Propagation and interaction of weakly nonlinear elastic plane waves in a cubic crystal

Domanski, Wlodzimierz

doi:10.1016/j.wavemoti.2007.07.011

Cited by 8 publications

(13 citation statements)

References 8 publications

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“…In the examples below we consider a cubic crystal of class m3m in which the strain energy W is characterized by three second order and six third order elastic constants (see [1,2,3,4]) c 12 , c 44 , c 111 , c 112 , c 144 , c 123 , c 166 , c 456 ).…”

Section: Examplesmentioning

confidence: 99%

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Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Domanski¹,

Norris²

2008

AIP Conference Proceedings

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Transverse elastic waves behave differently in nonlinear isotropic and anisotropic media. While in the former the quadratically nonlinear coupling in the evolution equations for wave amplitudes is not possible, such a coupling may occur for certain directions in anisotropic materials. We identify the expression responsible for the coupling and we derive coupled canonical evolution equations for transverse wave amplitudes in the case of the two-fold and three-fold symmetry acoustic axes. We illustrate our considerations by examples for a cubic crystal.

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Section: Examplesmentioning

confidence: 99%

“…Here we present the case where the propagation direction of the plane wave is a two-fold symmetry axis n = 1 √ 2 [1 1 0] which is not an acoustic axis. The speeds of shear waves are (see [1,2,3,4])…”

Section: Examplementioning

confidence: 99%

Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Domanski¹,

Norris²

2008

AIP Conference Proceedings

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“…It is known that a quadratic nonlinearity is present only in quasilongitudinal waves when we restrict ourselves to isotropic materials [1]. The "weakest" type of nonlinearity which is manifested by shear waves propagation is cubic in such materials [2,3]. However, anisotropy and in particular the presence of fibres changes this situation.…”

Section: Introductionmentioning

confidence: 99%

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

2021

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The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation.

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“…Our aim in this work is to derive equations similar to those in [2] but by a different method. Here, unlike the presentation in [2] (see also [3][4][5][6][7][8]), where the method of weakly nonlinear geometric optics was used, we will apply a double-scale expansion. Instead of introducing a new fast variable and applying geometric optics-type asymptotics, which leads to evolution equations with three independent variables, we introduce here just two new independent variables: a slow time variable τ and a characteristic variable θ .…”

Section: Introductionmentioning

confidence: 99%

The complex Burgers equation as a model for collinear interactions of weakly nonlinear shear plane waves in anisotropic elastic materials

Domanski

2014

J Eng Math

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Collinear interactions of weakly nonlinear quasi-shear plane waves in anisotropic (in particular fiberreinforced) compressible elastic materials are analyzed. Evolution equations for quasi-shear wave amplitudes are derived with the help of the asymptotic method of a double-scale expansion. It is shown that quadratically nonlinear coupling is possible when shear waves propagate along a special fiber direction in anisotropic materials. The evolution equations are reduced to a single inviscid complex Burgers equation when the fiber direction is a threefold symmetry acoustic axis. Some properties of this equation are analyzed. General considerations are illustrated on examples of shear waves propagating along a threefold symmetry acoustic axis in a cubic crystal and in an icosahedral quasicrystal.

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Propagation and interaction of weakly nonlinear elastic plane waves in a cubic crystal

Cited by 8 publications

References 8 publications

Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

The complex Burgers equation as a model for collinear interactions of weakly nonlinear shear plane waves in anisotropic elastic materials

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