Internal inertia gradient effect on two mode magneto‐elastic wave propagation

Abstract

In this study, two-mode harmonic wave propagation for thick rods subjected to a transverse magnetic field is addressed. The analysis is based on the Mindlin-Herrmann rod theory and the unified strain/inertia gradient model. The effects of the strain and inertial gradient parameters on lower and upper modes under different magnetic field intensities are presented in detail and discussed. This study also includes the single-mode wave propagation analysis based on the Rayleigh-Bishop rod theory, as a special case. An overview of the analysis results shows that the wave propagations in the lower and the upper modes are affected by the strain and inertial gradient parameters, as expected. However, some significant differences are observed in the effects of the internal inertial gradient parameter on these modes. In addition, this analysis reveals the relationship between the inertial gradient and cut-off frequency. K E Y W O R D Scut-off frequency, inertia gradient, Mindlin-Herrmann, transverse magnetic field, two-mode wave propagation INTRODUCTIONGenerally, gradient elasticity theories are used effectively, in the various scientific and industrial fields, to remove strain singularities and to provide realistic dispersion predictions. The beginning of the gradient elasticity theories extends to the basic formulation of Mindlin, in essence, that consists of a simultaneous extension of the potential energy and kinetic energy. The higher-order strain and inertia gradients arise by the additions to the potential energy density and kinetic energy density, respectively. The higher-order terms based on Laplacian gradient elasticity theory are proportional to the Laplacian of the corresponding lower-order terms. Nowadays, since 1986 the simplified versions of this basic formulation, proposed by Aifantis and co-workers, with one parameter are popularly preferred and frequently used. The various formats of gradient elasticity and their performance in static dynamic applications were discussed by Askes and Aifantis [1]. Askes et al. [2] and Askes and Aifantis [3] reported that the direct(in original form) use of gradient elasticity theory for the dynamics field may lead to infinite phase velocities, that is not physically realistic. Metrikine and Askes [4] showed that the strain gradient elasticity model must include the internal inertia parameter in order to give a realistic description of wave dispersion in heterogeneous materials. Again, the same authors emphasized that "the internal inertia guarantees a bounded phase velocity for all wavenumbers, whereas the phase velocity for the higher wavenumbers becomes either infinitely large or imaginary in case the internal length is present but the internal inertia absent." Papaygrai-Beskou et al. [5] reported that the only micro-elastic properties are not enough to obtain realistic dispersion predictions and in addition, the micro-inertia properties should be located in the fundamental formulation of analysis. Hence, a unified strain/ inertia gradient theory with two parameters (by ...

Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

“…should be added to Equations ( 9) and (10). Where ⃗ 𝐻-magnetic field intensity, 𝜎-conductivity, 𝜀 𝑒 -absolute dielectric constant, and 𝜇 𝑒 -absolute magnetic permeability of the medium.…”

“…The vast majority of the problems of magnetoelasticity are considered in the linear approximation and in one-dimensional or two-dimensional formulation. Among recent publications, we would like to underline [10][11][12].…”

“…In [10] the two‐mode propagation of harmonic waves for thick rods in a transverse magnetic field was introduced. The analysis is based on the Mindlin‐Herrmann bar theory and a strain gradient model.…”