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Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
Schematic illustration of the two coupled infinite atomic chains made of identical atoms of mass m with a lattice parameter a. β
would correspond to the stiffness of the spring linking atoms in the two horizontal chains if these springs were linear. A degree of
nonlinearity can be introduced inside the model via parameters δ and ε (see text for definition). β′ is the stiffness of the linear spring
connecting together atoms n in the two horizontal chains.
Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
Band structure associated with the zeroth-order solutions, i.e., ω0−(k) (black line) and ω0+(k) (gray line). Parameters a, m, β, and β′
(see text for definitions) were chosen to be a=1 m, m=1 kg, β=1 N · m−1, β′=6.4 N · m−1.
Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
ω−(k) (black lines) and ω+(k) (gray lines) as functions of wave number for three values of the spring constant β′, namely,
β′=1 N · m−1 (a), 2 N · m−1 (b), and 7 N · m−1 (c). The dotted lines correspond to the linear–linear two-chain model. The system
parameters are a=1 m, m=1 kg, β=1 N · m−1,ε=1.6 N · m−2, α0+=α0−=0.17 m. The black dashed line indicates the value of the critical
wave number: kc=1.27 m−1 (a), kc=1.57 m−1 (b), and kc=2.63 m−1 (c).
Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
Complex pulsation ω− as function of the wave number around the critical wave number k c when the stiffness β′ of the spring
coupling together the two horizontal chains takes complex values with real part equals to 7 N · m-1 and imaginary part equals to
3.5×10−2 N · m-1 (black solid lines), 7.0×10−2 N · m-1 (gray solid lines), and 1.4×10−1 N · m−1 (black dotted lines). The top (resp.
bottom) panel represents the behavior of the real (resp. imaginary) part of the pulsation ω−. The other parameters are the same as
those of Fig. 3(c). The black dashed line indicates the value of the critical wave number: kc=2.63 m-1.
Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
Contributions of the lower and upper dispersion branches to the energy of the linear two-chain model (dotted lines), E0−(k) (black)
and E0+(k) (gray), and the linear–nonlinear model (solid lines), E−(k) (black) and E+(k) (gray) normalized to their respective total
energy, E0(k) and E(k). The system parameters are a=1 m, m=1 kg, β=1 N · m−1, β′=4 N · m−1,ε=1.6 N · m−2, α0+=α0−=0.17 m. The
black dashed line indicates the value of the critical wave number: kc=2.0 m−1.
Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
ω0−(k) (black solid line), ω0+(k) (gray solid line), and ω0NL(k+kc)=ω0−(kc)+ω0−(k) (black dotted line) as functions of wave number
for three values of the spring constant β′, namely, 5.4 N · m−1 (a), 6.4 N · m-1 (b), and 7.4 N · m−1 (c). The system parameters are
a=1 m, m=1 kg, β=1 N · m−1. kℓ=kc and ku=(2π/a)−2kc are the wave numbers at which the nonlinear branch (black dotted line)
intersects the lower and the upper linear bands. The black dashed lines indicate the values of the wave numbers: kℓ=2.27 m-1 and
ku=1.74 m-1 (a), kℓ=2.48 m-1 and ku=1.32 m−1 (b), and kℓ=2.75 m−1 and ku=0.78 m−1 (c).
Date of download: 5/15/2017
Copyright © ASME. All rights reserved.
From: Nonlinear Phonon Modes in Second-Order Anharmonic Coupled Monoatomic Chains
J. Vib. Acoust. 2016;138(4):041016-041016-10. doi:10.1115/1.4033457
Figure Legend:
SEDs in J.s calculated from the velocities of the atoms in the linear–nonlinear two chains model for three values of the coupling
elastic constant β′=5.4 N · m−1 (a), β′=6.4 N · m−1 (b), and β′=7.4 N · m−1 (c). The system parameters are
a=1 m, m=1 kg, β=1 N · m−1, ε=1.6 N · m−2, α0+=α0−=0.17m. The background scale corresponds to log10(SED). The white arrow
indicates the value of kuSED: kuSED ≈ 1.7 m−1 (a), kuSED ≈ 1.3 m−1 (b), and kuSED ≈ 0.7 m−1 (c). Meaning of the black and gray
arrows is given in the text.