## Abstract

In this work, a rigorous study is presented for the problem associated with a circular inclusion embedded in an infinite matrix in finite plane elastostatics where both the inclusion and matrix are comprised a harmonic material. The inclusion/matrix boundary is treated as a circumferentially inhomogeneous imperfect interface that is described by a linear spring-type imperfect interface model where in the tangential direction, the interface parameter is infinite in magnitude and in the normal direction, the interface parameter is finite in magnitude (the so-called non-slip interface condition). Through the repeated use of the technique of analytic continuation, the boundary value problem for four analytic functions is reduced to solve a single first-order linear ordinary differential equation with variable coefficients for a single analytic function defined within the inclusion. The unknown coefficients of said function are then found via various analyticity requirements. The method is illustrated, using a specific example of a particular class of inhomogeneous non-slip imperfect interface. The results from these calculations are then contrasted with the results from the homogeneous imperfect interface. These comparisons indicate that the circumferential variation of interface damage has a pronounced effect on the average boundary stress.

## 1. Introduction

Inclusion problems in linear elasticity have been studied extensively over the past 70 years. Research of linearly elastic inclusion problems ranges from the fundamental works of Eshelby [1] and Muskhelishvili [2] which used perfect bonding conditions and ellipsoidal geometries, to the introduction of arbitrary inclusion geometries [3–6], imperfect bonding models (as an example, see [7]) and complex interphase models (as an example, see [8]) in more recent years. On the contrary, inclusion problems, in finite elasticity, specifically harmonic materials, have not seen the same degree of interest nor success in research. The fundamental works of Fritz [9], Ogden & Isherwood [10], Varley & Cumberbach [11] and Knowles & Sternberg [12] laid a foundation for work in finite deformations but it was not until Ru [6] developed a more tractable form of the complex variable formulation for harmonic materials that research into inclusion problems experienced rapid growth. Following Ru's work, the finite elastic problems of elliptical inclusions with uniform internal stress fields [13], partially debonded circular inclusions [14], harmonic three phase circular inclusions [15], three phase elliptical inclusions with uniform hydrostatic stress [3], three phase inclusions of arbitrary shape with uniform hydrostatic stress [4], a circular inclusion with homogeneous imperfect interface [16] and most recently a circular inclusion with a circumferentially inhomogeneous imperfect interface [17] have been studied. Of particular interest is the case of an inhomogeneous imperfect interface as it likely that for many real-world scenarios a homogeneous imperfect interface is not a realistic assumption. This assertion is supported, in part, by the previous work [17] where it was concluded that for a specific class of inhomogeneous imperfect interface, where the same degree of imperfection is realized in both the normal and tangential component directions of the boundary curve, the homogeneous model is insufficient in accurately predicting even the mean stress with up to 60% error (for a specific loading scenario).

The objective of this study was to model the case of the so-called non-slip (rough) imperfect interface condition. The linear spring model for an imperfect interface (where the interphase layer is modelled as a two-dimensional curve of vanishing thickness and the material properties of the interphase layer are given as spring-type interface parameters) will be employed where, in contrast to [17], we assume that the interfacial bonding is characterized by *m*(*θ*)=finite and *m*(*θ*) and *n*(*θ*) are the normal and tangential imperfect interface parameters, respectively, and *θ* is the polar angle). Physically, this type of imperfect interface condition is thought to be a potential by-product of manufacturing techniques where an abnormal degree of surface roughness along the interphase layer can occur. The presence of asperities and/or interdigitations allows for no relative shear displacement along the entire interface, but a certain relative normal displacement which is linearly proportional to the corresponding traction vector is permitted across the interface (i.e. a mechanical interlock is formed). Use of the linear spring model for the case of finite deformation is suitable for applications where type 1 harmonic materials are considered as it was demonstrated by Varley *et al.* [11] that for type 1 harmonic materials the differences in the principal Piola stresses are linearly proportional to the difference in the stretch ratios.

The work begins with §2 where the fundamental equations of type 1 harmonic materials are presented. Following §2, §3 discusses the formulation of the problem and the non-slip boundary conditions where eventually a first-order linear ordinary differential equation with variable coefficients is developed for the inclusion function. Section 4 illustrates the analysis for a specific class of imperfect interface and in §5, an example is given for the average mean stress on the boundary and the result is compared with the corresponding homogeneous imperfect interface. In §6, the solution for a homogeneously imperfect interface is presented in a more tractable form, and finally in §7, some results are presented.

## 2. Mathematical preliminaries

Consider a single simply connected domain bounded by a continuous circular curve ∂*D*_{1}, embedded in an infinite matrix in *x*_{1}*x*_{2} plane. Let *z*=*x*_{1}+*ix*_{2} be the Lagrangian coordinates of a particle in the reference configuration, and let *w*(*z*)=*y*_{1}(*z*)+*iy*_{2}(*z*) be the Eulerian coordinates of a particle in the current configuration. The inclusion is denoted by *D*_{1} and endowed with material properties *μ*_{1},*α*_{1},*β*_{1}. The matrix is denoted by domain *D*_{2} with material properties *μ*_{2},*α*_{2},*β*_{2}, where *W*(*I*,*J*) defined as follows
*I* and *J* are the scalar invariants of **FF**^{T} and are given by
*λ*_{1},*λ*_{2} are the principal stretches. According to Ru [6], the deformation map and the Piola stress function can be given in terms of two complex potential functions *ϕ*_{k}(*z*) and *ψ*_{k}(*z*) as follows
_{,j} represents differentiation with respect to the *j*th coordinate. Equation (2.4) may be transformed into polar coordinates if we allow for the assumption that either the shear components of the Cauchy stress tensor are zero or the principal stretches of the deformation gradient *λ*_{1},*λ*_{2} are equal, either of which implies that the Piola stress is symmetric. The primary motivation of either assumption is that symmetry of the Piola stress greatly simplifies the expressions for the traction in a polar coordinate setting. The polar form of (2.4) is then given by
*z*.

## 3. Formulation

Assuming that the inclusion is imperfectly bonded to the matrix along ∂*D*_{1}, using the notation of [18], the general imperfect interface conditions are given by
*m*(*θ*) and *n*(*θ*) are two non-negative imperfect interface parameters, and ∥.∥=(.)_{2}−(.)_{1} is the quantitative jump across ∂*D*_{1}. It is assumed that the potential functions *ϕ*_{2}(*z*) and *ψ*_{2}(*z*) exhibit the following asymptotic behaviour as *A* and *B* are complex constants that reflect the far-field loading and are given by [17]
*O*(1) are some first-order constant terms. Furthermore, it is assumed that the potentials *ϕ*_{k}(*z*) and *ψ*_{k}(*z*), *k*=1,2 admit the following series expansions

### Remark 3.1

From (3.5), we require that *X*_{1}≠0 for |*z*|≤*R* and *A*≠0 for |*z*|≥*R*. These assumptions guarantee that

Given that (2.5) is contingent on having a symmetric Piola stress, it is reasonable to assume that both *X*_{0},*Y* _{0}=0, and hence the continuity of traction condition from (3.1) gives
*Γ*=*μ*_{1}/*μ*_{2} into the above yields
*z*∈*D*_{1}, and the RHS is analytic for *z*∈*D*_{2}. Recalling the results of performing analytic continuation on (3.7), from [17], we have

Thus, the problem is now reduced to determine two unknown analytic functions *ϕ*_{1}(*z*) and *ϕ*_{2}(*z*) complying with the interface condition and the asymptotic condition for *ϕ*_{2}(*z*).

### (a) Solution for homogeneous imperfect interface

In this section, we briefly examine the homogeneous imperfect interface where the parameters *m* and *n* appearing in (3.1) are assumed to be constant along ∂*D*_{1}. Although a similar problem has been investigated by Wang in [16], for ease of comparison, we present a solution that is more amenable to validating this work.

The general form of the imperfect interface condition is given as
*A* is purely imaginary, in [16].

Noting that as either *χ*_{2}(*z*) and *w*_{k}(*z*), *k*=1,2 into (3.16) and comparing coefficients of like powers of *z* on the LHS and RHS gives the following

If we then input the compatibility condition from (3.13) into (3.17) for the case of *X*_{1}

### (b) Circumferentially inhomogeneous imperfect interface

The non-slip imperfect interface boundary conditions can be written in the following form
*z*=0. Similarly, we observe that the right-hand side of (3.25) has the following asymptotic behaviour as *D*(*z*) defined as follows (noting that *D*(*z*) given by (3.27) is well defined and analytic in the entire plane, Louisville's theorem states that *D*(*z*) must be a constant. In fact, because *D*(*z*)=0 throughout the entire plane and we arrive at the following two equations

Let us now consider the radial stress–displacement condition of (3.25) which we shall rewrite (from the matrix side) as
*ϕ*_{1}(*z*). Unlike the homogeneous imperfect interface condition examined in §3a where the conventional power series method led to a finite form solution, the variable parameter *m*(*θ*), in the present case, prevents the exact solution of (3.33) when the power series is adopted. To overcome this obstacle, the technique of analytic continuation is employed to reduce (3.33) to a first-order linear ordinary differential equation with variable coefficients for *ϕ*_{1}(*z*). In doing so, the finite form of the solution for a circumferentially inhomogeneous non-slip interface can be obtained.

First, let us introduce, for convenience, a new interface parameter *δ*(*θ*) to replace *m*(*θ*) defined as follows
*f*(*θ*) is a real and 2*π* periodic function on ∂*D*_{1}. Note that as *f*(*θ*) is periodic on ∂*D*_{1}, it admits a Fourier-series expansion as follows
*s* is non-negative natural number and *a*_{k},*b*_{k},(*k*=1,2,…,*s*) are given real constants. Moreover, on the interface ∂*D*_{1}, we can rewrite (3.35) as a function of the complex variable *z* as follows

### (c) The differential equation for *ϕ*_{1}(*z*)

To derive the differential equation for *ϕ*_{1}(*z*), we rearrange and rewrite (3.33) using (3.34) and 3.36 as follows

In (3.37), the left-hand side is analytic in *D*_{1,} and the right-hand side is analytic in *D*_{2} except for possibly the points *z*=0 and *L*(*z*) is given by
*D*_{1} and *D*_{2} including the point at infinity where it approaches zero. Thus, by Liouville's theorem, we conclude that *E*(*z*) is identically equal to zero. Hence, we obtain the following two equations
*ϕ*_{1}(*z*) determined from (3.43) must be compatible with that obtained from (3.44). It can be readily shown by allowing *D*_{1}(*z*), (3.43) can be rearranged to yield
*P*(*z*) is given by
*ϕ*_{1}(*z*). The general solution of which is given by
*z*_{I} is an arbitrary point in *D*_{1} and *C*_{0} is an arbitrary constant of integration.

Because the right-hand side in (3.49) contains the (*s*+1) undetermined coefficient *X*_{k},(*k*=1,2,…*s*+1) any admissible solution *ϕ*_{1}(*z*) of (3.49) must satisfy the following consistency condition
*ϕ*_{1}(*z*) has a Taylor series expansion in *D*_{1} given by
*z*, we arrive at the following
*s*. However, for the case of *X*_{1} and implies (3.51) is not automatically satisfied for *k*=1. Hence, we must impose the additional requirement that

In general, the solution for *ϕ*_{1}(*z*) captured by (3.49) will not be holomorphic in the uncut region *D*_{1} owing to the presence of multi-valued logarithmic functions resulting from the integration of *e*^{T(z)}*P*(*z*) and isolated singular points from the zeros of the interface function 1+*f*(*z*). In order to ensure the holomorphicity of *ϕ*_{1}(*z*), the domain *D*_{1} must be cut appropriately such that *ϕ*_{1}(*z*) is bounded at all isolated singular points and continuous across all branch cuts.

## 4. A specific class of inhomogeneous interface

To illustrate an example, we shall consider a specific form of the interface function *δ*(*θ*) as follows
*s*
*s* roots of (4.2), *s* will lie inside *D*_{1} and the remaining *s* will lie in *D*_{2}. Let the *s* roots inside *D*_{1} be denoted by
*ρ** is real and given by
*ρ**<1, and the remaining *s* roots in *D*_{2} are given by 1/*ρ*_{1},1/*ρ*_{2},…,1/*ρ*_{s}. As a consequence of the above interface definitions, we make note of the following
*s* branch points. In addition, to ensure the boundedness of *ϕ*_{1}(*z*) at *z*=*Rρ*_{k}, we set *C*_{0}=0, and we also require that
*ϕ*_{1}(*z*) at any of the potential isolated singular points *Rρ*_{k}, *k*=2,3,…,*s* in *D*_{1}. Additionally, by taking the difference
*z*^{+} denotes values of *z* from above the branch cut and *z*^{−} denotes values of *z* below the branch cut. We may prove that (4.6) is continuous across any of the *s* branch cuts by noting that, owing to the sign change of the exponents in and outside of the integral, any increments in the multivalued logarithmic terms that will arise from inside the integral will be nullified from which (4.8) is easily confirmed. The remaining irregular point to be considered is when *z*=0. Closer inspection of (4.6) reveals that there are three cases to be considered as

### (a) Case one: Ω > 1 2

When *ϕ*_{1}(*z*), we must ensure that *ϕ*_{1}(*z*) is continuous across the branch cut formed from *z*=*Rρ** along the real axis inside *D*_{1}. Closer inspection of (4.6) reveals the presence of an unintegrable singularity at *z*=0. Hence, we must define a new path of integration, *L**, to skirt around a neighbourhood of *z*=0 and set *z*=*z**, where *z** is any particular point on the branch cut from *z*=0, to compensate for this change. In this way, the continuity condition becomes
*X*_{s+1} unknown coefficients using (3.30), (3.45), (4.9) and in the case of *s*>1, (4.7).

### (b) Case two: Ω < 1 2

For this case, we shall rewrite (4.6) in the form
*X*_{0}=0, the LHS of (4.10) is analytic within *D*_{1}. As a consequence, *ϕ*_{1}(*z*)/(*z*/*R*) must be bounded at *z*=0 and because *t*/*R*)^{−2Ω} for *ρ** and is thusly integrable along such a domain [20]. We may then solve for the *X*_{s+1} unknown coefficients using (3.30), (3.45), (4.11) and in cases of *s*>1, (4.7).

### (c) Case three: Ω = 1 2

In this case, from (4.6), we see that *z*=0 is not a singular point of *ϕ*_{1}(*z*) and hence *ϕ*_{1}(0)=0. The consistency condition of (3.54) gives
*s*+1 unknown coefficients are then determined from (3.30), (3.45), (4.13) and in the case of *s*>1, (4.7).

## 5. Example

For ease of analysis in illustrating the method, we shall assume that *s*=1. From these preliminaries, we may evaluate (4.6) as

The coefficients *X*_{1},*X*_{2} in (5.3) and (5.4), we may then calculate the stress in the inclusion via (2.4) or (2.5).

We must now validate the expressions given in (5.3) and (5.4) for *X*_{1} and *X*_{2}. This is done in part by considering the case where *δ*_{0}/2=*mR*/2*μ*_{2}, (5.5) is identical to (3.22).

## 6. Results

Having verified the formulation, we may now proceed to compare the homogeneous imperfect interface to the inhomogeneous one. For the purpose of this example, we will compare the inhomogeneous interface of the form
*C*_{∂D1} is the circumference of the boundary ∂*D*_{1}. If we compute the ratio of the inhomogeneous version of (6.3) to the homogeneous version using the corresponding definitions of *X*_{1} for the inhomogeneous and homogeneous cases given in (5.3) and (3.22), then we observe the trends shown in figure 2. Figure 2 clearly demonstrates that the inhomogeneous interface parameter *ρ** has a significant effect on the estimation of the average mean stress on the inclusion boundary and at its peak reaches an error of 80%. In the linear analogue, Sudak *et al.* [21] reported an error of up to 400% between the inhomogeneous and homogeneous cases. While this work does not reach errors of a similar magnitude, these results clearly demonstrate that when analysing a circular inclusion in finite elasticity with non-slip interfacial boundary conditions, the traditional homogeneous imperfect interface model is, in general, not sufficient when calculating the average mean stress on the boundary.

## 7. Conclusion

A general solution has been developed for the case of an inhomogeneous imperfect interface with so-called non-slip boundary conditions which is captured by setting *m*(*θ*)=finite and *ρ**. From the results, it was observed that the imperfect interface parameter *ρ** has a significant impact on the average boundary stress when compared with the homogeneous case and, when the case of a non-slip boundary is warranted, the homogeneous imperfect interface model is insufficient.

## Authors' contributions

D.R.M carried out the derivation in consultation with L.J.S, who conceived the problem. Both authors contributed to editing and final approval for submission.

## Competing interests

There are no competing interests.

## Funding

This work is supported by the Natural Sciences and Engineering Research Council of Canada through NSERC grant no. 249516.

## Acknowledgements

We thank the reviewers for providing valuable comments on the work.

- Received April 25, 2016.
- Accepted May 27, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.