This paper explores the optimization macrostructure to reach a stable low coefficient of thermal expansion αx of a composite with carbon fibers. To limit the search area, a necessary condition for the existence of αx local minima is proposed, expressed in terms of the radii of hyperspheres in the design space of the angular orientation of the layers transformed by the PСA algorithm. The analysis of the structure variants characterized by low αx shows different sustainability to lamina properties variability. Multi-criteria optimization was carried out. The objective functions are expectation E(αx) and variance Var(αx). The analysis of Pareto fronts and probability density functions make it possible to estimate the reachability of the calculated αx under given conditions of lamina properties variability. The reduction variance opportunity of αx distribution by modifying the polymer matrix with MWCNTs under conditions of reinforcing fibers disorientation and lamina properties variability is investigated. The microstructure modification of the polymer composite material allows to reduce the Var(αx) by 91.61 % with a volume ratio of MWCNTs up to 1 %. Requirement thermomechanical properties are reached by determining the orientation of anisotropic layers. Based on the obtained optimal structures, specimens of CFRP with 0, 1 and 2 vol.% MWCNTs were made. Scanning electron microscopy using FE–SEM Hitachi S–5500 was performed to check the uniformity of distribution and compatibility of the epoxy matrix and MWCNTs. The measurement of αx is determined using a TAInstrumentsQ400 thermomechanical analyzer. Measured αx of specimens is in the range from 6.2·10-8 to 1.98·10-7 1/K. The structure optimization approach proposed in this paper makes it possible to obtain a set of solutions with a consistently low αx in the range up to 1·10-7 1/K. The transformation of the design space of the layers’ orientation angles and the limitation of the search area allowed to reduce the range of solutions under consideration by 83.9 %.

Full Text

The size changes control is very important in some applications, for example, in space structures. Zero thermal expansion materials is needed in structures subject to temperature changes such as a backplane support structure for a large space telescope, antenna booms, solar array frames and etc. Unlike classical materials, composite laminates can be designed in such a way as to reduce the coefficients of thermal expansion (CTE) in the desired direction to a specified value. This can be done by the appropriate sequence of laminate layers, the angular orientation of each layer and the microstructure properties control. The structure optimization problem of a composite material to obtain the specified mechanical and thermomechanical properties has repeatedly attracted the attention of researchers [1–9]. To reduce the CTE value of CFRP, the researchers design a new kind of multi-functional curing agent for epoxy resin [10] or add inorganic particles such as carbon nanotubes, aluminum nitride, silicon dioxide, and so on [11–18]. There is increasing interest in combining reinforcement scales of nanoscale reinforcements with traditional micron-sized fibers [19; 20]. The authors of the work [21] evaluated the feasibility of using multi-walled carbon nanotubes (MWCNTs) for the control of coefficients of thermal expansion of composite materials. The use of MWCNTs aligned axially was shown to be effective at controlling αx of polymer composites. The authors theoretically calculated the MWCNT volume fraction at which αx of composites based on different type of material and containing MWCNTs become zero. In particular, the authors determined the MWCNT content necessary for zero coefficients of thermal expansion to be about 10 vol.% in the polymer materials at temperature range of −5 °C to 85 °C. Inorganic compounds with negative thermal expansion (ZrW2O8, ScF3, Mn0.98CoGe, Sm2.75C60) can be used to compensate and control the CTEs of CFRP [22–29]. But for many inorganic NTE compounds, their CTE are small in magnitude or the effective temperature window is narrow [30; 31]. The orientation of fibers in composites may result in anisotropic thermal expansion has a small or even negative CTE value [33; 34]. Although the task of the composite’s macrostructure optimization is complicated by the high sensitivity of the theoretically achieved near-zero coefficients of thermal expansion to the lamina properties variability, which inevitably present in practice due to disorientation of reinforcing fibers, and various degrees of cure, volume fraction, residual moisture and etc. However, to date, the literature has not considered the possibility of obtaining a hybrid composite with zero αx, reinforced with carbon fibers in combination with MWCNTs, with an estimate of the density function of the probability distribution of αx under the conditions of lamina properties variability and its disorientation. This paper first describes the methodology of obtaining a layered composite with near zero αx based on estimate E(αx) and Var(αx) as target parameters. Possibilities of reducing the variance of the target parameter by means of an integrated approach to determining the optimal orientation of layers of hybrid composites with different volume content of MWCNT is considered. A detailed description of the calculation procedure for the target parameters using randomly generated lamina’s orientation angle error and lamina properties variability is presented in the next session. The results section discusses the possibility of reducing the Var(αx) using MWCNTs, analyzing the Pareto-optimum front curves. Further, the experimental estimates of αx of the composite are reported in detail.

About the authors

K. A. Pasechnik

Reshetnev University

I. V. Obvertkin

Reshetnev University

A. Y. Vlasov

Reshetnev University


  1. Anaya L., Vicente W., Pavanello R. Minimization of the Effective Thermal Expansion Coefficient of Composite Material Using a Multi-scale Topology Optimization Method. EngOpt 2018 Proceedings of the 6th International Conference on Engineering Optimization / ed. Rodrigues H.C. et al. Cham: Springer International Publishing, 2019, pp. 1055–1060.
  2. Zhengchun D. et al. Design and application of composite platform with extreme low thermal deformation for satellite. Compos. Struct. Elsevier Ltd, 2016, Vol. 152, pp. 693–703.
  3. Catapano A., Desmorat B., Vannucci P. Stiffness and Strength Optimization of the Anisotropy Distribution for Laminated Structures. J. Optim. Theory Appl. 2015, Vol. 167, № 1, pp. 118–146.
  4. Kim D. et al. Topology optimization of functionally graded anisotropic composite structures using homogenization design method. Comput. Methods Appl. Mech. Eng. Elsevier B.V., 2020, Vol. 369, pp. 113220.
  5. Schaedler de Almeida F. Optimization of laminated composite structures using harmony search algorithm. Compos. Struct. Elsevier Ltd, 2019, Vol. 221, pp. 110852.
  6. Peng X. et al. Multiple-scale uncertainty optimization design of hybrid composite structures based on neural network and genetic algorithm. Compos. Struct. Elsevier Ltd, 2020, Vol. 262, pp. 113371.
  7. Hao P. et al. Efficient reliability-based design optimization of composite structures via isogeometric analysis // Reliab. Eng. Syst. Saf. Elsevier Ltd, 2021, Vol. 209, pp. 107465.
  8. das Neves Carneiro G., Conceição António C. Dimensional reduction applied to the reliability-based robust design optimization of composite structures. Compos. Struct. Elsevier Ltd, 2021, Vol. 255, pp. 112937.
  9. Sigmund O., Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. J. Mech. Phys. Solids. Elsevier Ltd, 1997, Vol. 45, № 6, pp. 1037–1067.
  10. Q.S. Sun, Y.D. Feng, J. Guo, et al. High performance epoxy resin with ultralow coefficient of thermal expansion cured by conformation-switchable multi-functional agent. Chemical Engineering Journal, 2022, Vol. 450, pp. 138295.
  11. S.R. Wang, Z.Y. Liang, P. Gonnet, Y.H. Liao, B. Wang, C. Zhang. Effect of nanotube functionalization on the coefficient of thermal expansion of nanocomposites. Adv. Funct. Mater., 2007, 17(1), pp. 87-92.
  12. J.K. Ma, T.Y. Shang, L.L. Ren, Y.M. Yao, T. Zhang, J.Q. Xie, B.T. Zhang, X.L. Zeng, R. Sun, J.B. Xu, C.P. Wong. Through-plane assembly of carbon fibers into 3D skeleton achieving enhanced thermal conductivity of a thermal interface material. Chem. Eng. J., 2020, 380, p. 8.
  13. Obvertkin, I., K. Pasechnik, и A. Vlasov. The potential of using SWCNTs, MWCNTs and CNFs capable of increasing the composite material dimensional and technological stability as modifiers of a polymer matrix. PNRPU Mechanics Bulletin, 2021, № 4, pp. 98-110.
  14. K.C. Yung, B.L. Zhu, T.M. Yue, C.S. Xie. Effect of the Filler Size and Content on the Thermomechanical Properties of Particulate Aluminum Nitride Filled Epoxy Composites. J. Appl. Polym. Sci., 2010, 116 (1), pp. 225-236.
  15. C.J. Huang, S.Y. Fu, Y.H. Zhang, B. Lauke, L.F. Li, L. Ye. Cryogenic properties of SiO2/epoxy nanocomposites. Cryogenics, 2005, 45 (6), pp. 450-454.
  16. Ghasemi A.R., Mohammadi M.M., Mohandes M. The role of carbon nanofibers on thermo-mechanical properties of polymer matrix composites and their effect on reduction of residual stresses. Compos Part B Eng, 2015, no. 77, pp. 519-27.
  17. Shokrieh M.M., Akbari S., Daneshvar A. Reduction of residual stresses in polymer composites using nano-additives. Residual Stress Compos Mater, 2014, pp. 350-73,
  18. Pan J., Bian L. A physics investigation for influence of carbon nanotube agglomeration on thermal properties of composites. Mater ChemPhys, 2019, №. 236,
  19. Green K.J. et al. Multiscale fiber reinforced composites based on a carbon nanofiber/epoxy nanophased polymer matrix: Synthesis, mechanical, and thermomechanical behavior. Compos. Part A Appl. Sci. Manuf. Elsevier, 2009, Vol. 40, № 9, pp. 1470–1475.
  20. Fu S. et al. Some basic aspects of polymer nanocomposites: A critical review. Nano Mater. Sci. Elsevier BV, 2019. Vol. 1, № 1, pp. 2–30.
  21. Shirasu K. et al. Negative axial thermal expansion coefficient of carbon nanotubes: Experimental determination based on measurements of coefficient of thermal expansion for aligned carbon nanotube reinforced epoxy composites. Carbon N. Y. Elsevier Ltd, 2015, Vol. 95, pp. 904–909.
  22. J.C. Lin, P. Tong, K. Zhang, H.Y. Tong, X.G. Guo, C. Yang, Y. Wu, M. Wang, S. Lin, L. Chen, W.H. Song, Y.P. Sun. Colossal negative thermal expansion with an extended temperature interval covering room temperature in fine-powdered Mn0.98CoGe. Appl. Phys. Lett., 2016, 109 (24), p. 5.
  23. V.K. Thakur, Y.Z. Li, H.C. Wu, M.R. Kessler. Synthesis, characterization, and functionalization of zirconium tungstate (ZrW2O8) nano-rods for advanced polymer nanocomposites. Polym. Adv. Technol., 2017, 28 (11), pp. 1375-1381.
  24. T.A. Mary, J.S.O. Evans, T. Vogt, A.W. Sleight. Negative Thermal Expansion from 0.3 to 1050 Kelvin in ZrW2O8, Science, 1996, 272, p. 90.
  25. B.K. Greve, K.L. Martin, P.L. Lee, P.J. Chupas, K.W. Chapman, A.P. Wilkinson. Pronounced Negative Thermal Expansion from a Simple Structure: Cubic ScF3. J. Am. Chem. Soc., 2010, 132, p. 15496.
  26. Zheng, X., Kubozono, H., Yamada, H. et al. Giant negative thermal expansion in magnetic nanocrystals. Nature Nanotech 3, 724–726 (2008).
  27. X. Chu, Z. Wu, C. Huang, R. Huang, Y. Zhou, L. Li. ZrW2O8-doped epoxy as low thermal expansion insulating materials for superconducting feeder system. Cryogenics, 2012, 52(12), pp. 638-641.
  28. P. Badrinarayanan, M. Rogalski, H. Wu, X. Wang, W. Yu, M.R. Kessler. Epoxy Composites Reinforced with Negative-CTE ZrW2O8 Nanoparticles for Electrical Applications. Macromol. Mater. Eng., 2013, 298 (2), pp. 136-144.
  29. Y.Y. Zhao, F.X. Hu, L.F. Bao, J. Wang, H. Wu, Q.Z. Huang, R.R. Wu, Y. Liu, F.R. Shen, H. Kuang, M. Zhang, W.L. Zuo, X.Q. Zheng, J.R. Sun, B.G. Shen. Giant Negative Thermal Expansion in Bonded MnCoGe-Based Compounds with Ni2In-Type Hexagonal Structure. J. Am. Chem. Soc., 2015, 137 (5), pp. 1746-1749.
  30. L.A. Neely, V. Kochergin, E.M. See, H.D. Robinson. Negative thermal expansion in a zirconium tungstate/epoxy composite at low temperatures. J. Mater. Sci., 2014, 49 (1), pp. 392-396.
  31. J. Arvanitidis, K. Papagelis, S. Margadonna, K. Prassides, A.N. Fitch. Temperature-induced valence transition and associated lattice collapse in samarium fulleride. Nature, 2003, 425, p. 599.
  32. K. Shirasu, A. Nakamura, G. Yamamoto, T. Ogasawara, Y. Shimamura, Y. Inoue, T. Hashida. Potential use of CNTs for production of zero thermal expansion coefficient composite materials: An experimental evaluation of axial thermal expansion coefficient of CNTs using a combination of thermal expansion and uniaxial tensile tests. Compos. Pt. A-Appl. Sci. Manuf., 95 (2017), pp. 152-160.
  33. R.P. Zhu, C.T. Sun. Effects of Fiber Orientation and Elastic Constants on Coefficients of Thermal Expansion in Laminates. Mech. Adv. Mater. Struct., 10 (2) (2003), pp. 99-107.
  34. Yoon KJ, Kim J-S. Prediction of Thermal Expansion Properties of Carbon/Epoxy Laminates for Temperature Variation. Journal of Composite Materials. 2000;34(2):90-100.
  35. Polymer nanocomposites. MRS Bull, 2007, № 32, рр. 314-319.
  36. Chang T., Gao H. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J MechPhys Solids, 2003, №. 51, pp. 1059-74. .
  37. J. Blank and K. Deb, pymoo: Multi-Objective Optimization in Python. IEEE Access, Vol. 8, pp. 89497-89509, 2020. Polym. Adv. Technol., 2017, 28 (11), pp. 1375-1381.
  38. Blank, Julian and Kalyanmoy Deb. A. Running Perfor-mance Metric and Termination Criterion for Evaluating Evolu-tionary Multi- and Many-objective Optimization Algo-rithms. 2020 IEEE Congress on Evolutionary Computation (CEC), 2020, pp. 1-8.



Abstract - 23

PDF (Russian) - 23



Copyright (c) 2023 Pasechnik K.A., Obvertkin I.V., Vlasov A.Y.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies