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A high-order continuation for bifurcation analysis of functionally graded material sandwich plates

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Abstract

This work deals with nonlinear geometric functionally graded material (FGM) sandwich plates under various nonuniform compressions in the context of the high-order shear deformation theory. To do so, an efficient numerical model named a High Order Continuation with Finite Element Method (HOC-FEM) is used. The HOC-FEM has become an important tool to study and predict the behavior of an FGM sandwich plates, especially to analyze the nonlinear buckling and post-buckling phenomena. This numerical tool has an adaptive step length, which is effective especially for solving nonlinear problems and detecting bifurcation points. Thereafter, the procedure of numerical technique is based on the use of a Taylor series expansion, a finite element method, and a continuation procedure. Indeed, the Taylor series expansion permits to transform the nonlinear problem into a sequence of linear ones. Then, the use of the finite element method consists of interpolating the unknown of the obtained linear systems. The continuation procedure is introduced to compute the whole solution step by step manner. The accuracy and efficiency of the HOC-FEM are illustrated on numerical examples of an FGM plate and then an FGM sandwich-type.

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References

  1. Koizumi, M., Niino, M.: Overview of FGM research in Japan. MRS Bull. 20(1), 19–21 (1995)

    Article  Google Scholar 

  2. Koizumi, M.: FGM activities in Japan. Compos. B Eng. 28(1–2), 1–4 (1997)

    Article  Google Scholar 

  3. Bever, M., Duwez, P.: Gradients in composite materials. Mater. Sci. Eng. 10, 1–8 (1972)

    Article  Google Scholar 

  4. Sator, L., Sladek, V., Sladek, J.: Consistent 2D formulation of thermoelastic bending problems for FGM plates. Compos. Struct. 212, 412–422 (2019)

    Article  MATH  Google Scholar 

  5. Zhang, J., Chen, L., Lv, Y.: Elastoplastic thermal buckling of functionally graded material beams. Compos. Struct. 224, 111014 (2019)

    Article  Google Scholar 

  6. Aria, A., Friswell, M.: A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos. B Eng. 166, 233–246 (2019)

    Article  Google Scholar 

  7. Sator, L., Sladek, V., Sladek, J.: Coupling effects in elastic analysis of FGM composite plates by mesh-free methods. Compos. Struct. 115, 100–110 (2014)

    Article  MATH  Google Scholar 

  8. Bui, T., Khosravifard, A., Zhang, C., Hematiyan, M., Golub, M.: Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method. Eng. Struct. 47, 90–104 (2013)

    Article  Google Scholar 

  9. Chen, Y., Jin, G., Zhang, C., Ye, T., Xue, Y.: Thermal vibration of FGM beams with general boundary conditions using a higher-order shear deformation theory. Compos. B Eng. 153, 376–386 (2018)

    Article  Google Scholar 

  10. Civalek, Ö., Baltacıoglu, A.K.: Free vibration analysis of laminated and FGM composite annular sector plates. Compos. B Eng. 157, 182–194 (2019)

    Article  Google Scholar 

  11. Liu, Y., Su, S., Huang, H., Liang, Y.: Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane. Compos. B Eng. 168, 236–242 (2019)

    Article  Google Scholar 

  12. Van Do, V.N., Tran, M.T., Lee, C.-H.: Nonlinear thermal buckling analyses of functionally graded plates by a mesh-free radial point interpolation method. Eng. Anal. Bound. Elem. 87, 153–164 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Swaminathan, K., Naveenkumar, D.: Higher order refined computational models for the stability analysis of FGM plates-analytical solutions. Eur. J. Mech. A/Solids 47, 349–361 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.: An analytical solution for bending, buckling and vibration responses of FGM sandwich plates. J. Sandw. Struct. Mater. 21(2), 727–757 (2019)

    Article  Google Scholar 

  15. Yanga, J., Shen, H.-S.: Non-linear analysis of functionally graded plates under transverse and in-plane loads. Int. J. Non-Linear Mech. 38(4), 467–482 (2003)

    Article  MATH  Google Scholar 

  16. Zenkour, A.: A comprehensive analysis of functionally graded sandwich plates: part 2-buckling and free vibration. Int. J. Solids Struct. 42(18–19), 5243–5258 (2005)

    Article  MATH  Google Scholar 

  17. Bourihane, O., Mhada, K., Sitli, Y.: New finite element model for the stability analysis of a functionally graded material thin plate under compressive loadings. Acta Mech. 231(4), 1587–1601 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sitli, Y., Mhada, K., Bourihane, O., Rhanim, H.: Buckling and post-buckling analysis of a functionally graded material (FGM) plate by the asymptotic numerical method. Structures 31, 1031–1040 (2021)

    Article  Google Scholar 

  19. Shariat, B.S., Javaheri, R., Eslami, M.: Buckling of imperfect functionally graded plates under in-plane compressive loading. Thin-walled Struct. 43(7), 1020–1036 (2005)

    Article  Google Scholar 

  20. Reddy, J.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47(1–3), 663–684 (2000)

    Article  MATH  Google Scholar 

  21. Santos, H., Soares, C.M.M., Soares, C.A.M., Reddy, J.: A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock. Compos. Struct. 86(1–3), 10–21 (2008)

    Article  Google Scholar 

  22. Santos, H., Soares, C.M.M., Soares, C.A.M., Reddy, J.: A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials. Compos. Struct. 91(4), 427–432 (2009)

    Article  Google Scholar 

  23. Ferreira, A., Batra, R., Roque, C., Qian, L., Jorge, R.: Natural frequencies of functionally graded plates by a meshless method. Compos. Struct. 75(1–4), 593–600 (2006)

    Article  Google Scholar 

  24. Thai, C.H., Ferreira, A., Tran, T., Phung-Van, P.: Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation. Compos. Struct. 220, 749–759 (2019)

    Article  Google Scholar 

  25. Javaheri, R., Eslami, M.: Buckling of functionally graded plates under in-plane compressive loading. ZAMM J. Appl. Math. Mech./Z. Angew. Math. Mech. Appl. Math. Mech. 82(4), 277–283 (2002)

    Article  MATH  Google Scholar 

  26. Adhikari, B., Dash, P., Singh, B.: Buckling analysis of porous FGM sandwich plates under various types nonuniform edge compression based on higher order shear deformation theory. Compos. Struct. 251, 112597 (2020)

    Article  Google Scholar 

  27. Van Do, V.N., Lee, C.-H.: Numerical investigation on post-buckling behavior of FGM sandwich plates subjected to in-plane mechanical compression. Ocean Eng. 170, 20–42 (2018)

    Article  Google Scholar 

  28. Pandya, B., Kant, T.: Higher-order shear deformable theories for flexure of sandwich plates-finite element evaluations. Int. J. Solids Struct. 24(12), 1267–1286 (1988)

    Article  MATH  Google Scholar 

  29. Boutyour, E., Zahrouni, H., Potier-Ferry, M., Boudi, M.: Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants. Int. J. Numer. Methods Eng. 60(12), 1987–2012 (2004)

    Article  MATH  Google Scholar 

  30. Askour, O., Tri, A., Braikat, B., Zahrouni, H., Potier-Ferry, M.: Bifurcation indicator for geometrically nonlinear elasticity using the method of fundamental solutions. C. R. Méc. 347(2), 91–100 (2019)

    Article  MATH  Google Scholar 

  31. Rammane, M., Mesmoudi, S., Tri, A., Braikat, B., Damil, N.: Bifurcation points and bifurcated branches in fluids mechanics by high-order mesh-free geometric progression algorithms. Int. J. Numer. Methods Fluids 93(3), 834–852 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tian, H., Potier-Ferry, M., Abed-Meraim, F.: A numerical method based on Taylor series for bifurcation analyses within föppl-von karman plate theory. Mech. Res. Commun. 93, 154–158 (2018)

    Article  Google Scholar 

  33. Azrar, L., Cochelin, B., Damil, N., Potier-Ferry, M.: An asymptotic-numerical method to compute the postbuckling behaviour of elastic plates and shells. Int. J. Numer. Methods Eng. 36(8), 1251–1277 (1993)

    Article  MATH  Google Scholar 

  34. Cochelin, B.: A path-following technique via an asymptotic-numerical method. Comput. Struct. 53(5), 1181–1192 (1994)

    Article  MATH  Google Scholar 

  35. Cochelin, B., Damil, N., Potier-Ferry, M.: Asymptotic-numerical methods and Pade approximants for non-linear elastic structures. Int. J. Numer. Methods Eng. 37(7), 1187–1213 (1994)

    Article  MATH  Google Scholar 

  36. Damil, N., Potier-Ferry, M.: A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures. Int. J. Eng. Sci. 28(9), 943–957 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mottaqui, H., Braikat, B., Damil, N.: Discussion about parameterization in the asymptotic numerical method: application to nonlinear elastic shells. Comput. Methods Appl. Mech. Eng. 199(25–28), 1701–1709 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bourihane, O., Hilali, Y., Mhada, K.: Nonlinear dynamic response of functionally graded material plates using a high-order implicit algorithm. ZAMM J. Appl. Math. Mech./Z. Angew. Math. Mech. 100(12), e202000087 (2020)

    MathSciNet  Google Scholar 

  39. Mesmoudi, S., Timesli, A., Braikat, B., Lahmam, H., Zahrouni, H.: A 2D mechanical-thermal coupled model to simulate material mixing observed in friction stir welding process. Eng. Comput. 33(4), 885–895 (2017)

    Article  Google Scholar 

  40. Askour, O., Tri, A., Braikat, B., Zahrouni, H., Potier-Ferry, M.: Method of fundamental solutions and high order algorithm to solve nonlinear elastic problems. Eng. Anal. Bound. Elem. 89, 25–35 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mesmoudi, S., Braikat, B., Lahmam, H., Zahrouni, H.: Three-dimensional numerical simulation of material mixing observed in FSW using a mesh-free approach. Eng. Comput. 36, 13–27 (2018)

    Article  Google Scholar 

  42. Yang, J., Potier-Ferry, M., Hu, H.: Solving the stationary Navier–Stokes equations by using Taylor meshless method. Eng. Anal. Bound. Elem. 98, 8–16 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rammane, M., Mesmoudi, S., Tri, A., Braikat, B., Damil, N.: Solving the incompressible fluid flows by a high order mesh-free approach. Int. J. Numer. Methods Fluids 92, 422–435 (2020)

  44. Fouaidi, M., Jamal, M., Belouaggadia, N.: Nonlinear bending analysis of functionally graded porous beams using the multiquadric radial basis functions and a Taylor series-based continuation procedure. Compos. Struct. 252, 112593 (2020)

    Article  Google Scholar 

  45. Askour, O., Mesmoudi, S., Braikat, B.: On the use of radial point interpolation method (RPIM) in a high order continuation for the resolution of the geometrically nonlinear elasticity problems. Eng. Anal. Bound. Elem. 110, 69–79 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Askour, O., Mesmoudi, S., Tri, A., Braikat, B., Zahrouni, H., Potier-Ferry, M.: Method of fundamental solutions and a high order continuation for bifurcation analysis within Föppl-von karman plate theory. Eng. Anal. Bound. Elem. 120, 67–72 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rammane, M., Mesmoudi, S., Tri, A., Braikat, B., Damil, N.: A dimensionless numerical mesh-free model for the compressible fluid flows. Comput. Fluids 221, 104845 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Modélisation des Structures et Systèmes Mécaniques (M2SM), École National Supérieure d’Art et de Métier (ENSAM), University Mohammed V of Rabat, B.P., 6207, Avenue des Forces Armées Royales, 10100, Rabat, Morocco

    Hamza Chaabani, Lhoucine Boutahar & Khalid El Bikri

  2. LISA Laboratory, Ecole Nationale des Sciences Appliquées, Hassan First University of Settat, 26100, Berrechid, Morocco

    Said Mesmoudi

Authors
  1. Hamza Chaabani
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  2. Said Mesmoudi
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  4. Khalid El Bikri
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Correspondence to Hamza Chaabani.

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Appendices

Appendix A: Matrices H,A, \({\hat{S}}_0\)

$$\begin{aligned} {[}H]= & {} \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 1&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 1&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 2&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 2&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 3&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 3&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \end{array} \right] \end{aligned}$$
(25)
$$\begin{aligned} {[}A(\{\theta \})]= & {} \left[ \begin{array}{llllllllllllllllllllllllllllllllll} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad w_{,x} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad w_{,y} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad w_{,y} &{} \quad w_{,x} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{array} \right] \end{aligned}$$
(26)
$$\begin{aligned} {[}{\hat{S}}_0]= & {} \left[ \begin{array}{cccccccccccccccccccccccccccccc} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad N_{0xx} &{} \quad N_{0xy} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad N_{0xy} &{} \quad N_{0yy} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ \end{array} \right] \nonumber \\ \end{aligned}$$
(27)

Appendix B: Boundary conditions

$$\begin{aligned}&\text {CCCC: } w=\phi _x=\phi _y=\psi _x=\psi _y=\beta _x=\beta _y=0 \quad \text {on: } x=0;\;x=L_x\;\text {and}\;y=0;\;y=L_y \nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0; \;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(28)
$$\begin{aligned}&\text {CSCS: } w=\phi _x=\phi _y=\psi _x=\psi _y=\beta _x=\beta _y=0; \quad \text {on: }\; x=0;\;x=L_x \nonumber \\&\quad w=\phi _x=\psi _x=\beta _x=0; \;\text {on: }\; y=0;\;y=L_y \nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0; \;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(29)
$$\begin{aligned}&\text {CFCF: } w=\phi _x=\phi _y=\psi _x=\psi _y=\beta _x=\beta _y=0; \quad \text {on: } x=0;\;x=L_x \nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0; \;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(30)
$$\begin{aligned}&\text {SSSS: } w=\phi _x=\psi _x=\beta _x=0; \; \text {on: } y=0;\;y=L_y \nonumber \\&\quad w=\phi _y=\psi _y=\beta _y=0;\; \text {on: } x=0;\;x=L_x \nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0\;;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(31)
$$\begin{aligned}&\text {SCSC: } w=\phi _y=\psi _y=\beta _y=0;\quad \text {on: } x=0;\;x=L_x \nonumber \\&\quad w=\phi _x=\phi _y=\psi _x=\psi _y=\beta _x=\beta _y=0;\; \text {on: } y=0;\;y=L_y \nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0; \;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(32)
$$\begin{aligned}&\text {SFSF: } w=\phi _y=\psi _y=\beta _y=0; \quad \text {on: } x=0;\;x=L_x\nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0; \;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(33)
$$\begin{aligned}&{C^\circledast C^\circledast C^\circledast C^\circledast : } w=\phi _y=\psi _y=\beta _y=0; \; \text {on: } x=0;\;x=L_x\nonumber \\&\quad w=\phi _x=\psi _x=\beta _x=0;\; \text {on: } y=0;\;y=L_y\nonumber \\&\quad u=0;\; \text {on: } x=\dfrac{L_x}{2}\; \text {and}\;v=0; \;\text {on: } y=\dfrac{L_y}{2} \end{aligned}$$
(34)

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Chaabani, H., Mesmoudi, S., Boutahar, L. et al. A high-order continuation for bifurcation analysis of functionally graded material sandwich plates. Acta Mech 233, 2125–2147 (2022). https://doi.org/10.1007/s00707-022-03216-4

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