Mechanics Research Communications 52 (2013) 88–91
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Misﬁt dislocation dipoles in coated ﬁbrous composites Xu Wang a,∗∗ , Kun Zhou b,∗ a b
School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
a r t i c l e
i n f o
Article history: Received 4 September 2012 Received in revised form 24 June 2013 Accepted 4 July 2013 Available online 16 July 2013 Keywords: Misﬁt dislocation dipole Fibrous composite Inclusion Equilibrium position
a b s t r a c t We investigate the stability of a misﬁt screw dislocation dipole in a coated ﬁbrous composite. The ﬁber is stiffer whereas the coating is more compliant than the matrix. A critical coating thickness is identiﬁed for given material parameters of the composite. For a compliant coating below the critical thickness, there exist an inner unstable and an outer stable equilibrium position in the matrix for the dislocation dipole. However, for a compliant coating above the critical thickness, the dislocation dipole will be attracted to the coating–matrix interface. If the coating thickness is greater than the inclusion radius, the dislocation dipole cannot lodge in the matrix no matter what values of the material parameters are chosen. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Investigation of dislocations in ﬁber/particle reinforced composites is crucial for optimal design of advanced materials for aerospace, automobile, offshore engineering and many other applications. This is because the mobility of dislocations and their interactions with the interfaces and constituents of a composite can signiﬁcantly affect its strengthening and failure mechanisms (Hirth and Lothe, 1982). In the design of ﬁbrous composites, coating of ﬁbers is usually employed to improve the performance of the composites (see e.g., Walpole, 1978; Mikata and Taya, 1985; Qiu and Weng, 1991; Ru, 1999; Wang and Shen, 2000; Feng et al., 2003, 2004; Liu et al., 2004; Wang, 2006; Wang and Gao, 2011; Hoh et al., 2012; Zhou, 2012; Wang and Zhou, 2012, 2013). As a result, it is essential to fully understand the inﬂuence of coating on the mobility and stability of misﬁt dislocations and dislocation dipoles. Xiao and Chen (2000) studied a screw dislocation near a coated ﬁber, and observed that when the ﬁber is stiffer whereas the coating is more compliant than the surrounding matrix, there exists an unstable equilibrium position near the coating–matrix interface. Recently, Fang et al. (2008) considered the generation of a misﬁt screw dislocation dipole in an inﬁnite compliant matrix reinforced by a stiff inclusion in the absence of coating. They identiﬁed a stable equilibrium
∗ Corresponding author. Tel.: +65 67905499; fax: +65 67924062. ∗∗ Corresponding author. Tel.: +86 21 64251805, fax: +86 21 64251805. E-mail addresses: [email protected]
(X. Wang), [email protected]
(K. Zhou). 0093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2013.07.004
position for the dislocation dipole. If the coating effect is taken into consideration, more complex and intriguing behaviors of the misﬁt dislocation dipole are expected to be observed. More studies of inclusions and their applications in composites can be found in a recent review by Zhou et al. (2013). Thus, the present work aims to investigate the stability of a misﬁt screw dislocation dipole near a coated ﬁber.
2. Stability of the misﬁt dislocation dipole Consider a ﬁbrous composite in which a circular elastic inclusion (the ﬁber) bonded to an inﬁnite elastic matrix through an annulus coating, as shown in Fig. 1. Let S1 , S2 and S3 denote the internal inclusion, the intermediate coating, and the surrounding matrix, respectively, all of which are perfectly bonded across two concentric circles of radii a and b with b > a, respectively. Thus, the thickness of the coating is given by H = b − a. The inclusion, coating and matrix are assumed to be elastically isotropic with the shear moduli 1 , 2 , and 3 , respectively. This study will focus on the practical situation in which the inclusion is stiffer whereas the coating is more compliant than the matrix (i.e., 1 > 3 > 2 ). We will derive the elastic ﬁeld induced by a misﬁt screw dislocation dipole in the ﬁbrous composite and discuss the stability of the dislocation dipole due to its interaction with the coated inclusion. Speciﬁcally, the misﬁt dislocation dipole is composed of two screw dislocations at (ˆx, 0) and (−ˆx, 0) (ˆx > b) with their respective Burgers vectors bz and −bz (bz > 0) in the matrix and their lines are parallel to the axis of the cylindrical inclusion.
X. Wang, K. Zhou / Mechanics Research Communications 52 (2013) 88–91
Fig. 1. A misﬁt screw dislocation dipole in a coated ﬁbrous composite.
Under anti-plane shear deformation, the out-of-plane displacement w, the stress function , and the stress components zy and zx can be expressed in terms of an analytic function f(z) of the complex variable z = x + iy as (Muskhelishvili, 1953) −1 + iw = f (z),
zy + izx = f (z),
Fig. 2. The normalized image force F ∗ = bF/3 b2z acting on the right dislocation of the misﬁt screw dislocation dipole for = 0.7, 0.8 and 0.9 with 1 = 0.5 and 3 = 0.891.
acting on the right screw dislocation of the dipole located at (ˆx, 0) is obtained as b 3 b2z
1 + 4
2n−1 (1 − 1 )(1 + 3 ) − (1 + 1 )(1 − 3 ) , 4n−1 [(1 + 1 )(1 + 3 ) − 2n−1 (1 − 1 )(1 − 3 )]
where is the shear modulus, and the two stress components can be expressed in terms of the stress function as
where the image force F is along the x-direction, and
zy = ,x ,
zx = −,y
By enforcing the continuity conditions of displacement and traction across the inner and outer circular interfaces of radii a and b, the analytic functions deﬁned for the inclusion, coating and the surrounding matrix can be ﬁnally derived as f1 (z) =
An z 2n−1 ,
(|z| < a)
f2 (z) =
∞ 1 + 1 n=1
An z 2n−1 +
1 − 1 An a4n−2 z −2n+1 , 21
(a < |z| < b)
xˆ > 1. b
Here we are much interested in probing the existence of equilibrium positions where the image force F is zero for the misﬁt dislocation dipole. The numerical results indicate that there exists a critical value cr of the thickness parameter for given material parameters 1 and 3 (or equivalently there exists a critical −1/2 thickness Hcr = a(cr − 1)). When > cr , which implies that the coating is so thin that Hcr < H, there exist an inner unstable and an outer stable equilibrium position for the misﬁt dislocation dipole. Only when the misﬁt dislocation dipole is located within the inner unstable equilibrium position, will the dipole be attracted to the circular coating–matrix interface of radius b; otherwise the dipole will ﬁnally lodge at the outer stable equilibrium position. When cr < , which implies that the coating is thick enough such that
bz z − xˆ 2n−1 (1 − 1 )(1 + 3 ) − (1 + 1 )(1 − 3 ) An b4n−2 z −2n+1 , ln + 2 41 z + xˆ ∞
f3 (z) =
(|z| > b)
where the slip planes are deﬁned by y = 0 and |x| > xˆ , and 1 =
2 , 1
An = −
2 , 3
a 2 b
41 xˆ −2n+1 bz , (2n − 1)[(1 + 1 )(1 + 3 ) − 2n−1 (1 − 1 )(1 − 3 )]
(n = 1, 2, · · ·, +∞).
The development of Eqs. (3)–(5) is explained in Appendix A. Since 1 > 3 > 2 , we will have the inequality 1 < 3 < 1. The parameter (0 < < 1) reﬂects the dimensionless thickness of the coating: → 0 means that the coating is extremely thick; whilst → 1 means an extremely thin coating. By using the Peach–Koehler formula (see Eqs. (3.5) and (3.6) in Dundurs, 1969), the image force
H > Hcr , the misﬁt dislocation dipole will always be attracted to the coating–matrix interface of radius b. The above observation is clearly illustrated in Fig. 2 for the three values 0.7, 0.8 and 0.9 of with = 0.5 and 3 = 0.891. This example has cr = 0.8. When = 0.9 > cr , there exist an inner unstable equilibrium position at = 1.0146 and an outer stable equilibrium position at = 1.2213. When = cr = 0.8, a saddle position at = 1.08 which is neither stable nor unstable is observed. When = 0.7 < cr , the image force is always negative, which implies that the misﬁt dislocation dipole is attracted to the coating–matrix interface. It is easily found from Eq. (8) that cr should be a function of 1 and 3 . Fig. 3 shows the variations of 1 and 3 for the seven values 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.99 of cr . If the chosen pair ( 1 , 3 ) lies above the curve for a ﬁxed value of cr and meanwhile = cr , two equilibrium positions of different natures for the
X. Wang, K. Zhou / Mechanics Research Communications 52 (2013) 88–91
3. Conclusions We investigate in detail a misﬁt screw dislocation dipole near a coated ﬁber embedded in the matrix. When the coating is considered in the dislocation dipole-inclusion interaction problem, the behavior of the misﬁt screw dislocation dipole becomes much more complicated than that observed by Fang et al. (2008) in the absence of the coating. The complexity lies in that depending on the thickness of the compliant coating, there exist two equilibrium positions of different natures or there even exists no equilibrium position for the dislocation dipole. We also derive a very simple approximate closed-form expression in Eq. (11) of the image force in addition to the exact series form expression in Eq. (8). It is expected that similar observation can be observed when a misﬁt screw dislocation dipole interacts with a circular elastic inclusion with a spring-type imperfect interface of vanishing thickness. Acknowledgements
Fig. 3. Variations of 1 and 3 for different values of cr .
misﬁt dislocation dipole can be found. On the other hand, if the chosen pair ( 1 , 3 ) lies below the curve for a ﬁxed value of cr and above the straight line ( 1 = 3 ) and meanwhile = cr , there is no equilibrium position for the misﬁt dislocation dipole and the dislocation dipole will be attracted to the coating–matrix interface. It is observed from Fig. 3 that as the value of cr decreases and = cr is set (now the coating becomes thicker), the ( 1 , 3 ) zone where there exist equilibrium positions shrinks whereas the free-of-equilibrium-position zone enlarges. The numerical results also indicate that cr ≥ 0.2364. This inequality of cr implies that if H > 1.0567a ≈ a (roughly speaking, the coating thickness is greater than the inclusion radius), the misﬁt screw dislocation dipole cannot lodge in the matrix no matter what values of 1 and 3 ( 1 < 3 < 1) are chosen. This simple fact can be utilized to control the nucleation of misﬁt dislocation dipoles in the matrix. Furthermore, we will present a simple approximate closedform expression of the image force on the right dislocation of the misﬁt screw dislocation dipole. Since the material and thickness parameters satisfy the restrictions 0 < 1 < 3 < 1 and 0< < 1, the following approximate relationship can be established: (1 + 1 )(1 + 3 ) − 2n−1 (1 − 1 )(1 − 3 ) ≈ (1 + 1 )(1 + 3 ) for n = 1, 2, . . ., +∞
Consequently, the image force on the right dislocation of the dipole can be approximately given as b 3 b2z
1 1 − 3 1 − 1 . − + 1 + 1 4 − 2 1 + 3 4 − 1 4
The numerical results show that the accuracy of the above approximate expression is quite satisfactory. Possible equilibrium positions can then be determined by solving the following quartic equation in 2 8 + 4( 3 − 1 ) 6 − (1 + 2 ) 4 + 4( 1 − 3 ) 2 + 2 = 0,
where 1 and 3 are two mismatch parameters deﬁned by 1 =
1 − 1 , 1 + 1
1 − 3 , 1 + 3
(0 < 3 < 1 < 1).
For example, if 1 = 0.5, 3 = 0.891 and = 0.9, the two equilibrium positions can be obtained from Eq. (12) as = 1.0147 and = 1.2165, which are very close to the exact values of = 1.0146 and = 1.2213.
This work is supported by the National Natural Science Foundation of China (grant no.: 11272121), Innovation Program of Shanghai Municipal Education Commission, China (grant no.: 12ZZ058) and Agency for Science, Technology and Research, Singapore (SERC grant no.: 112 290 4015). Appendix A. Eqs. (3)–(5) are derived as follows. First, Eq. (3) can be assumed as the analytic function deﬁned for the circular inclusion. Then, Eq. (4) can be obtained through the satisfaction of continuity conditions of displacement and traction across the inner circular interface |z| = a. Next, the satisfaction of the continuity conditions across the out circular interface |z| = b leads to an expression of f3 (z) which is convergent in the annulus b < |z| < xˆ . Meanwhile f3 (z) should take the following form: z − xˆ bz + ln Cn z −2n+1 , 2 z + xˆ ∞
f3 (z) =
(|z| > b),
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