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Mathematical Models and Finite Element Approaches for Nanosized Piezoelectric Bodies with Uncoulped and Coupled Surface Effects

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Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 59))

Abstract

In this chapter the dynamic problems for piezoelectric nanosized bodies with account for coupled damping and surface effects are considered. For these problems we propose new mathematical model which generalizes the models of the elastic medium with damping in sense of the Rayleigh approach and with surface effects for the cases of piezoelectric materials. Our model of attenuation and surface effects has coupling properties between mechanical and electric fields, both for the damping terms and constitutive equations for piezoelectric materials on the surface. For solving the problems stated the finite element approximations are discussed. A set of effective finite element schemes is examined for finding numerical solutions of week statements for nonstationary problems, steady-state oscillation problems, modal problems and static problems within the framework of modelling of piezoelectric nanosized materials with damping and surface effects. For transient and harmonic problems, we demonstrate that the proposed models allow the use of the mode superposition method. In addition, we note that for transient and static problems we can use efficient finite element algorithms for solving the systems of linear algebraic equations with symmetric quasi-definite matrices both in the case of uncoupled surface effects and in the case of coupled surface effects.

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Acknowledgements

This work was supported by the Russian Science Foundation (grant number 15-19-10008).

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Authors and Affiliations

  1. Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Milchakova Street 8a, Rostov-on-Don, 344090, Russia

    Victor A. Eremeyev & A. V. Nasedkin

  2. Rzeszów University of Technology, al. Powstańców Warszawy 8, 35959, Rzeszów, Poland

    Victor A. Eremeyev

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  1. Victor A. Eremeyev
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  2. A. V. Nasedkin
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Correspondence to Victor A. Eremeyev .

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  1. Insti. for Math., Mech. and Comp. Sci., Southern Federal University Insti. for Math., Mech. and Comp. Sci., Rostov-on-Don, Russia

    Mezhlum A. Sumbatyan

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Eremeyev, V.A., Nasedkin, A.V. (2017). Mathematical Models and Finite Element Approaches for Nanosized Piezoelectric Bodies with Uncoulped and Coupled Surface Effects. In: Sumbatyan, M. (eds) Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials . Advanced Structured Materials, vol 59. Springer, Singapore. https://doi.org/10.1007/978-981-10-3797-9_1

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  • DOI: https://doi.org/10.1007/978-981-10-3797-9_1

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