Application of an improved three-phase model to calculate effective characteristics for a composite with cylindrical inclusions

A modified three-phase composite model yielding reliable effective characteristics of composite structures has been proposed. In particular, the problem of effective heat transfer coefficient of the composite structure with periodically located inclusions of circular cross-sections located on a square net is solved. Advantages of the proposed model in comparison to the classical three-phase model are illustrated and discussed.


INTRODUCTION
The three-phase model of a composite (TPhM) has been used in references [2][3][4][5]10,11,[15][16][17] in order to define effective characteristics of composite structures.The physical meaning of the idealization introduced by TPhM relies on substitution of the periodic structure being studied by its counterpart homogeneous homogenized structure with the equivalently reduced parameters (to be defined) except of only one characteristic structure cell.Further step of the solution requires derivation of mathematical formulas regarding homogenized coefficients through the application of physical understanding and mainly via either the energy principle [4,5,10,11,17] or structure geometry [2,3,15,16].
In particular, in the problem devoted to determination of the equivalent heat transfer coefficient of a two-phase composite structure with periodic cylindrical inclusions of circular cross-sections located on a square net the analysis revealed the following essential features: (i) For small inclusions size 1 a heat transfer parameter is validated for its arbitrary values; (ii) For large inclusion size 1 a the TPhM model yields reliable results assuming that heat transfer of the inclusions is of the order of structure matrix ~1 ; Latin American Journal of Solids and Structures 10(2013) 197 -222 (iii) ТPhM may yield even a qualitatively wrong result for either large inclusion size 1 a or large/small conductivity properties, i.e. for and 0 .
A construction of the first order TPhM approximation using the method of variations of boundary shapes of the studied structure does not allow us to overcome the earlier mentioned problems, in particular with respect to the possible wide spectrum of applications.Furthermore, a solution being limited to the first inclusion approximation does not describe the structure properties adequately for large values of the inclusion size.Namely, it does not allow achieving even a qualitative picture of the processes which occur in the composite (for example, a validation of the infinite cluster occurrence).The latter commentary can also be expressed mathematically.Namely, the first terms of the asymptotic series do not influence the further asymptotic sequence being defined through a zero order approximation.
In this work novel algorithms associated with the application of the composite TPhM are developed.The main idea and advantages of our proposal are illustrated and discussed taking as an example a solution to the heat transfer problem of the composite structure with cylindrical inclusions of circular cross-sections.

SOLVING THE HEAT TRANSFER PROBLEM WITH THE USE OF THE IMPROVED COMPOS-ITE TPHM
The problem of determination of effective coefficients of the micro non-homogeneous material consisting of a continuous matrix and periodically located cylindrical inclusions with circular crosssections is solved in this work.In our study structure size in the direction of the fibre length essentially exceed the remaining size, i.e.

L
(Figure 2.1).We assume that the studied structure is two-periodic with the same period in both directions and the inclusions are located within the square net.Period 2b is small in comparison with the characteristic diameter of the composite cross-section, i.e.
Phases of the composite have different heat transfer coefficients and in the matrix (area  Material behaviour in the area of i and i is governed by the Poisson equations of the following form uF in i ; (2.1) where u , u are the functions of temperature distributions regarding the mentioned areas; F stands for the density of heat sources.
In the interface of the matrix with inclusions the following compatibility conditions hold: where n denotes a contour normal to the inclusion.Solution to the boundary value problems (2.1)-(2.4),owing to the homogenization method [12], can be presented in the form of the asymptotic series with respect to a small parameter of the following form: , are fast variables, describing the problem on the structure cell and 1 is the small parameter characterizing the composite periodicity.
1. Solution to the problem is constructed using the modernized three-phase composite model (ТPhMM) being characterised by the following properties: the whole composite structure , except one cell, is substituted by the equivalent homogeneous medium having the known (to be found) heat transfer coefficient .Furthermore, we introduce two circles to describe the square matrix cell contour with the following radii (see Figure 2.3): (2.5) The cell problem in polar coordinates , r can be cast to the following form [16]: the integration is carried out using Eq.(2.5), i.e. in formula (2.13) we take In Figure 2.4 a quadrant of the three phase area is shown, where the integration via the TPhMM is carried out.After integration using Eqs.(2.13), (2.14) and the following formula ,1 a and 0, 1 a .In the latter case it is worthwhile to use the asymptotic representations, which can be obtained on the basis of relations (2.15) and (2.16).

ASYMPTOTIC RELATIONS FOR THE EQUIVALENT HEAT TRANSFER PARAMETER
1) The equivalent heat transfer parameter obtained from the transcendental equations (2.15)-(2.16)satisfies Keller's theorem [9]: Indeed, we have Furthermore, the structure of equations (2.15) and (2.16) allows for a direct use of as a natural small parameter 01 for arbitrary values of 0 and 01 a .Now, having this parameter one may define and investigate further the limiting transitions.
2) Let us investigate a composite having heat transfer of the matrix and inclusions of the same order, i.e.

1.
In this case for arbitrary values of the inclusion size we have: It implies that transcendental Eqs.(2.15) and (2.16) can be reduced to the following forms: 2.1) For inclusions of small geometric size 0 a : 3) Let us study the case of the absolute heat transfer, i.e. when .The physical meaning of the problem implies that in the case of inclusions with infinitely high heat transfer properties and geometric size close to the limiting large ones of the order ~1 a , the homogenized heat transfer coefficient is also infinitely large, i.e.

.
Therefore, transcendental Eqs.(2.15) and (2.16) are transformed to the following form , reported in [13] (with accuracy of the normalisation introduced in [13]) for the effective heat transfer of a composite with circular cross-sections of large cylindrical inclusions having absolute heat transfer properties.
4) Below, we study the entirely resistant heat transfer inclusions, i.e. these with 0 .
4.1) For small additives 0 a we have ~1 1 , and consequently Formula (3.4) for 0 coincides (with accuracy up to the terms of the order of 2 a ) with the results obtained in [1] for the effective heat transfer of the composite consisting of inclusions without heat transfer property.In order to compare the obtained results, the Hashin-Shtrickman (H-S) boundaries are also reported [6,7,18] which are defined through the following relations (2.15) and (2.16)

RESULTS OF COMPUTATION OF THE EFFECTIVE HEAT TRANSFER COEFFICIENT FOUND USING TPHMM
whereas that obtained in [9,13]

15) and (3.7)
  аsympt q  [13]      7) Table 4.3 gives computational results of the homogenized heat transfer coefficient obtained using the TPhMM and their comparison with the analytical solution given in reference [13] for the case of the inclusion size 01 a and large heat conductivity 0 .

8)
In reference [1] for small inclusion size 1, a the formula for equivalent heat transfer coefficient obtained using the Schwarz method of successive approximations is reported.Table 4.4 gives the computational results of homogenized heat transfer coefficient using ThPM [16] as well as the Schwarz method [1] and the TPhMM for the case of various heat transfer and small values of the inclusion size a .

CONCLUSIONS
One may conclude from the data reported in Table 4.1 and from the paper text body that our results constructed through the proposed improved TPhM allow us to improve the results obtained via the classical TPhM approach regarding the estimation of effective heat transfer parameter of the Numerical computation [13] Asymptotics [13] 0.5899 0.7884 4. Inclusions: Large size : 0.9 995 a Large conductivity: 25  10 10 Asymptotics [13] 3.1354

i
) and inclusions (area i ), respectively, where .The characteristic of the periodically repeated cell composite is shown in Figure 2.2.

3. 1 ) 2 )
If size of the inclusions are small, i.e. when We study the case of inclusions of large geometric size 01 a .Latin American Journal of Solids and Structures 10(2013) 197 -222 emphasized that the main term of the series development (3.3) coincides with the asymptotic representation asympt q noting that the solution obtained via the TPhM identically coincides with the lower boundary of the H-S estimation for 1 and regarding the upper boundary for 01 .TPhMM q

Table 4 .
1. Numerical results of estimation of heat transfer coefficient (absolutely conductive inclusions).

Table 4 .
2. Numerical and analytical results of the estimation of heat transfer coefficient (absolutely conductive inclusions).

Table 4 .
3. Computational results of the estimation of effective heat transfer coefficient (large size and large conductivity of inclusions).

Table 4 .
4. Computational results of the estimation of effective heat transfer coefficient (small size of inclusions).

Table 4 .
5gives the mean absolute discrepancy of the computation of effective heat transfer coefficient using the TPhMM versus known results obtained by other authors.

Table 4 .
5. Mean value of the absolute discrepancy of the estimation of heat transfer coefficient using the TPhMM and results obtained by other authors in %