Degenerate weakly non-linear elastic plane waves

Domanski, Wlodzimierz; Norris, Andrew N.

doi:10.1016/j.ijnonlinmec.2008.12.009

Cited by 6 publications

(8 citation statements)

References 14 publications

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“…Since n · k 1 = 0, hence the last term in (41) disappears and we obtain (47) and (48); see [2] for more details.…”

Section: Lemmamentioning

confidence: 74%

“…In this case we obtain that 1 + 1 2 = 0 and that 2 = 2 1 = 0 (see [6] or [2] for details). Therefore, we can formulate the following lemma.…”

Section: Threefold Axismentioning

confidence: 83%

“…Only one coefficient, 2 , determines the coupling between the evolution equations when the propagation direction is a threefold symmetry axis.…”

Section: Lemma 2 the Evolution Equations For The Amplitudes Of Shear mentioning

confidence: 99%

“…Using the expansion of weakly nonlinear geometric optics we have shown [2] what the quadratically nonlinear coupled equations for quasi-shear waves in anisotropic materials look like. Our aim in this work is to derive equations similar to those in [2] but by a different method.…”

Section: Introductionmentioning

confidence: 99%

“…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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“…Since n · k 1 = 0, hence the last term in (41) disappears and we obtain (47) and (48); see [2] for more details.…”

Section: Lemmamentioning

confidence: 74%

“…In this case we obtain that 1 + 1 2 = 0 and that 2 = 2 1 = 0 (see [6] or [2] for details). Therefore, we can formulate the following lemma.…”

Section: Threefold Axismentioning

confidence: 83%

“…Only one coefficient, 2 , determines the coupling between the evolution equations when the propagation direction is a threefold symmetry axis.…”

Section: Lemma 2 the Evolution Equations For The Amplitudes Of Shear mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Self Cite

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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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Non-linear elastic micro-dilatation theory: Matrix exponential function paradigm

Ramézani

Jeong²

2015

International Journal of Solids and Structures

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a b s t r a c tIn the present paper, the micro-dilatation theory or void elasticity is extended to both large displacement and large dilatation. Firstly, the deformation gradient tensor has been freshly defined by means of the matrix exponential function. The newly defined relation for the deformation gradient has painstakingly investigated for the uniqueness, decomposition issues as well as objectivity and isotropy considerations. The relation of the displacement gradient and deformation gradient tensor is brought via the matrix logarithm function. The micro-dilatation theory constitutive laws are derived using the thermodynamic principles under the zero-centrosymmetric, weakly-centrosymmetric and fully-centrosymmetric cases. These cases have been derived and scrutinized by the numerical experiments. To achieve this assignment, the basic loadings are taken into account, e.g. the hydrostatic loading, simple traction and shear. Some conclusions and outlook pertaining to the above-mentioned cases and variable bulk density have thereafter discussed.

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Degenerate weakly non-linear elastic plane waves

Cited by 6 publications

References 14 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

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

Non-linear elastic micro-dilatation theory: Matrix exponential function paradigm

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