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

Domanski, Wlodzimierz; Jemioło, Stanisław; Franus, Aleksander

doi:10.1007/s10665-021-10093-8

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…Remark. Here, we have to point out that the results obtained in equations ( 11) and ( 14) can be associated with any strain-energy densities in the form here considered and not only to equation (6). Indeed, if we start from a general strain-energy density W = W (I 1 À 3, I 2 À 3, I 4 À 1) to obtain the previous asymptotic equations we have to compute its Taylor expansion with respect to I 1 À 3, I 2 À 3, and I 4 À 1 up to the second order.…”

Section: Shear Wavesmentioning

confidence: 94%

“…In recent times, to the best of our knowledge, the interest of the scientific community has been mainly focused on the theory of weakly nonlinear elasticity. This is the case of an interesting series of papers by Domański, see, e.g., Domański and his colleagues [5,6], and a recent paper [7] about the interaction of shear waves in transversely isotropic soft solids. Less common is to come across studies where the full theory of continuum mechanics is employed to investigate dynamic problems for anisotropic materials.…”

Section: Introductionmentioning

confidence: 99%

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On shear motions in nonlinear transverse isotropic elastodynamics

Amendola

Saccomandi

Speranzini

2022

Mathematics and Mechanics of Solids

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We study the propagation of shear waves in an anisotropic incompressible medium composed of an elastic fiber-reinforced material. We first derive the general determining equations of such motions when the fibers are arbitrarily arranged with respect to the direction of propagation of the waves. Then, we examine the “standard reinforced model” so as to root the nonlinearity of the dynamic problem directly to the presence of the reinforcing fibers, and finally, we consider an asymptotic model in terms of the limit of finite but small amplitude of the waves. Clearly, if the fibers are arranged along the direction of propagation of the waves, the mechanical behavior of the material will be isotropic. When the fibers are tilted an amount of the same order of the amplitude of the wave with respect to this direction, the asymptotic first-order reduced system is characterized by a peculiar mixed second-/third-order nonlinearity. This contrasts to what occurs for larger tilting angles where only quadratic nonlinearities must be considered. In the latter case, as we may expect, in the asymptotic limit, we reduce the system to an inviscid classical Burger’s equation. Our results are fundamental to the study of the dynamics of fiber-reinforced materials and equally when we consider dissipative and/or dispersive effects.

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Section: Shear Wavesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

On shear motions in nonlinear transverse isotropic elastodynamics

Amendola

Saccomandi

Speranzini

2022

Mathematics and Mechanics of Solids

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“…This concerns particularly transverse waves. Unlike as it is in the isotropic case, anisotropy may imply, e.g., the coupling on the quadratically nonlinear level in the evolution equations for the shear waves [4][5][6][7]. The information about the coupling can be deduced from the knowledge of interaction coefficients.…”

Section: Introductionmentioning

confidence: 99%

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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Plane nonlinear shear wave propagation in transversely isotropic soft solids

Cormack

2021

The Journal of the Acoustical Society of America

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Nonlinear wave equations are obtained for the two plane shear wave modes in a transversely isotropic soft solid. The material is modeled using a general expansion of the strain energy density up to fourth order in strain. Whereas, in an isotropic soft solid, leading order nonlinearity for plane wave propagation appears at cubic order in strain, elastic anisotropy in a transversely isotropic material introduces nonlinear effects at quadratic order, including interaction between the modes of a wave with two displacement components. Expressions for second harmonic generation in an elliptically polarized wave field illustrate the low efficiency of nonlinear interactions between the two displacement components, which results from the disparity between propagation speeds of the two shear wave modes. Coupled wave equations with up to cubic nonlinearity are presented and then approximated to describe linearly polarized waves by neglecting interaction between modes. Evolution equations are obtained for linearly polarized progressive waves, and explicit expressions are given in terms of elastic moduli and propagation direction for the coefficients of leading order nonlinearity. Expressions are presented for up to third harmonic generation from a time-harmonic source.

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

Cited by 4 publications

References 14 publications

On shear motions in nonlinear transverse isotropic elastodynamics

On shear motions in nonlinear transverse isotropic elastodynamics

Wave interaction coefficients for a cubic crystal

Plane nonlinear shear wave propagation in transversely isotropic soft solids

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