## Abstract

A three-dimensional, generally curved, constant thickness interphase whose properties vary across its thickness is considered. This graded interphase is modelled by an interface that is located at its mid-surface and which comes now into direct contact with the media that were adjacent to it. The derivation is carried out in the setting of heat conduction, and appropriate jump conditions for the temperature and normal heat flux across the interface are derived, which characterize the interface model. The inhomogeneity of the interphase becomes a major motivation for its replacement by an interface. This representation is expected to simplify significantly the solution of boundary-value problems that involve thin inhomogeneous regions separating two media. Analytical solutions that are not available for a large class of graded interphases can become readily accessible with the derived interface model.

## 1. Introduction

The present paper is a continuation of two previous studies by the first author, in which an interface model of an interphase between two media was developed (Benveniste (2006*a*,*b*), to be denoted here as B1 and B2, respectively). In those works, a thin interphase of constant thickness *h* lying between two media was considered. This interphase was modelled by an interface that is located at its mid-surface and which comes now into direct contact with the media that were adjacent to it (see figure 1*a*,*b*). Appropriate interface conditions were designed on that new interface to make the described replacement possible. The analysis in B1 is correct to *O*(*h*) accuracy, and it was developed in the setting of anisotropic non-steady thermal conduction and dynamic elasticity. The analysis in B2 is correct to *O*(*h*^{N}) accuracy, where *N* is an arbitrarily assigned integer, and it was carried out in the setting of steady thermal conduction in isotropic media. Both papers are closely related to previous studies by Bövik & Olsson (1992) and Bövik (1994).

The investigation of interface phenomena in solids has gained a recent impetus, owing to a growing interest in nanoscale phenomena. Interface effects become specially pronounced in nanocomposites, since for a given volume fraction of host particles in a matrix, the smaller the particle size, the larger will be the total contact surface between the particles and the matrix. For a list of references on the subject, see Miller & Shenoy (2000), Sharma *et al*. (2003), Dingreville *et al*. (2005) and Duan *et al*. (2005). A rather comprehensive literature survey on the modelling of interphases by an interface can be found in B1 and B2. The majority of the papers on the subject considers thin interphases that are homogeneous. However, it is well known that graded interphases in which the material properties vary across the thickness of the interphase are often encountered in physical systems (see Lutz & Zimmerman 1996, 2005). Interface models of such interphases are useful in composite media having graded constituents. For such composites, see the recent works by Dong *et al*. (2004) and Wang *et al*. (2006). Analytical solutions for a large class of graded interphases are non-existent even for simple geometries like a cylindrical interphase or a spherical interphase, but can be conveniently carried out in the setting of the introduced representation. Numerical solutions of configurations involving an inhomogeneous interphase may also be simplified with the interface model.

The present paper extends the *O*(*h*^{N}) isotropic analysis of B2 so as to incorporate the presence of a graded anisotropic interphase whose properties vary across the thickness of the interphase. The derivation of the interface model is given here in the setting of non-steady thermal conduction. In B2, the formal statement of the interface conditions is given in terms of a hierarchy of surface differential operators affiliated to the normal derivatives of the temperature and normal heat flux to the interface. In §2, we point out that this formal statement remains unchanged in the present study. However, the specific form of the differential operators is strongly affected by the inhomogeneity and anisotropy of the interphase. They are derived in §3 for the *O*(*h*^{2}) theory in the setting of a general orthogonal curvilinear parallel coordinate system and non-steady conduction, and they can be recursively obtained for higher-order theories using a similar procedure. In the electronic supplementary material A, the explicit expressions of the surface differential forms for the *O*(*h*^{3}) theory are given under steady-state conditions in the setting of a cylindrical coordinate system for the use in the example of §4. In §4*b*, the developed interface model is implemented to transverse steady-state heat conduction in a coated fibre, which is embedded in an infinite medium. All three media are cylindrically orthotropic: the fibre and the matrix are homogeneous, and the coating is graded. Since the purpose of §4*b* is to study the performance of the interface model vis-à-vis, an exact solution of the three-phase configuration, a specific variation of the conductivity tensor in the coating, has been chosen for which an analytical solution is available. Section 4 concludes with a comparison of the predictions of the *O*(*h*), *O*(*h*^{2}), *O*(*h*^{3}) theories with the exact solution.

## 2. The interface model of a curvilinearly anisotropic graded interphase

Consider an arbitrarily curved thin three-dimensional interphase of constant thickness *h* lying between two media, as described in figure 1*a*. This interphase is mapped by a parallel orthogonal curvilinear system (*v*_{1}, *v*_{2}, *v*_{3}), where *v*_{1} and *v*_{2} are the two parametric curves that define parallel surfaces within the interphase and *v*_{3} is the linear coordinate that is along the common normal to the surfaces, with *v*_{3}=0, denoting the mid-surface *S*_{0}. 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. A detailed description of this coordinate system is given in the introductory part of Appendix A in B1. The interphase will be assumed to be curvilinearly anisotropic, with the nature of anisotropy being specified and remaining the same with respect to a local triad of unit vectors tangent to the *v*_{1}, *v*_{2}, *v*_{3} curves. It is also graded with its material properties being dependent on *v*_{3}.

Heat conduction is governed by(2.1)where *q*_{i} are the components of the heat flux vector; *k*_{ij} is the conductivity tensor, satisfying *k*_{ij}=*k*_{ji}; *ρ* is the density; *C*_{p} is the specific heat; and *t* denotes the time. The gradient of the scalar *ϕ* and the divergence of the vector ** q** in the orthogonal parallel curvilinear system are defined in (A.4) and (A.22) of B1.

Across a surface *Γ* separating two solids *α* and *β*, and under ideal contact conditions, the following continuity conditions prevail:(2.2)where the superscript in ( )^{(r)} indicates that the relevant quantity pertains to media *r* with *r*=*α*, *β*; and *n*_{i} denote the components of the unit normal to *Γ*, taken (say) from medium *α* to medium *β*.

As indicated in §1, the purpose of the present study is to represent the anisotropic and inhomogeneous interphase by an interface lying between the inner and the outer media, 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}, which comes now into a direct contact with the media which were adjacent to it (figure 1*b*). The derived interface model of the interphase will be correct to *O*(*h*^{N}), where *N* is an arbitrarily assigned integer.

The interface model of B2 for an isotropic and homogeneous interphase 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. The derivation in B2 consists of two stages. In the first stage, the temperature *ϕ* and the normal heat flux *q*_{3} are expanded in Taylor expansions in *v*_{3} about the mid-surface of the interphase. In parallel, use is made of a representation of the normal derivatives of *ϕ* and *q*_{3} with respect to *v*_{3} at a surface *Γ* (which is a member of the parallel coordinate surfaces) in terms of the surface differential forms along *Γ*. The first stage concludes with a description of the jumps for the temperature and normal heat flux across the interface in terms of a hierarchy of surface differential forms. Explicit expressions for these differential forms are derived in the second stage of the derivation. Scrutiny of the first stage, as described in B2, reveals that it remains formally unchanged in the context of the present study. On the other hand, the inhomogeneity of the interphase introduced here affects the derivation of the differential forms that enter in the interface model. In the present short section, we give the definition of the surface differential operators which are central to the analysis, and then state the form of the interface model. The derivation of the surface differential forms for the graded interphase model is carried out in detail in §3.

Consider a system of parallel surfaces in a medium *α*. It is shown in B2 that on one such surface defined by *Γ*, one can express the normal derivatives of the temperature *ϕ* and the normal heat flux *q*_{3} in the form(2.3)where are surface differential operators, which depend on the conductivity tensor of the solid and contain surface derivatives only. This formal representation that was established for an isotropic and homogeneous solid in B2 is shown in the present §3 to remain valid for an anisotropic and inhomogeneous solid. Next, suppose that the specific surface *Γ* separates now two solids *α* and *β*. Since the expressions , , involve surface derivatives only, one can envisage them as being evaluated on either side of *Γ*. The construction of the interface model in B2 makes use of the derivatives of the type , where the subscript *β* indicates that differentiation of has been performed in medium *β*. Those derivatives have been shown in B2 to have a representation(2.4)where and are additional surface operators. Similarly, the operators , , affiliated to are defined analogously to (2.4). It was shown in B2 that the formal statement of the *O*(*h*^{N}) interface model in terms of the above defined surface operators is (2.5)(2.6)where (.)_{+} indicates that the field (.) is evaluated at the interface on the side of medium ‘2’ and (.)_{−} stands for the evaluation at the interface on the side of medium ‘1’ in the configuration of figure 1*b*.

## 3. The derivation of the surface differential forms for an anisotropic graded interphase

In this section, explicit expressions will be derived for the differential forms entering in the *O*(*h*^{2}) model of a graded interphase in the setting of a general orthogonal curvilinear parallel coordinate system. The differential forms entering in the higher-order models can be derived in a recursive manner.

Consider a surface *Γ* in a conducting solid *α* with a constitutive law of the type (2.1). Let the anisotropic law be curvilinearly anisotropic and inhomogeneous in the direction, with . Although a dependence on *v*_{3} will be admitted only for the interphase (*α*=0), it is found convenient in the derivation to assume that such a dependence exists for all of the three media. The constitutive law in (2.1) with *i*=3 is now invoked, and use is made of the definition of surface gradient of a scalar, given in (A.6) of B1 with the property of(3.1)where *δ*_{ij} is the Kronecker delta. This provides(3.2)Equation (3.2) allows one to extract the explicit expressions of the differential forms and , which have been introduced in (2.3),(3.3)where and *f*^{(α)} have been defined as(3.4)

To derive the surface operators and , the constitutive law in (2.1) is first rewritten as(3.5)where equation (3.1) has been used. Substitution of equation (3.2) into (3.5) yields(3.6)where(3.7)The dependence of the material parameters on *v*_{3} has explicitly been indicated in the above equations, but will be omitted in those which follow, for simplicity of notation.

The next step is to rewrite the balance law in (2.1) in accordance with (A.24) of B1 as(3.8)where div_{S} ** q** has been defined in (A 23) of B1. This makes possible to write(3.9)where the components of the vector

**p**^{(α)}

**grad**

_{S}

*ϕ*in the curvilinear coordinate system are indicated by and those of the vector

**m**^{(α)}

*q*

_{3}by . The differential forms , defined in (2.3) can now be readily identified from (3.9)(3.10)where the vectors

**U**^{(α)}(

*ϕ*) and

**F**^{(α)}(

*q*

_{3}) have been defined as(3.11)

We proceed now to the derivation of . In order to obtain and , one starts with (3.2) and differentiates it with respect to *v*_{3} in medium *α*. This provides(3.12)where has been defined for convenience as(3.13)

Next, the representation ∂*ϕ*/∂*v*_{3} and ∂*q*_{3}/∂*v*_{3} in terms of , , , , as described in (3.3) and (3.10), is invoked and substituted in (3.13). From this last step, after some manipulation, one can now identify the differential forms , as follows:(3.14)

The derivation of , is carried out along the same lines, but it is more elaborated. One starts with (3.9) and (3.10) and differentiates it with respect to *v*_{3}. This gives rise to the terms ∂/∂*v*_{3}({grad_{S}**U**^{(α)}}_{kk}) and ∂/∂*v*_{3}({grad_{S}**F**^{(α)}}_{kk}), which can be developed further. Making use of the definition of grad_{S}* V* (where

*is a vector) in (A.14) of B1 and invoking the property of(3.15)valid in the parallel coordinate system, allows one to write(3.16)where(3.17)with similar expressions being valid for*

**V**

**U**^{(α)}and

**F**^{(α)}. Furthermore, using the definitions of the

**U**^{(α)}and

**F**^{(α)}given in (3.11) and recalling the representation of the normal derivatives of

*ϕ*and

*q*

_{3}in terms of surface differential forms, one can show after some manipulations that (∂

**U**^{(α)}/∂

*v*

_{3}) and (∂

**F**^{(α)}/∂

*v*

_{3}) are given by(3.18)Finally, the use of (3.16) and (3.18) in the second derivative of

*q*

_{3}with respect to

*v*

_{3}(obtained by differentiating (3.9)) makes it possible to identify the following expressions for and :(3.19)

The remaining differential forms which complete the formulation of the *O*(*h*^{2}) theory are , , , and , , , . Their derivation proceeds along the same lines described in B2 and will not be given here for the sake of brevity. The final results are(3.20)The above derived differential forms can in principle be applied to a setting in which all of the three media are curvilinearly inhomogeneous in the -direction. For a graded interphase between two curvilinearly homogeneous media, the dependence on *v*_{3} is present only for the interphase (*α*=0).

## 4. Example: steady-state transverse heat conduction in a matrix-embedded fibre that is coated with a graded cylindrically orthotropic layer

In this section, the derived interface model will be implemented in the simple setting of steady-state transverse heat conduction in an infinitely long coated fibre embedded in a matrix. A graded interphase is chosen, for which an analytical solution is available in the three-phase configuration, with which the predictions of the interface model can be conveniently compared. Let the radius of the fibre be *a*, the thickness of the coating be *h* and the outer radius of the coating be *b*. The mid-surface of the coating will be denoted by *R*=(*a*+*b*)/2. A cylindrical coordinate system is defined by *v*_{1}=*θ*, *v*_{2}=*z*, *v*_{3}=*r* and transverse heat conduction in which ∂/∂*z*=0 is considered. The fibre, coating and matrix are assumed to be cylindrically orthotropic, with the fibre and the matrix being homogeneous and the coating graded. Specifically, the constitutive laws are(4.1)where the superscripts 1, 0, 2 denote the fibre, coating and matrix, respectively. The conductivities *k*_{r} and *k*_{θ} will be taken to be constant in the fibre and the matrix, but dependent on *r* in the graded coating. In steady-state conditions, the balance law for the temperature as implied by (2.1) becomes(4.2)The fibre–coating–matrix assembly is subjected at infinity to a uniform temperature gradient, i.e. .

This section contains two parts. In §4*a*, an exact analytical solution in the three-phase configuration is developed for a specific variation of *k*_{r}(*r*) and *k*_{θ}(*r*) within the coating. In §4*b*, the interface model is implemented to the described problem and its predictions are compared with those of the exact solution.

### (a) The exact analytical solution for the three-phase configuration

We indicate at the outset that we do not aim at choosing a variation of *k*_{r}(*r*) and *k*_{θ}(*r*), which may be encountered in real systems. Here, the purpose is to choose a graded interphase for which a simple analytical solution is possible. Assuming a temperature dependence of the form *ϕ*(*r*, *θ*)=*Φ*(*r*)cos *θ* and substituting in (4.2) provides(4.3)The question is then asked, under what conditions on *k*_{r}(*r*) and *k*_{θ}(*r*) does equation (4.3) transforms to the so-called ‘equidimentional equation’ (e.g. Hildebrand (1962), p. 13)(4.4)where *C* and *D* are constants? It can be readily checked that this occurs when , , where denotes the conductivity *k*_{r} at the mid-surface *R* and *g* and *p* are constants. Following the standard procedures of solving the equidimentional equation, one can show that the solution of (4.4) becomes(4.5)where *C*_{1} and *C*_{2} are constants. Choosing *p*=0, one recovers the solution of a homogeneous cylindrically orthotropic medium. On the other hand, letting *g*=1 provides the isotropic case. The exact solution for the temperature in the fibre–coating–matrix assembly can now be readily obtained by using the form of the solution in equation (4.5). The unknown constants are determined by applying the continuity of the temperature and normal heat flux at *r*=*a*,*b*, the fulfilment of the condition at infinity and the demand that temperature remains bounded at *r*=0.

### (b) The interface model

The implementation of the interface model to the above described geometry is now carried out. We choose to implement the *O*(*h*^{3}) theory as characterized by the differential forms given in the electronic supplementary material in conjunction with the interface conditions (2.5) and (2.6) (the *O*(*h*^{3}) equations are summarized in (4.1)–(4.5) of B2). The resulting interface conditions are given below. It should be noted that in these equations, we have set ∂/∂*z*=0 and denoted , with *α*=1, 0, 2, simply by *g*^{(α)}. Moreover, the equations below are written for the special case in which *g*^{(0)}=const. (which has permitted the exact solution of the three-phase configuration in §4*a*). Note also that the fibre and the matrix are taken to be homogeneous. Finally, the parameters , and their derivatives with respect to *r*, are to be evaluated at *r*=*R*, (4.6)

(4.7)

The configuration is now that of the fibre being separated from the matrix by the interface located at *r*=*R*. The temperature fields in both media are determined by using the form of the solution (4.5) in the fibre and the matrix, with *p*=0. The implementation of the interface conditions (4.6) and (4.7), the fulfilment of the condition at infinity and the demand that the temperature be bounded at *r*=0 allows the determination of the unknown constants.

First, we consider the case of isotropic constituents and an inhomogeneous interphase. With *g*^{(1)}=*g*^{(0)}=*g*^{(2)}=1, *ϵ*=*h*/*R*=0.1 and *β*=*k*^{(2)}/*k*^{(1)}=0.1, *p*=25, we obtain the results given in figure 2, where the non-dimensionalized temperatures *Φ*(*a*)/*A*^{*} and *Φ*(*b*)/*A*^{*} are plotted versus . In order to study further the effect of a graded interphase, we let *p*=100, with the remaining parameters being the same, and obtain the results of figure 3. With such a high value of the parameter *p*, we observe a somewhat inconsistent behaviour reflected in the fact that the *O*(*h*) model performs better than the *O*(*h*^{2}) model for certain ranges of *λ*. The difficulty of the model in this case may be explained by the fact that an interphase with *h*/*R*=0.1 ceases to behave as a thin interphase for a strong inhomogeneity through the thickness characterized by *p*=100. In fact, reducing the thickness to *ϵ*=*h*/*R*=0.01 while keeping *p*=100 results in the excellent performance illustrated in figure 4. Finally, the effect of anisotropy of the interphase is illustrated in figure 5 by choosing *ϵ*=*h*/*R*=0.1, *β*=1, *g*^{(1)}=1, *g*^{(0)}=200, *g*^{(2)}=1 and *p*=0.

A fundamental question which is pertinent to the performance of any approximate model is applicable here as well. For a given set of parameters, how is one to judge the results of the interface model for instances in which no exact solution is available? First, it should be noted that even if we were not in possession of the exact solution, the *O*(*h*^{2}) and *O*(*h*^{3}) results of figure 2, for example, could be accepted with some confidence, since they are close to each other when compared with the *O*(*h*) results. Second, it is emphasized here that given a set of parameters which describe the problem, our numerical experiments have shown that making the interphase thickness thinner and thinner (as in passing from figures 3 to 4) always yields satisfactory results for a large spectrum of the conductivity of the interphase. Thus, before producing results for a given combination of parameters and a certain interphase thickness, it is advisable to make numerical experiments for thinner and thinner interphases to get a familiarity with the correct behaviour of the solution.

## Acknowledgments

The authors wish to thank the editor and two anonymous referees for recommending a reorganization and shortening of a previous version of the paper. Their advice has led to the present compact form of the study.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2006.1777 or via http://www.journals.royalsoc.ac.uk.

- Received August 11, 2006.
- Accepted September 6, 2006.

- © 2006 The Royal Society