Finite element method is effectively used to homogenize the thermal conductivity of FRP composites consisting of hybrid materials and fibre-matrix debonds at some of the fibres. The homogenized result at microlevel is used to determine the property of the layer using macromechanics principles; thereby, it is possible to minimize the computational efforts required to solve the problem as in state through only micromechanics approach. The working of the proposed procedure is verified for three different problems: (i) hybrid composite having two different fibres in alternate layers, (ii) fibre-matrix interface debond in alternate layers, and (iii) fibre-matrix interface debond at one fibre in a group of four fibres in one unit cell. It is observed that the results are in good agreement with those obtained through pure micro-mechanics approach.
Composite materials are extensively used in many fields of engineering such as aerospace, electronic packaging, reactor vessels, and turbines, due to light weight, high strength, long durability, stability against chemical reaction, tailorable properties, and so forth. FRP composites can be designed as heat conductors for enhancing heat transfer rate and also for insulation purpose; this depends on the thermal properties, volume fraction, orientation, and so forth, of each constituent of the composite. The effective thermal conductivity and other thermophysical properties of composites have been a topic of considerable theoretical, experimental, and numerical interest from long period.
Composite materials are nonhomogeneous and exhibit anisotropic response due to structural and thermal loads. Analysis of a composite structure as in state of heterogeneity by providing the material properties of constituent materials is mathematically complex, and therefore theories such as micromechanics and macromechanics are developed for the theoretical analysis. The homogenized properties of a composite lamina obtained from micromechanical theories are used for the macromechanical analysis of a composite made of several individual laminas stacked in a specified manner.
The micromechanical theories select a particular portion of the composite known as “Representative Volume Elements” (RVE) and find the properties of RVE which are considered to be lamina properties. In this approach, there are many assumptions such as fibres which are arranged in a particular pattern (square/hexagonal) in a matrix, no voids in the matrix, all fibres are of uniform cross-section and perfectly aligned, the interface between the fibre and matrix is perfectly or totally debond; this leads to much deviation of theoretical and experimental results. Numerical approaches such as Finite Element Method (FEM) are developed to overcome some of the assumptions of micromechanical theories but still not explored in addressing many complexities in micromechanical approach. Though FEM is an approximate method, it can be effectively used after proper mesh refinement and validation.
Aligned fibre composite laminates are frequently used in beam, plate, or shell form. The axial thermal conductivity (in the fibre direction) of each lamina is satisfactorily predicted by a simple rule of mixtures Chawla [
Earlier several researchers studied thermal conductivities of composites by experimental, theoretical, and numerical approaches. Prediction of effective transverse thermal conductivity of fibre reinforced composites is made for several models, such as experimental determination of effective thermal conductivity of aligned fibre composite of Chamis [
In the present analysis, a 3D FEM is proposed to address various nonsimilarities in the unit cells at microlevel and developing equivalence between micro- and macromechanical approaches through some of the examples of heat conduction. Three cases are considered for present study: case (i) hybrid composite constituting two different thermal conductivity fibres and matrix, case (ii) composite with alternate layers of fibres fully debonded, and case (iii) composite with one in set of four fibres of a unit cell is fully debonded.
Figures
Concept of square unit cell for aligned pattern of hybrid composite with two different types of fibres.
One-fourth model of unit cell for aligned pattern of fibres.
Concept of square unit cell for aligned pattern of composite with alternate layers of debonded fibres.
One-fourth model of unit cell for composite with alternate layers of debonded fibres.
Concept of square unit cell for aligned pattern of composite with one in set of four fibres of unit cell totally debonded.
Full model for aligned pattern of composite with one in set of four fibres of unit cell totally debonded.
In macromechanics approach for first two cases, two blocks are modelled one over the other. For case (i) one block represents fibre1 and matrix portion and the other block is for fibre2 and matrix portion. Case (ii) one block represents total debonded fibre matrix and other perfectly bonded fibre matrix, whereas for case (iii) four blocks are modelled in square pattern. One block represents debonded fibre and matrix portion and other three blocks for perfectly bonded fibre matrix. Surfaces at the junction of blocks are merged for heat transfer connectivity without any interfacial thermal barrier.
The problem is modelled in commercial Finite Element Software ANSYS 12 [
Geometry and FE mesh of a 3D model for case (i) and case (ii).
Geometry and FE mesh of a 3D model for case (iii).
Temperature difference (
For the validation of the models developed in the present analysis, the following properties of fibres and matrix [
Polyimide matrix with thermal conductivity
(T-300) Carbon fibre with thermal conductivity
For all the three cases of the study, matrix is polyimide matrix.
For the hybrid model, fibre1 : fibre2 : matrix thermal conductivity ratios used are 100 : 10 : 1, 80 : 8 : 1, 60 : 6 : 1, and 40 : 4 : 1.
For the debond models, fibre thermal conductivities range from 0.14 W/mK to
The analytical solution for 1D heat conduction in homogeneous slabs is readily available. However, the analytical solution for 1D heat conduction is quite complex for heterogeneous materials such as fibre reinforced composite materials. Thus, the numerical finite element models have been developed to suit the different cases under consideration for this study. The models are first tested for mesh-independent solution by imposing earlier stipulated boundary conditions; then with the heat flux obtained from ANSYS software, effective thermal conductivity of composite is found by Fourier’s law of heat conduction [
FE model developed in ANSYS is first validated with the Hasselman and Johnson model (H-J) [
Variation in transverse thermal conductivity with respect to Vf for perfect bond and total debond conditions.
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Variation in
Variation in
Variation in
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Variation in
Variation in
Variation in
Figures
Variation in
Variation of
Table
% Error between macro- and micromechanics approach results for case-(i) hybrid composite.
Vf | 100 : 10 : 01 | 80 : 08 : 01 | 60 : 06 : 01 | 40 : 04 : 01 | ||||
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% Error | % Error | % Error | % Error | % Error | % Error | % Error | % Error | |
0.1 | 0.025 | 0.022 | 0.008 | 0.010 | 0.000 | 0.003 | 0.000 | 0.010 |
0.2 | 0.021 | 0.013 | 0.007 | 0.024 | 0.028 | 0.006 | 0.029 | 0.022 |
0.3 | 0.017 | 0.009 | 0.029 | 0.016 | 0.065 | 0.004 | 0.074 | 0.057 |
0.4 | 0.037 | 0.028 | 0.057 | 0.038 | 0.093 | 0.060 | 0.160 | 0.095 |
0.5 | 0.085 | 0.037 | 0.125 | 0.046 | 0.173 | 0.097 | 0.294 | 0.177 |
0.6 | 0.136 | 0.103 | 0.193 | 0.143 | 0.269 | 0.230 | 0.440 | 0.389 |
0.7 | 0.212 | 0.369 | 0.273 | 0.502 | 0.371 | 0.708 | 0.521 | 1.092 |
Table
% Error between macro- and micromechanics approaches for composite with alternate layers of fibres fully debonded.
Vf |
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% Error | % Error | % Error | % Error | % Error | % Error | % Error | % Error | % Error | % Error | |
0.1 | 0.01 | 0.02 | 0.10 | 0.10 | 0.07 | 0.12 | 0.09 | 0.11 | 0.09 | 0.10 |
0.2 | 0.01 | 0.13 | 0.37 | 0.34 | 0.35 | 0.43 | 0.35 | 0.43 | 0.34 | 0.44 |
0.3 | 0.06 | 0.26 | 0.82 | 0.77 | 0.82 | 0.88 | 0.82 | 0.88 | 0.82 | 0.88 |
0.4 | 0.20 | 0.41 | 1.37 | 1.34 | 1.38 | 1.51 | 1.39 | 1.51 | 1.40 | 1.51 |
0.5 | 0.31 | 0.88 | 1.89 | 1.93 | 1.92 | 2.27 | 1.93 | 2.28 | 1.94 | 2.28 |
0.6 | 0.75 | 0.87 | 2.24 | 2.30 | 2.28 | 2.53 | 2.30 | 2.54 | 2.30 | 2.54 |
0.7 | 1.45 | 1.82 | 2.03 | 2.08 | 1.97 | 3.08 | 1.96 | 3.09 | 1.97 | 3.09 |
Table
% Error between macro- and micromechanics approach results for composite with one out of four fibres of unit cell fully debonded.
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% Error | % Error | % Error | % Error | % Error | |
0.1 | 0.01 | 0.01 | 0.03 | 0.01 | 0.01 |
0.2 | 0.01 | 0.02 | 0.06 | 0.06 | 0.08 |
0.3 | 0.03 | 0.10 | 0.15 | 0.15 | 0.15 |
0.4 | 0.15 | 0.26 | 0.36 | 0.37 | 0.35 |
0.5 | 0.34 | 0.61 | 0.79 | 0.79 | 0.78 |
0.6 | 0.93 | 1.27 | 1.54 | 1.56 | 1.56 |
0.7 | 2.08 | 2.65 | 3.23 | 3.28 | 3.28 |
Effective longitudinal thermal conductivity for three cases matches with the simple rule of mixtures and increases in linear manner with the increase in Vf (Figures
Transverse thermal conductivities
For case (ii) inplane thermal conductivity
For case (ii) through-thickness thermal conductivity
Applicability of homogenization technique by using finite element method is successfully tested for predicting effective thermal conductivities of a hybrid and thermal contact resistance models for different volume fractions within practically possible range of 10% to 70% and thermal conductivity mismatch ratio 0.74 to