International Journal of Mechanical Sciences 74 (2013) 173–184
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Edge misfit dislocations in core–shell nanowire with surface/interface effects and different elastic constants Y.X. Zhao a, Q.H. Fang a,b,n, Y.W. Liu a a b
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan Province 410082, PR China School of Mechanical and Manufacturing Engineering, The University of New South Wales, NSW 2052, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 1 June 2012 Received in revised form 28 October 2012 Accepted 29 May 2013 Available online 6 June 2013
A model of the generation for an edge misfit dislocation in the system of a nanowire surrounded by a co-axial film with surface/interface effects is investigated. The critical conditions of an edge misfit dislocation formation at the interface are analyzed, under considering the influence of the material elastic dissimilarity, misfit strains, the radius of the nanowire, core radius of the misfit dislocation and the surface/interface effects. The results show that the critical film thickness reduces with increment of the misfit strains, nanowire radius and core radius of the edge misfit dislocation, below the critical values of which, the misfit dislocation is energetically unfavorable whatever the film thickness. Critical film thickness first decreases and then increases with increasing the ratio of the shear modulus. There exists a critical film thickness below which no interfacial misfit dislocation could be introduced whatever the ratio of the shear modulus. There also exists a critical value of the ratio of the shear modulus, above which edge misfit dislocation does not form at any film thickness. The negative (positive) surface/ interface stress can decrease (increase) the formation energy of the edge misfit dislocation. The positive (negative) surface/interface stress would increase (decrease) the critical film thickness, critical misfit strains and critical nanowire radius. The positive (negative) surface/interface stress would decrease (increase) the range of the film thickness and the critical ratio of the shear modulus. The larger the values of the surface/interface stress qualities, the greater the influence of the surface/interface stress on critical parameters. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Edge misfit dislocation Nanowire Critical film thickness Surface/interface effects
1. Introduction In general, the stability of both the structures and properties of crystalline nanocomposites are strongly influenced by misfit stresses arising due to a misfit between the crystal lattices of the adjacent component phases at interface boundaries. A partial relaxation of misfit stresses often occurs by nucleation and evolution of misfit dislocations (MDs) at interface boundaries in thin-film and bulk nanocomposites [1–5]. Dislocation generation mechanism is also considered to be an effective way in plastic deformation of composite materials [6,7]. Additionally, experiments studied the critical conditions of the generation for the misfit defects at the interface boundary [8]. During recent years, great effort has focused on the synthesis, fabrication and characterization of so-called core/shell nanostructures [9,10]. The investigation of MD in composite cylinder (core–shell nanowire) was motivated by the synthesis of these solid nanowires. The lattice mismatch between adjacent materials results in internal strains during the syntheses of the core–shell nanowires, which could be partially/fully relaxed via the formation of misfit/threading dislocations. These defects can negatively affect the physical behavior, e.g., mechanical strength, features of the electric and optical spectra [1]. A first approximation model of MDs in film/substrate composites of wire form was suggested Gutkin et al. [11], and the effects of geometric parameters of such composites on generation of MDs were theoretically analyzed by methods of elasticity theory of defects in solids. The results showed that in considering misfit composite structure of cylindrical (wire) form, the set of geometric parameters crucially affecting the generation of MDs contains the wire composite radius, film thickness and misfit strain. In order to simplify the analysis, the most studies on the critical condition for the formation of MDs or dislocation loops in the strained core–shell structures [12,13], the cylinder (core) and surface film (shell) have the same values of elastic
n Corresponding author at: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan Province 410082, PR China. Tel.: +86 731 89822841; fax: +86 731 88822330. E-mail addresses:
[email protected],
[email protected] (Q.H. Fang).
0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.05.013
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constants, which cannot reflect the real properties of the elastic mismatch between the substrate and the surrounding film. Fang et al. [14–16] examined the influence of the difference of elastic constants on the critical thickness of the film and critical radius of the strained inhomogeneity for screw misfit dislocation formation. Yi et al. [17] have extended the 3D Eshelby formalism to multi-shell nano-onions and applied to quantum dots of uniform and non-uniform compositions. Duan et al. [18,19] have considered quantum dots with incompatible strain fields and given conditions when misfit dislocations (MD) are emitted. Zhao et al. [20] investigated the model of an edge misfit dislocation at the interface of the hollow nanopore and the infinite substrate with surface/interface stress. The influence of the ratio of the shear modulus between the film and the infinite substrate, the misfit strain, the radius of the nanopore and the surface/ interface stress on the critical thickness of the film was discussed. Additionally, interfacial bonding condition is one of the important factors that control the local elastic fields and the overall properties of composites. When the size of the inclusion is of the order of nanometer, the inclusion–matrix interface energy cannot be neglected because of the increased contribution to the total energy from the surface/interface [21,22]. In recent decade, the theory of the surface/ interface elasticity has been used to analyze many size-dependent elastic problems on the nanoscale. The dependence of the effective elastic or thermoelastic properties on the size of the particulate composite materials highlights the importance of the surface/interface in analyzing the deformation of nano-scale structures [23,24]. The influences of particle size and interface energy on the void nucleation and evolution mechanism were investigated [25,26]. One-dimensional coherent nanowires with tilted internal twins exhibit anisotropic size effect: their strengths show no apparent change if only their thicknesses reduce, but become stronger as the sample sizes are reduced proportionally, which could help to develop straightforward understanding on the origin of size effect in strength [27,28]. A classical continuum model for elastic solids incorporating surface/interface energy (surface/interface stress) was first formulated by Gurtin and Murdoch [29]. This model (namely, surface/interface stress model) has been adopted by some authors in studying nanoscale structures, thin films, nanocomposites and quantum dots [30–40]. As will be seen, the surface/interface effects are a critical factor in the physical behavior of the materials containing nanowire of a sufficiently small size. In this paper, we consider a two-phase misfit nanowire system which consists of a cylindrical nanowire and a co-axial cylindrical film, and suppose the misfit strain is accommodated through the generation of edge MD at the interface boundary. Critical film thickness for the generation of the first edge MD at the interface (film/nanowire) boundary to be energetically favorable is calculated, especially considering the influence of the surface/interface effects and the material elastic dissimilarity. The presence of MDs has a detrimental effect on the performance of the strained material systems. Therefore, understanding and controlling the generation of MDs are very important. Such control is necessary in manufacturing (e.g., microelectronics) where defects must be minimized [1,41]. Due to these factors, the investigation of the critical conditions for the generation of MDs becomes significant.
2. Critical condition for generation of edge MD Here the model considered is that the cylinder consists of a nanowire of radius R1 and film of thickness h ¼ R2 −R1 , as shown in Fig. 1. The elastic constants of the nanowire and film indicate μ1 ,υ1 and μ2 , υ2 , where μi is shear modulus of the material and υi is Poisson’s ratio. In the following analysis, we will use the subscripts 1 and 2 to identify the corresponding physical quantities in the region of the nanowire and the film, respectively. The edge MD with Burgers vector ð0; by Þ at the interphase (film/nanowire) boundary and its lines are parallel with the axis of the nanowire. The lattice mismatch strains would be expressed as uniform strains ε0x , ε0y and ε0xy . In view of the works of Gutkin et al. [2,11], and Freund and Suresh [1], the criterion for the generation of the first edge MD at the interface (film/nanowire) boundary to be energetically favorable is as follows: ΔW ¼ W d þ W m þ W c ≤0
ð1Þ
Where W d denotes the elastic energy of the edge MD in the nanowire and film system, W m is the elastic energy associated with the elastic interaction between the edge MD and the misfit stress and W c is the energy of the MD core. Hereon, in order to fully and accurately analyze the influence of various parameters on the edge MD formation and the critical geometries, the energy of the MD core is 2 considered. According to the Hirth and Lothe [42], the energy of the MD core W c is about μ2 by =½4π ð1−υ2 Þ. The elastic energy of an edge dislocation (the energy per its unit length) under consideration can be expressed as follows [42]: Z by R2 s ðx; 0Þdx ð2Þ Wd ¼ 2 R1 þr0 θθd Where sθθd ðx; 0Þ is the stress component of the edge MD in the nanowire and film system and r 0 is the core radius of the edge dislocation.
Fig. 1. Edge MD at the interface boundary in a nanowire composite with surface/interface effects.
Y.X. Zhao et al. / International Journal of Mechanical Sciences 74 (2013) 173–184
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For plane strain problem, the stress components in the bulk solid can be expressed in terms of two Muskhelishvili complex potentials ΦðzÞ and Ψ ðzÞ [43] ð3Þ srr þ sθθ ¼ 2 ΦðzÞ þ ΦðzÞ srr þ isrθ ¼ ΦðzÞ þ ΦðzÞ−zΦ′ðzÞ−ðz=zÞΨ ðzÞ
ð4Þ
By using the complex potential method, the stress component sθθd ðx; 0Þ can be expressed easily by the complex potentials ΦðzÞ and Ψ ðzÞ in Appendix A, and then substitute the above obtained stress component into Eq. (2), the elastic strain energy of the edge dislocation in the interface boundary can be derived. The elastic energy of the interaction between the edge MD and the misfit stress field is given by Z R2 −r0 W m ¼ −by sθθm ðx; 0Þdx ð5Þ R1
where sθθm ðx; 0Þ denotes the misfit stress component. By using the complex potential method, the stress component sθθm ðx; 0Þ can be expressed easily by the complex potentials ΦðzÞ and Ψ ðzÞ in Appendix B. Substituting of the above obtained stress component into Eq. (5), the elastic energy of the interaction between the edge MD and the misfit stress field can be calculated. In view of the complicated expressions, the expressions of the above two elastic strain energies would be omitted here. According to Eqs. (1), (2) and (5), the total energy variation ΔW can be obtained. So the critical conditions of an edge misfit dislocation formation can be analyzed, under considering the influence of the material elastic dissimilarity, misfit strains, the radius of the nanowire, core radius of the misfit dislocation and the surface/interface effects, according to the criterion for the generation of an edge MD in the interphase (nanowire/film) boundary to be energetically favorable.
3. Discussion In order to convenient and handy the analysis, the following normalized physical quantities illustrated: the edge MD formation energy 2 Δw ¼ ΔW=ðμ2 by Þ, the ratio of the shear modulus of the nanowire and the film β2 ¼ 2μ02 þ λ02 −τ02 = 2R2 λ2 þ μ2 , the radius of the nanowire b ¼ R1 =by , and the thickness of the film c ¼ h=by . For the description of the nanoscale surface/interface properties, one needs the surface/interface elastic constants μ0 , λ0 and the residual surface/interface tension τ0 . The normalized intrinsic lengths become w1 ¼ μ01 =μ2 , w2 ¼ λ01 =μ2 , w3 ¼ μ02 =μ2 , w4 ¼ λ02 =μ2 , w5 ¼ τ01 =μ2 and w6 ¼ τ02 =μ2 . Former studies have shown that the absolute values of intrinsic lengths w1 , w2 , w3 , w4 , w5 and w6 are nearly 0.1 nm [21]. Let k1 ¼ 2w1 þ w2 and k2 ¼ 2w3 þ w4 , the sign and value of whose qualities are important for characterizing the nanoscale surface/interface properties [37]. In addition, let υ1 ¼ υ2 ¼ 0:3. In the following, the influence of the ratio of the shear modulus a, misfit strains ε0x , ε0y and ε0xy , radius of the nanowire b, core radius of the edge MD r 0 , the surface/interface elastic constants k1 ,k2 , and the residual surface/interface tensions w5 , w6 on the critical geometries (Δw ¼ 0) is studied. As a starting point, we examine the shear modulus effects of the nanowire and the film a ¼ μ2 =μ1 on critical film thickness hc for the generation of the edge MD in Figs. 2–4. Fig. 2 shows when the radius of the nanowire b ¼ R1 =by is relatively large, the variation of the edge MD formation energy Δw versus the film thickness c without the surface/interface stress. It is shown that when the ratio of the shear modulus a is given, the formation energy increases to a maximum and then decreases with increasing film thickness. Critical film thickness first decreases and then increases with increasing the ratio of the shear modulus. Such plots indicate that there exists critical film thickness below which no interfacial edge MD should be introduced whatever the ratio of the shear modulus. There also exists a critical value of the ratio of the shear modulus, above which edge MD does not form at any film thickness. At the initial stage, when the ratio of the shear modulus a is relatively small, critical film thickness m9 ¼ 4μ1 R21 ðε2 þ iε3 Þ(Δw ¼ 0) for the formation of the edge MD decreases slightly with the increment of the ratio of the shear modulus a. But if the ratio of the shear modulus a is reached to a certain value, critical film thickness m9 ¼ 4μ1 R21 ðε2 þ iε3 Þ(Δw ¼ 0) increases remarkably with the increment of the ratio of the shear modulus a. However, if the ratio of the shear modulus a increases to a certain value, the generation of the edge MD is energetically favorable when the film thickness c is in the range hc1 oc o hc2 (Δw o 0). In addition, when the ratio of the shear modulus a is large enough, for example a ¼ 2:4, the variation of the formation energy Δw is always positive and the generation of the edge MD is energetically unfavorable at any value of the film thickness. In other words, the edge MD 0.4
a=0.2 a=0.4 a=0.7 a=0.9
0.2
a=1.2
w
a=1.7 a=2.4
0.0
hc2 hc
-0.2
0
20
hc1 40
c
60
80
Fig. 2. Formation energy Δw versus film thickness c with the different shear moduli a ¼ μ2 =μ1 for b ¼ 100, εx ¼ εy ¼ εxy ¼ 0:013, k1 ¼ k2 ¼ w5 ¼ w6 ¼ 0, and r 0 ¼ 0:5by .
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0.12 0.16 k 1=k 2= 0 ,w 5=w 6= 0
0.12
k 1=k 2= 0 ,w 5=w 6= 0.1
0.08
k 1=k 2=0 ,w 5=-w 6= 0.1
0.08
k1=k2=-0.3 ,w5=w6=0
k1=k2=0.1 ,w5=w6=0
k1=k2=0,w5=w6=-0.1
0.04
0.04
0.02 0.00
0.00 -0.04
k1=k2=0,w5=w6=0
0.06
Δw
Δw
k 1=k 2= 0.3 ,w 5=w 6= 0
0.10
-0.02 0
5
10
15
c
20
0
2
4
6
8
10
c
12
14
16
18
20
22
Fig. 3. Formation energy Δw versus film thickness c with b ¼ 15, εx ¼ εy ¼ εxy ¼ 0:05, r 0 ¼ 0:5by for (a) a ¼ 0:5 and (b) a ¼ 0:9.
60 50
k 1=k 2= 0 ,w 5=w 6= 0
40
hc
k 1=k 2= 0.3 ,w 5=w 6=0
hc2
k 1=k 2= 0.1 ,w 5=w 6=0
30
k 1=k 2= 0 ,w 5=w 6=0.1
hc1
20 10 0
1
2
a ac4
3
4 a c2 a c3 a c1
Fig. 4. Critical film thickness hc versus the shear modulus a with the different surface/interface stresses for εx ¼ εy ¼ εxy ¼ 0:025, b ¼ 60 and r 0 ¼ 0:5by .
cannot be generated with the ratio of the shear modulus a larger than a certain critical value, whatever the film thickness. Here the influence of the ratio of the shear modulus of the nanowire and film on critical film thickness for the edge MD formation is different from that of the edge MD formation at the interface of the hollow nanopore and the infinite substrate in Zhao et al. [20]. The major reason would be the influence of the increasingly considered energy of the MD core and the different physical models, which show the different effect of the shear modulus of the materials on the first interfacial edge MD that forms. Fig. 3 shows when the radius of the nanowire p11 ¼ e1 m9 is relatively small, the variation of the edge MD formation energy Δw versus the film thickness c with the different surface/interface stresses. It can be found from two plots that the formation energy first increases to a maximum, and decreases to a minimum, and then increases with increasing the film thickness c, which appears a range of the film thickness for the generation of the edge MD. The phenomenon shown in Fig. 3 is different from that considering the relatively large nanowire radius without the influence of the surface/interface effects shown in Fig. 2. As an estimate in detail, it appears that the positive (negative) surface/interface stress would shrink (widen) the range of the film thickness c for the generation of the edge MD. Furthermore, the effects of the residual surface/interface tensions are greater than the surface/interface elastic constants. When the residual surface/ interface tensions take the same sign compared with the opposite sign, the impact of surface/interface stress on the range of the film thickness c for the generation of the edge MD is far greater. Due to the positive residual surface/interface tensions, the core–shell nanowire heterostructures may exhibit defect-free interfaces under other same conditions, i.e., k1 ¼ k2 ¼ 0, w5 ¼ w6 ¼ 0:1. Therefore, there is good reason for the nucleation of new MD to not occur in nanowire geometries. While there will be introduced the edge MD owing to the negative surface/interface elastic constants or residual surface/interface tensions, i.e., k1 ¼ k2 ¼ −0:3, w5 ¼ w6 ¼ 0 or k1 ¼ k2 ¼ 0, w5 ¼ w6 ¼ −0:1. Fig. 4 shows critical film thickness hc versus the shear modulus a ¼ μ2 =μ1 with the different surface/interface stresses. It can be found that at the early stage, critical film thickness for the formation of the edge MD first decreases with increasing the ratio of the shear modulus a. The formation of the edge MD is energetically favorable when the film thickness c is larger than the critical thickness hc . If the ratio of the shear modulus a is in some range, the formation of the edge MD is energetically favorable when the film thickness c is in a range q4 ¼ e3 n2 (Δw o 0) for a given value of the ratio of the shear modulus a. It also illustrates that the formation of the edge MD is energetically unfavorable when the ratio of the shear modulus a is larger than a critical value ac . As shown in Fig. 4, the positive surface/ interface elastic constants and residual surface/interface tensions would decrease the range between two critical film thicknesses hc1 and hc2 , and the critical ratio of the shear modulus ac . Furthermore, the larger the values of the surface/interface stress qualities, the greater the influence of the surface/interface stress on the shrink of the range between two critical film thicknesses and the critical ratio of the shear modulus. The influence of the residual surface/interface tensions is greater than that of the surface/interface elastic constants. It shows clearly that the surface/interface stresses change dramatically not only the critical value of the film thickness hc , in which the formation of the edge MD is energetically favorable, but also the critical value of the ratio of the shear modulus ac . In Figs. 5 and 6, we analyze the misfit strains ε0x , ε0y and ε0xy effects on critical film thickness hc for the generation of the edge MD. Fig. 5 shows when the radius of the nanowire b ¼ R1 =by is relatively large, the variation of formation energy Δw versus the film thickness c
Y.X. Zhao et al. / International Journal of Mechanical Sciences 74 (2013) 173–184
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0.4
w
0.2
0.0
-0.2 0
20
40
60
c
80
Fig. 5. Formation energy Δw versus film thickness c with the different misfit strains ε0x , ε0y and ε0xy for a ¼ 0:8, b ¼ 100 and k1 ¼ k2 ¼ w5 ¼ w6 ¼ 0.
0.14 0.12 0.10 0.08
w
0.06 0.04 0.02 0.00
hc1
-0.02
hc 2
-0.04 -0.06 -0.08
0
5
10
c
15
20
25
Fig. 6. Formation energy Δw versus film thickness c with the different misfit strains ε0x , ε0y , ε0xy and surface/interface stresses for a ¼ 0:9, b ¼ 15 and r 0 ¼ 0:5by .
without the surface/interface stress for the different core radii of the edge MD r 0 . It is shown that the formation energy may decrease with the increase of the misfit strains. If the misfit strains reach a relatively large value, the generation of the edge MD is energetically favorable, and the critical film thickness hc reduces significantly with the increase of the misfit strains. Edge MD formation is not energetically favorable for misfit strains below some critical values, whatever the film thickness. The influence of the misfit strains on the generation of the edge MD is in agreement with the results of Gutkin et al. [11]. It is also found that the formation energy decreases with increasing core radius for given misfit strains. Critical film thickness hc increases with the decrease in core radius of the edge MD. When the core radius of the edge MD introduced reduces to a certain extent, edge MD is never energetically favorable, whatever the film thickness, i.e., r 0 ¼ 0:1by and εx ¼ εy ¼ εxy ¼ 0:012. Fig. 6 shows when the radius of the nanowire b is relatively small, the variation of the edge MD formation energy Δw versus the film thickness c with the surface/interface stresses. It can be seen that when the radius of the nanowire is relatively small, the influence of the surface/interface stress is significant and evident. The negative (positive) surface/interface stress can decrease (increase) the formation energy and widen (shrink) the range of critical film thickness. In addition, if considering the surface/interface stress, the generation of edge MD may be energetically favorable when only the film thickness h is in some range for given misfit strains, that is hc1 o h o hc2 , i.e., εx ¼ εy ¼ εxy ¼ 0:05, k1 ¼ k2 ¼ −0:3, and w5 ¼ w6 ¼ 0. Fig. 7 shows the formation energy Δw versus the film thickness c with the different radii of the nanowire b ¼ R1 =by and surface/ interface stresses. It can be seen that when the radius of the nanowire and the film thickness increase to some certain values, the generation of edge MD is just energetically favorable. If the film thickness is relatively small, both the radius of the nanowire and the surface/interface stress have little effects on the formation energy. Critical film thickness increases with decreasing the radius of the nanowire. The positive (negative) surface/interface stress can increase (decrease) the formation energy and shrink (widen) the range of critical film thickness. The smaller the radius of the nanowire is, the greater the influence of the surface/interface stress is. If considering the surface/interface stress, the generation of edge MD is not energetically favorable for given radius of the nanowire, i.e., b ¼ 50, k1 ¼ k2 ¼ 0, and w5 ¼ w6 ¼ 0:1. Fig. 8 shows critical film thickness hc versus the misfit strains εx ¼ εy ¼ εxy ¼ ε with the different surface/interface stresses and the radii of the nanowire b ¼ R1 =by . It shows that critical film thickness decreases with the increase of the misfit strains ε. For a given misfit strains ε, critical film thickness decreases with increasing radius of the nanowire b. Fig. 8 also demonstrates that with the decrease of critical film thickness, there is a stronger decrease for smaller misfit strains and a significantly weaker decrease for larger misfit strains. Critical film thickness is infinite below some critical misfit strains values, meaning that edge MD is never energetically favorable, which is in agreement with the results in Gutkin et al. [11]. The positive surface/interface elastic constants and the residual surface/interface tensions would increase critical film thickness and critical misfit strains. Furthermore, the effects of the residual surface/interface tensions are greater than the surface/interface elastic constants. The influence of the surface/interface stress on critical film thickness is little increasing with decreasing radius of the nanowire. In addition, at sufficiently small values of the misfit strains, the edge MD is not generated at any film thickness with or without the surface/interface stress. Fig. 9 shows critical film thickness hc versus the nanowire radius b ¼ R1 =by with the different surface/interface stresses and the misfit strains εx ¼ εy ¼ εxy ¼ ε. The plots indicate that critical film thickness decreases monotonically and tends to a stable value with increasing
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0.4 0.3
Δw
0.2 0.1 0.0 -0.1 -0.2 0
20
40
60
c
80
Fig. 7. Formation energy Δw versus film thickness c with the different radii of the nanowire b ¼ R1 =by and surface/interface stresses for a ¼ 0:8, εx ¼ εy ¼ εxy ¼ 0:02 and r 0 ¼ 0:5by .
80
hc
60
40
20
0
0.00
0.01
0.02
0.03
ε
0.04
0.05
Fig. 8. Critical film thickness hc versus misfit strains εx ¼ εy ¼ εxy ¼ ε with the different surface/interface stresses and radii of the nanowire b ¼ R1 =by for a ¼ 0:5 and r 0 ¼ 0:5by .
40
hc
30
20
10
0 20
30
40
50
60
70
b Fig. 9. Critical film thickness hc versus nanowire radius b ¼ R1 =by with the different surface/interface stresses and the misfit strains εx ¼ εy ¼ εxy ¼ ε for a ¼ 0:5 and r 0 ¼ 0:5by .
nanowire radius for a given misfit strain ε. The sharper the critical film thickness decreases, the smaller the nanowire radius is. There exists a critical value of nanowire radius, below which MD does not form at any film thickness. The larger the misfit strains, the smaller the critical values of nanowire radii. The positive (negative) surface/interface elastic constants and residual surface/interface tensions would increase (decrease) critical film thickness and critical nanowire radius. Furthermore, the larger the values of the surface/interface stress qualities, the greater the influence of the surface/interface stress on them. If the values of the nanowire radius are small enough, the edge MD is not generated with or without the surface/interface stress, whatever the film thickness. Thus our analysis not only is very valid when determining the critical size at which edge MD is first introduced but also is extremely useful in guiding interface engineering and improving interface quality. 4. Conclusions A model of the generation for an edge MD at the interface boundary in core–shell nanowires composite with surface/interface effects and different elastic constants is investigated. The critical conditions of an edge MD formation are analyzed. The influence of the material
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elastic dissimilarity, misfit strains, the radius of the nanowire, the core radius of the edge MD and the surface/interface effects is considered. Some conclusions drawn from above analyses are summarized as: 1. The formation energy decreases with the increase of the ratio of the shear modulus, the misfit strains, the radius of the nanowire, and the core radius of edge MD. Critical film thickness reduces with increasing the misfit strains, nanowire radius and core radius of the edge MD, below the critical values of which, edge MD is never energetically favorable whatever the film thickness. Critical film thickness first decreases and then increases with increasing the ratio of the shear modulus. There exists critical film thickness below which no interfacial edge MD should be introduced whatever the ratio of the shear modulus. There also exists a critical value of the ratio of the shear modulus, above which edge MD does not form at any film thickness. 2. When the ratio of the shear modulus is relatively small to a certain value, the misfit strains and the radii of the nanowire are relatively large to certain values, the generation of the edge MD is energetically favorable when the film thickness is in the range. In addition, when the ratio of the shear modulus is large enough, the misfit strains and the radii of the nanowire are small enough, the generation of the edge MD is energetically unfavorable at any value of the film thickness. The larger misfit strains, the smaller critical values of nanowire radii. 3. The negative (positive) surface/interface stress can decrease (increase) the formation energy of the edge MD. The positive (negative) surface/interface stress would increase (decrease) critical film thickness, critical misfit strains, and critical nanowire radius. The positive (negative) surface/interface stress would decrease (increase) the range of the film thickness and the critical ratio of the shear modulus. The influence of the residual surface/interface tensions is greater than that of the surface/interface elastic constants. When the residual surface/interface tensions take the same sign compared with the opposite sign, the impact of surface/interface stress on critical parameters for the generation of the edge MD is far greater. Furthermore, the larger the values of the surface/interface stress qualities, the greater the influence of the surface/interface stress on critical parameters. 4. If considering the surface/interface stress, the generation of edge MD may be energetically favorable when only the film thickness is in some range for given misfit strains, while the generation of edge MD may be energetically unfavorable for given radius of the nanowire. In addition, at sufficiently small values of the misfit strains and nanowire radius, the edge MD is not generated with or without the surface/interface stress, whatever the film thickness.
Acknowledgments The authors would like to deeply appreciate the support from the NNSFC (11172094, and 11172095) and the NCET-11-0122. The work was also supported by the Fundamental Research Funds for the Central Universities, Hunan University.
Appendix A For studying the model of an edge dislocation in the system of a nanowire surrounded by a co-axial film with surface/interface effects, the boundary conditions at the inner and outer surface/interface can be expressed as follows: ux1 ðtÞ þ iuy1 ðtÞ ¼ ux2 ðtÞ þ iuy2 ðtÞ ðA:1Þ t ¼ R1 ∂s0 ðtÞ ½sr2 ðtÞ þ isrθ2 ðtÞ2 −½sr1 ðtÞ þ isrθ1 ðtÞ1 ¼ s0θθ2 ðtÞ−i θθ2 =R1 ∂θ ½sr2 ðtÞ þ isrθ2 ðtÞ2 ¼ −
∂s0 ðtÞ 1 s0θθ2 ðtÞ−i θθ2 R2 ∂θ
t ¼ R2
t ¼ R1
ðA:2Þ
ðA:3Þ
An edge dislocation with Burgers vectors ðbx ; by Þ locates in the film. According to Muskhelishvili [43], two complex potentials Φ1 ðzÞ and Ψ 1 ðzÞ are holomorphic in the nanowire, two complex potentials Φ2 ðzÞ and Ψ 2 ðzÞ in the film can be taken the following forms: γ ðA:4Þ Φ2 ðzÞ ¼ 2 þ Φ20 ðzÞ z−z0 Ψ 2 ðzÞ ¼
γ2 γ z0 þ 2 þ Ψ 20 ðzÞ z−z0 ðz−z0 Þ2
ðA:5Þ
Where γ 2 ¼ ðμ2 =πð1 þ κ 2 ÞÞðby −ibx Þ, Φ20 ðzÞ and Ψ 20 ðzÞ which are holomorphic in the film can be expanded as a Laurent series in the annular region ∞
∞
k¼0
k¼1
Φ20 ðzÞ ¼ ∑ ak zk þ ∑ bk z−k ∞
∞
k¼0
k¼1
Ψ 20 ðzÞ ¼ ∑ ck zk−2 þ ∑ dk z−k−2
ðA:6Þ ðA:7Þ
For the convenience, introduce the following new analytical functions in the corresponding regions according to the Schwarz symmetry principle. ! ! ! R1 2 R1 2 0 R1 2 R1 2 R1 2 Φ1 Φn1 ðzÞ ¼ −Φ1 þ þ 2 Ψ1 ðA:8Þ z z z z z
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n
Φ2 ðzÞ ¼ −Φ2
nn
Φ2 ðzÞ ¼ −Φ2
R1 2 z
!
R2 2 z
R1 2 0 Φ2 þ z
!
R2 2 0 Φ2 þ z
R1 2 z
!
R2 2 z
R1 2 R1 2 þ 2 Ψ2 z z
!
! ðA:9Þ
R2 2 R2 2 þ 2 Ψ2 z z
! ðA:10Þ
Under the assumption that the interface adheres to the bulk without slipping, and in the absence of body forces, according to Refs. [44,45], the constitutive equations in the surface/interface region is given as s0θθ ¼ τ0 þ ð2μ0 þ λ0 −τ0 Þε0θθ
ðA:11Þ
where s0θθ and ε0θθ denote surface/interface stress and strain, μ0 and λ0 are surface/interface Lame constants, and τ0 is the residual surface/ interface tension. Under above assumption the surface/interface stress is related to deformations that maintain the coherency of the material in the surface/interface plane with equal tangential strain in both phases. So for a coherent surface/interface, the surface/interface strain ε0θθ is equal to the associated tangential strain in the abutting bulk materials. In view of Eqs. (A4), (A8), (A9), and (A11), and the displacement and stress interface conditions on the entire inner interface, Eqs. (A.(1) and A.2) can be rewritten as
κ1 1 Φ1 ðtÞ− Φn2 ðtÞ μ2 μ1
h
1
¼
n
κ2 1 Φ2 ðtÞ− Φn1 ðtÞ μ1 μ2
n0
Φ1 ðtÞ þ ð1 þ α1 ÞΦ2 ðtÞ þ α1 tΦ2 ðtÞ þ ðα1 þ β1 ÞΦ2
2
2
R1 t
−ðα1 þ
t ¼ R1
R2 β1 Þ t1
Φ20
ðA:12Þ
2 i1 R1 t
" ¼
n
ð1−α1 −β1 ÞΦ2 ðtÞ þ Φ1 ðtÞ−ðα1 þ
β1 ÞtΦ20 ðtÞ−α1 Φn2
R21 t
!
t ¼ R1
R2 0 R21 þ α1 1 Φn2 t t
!#2 −
τ01 R1
ðA:13Þ
In view of Eqs. (A4), (A10), and (A11), and the displacement and stress surface conditions on the entire outer surface, Eq. (A.3) can be rewritten as h 2
2 0 i3 τ R2 R2 R nn0 þ ðα2 þ β2 Þ t2 Φ20 t2 − R22 ¼ ð1−α2 ÞΦnn 2 ðtÞ−α2 tΦ2 ðtÞ−ðα2 þ β 2 ÞΦ2 t
" ð1 þ α2 þ β2 ÞΦ2 ðtÞ þ ðα2 þ
β2 ÞtΦ20 ðtÞ
t ¼ R2
nn
þ α2 Φ2
! !#2 R22 R22 nn0 R22 −α2 Φ2 t t t
ðA:14Þ
By using the generalized Liouville theorem [43], the two expressions for Φ20 ðzÞ and Ψ 20 ðzÞ would be obtained, respectively, according to the inner and outer surface/interface conditions. In order to simultaneously satisfy all of the boundary conditions, the two expressions for Φ20 ðzÞ and Ψ 20 ðzÞ obtained in the above two parts must be compatible to each other. So the unknown coefficients ak , bk , ck and dk of Φ20 ðzÞ and Ψ 20 ðzÞ can be determined. The expressions of all the unknown coefficients would be listed as follows: a0 ¼
ðe02 e09 −e03 e08 Þðe06 m01 −e03 m02 Þ−ðe02 e06 −e03 e05 Þðe09 m01 −e03 m03 Þ ; ðe01 e06 −e03 e04 Þðe02 e09 −e03 e08 Þ−ðe02 e06 −e03 e05 Þðe01 e09 −e03 e07 Þ
c0 ¼
ðe01 e09 −e03 e07 Þðe06 m01 −e03 m02 Þ−ðe01 e06 −e03 e04 Þðe09 m01 −e03 m03 Þ ; ðe02 e06 −e03 e05 Þðe01 e09 −e03 e07 Þ−ðe01 e06 −e03 e04 Þðe02 e09 −e03 e08 Þ
a1 ¼
e12 m13 −e110 m14 ; e19 e112 −e110 e111
ak ¼
ðek11 ek18 þ ek12 ek17 Þðek15 mk4 þ ek12 mk5 Þ−ðek11 ek15 þ ek12 ek14 Þðek18 mk4 þ ek12 mk6 Þ ðek10 ek15 þ ek12 ek13 Þðek11 ek18 þ ek12 ek17 Þ−ðek10 ek18 þ ek12 ek16 Þðek11 ek15 þ ek12 ek14 Þ
bk ¼
ðek3 ek8 −ek2 ek9 Þðek5 mk1 −ek2 mk2 Þ−ðek3 ek5 −ek2 ek6 Þðek8 mk1 −ek2 mk3 Þ ðek1 ek5 −ek2 ek4 Þðek3 ek8 −ek2 ek9 Þ−ðek1 ek8 −ek2 ek7 Þðek3 ek5 −ek2 ek6 Þ
ck ¼
ðek10 ek18 þ ek12 ek16 Þðek15 mk4 þ ek12 mk5 Þ−ðek10 ek15 þ ek12 ek13 Þðek18 mk4 þ ek12 mk6 Þ ðek11 ek15 þ ek12 ek14 Þðek10 ek18 þ ek12 ek16 Þ−ðek11 ek18 þ ek12 ek17 Þðek10 ek15 þ ek12 ek13 Þ
dk ¼
ðek1 ek8 −ek2 ek7 Þðek5 mk1 −ek2 mk2 Þ−ðek1 ek5 −ek2 ek4 Þðek8 mk1 −ek2 mk3 Þ ðek3 ek5 −ek2 ek6 Þðek1 ek8 −ek2 ek7 Þ−ðek3 ek8 −ek2 ek9 Þðek1 ek5 −ek2 ek4 Þ
b1 ¼
e18 m11 þ e16 m12 ; e15 e18 þ e16 e17
c1 ¼
e11 m13 −e19 m14 ; e110 e111 −e19 e112
d1 ¼
e15 m12 −e17 m11 ; e15 e18 þ e16 e17 ðk≥2Þ;
ðk≥2Þ;
ðk≥2Þ
ðk≥2Þ;
Y.X. Zhao et al. / International Journal of Mechanical Sciences 74 (2013) 173–184
181
where e01 ¼ x01 x06 −x02 x05 ;
e02 ¼ x03 x06 −x02 x07 ;
e06 ¼ x04 x09 þ x02 x011 ; m02 ¼ x09 s01 −x02 s03 ;
e03 ¼ x04 x06 −x02 x08 ;
e07 ¼ x01 x013 −x02 x014 ; m03 ¼ x013 s01 −x02 s04 ;
e11 ¼ s11 =x12 ;
e15 ¼ x13 ðx122 −x123 e12 Þ þ x121 ðx14 þ x15 e12 Þ; e18 ¼ x13 x128 þ x16 x125 ;
e112 ¼ x120 ðx19 −x18 e14 Þ þ x110 ðx119 þ x118 e14 Þ;
e05 ¼ x03 x09 þ x02 x012 ;
e09 ¼ x04 x013 þ x02 x015 ;
e12 ¼ x11 =x12 ;
e16 ¼ x13 x124 þ x16 x121 ;
e19 ¼ x17 x114 −x110 x111 ;
m12 ¼ x125 ðs12 −x15 e11 Þ−x13 ðs18 −x127 e11 Þ;
e04 ¼ x01 x09 −x02 x010 ;
e08 ¼ x03 x013 þ x02 x016 ;
m01 ¼ x06 s01 −x02 s02 ;
e13 ¼ s15 =x15 ; e14 ¼ x16 =x15 ;
e17 ¼ x13 ðx126 þ x127 e12 Þ−x125 ðx14 þ x15 e12 Þ;
e110 ¼ x114 ðx19 −x18 e14 Þ−x110 ðx113 þ x112 e14 Þ;
e111 ¼ x17 x120 þ x117 x110 ;
m11 ¼ x13 ðs17 −x123 e11 Þ−x121 ðs12 −x15 e11 Þ;
m13 ¼ x114 ðs13 −x18 e13 Þ−x110 ðs14 þ x112 e13 Þ;
m14 ¼ x120 ðs13 −x18 e13 Þ þ x110 ðs16 þ x118 e13 Þ; ek1 ¼ xk2 xk5 −xk1 xk6 ; ek3 ¼ xk4 xk5 −xk1 xk8 ; ek4 ¼ xk2 xk25 þ xk1 xk26 ; ek5 ¼ xk3 xk25 −xk1 xk27 ; ek6 ¼ xk4 xk25 þ xk1 xk28 ; ek7 ¼ xk2 xk29 þ xk1 xk30 ; ek8 ¼ xk3 xk29 −xk1 xk31 ; ek9 ¼ xk4 xk29 −xk1 xk32 ; ek10 ¼ xk9 xk14 −xk10 xk13 ;
ek11 ¼ xk11 xk14 −xk10 xk15 ;
ek14 ¼ xk11 xk18 þ xk10 xk19 ;
ek12 ¼ xk12 xk14 þ xk10 xk16 ;
ek15 ¼ xk10 xk20 −xk12 xk18 ;
ek18 ¼ xk10 xk24 −xk12 xk22 ;
mk1 ¼ xk5 sk1 −xk1 sk2 ;
mk2 ¼ xk25 sk1 −xk1 sk7 ;
mk5 ¼ xk18 sk3 þ xk10 sk5 ;
mk6 ¼ xk22 sk3 þ xk10 sk6 ;
x01 ¼ η3 p7 −η3 p9 ;
x03 ¼ η3 p9 R−2 1 ; x09 ¼ x01 ;
x04 ¼ η4 q7 R−2 2 ;
x010 ¼ x02 ;
x05 ¼ R21 ðg 11 −g 17 Þ þ R22 ðh18 −h12 Þ;
x011 ¼ x03 ;
x012 ¼ x04 ;
ek13 ¼ xk9 xk18 þ xk10 xk17 ;
ek16 ¼ xk9 xk22 þ xk10 xk21 ;
x013 ¼ x05 ;
ek17 ¼ xk11 xk22 þ xk10 xk23 ;
mk3 ¼ xk29 sk1 −xk1 sk8 ;
mk4 ¼ xk14 sk3 −xk10 sk4 ;
x02 ¼ η3 p10 þ η4 q7 ; x06 ¼ R21 g 14 þ R22 h11 ;
x014 ¼ x06 ;
x015 ¼ x07 ;
x07 ¼ q17 −h18 ;
x08 ¼ h11 ;
x016 ¼ x08 ;
s01 ¼ η3 p6 þ η4 q1 znn−1 þ η4 q2 znn−2 −η4 τ02 =R2 ; s02 ¼ −R21 ðg 6 −g 8 −γ 2 =z0 −η5 τ01 =R1 Þ þ R22 ð−h1 znn−1 þ h2 znn−2 −γ 2 =z0 −η4 τ02 =R2 Þ; x11 ¼ −η3 p10 −η4 q6 ;
x12 ¼ η3 p8 −η3 p9 ;
s03 ¼ s01 ;
s04 ¼ s02 ;
x13 ¼ ðg 14 þ g 15 þ g 20 ÞR41 ;
x14 ¼ 2R21 ðg 16 þ 2g 17 þ g 18 þ g 20 Þ þ R22 ðh10 −2h18 þ h9 Þ;
x15 ¼ R21 ðg 12 −g 13 −2g 17 Þ;
q8 ÞR−2 2 ;
x19 ¼ 2η4 q9 R−2 x17 ¼ η3 p7 þ η4 q8 ; x18 ¼ 2η4 ðq7 −q9 þ 2 ; x112 ¼ −2h11 −2h15 þ h16 −h17 þ 3h20 ; x113 ¼ g 19 þ h14 −2h13 ;
x16 ¼ 2g 17 þ g 18 −h18 ;
x111 ¼ R21 ðg 11 −g 21 Þ;
x114 ¼ ðh11 þ h15 ÞR−2 x115 ¼ x11 ; x116 ¼ x12 ; x117 ¼ x13 ; x118 ¼ x14 ; x119 ¼ x15 ; x120 ¼ x16 ; 2 ; x121 ¼ x17 ; x123 ¼ x19 ; x124 ¼ x110 ; x125 ¼ x111 ; x126 ¼ x112 ; x127 ¼ x113 ; x128 ¼ x114 ; s11 ¼ −η3 p1 þ η3 p2 −η3 p4 þ η3 p6 zn ;
s12 ¼ −R21 g 1 þ g 2 þ g 4 þ g 9 þ ðg 6 −2g 8 Þzn ;
s13 ¼ η4 ðq1 −q3 Þznn−2 þ ð2η4 q2 −η4 q4 Þznn−3 −η4 q5 znn−1 −2η4 q9 γ 2 R−2 2 ; s14 ¼ −R21 g 10 þ R22 ð−h1 þ h3 Þznn−2 þ R22 ð2h2 −h4 Þznn−3 −R22 h5 znn−1 þ h6 R22 þ ð2h13 −h14 Þγ 2 ; s15 ¼ s11 ; s16 ¼ s12 ; s17 ¼ s13 ; s18 ¼ s14 ; xk1 ¼ η3 p8 ðk−1Þ−η3 p9 kðk−1Þ þ η3 p10 ð1−kÞ R2k 1 ; xk3 ¼ xk5 ¼
xk2 ¼ −η3 p10 k þ η3 p9 ðk−1Þðk 2ðk−1Þ −2 ðη3 p8 −η3 p9 kÞR1 ; xk4 ¼ η3 p9 ð1−kÞR1 ; R21ðk−1Þ g 12 ðk−1Þ−g 13 kðk−1Þ þ g 14 þ g 15 k−g 17 kðk−1Þðk þ 1Þ−g 20 kðk−2Þ ;
þ 1Þ−η4 q6 ;
h i 2 xk6 ¼ R21 −g 16 ðk þ 1Þ þ g 17 ðk þ 1Þðk −2k−1Þ−g 18 kðk þ 1Þ−g 20 kðk þ 1Þ −R22 h10 −h18 ðk þ 1Þ þ h19 k ; 2 xk7 ¼ g 12 −g 13 k−g 17 kðk þ 1Þ R2k xk8 ¼ −g 17 ðk −2k−1Þ þ g 18 k−h18 ; 1 ; xk9 ¼ η3 p7 þ η4 q8 k þ η4 q9 ðk−1Þðk þ 1Þ; xk10 ¼ −η4 q7 ðk þ 1Þ þ η4 q9 kðk þ 1Þ−η4 q8 ðk þ 1Þ R−2k 2 ; ðkþ1Þ xk11 ¼ η4 q9 ðk þ 1ÞR−2 xk12 ¼ ðη4 q7 −η4 q9 kÞR−2 ; 2 ; 2 h i 2 2 2 xk13 ¼ R1 g 11 þ g 19 k−1−g 21 k −R2 h12 ð1−kÞ þ h13 ðk−1Þðk þ 2k−1Þ−h14 kðk−1Þ þ h20 kðk−1Þ ; ; xk14 ¼ −h11 ðk þ 1Þ−h15 kðk þ 1Þ þ h16 −h17 k−h13 kðk−1Þðk þ 1Þ þ h20 kðk þ 2Þ R−2kþ2 2 −2k 2 xk15 ¼ g 19 −h13 ðk þ 2k−1Þ þ h14 k; xk16 ¼ − h11 þ h15 k þ h13 kðk−1Þ R2 ;
xk17 ¼ xk1 ; xk25 ¼ xk9 ;
xk18 ¼ xk2 ; xk19 ¼ xk3 ; xk20 ¼ xk4 ; xk21 ¼ xk5 ; xk22 ¼ xk6 ; xk23 ¼ xk7 ; xk24 ¼ xk8 ; xk26 ¼ xk10 ; xk27 ¼ xk11 ; xk28 ¼ xk12 ; xk29 ¼ xk13 ; xk30 ¼ xk14 ; xk31 ¼ xk15 ; xk32 ¼ xk16 ;
sk1 ¼ ðη3 p2 −η3 p4 kÞznðk−1Þ − η3 p3 ðk−1Þ−0:5η3 p5 ðk−1Þk znðk−2Þ þ η3 p6 znk ; sk2 ¼ −R21 g 2 þ g 4 k þ 0:5g 9 kðk þ 1Þ znðk−1Þ þ g 6 −g 8 ðk þ 1Þ znk þ g 3 ðk−1Þ þ 0:5g 5 kðk−1Þ−1=6g 7 ðk−1Þkðk þ 1Þ znðk−2Þ ; sk3 ¼ η4 q1 −η4 q3 k znn−ðkþ1Þ þ η4 q2 ðk þ 1Þ−0:5η4 q4 kðk þ 1Þ znn−ðkþ2Þ −η4 q5 znn−k ; nn−ðkþ1Þ ( ) −h1 þ h3 k þ 0:5h7 kðk−1Þ z þ −h5 þ h9 ðk−1Þ znn−k nn−ðkþ2Þ ; sk5 ¼ sk1 ; sk6 ¼ sk2 ; sk7 ¼ sk3 ; sk8 ¼ sk4 ; sk4 ¼ R22 þ ðk þ 1Þh2 −0:5h4 kðk þ 1Þ þ 1=6h8 ðk−1Þkðk þ 1Þ z α1 ¼ ð2μ01 þ λ01 −τ01 Þ=ð4R1 μ2 Þ;
β1 ¼ ð2μ01 þ λ01 −τ01 Þ=2R1 ðλ2 þ μ2 Þ; α2 ¼ ð2μ02 þ λ02 −τ02 Þ=ð4R2 μ2 Þ; i i h h β2 ¼ ð2μ02 þ λ02 −τ02 Þ=2R2 ðλ2 þ μ2 Þ; η1 ¼ γ 2 zn ðz0 −zn Þ = z0 ðz−zn Þ2 ; η2 ¼ γ 2 znn ðz0 −znn Þ = z0 ðz−znn Þ2 ; η3 ¼ 1=ðμ1 κ2 =μ2 þ 1−α1 −β1 Þ; p1 ¼ ð1−μ1 =μ2 Þγ 2 ;
η4 ¼ 1=ð1 þ α2 þ β2 Þ;
p2 ¼ ð1 þ α1 −μ1 =μ2 Þγ 2 ;
η5 ¼ 1=ðμ1 =κ 1 μ2 þ 1 þ α1 Þ;
p3 ¼ ð1 þ α1 −μ1 =μ2 Þη1 ;
η6 ¼ 1=ð1−α2 Þ;
p4 ¼ α1 γ 2 þ ðα1 þ β1 Þγ 2 R21 =z0 2 ;
p5 ¼ 2α1 η1 ;
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p6 ¼ ðα1 þ β1 Þγ 2 =z0 ;
p7 ¼ μ1 κ 2 =μ2 þ 1−α1 −β1 ;
p11 ¼ ðμ1 =ðκ1 μ2 Þ þ 1 þ α1 Þγ 2 =R21 ; p14 ¼ 2α1 γ 2 ;
p15 ¼ α1 γ 2 ;
p8 ¼ 1 þ α1 −μ1 =μ2 ;
p16 ¼ α1 η1 ;
p17 ¼ 2α1 η1 ;
p20 ¼ ð1−μ1 =μ2 Þγ 2 ; p21 ¼ 2ð1 þ 2α1 −μ1 =μ2 Þη1 ; h i p25 ¼ ð1 þ 2α1 −μ1 =μ2 Þγ 2 þ ðα1 þ β1 Þγ 2 R21 =z0 2 ; p27 ¼ μ1 κ 2 =μ2 þ 1−α1 −β1 ; g 1 ¼ η3 p1 −η5 p14 þ η3 p20 ;
g 2 ¼ −η3 p2 þ η5 p15 ;
g 11 ¼ η3 p7 ;
g 12 ¼ η3 p8 −η5 p9 ;
g 18 ¼ −2η3 p9 −η5 p9 ;
q13 ¼ α2 γ 2 ;
q2 ¼ ð1−α2 Þη2 ; q7 ¼ 1−α2 ;
q14 ¼ α2 η2 ;
g 19 ¼ η5 p18 ;
q8 ¼ α2 þ β2 ;
h13 ¼ η4 q9 ;
h19 ¼ η4 q25 ;
h20 ¼ η4 q8
q9 ¼ α 2 ;
q16 ¼ 1−2α2 ;
g 8 ¼ η3 p24 ;
g 14 ¼ η3 p10 þ η5 p19 ;
g 20 ¼ η3 p10 ;
g 10 ¼ η5 p11 −γ 2 =R21 ;
g 9 ¼ η3 p26 ;
g 15 ¼ −2η3 p10 −η5 p10 ;
g 16 ¼ η5 p10 ;
g 21 ¼ η3 p27 ; q4 ¼ 2α2 η2 ;
q5 ¼ ðα2 þ β2 Þγ 2 =z0 ;
q10 ¼ ð1 þ α2 þ β2 Þγ 2 =z0 ; q17 ¼ 1 þ α2 þ β2 ;
q20 ¼ 2ð1−2α2 Þη2 ;
q11 ¼ 0;
q12 ¼ ðα2 þ β2 Þγ 2 R22 =z0 2 ;
q18 ¼ 1−α2 ;
q21 ¼ 2ðα2 γ 2 þ ðα2 þ β2 Þγ 2 R22 =z0 2 Þ;
q22 ¼ 6α2 η2 ;
q25 ¼ 1 þ α2 þ β2 ;
h2 ¼ η4 q2 þ η6 q14 ;
h12 ¼ η6 q8 ;
p19 ¼ 1−α1 −β1 −μ1 κ 2 =ðκ 1 μ2 Þ;
g 4 ¼ η3 p4 þ η5 p13 −η3 p25 ;
g 7 ¼ η3 p22 ;
q3 ¼ α2 γ 2 þ ðα2 þ β2 Þγ 2 R22 =z0 2 ;
q15 ¼ 2α2 η2 ;
q23 ¼ q24 ¼ ðα2 þ β2 Þγ 2 =z0 ; h1 ¼ −η4 q1 −η6 q13 ;
g 3 ¼ η3 p3 −η5 p16 ;
g 13 ¼ η3 p9 þ η5 p9 −η3 p28 ;
q19 ¼ ð1−2α2 Þγ 2 −ðα2 þ β2 Þγ 2 R22 =z0 2 ;
h5 ¼ η4 ðq5 −q23 Þ−η6 q10 ;
p18 ¼ μ1 =ðκ1 μ2 Þ þ 1 þ α1 ;
p22 ¼ 6α1 η1 ; p23 ¼ p24 ¼ ðα1 þ β1 Þγ 2 =z0 ; h i p26 ¼ 2 α1 γ 2 þ ðα1 þ β1 Þγ 2 R21 =z0 2 ;
g 6 ¼ −η3 p6 −η5 p12 þ η3 p23 ;
g 17 ¼ η3 p9 ;
q6 ¼ 1 þ α 2 þ β 2 ;
p10 ¼ α1 þ β1 ;
p13 ¼ α1 γ 2 þ ðα1 þ β1 Þγ 2 R21 =z0 2 ;
p28 ¼ 1 þ 2α1 −μ1 =μ2
g 5 ¼ −η3 p5 −η5 p17 þ η3 p21 ;
q1 ¼ ð1−α2 Þγ 2 ;
p9 ¼ α 1 ;
p12 ¼ ð1−α1 −β1 −μ1 κ 2 =ðκ 1 μ2 ÞÞγ 2 =z0 ;
h3 ¼ −η4 ðq3 þ q19 Þ−η6 ðq12 þ q13 Þ;
h6 ¼ η6 q11 −γ 2 =R22 ; h14 ¼ 2η4 q9 þ η6 q9 ;
h7 ¼ η4 q21 ;
h8 ¼ η4 q22 ;
h15 ¼ −η4 q9 −η6 q9 ;
h4 ¼ η4 ðq4 þ q20 Þ;
h9 ¼ η4 q24 ;
h10 ¼ η4 q6 ;
h16 ¼ −η4 q8 þ η6 q17 ;
h11 ¼ η4 q7 þ η6 q9 ;
h17 ¼ 2η4 q8 þ η6 q8 ;
h18 ¼ η6 q18 ;
Appendix B For studying the misfit stress in the system of a nanowire surrounded by a co-axial film with surface/interface effects, the boundary conditions at the inner and outer surface/interfaces can be expressed as follows: ½ur2 ðtÞ þ iuθ2 ðtÞ2 −½ur1 ðtÞ þ iuθ1 ðtÞ1 ¼ u0r þ iu0θ ðB:1Þ t ¼ R1 ∂s0 ðtÞ ½sr2 ðtÞ þ isrθ2 ðtÞ2 −½sr1 ðtÞ þ isrθ1 ðtÞ1 ¼ s0θθ2 ðtÞ−i θθ2 =R1 ∂θ ∂s0 ðtÞ ½sr2 ðtÞ þ isrθ2 ðtÞ2 ¼ − s0θθ2 ðtÞ−i θθ2 =R2 ∂θ
t ¼ R1
ðB:2Þ
t ¼ R2
ðB:3Þ n
o
Where u0r and u0θ are the displacement induced by the normal and shear uniform eigenstrains ε0x ; ε0y ; ε0xy . According to Gao [46], the displacement produced by the uniform eigenstrains of the nano inhomogeneity can be expressed as 0 ur þ iu0θ ¼ R1 ε1 þ R31 ðε2 þ iε3 Þ=t 2 ðB:4Þ t ¼ R1 Where ε1 ¼ ðε0x þ ε0y Þ=2, ε2 ¼ ðε0x −ε0y Þ=2, ε3 ¼ ε0xy . There is the prescribed uniform eigenstrain inside the film, so the two complex potentials Φ1 ðzÞ and Ψ 1 ðzÞ are holomorphic in the nanowire, two complex potentials Φ2 ðzÞ and Ψ 2 ðzÞ due to the prescribed uniform eigenstrain inside the film can be taken the following forms: ∞
∞
k¼0
k¼1
Φ2 ðzÞ ¼ ∑ ak zk þ ∑ bk z−k ∞
∞
k¼0
k¼1
Ψ 2 ðzÞ ¼ ∑ ck zk−2 þ ∑ dk z−k−2
ðB:5Þ
ðB:6Þ
For the convenience, in the corresponding regions according to the Schwarz symmetry principle introduce the following analytical functions Φn1 ðzÞ, Φn2 ðzÞ and Φnn 2 ðzÞ, which are the same with Eqs. (A.8)–(A.10) in Appendix A. In view of Eqs. (B.(4) and B.5), and the displacement and stress interface conditions on the entire inner interface, Eqs. (B.(1) and B.2) can be rewritten as 1 2 κ 1 Φ1 ðtÞ=μ1 −Φn2 ðtÞ=μ2 − κ2 Φ2 ðtÞ=μ2 −Φn1 ðtÞ=μ1 ¼ −2ε1 þ 2R21 ðε2 þ iε3 Þ=t 2 ðB:7Þ t ¼ R1 ∂ðsθ2 −sr2 Þ μ2 ∂ðsθ2 þ sr2 Þ ½sr2 þ isrθ2 2 −½sr1 þ isrθ1 1 ¼ τ01 =R1 þ ð2μ01 þ λ01 −τ01 Þ=ð4R1 μ2 Þ ðsθ2 −sr2 Þ þ μ2 ðsθ2 þ sr2 Þ=ðλ2 þ μ2 Þ−i −i ∂θ ∂θ λ2 þ μ2
t ¼ R1 ðB:8Þ
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183
In view of Eqs. (B.(4) and B.5) and the displacement and stress surface conditions on the entire outer surface, Eq. (B.3) can be rewritten as " ! !#2 2 h 2
2 0 i3 R22 nn0 R22 τ2 R2 R22 R2 0 nn R2 nn nn0 0 −α2 Φ2 ð1−α2 ÞΦ2 ðtÞ−α2 tΦ2 ðtÞ−ðα2 þ β2 ÞΦ2 t þ ðα2 þ β2 Þ t Φ2 t − R2 ¼ ð1 þ α2 þ β2 ÞΦ2 ðt Þ þ ðα2 þ β2 ÞtΦ2 ðtÞ þ α2 Φ2 t t t ðB:9Þ t ¼ R2 By using the generalized Liouville theorem, the two expressions for Φ20 ðzÞ and Ψ 20 ðzÞ would be obtained, respectively, according to the inner and outer surface/interface conditions. In order to simultaneously satisfy all of the boundary conditions, the two expressions for Φ20 ðzÞ and Ψ 20 ðzÞ obtained in the above two parts must be compatible to each other. So the unknown coefficients ak , bk , ck and dk of Φ20 ðzÞ and Ψ 20 ðzÞ can be determined. The expressions of all the unknown coefficients would be expressed as follows: ðt 02 t 09 −t 03 t 08 Þðt 06 v01 −t 03 v02 Þ−ðt 02 t 06 −t 03 t 05 Þðt 09 v01 −t 03 v03 Þ ; ðt 01 t 06 −t 03 t 04 Þðt 02 t 09 −t 03 t 08 Þ−ðt 02 t 06 −t 03 t 05 Þðt 01 t 09 −t 03 t 07 Þ ðt 01 t 09 −t 03 t 07 Þðt 06 v01 −t 03 v02 Þ−ðt 01 t 06 −t 03 t 04 Þðt 09 v01 −t 03 v03 Þ c0 ¼ ; ðt 02 t 06 −t 03 t 05 Þðt 01 t 09 −t 03 t 07 Þ−ðt 01 t 06 −t 03 t 04 Þðt 02 t 09 −t 03 t 08 Þ ðt 22 t 29 −t 23 t 28 Þðt 26 v21 −t 23 v22 Þ−ðt 22 t 26 −t 23 t 25 Þðt 29 v21 −t 23 v23 Þ ; a2 ¼ ðt 21 t 26 −t 23 t 24 Þðt 22 t 29 −t 23 t 28 Þ−ðt 21 t 29 −t 23 t 27 Þðt 22 t 26 −t 23 t 25 Þ ðt 212 t 217 −t 211 t 218 Þðt 214 v24 −t 211 v25 Þ−ðt 212 t 214 −t 211 t 215 Þðt 217 v24 −t 211 v26 Þ b2 ¼ ; ðt 210 t 214 −t 211 t 213 Þðt 212 t 217 −t 211 t 218 Þ−ðt 210 t 217 −t 211 t 216 Þðt 212 t 214 −t 211 t 215 Þ ðt 21 t 29 −t 23 t 27 Þðt 26 v21 −t 23 v22 Þ−ðt 21 t 26 −t 23 t 24 Þðt 29 v21 −t 23 v23 Þ c2 ¼ ; ðt 21 t 29 −t 23 t 27 Þðt 22 t 26 −t 23 t 25 Þ−ðt 21 t 26 −t 23 t 24 Þðt 22 t 29 −t 23 t 28 Þ ðt 210 t 217 −t 211 t 216 Þðt 214 v24 −t 211 v25 Þ−ðt 210 t 214 −t 211 t 213 Þðt 217 v24 −t 211 v26 Þ d2 ¼ ; ðt 210 t 214 −t 211 t 213 Þðt 212 t 217 −t 211 t 218 Þ−ðt 210 t 217 −t 211 t 216 Þðt 212 t 214 −t 211 t 215 Þ a0 ¼
where t 01 ¼ x01 x06 −x02 x05 ; t 02 ¼ x03 x06 −x02 x07 ; t 03 ¼ x04 x06 −x02 x08 ; t 04 ¼ x01 x09 −x02 x010 ; t 05 ¼ x03 x09 þ x02 x012 ; t 06 ¼ x04 x09 þ x02 x011 ; t 07 ¼ x01 x013 −x02 x014 ; t 08 ¼ x03 x013 þ x02 x016 ; t 09 ¼ x04 x013 þ x02 x015 ; v01 ¼ x06 s01 −x02 s02 ; v02 ¼ x09 s01 −x02 s03 ; v03 ¼ x013 s01 −x02 s04 ; x01 ¼ e1 m4 −e1 m2 ; x02 ¼ e1 m3 þ e3 n3 ; x03 ¼ e1 m2 ; x04 ¼ e3 n3 ; x05 ¼ R21 ðp5 −p7 Þ þ R22 ðq14 −q9 Þ; x012 ¼ x04 ;
x013 ¼ x05 ;
x06 ¼ R21 p3 þ R22 q5 ;
x014 ¼ x06 ;
x015 ¼ x07 ;
s02 ¼ ðq1 þ e4 n6 Þlnðz1 =z2 Þ þ R21 p12 −R22 q12 ;
x07 ¼ p7 −q14 ;
s03 ¼ s01 ;
x08 ¼ q5 ;
x09 ¼ x01 ;
x010 ¼ x02 ;
x011 ¼ x03 ;
s01 ¼ e3 n1 lnðz1 =z2 Þ−e3 τ02 =R2 ;
x016 ¼ x08 ; s04 ¼ s02 ;
t 21 ¼ x21 x26 −x22 x25 ;
t 22 ¼ x23 x26 −x22 x27 ;
t 23 ¼ x24 x26 þ x22 x28 ; t 24 ¼ x21 x226 −x22 x225 ; t 25 ¼ x23 x226 −x22 x227 ; t 26 ¼ x24 x226 þ x22 x228 ; t 27 ¼ x21 x230 −x22 x229 ; t 28 ¼ x23 x230 −x22 x231 ; t 29 ¼ x24 x230 −x22 x232 ; t 210 ¼ x210 x213 −x29 x214 ; t 211 ¼ x211 x213 −x29 x215 ; t 212 ¼ x212 x213 þ x29 x216 ; t 213 ¼ x210 x217 −x29 x218 ; t 214 ¼ x211 x217 −x29 x219 ; t 215 ¼ x212 x217 þ x29 x220 ; t 216 ¼ x210 x221 −x29 x222 ; t 217 ¼ x211 x221 −x29 x223 ; t 218 ¼ x212 x221 −x29 x224 ; v21 ¼ x26 s21 −x22 s22 ; v22 ¼ x226 s21 −x22 s27 ; v23 ¼ x230 s21 −x22 s28 ; v24 ¼ x213 s23 −x29 s24 ; v25 ¼ x217 s23 −x29 s25 ; v26 ¼ x221 s23 −x29 s26 ; x21 ¼ e1 m4 þ 2e3 n5 þ 3e3 n4 ; x22 ¼ 3e3 n3 −6e3 n4 þ 3e3 n5 ; x25 ¼ R21 ð−p5
þ p10 Þ þ
R22 ðq9
x23 ¼ 3e3 n4 R−2 2 ;
x24 ¼ ð2e3 n4 −e3 n3 ÞR−6 2 ; x26 ¼ ð3q5 þ 6q6 −q7 þ 2q8 þ 6q10 −8q15 ÞR−2 2 ;
þ 2q11 −2q15 −7q10 Þ;
x28 ¼ q5 þ 2q6 þ 2q10 ; x29 ¼ ðe1 m1 −2e1 m2 −e1 m3 ÞR41 ; x210 2 x211 ¼ ðe1 m1 −2e1 m2 ÞR1 ; x212 ¼ e1 m2 R−2 x213 ¼ ðp1 −2p2 þ p3 þ 2p4 −6p7 ÞR61 ; 1 ; 2 2 x214 ¼ R1 ð−6p8 −3p6 −6p13 −3p7 Þ þ R2 ð−3q4 þ 3q14 Þ; x215 ¼ ðp1 −2p2 −6p7 ÞR41 ; x216 ¼ x27 ¼ p10 −7q10 þ 2q11 ;
x217 ¼ x21 ;
x218 ¼ x22 ;
x227 ¼ x211 ;
x228 ¼ x212 ;
x219 ¼ x23 ;
x220 ¼ x24 ;
x229 ¼ x213 ;
x221 ¼ x25 ;
x230 ¼ x214 ;
−2 −2 −2 s22 ¼ R22 ð−0:5q1 z−2 1 þ 0:5q1 z2 þ q2 z1 −q3 z2 Þ;
x222 ¼ x26 ;
x231 ¼ x215 ;
s23 ¼ e1 m5 ;
¼ −2e1 m3 þ 3e1 m2 −e3 n2 ;
2p8 þ p7 −q14 ; x223 ¼ x27 ; x224 ¼ x28 ;
x232 ¼ x216 ;
s24 ¼ R21 ðp9 þ p11 Þ;
x225 ¼ x29 ;
x226 ¼ x210 ;
−2 s21 ¼ 0:5e3 n1 ðz−2 2 −z1 Þ;
s25 ¼ s21 ;
s26 ¼ s22 ;
s27 ¼ s23 ; s28 ¼ s24
α1 ¼ ð2μ01 þ λ01 −τ01 Þ=ð4R1 μ2 Þ; β1 ¼ ð2μ01 þ λ01 −τ01 Þ= 2R1 ðλ2 þ μ2 Þ ; α2 ¼ ð2μ02 þ λ02 −τ02 Þ=ð4R2 μ2 Þ; β2 ¼ ð2μ02 þ λ02 −τ02 Þ= 2R2 ðλ2 þ μ2 Þ ; e1 ¼ 1=ðμ1 κ 2 =μ2 þ 1−α1 −β1 Þ; e2 ¼ 1=ðμ1 =κ 1 μ2 þ 1 þ α1 Þ; e3 ¼ 1=ð1 þ α2 þ β2 Þ; e4 ¼ 1−α2 ; m1 ¼ 1 þ α1 −μ1 =μ2 ; m2 ¼ α 1 ;
m3 ¼ α 1 þ β 1 ;
m4 ¼ μ1 κ 2 =μ2 þ 1−α1 −β1 ;
m5 ¼ 2μ1 R21 ðε2 þ iε3 Þ;
m6 ¼ μ1 =ðκ 1 μ2 Þ þ 1 þ α1 ;
4μ1 R21 ðε2
m7 ¼ 1−α1 −β1 −μ1 κ 2 =ðκ 1 μ2 Þ; m8 ¼ 1 þ 2α1 −μ1 =μ2 ; m9 ¼ þ iε3 Þ; p1 ¼ e1 m1 −e2 m2 ; p2 ¼ e1 m2 þ e2 m2 −e1 m8 ; p3 ¼ e1 m3 þ e2 m7 ; p4 ¼ −2e1 m3 −e2 m3 ; p5 ¼ e1 m4 ; p6 ¼ e2 m3 ;
p7 ¼ e1 m2 ;
p8 ¼ −2e1 m2 −e2 m2 ; p9 ¼ e1 m5 ; p10 ¼ e2 m6 ; p11 ¼ e1 m9 ; p12 ¼ e2 ðτ01 =R1 −2ε1 μ1 =κ 1 Þ; p13 ¼ e1 m3 ; n1 ¼ 0; n2 ¼ 1 þ α2 þ β2 ; n3 ¼ 1−α2 ; n4 ¼ α2 ; n5 ¼ α2 þ β2 ; n6 ¼ 0; n7 ¼ 0; n8 ¼ 0; n9 ¼ 0; n10 ¼ 1 þ α2 þ β2 ; n11 ¼ 1−α2 ; q1 ¼ e3 n1 −e4 n6 ; q2 ¼ e4 n7 þ e3 n1 ; q3 ¼ e4 n8 þ e3 n1 ; q4 ¼ e3 n2 ; q5 ¼ e3 n3 þ e4 n4 ; q6 ¼ −e4 n4 −e3 n3 ; q7 ¼ −e3 n5 þ e4 n10 ; q8 ¼ 2e3 n5 þ e4 n5 ; q9 ¼ e4 n5 ; q10 ¼ e3 n4 ; q13 ¼ e4 n9 ; q14 ¼ e4 n11 ; q15 ¼ e3 n5
q11 ¼ 2e3 n4 þ e4 n4 ;
q12 ¼ e3 τ02 =R2 −e4 n6 lnðz1 =z2 Þ;
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