Multiple cracking of magnetoelectroelastic materials in coupling thermo–electro–magneto-mechanical loading environments

Computational Materials Science 39 (2007) 291–304 www.elsevier.com/locate/commatsci

Multiple cracking of magnetoelectroelastic materials in coupling thermo–electro–magneto-mechanical loading environments B.L. Wang, J.C. Han

*

Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China Received 30 May 2006; accepted 20 June 2006

Abstract Magnetoelectroelastic materials are inherently brittle and prone to fracture. Therefore, it is important to model evaluate the fracture behavior of these advanced materials. In this paper, a periodic array of cracks in a magnetoelectroelastic material is investigated. The problem is solved by integral transform and integral equation technique. Impermeable and permeable crack-face electromagnetic boundary condition assumptions are studied. The stress, electric displacement and magnetic induction and their intensity factors are obtained. Effect of the crack spacing on these quantities is investigated in details. The problem of transient fracture by a transient temperature variation along the crack lines is also investigated. Parameters governing the transient thermo–electro–magneto-elastic fields are identified.  2006 Elsevier B.V. All rights reserved. Keywords: Magnetoelectroelastic materials; Fracture mechanics; Cracks, Integral equation method

1. Introduction Magnetoelectroelastic materials possess the features that the application of electric field induces magnetization and magnetic field induces electric polarization [1]. This makes them useful in the development of smart/intelligent structures. The magnetoelectric (ME) effect of magnetoelectric composites was studied by van den Boomgaard et al. [2] and van Run et al. [3]. From early analysis on magnetoelectroelasticity, Parton and Kudryavtsev [4] came up with the linear theory. Since these pioneering works, numerous investigations have been devoted to predicting and determining the ME effect of magnetoelectric composites [5– 11]. In the present paper of Fiebig [12], a detailed review of the research topics for the magnetoelectric materials was given. When subjected to loads, magnetoelectroelastic materials can fail prematurely owing to their brittle nature. *

Corresponding author. E-mail address: [email protected] (J.C. Han).

0927-0256/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.06.008

Therefore, efforts have been made to analyze the fracture behavior of magnetoelectroelastic materials with defects (e.g., cracks, holes, inclusions, etc.). A crack in a magnetoelectroelastic material can be impermeable or permeable, with respect to electric and magnetic fields. Huang and Kuo [13] and Li and Dunn [14] solved the inclusion problems encountered in magnetoelectroelastic materials with the aid of the three-dimensional magnetoelectroelastic Green’s functions. Liu et al. [15] gave the Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. Wu et al. [16] developed the elasticity solution to the problem of a magnetoelectroelastiic hyperboloidal notch subjected to axial loading conditions at infinity. Qin [17] solved Green’s functions for magnetoelectroelastic medium with an arbitrarily oriented half-plane or bi-material interface. Feng et al. [18] examined the dynamic response of an interface crack between two dissimilar magnetoelectroelastic materials subjected to anti-plane mechanical, and in-plane electric and magnetic impacts. Hu and Li [19] analyzed the magnetoelectroelastic problem of a finite crack in an orthotropic piezoelectromagnetic

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B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

strip under longitudinal shear. Tian and Rajapakse [20] presented theoretical study of a magnetoelectroelastic composite plane with a single conducting crack. Zhou et al. [21] investigated the behavior of a crack in functionally graded piezoelectric/piezomagnetic materials subjected to an anti-plane shear loading. Gao et al. [22] investigated an anti-plane crack in a magnetoelectroelastic solid. Gao and Noda [23] studied the interface crack problems for the infinite magnetoelectroelastic media. Recently, Chue and Liu [24] performed a magnetoelectroelastic anti-plane analysis for a BaTiO3–CoFe2O4 composite wedge with an interface crack. Hao and Liu [25] investigated interaction between a screw dislocation and a semi-infinite interfacial crack in a transversely isotropic magnetoelectroelastic bi-material system. Tian and Gabbert [26] investigated a parallel crack near the interface of magnetoelectroelastic bi-materials. The thermal analyses for magnetoelectroelastic media are very important in industrial applications. Many structural components containing magnetoelectroelastic materials are subjected to thermal loads. Typical examples are aerospace structures operating in cooling and/or heating environments, and smart/intelligent materials sensing temperature changes. Thus, it is necessary to study thermal effects in magnetoelectroelastic materials. Aboudi [7] carried out micromechanical analysis of fully coupled electro–magneto–thermo-elastic multiphase composites. Ootao and Tanigawa [27] investigated multilayered magneto–electro–thermo-elastic strip due to unsteady and nonuniform heat supply. Gao et al. [28] analyzed fully coupled electro–magneto–thermo-elastic medium with collinear permeable-cracks. Niraula and Wang [29] obtained the exact solution of penny-shaped crack problem in an infinite magnetoelectroelastic medium under uniform heat flow. Because of mathematical difficulty, research on the problem of multiple crack interaction in magnetoelectroelastic materials is very limited. To our best understanding, only the problems of two collinear permeable cracks [28,30] and a row of collinear cracks [31] were investigated. This paper investigates a periodic array of non-collinear cracks in magnetoelectroelastic materials. A general analytical model for the magneto–electro-mechanical coupling is established. The crack faces are subjected to any mechanical, electrical and magnetic loads. The stress, electric displacement and magnetic induction and their intensity factors are discussed. The crack-face electromagnetic boundary conditions are assumed to be impermeable or permeable. Effect of crack spacing and the crack-face electromagnetic boundary conditions are studied. In numerical examples, the problem of a transient temperature change parallel to the crack line is also solved. Material parameters governing the magnitudes of the stress, electric displacement, and magnetic induction in the un-cracked medium are identified. The model developed in this paper may be useful to researchers in their future investigation of the mechanics and physics of magnetoelectroelastic materials.

2. Basic thermo–magneto–electro-elasticity equations and expressions We consider a two-dimensional problem such that all the field variables are only functions of the coordinates (x, z), which are coincident with the principal axes of the material symmetry. Denote the symbols r, D and B as the stress, the electric displacement and the magnetic induction, respectively. The displacement components along the x and z directions are u and w, respectively. The electric potential is denoted as / and the magnetic potential is denoted as u. The basic expressions of magnetoelectroelasticity have been given by Parton and Kudryavtsev [4], Nan [5], Li and Dunn [11,14], Aboudi [8] etc. The constitutive equations are D ¼ eT e þ E þ bH;

r ¼ ce  eE  hH; B ¼ hT e þ bE þ cH;

ð1Þ

where r, e, D, E, B and H are vectors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field, respectively. These vectors are defined by T

r ¼ ðrxx ; rzz ; rxz Þ ;  T ou ow ou ow ; ; þ e¼ ; ox oz oz ox  T o/ o/ T ; D ¼ ðDx ; Dz Þ ; E ¼  ; ox oz  T ou ou T ; B ¼ ðBx ; Bz Þ ; H ¼  : ox oz

ð2aÞ ð2bÞ ð2cÞ ð2dÞ

For a magnetoelectroelastic medium whose poling direction coincides with the positive z-axis, the material constant matrices c, e, h, e, b and c have the following forms: 2 3 2 3 0 e31 c11 c13 0 6 7 6 7 c ¼ 4 c13 c33 0 5; e ¼ 4 0 e33 5; 2

0

0

3

c44

0 h31 6 7 h ¼ 4 0 h33 5; h15 0   b11 0 b¼ ; 0 b33

e15 

11 ¼ 0  c¼

c11 0

 0 ; 33

0 ð3Þ

 0 ; c33

where cijkl, ekij, hkij, ij, bij, and cij are respectively, elastic coefficients, electro-mechanical coupling coefficients, magneto-mechanical coupling coefficients, dielectric permeability coefficients, magnetoelectric coupling coefficients, and magnetic permeability coefficients. Equations for the displacements, electric potential and magnetic potential can be established by substituting Eqs. (1)–(3) into the following equilibrium equations: rxx;x þ rxz;z ¼ 0; rxz;x þ rzz;z ¼ 0; Dx;x þ Dz;z ¼ 0; Bx;x þ Bz;z ¼ 0:

ð4aÞ ð4bÞ

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

Here, the body force, charge density and current density are absent. The system governing equations (1)–(4) must be solved under prescribed magneto–electro-mechanical boundary conditions. 3. Magneto–electro-mechanical model of a periodic array of cracks To solve the crack problem in linear problems, the superposition technique is usually used. Thus, we first solve the problem without cracks in the medium under thermo– electro–magneto-mechanical loads. Then, we use equal and opposite stresses, electric displacements and magnetic induction as the crack surface magneto–electro-mechanical loads and solve the crack problem. For this, it is necessary to establish a model for cracks subjected to magneto–electro-mechanical loads. As shown in Fig. 1, there is a periodic array of cracks in a magnetoelectroelastic material. The crack length is 2a. Crack spacing is 2c. If we are interested in crack initiation behavior, the crack can be assumed to be infinitesimal (i.e., a  b, in which b is the horizontal size of the medium). Usually, crack surfaces are traction free. However, because air or vacuum allows some penetrations of electric flux and magnetic flux through crack interior, the normal (along the z-axis) components of the electric displacement vector and the magnetic induction vector are not zero inside the crack, which are denoted as D0 and B0, respectively. Since the crack problem is solved by superposition technique, the z-direction components of the stress, electric displacement, and magnetic induction, obtained for the un-cracked medium, with opposite signs are added to the crack surfaces for mechanical, electrical and magnetic loads. Thus, there are z

2c

x 2a

Fig. 1. A periodic array of Griffith cracks in a magnetoelectroelastic material (2c: crack spacing; a: half crack length).

293

mixed boundary conditions at the cracked planes. For example, on the z = 0 plane, we have 9 rzz ðx; 0Þ ¼ p01 ðxÞ ¼ r0 ðxÞ; z ¼ 0; jxj 6 a > = Dz ðx; 0Þ ¼ p02 ðxÞ ¼ d 0 ðxÞ  D0 ðxÞ; z ¼ 0; jxj 6 a ð5Þ > ; Bz ðx; 0Þ ¼ p03 ðxÞ ¼ b0 ðxÞ  B0 ðxÞ; z ¼ 0; jxj 6 a and wðx; 0Þ ¼ 0;

/ðx; 0Þ ¼ 0;

uðx; 0Þ ¼ 0;

z ¼ 0;

jxj P a: ð6Þ

In Eq. (5), r0(x), D0(x) and B0(x) are, respectively, the stress, electric displacement, and magnetic induction parallel to the z-axis of the un-cracked medium. Because of symmetry and periodicity, the problem can be considered for a representative element (0 6 z 6 c, 0 6 x < 1), subjected to the following homogeneous boundary conditions: rxz ðx; 0Þ ¼ rxz ðx; cÞ ¼ 0; wðx; cÞ ¼ 0; /ðx; cÞ ¼ 0;

ð7Þ ð8Þ

uðx; cÞ ¼ 0:

The solution of the representative element is expressed in terms of some unknown coefficients. These unknown coefficients are then determined from the boundary conditions of the problem. 3.1. Magnetoelectroelastic fields The constitutive equations of magnetoelectroelastic coupling have been given by Eqs. (1)–(3). The governing equations of magnetoelectroelastic media in the absence of body force, electric charge density and current density are presented by Eq. (4). The solution of the governing equations can be obtained in a straightforward manner by applying Fourier transform 9 8 9 8 uðx; zÞ > a1m sinðsxÞ > > > > > > > > = 2Z 1X < wðx; zÞ > = < a cosðsxÞ > 8 > 2m ¼ expðskm zÞAm ds; > p 0 m¼1 > /ðx; zÞ > a3m cosðsxÞ > > > > > > > > > ; : ; : uðx; zÞ a4m cosðsxÞ ð9Þ in which Am(s) stand for unknown functions of s. The constants aim and parameters km (m = 1, 2, 3, 4) appearing in Eq. (9) are determined from the following eigenvalue problem: 2 3 c11  c44 k2m ðc13 þ c44 Þkm ðe31 þ e15 Þkm ðh31 þ h15 Þkm 6 ðc þ c Þk c33 k2  c44 e33 k2m  e15 h33 k2m  h15 7 44 m 6 13 7 6 7 4 ðe31 þ e15 Þkm e33 k2m  e15 11  33 k2 b11  b33 k2m 5 ðh31 þ h15 Þkm h33 k2m  h15 9 8 a1m > > > > > = 2m ¼ 0:  > a3m > > > > > ; : a4m

b11  b33 k2

c11  c33 k2m

ð10Þ

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In Eq. (10), nontrivial aim (j = 1, 2, 3, 4) exist if k is a root of the determinant. Obviously, there are eight roots for km. It can be shown that if [km, (a1m, a2m, a3m, a4m)T] is an eigensolution of Eq. (10), then [km, (a1m, a2m, a3m, a4m)T] is also an eigen-solution of Eq. (10). In what follows, the order of roots km are arranged such that Re(k1) < 0, Re(k2) < 0, Re(k3) < 0, Re(k4) < 0, and k5 = k1, k6 = k2, k7 = k3, k8 = k4. The stresses, electric displacements, and the magnetic inductions associated with Eq. (9) are obtained from the constitutive equations (1)–(3) as follows: 9 9 8 8 > 8 Z 1 > = = 2X < C 1m > < rzz ðx; zÞ > Dz ðx; zÞ ¼ s C 2m cosðsxÞ expðskm zÞAm ds; > > > ; ; p m¼1 0 > : : Bz ðx; zÞ C 3m ð11Þ 9 9 8 8 > = = 2X < C 4m > 8 Z 1 > < rxz ðx; zÞ > Dx ðx; zÞ ¼ s C 5m sinðsxÞ expðskm zÞAm ds; > > > ; ; p m¼1 0 > : : Bx ðx; zÞ C 6m ð12Þ where the constants Cjm (j = 1, . . . , 6; m = 1, . . . , 8) are given in Appendix A.

Now, we consider the continuity conditions (6), it follows that Z 1 gj ðsÞ cosðsxÞds ¼ 0 ðj ¼ 1; 2; 3Þ; x P a: ð17Þ 0

Hence, gi (i = 1, 2, 3) have the solutions of the form Z a Uj ðlÞJ 0 ðslÞ dl ðj ¼ 1; 2; 3Þ; gj ðsÞ ¼

provided that limt!0t1Uj(t) = 0. In Eq. (18), J0 is the Bessel function of the first kind of order zero. The functions Uj(x) will be determined from the crack-face boundary conditions through a system of Fredholm integral equations of the second kind. 3.3. The integral equations The system of integral equations can be obtained by substituting Eqs. (14) and (18) into Eq. (11), and using crack-face boundary conditions (5). As a result, we obtain "Z # Z 3 a X 2 1 cosðsxÞ sJ 0 ðslÞ Kij ðsÞUj ðlÞ dl ds ¼ p0i ðxÞ p 0 0 j¼1 ði ¼ 1; 2; 3Þ:

ð19Þ

3.2. Satisfying the symmetry and periodicity conditions

The contractions

The unknown coefficients Am involved in the magnetoelectroelastic fields will be determined from the symmetry and periodicity conditions (7) and (8), as well as the mixed mode conditions (5) and (6). To this end, three auxiliary functions (g1, g2, g3) on the cracked plane are introduced as follows:

Kij ðsÞ ¼

gi ðsÞ ¼

8 X

Am aðiþ1Þm

ði ¼ 1; 2; 3Þ:

ð13Þ

ð18Þ

0

8 X

C im bmj ðsÞ ¼ 

m¼1

4 X

C im Bmj cothðsckm Þ ði; j ¼ 1; 2; 3Þ

m¼1

ð20Þ

have been made. In order to determine the possible singular behavior of Eq. (19), the behavior of the kernels Kij for large values of s needs to be examined. It can be seen from Eq. (20) that as s approaches infinity, the quantities Kij(s) become constants

m¼1

It follows from the symmetry and periodic conditions (7) and (8), and with substitution of Eq. (9) into Eq. (13), Am can be expressed in terms of (g1, g2, g3) as Am ðnÞ ¼

3 X

bmj ðnÞgj ðsÞ;

ð14Þ

j¼1

where bmj, (m = 1, . . . , 8; j = 1, 2, 3) are known coefficients 1 Bmj ; 1  expð2sckm Þ ðm ¼ 1; 2; 3; 4Þ;

bmj ¼

and where Bmj are 2 a21 a22 6a 6 31 a32 ½B ¼ 6 4 a41 a42 C 41

C 42

C 44

4 X

ð21Þ

C im Bmj :

m¼1

By adding and subtracting the asymptotic values of K0ij to and from Kij(s) in Eq. (20), Eq. (19) can be re-written as "Z # Z 3 a X 2 1 0 cosðsxÞ sJ 0 ðslÞ Kij Uj ðlÞ dl ds p 0 0 j¼1 Z a wij ðx; rÞUj ðrÞ dr ¼ p0i ðxÞ; ð22Þ þ 0

bðmþ4Þj ¼ expð2sckm Þbmj

where Wij are functions of x and r ð15Þ

the elements of the following matrix: 31 a23 a24 a33 a34 7 7 ð16Þ 7 : a43 a44 5 C 43

Kij ð1Þ ¼ K0ij ¼

wij ðx;rÞ ¼

4 2X C im Bmj p m¼1

Z

1

cosðsxÞ 0

2s expð2sckm Þ J 0 ðsrÞds: 1  expð2sckm Þ ð23Þ

For c ! 1 (infinite crack spacing) or r ! x, Eq. (23) gives Wij = 0. Accordingly, integral Eq. (22) reduces to the single crack solution (for infinite c).

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

The integral equation (22) can be further transferred to  Z a K0ij Uj ðlÞ þ K ij ðl; rÞUj ðrÞ dr ¼ Qi ðlÞ; ð24Þ

3  X

0

j¼1

where (i = 1, 2, 3), the integral kernels Kij and the generalized loads Qi are Z 1 4 X s expð2sckm Þ J 0 ðsrÞJ 0 ðslÞ ds C im Bmj K ij ðl; rÞ ¼ 2l 1  expð2sckm Þ 0 m¼1 ð25Þ and Qi ðlÞ ¼ l

Z 0

l

p0i ðxÞ ffi dx; pffiffiffiffiffiffiffiffiffiffiffiffiffi l2  x 2

ð26Þ

respectively. Eq. (24) is the desired system of integral equations, which can be solved by a standard method (e.g., by collocation technique). Since Eq. (26) contains unknown constants d0 and b0, which are, respectively, the normal component of the electric displacement vector and the normal component of the magnetic induction vector inside the crack (see Eq. (5)). To obtain d0, b0 and Uj (j = 1, 2, 3) from Eq. (24), additional assumptions are needed. This will be discussed in Section 3.5. Once d0, b0, and Uj (j = 1, 2, 3) become known, the full-field solution is obtained. 3.4. Crack tip field intensity factors, the stress, electric displacement, and magnetic induction The field intensity factors for stress k1, electric displacement kD and magnetic induction kB at the crack tip defined by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 ¼ lim 2pðr  aÞrzz ðr; 0Þ; ð27aÞ r!a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k D ¼ lim 2pðr  aÞDz ðr; 0Þ; ð27bÞ r!a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k B ¼ lim 2pðr  aÞBz ðr; 0Þ; ð27cÞ r!a

can be expressed in term of the functions Uj (j = 1, 2, 3) by rffiffiffiffiffiffi 3 1 X 0 K Uj ðaÞ; k 1 ¼ 2 pa j¼1 1j rffiffiffiffiffiffi 3 1 X 0 k D ¼ 2 K Uj ðaÞ; ð28Þ pa j¼1 2j rffiffiffiffiffiffi 3 1 X 0 k B ¼ 2 K Uj ðaÞ: pa j¼1 3j With the substitution of Eqs. (14) and (18), the normal displacement, the electric potential, and the magnetic potential on the upper surface of the crack can be determined from Eq. (9) as 9 9 8 8 U1 ðlÞ > > > = 2Z a = < wðx; 0Þ > < 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U2 ðlÞ dl: ð29Þ /ðx; 0Þ ¼ > > > l2  x 2 > ; p x ; : : uðx; 0Þ U3 ðlÞ

295

The energy release rate of the penny-shaped cracks can be obtained from virtual crack closure integral technique as 1 G ¼ ðK I ; K D ; K B Þ½KðK I ; K D ; K B ÞT ; 2

ð30Þ

where ½K is a three by three matrix, which is the inversion of the matrix [K0]. The elements of the matrix [K0] have been defined by Eq. (21). Therefore, ½K ¼ ½K0 :

ð31Þ

Three quantities of considerable practical interest are the stress rzz(x, z), the electric displacement Dz(x, z), and the magnetic induction Bz(x, z) as they may have important influence on further cracking. It is expected that the overall stress is maximized at the z = c plane. Substituting Eqs. (14) and (18) into Eq. (11), the stress, the electric displacement and the magnetic induction at z = c can be obtained as follows: 9 9 8 8 > =Z a = 4X < C 1m > 4 > 3 < rzz ðx; cÞ > X Dz ðx; cÞ ¼ C 2m Rm ðx; lÞ Bmj Uj ðlÞ dl; > > > ; 0 ; p m¼1 > : : j¼1 Bz ðx; cÞ C 3m ð32Þ where Rm ðx; lÞ ¼

Z

1

cosðsxÞ 0

s expðsckm Þ J 0 ðslÞ ds: 1  expð2sckm Þ

ð33Þ

Eq. (32) would give the stress, the electric displacement and the magnetic induction for the perturbation problem solved under the conditions (5)–(8). To obtain the correct stress, electric displacement, and magnetic induction, the solution of the un-cracked medium under prescribed external loads must be added to those given by Eq. (32). The solutions given in this subsection are expressed in terms of the functions Uj (j = 1, 2, 3), which depend on the electric displacement d0 and the magnetic induction b0 inside the cracks. To determine Uj (j = 1, 2, 3), d0 and b0 simultaneously from the integral Eq. (24), the crack-face electromagnetic boundary conditions must be considered properly. For further analysis, the ideal electromagnetic boundary condition assumptions (i.e., the fully impermeable and fully permeable crack-face electromagnetic boundary conditions) are considered. 3.5. Crack face electromagnetic boundary conditions Generally, in magnetoelectroelastic fracture, two kinds of idealized electromagnetic boundary conditions on the crack faces can be used. One boundary condition is the specification that the normal components of electric displacement and magnetic induction on the crack surfaces equals zero (impermeable boundary condition). Another boundary condition treats the cracks as being electrically and magnetically conductive (permeable boundary condition).

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3.5.1. Impermeable crack-face boundary condition In this case, the cracks are fully insulated to the electromagnetic fields. Thus, the normal components of electric displacement d0 and magnetic induction b0 inside the cracks are zero. The equivalent crack-face electromagnetomechanical loads are p01 = r0, p02 = D0 and p03 =  B0, where r0, D0 and B0 are obtained from the solution of the same medium without cracks. Therefore, Eq. (26) is known and the functions Uj (j = 1, 2, 3) can be determined directly from the solution of Eq. (24). 3.5.2. Permeable crack-face boundary condition Another extreme case is that the cracks are electrically and magnetically conductive. Thus, the electric potential and magnetic potential jumps across the crack are zero but the electric displacement and the magnetic induction inside the crack remain unknown. In this case, U2 = 0 and U3 = 0 hold. The unknown function U1 can be determined from the first equation of Eq. (24) Z a Z 2 l xp01 ðxÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx: ð34Þ K11 U1 ðlÞ þ K 11 ðl; rÞU1 ðrÞ dr ¼ p 0 0 l2  x 2 Obviously, the applied electric and magnetic loads have no effect on the solution of the problem of permeable cracks. The field intensity factors for the stress, the electric displacement and the magnetic induction obtained from Eq. (28) are as follows: rffiffiffiffiffiffi 1 0 K0 K031 k 1 ¼ 2 k ; k ¼ k 1 : ð35Þ K11 U1 ðaÞ; k D ¼ 21 1 B pa K011 K011 The displacement on the upper surface of the electrically and magnetically permeable crack is the same as the first of Eq. (29). Since the cracks are permeable to the electric and magnetic fields, the electric potential and magnetic potential are continuous across the cracks. However, the electric displacement and the magnetic induction inside the cracks are non-zero. It follows from Eqs. (24) and (26) that d0 and b0 satisfy the equations Z a K j1 ðl; rÞU1 ðrÞ dr K0j1 U1 ðlÞ þ 2 ¼ p

0

Z

l 0

xp0j ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dx ðj ¼ 2; 3Þ; l2  x 2

1 d x dx

1 d b0 ðxÞ ¼ B0 ðxÞ þ x dx

Z

x

0

Z

ð36Þ

0

  Z a t pffiffiffiffiffiffiffiffiffiffiffiffiffi K021 U1 ðtÞ þ K 21 ðt; rÞU1 ðrÞdr dt; x2  t 2 0

ð37aÞ

x

4. Infinite crack spacing The analytical model established in Section 3 is for the case of finite crack spacing c. In the case of infinite crack spacing, the integral kernels wij of Eq. (23) and Kij of Eq. (25) vanish. Consequently, the system of integral equation (24) can be solved in closed-form. Giving Uj ðlÞ ¼ 

3 X

  Z a t pffiffiffiffiffiffiffiffiffiffiffiffiffi K031 U1 ðtÞ þ K 31 ðt; rÞU1 ðrÞdr dt: x2  t 2 0

ð37bÞ The electric displacement inside the crack d0 consists of two parts. The first part is the first term on the right-hand side of Eq. (37a), which equals the applied electric load D0. The second part is the second term on the right-hand side of Eq.

Kji Qi ðlÞ ðj ¼ 1; 2; 3Þ;

ð38Þ

i¼1

in which Kji have been defined in Eq. (31). For constant equivalent loads applied on the crack faces, substituting Eqs. (26) and (38) into Eq. (29), the normal component of the displacement vector, the electric potential and the magnetic potential on the upper surface of the crack are obtained as 9 9 8 8 > 3 > = = < wðx; 0Þ > < K1i > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi X /ðx; 0Þ ¼  a2  x2 ð39Þ K2i p0i ; jxj 6 a; > > > > ; ; : i¼1 : uðx; 0Þ K3i where p01 = r0, p02 = d0  D0 and p03 = b0  B0. For impermeable crack assumption, the crack tip field intensity factors can be obtained in closed-form from Eqs. (28) and (38) as rffiffiffiffiffiffi rffiffiffiffiffiffi 1 1 k 1 ¼ 2 Q ðaÞ; k D ¼ 2 Q ðaÞ; pa 1 pa 2 ð40Þ rffiffiffiffiffiffi 1 k B ¼ 2 Q ðaÞ; pa 3 where Qi ðaÞ ¼ a

Z 0

where p02 = d0  D0 and p03 = b0  B0. Eq. (36) is the Abel type integral equation, which gives the solution d 0 ðxÞ ¼ D0 ðxÞ þ

(37a), which is produced by the applied mechanical load r0 on the crack faces. Similarly, the magnetic induction inside the crack b0 consists of two parts. The first part is the first term on the right-hand side of Eq. (37b), which equals the applied magnetic load B0. The second part is the second term on the right-hand side of Eq. (37b), which is produced by the applied mechanical load r0 on the crack faces.

a

p0i ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx a2  x 2

ði ¼ 1; 2; 3Þ:

ð41Þ

For permeable crack assumption, the crack tip field intensity factors can be obtained in closed-form from Eq. (35) as rffiffiffiffiffiffi 1 K0 K031 k 1 ¼ 2 k ; k ¼ k1: ð42Þ Q1 ðaÞ; k D ¼ 21 1 B pa K011 K011 The function Q1 has been defined in Eq. (41). The electric displacement and the magnetic induction inside the permeable cracks can also be determined in closed-form. Due to the fact that U2 and U3 are zero for permeable cracks, Eq. (24) becomes Z 2 l xp0j ðxÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ðj ¼ 1; 2; 3Þ; Kj1 U1 ðlÞ ¼ ð43Þ p 0 l2  x 2

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

where p01 = r0, p02 = d0  D0 and p03 = b0  B0. The Abel type integral equation (43) gives the solutions Z x 1 d t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U1 ðtÞ dt; 0 ¼ r0 ðxÞ þ K011 ð44aÞ 2 x dx 0 x  t2 Z x 1 d t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U1 ðtÞ dt; ð44bÞ d 0 ðxÞ ¼ D0 ðxÞ þ K021 x dx 0 x2  t 2 Z x 1 d t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi U1 ðtÞ dt: ð44cÞ b0 ðxÞ ¼ B0 ðxÞ þ K031 2 x dx 0 x  t2 From Eqs. (44a)–(44c) we know that K0 r0 ðxÞ; d 0 ðxÞ ¼ D0 ðxÞ  21 K011

0.8 0.6

Permeable crack solution

kI/k0

0.4

Uncoupled (eij =0, hij =0,) solution σzz(0,c)/σ0

0.2

-0.2 0

Therefore, the electric displacement and magnetic induction inside the crack are solely determined from the applied magneto–electro-mechanical loads and the material properties. It can be shown that if the applied loads (i.e., r0, D0, B0) are constants, then the electric displacement and magnetic induction inside the crack are also constants. 5. Numerical results In this section, numerical results are given for a periodic array of cracks in a BaTiO3–CoFe2O4 magnetoelectroelastic composite material, in which BaTiO3 to CoFe2O4 ratio is roughly 50:50. The material properties for the composite material are [32]: c11 = 22.6 · 1010 N/m2, c13 = 12.4 · 1010 N/m2, c33 = 21.6 · 1010 N/m2, c44 = 4.4 · 1010 N/m2, c12 = 12.5 · 1010 N/m2, e31 = 2.2 C/m2, e33 = 9.3 C/m2, e15 = 5.8 C/m2, 11 = 56.4 · 1010 C2/(N m2), 33 = 63.5 · 1010 C2/ (N m2), c11 = 810,000 · 1010 N s2/C2, c33 = 835,000 · 1010 N s2/C2, h31 = 290.2 N/(A m), h33 = 350 N/(A m), h15 = 275 N/(A m). Using these material properties, the key material property matrix, ½K of Eq. (31), are obtained as follows: 2 3 0:01106 2:859  105 1:272  1011 6 7 ½K ¼ 4 0:01106 1:483  108 4:177  104 5: 4:177  104

Impermeable crack solution

1.0

0.0

K0 b0 ðxÞ ¼ B0 ðxÞ  31 r0 ðxÞ: K011 ð45Þ

2:859  105

297

1:202  104

ð46Þ In Sections 5.1–5.3, the medium is subjected independently to a uniform tension, a uniform electric displacement, and a uniform magnetic induction, remote from the cracks (i.e., at z = ±1). Since the system is linear, the solutions for any combination of electro–magnetomechanical loads can be determined directly from these independent solutions. In Section 5.4, a transient thermal load will be considered. 5.1. A constant stress load rzz(x, ±1) = r0 applied at infinity Plotted in Fig. 2 are the normalized stress intensity factor k1 as a function of the crack spacing. Results based on

1

2

3

4

5

Normalized half crack spacing c/a Fig. 2. Axial stress rzz at (x, z) = (0, c) and stress intensity factor k1 as a function of crack spacing for the far-field constant stress load r0, pffiffiffiffiffiffi ðk 0 ¼ r0 pa).

impermeable and permeable crack assumptions are given. To explore the effect of electro-mechanical and magnetomechanical coupling coefficients, the un-coupled solutions (this is, for eij = 0 and hij = 0) are also presented. Obviously, the effect of the crack-face electromagnetic boundary condition assumption on the stress intensity factors is quite insignificant. Effect of electro-mechanical and magnetomechanical coupling coefficients on k1 is also negligible. Therefore, stress intensity factors for the magnetoelectroelastic materials are same as those for the traditional orthotropic materials with the same material constants c11, c13, c33 and c44 and under the same mechanical loading condition. (Note that, because the material is orthotropic in the x  z plane, the stress intensity factor is different from that of the isotropic materials.) It can be seen that multiple cracking has a significant tendency to reduce the stress intensity factor. If the crack spacing approaches to zero, the stress intensity factor also approaches to zero. k1 increases monotonously with increasing crack spacing. The stress factor approaches to the known result pffiffiffiffiffiintensity ffi k 1 ¼ r0 pa as the crack spacing becomes infinity. The normal stress rzz(0, c) as a function of the normalized crack spacing c/a is also plotted in Fig. 2. It can be shown that results from the impermeable and permeable crack assumptions are almost identical and the effect of the coupling coefficients eij and hij on the stress is negligible. Unlike the stress intensity factor, the dependency of the stress on the crack spacing is relatively complicated. When the crack spacing is infinite, the stress rzz(0, c) is maximum and is equal to the applied stress load r0. At first, rzz(0, c) decreases monotonously with decreasing crack spacing. When the crack spacing approaches c/a  1, the stress rzz(0, c) becomes zero. When the crack spacing further decreases, rzz(0, c) becomes negative. rzz(0, c) finally approaches to zero as the crack spacing approaches to zero. After the stress intensity factors are obtained, the electric displacement and magnetic induction intensity factors for the permeable cracks can be obtained directly from

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

the relationship given by Eq. (35). In Figs. 3 and 4, the electric displacement and magnetic induction intensity factors, and the electric displacement and magnetic induction at the mid-point (x, z) = (0, c) are plotted for the impermeable crack-face electromagnetic boundary condition assumption. It is found that kD, Dz(0, c), kB and Bz(0, c) are very complicate functions of the crack spacing. Generally, each of the magnitudes of kD and kB increases with c/a, to a peak, and then decreases monotonously. If there is only one single crack in the medium (this is, for c/a ! 1), the values of kD, Dz(0, c), kB and Bz(0, c) become zero and do not depend on the crack-face electromagnetic boundary condition assumptions. Therefore, at the cracked plane ahead of the crack tip, the stress, the electric displacement and the magnetic induction are un-coupled only when the crack spacing is sufficiently large. Some additional results for the normal stress rzz on the z = c plane is displayed in Fig. 5. It can be seen that for each given crack spacing, the stress is nearly a monotonously increasing function of the horizontal coordinate x. 0.00 -0.05

1.2

Normalized stress σzz(x,c)/σ0

298

1.0 0.8 0.6

c/a=2

0.4

c/a=1.5

0.2

c/a=1 c/a=0.25

0.0

c/a=0.5

-0.2 0.00

0.25

0.50

0.75

1.00

1.25

1.50

Normalized position x/a Fig. 5. Stress distribution on the z = c plane for the far-field constant stress load.

The stress is minimum at the central axis of the medium (i.e., at x = 0). As x becomes sufficiently larger, the solution for the un-cracked medium is recovered (this is, rzz = r0 for x  a). 5.2. A constant electric displacement load Dz(x, ±1) = D0 applied at infinity

kD/k0

-0.10 -0.15

Dz(0,c)/σ0

-0.20 -0.25 -0.30 0

1

2

3

4

5

Normalized half crack spacing c/a Fig. 3. Electric displacement Dz at (x, z) = (0, c) and electric displacement intensity factor kD as a function of crack spacing for the far-field constant pffiffiffiffiffiffi stress load r0 (impermeable crack assumption; k 0 ¼ r0 pa).

2

In most cases, the electromagnetic loads and the mechanical load are applied simultaneously to the medium. Because of the fact that the permeable cracks do not obstruct any electromagnetic fields, the field intensity factors are zero for electrically and magnetically permeable cracks under pure electromagnetic loads. Hence, only the impermeable crack solution needs to be considered for the pure electromagnetic loads. For an applied electric displacement load D0 applied at z = ±1, the stress and the stress intensity factor, the electric displacement and the electric displacement intensity factor, the magnetic induction and the magnetic intensity factor, and the stress at the z = c plane are plotted, respectively in Figs. 6–9. 0.001

kB/k0 1

0.000

0

-0.001

k1/k0 Bz(0,c)/σ0 -1

-0.002

-2

-0.003 σzz(0,c)/D0

-3

-0.004 0

1

2

3

4

5

Normalized half crack spacing c/a Fig. 4. Magnetic induction Bz at (x, z) = (0, c) and magnetic induction displacement intensity factor kB as a function of crack spacing for the farpffiffiffiffiffiffi field constant stress load r0 (impermeable crack assumption; k 0 ¼ r0 pa).

0

1

2

3

4

5

Normalized half crack spacing c/a Fig. 6. Axial stress rzz at (x, z) = (0, c) and stress intensity factor k1 as a function of crack spacing for the far-field constant electric displacement pffiffiffiffiffiffi load D0 (impermeable crack assumption; k 0 ¼ D0 pa).

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304 1.0

kD/k0

0.8 Dz(0,c)/D0 0.6 0.4 0.2 0.0 0

1

2

3

4

5

Normalized half crack spacing c/a Fig. 7. Electric displacement Dz at (x, z) = (0, c) and electric displacement intensity factor kD as a function of crack spacing for the far-field constant electric displacement load D0 (impermeable crack assumption; k 0 ¼ pffiffiffiffiffiffi D0 pa).

1.0 Bz(0,c)/D0 0.8

299

of pure mechanical load. In particular, the normalized values of the electric displacement and the electric displacement intensity factor become the corresponding impermeable crack solution as the crack spacing becomes sufficiently large (e.g., for c > 5a). The stress, the stress intensity factor, the magnetic induction and the magnetic induction intensity factor becomes equal to zero for the extreme cases of the crack spacing (i.e., for c  a and c  a). From the analytical results, it can be shown that the applied electrical load can only produce very small stress and stress intensity factor. We quantitatively discuss this in the following. If the applied are the stress rzz = r0, the electric field Ez = E0 and the magnetic field Hz = H0 at z = ±1, then rxx = 0, rxz = 0, Dx = 0 and Bx = 0. From the constitutive equation (1), the equivalent electric displacement load Dz = D0 and magnetic induction load Bz = B0 can be obtained as c11 e33  c13 e31 D0 ¼ r0 c11 c33  c213   c11 e233  2c13 e31 e33 þ c33 e231 þ 33 þ E0 c11 c33  c213   ðc11 e33  c13 e31 Þh33 þ ðc33 e31  c13 e33 Þh31 þ b33 þ H0 c11 c33  c213

0.6

ð47aÞ and

0.4

kB/k0

B0 ¼

0.2 0.0 0

1

2

3

4

5

Normalized half crack spacing c/a Fig. 8. Magnetic induction Bz at (x, z) = (0, c) and magnetic induction intensity factor kB as a function of crack spacing for the far-field constant electric displacement load D0 (impermeable crack assumption; k 0 ¼ pffiffiffiffiffiffi D0 pa).

c11 h33  c13 h31 r0 c11 c33  c213   ðc11 e33  c13 e31 Þh33 þ ðc33 e31  c13 e33 Þh31 þ b33 þ E0 c11 c33  c213   c11 h233  2c13 h31 h33 þ c33 h231 þ c33 þ H 0; c11 c33  c213 ð47bÞ

which are D0 ¼ ð0:710r0 þ 71:2E0 þ 107:2H 0 Þ  1010 C=m2

Normalized stress σzz(x,c)/D0

0.002

ð48aÞ

and B0 ¼ ð12:9r0 þ 107:2E0 þ 8:41  105 H 0 Þ  1010 N=ðA mÞ

0.001 c/a=0.5

c/a=0.25

ð48bÞ

0.000

for the present magnetoelectroelastic material. Suppose there is a sufficiently large electric field E0 = 1 MV/m applied along the negative z direction on the medium, then the maximum stress produced by this electric field can be found from Fig. 6 and Eq. (48a). It is found that the peak values of the stress and stress intensity factor

-0.001 -0.002

c/a=1

-0.003

c/a=2 c/a=1.5

-0.004 0.00

2

0.25

0.50

0.75

1.00

1.25

1.50

Normalized position x/a Fig. 9. Stress distribution on the z = c plane for the far-field constant electric displacement load D0 (impermeable crack assumption).

Explanations for the effect of the crack spacing on the stress and the field intensity factor are similar to the case

rmax  0:00369  71:2  106 N=m  0:26 MPa

ð49aÞ

and pffiffiffi pffiffiffiffiffiffi k 1max  0:00189  71:2  106 pa  0:24 MPa a

ð49bÞ

appear approximately at c/a = 1.625 and c/a = 1.125, respectively. Comparing with the strength limit and the fracture toughness of the common materials, these values

300

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

are not significant (for example, for a crack of radius a = pffiffiffiffi 1 cm, Eq. (49b) gives k 1max ¼ 0:024 MPa m. This is a small value, compared with the toughness of common ceramic materials). Because of the coupling between the electric and magnetic fields in magnetoelectroelastic materials, the electrical load can produce the magnetic induction and the magnetic induction intensity factor. Such a fact can be seen from Fig. 8, where Bz and kB are plotted as a function of the crack spacing. It is interesting to know how significant of the coupling between the electric field and the magnetic induction. From Eq. (48a) and Fig. 8 we know that an electric field Ez = 1 V/m can produce a magnetic induction of the magnitude Bz ¼ 71:2  1010  0:992 ¼ 70:6  1010 N=ðA mÞ:

ð50Þ

This reflects the coupling coefficient between the applied electric field and the induced magnetic induction. 5.3. A constant magnetic induction load Bz(x, ±1) = D0 applied at infinity

2.0x10-5 0.0 k1/k0

-2.0x10-5

4.0x10-5

2.0x10-5 Dz(0,c)/B0 0.0 kD/k0 -5

-2.0x10

0

-4.0x10-5 -6.0x10-5

1

2

3

4

5

Normalized half crack spacing c/a Fig. 11. Electric displacement Dz at (x, z) = (0, c) and electric displacement intensity factor kD as a function of crack spacing for the far-field constant pffiffiffiffiffiffi magnetic induction load B0 (impermeable crack assumption; k 0 ¼ B0 pa).

1.0x100

kB/k0

8.0x10-1

Bz(0,c)/B0

6.0x10-1 4.0x10-1 2.0x10-1 0.0 0

1

2

3

4

5

Normalized half crack spacing c/a Fig. 12. Magnetic induction Bz at (x, z) = (0, c) and magnetic induction intensity factor kB as a function of crack spacing for the far-field constant pffiffiffiffiffiffi magnetic induction load B0 (impermeable crack assumption; k 0 ¼ B0 pa).

4.0x10-5

Normalized stress σzz(x,c)/B0

In Figs. 10–13, the results are plotted for the applied magnetic induction load. Again, only the impermeable crack condition is considered since the permeable cracks do not obstruct any magnetic field. Effect of the magnetic induction on the stress and stress intensity factors is very similar to the case the electric displacement load. The normalized values of the magnetic induction and the magnetic induction intensity factor become the corresponding impermeable crack solution if crack spacing becomes large enough (e.g., for c > 5a). The stress, stress intensity factor, electric displacement and electric displacement intensity factor are zero when crack spacing is sufficiently larger than crack length (c  a) or when the crack spacing is sufficiently smaller than crack length (c  a). Similar to the case of electric field load E0, it is also possible to understand that the applied magnetic field load H0 can only pro-

6.0x10-5

2.0x10-5

c/a=0.25

c/a=0.5

0.0 -2.0x10-5 -4.0x10-5 -6.0x10-5

c/a=1

-8.0x10-5

c/a=2

-4

-1.0x10

c/a=1.5

-4

-1.2x10

-5

0.00

σzz(0,c)/B0

-8.0x10

1

2

3

0.50

0.75

1.00

1.25

1.50

Normalized position x/a

-1.0x10-4 0

0.25

4

5

Normalized half crack spacing c/a Fig. 10. Axial stress rzz at (x, z) = (0, c) and stress intensity factor k1 as a function of crack spacing for the far-field constant magnetic induction pffiffiffiffiffiffi load B0 (impermeable crack assumption; k 0 ¼ B0 pa).

Fig. 13. Stress distribution on the z = c plane for the far-field constant magnetic induction load B0 (impermeable crack assumption).

duce small stress and stress intensity factor. In fact, from Fig. 10 and Eq. (48b) we know that if there is a sufficiently

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

large magnetic field H0 = 2000GS (1GS = 1A m/m2, 2000GS is the safety limit of the magnetic field that human can sustain) applied along the negative z direction on the medium, then the maximum stress and stress intensity factor produced by this magnetic field are

301

z

rmax  0:000101  8:41  105  2000 N=m2  0:17 MPa ð51aÞ

T0H(t) 2c

and k 1max

T0H(t) x

pffiffiffi pffiffiffiffiffiffi  0:000052  8:41  105  2000 pa  0:16 MPa a;

2a

ð51bÞ respectively, which appear approximately at c/a = 1.625 and c/a = 1.125, respectively. Comparing with the strength limit and the fracture toughness of the common ceramic materials, these values are not significant (for example, for a crackpof ffiffiffiffi radius a = 1 cm, Eq. (51b) gives k 1max ¼ 0:016 MPa m. This value is small, compared with the toughness of common ceramic materials). 5.4. Thermo–magneto–electro-elasticity cracking Under thermal conditions, the system of equilibrium equations is the same as (4). The constitutive equations, after taking into account the temperature field, become r ¼ ce  eE  hH  rT ;

Fig. 14. A magnetoelectroelastic layer under a sudden temperature change on its outer surface (because we consider the crack initiating behavior, the width of the medium is considerably larger than crack length. This is, 2b  2a.).

flow in this arrangement. The temperature field T is then governed by [33] !     1 X sinðmp=2Þ mp 2 t mp exp  T ðx; tÞ ¼ T 0 1  4 cos x ; mp 2 t0 2b m¼1;3;5

D ¼ eT e þ E þ bH  DT ;

T

B ¼ h e þ bE þ cH  B T ;

ð55Þ

ð52Þ

where r, e, D, E, B and H are vectors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field, respectively, defined in Eq. (2), rT, DT and BT are temperature-related vectors of stress, electric displacement and magnetic induction, defined by T

T

T

rT ¼ ðk11 T ; k33 T ; 0Þ ; DT ¼ ðs1 T ; s3 T Þ ; BT ¼ ðq1 T ; q3 T Þ ; ð53Þ where kij, si and qi are respectively, temperature-stress, pyroelectric and pyromagnetic coefficients, T is the temperature change. The material constant matrices c, e, h, e, b and c have the same forms as Eq. (3). 5.4.1. Solution in the absence of cracks In this section, we consider such a thermal load: initially the magnetoelectroelastic medium is at a uniform temperature zero, at the time t = 0 and far away from the crack region there is a sudden temperature change T0. Referring to Fig. 14, the thermal conditions can be described as T ðx; 0Þ ¼ 0;

2b

T ðb; tÞ ¼ T 0 H ðtÞ;

ð54Þ

where 2b is the width of the medium. Eq. (54) defines a thermal shock condition on the medium where H(t) is the Heaviside function. In the problem considered, the heat conduction is one-dimensional along the x direction, and straight cracks in this direction do not obstruct the heat

where t0 = qcvb2/k2, k2 is the coefficient of thermal conductivity along the x direction, q is the density, and cv is the specific heat. It can be seen that the thermal diffusivity k2/qcv dictates the time scale for the transient temperature distribution and does not affect the level of the temperature. Before going to the analysis of the crack problem, we identify the thermally induced magnetoelectroelastic fields in the absence of any cracks. Once these fields are obtained for non-cracked medium, the crack problem can be solved by superposition technique and the equal and opposite values of the stress, electric displacement and magnetic induction are added to the crack faces to form the equivalent external loads. Assume that the plate is infinite along y-axis and z-axis, (i.e., 1 < (y, z) < 1), free of surface tractions, electric charges, and electric currents at x = ±b, it may be shown that rxx ¼ 0;

rxz ¼ 0;

Dx ¼ 0;

Bx ¼ 0;

ð56Þ

hold for all values of x, y and z, and all non-vanishing field quantities are independent of y and z. By solving exx from the first of Eq. (52), in terms of ezz, Ez, Hz and T, and then substituting it into the rest of Eq. (52), it is found that rzz ¼ c33 ezz  e33 Ez  h33 H z  k33 T ; 33 H z  s3 T ; Dz ¼ e33 ezz þ 33 Ez þ b

ð57bÞ

33 Ez þ c33 H z  q 3 T ; Bz ¼ h33 ezz þ b

ð57cÞ

ð57aÞ

302

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

magneto–electro-mechanical field in the absence of cracks is obtained by substituting Eq. (55) into Eq. (63). This gives 9 9 8 9 8 8  > > = = > = < k33 > < r0 ðx; tÞ > < rzz ðx; tÞ > D0 ðx; tÞ ¼ Dz ðx; tÞ ¼ f ðx; tÞ s3 T 0 : ð64Þ > > > > > ; ; > ; : : : B0 ðx; tÞ Bz ðx; tÞ 3 q

where c33 ¼ c33  c213 =c11 ; e33 ¼ e33  c13 e31 =c11 ;  k33 ¼ k33  c13 k11 =c11 ; h33 ¼ h33  c13 h31 =c11 ;  33 ¼ 33 þ e231 =c11 ;

33 ¼ b þ e31 h31 =c11 ; b 33

s3 ¼ s3  e31 k11 =c11 ; c33 ¼ c33 þ h231 =c11 ; 3 ¼ q3  h31 k11 =c11 : q

ð58Þ

f ðx;tÞ

The compatibility conditions are ð59aÞ ð59bÞ

oH z =ox ¼ oH x =oz ¼ 0:

ð59cÞ

This gives Ez ¼ constant;

H z ¼ constant;

ð60Þ

where A, B, Ez and Hz are unknown constants to be obtained from magnetoelectromechanical equilibrium conditions of the medium Z

b

b Z b

rzz ðx; tÞ dx ¼ 0; Dz ðx; tÞ dx ¼ 0;

b

Z

b

b Z b

rzz ðx; tÞx dx ¼ 0; Bz ðx; tÞ dx ¼ 0:

ð61Þ

b

The results are 2

c33 6 4 e33 h33

e33 33 33 b

9 38 9 8   h33 > = > = 1 Z b < B > < k33 > 7  ¼ E T ðx; tÞ dx; b33 5 s3 z > > ; > ; 2b b : > : Hz c33 3 q

Z b k33 3  T ðx; tÞx dx: A¼ 2 b3c33 b

!     mp 2 t sinðmp=2Þ mp 2 ¼4 cos x  exp  2 2 t0 mp 2b ðmpÞ m¼1;3;5 1 X

d2 ezz =dx2 ¼ 0; oEz =ox ¼ oEx =oz ¼ 0;

ezz ¼ Ax þ B;

The function

ð62aÞ ð62bÞ

ð65Þ describes the time and space variations of the magnetoelectroelastic field in the absence of cracks. Thus, among many material constants, the parameters that govern the thermal stress, the electric displacement and the magnetic induction in the axial direction are identified as, respectively  k33 , s3 , 3 . The equivalent material constants k33 , s3 , and q 3 and q have been given in Eq. (58). 5.4.2. Thermal crack problem To solve the crack tip field, the thermal stresses, electric displacements and magnetic inductions for the un-cracked medium, obtained in Section 5.4.1, with opposite sign, are added to the crack faces to form the equivalent crack-face magneto–electro-mechanical loads. Then the problem is solved with the general model developed in Section 3. The equivalent crack-face loads are obtained from the substituting of Eqs. (64) and (65) into Eq. (26). Since l/b is small, cos[mpx/(2b)]  1. It follows that Q1 ðlÞ ¼ ðp=2Þlk33 T 0 F ðtÞ;

in which F ðtÞ ¼ 4

1 X m¼1;3;5

Substituting Eqs. (62a) and (62b) into Eq. (57), the stress, electric displacement and magnetic induction along the z direction are obtained as 9 9 8 8 > Z b =3  = > < c33 > < rzz ðx; tÞ > k33  Dz ðx; tÞ ¼ e33 x T ðx; tÞx dx 3 > > ; 2 b c33 b ; > : > : Bz ðx; tÞ h33 9 9 8 8   > > = 1 Z b = < k33 > < k33 > þ s3 T ðx; tÞ dx  s3 T ðx; tÞ: > > > > ; 2b b ; : : 3 3 q q ð63Þ Eq. (63) is valid for any temperature distribution T(x, t), and for any magnetoelectroelastic materials. For the temperature field considered in this section, the associated

Q2 ðlÞ ¼ ðp=2Þls33 T 0 F ðtÞ;

Q3 ðlÞ ¼ ðp=2Þl q33 T 0 F ðtÞ;

ð66Þ

!     mp 2 t sinðmp=2Þ 2  exp  2 2 t0 mp ðmpÞ

ð67Þ is a function of time t and is independent of material properties. In the problem under consideration time t enters into the analysis through Qi(l) only. Hence, the results of Sections 5.1–5.3 can be directly used to obtain the thermal crack tip field. In fact, the crack-face loads considered in Sections 5.1–5.3 are Q1 ðlÞ ¼ ðp=2Þlr0 ; Q2 ðlÞ ¼ ðp=2ÞlD0 ; Q3 ðlÞ ¼ ðp=2ÞlB0 : ð68Þ

From Eqs. (66) and (68), we immediately know that by 33 T 0 , replacing r0 with k33 T 0 , D0 with s33 T 0 , and B0 with q and multiple the results of Sections 5.1–5.3 by F(t), the transient thermal crack tip fields are obtained. The time function F(t) is depicted in Fig. 15. Clearly, the transient thermally induced electro-mechanical field is zero

B.L. Wang, J.C. Han / Computational Materials Science 39 (2007) 291–304

(4) The magnitude of the transient thermal stress, electric displacement and magnetic induction medium depends on a number of parameters. Among these parameters, the thermal stresses, the thermal electric displacements and the thermal magnetic inductions in the un-cracked medium are only controlled, respec3 , which tively, by equivalent parameters k33 , s3 , and q have been identified and given in Eq. (58).

0.35 0.30 0.25

F (t)

303

0.20 0.15 0.10 0.05 0.00 0.0

Appendix A 0.2

0.4

0.6

0.8

1.0

t/t0 Fig. 15. Time function F(t).

initially. F(t) increases to a peak as time goes on, and then decreases to zero as time approaches infinity. The function F(t) is positive for all values of t. Its maximum absolute value is fmax ¼ 0:31;

ð69Þ

which is attained at the time t/t0 = 0.12. At this time, the transient crack tip field is maximized. Time-varying behavior of F(t) reflects the time-dependence of the electro–magneto-elastic field and their intensity factors in the cracked medium. 6. Conclusions

Material constants Cjm (j = 1, . . . ,6; m = 1, . . . , 8) are as follows: 9 8 9 9 8 8 > > = > = = < C 1m > < c13 > < c33 > C 2m ¼ e31 a1m þ e33 km a2m > > > > > ; > ; ; : : : C 3m h31 h33 9 9 8 8 > > = = < e33 > < h33 > þ 33 km a3m þ b33 km a4m ; ðA:1Þ > > > > ; ; : : b33 c33 9 8 9 9 8 8 > > = > = = < C 4m > < c44 > < c44 > C 5m ¼ e15 km a1m  e15 a2m > > > > > ; > ; ; : : : C 6m h15 h15 9 9 8 8 > > = = < e15 > < h15 > a3m þ þ 11 b11 a4m : ðA:2Þ > > > > ; ; : : b11 c11 These coefficients have the relationship

This paper develops an analytical model for the interactions of an infinite periodic array of cracks in a magnetoelectroelastic material. The poling direction of the material is perpendicular to the crack planes. The crack length is considerably smaller than the size of the magnetoelectroelastic medium. The most outstanding parameter that dramatically affects the thermo–magneto–electro-elastic fields is the crack spacing. The following conclusions can be drawn: (1) The crack-face electromagnetic boundary conditions, the electro-mechanical coupling coefficients (eij) and the magnetic-mechanical coupling coefficients (hij) have little influence on the stress and the stress intensity factors. (2) Because of the interactions between the cracks, an applied mechanical load alone can produce the electric displacement and magnetic induction at the cracked plane ahead of the crack tips. (3) Because of the interactions between the cracks, an electric load or a magnetic load alone can produce stresses and stress intensity factors in the medium. However, the stresses and the stress intensity factors induced by the electric and/or magnetic loads are negligible, compared with the strength limit and the fracture toughness of the common materials.

C 1ð4þiÞ ¼ C 1i ; C 4ð4þiÞ ¼ C 4i ; C 6ð4þiÞ ¼ C 6i

C 2ð4þiÞ ¼ C 2i ;

C 3ð4þiÞ ¼ C 3i ;

C 5ð4þiÞ ¼ C 5i ; ði ¼ 1; 2; 3; 4Þ:

ðA:3Þ

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