3-D finite element models for the prediction of effective transverse thermal conductivity of unidirectional fibre reinforced composites.

Rao, G. Sambasiva ; Subramanyam, T. ; Murthy, V. Balakrishna 等

Introduction

Prediction of thermo-physical properties of composite material is of great concern as these materials are widely used in aero-space applications. Considerable research is being carried out in this area using analytical, experimental and numerical models. Much effort, time and expense would be saved with the use of latest available software as modelling and analysis becomes much simpler and flexible.

The effective microscopic thermal properties of a composite will in general be dependent on the individual properties of the constituents, their relative volume fractions and their micro-structural arrangement. When the reinforcement consists of aligned long continuous fibres then the effective properties become anisotropic. The fibre direction effective conductivity (k1) of each lamina is satisfactorily predicted by a simple rule of mixtures and is independent of geometrical arrangement, since both fibre and matrix present continuous path for the conduction of heat.

In transverse direction, due to geometry and orientation of fibres, heat transfer path may not be continuous and prediction of effective transverse thermal conductivity ([k.sub.2]) is not straight forward. Although most composites possesses fibres of random distributions, great insight of microstructure on the effective properties can be gained from investigation of composites with periodic structures, thus it is sufficient to draw conclusions for the whole system considering only unit cell.

Analysis of periodic systems date back to the work of Rayleigh [1], who considered the effective electrical conductivity of a composite material with equal sized spherical inclusions arranged in a simple cubic array. The literature dealing with theoretical prediction of the transverse thermal conductivity of composite materials is considerable and several reviews have been published [2-5]. Perrins [6] extended Rayleigh [1] to enable the calculation of the transport properties of circular cylinders in square and hexagonal arrays of unit cells. Grove [7] computed transverse thermal conductivity in continuous unidirectional fibre composite materials using both 2-D FEM and statistical techniques for a range of fibre volume fractions upto 0.5 and [k.sub.f]/[k.sub.m] between 2-500. Yuan Lu [8] used boundary collocation scheme for calculation of transverse effective thermal conductivity of 2-dimensional periodic arrays of long circular and square cylinders with square array and long circular cylinders with hexagonal array for a complete range of fibre volume fractions and for [k.sub.f]/[k.sub.m] between 0 to 8. Islam [9] used 2-dimensional FEM to predict the transverse thermal conductivity of both square and circular cross section fibres for perfect bonding at fibre-matrix interface as well as with interfacial barrier by using four different sets of thermal boundary conditions. Tai [10] proposed two circular filaments in a unit cell of square packing array and obtained the transverse thermal conductivity of unidirectional fibres.

In this paper, 3-dimensional finite element models for the array of square and hexagonal unit cells of long circular cylinders are developed with appropriate thermal boundary conditions at all faces. The models are tested for entire range of fibre volume fractions and [k.sub.f]/[k.sub.m] of 5 to 8. Output is compared with experimental and analytical results presented by various researchers.

Finite element model

Schematic diagrams of the unidirectional fibre composite are shown in 'figures 1& 2', where the fibres are arranged in square and hexagonal arrays respectively. A Representative Volume Element (R.V.E.) in the form of a square and hexagonal unit cells are adopted for the present analysis. The cross-sectional area of fibre relative to the total cross-sectional area of the unit cell ('figures 3 & 4') is a measure of the volume of fibre relative to the total volume of the composite. This fraction is an important parameter in composite materials and is called fibre volume fraction ([V.sub.f]).

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The 1-2-3 coordinate system shown in 'figures 3 & 4' is used to study the behaviour of a unit cell (The direction 1 is along the fibre axis and normal to the plane of the 2D figures shown). The isolated unit cell behaves as a part of a larger array of unit cells.

It is assumed that the geometry, material and loading of the unit cell are symmetrical with respect to 1-2-3 coordinate system. Therefore, a one forth portion of the unit cells are modelled ('figures 5 & 6') for the prediction of mechanical properties. The 3D Finite Element mesh on one forth portion of the unit cell is shown in 'figures 5 & 6'. Width of the unit cell is taken as '2a' and length is taken sufficiently long. Height of the hexagonal unit cell is taken as 1.732 times width of the unit cell.

The element SOLID90 of ANSYS 10.0 is used in the present analysis which has 20 nodes with a single degree of freedom i.e. temperature at each node. The 20-node elements have compatible temperature shapes and are well suited to model curved boundaries.

Temperature boundary conditions for one-fourth model are as follows: T(0,y) = [T.sub.1]; T(a,y) = [T.sub.2]

All other faces are subjected to adiabatic boundary conditions.

Results and discussion

The effective transverse thermal conductivity is calculated using the equation

[q.sub.x] = -[k.sub.2] [delta]T/[delta]x

Heat flux and the temperature gradient in the above equation are obtained from the finite element solution

'Figures 7 & 8' show the comparison of the results with Springer and Tsai [11] and Thornburg and Pears [12] at [k.sub.f]/[k.sub.m] values of 666 and 4.4 for square model. In both cases the proposed model is predicting higher [k.sub.2] than [11]. However at lower fibre volume fractions, reasonably close agreement is observed. In the case of [k.sub.f]/[k.sub.m] = 666, experimental values [12] are higher than the proposed model. This may be due to random distribution of fibres for actual model where some of the fibres may be packed very closely, touching each other which lead to higher effective transverse thermal conductivity.

'Table 1' shows comparison of present square model with the analytical solutions of [6] for [k.sub.f]/[k.sub.m] values of 5, 10, 50, and [infinity] at fibre volume fractions between 0.1 and 0.785. 'Table 2' shows comparison of hexagonal model with analytical solutions of [6] for [k.sub.f]/[k.sub.m] values of 5, 10, 50, and [infinity] at fibre volume fractions between 0.1 and 0.905. It has been observed that the output of present model is in excellent agreement with [6] for all values of [k.sub.f]/[k.sub.m] at all volume fractions in the range.

It is proved that the proposed model is capable of predicting the transverse thermal conductivities as effectively as analytical solutions proposed by various researchers [6,8,9] for entire range of fibre volume fractions. This model can also be extended to predict thermal conductivities in instances where i) fibre anisotropy, ii) voids and/ or cracks in matrix, iii) cracks in fibre and iv) fibre -matrix debond.

The proposed finite element models are applied to predict transverse thermal conductivities of two different composites for a range of matrix conductivities between 0.1 - 1.0 W/m-K which covers different matrix materials commonly in use. The fibres are so selected that one composite (P120, [k.sub.f] = 640 W/m-K) gives higher [k.sub.f]/[k.sub.m] and other (E-Glass, [k.sub.f] = 1.3 W/m-K) with low [k.sub.f]/[k.sub.m].

'Figure 9' shows the effective thermal conductivity of T300 fibre and thermosetting matrix composites for rage of matrix conductivities between 0.1 - 1.0 W/m-K for square array. For lower values of [k.sub.m] a steep raise in [k.sub.2] is observed at higher volume fractions beyond 0.65 and a gradual raise in [k.sub.2] is found with increase in [V.sub.f] as [k.sub.m] increases. However for the total range of [k.sub.m] a liner raise in [k.sub.2] is seen for volume fractions up to 0.65. Fox hexagonal array ('figure 10') similar trend is observed as of square array at fibre volume fraction of 0.7. 'Figure 11' shows comparison of [k.sub.2] for square and hexagonal arrays. For the values of [V.sub.f] up to 0.6 and at maximum possible [V.sub.f] both models are predicting same value. However beyond 0.6 [V.sub.f] square model predicts higher values as it is reaching maximum possible [V.sub.f] earlier than hexagonal model.

'Figures 12 and 13' are showing the variation of thermal conductivities of E-Glass polymer composite. Liner variation of [k.sub.2] is observed for higher values of [k.sub.m] and nonlinearity is seen for lower values of [k.sub.m] beyond 0.6 [V.sub.f] for both square and hexagonal arrays. It is observed from 'figure 14' that both square and hexagonal arrays predict same value of [k.sub.2] for [k.sub.m] beyond 0.5 and at lower values of [k.sub.m] hexagonal array predicts lower [k.sub.2] for [V.sub.f] beyond 0.6. However maximum value of [k.sub.2] is seen in hexagonal array at maximum possible [V.sub.f].

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Conclusions

Three-dimensional finite element models for the array of square and hexagonal unit cells of long circular cylinders are developed with appropriate thermal boundary conditions to predict the thermal conductivity of unidirectional Fibre Reinforced composite lamina. The models are validated for entire range of fibre volume fractions and [k.sub.f]/[k.sub.m] of 5 to 8 with experimental and analytical results presented by various researchers. The transverse thermal conductivity of two different fibre-matrix combinations is evaluated from the developed models. As the proposed models are predicting transverse thermal conductivity at par with analytical solutions, they can be conveniently extended to predict the transverse thermal conductivity of the composite with imperfections which is otherwise difficult using analytical expressions.

References

[1] Rayleigh, L., 1892, "On the influence of obstacles arranged in rectangular order upon the properties of a medium" Philosophy Magazine, 34, pp.481-502.

[2] Hale, D.K., 1976, "The Physical Properties of Composite Materials", Journal of Material Science, 11, pp.2105-2141.

[3] Progelhof, R. C., Throne, J. L. and Ruetsch, R. R., 1976, "Methods for Predicting the Thermal Conductivity of Composite Systems: A Review", Polym. Eng. Sci., 16, pp. 615-625.

[4] Dawson, D.M., and Briggs, A., 1981, "Prediction of the Thermal Conductivities of Insulation Materials", Journal of Material Science, 16, pp. 3346-3356.

[5] Hasim, Z. S., 1983, "Analysis of Composite Materials--A Survey", Trans. ASME Jou. Appl. Mech., 50, pp.481-505.

[6] Perrins, W. T., McKenzie, D. R. and McPhedran, R. C., 1979, "Transport Properties of Regular Arrays of Cylinders", Proc. R. Soc., Lond., A369, pp. 207-223.

[7] Grove, S. M., 1990, "A Model of Transverse Thermal Conductivity in Unidirectional Fibre-Reinforced Composites", Comp. Sci. and Tech., 38, pp.199-209.

[8] Yuan Lu-Shih, 1995, "The Effective Thermal Conductivities of Composites with 2-D arrays of circular and square Cylinders", Jou. of Comp. Materials, 29, pp.483-505.

[9] Islam, R. Md. and Pramila, A., 1999, "Thermal Conductivity of Fibre reinforced Composites by the FEM", Jou. of Comp. Materials, 33, pp.1699-1715.

[10] Tsai, H., 2002, "On the Thermal Model of Transverse Flow of Unidirectional Materials", NASA/TM-2002-211649.

[11] Springer, G. S. and Tsai, S. W., 1967, "Thermal Conductivities of Unidirectional Materials", Jou. of Comp. Materials, 1, pp.166.

[12] Thornburg, J.D. and Pears, C.D., 1965, "Prediction of the Thermal Conductivity of Filled and Reinforced Plastics", ASME Paper 65-WA/HT-4.

G. Sambasiva Rao (1), T. Subramanyam (2) and V. Balakrishna Murthy (1)

(1) Dept. of Mech. Engg., PVP Siddhartha Institute of Technology, Vijayawada-520007, Andhra Pradesh, India

(2) Prof. & Head, Dept. of Mech. Engg., A.U. College of Engineering, Visakhapatnam -530003, Andhra Pradesh, India E-mail ID:intjou@yahoo.co.in

Table 1 : Comparison of the present Square model with analytical
results.

            [k.sub.f]/         [k.sub.f]/         [k.sub.f]/
            [k.sub.m] = 5      [k.sub.m] = 10     [k.sub.m] = 50

[V.sub.f]   Ref[6]   Present   Ref[6]   Present   Ref[6]   Present

0.1         1.143    1.143     1.178    1.178      1.213    1.213
0.2         1.308    1.308     1.392    1.391      1.476    1.476
0.3         1.501    1.501     1.652    1.652      1.813    1.813
0.4         1.731    1.731     1.981    1.981      2.263    2.263
0.5         2.013    2.013     2.416    2.416      2.915    2.915
0.6         2.374    2.374     3.037    3.037      3.988    3.988
0.7         2.871    2.871     4.062    4.062      6.336    6.336
0.75        3.210    3.210     4.944    4.944      9.536    9.536
0.76        3.290    3.290     5.187    5.189     10.819   10.829
0.77        3.375    3.375     5.467    5.468     12.741   12.745
0.78        3.466    3.466     5.804    5.804     16.310   16.312
0.785       3.515    3.515     6.004    6.010     20.500   20.492

            [k.sub.f]/
            [k.sub.m] =
            [infinity]

[V.sub.f]   Ref[6]   Present

0.1          1.222     1.222
0.2          1.500     1.500
0.3          1.860     1.860
0.4          2.351     2.351
0.5          3.080     3.080
0.6          4.342     4.342
0.7          7.433     7.433
0.75        12.751    12.752
0.76        15.441    15.463
0.77        20.433    20.441
0.78        35.934    35.945
0.785       --        137.608

Table 2 : Comparison of the present Hexagonal model with analytical
results

            [k.sub.f]/         [k.sub.f]/        [k.sub.f]/
            [k.sub.m] = 5      [k.sub.m] = 10    [k.sub.m] = 50

[V.sub.f]   Ref[6]   Present   Ref6]   Present   Ref[6]   Present

0.1         1.143    1.143     1.178   1.178      1.213    1.213
0.2         1.308    1.308     1.391   1.391      1.476    1.476
0.3         1.500    1.500     1.651   1.651      1.810    1.810
0.4         1.727    1.727     1.973   1.973      2.249    2.249
0.5         2.001    2.001     2.387   2.387      2.853    2.853
0.6         2.337    2.337     2.938   2.938      3.743    3.743
0.65        2.536    2.536     3.291   3.291      4.374    4.374
0.70        2.763    2.763     3.719   3.719      5.215    5.215
0.73        2.915    2.915     4.025   4.024      5.873    5.873
0.76        3.082    3.082     4.378   4.378      6.708    6.708
0.78        3.202    3.202     4.647   4.647      7.404    7.404
0.80        3.331    3.331     4.949   4.949      8.260    8.260
0.82        3.470    3.470     5.292   5.292      9.348    9.348
0.84        3.620    3.620     5.689   5.689     10.793   10.793
0.85        3.700    3.700     5.912   5.912     11.720   11.721
0.86        3.784    3.784     6.155   6.155     12.850   12.849
0.87        3.872    3.872     6.424   6.424     14.273   14.273
0.88        3.965    3.964     6.723   6.722     16.157   16.156
0.89        4.062    4.062     7.061   7.061     18.853   18.852
0.895       4.114    4.114     7.249   7.249     20.761   20.761
0.90        4.167    4.166     7.454   7.454     23.397   23.395
0.905       4.221    4.221     7.679   7.680     27.680   27.675

            [k.sub.f]/
            [k.sub.m] =
            [infinity]

[V.sub.f]   Ref[6]    Present

0.1           1.222     1.222
0.2           1.500     1.500
0.3           1.857     1.857
0.4           2.334     2.334
0.5           3.005     3.005
0.6           4.027     4.027
0.65          4.776     4.776
0.70          5.811     5.811
0.73          6.652     6.652
0.76          7.760     7.760
0.78          8.722     8.723
0.80          9.959     9.958
0.82         11.621    11.621
0.84         14.009    14.009
0.85         15.666    15.665
0.86         17.838    17.837
0.87         20.857    20.857
0.88         25.451    25.449
0.89         33.701    33.697
0.895        41.341    41.336
0.90         56.229    56.219
0.905       112.800   112.701