqs. (46 –49 ) can be written as ∫ +1

−1

m

(0) 1

(ς )

ς − τdς = 0 , −1 < τ < +1 , (60)

∫ 1

−1

m

(0) 1

(ς ) dς =

2 ( M − M F )

d 0 , (61)

in terms of m

(0) 1

(ς ) ,

∫ +1

−1

q (0) 1

(ς )

ς − τdς = 0 , −1 < τ < +1 , (62)

∫ 1

−1

q (0) 1

(ς ) dς =

2 ( Q − Q F )

d 0 , (63)

in terms of q (0) 1

(ς ) , and

∫ +1

−1

� j1 p (0) (ς ) + � j2 q

(0) 2

(ς ) + � j3 m

(0) 2

(ς )

ς − τdς =

⎧ ⎨

⎩

−π

R

, j = 1

0 , j = 2

0 , j = 3

, −1 < τ < +1 , (64)

∫ 1

−1

p (0) (ς ) dς =

2 P

d 0 , (65)

∫ 1

−1

q (0) 2

(ς )(ς ) dς =

2 Q F

d 0 , (66)

∫ 1

−1

m

(0) 2

(ς ) dς =

2 M F

d 0 , (67)

in terms of p (0) ( ς ), q (0) 2

(ς ) and m

(0) 2

(ς ) , and Q F and M F denote the electric charge and the magnetic induction due to the

total indentation force P .

The exact solutions of integral Eqs. (60 –67 ) take the form

m

(0) 1

(ς ) =

2 ( M − M F )

d 0 π√

1 − ς

2 , (68)

q (0) 1

(ς ) =

2 ( Q − Q F )

d 0 π√

1 − ς

2 , (69)


p (0) (ς ) =

1

�P R

√

1 − ς

1 + ς

, q (0) 2

( ς ) =

1

�Q R

√

1 − ς

1 + ς

,

m

(0) 2

(ς ) =

1

�M

R

√

1 − ς

1 + ς

,

(70)

where �P , �Q and �M

are given as

�P =

det (( �mn ) 3 ×3

)�22 �33 − �32 �23

,

�Q =

det (( �mn ) 3 ×3

)�23 �31 − �21 �33

,

�M

=

det (( �mn ) 3 ×3

)�21 �32 − �22 �31

.

(71)

Then, the contact stress, electric displacement and magnetic induction inside the contact region are given as follows:

p(x ) =

| x | �P R

√

( l 2 ) 2 − x 2

x 2 − ( l 1 ) 2 , l 1 < | x | < l 2 , (72)

q (x ) =

| x | �Q R

√

( l 2 ) 2 − x 2

x 2 − ( l 1 ) 2

+

2 ( Q − Q F ) | x | π

√ [( l 2 )

2 − x 2 ][

x 2 − ( l 1 ) 2 ] , l 1 < | x | < l 2 , (73)

m (x ) =

| x | �M

R

√

( l 2 ) 2 − x 2

x 2 − ( l 1 ) 2

+

2 ( M − M F ) | x | π

√ [( l 2 )

2 − x 2 ][

x 2 − ( l 1 ) 2 ] , l 1 < | x | < l 2 . (74)

From the first terms in the right hand side of Eqs. (72 –74 ), it can be seen that the surface contact stress, the surface

electric displacement and the surface magnetic induction are always singular at the inner edges of the semi-cylindrical

indenter because of the physical nature that the inner edge is a sharp corner. Moreover, the surface electric displacement

and the surface magnetic induction may also have singularities at the outer edges of the semi-cylindrical indenter because of

the external loadings as shown in the second terms in Eqs. (73) and ( 74 ), which can be suppressed by choosing appropriate

combinations of the multi-field loadings. Later as seen in Eq. (75) , the singular case of the surface in-plane stress is similar

with that of the surface electric displacement and the surface magnetic induction, and the outer-edge singularities can also

be suppressed by choosing appropriate combinations of the multi-field loadings.

Considering Eqs. (34) , ( 35 ), ( 72–74 ) results in the surface stresses, surface electric displacements and surface magnetic

inductions as follows:

σxx ( x, 0 ) =

{

ϒ(P) xx p(x ) + ϒ(Q )

xx q (x ) − ϒ(M) xx m (x ) , l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2 , (75)

σzz ( x, 0 ) =

{

ϒ(P) zz p(x ) + ϒ(Q )

zz q (x ) − ϒ(M) zz m (x ) , l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2 , (76)

σxz ( x, 0 ) =

⎧ ⎨

⎩

(ϒ(P)

xz

�P R

+

ϒ(Q ) xz

�Q R

− ϒ(M) xz

�M

R

)x, l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2

, (77)

D x ( x, 0 ) =

⎧ ⎪ ⎨

⎪ ⎩

(

ϒ(P) dx

�P R

+

ϒ(Q ) dx

�Q R

− ϒ(M) dx

�M

R

)

x, l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2

, (78)

D z ( x, 0 ) =

{

ϒ(P) dz

p(x ) + ϒ(Q ) dz

q (x ) − ϒ(M) dz

m (x ) , l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2 , (79)


B x ( x, 0 ) =

⎧ ⎪ ⎨

⎪ ⎩

(

ϒ(P) bx

�P R

+

ϒ(Q ) bx

�Q R

− ϒ(M) bx

�M

R

)

x, l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2

, (80)

B z ( x, 0 ) =

{

ϒ(P) bz

p(x ) + ϒ(Q ) bz

q (x ) − ϒ(M) bz

m (x ) , l 1 < | x | < l 2

0 , | x | < l 1 or | x | > l 2 , (81)

where

ϒ(P) j

=

4 ∑

k =1

ϒ(k ) j ( ω, 0 )

(−1) k C (1 k )

B

det ( C B ) , ϒ(Q )

j =

4 ∑

k =1

ϒ(k ) j ( ω, 0 )

(−1) k C (3 k )

B

det ( C B ) ,

ϒ(M) j

=

4 ∑

k =1

ϒ(k ) j ( ω, 0 )

(−1) k C (4 k )

B

det ( C B ) , j = xx, zz, xz, dz, dx, bz, bx.

(82)

Employing the properties about minors of the matrix C B = [ C lk ] in Eq. (45) arrives at

ϒ(P) zz = −1 , ϒ(Q )

zz = 0 , ϒ(M) zz = 0 , (83)

ϒ(P) xz = 0 , ϒ(Q )

xz = 0 , ϒ(M) xz = 0 , (84)

ϒ(P) dz

= 0 , ϒ(Q ) dz

= −1 , ϒ(M) dz

= 0 , (85)

ϒ(P) bz

= 0 , ϒ(Q ) bz

= 0 , ϒ(M) bz

= 1 . (86)

Then, one may find the required boundary conditions Eqs. (15 –17 ) and ( 21 ) hold.

Substituting Eq. (70) into Eqs. (65 –67 ), one can determine the width of the contact region with the singular end-points

| x | = l 1 known a priori, Q F and M F as follows:

l 2 =

√

4 P �P R

π+ ( l 1 )

2 , (87)

Q F =

�P P

�Q

, (88)

M F =

�P P

�M

. (89)

One may define the following stress, electric displacement and magnetic induction intensity factors to measure the sin-

gular behavior at the ends of the semi-cylindrical indenters:

K I ( ±l 1 ) = lim

x →±l 1

√

2 π | x ∓ l 1 | p(x ) , (90)

K D ( ±l 1 ) = lim

x →±l 1

√

2 π | x ∓ l 1 | q (x ) , (91)

K B ( ±l 1 ) = lim

x →±l 1

√

2 π | x ∓ l 1 | m (x ) . (92)

Considering Eqs. (68– 70 ), one can rewrite Eqs. (90 –92 ) as follows:

K I ( ±l 1 ) =

√

π l 1 [( l 2 )

2 − ( l 1 ) 2 ]

�P R

, (93)

K D ( ±l 1 ) =

√

π l 1 [( l 2 )

2 − ( l 1 ) 2 ]

�Q R

+ 2 ( Q − Q F )

√

l 1

π[( l 2 )

2 − ( l 1 ) 2 ] , (94)

K B ( ±l 1 ) =

√

π l 1 [( l 2 )

2 − ( l 1 ) 2 ]

�M

R

+ 2 ( M − M F )

√

l 1

π[( l )

2 − ( l ) 2 ] . (95)

2 1


5.2. Verification of the exact analysis

To validate the theoretical derivation for two semi-cylindrical indenters, one can set l 1 = 0. In such a special case, two

semi-cylindrical indenters each with the accumulated electric charge Q , accumulated magnetic induction M and total in-

dentation force P become a single cylindrical indenter with the contact region ( − l 2 , l 2 ), accumulated electric charge 2 Q ,

accumulated magnetic induction 2 M and total indentation force 2 P with considering Eqs. (18 –20 ), i.e. ∫ l 2

−l 2

q (x ) dx =

∫ 0

−l 2

q (x ) dx +

∫ l 2

0

q (x ) dx = Q + Q = 2 Q , (96)

∫ l 2

−l 2

m (x ) dx =

∫ 0

−l 2

m (x ) dx +

∫ l 2

0

m (x ) dx = M + M = 2 M , (97)

∫ l 2

−l 2

p(x ) dx =

∫ 0

−l 2

p(x ) dx +

∫ l 2

0

p(x ) dx = P + P = 2 P . (98)

Setting l 1 = 0 in Eqs. (72 –74 ) and ( 87 ), one has

p(x ) =

1

�P R

√

( l 2 ) 2 − x 2 , x ∈ ( −l 2 , l 2 ) , (99)

q (x ) =

1

�Q R

√

( l 2 ) 2 − x 2 +

2 ( Q − Q F )

π√

( l 2 ) 2 − x 2

, x ∈ ( −l 2 , l 2 ) , (100)

m (x ) =

1

�M

R

√

( l 2 ) 2 − x 2 +

2 ( M − M F )

π√

( l 2 ) 2 − x 2

, x ∈ ( −l 2 , l 2 ) , (101)

l 2 = 2

√

P �P R

π. (102)

Eqs. (99 –102 ) are identical to those given by Zhou and Lee [20] for a single cylindrical indenter with the contact region

( − l 2 , l 2 ), accumulated electric charge 2 Q , accumulated magnetic induction 2 M and total indentation force 2 P . Thus, the

solutions for a single cylindrical indenter are recovered by those for two semi-cylindrical indenters.

For a single cylindrical indenter, the full-field stresses, electric displacements and magnetic inductions of MEE can be

given explicitly besides those on the surface given in Eqs. (75 –81 ). Various stresses, electric displacements and magnetic

induction of MEE take the form (σxx σzz σxz D z D x B z B x

)T =

4 ∑

k =1

A k

(I (a,k ) xx I (a,k )

zz I (a,k ) xz I (a,k )

dz I (a,k ) dx

I (a,k ) bz

I (a,k ) bx

)T

+

4 ∑

k =1

B k

(I (b,k ) xx I (b,k )

zz I (b,k ) xz I (b,k )

dz I (b,k ) dx

I (b,k ) bz

I (b,k ) bx

)T ,

(103)

where

A k =

(−1) k 2

[( Q − Q F ) C

(3 k ) B

− ( M − M F ) C (4 k ) B

]det ( C B ) π

, (104)

B k =

(−1) k l 2

R det ( C B )

(C (1 k )

B

�P

+

C (3 k ) B

�Q

− C (4 k ) B

�M

). (105)

In Eq. (103) , I (a,k ) j

(x, z) and I (b,k ) j

(x, z) ( l = xx , zz , xz , dx , dz , bx , bz ) are given as

I (k,m ) r (x, z) = γr ( ζm

) �(k ) 1

(x, z, ζm

) , k = a, b, m = 1 , . . . , K 1 , r = xx, zz, dz, bz, (106)

I (k, K 1 +2 n −1) r (x, z) =

1

2

[γ (C)

r ( αn , βn ) �(k ) 1 ( x, z, αn , βn ) − γ (S)

r ( αn , βn ) �(k ) 1 ( x, z, αn , βn )

]k = a, b, n = 1 , . . . , K 2 , r = xx, zz, dz, bz, (107)

I (k, K 1 +2 n ) r (x, z) =

1

2

[γ (S)

r ( αn , βn ) �(k ) 1 ( x, z, αn , βn ) + γ (C)

r ( αn , βn ) �(k ) 1 ( x, z, αn , βn )

],

k = a, b, n = 1 , . . . , K 2 , r = xx, zz, dz, bz, (108)


I (k,m ) t (x, z) = γt ( ζm

) �(k ) 2

(x, z, ζm

) , k = a, b, m = 1 , . . . , K 1 , t = xz, dx, bx, (109)

I (k, K 1 +2 n −1) t (x, z) =

1

2

[γ (C)

t ( αn , βn ) �(k ) 2 ( x, z, αn , βn )

−γ (S) t ( αn , βn ) �

(k ) 2 ( x, z, αn , βn )

], k = a, b, n = 1 , . . . , K 2 , t = xz, dx, bx,

(110)

I (k, K 1 +2 n ) t (x, z) =

1

2

[γ (S)

t ( αn , βn ) �(k ) 2 ( x, z, αn , βn )

+ γ (C) t ( αn , βn ) �

(k ) 2 ( x, z, αn , βn )

], k = a, b, n = 1 , . . . , K 2 , t = xz, dx, bx,

(111)

where �(k ) i

( x, z, αn , βn ) and �(k ) i

( x, z, αn , βn ) ( k = a , b , i = 1, 2, m = 1, …, K 1 , n = 1, …, K 2 ) are given as

�(k ) 1 ( x, z, αn , βn ) = �(k )

1 ( x 1 ( ξn ) , z, ξn ) + �(k ) 1 ( x 2 ( ξn ) , z, ξn ) ,

�(k ) 1 ( x, z, αn , βn ) = �(k )

2 ( x 1 ( ξn ) , z, ξn ) + �(k ) 2 ( x 2 ( ξn ) , z, ξn ) ,

k = a, b, n = 1 , . . . , K 2 ,

(112)

�(k ) 2 ( x, z, αn , βn ) = �(k )

2 ( x 1 ( ξn ) , z, ξn ) − �(k ) 2 ( x 2 ( ξn ) , z, ξn ) ,

�(k ) 2 ( x, z, αn , βn ) = −�(k )

1 ( x 1 ( ξn ) , z, ξn ) + �(k ) 1 ( x 2 ( ξn ) , z, ξn ) ,

k = a, b, n = 1 , . . . , K 2 ,

(113)

where

�(a ) 1

(x, z, υ) =

√

[ �2 ( x, Re (υ) ) ] 2 − x 2

[ �2 ( x, Re (υ) ) ] 2 − [ �1 ( x, Re (υ) ) ]

2 , (114)

�(a ) 2

(x, z, υ) =

sgn (x ) √

x 2 − [ �1 ( x, Re (υ) ) ] 2

[ �2 ( x, Re (υ) ) ] 2 − [ �1 ( x, Re (υ) ) ]

2 , (115)

�(b) 1

(x, z, υ) =

√

[ �2 ( x, Re (υ) ) ] 2 − x 2 + Re (υ) z

l 2 , (116)

�(b) 2

(x, z, υ) =

x − sgn (x ) √

x 2 − [ �1 ( x, Re (υ) ) ] 2

l 2 , (117)

�1 ( s, t ) =

1

2

[√

(s + l 2 ) 2 + (tz)

2 −√

(s − l 2 ) 2 + (tz)

2

], (118)

�2 ( s, t ) =

1

2

[√

(s + l 2 ) 2 + (tz)

2 +

√

(s − l 2 ) 2 + (tz)

2

], (119)

where sgn (·) is the sign function, and

x 1 ( υ) = Im ( υ) z + x, x 2 ( υ) = Im ( υ) z − x. (120)

One can prove that the surface stresses, surface electric displacements and surface magnetic inductions for l 1 = 0 given

in Eqs. (75 –82 ) are equal to those given in Eq. (103) when z = 0.

6. Numerical results

Table 1 gives the piezoelectric and piezomagnetic constants used in numerical computation with all absent material co-

efficients being zero. The values given in Table 1 are obtained by applying the simple mixture rule [21] with the assumption

that the volume fraction for PZT-5A in PZT-5A–CoFe 2 O 4 composites [21,22] is 0.7.

As mentioned before, one can determine the width of the contact region, l 2 − l 1 , from Eq. (87) considering that the

singular end-points | x | = l 1 are known a priori. In practical analysis, the location at which the maximum penetration depth

arrives seems to be the singular end point for some kinds of indenters, such as semi-cylindrical indenter and wedge-shaped

indenter. Eq. (87) shows that the contact width l 2 − l 1 is a monotonically increasing function of the external loading P . Thus,

the contact region becomes narrower with the external loading P decreasing, and vanishes with P = 0 as shown in Fig. 3 .

The interaction between two semi-cylindrical indenters also contributes to the contact width. As the value of l 1 become

smaller, the contact region becomes wider. For a flat indenter with a constant penetration depth, the interaction effect can


Table 1

Material properties of MEE ( c mn in 10 9 N/ m

2 , e mn in C/ m

2 , h mn in

N/ ( Am ) , εmn in 10 −9 C 2 / ( N m

2 ) , μmn in 10 −6 N S 2 / C 2 ).

c 11

155 . 2407

c 13

86 . 6946

c 33

141 . 6492

c 44

28 . 36

e 31

−5 . 0463

e 33

10 . 5826

e 15

8 . 6324

h 31

174 . 09

h 33

209 . 91

h 15

165

ε 11

10 . 734

ε 33

10 . 5279

μ11

177

μ33

47 . 1

Fig. 3. Contact width vs. external loading P with R = 0.09 m .

Fig. 4. Effect of the interaction between two semi-cylindrical indenters on the surface normal stress σ zz ( x , 0) with P = 10 7 N/m and R = 0.09 m in which the

unit of distance l 1 is m .

be ignored when the two indenters became far away from each other. For the present two semi-cylindrical indenters with

profile given in Eq. (53) , the maximum penetration depth varies with the values of l 1 changing. Thus, the interaction effect

has a contribution to the contact behavior even when the two indenters become far away.

Fig. 4 shows how the interaction between two semi-cylindrical indenters affects the surface normal stress σ zz ( x , 0). At

the outer edges | x | = l 2 , the surface normal stress remains free, while has singularities, stress concentrations, at the inner

edges | x | = l 1 for two separated semi-cylindrical indenters. With the value of l 1 becoming smaller, the strength of the stress

singularity is relieved, and the stress singularity vanishes when l 1 = 0.

Fig. 5 reveals the influence of the interaction between two semi-cylindrical indenters on the surface electric displacement

D z ( x , 0), the surface magnetic induction B z ( x , 0) and the surface in-plane stress σ xx ( x , 0). Different from the surface normal

stress, these quantities may have singularities not only at the inner edges | x | = l but also at the outer edges | x | = l . The
1 2


Fig. 5. Influence of the interaction between two semi-cylindrical indenters with P = 10 7 N/m and R = 0.09 m on: a) the surface electric displacement D z ( x , 0)

when Q = 0 . 5 C/m , b) the surface magnetic induction B z ( x , 0) when M = 60 N/A , and c) the surface in-plane stress σ xx ( x , 0) when Q = 0 . 5 C/m and M = 60 N/A

in which the unit of distance l 1 is m .

strength of these singularities weakens as two semi-cylindrical indenters stay near. The singularity strength at the outer

edges | x | = l 2 is stronger than that around the inner edges | x | = l 1 . It is the multi-field coupling possessed by MEE that causes

the singularities at the outer edges | x | = l 2 . More specifically, the singularity for the surface electric displacement is generated

by the additional electric charge Q − Q F as seen from Eq. (73) or Eq. (78) , by the additional magnetic induction M − M F for

the surface magnetic induction as seen from Eq. (74) or Eq. (79) , and by both the electric charge and the additional magnetic

induction for the surface in-plane stress as seen from Eq. (75) .


Fig. 6. Effect of the combinations of multi-field loadings when R = 0.09 m and l 1 = 10 − 3 m on: a) the surface electric displacement D z ( x , 0), b) the surface

magnetic induction B z ( x , 0), and c) the surface in-plane stress σ xx ( x , 0) in which the units of P , Q and M are N/m , 0 . 5 C/m and N/A .

The singularities of the surface normal and in-plane stresses, the surface electric displacement and the surface magnetic

induction as seen from Figs. 4 and 5 can cause crack initiation on the surface of the substrate, which may cause materials

to damage more quickly. So engineers must minimize these concentrations.

By setting Q = Q F and/or M = M F in Eqs. (73 –75 ) and calculating the electric charge and the magnetic induction due to

the total indentation force P , i.e. Q and M , from Eqs. (88) and ( 89 ), the singularities at the outer edges | x | = l for the
F F 2


surface electric displacement, the surface magnetic induction and the surface in-plane stress can be suppressed because of

the multi-field coupling as shown in Fig. 6 . These quantities only have singularities at the inner edges | x | = l 1 , and are free at

the outer edges | x | = l 2 . As the combinations of the multi-field loadings escalate, the singularity strength at the inner edges

| x | = l 1 intensifies.

7. Conclusions

Indentation problem of magnetoelectroelastic materials under two electrically-conducting and magnetically-conducting

indenters is investigated. For two perfectly conducting semi-cylindrical indenters, the closed-form solutions for the stresses,

electric displacement and magnetic induction on the surface are obtained, and the stress, electric displacement and mag-

netic induction intensity factors are defined. Numerical results are calculated to show the influences of the interaction of

two semi-cylindrical indenters on the contact behaviors under different multi-field loadings. The obtained results reveal that

the inner and outer edges are the most likely crack initiation locations because of the singularities for the surface electric

displacement, the surface magnetic induction and the surface in-plane stress, which may explain why surface damage oc-

curs for magnetoelectroelastic materials. The singularities at the outer edges can be suppressed by choosing appropriate

combinations of the multi-field loadings due to the multi-field coupling to alleviate the surface damage.

The present article studies the contact problem of homogeneous magnetoelectroelastic materials. Wang and Kuna did

excellent work with deriving the analytical solutions of the static screw dislocation [23] and time-harmonic dynamic Green’s

functions [24] of the functionally graded magnetoelectroelastic solids, and finding that the inhomogeneity has quite different

influences from the homogeneous magnetoelectroelastic materials. The effect of inhomogeneity of magnetoelectroelastic

materials on contact behaviors especially on the dynamic contact behaviors needs to be further revealed in the future work.

Acknowledgments

This work was supported by the National Natural Science Foundation of China ( 11472193 , 11572227 and 11261042 ), and

the Fundamental Research Funds for the Central Universities ( 1330219162 ). Dr. Yue-Ting Zhou thanks the Research Institute

for Sustainable Urban Development and Dr. Haimin Yao at The Hong Kong Polytechnic University for offering a visiting

fellowship.

Appendix

1. Expressions of the operator matrix K related to Eq. (24)

K =

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

c 11 ∂ 2

∂ 2 x + c 44

∂ 2

∂ 2 z ( c 13 + c 44 )

∂ 2

∂ x∂ z ( e 31 + e 15 )

∂ 2

∂ x∂ z ( h 31 + h 15 )

∂ 2

∂ x∂ z

( c 13 + c 44 ) ∂ 2

∂ x∂ z c 44

∂ 2

∂ 2 x + c 33

∂ 2

∂ 2 z e 15

∂ 2

∂ 2 x + e 33

∂ 2

∂ 2 z h 15

∂ 2

∂ 2 x + h 33

∂ 2

∂ 2 z

( e 31 + e 15 ) ∂ 2

∂ x∂ z e 15

∂ 2

∂ 2 x + e 33

∂ 2

∂ 2 z −ε 11

∂ 2

∂ 2 x − ε 33

∂ 2

∂ 2 z −d 11

∂ 2

∂ 2 x − d 33

∂ 2

∂ 2 z

( h 31 + h 15 ) ∂ 2

∂ x∂ z h 15

∂ 2

∂ 2 x + h 33

∂ 2

∂ 2 z −d 11

∂ 2

∂ 2 x − d 33

∂ 2

∂ 2 z −μ11

∂ 2

∂ 2 x − μ33

∂ 2

∂ 2 z

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. (A.1)

2. Expressions of kn ( k , n = 1, 2, 3, 4) appearing in Eq. (25)

11 = ( c 13 + c 44 ) (d 2 11 − ε 11 μ11

)+ ( h 15 + h 31 ) ( e 15 d 11 − h 15 ε 11 ) + ( e 15 + e 31 ) ( h 15 d 11 − e 15 μ11 ) ,

12 = ( c 13 + c 44 ) ( 2 d 11 d 33 − ε 11 μ33 − ε 33 μ11 ) + ( h 15 + h 31 ) ( e 15 d 33 + e 33 d 11 − h 15 ∈ 33 − h 33 ε 11 ) + ( e 15 + e 31 ) ( h 15 d 33 + h 33 d 11 − e 15 μ33 − e 33 μ11 ) ,

13 = ( c 13 + c 44 ) (d 2 33 − ε 33 μ33

)+ ( h 15 + h 31 ) ( e 33 d 33 − h 33 ε 33 ) + ( e 15 + e 31 ) ( h 33 d 33 − e 33 μ33 ) .

(A.2)

21 = c 11 ε 11 μ11 − c 11 d 2 11 ,

22 = c 11 ε 11 μ33 + c 11 ε 33 μ11 + c 44 ε 11 μ11 − 2 d 11 ( e 15 + e 31 ) ( h 15 + h 31 ) + ε 11 ( h 15 + h 31 ) 2

+ μ11 ( e 15 + e 31 ) 2 −(2 c 11 d 11 d 33 + c 44 d

2 11

),

23 = c 11 ε 33 μ33 + c 44 ε 11 μ33 + c 44 ε 33 μ11 − 2 d 33 ( e 15 + e 31 ) ( h 15 + h 31 ) + ε 33 ( h 15 + h 31 ) 2

+ μ33 ( e 15 + e 31 ) 2 −(c 11 d

2 33 + 2 c 44 d 11 d 33

),

24 = c 44 ε 33 μ33 − c 44 d 2 .

(A.3)

33


31 = e 15 c 11 μ11 − h 15 c 11 d 11 ,

32 = ( c 11 e 15 μ33 + c 11 e 33 μ11 + c 44 e 15 μ11 ) − h 15 ( h 15 + h 31 ) ( e 15 + e 31 ) + d 11 ( h 15 + h 31 ) ( c 13 + c 44 )

+ e 15 ( h 15 + h 31 ) 2 − μ11 ( c 13 + c 44 ) ( e 15 + e 31 ) − ( c 11 h 15 d 33 + c 11 h 33 d 11 + c 44 h 15 d 11 ) ,

33 = ( c 11 e 33 μ33 + c 44 e 15 μ33 + c 44 e 33 μ11 ) − h 33 ( h 15 + h 31 ) ( e 15 + e 31 ) + d 33 ( h 15 + h 31 ) ( c 13 + c 44 )

+ e 33 ( h 15 + h 31 ) 2 − μ33 ( c 13 + c 44 ) ( e 15 + e 31 ) − ( c 11 h 33 d 33 + c 44 h 33 d 11 + c 44 h 15 d 33 ) ,

34 = c 44 e 33 μ33 − c 44 h 33 d 33 .

(A.4)

41 = c 11 h 15 ε 11 − c 11 e 15 d 11 ,

42 = ( c 11 h 15 ε 33 + c 11 h 33 ε 11 + c 44 h 15 ε 11 ) − ( c 11 e 15 d 33 + c 11 e 33 d 11 + c 44 e 15 d 11 ) + h 15 ( e 15 + e 31 ) 2

−ε 11 ( c 13 + c 44 ) ( h 15 + h 31 ) − e 15 ( h 15 + h 31 ) ( e 15 + e 31 ) + d 11 ( c 13 + c 44 ) ( e 15 + e 31 ) ,

43 = ( c 11 h 33 ε 33 + c 44 h 15 ε 33 + c 44 h 33 ε 11 ) − ( c 11 e 33 d 33 + c 44 e 15 d 33 + c 44 e 33 d 11 ) + h 33 ( e 15 + e 31 ) 2

−ε 33 ( c 13 + c 44 ) ( h 15 + h 31 ) − e 33 ( h 15 + h 31 ) ( e 15 + e 31 ) + d 33 ( c 13 + c 44 ) ( e 15 + e 31 ) , 44 = c 44 h 33 ε 33 − c 44 e 33 d 33 .

(A.5)

3. Expressions of ϖj ( j = 1, …, 5) appearing in Eq. (27)

� 1 = c 44 c 33 ε 33 μ33 − 2 c 44 e 33 h 33 d 33 + c 44 e 2 33 μ33 − c 44 c 33 d

2 33 + c 44 h

2 33 ε 33 , (A.6)

� 2 = −c 11 c 33 ε 33 μ33 + 2 c 11 e 33 h 33 d 33 − c 44 c 33 ε 11 μ33 − c 44 c 33 ε 33 μ11 +2 c 44 c 33 d 11 d 33 − 2 c 44 e 15 e 33 μ33 + 2 c 44 e 33 h 15 d 33

+2 c 44 e 15 h 33 d 33 + 2 c 44 e 33 h 33 d 11 − 2 c 44 h 15 h 33 ε 33 +2 c 1344 e 33 e 3115 μ33 − 2 c 1344 e 33 h 3115 d 33 − 2 c 1344 h 33 e 3115 d 33

+2 c 1344 h 33 h 3115 ε 33 +2 e 3115 c 33 h 3115 d 33 + 2 e 3115 h 33 h 3115 e 33 − c 2 1344 d 2 33 − h

2 3115 e

2 33 + c 2 44 d

2 33 − e 2 3115 h

2 33 − c 11 e

2 33 μ33

+ c 11 c 33 d 2 33 − c 11 h

2 33 ε 33 −c 2 44 ε 33 μ33 − c 44 e

2 33 μ11 − c 44 h

2 33 ε 11 + c 2 1344 ε 33 μ33 −e 2 3115 c 33 μ33 − h

2 3115 c 33 ε 33 ,

(A.7)

� 3 = −c 11 c 44 d 2 33 + c 11 e

2 33 μ11 + c 11 h

2 33 ε 11 + c 2 44 ε 11 μ33 + c 2 44 ε 33 μ11 −2 c 2 44 d 11 d 33 − c 44 c 33 d

2 11 + c 44 e

2 15 μ33 + c 44 h

2 15 ε 33

−c 2 1344 ε 11 μ33 − c 2 1344 ε 33 μ11 + 2 c 2 1344 d 11 d 33 + e 2 3115 c 44 μ33 + e 2 3115 c 33 μ11 + 2 e 2 3115 h 15 h 33 + h

2 3115 c 44 ε 33

+ h

2 3115 c 33 ε 11 + 2 h

2 3115 e 15 e 33 −2 c 1344 e 15 e 3115 μ33 + 2 c 1344 e 15 h 3115 d 33 −2 c 1344 e 33 e 3115 μ11 + 2 c 1344 e 33 h 3115 d 11

+2 c 1344 h 15 e 3115 d 33 − 2 c 1344 h 15 h 3115 ε 33 + 2 c 1344 h 33 e 3115 d 11 −2 c 1344 h 33 h 3115 ε 11 − 2 e 3115 c 44 h 3115 d 33

−2 e 3115 c 33 h 3115 d 11 − 2 e 3115 h 15 h 3115 e 33 −2 e 3115 h 33 h 3115 e 15 + c 11 c 44 ε 33 μ33 + c 11 c 33 ε 11 μ33 + c 11 c 33 ε 33 μ11

−2 c 11 c 33 d 11 d 33 + 2 c 11 e 15 e 33 μ33 − 2 c 11 e 15 h 33 d 33 − 2 c 11 e 33 h 33 d 11 − 2 c 11 e 33 h 15 d 33 + 2 c 11 h 15 h 33 ε 33

+ c 44 c 33 ε 11 μ11 + 2 c 44 e 15 e 33 μ11 −2 c 44 e 15 h 15 d 33 − 2 c 44 e 15 h 33 d 11 − 2 c 44 e 33 h 15 d 11 +2 c 44 h 15 h 33 ε 11 ,

(A.8)

� 4 = c 11 c 33 d 2 11 − c 11 e

2 15 μ33 − c 11 h

2 15 ε 33 − c 44 e

2 15 μ11 − c 44 h

2 15 ε 11 −c 2 44 ε 11 μ11 + c 2 1344 ε 11 μ11 − e 2 3115 c 44 μ11 − h

2 3115 c 44 ε 11

−c 11 c 44 ε 11 μ33 − c 11 c 44 ε 33 μ11 + 2 c 11 c 44 d 11 d 33 −c 11 c 33 ε 11 μ11 − 2 c 11 e 15 e 33 μ11 + 2 c 11 e 15 h 15 d 33 +2 c 11 e 33 h 15 d 11

+2 c 11 e 15 h 33 d 11 − 2 c 11 h 15 h 33 ε 11 + 2 c 44 e 15 h 15 d 11 + 2 c 1344 e 15 e 3115 μ11 − 2 c 1344 e 15 h 3115 d 11 − 2 c 1344 h 15 e 3115 d 11

+2 c 1344 h 15 h 3115 ε 11 +2 e 3115 c 44 h 3115 d 11 + 2 e 3115 h 15 h 3115 e 15 − h

2 3115 e

2 15 − e 2 3115 h

2 15 −c 2 1344 d

2 11 + c 2 44 d

2 11 ,

(A.9)

� 5 = −c 11

(−c 44 ε 11 μ11 + 2 e 15 h 15 d 11 − e 2 15 μ11 + c 44 d

2 11 − h

2 15 ε 11

), (A.10)

where

c 1344 = c 13 + c 44 , (A.11)

e 3155 = e 31 + e 15 , (A.12)

h 3155 = h 31 + h 15 . (A.13)

4. Expressions of γ l ( ζ m

) ( m = 1, …, K 1 , l = u , w , φ, ψ , xx , zz , xz , dx , dz , bx , bz ) appearing in Eq. (36)

γu ( ζm

) = −ζm

( 11 − 12 ζ

2 m

+ 13 ζ4 m

), (A.14)

γw

( ζm

) = − 21 + 22 ζ2 m

− 23 ζ4 m

+ 24 ζ6 m

, (A.15)

γϕ ( ζm

) = − 31 + 32 ζ2 m

− 33 ζ4 m

+ 34 ζ6 m

, (A.16)


γψ

( ζm

) = − 41 + 42 ζ2 m

− 43 ζ4 m

+ 44 ζ6 m

, (A.17)

γxx ( ζm

) = c 11 γu ( ζm

) + c 13 ζm

γw

( ζm

) + e 31 ζm

γϕ ( ζm

) + h 31 ζm

γψ

( ζm

) , (A.18)

γzz ( ζm

) = c 13 γu ( ζm

) + c 33 ζm

γw

( ζm

) + e 33 ζm

γϕ ( ζm

) + h 33 ζm

γψ

( ζm

) , (A.19)

γxz ( ζm

) = c 44 [ ξm

γu ( ζm

) − γw

( ζm

) ] − e 15 γϕ ( ζm

) − h 15 γψ

( ζm

) , (A.20)

γdx ( ζm

) = e 15 [ ξm

γu ( ζm

) − γw

( ζm

) ] + ε 11 γϕ ( ζm

) + d 11 γψ

( ζm

) , (A.21)

γdz ( ζm

) = e 31 γu ( ζm

) + e 33 ζm

γw

( ζm

) − ε 33 ζm

γϕ ( ζm

) − d 33 ζm

γψ

( ζm

) , (A.22)

γbx ( ζm

) = h 15 [ ξm

γu ( ζm

) − γw

( ζm

) ] + d 11 γϕ ( ζm

) + μ11 γψ

( ζm

) , (A.23)

γbz ( ζm

) = h 31 γu ( ζm

) + h 33 ζm

γw

( ζm

) − d 33 ζm

γϕ ( ζm

) − μ33 ζm

γψ

( ζm

) . (A.24)

5. Expressions of γ (C) l

( αn , βn ) and γ (S) l

( αn , βn ) ( n = 1, …, K 2 , l = u , w , φ, ψ , xx , zz , xz , dx , dz , bx , bz ), which are values of

functions γ (C) l

( α, β) and γ (S) l

( α, β) at the point, ( αn , βn ) appearing in Eqs. (37) and ( 38 )

γ (C) u ( α, β) = α

[− 11 + 12

(α2 − 3 β2

)− 13

(α4 − 10 α2 β2 + 5 β4

)], (A.25)

γ (S) u ( α, β) = β

[− 11 + 12

(3 α2 − β2

)− 13

(5 α4 − 10 α2 β2 + β4

)], (A.26)

γ (C) w

( α, β) = − 21 + 22

(α2 − β2

)− 23

(α4 − 6 α2 β2 + β4

)+ 24

(α6 − 15 α4 β2 + 15 α2 β4 − β6

), (A.27)

γ (S) w

( α, β) = 2 αβ[ 22 − 2 23

(α2 − β2

)+ 24

(3 α4 − 10 α2 β2 + 3 β4

)], (A.28)

γ (C) ϕ ( α, β) = − 31 + 32

(α2 − β2

)− 33

(α4 − 6 α2 β2 + β4

)+ 34

(α6 − 15 α4 β2 + 15 α2 β4 − β6

), (A.29)

γ (S) ϕ ( α, β) = 2 αβ

[ 32 − 2 33

(α2 − β2

)+ 34

(3 α4 − 10 α2 β2 + 3 β4

)], (A.30)

γ (C) ψ

( α, β) = − 41 + 42

(α2 − β2

)− 43

(α4 − 6 α2 β2 + β4

)+ 44

(α6 − 15 α4 β2 + 15 α2 β4 − β6

), (A.31)

γ (S) ψ

( α, β) = 2 αβ[ 42 − 2 43

(α2 − β2

)+ 44

(3 α4 − 10 α2 β2 + 3 β4

)], (A.32)

γ (C) xx ( α, β) = c 11 γ

(C) u ( α, β) + c 13

[γ (C)

w

( α, β) α − γ (S) w

( α, β) β]

+ e 31

[γ (C)

ϕ ( α, β) α − γ (S) ϕ ( α, β) β

]+ h 31 [ γ

(C) ψ

( α, β) α − γ (S) ψ

( α, β) β] , (A.33)

γ (S) xx ( α, β) = c 11 γ

(S) u ( α, β) + c 13

[γ (C)

w

( α, β) β + γ (S) w

( α, β) α]

+ e 31

[γ (C)

ϕ ( α, β) β + γ (S) ϕ ( α, β) α

]+ h 31

[ γ (C)

ψ

( α, β) β + γ (S) ψ

( α, β) α] , (A.34)

γ (C) zz ( α, β) = c 13 γ

(C) u ( α, β) + c 33

[γ (C)

w

( α, β) α − γ (S) w

( α, β) β]

+ e 33

[γ (C)

ϕ ( α, β) α − γ (S) ϕ ( α, β) β

]+ h 33

[ γ (C)

ψ

( α, β) α − γ (S) ψ

( α, β) β] , (A.35)

γ (S) zz ( α, β) = c 13 γ

(S) u ( α, β) + c 33

[γ (C)

w

( α, β) β + γ (S) w

( α, β) α]

+ e 33

[γ (C)

ϕ ( α, β) β + γ (S) ϕ ( α, β) α

]+ h 33

[ γ (C)

ψ

( α, β) β + γ (S) ψ

( α, β) α] , (A.36)

γ (C) xz ( α, β) = c 44

[γ (C)

u ( α, β) α − γ (S) u ( α, β) β − γ (C)

w

( α, β) ]

− e 15 γ(C) ϕ ( α, β) − h 15 γ

(C) ψ

( α, β) , (A.37)

γ (S) xz ( α, β) = c 44

[γ (C)

u ( α, β) β + γ (S) u ( α, β) α − γ (S)

w

( α, β) ]

− e 15 γ(S) ϕ ( α, β) − h 15 γ

(S) ψ

( α, β) , (A.38)

γ (C) dx

( α, β) = e 15

[γ (C)

u ( α, β) α − γ (S) u ( α, β) β − γ (C)

w

( α, β) ]

+ ε 11 γ(C) ϕ ( α, β) + d 11 γ

(C) ψ

( α, β) , (A.39)

γ (S) dx

( α, β) = e 15

[γ (C)

u ( α, β) β + γ (S) u ( α, β) α − γ (S)

w

( α, β) ]

+ ε 11 γ(S) ϕ ( α, β) + d 11 γ

(S) ψ

( α, β) , (A.40)

γ (C) dz

( α, β) = e 31 γ(C)

u ( α, β) + e 33

[γ (C)

w

( α, β) α − γ (S) w

( α, β) β]

− ε 33

[γ (C)

ϕ ( α, β) α − γ (S) ϕ ( α, β) β

]


−d 33

[ γ (C)

ψ

( α, β) α − γ (S) ψ

( α, β) β] , (A.41)

γ (S) dz

( α, β) = e 31 γ(S)

u ( α, β) + e 33

[γ (C)

w

( α, β) β + γ (S) w

( α, β) α]

− ε 33

[γ (C)

ϕ ( α, β) β + γ (S) ϕ ( α, β) α

]−d 33 [ γ

(C) ψ

( α, β) β + γ (S) ψ

( α, β) α] , (A.42)

γ (C) bx

( α, β) = h 15

[γ (C)

u ( α, β) α − γ (S) u ( α, β) β − γ (C)

w

( α, β) ]

+ d 11 γ(C) ϕ ( α, β) + μ11 γ

(C) ψ

( α, β) , (A.43)

γ (S) bx

( α, β) = h 15

[γ (C)

u ( α, β) β + γ (S) u ( α, β) α − γ (S)

w

( α, β) ]

+ d 11 γ(S) ϕ ( α, β) + μ11 γ

(S) ψ

( α, β) , (A.44)

γ (C) bz

( α, β) = h 31 γ(C)

u ( α, β) + h 33

[γ (C)

w

( α, β) α − γ (S) w

( α, β) β]

− d 33

[γ (C)

ϕ ( α, β) α − γ (S) ϕ ( α, β) β

]−μ33

[ γ (C)

ψ

( α, β) α − γ (S) ψ

( α, β) β] , (A.45)

γ (S) bz

( α, β) = h 31 γ(S)

u ( α, β) + h 33

[γ (C)

w

( α, β) β + γ (S) w

( α, β) α]

− d 33

[γ (C)

ϕ ( α, β) β + γ (S) ϕ ( α, β) α

]−μ33

[ γ (C)

ψ

( α, β) β + γ (S) ψ

( α, β) α] . (A.46)

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