Journal of Mathematical Sciences, Vol. 162, No. 2, 2009
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA S. A. Kaloerov and A. V. Petrenko
UDC 539.3
We propose a method for the solution of connected two-dimensional and plane problems of electromagnetoelasticity for multiply connected domains. Basic relations of two-dimensional and plane problems are obtained. Generalized complex potentials of electromagnetoelasticity are introduced and investigated. Boundary conditions for their determination and, using them, expressions of main characteristics of the electromagnetoelastic state (stresses, displacements, electromagnetic field intensity vectors, induction vectors, potentials of the electric and magnetic fields) are obtained. We present a solution of the problem for a plate with an elliptic hole or a crack.
Due to the extensive use of piezoelectric and piezomagnetic materials in different fields of engineering in the last few decades, interest in problems on the elastic equilibrium of bodies of such materials under the action of mechanical forces and electromagnetic fields and to the solution of different problems of the engineering practice has increased [1, 2, 12–15]. Elements of real structures of the modern engineering often have holes, foreign inclusions, and cracks. In this connection, it is necessary to develop methods for the investigation of the electromagnetic state of bodies in the case of multiply connected domains. For such bodies, methods for the solution of two-dimensional and plane problems of electroelasticity were proposed in [8, 16], and methods for the solution of magnetoelasticity problems were presented in [6, 7]. In the present work, the indicated approaches are extended to the case of electromagnetoelasticity when, in equations of the electromagnetoelastic state, both electric and magnetic properties of a material are taken into account. 1. Statement of the Problem Let us consider an anisotropic cylindrical body of a piezomaterial weakened with L longitudinal cavities with generators parallel to the axis of the cylinder. Let the body subjected to the action of external mechanical forces and an electromagnetic field be in the two-dimensional electromagnetoelastic state, which does not change along the axis of the cylinder taken as the axis Oz of a rectilinear coordinate system Oxyz . Volume forces, electric charges, and the initial magnetization are absent. Distributed force, electric, and magnetic actions that do not change along the axis Oz are applied to the cylindrical surfaces; concentrated forces, electric charges, and magnetization act along internal lines parallel to the axis Oz . The solution of the problem of determination of the electromagnetoelastic state of the considered body is reduced to the integration of a system of equations [3 – 9, 11, 13, 16] which consists of – equilibrium equations
xy x + = 0, x y
xy x
+
y y
= 0,
yz xz + = 0; x y
(1)
Donetsk National University, Donetsk, Ukraine. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 2, pp. 208–221, April–June, 2008. Original article submitted March 29, 2008. 254
1072-3374/09/1622–0254
© 2009
Springer Science+Business Media, Inc.
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
255
– equations of electromagnetostatics
D y Dx + = 0, x y
E y E x = 0, y x
(2)
B y Bx + = 0, x y
H y H x = 0; y x
(3)
– equations of electromagnetoelastic state DB DB DB DB DB DB x = s11 x + s12 y + s13 z + s14 yz + s15 xz + s16 xy
,D ,D ,D ,B ,B ,B + g11 Dx + g21 D y + g31 Dz + p11 Bx + p21 B y + p31 Bz ,
DB DB DB DB DB DB y = s12 x + s22 y + s23 z + s24 yz + s25 xz + s26 xy
,D ,D ,D ,B ,B ,B + g12 Dx + g22 Dy + g32 Dz + p12 Bx + p22 By + p32 Bz ,
DB DB DB DB DB DB z = s13 x + s23 y + s 33 z + s 34 yz + s 35 xz + s 36 xy
,D ,D ,D ,B ,B ,B + g13 Dx + g23 Dy + g33 Dz + p13 Bx + p23 By + p33 Bz ,
DB DB DB DB DB DB yz = s14 x + s24 y + s 34 z + s 44 yz + s 45 xz + s 46 xy
,D ,D ,D ,B ,B ,B + g14 Dx + g24 Dy + g34 Dz + p14 Bx + p24 By + p34 Bz ,
DB DB DB DB DB DB xz = s15 x + s25 y + s 35 z + s 45 yz + s55 xz + s56 xy
,D ,D ,D ,B ,B ,B + g15 Dx + g25 Dy + g35 Dz + p15 Bx + p25 By + p35 Bz ,
DB DB DB DB DB DB xy = s16 x + s26 y + s 36 z + s 46 yz + s56 xz + s66 xy
,D ,D ,D ,B ,B ,B + g16 Dx + g26 Dy + g36 Dz + p16 Bx + p26 By + p36 Bz ,
256
S. A. KALOEROV
AND
A. V. PETRENKO
,D ,D ,D ,D ,D E x = g11 x g12 y g13 z g14 yz g15 xz
,D g16 xy + 11 Dx + 12 Dy + 13 Dz + 11 Bx + 12 By + 13 Bz ,
,D ,D ,D ,D ,D E y = g21 x g22 y g23 z g24 yz g25 xz
,D g26 xy + 12 Dx + 22 D y + 23 Dz + 12 Bx + 22 B y + 23 Bz ,
,D ,D ,D ,D ,D E z = g31 x g32 y g33 z g34 yz g35 xz
,D g36 xy + 13 Dx + 23 D y + 33 Dz + 13 Bx + 23 B y + 33 Bz ,
,B ,B ,B ,B ,B H x = p11 x p12 y p13 z p14 yz p15 xz
,B p16 xy + 11 Dx + 12 D y + 13 Dz + 11 Bx + 12 B y + 13 Bz ,
,B ,B ,B ,B ,B H y = p21 x p22 y p23 z p24 yz p25 xz
,B p26 xy + 12 Dx + 22 D y + 23 Dz + 12 Bx + 22 B y + 23 Bz ,
,B ,B ,B ,B ,B ,B H z = p31 x p32 y p33 z p34 yz p35 xz p36 xy
+ 13 Dx + 23 D y + 33 Dz + 13 Bx + 23 B y + 33 Bz ;
(4)
– Cauchy relations
x = u , x xz = w , x E z = 0,
y = v , y
xy = u + v , y x Hx =
, x
yz = w , y
z = 0,
Ex =
Hy =
, x
, y
Ey =
Hz = 0 .
, y
(5)
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
257
Here x , y , z , yz , xz , xy and x , y , z , yz , xz , xy are the components of the stress and strain tensor, Dx , Dy , Dz , E x , E y , E z , and are the components of the induction vector, electric field strength vector, and potential of the electric field, Bx , By , Bz , H x , H y , H z , and are the components are the strain of the induction vector, magnetic field strength vector, and potential of the magnetic field, s DB ij coefficients of the material of the body measured at constant inductions of the electric and magnetic fields, ,D gki
and
,B pki
are the piezoelectric and piezomagnetic strain coefficients, and field strength coefficients
measured at constant stress and constant induction, k and k are the dielectric susceptibility coefficients and magnetic susceptibility coefficients measured at constant stress, and k are the electromagnetic susceptibility coefficients measured at constant stress. In this case, Saint-Venant compatibility relations are satisfied:
2 x y
2
+
2 y x
2
2 xy xy
yz
= 0,
x
xz = 0. y
System (1)–(5) must be integrated under the boundary conditions given on the boundary surfaces. For instance, if the mechanical forces and the electric and magnetic inductions are given on the boundary, these conditions have the form
x cos (nx) + xy cos (ny) + xz cos (nz) = X n , xy cos (nx) + y cos (ny) + yz cos (nz) = Yn , xz cos (nx) + yz cos (ny) + z cos (nz) = Z n ,
Dx cos (nx) + D y cos (ny) + Dz cos (nz) = Dn , Bx cos (nx) + B y cos (ny) + Bz cos (nz) = Bn .
(6)
Taking into account that, for the two-dimensional problem, z = 0 , E z = 0 , and H z = 0 , and solving a system which consists of the third, ninth, and twelfth equations of Eqs. (4) for z , Dz , and Bz , we find
(
)
(
)
,B ,D ,B ,D DB DB z = p31 A1 + g31 A2 + s13 A4 x + p32 A1 + g32 A2 + s23 A4 y
(
)
(
)
(
)
(
)
,B ,D ,B ,D DB DB + p34 A1 + g34 A2 + s34 A4 yz + p35 A1 + g35 A2 + s35 A4 xz ,B ,D ,D DB + p36 A1 + g36 A2 + s36 A4 xy + 13 A1 13 A2 + g13 A4 Dx
258
S. A. KALOEROV
(
)
(
)
(
AND
A. V. PETRENKO
)
,D ,B + 23 A1 23 A2 + g23 A4 D y + 13 A1 13 A2 + p13 A4 Bx
,B + 23 A1 23 A2 + p23 A4 B y 1 , D
(
DB Dz = s13 A2 + p
,B 31 A3
)
(
)
,D ,B ,D DB g31 A5 x + s23 A2 + p32 A3 g32 A5 y
(
)
(
(
)
(
)
,B ,D ,B ,D DB DB + s34 A2 + p34 A3 g34 A5 yz + s35 A2 + p35 A3 g35 A5 xz
)
,B ,D ,D DB + s36 A2 + p36 A3 g36 A5 xy + g13 A2 13 A3 + 13 A5 Dx
(
)
(
)
(
)
,D ,B + g23 A2 23 A3 + 23 A5 D y + p13 A2 13 A3 + 13 A5 Bx
,B + p23 A2 23 A3 + 23 A5 B y 1 , D
(
)
(
)
(
)
(
)
(
)
(
,D ,B ,D ,B DB DB Bz = s13 A1 + g31 A3 p31 A6 x + s23 A1 + g32 A3 p32 A6 y ,D ,B ,D ,B DB DB + s34 A1 + g34 A3 p34 A6 yz + s35 A1 + g35 A3 p35 A6 xz
)
,D ,B ,D DB + s36 A1 + g36 A3 p36 A6 xy + g13 A1 13 A3 + 13 A6 Dx
(
)
(
)
(
)
,D ,B + g23 A1 23 A3 + 23 A6 D y + p13 A1 13 A3 + 13 A6 Bx
,B + p23 A1 23 A3 + 23 A6 B y 1 , D
(7)
where ,B ,D ,D ,B DB D = p33 A1 + g33 A2 + s33 A4 = g33 A2 33 A3 + 33 A5 = p33 A1 33 A3 + 33 A6 , ,B ,D A1 = 33 p33 33 g33 , ,B ,D DB s33 + p33 g33 , A3 = 33 ,B 2 DB s33 + ( p33 ) , A5 = 33
,D ,B A2 = 33 g33 33 p33 , 2 A4 = 33 33 (33 ) , ,D 2 DB A6 = 33 s33 + (g33 ) .
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
259
Substituting (7) in the system of equalities (4), we get
x = a11 x + a12 y + a14 yz + a15 xz + a16 xy + b11 Dx + b21 D y + d11 Bx + d 21 B y , y = a12 x + a22 y + a24 yz + a25 xz + a26 xy + b12 Dx + b22 D y + d12 Bx + d 22 B y , yz = a14 x + a24 y + a44 yz + a45 xz + a46 xy + b14 Dx + b24 D y + d14 Bx + d 24 B y , xz = a15 x + a25 y + a45 yz + a55 xz + a56 xy + b15 Dx + b25 D y + d15 Bx + d 25 B y , xy = a16 x + a26 y + a46 yz + a56 xz + a66 xy + b16 Dx + b26 D y + d16 Bx + d 26 B y , E x = b11 x b12 y b14 yz b15 xz b16 xy + c11 Dx + c12 D y + e11 Bx + e12 B y , E y = b21 x b22 y b24 yz b25 xz b26 xy + c12 Dx + c22 D y + e12 Bx + e22 B y , H x = d11 x d12 y d14 yz d15 xz d16 xy + e11 Dx + e12 D y + f11 Bx + f12 B y , H y = d 21 x d 22 y d 24 yz d 25 xz d 26 xy + e12 Dx + e22 D y + f12 Bx + f22 B y , where
(
)
(
)
,D DB DB aij = sijDB si3 p3,B j A1 + g3 j A2 + s j3 A4
(
)
,D DB ,D ,B DB ,B 1 s j3 A2 + p3,B s j3 A1 + g3,D + g3i j A3 g3 j A5 + p3i j A3 p3 j A6 D ,
(
,D ,D bmj = gmj s DB j3 m3 A1 m3 A2 + gm3 A4
(
)
(
)
) (
)
,D ,D gm3 A2 m3 A3 + m3 A5 + p3,B gm3 A1 m3 A3 + m3 A6 1 , + g3,D j j D
,D cnm = nm gn,D 3 m 3 A1 + m 3 A2 g m 3 A4
(
)
(
)
,D ,D + n3 gm3 A2 m3 A3 + m3 A5 + n3 gm3 A1 m3 A3 + m3 A6 1 , D
(
,B ,B d mj = pmj s DB j 3 m 3 A1 m 3 A2 + p m 3 A4
(
)
) (
)
,B ,B + g3,D pm3 A2 m3 A3 + m3 A5 + p3,B pm3 A1 m3 A3 + m3 A6 1 , j j D
(8)
260
S. A. KALOEROV
(
,B enm = nm gn,D 3 m 3 A1 + m 3 A2 p m 3 A4
(
AND
A. V. PETRENKO
)
)
(
)
(
)
,B ,B + n3 pm3 A2 m3 A3 + m3 A5 + n3 pm3 A1 m3 A3 + m3 A6 1 , D
(
,B fnm = nm pn,B 3 m 3 A1 + m 3 A2 p m 3 A4
(
)
)
,B ,B A2 m3 A3 + m3 A5 + n3 pm3 A1 m3 A3 + m3 A6 1 . + n3 pm3 D
From the aforesaid it follows that the two-dimensional problem of electromagnetoelasticity is reduced to the solution of the system of equations which consists of Eqs. (1) – (3), (5), and (8) under conditions given on the boundary, e.g., conditions of type (6). 2. Complex Potentials of the Problem Let us introduce the functions of stresses, the electric induction, and the magnetic induction in such a way that equalities (1) and the first equations of systems (2) and (3) are satisfied identically by setting [5 – 8] 2 x = F , y 2
2 y = F , x 2
2 xy = F , xy
xz = , y
yz = , x
Dx = , y
D y = , x
Bx = , y
B y = . x
(9)
Then, taking into account (8), from relations (5) and the second equations of (2) and (3), we obtain a system of differential equations of the type
L4a F + L3a + L3b + L3d = 0 , L3a F + L2a + L2b + L2d = 0 , L3b F + L2b + L2c + L2e = 0 , L 3d F + L2d + L2e + L2 f = 0 , where
(10)
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
261
4 4 4 4 4 L4a = a22 4 2a26 3 + (2a12 + a66 ) 2 2 2a16 3 + a11 4 , x x y x y xy y
3 3 3 3 L3a = a24 3 + (a25 + a46 ) 2 (a14 + a56 ) 2 + a15 3 , x x y xy y
2 2 2 L2a = a44 2 2a45 + a55 2 , x y x y
3 3 3 3 L3b = b22 3 + (b12 + b26 ) 2 (b21 + b16 ) 2 + b11 3 , x x y xy y
2 2 2 L2b = b24 2 (b14 + b25 ) + b15 2 , x y x y
3 3 3 3 L3d = d 22 3 + (d12 + d 26 ) 2 (d 21 + d16 ) 2 + d11 3 , x x y x y y
2 2 2 L2d = d 24 2 (d14 + d 25 ) + d15 2 , x y x y
2 2 2 L2c = c22 2 + 2c12 c11 2 , x y x y
2 2 2 L2e = e22 2 + 2e12 e11 2 , x y x y
2 2 2 L2 f = f22 2 + 2 f12 f11 2 . x y x y
Solving system (10), we get 5
F(x, y) = 2 Re Fk (z k ), k =1 5
(x, y) = 2 Re k (z k ), k =1
5
(x, y) = 2 Re k (z k ) , k =1 5
(x, y) = 2 Re k (z k ) . k =1
(11)
262
S. A. KALOEROV
AND
A. V. PETRENKO
Here, Fk (z k ) , k (z k ) , k (z k ) , and k (z k ) are arbitrary analytic functions of the generalized complex variables
z k = x + k y = x k + iyk ,
(12)
and k = k + i k , k > 0 , are roots of the characteristic equation 4a ()
3a ()
3b ()
3d ()
3a ()
2a ()
2b ()
2d ()
3b ()
2b ()
2c ()
2e ()
3d ()
2d ()
2e ()
2 f ()
= 0,
(13)
where 4a () = a11 4 2a16 3 + (2a12 + a66 ) 2 2a26 + a22 , 3a () = a15 3 (a14 + a56 ) 2 + (a25 + a46 ) a24 , 2a () = a55 2 2a45 + a44 , 2b () = b15 2 (b14 + b25 ) + b24 , 3b () = b11 3 (b21 + b16 ) 2 + (b12 + b26 ) b22 , 2c () = c11 2 + 2c12 c22 , 3d () = d11 3 (d 21 + d16 ) 2 + (d12 + d 26 ) d 22 , 2d () = d15 2 (d14 + d 25 ) + d 24 , 2e () = e11 2 + 2e12 e22 , 2 f () = f11 2 + 2 f12 f22 .
Since the functions Fk (z k ) , k (z k ) , k (z k ) , and k (z k ) must satisfy the initial system of equations (10), it is easy to understand that bonds of the following type exist between them
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
k (z k ) = k Fk (z k ),
k (z k ) = k Fk (z k ),
263
k (z k ) = k Fk (z k ) ,
(14)
where k , k , and k are constants. Substituting expressions (14) in the equations of system (10) and taking into account (1), we obtain 4 a + k 3a + k 3b + k 3d = 0 , 3a + k 2a + k 2b + k 2d = 0 , 3b + k 2b + k 2c + k 2e = 0 , 3d + k 2d + k 2e + k 2 f = 0,
k = 1, … , 5 .
(15)
From the second, third, and fourth equations of system (15), we find
k =
1k , 0k
k =
2k , 0k
k =
3k , 0k
k = 1, … , 5 ,
(16)
where
0k =
2k =
2a ( k )
2b ( k )
2d ( k )
2b ( k )
2c ( k )
2e ( k ) ,
2d ( k )
2e ( k )
2 f ( k )
2a ( k )
3a ( k ) 2d ( k )
2b ( k )
3b ( k )
2d ( k )
3d ( k ) 2 f ( k )
3a ( k ) 2b ( k ) 2d ( k ) 1k =
2e ( k ) ,
3b ( k ) 2c ( k )
2e ( k ) ,
3d ( k ) 2e ( k ) 2 f ( k )
3k =
2a ( k )
2b ( k )
3a ( k )
2b ( k )
2c ( k )
3b ( k ) .
2d ( k )
2e ( k )
3d ( k )
Substituting (16) in the first equation of system (15), we get 4 a 0 k + 3a 1k + 3b 2 k + 3d 3k = 0 .
The last relation on the basis of (13) is satisfied identically In the consideration of specific cases of anisotropy (when the two-dimensional problem splits into problems of plain and antiplane deformations), the use of 5 , calculated by formula (16), leads to some inconveniences. That is why, by analogy with conclusions made by Lekhnitskii [10] for the two-dimensional problem of elasticity, we determine 5 by solving a system that consists of the first and third equations of (15). Then
5 =
5 , 6
(17)
264
S. A. KALOEROV
AND
A. V. PETRENKO
where
5 =
4 a ( 5 )
3b ( 5 )
3d ( 5 )
3b ( 5 )
2c ( 5 )
2e ( 5 ) ,
3d ( 5 )
2e ( 5 )
2 f ( 5 )
6 =
3a ( 5 )
3b ( 5 )
3d ( 5 )
2b ( 5 )
2c ( 5 )
2e ( 5 ) .
2d ( 5 )
2e ( 5 )
2 f ( 5 )
For practical purposes [5 – 8, 10, 16], it is more convenient to deal with the reciprocal of (17). Denoting the new value again by 5 , we get
5 =
6 . 5
(18)
In connection with the foregoing, hereinafter, by 1 , 2 , 3 , 4 , k , k , k = 1, … , 5 , we mean quantities (16), and by 5 we mean quantity (18). Taking expressions (14) into account, we get 5
F(x, y) = 2 Re Fk (z k ), k =1
4
(x, y) = 2 Re j Fj (z j ) + 1 F5 (z5 ) , 5 j =1
5
5
(x, y) = 2 Re k Fk (z k ),
(x, y) = 2 Re k Fk (z k ) .
k =1
(19)
k =1
Substituting (19) in equalities (9), we get 5
( x , y , yz , xz , xy ) = 2 Re (1k , 2 k , 4 k , 5 k , 6 k ) k (z k ) ,
(20)
k =1
5
(u, v, w, , ) = 2 Re ( pk , qk , s k0 , rk0 , hk0 ) k (z k ) + ( 3 y + u 0 , 3 x + v0 , w0 , 0 , 0 ) ,
(21)
k =1
5
(Dx , Dy ) = 2 Re ( 7 k , 8 k ) k (z k ) ,
(22)
k =1 5
(E x , E y ) = 2 Re (rk0 , k rk0 ) k (z k ) ,
(23)
k =1
5
(Bx , By ) = 2 Re ( 9 k , 10 k ) k (z k ) , k =1
(24)
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
265
5
(H x , H y ) = 2 Re (hk0 , k hk0 ) k (z k ) .
(25)
k =1
Here,
1 j = 2j , 7 j = j j , 25 = 5 ,
2 j = 1, 8 j = j , 45 = 1, 85 = 5 5 ,
4 j = j ,
5 j = j j ,
9 j = j j , 55 = 5 ,
15 = 5 25 ,
10 j = j ,
65 = 5 5 ,
95 = 5 5 5 ,
6 j = j ,
75 = 5 5 5 ,
105 = 5 5 ,
p j = a11 2j a16 j + a12 + (a15 j a14 ) j + (b11 j b21 ) j ,
j = 1, … , 4 ,
q j = a12 j a26 +
a22 a b d + a25 24 j + b12 22 j + d12 22 j , j j
j
j
s 0j = a14 j a46 +
a24 a b d + a45 44 j + b14 24 j + d14 24 j , j j
j
j
rj0 = b11 2j b16 j + b12 + (b15 j b14 ) j (c11 j c12 ) j (e11 j e12 ) j , h 0j = d11 2j d16 j + d12 + (d15 j d14 ) j (e11 j e12 ) j ( f11 j f12 ) j , p5 = (a11 25 a16 5 + a12 ) 5 + a15 5 a14 + (b11 5 b21 ) 5 5 + (d11 5 d 21 ) 5 5 ,
a a b d q5 = a12 5 a26 + 22 5 + a25 24 + b12 22 5 5 + d12 22 5 5 , 5
5 5
5
a a b b s50 = a14 5 a46 + 24 5 + a45 44 + b14 24 5 5 + d14 d 24 5 5 , 5
5 5
5
r50 = (b11 25 b16 5 + b12 ) 5 + b15 5 b14 (c11 5 c12 ) 5 5 (e11 5 e12 ) 5 5 ,
266
S. A. KALOEROV
AND
A. V. PETRENKO
h50 = (d11 25 d16 5 + d12 ) 5 + d15 5 d14 (e11 5 e12 ) 5 5 ( f11 5 f12 ) 5 5 ,
j (z j ) = Fj (z j ),
j = 1, … , 4,
5 (z5 ) =
1 F (z ) , 5 5 5
3 y + u 0 , 3 x + v0 , and w0 are rigid displacements of the body as a whole, 0 and 0 are zero levels of the potentials of the electrostatic and magnetic fields. Using the same procedure as in [5 – 8, 16], we establish that the complex potentials k (z k ) are defined in the domains Sk obtained from the given domain S with affine transforms (12) and can be represented in the form
k (z k ) = k z k +
L
J
Ak ln (zk zk ) + Akj0 ln (zk zkj0 ) +
=1
j =1
k 0 (z k ) ,
(26)
where k are constants that are equal to zero in the case of the finite domain S and determined for an infinite domain from the systems 5
2 Re (1k , 2 k , 4 k , 5 k , 6 k , 7 k , 8 k , 9 k , 10 k , qk k pk ) k k =1
= ( x , y , yz , xz , xy , D x , D y , Bx , By , 2 3 ) ,
(27)
if, at infinity, mechanical forces and inductions are given or 5
2 Re (1k , 2 k , 4 k , 5 k , 6 k , rk0 , k rk0 , hk0 , k hk0 , qk k pk ) k k =1
= ( x , y , yz , xz , xy , E x , E y , H x , H y , 2 3 ) ,
(28)
when, instead of the induction, the electric field strength and magnetic field strength are given, Ak are coefficients satisfying the system 5 Q X Y Z Q 2 Re ( 6 k , 2 k , 4 k , 8 k , 10 k , pk , qk , rk0 , s k0 , hk0 )iAk = , , , e , m , 0, 0, 0, 0, 0 , (29) 2 2 2 2 2 k =1
X , Y , Z , Qe , and Qm are the components of the principal vector of external forces and the total fluxes 0 of electric and magnetic charges acting on the contour L , Akj are coefficients satisfying the system obtained 0 0 0 , X 0 , Y0 , Z 0 , Qe , and Qm from (29) by replacing Akj , X , Y , Z , Qe , and Qm by Akj , re-
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
267
0 0 spectively, X 0 , Y0 , Z 0 , Qe , and Qm are the components of the concentrated force and concentrated
electric and magnetic charges acting along lines that, in the cross-section, correspond to the internal point z 0j of the domain S , and k 0 (z k ) are functions that are holomorphic in the multiply connected domains Sk , in0 that correspond to z 0j at the indicated affine transformations. cluding the points z kj
The boundary conditions for the determination of complex potentials follow from the corresponding boundary conditions, in particular from conditions (6). Using the same procedure as in [5 – 8, 16] for the determination of k (z k ) , we obtain the following boundary conditions: 5
0 2 Re gki k (t k ) = Fi (t ),
i = 1, … , 5 ,
k =1
where, in the case of giving mechanical forces on the boundary 0 (gk1 , gk02 , gk05 ) = ( 6 k , 2 k , 4 k ) ,
s
(F1 , F2 , F5 ) = (X n , Yn , Z n )ds + (c1 , c2 , c5 ) , 0
and in giving displacements 0 (gk1 , gk02 , gk05 ) = ( pk , qk , s k0 ) ,
(F1 , F2 , F5 ) = (u + 3 y u 0 , v 3 x v0 , w w0 ) ; for the electric and magnetic boundary conditions ( i = 3, 4 ) in (30) s
gk0 3 = 8 k ,
F3 = Dn ds + c3 , 0
s
gk0 4 = 10 k ,
F4 = Bn ds + c4 , 0
if, on the boundary, inductions are given, and
gk0 3 = rk0 ,
F3 = (z) + c3 ,
gk0 4 = hk0 ,
F4 = (z) + c4
in the case of giving potentials on the boundary.
(30)
268
S. A. KALOEROV
AND
A. V. PETRENKO
Fig. 1
3. Solution of the Problem for a Body with an Elliptic Hole Let us consider an infinite body with an elliptic hole to which, in the cross-section hole, an ellipse L with semiaxes a and b along the axes Ox and Oy corresponds (Fig. 1). At infinity, a homogeneous electro magnetoelastic state, characterized by the quantities x , y , yz , xz , xy , D x , D y (or E x , E y ),
Bx , By (or H x , H y ), 3 = 0 , is given; on the contour of the hole, mechanical forces, electric, and magnetic actions are absent. In the considered case, the complex potentials (26) have the form
k (z k ) = k z k + k 0 (z k ) ,
(31)
where k are constants determined from system (27) or (28), and k 0 (z k ) are functions holomorphic outside ellipses L k obtained from L with the affine transforms (12). Let us map conformally the exterior of a unit circle k 1 on the exterior of the ellipses L k [8]:
m z k = Rk k + k . k Here,
Rk = 1 (a i k b), 2
mk =
a + i k b . a i k b
Substituting expression (31) in the boundary conditions (30) and using the method of expansion in series, we get
k (z k ) = k z k + (ak1 k Rk m k ) 1 , k ak1 = rk Rk k sk +1 Rk +1 k +1 ek +2 Rk +2 k +2 nk + 3 Rk + 3 k + 3 m k + 4 Rk + 4 k + 4 , where
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
269
sk +1 = 6,k +1 M 6 k + 2,k +1 M 2 k + 8,k +1 M 8 k + 4,k +1 M 4 k + 10,k +1 M 10 k 1 , k ek +2 = 6,k +2 M 6 k + 2,k +2 M 2 k + 8,k +2 M 8 k + 4,k +2 M 4 k + 10,k +2 M 10 k 1 , k nk + 3 = 6,k + 3 M 6 k + 2,k + 3 M 2 k + 8,k + 3 M 8 k + 4,k + 3 M 4 k + 10,k + 3 M 10 k 1 , k m k + 4 = 6,k + 4 M 6 k + 2,k + 4 M 2 k + 8,k + 4 M 8 k + 4,k + 4 M 4 k + 10,k + 4 M 10 k 1 , k rk = 6 k M 6 k + 2 k M 2 k + 8 k M 8 k + 4 k M 4 k + 10 k M 10 k 1 , k
k +1 = k ,
k = 6 k M 6 k + 2 k M 2 k + 8 k M 8 k + 4 k M 4 k + 10 k M 10 k ,
M 2k =
M 4k =
M 6k =
M 8k =
6,k +1
6,k +2
6,k + 3
6,k + 4
8,k +1
8,k +2
8,k + 3
8,k + 4
4,k +1
4,k +2
4,k + 3
4,k + 4
10,k +1
10,k +2
10,k + 3
10,k + 4
6,k +1
6,k +2
6,k + 3
6,k + 4
2,k +1
2,k +2
2,k + 3
2,k + 4
8,k +1
8,k +2
8,k + 3
8,k + 4
10,k +1
10,k +2
10,k + 3
10,k + 4
2,k +1
2,k +2
2,k + 3
2,k + 4
8,k +1
8,k +2
8,k + 3
8,k + 4
4,k +1
4,k +2
4,k + 3
4,k + 4
10,k +1
10,k +2
10,k + 3
10,k + 4
6,k +1
6,k +2
6,k + 3
6,k + 4
2,k +1
2,k +2
2,k + 3
2,k + 4
4,k +1
4,k +2
4,k + 3
4,k + 4
10,k +1
10,k +2
10,k + 3
10,k + 4
,
,
,
,
(32)
270
S. A. KALOEROV
M 10 k =
6,k +1
6,k +2
6,k + 3
6,k + 4
2,k +1
2,k +2
2,k + 3
2,k + 4
8,k +1
8,k +2
8,k + 3
8,k + 4
4,k +1
4,k +2
4,k + 3
4,k + 4
AND
A. V. PETRENKO
,
k is an index taking the values 1, 2, 3, 4, and 5; the value of the index k + j , when it is larger than 5, is formally set equal to k + j 5 . Passing to the variable z k and differentiating the obtained function, we have k (z k ) = ±
d k1 z k z k2 4 Rk2 m k
d k1 + k ,
d k1 =
k Rk m k ak1 . 2Rk m k
If the semiaxis b = 0 (the body with a crack of length 2a ), then m k = 1 , Rk = a / 2 . In the vicinity of the ends of the notch, z k = ± (a + z k ) , where z k is a small value. Then
k (z k ) = ±
Mk 2 2z k
+ O(1) ,
(33)
M k = ( k + rk k + sk +1 k +1 + ek +2 k +2 + nk + 3 k + 3 + m k + 4 k + 4 ) a ,
O(1) is a bounded quantity. Substituting (32) in (31) and collecting combinations of k in the obtained expression according to formulas (27) and (28), we have
1 M k = M 2 k y + M 6 k xy + M 4 k yz + M 8 k D y + M 10 k By a . k Analogously, we find
1 EH EH EH EH M kEH = M 2EH k y + M 6 k xy + M 4 k yz + M 8 k D y + M 10 k By a . k Replacing y a , xy a , yz a , D y a , E y a , By a , and H y a by the intensity coefficients of
normal fracture k1 , tangent shear k2 , longitudinal shear k 3 , electric induction k D , magnetic induction
k B , electric field strength k E , and magnetic field strength k H , we obtain M k = [ M 2 k k1 + M 6 k k2 + M 4 k k 3 + M 8 k k D + M 10 k k B ] 1 , k
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
271
EH EH EH EH M kEH = M 2EH k k1 + M 6 k k 2 + M 4 k k 3 + M 8 k k E + M 10 k k H
1 .
EH k
On the basis of formulas (20) – (25), and (33), for the main characteristics of the electromagnetoelastic state in the vicinity of the ends of the notch, we find 5
( x , y , yz , xz , xy ) =
1 Re ( , , , , ) 1k 2 k 4 k 5 k 6 k 2r k =1 5
1 Re ( , ) 7k 8k 2r k =1
(Dx , Dy ) =
1 Re ( r 0 , r 0 ) k kk 2r k =1
cos + k sin
cos + k sin
Mk cos + k sin
1 Re ( h 0 , h 0 ) k k k 2r k =1
,
,
M kEH
5
(H x , H y ) =
,
,
M kEH
5
1 Re ( , ) 9 k 10 k 2r k =1
(Bx , By ) =
cos + k sin
Mk
5
(E x , E y ) =
Mk
cos + k sin
,
5
(u, v, w, , ) =
2r Re ( pk , qk , s k0 , rk0 , hk0 ) M k cos + k sin . k =1
In this case, the following equalities take place: 5
(k1, k2 , k 3 ) = 2 Re ( 2 k , 6 k , 4 k )M k =
a ( y , xy , yz ) ,
k =1
5
k D = Re 8 k M k = k =1
5
k B = Re 10 k M k = k =1
5
a Dy ,
k E = Re rk M kEH =
a By ,
k H = Re hk M kEH =
k =1 5
k =1
a E y ,
a H y .
For a plate with a circular hole in its unilateral tension or under the action of a homogeneous electric field with a given strength, numerical investigations of the stress distribution, field strength, and induction were performed. As a material of the plate, we considered a BaTiO 3 CoFe 2 O 4 alloy polarized in the direction of the axis Oy . For it [17], we have
272
S. A. KALOEROV
AND
EH c11 = 2.26 ,
EH c22 = 2.16 ,
EH c33 = 2.26 ,
EH c44 = 0.44 ,
EH c55 = 0.55 ,
EH c66 = 0.44 ,
EH c12 = 1.24 ,
EH c13 = 1.25 ,
,E e12 = 2.2 ,
,E e22 = 9.3 ,
EH c23 = 1.25 ,
,E e16 = 5.8 ,
,E e32 = 2.2 ,
,E e34 = 5.8 ,
,H f22 = 350 ,
,H f 32 = 290.2 ,
22 = 6.35 , 33 = 2.97 ,
33 = 5.64 , 11 = 5.367 ,
= 10 3 MPa , = 10 9 C/(N ·m 2 ) ,
,H f16 = 275 , ,H f 34 = 275 ; 11 = 2.97 ,
22 = 2737.5 ,
= 1 C/m 2 , = 10 4 H/m ,
A. V. PETRENKO
,H f12 = 290.2 , 11 = 5.64 ,
22 = 0.835 , 33 = 5.367 ,
= 1 Wb/m 2 , = 10 12 N ·s / (Wb ·C) .
To find the coefficients of state equation (4), we used formulas of recalculation [5]. Numerical investigations were carried out for four cases, namely in the solution of the problem of the theory of elasticity (in the state equations, the “electromagnetic constants” were dropped with the corresponding changes in other equations), problem of electroelasticity (“magnetic constants” were discarded), problem of magnetoelasticity (“electric constants” were discarded), and problem of electromagnetoelasticity (in the equations, all constants were retained). The results of these investigations show that, under the action of mechanical forces, in the plate, not only stresses but also an electromagnetic field (though its strength is insignificant) arises, which cannot be neglected in the investigation of the electromagnetoelastic state. Under the action of a field of constant strength, the developed stresses are substantial and cannot be disregarded in the investigation of the electromagnetoelastic state. These stresses can be found only by solving the general problem of electromagnetoelasticity, which is given in the present work. REFERENCES 1. Ya. Yo. Burak, A. R. Gachkevych, and R. F. Terlets’kyi, Thermomechanics of Multicomponent Bodies of Low Electroconductivity [in Ukrainian], in Vol. 5: Modeling and Optimization in Thermomechanics of Inhomogeneous Bodies, Spolom, Lviv (2006). 2. A. R. Gachkevich, Thermomechanics of Electroconductive Bodies under the Action of Quasi-Steady Electromagnetic Fields [in Russian], Naukova Dumka, Kiev (1992). 3. V. T. Grinchenko, A. F. Ulitko, and N. A. Shul’ga, Electroelasticity [in Russian], Vol. 5: Mechanics of Connected Fields in Structural Elements, Naukova Dumka, Kiev (1989). 4. I. S. Zheludev, Physics of Crystalline Dielectrics [in Russian], Nauka, Moscow (1968). 5. S. A. Kaloerov, A. I. Baeva, and O. I. Boronenko, Two-Dimensional Problems of Electro- and Magnetoelasticity for Multiply Connected Media [in Russian], Yugo-Vostok, Donetsk (2007). 6. S. A. Kaloerov and O. I. Boronenko, “A two-dimensional problem of magnetoelasticity for a multiply connected body,” Prikl. Mekh., 41, No. 10, 64–74 (2005).
TWO-DIMENSIONAL PROBLEM OF ELECTROMAGNETOELASTICITY FOR MULTIPLY CONNECTED MEDIA
273
7. S. A. Kaloerov and O. I. Boronenko, “A two-dimensional and a plane problem for a piezomagnetic body with holes and cracks,” Teor. Prikl. Mekh., Issue 41, 111–123 (2005). 8. S. A. Kaloerov and E. S. Goryanskaya, “Two-dimensional stress-strain state of a multiply connected anisotropic body,” in: A. N. Guz’ (editor), Mechanics of Composites [in Russian], Vol. 7: Stress Concentration, A.S.K., Kiev (1998), pp. 10–26. 9. L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982). 10. S. G. Lekhnitskii, Theory of Elasticity of Anisotropic Bodies [in Russian], Nauka, Moscow (1977). 11. N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Nauka, Moscow (1966); English translation: Springer, New York (1977). 12. Ya. S. Pidstryhach, Ya. Yo. Burak, and V. F. Kondrat, Magnetoelasticity of Electroconductive Bodies [in Russian], Naukova Dumka, Kiev (1982). 13. L. P. Khoroshun, B. P. Maslov, and P. V. Lyashenko, Prediction of Efficient Piezoactive Composite Materials [in Russian], Naukova Dumka, Kiev (1989). 14. N. A. Shul’ga, “Efficient physicomechanical properties of thin-layered piezoelectric and piezomagnetic materials,” Soprot. Mater. Teor. Sooruzh., Issue 48, 43–45 (1986). 15. D. A. Berlincourt, D. R. Curran, and H. Jaffe, “Piezoelectric and piezomagnetic materials and their function in transducers,” Phys. Acoust., No. 1, 169–270 (1964). 16. S. A. Kaloerov, A. I. Baeva, and Yu. A. Glushchenko, “Two-dimensional electroelastic problem for a multiply connected piezoelectric body,” Int. Appl. Mech., 39, No. 1, 77–84 (2003). 17. M. H. Zhao, H. Wang, F. Yang, and T. A. Liu, “A magnetoelectroelastic medium with an elliptical cavity under combined mechanical–electric–magnetic loading,” Theor. Appl. Fract. Mech., 45, 227–237 (2006).
