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Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

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Abstract

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work, the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is established to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method (FEM). The convergent stresses have good agreements with those results obtained by three dimensional (3D) FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kantorovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.

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References

  1. Herakovich, C.T.: Mechanics of composites: a historical review. Mech. Res. Commun. 41, 1–20 (2012)

    Article  Google Scholar 

  2. Wang, S.S., Choi, I.: Boundary-layer effects in composite laminates: part 1—free-edge stress singularities. J. Appl. Mech. 49, 541–548 (1982)

    Article  MATH  Google Scholar 

  3. Wang, S.S., Choi, I.: Boundary-layer effects in composite laminates: part 2—free-edge stress solutions and basic characteristics. J. Appl. Mech. 49, 549–560 (1982)

    Article  MATH  Google Scholar 

  4. Spilker, R.L., Chou, S.C.: Edge effects in symmetric composite laminates: importance of satisfying the traction-free-edge condition. J. Compos. Mater. 14, 2–20 (1980)

    Google Scholar 

  5. Ghugal, Y.M., Shimpi, R.P.: A review of refined shear deformation theories of isotropic and anisotropic laminated plates. J. Reinf. Plast. Compos. 21, 775–813 (2002)

    Article  Google Scholar 

  6. Kassapoglou, C., Lagace, P.A.: An efficient method for the calculation of interlaminar stresses in composite materials. J. Appl. Mech. 53, 744–750 (1986)

    Article  MATH  Google Scholar 

  7. Zhu, C., Lam, Y.C.: A Rayleigh-Ritz solution for local stresses in composite laminates. Compos. Sci. Technol. 58, 447–461 (1998)

    Article  Google Scholar 

  8. Konieczny, S., Woźniak, C.: Corrected 2D-theories for composite plates. Acta Mech. 103, 145–155 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, Y.Y., Lam, K.Y., Liu, G.R. et al.: A strip element method for bending analysis of orthotropic plates. JSME Int. J. 40, 398–406 (1997)

  10. Kapuria, S., Dube, G.P., Dumir, P.C.: First-order shear deformation theory solution for a circular piezoelectric composite plate under axisymmetric load. Smart Mater. Struct. 12, 417 (2003)

    Article  Google Scholar 

  11. Thai, H.T., Choi, D.H.: A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Compos. Struct. 101, 332–340 (2013)

    Article  Google Scholar 

  12. Reddy, J.N., Liu, C.F.: A higher-order shear deformation theory of laminated elastic shells. Int. J. Eng. Sci. 23, 319–330 (1985)

    Article  MATH  Google Scholar 

  13. Cho, M., Parmerter, R.: Efficient higher order composite plate theory for general lamination configurations. AIAA J. 31, 1299–1306 (1993)

    Article  MATH  Google Scholar 

  14. Chattopadhyay, A., Gu, H.: New higher order plate theory in modeling delamination buckling of composite laminates. AIAA J. 32, 1709–1716 (1994)

    Article  MATH  Google Scholar 

  15. Robbins, D.H., Reddy, J.N.: Modelling of thick composites using a layerwise laminate theory. Int. J. Numer. Methods Eng. 36, 655–677 (1993)

    Article  MATH  Google Scholar 

  16. Zhou, X., Chattopadhyay, A., Kim, H.S.: An efficient layerwise shear-deformation theory and finite element implementation. J. Reinf. Plast. Compos. 23, 131–152 (2004)

    Article  Google Scholar 

  17. Oh, J.H., Cho, M., Kim, J.S.: Enhanced lower-order shear deformation theory for fully coupled electro-thermo-mechanical smart laminated plates. Smart Mater. Struct. 16, 2229–2241 (2007)

    Article  Google Scholar 

  18. Kim, J.S.: Free vibration of laminated and sandwich plates using enhanced plate theories. J. Sound Vib. 308, 268–286 (2007)

    Article  Google Scholar 

  19. Mitchell, J.A., Reddy, J.N.: A refined hybrid plate theory for composite laminates with piezoelectric laminae. Int. J. Solids Struct. 32, 2345–2367 (1995)

    Article  MATH  Google Scholar 

  20. Detwiler, D.T., Shen, M.H., Venkayya, V.B.: Finite element analysis of laminated composite structures containing distributed piezoelectric actuators and sensors. Finite Elem. Anal. Des. 20, 87–100 (1995)

    Article  MATH  Google Scholar 

  21. Artel, J., Becker, W.: Coupled and uncoupled analyses of piezoelectric free-edge effect in laminated plates. Compos. Struct. 69, 329–335 (2005)

    Article  Google Scholar 

  22. Oh, J.H., Cho, M., Kim, J.S.: Buckling analysis of a composite shell with multiple delaminations based on a higher order zig-zag theory. Finite Elem. Anal. Des. 44, 675–685 (2008)

    Article  Google Scholar 

  23. Lee, J., Cho, M., Kim, H.S.: Bending analysis of a laminated composite patch considering the free-edge effect using a stress-based equivalent single-layer composite model. Int. J. Mech. Sci. 53, 606–616 (2011)

    Article  Google Scholar 

  24. Kim, H.S., Lee, J., Cho, M.: Free-edge interlaminar stress analysis of composite laminates using interface modeling. J. Eng. Mech. 138, 973–983 (2012)

    Article  Google Scholar 

  25. Huang, B., Kim, H.S.: Free-edge interlaminar stress analysis of piezo-bonded composite laminates under symmetric electric excitation. Int. J. Solids Struct. 51, 1246–1252 (2014)

    Article  Google Scholar 

  26. Kim, H.S., Cho, M., Lee, J., et al.: Three dimensional stress analysis of a composite patch using stress functions. Int. J. Mech. Sci. 52, 1646–1659 (2010)

  27. Flanagan, G.: An efficient stress function approximation for the free-edge stresses in laminates. Int. J. Solids Struct. 31, 941–952 (1994)

  28. Yin, W.L.: Free-edge effects in anisotropic laminates under extension, bending and twisting, Part I: a stress-function-based variational approach. J. Appl. Mech. 61, 410–415 (1994)

    Article  MATH  Google Scholar 

  29. Yin, W.L.: Free-edge effects in anisotropic laminates under extension, bending, and twisting, Part II: eigenfunction analysis and the results for symmetric laminates. J. Appl. Mech. 61, 416–421 (1994)

    Article  Google Scholar 

  30. Kerr, A.D., Alexander, H.: An application of the extended Kantorovich method to the stress analysis of a clamped rectangular plate. Acta Mech. 6, 180–196 (1968)

    Article  MATH  Google Scholar 

  31. Kerr, A.D.: An extended Kantorovich method for the solution of eigenvalue problems. Int. J. Solids Struct. 5, 559–572 (1969)

    Article  MATH  Google Scholar 

  32. Tahani, M., Andakhshideh, A.: Interlaminar stresses in thick rectangular laminated plates with arbitrary laminations and boundary conditions under transverse loads. Compos. Struct. 94, 1793–1804 (2012)

    Article  Google Scholar 

  33. Tahani, M., Naserian-Nik, A.M.: Bending analysis of piezolaminated rectangular plates under electromechanical loadings using multi-term extended Kantorovich method. Mech. Adv. Mater. Struct. 20, 415–433 (2013)

    Article  Google Scholar 

  34. Kapuria, S., Kumari, P.: Extended Kantorovich method for coupled piezoelasticity solution of piezolaminated plates showing edge effects. Proc. R. Soc. A Math. Phys. 469, 20120565 (2013)

    Article  MathSciNet  Google Scholar 

  35. Kapuria, S., Kumari, P.: Extended Kantorovich method for three-dimensional elasticity solution of laminated composite structures in cylindrical bending. Journal of Applied Mechanics 78, 1774–1800 (2011)

    Article  Google Scholar 

  36. Cho, M., Kim, H.S.: Iterative free-edge stress analysis of composite laminates under extension, bending, twisting and thermal loadings. Int. J. Solids Struct. 37, 435–459 (2000)

    Article  MATH  Google Scholar 

  37. Kim, H.S., Cho, M., Kim, G.I.: Free-edge strength analysis in composite laminates by the extended Kantorovich method. Compos. Struct. 49, 229–235 (2000)

    Article  Google Scholar 

  38. Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Body. Holden-Day, San Francisco (1963)

  39. Tahani, M., Nosier, A.: Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates. Mater. Des. 24, 121–130 (2003)

    Article  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants 11372145, 11372146, and 11272161), the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant MCMS-0516Y01), Zhejiang Provincial Top Key Discipline of Mechanics Open Foundation (Grant xklx1601), and the K. C. Wong Magna Fund through Ningbo University.

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Authors and Affiliations

  1. Piezoelectric Device Laboratory, School of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, 315211, China

    Bin Huang, Ji Wang, Jianke Du, Tingfeng Ma & Lijun Yi

  2. College of Science & Technology, Ningbo University, Ningbo, 315211, China

    Yan Guo

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  1. Bin Huang
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  2. Ji Wang
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  3. Jianke Du
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Correspondence to Bin Huang.

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Huang, B., Wang, J., Du, J. et al. Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions. Acta Mech. Sin. 32, 854–865 (2016). https://doi.org/10.1007/s10409-016-0570-6

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  • DOI: https://doi.org/10.1007/s10409-016-0570-6

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