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Mass loading effect on surface wave in piezoelectric–flexoelectric dielectric plate clamped on fiber-reinforced rigid base

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Abstract

Present study elucidates shear surface wave dispersion and attenuation in a piezoelectric-flexoelectric micro layer bedded over a heterogeneous initially stressed fiber-reinforced host structure with or with out a thin mass loading layer (ZnO layer) at the top surface of the micro-layer. Mechanical imperfection between host structure and electrically active layer is modelled with the help of Shear-lag model. Dual electromechanical coupled (flexoelectric and piezoelectric) field equations are solved by means of analytical technique and dispersion relations are obtained for electroded and non-electroded surface, separately when the infinitesimal mass loading layer is associated or not associated with piezo-flexo layer. With the help of suitable numerical example phase velocity curves and dissipation curves are plotted to illuminate the parametric responses of flexoelectricity, piezoelectricity, dielectricity, imperfection, reinforcement anisotropy and initial tensile stress. Detailed discussions about electromechanical coupling are done for different cases. Influence of piezoelectricity, dielectricity and flexoelectricity on mass loading sensitivity are also expatiated. The study may provide theoretical guidelines in investigations about mass loading sensitivity of SAW sensors with piezo-flexo coupling.

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  1. Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India

    Sanjeev A. Sahu & Mahargha Biswas

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  1. Sanjeev A. Sahu
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Appendix

Appendix

1.1 Appendix A (without mass loading)

1.1.1 For electrically open case:

\(B_{11} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{-m^f _1 h_1},\) \(B_{12} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{m^f _1 h_1},\) \(B_{13} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1}\) \(B_{14} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1},\)

\(B_{21}=(-a^f_{11}+se^f_{15}+ik_1 s(h_{41}+\frac{h_{52}}{2}))m^f _1 e^{-m^f _1 h_1},\) \(B_{22}=-(-a^f_{11}+se^f_{15}+ik_1 s(h_{41}+\frac{h_{52}}{2}))m^f _1 e^{m^f _1 h_1},\) \(B_{23}=(-a^f_{11}+te^f_{15}+ik_1 t(h_{41}+\frac{h_{52}}{2}))m^f _2 e^{-m^f _2 h_1},\) \(B_{24}=-(-a^f_{11}+te^f_{15}+ik_1 t(h_{41}+\frac{h_{52}}{2}))m^f _2 e^{m^f _2 h_1},\)

\(B_{31} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(B_{32} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(B_{33} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2,\) \(B_{34} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2,\) \(B_{35}=-(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})),\)

\(B_{41}=-\chi s,\) \(B_{42}=-\chi s,\) \(B_{43}=-\chi t,\) \(B_{44}=-\chi t,\) \(B_{45}=(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})-\chi , \)

\(B_{51}=1, \) \(B_{52}=1,\) \(B_{53}=1,\) \(B_{54}=1\)

1.1.2 For electrically short case:

\(H_{11} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{-m^f _1 h_1},\) \(H_{12} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{m^f _1 h_1},\) \(H_{13} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1},\) \(H_{14} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1},\)

\(H_{21}= e^{-m^f _1 h_1},\) \(H_{22}=e^{m^f _1 h_1},\) \(H_{23}=e^{-m^f _2 h_1},\) \(H_{24}=e^{m^f _2 h_1},\)

\(H_{31} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(H_{32} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(H_{33} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2\) \(H_{34} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2,\) \(H_{35}=-(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})),\)

\(H_{41}=-\chi s,\) \(H_{42}=-\chi s,\) \(H_{43}=-\chi t,\) \(H_{44}=-\chi t,\) \(H_{45}=(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})-\chi , \)

\(H_{51}=1, \) \(H_{52}=1,\) \(H_{53}=1,\) \(H_{54}=1\)

1.2 Appendix B (with mass loading)

1.2.1 For electrically open case:

\(B_{11} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{-m^f _1 h_1} -\rho ^m H {k_1}^2 c^2 m^f _1 e^{-m^f _1 h_1},\) \(B_{12} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{m^f _1 h_1} -\rho ^m H {k_1}^2 c^2 m^f _1 e^{m^f _1 h_1},\) \(B_{13} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1}-\rho ^m H {k_1}^2 c^2 m^f _2 e^{-m^f _2 h_1}\) \(B_{14} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1}-\rho ^m H {k_1}^2 c^2 m^f _1 e^{m^f _2 h_1}\) \(B_{21}=(-a^f_{11}+se^f_{15}+ik_1 s(h_{41}+\frac{h_{52}}{2}))m^f _1 e^{-m^f _1 h_1},\) \(B_{22}=-(-a^f_{11}+se^f_{15}+ik_1 s(h_{41}+\frac{h_{52}}{2}))m^f _1 e^{m^f _1 h_1},\) \(B_{23}=(-a^f_{11}+te^f_{15}+ik_1 t(h_{41}+\frac{h_{52}}{2}))m^f _2 e^{-m^f _2 h_1},\) \(B_{24}=-(-a^f_{11}+te^f_{15}+ik_1 t(h_{41}+\frac{h_{52}}{2}))m^f _2 e^{m^f _2 h_1},\)

\(B_{31} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(B_{32} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(B_{33} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2\) \(B_{34} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2,\) \(B_{35}=-(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})),\)

\(B_{41}=-\chi s,\) \(B_{42}=-\chi s,\) \(B_{43}=-\chi t,\) \(B_{44}=-\chi t,\) \(B_{45}=(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})-\chi , \)

\(B_{51}=1, \) \(B_{52}=1,\) \(B_{53}=1,\) \(B_{54}=1\)

1.2.2 For electrically short case:

\(H_{11} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{-m^f _1 h_1} -\rho ^m H {k_1}^2 c^2 m^f _1 e^{-m^f _1 h_1},\) \(H_{12} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1 e^{m^f _1 h_1} -\rho ^m H {k_1}^2 c^2 m^f _1 e^{m^f _1 h_1},\) \(H_{13} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1}-\rho ^m H {k_1}^2 c^2 m^f _2 e^{-m^f _2 h_1}\) \(H_{14} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2 e^{m^f _2 h_1}-\rho ^m H {k_1}^2 c^2 m^f _1 e^{m^f _2 h_1}\)

\(H_{21}= e^{-m^f _1 h_1},\) \(H_{22}=e^{m^f _1 h_1},\) \(H_{23}=e^{-m^f _2 h_1},\) \(H_{24}=e^{m^f _2 h_1},\)

\(H_{31} = (sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(H_{32} = -(sc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _1,\) \(H_{33} = (tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2\) \(H_{34} = -(tc^f_{44}+e^f_{15}-ik_1 \frac{h_{41}}{2})m^f _2,\) \(H_{35}=-(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})),\)

\(H_{41}=-\chi s,\) \(H_{42}=-\chi s,\) \(H_{43}=-\chi t,\) \(H_{44}=-\chi t,\) \(H_{45}=(\mu '_{T}\xi _1+a^2_1(\mu '_{T}-\mu '_{L})\xi _1+a_1 a_2 ik_1 (\mu '_{T}-\mu '_{L})-\chi , \)

\(H_{51}=1, \) \(H_{52}=1,\) \(H_{53}=1,\) \(H_{54}=1\)

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Sahu, S.A., Biswas, M. Mass loading effect on surface wave in piezoelectric–flexoelectric dielectric plate clamped on fiber-reinforced rigid base. Int J Mech Mater Des 18, 919–938 (2022). https://doi.org/10.1007/s10999-022-09613-w

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