# An Upper Bound of Longitudinal Elastic Modulus for Unidirectional Fibrous Composites as Obtained from Strength of Materials Approach

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## Abstract

**:**

## 1. Introduction

## 2. Towards the Determination of an Upper Bound for Longitudinal Modulus of the Composite

- The fibre arrangement inside the matrix is uniform.
- The adhesion efficiency is perfect.
- Matrix is free of voids.
- The applied loads are either parallel or perpendicular to the fibre direction.
- The entire material is initially in a stress free condition, a fact that implies that no residual stresses occur.
- Fibre and matrix behave as linearly elastic isotropic materials.

- Approximation of the stiffness of interphase layer by a polynomial function or any other arbitrary continuous function with respect to the radius of the coaxial cylindrical three layer model.
- Estimation of the averaging values of stiffness for the interphase zone. This procedure takes place to accommodate the calculations, as can be observed in a past paper [15].
- Measurement of the thickness of interphase zone by means of Differential Scanning Calorimetry (DSC) experiments.

## 3. Results and Discussion

_{f}= 72 GN/m

^{2}, and E

_{m}= 3.5 GN/m

^{2}whereas the Poisson’s ratios of fibre and matrix are ν

_{f}= 0.2 and ν

_{m}= 0.35, respectively. In the same table, the theoretical values obtained from Theocaris et al. [11], according to linear, hyperbolic, and parabolic variation laws also occur. Besides, the theoretical values yielded by two and three term unfolding models derived from Theocaris [14] are presented. Finally, the theoretical values achieved by the expression for apparent Young’s modulus in the direction of fibres obtained from Sideridis et al. [15], according to three variation laws are exhibited.

_{L}arising from Equation (21) by the aid of (20) are very close to those yielded by all the other theoretical formulae used for comparison. In addition, they are in good agreement with the experimental values obtained from Sideridis [13]. Next, for medium fibre concentrations by volume one may point out a deviation, especially between the values obtained from Equation (21), as well as from Sideridis’ upper bound [15] with the predictions given by Theocaris’ two and three term unfolding model. Yet, a comparison between the values obtained from Equation (21) and those arising from Theocaris et al. [11] yields a smaller deviation.

_{L}in the final expression of longitudinal modulus, although it was observed [25] that the parabolic variation law generally yields the lowest values for the interphase stiffness of unidirectional fibrous composites when compared with other laws that are commonly used. On the other hand, one may also pinpoint that since according to strength of materials approach fibres, matrix and interphase are supposed somewhat as solid blocks the volumes of which are proportional to their relative abundance in the overall material instead of the modified form of Hashin–Rosen cylinder assemblage model presented in Figure 1 one may adopt the following simplified model (see Figure 4) to simulate the microstructure of the unidirectional fibrous composite.

## 4. Conclusions

- Theoretical approximation of interphase stiffness,
- Estimation of its averaging values,
- Measurement of interphase thickness via DSC experiments.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

- Paul, B. Prediction of Elastic Constants of Multiphase Materials, Transactions of the Metallurgical Society of AIME; American Institute of Mining, Metallurgical, and Petroleum Engineers: New York, NY, USA, 1960; p. 36. [Google Scholar]
- Hill, R. Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids
**1963**, 11, 357. [Google Scholar] [CrossRef] - Hashin, Z.; Rosen, B.W. The Elastic Moduli of Fiber-Reinforced Materials. J. Appl. Mech. BIE
**1964**, 31, 223–232. [Google Scholar] [CrossRef] - Chamis, C.C.; Sendeckyj, G.P. Critique on Theories predicting properties of Fibrous Composites. J. Compos. Mater.
**1968**, 2, 332–358. [Google Scholar] [CrossRef] - Hashin, Z. The Elastic Moduli of Heterogeneous Materials. J. Appl. Mech.
**1962**, 29, 143–150. [Google Scholar] [CrossRef][Green Version] - Lipatov, Y. Physical Chemistry of Filled Polymers; Khimiya: Moscow, Russia, 1977. [Google Scholar]
- 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 fibres. Fibre Sci. Techol.
**1979**, 12, 421–433. [Google Scholar] [CrossRef] - Papanicolaou, G.C.; Theocaris, P.S.; Spathis, G.D. Adhesion efficiency between phases in fibre-reinforced polymers by means of the concept of boundary interphase. Colloid Polym. Sci.
**1980**, 258, 1231–1237. [Google Scholar] [CrossRef] - Theocaris, P.S. The Adhesion Quality and the Extent of the Mesophase in Particulates. J. Reinf. Plast. Compos.
**1984**, 3, 204–231. [Google Scholar] [CrossRef] - Theocaris, P.S.; Sideridis, E.P.; Papanicolaou, G.C.; Reinf, J. The elastic longitudinal modulus and Poisson’s ratio for fiber composites. J. Reinf. Plast. Compos.
**1985**, 4, 396–418. [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] - Sideridis, E. Study of the Thermomechanical Properties of (Iron) Particle and (Glass) Fibre Reinforced (Epoxy) Composite Materials by the Concept of Interphase. Ph.D. Thesis, National Technical University of Athens (NTUA), Athens, Greece, 1996. (In Greek). [Google Scholar]
- Theocaris, P.S. The unfolding model for the representation of the mesophase layer in composites. J. Appl. Polym. Sci.
**1985**, 30, 621–645. [Google Scholar] [CrossRef] - Sideridis, E.; Papadopoulos, G.A.; Kyriazi, E. Strength of Materials and Elasticity Approach to Stiffness of Fibrous Composites Using the Concept of Interphase. J. Appl. Polym. Sci.
**2005**, 95, 1578–1588. [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] - Huang, Z.M. Simulation of the mechanical properties of fibrous composites by the bridging micromechanics model. Compos. Part A Appl. Sci. Manuf.
**2001**, 32, 143–172. [Google Scholar] [CrossRef] - Bonnet, G. Effective properties of elastic periodic composite media with fibers. J. Mech. Phys. Solids
**2007**, 55, 881–899. [Google Scholar] [CrossRef] - Saidi, M.; Safi, B.; Benmounah, A.; Aribi, C. Effect of size and stacking of glass fibers on the mechanical properties of the fiber-reinforced-mortars (FRMs). Int. J. Phys. Sci.
**2011**, 6, 1569–1582. [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] - Xu, Y.; Du, S.; Xiao, J. Evaluation of the effective elastic properties of long fiber reinforced composites with interphases. Comput. Mater. Sci.
**2012**, 61, 34–41. [Google Scholar] [CrossRef] - Riano, L.; Belec, L.; Chailan, J.; Joliff, Y. Effect of interphase region on the elastic behavior of unidirectional glass-fiber/epoxy composites. Compos. Struct.
**2018**, 198, 109–116. [Google Scholar] [CrossRef] - Theocaris, P.S. On the Evaluation of Adhesion Between Phases in Fiber Composites. Colloid Polym. Sci.
**1984**, 262, 929–938. [Google Scholar] [CrossRef] - Theocaris, P.S. The Mesophase Concept in Composites. In Polymers—Properties and Applications; Henrici-Olivé, G., Olivé, S., Eds.; Springer: Berlin, Germany, 1987; Volume 11, ISBN 978-3-642-70184-9, 978-3-642-70182-5. [Google Scholar]
- Sideridis, E.; Theotokoglou, E.; Giannopoulos, I. Analytical and computational study of the moduli of fiber-reinforced composites and comparison with experiments. Compos. Interfaces
**2015**, 22, 563–578. [Google Scholar] [CrossRef]

**Table 1.**Theoretical and experimental values of longitudinal modulus with respect to fibre concentration.

E_{L} GN/m^{2} | Authors | Theocaris et al. [11] Linear Law | Theocaris et al. [11] Hyperbolic Law | Theocaris et al. [11] Parabolic Law | Sideridis et al. [15] Linear Law | Sideridis et al. [15] Hyperbolic Law | Sideridis et al. [15] Parabolic Law | Two Term Unfolding Model [14] | Three Term Unfolding Model [14] | Clements and Moore [12] | Sideridis Exp. [13] | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

U_{f} | ||||||||||||

0 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.15 | 3.5 | 3.5 | 3.45 | ||

0.1 | 10.47845 | 10.39 | 10.385 | 10.372 | 10.1126 | 10.1089 | 10.0821 | 10.28992 | 10.7423438 | 10.3 | ||

0.2 | 17.71406 | 17.37 | 17.37 | 17.31 | 17.4122 | 17.2385 | 17.1024 | 17.42867 | 17.9270779 | 17.26 | ||

0.3 | 25.207991 | 24.43 | 24.43 | 24.3 | 24.8958 | 24.3515 | 24.107 | 23.28899 | 24.2137138 | |||

0.4 | 32.963901 | 31.57 | 31.57 | 31.35 | 32.4742 | 32.1485 | 32.104 | 28.26454 | 30.2908271 | |||

0.5 | 40.993581 | 38.8 | 38.79 | 38.45 | 40.171 | 40.1509 | 39.988 | 34.09573 | 36.2452983 | 38.14 | ||

0.6 | 49.339712 | 46.11 | 46.09 | 45.6 | 48.7423 | 48.3837 | 47.124 | 40.14736 | 43.3375487 | 38.28 | 45.1 | |

0.65 | 53.67567 | 49.75 | 49.73 | 49.13 | 52.798 | 52.2038 | 51.0952 | 45.1047 | 48.0763212 | 45.24 | 48.5 |

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**MDPI and ACS Style**

Venetis, J.; Sideridis, E. An Upper Bound of Longitudinal Elastic Modulus for Unidirectional Fibrous Composites as Obtained from Strength of Materials Approach. *Fibers* **2018**, *6*, 57.
https://doi.org/10.3390/fib6030057

**AMA Style**

Venetis J, Sideridis E. An Upper Bound of Longitudinal Elastic Modulus for Unidirectional Fibrous Composites as Obtained from Strength of Materials Approach. *Fibers*. 2018; 6(3):57.
https://doi.org/10.3390/fib6030057

**Chicago/Turabian Style**

Venetis, John, and Emilio Sideridis. 2018. "An Upper Bound of Longitudinal Elastic Modulus for Unidirectional Fibrous Composites as Obtained from Strength of Materials Approach" *Fibers* 6, no. 3: 57.
https://doi.org/10.3390/fib6030057