## Abstract

Imperfect bonding between the constitutive components can greatly affect the properties of composite structures. We propose an asymptotic analysis of different types of imperfect interfaces arising in the problem of conduction through a simple cubic array of spherical inclusions. The performed study is based on the two-scale asymptotic homogenization method. The microscopic problem on the unit cell is solved using the underlying principles of the boundary-shape perturbation technique. The influence of the interface properties on the effective conductivity and on the local potential and flux fields is studied.

## 1. Introduction

Physical characteristics of particle-reinforced composite materials depend crucially upon the bonding conditions at the interface between the matrix and inclusions. In real composites, properties of the interface may be subjected to various factors such as presence of a thin coating layer, chemical reactions and mechanical damages. Imperfect interfacial bonding can greatly affect the overall behaviour of a heterogeneous solid. Strengthening the interface increases effective coefficients (rigidity, conductivity, etc.) At the same time, it induces high gradients of the local fields (stresses, fluxes) that can significantly decrease the bearing capacity of the composite structure. Weakening the interfacial bonding results in lower effective moduli, but provides a more uniform redistribution of the local fields, which may help to avoid a fracture. However, in mechanics, the weak-interface response often leads to the development of dislocations and voids, which also decreases the strength of the entire solid. Deliberate tuning of the parameters of the interface allows one to achieve the optimal properties of the composite material to provide better performance and reliability.

In the present paper, we propose an asymptotic analysis of imperfect interfaces in conduction-type problems (i.e. related to electric or heat transfer, diffusion, magnetic permeability, etc.) At the perfect interface, the potential *u* and the normal flux *q* are continuous. Generally, any interface that does not meet these criteria is imperfect. In practice, there can be an unlimited variety of imperfect interfaces arising for different production and technological reasons. For a detailed review of previous work on imperfect interfaces in conductivity of heterogeneous media, we refer to Miloh & Benveniste (1999).

The basic idea of the present study is to introduce between the components (a matrix of conductivity *k*^{(1)} and inclusions of conductivity *k*^{(2)}) an intermediate coating layer of conductivity *k*^{(3)} and non-dimensional thickness *t*, derive a solution for the given three-component medium and finally set *t*→0. In the asymptotic limit, the perfect interface conditions imposed originally at the ‘matrix-coating’ and ‘inclusion-coating’ boundaries transform to imperfect interface conditions acting at the ‘matrix-inclusion’ boundary ∂*Ω*.

Let us introduce the conductivity of the coating as follows:
1.1
where *α* and *β* are the bonding constants and *t*=*b*/*a*, where *a* is the radius of the inclusion and *b* is the thickness of the coating.

The parameter *β* determines a qualitative type of asymptotic behaviour of the interface, while the parameter *α* provides a quantitative description of the interface properties. The case *β*<−1 describes a superconducting interface, when the composite acts like a material with perfectly conducting inclusions. In the region −1<*β*<1, we obtain perfect bonding conditions. The case *β*>1 corresponds to complete separation of the components, so the inclusions play a role of perfect insulators (e.g. holes). At the ‘bifurcation’ points *β*=±1, the proposed interface model ‘switches’ between different scenarios of the limiting behaviour.

Setting *β*=1 and varying the bonding constant *α*, we simulate the low-conducting (LC) interface. It describes a partial debonding (up to complete separation) of the components. At the matrix-inclusion boundary ∂*Ω*, there is a jump of the potential *u*, while the normal flux *q* is continuous. In the case of a linear interface, the jump of *u* is proportional to *q*. Such a model is often referred to as the interface with Kapitza thermal resistance (Kapitza 1965). The LC interfaces were considered by Sanchez-Palencia (1970), Benveniste & Miloh (1986), Benveniste (1987), Hasselman & Johnson (1987), Torquato & Rintoul (1995), Lipton (1996), Lipton & Vernescu (1996*a*,*b*), Cheng & Torquato (1997*a*) and Danishevs’kyy (in press). Similar bonding conditions in elasticity, when a displacement jump across the interface is proportional to the interface traction in terms of a spring-type parameter, are called soft- (or weak-) interface models. These were initiated by Goland & Reissner (1944) and Mal & Bose (1975), and further employed in many papers (Benveniste 1985; Achenbach & Zhu 1989, 1990; Hashin 1990, 1991*a*,*b*; Klarbing 1991; Geymonat *et al*. 1999; Lenci 2000; Lenci & Menditto 2000; Andrianov *et al*. 2007, 2008). Variational formulations with such imperfect bonding conditions were given by Hashin (1992) and Lipton & Vernescu (1995). Nonlinear interfaces were studied by Levy (1996, 2000) and Levy & Dong (1998).

Setting *β*=−1, we examine the high-conducting (HC) interface. The potential *u* is continuous, but the normal flux *q* undergoes a jump that is proportional to the surface Laplacian of *u* at ∂*Ω*. The latter was rigorously shown by Pham Huy & Sanchez-Palencia (1974) for planar boundaries and by Miloh & Benveniste (1999) for generally curved boundaries. The HC interfaces were studied by Torquato & Rintoul (1995), Cheng & Torquato (1997*b*), Lipton (1997*a*,*b*, 1998) and Hashin (2001). An important physical example of the HC interface concerns the effect of ionic diffusion in concrete and cement-based composites (Garboczi & Bentz 1992; Bentz *et al*. 1994). An analogous bonding model in elasticity, called the stiff (or strong) interface, implies a jump of the stress field across the contact boundary ∂*Ω*. Depending on the properties of the interface (which is generally anisotropic), several different types of bonding conditions may arise (Caillerie 1978). This is, evidently, a more complicated problem; there are only a few advances in the field up to now (Benveniste & Miloh 2001; Hashin 2002; Benveniste 2006).

In the present paper, a conduction problem for a granular composite with a simple cubic array of spherical inclusions is considered. The effects of the interface properties on the effective conductivity and on the local potential and flux fields are studied. The performed analysis is based on the two-scale asymptotic homogenization method (Bakhvalov & Panasenko 1989), which allows macro- and microscopic components of the solution to be separated, and to pass from the input boundary-value problem (BVP) in a multi-connected domain to a recurrent sequence of local BVPs, formulated within a distinguished unit cell of the composite structure.

Solution of the cell problem may be obtained by different approaches. Usually, the main computational difficulties arise in the case of high-contrast densely packed composites. Interactions between neighbouring inclusions induce rapid oscillations of the fields on a microlevel. As the inclusion volume fraction and the contrast between the component properties increase, the local gradients can grow significantly. Then, many of the commonly used methods may lack convergence; analytical approaches that represent physical fields by infinite series expansions need to calculate a number of additional terms of the series and finite-element methods require a drastic increase in the density of the discretization mesh, etc. We develop an approximate analytical solution of the cell problem using the underlying principles of the boundary-shape perturbation technique (Guz & Nemish 1987; Andrianov *et al*. 2005; Henry 2005).

## 2. Input problem

We consider a composite material consisting of an infinite matrix *Ω*^{(1)} and spherical inclusions *Ω*^{(2)}, separated by a coating layer *Ω*^{(3)} (figure 1). The inclusions are distributed periodically in the form of a simple cubic array determined by the fundamental translation vectors *l*_{s}, *s*=1,2,3. The characteristic dimension *l* of the internal microstructure is assumed to be much smaller than the macroscopic size *L* of the entire composite solid: *l*≪*L*.

The governing steady-state conduction equation can be written as follows:
2.1
where *u*^{(a)} are potentials, *k*^{(a)} are conductivities of the components, *f*^{(a)} are densities of the volume sources and . The superscript ‘*a*’ denotes different components of the composite medium, *a*=1,2,3.

At the matrix-coating boundary ∂*Ω*_{1} and inclusion-coating boundary ∂*Ω*_{2}, perfect bonding conditions are accepted corresponding to the equalities of potentials and normal fluxes,
2.2
and
2.3
where *q*^{(a)}=−*k*^{(a)}∂*u*^{(a)}/∂** n**, ∂/∂

**is the normal derivative to ∂**

*n**Ω*

_{1}and ∂

*Ω*

_{2}.

## 3. Asymptotic homogenization procedure

We start with the analysis of the input BVPs (2.1)–(2.3) by the asymptotic homogenization method (Bakhvalov & Panasenko 1989). Let us define a natural small parameter,
characterizing the rate of heterogeneity of the composite structure. Instead of the original coordinates *x*_{s}, we introduce the so-called slow *x*_{s} and fast *y*_{s} variables,
and represent the potential field as a two-scale asymptotic expansion
where and , *e*_{s} are the unit Cartesian vectors. The first term *u*_{0} is the homogenized part of the solution, which does not depend on the fast coordinates: ∂*u*_{0}/∂*y*_{s}=0. The next terms , *i*=1,2,3,…, provide corrections of the orders *ε*^{i} and describe local variations of the potential on the microscale. The spatial periodicity of the medium induces the same periodicity for with respect to *y*_{s},
3.1
where *L*_{p}=*ε*^{−1}*l*_{p}, and *p*_{s}=0,±1,±2,…

The differential operators read where and .

Splitting the input BVPs (2.1)–(2.3) with respect to *ε* leads to a recurrent sequence of cell BVPs involving microscopic conduction equations
3.2
where , for *i*=2, for *i*≠2, and microscopic perfect bonding conditions
3.3
3.4
where ∂/∂** m** is the normal derivative to ∂

*Ω*

_{1}and ∂

*Ω*

_{2}written in fast variables. Owing to the periodicity of in (3.1), the BVPs (3.2)–(3.4) can be considered within only one periodically repeated unit cell (figure 2) of the composite structure.

Solution of the cell BVPs (2.3) and (3.2)–(3.4) at *i*=1 allows to evaluate the term . Then, in order to determine the effective conductivity *k*_{0}, we apply to equation (3.2) at *i*=2 the homogenizing operator over the unit-cell domain , where *V* _{0}=*L*^{3} is the volume of the unit cell and d*V* =d*y*_{1}d*y*_{2}d*y*_{3}. The terms are eliminated by means of Green’s theorem that, together with the periodicity relation (3.1) and the boundary conditions (3.4), implies
As a result, the homogenized conduction equation of the order *ε*^{0} is obtained as follows:
3.5
where *c*^{(a)} are volume fractions of the components, *c*^{(1)}=1−*c*^{(2)}−*c*^{(3)}, , *c*^{(3)}=4*π*[(*A*+*B*)^{3}−*A*^{3}]/(3*V* _{0})=*c*^{(2)}(3*t*+3*t*^{2}+*t*^{3}), *t*=*b*/*a*, *A*=*ε*^{−1}*a* and *B*=*ε*^{−1}*b*.

Substituting into equation (3.5) expressions evaluated below, we shall come to a macroscopic conduction equation
where *f*_{0}=*c*^{(1)}*f*^{(1)}+*c*^{(2)}*f*^{(2)}+*c*^{(3)}*f*^{(3)} is the homogenized density of the volume sources. The effective conductivity *k*_{0} can be determined after calculation of the integrals in equation (3.5). In the present work, the numerical integration was performed with the use of standard procedures of the program Maple.

## 4. Solution of the cell problem

A critical difficulty in practical applications of the asymptotic homogenization procedure consists of the solution of cell problems. In the present paper, we find an approximate analytical solution using the underlying principles of the method of perturbation of the boundary shape (Guz & Nemish 1987; Henry 2005; Andrianov *et al*. 2005).

Let us introduce in the unit cell (figure 2) the spherical coordinate system , and . Equations (3.2)–(3.4) at *i*=1 read
4.1
4.2
4.3

For axially symmetric problems, conditions (3.1) of periodic continuation for and can be equivalently replaced by appropriate boundary conditions at the centre and on the outer boundary ∂*Ω*_{0} of the unit cell (Bakhvalov & Panasenko 1989). For , we obtain
4.4
Despite the effective conductivity *k*_{0} of the simple cubic array being isotropic, boundary conditions for will depend on the direction of the macroscopic flux. If, for example, the gradient of *u*_{0} is parallel to the axis *y*_{1}, we shall obtain zero microscopic potential on two of the six sides of the outer boundary ∂*Ω*_{0},
4.5
and zero microscopic flux on the other four sides,
4.6

In order to simplify the cell problem, we propose to substitute equations (4.6) by equation (4.5), so on all six sides of ∂*Ω*_{0}, we accept zero boundary conditions for ,
4.7
where the function *R*(*θ*,*φ*) describes the cubical shape of ∂*Ω*_{0} in spherical coordinates. For the eighth part of the unit cell (0≤*θ*,*φ*≤*π*/2), it is given by the formulae
4.8
where *R*_{0}=*L*/2 is the radius of the inscribed circle. At other values of the polar angles *θ* and *φ* the function *R*(*θ*,*φ*) continues periodically.

From the physical point of view, approximation (4.7) can provide an upper bound for the effective conductivity. This could be seen if the cell problem is reformulated variationally (Milton 2002). The numerical results (§5*b*) confirm that the obtained approximate solution presents an upper bound for *k*_{0} and, moreover, shows a good accuracy at all values of conductivities and volume fractions of the components.

Next, following the method of the boundary-shape perturbation, we replace the contour ∂*Ω*_{0} by a sphere of constant radius and assume *R*(*θ*,*φ*)=*R*_{0}. With account of this simplification, the explicit analytical solution of the cell problem (equations (4.1)–(4.4) and (4.7)) can be obtained as follows:
where and are some functions of *θ* and *φ*.

After that, we set up the dependence of the radius *R*(*θ*,*φ*) upon the polar angles *θ* and *φ* in such a way that restores the original cubical shape of the contour ∂*Ω*_{0}. Thus, in the final solution, the constant *R*_{0} is substituted by the function *R*(*θ*,*φ*) in accordance with equations (4.8). In this case, the boundary conditions (4.2)–(4.4) and (4.7) are satisfied exactly, but some residual discrepancy is introduced into equation (4.1). Then, we obtain
and
where *λ*^{(2)}=*k*^{(2)}/*k*^{(1)} and *λ*^{(3)}=*k*^{(3)}/*k*^{(1)} are the non-dimensional conductivities, respectively, of the inclusion and the coating, , is the maximally possible volume fraction of the inclusions, , if *c*^{(3)}=0.

Knowing , it is possible to calculate the microscopic fields. In particular, the local potentials and and the normal fluxes and at the boundaries ∂*Ω*_{1} and ∂*Ω*_{2} can be determined as follows:
4.9
4.10
where *q*_{0}=−*k*_{0}∂*u*_{0}/∂** n** is the macroscopic (homogenized) flux and

*λ*

_{0}=

*k*

_{0}/

*k*

^{(1)}is the non-dimensional effective conductivity.

Finally, in the obtained solution, we substitute the conductivity *k*^{(3)} of the coating according to the interface model (1.1), i.e. we set *λ*^{(3)}=*αt*^{β}.

## 5. Numerical results

### (a) Limiting states of the interface

Varying the parameter *β* allows different asymptotic behaviour of the interface to be examined. Analysing the obtained solution (equations (4.9) and (4.10)) at *t*→0, we can distinguish three limiting states: the superconducting interface (*β*<−1), the perfect interface (−1<*β*<1) and the completely insulated interface (*β*>1). In these limits, the solution is independent of the bonding constant *α*. Qualitative results for the local potentials and and for the normal fluxes and on both sides of the interface are displayed in figures 3 and 4.

The points of interest are *β*=1 and *β*=−1 that correspond, respectively, to the LC and HC interface models. In these cases, the magnitude of the parameter *α* determines the rate of the imperfect bonding.

### (b) Perfect interface

Let us firstly consider the perfect interface: −1<*β*<1, *t*→0. Both the local potential *U*^{(3)} and the normal flux *q*^{(3)} at the matrix-inclusion boundary ∂*Ω* are continuous,

Numerical results for the effective conductivity *λ*_{0} in the case of perfectly conducting inclusions () are given in table 1. We can observe that the present solution provides an upper bound for *λ*_{0} and shows a good agreement with the convergent-proved data of McPhedran & McKenzie (1978), obtained by the method of Rayleigh multi-pole expansions. The asymptotic behaviour of *λ*_{0} at and is presented in figure 5. The obtained solution is compared with the asymptotic formula of Batchelor & O’Brien (1977), derived for a pair of highly conducting spheres separated by a narrow matrix gap. Dependencies of *λ*_{0} upon *c*^{(2)} at different values of *λ*^{(2)} are displayed in figure 6.

Figure 7 shows the distribution of the normal flux *q*^{(3)} along the surface of the inclusion for the 1/16 part of the unit cell (0≤*θ*≤*π*/2, 0≤*φ*≤*π*/4, *λ*^{(2)}=20, *c*^{(2)}=0.4). It can be easily seen that *q*^{(3)} reaches a maximum at the points *θ*=*φ*=*πn*/2, *n*=0,1,2,…, where the distance between neighbouring inclusions is minimal (i.e. the interaction between the inclusions is maximal).

### (c) Low-conducting interface

In order to examine the LC interface, let us set *β*=1. In the asymptotic limit *t*→0, expressions (4.9) and (4.10) result in the following bonding conditions at ∂*Ω*:
where the normal flux *q*^{(3)} is linearly proportional to the potential jump Δ*U*^{(3)}.

Figures 8–10 display the influence of the bonding constant *α* on the effective conductivity *λ*_{0}, on the local potentials and , and on the normal fluxes and . Here and in the sequel, in the numerical examples, we accept *λ*^{(2)}=20, *c*^{(2)}=0.4 and *θ*=*φ*=*πn*/2. Dashed lines indicate the values of *λ*_{0}, *U*^{(3)} and *q*^{(3)} obtained at the perfect interface for *λ*^{(2)}=0, *λ*^{(2)}=20 and . The case corresponds to perfect bonding, the case *α*→0 to the complete separation of the matrix and inclusions. The following asymptotic relations take place:
and

We can observe that weakening the bonding results in lower effective conductivity and also reduces the local flux across the interface.

### (d) High-conducting interface

The bonding conditions of the HC interface can be derived by setting *β*=−1. When *t*→0, expressions (4.9) and (4.10) give
The potential is continuous, but the normal flux undergoes a jump Δ*q*^{(3)}.

The influence of the bonding constant *α* on the effective conductivity *λ*_{0}, on the local potentials and , and on the normal fluxes and is considered in figures 11–13. Dashed lines indicate the solutions obtained at the perfect interface. The case *α*→0 corresponds to the perfect bonding conditions and the case describes the superconducting interface. The following asymptotic relations take place:
and

Obtained results show that the increase in the interface conductivity allows one to achieve higher values of the effective coefficient *λ*_{0}. At the same time, it leads to the growth of the local flux across the matrix-interface boundary. For high-gradient thermal processes (e.g. heat impact or explosion), this may result in a local fracture of the composite solid owing to a rapid and non-uniform thermal expansion.

## 6. Conclusions

An asymptotic model of the imperfect interface for the problem of conduction through a simple cubic array of spherical inclusions is proposed. The solution is evaluated by the two-scale asymptotic homogenization method. The microscopic problem on the unit cell is solved using the basic principles of the boundary-shape perturbation technique. The entire range of the interface conductivity is examined, and different asymptotic limits are analysed (from perfect insulation to superconductivity). The effects of the interface properties on the effective conductivity and on the local potential and flux fields are predicted. The advantage of the obtained asymptotic solutions is that they are suitable for all values of conductivities and volume fractions of the components, including the case of perfectly conducting densely packed inclusions.

## Acknowledgements

This work is supported by the Alexander von Humboldt Foundation (Institutional academic cooperation programme, grant no. 3.4-Fokoop-UKR/1070297) and by the German Research Foundation (DFG grant no. WE 736/25-1). We are grateful to anonymous referees, whose valuable comments and suggestions helped to improve this paper.

## Footnotes

- Received February 2, 2010.
- Accepted March 3, 2010.

- © 2010 The Royal Society