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

doi:10.1007/s10483-026-3340-7

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)

2010 MSC Number:

TB332

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

TrendMD

Figures/Tables 10

Fig. 1

Fig. 2

Table 1

Mechanical and geometrical properties of sandwich FG auxetic beam"

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/ $(rad ⋅ s − 1)$	$ω 0 = 100$

Table 1

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

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