- Email: [email protected]
Fracture assessment of an interface crack between two dissimilar magnetoelectroelastic materials under heat flow and magnetoelectromechanical loadings
Acta Mechanica Solida Sinica, Vol. 24, No. 5, October, 2011 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
FRACTURE ASSESSMENT OF AN INTERFACE CRACK BETWEEN TWO DISSIMILAR MAGNETOELECTROELASTIC MATERIALS UNDER HEAT FLOW AND MAGNETOELECTROMECHANICAL LOADINGS Peng Ma1
Wenjie Feng1
Ray Kai-Leung Su2
1
( Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China) (2 Department of Civil Engineering, The University of Hong Kong, China)
Received 9 April 2011, revision received 23 June 2011
ABSTRACT A magnetoelectrically permeable interface crack between two semi-infinite magnetoelectroelastic planes under the action of a heat flow and remote magnetoelectromechanical loadings is considered, where the assumption of frictionless contact between two dissimilar halfplanes is adopted. Not only the solutions of the interface crack problem are presented in an explicit form, but also the general condition for the transition from a perfect thermal contact of two magnetoelectroelastic bodies to their separation is given.
KEY WORDS fracture, interface crack, magnetoelectrically permeable crack, frictionless interface, heat flow
I. INTRODUCTION If a body conducting heat contains a crack, the crack will act as an obstruction to the thermal flux, and produce a local perturbation in the temperature field. Thermal stresses will therefore be produced. And if the crack occurs at the interface between dissimilar materials, the complications will be further introduced. In the last ten years, although the crack problems of magnetoelectroelastic materials under temperature loadings are widely investigated[1–6] , because of the mathematical complexity, the reports related to the corresponding interface crack problems of a magnetoelectroelastic bimaterial are very limited[7, 8] . Among others, Gao and Noda[7] presented an explicitly analytic solution for a generalized two-dimensional problem of an interface crack between two dissimilar magnetoelectroelastic materials under uniform heat flow, where the crack was assumed to be magnetoelectrically permeable. It was shown that when uniform heat flow was applied at infinity, the structure of singularities was the same as that in a purely elastic bi-material system with interface cracks. Only recently, Zhu et al.[8] further investigated the mixed-mode stress intensity factors of 3D interface cracks in fully coupled magneto
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 10772123 and 11072160), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0971) and the Natural Science Fund for Outstanding Younger of Hebei Province (A2009001624), China.
· 430 ·
ACTA MECHANICA SOLIDA SINICA
2011
electrothermoelastic multiphase composites by using the extended hypersingular intergro-differential equation (E-HIDE) method. It is the purpose of the present work to analyse a 2D plane strain problem of an interface crack between two dissimilar magnetoelectroelastic materials under uniform heat flow and magnetoelectromechanical loadings at infinity. Although the crack surfaces are also assumed to be magnetoelectrically permeable, different from Cao and Noda[7] , in this study, the assumption of frictionless contact between the two dissimilar half-planes is adopted. Not only the solutions in a magnetoelectroelastic bimaterial system are presented in an explicit form, but also the general condition for the transition from a perfect thermal contact of two magnetoelectroelastic bodies to their separation is given.
II. BASIC RELATIONS FOR A MAGNETOELECTROELASTIC SOLID For a stationary process, in the absence of body forces and free charges, the gorverning equations for a linear magnetoelectrothermalelastic material can be presented in the form[7] ΠiJ,i = 0 qi,i = 0
(1) (2)
ΠiJ = EiJKl VK,l − βiJ T
(3)
qi = −λij T,j
(4)
where
⎧ ⎨ σij ΠiJ = Di ⎩ Bi ⎧ ⎨ uk VK = ϕ ⎩ ψ
with
and
EiJKl
⎧ cijkl ⎪ ⎪ ⎪ ⎪ elij ⎪ ⎪ ⎪ ⎪ eikl ⎪ ⎪ ⎨ flij = f ⎪ ikl ⎪ ⎪ ⎪ −εil ⎪ ⎪ ⎪ ⎪ −g ⎪ il ⎪ ⎩ −μil
(J = 1, 2, 3) (J = 4) (J = 5)
(5)
(K = 1, 2, 3) (K = 4) (K = 5)
(6)
(J, K = 1, 2, 3) (J = 1, 2, 3; K = 4) (J = 4; K = 1, 2, 3) (J = 1, 2, 3; K = 5) (J = 5; K = 1, 2, 3) (J, K = 4) (J = 4; K = 5, J = 5; K = 4) (J, K = 5)
(7)
In Eqs.(3)-(7), uk , ϕ, ψ and T are the elastic displacements, electric potential, magnetic potential and temperature change, respectively; σij , Di , Bi and q are the stresses, electric displacements, magnetic inductions and thermal flux, respectively; cijlm , εij and μij are the elastic tensors, dielectric and magnetic permeability tensors, respectively; eijk , fijk and gij are the piezoelectric, piezomagnetic and magnetoelectric coefficients, respectively; βiJ are the stress-temperature coefficients for J = 1, 2, 3, and βi4 and βi5 are the pyroelectric and pyromagnetic constants, respectively. In addition, a subscript comma denotes the partial differentiation with respect to the coordinates (i.e., x1 , x2 , x3 or x, y, z), and summation from 1 to 3 (1 to 5) over repeated lowercase (uppercase) subscripts is assumed. It should also be pointed out that the following symmetry relations in Eq.(7) hold true. cijlm = cjilm = clmij ,
ekji = ekij ,
fkji = fkij ,
εij = εji ,
gij = gji ,
μij = μji
(8)
Assuming all fields are independent on the coordinate x2 , one can obtain the following general solution of Eq.(2) (see also Cao and Noda[7] , or Herrmann and Loboda[9] , et al.) ¯ (¯ zt ) T = χ (zt ) + χ
(9)
Vol. 24, No. 5
· 431 ·
Peng Ma et al.: Fracture Assessment of an Interface Crack
qi = − (λi1 + τ λi2 ) χ (zt ) − (λi1 + τ¯λi2 ) χ ¯ (¯ zt )
(10)
where zt = x1 + τ x3 , the prime ( ) denotes differentiation with respect to the argument, the overbar stands for the complex conjugate, and τ is a root with a positive imaginary part of the equation λ33 τ 2 + (λ13 + λ31 ) τ + λ11 = 0
(11)
In the case of transverse isotropy and plane strain in the plane (x1 , x3 ), which has an essential practical significance, a general solution of Eq.(1) by using the Lekhnitskii-Eshelby-Stroh representation and its application to magnetoelectrothermoelastic materials can be presented in the form[7] ¯f¯(¯ V = Af (z) + cχ (zt ) + A z ) + c¯χ ¯ (¯ zt ) ¯ ¯ ¯ f (¯ z ) + dχ ¯ (¯ zt ) t = Bf (z) + dχ (zt ) + B
(12) (13)
where T
V = { u1 , u3 , ϕ, ψ }
(14) T
t ≡ t3 = { σ31 , σ33 , D3 , B3 }
(15)
f (z) = { f1 (z1 ) , f2 (z2 ) , f3 (z3 ) , f4 (z4 ) } T A = A1 , A2 , A3 , A4
T
(16) (17) T
zj = x1 + pj x3 (j = 1, 2, 3, 4), and for a fixed j, Aj = { a1j , a2j , a3j , a4j } and pj (j = 1, 2, 3, 4) are, respectively, an eigenvector and eigenvalue of the system
Q + pj R + RT + p2j T Aj = 0 (18) with QJK = E1JK1 ,
RJK = E1JK3 ,
TJK = E3JK3
which can also be expressed in contracted notation as follows: ⎡ ⎤ ⎡ ⎤ c11 c15 e11 f11 c15 c13 e31 f31 ⎢ c15 c55 e15 f15 ⎥ ⎢ c55 c53 e35 f35 ⎥ ⎥ ⎥ ⎢ Q=⎢ ⎣ e11 e15 −ε11 −g11 ⎦ , R = ⎣ e15 e13 −ε13 −g13 ⎦ , f11 f15 −g11 −μ11 f15 f13 −g13 −μ13
⎡
c55 ⎢ c35 T =⎢ ⎣ e35 f35
(19) c53 c33 e33 f33
e35 e33 −ε33 −g33
⎤ f35 f33 ⎥ ⎥ −g33 ⎦ −μ33
The vector c is defined from the equation Q + τ R + RT + τ 2 T c = N 1 + τ N 2
(20)
(21)
T
with N m = { βm1 , βm3 , βm4 , βm5 } (m = 1, 2), and the 4 × 4 matrix B and the vector d can be found by the formulas B = RT A + T AP , d = RT + τ T c − N 2 (22) with P = diag p1 , p2 , p3 , p4 . Furthermore, T ¯ P¯ f¯ (¯ ¯τ χ z ) − d¯ ¯ (¯ zt ) t1 = { σ11 , σ13 , D1 , B1 } = −BP f (z) − dτ χ (zt ) − B
(23)
It is worth to mention that χ (zt ) in Eqs.(9), (10), (12) and (13) is an arbitrary analytic function which can be determined later, and f (z) in Eqs.(12) and (13) is an arbitrary analytic vector function with four components determined later as well.
III. AN INTERFACE CRACK BETWEEN TWO DISSIMILAR MAGNETOELECTROTHERMOELASTIC PLANES As shown in Fig.1, a soft thermal insulator located in the region (−a, a) lies on the interface between two different magnetoelectroelastic semi-infinite planes x3 > 0 and x3 < 0 with material properties
· 432 ·
ACTA MECHANICA SOLIDA SINICA (1)
(1)
(1)
(2)
(2)
2011
(2)
defined by EiJKl , λij , βiJ and EiJKl , λij , βiJ , respectively. The thermal insulator is not assumed to cause stresses. At the same time, the component q3 of the thermal flux vector and the vector t are assumed to be continuous across the whole bimaterial interface; and the contact between the halfplanes is frictionless. On the other hand, from the point of view of magnetoelectric fields, the insulator is assumed to be a magnetoelectrically permeable interface crack. Thus, the boundary conditions at the interface can be presented in the form q3± = 0 (m) (m) σ13 (x1 , 0) = 0, σ33 (x1 , 0) = 0 [ϕ (x1 )] = 0, [ψ (x1 )] = 0 [D3 (x1 )] = 0, [B3 (x1 )] = 0 [T ] = 0,
(24)
[q3 ] = 0
(m) (x1 , 0) = 0, σ33 (x) = 0 [u3 (x1 )] = 0 [ϕ (x1 )] = 0, [ψ (x1 )] = 0 [D3 (x1 )] = 0, [B3 (x1 )] = 0 (m) σ13
(|x1 | ≤ a)
(|x1 | > a)
(25)
where
[Υ (x1 )] = Υ + (x1 , 0) − Υ − (x1 , 0) , Υ = T, u3 , σ33 , ϕ, ψ, D3 , B3 (26) In Eq.(26), the signs “+” and “−” denote the upper and lower parts of the interface. Certainly, the boundary conditions at infinity should also be satisfied.
Fig. 1. Magnetoelectroelastic plane with an interface crack under heat flow and magnetoelectromechanical loadings.
3.1. The Thermal Solution Introducing auxiliary functions ˜i (zt ) χi (zt ) = q3∞ + χ [9]
(i = 1, 2)
(27)
⎧ k (1) ⎪ ⎪ ˜1 (zt ) (x3 > 0) ⎨ 1 + (2) χ k (28) θ (zt ) = (2) k ⎪ ⎪ ⎩ 1 + (1) χ ˜2 (zt ) (x3 < 0) k √ with k = λ33 (τ − τ¯)/ (2i) i = −1 , k(1) , χ1 (zt ), χ ˜1 (zt ) and k(2) , χ2 (zt ), χ ˜2 (zt ), respectively, being related to the upper and lower half-planes, one can finally obtain (1) q3 (x1 , 0) = q3∞ − ik0 θ+ (x1 ) + θ− (x1 ) (29) and
Vol. 24, No. 5
Peng Ma et al.: Fracture Assessment of an Interface Crack
· 433 ·
where
k (1) k (2) (30) k (1) + k (2) It should be pointed out that in the process of deriving Eq.(29), the boundary condition q3 (z)|z→∞ = q3∞ has been used. Substituting Eq.(29) into Eq.(24)1 yields the following equation k0 =
θ+ (x1 ) + θ− (x1 ) = −
iq3∞ , k0
x1 ∈ (−a, a)
The solution of Eq.(31) disappearing at infinity can be presented in the form z iq ∞ √ −1 θ (z) = 3 2k0 z 2 − a2
(31)
(32)
integrating which gives the following expression: θ (z) =
iq3∞ 2 z − a2 − z 2k0
(33)
It is remarked that Eq.(33) satisfies the condition θ (z)|z→∞ = 0. Therefore, the temperature jump across the material interface for −a < x1 < a and the thermal flux for x1 > a can be presented in the following form: q3∞ (x1 + a)(a − x1 ) k0 x 1 −1 q3 (x1 , 0) = q3∞ 2 x1 − a2
[T (x1 )] = −
(34a)
(34b)
which completely defines the temperature jump and the thermal flux in the bimaterial system for any position of point a. Equations (34) also reveal that the temperature jump across the interface crack depends on not only the heat flow applied at infinity but also the related material properties, and that the thermal flux, however, is independent of √ material constants. In addition, it is easily obtained from 0.5 Eq.(34b) that lim q3 (x1 , 0) = q3∞ (0.5a) / x1 − a − q3∞ , which implies that the thermal flux at the x1 →a
crack tip has a square root singularity. 3.2. The Magnetoelectroelastic Solution Carrying out a derivation similar to the one by Herrmann and Loboda[9] , from Eqs.(12), (13), (24)2 , (25)2 , the following expressions at the interface are obtained: [V (x1 )] = W + (x1 ) − W − (x1 ) ¯ + (x1 ) − g (x1 ) t(1) (x1 , 0) = GW + (x1 ) − GW
(35) (36)
where W (z) = { W1 (z) , W2 (z) , W3 (z) , W4 (z) }T is an introduced unknown vector function, and W + (x1 ) = W (x1 + i0) , W − (x1 ) = W (x1 − i0) ¯ − (x1 ) g (x1 ) = hθ+ (x1 ) − hθ G=B
(1)
D
−1
(37) (38) (39)
with ¯ (1) D = A(1) − LB ∗ 1 ¯ − c∗ − k (2) d(1) h = (1) −G Ld k + k (2) and
−1 L = A(2) − B (2) ,
c∗ = k (2) c(1) + k (1) c¯(2) ,
d∗ = k (2) d(1) + k (1) d¯(2)
(40) (41)
(42)
· 434 ·
ACTA MECHANICA SOLIDA SINICA
2011
... ... It is worth to note that the related to the matrix H (or N ) and the ... matrix G and the vector W (z) are ... vector function ω (z) (or Φ (z)) of the paper of Gao and Noda[7] (or of the paper of Li and Kardomateas[10] ... ... −1 ... ... as iG−1 = H = N , W (z) = −i ω (z) = Φ (z), respectively, where all the quantities with the sign ‘. . .’ denote the corresponding quantities in the papers of Gao and Noda[7] and Li and Kardomateas[10]. Thus, Eqs.(35) and (36) have no distinct differences from the corresponding expressions given by Gao and Noda[7] . Furthermore, as demonstrated by Feng et al.[11] before, for a magnetoelectroelastic bimaterial system, the matrix G in Eq.(36) has the following form: ⎡ ⎤ ig11 g12 g13 g14 ⎢ g21 ig22 ig23 ig24 ⎥ ⎥ G=⎢ (43) ⎣ g31 ig32 ig33 ig34 ⎦ g41 ig42 ig43 ig44 where Im (gij ) = 0 (i, j = 1, 2, 3, 4), and g21 = −g12 , g31 = −g13 , g41 = −g14 , g32 = g23 , g42 = g24 , g43 = g34 . By the way, in the present study, the matrix G has been further calculated for one group of material combination, and the numerical results are displayed in Appendix. Also, as discussed by Herrmann and Loboda[9] for piezoelectric interfacial crack problems, one gets h = { iϑ1 ,
ϑ2 ,
ϑ3 ,
ϑ4 }T
(44)
[ψ]}T
(45)
where ϑi (i = 1, 2, 3, 4) are all real (e.g., see Appendix). Introducing the vectors (1)
S = {σ13 , [u3 ] , [ϕ] , Π=
{[u1 ] ,
(1) σ33 ,
(1) D3 ,
(1) B3 }T
(46)
the following relations can be easily found by means of Eqs.(35) and (36): ¯ W − (x1 ) + m∗ θ+ (x1 ) − m ¯ ∗ θ− (x1 ) S (x1 ) = M W + (x1 ) − M ¯ W − (x1 ) + n∗ θ+ (x1 ) − n ¯ ∗ θ− (x1 ) Π (x1 ) = N W + (x1 ) − N where the matrices M and N and the vectors m∗ and n∗ are defined as ⎡ ⎤ ⎡ ig11 g12 g13 g14 1 0 0 ⎢ 0 ⎥ ⎢ g21 ig22 ig23 1 0 0 ⎥, N = ⎢ M =⎢ ⎣ 0 ⎣ g31 ig32 ig33 0 1 0 ⎦ 0 0 0 1 g41 ig42 ig43 m∗ = { −h1 , 0, 0, 0 }T ,
⎤ 0 ig24 ⎥ ⎥ ig34 ⎦ ig44
n∗ = { 0, −h2 , −h3 , −h4 }T
By further introducing another auxiliary vector-function Ξ (z) (x3 > 0) M W (z) + m∗ θ (z) Ξ (z) = ¯ W (z) + m M ¯ ∗ θ (z) (x3 < 0)
(47) (48)
(49)
(50)
(51)
one can get S (x1 ) = Ξ + (x1 ) − Ξ − (x1 ) ¯ − (x1 ) − eθ+ (x1 ) + e¯θ − (x1 ) Π (x1 ) = ΩΞ + (x1 ) − ΩΞ
(52) (53)
with Ω = N M −1 and e = Ωm∗ − n∗ . It is easily seen that the vector function Ξ (z) is analytic in the whole plane with a cut along (|x1 | ≤ a, x3 = 0). The numerical analysis shows that for all considered material combinations, the matrix Ω is pure ¯ = −Ω, e imaginary and the vector e is real, i.e., Ω ¯ = e (by the way, in Appendix, both Ω and e are also numerically displayed for the considered material combination). Thus, Eq.(53) can be written in the form Π (x1 ) = Ω Ξ + (x1 ) + Ξ − (x1 ) − e θ+ (x1 ) − θ− (x1 ) (54)
Vol. 24, No. 5
Peng Ma et al.: Fracture Assessment of an Interface Crack
· 435 ·
(1)
Moreover, it follows from the relation (52) and the equations σ13 (x1 , 0) = 0, [ϕ (x1 )] = [ψ (x1 )] = 0 for |x1 | ≤ a that the functions Ξ1 (z), Ξ3 (z)and Ξ4 (z) are analytic in the whole plane. Noting the conditions at infinity one can conclude that Ξ1 (z) = C1 , Ξ3 (z) = C3 and Ξ4 (z) = C4 (C1 , C3 and C4 are, respectively, different constants to be determined). From Eqs.(46) and (24)2 , one can further get Π2 (x1 ) = 0
|(x1 | ≤ a)
(55)
where Π2 (x1 ) only involves function Ξ2 (x1 ). By virtue of [u1 (x1 )]|x1 →∞ = 0, the conditions at infinity can be written by means of Eq.(54) as follows: ⎧ ⎫ ⎫ ⎡ ⎤⎧ 0 ⎪ Ω11 Ω12 Ω13 Ω14 ⎪ C1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∞⎪ ⎬ ⎬ ⎢ Ω21 Ω22 Ω23 Ω24 ⎥ ⎨ Ξ2 | σ z→∞ ⎢ ⎥ = 0.5 (56) ∞ ⎣ Ω31 Ω32 Ω33 Ω34 ⎦ ⎪ C3 ⎪ D ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∞⎪ ⎭ ⎭ ⎩ Ω41 Ω42 Ω43 Ω44 C4 B Thus, C1 , C3 , C4 and Ξ2 |z→∞ can be determined by ⎧ ⎫ ⎡ C1 ⎪ Ω11 Ω12 ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ Ω21 Ω22 Ξ2 |z→∞ = 0.5 ⎢ ⎣ Ω31 Ω32 C3 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ C4 Ω41 Ω42
Ω12 Ω23 Ω33 Ω43
⎫ ⎤−1 ⎧ 0 ⎪ Ω14 ⎪ ⎪ ⎨ ∞⎪ ⎬ σ Ω24 ⎥ ⎥ ∞ Ω34 ⎦ ⎪ D ⎪ ⎪ ⎩ ∞⎪ ⎭ Ω44 B
By use of Eqs.(54) and (33), Eq.(55) can be written in the form e2 q3∞ 2Ω 21 2Ω 23 2Ω 24 Ξ2+ (x1 ) + Ξ2− (x1 ) = x21 − a2 − C1 − C3 − C4 Im (Ω22 ) k0 Ω22 Ω22 Ω22
(57)
(|x1 | ≤ a)
(58)
The solution of the Riemann problem (54) under the condition Ξ2 at infinity given in Eq.(57) can be written as z 2az σ∞ e2 q3∞ z−a √ √ + Ξ2 (z) = + z 2 − a2 ln 2 2 2 2πik0 Im(Ω22 ) z+a 2Ω22 z − a2 z −a Q Q Q − 21 C1 − 23 C3 − 24 C4 (59) Q22 Q22 Q22 The normal stress and the derivative of the normal displacement jump at the interface by use of Eqs.(59), (54) and (52), read then as follows: e2 q3∞ σ ∞ x1 2ax1 x1 − a (1) 2 2 + 2 σ33 (x1 , 0) = + x1 − a ln (|x1 | > a) (60) 2 πk 0 x1 + a x1 − a2 x1 − a2 ∞ ∞ q x e 2ax σ a − x 2 1 1 1 −1 3 − − (|x1 | < a) [u3 (x1 )] = [Im (Ω22 )] + a2 − x21 ln πk0 a + x1 a2 − x21 a2 − x21 (61) The formulas (60) and (61) are valid for any values of q3∞ and σ ∞ , and it is seen that similar to (1) thermal flux (see Eq.(34b)), both σ33 (x1 , 0) and [u3 (x1 )] pose a square root singularity at the edges of the zone of thermal insulation as well. As usual, by defining mode-I stress intensity factor (SIF) KI and crack open displacement (COD) intensity factor KCOD at the crack tip a as follows: (1) KI = lim + 2π (x1 − a)σ33 (x1 , 0) (62) x1 →a
KCOD = lim x1 →a−
[u3 (x1 )]
(63)
2π (a − x1 )
we can easily obtain from Eqs.(60) and (61) that KI =
2a1.5 e2 ∞ 0.5 q + σ ∞ (πa) , π 0.5 k0 3
KCOD =
1 −1 [Im (Ω22 )] KI π
(64)
· 436 ·
ACTA MECHANICA SOLIDA SINICA
2011
It is worth to note that, as expected, neither the SIF nor the COD intensity factor depends on the electric displacement D∞ and/or magnetic induction B ∞ , and that both the SIF and the COD intensity factor consist two independent parts, the first is induced by the mechanical loading σ ∞ at infinity, and the second is induced by the heat flow q3∞ . And as shown in Eqs.(64), the SIF induced by σ ∞ is independent of material properties of magnetoelectroelastic biomaterial system. However, the SIF induced by q3∞ simultaneously depends on the corresponding material properties. On the other hand, the COD intensity factor induced by either σ ∞ or q3∞ directly depends the material constants composed of them. In addition, form Eq.(64) (or Eqs.(60) and (61)), when q3∞ and σ ∞ satisfy the following relation: q3∞ = −
πk0 ∞ σ 2ae2
(65)
(1)
the singularities of both σ33 (x1 , 0) and [u3 (x1 )] can be absent. In this case, Eqs.(60) and (61) take the following form: e2 q3∞ x1 − a (1) σ33 (x1 , 0) = (|x1 | > a) x21 − a2 ln (66) πk0 x1 + a ∞ a − x1 −1 e2 q3 a2 − x21 ln (|x1 | < a) (67) [u3 (x1 )] = [Im (Ω22 )] πk0 a + x1 3.3. On an Admissible Direction of the Thermal Flux Because the logarithmic functions in Eq.(66) and in Eq.(67) are negative for point x1 situated in the vicinity of point a, and according to the numerical calculation the inequality Im (Ω22 ) > 0, it follows (1) that in the vicinity of point a, σ33 (x1 , 0) < 0 and [u3 (x1 )] > 0 hold true if and only if e2 q3∞ > 0
(68)
Noting the definition of e2 and Eqs.(49) and (50), the inequality (68) can be also written in the following form: g11 ϑ2 + g12 ϑ1 ∞ q3 > 0 (69) g11 The inequality (69) defines the direction of the thermal flux q3 for which a transition from a perfect thermal contact of two magnetoelectroelastic bodies to their separation is possible. Moreover, because (1) the Eqs.(66) and (67) define the asymptotic behavior of σ33 (x1 , 0) and [u3 (x1 )] at the transition point a, respectively, the inequality virtually can be treated as a general condition for the perfect thermal contact. In addition, it should be pointed out that if the inequality (68) (or (69)) is not valid, Eqs.(25)1 and (25)2 cannot be used, then an imperfect thermal contact should be introduced.
IV. BRIEF CONCLUSIONS (1) Different from the temperature jump across the interface crack, the thermal flux at the interface depends on only the heat flow applied at infinity, and has a square root singularity at the crack tip. (2) Both the normal stress and the derivative of the normal displacement jump at the interface induced by either heat flow or mechanical loading pose a square root singularity at the crack tip. (3) Both magnetical and electrical loadings have no effects on mode-I SIF and COD intensity factor. (1) (4) The singularities of both σ33 (x1 , 0) and [u3 (x1 )] are absent as q3∞ = −πk0 σ ∞ /(2ae2 ). (5) For perfect thermal contact, the general condition (g11 ϑ2 + g12 ϑ1 )q3∞ /g11 > 0 should be satisfied.
References [1] Gao,C.F., Kessler,H. and Balke,H., Fracture analysis of electromagnetic thermoelastic solids. European Journal of Mechanics A-Solids, 2003, 22: 433-442. [2] Niraula,O.P. and Wang,B.L., Thermal stress analysis in magneto-electro-thermoelasticity with a pennyshaped crack under uniform heat flow. Journal of Thermal Stresses, 2006, 29: 423-437. [3] Wang,B.L. and Niraula,O.P., Transient thermal fracture analysis of transversely isotropic magneto-electroelastic materials. Journal of Thermal Stresses, 2007, 30: 297-317.
Vol. 24, No. 5
Peng Ma et al.: Fracture Assessment of an Interface Crack
· 437 ·
[4] Feng,W.J., Pan,E. and Wang,X., Stress analysis of a penny-shaped crack in a magneto-electro-thermoelastic layer under uniform heat flow and shear loads. Journal of Thermal Stresses, 2008, 31: 497-514. [5] Chen,X.H., On magneto-thermo-viscoelastic deformation and fracture. International Journal of Non-linear Mechanics, 2009, 44: 244-248. [6] Sladek,J., Sladek,V., Solek,P. and Zhang,C., Fracture analysis in continuously nonhomogeneous magnetoelectro-elastic solids under a thermal load by the MLPG. International Journal of Solids and Structures, 2010, 47: 1381-1391. [7] Gao,C.F. and Noda,N., Thermal-induced interfacial cracking of magnetoelectroelastic material. International Journal of Engineering Science, 2004, 42: 1347-1360. [8] Zhu,B.J., Shi,Y.L., Qin,T.Y., Sukop,M., Yu,S.H. and Li,Y.B., Mixed-mode stress intensity factors of 3D interface crack in fully coupled electromagnetothermoelastic multiphase composites. International Journal of Solids and Structures, 2009, 46: 2669-2679. [9] Herrmann,K.P. and Loboda,V.V., Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings I. Electrically Permeable interface cracks. International Journal of Solids and Structures, 2003, 40: 4191-4217. [10] Li,R. and Kardomateas,G.A., The mixed mode I and II interface crack in piezoelectromagneto-elastic anisotropic bimaterials. ASME Journal of Applied Mechanics, 2007, 74: 614-627. [11] Feng,W.J., Li,Y.S. and Xu,Z.H., Transient response of an interfacial crack between dissimilar magnetoelectroelastic layers under magnetoelectromechanical impact loadings: Mode-I problem. International Journal of Solids and Structures, 2009, 46: 3346-3356. [12] Sih,G.C. and Song,Z.F., Magnetic and electric poling effects associated with crack growth in BaTiO3 – CoFe2 O4 composite. Theoretical and Applied Fracture Mechanics, 2003, 39: 209–227. [13] Herrmann,K.P., Loboda,V.V. and Khodanen,T.V., An interface crack with contact zones in a piezoelectric/piezomagnetic biomaterial. Archive of Applied Mechanics, 2010, 80: 651-670. [14] Hou,P.F., Teng,G.H. and Chen,H.R., Three-dimensional Green’s function for a point heat source in twophase transversely isotropic magneto-electro-thermo-elastic material. Mechanics of Materials, 2009, 41: 329-338.
APPENDIX A NUMERICAL RESULTS OF SOME MATRICES AND/OR VECTORS According to the material properties given in Table 1, the matrix G and vector h in Eqs.(43) and (44) have, respectively, the following forms: ⎤ ⎡ 0.1419i −0.0124 0.1896 0.0041 ⎢ 0.0124 0.1386i 0.6196i 0.0491i ⎥ ⎥ (70) G=⎢ ⎣ −0.1896 0.6196i −94.7049i −0.0685i ⎦ 0.0041 0.0491i −0.0685i −23.9537i h = { 0.1061i,
0.0779,
−0.5572,
0.0306 }T
Ω and e in Eq.(53) can be expressed by the following forms: ⎡ −7.0458i −0.0875i 1.3362i ⎢ −0.0875i 0.1375i 0.6362i ⎢ Ω=⎣ 1.3362i 0.6362i −94.9583i 0.0291i 0.0495i −0.0740i e = { −0.7475,
−0.0872,
−0.4155,
⎤ 0.0291i 0.0495i ⎥ ⎥ −0.0740i ⎦ −23.9538i 0.0337 }T
(71)
(72)
(73)
It is easily seen that not only the matrices G and Ω but also the vectors h and e satisfy the properties discussed in the text.
· 438 ·
ACTA MECHANICA SOLIDA SINICA Table 1. Material property∗
Material constants Material 1 Material 2 c11 (GPa) 274 178.0 c13 (GPa) 161 87.2 c33 (GPa) 259 172.8 c44 (GPa) 45 43.2 −4.4 −3.96 e31 C/m2 e33 C/m2 1.86 16.74 e15 C/m2 1.16 10.44 ε11 (×10−10 C2 /N · m2 ) 11.9 100.9 ε33 (×10−10 C2 /N · m2 ) 13.4 113.5 f31 (N/A · m) 522.3 58.03 f33 (N/A · m) 629.7 69.97 f15 (N/A · m) 495.0 55.00 531.5 63.5 μ11 ×10−6 N · s2 /C2 2 μ33 ×10−6 N · s2 /C 142.3 24.7 β11 ×105 N/Km2 6.21 4.738 β33 ×105 N/Km2 5.51 4.529 β34 ×10−6 C/N −2.94 25.0 β35 ×10−6 N/AmK 5.187 5.187 λ11 (W/Km) 9 1.2 λ33 (W/Km) 9 1.5 ∗ The corresponding material properties including elastic, dielectric, magnetic permeability constants and piezoelectric, piezomagnetic, magnetoelectric coefficients are taken from Sih and Song[12] and Herrmann et al.[13] . Besides, the stresstemperature coefficients, pyroelectric and pyromagnetic constants, heat conduction coefficients are taken from Hou et al.[14] .
2011
ISSN 0894-9166
FRACTURE ASSESSMENT OF AN INTERFACE CRACK BETWEEN TWO DISSIMILAR MAGNETOELECTROELASTIC MATERIALS UNDER HEAT FLOW AND MAGNETOELECTROMECHANICAL LOADINGS Peng Ma1
Wenjie Feng1
Ray Kai-Leung Su2
1
( Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China) (2 Department of Civil Engineering, The University of Hong Kong, China)
Received 9 April 2011, revision received 23 June 2011
ABSTRACT A magnetoelectrically permeable interface crack between two semi-infinite magnetoelectroelastic planes under the action of a heat flow and remote magnetoelectromechanical loadings is considered, where the assumption of frictionless contact between two dissimilar halfplanes is adopted. Not only the solutions of the interface crack problem are presented in an explicit form, but also the general condition for the transition from a perfect thermal contact of two magnetoelectroelastic bodies to their separation is given.
KEY WORDS fracture, interface crack, magnetoelectrically permeable crack, frictionless interface, heat flow
I. INTRODUCTION If a body conducting heat contains a crack, the crack will act as an obstruction to the thermal flux, and produce a local perturbation in the temperature field. Thermal stresses will therefore be produced. And if the crack occurs at the interface between dissimilar materials, the complications will be further introduced. In the last ten years, although the crack problems of magnetoelectroelastic materials under temperature loadings are widely investigated[1–6] , because of the mathematical complexity, the reports related to the corresponding interface crack problems of a magnetoelectroelastic bimaterial are very limited[7, 8] . Among others, Gao and Noda[7] presented an explicitly analytic solution for a generalized two-dimensional problem of an interface crack between two dissimilar magnetoelectroelastic materials under uniform heat flow, where the crack was assumed to be magnetoelectrically permeable. It was shown that when uniform heat flow was applied at infinity, the structure of singularities was the same as that in a purely elastic bi-material system with interface cracks. Only recently, Zhu et al.[8] further investigated the mixed-mode stress intensity factors of 3D interface cracks in fully coupled magneto
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 10772123 and 11072160), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0971) and the Natural Science Fund for Outstanding Younger of Hebei Province (A2009001624), China.
· 430 ·
ACTA MECHANICA SOLIDA SINICA
2011
electrothermoelastic multiphase composites by using the extended hypersingular intergro-differential equation (E-HIDE) method. It is the purpose of the present work to analyse a 2D plane strain problem of an interface crack between two dissimilar magnetoelectroelastic materials under uniform heat flow and magnetoelectromechanical loadings at infinity. Although the crack surfaces are also assumed to be magnetoelectrically permeable, different from Cao and Noda[7] , in this study, the assumption of frictionless contact between the two dissimilar half-planes is adopted. Not only the solutions in a magnetoelectroelastic bimaterial system are presented in an explicit form, but also the general condition for the transition from a perfect thermal contact of two magnetoelectroelastic bodies to their separation is given.
II. BASIC RELATIONS FOR A MAGNETOELECTROELASTIC SOLID For a stationary process, in the absence of body forces and free charges, the gorverning equations for a linear magnetoelectrothermalelastic material can be presented in the form[7] ΠiJ,i = 0 qi,i = 0
(1) (2)
ΠiJ = EiJKl VK,l − βiJ T
(3)
qi = −λij T,j
(4)
where
⎧ ⎨ σij ΠiJ = Di ⎩ Bi ⎧ ⎨ uk VK = ϕ ⎩ ψ
with
and
EiJKl
⎧ cijkl ⎪ ⎪ ⎪ ⎪ elij ⎪ ⎪ ⎪ ⎪ eikl ⎪ ⎪ ⎨ flij = f ⎪ ikl ⎪ ⎪ ⎪ −εil ⎪ ⎪ ⎪ ⎪ −g ⎪ il ⎪ ⎩ −μil
(J = 1, 2, 3) (J = 4) (J = 5)
(5)
(K = 1, 2, 3) (K = 4) (K = 5)
(6)
(J, K = 1, 2, 3) (J = 1, 2, 3; K = 4) (J = 4; K = 1, 2, 3) (J = 1, 2, 3; K = 5) (J = 5; K = 1, 2, 3) (J, K = 4) (J = 4; K = 5, J = 5; K = 4) (J, K = 5)
(7)
In Eqs.(3)-(7), uk , ϕ, ψ and T are the elastic displacements, electric potential, magnetic potential and temperature change, respectively; σij , Di , Bi and q are the stresses, electric displacements, magnetic inductions and thermal flux, respectively; cijlm , εij and μij are the elastic tensors, dielectric and magnetic permeability tensors, respectively; eijk , fijk and gij are the piezoelectric, piezomagnetic and magnetoelectric coefficients, respectively; βiJ are the stress-temperature coefficients for J = 1, 2, 3, and βi4 and βi5 are the pyroelectric and pyromagnetic constants, respectively. In addition, a subscript comma denotes the partial differentiation with respect to the coordinates (i.e., x1 , x2 , x3 or x, y, z), and summation from 1 to 3 (1 to 5) over repeated lowercase (uppercase) subscripts is assumed. It should also be pointed out that the following symmetry relations in Eq.(7) hold true. cijlm = cjilm = clmij ,
ekji = ekij ,
fkji = fkij ,
εij = εji ,
gij = gji ,
μij = μji
(8)
Assuming all fields are independent on the coordinate x2 , one can obtain the following general solution of Eq.(2) (see also Cao and Noda[7] , or Herrmann and Loboda[9] , et al.) ¯ (¯ zt ) T = χ (zt ) + χ
(9)
Vol. 24, No. 5
· 431 ·
Peng Ma et al.: Fracture Assessment of an Interface Crack
qi = − (λi1 + τ λi2 ) χ (zt ) − (λi1 + τ¯λi2 ) χ ¯ (¯ zt )
(10)
where zt = x1 + τ x3 , the prime ( ) denotes differentiation with respect to the argument, the overbar stands for the complex conjugate, and τ is a root with a positive imaginary part of the equation λ33 τ 2 + (λ13 + λ31 ) τ + λ11 = 0
(11)
In the case of transverse isotropy and plane strain in the plane (x1 , x3 ), which has an essential practical significance, a general solution of Eq.(1) by using the Lekhnitskii-Eshelby-Stroh representation and its application to magnetoelectrothermoelastic materials can be presented in the form[7] ¯f¯(¯ V = Af (z) + cχ (zt ) + A z ) + c¯χ ¯ (¯ zt ) ¯ ¯ ¯ f (¯ z ) + dχ ¯ (¯ zt ) t = Bf (z) + dχ (zt ) + B
(12) (13)
where T
V = { u1 , u3 , ϕ, ψ }
(14) T
t ≡ t3 = { σ31 , σ33 , D3 , B3 }
(15)
f (z) = { f1 (z1 ) , f2 (z2 ) , f3 (z3 ) , f4 (z4 ) } T A = A1 , A2 , A3 , A4
T
(16) (17) T
zj = x1 + pj x3 (j = 1, 2, 3, 4), and for a fixed j, Aj = { a1j , a2j , a3j , a4j } and pj (j = 1, 2, 3, 4) are, respectively, an eigenvector and eigenvalue of the system
Q + pj R + RT + p2j T Aj = 0 (18) with QJK = E1JK1 ,
RJK = E1JK3 ,
TJK = E3JK3
which can also be expressed in contracted notation as follows: ⎡ ⎤ ⎡ ⎤ c11 c15 e11 f11 c15 c13 e31 f31 ⎢ c15 c55 e15 f15 ⎥ ⎢ c55 c53 e35 f35 ⎥ ⎥ ⎥ ⎢ Q=⎢ ⎣ e11 e15 −ε11 −g11 ⎦ , R = ⎣ e15 e13 −ε13 −g13 ⎦ , f11 f15 −g11 −μ11 f15 f13 −g13 −μ13
⎡
c55 ⎢ c35 T =⎢ ⎣ e35 f35
(19) c53 c33 e33 f33
e35 e33 −ε33 −g33
⎤ f35 f33 ⎥ ⎥ −g33 ⎦ −μ33
The vector c is defined from the equation Q + τ R + RT + τ 2 T c = N 1 + τ N 2
(20)
(21)
T
with N m = { βm1 , βm3 , βm4 , βm5 } (m = 1, 2), and the 4 × 4 matrix B and the vector d can be found by the formulas B = RT A + T AP , d = RT + τ T c − N 2 (22) with P = diag p1 , p2 , p3 , p4 . Furthermore, T ¯ P¯ f¯ (¯ ¯τ χ z ) − d¯ ¯ (¯ zt ) t1 = { σ11 , σ13 , D1 , B1 } = −BP f (z) − dτ χ (zt ) − B
(23)
It is worth to mention that χ (zt ) in Eqs.(9), (10), (12) and (13) is an arbitrary analytic function which can be determined later, and f (z) in Eqs.(12) and (13) is an arbitrary analytic vector function with four components determined later as well.
III. AN INTERFACE CRACK BETWEEN TWO DISSIMILAR MAGNETOELECTROTHERMOELASTIC PLANES As shown in Fig.1, a soft thermal insulator located in the region (−a, a) lies on the interface between two different magnetoelectroelastic semi-infinite planes x3 > 0 and x3 < 0 with material properties
· 432 ·
ACTA MECHANICA SOLIDA SINICA (1)
(1)
(1)
(2)
(2)
2011
(2)
defined by EiJKl , λij , βiJ and EiJKl , λij , βiJ , respectively. The thermal insulator is not assumed to cause stresses. At the same time, the component q3 of the thermal flux vector and the vector t are assumed to be continuous across the whole bimaterial interface; and the contact between the halfplanes is frictionless. On the other hand, from the point of view of magnetoelectric fields, the insulator is assumed to be a magnetoelectrically permeable interface crack. Thus, the boundary conditions at the interface can be presented in the form q3± = 0 (m) (m) σ13 (x1 , 0) = 0, σ33 (x1 , 0) = 0 [ϕ (x1 )] = 0, [ψ (x1 )] = 0 [D3 (x1 )] = 0, [B3 (x1 )] = 0 [T ] = 0,
(24)
[q3 ] = 0
(m) (x1 , 0) = 0, σ33 (x) = 0 [u3 (x1 )] = 0 [ϕ (x1 )] = 0, [ψ (x1 )] = 0 [D3 (x1 )] = 0, [B3 (x1 )] = 0 (m) σ13
(|x1 | ≤ a)
(|x1 | > a)
(25)
where
[Υ (x1 )] = Υ + (x1 , 0) − Υ − (x1 , 0) , Υ = T, u3 , σ33 , ϕ, ψ, D3 , B3 (26) In Eq.(26), the signs “+” and “−” denote the upper and lower parts of the interface. Certainly, the boundary conditions at infinity should also be satisfied.
Fig. 1. Magnetoelectroelastic plane with an interface crack under heat flow and magnetoelectromechanical loadings.
3.1. The Thermal Solution Introducing auxiliary functions ˜i (zt ) χi (zt ) = q3∞ + χ [9]
(i = 1, 2)
(27)
⎧ k (1) ⎪ ⎪ ˜1 (zt ) (x3 > 0) ⎨ 1 + (2) χ k (28) θ (zt ) = (2) k ⎪ ⎪ ⎩ 1 + (1) χ ˜2 (zt ) (x3 < 0) k √ with k = λ33 (τ − τ¯)/ (2i) i = −1 , k(1) , χ1 (zt ), χ ˜1 (zt ) and k(2) , χ2 (zt ), χ ˜2 (zt ), respectively, being related to the upper and lower half-planes, one can finally obtain (1) q3 (x1 , 0) = q3∞ − ik0 θ+ (x1 ) + θ− (x1 ) (29) and
Vol. 24, No. 5
Peng Ma et al.: Fracture Assessment of an Interface Crack
· 433 ·
where
k (1) k (2) (30) k (1) + k (2) It should be pointed out that in the process of deriving Eq.(29), the boundary condition q3 (z)|z→∞ = q3∞ has been used. Substituting Eq.(29) into Eq.(24)1 yields the following equation k0 =
θ+ (x1 ) + θ− (x1 ) = −
iq3∞ , k0
x1 ∈ (−a, a)
The solution of Eq.(31) disappearing at infinity can be presented in the form z iq ∞ √ −1 θ (z) = 3 2k0 z 2 − a2
(31)
(32)
integrating which gives the following expression: θ (z) =
iq3∞ 2 z − a2 − z 2k0
(33)
It is remarked that Eq.(33) satisfies the condition θ (z)|z→∞ = 0. Therefore, the temperature jump across the material interface for −a < x1 < a and the thermal flux for x1 > a can be presented in the following form: q3∞ (x1 + a)(a − x1 ) k0 x 1 −1 q3 (x1 , 0) = q3∞ 2 x1 − a2
[T (x1 )] = −
(34a)
(34b)
which completely defines the temperature jump and the thermal flux in the bimaterial system for any position of point a. Equations (34) also reveal that the temperature jump across the interface crack depends on not only the heat flow applied at infinity but also the related material properties, and that the thermal flux, however, is independent of √ material constants. In addition, it is easily obtained from 0.5 Eq.(34b) that lim q3 (x1 , 0) = q3∞ (0.5a) / x1 − a − q3∞ , which implies that the thermal flux at the x1 →a
crack tip has a square root singularity. 3.2. The Magnetoelectroelastic Solution Carrying out a derivation similar to the one by Herrmann and Loboda[9] , from Eqs.(12), (13), (24)2 , (25)2 , the following expressions at the interface are obtained: [V (x1 )] = W + (x1 ) − W − (x1 ) ¯ + (x1 ) − g (x1 ) t(1) (x1 , 0) = GW + (x1 ) − GW
(35) (36)
where W (z) = { W1 (z) , W2 (z) , W3 (z) , W4 (z) }T is an introduced unknown vector function, and W + (x1 ) = W (x1 + i0) , W − (x1 ) = W (x1 − i0) ¯ − (x1 ) g (x1 ) = hθ+ (x1 ) − hθ G=B
(1)
D
−1
(37) (38) (39)
with ¯ (1) D = A(1) − LB ∗ 1 ¯ − c∗ − k (2) d(1) h = (1) −G Ld k + k (2) and
−1 L = A(2) − B (2) ,
c∗ = k (2) c(1) + k (1) c¯(2) ,
d∗ = k (2) d(1) + k (1) d¯(2)
(40) (41)
(42)
· 434 ·
ACTA MECHANICA SOLIDA SINICA
2011
... ... It is worth to note that the related to the matrix H (or N ) and the ... matrix G and the vector W (z) are ... vector function ω (z) (or Φ (z)) of the paper of Gao and Noda[7] (or of the paper of Li and Kardomateas[10] ... ... −1 ... ... as iG−1 = H = N , W (z) = −i ω (z) = Φ (z), respectively, where all the quantities with the sign ‘. . .’ denote the corresponding quantities in the papers of Gao and Noda[7] and Li and Kardomateas[10]. Thus, Eqs.(35) and (36) have no distinct differences from the corresponding expressions given by Gao and Noda[7] . Furthermore, as demonstrated by Feng et al.[11] before, for a magnetoelectroelastic bimaterial system, the matrix G in Eq.(36) has the following form: ⎡ ⎤ ig11 g12 g13 g14 ⎢ g21 ig22 ig23 ig24 ⎥ ⎥ G=⎢ (43) ⎣ g31 ig32 ig33 ig34 ⎦ g41 ig42 ig43 ig44 where Im (gij ) = 0 (i, j = 1, 2, 3, 4), and g21 = −g12 , g31 = −g13 , g41 = −g14 , g32 = g23 , g42 = g24 , g43 = g34 . By the way, in the present study, the matrix G has been further calculated for one group of material combination, and the numerical results are displayed in Appendix. Also, as discussed by Herrmann and Loboda[9] for piezoelectric interfacial crack problems, one gets h = { iϑ1 ,
ϑ2 ,
ϑ3 ,
ϑ4 }T
(44)
[ψ]}T
(45)
where ϑi (i = 1, 2, 3, 4) are all real (e.g., see Appendix). Introducing the vectors (1)
S = {σ13 , [u3 ] , [ϕ] , Π=
{[u1 ] ,
(1) σ33 ,
(1) D3 ,
(1) B3 }T
(46)
the following relations can be easily found by means of Eqs.(35) and (36): ¯ W − (x1 ) + m∗ θ+ (x1 ) − m ¯ ∗ θ− (x1 ) S (x1 ) = M W + (x1 ) − M ¯ W − (x1 ) + n∗ θ+ (x1 ) − n ¯ ∗ θ− (x1 ) Π (x1 ) = N W + (x1 ) − N where the matrices M and N and the vectors m∗ and n∗ are defined as ⎡ ⎤ ⎡ ig11 g12 g13 g14 1 0 0 ⎢ 0 ⎥ ⎢ g21 ig22 ig23 1 0 0 ⎥, N = ⎢ M =⎢ ⎣ 0 ⎣ g31 ig32 ig33 0 1 0 ⎦ 0 0 0 1 g41 ig42 ig43 m∗ = { −h1 , 0, 0, 0 }T ,
⎤ 0 ig24 ⎥ ⎥ ig34 ⎦ ig44
n∗ = { 0, −h2 , −h3 , −h4 }T
By further introducing another auxiliary vector-function Ξ (z) (x3 > 0) M W (z) + m∗ θ (z) Ξ (z) = ¯ W (z) + m M ¯ ∗ θ (z) (x3 < 0)
(47) (48)
(49)
(50)
(51)
one can get S (x1 ) = Ξ + (x1 ) − Ξ − (x1 ) ¯ − (x1 ) − eθ+ (x1 ) + e¯θ − (x1 ) Π (x1 ) = ΩΞ + (x1 ) − ΩΞ
(52) (53)
with Ω = N M −1 and e = Ωm∗ − n∗ . It is easily seen that the vector function Ξ (z) is analytic in the whole plane with a cut along (|x1 | ≤ a, x3 = 0). The numerical analysis shows that for all considered material combinations, the matrix Ω is pure ¯ = −Ω, e imaginary and the vector e is real, i.e., Ω ¯ = e (by the way, in Appendix, both Ω and e are also numerically displayed for the considered material combination). Thus, Eq.(53) can be written in the form Π (x1 ) = Ω Ξ + (x1 ) + Ξ − (x1 ) − e θ+ (x1 ) − θ− (x1 ) (54)
Vol. 24, No. 5
Peng Ma et al.: Fracture Assessment of an Interface Crack
· 435 ·
(1)
Moreover, it follows from the relation (52) and the equations σ13 (x1 , 0) = 0, [ϕ (x1 )] = [ψ (x1 )] = 0 for |x1 | ≤ a that the functions Ξ1 (z), Ξ3 (z)and Ξ4 (z) are analytic in the whole plane. Noting the conditions at infinity one can conclude that Ξ1 (z) = C1 , Ξ3 (z) = C3 and Ξ4 (z) = C4 (C1 , C3 and C4 are, respectively, different constants to be determined). From Eqs.(46) and (24)2 , one can further get Π2 (x1 ) = 0
|(x1 | ≤ a)
(55)
where Π2 (x1 ) only involves function Ξ2 (x1 ). By virtue of [u1 (x1 )]|x1 →∞ = 0, the conditions at infinity can be written by means of Eq.(54) as follows: ⎧ ⎫ ⎫ ⎡ ⎤⎧ 0 ⎪ Ω11 Ω12 Ω13 Ω14 ⎪ C1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∞⎪ ⎬ ⎬ ⎢ Ω21 Ω22 Ω23 Ω24 ⎥ ⎨ Ξ2 | σ z→∞ ⎢ ⎥ = 0.5 (56) ∞ ⎣ Ω31 Ω32 Ω33 Ω34 ⎦ ⎪ C3 ⎪ D ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∞⎪ ⎭ ⎭ ⎩ Ω41 Ω42 Ω43 Ω44 C4 B Thus, C1 , C3 , C4 and Ξ2 |z→∞ can be determined by ⎧ ⎫ ⎡ C1 ⎪ Ω11 Ω12 ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ Ω21 Ω22 Ξ2 |z→∞ = 0.5 ⎢ ⎣ Ω31 Ω32 C3 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ C4 Ω41 Ω42
Ω12 Ω23 Ω33 Ω43
⎫ ⎤−1 ⎧ 0 ⎪ Ω14 ⎪ ⎪ ⎨ ∞⎪ ⎬ σ Ω24 ⎥ ⎥ ∞ Ω34 ⎦ ⎪ D ⎪ ⎪ ⎩ ∞⎪ ⎭ Ω44 B
By use of Eqs.(54) and (33), Eq.(55) can be written in the form e2 q3∞ 2Ω 21 2Ω 23 2Ω 24 Ξ2+ (x1 ) + Ξ2− (x1 ) = x21 − a2 − C1 − C3 − C4 Im (Ω22 ) k0 Ω22 Ω22 Ω22
(57)
(|x1 | ≤ a)
(58)
The solution of the Riemann problem (54) under the condition Ξ2 at infinity given in Eq.(57) can be written as z 2az σ∞ e2 q3∞ z−a √ √ + Ξ2 (z) = + z 2 − a2 ln 2 2 2 2πik0 Im(Ω22 ) z+a 2Ω22 z − a2 z −a Q Q Q − 21 C1 − 23 C3 − 24 C4 (59) Q22 Q22 Q22 The normal stress and the derivative of the normal displacement jump at the interface by use of Eqs.(59), (54) and (52), read then as follows: e2 q3∞ σ ∞ x1 2ax1 x1 − a (1) 2 2 + 2 σ33 (x1 , 0) = + x1 − a ln (|x1 | > a) (60) 2 πk 0 x1 + a x1 − a2 x1 − a2 ∞ ∞ q x e 2ax σ a − x 2 1 1 1 −1 3 − − (|x1 | < a) [u3 (x1 )] = [Im (Ω22 )] + a2 − x21 ln πk0 a + x1 a2 − x21 a2 − x21 (61) The formulas (60) and (61) are valid for any values of q3∞ and σ ∞ , and it is seen that similar to (1) thermal flux (see Eq.(34b)), both σ33 (x1 , 0) and [u3 (x1 )] pose a square root singularity at the edges of the zone of thermal insulation as well. As usual, by defining mode-I stress intensity factor (SIF) KI and crack open displacement (COD) intensity factor KCOD at the crack tip a as follows: (1) KI = lim + 2π (x1 − a)σ33 (x1 , 0) (62) x1 →a
KCOD = lim x1 →a−
[u3 (x1 )]
(63)
2π (a − x1 )
we can easily obtain from Eqs.(60) and (61) that KI =
2a1.5 e2 ∞ 0.5 q + σ ∞ (πa) , π 0.5 k0 3
KCOD =
1 −1 [Im (Ω22 )] KI π
(64)
· 436 ·
ACTA MECHANICA SOLIDA SINICA
2011
It is worth to note that, as expected, neither the SIF nor the COD intensity factor depends on the electric displacement D∞ and/or magnetic induction B ∞ , and that both the SIF and the COD intensity factor consist two independent parts, the first is induced by the mechanical loading σ ∞ at infinity, and the second is induced by the heat flow q3∞ . And as shown in Eqs.(64), the SIF induced by σ ∞ is independent of material properties of magnetoelectroelastic biomaterial system. However, the SIF induced by q3∞ simultaneously depends on the corresponding material properties. On the other hand, the COD intensity factor induced by either σ ∞ or q3∞ directly depends the material constants composed of them. In addition, form Eq.(64) (or Eqs.(60) and (61)), when q3∞ and σ ∞ satisfy the following relation: q3∞ = −
πk0 ∞ σ 2ae2
(65)
(1)
the singularities of both σ33 (x1 , 0) and [u3 (x1 )] can be absent. In this case, Eqs.(60) and (61) take the following form: e2 q3∞ x1 − a (1) σ33 (x1 , 0) = (|x1 | > a) x21 − a2 ln (66) πk0 x1 + a ∞ a − x1 −1 e2 q3 a2 − x21 ln (|x1 | < a) (67) [u3 (x1 )] = [Im (Ω22 )] πk0 a + x1 3.3. On an Admissible Direction of the Thermal Flux Because the logarithmic functions in Eq.(66) and in Eq.(67) are negative for point x1 situated in the vicinity of point a, and according to the numerical calculation the inequality Im (Ω22 ) > 0, it follows (1) that in the vicinity of point a, σ33 (x1 , 0) < 0 and [u3 (x1 )] > 0 hold true if and only if e2 q3∞ > 0
(68)
Noting the definition of e2 and Eqs.(49) and (50), the inequality (68) can be also written in the following form: g11 ϑ2 + g12 ϑ1 ∞ q3 > 0 (69) g11 The inequality (69) defines the direction of the thermal flux q3 for which a transition from a perfect thermal contact of two magnetoelectroelastic bodies to their separation is possible. Moreover, because (1) the Eqs.(66) and (67) define the asymptotic behavior of σ33 (x1 , 0) and [u3 (x1 )] at the transition point a, respectively, the inequality virtually can be treated as a general condition for the perfect thermal contact. In addition, it should be pointed out that if the inequality (68) (or (69)) is not valid, Eqs.(25)1 and (25)2 cannot be used, then an imperfect thermal contact should be introduced.
IV. BRIEF CONCLUSIONS (1) Different from the temperature jump across the interface crack, the thermal flux at the interface depends on only the heat flow applied at infinity, and has a square root singularity at the crack tip. (2) Both the normal stress and the derivative of the normal displacement jump at the interface induced by either heat flow or mechanical loading pose a square root singularity at the crack tip. (3) Both magnetical and electrical loadings have no effects on mode-I SIF and COD intensity factor. (1) (4) The singularities of both σ33 (x1 , 0) and [u3 (x1 )] are absent as q3∞ = −πk0 σ ∞ /(2ae2 ). (5) For perfect thermal contact, the general condition (g11 ϑ2 + g12 ϑ1 )q3∞ /g11 > 0 should be satisfied.
References [1] Gao,C.F., Kessler,H. and Balke,H., Fracture analysis of electromagnetic thermoelastic solids. European Journal of Mechanics A-Solids, 2003, 22: 433-442. [2] Niraula,O.P. and Wang,B.L., Thermal stress analysis in magneto-electro-thermoelasticity with a pennyshaped crack under uniform heat flow. Journal of Thermal Stresses, 2006, 29: 423-437. [3] Wang,B.L. and Niraula,O.P., Transient thermal fracture analysis of transversely isotropic magneto-electroelastic materials. Journal of Thermal Stresses, 2007, 30: 297-317.
Vol. 24, No. 5
Peng Ma et al.: Fracture Assessment of an Interface Crack
· 437 ·
[4] Feng,W.J., Pan,E. and Wang,X., Stress analysis of a penny-shaped crack in a magneto-electro-thermoelastic layer under uniform heat flow and shear loads. Journal of Thermal Stresses, 2008, 31: 497-514. [5] Chen,X.H., On magneto-thermo-viscoelastic deformation and fracture. International Journal of Non-linear Mechanics, 2009, 44: 244-248. [6] Sladek,J., Sladek,V., Solek,P. and Zhang,C., Fracture analysis in continuously nonhomogeneous magnetoelectro-elastic solids under a thermal load by the MLPG. International Journal of Solids and Structures, 2010, 47: 1381-1391. [7] Gao,C.F. and Noda,N., Thermal-induced interfacial cracking of magnetoelectroelastic material. International Journal of Engineering Science, 2004, 42: 1347-1360. [8] Zhu,B.J., Shi,Y.L., Qin,T.Y., Sukop,M., Yu,S.H. and Li,Y.B., Mixed-mode stress intensity factors of 3D interface crack in fully coupled electromagnetothermoelastic multiphase composites. International Journal of Solids and Structures, 2009, 46: 2669-2679. [9] Herrmann,K.P. and Loboda,V.V., Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings I. Electrically Permeable interface cracks. International Journal of Solids and Structures, 2003, 40: 4191-4217. [10] Li,R. and Kardomateas,G.A., The mixed mode I and II interface crack in piezoelectromagneto-elastic anisotropic bimaterials. ASME Journal of Applied Mechanics, 2007, 74: 614-627. [11] Feng,W.J., Li,Y.S. and Xu,Z.H., Transient response of an interfacial crack between dissimilar magnetoelectroelastic layers under magnetoelectromechanical impact loadings: Mode-I problem. International Journal of Solids and Structures, 2009, 46: 3346-3356. [12] Sih,G.C. and Song,Z.F., Magnetic and electric poling effects associated with crack growth in BaTiO3 – CoFe2 O4 composite. Theoretical and Applied Fracture Mechanics, 2003, 39: 209–227. [13] Herrmann,K.P., Loboda,V.V. and Khodanen,T.V., An interface crack with contact zones in a piezoelectric/piezomagnetic biomaterial. Archive of Applied Mechanics, 2010, 80: 651-670. [14] Hou,P.F., Teng,G.H. and Chen,H.R., Three-dimensional Green’s function for a point heat source in twophase transversely isotropic magneto-electro-thermo-elastic material. Mechanics of Materials, 2009, 41: 329-338.
APPENDIX A NUMERICAL RESULTS OF SOME MATRICES AND/OR VECTORS According to the material properties given in Table 1, the matrix G and vector h in Eqs.(43) and (44) have, respectively, the following forms: ⎤ ⎡ 0.1419i −0.0124 0.1896 0.0041 ⎢ 0.0124 0.1386i 0.6196i 0.0491i ⎥ ⎥ (70) G=⎢ ⎣ −0.1896 0.6196i −94.7049i −0.0685i ⎦ 0.0041 0.0491i −0.0685i −23.9537i h = { 0.1061i,
0.0779,
−0.5572,
0.0306 }T
Ω and e in Eq.(53) can be expressed by the following forms: ⎡ −7.0458i −0.0875i 1.3362i ⎢ −0.0875i 0.1375i 0.6362i ⎢ Ω=⎣ 1.3362i 0.6362i −94.9583i 0.0291i 0.0495i −0.0740i e = { −0.7475,
−0.0872,
−0.4155,
⎤ 0.0291i 0.0495i ⎥ ⎥ −0.0740i ⎦ −23.9538i 0.0337 }T
(71)
(72)
(73)
It is easily seen that not only the matrices G and Ω but also the vectors h and e satisfy the properties discussed in the text.
· 438 ·
ACTA MECHANICA SOLIDA SINICA Table 1. Material property∗
Material constants Material 1 Material 2 c11 (GPa) 274 178.0 c13 (GPa) 161 87.2 c33 (GPa) 259 172.8 c44 (GPa) 45 43.2 −4.4 −3.96 e31 C/m2 e33 C/m2 1.86 16.74 e15 C/m2 1.16 10.44 ε11 (×10−10 C2 /N · m2 ) 11.9 100.9 ε33 (×10−10 C2 /N · m2 ) 13.4 113.5 f31 (N/A · m) 522.3 58.03 f33 (N/A · m) 629.7 69.97 f15 (N/A · m) 495.0 55.00 531.5 63.5 μ11 ×10−6 N · s2 /C2 2 μ33 ×10−6 N · s2 /C 142.3 24.7 β11 ×105 N/Km2 6.21 4.738 β33 ×105 N/Km2 5.51 4.529 β34 ×10−6 C/N −2.94 25.0 β35 ×10−6 N/AmK 5.187 5.187 λ11 (W/Km) 9 1.2 λ33 (W/Km) 9 1.5 ∗ The corresponding material properties including elastic, dielectric, magnetic permeability constants and piezoelectric, piezomagnetic, magnetoelectric coefficients are taken from Sih and Song[12] and Herrmann et al.[13] . Besides, the stresstemperature coefficients, pyroelectric and pyromagnetic constants, heat conduction coefficients are taken from Hou et al.[14] .
2011