## Abstract

We propose an elegant and concise general method for the solution of a problem involving the interaction of a screw dislocation and a nano-sized, arbitrarily shaped, elastic inhomogeneity in which the contribution of interface/surface elasticity is incorporated using a version of the Gurtin–Murdoch model. The analytic function inside the arbitrarily shaped inhomogeneity is represented in the form of a Faber series. The real periodic function arising from the contribution of the surface mechanics is then expanded as a Fourier series. The resulting system of linear algebraic equations is solved through the use of simple matrix algebra. When the elastic inhomogeneity represents a hole, our solution method simplifies considerably. Furthermore, we undertake an analytical investigation of the challenging problem of a screw dislocation interacting with two closely spaced nano-sized holes of arbitrary shape in the presence of surface stresses. Our solutions quite clearly demonstrate that the induced elastic fields and image force acting on the dislocation are indeed size-dependent.

## 1. Introduction

The elastic interaction between dislocations and fibres (often represented as elastic inhomogeneities) is a classical topic in micromechanics [1]. When the diameters of the fibres are in the nanoscale range (e.g. less than 100 nm), surface/interface stresses, surface tension and surface energies become significant when formulating and analysing the corresponding interaction problem [2,3].

One of the most celebrated models of surface/interface elasticity is the Gurtin–Murdoch surface model, originally proposed by Gurtin, Murdoch and co-workers [4,5] and most recently elucidated by Ru [6]. Many size-dependent phenomena present at the nanoscale have been explained by incorporating the Gurtin–Murdoch model into the description of deformation, for example, the corresponding Eshelby inclusion problem [7], the inhomogeneity problem [8–11], the crack problem [12–14], the study of dislocations in nano-composites [2,3], wave scattering phenomena at the nanoscale [15] and the effect of a nano-sized elliptical hole in the deformation of an elastic isotropic or anisotropic solid [16,17].

The present research endeavours to incorporate the Gurtin–Murdoch model of surface/interface elasticity into the description of the interaction between a screw dislocation and a nano-sized elastic inhomogeneity of arbitrary shape in the presence of remote uniform anti-plane stresses and uniform anti-plane eigenstrains imposed on the inhomogeneity. We propose and implement an elegant method based on Faber and Fourier series expansions together with the use of simple matrix algebra. The Gurtin–Murdoch model is further incorporated into the analytical study of a screw dislocation interacting with two closely spaced nano-sized holes of arbitrary shape in the presence of remote uniform anti-plane stresses.

## 2. Preliminaries and basic formulation

### (a) Bulk and interface elasticity

The equilibrium and constitutive equations of the isotropic bulk solid are given by
*i*,*j*,*k*=1,2,3; λ and *μ* are Lame constants; *σ*_{ij} and *ε*_{ij} represent, respectively, the stress and strain tensors in the bulk material; *u*_{i} is the *i*th component of the displacement vector **u** and *δ*_{ij} is the Kronecker delta.

The equilibrium conditions on the interface incorporating interface/surface elasticity can be expressed as [4,6,12–14]
*α*,*β*=1,2; *n*_{i} represents the unit normal vector of the interface, [*] denotes the jump of the quantities across the interface; *κ*_{αβ} describes the curvature tensor of the surface. In addition, the constitutive equations on the isotropic interface are given by (for detailed derivations, see [4–6])
*σ*_{0} is the surface tension, λ_{s} and *μ*_{s} are surface Lame constants and ∇_{s} is the surface gradient.

### (b) Complex variable formulation

For the anti-plane shear deformation of an isotropic elastic material, the two shear stress components *σ*_{31} and *σ*_{32}, out-of-plane displacement *w*=*u*_{3} and stress function *ϕ* can be expressed in terms of a single analytic function *f*(*z*) of the complex variable *z*=*x*_{1}+*ix*_{2} as [18]
*f*^{′}(*z*)=d*f*(*z*)/d*z*.

Let *t*_{3} be the only non-zero traction component along the *x*_{3}-direction on a boundary *L*. If *s* is the arc-length measured along *L* such that, when facing the direction of increasing *s*, the material is on the left-hand side, it is readily shown that [18]

## 3. Interaction between a screw dislocation and a nano-sized inhomogeneity of arbitrary shape

As shown in figure 1, we consider a domain in *R*^{2}, infinite in extent, containing an arbitrarily shaped elastic inhomogeneity, with its elastic properties distinct from those of the surrounding matrix. The linearly elastic materials occupying the inhomogeneity and the matrix are assumed to be homogeneous and isotropic with the associated shear moduli *μ*_{1} and *μ*_{2}, respectively. The matrix is subjected to uniform anti-plane shear stresses *b*_{z} is lodged at *z*=*z*_{0} in the matrix. We represent the matrix by the domain *S*_{2}, and assume that the inhomogeneity occupies the region *S*_{1}. The inhomogeneity–matrix interface is denoted by *L*. In what follows, the subscripts 1 and 2 (or the superscripts (1) and (2)) will refer to the regions *S*_{1} and *S*_{2}, respectively.

If we assume that the interface *L* is a coherent one (i.e. *L* can be described specifically by the equations
_{s} is the surface Laplacian operator, and

In view of equation (2.5), the expression (3.1) can be equivalently written as
*s* increases counterclockwise around *L*.

In order to solve the boundary value problem, we first introduce the following conformal mapping function
*R* is a real scaling constant and *m*_{n} *R*| can be considered as a parameter measuring the size of the inhomogeneity. The mapping function (3.3) conformally maps the region occupied by the surrounding matrix *S*_{2} onto |*ξ*|≥1 in the mapped *ξ*-plane.

The analytic function *f*_{1}(*z*) within the inhomogeneity can be expanded into the following Faber series [19,20]
*a*_{n}, *P*_{k}(*z*) is the *k*th degree Faber polynomial which can be expressed explicitly as
*β*_{k,n} are determined by the following recurrence relations [19]
*f*_{1}(*z*)=*f*_{1}(*ω*(*ξ*))=*f*_{1}(*ξ*) can be expressed as
*f*_{2}(*z*)=*f*_{2}(*ω*(*ξ*))=*f*_{2}(*ξ*), the continuity condition of displacement across the interface |*ξ*|=1 in equation (3.2)_{1} can then be expressed as
*ξ*|=1, respectively.

The singular part *f*_{2s}(*z*) of *f*_{2}(*z*) at *z*=*z*_{0} is *f*_{2}(*z*) at infinity is *f*_{2s}(*ξ*) of *f*_{2}(*ξ*) at *ξ*=*ξ*_{0}=*ω*^{−1}(*z*_{0}) is *f*_{2}(*ξ*) at infinity is *f*_{2}(*ξ*) from equation (3.9)
_{2}, the interface condition (3.2)_{2} on the interface |*ξ*|=1 can then be expressed in terms of *f*_{1}(*ξ*) and *f*_{2}(*ξ*) as
*Γ*=*μ*_{1}/*μ*_{2}.

Inserting the expressions for *f*_{1}(*ξ*) and *f*_{2}(*ξ*) from equations (3.7) and (3.10) into equation (3.11), we obtain
*γ*=(*μ*_{s}−*σ*_{0})/(|*R*|*μ*_{2}) is a size-dependent dimensionless parameter for the interface *L*, *b*_{0} is an unknown real constant which should be taken into consideration in the analysis, and
*θ* of period 2*π*, then |*ω*^{′}(*ξ*)|/|*R*| can be expanded into the following Fourier series
*ξ* in equation (3.18), we arrive at the following set of linear algebraic equations
*N*, we obtain 4*N*+1 independent linear algebraic equations for the 4*N*+1 unknowns *b*_{0}, *a*_{n}, *b*_{n}, *n*=1,2,…,*N*). These unknown coefficients can then be uniquely determined. In the following, we rewrite (3.13) and (3.19) in matrix form to facilitate the analysis. Equation (3.13) is written in matrix form as
**j**_{b}, **j**_{σ} and **j**_{ε} being three loading vectors arising from the screw dislocation, remote loading and imposed eigenstrains. These three loading vectors are explicitly given by
*b*_{0}, we arrive at
*N*-dimensional vector **a** as
**G**_{1}, **G**_{2} and **h** are defined by
**G**_{1} and **G**_{2} contain the size-dependent parameter *γ*, the vector **a** containing the coefficients in the Faber series is also size-dependent. This fact implies that the induced elastic fields both in the inhomogeneity and in the surrounding matrix are size-dependent. The above analysis also indicates that when discussing an inhomogeneity with interface stresses, there is no correspondence principle for the internal stress field between remote loading and imposed eigenstrains on the inhomogeneity unlike the case of an elliptical inhomogeneity with spring-type imperfect interface [21]. This fact can be clearly observed from the additional terms on the right-hand side of equation (3.11) which are due to imposed eigenstrains. In fact, our analysis also suggests that the correspondence principle in Shen *et al.* [21] is no longer valid for a non-elliptical inhomogeneity with a homogeneous spring-type imperfect interface.

Using the Peach–Koehler formula [1], the image force acting on the screw dislocation in the presence of remote uniform loading and imposed eigenstrains can be simply derived from equation (3.10) as
*F*_{1} and *F*_{2} are the *x*_{1} and *x*_{2} components of the image force. Equation (3.32) indicates that both the magnitude and direction of the image force are size-dependent.

## 4. Interaction between a screw dislocation and a nano-sized hole of arbitrary shape

The problem of a screw dislocation interacting with a nano-sized arbitrary shaped hole with surface stresses is also of physical relevance. The results in §3 continue to apply to the present in the limit as *L* can be written as
*ϕ* and *w* pertain to the surrounding matrix.

As a result, it follows from equations (2.4)_{2} and (4.1) that the analytic function *f*(*z*)=*f*(*ω*(*ξ*))=*f*(*ξ*) defined in |*ξ*|≥1 should satisfy the following boundary condition
*f*(*ξ*) as follows
*c*_{n},

Inserting equation (4.3) for the expression of *f*(*ξ*) into equation (4.2), we obtain
*γ*=(*μ*_{s}−*σ*_{0})/(|*R*|*μ*) is the size-dependent dimensionless parameter for the surface *L*, *d*_{0} is an unknown real constant, and
*ξ*, we can finally arrive at the following set of linear algebraic equations
**j**_{b}, **j**_{σ}, **e**, ** Λ**,

**C**and

**D**have been defined by equations (3.23)

_{1,2}and (3.26), and

*d*

_{0}, we arrive at

*N*-dimensional vector

**c**as

*ω*

^{′}(e

^{iθ})| is determined by equation (3.14), the vector

**c**can be determined; (ii) the induced stress field in the matrix and the magnitude and direction of the image force acting on the screw dislocation are size-dependent.

## 5. Interaction between a screw dislocation and two nano-sized holes of arbitrary shaped

In §4, we have discussed a screw dislocation interacting with an arbitrary shaped hole with surface stresses. Here, we further consider the more challenging problem of an infinite matrix weakened by two closely spaced arbitrary shaped holes with surfaces stresses, as shown in figure 2. The left and right surfaces are denoted by *L*_{1} and *L*_{2}, respectively. In addition, a screw dislocation with Burgers vector *b*_{z} is located at *z*=*z*_{0} in the matrix, and the matrix is also subjected to uniform anti-plane stresses

We consider the following conformal mapping function
*R* is a real scaling constant and *p*_{n}, *p*_{−n}, *ξ*|≤*ρ*. The left surface *L*_{1} is mapped to the inner unit circle |*ξ*|=1, whereas the right surface *L*_{2} is mapped to the outer circle |*ξ*|=*ρ*, *ξ*=*δ*.

*f*(*ξ*) is assumed to take the following form
*ξ*_{0}=*ω*^{−1}(*z*_{0}), *k*_{0}, *k*_{n}, *k*_{−n},

As in equation (4.1), the boundary conditions on the two surfaces *L*_{1} and *L*_{2} can be written as
*L*_{1} and *L*_{2}, respectively.

The boundary condition on *L*_{1} in equation (5.3) can be expressed in terms of *f*(*ξ*), (1≤|*ξ*|≤*ρ*) as
*L*_{2} in equation (5.4) can be expressed in terms of *f*(*ξ*), (1≤|*ξ*|≤*ρ*) as
*L*_{1}, *q*_{0} is an unknown real constant, and
*ω*^{′}(e^{iθ})|/|*R*| is a periodic real function of *θ* of period 2*π*, then it can be expanded into the following Fourier series
*ξ* in equation (5.11), we finally obtain the following set of linear algebraic equations
**e**, ** Λ**,

**C**and

**D**are identical to those given by equation (3.26), and

*L*

_{2},

*s*

_{0}is an unknown real constant, and

*ω*

^{′}(

*ρ*e

^{iθ})|/|

*R*| is a periodic real function of

*θ*of period 2

*π*, we can write the following Fourier series

*ξ*in equation (5.22), we finally obtain the following set of linear algebraic equations

*N*-dimensional vector

**k**

The analysis carried out in this section indicates that the induced stress field in the matrix is dependent on two size-dependent parameters *γ*_{1} and *γ*_{2}. It is clear that the image force acting on the screw dislocation is also dependent on the two size-dependent parameters. If the two surfaces *L*_{1} and *L*_{2} possess the same elastic properties, i.e. *γ*_{1}=*γ*_{2}.

Finally, as an illustration, we consider the specific mapping function
*α* is a complex parameter. The above mapping function describes two closely spaced holes of identical shape. It is rigorously verified that if *α* is a real number, |*ω*^{′}(e^{iθ})|/|*R*|≡*ρ*|*ω*^{′}(*ρ*e^{iθ})|/|*R*|. The condition *ω*^{′}(*ξ*)≠0 for 1≤|*ξ*|≤*ρ* should be satisfied in order to ensure that the mapping (5.32) is one to one. This condition is equivalent to the fact that all four roots of the quartic equation
*ξ*|<1 or |*ξ*|>*ρ*. If *α* is a real number, they must lie in the region bounded by the two curves shown in figure 3. It is observed that −0.5<*α*<1 no matter what value of *ρ*(>1) is chosen. Figures 4 and 5 illustrate the two identical hole shapes obtained by setting *ρ*=1.5, *α*=0.3 and *ρ*=1.5, *α*=0.63i, respectively, in equation (5.32). The two holes in figure 4 are smooth, whereas those in figure 5 have two needle-shaped tips. We show in figure 6 variations of |*ω*^{′}(e^{iθ})|/|*R*| (solid line) and its Fourier series expansion (circles) with *ρ*=1.5 and *α*=0.3 in equation (5.32). The Fourier series expansion in equation (5.19) is truncated at *n*=100. It is clear from figure 6 that the Fourier series expansion matches quite well the exact values of |*ω*^{′}(e^{iθ})|/|*R*| at most of the points except at *θ*=0.

## 6. Conclusion

In this paper, we have incorporated the effect of interface elasticity via the continuum-based Gurtin–Murdoch model into the interaction problem of a screw dislocation near an elastic inhomogeneity of arbitrary shape. The complexities of the discussed boundary value problem lie in the following aspects: (i) the elastic inhomogeneity is of arbitrary shape; (ii) the interface/surface elasticity is taken into account; (iii) the loadings include a screw dislocation in the matrix, remote uniform stresses and uniform eigenstrains imposed on the inhomogeneity. Despite these complexities, an elegant analytical solution has been obtained by means of complex variable techniques and matrix algebra. The size dependency of the elastic fields and image force on the dislocation have been clearly demonstrated from the solutions obtained. We have also incorporated the Gurtin–Murdoch model into the interaction problem of a screw dislocation in the vicinity of two closely spaced holes of irregular shape. An elegant solution to this interaction problem has been derived. The induced stress field and image force rely on two size-dependent parameters *γ*_{1} and *γ*_{2}.

## Funding statement

This work is supported by the National Natural Science Foundation of China (grant no. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

- Received April 15, 2014.
- Accepted July 4, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.