Skip to main content
SpringerLink
Log in

Far-field analytical solution of composite materials considering steigmann-ogden surface

考虑Steigmann-Ogden界面的复合材料远场解析解

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

This paper presents an analytical solution for two-dimensional heterogeneous materials containing nano-fibers or pores, taking into account the Steigmann-Ogden elastic surface under far-field loading. The solution is validated against numerical results from the complex function method in recent literature, and the closed-form expressions for specific displacement and stress fields are provided. The effects of surface elasticity parameters, surface residual stress, fiber/pore size, and far-field load on local stress distribution are numerically investigated. Results show that surface elasticity parameters can disturb internal stresses within the fiber domain, while surface bending stiffness parameters significantly impact stress concentrations, which is different from the uniform stress distribution in the classical Eshelby problem. The analytical expressions reveal interesting phenomena, e.g., the stress/displacement fields of fiber composites under hydrostatic load are only related to the surface Lamé parameters, and the non-constant coefficient in the analytical expression under shear load is kept twice of that under uniaxial tensile load, which are first reported in this paper. The developed solution is crucial for accurately capturing the mechanical responses of nanocomposites with significant surface effects.

摘要

本文给出了含有纳米纤维或孔隙的二维异质材料在远场载荷作用下的Steigmann-Ogden弹性面的解析解. 该求解与最近文献中 复函数法的数值结果进行了验证, 并提供了具体位移和应力场的闭式表达式. 数值研究了表面弹性参数、表面残余应力、纤维/孔径和 远场载荷对局部应力分布的影响. 结果表明, 表面弹性参数会扰乱纤维域内的内应力, 而表面弯曲刚度参数会显著影响应力集中, 这与 经典Eshelby问题中的均匀应力分布不同. 分析表达式揭示了一些有趣的现象, 例如, 纤维复合材料在静水载荷作用下的应力/位移场仅 与表面Lamé参数有关, 剪切载荷作用下分析表达式中的非常数系数保持为单轴拉伸载荷作用下的两倍, 这些都是本文首次报道的. 所 建立的解决方案对于准确捕捉具有显著表面效应的纳米复合材料的力学响应至关重要.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. J. W. Gibbs, The Scientific Papers of J. Willard Gibbs (Longmans, Green and Company, New York, 1906).

    MATH  Google Scholar 

  2. M. E. Gurtin, and A. Ian Murdoch, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57, 291 (1975).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. M. E. Gurtin, and A. Ian Murdoch, Surface stress in solids, Int. J. Solids Struct. 14, 431 (1978).

    Article  MATH  Google Scholar 

  4. A. A. Abdelrahman, N. A. Mohamed, and M. A. Eltaher, Static bending of perforated nanobeams including surface energy and microstructure effects, Eng. Comput. 38, 415 (2022).

    Article  Google Scholar 

  5. Q. Jin, Y. Ren, H. Jiang, and L. Li, A higher-order size-dependent beam model for nonlinear mechanics of fluid-conveying FG nanotubes incorporating surface energy, Compos. Struct. 269, 114022 (2021).

    Article  Google Scholar 

  6. M. Hashemian, S. Foroutan, and D. Toghraie, Comprehensive beam models for buckling and bending behavior of simple nanobeam based on nonlocal strain gradient theory and surface effects, Mech. Mater. 139, 103209 (2019).

    Article  Google Scholar 

  7. G. Wang, Q. Chen, Z. He, and M. J. Pindera, Homogenized moduli and local stress fields of unidirectional nano-composites, Compos. Part B-Eng. 138, 265 (2018).

    Article  Google Scholar 

  8. Z. He, G. Wang, and M. J. Pindera, Multiscale homogenization and localization of materials with hierarchical porous microstructures, Compos. Struct. 222, 110905 (2019).

    Article  Google Scholar 

  9. Q. Chen, G. Chatzigeorgiou, and F. Meraghni, Extended mean-field homogenization of unidirectional piezoelectric nanocomposites with generalized Gurtin-Murdoch interfaces, Compos. Struct. 307, 116639 (2023).

    Article  Google Scholar 

  10. D. J. Steigmann, and R. W. Ogden, Plane deformations of elastic solids with intrinsic boundary elasticity, Proc. R. Soc. Lond. A 453, 853 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  11. D. J. Steigmann, and R. W. Ogden, Elastic surface—substrate interactions, Proc. R. Soc. Lond. A 455, 437 (1999).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. A. Y. Zemlyanova, and S. G. Mogilevskaya, Circular inhomogeneity with Steigmann-Ogden interface: Local fields, neutrality, and Maxwell’s type approximation formula, Int. J. Solids Struct. 135, 85 (2018).

    Article  Google Scholar 

  13. M. Manav, P. Anilkumar, and A. S. Phani, Mechanics of polymer brush based soft active materials—theory and experiments, J. Mech. Phys. Solids 121, 296 (2018).

    Article  MathSciNet  ADS  Google Scholar 

  14. X. Li, and C. Mi, Nanoindentation hardness of a Steigmann-Ogden surface bounding an elastic half-space, Math. Mech. Solids 24, 2754 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Y. Zemlyanova, and S. G. Mogilevskaya, On spherical inhomogeneity with Steigmann-Ogden interface, J. Appl. Mech. 85, 121009 (2018).

    Article  ADS  Google Scholar 

  16. Y. Ban, and C. Mi, Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann-Ogden surface model, Math. Mech. Solids 25, 1904 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Nazarenko, H. Stolarski, and H. Altenbach, Effective properties of particulate nano-composites including Steigmann-Ogden model of material surface, Comput. Mech. 68, 651 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Wang, P. Yan, L. Dong, and S. N. Atluri, Spherical nano-inhomogeneity with the Steigmann-Ogden interface model under general uniform far-field stress loading, Int. J. Solids Struct. 185–186, 311 (2020).

    Article  Google Scholar 

  19. J. Wang, P. Yan, L. Dong, and S. N. Atluri, Direct numerical simulation of complex nano-structured composites, considering interface stretching and bending effects, using nano-computational grains, Int. J. Numer. Methods Eng. 122, 1476 (2021).

    Article  MathSciNet  Google Scholar 

  20. S. G. Mogilevskaya, A. Y. Zemlyanova, and V. I. Kushch, Fiber- and particle-reinforced composite materials with the Gurtin-Murdoch and Steigmann-Ogden surface energy endowed interfaces, Appl. Mech. Rev. 73, 050801 (2021).

    Article  ADS  Google Scholar 

  21. M. Dai, A. Gharahi, and P. Schiavone, Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations, Appl. Math. Model. 55, 160 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  22. Z. Han, S. G. Mogilevskaya, and D. Schillinger, Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and Steigmann-Ogden interfaces, Int. J. Solids Struct. 147, 166 (2018).

    Article  Google Scholar 

  23. S. G. Mogilevskaya, V. I. Kushch, and A. Y. Zemlyanova, Displacements representations for the problems with spherical and circular material surfaces, Q. J. Mech. Appl. Math. 72, 449 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  24. R. M. Christensen, and K. H. Lo, Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids 27, 315 (1979).

    Article  MATH  ADS  Google Scholar 

  25. H. L. Duan, J. Wang, Z. P. Huang, and B. L. Karihaloo, Eshelby formalism for nano-inhomogeneities, Proc. R. Soc. A 461, 3335 (2005).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  26. H. L. Duan, X. Yi, Z. P. Huang, and J. Wang, Eshelby equivalent inclusion method for composites with interface effects, Key Eng. Mater. 312, 161 (2006).

    Article  Google Scholar 

  27. G. Wang, and M. J. Pindera, Locally-exact homogenization of unidirectional composites with coated or hollow reinforcement, Mater. Des. 93, 514 (2016).

    Article  Google Scholar 

  28. G. Wang, The elastic solutions of separable problems with the applications to multilayered structures, Arch. Appl. Mech. 88, 1525 (2018).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12002303), the National Key Research and Development Program of China (Grant No. 2020YFA0711700), ZJU-ZCCC Institute of Collaborative Innovation (Grant No. ZDJG2021002), and the Scientific Research Program of the Department of Education of Hunan Province, China (Grant No. 22B0022).

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, China

    Mengyuan Gao  (高梦园), Ougbe Anselme Ahehehinnou & Guannan Wang  (王冠楠)

  2. Center for Balance Architecture, Zhejiang University, Hangzhou, 310007, China

    Mengyuan Gao  (高梦园) & Guannan Wang  (王冠楠)

  3. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, China

    Zhelong He  (贺哲龙)

Authors
  1. Mengyuan Gao  (高梦园)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhelong He  (贺哲龙)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ougbe Anselme Ahehehinnou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guannan Wang  (王冠楠)
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Author contributionsGuannan Wang and Mengyuan Gao designed the research. Mengyuan Gao and Guannan Wang wrote the first draft of the manuscript. Ougbe Anselme Ahehehinnou helped organize the manuscript. Guannan Wang and Zhelong He revised and edited the final version.

Corresponding author

Correspondence to Guannan Wang  (王冠楠).

Ethics declarations

Conflict of interestOn behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, M., He, Z., Ahehehinnou, O.A. et al. Far-field analytical solution of composite materials considering steigmann-ogden surface. Acta Mech. Sin. 40, 123196 (2024). https://doi.org/10.1007/s10409-023-23196-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-023-23196-x

Search

Navigation