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Non-continuous curves phenomenon in stressed magneto-electro-elastic plates with negative magnetic permeability

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Abstract

Several numerical methods allow the prediction of guided dispersion curves. Due to some drawbacks of such methods, e.g., the challenge of divergence, which heavily depends on partial waves, the Legendre polynomial (LP) approach is applied to the magneto-electro-elastic (MEE) alloy. This approach is based on polynomial approximation, which replaces the problem of solving a transcendental dispersion equation with an eigenvalue problem. To apply the LP method for MEE alloy, this paper presents a numerical solution to calculate the waveguide dispersion curves. The roots prediction overlaps with the convergence character of this polynomial approach, leading to accurate phase velocities of guided waves. The results indicate the existence of the obvious physical phenomenon: non-continuous furcation in guided dispersion curves. It is thus now possible to confirm this physical phenomenon, differently reported in available data. However, with an incremental change of structure from bilayer plate to alloy, non-continuous furcation for SH modes appears more obviously, which was not discussed before. With such a phenomenon, the uniaxial residual stresses can affect these new trajectories, where these stress states could play a useful role in controlling the dispersion curves. Thereby, a comparison of the guided dispersion curves in stressed and unstressed 40%BaTiO3-60%CoFe2O4 alloy answers this need and sets forth to explore the effects of initial stress. Results show that the effects of initial stresses on phase velocity change are significant and give useful information, although their influence on the non-continuous furcation phenomenon is neglected.

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Acknowledgements

This work was supported by Tunisian Ministry of Higher Education, Scientific Research (LPM laboratory).

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Authors and Affiliations

  1. Chair of Acoustics and Haptic Engineering, Dresden University of Technology, 01062, Dresden, Germany

    Cherif Othmani

  2. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, China

    Cherif Othmani

  3. Laboratory of Physics of Materials, Faculty of Sciences of Sfax, University of Sfax, PB 815, 3018, Sfax, Tunisia

    Samiha Karmi, Farid Takali & Anouar Njeh

  4. National School of Engineers of Sfax (ENIS), Street Soukra Km 4, 3018, Sfax, Tunisia

    Farid Takali

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  1. Samiha Karmi
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Appendix: The elements of \({\varvec{A}}^{{{\varvec{jm}}}}\), \(\aleph^{{{\varvec{jm}}}}\) and \({\varvec{M}}^{{{\varvec{jm}}}}\)

Appendix: The elements of \({\varvec{A}}^{{{\varvec{jm}}}}\), \(\aleph^{{{\varvec{jm}}}}\) and \({\varvec{M}}^{{{\varvec{jm}}}}\)

The elements of \({\varvec{A}}^{{{\varvec{jm}}}}\), \(\aleph^{{{\varvec{jm}}}}\) and \({\varvec{M}}^{{{\varvec{jm}}}}\) in Eq. (22) are expressed as:

$$M^{jm} = \left[ {\vartheta_{1} \left( {m,j,0} \right) \times \rho \times \pi_{0 - kH} } \right]$$
(A.1)
$$A_{11}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \left[ {C_{11} + \sigma_{0}^{X} } \right] \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times C_{55} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times C_{55} \times \pi_{0 - kH} } \right)} \right]$$
(A.2)
$$A_{13}^{jm} = \left[ {\begin{array}{*{20}c} {\left( {i\vartheta_{1} \left( {m,j,1} \right) \times \left[ {C_{55} + C_{13} } \right] \times \pi_{0 - kH} } \right)} \\ { + \left( {i\vartheta_{2} \left( {m,j,0} \right) \times C_{55} \times \pi_{0 - kH} } \right)} \\ \end{array} } \right]$$
(A.3)
$$A_{14}^{jm} = \left[ {\begin{array}{*{20}c} {\left( {i\vartheta_{1} \left( {m,j,1} \right) \times \left[ {e_{15} + e_{31} } \right] \times \pi_{0 - kH} } \right)} \\ { + \left( {i\vartheta_{2} \left( {m,j,0} \right) \times e_{15} \times \pi_{0 - kH} } \right)} \\ \end{array} } \right]$$
(A.4)
$$A_{15}^{jm} = \left[ {\begin{array}{*{20}c} {\left( {i\vartheta_{1} \left( {m,j,1} \right) \times \left[ {q_{15} + q_{31} } \right] \times \pi_{0 - kH} } \right)} \\ { + \left( {i\vartheta_{2} \left( {m,j,0} \right) \times q_{15} \times \pi_{0 - kH} } \right)} \\ \end{array} } \right]$$
(A.5)
$$A_{22}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \left[ {C_{66} + \sigma_{0}^{X} } \right] \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times C_{44} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times C_{44} \times \pi_{0 - kH} } \right)} \right]$$
(A.6)
$$A_{24}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times e_{16} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times e_{34} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times e_{34} \times \pi_{0 - kH} } \right)} \right]$$
(A.7)
$$A_{25}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times q_{16} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times q_{34} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times q_{34} \times \pi_{0 - kH} } \right)} \right]$$
(A.8)
$$A_{31}^{jm} = \left[ {\begin{array}{*{20}c} {\left( {i\vartheta_{1} \left( {m,j,1} \right) \times \left[ {C_{55} + C_{31} } \right] \times \pi_{0 - kH} } \right)} \\ { + \left( {i\vartheta_{2} \left( {m,j,0} \right) \times C_{31} \times \pi_{0 - kH} } \right)} \\ \end{array} } \right]$$
(A.9)
$$A_{33}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \left[ {C_{55} + \sigma_{0}^{X} } \right] \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times C_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times C_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.10)
$$A_{34}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times e_{15} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times e_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times e_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.11)
$$A_{35}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times q_{15} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times q_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times q_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.12)
$$A_{41}^{jm} = \left[ {\begin{array}{*{20}c} {\left( {i\vartheta_{1} \left( {m,j,1} \right) \times \left[ {e_{31} + e_{15} } \right] \times \pi_{0 - kH} } \right)} \\ { + \left( {i\vartheta_{2} \left( {m,j,0} \right) \times e_{31} \times \pi_{0 - kH} } \right)} \\ \end{array} } \right]$$
(A.13)
$$A_{42}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times e_{16} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times e_{34} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times e_{34} \times \pi_{0 - kH} } \right)} \right]$$
(A.14)
$$A_{43}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times e_{15} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times e_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times e_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.15)
$$A_{44}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \varepsilon_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times \varepsilon_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times \varepsilon_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A16)
$$AA_{44}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \varepsilon_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times \varepsilon_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times \varepsilon_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.17)
$$A_{45}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times g_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times g_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times g_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.18)
$$AA_{45}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times g_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times g_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times g_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.19)
$$A_{51}^{jm} = \left[ {\begin{array}{*{20}c} {\left( {i\vartheta_{1} \left( {m,j,1} \right) \times \left[ {q_{31} + q_{15} } \right] \times \pi_{0 - kH} } \right)} \\ { + \left( {i\vartheta_{2} \left( {m,j,0} \right) \times q_{31} \times \pi_{0 - kH} } \right)} \\ \end{array} } \right]$$
(A.20)
$$A_{52}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times q_{16} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times q_{34} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times q_{34} \times \pi_{0 - kH} } \right)} \right]$$
(A.21)
$$A_{53}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times q_{15} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times q_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times q_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.22)
$$A_{54}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times g_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times g_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times g_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.23)
$$AA_{54}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times g_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times g_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times g_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.24)
$$A_{55}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \mu_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times \mu_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times \mu_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.25)
$$AA_{55}^{jm} = \left[ {\left( { - \vartheta_{1} \left( {m,j,0} \right) \times \mu_{11} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{1} \left( {m,j,2} \right) \times \mu_{33} \times \pi_{0 - kH} } \right) + \left( {\vartheta_{2} \left( {m,j,1} \right) \times \mu_{33} \times \pi_{0 - kH} } \right)} \right]$$
(A.26)
$$\vartheta_{1} \left( {m,j,n} \right) = \mathop \int \limits_{0}^{kH} Q_{j}^{*} \left( \beta \right)\frac{{\partial^{n} Q_{m} \left( \beta \right)}}{\partial \beta }d\beta$$
(A.27)
$$\vartheta_{2} \left( {m,j,n} \right) = \mathop \int \limits_{0}^{kH} Q_{j}^{*} \left( \beta \right)\left( {\delta \left( {\beta = 0} \right) - \delta \left( {\beta = kH} \right)} \right)\frac{{\partial^{n} Q_{m} \left( \beta \right)}}{\partial \beta }d\beta$$
(A.28)

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Karmi, S., Othmani, C., Takali, F. et al. Non-continuous curves phenomenon in stressed magneto-electro-elastic plates with negative magnetic permeability. Acta Mech 234, 1599–1618 (2023). https://doi.org/10.1007/s00707-022-03460-8

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