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Investigation of the stress singularity of a magnetoelectroelastic bonded antiplane wedge
Applied Mathematical Modelling 31 (2007) 2313–2331 www.elsevier.com/locate/apm
Investigation of the stress singularity of a magnetoelectroelastic bonded antiplane wedge W.C. Sue a, J.Y. Liou b, J.C. Sung a
a,*
Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC b Department of Civil Engineering, Kao Yuan University, Kaohsiung 821, Taiwan, ROC
Received 1 November 2004; received in revised form 1 August 2006; accepted 6 September 2006 Available online 30 October 2006
Abstract The order of the stress singularity of a magnetoelectroelastic bonded antiplane wedge is analyzed by complex potential function and eigenfunction expansion method. Contrary to the familiar problem of elastic anisotropic bonded wedges which always produce real values for the order of singularity, the results of the magnetoelectroelastic bonded wedges may be real or complex. Numerical results are presented for problems with different boundary conditions. In particular, special behaviors of the order of the stress singularity for some degenerate composite materials and for some special wedge angles are noted. 2006 Elsevier Inc. All rights reserved. Keywords: Magnetoelectroelastic; Stress singularity; Wedge; Eigenfunction expansion
1. Introduction The investigations of the stress singularities at the apex of the wedge generated by the discontinuities of material properties and geometry are one of the typical fracture mechanics problems. In the literature, the stress singularities near the tip for either isotropic or anisotropic elastic wedges have been widely investigated (e.g., [1–6]). Due to the important applications of the piezoelectric materials, the stress singularity problems near the tip of the piezoelectric bonded wedge have been studied recently, for example, Xu and Rajapakse [7], Chue and Chen [8] and Chen and Chue [9]. Recently, cracks in a magnetoelectroelastic material under antiplane shear loading and in-plane electrical and magnetic loadings have been treated by Hu and Li [10,11]. Because of a unique property of magnetoelectric coupling effect resulting from the interaction between magnetism, electricity, and mechanism, piezoelectric–piezomagnetic composite materials or magnetoelectroelastic materials are used in a remarkably wide variety of application including intelligent or active structures, magnetoelectromechanical transducers, magnetic sensors, and acoustic actuators. Due to the singular stress fields
*
Corresponding author. Tel.: +886 2757575; fax: +886 6 2358542. E-mail address: [email protected] (J.C. Sung).
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.09.003
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
near the tip of the wedge inside materials or structures, fractures occurred at the stress singularity point make the disadvantage of the function. The singular stress fields can be expressed as rij ðr; hÞ ¼ krk1 fij ðhÞ;
ð1:1Þ
where (k 1) represents the orders of the stress singularity, k is the stress intensity factor, and fij(h) is the angular function. This paper attempts to explore the orders of the stress singularity near the tip of the magnetoelectroelastic bonded antiplane wedge by complex potential function and eigenfunction expansion method. Following Section 2, the basic equations for magnetoelectroelastic media are summarized first. In Section 3, the solution for dissimilar magnetoelectroelastic bonded wedge is presented. In Sections 4, the numerical results are discussed. Finally, this paper is concluded in Section 5. 2. Basic equations for magnetoelectroelastic media In a fixed rectangular coordinate systems xi (i = 1, 2, 3), the basic equations for a linear magnetoelectroelastic solid are given by [12–14] Constitutive equations rij ¼ cijkl skl ekij Ek qkij H k ; ð2:1Þ
Di ¼ eikl skl þ eik Ek þ d ik H k ; Bi ¼ qikl skl þ d ik Ek þ lik H k The extended strain–displacement relation 1 sij ¼ ðui;j þ uj;i Þ; 2
Ei ¼ /;i ;
H i ¼ u;i :
ð2:2Þ
Governing equations for no body force, electric charge density and electric current density rij;j ¼ 0;
Di;i ¼ 0;
Bi;i ¼ 0:
ð2:3Þ
In Eqs. (2.1)–(2.3), a comma followed by i (i = 1, 2, 3) denotes partial differentiation with respect to the ith spatial coordinate; the repeated indices imply summation over that indices. ui, / and u are the elastic displacements, electric potential and magnetic potential, respectively. rij, skl, Di, Ek, Bi and Hk are the stress, strain, electric displacement, electric field, magnetic induction and magnetic field, respectively. cijkl, eikl, qikl, eik, dik and lik are the elastic, piezoelectric, piezomagnetic, dielectric, magnetoelectric and magnetic permeability constants, respectively. These material constants satisfy the following symmetric relations: cijkl ¼ cklij ¼ cijlk ¼ cjikl ; qkij ¼ qkji ;
eik ¼ eki ;
d ik ¼ d ki ;
lik ¼ lki :
ekij ¼ ekji ; ð2:4Þ
Since we are interested only in the problems of antiplane displacement component coupled with the inplane electric field components and inplane magnetic field components, the displacement, electric field and magnetic field will take the following special forms, which are expressed as u1 ¼ u2 ¼ 0;
u3 ¼ wðx1 ; x2 Þ;
E1 ¼ E1 ðx1 ; x2 Þ; E2 ¼ E2 ðx1 ; x2 Þ; E3 ¼ 0; H 1 ¼ H 1 ðx1 ; x2 Þ; H 2 ¼ H 2 ðx1 ; x2 Þ; H 3 ¼ 0: With above special forms, then Eqs. (2.2) and (2.3) may simplify to
ð2:5Þ
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
1 s31 ¼ w;1 ; 2 E1 ¼ /;1 ;
1 s32 ¼ w;2 ; 2 E2 ¼ /;2 ;
H 1 ¼ u;1 ;
H 2 ¼ u;2 :
2315
ð2:6Þ
r31;1 þ r32;2 ¼ 0; ð2:7Þ
D1;1 þ D2;2 ¼ 0; B1;1 þ B2;2 ¼ 0:
For a special case of a transversely isotropic magnetoelectroelastic material, the constitutive Eq. (2.1) take the form [14] 9 2 8 c11 r11 > > > > > > > > 6 > r22 > > > 6 c12 > > > 6 > < r33 = 6 c13 ¼6 6 > > 6 0 > r32 > > > > > 6 > > 4 0 > r31 > > > > > ; : 0 r12 2
9 38 s11 > > > > > > s > 7> c11 c13 0 0 0 22 > > > 7> > > > 7> < 7 s33 = c13 c33 0 0 0 7 7> 2s > 0 0 c44 0 0 32 > 7> > > > 7> > > > 5> 0 0 0 c44 0 2s > > 31 > > ; : 0 0 0 0 ðc11 c12 Þ=2 2s12 3 3 2 0 0 e31 0 0 q31 6 0 6 9 0 e31 7 0 q31 7 78 9 6 0 78 6 E H1 > > 7> 7> 6 6 1 = = < < 7 7 6 0 6 0 e33 0 0 q33 7 E2 6 7 H2 ; 6 7 7 6 0 e 6 > 0 7> 15 ; 6 0 q15 0 7> ; : > : 6 7 E3 7 H3 6 6 4 e15 0 4 q15 0 0 5 0 5 c12
0
0
c13
0
0
0
0
0 0 9 8 s 11 > > > > > > > 8 9 2 > 3> s 22 > > > > > D 0 0 0 0 e 0 > > > > 1 15 = < < = 6 7 s33 0 0 e15 0 0 5 D2 ¼ 4 0 > > > > ; : > > > 2s32 > e31 e31 e33 0 D3 0 0 > > > > > 2s31 > > > > > ; : 2s12 2 38 9 2 e11 0 d 11 0 0 > = < E1 > 6 7 6 þ 4 0 e11 0 5 E2 þ 4 0 d 11 > ; : > 0 0 e33 0 0 E3 9 8 s11 > > > > > > > > 8 9 2 > 3> s22 > > > > B 0 0 0 0 q 0 > > > 15 = < = < 1> s 33 6 7 0 0 q15 0 0 5 B2 ¼ 4 0 > > > > 2s32 > ; : > > q31 q31 q33 0 B3 0 0 > > > > 2s > > > > > 31 > > ; : 2s12 2 38 9 2 d 11 0 l11 0 0 > = < E1 > 6 7 6 þ 4 0 d 11 0 5 E2 þ 4 0 l11 > ; : > 0 0 d 33 0 0 E3
ð2:8Þ
0
9 38 0 > = < H1 > 7 0 5 H2 ; > > ; : d 33 H3
9 38 0 > = < H1 > 7 0 5 H2 ; > > ; : l33 H3
ð2:9Þ
ð2:10Þ
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
where the contracted notations have been used to describe the material constants. Combining Eqs. (2.6)–(2.9) and (2.10) may yield c44 r2 w þ e15 r2 / þ q15 r2 u ¼ 0; e15 r2 w e11 r2 / d 11 r2 u ¼ 0; 2
2
ð2:11Þ
2
q15 r w d 11 r / l11 r u ¼ 0 or in matrix notation Cr2 u ¼ 0;
ð2:12Þ
where r2 ¼ o2 =ox21 þ o2 =ox22 is a two-dimensional Laplacian operator and 8 9 2 3 c44 e15 q15 > = 6 7 C ¼ 4 e15 e11 d 11 5; u ¼ / : > ; : > u q15 d 11 l11
ð2:13Þ
In polar coordinate systems (r, h), the generalized strain (shear strain, electric field and magnetic field) can be expressed in terms of u as 9 9 8 8 2s3h > > > = = < 2s3r > < o o sr ¼ Er ¼ u; sh ¼ Eh ¼ u: ð2:14Þ > > > > ; or ; r oh : : H r H h Similarly, the components of generalized stress (shear stress, electric displacement and magnetic induction) in polar coordinates can be expressed in terms of u as 9 9 8 8 r3h > > > = = < r3r > < o o tr ¼ Dr ¼ C u; th ¼ Dh ¼ C u: ð2:15Þ > > > > or r oh ; ; : : Br Bh It is known that the general solution of Eq. (2.12) is 9 8 > = < fw ðzÞ > uðx1 ; x2 Þ ¼ Re f/ ðzÞ ¼ Re½f ðzÞ; > > ; : fu ðzÞ
ð2:16Þ
where fw(z), f/(z) and fu(z) are three undetermined analytical complex functions of a complex variable z = x1 + ix2 = reih and Re denotes the real part. With the general solution expressed above, the generalized strain and generalized stress may be evaluated simply by the following equations sr ¼ Re½eih f 0 ðzÞ; tr ¼ C Re½eih f 0 ðzÞ;
sh ¼ Re½ieih f 0 ðzÞ; th ¼ C Re½ieih f 0 ðzÞ:
ð2:17Þ
To determine the stress singularities, we may use the eigenfunction expansion method by adopting the complex functions as fw ðzÞ ¼ cw zk ¼ rk ½c1 cosðkhÞ þ c2 sinðkhÞ; f/ ðzÞ ¼ c/ zk ¼ rk ½c3 cosðkhÞ þ c4 sinðkhÞ; k
ð2:18Þ
k
fu ðzÞ ¼ cu z ¼ r ½c5 cosðkhÞ þ c6 sinðkhÞ; where ci (i = 1–6) are undetermined complex constants; the constant k is the eigenvalue. For the present problem under consideration, constants in Eq. (2.18) are related in the following manner: c1 = cw, c2 = icw, c3 = c/, c4 = ic/, c5 = cu and c6 = icu. Substituting Eq. (2.18) into Eqs. (2.13)–(2.17) yields
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
9 8 8 9 > > = = < 2s3r > u ¼ / ¼ Re½rk c g; sr ¼ Er ¼ Re½krk1 c g; > > > ; ; : : > u H r 9 9 8 8 > > = = < 2s3h > < r3r > sh ¼ Eh ¼ Re½krk1 c g; tr ¼ Dr ¼ CRe½krk1 c g; > > > > ; ; : : H h Br 9 8 > = < r3h > th ¼ Dh ¼ CRe½krk1 c g; > > ; : Bh
2317
ð2:19Þ
where 2
c1
6 c ¼ 4 c 3 c5
c2
3
7 c4 5 ; c6
2
c2
6 c ¼ 4 c4 c6
c1
3
7 c3 5; c5
g¼
cosðkhÞ : sinðkhÞ
ð2:20Þ
In the following, the power of the stress singularities k 1 will be investigated for the magnetoelectroelastic bonded wedge. Those d = k 1 in the region 1 < ReðdÞ < 0
ð2:21Þ
will be of interest and will be presented. 3. Characteristic equations for the dissimilar magnetoelectroelastic bonded wedge Consider a two-bonded magnetoelectroelastic wedge as shown in Fig. 1. The wedge angles of magnetoelectroelastic materials 1 and 2 are denoted by a and b, respectively. Let the bonded interface be lie on the x1-axis. The characteristic equations for the determination of the order of stress singularities are derived in the following for three different boundary conditions prescribed on the surfaces of the wedge. 3.1. Case 1: The traction-free, electrically and magnetically insulated–traction-free, electrically and magnetically insulated wedge The first conditions considered at the boundary edges are assumed to be traction-free, electrically and magnetically insulated:
Fig. 1. Two-material bonded wedge.
2318
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331 ð1Þ
ð2Þ
r3h ðr; aÞ ¼ r3h ðr; bÞ ¼ 0; ð1Þ
ð2Þ
ð3:1Þ
Dh ðr; aÞ ¼ Dh ðr; bÞ ¼ 0; ð1Þ Bh ðr; aÞ
¼
ð2Þ Bh ðr; bÞ
¼ 0:
The conditions at the bonded interface are assumed to be perfectly bonded: ð1Þ
ð2Þ
wð1Þ ðr; 0Þ ¼ wð2Þ ðr; 0Þ;
r3h ðr; 0Þ ¼ r3h ðr; 0Þ; ð1Þ
ð2Þ
Erð1Þ ðr; 0Þ ¼ Eð2Þ r ðr; 0Þ;
Dh ðr; 0Þ ¼ Dh ðr; 0Þ; ð1Þ Bh ðr; 0Þ
¼
ð2Þ Bh ðr; 0Þ;
ð3:2Þ
ð2Þ H ð1Þ r ðr; 0Þ ¼ H r ðr; 0Þ:
The superscripts (1) and (2) in Eqs. (3.1) and (3.2) denote the materials 1 and 2, respectively. Substituting Eq. (2.19) into Eqs. (3.1) and (3.2), it will give rise to a 12 · 12 homogeneous system of linear equations as ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c1 c44 sinðkaÞ þ c2 c44 cosðkaÞ c3 e15 sinðkaÞ þ c4 e15 cosðkaÞ ð1Þ ð1Þ
ð1Þ ð1Þ
c5 q15 sinðkaÞ þ c6 q15 cosðkaÞ ¼ 0; ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c1 e15 sinðkaÞ þ c2 e15 cosðkaÞ þ c3 e11 sinðkaÞ c4 e11 cosðkaÞ ð1Þ ð1Þ
ð1Þ ð1Þ
þ c5 d 11 sinðkaÞ c6 d 11 cosðkaÞ ¼ 0; ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c1 q15 sinðkaÞ þ c2 q15 cosðkaÞ þ c3 d 11 sinðkaÞ c4 d 11 cosðkaÞ ð1Þ ð1Þ
ð1Þ ð1Þ
þ c5 l11 sinðkaÞ c6 l11 cosðkaÞ ¼ 0; ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c2 c44 þ c4 e15 þ c6 q15 c2 c44 c4 e15 c6 q15 ¼ 0; c2 e15 c4 e11 c6 d 11 c2 e15 þ c4 e11 þ c6 d 11 ¼ 0; ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
c2 q15 c4 d 11 c6 l11 c2 q15 þ c4 d 11 þ c6 l11 ¼ 0; ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
c1 c1 ¼ 0; c3 c3 ¼ 0; c5 c5 ¼ 0; ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
c1 c44 sinðkbÞ þ c2 c44 cosðkbÞ þ c3 e15 sinðkbÞ þ c4 e15 cosðkbÞ þ c5 q15 sinðkbÞ þ c6 q15 cosðkbÞ ¼ 0; c1 e15 sinðkbÞ þ c2 e15 cosðkbÞ c3 e11 sinðkbÞ c4 e11 cosðkbÞ c5 d 11 sinðkbÞ c6 d 11 cosðkbÞ ¼ 0; ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
c1 q15 sinðkbÞ þ c2 q15 cosðkbÞ c3 d 11 sinðkbÞ c4 d 11 cosðkbÞ c5 l11 sinðkbÞ c6 l11 cosðkbÞ ¼ 0 ð3:3Þ or in matrix notation Kc ¼ 0;
ð3:4Þ
where ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
c ¼ fc1 ; c2 ; c3 ; c4 ; c5 ; c6 ; c1 ; c2 ; c3 ; c4 ; c5 ; c6 gT :
ð3:5Þ
For a non-trivial solution of c in Eq. (3.4) k must be a root of the following characteristic equation det K ¼ jKj ¼ 0:
ð3:6Þ
Eq. (3.6) can be expanded and simplified as b1 cos3 ðbkÞ sin3 ðakÞ þ b2 cos2 ðbkÞ sin2 ðakÞ cosðakÞ sinðbkÞ þ b3 cosðbkÞ sinðakÞ cos2 ðakÞ sin2 ðbkÞ þ b4 cos3 ðakÞ sin3 ðbkÞ ¼ 0;
ð3:7Þ
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2319
where ð1Þ
ð1Þ
ð1Þ ð1Þ ð1Þ
ð2Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ ð1Þ ð1Þ
b1 ¼ c44 ½d 11 2 þ 2d 11 e15 q15 ½q15 2 e11 ½e15 2 l11 c44 e11 l11 ; ð2Þ
ð2Þ ð1Þ ð1Þ
ð1Þ ð2Þ ð1Þ
ð1Þ ð1Þ ð2Þ
ð1Þ ð2Þ ð1Þ
b2 ¼ c44 ½d 11 2 þ 2c44 d 1ð11Þ d ð11Þ þ 2d 11 e15 q15 þ 2d 11 e15 q15 þ 2d 11 e15 q15 2q15 q15 e11 ; ð1Þ
ð2Þ
ð1Þ ð2Þ ð1Þ
ð2Þ ð1Þ ð1Þ
ð1Þ ð2Þ ð1Þ
ð1Þ
ð2Þ
ð1Þ ð1Þ ð2Þ
½q15 2 e11 2e15 e15 l11 c44 e11 l11 c44 e11 l11 ½e15 2 l11 c44 e11 l11 ; ð1Þ
ð2Þ
ð2Þ ð1Þ ð2Þ
ð2Þ ð2Þ ð1Þ
ð1Þ ð1Þ ð2Þ
ð1Þ ð2Þ ð2Þ
ð1Þ ð2Þ ð2Þ
b3 ¼ c44 ½d 11 2 þ 2c44 d 11 d ð11Þ þ 2d 11 e15 q15 þ 2d 11 e15 q15 þ 2d 11 e15 q15 2q15 q15 e11 ; ð2Þ 2 ð1Þ
ð1Þ ð2Þ ð2Þ
ð2Þ ð1Þ ð2Þ
ð1Þ ð2Þ ð2Þ
ð2Þ 2 ð1Þ
ð2Þ ð2Þ ð1Þ
½q15 e11 2e15 e15 l11 c44 e11 l11 c44 e11 l11 ½e15 l11 c44 e11 l11 ; ð2Þ
ð2Þ
ð2Þ ð2Þ ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ ð2Þ ð2Þ
b4 ¼ c44 ½d 11 2 þ 2d 11 e15 q15 ½q15 2 e11 ½e15 2 l11 c44 e11 l11 :
Α1 Α2 Α3
1
Re[Ai]
0.5
0
-0.5
-1 0
20
40
60
80
100
κ Fig. 2. Variation of the real part of Ai (i = 1, 2, 3) when 0 < j 6 100.
Α1 Α2 Α3
Im[Ai]
0.01
0
-0.01
0
20
40
κ
60
80
Fig. 3. Variation of the image part of Ai (i = 1, 2, 3) when 0 < j 6 100.
100
ð3:8Þ
2320
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
Eq. (3.7) can be further rewritten into the following three equations: sin½ða þ bÞk þ A1 sin½ða bÞk ¼ 0; sin½ða þ bÞk þ A2 sin½ða bÞk ¼ 0; sin½ða þ bÞk þ A3 sin½ða bÞk ¼ 0;
ð3:9Þ
where pffiffiffi 3ðg fÞ gþf A1 ¼ ; þi 2 pffiffiffi 2 3ðg fÞ gþf A2 ¼ ; i 2 2 A3 ¼ g þ 1
ð3:10Þ
with g¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffi 3 þ D; 2
f¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffi 3 D; 2
q2 p 3 a2 þ ; p ¼ 1 þ a2 ; 4 27 3 3b1 þ b2 b3 3b4 ; a1 ¼ b1 þ b2 þ b3 þ b4 3b1 b2 b3 þ 3b4 ; a2 ¼ b1 þ b2 þ b3 þ b4 b1 b2 þ b3 b4 : a3 ¼ b1 þ b2 þ b3 þ b4
q¼
D¼
ð3:11Þ 2a31 a1 a2 þ a3 ; 27 3
ð3:12Þ
ð3:13Þ
Note that g and f must satisfy gf = p/3 [15].
0
+
-0.1
κ =0.05 κ =0.256 κ =1 κ =2 κ =5 κ =10
*
-0.2
+
+ -0.3
+
δ1
-0.4 -0.5
+
*
+
*
*
*
+
+
*
+
*
+
*
+
*
+
*
+
*
+
x2
-0.6 -0.7
#1
α =180°
x1 -0.8
β #2
-0.9 -1
0
30
60
90
β (Degree)
120
150
Fig. 4. Variation of the singularity order d1 when 0 < j < 11.34.
180
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331 0
*+
x2
-0.1
+
x1
Re[δ 1 ]
-0.2
*
β
#2
-0.3
κ =12 κ =14 κ =16 κ =30 κ =50 κ =100
*
α=180o
#1
2321
+
* *
-0.4
+
*
*+ *
+ *
*
-0.5
*
0.004
+*
+*
0.002
+*
+ *
*+ +
+ +
*
*
*
* 0
*
+
*
+
+
*
*
*
*+ *+
*++ *
Im[δ 1 ]
*
+
+
*+ *
0.006
*
+
*+
+*
*+ 0
-0.002 -0.004 -0.006
30
60
90
120
150
180
β (Degree) Fig. 5. Variation of the singularity order d1 (real and imaginary part) when 11.34 < j 6 100.
0
*+
x2
+*
-0.1
*
+
β
#2
+
δ2
* +
* *
+
-0.3
κ =0.05 κ =0.5 κ =1 κ =2 κ =5 κ =10
* x1
+ -0.2
α
#1
=180°
+
*
+
*
+
-0.4
+*
+
*
+
*
-0.5 0
30
60
90
120
150
β (Degree) Fig. 6. Variation of the singularity order d2 when 0 < j < 11.34.
+
* +180
2322
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
3.2. Case 2: The fixed and grounded–fixed and grounded wedge The second conditions considered at the boundary edges are assumed to be fixed and grounded: wð1Þ ðr; aÞ ¼ wð2Þ ðr; bÞ ¼ 0; /ð1Þ ðr; aÞ ¼ /ð2Þ ðr; bÞ ¼ 0; ð1Þ
ð3:14Þ
ð2Þ
u ðr; aÞ ¼ u ðr; bÞ ¼ 0: The continuity conditions at the bonded interface are assumed to be the same as case 1, i.e., Eq. (3.2). The derivation of the characteristic equation is similar to that for the case 1 so that we only record the final result below. We have b4 cos3 ðbkÞ sin3 ðakÞ þ b3 cos2 ðbkÞ sin2 ðakÞ cosðakÞ sinðbkÞ þ b2 cosðbkÞ sinðakÞ cos2 ðakÞ sin2 ðbkÞ þ b1 cos3 ðakÞ sin3 ðbkÞ ¼ 0:
ð3:15Þ
0
*
κ =0.053 κ =0.5 κ =1 κ =5 κ =11.34 κ =100
* +
-0.1
*
+
*
-0.2
δ3
*
-0.3
+
*
x2
+
*
α=180°
#1
+
x1
+
*
-0.4 β
#2
+
*
+
*+
-0.5 0
30
60
90
120
150
*180
β (Degree) Fig. 7. Variation of the singularity order d3 when 0 < j 6 100. Table 1 The order of the generalized stress singularity when j < 11.34
d1 d2 d3
j 6 0.053
0.053 < j < 0.256
j = 0.256
0.256 < j < 11.34
1.0 to 0.5 0.5 to 0 0.5 to 0
1.0 to 0.5 0.5 to 0 0.5 to 0 when b > 90.0 (No singularity when b < 90.0)
0.5 0.5 to 0 0.5 to 0 when b > 90.0 (No singularity when b < 90.0)
0.5 to 0 0.5 to 0 0.5 to 0 when b > 90.0 (No singularity when b < 90.0)
Table 2 The order of the generalized stress singularity when j > 11.34 j > 11.34 d1 and d2 (conjugate complex) d3 (real)
1. Re[d1] = Re[d2]: 0.5 to 0 2. Im[d1] = Im[d2]: 0.0065 to 0.0065 0.5 to 0.16 when b > 124.0 (no singularity when b < 124.0)
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2323
It is noted that the form of this equation is quite similar to that for the case 1 problem, i.e., Eq. (3.7), except the appearing of the coefficients bi, i = 1, 2, 3, 4 which are expressed in Eq. (3.8), are placed in a different manner. Hence, Eq. (3.15) can also be rewritten into a form exactly the same as those stated in Eq. (3.9). The three coefficients A1, A2 and A3 are expressed by Eq. (3.10) again but the contents are different from those for case 1, i.e., a1 and a3 expressed in Eq. (3.13) should be replaced by a1 ¼
3b1 þ b2 b3 3b4 ; b1 þ b2 þ b3 þ b4
a3 ¼
b1 b2 þ b3 b 4 : b1 þ b2 þ b3 þ b 4
ð3:16Þ
No replacement is required for a2. 3.3. Case 3: The traction-free, electrically and magnetically insulated–fixed and grounded wedge The last case treated will be one of the conditions at the boundary edges is assumed to be traction-free, electrically and magnetically insulated, the other is assumed to be fixed and grounded: ð1Þ
ð1Þ
r3h ðr; aÞ ¼ 0;
ð1Þ
Dh ðr; aÞ ¼ 0;
wð2Þ ðr; bÞ ¼ 0;
Bh ðr; aÞ ¼ 0;
/ð2Þ ðr; bÞ ¼ 0;
ð3:17Þ
uð2Þ ðr; bÞ ¼ 0:
The continuity conditions at the bonded interface are again perfectly bonded, the same as case 1, i.e., Eq. (3.2). The final characteristic equation for the last case is b4 cos3 ðbkÞ cos3 ðakÞ þ b3 cos2 ðbkÞ cos2 ðakÞ sinðakÞ sinðbkÞ þ b2 cosðbkÞ cosðakÞ sin2 ðakÞ sin2 ðbkÞ þ b1 sin3 ðakÞ sin3 ðbkÞ ¼ 0;
ð3:18Þ
where bi, i = 1, 2, 3, 4 are defined by Eq. (3.8). It is interesting to note that the appearing of the coefficients bi, i = 1, 2, 3, 4 in above equation is exactly the same as those for the case 2 as shown in Eq. (3.15). This equation can also be rewritten in the following three equations:
0
*+
*+
*+ *+ *
*
+
-0.1
*
*
+
+ -0.2
+
*
δ1
+
* +
x2
-0.3
* +
α=180°
#1
x1
* +
-0.4
β
#2
κ =0.05 κ =0.256 κ =1 κ =2 κ =5 κ =10
* +
*
+
-0.5 0
30
60
90
120
150
β (Degree) Fig. 8. Variation of the singularity order d1 when 0 < j < 11.34.
180
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
cos½ða þ bÞk þ A1 cos½ða bÞk ¼ 0; cos½ða þ bÞk þ A2 cos½ða bÞk ¼ 0;
ð3:19Þ
cos½ða þ bÞk þ A3 cos½ða bÞk ¼ 0; where three coefficients A1, A2 and A3 are again expressed in Eq. (3.10). It is observed that for three different cases the derived characteristic equations all take similar forms. The common feature is that the dependence of these roots on the material constants is all through the parameters Ai, i = 1, 2, 3. These parameters are related to material constants as defined through Eqs. (3.10)–(3.13). It is also observed that the behavior of the roots may be conveniently discussed according to what values of the parameter D will take. Depending on the values of the parameter D, we summarized the behavior of the characteristic roots as follows: (a) D = 0: All three roots are real. Among them, two roots are actually repeated roots. (b) D < 0: All roots are real. (c) D > 0 and a = b (i.e., the extended angles of the wedge are equal): All roots are real. (d) D > 0 and either a = 0 or b = 0 (i.e., one of the extended angles is zero): All roots are real.
*+ *
x2
+*
*
+
-0.1
*
α=180°
#1
Re[δ 1 ]
*+
x1
*
-0.2
+* -0.3
κ =12 κ =14 κ =16 κ =30 κ =50 κ =100
* -0.4
+
β
#2
* +
* *+
-0.5
* *+
* +
0.004
+*
*+ *+
0.002
+*
*+
+
+
*
+
+
* 30
*
+
+
60
*
90
*0
-0.002
*
*
*++ *
*
+
+
* 0
+*
*+ *
*
Im[δ1 ]
0
β (Degree)
-0.004 120
150
180
Fig. 9. Variation of the singularity order d1 (real and imaginary part) when 11.34 < j 6 100.
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2325
(e) D > 0 and a 5 b and both extended angles are non-zero: One real root and the rest two are complex conjugate roots. According to above classifications, we note that among all cases there are only one case which may give rise to complex eigenvalues. That is the case of a non-vanished, unequaled extended angles (i.e., a 5 0, b 5 0 and a 5 b) wedge with D > 0. All the rest cases will possess only real roots. 4. Numerical results and discussions In this section, we will present some numerical results and discuss the effects of the material constants on the behaviors of the order of the generalized stress singularity di = ki 1 (i = 1, 2, 3). The materials considered are BaTiO3 and CoFe2O4. The properties are given as follows [16]: BaTiO3: 2
2
c44 ¼ 43 109 N=m ;
e15 ¼ 11:6 C=m ;
e11 ¼ 11:2 109 C2 =N m2 ;
ð4:1Þ
l11 ¼ 5:0 106 N s2 =C2 :
CoFe2O4: c44 ¼ 45:3 109 N=m2 ; e11 ¼ 0:08 10
9
q15 ¼ 550 N=Am; e15 ¼ 0;
2
ð4:2Þ
l11 ¼ 590 106 N s2 =C2 ;
2
C =N m ;
where N is the force in Newtons, C is the charge in Coulombs, A is the current in Amperes, s is the time in seconds and m is the length in meters. Let the material 1 of the magnetoelectroelastic bonded wedge be CoFe2O4 and material 2 be BaTiO3, respectively. For the purpose of the following discussions, we will choose the material property e15 of material 2 to be a multiple of the material BaTiO3, i.e., we let ð2Þ
e15 ¼ j 11:6
ð4:3Þ
with these material constants shown in Eqs. (4.1)–(4.3), those parameters defined by Eqs. (3.12) and (3.13) may be evaluated and especially the D becomes 0
*+ +
*
*
+
-0.1
+ +
*
*
-0.2
+ +
δ2
*
-0.3
+
*
x2 #1
+
*
α=180°
+
*
x1
-0.4
#2
κ =0.05 κ =0.5 κ =1 κ =2 κ =5 κ =10
+
*
+
*
β
+
*
-0.5 0
30
60
90
120
150
β (Degree) Fig. 10. Variation of the singularity order d2 when 0 < j < 11.34.
+
* +180
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
D ¼ ð5932:33 4852:35 j2 1419:69 j4 173:04 j6 69712 j8 þ 0:0653025 j10 6
1:95345 1017 j12 Þ=ð7:35873 þ j2 Þ :
ð4:4Þ
It can be shown that D = 0 when the value j is equal to 11.34. The variations of the parameters Ai (i = 1, 2, 3) defined by Eqs. (3.10)–(3.13) versus j are plotted in Figs. 2 and 3. It is observed that A3 is always real for allj while A1 and A2 are real only if j 6 11.34 (i.e., D 6 0) and they both will be complex conjugate if j P 11.34 (i.e., D P 0). It may be verified that the value j is equal to 0.256 and 0.053 when A1 = 1 and A3 = 1, 0
κ =0.01 κ =0.1 κ =1 κ =5 κ =11.34 κ =100
x2
-0.1
*
α=180°
#1
-0.2
x1
-0.3
+
β
#2
δ3
-0.4 -0.5
* * * * * * *+ *+ * + * + * +* +* *+ +
+
-0.6
+
-0.7
+
-0.8 -0.9 -1
+
*0
30
60
90
120
150
180
β (Degree) Fig. 11. Variation of the singularity order d3 when 0 < j 6 100.
0
*
x2
-0.1 #1
α=180°
x1
-0.2 -0.3
#2
*
δ3
-0.4 -0.5
+
-0.6
* +
β
κ =0.01 κ =0.04 κ =0.053 κ =0.06 κ =0.08 κ =0.1
* * * * *+ *+ *+ *+ *+ *+ *+ * + * + + + +
-0.7 -0.8 -0.9 -1 0
+
0.02
0.04
0.06
β (Degree)
0.08
Fig. 12. Variation of the singularity order d3 when j 6 0.1 and b is small angle.
0.1
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2327
respectively. In the following we will focus on the problem where the wedge angle a of the material 1 is equal to 180 while the wedge angle b of the material 2 may be varied, i.e. 0 < b < 180. 4.1. Case 1 We first present the order of singularity di (i = 1, 2, 3) for the conditions of the boundary edges are tractionfree, electrically and magnetically insulated. The numerical results are shown in Figs. 4–7. The common features shown in Figs. 5–7 may also be summarized in Tables 1 and 2 according to whether j > 11.34 (D > 0) or j < 11.34 (D < 0). First note that all the order of the generalized stress singularity di (i = 1, 2, 3) listed in Table 1 are real when j 6 11.34. It is also observed that there order of the stress singularity for d1 is square root when j = 0.256 and those for d1 will be higher than square root if j < 0.256 and lower than square root if j > 0.256. The order of singularity for d3 disappear if j > 0.053 and b < 90.0 which may be easily observed from Fig. 7. Table 3 The order of the generalized stress singularity when j < 11.34
d1
d2 d3
j < 0.053
j = 0.053
0.053 < j < 0.256
0.256 < j < 11.34
0.5 to 0 when b > 90.0 (No singularity when b < 90.0) 0.5 to 0 0.5 to 0
0.5 to 0 when b > 90.0 (No singularity when b < 90.0) 0.5 to 0 0.5
0.5 to 0 when b > 90.0 (No singularity when b < 90.0) 0.5 to 0 1 to 0.5
0.5 to 0
0.5 to 0 1 to 0.5
Table 4 The order of the generalized stress singularity when j > 11.34 j > 11.34 1. Re[d1] = Re[d2]: 0.5 to 0 2. Im[d1] = Im[d2]: 0.0045 to 0.0045 1 to 0.5 when b > 25.0 (no singularity when b < 25.0)
d1 and d2 (conjugate complex) d3 (real)
-0.5
*+
*+
*+ *
*
+
* +
* +
* +
*
*
*
*
+ + +
-0.55
δ1
+ + x2
-0.6
* +
κ =0.05 κ =0.256 κ =1 κ =2 κ =5 κ =10
#1
α=180°
x1 #2
β
-0.65 0
30
60
90
120
150
β (Degree) Fig. 13. Variation of the singularity order d1 when 0 < j < 11.34.
180
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
-0.5
*+
-0.52
*
+
*
*+
Re[δ 1 ]
-0.54
* *
+
*+
*
-0.56
x2
+ *
*
-0.58
α =180°
#1
κ =12 κ =14 κ =16 κ =30 κ =50 κ =100
-0.6
+
x1
* *+
β
-0.62
#2
*
-0.64
+
+
*
+
+
+
0.002 0.001 0
*
0
+
0.003
*
*
Im[δ 1 ]
*+
*+ *+
*+
*
*+
*
*
*
+
*
30
-0.001
+
*
+
+
*
60
+
*
90
+
*
*
120
-0.002
+
+
* -0.003
* 150
180
β (Degree) Fig. 14. Variation of the singularity order d1 (real and imaginary part) when 11.34 < j 6 100.
-0.5
*+
*
+
*
-0.55
+
+ +
*
+
*
δ2
-0.6
+ +
* #1
+
*
x2
-0.65
+
*
α=180°
-0.7
+ +
*
x1 #2
κ =0.05 κ =0.5 κ =1 κ =2 κ =5 κ =10
*
β
+ +
* *
-0.75 0
30
60
90
120
150
β (Degree) Fig. 15. Variation of the singularity order d2 when 0 < j < 11.34.
* 180
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2329
Listed in Table 2 are the results for j > 11.34. For this situation, it is noted that the order of d2 is found to be the complex conjugate of the order of d1, i.e., d2 ¼ d1 . Therefore the value of d2 for j > 11.34 may be obtained from those presented in Fig. 5. As to d3, which remains real, will possess no singularity at all when j > 11.34 and b < 124.0.
* *+ * +* +* *+ * +* +* *+ *+* +
0 -0.1
*
-0.2
+
κ =0.01 κ =0.1 κ =0.5 κ =1 κ =5 κ =100
-0.3
δ3
-0.4 -0.5
x2
*+
-0.6
α=180°
#1
x1 -0.7
β
#2
-0.8 -0.9 -1 0
30
60
90
120
150
180
β (Degree) Fig. 16. Variation of the singularity order d3 when 0 < j 6 100.
0
+
+
+
+
+
+
+
+
+
+
+
+
-0.1
κ =0.01 κ =0.04 κ =0.053 κ =0.054 κ =0.08 κ =0.1
*
-0.2
+
-0.3
+
+
-0.4
δ3
x2 -0.5
*+
α=180°
#1
-0.6
x1
-0.7
*
-0.8 -0.9
#2
β
* * * * * * * * * * * * *
-1 0
0.02
0.04
0.06
0.08
β (Degree) Fig. 17. Variation of the singularity order d3 when j 6 0.1 and b is small angle.
0.1
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
Table 5 The order of the generalized stress singularity when j < 11.34.
d1 d2 d3
j < 0.053
0.053 < j < 0.256
j = 0.256
0.053 < j < 11.34
0.5 to 0.45 when b < 171.0 (No singularity when b > 171.0) 0.75 to 0.5 1.0 to 0.5
0.5 to0.45 when b < 171.0 (No singularity when b > 171.0) 0.75 to0.5 0.5 to 0 when b < 90 (No singularity when b > 90)
0.5
0.63 to 0.5
0.75 to 0.5 0.5 to 0 when b < 90 (No singularity when b > 90)
0.75 to 0.5 0.5 to 0 when b < 90 (No singularity when b > 90)
Table 6 The order of the generalized stress singularity when j > 11.34 j > 11.34 d1 and d2 (conjugate complex) d3 (real)
1. Re[d1] = Re[d2]: 0.64 to 0.5 2. Im[d1] = Im[d2]: 0.0033 to 0.0033 0.5 to 0 when b < 90 (no singularity when b > 90)
4.2. Case 2 The results for the second case where the boundary conditions are fixed and grounded are shown in Figs. 8–12. Again the results shown in the figures may be summarized in Tables 3 and 4 according to whether j > 11.34 (D > 0) or j < 11.34 (D < 0). First note again that all di (i = 1, 2, 3) listed in Table 3 are real when j 6 11.34 since D < 0. The boundary conditions which are fixed and grounded for the present case do play a significant effect on di (i = 1, 2, 3). For the present case there will be stronger order of the stress singularity for d3 when j > 0.053 and the singularity for d1 will disappear when j < 0.256 and b < 90.0. Listed in Table 4 are the results for j > 11.34. The generalized stress singularities for d1 and d2 are complex conjugate to each other again, hence, for j > 11.34 the order of d2 may be obtained from those for d1 which are given in Fig. 9. Finally we see that the order d3 has no singularity when b < 25.0. 4.3. Case 3 The last case treated is the conditions of the boundary edges one of them is traction-free, electrically insulated and magnetically insulated, and the other the other is fixed, grounded. Numerical results are presented in Figs. 13–17. Results for the last case are again summarized in Tables 5 and 6. The stress singularity di (i = 1, 2, 3) listed in Table 5 are all real since j 6 11.34. And there will be stronger order of the stress singularity for d2 and d3 when j < 0.053. The order of d1 will disappear if j < 0.256 and b > 171.0 (see Fig. 12) while the order of d3 will have no singularity when j > 0.053 and b > 90 (see Figs. 16 and 17). Listed in Table 6 shows that the feature of d2 ¼ d1 preserves again when j > 11.34, which indicates that the order of d2 for j > 11.34 may be obtained from Fig. 14. Finally we see that the order of d3 disappear when b > 90. 5. Conclusions In this paper, we deal with the problems of the stress singularity of a magnetoelectroelastic bonded antiplane wedge by complex potential function and eigenfunction expansion method. The orders of stress singularity are real and conjugate complex when D > 0, a 5 0 (the wedge angle of magnetoelectroelastic material 1), b 5 0 (the wedge angle of magnetoelectroelastic material 2) and a 5 b, all the others are real. Due to the magnetoelectroelastic effects, the singular behavior is different from that of an elastic bonded wedge and is similar to that for a piezoelectric bonded wedge. The numerical results show that the stronger order of the generalized stress singularity d which range is 1 to 0.5 may be generated for some special composite materials when the wedge angle a = 180 and b is small angle.
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References [1] D.B. Bogy, Edge-bonded dissimilar orthogonal elastic wedge under normal and shear loading, ASME J. Appl. Mech. 35 (1968) 460– 466. [2] D.B. Bogy, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions, ASME J. Appl. Mech. 38 (1971) 377–386. [3] C.C. Ma, B.L. Hour, Analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation, Int. J. Solids Struct. 25 (1989) 1295–1309. [4] P.S. Theocaris, The order of singularity at a multi-wedge corner of a composite plate, Int. J. Eng. Sci. 12 (1974) 107–120. [5] D.H. Chen, H. Nisitani, Singular stress field in two bonded wedges, Trans. Jpn. Soc. Mech. Eng. A58 (1992) 457–464. [6] D.H. Chen, H. Nisitani, Singular stress field near the corner of jointed dissimilar materials, ASME J. Appl. Mech. 60 (1993) 607–613. [7] X.L. Xu, R.K.N.D. Rajapakse, On Singularities in composite piezoelectric wedge and junctions, Int. J. Solids Struct. 37 (2000) 3253– 3275. [8] C.H. Chue, C.D. Chen, Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problem, Int. J. Solids Struct. 39 (2002) 3131–3158. [9] C.D. Chen, C.H. Chue, Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear, Int. J. Solids Struct. 40 (2003) 6513–6526. [10] K.Q. Hu, G.Q. Li, Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading, Int. J. Solids Struct. 42 (2005) 2823–2835. [11] K.Q. Hu, G.Q. Li, Electro-magneto-elastic analysis of a piezoelectromagnetic strip with a finite crack under longitudinal shear, Mech. Mater. 37 (2005) 925–934. [12] J.Y. Li, Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials, Int. J. Eng. Sci. 38 (1993) 1993–2011. [13] J.X. Liu, X.G. Liu, Y.B. Zhao, Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack, Int. J. Eng. Sci. 39 (2001) 1405–1418. [14] E. Pan, Exact solution for simply support and multilayered magneto-electro-elastic plates, ASME J. Appl. Mech. 68 (2001) 608–618. [15] P.G. Ciarlet, J.L. Lions Handbook of Numerical Analysis, vol. 3, Elsevier, North-Holland, 1989. [16] J.H. Huang, H.K. Liu, W.L. Dai, The optimized fiber volume fraction for magnetoelectric coupling effect in piezoelectric– piezomagnetic continuous fiber reinforced composite, Int. J. Eng. Sci. 38 (2000) 1207–1217.
Investigation of the stress singularity of a magnetoelectroelastic bonded antiplane wedge W.C. Sue a, J.Y. Liou b, J.C. Sung a
a,*
Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC b Department of Civil Engineering, Kao Yuan University, Kaohsiung 821, Taiwan, ROC
Received 1 November 2004; received in revised form 1 August 2006; accepted 6 September 2006 Available online 30 October 2006
Abstract The order of the stress singularity of a magnetoelectroelastic bonded antiplane wedge is analyzed by complex potential function and eigenfunction expansion method. Contrary to the familiar problem of elastic anisotropic bonded wedges which always produce real values for the order of singularity, the results of the magnetoelectroelastic bonded wedges may be real or complex. Numerical results are presented for problems with different boundary conditions. In particular, special behaviors of the order of the stress singularity for some degenerate composite materials and for some special wedge angles are noted. 2006 Elsevier Inc. All rights reserved. Keywords: Magnetoelectroelastic; Stress singularity; Wedge; Eigenfunction expansion
1. Introduction The investigations of the stress singularities at the apex of the wedge generated by the discontinuities of material properties and geometry are one of the typical fracture mechanics problems. In the literature, the stress singularities near the tip for either isotropic or anisotropic elastic wedges have been widely investigated (e.g., [1–6]). Due to the important applications of the piezoelectric materials, the stress singularity problems near the tip of the piezoelectric bonded wedge have been studied recently, for example, Xu and Rajapakse [7], Chue and Chen [8] and Chen and Chue [9]. Recently, cracks in a magnetoelectroelastic material under antiplane shear loading and in-plane electrical and magnetic loadings have been treated by Hu and Li [10,11]. Because of a unique property of magnetoelectric coupling effect resulting from the interaction between magnetism, electricity, and mechanism, piezoelectric–piezomagnetic composite materials or magnetoelectroelastic materials are used in a remarkably wide variety of application including intelligent or active structures, magnetoelectromechanical transducers, magnetic sensors, and acoustic actuators. Due to the singular stress fields
*
Corresponding author. Tel.: +886 2757575; fax: +886 6 2358542. E-mail address: [email protected] (J.C. Sung).
0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2006.09.003
2314
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
near the tip of the wedge inside materials or structures, fractures occurred at the stress singularity point make the disadvantage of the function. The singular stress fields can be expressed as rij ðr; hÞ ¼ krk1 fij ðhÞ;
ð1:1Þ
where (k 1) represents the orders of the stress singularity, k is the stress intensity factor, and fij(h) is the angular function. This paper attempts to explore the orders of the stress singularity near the tip of the magnetoelectroelastic bonded antiplane wedge by complex potential function and eigenfunction expansion method. Following Section 2, the basic equations for magnetoelectroelastic media are summarized first. In Section 3, the solution for dissimilar magnetoelectroelastic bonded wedge is presented. In Sections 4, the numerical results are discussed. Finally, this paper is concluded in Section 5. 2. Basic equations for magnetoelectroelastic media In a fixed rectangular coordinate systems xi (i = 1, 2, 3), the basic equations for a linear magnetoelectroelastic solid are given by [12–14] Constitutive equations rij ¼ cijkl skl ekij Ek qkij H k ; ð2:1Þ
Di ¼ eikl skl þ eik Ek þ d ik H k ; Bi ¼ qikl skl þ d ik Ek þ lik H k The extended strain–displacement relation 1 sij ¼ ðui;j þ uj;i Þ; 2
Ei ¼ /;i ;
H i ¼ u;i :
ð2:2Þ
Governing equations for no body force, electric charge density and electric current density rij;j ¼ 0;
Di;i ¼ 0;
Bi;i ¼ 0:
ð2:3Þ
In Eqs. (2.1)–(2.3), a comma followed by i (i = 1, 2, 3) denotes partial differentiation with respect to the ith spatial coordinate; the repeated indices imply summation over that indices. ui, / and u are the elastic displacements, electric potential and magnetic potential, respectively. rij, skl, Di, Ek, Bi and Hk are the stress, strain, electric displacement, electric field, magnetic induction and magnetic field, respectively. cijkl, eikl, qikl, eik, dik and lik are the elastic, piezoelectric, piezomagnetic, dielectric, magnetoelectric and magnetic permeability constants, respectively. These material constants satisfy the following symmetric relations: cijkl ¼ cklij ¼ cijlk ¼ cjikl ; qkij ¼ qkji ;
eik ¼ eki ;
d ik ¼ d ki ;
lik ¼ lki :
ekij ¼ ekji ; ð2:4Þ
Since we are interested only in the problems of antiplane displacement component coupled with the inplane electric field components and inplane magnetic field components, the displacement, electric field and magnetic field will take the following special forms, which are expressed as u1 ¼ u2 ¼ 0;
u3 ¼ wðx1 ; x2 Þ;
E1 ¼ E1 ðx1 ; x2 Þ; E2 ¼ E2 ðx1 ; x2 Þ; E3 ¼ 0; H 1 ¼ H 1 ðx1 ; x2 Þ; H 2 ¼ H 2 ðx1 ; x2 Þ; H 3 ¼ 0: With above special forms, then Eqs. (2.2) and (2.3) may simplify to
ð2:5Þ
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
1 s31 ¼ w;1 ; 2 E1 ¼ /;1 ;
1 s32 ¼ w;2 ; 2 E2 ¼ /;2 ;
H 1 ¼ u;1 ;
H 2 ¼ u;2 :
2315
ð2:6Þ
r31;1 þ r32;2 ¼ 0; ð2:7Þ
D1;1 þ D2;2 ¼ 0; B1;1 þ B2;2 ¼ 0:
For a special case of a transversely isotropic magnetoelectroelastic material, the constitutive Eq. (2.1) take the form [14] 9 2 8 c11 r11 > > > > > > > > 6 > r22 > > > 6 c12 > > > 6 > < r33 = 6 c13 ¼6 6 > > 6 0 > r32 > > > > > 6 > > 4 0 > r31 > > > > > ; : 0 r12 2
9 38 s11 > > > > > > s > 7> c11 c13 0 0 0 22 > > > 7> > > > 7> < 7 s33 = c13 c33 0 0 0 7 7> 2s > 0 0 c44 0 0 32 > 7> > > > 7> > > > 5> 0 0 0 c44 0 2s > > 31 > > ; : 0 0 0 0 ðc11 c12 Þ=2 2s12 3 3 2 0 0 e31 0 0 q31 6 0 6 9 0 e31 7 0 q31 7 78 9 6 0 78 6 E H1 > > 7> 7> 6 6 1 = = < < 7 7 6 0 6 0 e33 0 0 q33 7 E2 6 7 H2 ; 6 7 7 6 0 e 6 > 0 7> 15 ; 6 0 q15 0 7> ; : > : 6 7 E3 7 H3 6 6 4 e15 0 4 q15 0 0 5 0 5 c12
0
0
c13
0
0
0
0
0 0 9 8 s 11 > > > > > > > 8 9 2 > 3> s 22 > > > > > D 0 0 0 0 e 0 > > > > 1 15 = < < = 6 7 s33 0 0 e15 0 0 5 D2 ¼ 4 0 > > > > ; : > > > 2s32 > e31 e31 e33 0 D3 0 0 > > > > > 2s31 > > > > > ; : 2s12 2 38 9 2 e11 0 d 11 0 0 > = < E1 > 6 7 6 þ 4 0 e11 0 5 E2 þ 4 0 d 11 > ; : > 0 0 e33 0 0 E3 9 8 s11 > > > > > > > > 8 9 2 > 3> s22 > > > > B 0 0 0 0 q 0 > > > 15 = < = < 1> s 33 6 7 0 0 q15 0 0 5 B2 ¼ 4 0 > > > > 2s32 > ; : > > q31 q31 q33 0 B3 0 0 > > > > 2s > > > > > 31 > > ; : 2s12 2 38 9 2 d 11 0 l11 0 0 > = < E1 > 6 7 6 þ 4 0 d 11 0 5 E2 þ 4 0 l11 > ; : > 0 0 d 33 0 0 E3
ð2:8Þ
0
9 38 0 > = < H1 > 7 0 5 H2 ; > > ; : d 33 H3
9 38 0 > = < H1 > 7 0 5 H2 ; > > ; : l33 H3
ð2:9Þ
ð2:10Þ
2316
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
where the contracted notations have been used to describe the material constants. Combining Eqs. (2.6)–(2.9) and (2.10) may yield c44 r2 w þ e15 r2 / þ q15 r2 u ¼ 0; e15 r2 w e11 r2 / d 11 r2 u ¼ 0; 2
2
ð2:11Þ
2
q15 r w d 11 r / l11 r u ¼ 0 or in matrix notation Cr2 u ¼ 0;
ð2:12Þ
where r2 ¼ o2 =ox21 þ o2 =ox22 is a two-dimensional Laplacian operator and 8 9 2 3 c44 e15 q15 > =
ð2:13Þ
In polar coordinate systems (r, h), the generalized strain (shear strain, electric field and magnetic field) can be expressed in terms of u as 9 9 8 8 2s3h > > > = = < 2s3r > < o o sr ¼ Er ¼ u; sh ¼ Eh ¼ u: ð2:14Þ > > > > ; or ; r oh : : H r H h Similarly, the components of generalized stress (shear stress, electric displacement and magnetic induction) in polar coordinates can be expressed in terms of u as 9 9 8 8 r3h > > > = = < r3r > < o o tr ¼ Dr ¼ C u; th ¼ Dh ¼ C u: ð2:15Þ > > > > or r oh ; ; : : Br Bh It is known that the general solution of Eq. (2.12) is 9 8 > = < fw ðzÞ > uðx1 ; x2 Þ ¼ Re f/ ðzÞ ¼ Re½f ðzÞ; > > ; : fu ðzÞ
ð2:16Þ
where fw(z), f/(z) and fu(z) are three undetermined analytical complex functions of a complex variable z = x1 + ix2 = reih and Re denotes the real part. With the general solution expressed above, the generalized strain and generalized stress may be evaluated simply by the following equations sr ¼ Re½eih f 0 ðzÞ; tr ¼ C Re½eih f 0 ðzÞ;
sh ¼ Re½ieih f 0 ðzÞ; th ¼ C Re½ieih f 0 ðzÞ:
ð2:17Þ
To determine the stress singularities, we may use the eigenfunction expansion method by adopting the complex functions as fw ðzÞ ¼ cw zk ¼ rk ½c1 cosðkhÞ þ c2 sinðkhÞ; f/ ðzÞ ¼ c/ zk ¼ rk ½c3 cosðkhÞ þ c4 sinðkhÞ; k
ð2:18Þ
k
fu ðzÞ ¼ cu z ¼ r ½c5 cosðkhÞ þ c6 sinðkhÞ; where ci (i = 1–6) are undetermined complex constants; the constant k is the eigenvalue. For the present problem under consideration, constants in Eq. (2.18) are related in the following manner: c1 = cw, c2 = icw, c3 = c/, c4 = ic/, c5 = cu and c6 = icu. Substituting Eq. (2.18) into Eqs. (2.13)–(2.17) yields
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
9 8 8 9 > > = = < 2s3r >
2317
ð2:19Þ
where 2
c1
6 c ¼ 4 c 3 c5
c2
3
7 c4 5 ; c6
2
c2
6 c ¼ 4 c4 c6
c1
3
7 c3 5; c5
g¼
cosðkhÞ : sinðkhÞ
ð2:20Þ
In the following, the power of the stress singularities k 1 will be investigated for the magnetoelectroelastic bonded wedge. Those d = k 1 in the region 1 < ReðdÞ < 0
ð2:21Þ
will be of interest and will be presented. 3. Characteristic equations for the dissimilar magnetoelectroelastic bonded wedge Consider a two-bonded magnetoelectroelastic wedge as shown in Fig. 1. The wedge angles of magnetoelectroelastic materials 1 and 2 are denoted by a and b, respectively. Let the bonded interface be lie on the x1-axis. The characteristic equations for the determination of the order of stress singularities are derived in the following for three different boundary conditions prescribed on the surfaces of the wedge. 3.1. Case 1: The traction-free, electrically and magnetically insulated–traction-free, electrically and magnetically insulated wedge The first conditions considered at the boundary edges are assumed to be traction-free, electrically and magnetically insulated:
Fig. 1. Two-material bonded wedge.
2318
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331 ð1Þ
ð2Þ
r3h ðr; aÞ ¼ r3h ðr; bÞ ¼ 0; ð1Þ
ð2Þ
ð3:1Þ
Dh ðr; aÞ ¼ Dh ðr; bÞ ¼ 0; ð1Þ Bh ðr; aÞ
¼
ð2Þ Bh ðr; bÞ
¼ 0:
The conditions at the bonded interface are assumed to be perfectly bonded: ð1Þ
ð2Þ
wð1Þ ðr; 0Þ ¼ wð2Þ ðr; 0Þ;
r3h ðr; 0Þ ¼ r3h ðr; 0Þ; ð1Þ
ð2Þ
Erð1Þ ðr; 0Þ ¼ Eð2Þ r ðr; 0Þ;
Dh ðr; 0Þ ¼ Dh ðr; 0Þ; ð1Þ Bh ðr; 0Þ
¼
ð2Þ Bh ðr; 0Þ;
ð3:2Þ
ð2Þ H ð1Þ r ðr; 0Þ ¼ H r ðr; 0Þ:
The superscripts (1) and (2) in Eqs. (3.1) and (3.2) denote the materials 1 and 2, respectively. Substituting Eq. (2.19) into Eqs. (3.1) and (3.2), it will give rise to a 12 · 12 homogeneous system of linear equations as ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c1 c44 sinðkaÞ þ c2 c44 cosðkaÞ c3 e15 sinðkaÞ þ c4 e15 cosðkaÞ ð1Þ ð1Þ
ð1Þ ð1Þ
c5 q15 sinðkaÞ þ c6 q15 cosðkaÞ ¼ 0; ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c1 e15 sinðkaÞ þ c2 e15 cosðkaÞ þ c3 e11 sinðkaÞ c4 e11 cosðkaÞ ð1Þ ð1Þ
ð1Þ ð1Þ
þ c5 d 11 sinðkaÞ c6 d 11 cosðkaÞ ¼ 0; ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c1 q15 sinðkaÞ þ c2 q15 cosðkaÞ þ c3 d 11 sinðkaÞ c4 d 11 cosðkaÞ ð1Þ ð1Þ
ð1Þ ð1Þ
þ c5 l11 sinðkaÞ c6 l11 cosðkaÞ ¼ 0; ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð1Þ ð1Þ
ð1Þ ð1Þ
c2 c44 þ c4 e15 þ c6 q15 c2 c44 c4 e15 c6 q15 ¼ 0; c2 e15 c4 e11 c6 d 11 c2 e15 þ c4 e11 þ c6 d 11 ¼ 0; ð1Þ ð1Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
c2 q15 c4 d 11 c6 l11 c2 q15 þ c4 d 11 þ c6 l11 ¼ 0; ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
c1 c1 ¼ 0; c3 c3 ¼ 0; c5 c5 ¼ 0; ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
c1 c44 sinðkbÞ þ c2 c44 cosðkbÞ þ c3 e15 sinðkbÞ þ c4 e15 cosðkbÞ þ c5 q15 sinðkbÞ þ c6 q15 cosðkbÞ ¼ 0; c1 e15 sinðkbÞ þ c2 e15 cosðkbÞ c3 e11 sinðkbÞ c4 e11 cosðkbÞ c5 d 11 sinðkbÞ c6 d 11 cosðkbÞ ¼ 0; ð2Þ ð2Þ
ð2Þ ð2Þ
ð2Þ ð2Þ
c1 q15 sinðkbÞ þ c2 q15 cosðkbÞ c3 d 11 sinðkbÞ c4 d 11 cosðkbÞ c5 l11 sinðkbÞ c6 l11 cosðkbÞ ¼ 0 ð3:3Þ or in matrix notation Kc ¼ 0;
ð3:4Þ
where ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
c ¼ fc1 ; c2 ; c3 ; c4 ; c5 ; c6 ; c1 ; c2 ; c3 ; c4 ; c5 ; c6 gT :
ð3:5Þ
For a non-trivial solution of c in Eq. (3.4) k must be a root of the following characteristic equation det K ¼ jKj ¼ 0:
ð3:6Þ
Eq. (3.6) can be expanded and simplified as b1 cos3 ðbkÞ sin3 ðakÞ þ b2 cos2 ðbkÞ sin2 ðakÞ cosðakÞ sinðbkÞ þ b3 cosðbkÞ sinðakÞ cos2 ðakÞ sin2 ðbkÞ þ b4 cos3 ðakÞ sin3 ðbkÞ ¼ 0;
ð3:7Þ
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2319
where ð1Þ
ð1Þ
ð1Þ ð1Þ ð1Þ
ð2Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ ð1Þ ð1Þ
b1 ¼ c44 ½d 11 2 þ 2d 11 e15 q15 ½q15 2 e11 ½e15 2 l11 c44 e11 l11 ; ð2Þ
ð2Þ ð1Þ ð1Þ
ð1Þ ð2Þ ð1Þ
ð1Þ ð1Þ ð2Þ
ð1Þ ð2Þ ð1Þ
b2 ¼ c44 ½d 11 2 þ 2c44 d 1ð11Þ d ð11Þ þ 2d 11 e15 q15 þ 2d 11 e15 q15 þ 2d 11 e15 q15 2q15 q15 e11 ; ð1Þ
ð2Þ
ð1Þ ð2Þ ð1Þ
ð2Þ ð1Þ ð1Þ
ð1Þ ð2Þ ð1Þ
ð1Þ
ð2Þ
ð1Þ ð1Þ ð2Þ
½q15 2 e11 2e15 e15 l11 c44 e11 l11 c44 e11 l11 ½e15 2 l11 c44 e11 l11 ; ð1Þ
ð2Þ
ð2Þ ð1Þ ð2Þ
ð2Þ ð2Þ ð1Þ
ð1Þ ð1Þ ð2Þ
ð1Þ ð2Þ ð2Þ
ð1Þ ð2Þ ð2Þ
b3 ¼ c44 ½d 11 2 þ 2c44 d 11 d ð11Þ þ 2d 11 e15 q15 þ 2d 11 e15 q15 þ 2d 11 e15 q15 2q15 q15 e11 ; ð2Þ 2 ð1Þ
ð1Þ ð2Þ ð2Þ
ð2Þ ð1Þ ð2Þ
ð1Þ ð2Þ ð2Þ
ð2Þ 2 ð1Þ
ð2Þ ð2Þ ð1Þ
½q15 e11 2e15 e15 l11 c44 e11 l11 c44 e11 l11 ½e15 l11 c44 e11 l11 ; ð2Þ
ð2Þ
ð2Þ ð2Þ ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ ð2Þ ð2Þ
b4 ¼ c44 ½d 11 2 þ 2d 11 e15 q15 ½q15 2 e11 ½e15 2 l11 c44 e11 l11 :
Α1 Α2 Α3
1
Re[Ai]
0.5
0
-0.5
-1 0
20
40
60
80
100
κ Fig. 2. Variation of the real part of Ai (i = 1, 2, 3) when 0 < j 6 100.
Α1 Α2 Α3
Im[Ai]
0.01
0
-0.01
0
20
40
κ
60
80
Fig. 3. Variation of the image part of Ai (i = 1, 2, 3) when 0 < j 6 100.
100
ð3:8Þ
2320
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
Eq. (3.7) can be further rewritten into the following three equations: sin½ða þ bÞk þ A1 sin½ða bÞk ¼ 0; sin½ða þ bÞk þ A2 sin½ða bÞk ¼ 0; sin½ða þ bÞk þ A3 sin½ða bÞk ¼ 0;
ð3:9Þ
where pffiffiffi 3ðg fÞ gþf A1 ¼ ; þi 2 pffiffiffi 2 3ðg fÞ gþf A2 ¼ ; i 2 2 A3 ¼ g þ 1
ð3:10Þ
with g¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffi 3 þ D; 2
f¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffi 3 D; 2
q2 p 3 a2 þ ; p ¼ 1 þ a2 ; 4 27 3 3b1 þ b2 b3 3b4 ; a1 ¼ b1 þ b2 þ b3 þ b4 3b1 b2 b3 þ 3b4 ; a2 ¼ b1 þ b2 þ b3 þ b4 b1 b2 þ b3 b4 : a3 ¼ b1 þ b2 þ b3 þ b4
q¼
D¼
ð3:11Þ 2a31 a1 a2 þ a3 ; 27 3
ð3:12Þ
ð3:13Þ
Note that g and f must satisfy gf = p/3 [15].
0
+
-0.1
κ =0.05 κ =0.256 κ =1 κ =2 κ =5 κ =10
*
-0.2
+
+ -0.3
+
δ1
-0.4 -0.5
+
*
+
*
*
*
+
+
*
+
*
+
*
+
*
+
*
+
*
+
x2
-0.6 -0.7
#1
α =180°
x1 -0.8
β #2
-0.9 -1
0
30
60
90
β (Degree)
120
150
Fig. 4. Variation of the singularity order d1 when 0 < j < 11.34.
180
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331 0
*+
x2
-0.1
+
x1
Re[δ 1 ]
-0.2
*
β
#2
-0.3
κ =12 κ =14 κ =16 κ =30 κ =50 κ =100
*
α=180o
#1
2321
+
* *
-0.4
+
*
*+ *
+ *
*
-0.5
*
0.004
+*
+*
0.002
+*
+ *
*+ +
+ +
*
*
*
* 0
*
+
*
+
+
*
*
*
*+ *+
*++ *
Im[δ 1 ]
*
+
+
*+ *
0.006
*
+
*+
+*
*+ 0
-0.002 -0.004 -0.006
30
60
90
120
150
180
β (Degree) Fig. 5. Variation of the singularity order d1 (real and imaginary part) when 11.34 < j 6 100.
0
*+
x2
+*
-0.1
*
+
β
#2
+
δ2
* +
* *
+
-0.3
κ =0.05 κ =0.5 κ =1 κ =2 κ =5 κ =10
* x1
+ -0.2
α
#1
=180°
+
*
+
*
+
-0.4
+*
+
*
+
*
-0.5 0
30
60
90
120
150
β (Degree) Fig. 6. Variation of the singularity order d2 when 0 < j < 11.34.
+
* +180
2322
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
3.2. Case 2: The fixed and grounded–fixed and grounded wedge The second conditions considered at the boundary edges are assumed to be fixed and grounded: wð1Þ ðr; aÞ ¼ wð2Þ ðr; bÞ ¼ 0; /ð1Þ ðr; aÞ ¼ /ð2Þ ðr; bÞ ¼ 0; ð1Þ
ð3:14Þ
ð2Þ
u ðr; aÞ ¼ u ðr; bÞ ¼ 0: The continuity conditions at the bonded interface are assumed to be the same as case 1, i.e., Eq. (3.2). The derivation of the characteristic equation is similar to that for the case 1 so that we only record the final result below. We have b4 cos3 ðbkÞ sin3 ðakÞ þ b3 cos2 ðbkÞ sin2 ðakÞ cosðakÞ sinðbkÞ þ b2 cosðbkÞ sinðakÞ cos2 ðakÞ sin2 ðbkÞ þ b1 cos3 ðakÞ sin3 ðbkÞ ¼ 0:
ð3:15Þ
0
*
κ =0.053 κ =0.5 κ =1 κ =5 κ =11.34 κ =100
* +
-0.1
*
+
*
-0.2
δ3
*
-0.3
+
*
x2
+
*
α=180°
#1
+
x1
+
*
-0.4 β
#2
+
*
+
*+
-0.5 0
30
60
90
120
150
*180
β (Degree) Fig. 7. Variation of the singularity order d3 when 0 < j 6 100. Table 1 The order of the generalized stress singularity when j < 11.34
d1 d2 d3
j 6 0.053
0.053 < j < 0.256
j = 0.256
0.256 < j < 11.34
1.0 to 0.5 0.5 to 0 0.5 to 0
1.0 to 0.5 0.5 to 0 0.5 to 0 when b > 90.0 (No singularity when b < 90.0)
0.5 0.5 to 0 0.5 to 0 when b > 90.0 (No singularity when b < 90.0)
0.5 to 0 0.5 to 0 0.5 to 0 when b > 90.0 (No singularity when b < 90.0)
Table 2 The order of the generalized stress singularity when j > 11.34 j > 11.34 d1 and d2 (conjugate complex) d3 (real)
1. Re[d1] = Re[d2]: 0.5 to 0 2. Im[d1] = Im[d2]: 0.0065 to 0.0065 0.5 to 0.16 when b > 124.0 (no singularity when b < 124.0)
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2323
It is noted that the form of this equation is quite similar to that for the case 1 problem, i.e., Eq. (3.7), except the appearing of the coefficients bi, i = 1, 2, 3, 4 which are expressed in Eq. (3.8), are placed in a different manner. Hence, Eq. (3.15) can also be rewritten into a form exactly the same as those stated in Eq. (3.9). The three coefficients A1, A2 and A3 are expressed by Eq. (3.10) again but the contents are different from those for case 1, i.e., a1 and a3 expressed in Eq. (3.13) should be replaced by a1 ¼
3b1 þ b2 b3 3b4 ; b1 þ b2 þ b3 þ b4
a3 ¼
b1 b2 þ b3 b 4 : b1 þ b2 þ b3 þ b 4
ð3:16Þ
No replacement is required for a2. 3.3. Case 3: The traction-free, electrically and magnetically insulated–fixed and grounded wedge The last case treated will be one of the conditions at the boundary edges is assumed to be traction-free, electrically and magnetically insulated, the other is assumed to be fixed and grounded: ð1Þ
ð1Þ
r3h ðr; aÞ ¼ 0;
ð1Þ
Dh ðr; aÞ ¼ 0;
wð2Þ ðr; bÞ ¼ 0;
Bh ðr; aÞ ¼ 0;
/ð2Þ ðr; bÞ ¼ 0;
ð3:17Þ
uð2Þ ðr; bÞ ¼ 0:
The continuity conditions at the bonded interface are again perfectly bonded, the same as case 1, i.e., Eq. (3.2). The final characteristic equation for the last case is b4 cos3 ðbkÞ cos3 ðakÞ þ b3 cos2 ðbkÞ cos2 ðakÞ sinðakÞ sinðbkÞ þ b2 cosðbkÞ cosðakÞ sin2 ðakÞ sin2 ðbkÞ þ b1 sin3 ðakÞ sin3 ðbkÞ ¼ 0;
ð3:18Þ
where bi, i = 1, 2, 3, 4 are defined by Eq. (3.8). It is interesting to note that the appearing of the coefficients bi, i = 1, 2, 3, 4 in above equation is exactly the same as those for the case 2 as shown in Eq. (3.15). This equation can also be rewritten in the following three equations:
0
*+
*+
*+ *+ *
*
+
-0.1
*
*
+
+ -0.2
+
*
δ1
+
* +
x2
-0.3
* +
α=180°
#1
x1
* +
-0.4
β
#2
κ =0.05 κ =0.256 κ =1 κ =2 κ =5 κ =10
* +
*
+
-0.5 0
30
60
90
120
150
β (Degree) Fig. 8. Variation of the singularity order d1 when 0 < j < 11.34.
180
2324
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
cos½ða þ bÞk þ A1 cos½ða bÞk ¼ 0; cos½ða þ bÞk þ A2 cos½ða bÞk ¼ 0;
ð3:19Þ
cos½ða þ bÞk þ A3 cos½ða bÞk ¼ 0; where three coefficients A1, A2 and A3 are again expressed in Eq. (3.10). It is observed that for three different cases the derived characteristic equations all take similar forms. The common feature is that the dependence of these roots on the material constants is all through the parameters Ai, i = 1, 2, 3. These parameters are related to material constants as defined through Eqs. (3.10)–(3.13). It is also observed that the behavior of the roots may be conveniently discussed according to what values of the parameter D will take. Depending on the values of the parameter D, we summarized the behavior of the characteristic roots as follows: (a) D = 0: All three roots are real. Among them, two roots are actually repeated roots. (b) D < 0: All roots are real. (c) D > 0 and a = b (i.e., the extended angles of the wedge are equal): All roots are real. (d) D > 0 and either a = 0 or b = 0 (i.e., one of the extended angles is zero): All roots are real.
*+ *
x2
+*
*
+
-0.1
*
α=180°
#1
Re[δ 1 ]
*+
x1
*
-0.2
+* -0.3
κ =12 κ =14 κ =16 κ =30 κ =50 κ =100
* -0.4
+
β
#2
* +
* *+
-0.5
* *+
* +
0.004
+*
*+ *+
0.002
+*
*+
+
+
*
+
+
* 30
*
+
+
60
*
90
*0
-0.002
*
*
*++ *
*
+
+
* 0
+*
*+ *
*
Im[δ1 ]
0
β (Degree)
-0.004 120
150
180
Fig. 9. Variation of the singularity order d1 (real and imaginary part) when 11.34 < j 6 100.
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2325
(e) D > 0 and a 5 b and both extended angles are non-zero: One real root and the rest two are complex conjugate roots. According to above classifications, we note that among all cases there are only one case which may give rise to complex eigenvalues. That is the case of a non-vanished, unequaled extended angles (i.e., a 5 0, b 5 0 and a 5 b) wedge with D > 0. All the rest cases will possess only real roots. 4. Numerical results and discussions In this section, we will present some numerical results and discuss the effects of the material constants on the behaviors of the order of the generalized stress singularity di = ki 1 (i = 1, 2, 3). The materials considered are BaTiO3 and CoFe2O4. The properties are given as follows [16]: BaTiO3: 2
2
c44 ¼ 43 109 N=m ;
e15 ¼ 11:6 C=m ;
e11 ¼ 11:2 109 C2 =N m2 ;
ð4:1Þ
l11 ¼ 5:0 106 N s2 =C2 :
CoFe2O4: c44 ¼ 45:3 109 N=m2 ; e11 ¼ 0:08 10
9
q15 ¼ 550 N=Am; e15 ¼ 0;
2
ð4:2Þ
l11 ¼ 590 106 N s2 =C2 ;
2
C =N m ;
where N is the force in Newtons, C is the charge in Coulombs, A is the current in Amperes, s is the time in seconds and m is the length in meters. Let the material 1 of the magnetoelectroelastic bonded wedge be CoFe2O4 and material 2 be BaTiO3, respectively. For the purpose of the following discussions, we will choose the material property e15 of material 2 to be a multiple of the material BaTiO3, i.e., we let ð2Þ
e15 ¼ j 11:6
ð4:3Þ
with these material constants shown in Eqs. (4.1)–(4.3), those parameters defined by Eqs. (3.12) and (3.13) may be evaluated and especially the D becomes 0
*+ +
*
*
+
-0.1
+ +
*
*
-0.2
+ +
δ2
*
-0.3
+
*
x2 #1
+
*
α=180°
+
*
x1
-0.4
#2
κ =0.05 κ =0.5 κ =1 κ =2 κ =5 κ =10
+
*
+
*
β
+
*
-0.5 0
30
60
90
120
150
β (Degree) Fig. 10. Variation of the singularity order d2 when 0 < j < 11.34.
+
* +180
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
D ¼ ð5932:33 4852:35 j2 1419:69 j4 173:04 j6 69712 j8 þ 0:0653025 j10 6
1:95345 1017 j12 Þ=ð7:35873 þ j2 Þ :
ð4:4Þ
It can be shown that D = 0 when the value j is equal to 11.34. The variations of the parameters Ai (i = 1, 2, 3) defined by Eqs. (3.10)–(3.13) versus j are plotted in Figs. 2 and 3. It is observed that A3 is always real for allj while A1 and A2 are real only if j 6 11.34 (i.e., D 6 0) and they both will be complex conjugate if j P 11.34 (i.e., D P 0). It may be verified that the value j is equal to 0.256 and 0.053 when A1 = 1 and A3 = 1, 0
κ =0.01 κ =0.1 κ =1 κ =5 κ =11.34 κ =100
x2
-0.1
*
α=180°
#1
-0.2
x1
-0.3
+
β
#2
δ3
-0.4 -0.5
* * * * * * *+ *+ * + * + * +* +* *+ +
+
-0.6
+
-0.7
+
-0.8 -0.9 -1
+
*0
30
60
90
120
150
180
β (Degree) Fig. 11. Variation of the singularity order d3 when 0 < j 6 100.
0
*
x2
-0.1 #1
α=180°
x1
-0.2 -0.3
#2
*
δ3
-0.4 -0.5
+
-0.6
* +
β
κ =0.01 κ =0.04 κ =0.053 κ =0.06 κ =0.08 κ =0.1
* * * * *+ *+ *+ *+ *+ *+ *+ * + * + + + +
-0.7 -0.8 -0.9 -1 0
+
0.02
0.04
0.06
β (Degree)
0.08
Fig. 12. Variation of the singularity order d3 when j 6 0.1 and b is small angle.
0.1
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2327
respectively. In the following we will focus on the problem where the wedge angle a of the material 1 is equal to 180 while the wedge angle b of the material 2 may be varied, i.e. 0 < b < 180. 4.1. Case 1 We first present the order of singularity di (i = 1, 2, 3) for the conditions of the boundary edges are tractionfree, electrically and magnetically insulated. The numerical results are shown in Figs. 4–7. The common features shown in Figs. 5–7 may also be summarized in Tables 1 and 2 according to whether j > 11.34 (D > 0) or j < 11.34 (D < 0). First note that all the order of the generalized stress singularity di (i = 1, 2, 3) listed in Table 1 are real when j 6 11.34. It is also observed that there order of the stress singularity for d1 is square root when j = 0.256 and those for d1 will be higher than square root if j < 0.256 and lower than square root if j > 0.256. The order of singularity for d3 disappear if j > 0.053 and b < 90.0 which may be easily observed from Fig. 7. Table 3 The order of the generalized stress singularity when j < 11.34
d1
d2 d3
j < 0.053
j = 0.053
0.053 < j < 0.256
0.256 < j < 11.34
0.5 to 0 when b > 90.0 (No singularity when b < 90.0) 0.5 to 0 0.5 to 0
0.5 to 0 when b > 90.0 (No singularity when b < 90.0) 0.5 to 0 0.5
0.5 to 0 when b > 90.0 (No singularity when b < 90.0) 0.5 to 0 1 to 0.5
0.5 to 0
0.5 to 0 1 to 0.5
Table 4 The order of the generalized stress singularity when j > 11.34 j > 11.34 1. Re[d1] = Re[d2]: 0.5 to 0 2. Im[d1] = Im[d2]: 0.0045 to 0.0045 1 to 0.5 when b > 25.0 (no singularity when b < 25.0)
d1 and d2 (conjugate complex) d3 (real)
-0.5
*+
*+
*+ *
*
+
* +
* +
* +
*
*
*
*
+ + +
-0.55
δ1
+ + x2
-0.6
* +
κ =0.05 κ =0.256 κ =1 κ =2 κ =5 κ =10
#1
α=180°
x1 #2
β
-0.65 0
30
60
90
120
150
β (Degree) Fig. 13. Variation of the singularity order d1 when 0 < j < 11.34.
180
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W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
-0.5
*+
-0.52
*
+
*
*+
Re[δ 1 ]
-0.54
* *
+
*+
*
-0.56
x2
+ *
*
-0.58
α =180°
#1
κ =12 κ =14 κ =16 κ =30 κ =50 κ =100
-0.6
+
x1
* *+
β
-0.62
#2
*
-0.64
+
+
*
+
+
+
0.002 0.001 0
*
0
+
0.003
*
*
Im[δ 1 ]
*+
*+ *+
*+
*
*+
*
*
*
+
*
30
-0.001
+
*
+
+
*
60
+
*
90
+
*
*
120
-0.002
+
+
* -0.003
* 150
180
β (Degree) Fig. 14. Variation of the singularity order d1 (real and imaginary part) when 11.34 < j 6 100.
-0.5
*+
*
+
*
-0.55
+
+ +
*
+
*
δ2
-0.6
+ +
* #1
+
*
x2
-0.65
+
*
α=180°
-0.7
+ +
*
x1 #2
κ =0.05 κ =0.5 κ =1 κ =2 κ =5 κ =10
*
β
+ +
* *
-0.75 0
30
60
90
120
150
β (Degree) Fig. 15. Variation of the singularity order d2 when 0 < j < 11.34.
* 180
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
2329
Listed in Table 2 are the results for j > 11.34. For this situation, it is noted that the order of d2 is found to be the complex conjugate of the order of d1, i.e., d2 ¼ d1 . Therefore the value of d2 for j > 11.34 may be obtained from those presented in Fig. 5. As to d3, which remains real, will possess no singularity at all when j > 11.34 and b < 124.0.
* *+ * +* +* *+ * +* +* *+ *+* +
0 -0.1
*
-0.2
+
κ =0.01 κ =0.1 κ =0.5 κ =1 κ =5 κ =100
-0.3
δ3
-0.4 -0.5
x2
*+
-0.6
α=180°
#1
x1 -0.7
β
#2
-0.8 -0.9 -1 0
30
60
90
120
150
180
β (Degree) Fig. 16. Variation of the singularity order d3 when 0 < j 6 100.
0
+
+
+
+
+
+
+
+
+
+
+
+
-0.1
κ =0.01 κ =0.04 κ =0.053 κ =0.054 κ =0.08 κ =0.1
*
-0.2
+
-0.3
+
+
-0.4
δ3
x2 -0.5
*+
α=180°
#1
-0.6
x1
-0.7
*
-0.8 -0.9
#2
β
* * * * * * * * * * * * *
-1 0
0.02
0.04
0.06
0.08
β (Degree) Fig. 17. Variation of the singularity order d3 when j 6 0.1 and b is small angle.
0.1
2330
W.C. Sue et al. / Applied Mathematical Modelling 31 (2007) 2313–2331
Table 5 The order of the generalized stress singularity when j < 11.34.
d1 d2 d3
j < 0.053
0.053 < j < 0.256
j = 0.256
0.053 < j < 11.34
0.5 to 0.45 when b < 171.0 (No singularity when b > 171.0) 0.75 to 0.5 1.0 to 0.5
0.5 to0.45 when b < 171.0 (No singularity when b > 171.0) 0.75 to0.5 0.5 to 0 when b < 90 (No singularity when b > 90)
0.5
0.63 to 0.5
0.75 to 0.5 0.5 to 0 when b < 90 (No singularity when b > 90)
0.75 to 0.5 0.5 to 0 when b < 90 (No singularity when b > 90)
Table 6 The order of the generalized stress singularity when j > 11.34 j > 11.34 d1 and d2 (conjugate complex) d3 (real)
1. Re[d1] = Re[d2]: 0.64 to 0.5 2. Im[d1] = Im[d2]: 0.0033 to 0.0033 0.5 to 0 when b < 90 (no singularity when b > 90)
4.2. Case 2 The results for the second case where the boundary conditions are fixed and grounded are shown in Figs. 8–12. Again the results shown in the figures may be summarized in Tables 3 and 4 according to whether j > 11.34 (D > 0) or j < 11.34 (D < 0). First note again that all di (i = 1, 2, 3) listed in Table 3 are real when j 6 11.34 since D < 0. The boundary conditions which are fixed and grounded for the present case do play a significant effect on di (i = 1, 2, 3). For the present case there will be stronger order of the stress singularity for d3 when j > 0.053 and the singularity for d1 will disappear when j < 0.256 and b < 90.0. Listed in Table 4 are the results for j > 11.34. The generalized stress singularities for d1 and d2 are complex conjugate to each other again, hence, for j > 11.34 the order of d2 may be obtained from those for d1 which are given in Fig. 9. Finally we see that the order d3 has no singularity when b < 25.0. 4.3. Case 3 The last case treated is the conditions of the boundary edges one of them is traction-free, electrically insulated and magnetically insulated, and the other the other is fixed, grounded. Numerical results are presented in Figs. 13–17. Results for the last case are again summarized in Tables 5 and 6. The stress singularity di (i = 1, 2, 3) listed in Table 5 are all real since j 6 11.34. And there will be stronger order of the stress singularity for d2 and d3 when j < 0.053. The order of d1 will disappear if j < 0.256 and b > 171.0 (see Fig. 12) while the order of d3 will have no singularity when j > 0.053 and b > 90 (see Figs. 16 and 17). Listed in Table 6 shows that the feature of d2 ¼ d1 preserves again when j > 11.34, which indicates that the order of d2 for j > 11.34 may be obtained from Fig. 14. Finally we see that the order of d3 disappear when b > 90. 5. Conclusions In this paper, we deal with the problems of the stress singularity of a magnetoelectroelastic bonded antiplane wedge by complex potential function and eigenfunction expansion method. The orders of stress singularity are real and conjugate complex when D > 0, a 5 0 (the wedge angle of magnetoelectroelastic material 1), b 5 0 (the wedge angle of magnetoelectroelastic material 2) and a 5 b, all the others are real. Due to the magnetoelectroelastic effects, the singular behavior is different from that of an elastic bonded wedge and is similar to that for a piezoelectric bonded wedge. The numerical results show that the stronger order of the generalized stress singularity d which range is 1 to 0.5 may be generated for some special composite materials when the wedge angle a = 180 and b is small angle.
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References [1] D.B. Bogy, Edge-bonded dissimilar orthogonal elastic wedge under normal and shear loading, ASME J. Appl. Mech. 35 (1968) 460– 466. [2] D.B. Bogy, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions, ASME J. Appl. Mech. 38 (1971) 377–386. [3] C.C. Ma, B.L. Hour, Analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation, Int. J. Solids Struct. 25 (1989) 1295–1309. [4] P.S. Theocaris, The order of singularity at a multi-wedge corner of a composite plate, Int. J. Eng. Sci. 12 (1974) 107–120. [5] D.H. Chen, H. Nisitani, Singular stress field in two bonded wedges, Trans. Jpn. Soc. Mech. Eng. A58 (1992) 457–464. [6] D.H. Chen, H. Nisitani, Singular stress field near the corner of jointed dissimilar materials, ASME J. Appl. Mech. 60 (1993) 607–613. [7] X.L. Xu, R.K.N.D. Rajapakse, On Singularities in composite piezoelectric wedge and junctions, Int. J. Solids Struct. 37 (2000) 3253– 3275. [8] C.H. Chue, C.D. Chen, Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problem, Int. J. Solids Struct. 39 (2002) 3131–3158. [9] C.D. Chen, C.H. Chue, Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear, Int. J. Solids Struct. 40 (2003) 6513–6526. [10] K.Q. Hu, G.Q. Li, Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading, Int. J. Solids Struct. 42 (2005) 2823–2835. [11] K.Q. Hu, G.Q. Li, Electro-magneto-elastic analysis of a piezoelectromagnetic strip with a finite crack under longitudinal shear, Mech. Mater. 37 (2005) 925–934. [12] J.Y. Li, Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials, Int. J. Eng. Sci. 38 (1993) 1993–2011. [13] J.X. Liu, X.G. Liu, Y.B. Zhao, Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack, Int. J. Eng. Sci. 39 (2001) 1405–1418. [14] E. Pan, Exact solution for simply support and multilayered magneto-electro-elastic plates, ASME J. Appl. Mech. 68 (2001) 608–618. [15] P.G. Ciarlet, J.L. Lions Handbook of Numerical Analysis, vol. 3, Elsevier, North-Holland, 1989. [16] J.H. Huang, H.K. Liu, W.L. Dai, The optimized fiber volume fraction for magnetoelectric coupling effect in piezoelectric– piezomagnetic continuous fiber reinforced composite, Int. J. Eng. Sci. 38 (2000) 1207–1217.