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Contribution to critical shear stress of nanocomposites produced by interaction of screw dislocation with nanoscale inclusion
Materials Letters 62 (2008) 3521–3523
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / m a t l e t
Contribution to critical shear stress of nanocomposites produced by interaction of screw dislocation with nanoscale inclusion Q.H. Fang a,b,⁎, Y. Liu a, B.Y. Huang a, Y.W. Liu b, P.H. Wen c a b c
State Key Laboratory for Powder Metallurgy, Central South University, Changsha 410083, PR China College of Mechanics and Aerospace, Hunan University, Changsha, 410082, PR China Department of Engineering, Queen Mary, University of London, London, EI 4NS, UK
A R T I C L E
I N F O
Article history: Received 2 November 2007 Accepted 14 March 2008 Available online 20 March 2008 Keywords: Defects Composite materials Screw dislocation Nanoscale inclusion Interface stress
A B S T R A C T The contribution to the critical shear stress of nanocomposites caused by the interaction between screw dislocations and nanoscale circular cylindrical inclusions with interface stresses is derived by means of the Mott and Nabarro's model. The effect of the radius of the nanoscale inclusion and the volume fraction of inclusions as well as the interface stress on the critical shear stress is examined. The most important finding is that, when the negative interface stress is considered, a critical value of the radius of the nanoscale inclusion or the volume fraction of inclusions may exist to gain the best strengthening effects for the secondphase strengthened composites which differs from the classical solution without considering the interface stress under same external conditions. © 2008 Elsevier B.V. All rights reserved.
1. Introduction For the understanding of strengthening mechanisms in a number of alloys and composites, the elementary interaction between a single inclusion and a dislocation on the nearby slip plane will be first considered which leads to an interaction force acting on the dislocation. Considering its importance, the interaction of a dislocation (screw or edge dislocation) with an inclusion had received much attention during the last several decades (see, for example, [1–8] and references cited therein). In the literature a great number of possible strengthening mechanisms had been discussed [9]. The modulus mismatch strengthening is an important strengthening mechanism which arises from the difference of elastic moduli in the matrix and the inclusion. Russell and Brown [10] gave an approximate expression for the critical resolved shear stress of the material strengthened by a modulus difference between matrix and precipitate. The interaction force acting on the dislocation due to the interaction between a spherical precipitate and a dislocation was calculated numerically by Nembach [11]. From the maximum and the range of this interaction force, the critical resolved shear stress was also derived. In the present paper, using the obtained complex potential of the interaction between a screw dislocation and a nanoscale cylindrical inclusion with interface stress [12], the interaction energy and the ⁎ Corresponding author. State Key Laboratory for Powder Metallurgy, Central South University, Changsha 410083, PR China. Tel./fax: +86 731 8822330. E-mail address: [email protected] (Q.H. Fang). 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.03.037
interaction force for this interaction are calculated. The contribution to the critical shear stress of the composite materials due to this interaction is derived by means of the obtained interaction force acting on the dislocation and the Mott and Nabarro's model [13]. The effect of interface stress on the critical shear stress is examined. For the interface stress model, a generic and mathematical exposition has been presented by Gurtin and Murdoch [14]. Utilizing this model, great effort has been made recently to understand some unusual phenomena related to the interface stress in nanocomposites [15–18]. These studies indicate that the interface effect is a critical factor in the physical behavior of the materials containing inclusions of a sufficiently small size. 2. Formulation Consider an infinite matrix containing a circular nano-inclusion of a radius R with different shear moduli (μ1 and μ2, the subscripts “1”and “2 ” refer to the inclusion and the matrix regions respectively). A screw dislocation with Burgers vector b is assumed to be located inside the matrix at arbitrary point z0. As the present problem is an anti-plane one, following Sharma et al. [19], the boundary condition for the abutting bulk solid can be given as: τrz1(t) −τrz2(t) = [∂τsθz/(∂θ)]/ R. Where τrz and τθz are stress components in polar coordinates r and θ, the superscript s denotes the interface region and |t|= R denotes the points on the circular arc interface L. In addition, the constitutive equation for the interface region is expressed as: τsθz =2(μs −τs)εsθz. Where τsθz and εsθz denote interfacial stress
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Q.H. Fang et al. / Materials Letters 62 (2008) 3521–3523
and strain, μs is the elastic constant of the interface and τs is residual interface tension. According to the work of Sharma et al. [19], the dimension of the interface elastic constant μs is N/m. For a coherent interface, the interfacial strain εsθz equals to the associated tangential strain in the abutting bulk materials. With a semi-coherent or incoherent interface, an additional measure of the interfacial strain is required [20]. In the following, we will study the case of a coherent interface. Referring to the work of Muskhelishvili [21], in the bulk solid, shear stresses τrz and τθz can be written in terms of an analytical function Φ(z) of the complex variable z = reiθ as follows srz ishz ¼ Aeih UðzÞ
ð1Þ
For the present problem, using Riemann–Schwarz's symmetry principle integrated with Laurent series expansion technique, we obtain the analytical function Φ(z) in the matrix region. b 1 1 1 UðzÞ ¼ þ P 2pi z z0 z ðR2 =z0 Þ z 2 kþ1 l b X 2A2 R zk2 jzjNR P s s 2pi k¼0 A1 þ A2 þ ð1 þ kÞðA s Þ=R z0 ð2Þ The components of the stress in the matrix region can be derived from Eq. (1) together with Eq. (2). The interaction energy can be formally written as [22] W¼
A2 b R Im KUðzÞdz 2
Fig. 2. The critical shear stress Δτc0 as a function of the volume fraction f with different values of intrinsic length α = μs / μ2 for μ1 / μ2 = 1.2, R = 5 nm and Burgers vector b = 0.25 nm.
The average distance of the screw dislocation in the matrix from the nearest of them is half the distance between inclusions or r0 = (N− 1/2) / 2. Thus if f =NπR2 is the volume fraction of inclusions, the mean force Frm acting on the screw dislocation can be obtained Frm ¼
" kþ1 # l X A2 b2 4f 3=2 2A2 1=2 4f f p Rp3=2 p 4f k¼0 A1 þ A2 þ ð1 þ kÞðAs ss Þ=R ð6Þ
ð3Þ
where “Im” stands for the imaginary part of a complex quantity, Λ represents the part of the complex potential after the dislocation singularity has been removed and Φ(z) is given by Eq. (2). The interaction energy for a screw dislocation in the point z0 = r0eiθ can be found by substituting Eq. (2) into Eq. (3). This leads to " 2kþ2 # l X r2 A b2 1 2A2 R ln 2 0 2 W¼ 2 s ss Þ=R r ð ð Þ 1 þ k 4p A þ A þ 1 þ k A r0 R 0 1 2 k¼0 ð4Þ The force Fr acting on the screw dislocation is defined as the negative gradient of the interaction energy with respect to the position of the dislocation, thus AW Ar0" !# l X A2 b2 R2 2A2 R2kþ2 ¼ 2p r0 r02 R2 A1 þ A2 þ ð1 þ kÞðAs ss Þ=R r02kþ3 k¼0
Fr ¼
ð5Þ According to the idea of the Mott and Nabarro's model, consider the composites containing N inclusions per unit area, each of radius R.
Referring to the work of Nembach [9], the critical shear stress caused by this elastic interaction (or the contribution to the critical shear stress of the material) can be derived from the mean force. Dsc ¼
" kþ1 # l X A2 b2 4f 3=2 2A2 1=2 4f f p Rp3=2 p 4f k¼0 A1 þ A2 þ ð1 þ kÞðAs ss Þ=R ð7Þ
The effect of the radius R of the inclusion and the volume fraction of inclusions f as well as the interface stress upon the contribution to the critical shear stress can be investigated through Eq. (7). 3. Results and discussion In subsequent numerical results we set the residual interface tension τs = 0 and Burgers vector b = 0.25 nm. For the description of the interface one needs the constant μs, according to the calculation results in Miller and Shenoy [23], the absolute value of the intrinsic length α = μs / μ2 is nearly 0.1 nm. The normalized critical shear stress Δτc0 is defined as Δτc0 = (Δτc ⁎ 104) / μ2. The normalized critical shear stress Δτc0 is plotted as a function of the radius R in Fig. 1 with different values of α for relative shear modulus of the inclusion and the matrix μ1/μ2 = 1.2. It can be found that the contribution to the critical shear stress will increase with the decrease of the inclusion radius R when the intrinsic length α = μs / μ2 = 0.1 nm and the volume fraction f = 0.1. However, if α = −0.1 nm and f = 0.1, the contribution to the critical shear stress first increases then decreases with the decrease of the inclusion radius R. In this case, there is a critical value of the inclusion radius R to determine the maximal contribution to the critical shear stress of the material when the volume fraction of the nano-inclusions keeps a constant. The phenomenon cannot be predicted by the classical solutions without considering the interface effects. The variation of the normalized critical shear stress Δτc0 with respect to the volume fraction f is depicted in Fig. 2 with different values of α for R = 5 nm and μ1 / μ2 = 1.2. An interesting result from Fig. 2 is that the contribution to the critical shear stress will first increase then decrease with the increment of the volume fraction f if α = −0.1 nm and the radius of the nano-inclusion is a constant. There also exists a critical volume fraction f to obtain the maximal contribution to the critical shear stress of the material.
4. Conclusions
Fig. 1. The critical shear stress Δτc0 as a function of the radius R with different values of intrinsic length α = μs / μ2 for μ1 / μ2 = 1.2, f = 0.1 and Burgers vector b = 0.25 nm.
In this paper, we study the influence of the interaction between a screw dislocation and a nanoscale inclusion with interface stress upon the mechanical characteristics of the materials. The interaction is produced by the difference in the elastic constants between the matrix and the inclusion as well as interface effects. The interaction
Q.H. Fang et al. / Materials Letters 62 (2008) 3521–3523
energy, the interaction force and the critical shear stress due to this interaction are calculated. The numerical results show that, if the volume fraction of the inclusions keeps a constant and the interface stress is negative, there is a critical value of the nanoscale inclusion radius to determine the maximal contribution to the critical shear stress under some conditions in nanocomposites. Additionally, if the radius of the inclusion keeps a constant and the interface stress is negative, there also exists a critical volume fraction of the inclusions to determine the maximal contribution to the critical shear stress of the material. These results indicate that the interface stress is a critical factor in the mechanical behavior of the materials containing nanoscale inclusions. Acknowledgements This project was supported by the Postdoctoral Science Foundation of Central South University, China Postdoctoral Science Foundation and National Natural Science Foundation of China. References [1] F.R.N. Nabarro (Ed.), Dislocation in Solids, Vol. 4, 1979, New York; North-Holland. [2] J. Dundurs, T. Mura, J. Mech. Phys. Solids 12 (1964) 177–189. [3] E. Smith, Int. J. Eng. Sci. 6 (1968) 129–143.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
3523
L. Stagni, R.J. Lizzio, Appl. Phys. A 30 (1983) 217–221. H.A. Luo, Y. Chen, ASME J. Appl. Mech. 58 (1991) 75–86. Z.M. Xiao, B.J. Chen, Int. J. Solids. Struct. 38 (2001) 2533–2548. Q.H. Fang, Y.W. Liu, C.P. Jiang, Phys. Status Solidi (b) 242 (2005) 2775–2794. Q.H. Fang, Y.W. Liu, Acta Mater. 54 (2006) 4213–4220. E. Nembach, Particle Strengthening of Metals and Alloys, Wiley, New York, 1996. K.C. Russell, L.M. Brown, Acta Metall. 20 (1972) 969–974. E. Nembach, Phys. Status Solidi (a) 78 (1983) 571–581. Q.H. Fang, Y.W. Liu, Scripta Mater. 55 (2006) 99–102. N.F. Mott, F.R.N. Nabarro, Proc. Phys. Soc. 52 (1940) 86–89. M.E. Gurtin, A.I. Murdoch, Arch. Ration. Mech. Anal. 57 (1975) 291–323. P. Sharma, S. Ganti, Phys. Status Solidi (b) 234 (2002) R10–R12. H.L. Duan, J. Wang, Z.P. Huang, B.L. Karihaloo, J. Mech. Phys. Solids 53 (2005) 1574–1596. C.W. Lim, Z.R. Li, L.H. He, Int. J. Solids Struct. 43 (2006) 5055–5065. T. Chen, G.J. Dvorak, Appl. Phys. Lett. 88 (2006) 211912–211915. P. Sharma, S. Ganti, N. Bhate, Appl. Phys. Lett. 82 (2003) 535–537. R.C. Cammarata, K. Sieradzki, F. Spaepen, J. Appl. Phys. 87 (2000) 1227–1234. N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Leyden, Noordhoff, 1975. S.X. Gong, S.A. Meguid, Int. J. Eng. Sci. 32 (1994) 1221–1228. R.E. Miller, V.B. Shenoy, Nanotechnology 11 (2000) 139–147.
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / m a t l e t
Contribution to critical shear stress of nanocomposites produced by interaction of screw dislocation with nanoscale inclusion Q.H. Fang a,b,⁎, Y. Liu a, B.Y. Huang a, Y.W. Liu b, P.H. Wen c a b c
State Key Laboratory for Powder Metallurgy, Central South University, Changsha 410083, PR China College of Mechanics and Aerospace, Hunan University, Changsha, 410082, PR China Department of Engineering, Queen Mary, University of London, London, EI 4NS, UK
A R T I C L E
I N F O
Article history: Received 2 November 2007 Accepted 14 March 2008 Available online 20 March 2008 Keywords: Defects Composite materials Screw dislocation Nanoscale inclusion Interface stress
A B S T R A C T The contribution to the critical shear stress of nanocomposites caused by the interaction between screw dislocations and nanoscale circular cylindrical inclusions with interface stresses is derived by means of the Mott and Nabarro's model. The effect of the radius of the nanoscale inclusion and the volume fraction of inclusions as well as the interface stress on the critical shear stress is examined. The most important finding is that, when the negative interface stress is considered, a critical value of the radius of the nanoscale inclusion or the volume fraction of inclusions may exist to gain the best strengthening effects for the secondphase strengthened composites which differs from the classical solution without considering the interface stress under same external conditions. © 2008 Elsevier B.V. All rights reserved.
1. Introduction For the understanding of strengthening mechanisms in a number of alloys and composites, the elementary interaction between a single inclusion and a dislocation on the nearby slip plane will be first considered which leads to an interaction force acting on the dislocation. Considering its importance, the interaction of a dislocation (screw or edge dislocation) with an inclusion had received much attention during the last several decades (see, for example, [1–8] and references cited therein). In the literature a great number of possible strengthening mechanisms had been discussed [9]. The modulus mismatch strengthening is an important strengthening mechanism which arises from the difference of elastic moduli in the matrix and the inclusion. Russell and Brown [10] gave an approximate expression for the critical resolved shear stress of the material strengthened by a modulus difference between matrix and precipitate. The interaction force acting on the dislocation due to the interaction between a spherical precipitate and a dislocation was calculated numerically by Nembach [11]. From the maximum and the range of this interaction force, the critical resolved shear stress was also derived. In the present paper, using the obtained complex potential of the interaction between a screw dislocation and a nanoscale cylindrical inclusion with interface stress [12], the interaction energy and the ⁎ Corresponding author. State Key Laboratory for Powder Metallurgy, Central South University, Changsha 410083, PR China. Tel./fax: +86 731 8822330. E-mail address: [email protected] (Q.H. Fang). 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.03.037
interaction force for this interaction are calculated. The contribution to the critical shear stress of the composite materials due to this interaction is derived by means of the obtained interaction force acting on the dislocation and the Mott and Nabarro's model [13]. The effect of interface stress on the critical shear stress is examined. For the interface stress model, a generic and mathematical exposition has been presented by Gurtin and Murdoch [14]. Utilizing this model, great effort has been made recently to understand some unusual phenomena related to the interface stress in nanocomposites [15–18]. These studies indicate that the interface effect is a critical factor in the physical behavior of the materials containing inclusions of a sufficiently small size. 2. Formulation Consider an infinite matrix containing a circular nano-inclusion of a radius R with different shear moduli (μ1 and μ2, the subscripts “1”and “2 ” refer to the inclusion and the matrix regions respectively). A screw dislocation with Burgers vector b is assumed to be located inside the matrix at arbitrary point z0. As the present problem is an anti-plane one, following Sharma et al. [19], the boundary condition for the abutting bulk solid can be given as: τrz1(t) −τrz2(t) = [∂τsθz/(∂θ)]/ R. Where τrz and τθz are stress components in polar coordinates r and θ, the superscript s denotes the interface region and |t|= R denotes the points on the circular arc interface L. In addition, the constitutive equation for the interface region is expressed as: τsθz =2(μs −τs)εsθz. Where τsθz and εsθz denote interfacial stress
3522
Q.H. Fang et al. / Materials Letters 62 (2008) 3521–3523
and strain, μs is the elastic constant of the interface and τs is residual interface tension. According to the work of Sharma et al. [19], the dimension of the interface elastic constant μs is N/m. For a coherent interface, the interfacial strain εsθz equals to the associated tangential strain in the abutting bulk materials. With a semi-coherent or incoherent interface, an additional measure of the interfacial strain is required [20]. In the following, we will study the case of a coherent interface. Referring to the work of Muskhelishvili [21], in the bulk solid, shear stresses τrz and τθz can be written in terms of an analytical function Φ(z) of the complex variable z = reiθ as follows srz ishz ¼ Aeih UðzÞ
ð1Þ
For the present problem, using Riemann–Schwarz's symmetry principle integrated with Laurent series expansion technique, we obtain the analytical function Φ(z) in the matrix region. b 1 1 1 UðzÞ ¼ þ P 2pi z z0 z ðR2 =z0 Þ z 2 kþ1 l b X 2A2 R zk2 jzjNR P s s 2pi k¼0 A1 þ A2 þ ð1 þ kÞðA s Þ=R z0 ð2Þ The components of the stress in the matrix region can be derived from Eq. (1) together with Eq. (2). The interaction energy can be formally written as [22] W¼
A2 b R Im KUðzÞdz 2
Fig. 2. The critical shear stress Δτc0 as a function of the volume fraction f with different values of intrinsic length α = μs / μ2 for μ1 / μ2 = 1.2, R = 5 nm and Burgers vector b = 0.25 nm.
The average distance of the screw dislocation in the matrix from the nearest of them is half the distance between inclusions or r0 = (N− 1/2) / 2. Thus if f =NπR2 is the volume fraction of inclusions, the mean force Frm acting on the screw dislocation can be obtained Frm ¼
" kþ1 # l X A2 b2 4f 3=2 2A2 1=2 4f f p Rp3=2 p 4f k¼0 A1 þ A2 þ ð1 þ kÞðAs ss Þ=R ð6Þ
ð3Þ
where “Im” stands for the imaginary part of a complex quantity, Λ represents the part of the complex potential after the dislocation singularity has been removed and Φ(z) is given by Eq. (2). The interaction energy for a screw dislocation in the point z0 = r0eiθ can be found by substituting Eq. (2) into Eq. (3). This leads to " 2kþ2 # l X r2 A b2 1 2A2 R ln 2 0 2 W¼ 2 s ss Þ=R r ð ð Þ 1 þ k 4p A þ A þ 1 þ k A r0 R 0 1 2 k¼0 ð4Þ The force Fr acting on the screw dislocation is defined as the negative gradient of the interaction energy with respect to the position of the dislocation, thus AW Ar0" !# l X A2 b2 R2 2A2 R2kþ2 ¼ 2p r0 r02 R2 A1 þ A2 þ ð1 þ kÞðAs ss Þ=R r02kþ3 k¼0
Fr ¼
ð5Þ According to the idea of the Mott and Nabarro's model, consider the composites containing N inclusions per unit area, each of radius R.
Referring to the work of Nembach [9], the critical shear stress caused by this elastic interaction (or the contribution to the critical shear stress of the material) can be derived from the mean force. Dsc ¼
" kþ1 # l X A2 b2 4f 3=2 2A2 1=2 4f f p Rp3=2 p 4f k¼0 A1 þ A2 þ ð1 þ kÞðAs ss Þ=R ð7Þ
The effect of the radius R of the inclusion and the volume fraction of inclusions f as well as the interface stress upon the contribution to the critical shear stress can be investigated through Eq. (7). 3. Results and discussion In subsequent numerical results we set the residual interface tension τs = 0 and Burgers vector b = 0.25 nm. For the description of the interface one needs the constant μs, according to the calculation results in Miller and Shenoy [23], the absolute value of the intrinsic length α = μs / μ2 is nearly 0.1 nm. The normalized critical shear stress Δτc0 is defined as Δτc0 = (Δτc ⁎ 104) / μ2. The normalized critical shear stress Δτc0 is plotted as a function of the radius R in Fig. 1 with different values of α for relative shear modulus of the inclusion and the matrix μ1/μ2 = 1.2. It can be found that the contribution to the critical shear stress will increase with the decrease of the inclusion radius R when the intrinsic length α = μs / μ2 = 0.1 nm and the volume fraction f = 0.1. However, if α = −0.1 nm and f = 0.1, the contribution to the critical shear stress first increases then decreases with the decrease of the inclusion radius R. In this case, there is a critical value of the inclusion radius R to determine the maximal contribution to the critical shear stress of the material when the volume fraction of the nano-inclusions keeps a constant. The phenomenon cannot be predicted by the classical solutions without considering the interface effects. The variation of the normalized critical shear stress Δτc0 with respect to the volume fraction f is depicted in Fig. 2 with different values of α for R = 5 nm and μ1 / μ2 = 1.2. An interesting result from Fig. 2 is that the contribution to the critical shear stress will first increase then decrease with the increment of the volume fraction f if α = −0.1 nm and the radius of the nano-inclusion is a constant. There also exists a critical volume fraction f to obtain the maximal contribution to the critical shear stress of the material.
4. Conclusions
Fig. 1. The critical shear stress Δτc0 as a function of the radius R with different values of intrinsic length α = μs / μ2 for μ1 / μ2 = 1.2, f = 0.1 and Burgers vector b = 0.25 nm.
In this paper, we study the influence of the interaction between a screw dislocation and a nanoscale inclusion with interface stress upon the mechanical characteristics of the materials. The interaction is produced by the difference in the elastic constants between the matrix and the inclusion as well as interface effects. The interaction
Q.H. Fang et al. / Materials Letters 62 (2008) 3521–3523
energy, the interaction force and the critical shear stress due to this interaction are calculated. The numerical results show that, if the volume fraction of the inclusions keeps a constant and the interface stress is negative, there is a critical value of the nanoscale inclusion radius to determine the maximal contribution to the critical shear stress under some conditions in nanocomposites. Additionally, if the radius of the inclusion keeps a constant and the interface stress is negative, there also exists a critical volume fraction of the inclusions to determine the maximal contribution to the critical shear stress of the material. These results indicate that the interface stress is a critical factor in the mechanical behavior of the materials containing nanoscale inclusions. Acknowledgements This project was supported by the Postdoctoral Science Foundation of Central South University, China Postdoctoral Science Foundation and National Natural Science Foundation of China. References [1] F.R.N. Nabarro (Ed.), Dislocation in Solids, Vol. 4, 1979, New York; North-Holland. [2] J. Dundurs, T. Mura, J. Mech. Phys. Solids 12 (1964) 177–189. [3] E. Smith, Int. J. Eng. Sci. 6 (1968) 129–143.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
3523
L. Stagni, R.J. Lizzio, Appl. Phys. A 30 (1983) 217–221. H.A. Luo, Y. Chen, ASME J. Appl. Mech. 58 (1991) 75–86. Z.M. Xiao, B.J. Chen, Int. J. Solids. Struct. 38 (2001) 2533–2548. Q.H. Fang, Y.W. Liu, C.P. Jiang, Phys. Status Solidi (b) 242 (2005) 2775–2794. Q.H. Fang, Y.W. Liu, Acta Mater. 54 (2006) 4213–4220. E. Nembach, Particle Strengthening of Metals and Alloys, Wiley, New York, 1996. K.C. Russell, L.M. Brown, Acta Metall. 20 (1972) 969–974. E. Nembach, Phys. Status Solidi (a) 78 (1983) 571–581. Q.H. Fang, Y.W. Liu, Scripta Mater. 55 (2006) 99–102. N.F. Mott, F.R.N. Nabarro, Proc. Phys. Soc. 52 (1940) 86–89. M.E. Gurtin, A.I. Murdoch, Arch. Ration. Mech. Anal. 57 (1975) 291–323. P. Sharma, S. Ganti, Phys. Status Solidi (b) 234 (2002) R10–R12. H.L. Duan, J. Wang, Z.P. Huang, B.L. Karihaloo, J. Mech. Phys. Solids 53 (2005) 1574–1596. C.W. Lim, Z.R. Li, L.H. He, Int. J. Solids Struct. 43 (2006) 5055–5065. T. Chen, G.J. Dvorak, Appl. Phys. Lett. 88 (2006) 211912–211915. P. Sharma, S. Ganti, N. Bhate, Appl. Phys. Lett. 82 (2003) 535–537. R.C. Cammarata, K. Sieradzki, F. Spaepen, J. Appl. Phys. 87 (2000) 1227–1234. N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Leyden, Noordhoff, 1975. S.X. Gong, S.A. Meguid, Int. J. Eng. Sci. 32 (1994) 1221–1228. R.E. Miller, V.B. Shenoy, Nanotechnology 11 (2000) 139–147.