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Porous flexoelectric cylindrical nanoshell based on the non-classical continuum theory

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Abstract

In this paper, the coupling equations dependent on the size of the functionally graded porous flexoelectric cylindrical nanoshell are extracted based on non-classical theory. The electromechanical coupling equations dependent on the size of the functionally graded porous flexoelectric cylindrical nanoshell are a purpose of this paper and these coupling equations are extracted with polarization (independent of electrical potential) which is the underlying innovation of the current study. When flexoelectricity and porosity are not considered, the shell model developed in this paper is reduced to the simple classical first-order shell model. The law of power distribution has been used to model the changes of the constituent material in the shell thickness, and three different porosity distributions have been considered. The effects of microstructure are considered using modified flexoelectric strain gradient theory. Finally, to investigate the application of the formulation performed in this paper, the problem is solved for a specific case such as free vibrations. In free vibration analysis, the geometric and material effects of porous flexoelectric nanotubes have been investigated. According to the obtained results, it can be noted that in the case of positive external voltage, by increasing the flexoelectric coefficient of the material, the natural frequency of the nanotube increases. Also, increasing the porosity of the flexoelectric cylindrical nanoshell increases the frequencies. Also, in the presence of the flexoelectric effect, the modified strain gradient theory, by considering three size parameters, predicts higher results numerically than the couple and classical stress theories.

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Funding

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Authors and Affiliations

  1. Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran

    Ashkan Ashrafi Dehkordi & Reza Jahanbazi Goojani

  2. Faculty of Engineering, Shahrekord University, Shahrekord, Iran

    Yaghoub Tadi Beni

  3. Nanotechnology Research Institute, Shahrekord University, Shahrekord, 8818634141, Iran

    Yaghoub Tadi Beni

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  1. Ashkan Ashrafi Dehkordi
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  2. Reza Jahanbazi Goojani
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Correspondence to Yaghoub Tadi Beni.

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Appendix

Appendix

$$\begin{aligned} A_{11} & = \int_{ - h/2}^{h/2} {\left( { - 2{\mkern 1mu} k(z) - 8/3{\mkern 1mu} \mu (z) - f_{1} } \right){\text{d}}z} \\ A_{12} & = \int_{ - h/2}^{h/2} {4/5{\mkern 1mu} \mu (z)\left( {5{\mkern 1mu} l_{0}^{2} + 2{\mkern 1mu} l_{1}^{2} } \right){\text{d}}z} \\ A_{13} & = \int_{ - h/2}^{h/2} {2/3{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} \left( {3{\mkern 1mu} R^{2} + 4{\mkern 1mu} l_{1}^{2} + 6{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{4} }}{\text{d}}z} \\ A_{14} & = \int_{ - h/2}^{h/2} {1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} + 16{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{4} }}{\text{d}}z} \\ A_{15} & = \int_{ - h/2}^{h/2} {\left( {1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} \left( {87{\mkern 1mu} l_{0}^{2} + 64{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }} - 8/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{R^{2} }}} \right){\text{d}}z} \\ A_{16} & = \int_{ - h/2}^{h/2} {\left( {2{\mkern 1mu} \frac{{\left( {R^{2} + 4/3{\mkern 1mu} l_{1}^{2} - l_{2}^{2} } \right)\mu (z){\mkern 1mu} }}{{R^{3} }} - 2/3{\mkern 1mu} \frac{{3{\mkern 1mu} k(z){\mkern 1mu} - 2{\mkern 1mu} \mu (z){\mkern 1mu} }}{R} + 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} + 8{\mkern 1mu} l_{1}^{2} - 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }} - \frac{{f_{1} }}{R}} \right){\text{d}}z} \\ A_{17} & = \int_{ - h/2}^{h/2} {\left( { - 8/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{R^{3} }} - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27l_{0}^{2} - 32{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }}} \right){\text{d}}z} \\ A_{18} & = \int_{ - h/2}^{h/2} {\left( { - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} - 32{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{R} - 8/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} l_{1}^{2} }}{R}} \right){\text{d}}z} \\ A_{19} & = \int_{ - h/2}^{h/2} {\left( { - 2/3{\mkern 1mu} \frac{{3{\mkern 1mu} k(z){\mkern 1mu} - 2{\mkern 1mu} \mu (z){\mkern 1mu} }}{R} - \frac{{f_{1} }}{R}} \right){\text{d}}z} \\ A_{110} & = \int_{ - h/2}^{h/2} {8/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{R}{\text{d}}z} \\ A_{111} & = \int_{ - h/2}^{h/2} {\left( {8/3{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {l_{1}^{2} - 3/4{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }} - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} + 32l_{1}^{2} - 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }} - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} - 32l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }}} \right){\text{d}}z} \\ A_{112} & = \int_{ - h/2}^{h/2} {\left( { - 4/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} l_{1}^{2} }}{R} - f_{1} {\mkern 1mu} z} \right){\text{d}}z} \\ A_{113} & = \int_{ - h/2}^{h/2} {\left( {8/3{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {l_{1}^{2} + 3/2{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }} + 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27l_{0}^{2} + 16l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }}} \right){\text{d}}z} \\ A_{114} & = \int_{ - h/2}^{h/2} {\left( {8/3{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {l_{1}^{2} - 3/4{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }} - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} + 8{\mkern 1mu} l_{1}^{2} - 15l_{2}^{2} } \right)}}{{R^{2} }} - \frac{{f_{{1{\mkern 1mu} }} z}}{R}} \right){\text{d}}z} \\ A_{115} & = - R \\ \end{aligned}$$
$$\begin{aligned} {B_{21}} & = \int_{ - h/2}^{h/2} {\left( { - 2 {{\left( {{R^2} + 4/3 {l_1}^2 - {l_2}^2} \right)\mu (z) } \over {{R^3}}} - 2/3 {{3 k(z) - 2 \mu (z) } \over R} + 1/15 {{\mu (z) \left( {27 {l_0}^2 + 8 {l_1}^2 - 15 {l_2}^2} \right)} \over {{R^3}}} - {{{f_1}} \over {{R^2}}}} \right){\rm{d}}z} \\ {B_{22}} & = \int_{ - h/2}^{h/2} {\left( { - 8/5 {{\mu (z) {l_1}^2} \over {{R^3}}} - 1/15 {{\mu (z) \left( {27 {l_0}^2 - 32{l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}} \right){\rm{d}}z} \\ {B_{23}} & = \int_{ - h/2}^{h/2} {\left( { - 1/15 {{\mu (z) \left( {27 {l_1}^2 - 32 {l_1}^2 + 15 {l_2}^2} \right)} \over R} - 8/5 {{\mu (z) {l_1}^2} \over R}} \right){\rm{d}}z} \\ {B_{24}} & = \int_{ - h/2}^{h/2} {1/15 {{\mu (z) \left( {30 {R^2} + 27 {l_0}^2 + 136 {l_1}^2 + 15{l_2}^2} \right)} \over {{R^4}}}{\rm{d}}z} \\ {B_{25}} & = \int_{ - h/2}^{h/2} {\left( { - 2/3 {{\mu (z) \left( {3 {R^2} + 4 {l_1}^2 + 6 {l_2}^2} \right)} \over {{R^2}}} - 2/15 {{\mu (z) \left( {27 {l_0}^2 - 4{l_1}^2 - 15 {l_2}^2} \right)} \over {{R^2}}}} \right){\rm{d}}z} \\ {B_{26}} & = \int_{ - h/2}^{h/2} {1/15 \mu (z) \left( {27 {l_0}^2 + 16 {l_1}^2 + 15 {l_2}^2} \right){\rm{d}}z} \\ {B_{27}} & = \int_{ - h/2}^{h/2} {\left( { - 2 {1 \over {{R^4}}}\left( {\left( {4/3 {R^2} + {{19{l_0}^2} \over 5} + {{128 {l_1}^2} \over {15}} + {l_2}^2} \right)\mu (z) + {R^2}k(z) } \right) - {{32 \mu (z) {l_1}^2} \over {5 {R^4}}} - {{{f_1}} \over {{R^3}}}} \right){\rm{d}}z} \\ {B_{28}} & = \int_{ - h/2}^{h/2} {4/5 {{\mu (z) \left( {5 {l_0}^2 + 2{l_1}^2} \right)} \over {{R^4}}}dz} \\ {B_{29}} & = \int_{ - h/2}^{h/2} {\left( {1/15 {{\mu (z) \left( {87 {l_0}^2 + 64{l_1}^2 + 15 {l_2}^2} \right)} \over {{R^2}}} - 8/5 {{\mu (z) {l_1}^2} \over {{R^2}}}} \right){\rm{d}}z} \\ {B_{210}} & = \int_{ - h/2}^{h/2} {\left( { - 2 {{\mu (z) } \over {{R^2}}} - 1/15 {{\mu (z) \left( {27 {l_0}^2 + 184 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^4}}} - 2 {1 \over {{R^4}}}\left( {\left( {k(z) + 4/3 \mu (z) } \right){R^2} + {{32 \mu (z) {l_1}^2} \over 5}} \right) - {{{f_1}} \over {{R^3}}}} \right)dz} \\ {B_{211}} & = \int_{ - h/2}^{h/2} {\left( {2/15 {{\mu (z) \left( {27 {l_0}^2 + 32{l_1}^2 + 15{l_2}^2} \right)} \over {{R^4}}} + {{24 \mu (z) {l_1}^2} \over {5 {R^4}}}} \right){\rm{d}}z} \\ {B_{212}} & = \int_{ - h/2}^{h/2} {\left( {1/30 {{\mu (z) \left( {81 {l_0}^2 - 32{l_1}^2 - 45 {l_2}^2} \right)} \over {{R^2}}} + 1/15 {{\mu (z) \left( {27 {l_0}^2 + 64 {l_1}^2 + 15{l_2}^2} \right)} \over {{R^2}}} + 8/3 {{\mu (z) \left( {{l_1}^2 + 3/2{l_2}^2} \right)} \over {{R^2}}} + 1/15 {{\mu (z) \left( {27 {l_0}^2 - 16{l_1}^2 - 15{l_2}^2} \right)} \over {{R^2}}}} \right){\rm{d}}z} \\ {B_{213}} & = \int_{ - h/2}^{h/2} {\left( {8/3 {{\mu (z) {l_1}^2} \over {{R^2}}} - 1/30 {{\mu (z) \left( {81 {l_0}^2 + 64{l_1}^2 - 45 {l_2}^2} \right)} \over {{R^2}}} - 2 {{\mu (z) {l_2}^2} \over {{R^2}}} - 1/15 {{\mu (z) \left( {27{l_0}^2 - 32 {l_1}^2 + 15{l_2}^2} \right)} \over {{R^2}}} - {{{f_{1 }}z} \over {{R^2}}}} \right){\rm{d}}z} \\ {B_{214}} & = \int_{ - h/2}^{h/2} {\left( { - 2 {{\mu (z) } \over R} - 1/15 {{\mu (z) \left( {27 {l_0}^2 + 136 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}} \right){\rm{d}}z} \\ {B_{215}} & = \int_{ - h/2}^{h/2} {\left( {8/3 {{\mu (z) \left( {{l_1}^2 - 3/4 {l_2}^2} \right)} \over R} + 1/5 {{\mu (z) \left( {9 {l_0}^2 - 5 {l_2}^2} \right)} \over R} - {{4 \mu (z) {l_1}^2} \over {15 R}}} \right){\rm{d}}z} \\ {B_{216}} & = \int_{ - h/2}^{h/2} {\left( { - 2/15 {{\mu (z) \left( {27 {l_0}^2 - 64 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}} + {{16 \mu (z) {l_1}^2} \over {5 {R^3}}} - {{{f_1} z} \over {{R^3}}}} \right){\rm{d}}z} \\ {B_{217}} & = - R \end{aligned}$$
$$\begin{aligned} {C_{31}} & = \int_{ - h/2}^{h/2} {2/3 {{\left( {3 k(z) - 2 \mu (z) } \right)} \over R}{\rm{d}}z} \\ {C_{32}} & = \int_{ - h/2}^{h/2} {8/5 {{\mu (z) {l_1}^2} \over R}{\rm{d}}z} \\ {C_{33}} & = \int_{ - h/2}^{h/2} {\left( { - 1/15 {{\mu (z) \left( {27 {l_0}^2 + 32 {l_1}^2 - 15 {l_2}^2} \right)} \over {{R^3}}} - 1/15 {{\mu (z) \left( {27 {l_0}^2 - 32 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}} \right){\rm{d}}z} \\ {C_{34}} & = \int_{ - h/2}^{h/2} { - 8/3 {{\mu (z) \left( {{l_1}^2 - 3/4 {l_2}^2} \right)} \over {{R^3}}}{\rm{d}}z} \\ {C_{35}} & = \int_{ - h/2}^{h/2} {\left( {2 {{\mu (z) } \over {{R^2}}} + 1/15 {{\mu (z) \left( {27 {l_0}^2 + 184 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^4}}} + 2 {1 \over {{R^4}}}\left( {\left( {k(z) + 4/3 \mu (z) } \right){R^2} + {{32 \mu (z) {l_1}^2} \over 5}} \right)} \right){\rm{d}}z} \\ {C_{36}} & = \int_{ - h/2}^{h/2} {\left( { - 2/15 {{\mu (z) \left( {27 {l_0}^2 + 32 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^4}}} - {{24 \mu (z) {l_1}^2} \over {5 {R^4}}}} \right){\rm{d}}z} \\ {C_{37}} & = \int_{ - h/2}^{h/2} {\left( { - 1/30 {{\mu (z) \left( {81 {l_0}^2 - 32 {l_1}^2 - 45 {l_2}^2} \right)} \over {{R^2}}} - 1/15 {{\mu (z) \left( {27 {l_0}^2 + 64 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^2}}} - 8/3 {{\mu (z) \left( {{l_1}^2 + 3/2 {l_2}^2} \right)} \over {{R^2}}} - 1/15 {{\mu (z) \left( {27 {l_0}^2 - 16 {l_1}^2 - 15 {l_2}^2} \right)} \over {{R^2}}}} \right){\rm{d}}z} \\ {C_{38}} & = \int_{ - h/2}^{h/2} {{2 \over {{R^4}}}\left( {\left( {4/3 {R^2} + 2 {l_0}^2 + {{24 {l_1}^2} \over 5}} \right)\mu (z) + {R^2}k(z) } \right){\rm{d}}z} \\ {C_{39}} & = \int_{ - h/2}^{h/2} {\left( { - 1/15 {{\mu (z) \left( {30 {R^2} + 87 {l_0}^2 + 64 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^2}}} + 8/5 {{\mu (z) {l_1}^2} \over {{R^2}}}} \right){\rm{d}}z} \\ {C_{310}} & = \int_{ - h/2}^{h/2} {1/15 \mu (z) \left( {27 {l_0}^2 + 16 {l_1}^2 + 15 {l_2}^2} \right){\rm{d}}z} \\ {C_{311}} & = \int_{ - h/2}^{h/2} {\left( { - 1/15 {{\mu (z) \left( {30 {R^2} + 87 {l_0}^2 + 256 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^4}}} - {{32 \mu (z) {l_1}^2} \over {5 {R^4}}}} \right){\rm{d}}z} \\ {C_{312}} & = \int_{ - h/2}^{h/2} {2/15 {{\mu (z) \left( {27 {l_0}^2 + 8 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^4}}}{\rm{d}}z} \\ {C_{313}} & = \int_{ - h/2}^{h/2} {\left( {8/3 {{\left( {{l_1}^2 + 3/2 {l_2}^2} \right)\mu (z) } \over {{R^2}}} + 1/15 {{\mu (z) \left( {81 {l_0}^2 - 8 {l_1}^2 - 45 {l_2}^2} \right)} \over {{R^2}}}} \right){\rm{d}}z} \\ {C_{314}} & = \int_{ - h/2}^{h/2} {\left( { - 1/15 {{\mu (z) \left( {30 {R^2} - 27 {l_0}^2 + 32 {l_1}^2 - 15 {l_2}^2} \right)} \over {{R^2}}} + 8/5 {{\mu (z) {l_1}^2} \over {{R^2}}}} \right){\rm{d}}z} \\ {C_{315}} & = \int_{ - h/2}^{h/2} { - 1/15 \mu (z) \left( {27 {l_0}^2 - 32 {l_1}^2 + 15 {l_2}^2} \right){\rm{d}}z} \\ {C_{316}} & = \int_{ - h/2}^{h/2} {\left( { - {{32 \mu (z) {l_1}^2} \over {5 {R^3}}} - 1/15 {{\mu (z) \left( {30 {R^2} + 27 {l_0}^2 + 184 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}} \right){\rm{d}}z} \\ {C_{317}} & = \int_{ - h/2}^{h/2} {\left( {8/3 {{\left( {{l_1}^2 - 3/4 {l_2}^2} \right)\mu (z) } \over {{R^2}}} - 1/30 {{\mu (z) \left( {81 {l_0}^2 + 16 {l_1}^2 - 45 {l_2}^2} \right)} \over {{R^2}}}} \right){\rm{d}}z} \\ {C_{318}} & = \int_{ - h/2}^{h/2} { - 2/15 {{\mu (z) \left( {27 {l_0}^2 - 16 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}{\rm{d}}z} \\ {C_{319}} & = \int_{ - h/2}^{h/2} {\left( { - 1/30 {{\mu (z) \left( {81 {l_0}^2 + 16 {l_1}^2 - 45 {l_2}^2} \right)} \over R} + 8/3 {{\left( {{l_1}^2 - 3/4 {l_2}^2} \right)\mu (z) } \over R}} \right){\rm{d}}z} \\ {C_{320}} & = - R \end{aligned}$$
$$\begin{aligned} D_{41} & = \int_{ - h/2}^{h/2} {\left( { - 4/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{R} - zf_{1} } \right){\text{d}}z} \\ D_{42} & = \int_{ - h/2}^{h/2} {\left( {8/3{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {l_{1}^{2} + 3/2{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }} + 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} + 16{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }}} \right){\text{d}}z} \\ D_{43} & = \int_{ - h/2}^{h/2} {\left( {8/3{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{R^{2} }} - 1/30{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {81{\mkern 1mu} l_{0}^{2} + 64{\mkern 1mu} l_{1}^{2} - 45{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }} - 2{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{2}^{2} }}{{R^{2} }} - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} - 32{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }} - \frac{{zf_{1} }}{R}} \right){\text{d}}z} \\ D_{44} & = \int_{ - h/2}^{h/2} {\left( {1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {30{\mkern 1mu} R^{2} - 27{\mkern 1mu} l_{0}^{2} + 32{\mkern 1mu} l_{1}^{2} - 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }} - 8/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{R^{2} }} - \frac{{zf_{1} }}{R}} \right){\text{d}}z} \\ D_{45} & = \int_{ - h/2}^{h/2} {1/15\mu (z){\mkern 1mu} {\mkern 1mu} \left( {27{\mkern 1mu} l_{0}^{2} - 32{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right){\text{d}}z} \\ D_{46} & = \int_{ - h/2}^{h/2} {\left( { - 8/3{\mkern 1mu} \frac{{\left( {l_{1}^{2} - 3/4{\mkern 1mu} l_{2}^{2} } \right)\mu (z){\mkern 1mu} }}{{R^{2} }} + 1/30{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {81{\mkern 1mu} l_{0}^{2} + 16{\mkern 1mu} l_{1}^{2} - 45{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }}} \right){\text{d}}z} \\ D_{47} & = \int_{ - h/2}^{h/2} {1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} \left( {30{\mkern 1mu} R^{2} + 27{\mkern 1mu} l_{0}^{2} + 16{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }}{\text{d}}z} \\ D_{48} & = \int_{ - h/2}^{h/2} {\left( { - 1/15{\mkern 1mu} \left( {87{\mkern 1mu} l_{0}^{2} + 64{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} + 40{\mkern 1mu} z^{2} } \right)\mu (z){\mkern 1mu} - 2{\mkern 1mu} z^{2} k(z){\mkern 1mu} - f_{1} {\mkern 1mu} z^{2} } \right){\text{d}}z} \\ D_{49} & = \int_{ - h/2}^{h/2} {4/5{\mkern 1mu} \mu (z){\mkern 1mu} {\mkern 1mu} \left( {5{\mkern 1mu} l_{0}^{2} + 2{\mkern 1mu} l_{1}^{2} } \right)z^{2} {\text{d}}z} \\ D_{410} & = \int_{ - h/2}^{h/2} { - 2{\mkern 1mu} \frac{{\left( {\left( {z^{2} + 4/3{\mkern 1mu} l_{1}^{2} + 2{\mkern 1mu} l_{2}^{2} } \right)R^{2} + 4/3{\mkern 1mu} l_{1}^{2} z^{2} } \right)\mu (z){\mkern 1mu} }}{{R^{4} }}{\text{d}}z} \\ D_{411} & = \int_{ - h/2}^{h/2} {1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} z^{2} \left( {27{\mkern 1mu} l_{0}^{2} + 16{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{4} }}{\text{d}}z} \\ D_{412} & = \int_{ - h/2}^{h/2} {\left( {1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} z^{2} \left( {87{\mkern 1mu} l_{0}^{2} + 64{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{2} }} - 8/5{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} z^{2} }}{{R^{2} }}} \right){\text{d}}z} \\ D_{413} & = \int_{ - h/2}^{h/2} { - 2{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} \left( {\left( {z^{2} + 4/3{\mkern 1mu} l_{1}^{2} - l_{2}^{2} } \right)R^{2} + 4/3{\mkern 1mu} l_{1}^{2} z^{2} } \right)}}{{R^{3} }} - 2{\mkern 1mu} \frac{1}{R}\left( {\left( { - 2/3{\mkern 1mu} z^{2} + \frac{{27{\mkern 1mu} l_{0}^{2} }}{20} - \frac{{8{\mkern 1mu} l_{1}^{2} }}{15} - 3/4{\mkern 1mu} l_{2}^{2} } \right)\mu (z){\mkern 1mu} + z^{2} k(z){\mkern 1mu} } \right) + \frac{{8{\mkern 1mu} \mu (z){\mkern 1mu} {\mkern 1mu} l_{1}^{2} z^{2} }}{{15{\mkern 1mu} R^{3} }} - \frac{{f_{1} {\mkern 1mu} z^{2} }}{R}{\text{d}}z} \\ D_{414} & = \int_{ - h/2}^{h/2} {\left( { - 8/5{\mkern 1mu} \frac{{l_{1}^{2} z^{2} }}{{R^{3} }} - 1/15{\mkern 1mu} \frac{{\mu (z){\mkern 1mu} {\mkern 1mu} z^{2} \left( {27{\mkern 1mu} l_{0}^{2} - 32{\mkern 1mu} l_{1}^{2} + 15{\mkern 1mu} l_{2}^{2} } \right)}}{{R^{3} }}} \right){\text{d}}z} \\ D_{415} & = \int_{ - h/2}^{h/2} {\left( { - 9/5{\mkern 1mu} \frac{{\left( {l_{0}^{2} + 5/9{\mkern 1mu} l_{2}^{2} } \right)z^{2} \mu (z){\mkern 1mu} }}{R} + \frac{{8{\mkern 1mu} \mu (z){\mkern 1mu} l_{1}^{2} z^{2} }}{{15{\mkern 1mu} R}}} \right){\text{d}}z} \\ D_{416} & = - 2R \\ \end{aligned}$$
$$\begin{aligned} {E_{51}} & = \int_{ - h/2}^{h/2} {\left( {8/3 {{\mu (z) \left( {{l_1}^2 - 3/4 {l_2}^2} \right)} \over {{R^2}}} - 1/15 {{\mu (z) \left( {27 {l_{{0^2}}} + 8 {l_1}^2 - 15 {l_2}^2} \right)} \over {{R^2}}} - {{z{f_1}} \over {{R^2}}}} \right){\rm{d}}z} \\ {E_{52}} & = \int_{ - h/2}^{h/2} {\left( { - 2 {{\mu (z) } \over R} - 1/15 {{\mu (z) \left( {27 {l_0}^2 + 136 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}} \right){\rm{d}}z} \\ {E_{53}} & = \int_{ - h/2}^{h/2} {\left( {8/3 {{\mu (z) \left( {{l_1}^2 - 3/4 {l_2}^2} \right)} \over R} + 1/5 {{\mu (z) \left( {9 {l_0}^2 - 5 {l_2}^2} \right)} \over R} - {{4 \mu (z) {l_1}^2} \over {15 R}}} \right){\rm{d}}z} \\ {E_{54}} & = \int_{ - h/2}^{h/2} {\left( { - 2/15 {{\mu (z) \left( {27 {l_0}^2 - 64 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}} + {{16 \mu (z) {l_1}^2} \over {5 {R^3}}} - {{z{f_1}} \over {{R^3}}}} \right){\rm{d}}z} \\ {E_{55}} & = \int_{ - h/2}^{h/2} {\left( {1/15 {{\mu (z) \left( {30 {R^2} + 27 {l_0}^2 + 184 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}} + {{32 \mu (z) {l_1}^2} \over {5 {R^3}}} - {{z{f_1}} \over {{R^3}}}} \right){\rm{d}}z} \\ {E_{56}} & = \int_{ - h/2}^{h/2} {2/15 {{\mu (z) \left( {27 {l_0}^2 - 16 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}{\rm{d}}z} \\ {E_{57}} & = \int_{ - h/2}^{h/2} {\left( { - 8/3 {{\mu (z) \left( {{l_1}^2 - 3/4 {l_2}^2} \right)} \over R} + 1/30 {{\mu (z) \left( {81 {l_0}^2 + 16 {l_1}^2 - 45 {l_2}^2} \right)} \over R}} \right){\rm{d}}z} \\ {E_{58}} & = \int_{ - h/2}^{h/2} {\left( {2 {{\mu (z) \left( {\left( {{z^2} + 4/3 {l_1}^2 - {l_2}^2} \right){R^2} + 4/3 {l_1}^2{z^2}} \right)} \over {{R^3}}} - 2 {1 \over R}\left( {\left( { - 2/3 {z^2} + {{27 {l_0}^2} \over {20}} - {{8 {l_1}^2} \over {15}} - 3/4 {l_2}^2} \right)\mu (z) + {z^2}k(z) } \right) + {{8 \mu (z) {l_1}^2{z^2}} \over {15 {R^3}}} - {{{f_{1 }}{z^2}} \over {{R^2}}}} \right){\rm{d}}z} \\ {E_{59}} & = \int_{ - h/2}^{h/2} {\left( { - 8/5 {{{l_1}^2{z^2}} \over {{R^3}}} - 1/15 {{\mu (z) {z^2}\left( {27 {l_0}^2 - 32 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^3}}}} \right){\rm{d}}z} \\ {E_{510}} & = \int_{ - h/2}^{h/2} {\left( { - 9/5 {{\left( {{l_0}^2 + 5/9 {l_2}^2} \right){z^2}\mu (z) } \over R} + {{8 \mu (z) {l_1}^2{z^2}} \over {15 R}}} \right){\rm{d}}z} \\ {E_{511}} & = \int_{ - h/2}^{h/2} {1/15 {{\mu (z) \left( {30 {R^4} + 27 {l_0}^2{R^2} + 136 {R^2}{l_1}^2 + 15 {R^2}{l_2}^2 + 136 {l_1}^2{z^2}} \right)} \over {{R^4}}}{\rm{d}}z} \\ {E_{512}} & = \int_{ - h/2}^{h/2} {\left( { - 4 {{\mu (z) \left( {\left( {{z^2} + 2/3 {l_1}^2 + {l_2}^2} \right){R^2} + 2/3 \left( {{l_1}^2 + 1/2 {l_2}^2} \right){z^2}} \right)} \over {{R^2}}} + {{8 \mu (z) {l_1}^2{z^2}} \over {15 {R^2}}}} \right){\rm{d}}z} \\ {E_{513}} & = \int_{ - h/2}^{h/2} {1/15 \mu (z) {z^2}\left( {27 {l_0}^2 + 16 {l_1}^2 + 15 {l_2}^2} \right){\rm{d}}z} \\ {E_{514}} & = \int_{ - h/2}^{h/2} {\left( {2 {1 \over {{R^4}}}\left( {\left( {\left( {4/3 {z^2} + {{19 {l_0}^2} \over 5} + {{32 {l_1}^2} \over {15}} + {l_2}^2} \right){R^2} + 2 {z^2}\left( {{l_0}^2 + {{64 {l_1}^2} \over {15}}} \right)} \right)\mu (z) + {R^2}k(z) {z^2}} \right) - {{32 \mu (z) {l_1}^2{z^2}} \over {5 {R^4}}} - {{{f_{1 }}{z^2}} \over {{R^3}}}} \right){\rm{d}}z} \\ {E_{515}} & = \int_{ - h/2}^{h/2} {4/5 {{\mu (z) {z^2}\left( {5 {l_0}^2 + 2 {l_1}^2} \right)} \over {{R^4}}}{\rm{d}}z} \\ {E_{516}} & = \int_{ - h/2}^{h/2} {\left( {1/15 {{\mu (z) {z^2}\left( {87 {l_0}^2 + 64 {l_1}^2 + 15 {l_2}^2} \right)} \over {{R^2}}} - 8/5 {{{l_1}^2{z^2}} \over {{R^2}}}} \right){\rm{d}}z} \\ {E_{517}} & = - 2R. \end{aligned}$$

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Ashrafi Dehkordi, A., Jahanbazi Goojani, R. & Tadi Beni, Y. Porous flexoelectric cylindrical nanoshell based on the non-classical continuum theory. Appl. Phys. A 128, 478 (2022). https://doi.org/10.1007/s00339-022-05584-z

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