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Magneto–electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials
Accepted Manuscript
Magneto-electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials Yue-Ting Zhou , Sheng-Jie Pang , Yong Hoon Jang PII: DOI: Reference:
S0307-904X(17)30479-1 10.1016/j.apm.2017.07.041 APM 11891
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
12 April 2017 1 July 2017 20 July 2017
Please cite this article as: Yue-Ting Zhou , Sheng-Jie Pang , Yong Hoon Jang , Magneto-electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.07.041
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Highlights Interaction of two perfectly conducting indenters is examined.
Various intensity factors are defined to measure singular behavior at the edges.
Degradation from two perfectly conducting indenters to one indenter is done.
Interaction between two semi-cylindrical indenters is revealed.
Singularities at the edges can be suppressed because of multi-field coupling.
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1
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Magneto-electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials Yue-Ting Zhou a ,* , a
Sheng-Jie Pang a , Yong Hoon Jang b
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092,
b
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P. R. China School of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea
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M
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ABSTRACT Within the theory of linear full-field magneto-electro-elasticity, magneto-electro interaction 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 employed to address the two-dimensional, mixed boundary-value problem explicitly. The surface stresses, electric displacement and magnetic induction and their respective intensity factors are obtained in closed forms for two perfectly conducting semi-cylindrical indenters. 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 interaction between two semi-cylindrical indenters on contact behaviors subjected to multi-field loadings.
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Keywords: Magneto-electro interaction; Magnetoelectroelastic materials; Electrically-conducting and magnetically-conducting; Closed forms; Singularity; Multi-field coupling.
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Introduction
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The indentation technique involves the application of a well-defined indenter to
deform the testing materials and characterize their mechanical performances [1]. For example, since residual stresses exert pronounced influences on materials’ mechanical _________ *
Corresponding author at. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China. E-mail addresses: [email protected] (Y.T. Zhou). 2
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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].
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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
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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]
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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]
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and Zhou and Kim [9] developed a basic theory of the sliding contact of
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magnetoelectroelastic materials subject to a rigid indenter with flat profile, parabolic profile, triangular profile or cylindrical profile illustrating that the indenter profile
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greatly affects the contact behavior. Recently, Li et al. [10] presented fundamental
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solutions for the contact problem of a magnetoelectroelastic half-space punched by a smooth and rigid half-infinite indenter. 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 detect the fracture toughness of Solid-phase-sintered Silicon Carbide Ceramic (SSiC) by the Vickers and Knoop indentation method [11]. 3
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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
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different effect on the contact behavior. Addressing two collinear indenters acting on piezoelectric materials, one kind of single-phase magnetoelectroelastic materials, Wang et al. [13] found that the indenter tip fields were greatly affected by the relative
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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,
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which makes the interaction effect between two indenters deserve to be studied. This article conducts an exact contact analysis of magnetoelectroelastic materials
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under two electrically-conducting and magnetically-conducting indenters. Singular
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integral equations with kernel like 1 ( 2 x 2 ) because of the interaction of two indenters are obtained and solved analytically, which makes the solutions have a quite
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different form. The fields disturbed by the two perfectly conducting semi-cylindrical
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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.
Formulation of the Problem There are two electrically-conducting and magnetically-conducting indenters acting symmetrically about z-axis on the surface of magnetoelectroelastic materials 4
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(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. Basic Equations The constitutive equations are [14]
σ cS eE hH ,
(1)
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D eT S E dH ,
(2)
B hT S dE H ,
(3)
where the superscript T denotes the transposition, and σ , S , D , E , B and H
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are, respectively, the vectors of the stress, strain, electric displacement, electric field, magnetic induction, and magnetic field, which are given as follows:
σ xx zz xz
T
, S u, x
D Dx
Dz ,
E , x
B Bx
Bz ,
H , x
w, z
, z , T
M
T
T
1 u, z w, x , 2
, z , T
(5) (6)
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T
(4)
where the comma stands for the differentiation with respect to the corresponding
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coordinate variables, u and w are the mechanical displacement components, and
and represent the electric potential and the magnetic potential.
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In Eqs. (1)-(3), c , e , h , , d and are the matrices of the elastic
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coefficients, electromechanical coupling coefficients, magnetomechanical coupling coefficients,
dielectric
permeability
coefficients,
magneto-electro
coupling
coefficients, and magnetic permeability coefficients, which take the following forms for linearly, transversely isotropic magnetoelectroelastic materials:
c11 c13 c c13 c33 0 0
0 0 , 2c44
0 e 0 2e15
e31 e33 , 0
5
(7)
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0 h 0 2h15
h31 h33 , 0
11 0
11 0
0 d d 11 , 0 d33
0 , 33
(8)
0 . 33
(9)
xx, x xz , z 0 ,
xz , x zz , z 0 ,
Dx, x Dz , z 0 ,
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The equilibrium equations free of any generalized body source are (10) (11)
Bx, x Bz , z 0 .
(12)
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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
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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
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and magnetoelectroelastic materials well contact in the contact region. Thus, it is assumed that there is no air inside the contact region, and perfectly conducting
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boundary conditions are chosen.
The x>0 part of the system is considered since the stated problem is symmetric
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with respect to x = 0. Denote the contact area as 0 l1 x l2 in x>0 part (Fig. 2). Since each indenter is electrically-conducting and magnetically-conducting, one
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has
( x,0) 1 ( x) ,
0 l1 x l2 ,
(13)
( x,0) 1 ( x) ,
0 l1 x l2 ,
(14)
with 1 ( x) and 1 ( x) denoting the electric potential and the magnetic potential inside the contact area 0 l1 x l2 . The perfectly conducting property makes the normal components of the electric 6
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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.
p( x), 0 l1 x l2 , 0 x l1 , x l2 0,
zz ( x, 0)
(15)
(16)
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q( x), 0 l1 x l2 Dz ( x, 0) , 0 x l1 , x l2 0, m( x), 0 l1 x l2 Bz ( x, 0) , 0 x l1 , x l2 0,
(17)
where q( x) , m( x) , and p( x) are unknown functions. The accumulated electric
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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)
l1
l1
l2
l2
l2
l2
q( x)dx q( x)dx Q , l1
l2
m( x)dx m( x)dx M , l1
l2
M
l1
p( x)dx p( x)dx P . l1
(18) (19) (20)
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The surface shear stress keeps zero (21)
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xz ( x,0) 0 .
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The indenter profile is described as
w( x,0) w1 ( x) , 0 l1 x l2 ,
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(22)
with w1 ( x) being a known function. The following regularity conditions should be satisfied at infinity: u( x, z), w( x, z ), ( x, z ), ( x, z ) 0 ,
x2 z 2 .
General Solutions Considering Eqs. (1)-(12), one may discover the general solutions as [16] 7
(23)
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u
and
T
T
k1
k2
k3
k4
( x, z ) ,
(24)
( x, z ) is an auxiliary function, which satisfies Eq. (26) as will be seen later, kn
k , n 1, 2,3, 4
are the cofactors of the operator matrix Κ given in the
Appendix. In the following, 21
2n
n 1, 2,3, 4
6 11 x5z
6 12 x3z 3
x 6
x 4z 2
will be used, which take the form
6 13 xz 5
n 1 ,
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where
w
(25)
where
kn
n1
6
n2
k , n 1, 2,3, 4
x 2z 4 6
n3
z 6 6
n4
are given in the Appendix.
n 2,3, 4 ,
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2n
6
In Eq. (24), the following relationship holds for the function
det Κ ( x, z ) 0 .
( x, z ) : (26)
Applying the Fourier cosine transform with respect to x to Eq. (26) arrives at
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6 4 2 8 2 4 6 5 8 0 , 2 3 4 z 8 z 6 z 4 z 2
where j ( j 1,
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1
,5) are given in the Appendix, and
, z
(27)
is defined as
, z 0 x, z cos x dx .
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(28)
, z e z , one can obtain the characteristic equation associated
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Considering with Eq. (27) as
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1 8 2 6 3 4 4 2 5 0 .
(29)
where represents the root. According to the eigenvalue properties of Eq. (29) and using the Almansi's
theorem [17], one may express
, z
as follows:
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4 k k , z , 1 2 3 4 k 1 4 2 k 3 , z k z e3 z , 1 2 3 4 k k k 3 k 1 2 1 2 3 4 0 , , z 2 k 1ek z 2 k zek z , k 1 4 k 2 1e1 z k z e2 z , 1 2 3 4 0 k 2 4 k 1 k z e1 z , 1 2 3 4 0 k 1
(30)
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where k (k 1, 2,3, 4) are the unknown functions to be determined from the k
, z
depend on whether the eigenvalues are real or
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boundary conditions, and
complex. For example, in the case of 1 2 3 4 , if there are K1 positive real roots expressed as m m ( m 1,
, K1 ) and K 2 pairs of conjugate complex roots
m
, z
can be given as
, z e z
M
with positive real parts expressed as n n n in ( n 1,
K1 2 n 1
, z e z cos n z n
, z e z sin n z
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K1 2 n
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m
m
n
, K2 ) with i 2 1 ,
( m 1,
, K1 ),
(31)
( n 1,
, K2 ),
(32)
( n 1,
, K2 ).
(33)
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Note that when 1 2 3 4 , one may obtain K1 0 and K 2 2 , or
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K1 1 and K 2 1 ,or K1 4 and K 2 0 since K1 2K2 4 is required. In the 1 2 3 4 case,
1
, z
and
2
, z
can be given in the
same way as shown in Eq. (31). Various field quantities can be given on the basis of consideration of the constitutive equations (1)-(3)
9
, z
with the
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Dx
(k ) u
Bx T
(k ) w
4
0
k 1
6
k
sin x (34)
(k ) T bx
(xzk ) (dxk )
w xx zz
2
( k )
Dz
Bz T
2
d , 4
0
k 1
k
6
cos x (35)
(k ) ( k ) (xxk ) (zzk ) (dzk ) bz d , T
where known functions l( k ) , z ( k 1, are given as ( m 1,
, K1 ),
, 4 , l u, w, , , xx, zz, xz, dz, dx, bz, bx )
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l( m) , z l m e m z
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u xz
(36)
l( K1 2n1) , z l(C ) n , n cos n z l( S ) n , n sin n z en z
( n 1,
, K2 ),
(37)
l( K1 2 n ) , z l( S ) n , n cos n z l(C ) n , n sin n z en z
, K2 ),
, K1 , l u, w, , , xx, zz, xz, dx, dz, bx, bz ), l(C ) n , n and
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where l m ( m 1,
M
( n 1,
(38)
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l( S ) n , n ( n 1, , K2 , l u, w, , , xx, zz, xz, dx, dz, bx, bz ) are given in the Appendix (Results for distinctive eigenvalues case are presented in what follows since
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the commercially available MEE, which are transversely isotropic, generate
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distinctive eigenvalues).
Singular Integral Equations for two Perfectly Conducting Indenters Considering boundary conditions Eqs. (22), (13) and (14) results in singular
integral equations as follows:
l2
l1
11 p( ) 12 q( ) 13m( ) w1 x d , l1 x l2 , 2 2 x 2 x x
10
(39)
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l2
l2
l1
l1
21 p( ) 22 q( ) 23m( ) 1 x , l1 x l2 , d 2 2 x 2 x x
(40)
31 p( ) 32 q( ) 33m( ) 1 x , l1 x l2 , d 2 2 x 2 x x
(41)
(wk ) ( , 0), m 1 C m1 (1) k 1 ( k ) ( , 0), m 2 , det CB ( k ) k 1 ( , 0), m 3 (1k ) B
4
m2
(wk ) ( , 0), m 1 (3 k ) C B (1) k 1 ( k ) ( , 0), m 2 , det CB ( k ) k 1 ( , 0), m 3
m3
(wk ) ( , 0), m 1 C (1) k ( k ) ( , 0), m 2 , det CB ( k ) k 1 ( , 0), m 3 (4 k ) B
4
(42)
(43)
(44)
M
, 4) are the complement minors of the matrix CB Clk ,
where CB(lk ) (l , k 1, (l, k 1,
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4
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where mn (m, n 1, 2,3) are given as
, 4) taking the following forms:
,0
(k 1,
C3k (dzk ) ,0 , (45)
, 4) .
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C4 k
(k ) bz
C2 k (xzk ) ,0 ,
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C1k (zzk ) ,0 ,
The singular integral equations, Eqs. (39) and (40), and Eqs. (18)-(20) can be
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normalized as 1
j1 p (0) ( ) j 2 q (0) ( ) j 3m(0) ( )
1
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d j ( ) , 1 1, (46)
j 1, 2,3 ,
1
1
1
1
p (0) ( )d
2P , d0
(47)
q (0) ( )d
2Q , d0
(48)
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1
1
m(0) ( )d
2M , d0
(49)
where
p( )
2 l0 d0
1 ( )
,
q (0) ( )
,
q( )
m(0) ( )
l l1 2 2
x 2 l0 , d0
1 w1 ( x) , x x
,
d0
2 ( )
2
1 1 ( x) , x x
m( )
2
,
(50)
l l1 2 2
,
l0
3 ( )
2
1 1 ( x) . x x
Exact Contact Analysis
,
(51)
(52)
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Two Perfectly Conducting Semi-Cylindrical Indenters
2
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p (0) ( )
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
l1 x l2 ,
(53)
(54)
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1 ( x) 0 ,
x2 , 2R
ED
w1 ( x) w0
M
obtain
1 ( x) 0 ,
(55)
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where w0 , 0 and 0 are constants, and R is the radius of the semi-cylindrical
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indenter.
Note that the first contact point ( x l1 ) is not on the apex of the semi-cylindrical
shape of the indenter, and the maximum penetration depths vary with different values of l1 . 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. 12
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Considering Eqs. (39)-(41), (46), (53)-(55), one has
1 , R
(56)
2 ( ) 3 ( ) 0 .
(57)
1 ( )
Since
q( x) q1 ( x) q2 ( x)
m( x) m1 ( x) m2 ( x)
and
for the conducting
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semi-cylindrical indenter, in which q1 ( x) and m1 ( x) are caused by the electric potential and the magnetic potential and q2 ( x) and m2 ( x) caused by the normal mechanical load P, one has q(0) ( ) q1(0) ( ) q2(0) ( ) ,
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(58)
m(0) ( ) m1(0) ( ) m2(0) ( ) .
(59)
Thus, Eqs. (46)-(49) can be written as
1
1
m1(0) ( )d
1
1
(62)
2 Q QF , d0
(63)
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1
q1(0) ( )d
(60)
(61)
q1(0) ( ) d 0 , 1 1,
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1
2M MF , d0
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in terms of m1(0) ( ) ,
M
1
m1(0) ( ) d 0 , 1 1,
ED
1
in terms of q1(0) ( ) , and
R , (0) (0) (0) 1 j1 p ( ) j 2 q2 ( ) j 3m2 ( ) d 0, 1 0,
13
j 1 j 2 , 1 1, j 3
(64)
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1
1
1
1
1
1
p (0) ( )d
2P , d0
q2(0) ( )( )d m2(0) ( )d
(65)
2QF , d0
(66)
2M F , d0
(67)
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in terms of p (0) ( ) , q2(0) ( ) and m2(0) ( ) , and QF and M F denote the electric charge and the magnetic induction due to the total indentation force P . The exact solutions of integral equations (60)-(67) take the form
p (0) ( )
,
d0 1 2 2 Q QF d0 1 2
,
1 1 , P R 1
q2(0) ( )
(68)
(69)
1 1 , Q R 1
(70)
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1 , m2(0) ( ) M R 1 1
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q1(0) ( )
2M MF
M
m1(0) ( )
det mn 33
,
22 33 32 23
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P
PT
where P , Q and M are given as
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Q
M
det mn 33
(71)
,
2331 2133 det mn 33 2132 22 31
.
Then, the contact stress, electric displacement and magnetic induction inside the contact region are given as follows:
x p ( x) P R
l2
2
x2
x 2 l1
2
, l1 x l2 , 14
(72)
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q ( x)
m( x )
x Q R
l2
2
x2
x l1
x M R
2
l2
2
2
x2
x 2 l1
2
2 Q QF x 2 2 l2 x 2 x 2 l1
, l1 x l2 ,
2M MF x 2 2 l2 x 2 x 2 l1
, l1 x l2 .
(73)
(74)
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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
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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
M
electric displacement and the surface magnetic induction, and the outer-edge
multi-field loadings.
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singularities can also be suppressed by choosing appropriate combinations of the
Considering Eqs. (34), (35), (72)-(74) results in the surface stresses, surface
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electric displacements and surface magnetic inductions as follows: l1 x l2 , x l1 or x l2
(75)
( P) (Q ) (M ) zz p( x) zz q( x) zz m( x), 0,
l1 x l2 , x l1 or x l2
(76)
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( P) (Q ) (M ) xx p( x) xx q( x) xx m( x), xx x, 0 0,
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zz x, 0
( P ) (Q ) ( M ) l1 x l2 xz xz xz x, xz x, 0 P R Q R M R , 0, x l1 or x l2 ( P ) (Q ) ( M ) l1 x l2 dx dx dx x, Dx x, 0 P R Q R M R , 0, x l1 or x l2
15
(77)
(78)
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( P) (Q ) (M ) dz p( x) dz q( x) dz m( x), Dz x, 0 0,
( P ) (Q ) ( M ) bx bx bx x, Bx x, 0 P R Q R M R 0,
l1 x l2 , x l1 or x l2
l1 x l2 x l1 or
( P) (Q ) (M ) bz p( x) bz q( x) bz m( x), Bz x, 0 0,
k 1
(1) k CB(4 k ) (jk ) , 0 , det CB k 1
4
(jQ ) (jk ) , 0 k 1
(81)
(1)k CB(3k ) , det CB
(82)
4
j xx, zz, xz, dz, dx, bz, bx .
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(jM )
(1) k CB(1k ) , det CB
x l2
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4
(80)
l1 x l2 , x l1 or x l2
where
(jP ) (jk ) , 0
,
(79)
Employing the properties about minors of the matrix CB Clk in Eq. (45)
(zzP ) 1 ,
(zzQ ) 0 ,
(xzP ) 0 ,
(xzQ ) 0 ,
(dzP ) 0 ,
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arrives at
M
(83)
(xzM ) 0 ,
(84)
(dzQ ) 1,
(dzM ) 0 ,
(85)
(Q ) bz 0,
(M ) bz 1.
(86)
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(P) bz 0,
(zzM ) 0 ,
Then, one may find the required boundary conditions Eqs. (15)-(17) and (21)
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hold.
Substituting Eq. (70) into Eqs. (65)-(67), one can determine the width of the
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contact region with the singular end-points x l1 known a priori, QF and M F as follows:
l2 QF
4 P P R
l1 , 2
(87)
P P , Q
(88)
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MF
P P . M
(89)
One may define the following stress, electric displacement and magnetic induction intensity factors to measure the singular behavior at the ends of the semi-cylindrical indenters: K I l1 lim 2 x l1 p( x) ,
(90)
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x l1
K D l1 lim 2 x l1 q( x) ,
(91)
x l1
K B l1 lim 2 x l1 m( x) .
(92)
x l1
P R 2 2 l1 l2 l1
K D l1
Q R
M R
2 Q QF
2M M F
(93)
l1
2 l2 l1 2
2 2 l1 l2 l1
,
,
(94)
l1
2 l2 l1 2
ED
K B l1
M
K I l1
2 2 l1 l2 l1
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Considering Eqs. (68)-(70), one can rewrite Eqs. (90)-(92) as follows:
.
(95)
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Verification of The Exact Analysis
To validate the theoretical derivation for two semi-cylindrical indenters, one can
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set l1 0 . In such a special case, two semi-cylindrical indenters each with the accumulated electric charge Q, accumulated magnetic induction M and total
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indentation force P become a single cylindrical indenter with the contact region
l2 , l2 ,
accumulated electric charge 2Q, accumulated magnetic induction 2M and
total indentation force 2P with considering Eqs. (18)-(20), i.e.
l2
l2
l2
l2
0
l2
l2
0
q( x)dx q( x)dx q( x)dx Q Q 2Q , 0
l2
l2
0
m( x)dx m( x)dx m( x)dx M M 2M ,
17
(96) (97)
ACCEPTED MANUSCRIPT
l2
l2
0
l2
l2
0
p( x)dx p( x)dx p( x)dx P P 2 P .
(98)
Setting l1 0 in Eqs. (72)-(74) and (87), one has 1 P R
l2
q ( x)
1 Q R
l2
m( x)
1
2
l2
M R
P P R
l2 2
2
x2 ,
x2
2
2 Q QF
l2
x2
x
2
,
2
2M MF
l2 x2 2
,
x l2 , l2 ,
(99)
x l2 , l2 ,
(100)
CR IP T
p( x)
x l2 , l2 ,
(102)
.
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(101)
Eqs. (99)-(102) are identical to those given by Zhou and Lee [20] for a single cylindrical indenter with the contact region
l2 , l2 , accumulated electric charge 2Q,
M
accumulated magnetic induction 2M and total indentation force 2P. Thus, the solutions for a single cylindrical indenter are recovered by those for two
ED
semi-cylindrical indenters.
PT
For a single cylindrical indenter, the full-field stresses, electric displacements and magnetic inductions of MEE can be given explicitly bedsides those on the surface
CE
given in Eqs. (75)-(81). Various stresses, electric displacements and magnetic
AC
induction of MEE take the form
xx zz xz
Dz
4
k 1
k
k 1
Bz
Bx T
( a ,k ) xx
(zza ,k )
(xza ,k )
(dza ,k )
(dxa ,k )
( a ,k ) bz
(b,k ) xx
(zzb,k )
(xzb,k )
(dzb,k )
(dxb,k )
(b, k ) bz
4
k
Dx
where 18
( a ,k ) bx T
( b, k ) bx , T
(103)
ACCEPTED MANUSCRIPT
k
(1)k 2 Q QF CB(3k ) M M F CB(4 k ) det CB
k
,
(104)
(1)k l2 CB(1k ) CB(3k ) CB(4 k ) . R det CB P Q M
(105)
In Eq. (103), (ja ,k ) ( x, z ) and (jb,k ) ( x, z ) ( l xx, zz, xz, dx, dz, bx, bz ) are given as
(rk , K1 2 n1) ( x, z )
1 (C ) r n , n 1( k ) x, z, n , n 2
n , n x, z, n , n ， k a, b , (k ) 1
(rk , K1 2 n ) ( x, z )
n 1,
1 (S ) r n , n 1( k ) x, z, n , n 2
, K1 , r xx, zz, dz, bz , (106)
(107)
, K2 , r xx, zz, dz, bz ,
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(S ) r
k a, b , m 1,
CR IP T
(rk ,m) ( x, z ) r m 1( k ) ( x, z, m ) ,
r(C ) n , n 1( k ) x, z, n , n ， k a, b , n 1, , K2 , r xx, zz, dz, bz , t( k ,m) ( x, z ) t m (2k ) ( x, z, m ) , t( k , K1 2 n1) ( x, z )
1 (C ) t n , n (2k ) x, z, n , n 2
M
n , n x, z, n , n ， k a, b , (k ) 2
t( k , K1 2 n ) ( x, z )
, K1 , t xz, dx, bx ,
n 1,
, K2 , t xz, dx, bx ,
1 (S ) t n , n (2k ) x, z, n , n 2
ED
(S ) t
k a, b , m 1,
t(C ) n , n (2k ) x, z, n , n ， k a, b , n 1, , K2 , t xz, dx, bx ,
(109)
(110)
(111)
, K1 ,
, K2 ) are given as
CE
n 1,
PT
where i( k ) x, z, n , n and i( k ) x, z, n , n ( k a, b , i 1, 2 , m 1,
(108)
AC
1( k ) x, z, n , n 1( k ) x1 (n ), z, n 1( k ) x2 (n ), z, n , 1( k ) x, z, n , n (2k ) x1 (n ), z, n (2k ) x2 (n ), z, n , k a, b , n 1,
(112)
, K2 ,
(2k ) x, z, n , n (2k ) x1 (n ), z, n (2k ) x2 (n ), z, n , (2k ) x, z, n , n 1( k ) x1 (n ), z, n 1( k ) x2 (n ), z, n , k a, b , n 1,
, K2 , 19
(113)
ACCEPTED MANUSCRIPT
where
2 x, Re( ) x 2 2
( x, z , ) (a) 2
2 x, Re( ) 1 x, Re( ) 2
sgn( x) x 2 1 x, Re( )
2
,
(114)
2
,
(115)
2
2 x, Re( ) 1 x, Re( ) 2
2 x, Re( ) x 2 Re( ) z ( x, z , ) , l2 2
(b ) 1
( x, z , ) (b ) 2
x sgn( x) x 2 1 x, Re( )
2
l2
,
where sgn() is the sign function, and
x2 Im z x .
M
x1 Im z x ,
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1 ( s l2 ) 2 (tz )2 ( s l2 ) 2 (tz ) 2 , 2 1 2 s, t ( s l2 )2 (tz )2 ( s l2 ) 2 (tz ) 2 , 2
1 s, t
CR IP T
1( a ) ( x, z, )
(116)
(117)
(118) (119)
(120)
ED
One can prove that the surface stresses, surface electric displacements and
PT
surface magnetic inductions for l1 0 given in Eqs. (75)-(82) are equal to those given in Eq. (103) when z=0.
CE
Numerical Results
AC
Table 1 gives the piezoelectric and piezomagnetic constants used in numerical
computation with all absent material coefficients 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–CoFe2O4 composites [21, 22] is 0.7.
20
ACCEPTED MANUSCRIPT 9
2
2
Table 1 Material properties of MEE ( cmn in 10 N m , emn in C m , hmn in N
mn in 109 C 2 Nm2 , mn in 106 NS 2 C 2 ) c11
c13
c33
c44
155.2407
86.6946
141.6492
28.36
e31
e33
e15
h31
h33
h15
5.0463 11
10.5826
8.6324
174.09
209.91 33
165
10.734
10.5279
11 177
47.1
CR IP T
33
Am ,
As mentioned before, one can determine the width of the contact region, l2 l1 , from Eq. (87) considering that the singular end-points x l1 are known a priori. In
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practical analysis, the location at which the maximum penetration depth arrives seems to be the singular end point for some kinds indenters, such as semi-cylindrical indenter and wedge-shaped indenter. Eq. (87) shows that the contact width l2 l1 is a
M
monotonically increasing function of the external loading P. Thus, the contact region
ED
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
PT
contributes to the contact width. As the value of l1 become smaller, the contact
CE
region becomes wider. For a flat indenter with a constant penetration depth, the interaction effect can be ignored when the two indenters became far away from each
AC
other. For the present two semi-cylindrical indenters with profile given in Eq. (53), the maximum penetration depth varies with the values of l1 changing. Thus, the interaction effect has a contribution to the contact behavior even when the two indenters become far away. Figure 4 shows how the interaction between two semi-cylindrical indenters
21
ACCEPTED MANUSCRIPT
affects the surface normal stress zz ( x,0) . At the outer edges x l2 , the surface normal stress remains free, while has singularities, stress concentrations, at the inner edges x l1 for two separated semi-cylindrical indenters. With the value of l1 becoming smaller, the strength of the stress singularity is relieved, and the stress
CR IP T
singularity vanishes when l1 0 . Figure 5 reveals the influence of the interaction between two semi-cylindrical indenters on the surface electric displacement Dz ( x,0) , the surface magnetic
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induction Bz ( x, 0) and the surface in-plane stress xx ( x, 0) . Different from the surface normal stress, these quantities may have singularities not only at the inner edges x l1 but also at the outer edges x l2 . The strength of these singularities weakens as two semi-cylindrical indenters stay near. The singularity strength at the
M
outer edges x l2 is stronger than that around the inner edges x l1 . It is the
ED
multi-field coupling possessed by MEE that causes the singularities at the outer edges
x l2 . More specifically, the singularity for the surface electric displacement is
PT
generated by the additional electric charge Q QF as seen from Eq. (73) or Eq. (78),
CE
by the additional magnetic induction M M F for the surface magnetic induction as
AC
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). 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. 22
ACCEPTED MANUSCRIPT
By setting Q=QF 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. QF and
M F , from Eqs. (88) and (89), the singularities at the outer edges x l2 for the surface electric displacement, the surface magnetic induction and the surface in-plane
CR IP T
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 l1 , and are free at the outer edges x l2 . As the combinations of the multi-field loadings escalate, the singularity
Conclusions Indentation
problem
of
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strength at the inner edges x l1 intensifies.
magnetoelectroelastic
materials
under
two
M
electrically-conducting and magnetically-conducting indenters is investigated. For two perfectly conducting semi-cylindrical indenters, the closed-form solutions for the
ED
stresses, electric displacement and magnetic induction on the surface are obtained,
PT
and the stress, electric displacement and magnetic induction intensity factors are defined. Numerical results are calculated to show the influences of the interaction of
CE
two semi-cylindrical indenters on the contact behaviors under different multi-field
AC
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 occurs 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 23
ACCEPTED MANUSCRIPT
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
CR IP T
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
be further revealed in the future work.
Acknowledgments
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materials on contact behaviors especially on the dynamic contact behaviors needs to
This work was supported by the National Natural Science Foundation of China
M
(11472193, 11572227 and 11261042), and the Fundamental Research Funds for the Central Universities (1330219162).
ED
Appendix
1. Expressions of the operator matrix Κ related to Eq. (24)
AC
CE
PT
2 2 2 2 2 e31 e15 h31 h15 c11 2 x c44 2 z c13 c44 xz xz xz 2 2 2 2 2 2 2 c c c c e e h h 44 2 33 2 15 2 33 2 15 2 33 2 13 44 xz x z x z x z Κ . 2 2 2 2 2 2 2 e31 e15 xz e15 2 x e33 2 z 11 2 x 33 2 z d11 2 x d33 2 z 2 2 2 2 2 2 2 h h 31 15 xz h15 2 x h33 2 z d11 2 x d33 2 z 11 2 x 33 2 z
2. Expressions of
kn
k , n 1, 2,3, 4
(A.1)
appearing in Eq. (25)
11
c13 c44 d112 1111 h15 h31 e15d11 h1511 e15 e31 h15d11 e15 11 ,
12
c13 c44 2d11d33 1133 3311 h15 h31 e15d33 e33d11 h15 33 h3311
e15 e31 h15d33 h33d11 e15 33 e33 11 , 24
(A.2)
ACCEPTED MANUSCRIPT
13
c13 c44 d332 33 33 h15 h31 e33d33 h33 33 e15 e31 h33d33 e33 33 .
21
c111111 c11d112 ,
22
c111133 c11 33 11 c441111 2d11 e15 e31 h15 h31 11 h15 h31 2
11 e15 e31 2c11d11d33 c44 d112 , 2
c11 33 33 c441133 c44 33 11 2d33 e15 e31 h15 h31 33 h15 h31
(A.3)
23
33 e15 e31 c11d332 2c44 d11d33 , 2
CR IP T
2
c44 33 33 c44 d332 .
31
e15c1111 h15c11d11 ,
32
c11e15 33 c11e3311 c44e15 11 h15 h15 h31 e15 e31 d11 h15 h31 c13 c44
AN US
24
e15 h15 h31 11 c13 c44 e15 e31 c11h15d33 c11h33d11 c44h15d11 , 2
33
c11e3333 c44e15 33 c44e3311 h33 h15 h31 e15 e31 d33 h15 h31 c13 c44
(A.4)
2
M
e33 h15 h31 33 c13 c44 e15 e31 c11h33d33 c44h33d11 c44h15d33 ,
c44e3333 c44h33d33 .
41
c11h1511 c11e15d11 ,
42
c11h15 33 c11h3311 c44h1511 c11e15d33 c11e33d11 c44e15d11 h15 e15 e31
PT
ED
34
2
c11h33 33 c44h15 33 c44h3311 c11e33d33 c44e15d33 c44e33d11 h33 e15 e31
(A.5)
2
AC
43
CE
11 c13 c44 h15 h31 e15 h15 h31 e15 e31 d11 c13 c44 e15 e31 ,
33 c13 c44 h15 h31 e33 h15 h31 e15 e31 d33 c13 c44 e15 e31 ,
44
c44 h33 33 c44e33d33 .
3. Expressions of j ( j 1,
,5) appearing in Eq. (27)
1 c44c33 3333 2c44e33h33d33 c44e332 33 c44c33d 233 c44h332 33 ,
(A.6)
2 c11c33 3333 2c11e33h33d33 c44c331133 c44c33 3311
(A.7)
25
ACCEPTED MANUSCRIPT
2c44c33d11d33 2c44e15e33 33 2c44e33h15d33 2c44e15h33d33 2c44e33h33d11 2c44 h15h33 33 2c1344e33e3115 33 2c1344e33h3115d33 2c1344 h33e3115d33 2c1344 h33h3115 33
2 2 2 c44 d332 e3115 h332 c11e33 33 c11c33d332 c11h332 33 2 2 c44 33 33 c44e332 11 c44 h332 11 c1344 33 33
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2 2 e3115 c33 33 h3115 c33 33 ,
CR IP T
2 2 2 2e3115c33h3115d33 2e3115h33h3115e33 c1344 d332 h3115 e33
2 2 3 c11c44 d332 c11e332 11 c11h332 11 c44 1133 c44 33 11
2 2 2c44 d11d33 c44c33d112 c44e152 33 c44h152 33 c1344 1133 2 2 2 2 c1344 33 11 2c1344 d11d33 e3115 c44 33 e3115 c33 11
M
2 2 2 2 2e3115 h15h33 h3115 c44 33 h3115 c3311 2h3115 e15e33
ED
2c1344e15e3115 33 2c1344e15h3115 d33
2c1344e33e3115 11 2c1344e33h3115d11 2c1344h15e3115d33
PT
2c1344 h15h3115 33 2c1344 h33e3115 d11 (A.8)
CE
2c1344 h33h311511 2e3115c44 h3115 d33
AC
2e3115c33h3115 d11 2e3115 h15h3115e33 2e3115h33h3115e15 c11c44 33 33 c11c331133
c11c33 33 11 2c11c33d11d33 2c11e15e33 33 2c11e15h33d33 2c11e33h33d11 2c11e33h15d33
2c11h15h33 33 c44c331111 2c44e15e33 11 2c44e15h15d33 2c44e15h33d11 2c44e33h15d11 26
ACCEPTED MANUSCRIPT
2c44 h15h3311 ,
4 c11c33d112 c11e152 33 c11h152 33 c44e152 11 c44 h152 11 2 2 2 2 c44 1111 c1344 1111 e3115 c44 11 h3115 c4411
c11c441133 c11c44 33 11 2c11c44d11d33
2c11e33h15d11 2c11e15h33d11 2c11h15h3311 2c44e15h15d11 2c1344e15e3115 11 2c1344e15h3115d11
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2c1344 h15e3115d11 2c1344 h15h311511
CR IP T
c11c331111 2c11e15e33 11 2c11e15h15d33 (A.9)
2 2 2e3115c44 h3115d11 2e3115h15h3115e15 h3115 e152 e3115 h152
2 2 c1344 d112 c44 d112 ,
5 c11 c441111 2e15h15d11 e152 11 c44 d112 h152 11 ,
M
where
ED
c1344 c13 c44 ,
(A.10)
(A.11) (A.12)
h3155 h31 h15 .
(A.13)
PT
e3155 e31 e15 ,
CE
4. Expressions of l m
, K1 , l u, w, , , xx, zz, xz, dx, dz, bx, bz )
( m 1,
appearing in Eq. (36)
AC
u m m
11
w m
21
22
m
31
32
m
41
42
m2
12
m2
23
m2
33
m2
43
m4 ,
(A.14)
13
m4
24
m6 ,
(A.15)
m4
34
m6 ,
(A.16)
m4
44
m6 ,
(A.17)
xx m c11 u m c13 m w m e31 m m h31 m m , 27
(A.18)
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zz m c13 u m c33 m w m e33 m m h33 m m ,
(A.19)
xz m c44 m u m w m e15 m h15 m ,
(A.20)
dx m e15 m u m w m 11 m d11 m ,
(A.21) (A.22)
bx m h15 m u m w m d11 m 11 m ,
(A.23)
bz m h31 u m h33 m w m d33 m m 33 m m .
(A.24)
5. Expressions
l(C ) n , n
of
and
l( S ) n , n
, K2 , l u, w, , , xx, zz, xz, dx, dz, bx, bz ), which are values of functions
AN US
( n 1,
CR IP T
dz m e31 u m e33 m w m 33 m m d33 m m ,
l(C ) , and l( S ) , at the point, n , n appearing in Eqs. (37) and (38)
u( S ) ,
24
21
12
11
12
3
22
6
w( S ) , 2
3 2
13
2
13
5
2
2
2
31
22
23
4
2
2
32
23
2
2
2
2
4
2 33
24 4
4
10 2 2 5 4 ,
(A.25)
10 2 2 4 ,
(A.26)
4
6 2 2 4
15 15 4
PT
CE
(C ) ,
2
ED
w(C ) ,
11
M
u(C ) ,
6
(A.27)
,
3
4
10 2 2 3 4 ,
6 2 2 4 (A.29)
6 4 2 2 4 6 34 15 15 ,
AC
( S ) , 2
(C ) , 44
41
32
( S ) , 2
6
2
42
33
2
2
2
2
43
34 4
42
2
43
2
2
2
4
2
3
4
10 2 2 3 4 ,
(A.30)
6 2 2 4
15 15 4
(A.28)
44
6
(A.31)
,
3
4
10 2 2 3 4 ,
(A.32)
xx(C ) , c11 u(C ) , c13 w(C ) , w( S ) , e31 (C ) , ( S ) , h31 (C ) , ( S ) , , 28
(A.33)
ACCEPTED MANUSCRIPT
xx( S ) , c11 u( S ) , c13 w(C ) , w( S ) , e31 (C ) , ( S ) , h31 (C ) , ( S ) , ,
(A.34)
zz(C ) , c13 u(C ) , c33 w(C ) , w( S ) ,
, , h33 , , ,
e33
(C )
(S )
(C )
(A.35)
(S )
e33
(C )
CR IP T
zz( S ) , c13 u( S ) , c33 w(C ) , w( S ) ,
, , h33 , , , (S )
(C )
(A.36)
(S )
xz(C ) , c44 u(C ) , u( S ) , w(C ) ,
(A.37)
e15 (C ) , h15(C ) , ,
e15
(S )
, h15 , , (S )
AN US
xz( S ) , c44 u(C ) , u( S ) , w( S ) ,
(A.38)
dx(C ) , e15 u(C ) , u( S ) , w(C ) ,
(A.39)
M
11 (C ) , d11(C ) , ,
11
(S )
ED
dx( S ) , e15 u(C ) , u( S ) , w( S ) , (A.40)
, d11 , , (S )
PT
dz(C ) , e31 u(C ) , e33 w(C ) , w( S ) ,
33
, , d33 , , , (S )
(C )
(A.41)
(S )
CE
(C )
AC
dz( S ) , e31 u( S ) , e33 w(C ) , w( S ) ,
33 (C ) , ( S ) , d33 (C ) , ( S ) , ,
(A.42)
bx(C ) , h15 u(C ) , u( S ) , w(C ) , d11
(C )
(A.43)
, 11 , , (C )
bx( S ) , h15 u(C ) , u( S ) , w( S ) , (A.44)
d11 ( S ) , 11( S ) , , 29
ACCEPTED MANUSCRIPT
bz(C ) , h31 u(C ) , h33 w(C ) , w( S ) , d33 (C ) , ( S ) , 33 (C ) , ( S ) , ,
(A.45)
bz( S ) , h31 u( S ) , h33 w(C ) , w( S ) , d33
(C )
, , 33 , , . (S )
(C )
(A.46)
(S )
CR IP T
References
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[1] M. Ahearne, Y. Yang, K.Y. Then, K.K. Liu, An indentation technique to characterize the mechanical and viscoelastic properties of human and porcine corneas, Ann. Biomed. Eng. 35 (2007) 1608-1616. [2] M. Morales, E. Xuriguera, M. Martinez, J.A. Padilla, J. Molera, N. Ferrer, M. Segarra, F. Espiell, Mechanical characterization of copper-copper wires joined by friction welding using instrumented indentation technique, J. Mater. Eng. Perform. 23 (2014) 3941-3948. [3] L. Xiao, D.Y. Ye, C.Y. Chen, A further study on representative models for calculating the residual stress based on the instrumented indentation technique, Comp. Mater.als Sci., 82 (2014) 476-482. [4] F. Dinzart, H. Sabar, Magneto-electro-elastic coated inclusion problem and its
48 (2011) 2393-2401.
M
application to magnetic-piezoelectric composite materials, Int. J. Solids Struct.
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[5] P.F. Hou, Y.T.L. Andrew, H.J. Ding, The elliptical Herzian contact of transversely isotropic magnetoelectroelastic bodies, Int. J. Solids Struct. 40
PT
(2003) 2833-2850.
[6] H.J. Ding, A.M. Jiang, Fundamental solutions for transversely isotropic
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magneto-electro-elastic media and boundary integral formulation, Sci. China Ser. E, 46 (2003) 607-619.
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[7] W.Q. Chen, E. Pan, H.M. Wang, Ch. Zhang, Theory of indentation on multiferroic composite materials, J. Mech. Phys. Solids 58 (2010) 1524-1551.
[8] Y.T. Zhou, K.Y. Lee, Theory of sliding contact for multiferroic materials indented by a rigid punch, Int. J. Mech. Sci. 66 (2013) 156-167. [9] Y.T. Zhou, T.W. Kim, An exact analysis of sliding frictional contact of a rigid punch over the surface of magneto-electro-elastic materials, Acta Mech. 225 (2014) 625-645. 30
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[10] X.Y. Li, R.F. Zheng, W.Q. Chen, Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch, Int. J. Solids Struct. 51 (2014) 164-178. [11] X. Yang, X.J. Liu, Z. Huang, Surface cracks of solid-phase-sintered silicon carbide ceramics and their influences on material strength, J. Inorganic Mater. 29 (2014) 438-442. [12] M. He, F.G. Li, J. Cai, B. Chen, An indentation technique for estimating the
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energy density as fracture toughness with Berkovich indenter for ductile bulk materials, Theor. Appl. Fract. Mech. 56 (2011) 104-111.
[13] B.L. Wang, J.C. Han, S.Y. Du, H.Y. Zhang, Y.G. Sun, Electromechanical behaviour of a finite piezoelectric layer under a flat punch. Int. J. Solids Struct.
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45 (2008) 6384-6398.
[14] B.L. Wang, Fracture and effective properties of finite magnetoelectroelastic media, J. Intel. Mat. Syst. Str. 23 (2012) 1699-1712.
[15] T.H. Hao, Z.Y. Shen, A new electric boundary condition of electric fracture
M
mechanics and its applications, Eng. Fract. Mech. 47 (1994) 793-802. [16] Q. Guan, S.R. He, Two-dimensional analysis of piezoelectric/piezomagnetic and
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elastic media, Compos. Struct. 69 (2005) 229-237. [17] H.J. Ding, B. Chen, J. Liang, On the general solutions for coupled equation for
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piezoelectric media, Int. J. Solids Struct. 33 (1996) 2283-2298. [18] C. Hwu, C.W Fan, Sliding punches with or without friction along the surface of
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an anisotropic elastic half-plane, Q. J. Mech. Appl. Math. 51 (1998) 159-177. [19] M.A. Guler, F. Erdogan, The frictional sliding contact problems of rigid
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parabolic and cylindrical stamps on graded coatings, Int. J. Mech. Sci. 49 (2007) 161-182.
[20] Y.T. Zhou, K.Y. Lee, Contact problem for magneto-electro-elastic half-plane materials indented by a moving punch. Part I: Closed-form solutions, Int. J. Solids Struct. 49 (2012) 3853-3865. [21] Z.F. Song, G.C. Sih, Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation, J. Theor. Appl. Frac. Mech. 39 (2003) 189-207. 31
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[22] S.S. Vel, R.C. Batra, Three-dimensional analytical solution for hybrid multilayered piezoelectric plates, J. Appl. Mech.-T. ASME 67 (2000) 558-567.
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[23] Y.Z. Wang, M. Kuna, Screw dislocation in functionally graded magnetoelectroelastic solid, Philos. Mag. Lett. 94 (2014) 72-79. [24] Y.Z. Wang, M. Kuna, Time-harmonic dynamic Green's functions for two-dimensional functionally graded magnetoelectroelastic materials, J. Appl. Phys. 115 (2014) 043518.
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Figures
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PT
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Fig. 1. Two indenters acting on Solid-phase-sintered Silicon Carbide Ceramic (SSiC) [11].
33
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AC
CE
PT
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Fig. 2. Schematic figure of magnetoelectroelastic materials under two symmetrical, perfectly conducting indenters – two semi-cylindrical indenters as example.
34
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0.4 0.3 0.2 l1 increasing
0.1
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Width of contact region (cm)
0.5
[0.3 0.6 0.9 1.2 1.5 1.8 2.1]*10
0.0 0.00
0.25
0.50 7
P (10 N/m)
0.75
-3
1.00
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CE
PT
ED
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Fig. 3. Contact width vs. external loading P with R 0.09m .
35
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0
-6 -9
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zz(x,0)(GPa)
-3
l1=0
-12 -15 -0.3
0.0 x(cm)
-3
l1=1.2*10
-3
l1=1.5*10
-3
0.3
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-18 -0.6
l1=0.9*10
0.6
Fig. 4. Effect of the interaction between two semi-cylindrical indenters on the surface normal stress zz ( x,0) with P 107 N m and R 0.09m in which the
AC
CE
PT
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M
unit of distance l1 is m.
36
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2
Dz(x,0)(10 C/m )
0.0
3
-0.3
-0.6
-0.9 -0.6
-0.3
l1=0.9*10
-3
l1=1.2*10
-3
l1=1.5*10
-3
0.0 x(cm)
0.3
Bz(x,0)(10 N/(Am))
0
4
b) l1=0
M
-6
-9 -0.6
-0.3
2
ED
PT
xx(x,0)(10 GPa)
-3
l1=1.2*10
-3
l1=1.5*10
-3
0.3
0.6
0.3
0.6
c) l1=0
2.0
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l1=0.9*10
0.0 x(cm)
2.5
AC
0.6
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-3
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a) l1=0
1.5
l1=0.9*10
-3
l1=1.2*10
-3
l1=1.5*10
-3
1.0 0.5 0.0 -0.6
-0.3
0.0 x(cm)
Fig. 5. Influence of the interaction between two semi-cylindrical indenters with P 107 N m and R 0.09m on: a) the surface electric displacement Dz ( x,0) when Q 0.5 C m , b) the surface magnetic induction Bz ( x, 0) when M 60 N A , and c) the surface in-plane stress xx ( x, 0) when Q 0.5 C m and M 60 N A in which the unit of distance l1 is m. 37
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Dz (x,0)(C/m )
0
-2
-3 -1.5 -1.0 -0.5
a) 7 -2 P=0.510 , Q=0.077 10 7 -2 P=210 , Q=0.30810 7 -2 P=610 , Q=0.924 10 7 -2 P=810 , Q=1.232 10
0.0 0.5 x (cm)
Bz (x,0)(10 N/(Am))
0.0
1.0
1.5
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4
-0.5
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2
-1
-1.0
M
-1.5
ED
-2.0 -1.5 -1.0 -0.5
b) 7 P=0.510 , Q=5.4342 7 P=210 , Q=21.7369 7 P=610 , Q=65.2107 7 P=810 , Q=86.9476
0.0 0.5 x (cm)
1.0
1.5
1.2
AC
CE
xx (x,0)(10GPa)
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c) 7 -2 P=0.510 , Q=0.077 10 , M=5.4342 7 -2 P=210 , Q=0.30810 , M=21.7369 7 -2 P=610 , Q=0.924 10 , M=65.2107 7 -2 P=810 , Q=1.232 10 , M=86.9476
0.9 0.6 0.3
0.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 x (cm)
Fig. 6. Effect of the combinations of multi-field loadings when R 0.09m and l1 103 m on: a) the surface electric displacement Dz ( x,0) , b) the surface magnetic induction Bz ( x, 0) , and c) the surface in-plane stress xx ( x, 0) in which the units of P, Q and M are N m , 0.5C m and N A . 38
Magneto-electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials Yue-Ting Zhou , Sheng-Jie Pang , Yong Hoon Jang PII: DOI: Reference:
S0307-904X(17)30479-1 10.1016/j.apm.2017.07.041 APM 11891
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
12 April 2017 1 July 2017 20 July 2017
Please cite this article as: Yue-Ting Zhou , Sheng-Jie Pang , Yong Hoon Jang , Magneto-electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.07.041
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Highlights Interaction of two perfectly conducting indenters is examined.
Various intensity factors are defined to measure singular behavior at the edges.
Degradation from two perfectly conducting indenters to one indenter is done.
Interaction between two semi-cylindrical indenters is revealed.
Singularities at the edges can be suppressed because of multi-field coupling.
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1
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Magneto-electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials Yue-Ting Zhou a ,* , a
Sheng-Jie Pang a , Yong Hoon Jang b
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092,
b
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P. R. China School of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea
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ABSTRACT Within the theory of linear full-field magneto-electro-elasticity, magneto-electro interaction 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 employed to address the two-dimensional, mixed boundary-value problem explicitly. The surface stresses, electric displacement and magnetic induction and their respective intensity factors are obtained in closed forms for two perfectly conducting semi-cylindrical indenters. 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 interaction between two semi-cylindrical indenters on contact behaviors subjected to multi-field loadings.
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Keywords: Magneto-electro interaction; Magnetoelectroelastic materials; Electrically-conducting and magnetically-conducting; Closed forms; Singularity; Multi-field coupling.
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Introduction
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The indentation technique involves the application of a well-defined indenter to
deform the testing materials and characterize their mechanical performances [1]. For example, since residual stresses exert pronounced influences on materials’ mechanical _________ *
Corresponding author at. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China. E-mail addresses: [email protected] (Y.T. Zhou). 2
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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].
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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
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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]
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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]
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and Zhou and Kim [9] developed a basic theory of the sliding contact of
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magnetoelectroelastic materials subject to a rigid indenter with flat profile, parabolic profile, triangular profile or cylindrical profile illustrating that the indenter profile
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greatly affects the contact behavior. Recently, Li et al. [10] presented fundamental
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solutions for the contact problem of a magnetoelectroelastic half-space punched by a smooth and rigid half-infinite indenter. 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 detect the fracture toughness of Solid-phase-sintered Silicon Carbide Ceramic (SSiC) by the Vickers and Knoop indentation method [11]. 3
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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
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different effect on the contact behavior. Addressing two collinear indenters acting on piezoelectric materials, one kind of single-phase magnetoelectroelastic materials, Wang et al. [13] found that the indenter tip fields were greatly affected by the relative
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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,
M
which makes the interaction effect between two indenters deserve to be studied. This article conducts an exact contact analysis of magnetoelectroelastic materials
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under two electrically-conducting and magnetically-conducting indenters. Singular
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integral equations with kernel like 1 ( 2 x 2 ) because of the interaction of two indenters are obtained and solved analytically, which makes the solutions have a quite
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different form. The fields disturbed by the two perfectly conducting semi-cylindrical
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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.
Formulation of the Problem There are two electrically-conducting and magnetically-conducting indenters acting symmetrically about z-axis on the surface of magnetoelectroelastic materials 4
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(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. Basic Equations The constitutive equations are [14]
σ cS eE hH ,
(1)
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D eT S E dH ,
(2)
B hT S dE H ,
(3)
where the superscript T denotes the transposition, and σ , S , D , E , B and H
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are, respectively, the vectors of the stress, strain, electric displacement, electric field, magnetic induction, and magnetic field, which are given as follows:
σ xx zz xz
T
, S u, x
D Dx
Dz ,
E , x
B Bx
Bz ,
H , x
w, z
, z , T
M
T
T
1 u, z w, x , 2
, z , T
(5) (6)
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T
(4)
where the comma stands for the differentiation with respect to the corresponding
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coordinate variables, u and w are the mechanical displacement components, and
and represent the electric potential and the magnetic potential.
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In Eqs. (1)-(3), c , e , h , , d and are the matrices of the elastic
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coefficients, electromechanical coupling coefficients, magnetomechanical coupling coefficients,
dielectric
permeability
coefficients,
magneto-electro
coupling
coefficients, and magnetic permeability coefficients, which take the following forms for linearly, transversely isotropic magnetoelectroelastic materials:
c11 c13 c c13 c33 0 0
0 0 , 2c44
0 e 0 2e15
e31 e33 , 0
5
(7)
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0 h 0 2h15
h31 h33 , 0
11 0
11 0
0 d d 11 , 0 d33
0 , 33
(8)
0 . 33
(9)
xx, x xz , z 0 ,
xz , x zz , z 0 ,
Dx, x Dz , z 0 ,
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The equilibrium equations free of any generalized body source are (10) (11)
Bx, x Bz , z 0 .
(12)
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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
M
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
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and magnetoelectroelastic materials well contact in the contact region. Thus, it is assumed that there is no air inside the contact region, and perfectly conducting
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boundary conditions are chosen.
The x>0 part of the system is considered since the stated problem is symmetric
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with respect to x = 0. Denote the contact area as 0 l1 x l2 in x>0 part (Fig. 2). Since each indenter is electrically-conducting and magnetically-conducting, one
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has
( x,0) 1 ( x) ,
0 l1 x l2 ,
(13)
( x,0) 1 ( x) ,
0 l1 x l2 ,
(14)
with 1 ( x) and 1 ( x) denoting the electric potential and the magnetic potential inside the contact area 0 l1 x l2 . The perfectly conducting property makes the normal components of the electric 6
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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.
p( x), 0 l1 x l2 , 0 x l1 , x l2 0,
zz ( x, 0)
(15)
(16)
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q( x), 0 l1 x l2 Dz ( x, 0) , 0 x l1 , x l2 0, m( x), 0 l1 x l2 Bz ( x, 0) , 0 x l1 , x l2 0,
(17)
where q( x) , m( x) , and p( x) are unknown functions. The accumulated electric
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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)
l1
l1
l2
l2
l2
l2
q( x)dx q( x)dx Q , l1
l2
m( x)dx m( x)dx M , l1
l2
M
l1
p( x)dx p( x)dx P . l1
(18) (19) (20)
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The surface shear stress keeps zero (21)
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xz ( x,0) 0 .
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The indenter profile is described as
w( x,0) w1 ( x) , 0 l1 x l2 ,
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(22)
with w1 ( x) being a known function. The following regularity conditions should be satisfied at infinity: u( x, z), w( x, z ), ( x, z ), ( x, z ) 0 ,
x2 z 2 .
General Solutions Considering Eqs. (1)-(12), one may discover the general solutions as [16] 7
(23)
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u
and
T
T
k1
k2
k3
k4
( x, z ) ,
(24)
( x, z ) is an auxiliary function, which satisfies Eq. (26) as will be seen later, kn
k , n 1, 2,3, 4
are the cofactors of the operator matrix Κ given in the
Appendix. In the following, 21
2n
n 1, 2,3, 4
6 11 x5z
6 12 x3z 3
x 6
x 4z 2
will be used, which take the form
6 13 xz 5
n 1 ,
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where
w
(25)
where
kn
n1
6
n2
k , n 1, 2,3, 4
x 2z 4 6
n3
z 6 6
n4
are given in the Appendix.
n 2,3, 4 ,
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2n
6
In Eq. (24), the following relationship holds for the function
det Κ ( x, z ) 0 .
( x, z ) : (26)
Applying the Fourier cosine transform with respect to x to Eq. (26) arrives at
M
6 4 2 8 2 4 6 5 8 0 , 2 3 4 z 8 z 6 z 4 z 2
where j ( j 1,
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1
,5) are given in the Appendix, and
, z
(27)
is defined as
, z 0 x, z cos x dx .
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(28)
, z e z , one can obtain the characteristic equation associated
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Considering with Eq. (27) as
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1 8 2 6 3 4 4 2 5 0 .
(29)
where represents the root. According to the eigenvalue properties of Eq. (29) and using the Almansi's
theorem [17], one may express
, z
as follows:
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4 k k , z , 1 2 3 4 k 1 4 2 k 3 , z k z e3 z , 1 2 3 4 k k k 3 k 1 2 1 2 3 4 0 , , z 2 k 1ek z 2 k zek z , k 1 4 k 2 1e1 z k z e2 z , 1 2 3 4 0 k 2 4 k 1 k z e1 z , 1 2 3 4 0 k 1
(30)
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where k (k 1, 2,3, 4) are the unknown functions to be determined from the k
, z
depend on whether the eigenvalues are real or
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boundary conditions, and
complex. For example, in the case of 1 2 3 4 , if there are K1 positive real roots expressed as m m ( m 1,
, K1 ) and K 2 pairs of conjugate complex roots
m
, z
can be given as
, z e z
M
with positive real parts expressed as n n n in ( n 1,
K1 2 n 1
, z e z cos n z n
, z e z sin n z
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K1 2 n
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m
m
n
, K2 ) with i 2 1 ,
( m 1,
, K1 ),
(31)
( n 1,
, K2 ),
(32)
( n 1,
, K2 ).
(33)
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Note that when 1 2 3 4 , one may obtain K1 0 and K 2 2 , or
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K1 1 and K 2 1 ,or K1 4 and K 2 0 since K1 2K2 4 is required. In the 1 2 3 4 case,
1
, z
and
2
, z
can be given in the
same way as shown in Eq. (31). Various field quantities can be given on the basis of consideration of the constitutive equations (1)-(3)
9
, z
with the
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Dx
(k ) u
Bx T
(k ) w
4
0
k 1
6
k
sin x (34)
(k ) T bx
(xzk ) (dxk )
w xx zz
2
( k )
Dz
Bz T
2
d , 4
0
k 1
k
6
cos x (35)
(k ) ( k ) (xxk ) (zzk ) (dzk ) bz d , T
where known functions l( k ) , z ( k 1, are given as ( m 1,
, K1 ),
, 4 , l u, w, , , xx, zz, xz, dz, dx, bz, bx )
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l( m) , z l m e m z
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u xz
(36)
l( K1 2n1) , z l(C ) n , n cos n z l( S ) n , n sin n z en z
( n 1,
, K2 ),
(37)
l( K1 2 n ) , z l( S ) n , n cos n z l(C ) n , n sin n z en z
, K2 ),
, K1 , l u, w, , , xx, zz, xz, dx, dz, bx, bz ), l(C ) n , n and
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where l m ( m 1,
M
( n 1,
(38)
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l( S ) n , n ( n 1, , K2 , l u, w, , , xx, zz, xz, dx, dz, bx, bz ) are given in the Appendix (Results for distinctive eigenvalues case are presented in what follows since
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the commercially available MEE, which are transversely isotropic, generate
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distinctive eigenvalues).
Singular Integral Equations for two Perfectly Conducting Indenters Considering boundary conditions Eqs. (22), (13) and (14) results in singular
integral equations as follows:
l2
l1
11 p( ) 12 q( ) 13m( ) w1 x d , l1 x l2 , 2 2 x 2 x x
10
(39)
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l2
l2
l1
l1
21 p( ) 22 q( ) 23m( ) 1 x , l1 x l2 , d 2 2 x 2 x x
(40)
31 p( ) 32 q( ) 33m( ) 1 x , l1 x l2 , d 2 2 x 2 x x
(41)
(wk ) ( , 0), m 1 C m1 (1) k 1 ( k ) ( , 0), m 2 , det CB ( k ) k 1 ( , 0), m 3 (1k ) B
4
m2
(wk ) ( , 0), m 1 (3 k ) C B (1) k 1 ( k ) ( , 0), m 2 , det CB ( k ) k 1 ( , 0), m 3
m3
(wk ) ( , 0), m 1 C (1) k ( k ) ( , 0), m 2 , det CB ( k ) k 1 ( , 0), m 3 (4 k ) B
4
(42)
(43)
(44)
M
, 4) are the complement minors of the matrix CB Clk ,
where CB(lk ) (l , k 1, (l, k 1,
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4
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where mn (m, n 1, 2,3) are given as
, 4) taking the following forms:
,0
(k 1,
C3k (dzk ) ,0 , (45)
, 4) .
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C4 k
(k ) bz
C2 k (xzk ) ,0 ,
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C1k (zzk ) ,0 ,
The singular integral equations, Eqs. (39) and (40), and Eqs. (18)-(20) can be
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normalized as 1
j1 p (0) ( ) j 2 q (0) ( ) j 3m(0) ( )
1
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d j ( ) , 1 1, (46)
j 1, 2,3 ,
1
1
1
1
p (0) ( )d
2P , d0
(47)
q (0) ( )d
2Q , d0
(48)
11
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1
1
m(0) ( )d
2M , d0
(49)
where
p( )
2 l0 d0
1 ( )
,
q (0) ( )
,
q( )
m(0) ( )
l l1 2 2
x 2 l0 , d0
1 w1 ( x) , x x
,
d0
2 ( )
2
1 1 ( x) , x x
m( )
2
,
(50)
l l1 2 2
,
l0
3 ( )
2
1 1 ( x) . x x
Exact Contact Analysis
,
(51)
(52)
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Two Perfectly Conducting Semi-Cylindrical Indenters
2
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p (0) ( )
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
l1 x l2 ,
(53)
(54)
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1 ( x) 0 ,
x2 , 2R
ED
w1 ( x) w0
M
obtain
1 ( x) 0 ,
(55)
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where w0 , 0 and 0 are constants, and R is the radius of the semi-cylindrical
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indenter.
Note that the first contact point ( x l1 ) is not on the apex of the semi-cylindrical
shape of the indenter, and the maximum penetration depths vary with different values of l1 . 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. 12
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Considering Eqs. (39)-(41), (46), (53)-(55), one has
1 , R
(56)
2 ( ) 3 ( ) 0 .
(57)
1 ( )
Since
q( x) q1 ( x) q2 ( x)
m( x) m1 ( x) m2 ( x)
and
for the conducting
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semi-cylindrical indenter, in which q1 ( x) and m1 ( x) are caused by the electric potential and the magnetic potential and q2 ( x) and m2 ( x) caused by the normal mechanical load P, one has q(0) ( ) q1(0) ( ) q2(0) ( ) ,
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(58)
m(0) ( ) m1(0) ( ) m2(0) ( ) .
(59)
Thus, Eqs. (46)-(49) can be written as
1
1
m1(0) ( )d
1
1
(62)
2 Q QF , d0
(63)
AC
1
q1(0) ( )d
(60)
(61)
q1(0) ( ) d 0 , 1 1,
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1
2M MF , d0
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in terms of m1(0) ( ) ,
M
1
m1(0) ( ) d 0 , 1 1,
ED
1
in terms of q1(0) ( ) , and
R , (0) (0) (0) 1 j1 p ( ) j 2 q2 ( ) j 3m2 ( ) d 0, 1 0,
13
j 1 j 2 , 1 1, j 3
(64)
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1
1
1
1
1
1
p (0) ( )d
2P , d0
q2(0) ( )( )d m2(0) ( )d
(65)
2QF , d0
(66)
2M F , d0
(67)
CR IP T
in terms of p (0) ( ) , q2(0) ( ) and m2(0) ( ) , and QF and M F denote the electric charge and the magnetic induction due to the total indentation force P . The exact solutions of integral equations (60)-(67) take the form
p (0) ( )
,
d0 1 2 2 Q QF d0 1 2
,
1 1 , P R 1
q2(0) ( )
(68)
(69)
1 1 , Q R 1
(70)
ED
1 , m2(0) ( ) M R 1 1
AN US
q1(0) ( )
2M MF
M
m1(0) ( )
det mn 33
,
22 33 32 23
CE
P
PT
where P , Q and M are given as
AC
Q
M
det mn 33
(71)
,
2331 2133 det mn 33 2132 22 31
.
Then, the contact stress, electric displacement and magnetic induction inside the contact region are given as follows:
x p ( x) P R
l2
2
x2
x 2 l1
2
, l1 x l2 , 14
(72)
ACCEPTED MANUSCRIPT
q ( x)
m( x )
x Q R
l2
2
x2
x l1
x M R
2
l2
2
2
x2
x 2 l1
2
2 Q QF x 2 2 l2 x 2 x 2 l1
, l1 x l2 ,
2M MF x 2 2 l2 x 2 x 2 l1
, l1 x l2 .
(73)
(74)
CR IP T
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
AN US
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
M
electric displacement and the surface magnetic induction, and the outer-edge
multi-field loadings.
ED
singularities can also be suppressed by choosing appropriate combinations of the
Considering Eqs. (34), (35), (72)-(74) results in the surface stresses, surface
PT
electric displacements and surface magnetic inductions as follows: l1 x l2 , x l1 or x l2
(75)
( P) (Q ) (M ) zz p( x) zz q( x) zz m( x), 0,
l1 x l2 , x l1 or x l2
(76)
CE
( P) (Q ) (M ) xx p( x) xx q( x) xx m( x), xx x, 0 0,
AC
zz x, 0
( P ) (Q ) ( M ) l1 x l2 xz xz xz x, xz x, 0 P R Q R M R , 0, x l1 or x l2 ( P ) (Q ) ( M ) l1 x l2 dx dx dx x, Dx x, 0 P R Q R M R , 0, x l1 or x l2
15
(77)
(78)
ACCEPTED MANUSCRIPT
( P) (Q ) (M ) dz p( x) dz q( x) dz m( x), Dz x, 0 0,
( P ) (Q ) ( M ) bx bx bx x, Bx x, 0 P R Q R M R 0,
l1 x l2 , x l1 or x l2
l1 x l2 x l1 or
( P) (Q ) (M ) bz p( x) bz q( x) bz m( x), Bz x, 0 0,
k 1
(1) k CB(4 k ) (jk ) , 0 , det CB k 1
4
(jQ ) (jk ) , 0 k 1
(81)
(1)k CB(3k ) , det CB
(82)
4
j xx, zz, xz, dz, dx, bz, bx .
AN US
(jM )
(1) k CB(1k ) , det CB
x l2
CR IP T
4
(80)
l1 x l2 , x l1 or x l2
where
(jP ) (jk ) , 0
,
(79)
Employing the properties about minors of the matrix CB Clk in Eq. (45)
(zzP ) 1 ,
(zzQ ) 0 ,
(xzP ) 0 ,
(xzQ ) 0 ,
(dzP ) 0 ,
ED
arrives at
M
(83)
(xzM ) 0 ,
(84)
(dzQ ) 1,
(dzM ) 0 ,
(85)
(Q ) bz 0,
(M ) bz 1.
(86)
PT
(P) bz 0,
(zzM ) 0 ,
Then, one may find the required boundary conditions Eqs. (15)-(17) and (21)
CE
hold.
Substituting Eq. (70) into Eqs. (65)-(67), one can determine the width of the
AC
contact region with the singular end-points x l1 known a priori, QF and M F as follows:
l2 QF
4 P P R
l1 , 2
(87)
P P , Q
(88)
16
ACCEPTED MANUSCRIPT
MF
P P . M
(89)
One may define the following stress, electric displacement and magnetic induction intensity factors to measure the singular behavior at the ends of the semi-cylindrical indenters: K I l1 lim 2 x l1 p( x) ,
(90)
CR IP T
x l1
K D l1 lim 2 x l1 q( x) ,
(91)
x l1
K B l1 lim 2 x l1 m( x) .
(92)
x l1
P R 2 2 l1 l2 l1
K D l1
Q R
M R
2 Q QF
2M M F
(93)
l1
2 l2 l1 2
2 2 l1 l2 l1
,
,
(94)
l1
2 l2 l1 2
ED
K B l1
M
K I l1
2 2 l1 l2 l1
AN US
Considering Eqs. (68)-(70), one can rewrite Eqs. (90)-(92) as follows:
.
(95)
PT
Verification of The Exact Analysis
To validate the theoretical derivation for two semi-cylindrical indenters, one can
CE
set l1 0 . In such a special case, two semi-cylindrical indenters each with the accumulated electric charge Q, accumulated magnetic induction M and total
AC
indentation force P become a single cylindrical indenter with the contact region
l2 , l2 ,
accumulated electric charge 2Q, accumulated magnetic induction 2M and
total indentation force 2P with considering Eqs. (18)-(20), i.e.
l2
l2
l2
l2
0
l2
l2
0
q( x)dx q( x)dx q( x)dx Q Q 2Q , 0
l2
l2
0
m( x)dx m( x)dx m( x)dx M M 2M ,
17
(96) (97)
ACCEPTED MANUSCRIPT
l2
l2
0
l2
l2
0
p( x)dx p( x)dx p( x)dx P P 2 P .
(98)
Setting l1 0 in Eqs. (72)-(74) and (87), one has 1 P R
l2
q ( x)
1 Q R
l2
m( x)
1
2
l2
M R
P P R
l2 2
2
x2 ,
x2
2
2 Q QF
l2
x2
x
2
,
2
2M MF
l2 x2 2
,
x l2 , l2 ,
(99)
x l2 , l2 ,
(100)
CR IP T
p( x)
x l2 , l2 ,
(102)
.
AN US
(101)
Eqs. (99)-(102) are identical to those given by Zhou and Lee [20] for a single cylindrical indenter with the contact region
l2 , l2 , accumulated electric charge 2Q,
M
accumulated magnetic induction 2M and total indentation force 2P. Thus, the solutions for a single cylindrical indenter are recovered by those for two
ED
semi-cylindrical indenters.
PT
For a single cylindrical indenter, the full-field stresses, electric displacements and magnetic inductions of MEE can be given explicitly bedsides those on the surface
CE
given in Eqs. (75)-(81). Various stresses, electric displacements and magnetic
AC
induction of MEE take the form
xx zz xz
Dz
4
k 1
k
k 1
Bz
Bx T
( a ,k ) xx
(zza ,k )
(xza ,k )
(dza ,k )
(dxa ,k )
( a ,k ) bz
(b,k ) xx
(zzb,k )
(xzb,k )
(dzb,k )
(dxb,k )
(b, k ) bz
4
k
Dx
where 18
( a ,k ) bx T
( b, k ) bx , T
(103)
ACCEPTED MANUSCRIPT
k
(1)k 2 Q QF CB(3k ) M M F CB(4 k ) det CB
k
,
(104)
(1)k l2 CB(1k ) CB(3k ) CB(4 k ) . R det CB P Q M
(105)
In Eq. (103), (ja ,k ) ( x, z ) and (jb,k ) ( x, z ) ( l xx, zz, xz, dx, dz, bx, bz ) are given as
(rk , K1 2 n1) ( x, z )
1 (C ) r n , n 1( k ) x, z, n , n 2
n , n x, z, n , n ， k a, b , (k ) 1
(rk , K1 2 n ) ( x, z )
n 1,
1 (S ) r n , n 1( k ) x, z, n , n 2
, K1 , r xx, zz, dz, bz , (106)
(107)
, K2 , r xx, zz, dz, bz ,
AN US
(S ) r
k a, b , m 1,
CR IP T
(rk ,m) ( x, z ) r m 1( k ) ( x, z, m ) ,
r(C ) n , n 1( k ) x, z, n , n ， k a, b , n 1, , K2 , r xx, zz, dz, bz , t( k ,m) ( x, z ) t m (2k ) ( x, z, m ) , t( k , K1 2 n1) ( x, z )
1 (C ) t n , n (2k ) x, z, n , n 2
M
n , n x, z, n , n ， k a, b , (k ) 2
t( k , K1 2 n ) ( x, z )
, K1 , t xz, dx, bx ,
n 1,
, K2 , t xz, dx, bx ,
1 (S ) t n , n (2k ) x, z, n , n 2
ED
(S ) t
k a, b , m 1,
t(C ) n , n (2k ) x, z, n , n ， k a, b , n 1, , K2 , t xz, dx, bx ,
(109)
(110)
(111)
, K1 ,
, K2 ) are given as
CE
n 1,
PT
where i( k ) x, z, n , n and i( k ) x, z, n , n ( k a, b , i 1, 2 , m 1,
(108)
AC
1( k ) x, z, n , n 1( k ) x1 (n ), z, n 1( k ) x2 (n ), z, n , 1( k ) x, z, n , n (2k ) x1 (n ), z, n (2k ) x2 (n ), z, n , k a, b , n 1,
(112)
, K2 ,
(2k ) x, z, n , n (2k ) x1 (n ), z, n (2k ) x2 (n ), z, n , (2k ) x, z, n , n 1( k ) x1 (n ), z, n 1( k ) x2 (n ), z, n , k a, b , n 1,
, K2 , 19
(113)
ACCEPTED MANUSCRIPT
where
2 x, Re( ) x 2 2
( x, z , ) (a) 2
2 x, Re( ) 1 x, Re( ) 2
sgn( x) x 2 1 x, Re( )
2
,
(114)
2
,
(115)
2
2 x, Re( ) 1 x, Re( ) 2
2 x, Re( ) x 2 Re( ) z ( x, z , ) , l2 2
(b ) 1
( x, z , ) (b ) 2
x sgn( x) x 2 1 x, Re( )
2
l2
,
where sgn() is the sign function, and
x2 Im z x .
M
x1 Im z x ,
AN US
1 ( s l2 ) 2 (tz )2 ( s l2 ) 2 (tz ) 2 , 2 1 2 s, t ( s l2 )2 (tz )2 ( s l2 ) 2 (tz ) 2 , 2
1 s, t
CR IP T
1( a ) ( x, z, )
(116)
(117)
(118) (119)
(120)
ED
One can prove that the surface stresses, surface electric displacements and
PT
surface magnetic inductions for l1 0 given in Eqs. (75)-(82) are equal to those given in Eq. (103) when z=0.
CE
Numerical Results
AC
Table 1 gives the piezoelectric and piezomagnetic constants used in numerical
computation with all absent material coefficients 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–CoFe2O4 composites [21, 22] is 0.7.
20
ACCEPTED MANUSCRIPT 9
2
2
Table 1 Material properties of MEE ( cmn in 10 N m , emn in C m , hmn in N
mn in 109 C 2 Nm2 , mn in 106 NS 2 C 2 ) c11
c13
c33
c44
155.2407
86.6946
141.6492
28.36
e31
e33
e15
h31
h33
h15
5.0463 11
10.5826
8.6324
174.09
209.91 33
165
10.734
10.5279
11 177
47.1
CR IP T
33
Am ,
As mentioned before, one can determine the width of the contact region, l2 l1 , from Eq. (87) considering that the singular end-points x l1 are known a priori. In
AN US
practical analysis, the location at which the maximum penetration depth arrives seems to be the singular end point for some kinds indenters, such as semi-cylindrical indenter and wedge-shaped indenter. Eq. (87) shows that the contact width l2 l1 is a
M
monotonically increasing function of the external loading P. Thus, the contact region
ED
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
PT
contributes to the contact width. As the value of l1 become smaller, the contact
CE
region becomes wider. For a flat indenter with a constant penetration depth, the interaction effect can be ignored when the two indenters became far away from each
AC
other. For the present two semi-cylindrical indenters with profile given in Eq. (53), the maximum penetration depth varies with the values of l1 changing. Thus, the interaction effect has a contribution to the contact behavior even when the two indenters become far away. Figure 4 shows how the interaction between two semi-cylindrical indenters
21
ACCEPTED MANUSCRIPT
affects the surface normal stress zz ( x,0) . At the outer edges x l2 , the surface normal stress remains free, while has singularities, stress concentrations, at the inner edges x l1 for two separated semi-cylindrical indenters. With the value of l1 becoming smaller, the strength of the stress singularity is relieved, and the stress
CR IP T
singularity vanishes when l1 0 . Figure 5 reveals the influence of the interaction between two semi-cylindrical indenters on the surface electric displacement Dz ( x,0) , the surface magnetic
AN US
induction Bz ( x, 0) and the surface in-plane stress xx ( x, 0) . Different from the surface normal stress, these quantities may have singularities not only at the inner edges x l1 but also at the outer edges x l2 . The strength of these singularities weakens as two semi-cylindrical indenters stay near. The singularity strength at the
M
outer edges x l2 is stronger than that around the inner edges x l1 . It is the
ED
multi-field coupling possessed by MEE that causes the singularities at the outer edges
x l2 . More specifically, the singularity for the surface electric displacement is
PT
generated by the additional electric charge Q QF as seen from Eq. (73) or Eq. (78),
CE
by the additional magnetic induction M M F for the surface magnetic induction as
AC
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). 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. 22
ACCEPTED MANUSCRIPT
By setting Q=QF 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. QF and
M F , from Eqs. (88) and (89), the singularities at the outer edges x l2 for the surface electric displacement, the surface magnetic induction and the surface in-plane
CR IP T
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 l1 , and are free at the outer edges x l2 . As the combinations of the multi-field loadings escalate, the singularity
Conclusions Indentation
problem
of
AN US
strength at the inner edges x l1 intensifies.
magnetoelectroelastic
materials
under
two
M
electrically-conducting and magnetically-conducting indenters is investigated. For two perfectly conducting semi-cylindrical indenters, the closed-form solutions for the
ED
stresses, electric displacement and magnetic induction on the surface are obtained,
PT
and the stress, electric displacement and magnetic induction intensity factors are defined. Numerical results are calculated to show the influences of the interaction of
CE
two semi-cylindrical indenters on the contact behaviors under different multi-field
AC
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 occurs 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 23
ACCEPTED MANUSCRIPT
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
CR IP T
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
be further revealed in the future work.
Acknowledgments
AN US
materials on contact behaviors especially on the dynamic contact behaviors needs to
This work was supported by the National Natural Science Foundation of China
M
(11472193, 11572227 and 11261042), and the Fundamental Research Funds for the Central Universities (1330219162).
ED
Appendix
1. Expressions of the operator matrix Κ related to Eq. (24)
AC
CE
PT
2 2 2 2 2 e31 e15 h31 h15 c11 2 x c44 2 z c13 c44 xz xz xz 2 2 2 2 2 2 2 c c c c e e h h 44 2 33 2 15 2 33 2 15 2 33 2 13 44 xz x z x z x z Κ . 2 2 2 2 2 2 2 e31 e15 xz e15 2 x e33 2 z 11 2 x 33 2 z d11 2 x d33 2 z 2 2 2 2 2 2 2 h h 31 15 xz h15 2 x h33 2 z d11 2 x d33 2 z 11 2 x 33 2 z
2. Expressions of
kn
k , n 1, 2,3, 4
(A.1)
appearing in Eq. (25)
11
c13 c44 d112 1111 h15 h31 e15d11 h1511 e15 e31 h15d11 e15 11 ,
12
c13 c44 2d11d33 1133 3311 h15 h31 e15d33 e33d11 h15 33 h3311
e15 e31 h15d33 h33d11 e15 33 e33 11 , 24
(A.2)
ACCEPTED MANUSCRIPT
13
c13 c44 d332 33 33 h15 h31 e33d33 h33 33 e15 e31 h33d33 e33 33 .
21
c111111 c11d112 ,
22
c111133 c11 33 11 c441111 2d11 e15 e31 h15 h31 11 h15 h31 2
11 e15 e31 2c11d11d33 c44 d112 , 2
c11 33 33 c441133 c44 33 11 2d33 e15 e31 h15 h31 33 h15 h31
(A.3)
23
33 e15 e31 c11d332 2c44 d11d33 , 2
CR IP T
2
c44 33 33 c44 d332 .
31
e15c1111 h15c11d11 ,
32
c11e15 33 c11e3311 c44e15 11 h15 h15 h31 e15 e31 d11 h15 h31 c13 c44
AN US
24
e15 h15 h31 11 c13 c44 e15 e31 c11h15d33 c11h33d11 c44h15d11 , 2
33
c11e3333 c44e15 33 c44e3311 h33 h15 h31 e15 e31 d33 h15 h31 c13 c44
(A.4)
2
M
e33 h15 h31 33 c13 c44 e15 e31 c11h33d33 c44h33d11 c44h15d33 ,
c44e3333 c44h33d33 .
41
c11h1511 c11e15d11 ,
42
c11h15 33 c11h3311 c44h1511 c11e15d33 c11e33d11 c44e15d11 h15 e15 e31
PT
ED
34
2
c11h33 33 c44h15 33 c44h3311 c11e33d33 c44e15d33 c44e33d11 h33 e15 e31
(A.5)
2
AC
43
CE
11 c13 c44 h15 h31 e15 h15 h31 e15 e31 d11 c13 c44 e15 e31 ,
33 c13 c44 h15 h31 e33 h15 h31 e15 e31 d33 c13 c44 e15 e31 ,
44
c44 h33 33 c44e33d33 .
3. Expressions of j ( j 1,
,5) appearing in Eq. (27)
1 c44c33 3333 2c44e33h33d33 c44e332 33 c44c33d 233 c44h332 33 ,
(A.6)
2 c11c33 3333 2c11e33h33d33 c44c331133 c44c33 3311
(A.7)
25
ACCEPTED MANUSCRIPT
2c44c33d11d33 2c44e15e33 33 2c44e33h15d33 2c44e15h33d33 2c44e33h33d11 2c44 h15h33 33 2c1344e33e3115 33 2c1344e33h3115d33 2c1344 h33e3115d33 2c1344 h33h3115 33
2 2 2 c44 d332 e3115 h332 c11e33 33 c11c33d332 c11h332 33 2 2 c44 33 33 c44e332 11 c44 h332 11 c1344 33 33
AN US
2 2 e3115 c33 33 h3115 c33 33 ,
CR IP T
2 2 2 2e3115c33h3115d33 2e3115h33h3115e33 c1344 d332 h3115 e33
2 2 3 c11c44 d332 c11e332 11 c11h332 11 c44 1133 c44 33 11
2 2 2c44 d11d33 c44c33d112 c44e152 33 c44h152 33 c1344 1133 2 2 2 2 c1344 33 11 2c1344 d11d33 e3115 c44 33 e3115 c33 11
M
2 2 2 2 2e3115 h15h33 h3115 c44 33 h3115 c3311 2h3115 e15e33
ED
2c1344e15e3115 33 2c1344e15h3115 d33
2c1344e33e3115 11 2c1344e33h3115d11 2c1344h15e3115d33
PT
2c1344 h15h3115 33 2c1344 h33e3115 d11 (A.8)
CE
2c1344 h33h311511 2e3115c44 h3115 d33
AC
2e3115c33h3115 d11 2e3115 h15h3115e33 2e3115h33h3115e15 c11c44 33 33 c11c331133
c11c33 33 11 2c11c33d11d33 2c11e15e33 33 2c11e15h33d33 2c11e33h33d11 2c11e33h15d33
2c11h15h33 33 c44c331111 2c44e15e33 11 2c44e15h15d33 2c44e15h33d11 2c44e33h15d11 26
ACCEPTED MANUSCRIPT
2c44 h15h3311 ,
4 c11c33d112 c11e152 33 c11h152 33 c44e152 11 c44 h152 11 2 2 2 2 c44 1111 c1344 1111 e3115 c44 11 h3115 c4411
c11c441133 c11c44 33 11 2c11c44d11d33
2c11e33h15d11 2c11e15h33d11 2c11h15h3311 2c44e15h15d11 2c1344e15e3115 11 2c1344e15h3115d11
AN US
2c1344 h15e3115d11 2c1344 h15h311511
CR IP T
c11c331111 2c11e15e33 11 2c11e15h15d33 (A.9)
2 2 2e3115c44 h3115d11 2e3115h15h3115e15 h3115 e152 e3115 h152
2 2 c1344 d112 c44 d112 ,
5 c11 c441111 2e15h15d11 e152 11 c44 d112 h152 11 ,
M
where
ED
c1344 c13 c44 ,
(A.10)
(A.11) (A.12)
h3155 h31 h15 .
(A.13)
PT
e3155 e31 e15 ,
CE
4. Expressions of l m
, K1 , l u, w, , , xx, zz, xz, dx, dz, bx, bz )
( m 1,
appearing in Eq. (36)
AC
u m m
11
w m
21
22
m
31
32
m
41
42
m2
12
m2
23
m2
33
m2
43
m4 ,
(A.14)
13
m4
24
m6 ,
(A.15)
m4
34
m6 ,
(A.16)
m4
44
m6 ,
(A.17)
xx m c11 u m c13 m w m e31 m m h31 m m , 27
(A.18)
ACCEPTED MANUSCRIPT
zz m c13 u m c33 m w m e33 m m h33 m m ,
(A.19)
xz m c44 m u m w m e15 m h15 m ,
(A.20)
dx m e15 m u m w m 11 m d11 m ,
(A.21) (A.22)
bx m h15 m u m w m d11 m 11 m ,
(A.23)
bz m h31 u m h33 m w m d33 m m 33 m m .
(A.24)
5. Expressions
l(C ) n , n
of
and
l( S ) n , n
, K2 , l u, w, , , xx, zz, xz, dx, dz, bx, bz ), which are values of functions
AN US
( n 1,
CR IP T
dz m e31 u m e33 m w m 33 m m d33 m m ,
l(C ) , and l( S ) , at the point, n , n appearing in Eqs. (37) and (38)
u( S ) ,
24
21
12
11
12
3
22
6
w( S ) , 2
3 2
13
2
13
5
2
2
2
31
22
23
4
2
2
32
23
2
2
2
2
4
2 33
24 4
4
10 2 2 5 4 ,
(A.25)
10 2 2 4 ,
(A.26)
4
6 2 2 4
15 15 4
PT
CE
(C ) ,
2
ED
w(C ) ,
11
M
u(C ) ,
6
(A.27)
,
3
4
10 2 2 3 4 ,
6 2 2 4 (A.29)
6 4 2 2 4 6 34 15 15 ,
AC
( S ) , 2
(C ) , 44
41
32
( S ) , 2
6
2
42
33
2
2
2
2
43
34 4
42
2
43
2
2
2
4
2
3
4
10 2 2 3 4 ,
(A.30)
6 2 2 4
15 15 4
(A.28)
44
6
(A.31)
,
3
4
10 2 2 3 4 ,
(A.32)
xx(C ) , c11 u(C ) , c13 w(C ) , w( S ) , e31 (C ) , ( S ) , h31 (C ) , ( S ) , , 28
(A.33)
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xx( S ) , c11 u( S ) , c13 w(C ) , w( S ) , e31 (C ) , ( S ) , h31 (C ) , ( S ) , ,
(A.34)
zz(C ) , c13 u(C ) , c33 w(C ) , w( S ) ,
, , h33 , , ,
e33
(C )
(S )
(C )
(A.35)
(S )
e33
(C )
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zz( S ) , c13 u( S ) , c33 w(C ) , w( S ) ,
, , h33 , , , (S )
(C )
(A.36)
(S )
xz(C ) , c44 u(C ) , u( S ) , w(C ) ,
(A.37)
e15 (C ) , h15(C ) , ,
e15
(S )
, h15 , , (S )
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xz( S ) , c44 u(C ) , u( S ) , w( S ) ,
(A.38)
dx(C ) , e15 u(C ) , u( S ) , w(C ) ,
(A.39)
M
11 (C ) , d11(C ) , ,
11
(S )
ED
dx( S ) , e15 u(C ) , u( S ) , w( S ) , (A.40)
, d11 , , (S )
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dz(C ) , e31 u(C ) , e33 w(C ) , w( S ) ,
33
, , d33 , , , (S )
(C )
(A.41)
(S )
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(C )
AC
dz( S ) , e31 u( S ) , e33 w(C ) , w( S ) ,
33 (C ) , ( S ) , d33 (C ) , ( S ) , ,
(A.42)
bx(C ) , h15 u(C ) , u( S ) , w(C ) , d11
(C )
(A.43)
, 11 , , (C )
bx( S ) , h15 u(C ) , u( S ) , w( S ) , (A.44)
d11 ( S ) , 11( S ) , , 29
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bz(C ) , h31 u(C ) , h33 w(C ) , w( S ) , d33 (C ) , ( S ) , 33 (C ) , ( S ) , ,
(A.45)
bz( S ) , h31 u( S ) , h33 w(C ) , w( S ) , d33
(C )
, , 33 , , . (S )
(C )
(A.46)
(S )
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References
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[1] M. Ahearne, Y. Yang, K.Y. Then, K.K. Liu, An indentation technique to characterize the mechanical and viscoelastic properties of human and porcine corneas, Ann. Biomed. Eng. 35 (2007) 1608-1616. [2] M. Morales, E. Xuriguera, M. Martinez, J.A. Padilla, J. Molera, N. Ferrer, M. Segarra, F. Espiell, Mechanical characterization of copper-copper wires joined by friction welding using instrumented indentation technique, J. Mater. Eng. Perform. 23 (2014) 3941-3948. [3] L. Xiao, D.Y. Ye, C.Y. Chen, A further study on representative models for calculating the residual stress based on the instrumented indentation technique, Comp. Mater.als Sci., 82 (2014) 476-482. [4] F. Dinzart, H. Sabar, Magneto-electro-elastic coated inclusion problem and its
48 (2011) 2393-2401.
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application to magnetic-piezoelectric composite materials, Int. J. Solids Struct.
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[5] P.F. Hou, Y.T.L. Andrew, H.J. Ding, The elliptical Herzian contact of transversely isotropic magnetoelectroelastic bodies, Int. J. Solids Struct. 40
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(2003) 2833-2850.
[6] H.J. Ding, A.M. Jiang, Fundamental solutions for transversely isotropic
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magneto-electro-elastic media and boundary integral formulation, Sci. China Ser. E, 46 (2003) 607-619.
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[7] W.Q. Chen, E. Pan, H.M. Wang, Ch. Zhang, Theory of indentation on multiferroic composite materials, J. Mech. Phys. Solids 58 (2010) 1524-1551.
[8] Y.T. Zhou, K.Y. Lee, Theory of sliding contact for multiferroic materials indented by a rigid punch, Int. J. Mech. Sci. 66 (2013) 156-167. [9] Y.T. Zhou, T.W. Kim, An exact analysis of sliding frictional contact of a rigid punch over the surface of magneto-electro-elastic materials, Acta Mech. 225 (2014) 625-645. 30
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[10] X.Y. Li, R.F. Zheng, W.Q. Chen, Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch, Int. J. Solids Struct. 51 (2014) 164-178. [11] X. Yang, X.J. Liu, Z. Huang, Surface cracks of solid-phase-sintered silicon carbide ceramics and their influences on material strength, J. Inorganic Mater. 29 (2014) 438-442. [12] M. He, F.G. Li, J. Cai, B. Chen, An indentation technique for estimating the
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energy density as fracture toughness with Berkovich indenter for ductile bulk materials, Theor. Appl. Fract. Mech. 56 (2011) 104-111.
[13] B.L. Wang, J.C. Han, S.Y. Du, H.Y. Zhang, Y.G. Sun, Electromechanical behaviour of a finite piezoelectric layer under a flat punch. Int. J. Solids Struct.
AN US
45 (2008) 6384-6398.
[14] B.L. Wang, Fracture and effective properties of finite magnetoelectroelastic media, J. Intel. Mat. Syst. Str. 23 (2012) 1699-1712.
[15] T.H. Hao, Z.Y. Shen, A new electric boundary condition of electric fracture
M
mechanics and its applications, Eng. Fract. Mech. 47 (1994) 793-802. [16] Q. Guan, S.R. He, Two-dimensional analysis of piezoelectric/piezomagnetic and
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elastic media, Compos. Struct. 69 (2005) 229-237. [17] H.J. Ding, B. Chen, J. Liang, On the general solutions for coupled equation for
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piezoelectric media, Int. J. Solids Struct. 33 (1996) 2283-2298. [18] C. Hwu, C.W Fan, Sliding punches with or without friction along the surface of
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an anisotropic elastic half-plane, Q. J. Mech. Appl. Math. 51 (1998) 159-177. [19] M.A. Guler, F. Erdogan, The frictional sliding contact problems of rigid
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parabolic and cylindrical stamps on graded coatings, Int. J. Mech. Sci. 49 (2007) 161-182.
[20] Y.T. Zhou, K.Y. Lee, Contact problem for magneto-electro-elastic half-plane materials indented by a moving punch. Part I: Closed-form solutions, Int. J. Solids Struct. 49 (2012) 3853-3865. [21] Z.F. Song, G.C. Sih, Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation, J. Theor. Appl. Frac. Mech. 39 (2003) 189-207. 31
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[22] S.S. Vel, R.C. Batra, Three-dimensional analytical solution for hybrid multilayered piezoelectric plates, J. Appl. Mech.-T. ASME 67 (2000) 558-567.
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[23] Y.Z. Wang, M. Kuna, Screw dislocation in functionally graded magnetoelectroelastic solid, Philos. Mag. Lett. 94 (2014) 72-79. [24] Y.Z. Wang, M. Kuna, Time-harmonic dynamic Green's functions for two-dimensional functionally graded magnetoelectroelastic materials, J. Appl. Phys. 115 (2014) 043518.
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Figures
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CE
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ED
M
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Fig. 1. Two indenters acting on Solid-phase-sintered Silicon Carbide Ceramic (SSiC) [11].
33
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ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
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Fig. 2. Schematic figure of magnetoelectroelastic materials under two symmetrical, perfectly conducting indenters – two semi-cylindrical indenters as example.
34
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0.4 0.3 0.2 l1 increasing
0.1
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Width of contact region (cm)
0.5
[0.3 0.6 0.9 1.2 1.5 1.8 2.1]*10
0.0 0.00
0.25
0.50 7
P (10 N/m)
0.75
-3
1.00
AC
CE
PT
ED
M
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Fig. 3. Contact width vs. external loading P with R 0.09m .
35
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0
-6 -9
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zz(x,0)(GPa)
-3
l1=0
-12 -15 -0.3
0.0 x(cm)
-3
l1=1.2*10
-3
l1=1.5*10
-3
0.3
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-18 -0.6
l1=0.9*10
0.6
Fig. 4. Effect of the interaction between two semi-cylindrical indenters on the surface normal stress zz ( x,0) with P 107 N m and R 0.09m in which the
AC
CE
PT
ED
M
unit of distance l1 is m.
36
ACCEPTED MANUSCRIPT
2
Dz(x,0)(10 C/m )
0.0
3
-0.3
-0.6
-0.9 -0.6
-0.3
l1=0.9*10
-3
l1=1.2*10
-3
l1=1.5*10
-3
0.0 x(cm)
0.3
Bz(x,0)(10 N/(Am))
0
4
b) l1=0
M
-6
-9 -0.6
-0.3
2
ED
PT
xx(x,0)(10 GPa)
-3
l1=1.2*10
-3
l1=1.5*10
-3
0.3
0.6
0.3
0.6
c) l1=0
2.0
CE
l1=0.9*10
0.0 x(cm)
2.5
AC
0.6
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-3
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a) l1=0
1.5
l1=0.9*10
-3
l1=1.2*10
-3
l1=1.5*10
-3
1.0 0.5 0.0 -0.6
-0.3
0.0 x(cm)
Fig. 5. Influence of the interaction between two semi-cylindrical indenters with P 107 N m and R 0.09m on: a) the surface electric displacement Dz ( x,0) when Q 0.5 C m , b) the surface magnetic induction Bz ( x, 0) when M 60 N A , and c) the surface in-plane stress xx ( x, 0) when Q 0.5 C m and M 60 N A in which the unit of distance l1 is m. 37
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Dz (x,0)(C/m )
0
-2
-3 -1.5 -1.0 -0.5
a) 7 -2 P=0.510 , Q=0.077 10 7 -2 P=210 , Q=0.30810 7 -2 P=610 , Q=0.924 10 7 -2 P=810 , Q=1.232 10
0.0 0.5 x (cm)
Bz (x,0)(10 N/(Am))
0.0
1.0
1.5
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4
-0.5
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2
-1
-1.0
M
-1.5
ED
-2.0 -1.5 -1.0 -0.5
b) 7 P=0.510 , Q=5.4342 7 P=210 , Q=21.7369 7 P=610 , Q=65.2107 7 P=810 , Q=86.9476
0.0 0.5 x (cm)
1.0
1.5
1.2
AC
CE
xx (x,0)(10GPa)
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c) 7 -2 P=0.510 , Q=0.077 10 , M=5.4342 7 -2 P=210 , Q=0.30810 , M=21.7369 7 -2 P=610 , Q=0.924 10 , M=65.2107 7 -2 P=810 , Q=1.232 10 , M=86.9476
0.9 0.6 0.3
0.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 x (cm)
Fig. 6. Effect of the combinations of multi-field loadings when R 0.09m and l1 103 m on: a) the surface electric displacement Dz ( x,0) , b) the surface magnetic induction Bz ( x, 0) , and c) the surface in-plane stress xx ( x, 0) in which the units of P, Q and M are N m , 0.5C m and N A . 38