Dynamic homogenization and wave propagation in a nonlinear 1D composite material

Andrianov, Igor V.; Danishevskyy, Vladyslav V.; Ryzhkov, Oleksandr I.; Weichert, Dieter

doi:10.1016/j.wavemoti.2012.08.013

Cited by 31 publications

(34 citation statements)

References 39 publications

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“…Most of the foregoing studies for this topic consider periodic layered structures consisting of nonlinear elastic materials to explore the amplitude-dependent dispersion and band-gap characteristics [38,39] or the possible occurrence of localized solutions [40][41][42]. The second-harmonic generation in multilayered structures due to material nonlinearity was analyzed by Yun et al [43].…”

Section: Introductionmentioning

confidence: 99%

Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

Biwa

Ishii

2016

Wave Motion

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The second-harmonic generation characteristics in the elastic wave propagation across an infinite layered structure consisting of identical linear elastic layers and nonlinear spring-type interlayer interfaces are analyzed theoretically. The interlayer interfaces are assumed to have identical linear interfacial stiffness but can have different quadratic nonlinearity parameters. Using a perturbation approach and the transfer-matrix method, an explicit analytical expression is derived for the second-harmonic amplitude when the layered structure is impinged by a monochromatic fundamental wave. The analysis shows that the second-harmonic generation behavior depends significantly on the fundamental frequency reflecting the band structure of the layered structure. Two special cases are discussed in order to demonstrate such dependence, i.e., the second-harmonic generation by a single nonlinear interface as well as by multiple consecutive nonlinear interfaces. In particular, when the second-harmonic generation occurs at multiple consecutive nonlinear interfaces, the cumulative growth of the second-harmonic amplitude with distance is only expected in certain frequency ranges where the fundamental as well as the double frequencies belong to the pass bands of the layered structure. Furthermore, a remarkable increase of the second-harmonic amplitude is found near the terminating edge of pass bands. Approximate expressions for the low-frequency range are also obtained, which show the cumulative growth of the second-harmonic amplitude with quadratic frequency dependence.

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Section: Introductionmentioning

confidence: 99%

Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

Biwa

Ishii

2016

Wave Motion

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“…The governing dynamical equation was obtained earlier by the method of classic higher-order asymptotic homogenization method [19,20]. In this paper, the asymptotic analysis of the problem with the help of the method of multiple time scales is employed.…”

Section: Discussionmentioning

confidence: 99%

“…In [19,20], for this problem, the higher-order asymptotic homogenization method was The displacement field u is searched as the asymptotic expansion The spatial periodicity of the composite induces the same periodicity for…”

Section: Input Problemmentioning

confidence: 99%

“…We are aimed at answering the following important question: How does the presence of the microstructure affect the processes of internal resonances and mode interactions? It should be mentioned that the governing macroscopic dynamical equation has been obtained earlier with the help of the asymptotic homogenization method [19,20], where the method of multiple time scales has been employed for the analysis of nonlinear dynamical behavior of the rod.…”

Section: Introductionmentioning

confidence: 99%

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Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

Andrianov

Awrejcewicz

Danishevskyy

et al. 2017

Journal of Sound and Vibration

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Abstract. In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized.Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge-Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement.

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“…A theoretical framework for the asymptotic theories of long wave motion in plates and layered media was developed by Rogerson et al (2006Rogerson et al ( , 2007Rogerson et al ( , 2009), Lutianov and Rogerson (2010), Mukhomodyarov and Rogerson (2012). Homogenization of nonlinear dynamic problems was considered by Andrianov et al (2011Andrianov et al ( , 2013.…”

Section: Measuring the Characteristics Of Nonlinear Waves Enables Detmentioning

confidence: 99%

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

Danishevskyy

Kaplunov

Rogerson

2015

International Journal of Non-Linear Mechanics

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Abstract. The propagation of nonlinear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural nonlinearity is considered, for which the nonlinear behaviour of the composite solid is caused by imperfect bonding at the "fibre-matrix" interface. A macroscopic wave equation accounting for the effects of nonlinearity and dispersion is derived using the higher-order asymptotic homogenization method.Explicit analytical solutions for stationary nonlinear strain waves are obtained.This type of nonlinearity has a crucial influence on the wave propagation mode: for soft nonlinearity, localised shock (kink) waves are developed, while for hard nonlinearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and nonlinear, long-and shortwave approaches discussed.

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Dynamic homogenization and wave propagation in a nonlinear 1D composite material

Cited by 31 publications

References 39 publications

Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

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