Finite amplitude elastic waves in a cubic crystal

Domanski, Wlodzimierz

doi:10.1063/1.1309215

Cited by 3 publications

(7 citation statements)

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“…The possibility of the appearance of quadratically nonlinear coupling in the evolution equations for shear elastic waves propagating along the threefold symmetry acoustic axis direction in a cubic crystal was shown for the first time in our paper [3]; see also [4,5]. Experimental confirmation of this fact was announced in [20].…”

Section: Discussionmentioning

confidence: 51%

“…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%

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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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Section: Discussionmentioning

confidence: 51%

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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“…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%

“…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%

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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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Wave interaction coefficients for a cubic crystal

Domanski

2022

Mathematics and Mechanics of Solids

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We study wave interaction coefficients in elastic solids. They describe locally on a quadratically nonlinear level which waves interact and measure how strong the interactions are. We derive general formulas for these coefficients in an arbitrary anisotropic material expressing them in terms of the second and the third derivatives of the strain energy. To illustrate our results, we analyse the most symmetric class of cubic crystals calculating the coefficients at the zero constant states. We first investigate the geometrically nonlinear and materially linear solid and then we study the influence of material nonlinearities on the interaction coefficients. The obtained formulas are expressed in terms of elastic constants.

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Finite amplitude elastic waves in a cubic crystal

Cited by 3 publications

References 0 publications

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

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

Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Wave interaction coefficients for a cubic crystal

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