# The Thermal Conductivities of Periodic Fibrous Composites as Defined by a Mathematical Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation of Fiber Arrangement

- (i)
- The fibers are perfectly cylindrical in shape.
- (ii)
- All cross-sectional areas of the cylindrical model have the same microstructure.
- (iii)
- The matrix and the fibers are elastic, isotropic and homogenous, and the fiber arrangement is uniform so that no agglomeration occurs.

`h`. Next, to create a unit cell capable of simulating unidirectional fibrous composites of a periodic structure, one may consider a second prism of side $2\ell $ so as to circumscribe the first one, as can be seen in Figure 2.

_{f}is estimated as:

_{f}$\le $ 92.25%.

_{f}$\le $ 74.99%.

_{f}$\le $ 84.91%.

## 3. The Concept of the Interphase: Towards a Seven-Phase Cylindrical Model

_{1}represents the first area of the filler. The second phase is the cylindrical shell with the inner radius r

_{1}and outer radius r

_{2}and represents the first region of the intermediate phase. Furthermore, the third phase is the cylindrical shell of inner radius r

_{2}and outer radius r

_{3}and shows the first region of the matrix. The fourth phase is the cylindrical shell with the inner radius r

_{3}and the outer radius r

_{4}and represents the second region of the intermediate phase. The fifth phase is the cylindrical shell of inner radius r

_{4}and outer radius r

_{5}and represents the second region of the filler. The sixth stage is the cylindrical shell of inner radius r

_{5}and outer radius r

_{6}representing the third region of the interphase. The seventh and last phase is the cylindrical shell with an inner radius r

_{6}and an outer radius r

_{7}and represents the second region of the matrix.

_{f}arising from the geometric constraints of the prismatic models introduced in the previous section, i.e., Inequalities (11), (12), (22), (23), also concern the new seven-phase model. In the sequel, one may evaluate the volume fractions of all phases in terms of the corresponding radii as follows.

## 4. Materials and Experimental Work

^{−5}m and were contained at a volume fraction of about 65%. The fiber content was determined, as customary, by igniting samples of the composite and weighing the residue, which gave the weight fraction of glass as: W

_{f}= 79.6 ± 0.28%. This and the measured values of the relative densities of Permaglass (p

_{f}= 2.55 g/cm

^{3}) and of the epoxy matrix (p

_{m}= 1.20 g/cm

^{3}) gave the value U

_{f}= 0.65. Furthermore, chip specimens with a 0.004-m diameter and thicknesses varying between 0.001 m and 0.0015 m made either of the fiber composite of different filler contents or of the matrix material were tested by the authors on a differential scanning calorimetry (DSC) thermal analyzer at the zone of the glass transition temperature for each mixture, in order to determine the specific heat capacity values.

_{f}= 1.2 W/m·K, whilst the thermal conductivity of the matrix is K

_{f}= 0.2 W/m·K.

## 5. Estimation of Thermal Conductivities

_{i}< K

_{f}.

_{i}. The corresponding mathematical expression of this requirement can be formulated as follows:

_{i}(r) as expressed by the following formula:

_{i}(r) emerges from the following expression:

_{i}(r) varies according to a generic exponential law in the following form.

## 6. Discussion

_{f}= 0.5) obtained from Mutnuri [15] almost coincides with the corresponding theoretical value yielded by the two-phase inverse mixtures law, whereas it is well below the graphs of Equation (65), the Hashin and Springer-Tsai formulae with respect to U

_{f}.

_{f}= 0.65, which generally constitutes the optimum fiber content above which the reinforcing action of the fibers is upset.

_{f}= 0, increases, reaching a unique peak value and then decreases tending to zero at U

_{f}→ 1, given that ${\mathrm{U}}_{\mathrm{f}}=1-{\mathrm{U}}_{\mathrm{m}}-{\mathrm{U}}_{i}$.

_{f}.

_{f}= 0 and ${\mathrm{U}}_{i}$ = 0 at U

_{f}→ 1

^{−}, which obviously denote the minimum value of the interphase volume fraction.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

- (i)
- Longitudinal thermal conductivity:
- (a)
- Rayleigh’s formula [1]:$${\mathrm{K}}_{c}={\mathrm{K}}_{\mathrm{m}}\left(1+\frac{{\mathrm{K}}_{\mathrm{f}}-{\mathrm{K}}_{\mathrm{m}}}{{\mathrm{K}}_{\mathrm{m}}}\right){\mathrm{U}}_{\mathrm{f}}$$
- (b)
- Springer–Tsai’s Formula [4]:$${\mathrm{K}}_{11}={\mathrm{K}}_{\mathrm{f}}{\mathrm{U}}_{\mathrm{f}}+{\mathrm{K}}_{\mathrm{m}}{\mathrm{U}}_{\mathrm{m}}$$

- (ii)
- Transverse thermal conductivity:
- (a)
- Inverse rule of mixtures:$${\mathrm{K}}_{\mathrm{T}c}=\frac{{\mathrm{K}}_{\mathrm{f}}{\mathrm{K}}_{\mathrm{m}}}{{\mathrm{K}}_{\mathrm{f}}{\mathrm{U}}_{\mathrm{m}}+{\mathrm{K}}_{\mathrm{m}}{\mathrm{U}}_{\mathrm{f}}}$$Here, fibers and matrix are assumed somewhat as solid blocks with volumes analogous to their relative abundance in the entire material.
- (b)
- Hashin’s formula:$${\mathrm{K}}_{\mathrm{T}c}=\frac{{\mathrm{K}}_{\mathrm{m}}{\mathrm{U}}_{\mathrm{m}}+{\mathrm{K}}_{\mathrm{f}}(1+{\mathrm{U}}_{\mathrm{f}})}{{\mathrm{K}}_{\mathrm{m}}(1+{\mathrm{U}}_{\mathrm{f}})+{\mathrm{K}}_{\mathrm{f}}{\mathrm{U}}_{\mathrm{m}}}$$
- (c)
- Springer-Tsai’s formula:$${\mathrm{K}}_{22}={\mathrm{K}}_{\mathrm{m}}\left[1-2\sqrt{\frac{{\mathrm{U}}_{\mathrm{f}}}{\mathsf{\pi}}}\right]+\frac{1}{2\left(\frac{{\mathrm{K}}_{\mathrm{m}}}{{\mathrm{K}}_{\mathrm{f}}}-1\right)}\left\{\mathsf{\pi}-\frac{4}{\sqrt{1-\left[\frac{{U}_{\mathrm{f}}}{\mathsf{\pi}}\left(4{\left(\frac{{\mathrm{K}}_{\mathrm{m}}}{{\mathrm{K}}_{\mathrm{f}}}-1\right)}^{2}\right)\right]}}\cdot \mathrm{arctan}\left[\frac{\sqrt{1-\left[\frac{{\mathrm{U}}_{\mathrm{f}}}{\mathsf{\pi}}\left(4{\left(\frac{{\mathrm{K}}_{\mathrm{m}}}{{\mathrm{K}}_{\mathrm{f}}}-1\right)}^{2}\right)\right]}}{1+2\left(\frac{{\mathrm{K}}_{\mathrm{m}}}{{\mathrm{K}}_{\mathrm{f}}}-1\right)\sqrt{\frac{{\mathrm{U}}_{\mathrm{f}}}{\mathsf{\pi}}}}\right]\right\}$$

## References

- Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena; John Wiley & Sons: New York, NY, USA, 2007. [Google Scholar]
- Hashin, Z.; Rosen, B.W. The elastic moduli of fiber reinforced materials. J. Appl. Mech.
**1964**, 46, 543. [Google Scholar] [CrossRef] - Hashin, Z. Analysis of properties of fiber composites with anisotropic constituents. ASME Appl. Mech.
**1979**, 46, 543–550. [Google Scholar] [CrossRef] - Springer, G.; Tsai, S. Thermal conductivity of unidirectional materials. J. Compos. Mater.
**1967**, 1, 166–173. [Google Scholar] [CrossRef] - Papanicolaou, G.C.; Paipetis, S.A.; Theocaris, P.S. The Concept of Boundary Interphase in Composite Mechanics. Colloid Polym. Sci.
**1978**, 256, 625–630. [Google Scholar] [CrossRef] - Theocaris, P.S.; Papanicolaou, G.C. The Effect of the Boundary Interphase on the Thermomechanical Behaviour of Composites Reinforced with Short Fibers. J. Fibre Sci. Technol.
**1979**, 12, 421–433. [Google Scholar] [CrossRef] - Papanicolaou, G.C.; Theocaris, P.S.; Spathis, G.D. Adhesion Efficiency between Phases in Fiber-Reinforced Polymers by Means of the Concept of Boundary Interphase. Colloid Polym. Sci.
**1980**, 258, 1231–1237. [Google Scholar] [CrossRef] - Kanaun, S.K.; Kudriavtseva, L.T. Spherically layered inclusions in a homogeneous elastic medium. Appl. Math. Mech.
**1983**, 50, 483–491. [Google Scholar] [CrossRef] - Kanaun, S.K.; Kudriavtseva, L.T. Elastic and thermoelastic characteristics of composites reinforced with unidirectional fibre layers. Appl. Math. Mech.
**1986**, 53, 628–636. [Google Scholar] [CrossRef] - Clements, L.L.; Moore, R.L. Composite Properties for E-Glass Fibers in a Room Temperature Curable Epoxy Matrix. Composites
**1978**, 9, 93–99. [Google Scholar] [CrossRef] - Caruso, J.; Chamis, C. Assessment of simplified composite micromechanics using three dimensional finite element analysis. J. Compos. Technol. Res.
**1986**, 8, 77–83. [Google Scholar] - Muralidhar, K. Equivalent conduction of a heterogeneous medium. Int. J. Heat Mass Transf.
**1989**, 33, 1759–1766. [Google Scholar] [CrossRef] - Gusev, A.A.; Hine, P.J.; Ward, I.M. Fiber packing and elastic properties of a transversely random unidirectional glass/epoxy composite. Compos. Sci. Technol.
**2000**, 6, 535–541. [Google Scholar] [CrossRef] - Huang, Z.M. Micromechanical prediction of ultimate strength of transversely isotropic fibrous composites. Mater. Lett.
**1999**, 40, 164–169. [Google Scholar] [CrossRef] - Mutnuri, B. Thermal Conductivity Characterization of Composite Materials. Master’s Thesis, West Virginia University, Morgantown, WV, USA, 2006. [Google Scholar]
- Lei, Z.; Li, X.; Qin, F.; Qiu, W. Interfacial Micromechanics in Fibrous Composites: Design, Evaluation, and Models. Sci. World J.
**2014**, 2014, 282436. [Google Scholar] [CrossRef] [PubMed] - Bonnet, G. Effective properties of elastic periodic composite media with fibers. J. Mech. Phys. Solids
**2007**, 55, 881–899. [Google Scholar] [CrossRef] - Selvadurai, A.P.S.; Nikopour, H. Transverse elasticity of a unidirectionally reinforced composite with an irregular fibre arrangement: Experiments, theory and computations. Compos. Struct.
**2012**, 94, 1973–1981. [Google Scholar] [CrossRef] - Shah, S.Z.H.; Choudhry, R.S.; Khan, L.A. Investigation of compressive properties of 3D fiber reinforced polymeric (FRP) composites through combined end and shear loading. J. Mech. Eng. Res.
**2015**, 7, 34–48. [Google Scholar] [CrossRef] - Guinovart-Díaz, R.; Rodríguez-Ramos, R.; Bravo-Castillero, J.; López-Realpozo, J.C.; Sabina, F.J.; Sevostianov, I. Effective elastic properties of a periodic fiber reinforced composite with parallelogram-like arrangement of fibers and imperfect contact between matrix and fibers. Int. J. Solids Struct.
**2013**, 50, 2022–2032. [Google Scholar] [CrossRef] - Venetis, J.; Sideridis, E. Elastic constants of fibrous polymer composite materials reinforced with transversely isotropic fibers. AIP Adv.
**2015**, 5, 037118. [Google Scholar] [CrossRef] - Li, X.; Wang, F. Effect of the Statistical Nature of Fiber Strength on the Predictability of Tensile Properties of Polymer Composites Reinforced with Bamboo Fibers: Comparison of Linear- and Power-Law Weibull Models. Polymers
**2016**, 8, 24. [Google Scholar] [CrossRef] - Venetis, J.; Sideridis, E. Thermal conductivity coefficients of unidirectional fiber composites defined by the concept of interphase. J. Adhes.
**2015**, 91, 262–291. [Google Scholar] [CrossRef] - Rodrigo, P.A.R.; Anuel, M.; Cruz, E. Computation of the effective thermal conductivity of unidirectional fibrous composites with an interfacial thermal resistance. Numer. Heat Transf. Part A
**2001**, 39, 179–203. [Google Scholar] [CrossRef] - Kytopoulos, V.N.; Sideridis, E.P. Thermal conductivity of particulate composites by a hexaphase model. JP J. Heat Mass Transf.
**2016**, 13, 395–407. [Google Scholar] [CrossRef] - Venetis, J.; Sideridis, E. A mathematical model for thermal conductivity of homogeneous composite materials. Indian J. Pure Appl. Phys.
**2016**, 54, 313–320. [Google Scholar] - Theocaris, P.S. The Mesophase Concept in Composites, Polymers-Properties and Applications; Henrici-Olivé, G., Olivé, S., Eds.; Springer: Berlin, Germany, 1987; Volume 11, ISBN 978-3-642-70184-9. [Google Scholar]
- Lipatov, Y.S. Physical Chemistry of Filled Polymers, published by Khimiya (Moscow 1977). Translated from the Russian by Moseley, R.J. Int. Polym. Sci. Technol.
**1997**, 22, 1–59. [Google Scholar] - Theocaris, P.S.; Sideridis, E.P.; Papanicolaou, G.C. The Elastic Longitudinal Modulus and Poisson’s Ratio of Fiber Composites. J. Reinf. Plast. Comp.
**1985**, 4, 396–418. [Google Scholar] [CrossRef] - Bigg, D.M. Thermally Conductive Polymer Compositions. Polym. Compos.
**1986**, 7, 125. [Google Scholar] [CrossRef]

**Figure 8.**Transformation of the three unit cells into a seven-phase cylindrical model, after the consideration of the interphase concept.

U_{f} | U_{i} | r_{1} (μm) | r_{2} (μm) | r_{3} (μm) | r_{4} (μm) | r_{5} (μm) | r_{6} (μm) | r_{7} (μm) |
---|---|---|---|---|---|---|---|---|

0.10 | 0.0012 | 6 | 6.062 | 25.21 | 25.24 | 27.9 | 27.926 | 42.426 |

0.20 | 0.00492 | 6 | 6.127 | 16.74 | 16.79 | 20.62 | 20.692 | 30 |

0.30 | 0.01107 | 6 | 6.160 | 12.75 | 12.92 | 17.54 | 17.667 | 24.495 |

0.40 | 0.01968 | 6 | 6.181 | 10.28 | 10.38 | 15.77 | 15.976 | 21.213 |

0.50 | 0.03075 | 6 | 6.169 | 8.329 | 8.49 | 14.6 | 14.91 | 18.974 |

0.60 | 0.04428 | 6 | 6.084 | 6.770 | 6.85 | 13.77 | 14.211 | 17.321 |

0.65 | 0.052 | 6 | 6.007 | 6.053 | 6.08 | 13.44 | 13.964 | 16.641 |

U_{1} | U_{2} | U_{3} | U_{4} | U_{5} | U_{6} | U_{7} |
---|---|---|---|---|---|---|

0.02 | 0.0005 | 0.331 | 0.0005 | 0.08 | 0.00076 | 0.56674 |

0.04 | 0.0017 | 0.283 | 0.0019 | 0.14 | 0.00324 | 0.52426 |

0.06 | 0.0032 | 0.214 | 0.0035 | 0.22 | 0.00771 | 0.47979 |

0.08 | 0.0048 | 0.162 | 0.0051 | 0.32 | 0.01468 | 0.43282 |

0.10 | 0.0058 | 0.087 | 0.00569 | 0.411 | 0.02506 | 0.38244 |

0.12 | 0.0061 | 0.0286 | 0.0062 | 0.48 | 0.04068 | 0.32682 |

0.13 | 0.0075 | 0.002 | 0.0073 | 0.52 | 0.05163 | 0.29587 |

${\mathbf{U}}_{\mathbf{f}}$ | ${\overline{\mathbf{K}}}_{\mathit{i}}$ (W/mK) Equation (62) | ${\mathbf{K}}_{\mathbf{L}\mathit{c}}$ (W/mK) Equation (63) | Rayleigh [1] | Springer-Tsai [4] | Clements and Moore Experiment Values [10] | Mutnuri Experiment Value [15] |
---|---|---|---|---|---|---|

0 | 0.2 | 0.2 | 0 | 0.2 | ||

0.1 | 0.192 | 0.329 | 0.22 | 0.3 | ||

0.2 | 0.208 | 0.441 | 0.44 | 0.4 | ||

0.3 | 0.248 | 0.573 | 0.66 | 0.5 | ||

0.4 | 0.312 | 0.687 | 0.88 | 0.6 | ||

0.5 | 0.4 | 0.796 | 1.1 | 0.7 | 0.471 | |

0.6 | 0.512 | 0.899 | 1.32 | 0.8 | 1.06 | |

0.65 | 0.577 | 0.970 | 1.43 | 0.85 | 1.14 |

U_{f} | ${\mathbf{K}}_{\mathbf{L}\mathit{c}}$ (W/mK) Equation (65) | Two-Phase Inverse Mixtures Law | Springer-Tsai [4] | Hashin [3] | Clements and Moore Experiment Values [10] | Mutnuri Experiment Value [15] |
---|---|---|---|---|---|---|

0 | 0.2 | 0.2 | 0.2 | 0.2 | ||

0.1 | 0.242 | 0.218 | 0.252 | 0.232 | ||

0.2 | 0.265 | 0.241 | 0.301 | 0.267 | ||

0.3 | 0.301 | 0.267 | 0.373 | 0.309 | ||

0.4 | 0.354 | 0.302 | 0.428 | 0.360 | ||

0.5 | 0.443 | 0.343 | 0.511 | 0.422 | 0.348 | |

0.6 | 0.522 | 0.411 | 0.592 | 0.501 | 0.55 | |

0.65 | 0.553 | 0.436 | 0.613 | 0.547 | 0.59 |

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Venetis, J.; Sideridis, E.
The Thermal Conductivities of Periodic Fibrous Composites as Defined by a Mathematical Model. *Fibers* **2017**, *5*, 30.
https://doi.org/10.3390/fib5030030

**AMA Style**

Venetis J, Sideridis E.
The Thermal Conductivities of Periodic Fibrous Composites as Defined by a Mathematical Model. *Fibers*. 2017; 5(3):30.
https://doi.org/10.3390/fib5030030

**Chicago/Turabian Style**

Venetis, John, and Emilio Sideridis.
2017. "The Thermal Conductivities of Periodic Fibrous Composites as Defined by a Mathematical Model" *Fibers* 5, no. 3: 30.
https://doi.org/10.3390/fib5030030