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A wedge disclination dipole interacting with a coated cylindrical inhomogeneity
Acta Mechanica Solida Sinica, Vol. 28, No. 1, February, 2015 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
A WEDGE DISCLINATION DIPOLE INTERACTING WITH A COATED CYLINDRICAL INHOMOGENEITY⋆⋆ Yingxin Zhao
Qihong Fang
Youwen Liu⋆
(State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China) (College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China) Received 6 May 2013, revision received 19 April 2014
ABSTRACT A three-phase composite cylinder model is utilized to study the interaction of a wedge disclination dipole with a coated cylindrical inhomogeneity. The explicit expression of the force acting on the wedge disclination dipole is calculated. The motilities and the equilibrium positions of the disclination dipole near the coated inhomogeneity are discussed for various material combinations, relative thicknesses of the coating layer and the features of the disclination dipole. The results show that the material properties of the coating layer have a major part to play in alteringi the strengthening effect or toughening effect produced by the coated inhomogeneity.
KEY WORDS three-phase composite, disclination dipole, coating layer, complex variable function method
I. INTRODUCTION Disclinations together with dislocations represent a class of linear defects in solids and are characterized by typical singularities and the property of multi-value of the fields of displacement and rotation associated with the defects[1] . They are considered negative (or positive) if material needs to be filled in to bridge any gap due to the relative displacement (or taken away if the faces interpenetrate)[2] . Disclinations in solids exist only in the form of screened configurations with lower self-energy, including dipoles, quadrupoles and disclination loops[3] . Indeed, experimental evidence of the disclination concept is also becoming available: disclination dipoles and multipoles have been observed directly via electron microscopy in metals such as copper, titanium and α-iron single crystals subjected to severe plastic deformation[4, 5]. Disclinations are also observed in thin films grown on substrates[6] and in pentagonal nanoparticles[7]. Increasing attention is being attracted to disclinations with various configurations because of their diverse applications in materials science and physics. Disclination models are extremely important for exploring the physical nature of the rotation deformation mode peculiar to crystalline materials[8–11] . They can model the geometry and elastic distortions in rotational defect structures, high angle grain boundaries in polycrystals and crystalline materials. Important additions to understanding deformation and hardening properties of crystalline materials can be delivered in the framework of Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11172094 and 11172095), the New Century Excellent Talents in University (NCET-11-0122) and Hunan Provincial Natural Science Foundation for Creative Research Groups of China (No. 12JJ7001). ⋆
⋆⋆
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disclination models, which accounts for the specific disclination defects at the junctions of nanograin boundaries[12] . Owing to the singular internal stress fields induced by disclinations[13] , relaxation of these stresses brings about various configurational changes in solids, the most common being crack or void nucleation in their vicinity[14, 15] . Disclinated cracks have been studied via continuum elasticity[16, 17] . In Ref.[16], local amorphization of the disclination core is suggested as a relaxation mechanism that can retard crack nucleation. Consequently, the various types of relaxed structures directly affect the physical properties such as strength, toughness, diffusion and electrical conductivity. Another possibility of diminishing the disclination energy depends on the screening of disclination elastic fields by traction-free surfaces. In this case screened disclinations can be important in subsurface layers, shells, thin films and small crystalline particles[18, 19] . It turned out that these studies have proved disclinations quite helpful to the analysis of deformation behavior of polycrystals, crystalline materials, and whatnot. In the study of various defects, such as dislocation, disclination, crack, and so on, interaction with inhomogeneities is motivated by the need to gain a better understanding of certain strengthening and hardening materials, especially composite materials. This is mainly due to the fact that the mobilities and stabilities of defects can be influenced by various inhomogeneities in materials. When the defect is not extremely close to the interface and the size of inhomogeneity is fairly large, elastic interaction is expected to be predominant[20, 21] . In most inhomogeneity-reinforced composite materials, control of the wetting, reaction and bonding of the matrix to inhomogeneity is critical to achieving the desired property goals. Therefore, the coating layer on inhomogeneities is often widely employed in order to increase the bonding strength between inhomogeneity and matrix. In a ceramic matrix composite material, load transfer may be of secondary importance. While in metal matrix and polymer composite materials, as the mechanical properties are greatly enhanced by efficient load transfer from the matrix to inhomogeneities, the bond quality between the inhomogeneity and the matrix is very important. The bond should be strong enough to yield high quality composites. A coating layer around inhomogeneity-matrix interface can help achieve controlled delamination of the interface and prevent cracks initiated external to the inhomogeneity from damaging the matrix. The strengthening and hardening behavior of inhomogeneity reinforced composites can also be well explained by the elastic interaction mechanisms between dislocations/disclinations/cracks and multi-phase inhomogeneities[22–33] . Seeing that disclination mobility and stability can greatly affect the mechanical behaviors of the composites, the problem of disclinations interacting with composites has also drawn many investigators’ attention to evaluation of structures and mechanical properties of solids, e.g., film-substrate composites and nanostructured materials[34–36] . Incidentally, report on the interaction of disclination dipole with the three-phase composite material seems unavailable probably for the complexity of the calculation. In the present paper, the problem of the interaction between a wedge disclination dipole and a coated cylindrical inhomogeneity is theoretically solved. By complex variable techniques, the analytic solution of complex potentials and the force exerted on the disclination dipole is derived. The impact of material properties, the parameters of the coating layer and the features of the disclination dipole on the force is examined and discussed. Our results are helpful in the understanding of the motion mechanism of the disclinations and relevant physical phenomena in three-phase composite materials. Although the disclinations are assumed to be located within matrix, the other cases of dislocation located inside the inhomogeneity or coating layer can be obtained by the same method. The information obtained is useful in design or fabrication technology of three-phase composite materials for better or specified conductivities.
II. MODEL In this paper, we consider a coated inhomogeneity embedded in an infinite isotropic matrix consisting of two cylindrical domains made of different materials, as shown in Fig.1. ∆1 and ∆2 indicate the interfaces between the inhomogeneity and the annular coating layer, and the annular coating layer and the matrix, respectively. The two cylindrical interfaces are assumed to be perfectly bound and coherent. The radii of the inhomogeneity surface and coating layer-matrix interface are denoted by R1 and R2 , respectively. Suppose that a wedge disclination dipole consisting of a positive disclination with strength ω at zp (= z0 − aeiϕ ) and a negative disclination with the same strength at zn (= z0 + aeiϕ ) is at an arbitrary position inside the infinite matrix domain, where 2a denotes the dipole arm and the center of the disclination dipole z0 = ρeiθ . Both disclination lines are perpendicular to the (x, y) plane.
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Hereinafter phase ‘i’ means the inhomogeneity media and has material properties µi and νi , occupying the region r < R1 ; phase ‘c’ refers to the annular coating layer and has material properties µc and νc , occupying the area R1 < r < R2 ; phase ‘m’ denotes the matrix and has properties µm and νm with the area r > R2 .
Fig. 1. A wedge disclination dipole interacting with the coated inhomogeneity.
For a plane strain problem under consideration, stress fields and displacement fields are expressed by following two complex potentials Φ(z) and Ψ (z)[37] h i σyy + σxx = 2 Φ (z) + Φ (z) (1) σyy − σxx + 2iσxy = 2 [zΦ′ (z) − Ψ (z)] # " zΨ (z) ′ ′ ′ 2µ ux + uy = iz κΦ (z) − Φ (z) + zΦ (z) + z
(2)
(3)
where u′x = ∂ux /∂θ, u′y = ∂uy /∂θ, ui and σij are the displacement and stress tensors in Cartesian coordinate, respectively; Φ′ (z) = d [Φ (z)]/dz, z = ρeiθ ; the overbar represents the complex conjugate; κ = 3 − 4ν for plane strain state, µ is the shear modulus of the bulk solid, ν is the Poisson’s ratio of the bulk solid. The emergence of compatibility in the form of stress fields and displacement fields at the coherent interfaces gives rise to compatibility tensors in the framework of elastic fields. Accordingly, the conventional stress and displacement continuity conditions are given as follows: ′ + − uxi (t) + iu′yi (t) = u′xc (t) + iu′yc (t) (4) [σxi (t) + iσyi (t)]+ = [σxc (t) + iσyc (t)]− − ′ + uxc (ς) + iu′yc (ς) = u′xm (ς) + iu′ym (ς) +
−
[σxc (ς) + iσyc (ς)] = [σxm (ς) + iσym (ς)]
(5)
(6) (7)
where the superscripts ‘+’ and ‘−’ denote the boundary quantities as z approaches the interior and exterior of a cylindrical interface, respectively. Here, |t| = R1 and |ς| = R2 . Regarding classical elasticity, for a wedge disclination dipole located at an arbitrary point z0 in the matrix, two complex potentials Φm (z) and Ψm (z) in the matrix are introduced[33] Dω [ln (z − zp ) − ln (z − zn )] + Φm0 (z) (8) Φm (z) = 2 Dω zn zp Ψm (z) = − + Ψm0 (z) (9) 2 z − zn z − zp
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where D = µm /[2π (1 − νm )], Φm0 (z) and Ψm0 (z) are holomorphic in the matrix. In this case, the complex potentials Φi (z) and Ψi (z) are holomorphic in inhomogeneity. And two holomorphic complex potentials Φc (z) and Ψc (z) in the coating layer can be given as ∞ ∞ X X Φc (z) = ak z k + bk z −k (10) k=0
Ψc (z) =
∞ X
k=1 ∞ X
ck z k−2 +
k=0
dk z −k−2
(11)
k=1
where ak , bk , ck and dk are unknown coefficients to be determined from the boundary conditions (4)-(7). ∞ ∞ P P ak z k and AN (z) = bk z −k . Furthermore, let AP (z) = k=0
k=1
For the purpose of easy treatment of the boundary conditions on the interfaces and convenient calculation, the following new analytical functions are introduced in the corresponding regions according to the Schwarz symmetry principle. 2 2 R12 ′ R12 R12 R1 R1 ∗ Ωi (z) = −Φi + Φi + 2 Ψi (12) z z z z z 2 R2 R2 R1 R12 R12 + 1 Φ′c + 21 Ψc (13) Ωc∗ (z) = −Φc z z z z z 2 R2 R2 R22 R2 R22 Ωc∗∗ (z) = −Φc + 2 Φ′c + 22 Ψc (14) z z z z z 2 2 2 R2 R2 R2 R2 R2 ∗ Ωm + 2 Φ′m + 22 Ψm (15) (z) = −Φm z z z z z Substituting Eqs.(8)-(11) into (12)-(15), we arrive at the expressions as Ωc∗ (z) = SN (z) + SP (z) Ωc∗∗ (z) = TN (z) + TP (z) zp − zp∗ zn − zn∗ zn − zp z − zn∗ Dω ∗ ∗ + − + ln + Ωm0 (z) Ωm (z) = 2 z − zp∗ z − zp∗ z − zn∗ z ∗ (z) is holomorphic in the region |z| < R2 . where zp∗ = R22 /zp , zn∗ = R22 /zn , and Ωm0 ∞ i h X 2(k−1) z −k SN (z) = (k − 1) ak R12k + ck R1
(16) (17) (18)
(19)
k=1
SP (z) = −a0 + c0 R1−2 +
∞ h i X −2(k+1) − (k + 1) bk R1−2k + dk R1 zk
(20)
k=1
TN (z) =
∞ h X
2(k−1)
(k − 1) ak R22k + ck R2
k=1
TP (z) = −a0 + c0 R2−2 +
i
z −k
∞ h i X −2(k+1) zk − (k + 1) bk R2−2k + dk R2
(21) (22)
k=1
In connection with Eqs.(3), (8), (12), (13), (16) and the displacement continuity condition on the entire circular interface ∆1 , Eq.(4) is simplified as + − κi 1 ∗ κc 1 ∗ Φi (t) − Ωc (t) = Φc (t) − Ωi (t) (|t| = R1 ) (23) µi µc µc µi Let κi Φi (z) Ωc∗ (z) g1 (z) = − (|z| < R1 ) (24) µi µc κc Φc (z) Ωi∗ (z) g1 (z) = − (R1 < |z| < R2 ) (25) µc µi
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With reference to the generalized Liouville theorem, we have g1 (z) =
κc AP (z) SN (z) − µc µc
(26)
Using Eqs.(1), (2), (8), (12), (13), (16) and the stress continuity condition on the entire circular interface ∆1 , we obtain the following equation from Eq.(5): +
[Φi (t) + Ωc∗ (t)] = [Φc (t) + Ωi∗ (t)]
−
(|t| = R1 )
(27)
Suppose g2 (z) = Φi (z) + Ωc∗ (z) g2 (z) = Φc (z) +
Ωi∗
(z)
(|z| < R1 )
(28)
(R1 < |z| < R2 )
(29)
With the aid of the generalized Liouville theorem, this leads to g2 (z) = AP (z) + SN (z)
(30)
If we combine Eqs.(24) and (28), the following complex expression is derived: Ωc∗ (z) =
g2 (z) − µi g1 (z)/κi µi /(κi µc ) + 1
(31)
By considering Eqs.(25) and (29), the following complex expression is given by Φc (z) =
µi g1 (z) + g2 (z) µi κc /µc + 1
(32)
Substitution of Eqs.(26) and (30) into (32) yields Φc (z) = AP (z) +
(1 − µi /µc ) SN (z) µi κc /µc + 1
(33)
Taking the complex conjugate of Eq.(13) and using Eqs.(26), (30), (31) and (33) leads to 2 R1 R12 1 − µi /µc ′ Ψc (z) = 2 AP (z) + [SN (z) − zSN (z)] + SN z µi κc /µc + 1 z 2 1 − µi κc /(µc κi ) R1 + AP − zA′P (z) µi /(κi µc ) + 1 z
(34)
By use of the above same cleaning solution, the displacement and stress continuity conditions on the entire circular interface ∆2 in Eqs.(6) and(7) can be rewritten as
∗ (ς) κc Φc (ς) Ωm − µc µm
+
κm Φm (ς) Ωc∗∗ (ς) = − µm µc
+
−
∗ [Φc (ς) + Ωm (ς)] = [Φm (ς) + Ωc∗∗ (ς)]
−
(|ς| = R2 )
(|ς| = R2 )
(35) (36)
We postulate that ∗ κc Φc (z) Ωm (z) − (z) (R1 < |z| < R2 ) µc µm κm Φm (z) Ωc∗∗ (z) g3 (z) = − (|z| > R2 ) µm µc ∗ g4 (z) = Φc (z) + Ωm (z) (R1 < |z| < R2 )
g3 (z) =
g4 (z) = Φm (z) +
Ωc∗∗
(z)
(|z| > R2 )
(37) (38) (39) (40)
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By means of the generalized Liouville theorem, this is reduced to zp − zp∗ Dω κm z − zp z − zn∗ zn − zn∗ zn − zp 1 g3 (z) = ln + − + ln − 2 µm z − zn µm z − zp∗ z − zp∗ z − zn∗ z 1 + [κc AN (z) − TP (z)] µc zp − zp∗ z − zp zn − zn∗ zn − zp z − zn∗ Dω ln + − + + AN (z) + TP (z) + ln g4 (z) = 2 z − zn z − zp∗ z − zp∗ z − zn∗ z
(41) (42)
By combining the definitions of Eqs.(37) and (39), and then considering Eqs.(41) and (42), the following complex expression turns out to be Φc (z) =
Dω z − zp κm + 1 1 − µm /µc ln TP (z) + AN (z) + 2 µm κc /µc + 1 z − zn µm κc /µc + 1
(43)
Taking the complex conjugate of Eq.(14) and according to Eqs.(14), (38), (40)-(43), we are able to express Ψc (z) as follows: R2 Dω 1/κm + 1 z − zp κm + 1 Ψc (z) = 22 − ln z 2 µm κc /µc + 1 µm /(µc κm ) + 1 z − zn " # ∗ zp − zp κm + 1 Dω 1/κm + 1 z + − ∗ 2 µm /(µc κm ) + 1 zp µm κc /µc + 1 z − zp ∗ κm + 1 Dω 1/κm + 1 zn − zn z + + ∗ 2 µm /(µc κm ) + 1 zn µm κc /µc + 1 z − zn zp Dω 1/κm + 1 zp − zn z + ln + 2 µm /(µc κm ) + 1 R22 zn 1 − µm /µc +AN (z) + [TP (z) − zTP′ (z)] − zA′N (z) µm κc + 1 2 2 R2 R2 1 − µm κc /(µc κm ) AN + TP (44) + µm /(µc κm ) + 1 z z In order to simultaneously satisfy the continuity conditions of displacement and traction at the interfaces ∆1 and ∆2 , the two expressions of Φc0 (z) and Ψc0 (z) obtained in the above part must be compatible, respectively[38, 39] . Specifically, the compatible conditions for Φc0 (z) and Ψc0 (z) imply that the stress field and displacement field in the coating layer are unique. Thus, by the help of Eqs.(33), (34), (43) and (44), the unknown coefficients ak , bk , ck and dk in Φc (z) and Ψc (z) can be determined and the detailed coefficient expressions are listed in Appendix. The complex potentials Φi (z) and Ωi∗ (z) can be determined by Eqs.(24)-(26). The complex potential Ψi (z) can be determined by Eq.(12). In the same method, the final forms of the complex potential functions Φm (z) and Ψm (z) for the matrix can be achieved. Therefore, the complex potentials for the three material regions are determined, and by substituting them into Eqs.(1)-(3), the stress fields and the displacement fields in the three material regions can be found. After such complicated manipulations, the analytical expressions of the complex potentials Φm0 (z) and Ψm0 (z) are calculated at length as zp − zp∗ z − zn∗ zn − zn∗ zn − zp κc µm µm 1 Dω Φm0 (z) = − ln + − + + AN (z) + TN (z) (45) ∗ ∗ ∗ κm 2 z − zp z − zp z − zn z κm µc κm µc ( " zp − zp∗ + z zp − zp∗ z R22 Dω 1 z − zn∗ zn − zn∗ + z + Ψm0 (z) = 2 κm − ln − − z 2 κm z − zp∗ κm (z − zn∗ ) κm z − z ∗ 2 κm z − zp∗ p # z − zp∗ zp (zn − zn∗ ) z 2 (zn − zp ) zp (zn − zp ) z z −κm ln + + ln + − κm ln + 2 (zp − zn ) 2 − zn κm z zn R22 z − zn∗ R2 κm (z − zn∗ ) 2 2 µm κc µm κc µm R2 µm R2 + [AN (z) − zA′N (z)] + [TN (z) − zTN′ (z)] + AP + TP (46) κm µc κm µc µc z µc z
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If we let µi = µc and R1 = R2 , the problem is tantamount to the interaction between a wedge disclination dipole and an inhomogeneity. The solutions obtained are consistent with the reduced results obtained by Luo and Liu[19] .
III. FORCE ON WEDGE DISCLINATION DIPOLE In connection with the aforementioned complex potentials, the force acting upon the wedge disclination dipole inside the matrix can be calculated with the generalized Peach-Koehler formula[40] . The force components fx and fy are given by the perturbation complex functions Φm0 (z0 ), Ψm0 (z0 ) and Φ′m0 (z0 ): h i fx − ify = 2ωaeiϕ Φm0 (z0 ) + Φm0 (z0 ) + 2ωae−iϕ [z0 Φ′m0 (z0 ) + Ψm0 (z0 )]
(47)
By referring to Eqs.(45) and (46), the perturbation force components fx and fy on the disclination dipole can be calculated.
IV. NUMERICAL EXAMPLES In the three-phase composite model, the main parameters are the material combinations, the thickness of the coating layer and the features of the disclination. So this work will focus on studying the influence of these parameters on the behavior of the wedge disclination dipole. For simplicity of description and analysis, the normalized force components on disclination dipole center are defined as fx0 = 4πfx /(µ3 ω 2 R2 ) and fy0 = 4πfy /(µ3 ω 2 R2 ). In addition, we define the relative shear modulus α = µ1 /µ3 and β = µ2 /µ3 , the relative central position of disclination dipole δ = ρ/R2 , the relative thickness of the coating layer λ = R2 /R1 , the relative semi length of dipole arm η = a/R2 . In the study of this section wedge disclination angle and dipole location angle are set to be located on the x-axis if it is not specially presented, which means ϕ = θ = 0. In this case fy0 is zero because of the symmetry of the problem. In Figs.2 and 3, the variation of the normalized force fx0 versus the relative central position of disclination dipole δ is depicted for different relative shear moduli and Poisson’s ratios. Figure 2 demonstrates the effect of relative shear moduli on the normalized force fx0 is considerable, especially when disclination dipole approaches the external interface of the coated inhomogeneity. It is evident that if the inhomogeneity is harder, as the coating layer is softer than the matrix, the disclination dipole may be always attracted by the coated inhomogeneity. Furthermore, the attractive force on the disclination dipole fast increases as it is close to the coated inhomogeneity. As the disclination dipole approaches the interface from a distance, the hard coating layer first attracts the disclination dipole and then repels it because of the hard inhomogeneity. There is a stable equilibrium position on the x-axis where the normalized force fx0 equals zero. However, by increasing the stiffness of the coating layer, this stable equilibrium position disappears. Clearly, if the inhomogeneity and the coating layer are softer than the matrix (e.g. α = 0.8 and β = 0.9), the disclination dipole is first attracted and then
Fig. 2. fx0 vs. δ for various α and β (λ = 1.1, η = 0.005, ϕ = θ = 0, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.3).
Fig. 3. fx0 vs. δ for various ν2 (α = 1.1, β = 0.7, λ = 1.1, η = 0.005, ϕ = θ = 0, ν1 = 0.3 and ν3 = 0.3).
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repelled by the coated inhomogeneity when the disclination dipole and the external interface of the coated inhomogeneity are far apart. Figure 3 displays the variation of fx0 with δ for various Poisson’s ratios ν2 . The results indicate that the force on the disclination dipole strongly depends on the Poisson’s ratio of the coating layer. When the Poisson’s ratio of the inhomogeneity equals that of the matrix, the smaller ν2 contributes to the attraction of the coated inhomogeneity. As ν2 increases, an unstable equilibrium position of the disclination dipole can be found, i.e., for ν2 = 0.25 or thereabouts at the position δ = 1.7. The equilibrium position of the disclination dipole will vanish with increasing Poisson’s ratio of the coating layer. From these figures, it can be seen that the shear modulus and Poisson’s ratio of the coating layer are able to change not only the location of the equilibrium position, but also the stability of the disclination dipole. The curves in Fig.4 are plotted for α = 1.5, β = 0.8, λ = 1.1, ϕ = θ = 0, ν1 = 0.2, ν2 = 0.1, ν3 = 0.2, µ3 = 2.7 GPa and R1 = 100 nm to study the effect of disclination strength ω and relative semi length of dipole arm η on the force fx acting on the disclination dipole. Since the inhomogeneity is harder and the coating layer is softer than the matrix in these calculations, the disclination dipole is always repelled first and then attracted by the coated inhomogeneity when the disclination dipole is going away from the external interface of the coated inhomogeneity. In particular, the effect of the coated inhomogeneity on the force acting on the disclination dipole increases with the increase in disclination strength and relative semilength of the dipole Fig. 4 fx vs. δ for various ω and η (α = 1.5, β = arm. Interestingly, various disclination strengths and rela- 0.8, λ = 1.1, ϕ = θ = 0, ν1 = 0.2, ν2 = 0.1, tive semilengths of the dipole arm have negligible impact ν3 = 0.2, µ3 = 2.7 GPa and R1 = 100 nm). on the equilibrium position. The effect of the relative thickness of the coating layer λ on the normalized force fx0 acting on disclination dipoles is illustrated in Fig.5. It is clearly shown that, for the case of hard inhomogeneity with soft coating layer, as λ increases, the repulsive force decays and then turns attractive. When the coating layer is thick enough, the elastic property of the hard inhomogeneity has no significant influence on the force acting on the disclination dipole, but is shielded by the soft coating layer. Additionally, a hard inhomogeneity may cause an equilibrium position for the disclination dipole in the vicinity of the interface because of better shielding of the soft coating layer. As the disclination dipole gets closer to the external interface of the coated inhomogeneity, the shield effect will become stronger. In other words, the larger the value of λ is, the less influence the inhomogeneity phase has on the disclination
Fig. 5. fx0 vs. δ for various λ (α = 1.1, β = 0.9, η = 0.005, ϕ = θ = 0, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.2).
Fig. 6. fx0 vs. ν2 for various λ (α = 0.8, β = 1.1, η = 0.005, ϕ = θ = 0, δ = 1.1, ν1 = 0.3 and ν3 = 0.3).
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dipole. Therefore the influence on the equilibrium and stability of the disclination dipole is gradually approaching the two-phase case obtained by Luo and Liu[19] . In Fig.6, keeping the normalized fixed position of the disclination dipole δ = 1.1, we vary relative thicknesses of the coating layer λ to estimate the dependence of the normalized force fx0 on the Poisson’s ratio of the coating layer ν2 . As is seen, the dependence of the force acting on the disclination dipole upon the influence of λ is more considerable and becomes stronger for bigger ν2 . There exists a critical value ν2c of the Poisson’s ratio of the coating layer to keep the disclination dipole in equilibrium, i.e., ν2c = 0.17 for λ = 1.2. This critical value decreases with thickening of the coating layer. Thus, it can be concluded that the material properties of the coating layer have a major part to alter the strengthening or toughening effect produced by the coated inhomogeneity. Figure 7(a) and (b) give the plots for the normalized forces fx0 and fy0 as a function of the relative center position of disclination dipole δ with different wedge disclination angles ϕ and dipole location angles θ for α = 1.2, β = 0.8, η = 0.005, λ = 1.1, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.2, respectively. It is evident that the effects of ϕ and θ are remarkable on fx0 and fy0 , especially near the external interface of the coated inhomogeneity. Owing to the effects of ϕ and θ, the equilibrium position and the stability of the disclination dipole change significantly. For instance, for fx0 , there exists a stable equilibrium position for ϕ = θ = 25◦ , while for ϕ = θ = 60◦ , the equilibrium position is unstable. It is well known that fx0 equals zero at ϕ = θ = 90◦ , while fy0 equals zero at ϕ = θ = 0◦ , whatever the relative center position of disclination dipole δ.
Fig. 7. (a) fx0 and (b) fy0 vs. δ for various ϕ and θ (α = 1.2, β = 0.8, η = 0.005, λ = 1.1, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.2).
V. CONCLUSIONS In our current work, the elastic behavior of a wedge disclination dipole in the three-phase composite material is carried out. By using the complex variable function method, the force acting on the wedge disclination dipole is derived analytically. The effects of the material combination, relative thickness of the coating layer and the features of the disclination dipole on the equilibrium position and stability of the disclination dipole near the coated inhomogeneity are studied and discussed in detail with numerical examples. The primary conclusions are as follows: (1) The effects of the shear modulus and Poisson’s ratio of the coating layer on the normalized force are considerable, especially when disclination dipole approaches the external interface of the coated inhomogeneity. They change not only the location of the equilibrium position, but also the stability of the disclination dipole. (2) The effect of the coated inhomogeneity on the force acting on the disclination dipole increases with the enlargement of disclination strength and relative semilength of the dipole arm. Various disclination strengths and relative semilengths of the dipole arm have negligible impact on the equilibrium position. (3) The larger the value of relative thickness of the coating layer is, the less influence the inhomogeneity phase has on the disclination dipole. The dependence of the force acting on the disclination dipole upon
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the influence of relative thickness of the coating layer is more considerable and becomes stronger for bigger Poisson’s ratios of the coating layer. There exists a critical value of the Poisson’s ratio of the coating layer to keep the disclination dipole in equilibrium and it decreases with thickening of the coating layer. (4) The effects of wedge disclination angle and dipole location angle are remarkable on the normalized forces, especially near the coated inhomogeneity. Owing to the effects of the wedge disclination angle and dipole location angle, the equilibrium position and the stability of the disclination dipole change significantly.
References [1] Romanov,A.E. and Kolesnikova,A.L., Application of disclination concept to solid structures. Progress in Materials Science, 2009, 54(6): 740-769. [2] Romanov,A.E. and Vladimirov,V.I., Dislocations in Solids. Nabarro, F.R.N., Ed., North-Holland: Amsterdam, 1992. [3] Romanov,A.E., Screened disclinations in solids. Materials Science and Engineering A, 1993, 164(1-2): 58-68. [4] Murayama,M., Howe,J.M., Hidaka,H. and Takaki,S., Atomic-level observation of disclination dipoles in mechanically milled, nanocrystalline Fe. Science, 2002, 295(5564): 2433-2435. [5] Lei,R.S., Wang,M.P., Li,Z. and Wei,H.G., Disclination dipoles observation and nanocrystallization mechanism in ball milled Cu-Nb powders. Materials Letters, 2011, 65(19-20): 3044-3046. [6] Jiang,X. and Jia,C.L., The coalescence of [001] diamond grains heteroepitaxially grown on (001) silicon. Applied Physics Letters, 1996, 69(25): 3902-3094. [7] Yacam´ an,M.J., Ascencio,J.A., Liu,H.B. and Gardea-Torresdey,J., Structure shape and stability of nanometric sized particles. Journal of Vacuum Science & Technology B, 2001, 19(4): 1091-1103. [8] Ovid’ko,I.A., Deformation of nanostructures. Science, 2002, 295(5564): 2386. [9] Morozov,N.F., Ovid’ko,I.A., Sheinerman,A.G. and Aifantis,E.C., Special rotational deformation as a toughening mechanism in nanocrystalline solids. Journal of the Mechanics and Physics of Solids, 2010, 58(8): 1088-1099. [10] Zhao,Y.X., Fang,Q.H., Liu,Y.W. and Jiang,C.Z., Shielding effects of disclinations on the elliptical blunt crack. International Journal of Engineering Science, 2013, 70: 91-101. [11] Zhao,Y.X., Fang,Q.H. and Liu,Y.W., Effect of cooperative nanograin boundary sliding and migration on dislocation emission from a blunt nanocrack tip in nanocrystalline materials. Philosophical Magazine, 2014, 94(7): 700-730. [12] Gryaznov,V.G., Gutkin,M.Yu., Romanov,A.E. and Trusov,L.I., On the yield stress of nanocrystals. Journal of Materials Science, 1993, 28(16): 4359-4365. [13] Romanov,A.E., Mechanics and physics of disclinations in solids. European Journal of Mechanics - A/Solids, 2003, 22(5): 727-741. [14] Rybin,V.V. and Zhukovskii,I.M., Disclination mechanism of microcrack formation. Soviet Physics. Solid State, 1978, 20(2): 1056-1059. [15] Wu,M.S., Zhou,K. and Nazarov,A.A., Crack nucleation at disclinated triple junctions. Physical Review B, 2007, 76(13): 134105. [16] Gutkin,M.Yu. and Ovid’ko,I.A., Disclinations, amorphization and microcrack generation at grain boundary junctions in polycrystalline solids. Philosophical Magazine A, 1994, 70(4): 561-575. [17] Bobylev,S.V. and Ovid’ko,I.A., Nanoscale amorphization at disclination quadrupoles in deformed nanomaterials and polycrystals. Applied Physics Letters, 2008, 93(6): 061904. [18] Zhou,K., Nazarov,A.A. and Wu,M.S., Atomistic simulations of the tensile strength of a disclinated bicrystalline nanofilm. Philosophical Magazine, 2008, 88(27): 3181-3191. [19] Luo,J. and Liu,F., Stress analysis of a wedge disclination dipole interacting with a circular nanoinhomogeneity. European Journal of Mechanics - A/Solids, 2011, 30(1): 22-32. [20] Zhou,K., Hoh,H.J., Wang,X., Keer,L.M., Pang,H.J.L., Song,B. and Wang,Q.J., A review of recent works on inclusions. Mechanics of Materials, 2013, 60: 144-158. [21] Chen,C.Q., Cui,J.Z., Duan,H.L., Feng,X.Q., et al., Perspectives in mechanics of heterogeneous solids. Acta Mechanica Solida Sinica, 2011, 24(1): 1-26. [22] Wang,X. and Shen,Y.P., Arc interface crack in a three-phase piezoelectric composite constitutive model. Acta Mechanica Solida Sinica, 2001, 14(2): 104-117. [23] Wang,X. and Shen,Y.P., An edge dislocation in a three-phase composite cylinder model with a sliding interface. Journal of Applied Mechanics, 2002, 69(4): 527-538.
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[24] Liu,Y.W., Fang,Q.H. and Jiang,C.P., A piezoelectric screw dislocation interacting with an interphase layer between a circular inclusion and the matrix. International Journal of Solids and Structures, 2004, 41(11-12): 3255-3274. [25] Fang,Q.H., Liu,Y.W. and Wen,P.H., Screw dislocations in a three-phase composite cylinder model with interface stress. Journal of Applied Mechanics, 2008, 75(4): 041019. [26] Gong,K.Z., Li,Z. and Qin,W.Z., Influence of loading rate on dynamic fracture behavior of fiber-reinforced composites. Acta Mechanica Solida Sinica, 2008, 21(5): 457-460. [27] Wang,X., Sudak,L.J. and Pan,E., Pattern instability of functionally graded and layered elastic films under van der Waals forces. Acta Mechanica, 2008, 198(1-2): 65-86. [28] Fang,Q.H., Liu,Y.W., Jin,B. and Wen,P.H., A piezoelectric screw dislocation in a three-phase composite cylinder model with an imperfect interface. International Journal of Engineering Science, 2009, 47(1): 39-49. [29] Wang,F., Zeng,X.G. and Zhang,J.Q., Predictive approach to failure of composite laminates with equivalent constraint model. Acta Mechanica Solida Sinica, 2010, 23(3): 240-247. [30] Zhou,Z.G., Zhang,P.W. and Wu,L.Z., Multiple parallel symmetric permeable model-III cracks in a piezoelectric piezomagnetic composite material plane. Acta Mechanica Solida Sinica, 2010, 23(4): 336-352. [31] Wang,X. and Zhou,K., Novel near-cloaking multicoated structures for screw dislocations. Mechanics of Materials, 2012, 55: 73-81. [32] Wang,X. and Zhou,K., Three-phase piezoelectric inclusions of arbitrary shape with internal uniform electroelastic field. International Journal of Engineering Science, 2013, 63: 23-29. [33] Wang,X. and Zhou,K., Long-range interaction of a line dislocation with multiple multicoated inclusions of arbitrary shape. Acta Mechanica, 2013, 224(1): 63-70. [34] Valiev,R.Z., Islamgaliev,R.K. and Alexandrov,I.V., Bulk nanostructured materials from severe plastic deformation. Progress in Materials Science, 2000, 45(2): 103-189. [35] Zhou,K., Nazarov,A.A. and Wu,M.S., Competing relaxation mechanisms in a disclinated nanowire: temperature and size effects. Physical Review Letters, 2007, 98(3): 035501. [36] Wang,T., Luo,J., Xiao,Z.M. and Chen,J.Q., On the nucleation of a Zener crack from a wedge disclination dipole in the presence of a circular inhomogeneity. European Journal of Mechanics - A/Solids, 2009, 28(4): 688-696. [37] Muskhelishvili,N.I., Some Basic Problems of Mathematical Theory of Elasticity. Leyden: Noordhoff, 1975. [38] England,A.H., Complex Variable Method in Elasticity. New York: John Wiley and Sons, 1971. [39] Luo,H.A. and Chen,Y.J., An edge dislocation in a three-phase composite cylinder model. Journal of Applied Mechanics, 1991, 58(1): 75-86. [40] Peach,M. and Koehler,J.S., The forces exerted on dislocations and the stress fields produced by them. Physical Review, 1950, 80(3): 436-439.
APPENDIX p01 (p01 − p02 p03 ) − p01 p01 p201 − p02 p04 [p02 (p04 s01 − p01 s02 ) + p201 − p02 p04 (s01 − p01 s01 ) − p01 p02 + p201 − p02 p04 p01 p02 · (p01 − p02 p03 ) (p04 s01 − p01 s02 ) − p201 − p02 p04 (p03 s01 − p01 s02 ) ÷ p02 (p04 − p01 p03 ) + p201 − p02 p04 (1 − p01 p01 ) [p01 (p01 − p02 p03 ) −p01 p01 p201 − p02 p04 − (p04 − p01 p03 ) (p01 − p02 p03 ) − p201 − p02 p04 (p03 − p01 p04 ) · p01 p02 + p201 − p02 p04 p01 p02 (48)
a0 =
c0 =
(p04 − p01 p03 ) (p01 − p02 p03 ) − p201 − p02 p04 (p03 − p01 p04 ) [p02 (p04 s01 − p01 s02 ) + p201 − p02 p04 (s01 − p01 s01 ) − p02 (p04 − p01 p03 ) + p201 − p02 p04 (1 − p01 p01 ) · (p01 − p02 p03 ) (p04 s01 − p01 s02 ) − p201 − p02 p04 (p03 s01 − p01 s02 ) ÷ (p04 − p01 p03 ) (p01 − p02 p03 ) − p201 − p02 p04 (p03 − p01 p04 ) · p01 p02 + p201 − p02 p04 p01 p02 − p02 (p04 − p01 p03 ) + p201 − p02 p04 (1 − p01 p01 ) · p01 (p01 − p02 p03 ) − p01 p01 p201 − p02 p04 (49)
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s11 + (p12 p16 − p11 p14 ) 1 − p12 p15 p14 s12 b1 = 1 − p14 p13 s12 c1 = 1 − p13 p14 p15 s11 + (p16 /p14 − p15 p11 ) d1 = 1 − p15 p12 a1 =
(50) (51) (52) (53)
ak = {(pk1 pk6 − pk2 pk8 ) (pk3 sk1 + pk1 sk2 ) − [pk1 − pk8 (pk2 pk3 + pk1 pk4 )] sk1 } ÷ {[pk3 − pk7 (pk2 pk3 + pk1 pk4 )] (pk1 pk6 − pk2 pk8 ) − (1 + pk1 pk5 − pk2 pk7 ) [pk1 − pk8 (pk2 pk3 + pk1 pk4 )]}
(k ≥ 2)
(54)
bk = {(pk5 pk2 + pk6 pk4 ) (pk6 pk7 − pk5 pk8 ) sk2 − [pk4 (pk6 pk7 − pk5 pk8 ) + pk5 ] (pk5 sk1 + pk6 sk2 )} ÷{[pk7 − pk3 (pk6 pk7 − pk5 pk8 )] (pk5 pk2 + pk6 pk4 ) + [pk4 (pk6 pk7 − pk5 pk8 ) + pk5 ] (pk6 pk3 − 1 − pk5 pk1 )} (k ≥ 2) ck = {(1 + pk1 pk5 − pk2 pk7 ) (pk3 sk1 + pk1 sk2 ) − [pk3 − pk7 (pk2 pk3 + pk1 pk4 )] sk1 }
(55)
÷{(1 + pk1 pk5 − pk2 pk7 ) [pk1 − pk8 (pk2 pk3 + pk1 pk4 )] − [pk3 − pk7 (pk2 pk3 + pk1 pk4 )] (pk1 pk6 − pk2 pk8 )} (k ≥ 2) (56) dk = {(1 + pk5 pk1 − pk6 pk3 ) (pk6 pk7 − pk5 pk8 ) sk2 − [pk7 − pk3 (pk6 pk7 − pk5 pk8 )] (pk5 sk1 + pk6 sk2 )} ÷{[pk7 − pk3 (pk6 pk7 − pk5 pk8 )] (pk5 pk2 + pk6 pk4 ) + [pk4 (pk6 pk7 − pk5 pk8 ) + pk5 ] (pk6 pk3 − 1 − pk5 pk1 )}
(k ≥ 2)
(57)
where 1 − µi /µc 1 − µi κc /(µc κi ) , m2 = µi κc /µc − 1 µi /(κi µc ) + 1 Dω κm + 1 1 − µm /µc Dω 1/κm + 1 1 − µm κc /(µc κm ) n1 = , n2 = , n3 = , n4 = 2 µm κc /µc + 1 µm κc /µc + 1 2 µm /(µc κm ) + 1 µm /(µc κm ) + 1 ∗ ∗ n3 z p − z p n3 z n − z n n3 (zp − zn ) , n7 = , n8 = n5 = n1 − n3 , n6 = ∗ ∗ R22 z p + n1 z n + n1 m1 =
p03 = R12 + R22 , p04 = m2 R12 + n2 R22 zp zp s02 = (n3 + n5 ) ln s01 = n1 ln , zn zn −2 −4 p11 = 2n2 R2 , p12 = n2 R2 , p13 = n4 , p14 = m1 , p15 = m2 R14 , −1 −1 2 −1 −1 s11 = n1 zn − zp , s12 = R2 (n6 − n5 ) zp + (n5 − n7 ) zn + n8 p01 = n2 ,
p02 = n2 R2−2 ,
pk1 = n2 (k + 1) R2−2k , pk4 = n2 (k − 1) R2−2k ,
p16 = 2m1 R12
pk2 = n2 R2−2k−2 , pk3 = [n4 + n2 (k − 1) (k + 1)] R2−2k+2 2k pk5 = m1 (k − 1) R1 , pk6 = m1 R12k−2
pk7 = [m1 (k − 1) (k + 1) + m2 ] R12k+2 , pk8 = m1 (k + 1) R12k h i n1 zn−k − zp−k n5 −k n5 , sk2 = R22 n6 − zp + − n7 zn−k sk1 = k k k
ISSN 0894-9166
A WEDGE DISCLINATION DIPOLE INTERACTING WITH A COATED CYLINDRICAL INHOMOGENEITY⋆⋆ Yingxin Zhao
Qihong Fang
Youwen Liu⋆
(State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China) (College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China) Received 6 May 2013, revision received 19 April 2014
ABSTRACT A three-phase composite cylinder model is utilized to study the interaction of a wedge disclination dipole with a coated cylindrical inhomogeneity. The explicit expression of the force acting on the wedge disclination dipole is calculated. The motilities and the equilibrium positions of the disclination dipole near the coated inhomogeneity are discussed for various material combinations, relative thicknesses of the coating layer and the features of the disclination dipole. The results show that the material properties of the coating layer have a major part to play in alteringi the strengthening effect or toughening effect produced by the coated inhomogeneity.
KEY WORDS three-phase composite, disclination dipole, coating layer, complex variable function method
I. INTRODUCTION Disclinations together with dislocations represent a class of linear defects in solids and are characterized by typical singularities and the property of multi-value of the fields of displacement and rotation associated with the defects[1] . They are considered negative (or positive) if material needs to be filled in to bridge any gap due to the relative displacement (or taken away if the faces interpenetrate)[2] . Disclinations in solids exist only in the form of screened configurations with lower self-energy, including dipoles, quadrupoles and disclination loops[3] . Indeed, experimental evidence of the disclination concept is also becoming available: disclination dipoles and multipoles have been observed directly via electron microscopy in metals such as copper, titanium and α-iron single crystals subjected to severe plastic deformation[4, 5]. Disclinations are also observed in thin films grown on substrates[6] and in pentagonal nanoparticles[7]. Increasing attention is being attracted to disclinations with various configurations because of their diverse applications in materials science and physics. Disclination models are extremely important for exploring the physical nature of the rotation deformation mode peculiar to crystalline materials[8–11] . They can model the geometry and elastic distortions in rotational defect structures, high angle grain boundaries in polycrystals and crystalline materials. Important additions to understanding deformation and hardening properties of crystalline materials can be delivered in the framework of Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11172094 and 11172095), the New Century Excellent Talents in University (NCET-11-0122) and Hunan Provincial Natural Science Foundation for Creative Research Groups of China (No. 12JJ7001). ⋆
⋆⋆
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disclination models, which accounts for the specific disclination defects at the junctions of nanograin boundaries[12] . Owing to the singular internal stress fields induced by disclinations[13] , relaxation of these stresses brings about various configurational changes in solids, the most common being crack or void nucleation in their vicinity[14, 15] . Disclinated cracks have been studied via continuum elasticity[16, 17] . In Ref.[16], local amorphization of the disclination core is suggested as a relaxation mechanism that can retard crack nucleation. Consequently, the various types of relaxed structures directly affect the physical properties such as strength, toughness, diffusion and electrical conductivity. Another possibility of diminishing the disclination energy depends on the screening of disclination elastic fields by traction-free surfaces. In this case screened disclinations can be important in subsurface layers, shells, thin films and small crystalline particles[18, 19] . It turned out that these studies have proved disclinations quite helpful to the analysis of deformation behavior of polycrystals, crystalline materials, and whatnot. In the study of various defects, such as dislocation, disclination, crack, and so on, interaction with inhomogeneities is motivated by the need to gain a better understanding of certain strengthening and hardening materials, especially composite materials. This is mainly due to the fact that the mobilities and stabilities of defects can be influenced by various inhomogeneities in materials. When the defect is not extremely close to the interface and the size of inhomogeneity is fairly large, elastic interaction is expected to be predominant[20, 21] . In most inhomogeneity-reinforced composite materials, control of the wetting, reaction and bonding of the matrix to inhomogeneity is critical to achieving the desired property goals. Therefore, the coating layer on inhomogeneities is often widely employed in order to increase the bonding strength between inhomogeneity and matrix. In a ceramic matrix composite material, load transfer may be of secondary importance. While in metal matrix and polymer composite materials, as the mechanical properties are greatly enhanced by efficient load transfer from the matrix to inhomogeneities, the bond quality between the inhomogeneity and the matrix is very important. The bond should be strong enough to yield high quality composites. A coating layer around inhomogeneity-matrix interface can help achieve controlled delamination of the interface and prevent cracks initiated external to the inhomogeneity from damaging the matrix. The strengthening and hardening behavior of inhomogeneity reinforced composites can also be well explained by the elastic interaction mechanisms between dislocations/disclinations/cracks and multi-phase inhomogeneities[22–33] . Seeing that disclination mobility and stability can greatly affect the mechanical behaviors of the composites, the problem of disclinations interacting with composites has also drawn many investigators’ attention to evaluation of structures and mechanical properties of solids, e.g., film-substrate composites and nanostructured materials[34–36] . Incidentally, report on the interaction of disclination dipole with the three-phase composite material seems unavailable probably for the complexity of the calculation. In the present paper, the problem of the interaction between a wedge disclination dipole and a coated cylindrical inhomogeneity is theoretically solved. By complex variable techniques, the analytic solution of complex potentials and the force exerted on the disclination dipole is derived. The impact of material properties, the parameters of the coating layer and the features of the disclination dipole on the force is examined and discussed. Our results are helpful in the understanding of the motion mechanism of the disclinations and relevant physical phenomena in three-phase composite materials. Although the disclinations are assumed to be located within matrix, the other cases of dislocation located inside the inhomogeneity or coating layer can be obtained by the same method. The information obtained is useful in design or fabrication technology of three-phase composite materials for better or specified conductivities.
II. MODEL In this paper, we consider a coated inhomogeneity embedded in an infinite isotropic matrix consisting of two cylindrical domains made of different materials, as shown in Fig.1. ∆1 and ∆2 indicate the interfaces between the inhomogeneity and the annular coating layer, and the annular coating layer and the matrix, respectively. The two cylindrical interfaces are assumed to be perfectly bound and coherent. The radii of the inhomogeneity surface and coating layer-matrix interface are denoted by R1 and R2 , respectively. Suppose that a wedge disclination dipole consisting of a positive disclination with strength ω at zp (= z0 − aeiϕ ) and a negative disclination with the same strength at zn (= z0 + aeiϕ ) is at an arbitrary position inside the infinite matrix domain, where 2a denotes the dipole arm and the center of the disclination dipole z0 = ρeiθ . Both disclination lines are perpendicular to the (x, y) plane.
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Hereinafter phase ‘i’ means the inhomogeneity media and has material properties µi and νi , occupying the region r < R1 ; phase ‘c’ refers to the annular coating layer and has material properties µc and νc , occupying the area R1 < r < R2 ; phase ‘m’ denotes the matrix and has properties µm and νm with the area r > R2 .
Fig. 1. A wedge disclination dipole interacting with the coated inhomogeneity.
For a plane strain problem under consideration, stress fields and displacement fields are expressed by following two complex potentials Φ(z) and Ψ (z)[37] h i σyy + σxx = 2 Φ (z) + Φ (z) (1) σyy − σxx + 2iσxy = 2 [zΦ′ (z) − Ψ (z)] # " zΨ (z) ′ ′ ′ 2µ ux + uy = iz κΦ (z) − Φ (z) + zΦ (z) + z
(2)
(3)
where u′x = ∂ux /∂θ, u′y = ∂uy /∂θ, ui and σij are the displacement and stress tensors in Cartesian coordinate, respectively; Φ′ (z) = d [Φ (z)]/dz, z = ρeiθ ; the overbar represents the complex conjugate; κ = 3 − 4ν for plane strain state, µ is the shear modulus of the bulk solid, ν is the Poisson’s ratio of the bulk solid. The emergence of compatibility in the form of stress fields and displacement fields at the coherent interfaces gives rise to compatibility tensors in the framework of elastic fields. Accordingly, the conventional stress and displacement continuity conditions are given as follows: ′ + − uxi (t) + iu′yi (t) = u′xc (t) + iu′yc (t) (4) [σxi (t) + iσyi (t)]+ = [σxc (t) + iσyc (t)]− − ′ + uxc (ς) + iu′yc (ς) = u′xm (ς) + iu′ym (ς) +
−
[σxc (ς) + iσyc (ς)] = [σxm (ς) + iσym (ς)]
(5)
(6) (7)
where the superscripts ‘+’ and ‘−’ denote the boundary quantities as z approaches the interior and exterior of a cylindrical interface, respectively. Here, |t| = R1 and |ς| = R2 . Regarding classical elasticity, for a wedge disclination dipole located at an arbitrary point z0 in the matrix, two complex potentials Φm (z) and Ψm (z) in the matrix are introduced[33] Dω [ln (z − zp ) − ln (z − zn )] + Φm0 (z) (8) Φm (z) = 2 Dω zn zp Ψm (z) = − + Ψm0 (z) (9) 2 z − zn z − zp
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where D = µm /[2π (1 − νm )], Φm0 (z) and Ψm0 (z) are holomorphic in the matrix. In this case, the complex potentials Φi (z) and Ψi (z) are holomorphic in inhomogeneity. And two holomorphic complex potentials Φc (z) and Ψc (z) in the coating layer can be given as ∞ ∞ X X Φc (z) = ak z k + bk z −k (10) k=0
Ψc (z) =
∞ X
k=1 ∞ X
ck z k−2 +
k=0
dk z −k−2
(11)
k=1
where ak , bk , ck and dk are unknown coefficients to be determined from the boundary conditions (4)-(7). ∞ ∞ P P ak z k and AN (z) = bk z −k . Furthermore, let AP (z) = k=0
k=1
For the purpose of easy treatment of the boundary conditions on the interfaces and convenient calculation, the following new analytical functions are introduced in the corresponding regions according to the Schwarz symmetry principle. 2 2 R12 ′ R12 R12 R1 R1 ∗ Ωi (z) = −Φi + Φi + 2 Ψi (12) z z z z z 2 R2 R2 R1 R12 R12 + 1 Φ′c + 21 Ψc (13) Ωc∗ (z) = −Φc z z z z z 2 R2 R2 R22 R2 R22 Ωc∗∗ (z) = −Φc + 2 Φ′c + 22 Ψc (14) z z z z z 2 2 2 R2 R2 R2 R2 R2 ∗ Ωm + 2 Φ′m + 22 Ψm (15) (z) = −Φm z z z z z Substituting Eqs.(8)-(11) into (12)-(15), we arrive at the expressions as Ωc∗ (z) = SN (z) + SP (z) Ωc∗∗ (z) = TN (z) + TP (z) zp − zp∗ zn − zn∗ zn − zp z − zn∗ Dω ∗ ∗ + − + ln + Ωm0 (z) Ωm (z) = 2 z − zp∗ z − zp∗ z − zn∗ z ∗ (z) is holomorphic in the region |z| < R2 . where zp∗ = R22 /zp , zn∗ = R22 /zn , and Ωm0 ∞ i h X 2(k−1) z −k SN (z) = (k − 1) ak R12k + ck R1
(16) (17) (18)
(19)
k=1
SP (z) = −a0 + c0 R1−2 +
∞ h i X −2(k+1) − (k + 1) bk R1−2k + dk R1 zk
(20)
k=1
TN (z) =
∞ h X
2(k−1)
(k − 1) ak R22k + ck R2
k=1
TP (z) = −a0 + c0 R2−2 +
i
z −k
∞ h i X −2(k+1) zk − (k + 1) bk R2−2k + dk R2
(21) (22)
k=1
In connection with Eqs.(3), (8), (12), (13), (16) and the displacement continuity condition on the entire circular interface ∆1 , Eq.(4) is simplified as + − κi 1 ∗ κc 1 ∗ Φi (t) − Ωc (t) = Φc (t) − Ωi (t) (|t| = R1 ) (23) µi µc µc µi Let κi Φi (z) Ωc∗ (z) g1 (z) = − (|z| < R1 ) (24) µi µc κc Φc (z) Ωi∗ (z) g1 (z) = − (R1 < |z| < R2 ) (25) µc µi
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With reference to the generalized Liouville theorem, we have g1 (z) =
κc AP (z) SN (z) − µc µc
(26)
Using Eqs.(1), (2), (8), (12), (13), (16) and the stress continuity condition on the entire circular interface ∆1 , we obtain the following equation from Eq.(5): +
[Φi (t) + Ωc∗ (t)] = [Φc (t) + Ωi∗ (t)]
−
(|t| = R1 )
(27)
Suppose g2 (z) = Φi (z) + Ωc∗ (z) g2 (z) = Φc (z) +
Ωi∗
(z)
(|z| < R1 )
(28)
(R1 < |z| < R2 )
(29)
With the aid of the generalized Liouville theorem, this leads to g2 (z) = AP (z) + SN (z)
(30)
If we combine Eqs.(24) and (28), the following complex expression is derived: Ωc∗ (z) =
g2 (z) − µi g1 (z)/κi µi /(κi µc ) + 1
(31)
By considering Eqs.(25) and (29), the following complex expression is given by Φc (z) =
µi g1 (z) + g2 (z) µi κc /µc + 1
(32)
Substitution of Eqs.(26) and (30) into (32) yields Φc (z) = AP (z) +
(1 − µi /µc ) SN (z) µi κc /µc + 1
(33)
Taking the complex conjugate of Eq.(13) and using Eqs.(26), (30), (31) and (33) leads to 2 R1 R12 1 − µi /µc ′ Ψc (z) = 2 AP (z) + [SN (z) − zSN (z)] + SN z µi κc /µc + 1 z 2 1 − µi κc /(µc κi ) R1 + AP − zA′P (z) µi /(κi µc ) + 1 z
(34)
By use of the above same cleaning solution, the displacement and stress continuity conditions on the entire circular interface ∆2 in Eqs.(6) and(7) can be rewritten as
∗ (ς) κc Φc (ς) Ωm − µc µm
+
κm Φm (ς) Ωc∗∗ (ς) = − µm µc
+
−
∗ [Φc (ς) + Ωm (ς)] = [Φm (ς) + Ωc∗∗ (ς)]
−
(|ς| = R2 )
(|ς| = R2 )
(35) (36)
We postulate that ∗ κc Φc (z) Ωm (z) − (z) (R1 < |z| < R2 ) µc µm κm Φm (z) Ωc∗∗ (z) g3 (z) = − (|z| > R2 ) µm µc ∗ g4 (z) = Φc (z) + Ωm (z) (R1 < |z| < R2 )
g3 (z) =
g4 (z) = Φm (z) +
Ωc∗∗
(z)
(|z| > R2 )
(37) (38) (39) (40)
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By means of the generalized Liouville theorem, this is reduced to zp − zp∗ Dω κm z − zp z − zn∗ zn − zn∗ zn − zp 1 g3 (z) = ln + − + ln − 2 µm z − zn µm z − zp∗ z − zp∗ z − zn∗ z 1 + [κc AN (z) − TP (z)] µc zp − zp∗ z − zp zn − zn∗ zn − zp z − zn∗ Dω ln + − + + AN (z) + TP (z) + ln g4 (z) = 2 z − zn z − zp∗ z − zp∗ z − zn∗ z
(41) (42)
By combining the definitions of Eqs.(37) and (39), and then considering Eqs.(41) and (42), the following complex expression turns out to be Φc (z) =
Dω z − zp κm + 1 1 − µm /µc ln TP (z) + AN (z) + 2 µm κc /µc + 1 z − zn µm κc /µc + 1
(43)
Taking the complex conjugate of Eq.(14) and according to Eqs.(14), (38), (40)-(43), we are able to express Ψc (z) as follows: R2 Dω 1/κm + 1 z − zp κm + 1 Ψc (z) = 22 − ln z 2 µm κc /µc + 1 µm /(µc κm ) + 1 z − zn " # ∗ zp − zp κm + 1 Dω 1/κm + 1 z + − ∗ 2 µm /(µc κm ) + 1 zp µm κc /µc + 1 z − zp ∗ κm + 1 Dω 1/κm + 1 zn − zn z + + ∗ 2 µm /(µc κm ) + 1 zn µm κc /µc + 1 z − zn zp Dω 1/κm + 1 zp − zn z + ln + 2 µm /(µc κm ) + 1 R22 zn 1 − µm /µc +AN (z) + [TP (z) − zTP′ (z)] − zA′N (z) µm κc + 1 2 2 R2 R2 1 − µm κc /(µc κm ) AN + TP (44) + µm /(µc κm ) + 1 z z In order to simultaneously satisfy the continuity conditions of displacement and traction at the interfaces ∆1 and ∆2 , the two expressions of Φc0 (z) and Ψc0 (z) obtained in the above part must be compatible, respectively[38, 39] . Specifically, the compatible conditions for Φc0 (z) and Ψc0 (z) imply that the stress field and displacement field in the coating layer are unique. Thus, by the help of Eqs.(33), (34), (43) and (44), the unknown coefficients ak , bk , ck and dk in Φc (z) and Ψc (z) can be determined and the detailed coefficient expressions are listed in Appendix. The complex potentials Φi (z) and Ωi∗ (z) can be determined by Eqs.(24)-(26). The complex potential Ψi (z) can be determined by Eq.(12). In the same method, the final forms of the complex potential functions Φm (z) and Ψm (z) for the matrix can be achieved. Therefore, the complex potentials for the three material regions are determined, and by substituting them into Eqs.(1)-(3), the stress fields and the displacement fields in the three material regions can be found. After such complicated manipulations, the analytical expressions of the complex potentials Φm0 (z) and Ψm0 (z) are calculated at length as zp − zp∗ z − zn∗ zn − zn∗ zn − zp κc µm µm 1 Dω Φm0 (z) = − ln + − + + AN (z) + TN (z) (45) ∗ ∗ ∗ κm 2 z − zp z − zp z − zn z κm µc κm µc ( " zp − zp∗ + z zp − zp∗ z R22 Dω 1 z − zn∗ zn − zn∗ + z + Ψm0 (z) = 2 κm − ln − − z 2 κm z − zp∗ κm (z − zn∗ ) κm z − z ∗ 2 κm z − zp∗ p # z − zp∗ zp (zn − zn∗ ) z 2 (zn − zp ) zp (zn − zp ) z z −κm ln + + ln + − κm ln + 2 (zp − zn ) 2 − zn κm z zn R22 z − zn∗ R2 κm (z − zn∗ ) 2 2 µm κc µm κc µm R2 µm R2 + [AN (z) − zA′N (z)] + [TN (z) − zTN′ (z)] + AP + TP (46) κm µc κm µc µc z µc z
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If we let µi = µc and R1 = R2 , the problem is tantamount to the interaction between a wedge disclination dipole and an inhomogeneity. The solutions obtained are consistent with the reduced results obtained by Luo and Liu[19] .
III. FORCE ON WEDGE DISCLINATION DIPOLE In connection with the aforementioned complex potentials, the force acting upon the wedge disclination dipole inside the matrix can be calculated with the generalized Peach-Koehler formula[40] . The force components fx and fy are given by the perturbation complex functions Φm0 (z0 ), Ψm0 (z0 ) and Φ′m0 (z0 ): h i fx − ify = 2ωaeiϕ Φm0 (z0 ) + Φm0 (z0 ) + 2ωae−iϕ [z0 Φ′m0 (z0 ) + Ψm0 (z0 )]
(47)
By referring to Eqs.(45) and (46), the perturbation force components fx and fy on the disclination dipole can be calculated.
IV. NUMERICAL EXAMPLES In the three-phase composite model, the main parameters are the material combinations, the thickness of the coating layer and the features of the disclination. So this work will focus on studying the influence of these parameters on the behavior of the wedge disclination dipole. For simplicity of description and analysis, the normalized force components on disclination dipole center are defined as fx0 = 4πfx /(µ3 ω 2 R2 ) and fy0 = 4πfy /(µ3 ω 2 R2 ). In addition, we define the relative shear modulus α = µ1 /µ3 and β = µ2 /µ3 , the relative central position of disclination dipole δ = ρ/R2 , the relative thickness of the coating layer λ = R2 /R1 , the relative semi length of dipole arm η = a/R2 . In the study of this section wedge disclination angle and dipole location angle are set to be located on the x-axis if it is not specially presented, which means ϕ = θ = 0. In this case fy0 is zero because of the symmetry of the problem. In Figs.2 and 3, the variation of the normalized force fx0 versus the relative central position of disclination dipole δ is depicted for different relative shear moduli and Poisson’s ratios. Figure 2 demonstrates the effect of relative shear moduli on the normalized force fx0 is considerable, especially when disclination dipole approaches the external interface of the coated inhomogeneity. It is evident that if the inhomogeneity is harder, as the coating layer is softer than the matrix, the disclination dipole may be always attracted by the coated inhomogeneity. Furthermore, the attractive force on the disclination dipole fast increases as it is close to the coated inhomogeneity. As the disclination dipole approaches the interface from a distance, the hard coating layer first attracts the disclination dipole and then repels it because of the hard inhomogeneity. There is a stable equilibrium position on the x-axis where the normalized force fx0 equals zero. However, by increasing the stiffness of the coating layer, this stable equilibrium position disappears. Clearly, if the inhomogeneity and the coating layer are softer than the matrix (e.g. α = 0.8 and β = 0.9), the disclination dipole is first attracted and then
Fig. 2. fx0 vs. δ for various α and β (λ = 1.1, η = 0.005, ϕ = θ = 0, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.3).
Fig. 3. fx0 vs. δ for various ν2 (α = 1.1, β = 0.7, λ = 1.1, η = 0.005, ϕ = θ = 0, ν1 = 0.3 and ν3 = 0.3).
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repelled by the coated inhomogeneity when the disclination dipole and the external interface of the coated inhomogeneity are far apart. Figure 3 displays the variation of fx0 with δ for various Poisson’s ratios ν2 . The results indicate that the force on the disclination dipole strongly depends on the Poisson’s ratio of the coating layer. When the Poisson’s ratio of the inhomogeneity equals that of the matrix, the smaller ν2 contributes to the attraction of the coated inhomogeneity. As ν2 increases, an unstable equilibrium position of the disclination dipole can be found, i.e., for ν2 = 0.25 or thereabouts at the position δ = 1.7. The equilibrium position of the disclination dipole will vanish with increasing Poisson’s ratio of the coating layer. From these figures, it can be seen that the shear modulus and Poisson’s ratio of the coating layer are able to change not only the location of the equilibrium position, but also the stability of the disclination dipole. The curves in Fig.4 are plotted for α = 1.5, β = 0.8, λ = 1.1, ϕ = θ = 0, ν1 = 0.2, ν2 = 0.1, ν3 = 0.2, µ3 = 2.7 GPa and R1 = 100 nm to study the effect of disclination strength ω and relative semi length of dipole arm η on the force fx acting on the disclination dipole. Since the inhomogeneity is harder and the coating layer is softer than the matrix in these calculations, the disclination dipole is always repelled first and then attracted by the coated inhomogeneity when the disclination dipole is going away from the external interface of the coated inhomogeneity. In particular, the effect of the coated inhomogeneity on the force acting on the disclination dipole increases with the increase in disclination strength and relative semilength of the dipole Fig. 4 fx vs. δ for various ω and η (α = 1.5, β = arm. Interestingly, various disclination strengths and rela- 0.8, λ = 1.1, ϕ = θ = 0, ν1 = 0.2, ν2 = 0.1, tive semilengths of the dipole arm have negligible impact ν3 = 0.2, µ3 = 2.7 GPa and R1 = 100 nm). on the equilibrium position. The effect of the relative thickness of the coating layer λ on the normalized force fx0 acting on disclination dipoles is illustrated in Fig.5. It is clearly shown that, for the case of hard inhomogeneity with soft coating layer, as λ increases, the repulsive force decays and then turns attractive. When the coating layer is thick enough, the elastic property of the hard inhomogeneity has no significant influence on the force acting on the disclination dipole, but is shielded by the soft coating layer. Additionally, a hard inhomogeneity may cause an equilibrium position for the disclination dipole in the vicinity of the interface because of better shielding of the soft coating layer. As the disclination dipole gets closer to the external interface of the coated inhomogeneity, the shield effect will become stronger. In other words, the larger the value of λ is, the less influence the inhomogeneity phase has on the disclination
Fig. 5. fx0 vs. δ for various λ (α = 1.1, β = 0.9, η = 0.005, ϕ = θ = 0, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.2).
Fig. 6. fx0 vs. ν2 for various λ (α = 0.8, β = 1.1, η = 0.005, ϕ = θ = 0, δ = 1.1, ν1 = 0.3 and ν3 = 0.3).
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dipole. Therefore the influence on the equilibrium and stability of the disclination dipole is gradually approaching the two-phase case obtained by Luo and Liu[19] . In Fig.6, keeping the normalized fixed position of the disclination dipole δ = 1.1, we vary relative thicknesses of the coating layer λ to estimate the dependence of the normalized force fx0 on the Poisson’s ratio of the coating layer ν2 . As is seen, the dependence of the force acting on the disclination dipole upon the influence of λ is more considerable and becomes stronger for bigger ν2 . There exists a critical value ν2c of the Poisson’s ratio of the coating layer to keep the disclination dipole in equilibrium, i.e., ν2c = 0.17 for λ = 1.2. This critical value decreases with thickening of the coating layer. Thus, it can be concluded that the material properties of the coating layer have a major part to alter the strengthening or toughening effect produced by the coated inhomogeneity. Figure 7(a) and (b) give the plots for the normalized forces fx0 and fy0 as a function of the relative center position of disclination dipole δ with different wedge disclination angles ϕ and dipole location angles θ for α = 1.2, β = 0.8, η = 0.005, λ = 1.1, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.2, respectively. It is evident that the effects of ϕ and θ are remarkable on fx0 and fy0 , especially near the external interface of the coated inhomogeneity. Owing to the effects of ϕ and θ, the equilibrium position and the stability of the disclination dipole change significantly. For instance, for fx0 , there exists a stable equilibrium position for ϕ = θ = 25◦ , while for ϕ = θ = 60◦ , the equilibrium position is unstable. It is well known that fx0 equals zero at ϕ = θ = 90◦ , while fy0 equals zero at ϕ = θ = 0◦ , whatever the relative center position of disclination dipole δ.
Fig. 7. (a) fx0 and (b) fy0 vs. δ for various ϕ and θ (α = 1.2, β = 0.8, η = 0.005, λ = 1.1, ν1 = 0.3, ν2 = 0.1 and ν3 = 0.2).
V. CONCLUSIONS In our current work, the elastic behavior of a wedge disclination dipole in the three-phase composite material is carried out. By using the complex variable function method, the force acting on the wedge disclination dipole is derived analytically. The effects of the material combination, relative thickness of the coating layer and the features of the disclination dipole on the equilibrium position and stability of the disclination dipole near the coated inhomogeneity are studied and discussed in detail with numerical examples. The primary conclusions are as follows: (1) The effects of the shear modulus and Poisson’s ratio of the coating layer on the normalized force are considerable, especially when disclination dipole approaches the external interface of the coated inhomogeneity. They change not only the location of the equilibrium position, but also the stability of the disclination dipole. (2) The effect of the coated inhomogeneity on the force acting on the disclination dipole increases with the enlargement of disclination strength and relative semilength of the dipole arm. Various disclination strengths and relative semilengths of the dipole arm have negligible impact on the equilibrium position. (3) The larger the value of relative thickness of the coating layer is, the less influence the inhomogeneity phase has on the disclination dipole. The dependence of the force acting on the disclination dipole upon
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the influence of relative thickness of the coating layer is more considerable and becomes stronger for bigger Poisson’s ratios of the coating layer. There exists a critical value of the Poisson’s ratio of the coating layer to keep the disclination dipole in equilibrium and it decreases with thickening of the coating layer. (4) The effects of wedge disclination angle and dipole location angle are remarkable on the normalized forces, especially near the coated inhomogeneity. Owing to the effects of the wedge disclination angle and dipole location angle, the equilibrium position and the stability of the disclination dipole change significantly.
References [1] Romanov,A.E. and Kolesnikova,A.L., Application of disclination concept to solid structures. Progress in Materials Science, 2009, 54(6): 740-769. [2] Romanov,A.E. and Vladimirov,V.I., Dislocations in Solids. Nabarro, F.R.N., Ed., North-Holland: Amsterdam, 1992. [3] Romanov,A.E., Screened disclinations in solids. Materials Science and Engineering A, 1993, 164(1-2): 58-68. [4] Murayama,M., Howe,J.M., Hidaka,H. and Takaki,S., Atomic-level observation of disclination dipoles in mechanically milled, nanocrystalline Fe. Science, 2002, 295(5564): 2433-2435. [5] Lei,R.S., Wang,M.P., Li,Z. and Wei,H.G., Disclination dipoles observation and nanocrystallization mechanism in ball milled Cu-Nb powders. Materials Letters, 2011, 65(19-20): 3044-3046. [6] Jiang,X. and Jia,C.L., The coalescence of [001] diamond grains heteroepitaxially grown on (001) silicon. Applied Physics Letters, 1996, 69(25): 3902-3094. [7] Yacam´ an,M.J., Ascencio,J.A., Liu,H.B. and Gardea-Torresdey,J., Structure shape and stability of nanometric sized particles. Journal of Vacuum Science & Technology B, 2001, 19(4): 1091-1103. [8] Ovid’ko,I.A., Deformation of nanostructures. Science, 2002, 295(5564): 2386. [9] Morozov,N.F., Ovid’ko,I.A., Sheinerman,A.G. and Aifantis,E.C., Special rotational deformation as a toughening mechanism in nanocrystalline solids. Journal of the Mechanics and Physics of Solids, 2010, 58(8): 1088-1099. [10] Zhao,Y.X., Fang,Q.H., Liu,Y.W. and Jiang,C.Z., Shielding effects of disclinations on the elliptical blunt crack. International Journal of Engineering Science, 2013, 70: 91-101. [11] Zhao,Y.X., Fang,Q.H. and Liu,Y.W., Effect of cooperative nanograin boundary sliding and migration on dislocation emission from a blunt nanocrack tip in nanocrystalline materials. Philosophical Magazine, 2014, 94(7): 700-730. [12] Gryaznov,V.G., Gutkin,M.Yu., Romanov,A.E. and Trusov,L.I., On the yield stress of nanocrystals. Journal of Materials Science, 1993, 28(16): 4359-4365. [13] Romanov,A.E., Mechanics and physics of disclinations in solids. European Journal of Mechanics - A/Solids, 2003, 22(5): 727-741. [14] Rybin,V.V. and Zhukovskii,I.M., Disclination mechanism of microcrack formation. Soviet Physics. Solid State, 1978, 20(2): 1056-1059. [15] Wu,M.S., Zhou,K. and Nazarov,A.A., Crack nucleation at disclinated triple junctions. Physical Review B, 2007, 76(13): 134105. [16] Gutkin,M.Yu. and Ovid’ko,I.A., Disclinations, amorphization and microcrack generation at grain boundary junctions in polycrystalline solids. Philosophical Magazine A, 1994, 70(4): 561-575. [17] Bobylev,S.V. and Ovid’ko,I.A., Nanoscale amorphization at disclination quadrupoles in deformed nanomaterials and polycrystals. Applied Physics Letters, 2008, 93(6): 061904. [18] Zhou,K., Nazarov,A.A. and Wu,M.S., Atomistic simulations of the tensile strength of a disclinated bicrystalline nanofilm. Philosophical Magazine, 2008, 88(27): 3181-3191. [19] Luo,J. and Liu,F., Stress analysis of a wedge disclination dipole interacting with a circular nanoinhomogeneity. European Journal of Mechanics - A/Solids, 2011, 30(1): 22-32. [20] Zhou,K., Hoh,H.J., Wang,X., Keer,L.M., Pang,H.J.L., Song,B. and Wang,Q.J., A review of recent works on inclusions. Mechanics of Materials, 2013, 60: 144-158. [21] Chen,C.Q., Cui,J.Z., Duan,H.L., Feng,X.Q., et al., Perspectives in mechanics of heterogeneous solids. Acta Mechanica Solida Sinica, 2011, 24(1): 1-26. [22] Wang,X. and Shen,Y.P., Arc interface crack in a three-phase piezoelectric composite constitutive model. Acta Mechanica Solida Sinica, 2001, 14(2): 104-117. [23] Wang,X. and Shen,Y.P., An edge dislocation in a three-phase composite cylinder model with a sliding interface. Journal of Applied Mechanics, 2002, 69(4): 527-538.
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[24] Liu,Y.W., Fang,Q.H. and Jiang,C.P., A piezoelectric screw dislocation interacting with an interphase layer between a circular inclusion and the matrix. International Journal of Solids and Structures, 2004, 41(11-12): 3255-3274. [25] Fang,Q.H., Liu,Y.W. and Wen,P.H., Screw dislocations in a three-phase composite cylinder model with interface stress. Journal of Applied Mechanics, 2008, 75(4): 041019. [26] Gong,K.Z., Li,Z. and Qin,W.Z., Influence of loading rate on dynamic fracture behavior of fiber-reinforced composites. Acta Mechanica Solida Sinica, 2008, 21(5): 457-460. [27] Wang,X., Sudak,L.J. and Pan,E., Pattern instability of functionally graded and layered elastic films under van der Waals forces. Acta Mechanica, 2008, 198(1-2): 65-86. [28] Fang,Q.H., Liu,Y.W., Jin,B. and Wen,P.H., A piezoelectric screw dislocation in a three-phase composite cylinder model with an imperfect interface. International Journal of Engineering Science, 2009, 47(1): 39-49. [29] Wang,F., Zeng,X.G. and Zhang,J.Q., Predictive approach to failure of composite laminates with equivalent constraint model. Acta Mechanica Solida Sinica, 2010, 23(3): 240-247. [30] Zhou,Z.G., Zhang,P.W. and Wu,L.Z., Multiple parallel symmetric permeable model-III cracks in a piezoelectric piezomagnetic composite material plane. Acta Mechanica Solida Sinica, 2010, 23(4): 336-352. [31] Wang,X. and Zhou,K., Novel near-cloaking multicoated structures for screw dislocations. Mechanics of Materials, 2012, 55: 73-81. [32] Wang,X. and Zhou,K., Three-phase piezoelectric inclusions of arbitrary shape with internal uniform electroelastic field. International Journal of Engineering Science, 2013, 63: 23-29. [33] Wang,X. and Zhou,K., Long-range interaction of a line dislocation with multiple multicoated inclusions of arbitrary shape. Acta Mechanica, 2013, 224(1): 63-70. [34] Valiev,R.Z., Islamgaliev,R.K. and Alexandrov,I.V., Bulk nanostructured materials from severe plastic deformation. Progress in Materials Science, 2000, 45(2): 103-189. [35] Zhou,K., Nazarov,A.A. and Wu,M.S., Competing relaxation mechanisms in a disclinated nanowire: temperature and size effects. Physical Review Letters, 2007, 98(3): 035501. [36] Wang,T., Luo,J., Xiao,Z.M. and Chen,J.Q., On the nucleation of a Zener crack from a wedge disclination dipole in the presence of a circular inhomogeneity. European Journal of Mechanics - A/Solids, 2009, 28(4): 688-696. [37] Muskhelishvili,N.I., Some Basic Problems of Mathematical Theory of Elasticity. Leyden: Noordhoff, 1975. [38] England,A.H., Complex Variable Method in Elasticity. New York: John Wiley and Sons, 1971. [39] Luo,H.A. and Chen,Y.J., An edge dislocation in a three-phase composite cylinder model. Journal of Applied Mechanics, 1991, 58(1): 75-86. [40] Peach,M. and Koehler,J.S., The forces exerted on dislocations and the stress fields produced by them. Physical Review, 1950, 80(3): 436-439.
APPENDIX p01 (p01 − p02 p03 ) − p01 p01 p201 − p02 p04 [p02 (p04 s01 − p01 s02 ) + p201 − p02 p04 (s01 − p01 s01 ) − p01 p02 + p201 − p02 p04 p01 p02 · (p01 − p02 p03 ) (p04 s01 − p01 s02 ) − p201 − p02 p04 (p03 s01 − p01 s02 ) ÷ p02 (p04 − p01 p03 ) + p201 − p02 p04 (1 − p01 p01 ) [p01 (p01 − p02 p03 ) −p01 p01 p201 − p02 p04 − (p04 − p01 p03 ) (p01 − p02 p03 ) − p201 − p02 p04 (p03 − p01 p04 ) · p01 p02 + p201 − p02 p04 p01 p02 (48)
a0 =
c0 =
(p04 − p01 p03 ) (p01 − p02 p03 ) − p201 − p02 p04 (p03 − p01 p04 ) [p02 (p04 s01 − p01 s02 ) + p201 − p02 p04 (s01 − p01 s01 ) − p02 (p04 − p01 p03 ) + p201 − p02 p04 (1 − p01 p01 ) · (p01 − p02 p03 ) (p04 s01 − p01 s02 ) − p201 − p02 p04 (p03 s01 − p01 s02 ) ÷ (p04 − p01 p03 ) (p01 − p02 p03 ) − p201 − p02 p04 (p03 − p01 p04 ) · p01 p02 + p201 − p02 p04 p01 p02 − p02 (p04 − p01 p03 ) + p201 − p02 p04 (1 − p01 p01 ) · p01 (p01 − p02 p03 ) − p01 p01 p201 − p02 p04 (49)
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s11 + (p12 p16 − p11 p14 ) 1 − p12 p15 p14 s12 b1 = 1 − p14 p13 s12 c1 = 1 − p13 p14 p15 s11 + (p16 /p14 − p15 p11 ) d1 = 1 − p15 p12 a1 =
(50) (51) (52) (53)
ak = {(pk1 pk6 − pk2 pk8 ) (pk3 sk1 + pk1 sk2 ) − [pk1 − pk8 (pk2 pk3 + pk1 pk4 )] sk1 } ÷ {[pk3 − pk7 (pk2 pk3 + pk1 pk4 )] (pk1 pk6 − pk2 pk8 ) − (1 + pk1 pk5 − pk2 pk7 ) [pk1 − pk8 (pk2 pk3 + pk1 pk4 )]}
(k ≥ 2)
(54)
bk = {(pk5 pk2 + pk6 pk4 ) (pk6 pk7 − pk5 pk8 ) sk2 − [pk4 (pk6 pk7 − pk5 pk8 ) + pk5 ] (pk5 sk1 + pk6 sk2 )} ÷{[pk7 − pk3 (pk6 pk7 − pk5 pk8 )] (pk5 pk2 + pk6 pk4 ) + [pk4 (pk6 pk7 − pk5 pk8 ) + pk5 ] (pk6 pk3 − 1 − pk5 pk1 )} (k ≥ 2) ck = {(1 + pk1 pk5 − pk2 pk7 ) (pk3 sk1 + pk1 sk2 ) − [pk3 − pk7 (pk2 pk3 + pk1 pk4 )] sk1 }
(55)
÷{(1 + pk1 pk5 − pk2 pk7 ) [pk1 − pk8 (pk2 pk3 + pk1 pk4 )] − [pk3 − pk7 (pk2 pk3 + pk1 pk4 )] (pk1 pk6 − pk2 pk8 )} (k ≥ 2) (56) dk = {(1 + pk5 pk1 − pk6 pk3 ) (pk6 pk7 − pk5 pk8 ) sk2 − [pk7 − pk3 (pk6 pk7 − pk5 pk8 )] (pk5 sk1 + pk6 sk2 )} ÷{[pk7 − pk3 (pk6 pk7 − pk5 pk8 )] (pk5 pk2 + pk6 pk4 ) + [pk4 (pk6 pk7 − pk5 pk8 ) + pk5 ] (pk6 pk3 − 1 − pk5 pk1 )}
(k ≥ 2)
(57)
where 1 − µi /µc 1 − µi κc /(µc κi ) , m2 = µi κc /µc − 1 µi /(κi µc ) + 1 Dω κm + 1 1 − µm /µc Dω 1/κm + 1 1 − µm κc /(µc κm ) n1 = , n2 = , n3 = , n4 = 2 µm κc /µc + 1 µm κc /µc + 1 2 µm /(µc κm ) + 1 µm /(µc κm ) + 1 ∗ ∗ n3 z p − z p n3 z n − z n n3 (zp − zn ) , n7 = , n8 = n5 = n1 − n3 , n6 = ∗ ∗ R22 z p + n1 z n + n1 m1 =
p03 = R12 + R22 , p04 = m2 R12 + n2 R22 zp zp s02 = (n3 + n5 ) ln s01 = n1 ln , zn zn −2 −4 p11 = 2n2 R2 , p12 = n2 R2 , p13 = n4 , p14 = m1 , p15 = m2 R14 , −1 −1 2 −1 −1 s11 = n1 zn − zp , s12 = R2 (n6 − n5 ) zp + (n5 − n7 ) zn + n8 p01 = n2 ,
p02 = n2 R2−2 ,
pk1 = n2 (k + 1) R2−2k , pk4 = n2 (k − 1) R2−2k ,
p16 = 2m1 R12
pk2 = n2 R2−2k−2 , pk3 = [n4 + n2 (k − 1) (k + 1)] R2−2k+2 2k pk5 = m1 (k − 1) R1 , pk6 = m1 R12k−2
pk7 = [m1 (k − 1) (k + 1) + m2 ] R12k+2 , pk8 = m1 (k + 1) R12k h i n1 zn−k − zp−k n5 −k n5 , sk2 = R22 n6 − zp + − n7 zn−k sk1 = k k k