Mono-scale and multi-scale formulations of gradient-enriched dynamic piezomagnetics

Xu, Ming; Askes, Harm; Gitman, Inna M.

doi:10.1016/j.wavemoti.2019.102402

Cited by 2 publications

(4 citation statements)

References 47 publications

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“…It can be seen that the wave fronts of all four fields becomes less and less steep as time progresses. This is an indication that the harmonic components of higher frequency travel with lower velocity, and confirms that the model is dispersive-see [44] and Section 6.1 above. The micro-scale fields are more oscillatory than the associated macro-scale fields, which can be understood from the equivalence with homogenisation as explained in Section 2.…”

Section: Wave Dispersionsupporting

confidence: 73%

“…Next, the effect of the various length scales ℓ 1 , ℓ 3 and ℓ 4 on the dispersive properties of the material is investigated in more detail. In [44] and in Section 6.1, we demonstrated that the ratio ℓ 4 /ℓ 1 is the dominant parameter in this context, with higher wave numbers travelling slower for increased values of ℓ 4 /ℓ 1 . In Figure 6 we have plotted the results of micro-scale and macro-scale strain for two ratios of ℓ 4 /ℓ 1 .…”

Section: Wave Dispersionmentioning

confidence: 58%

“…We will explore homogenisation techniques that allow us to rewrite the equations of generalised piezomagnetic continua as a coupled set of multi-scale partial differential equations whereby the microscale mechanical and magnetic fields appear alongside the macro-scale mechanical and magnetic fields. This paper is the follow-up of earlier work we reported on statics [45] and a previous article where we explored suitable formats of gradient-enriched piezomagnetics in a one-dimensional dynamics context [44]. Novel aspects of the present paper include (i) a new motivation of the model that relies on homogenisation rather than postulation of an enriched energy functional, (ii) the extension to multiple spatial dimensions, (iii) the development of a simple C 0 -continuous finite element implementation, and (iv) the derivation of an optimal ratio of time step to element size.…”

Section: Introductionmentioning

confidence: 87%

“…A more in-depth dispersion analysis, including a comparison with the equivalent mono-scale model, can be found in [44]. The purpose here is to compare the dispersion relation of the continuum with that of the discretised equations.…”

Section: Dispersion Relation Of the Continuummentioning

confidence: 99%

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Finite element implementation of a multi-scale dynamic piezomagnetic continuum model

Gitman

Wei

et al. 2020

Computers & Structures

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A gradient-enriched dynamic piezomagnetic model is presented. The gradient enrichment introduces a number of microstructural terms in the model that allow the description of dispersive wave propagation. A novel derivation based on homogenisation principles is shown to lead to a multi-scale formulation in which the micro-scale displacements and magnetic potential are included alongside the macro-scale displacements and magnetic potential. The multi-scale formulation of the model has the significant advantage that all higher-order terms are rewritten as second-order spatial derivatives. As a consequence, a standard C 0 -continuous finite element discretisation can be used. Details of the finite element implementation are given. A series of one and two-dimensional examples shows the effectiveness of the model to describe dispersive wave propagation and remove singularities in a coupled elasto-magnetic context.

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Section: Wave Dispersionsupporting

confidence: 73%

Section: Wave Dispersionmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 87%

Section: Dispersion Relation Of the Continuummentioning

confidence: 99%

See 2 more Smart Citations

Finite element implementation of a multi-scale dynamic piezomagnetic continuum model

Gitman

Wei

et al. 2020

Computers & Structures

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

Güven

2021

Z Angew Math Mech

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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 ...

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Mono-scale and multi-scale formulations of gradient-enriched dynamic piezomagnetics

Abstract: This is a repository copy of Mono-scale and multi-scale formulations of gradient-enriched dynamic piezomagnetics.

Cited by 2 publications

References 47 publications

Finite element implementation of a multi-scale dynamic piezomagnetic continuum model

Finite element implementation of a multi-scale dynamic piezomagnetic continuum model

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

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