Moderately large amplitude forced vibration of sandwich functionally graded auxetic beams: an analytical approach

doi:10.1007/s10483-026-3340-7

摘要/Abstract

Abstract:

Sandwich functionally graded (FG) auxetic beams are extensively utilized in aerospace, automotive, and biomedical industries due to their excellent strength-to-weight ratio, impact resistance, and tunable mechanical properties. The integration of FG materials with auxetic structures enhances their adaptability in advanced engineering applications. However, understanding their dynamic behavior under external excitations is essential for optimal design and structural reliability. Nonlinear interactions in such structures pose significant challenges in vibration analysis, necessitating robust analytical methods. This study presents a closed-form solution for the nonlinear forced vibration analysis of sandwich FG auxetic beams, offering an accurate and efficient method for predicting their dynamic response. The beam consists of two FG face sheets with material properties varying through the thickness and a re-entrant honeycomb auxetic core with an adjustable Poisson’s ratio. The governing nonlinear equations of motion are derived using the first-order shear deformation theory (FSDT), the modified Gibson model, and the von Kármán relations, formulated through Hamilton’s principle. A closed-form solution is obtained via the Galerkin method and multiple-scale technique. The results demonstrate that FG layers enable control of the overweight and dynamic response amplitude, with positive power law indexes reducing weight. Comparisons with finite element results confirm the accuracy of the proposed formulation.

Key words: functionally graded (FG) auxetic beam, nonlinear forced vibration, closed-form solution, multiple-scale method, first-order shear deformation theory (FSDT)

中图分类号:

TB332

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 99-114.

F. M. NASREKANI, H. EIPAKCHI. Moderately large amplitude forced vibration of sandwich functionally graded auxetic beams: an analytical approach[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 99-114.

图/表 10

Property	Quantity
Length of beam/m	$L = 1$
Width/mm	$b = 50$
Upper- and lower-layer thicknesses/mm	$h 1 = h 3 = 20$
Core layer thickness/mm	$h 2 = 40$
Ratio of length of vertical cell rib to length of inclined cell rib	$a q = 2$
Amplitude of lateral harmonic load/ $(kN ⋅ m − 1)$	$P 0 = 200$
Density/ $(kg ⋅ m − 3)$	$ρ = ρ 0 = 7 800$
Poisson’s ratio	$ν = 0.3$
Young’s modulus/GPa	$E = E 0 = 200$
Inclined angle/( $∘$ )	$φ = − 55$
Ratio of thickness of cell wall to length of inclined cell rib	$a p = 0.013 857 1$
Frequency of harmonic excitation/ $(rad ⋅ s − 1)$	$ω 0 = 100$

参考文献 33

[1]	ANSARI, R., SHOJAEI, M. F., MOHAMMADI, V., GHOLAMI, R., and SADEGHI, F. Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams. Composite Structures, 113, 316–327 (2014)
[2]	REN, Y. S., DU, C. G., and SHI, Y. Y. Nonlinear free and forced vibration behavior of shear-deformable composite beams with shape memory alloy fibers. Shock and Vibration, 2016(1), 1056087 (2016)
[3]	ARANI, A. G., POURJAMSHIDIAN, M., and AREFI, M. Non-linear free and forced vibration analysis of sandwich nano-beam with FG-CNTRC face-sheets based on nonlocal strain gradient theory. Smart Structures and Systems, 22(1), 105–120 (2018)
[4]	ALLAHKARAMI, F., NIKKHAH-BAHRAMI, M., and SARYAZDI, M. G. Nonlinear forced vibration of FG-CNT-reinforced curved microbeam based on strain gradient theory considering out-of-plane motion. Steel and Composite Structures, 26(6), 673–691 (2018)
[5]	PAUL, A. and DAS, D. Non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam in thermal environment subjected to harmonic excitation and resting on non-linear elastic foundation. Journal of Strain Analysis for Engineering Design, 53(6), 446–462 (2018)
[6]	KARIMIASL, M., EBRAHIMI, F., and MAHESH, V. Nonlinear free and forced vibration analysis of multiscale composite doubly curved shell embedded in shape-memory alloy fiber under hygrothermal environment. Journal of Vibration and Control, 25(13), 1945–1957 (2019)
[7]	SHAFIEI, H. and SETOODEH, A. R. An analytical study on the nonlinear forced vibration of functionally graded carbon nanotube-reinforced composite beams on nonlinear viscoelastic foundation. Archives of Mechanics, 72(2), 81–107 (2020)
[8]	XU, W. T., PAN, G. J., MORADI, Z., and SHAFIEI, N. Nonlinear forced vibration analysis of functionally graded non-uniform cylindrical microbeams applying the semi-analytical solution. Composite Structures, 275, 114395, (2021)
[9]	WU, J. Q., CHEN, L. T., WU, R. X., and CHEN, X. C. Nonlinear forced vibration of bidirectional functionally graded porous material beam. Shock and Vibration, 2021(1), 6675125 (2021)
[10]	CIVALEK, Ö., AKBAŞ, Ş. D., AKGÖZ, B., and DASTJERDI, S. Forced vibration analysis of composite beams reinforced by carbon nanotubes. Nanomaterials, 11(3), 571 (2021)
[11]	LI, H., ZOU, Z. Y., YAN, Y. Y., SHI, X. J., XIONG, J., ZHANG, H. Y., WANG, X. P., and HA, S. K. Free and forced vibrations of composite cylindrical-cylindrical shells with partial bolt loosening connections: theoretical and experimental investigation. Thin-Walled Structures, 179, 109671 (2022)
[12]	DONG, B. C., LI, H., WANG, X. P., SUN, W., LUO, Z., MA, H., QIN, Z. Y., and HAN, Q. K. Nonlinear forced vibration of hybrid fiber/graphene nanoplatelets/polymer composite sandwich cylindrical shells with hexagon honeycomb core. Nonlinear Dynamics, 110(4), 3303–3331 (2022)
[13]	MINH, T. Q., NAM, V. H., DUC, V. M., HUNG, V. T., LY, L. N., and PHUONG, N. T. Nonlinear vibration and dynamic buckling responses of stiffened functionally graded graphene-reinforced cylindrical, parabolic, and sinusoid panels using the higher-order shear deformation theory. Journal of Applied Mathematics and Mechanics, 104(3), e202300580 (2024)
[14]	NASREKANI, F. M. and EIPAKCHI, H. Geometrically nonlinear effect on forced vibrational behavior of superlight composite beams with auxetic core layer under harmonic excitation based on FSDT. Mechanics Based Design of Structures and Machines, 52(8), 5435–5456 (2024)
[15]	YOUZERA, H., SALEH, M. M. S., GHAZWANI, M. H., MEFTAH, S. A., TOUNSI, A., and CUONG-LE, T. Nonlinear damping and forced vibration analysis of sandwich functionally graded material beams with composite viscoelastic core layer. Mechanics Based Design of Structures and Machines, 52(7), 4191–4210 (2024)
[16]	LEZGI, M., NIKOO, M. Z., and GHADIRI, M. Nonlinear vibration of Timoshenko FG porous sandwich beams subjected to a harmonic axial load. Earthquake Engineering and Engineering Vibration, 23(3), 649–662 (2024)
[17]	SHASHIRAJ, PITCHAIMANI, J., and KATTIMANI, S. Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment. Mechanics Based Design of Structures and Machines, 53(4), 2870–2901 (2025)
[18]	GHAZWANI, M. H., ALNUJAIE, A., YOUZERA, H., MEFTAH, S. A., and TOUNSI, A. Nonlinear forced vibration investigation of the sandwich porous FGM beams with viscoelastic core layer. Acta Mechanica, 235(5), 2889–2904 (2024)
[19]	ALSALEH, R., NASIR, A., and ATIEH, N. Vibration response of Euler-Bernoulli-damped beam with appendages subjected to a moving mass. Earthquake Engineering and Engineering Vibration, 24(1), 223–234 (2025)
[20]	FORIERO, A., DE MAGISTRIS, F. S., and FABBROCINO, G. A novel approach to the dynamic response analysis of Euler-Bernoulli beams resting on a Winkler soil model and subjected to impact loads. Earthquake Engineering and Engineering Vibration, 23(2), 389–401 (2024)
[21]	NASREKANI, F. M. and EIPAKCHI, H. Geometrically nonlinear free vibration analysis of axially loaded super-light auxetic beams reinforced by functionally graded face sheets. Journal of Vibration Engineering & Technologies, 13(1), 102 (2025)
[22]	SOBHY, M. A comprehensive nonlinear analysis of asymmetric nanocomposite sandwich beams with a negative Poisson’s ratio core on nonlinear elastic foundation under hygrothermal conditions. Journal of Applied Mathematics and Mechanics, 105(5), e70045 (2025)
[23]	ZHANG, P. L. and WANG, J. F. Macro fiber composite-based active control of nonlinear forced vibration of functionally graded plate. Applied Mathematics and Mechanics (English Edition), 46(5), 869–884 (2025) https://doi.org/10.1007/s10483-025-3250-9
[24]	SAURABH, S., KIRAN, R., SINGH, D., VAISH, R., and CHAUHAN, V. S. A comprehensive investigation on nonlinear vibration and bending characteristics of bio-inspired helicoidal laminated composite structures. Applied Mathematics and Mechanics (English Edition), 46(1), 81–100 (2025) https://doi.org/10.1007/s10483-025-3200-7
[25]	LI, H., DENG, Y. C., ZHANG, Z. W., HOU, J. X., ZHOU, J., WANG, H. Z., ZHANG, H. Y., WANG, X. P., and GUAN, Z. W. High-velocity impact characteristics of 3D auxetic composite cylindrical shell panels: theory and experiment. Thin-Walled Structures, 206, 112648 (2025)
[26]	MIRSKY, I. and HERRMANN, G. Axially symmetric motions of thick cylindrical shells. Journal of Applied Mechanics, 25(1), 97–102 (1958)
[27]	SADD, M. H. Elasticity-Theory, Applications, and Numerics, Elsevier, Amsterdam (2009)
[28]	ELISHAKOFF, I. E., PENTARAS, D., and GENTILINI, C. Mechanics of Functionally Graded Material Structures, World Scientific Publishing Company, Singapore (2015)
[29]	GHANNAD, M. and GHAROONI, H. Displacements and stresses in pressurized thick FGM cylinders with varying properties of power function based on HSDT. Journal of Solid Mechanics, 4(4), 237–251 (2012)
[30]	GIBSON L. J. and ASHBY, M. F. Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge (1997)
[31]	NAYFEH, A. H. Introduction to Perturbation Techniques, Wiley, New York (1981)
[32]	EIPAKCHI, H. and NASREKANI, F. M. Nonlinear response and resonance analysis of beam with non-uniform cross-section under harmonic and impulse excitations: an analytical approach. Acta Mechanica, 235(5), 2845–2865 (2024)
[33]	HAGEDORN, P. and DASGUPTA, A. Vibrations and Waves in Continuous Mechanical Systems, Wiley, Chichester (2007)