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Static Response and Buckling Loads of Multilayered Composite Beams Using the Refined Zigzag Theory and Higher-Order Haar Wavelet Method

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Mechanics of Composite Materials Aims and scope

The paper presents a review of Haar wavelet methods and an application of the higher-order Haar wavelet method to study the behavior of multilayered composite beams under static and buckling loads. The Refined Zigzag Theory (RZT) is used to formulate the corresponding governing differential equations (equilibrium/stability equations and boundary conditions). To solve these equations numerically, the recently developed Higher-Order Haar Wavelet Method (HOHWM) is used. The results found are compared with those obtained by the widely used Haar Wavelet Method (HWM) and the Generalized Differential Quadrature Method (GDQM). The relative numerical performances of these numerical methods are assessed and validated by comparing them with exact analytical solutions. Furthermore, a detailed convergence study is conducted to analyze the convergence characteristics (absolute errors and the order of convergence) of the method presented. It is concluded that the HOHWM, when applied to RZT beam equilibrium equations in static and linear buckling problems, is capable of predicting, with a good accuracy, the unknown kinematic variables and their derivatives. The HOHWM is also found to be computationally competitive with the other numerical methods considered.

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  1. S. Abrate and M. Di Sciuva, “Equivalent single layer theories for composite and sandwich structures: A review,” Composite Structures, 179, 482-94 (2017).

    Article  Google Scholar 

  2. K. M. Liew, Z. Z. Pan, and L. W. Zhang, “An overview of layerwise theories for composite laminates and structures: Development, numerical implementation and application,” Composite Structures, 216, 240-259 (2019).

    Article  Google Scholar 

  3. S. Abrate and M. Di Sciuva, “Multilayer models for composite and sandwich structures,” In: P. W. R. Beaumont and C. H. Zweben (eds.), Comprehensive Composite Materials II, 1, 399-425 (2018).

  4. M. Di Sciuva, “Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: An evaluation of a new displacement model,” Journal of Sound and Vibration, 105, 425-442 (1986).

    Article  Google Scholar 

  5. M. Di Sciuva, “An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates,” J. Appl. Mech., 54, 589-596 (1987).

    Article  Google Scholar 

  6. M. Di Sciuva, “Multilayered anisotropic plate models with continuous interlaminar stresses,” Composite Structures, 22, 149-167 (1992).

    Article  Google Scholar 

  7. H. Murakami, “Laminated composite plate theory with improved in-plane responses,” J. Appl. Mech., 53, 661-666 (1986).

    Article  Google Scholar 

  8. M. Cho and R. R. Parmerter, “An efficient higher-order plate theory for laminated composites,” Composite Structures, 20, 113-123 (1992).

    Article  Google Scholar 

  9. A. Tessler, M. Di Sciuva, and M. Gherlone, “Refinement of Timoshenko beam theory for composite and sandwich beams using zigzag kinematics,” NASA/TP-2007-215086,1-45 (2007).

  10. A. Tessler, M. Di Sciuva, and M. Gherlone, “Refined zigzag theory for laminated composite and sandwich plates,” NASA/TP-2009-215561, 1-53 (2009).

  11. D. Versino, M. Gherlone, and M. Di Sciuva, “Four-node shell element for doubly curved multilayered composites based on the Refined Zigzag Theory,” Composite Structures, 118, 392-402 (2014).

    Article  Google Scholar 

  12. M. Gherlone, A, Tessler, and M. Di Sciuva, “C0 beam elements based on the Refined Zigzag Theory for multilayered composite and sandwich laminates,” Composite Structures, 93, 2882-2894 (2011).

    Article  Google Scholar 

  13. E. Oñate, A. Eijo, and S. Oller, “Simple and accurate two-noded beam element for composite laminated beams using a refined zigzag theory,” Computer Methods in Applied Mechanics and Engineering, 213-216, 362-82 (2012).

    Article  Google Scholar 

  14. A. Eijo, E. Oñate, and S. Oller, “A numerical model of delamination in composite laminated beams using the LRZ beam element based on the refined zigzag theory,” Composite Structures, 104, 270-80 (2013).

    Article  Google Scholar 

  15. M. Gherlone, “On the use of zigzag functions in equivalent Single Layer Theories for laminated composite and sandwich beams: A comparative study and some observations on external weak layers,” J. Appl. Mech., 80, 6, 061004 (19 pages), (2013).

  16. M. Di Sciuva, M. Gherlone, L. Iurlaro, and A. Tessler, “A class of higher-order C0 composite and sandwich beam elements based on the Refined Zigzag Theory,” Composite Structures, 132, 784-803 (2015).

  17. A. Tessler, “Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle,” Meccanica, 50, 2621-2648 (2015).

    Article  Google Scholar 

  18. R. M. J. Groh and P. M. Weaver, “On displacement-based and mixed-variational equivalent single layer theories for modelling highly heterogeneous laminated beams,” International Journal of Solids and Structures, 59, 147-170 (2015).

    Article  Google Scholar 

  19. R. M. Groh, P. M. Weaver, and A. Tessler, “Application of the refined zigzag theory to the modeling of delaminations in laminated composites,” NASA/TM-2015-218808, 1-22 (2015).

  20. L. Iurlaro, M. Gherlone, and M. Di Sciuva, “The (3,2)-Mixed Refined Zigzag Theory for generally laminated beams: Theoretical development and C0 finite element formulation,” International Journal of Solids and Structures, 73-74, 1-19 (2015).

  21. H. Wimmer and M. Gherlone, “Explicit matrices for a composite beam-column with refined zigzag kinematics,” Acta Mech., 228, 2107-2117 (2017).

    Article  Google Scholar 

  22. H. Wimmer, W. Hochhauser, and K. Nachbagauer, “Refined Zigzag Theory: an appropriate tool for the analysis of CLTplates and other shear-elastic timber structures,”Eur. J. Wood. Prod., 78, 1125-1135 (2020).

    Article  CAS  Google Scholar 

  23. M. Dorduncu, “Stress analysis of laminated composite beams using refined zigzag theory and peridynamic differential operator,” Composite Structures, 218, 193-203 (2019).

    Article  Google Scholar 

  24. L. Iurlaro, M. Gherlone, and M. Di Sciuva, “Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag Theory,” Journal of Sandwich Structures & Materials, 16, 669-699 (2014).

    Article  Google Scholar 

  25. M. Di Sciuva and M. Sorrenti, “Bending, free vibration and buckling of functionally graded carbon nanotube-reinforced sandwich plates, using the extended Refined Zigzag Theory,” Composite Structures, 227, 111324 (2019).

    Article  Google Scholar 

  26. A. Ascione and M. Gherlone, “Nonlinear static response analysis of sandwich beams using the Refined Zigzag Theory,” Journal of Sandwich Structures & Materials, 22, 7, 2250-2286, (2020).

    Article  Google Scholar 

  27. M. Di Sciuva and M. Sorrenti, “Bending and free vibration analysis of functionally graded sandwich plates: An assessment of the Refined Zigzag Theory, “Journal of Sandwich Structures & Materials, 1-43 (2019).

  28. C. F. Chen and C. H. Hsiao, “Haar wavelet method for solving lumped and distributed-parameter systems, “IEE Proceedings - Control Theory and Applications, 144, 87-94 (1977).

    Article  Google Scholar 

  29. Ü. Lepik, “Numerical solution of differential equations using Haar wavelets,” Mathematics and Computers in Simulation, 68, 127-43 (2005).

    Article  Google Scholar 

  30. Ü. Lepik, “Haar wavelet method for nonlinear integro-differential equations, “Applied Mathematics and Computation, 176, 324-333 (2006).

  31. Siraj-ul-Islam and I. Aziz, “New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets,” Journal of Computational and Applied Mathematics, 239, 333-345 (2013).

  32. I. Aziz, Siraj-ul-Islam, and F. Khan, “For the numerical solution of two-dimensional nonlinear integral equations, “Journal of Computational and Applied Mathematics, 272, 70-80 (2014).

  33. Siraj-ul-Islam, I. Aziz, and A. S. Al-Fhaid, “An improved method based on Haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders,” Journal of Computational and Applied Mathematics, 260, 449-469 (2014).

  34. Ö. Oruç, “A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations,” Computers & Mathematics with Applications, 77, 1799-1820 (2019).

    Article  Google Scholar 

  35. X. Xie, G. Jin, and Z. Liu, “Free vibration analysis of cylindrical shells using the Haar wavelet method,” International Journal of Mechanical Sciences, 77, 47-56 (2013).

    Article  Google Scholar 

  36. X. Xie, G. Jin, T. Ye, and Z. Liu, “Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method.” Applied Acoustics, 85, 130-142 (2014).

    Article  Google Scholar 

  37. J. Majak, M. Pohlak, and M. Eerme, “Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells,” Mechanics of Composite Materials, 45, 631-642 (2009).

    Article  Google Scholar 

  38. H. Hein and L. Feklistova, “Computationally efficient delamination detection in composite beams using Haar wavelets,” Mechanical Systems and Signal Processing, 25, 2257-2270 (2011).

    Article  Google Scholar 

  39. J. Majak, B. Shvartsman, K. Karjust, M. Mikola, A. Haavajõe, and M. Pohlak, “On the accuracy of the Haar wavelet discretization method,” Composites Part B: Engineering, 80, 321-327 (2015).

    Article  Google Scholar 

  40. J. Majak, B. S. Shvartsman, M. Kirs, M. Pohlak, and H. Herranen, “Convergence theorem for the Haar wavelet based discretization method,” Composite Structures, 126, 227-232 (2015).

    Article  Google Scholar 

  41. M. Kirs, K. Karjust, I. Aziz, E. Õunapuu, and E. Tungel, “Free vibration analysis of a functionally graded material beam: evaluation of the Haar wavelet method,” Proceedings of the Estonian Academy of Sciences, 67, No. 1, 1-9 (2018).

  42. J. Majak, M. Pohlak, K. Karjust, M. Eerme, J. Kurnitski, and B. S. Shvartsman, “New higher order Haar wavelet method: Application to FGM structures, Composite Structures, 201, 72-78 (2018).

    Article  Google Scholar 

  43. S. K. Jena, S. Chakraverty and M. Malikan, “Implementation of Haar wavelet, higher order Haar wavelet, and differential quadrature methods on buckling response of strain gradient nonlocal beam embedded in an elastic medium,” Engineering with Computers, (2019).

  44. J. Majak, M. Pohlak, M. Eerme, and B. Shvartsman, “Solving ordinary differential equations with higher order Haar wavelet method,” AIP Conference Proceedings, 2116, 330002 (2019).

    Article  Google Scholar 

  45. J. Majak, B. Shvartsman, M. Ratas, D. Bassir, M. Pohlak, and K. Karjust, et al., “Higher-order Haar wavelet method for vibration analysis of nanobeams,” Materials Today Communications, 25, 101290 (2020).

    Article  CAS  Google Scholar 

  46. K. Torabi and H. Afshari, “Generalized differential quadrature method for vibration analysis of cantilever trapezoidal FG thick plate,” Journal of Solid Mechanics, 8, 184-203 (2016).

    Google Scholar 

  47. F. Tornabene, S. Brischetto, N. Fantuzzi, and M. Bacciocchi, “Boundary conditions in 2D numerical and 3D exact models for cylindrical bending analysis of functionally graded structures,” Shock and Vibration, 2016, 1-17 (2016).

    Article  Google Scholar 

  48. S. A. M. Ghannadpour, M. Karimi, and F. Tornabene, “Application of plate decomposition technique in nonlinear and post-buckling analysis of functionally graded plates containing crack,” Composite Structures, 220,158-167 (2019).

    Article  Google Scholar 

  49. F. Tornabene and E. Viola, “2-D solution for free vibrations of parabolic shells using generalized differential quadrature method,” European Journal of Mechanics - A/Solids, 27, 1001-1025 (2008).

    Article  Google Scholar 

  50. F. Tornabene, E. Viola, and D. J. Inman, “2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures,” Journal of Sound and Vibration, 328, 259–290 (2009).

    Article  Google Scholar 

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Correspondence to M. Sorrenti.

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Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 57, No. 1, pp. 3-26, January-February, 2021.

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Sorrenti, M., Di Sciuva, M., Majak, J. et al. Static Response and Buckling Loads of Multilayered Composite Beams Using the Refined Zigzag Theory and Higher-Order Haar Wavelet Method. Mech Compos Mater 57, 1–18 (2021).

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