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Influence of non-uniformly periodic distribution of fibers in composites on the stress field and effective shear modulus under anti-plane shear

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Abstract

Based on the complex variable techniques combined with the boundary collocation method, a semi-analytical procedure is proposed to explore the anti-plane shear behavior of non-uniformly periodic fibrous composites under uniform remote shear loadings. Specific series with unknown coefficients are introduced to describe the complex potentials of the representative unit cell (RUC) of the composite. The unknown coefficients are determined from the continuity conditions on the interface, the periodic boundary conditions imposed on the edge of the unit cell and the remote loading conditions. Once the complex potentials are determined, the effective moduli of the composite are obtained according to the average-field theory. Extensive numerical examples are provided to investigate the influence of the periodic distribution of the inclusions, the moduli of each component, and the volume fraction of the inclusions on the local stress fields and effective shear moduli of the composite. Numerical results show that the inhomogeneous distribution of the inclusions may induce a continuous area with high equivalent stress in the matrix under certain loadings and material properties of each component. However, we could overcome this disadvantage by changing the components or the periodic arrangement of the inclusions. For the case of a RUC with double inclusions, the two inclusions with the same shear moduli arranged symmetrically in the direction of 45 degrees may induce the transverse isotropy of the composite. Nevertheless, when the two inclusions are horizontally arranged and the volume fraction of the inclusions is large enough (more than 10%), the composite generally shows orthotropic characteristics.

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References

  1. Aboudi, J.: Mechanics of Composite Materials: Unified Micromechanical Approach. Elsevier Science Publisher, Amsterdam (1991)

    MATH  Google Scholar 

  2. Needleman, A., Tvergaard, V.: Comparison of crystal plasticity and isotropic hardening predictions for metal-matrix composites. J. Appl. Mech. 60(1), 70–76 (1993)

    Article  Google Scholar 

  3. Xia, Z.H., Zhang, Y.F.: Ellyin F: A unified periodical boundary condition for representative volume elements of composites and applications. Int. J. Solids Struct. 40(8), 1907–1921 (2003)

    Article  Google Scholar 

  4. Hori, M., Nemat-Nasser, S.: On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mech. Mat. 31(10), 667–682 (1999)

    Article  Google Scholar 

  5. Bakhvalov, N.S., Panasenko, G.: Homogenisation: Averaging Processes in Periodic Media. Springer, Dordrecht (1989)

    Book  Google Scholar 

  6. Berger, H., Kari, S., Gabbert, U., et al.: An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int. J. Solids Struct. 42(21–22), 5692–5714 (2005)

    Article  Google Scholar 

  7. Wang, H.W., Zhou, H.W., Peng, R.D., et al.: Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept. Compos. Sci. Technol. 71(7), 980–988 (2011)

    Article  Google Scholar 

  8. Yvonnet, J., Quang, H.L., He, Q.C.: An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput. Mech. 42(1), 119131 (2008)

    Article  MathSciNet  Google Scholar 

  9. Xu, Y.L., Lo, S.H., Jiang, C.P., et al.: Electro-elastic behavior of doubly periodic piezoelectric fiber composites under anti-plane shear. Int. J. Solids Struct. 44(3–4), 976–995 (2007)

    Article  Google Scholar 

  10. Rodriguez-Ramos, R., Sabina, F.J., Guinovart-Dıaz, R.: Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents-I. Elastic and square symmetry. Mech. Mat. 33(4), 223–235 (2001)

    Article  Google Scholar 

  11. Bravo-Castillero, J., Guinovart-Díaz, R., Rodríguez-Ramos, R., et al.: Unified analytical formulae for the effective properties of periodic fibrous composites. Mat. Lett. 73, 68–71 (2012)

    Article  Google Scholar 

  12. Otero, J.A., Rodríguez-Ramos, R., Bravo-Castillero, J., et al.: Semi-analytical method for computing effective properties in elastic composite under imperfect contact. Int. J. Solids Struct. 50(3–4), 609622 (2013)

    Google Scholar 

  13. López-Realpozo, J.C., Rodríguez-Ramos, R., Guinovart-Díaz, R., et al.: Effective elastic shear stiffness of a periodic fibrous composite with non-uniform imperfect contact between the matrix and the fibers. Int. J. Solids Struct. 51(6), 12531262 (2014)

    Article  Google Scholar 

  14. Dai, M., Schiavone, P., Gao, C.F.: Prediction of the stress field and effective shear modulus of composites containing periodic inclusions incorporating interface effects in anti-plane shear. J. Elast. 125(2), 217–230 (2016)

    Article  MathSciNet  Google Scholar 

  15. Dai, M., Schiavone, P., Gao, C.F.: A new method for the evaluation of the effective properties of composites containing unidirectional periodic nanofibers. Arch. Appl. Mech. 87, 647665 (2017)

    Article  Google Scholar 

  16. Dai, M.: Design of periodic harmonic holes with surface tension in plane deformations. Math. Mech. Solids 24(7), 20602065 (2019)

    Article  MathSciNet  Google Scholar 

  17. Yang, H.B., Dai, M., Gao, C.F.: Influence of hole shape and surface elasticity on anti-plane shear properties of porous structures with periodic holes. Acta. Mech. 228(7), 25192531 (2017)

    Article  MathSciNet  Google Scholar 

  18. Yang, H.B., Huang, C., Yu, C.B.: Longitudinal shear behavior of composites with unidirectional periodic nanofibers of some regular polygonal shapes. J. Mech. Mat. Struct. 13(2), 155–170 (2018)

    Article  MathSciNet  Google Scholar 

  19. Yang, H.B., Dai, M., Gao, C.F.: Stress field in a porous material containing periodic arbitrarily-shaped holes with surface tension. Math. Mech. Solids 23(1), 120–130 (2018)

    Article  MathSciNet  Google Scholar 

  20. Yang, H.B., Wang, S.: Interface tension-induced stress field around periodic nano-inclusions of arbitrary shape. Math. Mech. Solids 00, 1–14 (2018)

    Google Scholar 

  21. Yang, H.B., Dai, M.: Influence of surface roughness on the stress field around a nanosized hole with surface elasticity. Z. für Angew. Math. Phys. 69(5), 127 (2018)

    Article  MathSciNet  Google Scholar 

  22. Yang, H.B., Wang, S., Yu, C.B.: Effective in-plane stiffness of unidirectional periodic nanoporous materials with surface elasticity. Z. für Ange. Math. Phys. 70(4), 1–11 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Dong, C.S., Davies, I.J.: Flexural and tensile strengths of unidirectional hybrid epoxy composites reinforced by S-2 glass and T700S carbon fibres. Mat. Des. 54, 955–966 (2014)

    Article  Google Scholar 

  24. Zhang, Y.L., Li, Y., Ma, H., Yu, T.: Tensile and interfacial properties of unidirectional flax/glass fiber reinforced hybrid composites. Comp. Sci. Technol, 88, 172–177 (2013)

    Article  Google Scholar 

  25. Muskhelishvili, N.I.: Some basic problems of the mathematical theory of elasticity: fundamental equations, plane theory of elasticity, torsion and bending. Springer Science + Business Media, Dordrecht (1977)

    Book  Google Scholar 

  26. Dai, M., Meng, L.C., Huang, C., Gao, C.F.: Electro-elastic fields around two arbitrarily-shaped holes in a finite electrostrictive solid. Appl. Math. Modell. 40(7–8), 4625–4639 (2016)

    Article  MathSciNet  Google Scholar 

  27. Luo, J., Wang, X.: On the anti-plane shear of an elliptic nano inhomogeneity. European Journal of Mechanics A/Solids 28(5), 926–934 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank support from the National Natural Science Foundation of China (11872203).

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Hui-Feng Yang & Cun-Fa Gao

  2. School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang, 212003, People’s Republic of China

    Hui-Feng Yang

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  1. Hui-Feng Yang
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  2. Cun-Fa Gao
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Correspondence to Cun-Fa Gao.

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Yang, HF., Gao, CF. Influence of non-uniformly periodic distribution of fibers in composites on the stress field and effective shear modulus under anti-plane shear. Acta Mech 232, 515–531 (2021). https://doi.org/10.1007/s00707-020-02841-1

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