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Dynamics of SH Wave Propagation in Al/BaTiO3 Composite Structure

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Mathematical Modelling and Scientific Computing with Applications (ICMMSC 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 308))

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Abstract

The present article manifests the transference of horizontally polarised shear (SH) wave in an aluminium (Al) plate overlying by a Functionally Graded Piezoelectric Material (FGPM) layer. The separation of variables method is adopted to find the solution. The phase velocity of the wave is calculated for both electrically open and short cases. The significant influence of various affecting parameters on the phase velocity curve is demonstrated through graphs. Findings may be applicable in the structural health monitoring, surface acoustic wave (SAW) devices and ultrasonic inspection techniques.

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Acknowledgements

The authors express their sincere thanks to the Science and Engineering Research Board (SERB), New Delhi, India for providing financial assistance through project No. YSS/2015/002057.

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Authors and Affiliations

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

    Sonal Nirwal & Sanjeev A. Sahu

Authors
  1. Sonal Nirwal
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  2. Sanjeev A. Sahu
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Corresponding author

Correspondence to Sonal Nirwal .

Editor information

Editors and Affiliations

  1. Discipline of Mathematics, Indian Institute of Technology Indore, Indore, Madhya Pradesh, India

    Santanu Manna

  2. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, USA

    Biswa Nath Datta

  3. Discipline of Mathematics, Indian Institute of Technology Indore, Indore, Madhya Pradesh, India

    Sk. Safique Ahmad

Appendices

Appendix 1

For electrically open circuit case

$$ \begin{aligned} & Q_{11} = \left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right)e^{{ - rh_{1} }} \left[ {r\text{Cos} \left( {\mu_{1} h_{1} } \right) + \mu_{1} \text{Sin} \left( {\mu_{1} h_{1} } \right)} \right], \\ & Q_{13\,} = m_{1} \,e_{15}^{F} \,e^{{ - m_{1} h_{1} }} ,Q_{14\,} = m_{2} \,e_{15}^{F} \,e^{{ - m_{2} h_{1} }} , \\ & Q_{12} = \left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right)e^{{ - rh_{1} }} \left[ { - r\,\text{Sin} \left( {\mu_{1} h_{1} } \right) + \mu_{1} \text{Cos} \left( {\mu_{1} h_{1} } \right)} \right], \\ & Q_{15} = 0,Q_{16} = 0,Q_{21} = - m_{1} \varepsilon_{11}^{F} e^{{ - m_{1} h_{1} }} , \\ & Q_{22} = - m_{2} \varepsilon_{11}^{F} \,e^{{ - m_{2} h_{1} }} ,Q_{23} = Q_{24} = Q_{25} = Q_{26} = 0, \\ & Q_{31} = r\left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right),Q_{32} = \mu_{1} \left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right), \\ & Q_{33} = e_{15}^{F} m_{1} ,Q_{34} = e_{15}^{F} \,m_{2} Q_{35} = - m_{3} c_{44}^{H} ,Q_{36} = m_{4} c_{44}^{H} , \\ & Q_{41} = 1,Q_{42} = Q_{43} = Q_{44} = 0,Q_{45} = - 1, \\ & Q_{46} - 1,Q_{51} = \frac{{e_{15}^{F} }}{{\varepsilon_{11}^{F} }},Q_{52} = 0,Q_{53} = 1,Q_{54} = 1, \\ & Q_{55} = Q_{56} = Q_{61} = Q_{62} = Q_{63} = Q_{64} = 0, \\ & Q_{65} = m_{3} c_{44}^{H} e^{{m_{3} h_{2} }} ,Q_{66} = m_{4} c_{44}^{H} e^{{ - m_{4} h_{2} }} . \\ \end{aligned} $$

Appendix 2

For electrically short circuit case

$$ \begin{aligned} & Q_{11} = \left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right)e^{{ - rh_{1} }} \left[ {r\text{Cos} \left( {\mu_{1} h_{1} } \right) + \mu_{1} \text{Sin} \left( {\mu_{1} h_{1} } \right)} \right], \\ & Q_{13\,} = m_{1} \,e_{15}^{F} \,e^{{ - m_{1} h_{1} }} ,Q_{14\,} = m_{2} \,e_{15}^{F} \,e^{{ - m_{2} h_{1} }} , \\ & Q_{12} = \left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right)e^{{ - rh_{1} }} \left[ { - r\,\text{Sin} \left( {\mu_{1} h_{1} } \right) + \mu_{1} \text{Cos} \left( {\mu_{1} h_{1} } \right)} \right], \\ & Q_{15} = 0,Q_{16} = 0,Q_{25} = 0,Q_{26} = 0, \\ & Q_{21} = \frac{{e_{15}^{F} }}{{\varepsilon_{11}^{F} }}e^{{ - rh_{1} }} \,\text{Cos} \left( {\mu_{1} h_{1} } \right),Q_{22} = - \frac{{e_{15}^{F} }}{{\varepsilon_{11}^{F} }}e^{{ - rh_{1} }} \text{Sin} \left( {\mu_{1} h_{1} } \right), \\ & Q_{23} = e^{{ - m_{1} h_{1} }} ,Q_{24} = e^{{ - m_{2} h_{1} }} \\ & Q_{31} = r\left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right),Q_{32} = \mu_{1} \left( {c_{44}^{F} + \frac{{\left( {e_{15}^{F} } \right)^{2} }}{{\varepsilon_{11}^{F} }}} \right), \\ & Q_{33} = e_{15}^{F} m_{1} ,Q_{34} = e_{15}^{F} \,m_{2} Q_{35} = - m_{3} c_{44}^{H} , \\ & Q_{36} = m_{4} c_{44}^{H} ,Q_{41} = 1,Q_{42} = Q_{43} = Q_{44} = 0, \\ & Q_{45} = - 1,Q_{46} - 1,Q_{51} = \frac{{e_{15}^{F} }}{{\varepsilon_{11}^{F} }},Q_{52} = 0,Q_{53} = 1, \\ & Q_{54} = 1,Q_{55} = Q_{56} = Q_{61} = Q_{62} = Q_{63} = Q_{64} = 0, \\ & Q_{65} = m_{3} c_{44}^{H} e^{{m_{3} h_{2} }} ,Q_{66} = m_{4} c_{44}^{H} e^{{ - m_{4} h_{2} }} . \\ \end{aligned} $$

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Nirwal, S., Sahu, S.A. (2020). Dynamics of SH Wave Propagation in Al/BaTiO3 Composite Structure. In: Manna, S., Datta, B., Ahmad, S. (eds) Mathematical Modelling and Scientific Computing with Applications. ICMMSC 2018. Springer Proceedings in Mathematics & Statistics, vol 308. Springer, Singapore. https://doi.org/10.1007/978-981-15-1338-1_8

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