## Abstract

A thin isotropic three-dimensional curved interphase of thickness *h* between two isotropic media is considered in the setting of thermal conduction. This interphase is modelled by a surface between the two neighbouring media, and appropriate interface conditions on it are derived for the temperature and normal heat flux fields. The derivation makes use of Taylor expansions for the fields and is correct to *O*(*h*^{N}), where *h* denotes the thickness of the interphase. The jumps for the temperature and normal heat flux in the interface model are given in terms of a hierarchy of surface differential forms, which depend on the conductivities of the interphase and surrounding media, and involve surface derivatives of the temperature and normal heat flux along the interface. The analysis is directly transferable to the analogous physical phenomena of electrical conduction, dielectrics, magnetism, diffusion, flow in porous media and anti-plane elasticity.

## 1. Introduction

The modelling of a thin interphase between two media by a so-called ‘imperfect interface’ on which appropriate interface conditions have been designed has been one of standing interest in mechanics literature. Often the described representation is convenient in the course of analytical or numerical solutions of boundary-value problems containing the three-phase configuration of an interphase between two bounding media. Today, there is a rich class of literature dealing with this topic. It spans a variety of studies ranging from the pioneering works of Sanchez-Palencia (1970) and Pham Huy & Sanchez-Palencia (1974) in conduction to asymptotic studies, for example, by Klarbring & Movchan (1998) in elasticity, and to recent developments using a discrete lattice model representation by Movchan *et al*. (2003). For imperfect interfaces in composites and their effect on their effective behaviour see, for example, Benveniste & Miloh (1986), Miloh & Benveniste (1999), Hasselman & Johnson (1987), Hashin (1991, 2001, 2002), Torquato & Rintoul (1995), Lipton & Vernescu (1996), Lipton (1997), Cheng and Torquato (1997*a*,*b*), Wang *et al*. (2005) and Duan *et al*. (2005*a*,*b*). The majority of the studies in the literature deal with homogeneous interphases. There are important instances, however, in which a thin interphase between particles and matrices in a composite possesses properties which vary smoothly across its thickness (for modelling of such interphases see, for example, Lutz & Zimmerman 1996, 2005). Wave propagation phenomena in systems containing thin layers have also been treated in the literature using effective interface or boundary conditions (see an early study by Mal & Bose 1974 and more recent articles by Datta *et al*. 1988; Olsson *et al*. 1990; Bövik & Olsson 1992; Niklasson *et al*. 2000*a*,*b* and references cited therein).

A rather comprehensive list of references on the modelling of interphases by an imperfect interface can be found in Benveniste & Miloh (2001), Rubin & Benveniste (2004) and Benveniste (2006). In the first of these papers, formal asymptotic expansions for the displacement and stresses in a thin curved two-dimensional elastic layer have been used in order to derive different regimes of imperfect interfaces. Each of these regimes has been shown to be applicable for a certain degree of softness (or stiffness) of the thin layer with respect to the neighbouring media. In the second paper, a Cosserat shell model of a thin interphase has been formulated and shown to model successfully in a unified manner the several regimes of imperfect interfaces described in the first paper. The construction of the Cosserat model necessitates, however, the postulation of an appropriate strain energy for the shell-like interphase. The third paper belongs to a different category of studies which employ Taylor expansions of the relevant fields within the interphase and aim also at its unified representation by an interface for the complete spectrum of its material properties. The present study belongs to that category which is reviewed below.

In Bövik & Olsson (1992) and Bövik (1994), Taylor expansions of the relevant physical fields within the thin layer were combined with an elegant use of surface differential operators on a curved surface in order to achieve the representation of a thin isotropic interphase by an interface. The idea of a Taylor expansion in deriving an interface model of an interphase was also employed by Hashin (1991) for thin and soft elastic interphases, by Miloh & Benveniste (1999) for highly conducting thin interphases and by Hashin (2001, 2002) for thin interphases of arbitrary conductivity and elastic moduli. The study of Bövik (1994) was generalized by Benveniste (2006) to a thin anisotropic three-dimensional curved interphase between two anisotropic media in the setting of conduction and elasticity. The interface models of these works are of *O*(*h*) accuracy, where *h* is the thickness of the interphase. In Niklasson *et al*. (2000*a*), an *O*(*h*^{2}) model has been constructed for a planar two-dimensional thin coating layer. In the present study, we derive an interface model of a three-dimensional thin interphase to *O*(*h*^{N}) accuracy, where *N* is an arbitrary integer. The model is formulated in the setting of steady thermal conduction in isotropic media. It is readily transferable to the analogous physical phenomena like dielectrics, and can, in principle, be extended to the more complex contexts of elasticity, coupled field effects, anisotropic constituents and time-dependent fields.

The interface model derived in §2 is characterized by two relations: an expression for the jump in the temperature across the interface and an expression for the jump in the normal heat flux. These are given in terms of a hierarchy of surface differential forms, which depend on the conductivities of the interphase and the surrounding media, and involve surface derivatives of the temperature and normal heat flux along the interface. Explicit forms of these surface differential forms are derived in a recursive manner in §3. An example is given in §4, where the *O*(*h*^{3}) model is applied to transverse thermal conduction in a coated cylindrical fibre embedded in an infinite medium, and its predictions are compared with the exact solution.

We conclude this introduction by giving some examples to situations in which an ‘imperfect interface’ model of an interphase is useful. One instance is a setting in which an interface model of the interphase facilitates the construction of an analytical solution of the boundary problem at hand. For example, consider a constant thickness thin interphase surrounding an ellipsoidal inhomogeneity, which is embedded in a matrix subjected at constant strain or heat intensity at infinity. As a three-phase configuration this is not a confocal ellipsoidal geometry, and an analytical treatment of it is difficult. Yet, if one lets the interphase coalesce onto an imperfect interface of ellipsoidal shape, then the mathematical tools of ellipsoidal coordinates and the associated procedures become readily accessible for an analytical solution (e.g. Walpole 1978 in the setting of elasticity and Miloh & Benveniste 1999 in that of heat conduction). Another instance is a setting in which the numerical solution of the boundary-value problem in its three-phase configuration with a thin interphase raises certain difficulties, like the necessity of devising special finite-element methods suitable for thin domains (e.g. Vu-Quoq & Tang 2003), and a numerical solution in the imperfect interface configuration offers a good alternative.

## 2. Taylor expansions of *O*(*h*^{N}) and the derivation of the interface model

Consider an arbitrarily curved thin interphase of constant thickness *h* lying between two media, as described in figure 1*a*. A parallel curvilinear system (*v*_{1}, *v*_{2}, *v*_{3}) is constructed within the interphase, where *v*_{1} and *v*_{2} are two parametric curves which define the parallel surfaces, and the coordinate *v*_{3} is chosen to be along the common normal to the surfaces. The curves *v*_{1} and *v*_{2} are chosen to be the lines of curvatures of the parallel surfaces, and are thus orthogonal to each other. The metric coefficients of this orthogonal parallel curvilinear coordinate system are denoted by *h*_{1}, *h*_{2}, *h*_{3} with *h*_{3}=1. Let be the unit vectors tangent to the parametric curves *v*_{1} and *v*_{2} on the parallel surfaces, and be the unit normal to these surfaces, along the linear *v*_{3} coordinate. Finally, the inner, middle and outer surface of the interphase will be denoted by *S*_{1}, *S*_{0}, *S*_{2}, with *v*_{3} assuming, respectively, the values of −*h*/2, 0, *h*/2 on them.

The purpose of this study is to represent the interphase by an interface lying between the inner and outer media, and on which appropriate interface conditions need to be derived. Specifically, the interphase between media 1 and 2 will be replaced by an interface located at the surface *S*_{0}, with the inner and outer media being extended up to that interface, see figure 1*b*. The derived interface model of the interphase will be correct to *O*(*h*^{N}), where *N* is an arbitrarily assigned integer.

Thermal conduction in isotropic media is governed by(2.1)where *q*_{i} are the components of the heat flux vector, *k* is the conductivity and *H*_{i} is the heat intensity vector defined by(2.2)where *ϕ* denotes the temperature and **grad** *ϕ* is given by(2.3)Under steady state conditions, the heat flux vector obeys the balance law(2.4)

Generally, across a surface *S* separating two solids ‘1’ and ‘2’, the following continuity conditions prevail under perfect contact conditions:(2.5)where the superscript in (.)^{(α)} indicates that the quantity (.) pertains to media *α*, with *α*=1, 2, and *n*_{i} denote the components of the unit normal to *S*, taken, say, from medium ‘1’ to medium ‘2’.

The interface model will be characterized by two relations: an expression for the jump in the temperature across the interface and an expression for the jump in the normal heat flux. We start by developing the expression for the jump in the temperature. In the configuration of figure 1*a*, the temperature at the mid-surface *S*_{0} can be expressed in terms of a Taylor expansion about the lower surface *S*_{1}, or alternatively by means of Taylor expansion about the upper surface *S*_{2}. These options provide(2.6)

(2.7)Subtracting (2.7) from (2.6) yields(2.8)

In §3, it will be shown that on a given surface *Γ* in a solid of conductivity *k*_{α}, one can express the normal derivatives of the temperature *ϕ* and of the normal heat flux *q*_{3} in the form of(2.9)where are surface differential operators, which depend on the conductivity *k*_{α} of the solid and contain surface derivatives only. The representation (2.9) is central to the development of the interface model. Using it in (2.8) provides(2.10)

We now recall the continuity property of the temperature and the normal heat flux across the surfaces *S*_{1} and *S*_{2} separating the interphase from its neighbouring media, and due to the nature of surface operators, cast equation (2.10) in the form(2.11)Before proceeding further with (2.11), a development parallel to that between (2.6) and (2.11) is carried out for the normal heat flux *q*_{3}. This leads to(2.12)where the operators and have been defined in (2.9).

Equations (2.11) and (2.12) are two relations connecting the temperature and normal flux fields at the locations *v*_{3}=*h*/2 and *v*_{3}=−*h*/2 in the configuration of figure 1*a*. In fact, if desired, they allow one to solve a boundary-value problem containing a thin interphase without solving for the fields in the interphase and carrying out the analysis in media ‘1’ and ‘2’ only. Alternatively, the aimed representation of figure 1*b* can be further achieved as follows. First, we demand that the temperature and normal flux in figure 1*b* at the locations *v*_{3}=*h*/2 and *v*_{3}=−*h*/2 be related to each other by (2.11) and (2.12), which prevailed in the configuration of figure 1*a*. This will ensure that the fields in figure 1*b* in the regions *v*_{3}≤−*h*/2 and *v*_{3}≥*h*/2 be the same as those in media ‘1’ and ‘2’ of figure 1*a*. Therefore, we will now make use of (2.11) and (2.12) in the setting of figure 1*b*, and transform them further as described below.

First, in the setting of figure 1*b*, the following Taylor expansions are made for the left-hand sides of (2.11) and (2.12):(2.13)where (.)_{+} indicates that the field (.) is evaluated at the interface on the side of medium ‘2’ and (.)_{−} stands for evaluation at the interface on the side of medium ‘1’ in the configuration of figure 1*b*. Second, Taylor expansions for the differential forms appearing in the right-hand sides of (2.11) and (2.12) need to be made. These will be expansions about the interface in the configuration of figure 1*b*. The order of these expansions should be fixed in a manner that will ensure that in the final transformed forms of (2.11) and (2.12) terms of *O*(*h*^{N+1}) will be discarded. It is convenient at this stage to leave the upper index of these expansions undetermined and denoted by *T* as follows:(2.14)It should be noted that the expressions in (2.14) contain terms like , which means that while the operator is affiliated to phase ‘0’, the *p* derivatives have been taken in phase 2 (the + side of *Γ*). In §3, it will be shown that the following representation is valid for the normal *p* derivatives (taken in phase *β*) of {}, {}, {}, {}:(2.15)where are new surface operators with the significance of the subscripts *rp* and superscripts (*α*, *β*) having already been made explicit above.

We point out that in substituting (2.14) into (2.13) care should be taken not to include terms of *O*(*h*^{N+1}) in the resulting final forms. This necessitates that the summation on *p* in the double sum , which results by substituting (2.14) into (2.13), should run up to *N*−*r*. Furthermore, it becomes apparent that in this double sum, the largest value of *r* should be *N*−1 (when *r*=*N*−1, the only possible value for *p* will be *p*=1). In view of these clarifications, substitution of (2.14) into (2.13) leads to the final forms given below:(2.16)(2.17)

## 3. The surface operators

Explicit expressions for the operators entering in the interface model (2.16) and (2.17) will be now derived in a recursive manner. From (2.1) and (2.2), with the normal derivative being taken in medium *α*, one has(3.1)so that are immediately identified as(3.2)In order to derive and , we first note that (2.1), (2.2) and (2.4) yield Laplace's equation for *ϕ*:(3.3)This readily provides the representation(3.4)with(3.5)where Δ_{S}(*ϕ*) is the surface Laplacian of *ϕ* and *κ* is the mean curvature of *Γ* given by(3.6)It is noted that the operator vanishes identically, is simply the constant scalar −1/*k*_{α}, is a linear differential operator with variable coefficients and, finally, is the non-constant scalar 2*κ*. It will now be shown that all the operators appearing in (2.16) and (2.17) can be derived on the basis of (3.2) and (3.5), and that, in general, they turn out to be linear operators with variable coefficients. In order to derive , let us start with(3.7)Differentiating (3.7) with respect to *v*_{3} provides(3.8)which allows one to identify and as(3.9)The derivation of is carried out along the same lines, but is more elaborate. One writes(3.10)and differentiates it with respect to *v*_{3} to get(3.11)where and are given by(3.12)with being defined as(3.13)It is noted from (3.5) and (3.13) that is generated from Δ_{S}(*ϕ*) simply by differentiating the non-constant coefficients in Δ_{S}(*ϕ*) with respect to *v*_{3}. Finally, making use of (3.7), (3.10) and (3.12) in (3.11) yields(3.14)so that the explicit forms of and are identified as(3.15)All the operators with *r*>2 can be obtained recursively in the same manner. For example, with *r*=3, one has(3.16)where is defined by(3.17)

Let us now pass to the derivation of the operators introduced in (2.15) and appearing in (2.16) and (2.17) as being affiliated to the interphase through the superscript ‘(0)’. Suppose the previous operators have already been obtained by the procedure described above. These expressions will contain the conductivity *k*_{0} of the medium to which they are affiliated. Consider now and differentiate these with respect to *v*_{3} in a medium of conductivity *k*_{β}. This provides(3.18)where are generated from by taking the derivative of the variable coefficients of this last set with respect to *v*_{3}. For example, for *r*=1, there is(3.19)whereas for *r*=2, one has(3.20)

From (3.18), and the definitions in (2.15), one can identify the following forms:(3.21)As indicated above, these expressions contain the conductivity *k*_{0}, because the operators in (3.18) are affiliated to the interphase by their superscript ‘(0)’; they also contain the conductivity *k*_{β}, because the derivatives ∂*ϕ*/∂*v*_{3}, ∂*q*_{3}/∂*v*_{3} in (3.18) have been evaluated either on the upper (*β*=2) or the lower (*β*=1) sides of the interface in figure 1*b*.

The differential forms with *p*>1 are recursively obtained in the same manner. For example, those with *p*=2 are given by(3.22)where are obtained from by taking the derivative of their non-constant coefficients with respect to *v*_{3}.

Finally, it is of interest to point out the presence of certain consistency conditions between some special cases of the derived differential forms. Consider the *r*th derivative of *ϕ* taken in medium *α*(3.23)Now, differentiate additional *p* times with respect to *v*_{3} *in the same medium α*,(3.24)so that and are identified as(3.25)Similarly, one also has(3.26)It can be readily verified that the expressions given in (3.9), (3.15), (3.16) and in (3.21), (3.22) (where one now takes *k*_{α}=*k*_{β}=*k*_{0}) do indeed satisfy (3.25) and (3.26).

## 4. Example. The *O*(*h*^{3}) interface model versus the exact solution: heat conduction in a coated fibre embedded in an infinite medium

In this section, an example will be given to the implementation of the interface conditions (2.16) and (2.17) by solving a coated fibre configuration to *O*(*h*^{3}) accuracy. First, we state the explicit form of the *O*(*h*^{3}) interface model described by (2.16) and (2.17). The resulting equations can be written compactly if the notation(4.1)is used, where *Γ* is identified with *S*_{0} in the configuration of figure 1*b*, and the following combination of the previously defined differential forms are introduced:(4.2)With this notation one has, for instance,(4.3)The *O*(*h*^{3}) interface conditions can be now written as(4.4)

(4.5)

In order to study the performance of this interface model, we consider transverse heat conduction in an infinite coated fibre embedded in an infinite matrix. At infinity, the following temperature field is imposed:(4.6)where *M* is an arbitrary constant, and controls the variation of the temperature in space. The case of *M*=1 corresponds to a uniform temperature gradient at infinity, and is known to be useful in the determination of the effective conductivity in composites. The case of *M*>1 is a hypothetical loading resulting in a more complex distribution of the fields in space, and has been used in order to study the performance of the interface model in the setting of a boundary-value problem which is not necessarily affiliated to effective property determination in composite media. This type of a temperature dependence on *θ* was applied by Rubin (2004) on the surface of a cylindrical shell for studying his Cosserat shell model in heat conduction.

Let the radius of the fibre be *a*, the thickness of the coating be *h* and the outer radius of the coating be *b*. We further define(4.7)The temperature field is sought throughout and, in particular, at the contact surfaces *r*=*a* and *b*. It will be obtained by using the *O*(*h*^{3}) interface model of (4.4) and (4.5) in the setting of figure 1*b*, and will be compared with the exact solution of the fibre/interphase/matrix configuration of figure 1*a*. It should be noted that in the loading (4.6), the larger the value of the integer *M*, the smaller will be the length-scale of the variation of the temperature in space. Thus, interphases with given *ϵ*=(*h*/*R*)≪1 will behave as thin interphases only for sufficiently low values of the integer *M*.

Let the cylindrical coordinate system be defined by *v*_{1}=*θ*, *v*_{2}=*z*, *v*_{3}=*r*, and consider the setting in which ∂/∂*z*=0, applicable to the chosen example. The metric coefficients are *h*_{1}=*h*_{θ}=*r*, *h*_{2}=*h*_{z}=1, *h*_{3}=*h*_{r}=1. All the differential forms entering in the *O*(*h*^{3}) equations, as obtained from the general expressions given in §3, are given in appendix A. With these explicit forms of the differential operators, the interface conditions (4.4) and (4.5) reduce to:(4.8)(4.9)

First, we develop the exact solution. In the fibre, coating and matrix, the temperature satisfies Laplace's equation(4.10)The following temperature fields,(4.11)fulfil (4.10) in the three domains, and result in the following heat flux fields:(4.12)The perfect contact interface conditions(4.13)yield the following equations for the unknown coefficients:(4.14)where(4.15)and the components of * A* and

*are given by(4.16)*

**Y**Next, we derive the solution for the configuration of medium 1/interface/medium 2 of figure 1*b*, with the interface being located at *r*=*R*. In this configuration, the temperature and flux fields in media 1 and 2 are given by(4.17)

(4.18)Implementing the interface conditions (4.8) and (4.9) at *r*=*R* result in the following solution for the unknown constants and :(4.19)where and(4.20)

A comparison of the predictions of the *O*(*h*) and *O*(*h*^{3}) interface models with the exact solution is given in figures 2–7. The *O*(*h*) model, which is in fact that of Bövik & Olsson (1992) and Bövik (1994), can be deduced from (4.4) and (4.5) by keeping there the *O*(*h*) terms only, and has a relatively simple form. In the same figures, the predictions of the present model are also contrasted with those of Hashin (2001), in which *O*(*h*) Taylor expressions for the fields in the interphase have been used about the *inner* surface *S*_{1}. (A detailed discussion of the present *O*(*h*) model and that of Hashin (2001) and a clarification of the distinction between them is given in the appendix B of Benveniste (2006).)

Let us define the temperature dependence of (4.11), (4.12) and (4.17) symbolically by(4.21)and exhibit the non-dimensional temperatures *Φ*(*a*)/*A*^{*} and *Φ*(*b*)/*A*^{*} as predicted by the *O*(*h*) and *O*(*h*^{3}) interface models, and compare them with the exact solution. As indicated above, in the configuration of figure 1*b*, these fields are obtained by making evaluations at *v*_{3}=−*h*/2 and *v*_{3}=*h*/2 in media 1 and 2, respectively. The non-dimensional temperatures *Φ*(*a*)/*A*^{*} and *Φ*(*b*)/*A*^{*} are plotted as a function of log(*k*_{0}/*k*_{1}) for the choice of the set of parameters: *ϵ*=*h*/*R*=0.001, 0.1; *β*=*k*_{2}/k_{1}=0.1, 10; and *M*=1, 5.

In figures 2 and 3, the results are given for the choice of the parameters *ϵ*=0.001, *β*=0.1, with *M*=1 being applicable to figure 2 and *M*=5 being applicable to figure 3. It is seen that for such a thin interphase, the predictions of all the models coincide with the exact solution.

In figures 4–7, results are exhibited for thicker interphases characterized by *ϵ*=0.1. The remaining parameters in these figures are: *β*=0.1, *M*=1 in figure 4; *β*=0.1, *M*=5 in figure 5; *β*=10, *M*=1 in figure 6; and, finally, *β*=10, *M*=5 in figure 7. It is seen that for such thicker interphases, the present *O*(*h*) model and that of Hashin (2001) perform poorly when *M*=5. The *O*(*h*^{3}) model, on the other hand, is seen to produce results which are indistinguishable from the exact theory for all the cases. It is finally mentioned that the *O*(*h*^{2}) theory gives results which fall between the *O*(*h*) and the *O*(*h*^{2}) models; these have been omitted here for the sake of brevity.

## Acknowledgments

The comments of two anonymous referees which helped to improve the exposition in certain parts of the paper are thankfully acknowledged.

## Footnotes

- Received November 29, 2005.
- Accepted January 6, 2006.

- © 2006 The Royal Society