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Thermal-induced interfacial cracking of magnetoelectroelastic materials
International Journal of Engineering Science 42 (2004) 1347–1360 www.elsevier.com/locate/ijengsci
Thermal-induced interfacial cracking of magnetoelectroelastic materials Cun-Fa Gao *, Naotake Noda Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, Shizuoka 432-8561, Japan Received 26 January 2004; received in revised form 19 February 2004; accepted 18 March 2004
Abstract In this paper, we present an explicitly analytic solution for a generalized two-dimensional problem of an interface crack between two dissimilar magnetoelectroelastic materials under uniform heat flow. The crack is assumed to be electrically permeable and thus the thermal-induced electric–magnetic fields within the crack need to be determined as part of the solution. According to the extended Stroh formalism, the thermal potential functions and the electric–magnetic–elastic potential functions are at first presented in concise form. Then, the thermal-induced electric–magnetic fields within the crack and the stress intensity factor at the crack tip are obtained, respectively, in closed form. It is shown that when uniform heat flow is applied at infinity, the thermal-induced electric–magnetic fields within the crack are very complicated, especially near the crack tip for general cases. In addition, the stress fields at the crack tip may be singular and oscillatory, but the structure of singularities is the same as that in a purely elastic bi-material system with interface cracks, that is, it is uniquely characterized by an inverse square root singularity and a pair of oscillatory singularities. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Magnetoelectroelastic solid; Interface crack; Thermal loading; Stress intensity factor
1. Introduction Piezoelectric and piezomagnetic composites have wide application in smart devices, particularly in the aerospace and automotive industries. Since such materials simultaneously possess piezoelectric, piezomagnetic and high magnetoelectric coupling effects, they are very sensitive to elastic, *
Corresponding author. Fax: +81-53-474-7499. E-mail address: [email protected] (C.-F. Gao).
0020-7225/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.03.005
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electric, magnetic and thermal fields in environment, and solid defects, e.g., cavities, cracks or inclusions where the local fields greatly concentrate. The locally enhanced fields may induce crack initiation, crack growth and finally lead to fracture or failure of devices. Due to the importance of the reliability of these devices, fracture and failure analyses of magnetoelectroelastic solids with holes or cracks have received considerable interest in the recent decade. Chung and Ting [1], Liu et al. [2] and Gao et al. [3] derived a 2D solution of an elliptical cavity or a crack in an infinite magnetoelectroelastic solid, respectively. Wang and Shen [4] and Pan [5] solved a 3D problem for an infinite magnetoelectroelastic body and a bi-material system, respectively. Li [6] and Huang et al. [7] studied magnetoelectroelastic inclusion problems in piezoelectric–piezomagnetic composites, respectively. Recently, Gao et al. [8,9] obtained several analytical results for multiple crack problems in magnetoelectroelastic materials. Relatively, little efforts have been dedicated to fracture and failure of magnetoelectroelastic media under thermal loading. Liu et al. [10] derived the complex potentials of an elliptical cavity in an electromagnetic thermoelastic solid. Gao et al. [11] made an explicit treatment on a series of collinear permeable crack in a homogeneous magnetoelectroelastic material. Eringen [12] established a continuum theory of micromorphic electromagnetic thermoelastic solids. To the best of our knowledge, however, the interface crack problem in magnetoelectroelastic materials under thermal loading remains unsolved, though this problem is also important with increasingly wide application of piezoelectric and piezomagnetic composites in high environment. It is therefore the purpose of this work to analyse a generalized 2D problem of an interface crack in a magnetoelectroelastic bi-material system under uniform heat flow at infinity. On the interfacial cracks in purely elastic anisotropic media, great developments have been made by Ting in a series of his works, e.g., in [13–15], and these developments were partly summarized in [16]. When the anisotropic media are subjected to piecewise uniform traction at infinity, explicit solutions for the entire field and closed-form expression of the oscillatory parameter can be found in [13–16]. In the present work, we shall extend Ting’s work [13,14] to the case of an interfacial crack in magnetoelectroelastic media, where the crack is assumed to be permeable. Thus, this means that the thermal-induced electromagneto fields inside the crack have to be determined simultaneously as part of the solution. To do this, we develop a concise and explicit method to reduce the present problem to an equivalent one to that in purely elastic anisotropic media, and then give the final solution of an interfacial crack in a magnetoelectroelastic bi-material system under uniform heat flow at infinity. Below is the plan of this work: Section 2 outlines the extended Stroh formalism for a magnetoelectroelastic solid under thermal loading. In Section 3, we introduce the thermal complex potentials, and then the electric–magnetic–elastic potential functions are presented in Section 4 in explicit form. In Section 5 we establish a simple relationship between the induced electric–magnetic fields inside the crack and the applied thermal loading at infinity in order to discuss the distribution of electric–magnetic fields along the crack faces. Section 6 gives an exact expression for the thermalinduced stress intensity factor at the crack tip, and finally Section 7 concludes the work.
2. Basic equations In a fixed rectangular coordinate system xi ði ¼ 1; 2; 3Þ, the governing equations of a linear magnetoelectroelastic solid subjected to thermal loading can be expressed as
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1349
rij ¼ cijkl ckl ekij Ek akij Hk bij T ; Di ¼ eikl ckl þ eik Ek þ lik Hk þ si T ;
ð1Þ
Bi ¼ aikl ckl þ lik Ek þ tik Hk þ qi T ; cij ¼ 12ðui;j þ uj;i Þ; rij;i ¼ 0;
Ei ¼ u;i ;
Di;i ¼ 0;
qi ¼ kij T;j ;
Hi ¼ w;i ;
Bi;i ¼ 0;
qi;i ¼ 0;
ð2Þ ð3Þ ð4Þ
where a comma denotes partial differentiation; rij , Di , Bi and qi are the stress, the electric displacement, the magnetic induction and heat flow, respectively. cij , Ek and Hk represent the strain, the electric field and magnetic field, respectively. ui , u, w and T denote the displacement, the electric potential, the magnetic potential and temperature change, respectively. kij are the heat conduction coefficients. cijkl , ekij , akij lik , eik and tik are the elastic stiffness tensor, piezoelectric, piezomagnetic magnetoelectric coupling tensor, the dielectric permittivities and the magnetic permeabilities, respectively; and bij , si and qi are the thermal stress constants, pyroelectric coefficients and pyromagnetic coefficients, respectively. Substituting (1) together with (2) into (3), and then substituting the first equation in (4) into the second one in (4), one has ðcijks uk þ esij u þ asij wÞ;si bij T;i ¼ 0; ðeiks uk eis u lis wÞ;si þ si T;i ¼ 0;
ð5Þ
ðaiks uk lis u tis wÞ;si þ qi T;i ¼ 0; kij T;ij ¼ 0:
ð6Þ
Consider a generalized 2D problem of a magnetoelectroelastic solid with geometry and loading independent of x3 . In this case, (6) becomes k11
o2 T o2 T o2 T þ 2k þ k ¼ 0: 12 22 ox1 x2 ox21 ox22
ð7Þ
The general solution of (6) is T ¼ 2Re½g0 ðzÞ;
z ¼ x1 þ pt x2 ;
ð8Þ
where Re means taking the real part; g0 ðzÞ is a complex function to be determined; the prime (0 ) indicates differentiation with respect to its argument, and pt is the heat eigenvalue which is determined from k22 pt2 þ 2k12 pt þ k11 ¼ 0:
ð9Þ
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The solution of pt with positive imaginary part is pt ¼ ðk12 þ ijÞ=k22 ;
ð10Þ
where i¼
pffiffiffiffiffiffiffi 1;
j ¼ ðk11 k22 k212 Þ1=2 ;
k11 k22 k212 > 0:
Inserting (8) into the first equation in (4) leads to q1 ¼ 2Re½ipt jg00 ðzÞ;
q2 ¼ 2Re½ijg00 ðzÞ:
ð11Þ
Introduce a generalized displacement function vector u and a generalized stress function vector / as T
T
/ ¼ ð/1 ; /2 ; /3 ; /4 ; /5 Þ :
u ¼ ðu1 ; u2 ; u3 ; u; wÞ ;
Then, (3) is automatically satisfied if one takes ðr11 ; r12 ; r13 ; D1 ; B1 ÞT ¼ /;2 ;
ðr21 ; r22 ; r23 ; D2 ; B2 ÞT ¼ /;1 :
ð12Þ
Thus, after T is determined from (8), u and / can be determined from (5). In general, the solution of (5) consists of two parts, one being a homogeneous solution of (5) corresponding to the isothermal case, and another being a particular solution of (5). The homogeneous solution has the following form: uh ¼ 2Re½AfðzÞ;
/h ¼ 2Re½BfðzÞ;
ð13Þ
with fðzÞ ¼ ½f1 ðz1 Þ; f2 ðz2 Þ; f3 ðz3 Þ; f4 ðz4 Þ; f5 ðz5 ÞT ; A ¼ ða1 ; a2 ; a3 ; a4 ; a5 Þ;
za ¼ x1 þ pa x2
ða ¼ 1 5Þ;
B ¼ ðb1 ; b2 ; b3 ; b4 ; b5 Þ;
where fa ðza Þ are complex potentials to be found; pa are the complex eigenvalues with positive imaginary parts. The pa and corresponding eigenvectors aa and ba can be obtained from the following equations: jS þ pðR þ RT Þ þ p2 Tj ¼ 0; bS þ pa ðR þ RT Þ þ pa2 Tcaa ¼ 0; ba ¼ ðRT þ pa TÞaa : The matrices S, R and T are related to the material constants by
C.-F. Gao, N. Noda / International Journal of Engineering Science 42 (2004) 1347–1360
2
c1jk1 4 S ¼ eT1k1 aT1k1
e1j1 e11 l11
3
a1j1 l11 5; t11
2
c1jk2 4 R ¼ eT1k2 aT1k2
e2j1 e12 l12
3
a2j1 l12 5; t12
2
c2jk2 4 T ¼ eT2k2 aT1k1
e2j2 e22 l22
1351
3
a2j2 l22 5: t22
On the other hand, the particular solution of (5) has the form up ¼ 2Re½cgðzÞ;
/p ¼ 2Re½dgðzÞ;
ð14Þ
where c and d are two constant vectors, and they can be determined from the following equations: bS þ pt ðR þ RT Þ þ pt2 Tcc ¼ b1 þ pt b2 ; d ¼ ðRT þ pt TÞc b1 ¼
1 ½ðS þ pt RÞc þ b2 ; pt
with b1 ¼ ðb11 ; b12 ; b13 ; s1 ; q1 ÞT ;
b2 ¼ ðb21 ; b22 ; b23 ; s2 ; q2 ÞT :
Finally, (13) plus (14) leads to the general solution of u and / as u ¼ 2Re½AfðzÞ þ cgðzÞ;
ð15Þ
/ ¼ 2Re½BfðzÞ þ dgðzÞ:
ð16Þ
From (15) and (16) we have u;1 ¼ 2Re½Af 0 ðzÞ þ cg0 ðzÞ;
ð17Þ
/;1 ¼ 2Re½Bf 0 ðzÞ þ dg0 ðzÞ;
ð18Þ
where u;1 ¼ ðu1;1 ; u2;1 ; u3;1 ; E1 ; H1 ÞT ;
ðr21 ; r22 ; r23 ; D2 ; B2 ÞT ¼ /;1 :
ð19Þ
For convenience, we introduce a new vector function as UðzÞ ¼ Bf 0 ðzÞ þ dg0 ðzÞ:
ð20Þ
Then, (17) and (18) can be rewritten as /;1 ¼ UðzÞ þ UðzÞ;
ð21Þ
u;1 ¼ 2Im½YUðzÞ Mg0 ðzÞ;
ð22Þ
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where Y ¼ iAB1 ;
M ¼ Yd ic:
3. Thermal potentials Consider an electrically permeable interface crack Lc between two dissimilar magnetoelectroelastic media, as shown in Fig. 1. Assume that the media are subjected to uniform heat flow q1 2 at infinity, while the crack is free of force, external charge, electric current and heat flow, but filled with air. In this case, the thermal boundary condition on the crack can be expressed as qþ 2 ðx1 Þ ¼ q2 ðx1 Þ ¼ 0;
x1 2 Lc :
ð23Þ
The thermal complex function g0 ðzt Þ takes form of g0 ðzÞ ¼ ct z þ g00 ðzÞ;
ð24Þ
where g00 ðzÞ is a holomorphic function except on Lc , ct is a known constant depending on the heat flow at infinity. The g00 ðzÞ can be solely solved only by considering the thermal boundary condition. The results are listed as follows 0 ðzÞ ¼ g10
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 q2 ðz z2 a2 Þ; 2ij1
z 2 sþ ;
ð25Þ
0 ðzÞ ¼ g20
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 q2 ðz z2 a2 Þ; 2ij2
z 2 s ;
ð26Þ
8
q2
S1
x2
Lc
Lb 2a
x1
S2
8
q2
Fig. 1. An interface crack in a magnetoelectroelastic bi-material under uniform heat flow.
C.-F. Gao, N. Noda / International Journal of Engineering Science 42 (2004) 1347–1360
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and 0 0 j1 g10 ðzÞ þ j2 g20 ðzÞ ¼ 0;
z 2 sþ ;
ð27Þ
0 0 ðzÞ þ j1 g10 ðzÞ ¼ 0; j2 g20
z 2 s :
ð28Þ
4. Electromagnetoelastic potentials On the crack faces, we have the following boundary conditions [17]: rþ 2j ¼ 0;
r 2j ¼ 0
ðj ¼ 1–3Þ
ð29Þ
on Lc ;
Dþ 2 ðx1 Þ ¼ D2 ðx1 Þ;
Bþ 2 ðx1 Þ ¼ B2 ðx1 Þ;
E1þ ðx1 Þ ¼ E1 ðx1 Þ;
H1þ ðx1 Þ ¼ H1 ðx1 Þ;
1 < x1 < þ1;
ð30aÞ
1 < x1 < þ1;
ð30bÞ
where E1 and H1 are the components of electric field and magnetic field in the x1 direction, respectively. Taking into account the conditions of zero stresses and electric fields at infinity, the complex potential f 0 ðzÞ has the form f 0 ðzÞ ¼ f 00 ðzÞ;
ð31Þ
where f 00 ðzÞ stands for a holomorphic function vector, and f 00 ð1Þ ¼ 0. Substituting (24) and (31) into (21) and (22), and then considering the condition that both stress and strain are bounded at infinity, we have /;1 ¼ U0 ðzÞ þ U0 ðzÞ;
ð32Þ
u;1 ¼ 2Im½YU0 ðzÞ Mg00 ðzÞ;
ð33Þ
where U0 ðzÞ ¼ Bf 00 ðzÞ þ dg00 ðzÞ: Obviously, U0 ðzÞ is holomorphic in the z-plane except on Lc , and U0 ð1Þ ¼ 0. The continuity condition of the generalized tractions on x1 requires /þ ;1 ðx1 Þ ¼ /;1 ðx1 Þ;
1 < x1 < þ1:
ð34Þ
Substituting (32) into (34) produces þ
½U10 ðx1 Þ U20 ðx1 Þ ½U20 ðx1 Þ U10 ðx1 Þ ¼ 0; The solution of (35) is [18]
1 < x1 < þ1:
ð35Þ
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U10 ðzÞ U20 ðzÞ ¼ 0;
z 2 s1 ;
ð36Þ
U20 ðzÞ U10 ðzÞ ¼ 0;
zs 2 s2 :
ð37Þ
Introduce a jump function Du;1 ðx1 Þ defined as Du;1 ðx1 Þ ¼ bu;1 ðx1 Þc1 bu;1 ðx1 Þc2 :
ð38Þ
Inserting (33) into (38) and then using (27) and (28) we obtain iDu;1 ðx1 Þ ¼ xþ ðx1 Þ x ðx1 Þ;
ð39Þ
where 8 0 < HU10 ðzÞ M1 j1 M2 g10 ðzÞ; j2 xðzÞ ¼ : HU20 ðzÞ M2 j2 M1 g0 ðzÞ; 20 j1
z 2 s1 ; z 2 s2 ;
ð40Þ
H ¼ Y1 þ Y2 : Since Du;1 ðx1 Þ ¼ 0 on Lb , Eqs. (39) and (40) show that xðzÞ is analytic in the z-plane except on the crack, and moreover xð1Þ ¼ 0. In addition, the value-singled condition of generalized displacements requires I Du;1 ðx1 Þ dx1 ¼ 0; ð41Þ C
where C is a clockwise contour enclosed the crack. Inserting (39) into (41) leads to I ½xþ ðx1 Þ x ðx1 Þ dx1 ¼ 0:
ð42Þ
C
On the whole x1 axis, the continuous conditions of E1 ðx1 Þ and H1 ðx1 Þ give bDu;1 ðx1 ÞcJ ¼ 0;
J ¼ 4; 5; 1 < x1 < 1:
ð43Þ
Substituting (39) into (43) leads to xþ J ðx1 Þ xJ ðx1 Þ ¼ 0;
J ¼ 4; 5; 1 < x1 < 1:
ð44Þ
The solution of (44) is [18] xJ ðzÞ ¼ 0;
J ¼ 4; 5:
ð45Þ
Based on (45), the unknown function xðzÞ defined in (40) takes the form xðzÞ ¼ ½x1 ðzÞ; x2 ðzÞ; x3 ðzÞ; 0; 0T :
ð46Þ
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On the other hand, one has by using (32) that /þ ;1 ðx1 Þ ¼ U10 ðx1 Þ þ U10 ðx1 Þ:
ð47Þ
Using (37), (47) becomes /þ ;1 ðx1 Þ ¼ U10 ðx1 Þ þ U20 ðx1 Þ:
ð48Þ
Eq. (48) can be rewritten as 1
1 /þ ;1 ðx1 Þ ¼ H ½HU10 ðx1 Þ þ H ½HU20 ðx1 Þ:
ð49Þ
Using (40), (49) can be rewritten as /þ ;1 ðx1 Þ
1
þ
1
¼ H x ðx1 Þ þ H x ðx1 Þ þ H
j1 1 0 M2 M1 g20 ðx1 Þ: þH j2
1
j1 0 M1 M2 g10 ðx1 Þ j2 ð50Þ
Substituting (25) and (26) into (50) yields 1
1 þ 0 1 00 1 /þ ;1 ðx1 Þ ¼ H x ðx1 Þ þ H x ðx1 Þ þ m q2 x1 þ m q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
where m0 and m00 are two material constants:
1 M2 M2 1 M1 1 M1 0 H ; H m ¼ 2i j1 j2 j1 j2
1 M2 M2 1 M1 1 M1 00 H m ¼ þH ; 2i j1 j2 j1 j2
ð51Þ
ð52Þ
ð53Þ
which shows that m0 is real and m00 is purely imaginary. For a homogeneous material, we have from (52) and (53) that m0 ¼
1 1 H ðM MÞ; ij
m00 ¼ 0:
ð54Þ
Let
H
1
h ¼ 33 h23
h32 ; h22
0
m ¼
m031 ; m021
00
m ¼
m0031 ; m0021
x3 ¼ ðx1 ; x2 ; x3 ÞT :
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Then, taking the first three rows and final two rows of (51), respectively, produces 0 1 00 1 ðr21 ; r22 ; r23 ÞT ¼ h33 xþ 3 ðx1 Þ þ h33 x3 ðx1 Þ þ m31 q2 x1 þ m31 q2
0 1 00 1 ðD2 ; B2 ÞT ¼ h23 xþ 3 ðx1 Þ þ h23 x3 ðx1 Þ þ m21 q2 x1 þ m21 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
ð55Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 :
ð56Þ
T
On the crack face, we have ðr21 ; r22 ; r23 Þ ¼ 0. Thus, (55) gives 0 1 00 1 h33 xþ 3 ðx1 Þ þ h33 x3 ðx1 Þ ¼ m31 q2 x1 m31 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
x1 2 Lc :
ð56Þ
Solving (56) can give the solution of x3 ðzÞ, and then the unknown function xðzÞ defined in (46) can completely be obtained. Finally, U10 ðzÞ and U20 ðzÞ can be determined from (40), and thus the problem under consideration is solved. Eq. (56) can be rewritten as 1 1 1 0 1 00 1 xþ 3 ðx1 Þ þ h33 h33 x3 ðx1 Þ ¼ h33 m31 q2 x1 h33 m31 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
x1 2 Lc :
ð57Þ
Letting Q be the eigenvector matrix h1 33 h33 , one has 2pe 2pe ; e ; 1Þ ¼ hhe2pea ii; Q1 h1 33 h33 Q ¼ diagðe
where the angular bracket hh ii stands for a diagonal matrix, and the positive and real constant e depends on the material properties and characterizes the oscillatory singularities. The e can be determined by h33 e2pea h1 33 ¼ 0; from which one can obtain an explicit expression for the ea following the work of Ting [16]. The result is ea ¼ ðe; e; 0Þ; where 1 1 1þb e ¼ tan1 b ¼ ln þ ; p 2p 1b
06b ¼
1 ^ 2 trðSÞ 2
1=2 ;
^¼D ^ 1 W; ^ S
^ iW ^ ¼ ^h33 : D
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The general solution of (57) can be expressed as [18] pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 0 1 00 1 z2 a2 QÞ ½m q z þ m q ðh Q x3 ðzÞ ¼ 33 31 2 31 2 1 þ e2pea 2 z Xa ðzÞ Xa ðzÞ 1 0 00 1 þ ðh33 QÞ ðm31 þ m31 Þq2 þ ðh33 QÞ1 1 þ e2pea 1 þ e2pea ðc1 z þ c0 Þ;
ð58Þ
where c1 and c0 are two constant vectors to be found, and
ie 1 za a ; a ¼ 1; 2; 3: Xa ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 a2 z þ a To find c1 and c0 , one can use (42) and the condition x3 ð1Þ ¼ 0:
ð59Þ
Noting (45), (42) becomes I ½xþ 3 ðx1 Þ x3 ðx1 Þ dx1 ¼ 0:
ð60Þ
C
The use of (59) and (58) gives c1 ¼ hh2iea iiðm031 þ m0031 Þaq1 2 :
ð61Þ
After c1 is obtained, c0 can be determined by inserting (58) into (60). The result is 2 1 1 00 c0 ¼ hhae2a 4e2a iiðm031 þ m0031 Þaq1 2D 2m31 a q2D :
ð62Þ
Substituting (61) and (62) into (58) one can give the final expression of xðzÞ. Up to here we have completed the task for determining the electromagnetoelastic potentials. 5. Thermal-induced electric–magnetic field inside the crack On the crack surface, one finds from (56) that þ 0 1 00 1 1 x 3 ðx1 Þ ¼ h33 bh33 x3 ðx1 Þ þ m31 q2 x1 þ m31 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 c;
x 1 2 Lc :
ð63Þ
Substituting (63) into (56) results in T
1 1 ðD2 ; B2 Þ ¼ g1 xþ 3 ðx1 Þ þ g2 q2 x1 þ g3 q2
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
x 1 2 Lc ;
ð64Þ
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g1 ¼ h23 h23 h1 33 h33 ; 0 g2 ¼ m h23 h1 m0 ; g3 ¼
21 m0021
ð65Þ
33 31 00 h23 h1 33 m31 :
Eq. (64) indicates that the distribution of thermal-induced electric–magnetic fields within the crack are related to three material parameters g1 , g2 and g3 . Below we discuss the effect of these parameters on the electric–magnetic fields within the crack. (1) For the case where H is real, h is real too and (65) degenerates into g1 ¼ 0; 0 g2 ¼ m021 h23 h 33 m31 ;
ð66Þ
00 g3 ¼ m0021 h23 h1 33 m31 :
One finds from (52), (53) and (66) that in this case, g2 is real and g3 is purely imaginary, and (64) becomes 1 ðD2 ; B2 ÞT ¼ g2 q1 2 x1 þ ig3 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x21 :
ð67Þ
Eq. (67) shows that the thermal-induced electric–magnetic fields inside the crack vary nonlinearly along the crack line. When approaching the crack tip from on the crack surface, the electric– magnetic fields are equal to a constant vector g2 aq1 2 . (2) For the case a homogeneous case, one has from (54) and (65) that g1 ¼ 0; 0 g2 ¼ m021 h23 h1 33 m31 ;
ð68Þ
g3 ¼ 0; and (64) becomes ðD2 ; B2 ÞT ¼ g2 q1 2 x1 :
ð69Þ
Eq. (69) shows that the thermal-induced electric–magnetic fields inside the crack vary linearly along the crack line. (3) For general cases where H is complex, Eq. (64) can be rewritten as 1 1 ðD2 ; B2 ÞT ¼ g01 Q1 xþ 3 ðx1 Þ þ g2 q2 x1 þ g3 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
ð70Þ
where g01 ¼ g1 Q. Inserting (58) into (70) one can obtain the final expression of electric–magnetic fields inside the crack. It can be shown that in general cases, the electric–magnetic fields may be singular and oscillatory at the crack tip.
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6. Thermal-induced stress intensity factors Similar to the case of isotropic media, the stress intensity factors can be defined as [19] pffiffiffiffiffiffiffi kr ¼ ðkII ; kI ; kIII ÞT ¼ lim 2pr½h33 Qdiagðriea Þ½h33 Q1 r2s ðrÞ; ð71Þ r!0
where r means the distance from the crack-tip and r2s ðrÞ stands for the singular principle part of the stress vector r2 ðx1 Þ ahead of the crack tip, that is r2s ðrÞ ¼ ðh33 þ h33 Þx3 ðrÞ; which is obtained from (55). Substituting (72) together with (58) into (71), we can finally obtain h pffiffiffiffiffiffi c0 i þ c þ : kr ¼ pa½h33 Qdiagðð2aÞiea Þ½h33 Q1 aðm031 þ m0031 Þq1 1 2 a
ð72Þ
ð73Þ
Furthermore, inserting (61) and (62) into (73) yields the final expression of the stress intensity factor vector as kr ¼
pffiffiffiffiffiffi ie 1 pa½KQdiagðð2aÞ a Þ½KQ k0 ðmÞaq1 2 ;
ð74Þ
where k0 ðmÞ ¼ hh1 þ 2iea þ e2a 4e2a =aiiðm031 þ m0031 Þ 12m0031 : For the case of a homogeneous material, we have from (74) that pffiffiffiffiffiffi kr ¼ pam031 aq1 2 :
ð75Þ
It can be confirmed that (75) is consistent with the result of Gao et al. [11].
7. Conclusions We study an interface crack problem in a magnetoelectroelastic bi-material under uniform heat flow at infinity. The crack is modeled as an electrically permeable slit with unknown electric– magnetic fields inside it. According to complex potential method, we derive analytical expressions for the thermal-induced electric–magnetic fields inside the crack and the stress intensity factors at the crack. It is shown that different from the case of a homogeneous material, the thermal-induced electric–magnetic fields inside the interface crack are nonlinear variable with position along the crack line. Especially at the crack tip, the induced fields may be singular and oscillatory. In addition, ahead of the crack line, the structure of stress singularities is the same as that in a purely elastic bi-material system with interface cracks, which is uniquely characterized by an inverse square root singularity and a pair of oscillatory singularities. This means that for an electrically
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permeable interface crack in magnetoelectroelastic materials, the electric–magnetic effects may change the amplitude of stresses at the crack tip, but not change the singular structure of stress fields.
Acknowledgements The authors would like to express their gratitude for the support of the Japan Society for the Promotion of Science (JSPS).
References [1] M.Y. Chung, T.C.T. Ting, The Green function for a piezoelectric piezomagnetic anisotropic elastic medium with an elliptic hole or rigid inclusion, Philos. Mag. Lett. 72 (1995) 405–410. [2] J.X. Liu, X.L. Liu, Y.B. Zhao, Green’s functions for anisotropic magnetoelectro-elastic solids with an elliptical cavity or a crack, Int. J. Eng. Sci. 39 (2001) 1405–1418. [3] C.F. Gao, H. Kessler, H. Balke, Crack problems in magnetoelectroelastic solids. Part I: Exact solution of a crack, Int. J. Eng. Sci. 41 (2003) 969–981. [4] X. Wang, Y.P. Shen, The general solution of three-dimensional problems in magnetoelectroelastic media, Int. J. Eng. Sci. 40 (2002) 1069–1080. [5] E. Pan, Three-dimensional Green’s functions in anisotropic magneto-electro-elastic bimaterails, Z. Angew. Math. Phys. 53 (2002) 815–838. [6] J.Y. Li, Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials, Int. J. Eng. Sci. 38 (2000) 1993–2011. [7] J.H. Huang, Y.H. Chiu, H.K. Liu, Magneto-electro-elastic Eshelby tensors for a piezoelectric–piezomagnetic composite reinforced by ellipsoidal inclusions, J. Appl. Phys. 83 (1998) 5364–5370. [8] C.F. Gao, H. Kessler, H. Balke, Crack problems in magnetoelectroelastic solids. Part II: General solution of collinear cracks, Int. J. Eng. Sci. 41 (2003) 983–994. [9] C.F. Gao, P. Tong, T.Y. Zhang, Interfacial crack problems in magneto-electroelastic solids, Int. J. Eng. Sci. 41 (2003) 2105–2121. [10] X.L. Liu, X.F. Liu, J.X. Liu, et al., Electromagnetic thermoelastic solids with an elliptical cavity or a crack under uniform heat flow, Key Eng. Mater. 183 (2000) 139–144. [11] C.F. Gao, H. Kessler, H. Balke, Fracture analysis of electromagnetic thermoelastic solids, Eur. J. Mech. A––Solid 22 (2003) 433–442. [12] A.C. Eringen, Continuum theory of micromorphic electromagnetic thermoelastic solids, Int. J. Eng. Sci. 41 (2003) 653–665. [13] T.C.T. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites, Int. J. Solids Struct. 22 (1986) 965–983. [14] T.C.T. Ting, Interface cracks in anisotropic bimaterials, J. Mech. Phys. Solids 38 (1990) 505–513. [15] T.C.T. Ting, Interface cracks in anisotropic elastic bimaterials––a decomposition principle, Int. J. Solids Struct. 29 (1992) 1989–2003. [16] T.C.T. Ting, Anisotropic Elasticity: Theory and Applications, Oxford Science Publications, New York, 1996. [17] V.Z. Parton, B.A. Kudryavtsev, Electromagnetoelasticity, Gordon and Breacn, New York, 1988. [18] N.I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoof, Leyden, 1975. [19] K.C. Wu, Stress intensity factors and energy release rate for interfacial cracks between dissimilar anisotropic materials, ASME J. Appl. Mech. 57 (1990) 882–886.
Thermal-induced interfacial cracking of magnetoelectroelastic materials Cun-Fa Gao *, Naotake Noda Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, Shizuoka 432-8561, Japan Received 26 January 2004; received in revised form 19 February 2004; accepted 18 March 2004
Abstract In this paper, we present an explicitly analytic solution for a generalized two-dimensional problem of an interface crack between two dissimilar magnetoelectroelastic materials under uniform heat flow. The crack is assumed to be electrically permeable and thus the thermal-induced electric–magnetic fields within the crack need to be determined as part of the solution. According to the extended Stroh formalism, the thermal potential functions and the electric–magnetic–elastic potential functions are at first presented in concise form. Then, the thermal-induced electric–magnetic fields within the crack and the stress intensity factor at the crack tip are obtained, respectively, in closed form. It is shown that when uniform heat flow is applied at infinity, the thermal-induced electric–magnetic fields within the crack are very complicated, especially near the crack tip for general cases. In addition, the stress fields at the crack tip may be singular and oscillatory, but the structure of singularities is the same as that in a purely elastic bi-material system with interface cracks, that is, it is uniquely characterized by an inverse square root singularity and a pair of oscillatory singularities. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Magnetoelectroelastic solid; Interface crack; Thermal loading; Stress intensity factor
1. Introduction Piezoelectric and piezomagnetic composites have wide application in smart devices, particularly in the aerospace and automotive industries. Since such materials simultaneously possess piezoelectric, piezomagnetic and high magnetoelectric coupling effects, they are very sensitive to elastic, *
Corresponding author. Fax: +81-53-474-7499. E-mail address: [email protected] (C.-F. Gao).
0020-7225/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.03.005
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electric, magnetic and thermal fields in environment, and solid defects, e.g., cavities, cracks or inclusions where the local fields greatly concentrate. The locally enhanced fields may induce crack initiation, crack growth and finally lead to fracture or failure of devices. Due to the importance of the reliability of these devices, fracture and failure analyses of magnetoelectroelastic solids with holes or cracks have received considerable interest in the recent decade. Chung and Ting [1], Liu et al. [2] and Gao et al. [3] derived a 2D solution of an elliptical cavity or a crack in an infinite magnetoelectroelastic solid, respectively. Wang and Shen [4] and Pan [5] solved a 3D problem for an infinite magnetoelectroelastic body and a bi-material system, respectively. Li [6] and Huang et al. [7] studied magnetoelectroelastic inclusion problems in piezoelectric–piezomagnetic composites, respectively. Recently, Gao et al. [8,9] obtained several analytical results for multiple crack problems in magnetoelectroelastic materials. Relatively, little efforts have been dedicated to fracture and failure of magnetoelectroelastic media under thermal loading. Liu et al. [10] derived the complex potentials of an elliptical cavity in an electromagnetic thermoelastic solid. Gao et al. [11] made an explicit treatment on a series of collinear permeable crack in a homogeneous magnetoelectroelastic material. Eringen [12] established a continuum theory of micromorphic electromagnetic thermoelastic solids. To the best of our knowledge, however, the interface crack problem in magnetoelectroelastic materials under thermal loading remains unsolved, though this problem is also important with increasingly wide application of piezoelectric and piezomagnetic composites in high environment. It is therefore the purpose of this work to analyse a generalized 2D problem of an interface crack in a magnetoelectroelastic bi-material system under uniform heat flow at infinity. On the interfacial cracks in purely elastic anisotropic media, great developments have been made by Ting in a series of his works, e.g., in [13–15], and these developments were partly summarized in [16]. When the anisotropic media are subjected to piecewise uniform traction at infinity, explicit solutions for the entire field and closed-form expression of the oscillatory parameter can be found in [13–16]. In the present work, we shall extend Ting’s work [13,14] to the case of an interfacial crack in magnetoelectroelastic media, where the crack is assumed to be permeable. Thus, this means that the thermal-induced electromagneto fields inside the crack have to be determined simultaneously as part of the solution. To do this, we develop a concise and explicit method to reduce the present problem to an equivalent one to that in purely elastic anisotropic media, and then give the final solution of an interfacial crack in a magnetoelectroelastic bi-material system under uniform heat flow at infinity. Below is the plan of this work: Section 2 outlines the extended Stroh formalism for a magnetoelectroelastic solid under thermal loading. In Section 3, we introduce the thermal complex potentials, and then the electric–magnetic–elastic potential functions are presented in Section 4 in explicit form. In Section 5 we establish a simple relationship between the induced electric–magnetic fields inside the crack and the applied thermal loading at infinity in order to discuss the distribution of electric–magnetic fields along the crack faces. Section 6 gives an exact expression for the thermalinduced stress intensity factor at the crack tip, and finally Section 7 concludes the work.
2. Basic equations In a fixed rectangular coordinate system xi ði ¼ 1; 2; 3Þ, the governing equations of a linear magnetoelectroelastic solid subjected to thermal loading can be expressed as
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1349
rij ¼ cijkl ckl ekij Ek akij Hk bij T ; Di ¼ eikl ckl þ eik Ek þ lik Hk þ si T ;
ð1Þ
Bi ¼ aikl ckl þ lik Ek þ tik Hk þ qi T ; cij ¼ 12ðui;j þ uj;i Þ; rij;i ¼ 0;
Ei ¼ u;i ;
Di;i ¼ 0;
qi ¼ kij T;j ;
Hi ¼ w;i ;
Bi;i ¼ 0;
qi;i ¼ 0;
ð2Þ ð3Þ ð4Þ
where a comma denotes partial differentiation; rij , Di , Bi and qi are the stress, the electric displacement, the magnetic induction and heat flow, respectively. cij , Ek and Hk represent the strain, the electric field and magnetic field, respectively. ui , u, w and T denote the displacement, the electric potential, the magnetic potential and temperature change, respectively. kij are the heat conduction coefficients. cijkl , ekij , akij lik , eik and tik are the elastic stiffness tensor, piezoelectric, piezomagnetic magnetoelectric coupling tensor, the dielectric permittivities and the magnetic permeabilities, respectively; and bij , si and qi are the thermal stress constants, pyroelectric coefficients and pyromagnetic coefficients, respectively. Substituting (1) together with (2) into (3), and then substituting the first equation in (4) into the second one in (4), one has ðcijks uk þ esij u þ asij wÞ;si bij T;i ¼ 0; ðeiks uk eis u lis wÞ;si þ si T;i ¼ 0;
ð5Þ
ðaiks uk lis u tis wÞ;si þ qi T;i ¼ 0; kij T;ij ¼ 0:
ð6Þ
Consider a generalized 2D problem of a magnetoelectroelastic solid with geometry and loading independent of x3 . In this case, (6) becomes k11
o2 T o2 T o2 T þ 2k þ k ¼ 0: 12 22 ox1 x2 ox21 ox22
ð7Þ
The general solution of (6) is T ¼ 2Re½g0 ðzÞ;
z ¼ x1 þ pt x2 ;
ð8Þ
where Re means taking the real part; g0 ðzÞ is a complex function to be determined; the prime (0 ) indicates differentiation with respect to its argument, and pt is the heat eigenvalue which is determined from k22 pt2 þ 2k12 pt þ k11 ¼ 0:
ð9Þ
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The solution of pt with positive imaginary part is pt ¼ ðk12 þ ijÞ=k22 ;
ð10Þ
where i¼
pffiffiffiffiffiffiffi 1;
j ¼ ðk11 k22 k212 Þ1=2 ;
k11 k22 k212 > 0:
Inserting (8) into the first equation in (4) leads to q1 ¼ 2Re½ipt jg00 ðzÞ;
q2 ¼ 2Re½ijg00 ðzÞ:
ð11Þ
Introduce a generalized displacement function vector u and a generalized stress function vector / as T
T
/ ¼ ð/1 ; /2 ; /3 ; /4 ; /5 Þ :
u ¼ ðu1 ; u2 ; u3 ; u; wÞ ;
Then, (3) is automatically satisfied if one takes ðr11 ; r12 ; r13 ; D1 ; B1 ÞT ¼ /;2 ;
ðr21 ; r22 ; r23 ; D2 ; B2 ÞT ¼ /;1 :
ð12Þ
Thus, after T is determined from (8), u and / can be determined from (5). In general, the solution of (5) consists of two parts, one being a homogeneous solution of (5) corresponding to the isothermal case, and another being a particular solution of (5). The homogeneous solution has the following form: uh ¼ 2Re½AfðzÞ;
/h ¼ 2Re½BfðzÞ;
ð13Þ
with fðzÞ ¼ ½f1 ðz1 Þ; f2 ðz2 Þ; f3 ðz3 Þ; f4 ðz4 Þ; f5 ðz5 ÞT ; A ¼ ða1 ; a2 ; a3 ; a4 ; a5 Þ;
za ¼ x1 þ pa x2
ða ¼ 1 5Þ;
B ¼ ðb1 ; b2 ; b3 ; b4 ; b5 Þ;
where fa ðza Þ are complex potentials to be found; pa are the complex eigenvalues with positive imaginary parts. The pa and corresponding eigenvectors aa and ba can be obtained from the following equations: jS þ pðR þ RT Þ þ p2 Tj ¼ 0; bS þ pa ðR þ RT Þ þ pa2 Tcaa ¼ 0; ba ¼ ðRT þ pa TÞaa : The matrices S, R and T are related to the material constants by
C.-F. Gao, N. Noda / International Journal of Engineering Science 42 (2004) 1347–1360
2
c1jk1 4 S ¼ eT1k1 aT1k1
e1j1 e11 l11
3
a1j1 l11 5; t11
2
c1jk2 4 R ¼ eT1k2 aT1k2
e2j1 e12 l12
3
a2j1 l12 5; t12
2
c2jk2 4 T ¼ eT2k2 aT1k1
e2j2 e22 l22
1351
3
a2j2 l22 5: t22
On the other hand, the particular solution of (5) has the form up ¼ 2Re½cgðzÞ;
/p ¼ 2Re½dgðzÞ;
ð14Þ
where c and d are two constant vectors, and they can be determined from the following equations: bS þ pt ðR þ RT Þ þ pt2 Tcc ¼ b1 þ pt b2 ; d ¼ ðRT þ pt TÞc b1 ¼
1 ½ðS þ pt RÞc þ b2 ; pt
with b1 ¼ ðb11 ; b12 ; b13 ; s1 ; q1 ÞT ;
b2 ¼ ðb21 ; b22 ; b23 ; s2 ; q2 ÞT :
Finally, (13) plus (14) leads to the general solution of u and / as u ¼ 2Re½AfðzÞ þ cgðzÞ;
ð15Þ
/ ¼ 2Re½BfðzÞ þ dgðzÞ:
ð16Þ
From (15) and (16) we have u;1 ¼ 2Re½Af 0 ðzÞ þ cg0 ðzÞ;
ð17Þ
/;1 ¼ 2Re½Bf 0 ðzÞ þ dg0 ðzÞ;
ð18Þ
where u;1 ¼ ðu1;1 ; u2;1 ; u3;1 ; E1 ; H1 ÞT ;
ðr21 ; r22 ; r23 ; D2 ; B2 ÞT ¼ /;1 :
ð19Þ
For convenience, we introduce a new vector function as UðzÞ ¼ Bf 0 ðzÞ þ dg0 ðzÞ:
ð20Þ
Then, (17) and (18) can be rewritten as /;1 ¼ UðzÞ þ UðzÞ;
ð21Þ
u;1 ¼ 2Im½YUðzÞ Mg0 ðzÞ;
ð22Þ
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where Y ¼ iAB1 ;
M ¼ Yd ic:
3. Thermal potentials Consider an electrically permeable interface crack Lc between two dissimilar magnetoelectroelastic media, as shown in Fig. 1. Assume that the media are subjected to uniform heat flow q1 2 at infinity, while the crack is free of force, external charge, electric current and heat flow, but filled with air. In this case, the thermal boundary condition on the crack can be expressed as qþ 2 ðx1 Þ ¼ q2 ðx1 Þ ¼ 0;
x1 2 Lc :
ð23Þ
The thermal complex function g0 ðzt Þ takes form of g0 ðzÞ ¼ ct z þ g00 ðzÞ;
ð24Þ
where g00 ðzÞ is a holomorphic function except on Lc , ct is a known constant depending on the heat flow at infinity. The g00 ðzÞ can be solely solved only by considering the thermal boundary condition. The results are listed as follows 0 ðzÞ ¼ g10
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 q2 ðz z2 a2 Þ; 2ij1
z 2 sþ ;
ð25Þ
0 ðzÞ ¼ g20
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 q2 ðz z2 a2 Þ; 2ij2
z 2 s ;
ð26Þ
8
q2
S1
x2
Lc
Lb 2a
x1
S2
8
q2
Fig. 1. An interface crack in a magnetoelectroelastic bi-material under uniform heat flow.
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1353
and 0 0 j1 g10 ðzÞ þ j2 g20 ðzÞ ¼ 0;
z 2 sþ ;
ð27Þ
0 0 ðzÞ þ j1 g10 ðzÞ ¼ 0; j2 g20
z 2 s :
ð28Þ
4. Electromagnetoelastic potentials On the crack faces, we have the following boundary conditions [17]: rþ 2j ¼ 0;
r 2j ¼ 0
ðj ¼ 1–3Þ
ð29Þ
on Lc ;
Dþ 2 ðx1 Þ ¼ D2 ðx1 Þ;
Bþ 2 ðx1 Þ ¼ B2 ðx1 Þ;
E1þ ðx1 Þ ¼ E1 ðx1 Þ;
H1þ ðx1 Þ ¼ H1 ðx1 Þ;
1 < x1 < þ1;
ð30aÞ
1 < x1 < þ1;
ð30bÞ
where E1 and H1 are the components of electric field and magnetic field in the x1 direction, respectively. Taking into account the conditions of zero stresses and electric fields at infinity, the complex potential f 0 ðzÞ has the form f 0 ðzÞ ¼ f 00 ðzÞ;
ð31Þ
where f 00 ðzÞ stands for a holomorphic function vector, and f 00 ð1Þ ¼ 0. Substituting (24) and (31) into (21) and (22), and then considering the condition that both stress and strain are bounded at infinity, we have /;1 ¼ U0 ðzÞ þ U0 ðzÞ;
ð32Þ
u;1 ¼ 2Im½YU0 ðzÞ Mg00 ðzÞ;
ð33Þ
where U0 ðzÞ ¼ Bf 00 ðzÞ þ dg00 ðzÞ: Obviously, U0 ðzÞ is holomorphic in the z-plane except on Lc , and U0 ð1Þ ¼ 0. The continuity condition of the generalized tractions on x1 requires /þ ;1 ðx1 Þ ¼ /;1 ðx1 Þ;
1 < x1 < þ1:
ð34Þ
Substituting (32) into (34) produces þ
½U10 ðx1 Þ U20 ðx1 Þ ½U20 ðx1 Þ U10 ðx1 Þ ¼ 0; The solution of (35) is [18]
1 < x1 < þ1:
ð35Þ
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U10 ðzÞ U20 ðzÞ ¼ 0;
z 2 s1 ;
ð36Þ
U20 ðzÞ U10 ðzÞ ¼ 0;
zs 2 s2 :
ð37Þ
Introduce a jump function Du;1 ðx1 Þ defined as Du;1 ðx1 Þ ¼ bu;1 ðx1 Þc1 bu;1 ðx1 Þc2 :
ð38Þ
Inserting (33) into (38) and then using (27) and (28) we obtain iDu;1 ðx1 Þ ¼ xþ ðx1 Þ x ðx1 Þ;
ð39Þ
where 8 0 < HU10 ðzÞ M1 j1 M2 g10 ðzÞ; j2 xðzÞ ¼ : HU20 ðzÞ M2 j2 M1 g0 ðzÞ; 20 j1
z 2 s1 ; z 2 s2 ;
ð40Þ
H ¼ Y1 þ Y2 : Since Du;1 ðx1 Þ ¼ 0 on Lb , Eqs. (39) and (40) show that xðzÞ is analytic in the z-plane except on the crack, and moreover xð1Þ ¼ 0. In addition, the value-singled condition of generalized displacements requires I Du;1 ðx1 Þ dx1 ¼ 0; ð41Þ C
where C is a clockwise contour enclosed the crack. Inserting (39) into (41) leads to I ½xþ ðx1 Þ x ðx1 Þ dx1 ¼ 0:
ð42Þ
C
On the whole x1 axis, the continuous conditions of E1 ðx1 Þ and H1 ðx1 Þ give bDu;1 ðx1 ÞcJ ¼ 0;
J ¼ 4; 5; 1 < x1 < 1:
ð43Þ
Substituting (39) into (43) leads to xþ J ðx1 Þ xJ ðx1 Þ ¼ 0;
J ¼ 4; 5; 1 < x1 < 1:
ð44Þ
The solution of (44) is [18] xJ ðzÞ ¼ 0;
J ¼ 4; 5:
ð45Þ
Based on (45), the unknown function xðzÞ defined in (40) takes the form xðzÞ ¼ ½x1 ðzÞ; x2 ðzÞ; x3 ðzÞ; 0; 0T :
ð46Þ
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1355
On the other hand, one has by using (32) that /þ ;1 ðx1 Þ ¼ U10 ðx1 Þ þ U10 ðx1 Þ:
ð47Þ
Using (37), (47) becomes /þ ;1 ðx1 Þ ¼ U10 ðx1 Þ þ U20 ðx1 Þ:
ð48Þ
Eq. (48) can be rewritten as 1
1 /þ ;1 ðx1 Þ ¼ H ½HU10 ðx1 Þ þ H ½HU20 ðx1 Þ:
ð49Þ
Using (40), (49) can be rewritten as /þ ;1 ðx1 Þ
1
þ
1
¼ H x ðx1 Þ þ H x ðx1 Þ þ H
j1 1 0 M2 M1 g20 ðx1 Þ: þH j2
1
j1 0 M1 M2 g10 ðx1 Þ j2 ð50Þ
Substituting (25) and (26) into (50) yields 1
1 þ 0 1 00 1 /þ ;1 ðx1 Þ ¼ H x ðx1 Þ þ H x ðx1 Þ þ m q2 x1 þ m q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
where m0 and m00 are two material constants:
1 M2 M2 1 M1 1 M1 0 H ; H m ¼ 2i j1 j2 j1 j2
1 M2 M2 1 M1 1 M1 00 H m ¼ þH ; 2i j1 j2 j1 j2
ð51Þ
ð52Þ
ð53Þ
which shows that m0 is real and m00 is purely imaginary. For a homogeneous material, we have from (52) and (53) that m0 ¼
1 1 H ðM MÞ; ij
m00 ¼ 0:
ð54Þ
Let
H
1
h ¼ 33 h23
h32 ; h22
0
m ¼
m031 ; m021
00
m ¼
m0031 ; m0021
x3 ¼ ðx1 ; x2 ; x3 ÞT :
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Then, taking the first three rows and final two rows of (51), respectively, produces 0 1 00 1 ðr21 ; r22 ; r23 ÞT ¼ h33 xþ 3 ðx1 Þ þ h33 x3 ðx1 Þ þ m31 q2 x1 þ m31 q2
0 1 00 1 ðD2 ; B2 ÞT ¼ h23 xþ 3 ðx1 Þ þ h23 x3 ðx1 Þ þ m21 q2 x1 þ m21 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
ð55Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 :
ð56Þ
T
On the crack face, we have ðr21 ; r22 ; r23 Þ ¼ 0. Thus, (55) gives 0 1 00 1 h33 xþ 3 ðx1 Þ þ h33 x3 ðx1 Þ ¼ m31 q2 x1 m31 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
x1 2 Lc :
ð56Þ
Solving (56) can give the solution of x3 ðzÞ, and then the unknown function xðzÞ defined in (46) can completely be obtained. Finally, U10 ðzÞ and U20 ðzÞ can be determined from (40), and thus the problem under consideration is solved. Eq. (56) can be rewritten as 1 1 1 0 1 00 1 xþ 3 ðx1 Þ þ h33 h33 x3 ðx1 Þ ¼ h33 m31 q2 x1 h33 m31 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
x1 2 Lc :
ð57Þ
Letting Q be the eigenvector matrix h1 33 h33 , one has 2pe 2pe ; e ; 1Þ ¼ hhe2pea ii; Q1 h1 33 h33 Q ¼ diagðe
where the angular bracket hh ii stands for a diagonal matrix, and the positive and real constant e depends on the material properties and characterizes the oscillatory singularities. The e can be determined by h33 e2pea h1 33 ¼ 0; from which one can obtain an explicit expression for the ea following the work of Ting [16]. The result is ea ¼ ðe; e; 0Þ; where 1 1 1þb e ¼ tan1 b ¼ ln þ ; p 2p 1b
06b ¼
1 ^ 2 trðSÞ 2
1=2 ;
^¼D ^ 1 W; ^ S
^ iW ^ ¼ ^h33 : D
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1357
The general solution of (57) can be expressed as [18] pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 0 1 00 1 z2 a2 QÞ ½m q z þ m q ðh Q x3 ðzÞ ¼ 33 31 2 31 2 1 þ e2pea 2 z Xa ðzÞ Xa ðzÞ 1 0 00 1 þ ðh33 QÞ ðm31 þ m31 Þq2 þ ðh33 QÞ1 1 þ e2pea 1 þ e2pea ðc1 z þ c0 Þ;
ð58Þ
where c1 and c0 are two constant vectors to be found, and
ie 1 za a ; a ¼ 1; 2; 3: Xa ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 a2 z þ a To find c1 and c0 , one can use (42) and the condition x3 ð1Þ ¼ 0:
ð59Þ
Noting (45), (42) becomes I ½xþ 3 ðx1 Þ x3 ðx1 Þ dx1 ¼ 0:
ð60Þ
C
The use of (59) and (58) gives c1 ¼ hh2iea iiðm031 þ m0031 Þaq1 2 :
ð61Þ
After c1 is obtained, c0 can be determined by inserting (58) into (60). The result is 2 1 1 00 c0 ¼ hhae2a 4e2a iiðm031 þ m0031 Þaq1 2D 2m31 a q2D :
ð62Þ
Substituting (61) and (62) into (58) one can give the final expression of xðzÞ. Up to here we have completed the task for determining the electromagnetoelastic potentials. 5. Thermal-induced electric–magnetic field inside the crack On the crack surface, one finds from (56) that þ 0 1 00 1 1 x 3 ðx1 Þ ¼ h33 bh33 x3 ðx1 Þ þ m31 q2 x1 þ m31 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 c;
x 1 2 Lc :
ð63Þ
Substituting (63) into (56) results in T
1 1 ðD2 ; B2 Þ ¼ g1 xþ 3 ðx1 Þ þ g2 q2 x1 þ g3 q2
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
x 1 2 Lc ;
ð64Þ
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g1 ¼ h23 h23 h1 33 h33 ; 0 g2 ¼ m h23 h1 m0 ; g3 ¼
21 m0021
ð65Þ
33 31 00 h23 h1 33 m31 :
Eq. (64) indicates that the distribution of thermal-induced electric–magnetic fields within the crack are related to three material parameters g1 , g2 and g3 . Below we discuss the effect of these parameters on the electric–magnetic fields within the crack. (1) For the case where H is real, h is real too and (65) degenerates into g1 ¼ 0; 0 g2 ¼ m021 h23 h 33 m31 ;
ð66Þ
00 g3 ¼ m0021 h23 h1 33 m31 :
One finds from (52), (53) and (66) that in this case, g2 is real and g3 is purely imaginary, and (64) becomes 1 ðD2 ; B2 ÞT ¼ g2 q1 2 x1 þ ig3 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 x21 :
ð67Þ
Eq. (67) shows that the thermal-induced electric–magnetic fields inside the crack vary nonlinearly along the crack line. When approaching the crack tip from on the crack surface, the electric– magnetic fields are equal to a constant vector g2 aq1 2 . (2) For the case a homogeneous case, one has from (54) and (65) that g1 ¼ 0; 0 g2 ¼ m021 h23 h1 33 m31 ;
ð68Þ
g3 ¼ 0; and (64) becomes ðD2 ; B2 ÞT ¼ g2 q1 2 x1 :
ð69Þ
Eq. (69) shows that the thermal-induced electric–magnetic fields inside the crack vary linearly along the crack line. (3) For general cases where H is complex, Eq. (64) can be rewritten as 1 1 ðD2 ; B2 ÞT ¼ g01 Q1 xþ 3 ðx1 Þ þ g2 q2 x1 þ g3 q2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 a2 ;
ð70Þ
where g01 ¼ g1 Q. Inserting (58) into (70) one can obtain the final expression of electric–magnetic fields inside the crack. It can be shown that in general cases, the electric–magnetic fields may be singular and oscillatory at the crack tip.
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6. Thermal-induced stress intensity factors Similar to the case of isotropic media, the stress intensity factors can be defined as [19] pffiffiffiffiffiffiffi kr ¼ ðkII ; kI ; kIII ÞT ¼ lim 2pr½h33 Qdiagðriea Þ½h33 Q1 r2s ðrÞ; ð71Þ r!0
where r means the distance from the crack-tip and r2s ðrÞ stands for the singular principle part of the stress vector r2 ðx1 Þ ahead of the crack tip, that is r2s ðrÞ ¼ ðh33 þ h33 Þx3 ðrÞ; which is obtained from (55). Substituting (72) together with (58) into (71), we can finally obtain h pffiffiffiffiffiffi c0 i þ c þ : kr ¼ pa½h33 Qdiagðð2aÞiea Þ½h33 Q1 aðm031 þ m0031 Þq1 1 2 a
ð72Þ
ð73Þ
Furthermore, inserting (61) and (62) into (73) yields the final expression of the stress intensity factor vector as kr ¼
pffiffiffiffiffiffi ie 1 pa½KQdiagðð2aÞ a Þ½KQ k0 ðmÞaq1 2 ;
ð74Þ
where k0 ðmÞ ¼ hh1 þ 2iea þ e2a 4e2a =aiiðm031 þ m0031 Þ 12m0031 : For the case of a homogeneous material, we have from (74) that pffiffiffiffiffiffi kr ¼ pam031 aq1 2 :
ð75Þ
It can be confirmed that (75) is consistent with the result of Gao et al. [11].
7. Conclusions We study an interface crack problem in a magnetoelectroelastic bi-material under uniform heat flow at infinity. The crack is modeled as an electrically permeable slit with unknown electric– magnetic fields inside it. According to complex potential method, we derive analytical expressions for the thermal-induced electric–magnetic fields inside the crack and the stress intensity factors at the crack. It is shown that different from the case of a homogeneous material, the thermal-induced electric–magnetic fields inside the interface crack are nonlinear variable with position along the crack line. Especially at the crack tip, the induced fields may be singular and oscillatory. In addition, ahead of the crack line, the structure of stress singularities is the same as that in a purely elastic bi-material system with interface cracks, which is uniquely characterized by an inverse square root singularity and a pair of oscillatory singularities. This means that for an electrically
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permeable interface crack in magnetoelectroelastic materials, the electric–magnetic effects may change the amplitude of stresses at the crack tip, but not change the singular structure of stress fields.
Acknowledgements The authors would like to express their gratitude for the support of the Japan Society for the Promotion of Science (JSPS).
References [1] M.Y. Chung, T.C.T. Ting, The Green function for a piezoelectric piezomagnetic anisotropic elastic medium with an elliptic hole or rigid inclusion, Philos. Mag. Lett. 72 (1995) 405–410. [2] J.X. Liu, X.L. Liu, Y.B. Zhao, Green’s functions for anisotropic magnetoelectro-elastic solids with an elliptical cavity or a crack, Int. J. Eng. Sci. 39 (2001) 1405–1418. [3] C.F. Gao, H. Kessler, H. Balke, Crack problems in magnetoelectroelastic solids. Part I: Exact solution of a crack, Int. J. Eng. Sci. 41 (2003) 969–981. [4] X. Wang, Y.P. Shen, The general solution of three-dimensional problems in magnetoelectroelastic media, Int. J. Eng. Sci. 40 (2002) 1069–1080. [5] E. Pan, Three-dimensional Green’s functions in anisotropic magneto-electro-elastic bimaterails, Z. Angew. Math. Phys. 53 (2002) 815–838. [6] J.Y. Li, Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials, Int. J. Eng. Sci. 38 (2000) 1993–2011. [7] J.H. Huang, Y.H. Chiu, H.K. Liu, Magneto-electro-elastic Eshelby tensors for a piezoelectric–piezomagnetic composite reinforced by ellipsoidal inclusions, J. Appl. Phys. 83 (1998) 5364–5370. [8] C.F. Gao, H. Kessler, H. Balke, Crack problems in magnetoelectroelastic solids. Part II: General solution of collinear cracks, Int. J. Eng. Sci. 41 (2003) 983–994. [9] C.F. Gao, P. Tong, T.Y. Zhang, Interfacial crack problems in magneto-electroelastic solids, Int. J. Eng. Sci. 41 (2003) 2105–2121. [10] X.L. Liu, X.F. Liu, J.X. Liu, et al., Electromagnetic thermoelastic solids with an elliptical cavity or a crack under uniform heat flow, Key Eng. Mater. 183 (2000) 139–144. [11] C.F. Gao, H. Kessler, H. Balke, Fracture analysis of electromagnetic thermoelastic solids, Eur. J. Mech. A––Solid 22 (2003) 433–442. [12] A.C. Eringen, Continuum theory of micromorphic electromagnetic thermoelastic solids, Int. J. Eng. Sci. 41 (2003) 653–665. [13] T.C.T. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites, Int. J. Solids Struct. 22 (1986) 965–983. [14] T.C.T. Ting, Interface cracks in anisotropic bimaterials, J. Mech. Phys. Solids 38 (1990) 505–513. [15] T.C.T. Ting, Interface cracks in anisotropic elastic bimaterials––a decomposition principle, Int. J. Solids Struct. 29 (1992) 1989–2003. [16] T.C.T. Ting, Anisotropic Elasticity: Theory and Applications, Oxford Science Publications, New York, 1996. [17] V.Z. Parton, B.A. Kudryavtsev, Electromagnetoelasticity, Gordon and Breacn, New York, 1988. [18] N.I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoof, Leyden, 1975. [19] K.C. Wu, Stress intensity factors and energy release rate for interfacial cracks between dissimilar anisotropic materials, ASME J. Appl. Mech. 57 (1990) 882–886.