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Nonlinear damped transient response of sandwich auxetic plates with porous magneto-electro-elastic facesheets

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Abstract

In this article, the effectiveness of the active controlled layer damping (ACLD) technique on attenuating the coupled nonlinear response of sandwich plates with auxetic core and porous functionally graded magneto-electro-elastic (PFGMEE) facings is numerically studied under the framework of finite element (FE) methods. The sandwich plate is subjected to multiphysics loading conditions, including mechanical, electrical and magnetic loads. A layerwise shear deformation theory using higher-order transverse deformation terms is used to arrive at the governing equations. This article aims to assess the integrated effects of porosity and auxetics on the nonlinear coupled response of sandwich structures. In addition, the detailed parametric study evaluates the influence of porosity distribution patterns, auxetic core's rib length ratio, inclination angle, ACLD patch positions, piezoelectric fibre (PE) fibre orientation angle, control gain and gradient index on the damped response of auxetic core/PFGMEE sandwich plates that have been presented. Emphasis has been made to understand the impact of applying open and closed circuits associated with the rest of the parameters on the controlled nonlinear behaviour of smart sandwich plates. The numerical results of this work make some major revelation about the implementation of auxetic cores in the smart structures, which make this work interesting.

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Acknowledgements

The financial support by the Royal Society, London, through the Newton International Fellowship (NIF\R1\212432) is sincerely acknowledged by the author Vinyas Mahesh

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Authors and Affiliations

  1. Department of Mechanical Engineering and Aeronautics, City, University of London, London, EC1V 0HB, UK

    Vinyas Mahesh & Sathiskumar A. Ponnusami

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Appendix

Appendix

The strain displacement matrices can be expressed as follows:

$$ \left[ {B_{{{\text{tb}}}} } \right] = \left[ {\begin{array}{*{20}c} {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 \\ 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & 0 \\ 0 & 0 & 0 \\ {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 \\ \end{array} } \right];\,\,\,\left[ {B_{{{\text{ts}}}} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} \\ 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} \\ \end{array} } \right] $$

\(\left[ {B_{{{\text{rb}}}} } \right] = \left[ {\begin{array}{*{20}c} {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} \\ \end{array} } \right],.....\left[ {B_{{{\text{rs}}}} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\partial \mathord{\left/ {\vphantom {\partial {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}}} & 0 & 0 & 0 & 0 \\ \end{array} } \right]\),

The different transformation matrices [Z1] – [Z5] can be elaborated and expressed as follows:

$$ \begin{gathered} \left[ {Z_{1} } \right] = \left[ {\begin{array}{*{20}c} z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & {2z} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & z & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \hfill \\ \left[ {Z_{2} } \right] = \left[ {\begin{array}{*{20}c} {{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}} & 0 & 0 & 0 & 0 & {h_{{\text{v}}} } & 0 & 0 & {\left( {z - h_{{\text{v}}} - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} & 0 & 0 \\ 0 & {{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}} & 0 & 0 & 0 & 0 & {h_{{\text{v}}} } & 0 & 0 & {\left( {z - h_{{\text{v}}} - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} & 0 \\ 0 & 0 & 0 & 1 & {2z} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}} & 0 & 0 & 0 & 0 & {h_{{\text{v}}} } & 0 & 0 & {\left( {z - h_{{\text{v}}} - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}} \right)} \\ \end{array} } \right] \hfill \\ \left[ {Z_{3} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & z & 0 & {z^{2} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & z & 0 & {z^{2} } \\ \end{array} } \right];\,\left[ {Z_{4} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 & 0 & 0 & z & 0 & {z^{2} } & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & z & 0 & {z^{2} } \\ \end{array} } \right] \hfill \\ \left[ {Z_{5} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 1 & 0 & z & 0 & {z^{2} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & z & 0 & z \\ \end{array} } \right] \hfill \\ \end{gathered} $$
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Mahesh, V., Ponnusami, S.A. Nonlinear damped transient response of sandwich auxetic plates with porous magneto-electro-elastic facesheets. Eur. Phys. J. Plus 137, 563 (2022). https://doi.org/10.1140/epjp/s13360-022-02756-x

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