Applied Mathematics and Mechanics (English Edition) ›› 2026, Vol. 47 ›› Issue (7): 1487-1510.doi: https://doi.org/10.1007/s10483-026-3404-6
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M. H. BAE1, J. H. PARK1, C. S. PARK1,2, H. LEE3, H. M. SEUNG1,2(
)
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RSSReceived:2026-01-27
Revised:2026-04-27
Published:2026-06-30
Contact:
H. M. SEUNG, E-mail: shm@kriss.re.krSupported by:2010 MSC Number:
M. H. BAE, J. H. PARK, C. S. PARK, H. LEE, H. M. SEUNG. Inherent nonreciprocity in double-alternatingnonlinear metamaterial. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1487-1510.
Fig. 1
(a) Linear reciprocal wave propagation in a linear asymmetric medium, where the internal mode distribution maintains a constant ratio between the odd and even amplitudes for both propagation directions despite the structural symmetry; (b) nonlinear nonreciprocal wave propagation in a nonlinear asymmetric medium, where direction-dependent wave transmission exists because of the change of the internal mode distribution, owing to the amplitude dependence (color online)"
Fig. 2
(a) Configured system with m1=1.20, m2=0.80, and k=1.00. (b) Three representative nonlinear stiffness configurations: Case I, hardening-hardening (Γ1=0.50, Γ2=1.00); Case II, softening-softening (Γ1=−0.50, Γ2=−1.00); Case III, hardening-softening (Γ1=1.00, Γ2=−1.00) (color online)"
Fig. 3
Linear zero-order solutions obtained from the eigen analysis: (a) acoustic and optical dispersion relation derived from the eigenvalue E(0) of the zeroth-order system with alternating masses; (b) amplitude ratios of the corresponding acoustic and optical modes along the wavevector q (color online)"
Fig. 4
Geometric interpretation of capturing dispersion shift with perturbed mode shapes: (a) projection of the nonlinear response onto the unperturbed linear eigenvector basis (ϕaco(0), ϕopt(0)), which captures only eigenvalue shifts while neglecting the rotation of basis from the unperturbed linear eigenvector basis; (b) proposed mode alignment procedure that updates the basis to the perturbed eigenvectors (ϕaco, ϕopt), thereby reconstructing the corrected eigenvalues and accurately capturing the full nonlinear dispersion shift (color online)"
Fig. 5
Bandgap shifts in double-alternating nonlinear metamaterial for the three representative cases under an incident amplitude A0 = 0.25 m: (a) Case I, acoustic branch 1.291→1.313, optical branch 1.581→1.609; (b) Case II, acoustic branch 1.291→1.268, optical branch 1.581→1.553; (c) Case III: acoustic branch 1.291→1.287, optical branch 1.581→1.585 (color online)"
Fig. 6
Bandgap shifts in double-alternating nonlinear metamaterial for the three representative cases under an incident amplitude A0 = 0.40 m: (a) Case I, acoustic branch 1.291→1.346, optical branch 1.581→1.652; (b) Case II, acoustic branch 1.291→1.230, optical branch 1.581→1.510; (c) Case III: acoustic branch 1.291→1.263, optical branch 1.581→1.604 (color online)"
Fig. 7
Inherent nonreciprocal dispersion in the double-alternating nonlinear metamaterial: (a) forward and backward boundary configurations under an identical excitation amplitude of A0 = 0.100 m; direction-dependent dispersion relations for (b) Case I: alternating hardening, (c) Case II: alternating softening, and (d) Case III: mixed hardening-softening, where the forward (purple dotted markers) and backward (blue dotted markers) branches exhibit pronounced asymmetry near the bandgap, demonstrating inherent nonreciprocity (color online)"
Fig. 8
Numerical simulation setup for (a) forward propagation and (b) backward propagation in the double-alternating nonlinear metamaterial (color online)"
Fig. 9
Transmission spectra for forward propagation (left) and backward propagation (right) across three nonlinear configurations: (a), (b), Case I; (c), (d), Case II; (e), (f), Case III, where the vertical dashed lines denote the boundaries of the identified nonlinear bandgap (color online)"
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