Magneto-electro-thermo-elastic frequency response of functionally graded saturated porous annular plates via trigonometric shear deformation theory

Abstract

Due to the unique specifications of porous materials, such as lightweight, researchers are encouraged to study more about them and use them in different engineering structures. Also, they can be integrated with other stiffer materials to overwhelm their main limitation, i.e., their low stiffness. So, this work considers the frequency response of a functionally graded (FG) porous material annular plate, which is coated with two piezo-electro-magnetic layers. The FG porous core is assumed to be saturated by fluid, and the pores' compressibility effect is examined. Besides, the pores' placement patterns are regarded as three different thickness-dependent functions that affect the core's mechanical properties. Since the coating layers have electromagnetic properties, therefore electromagnetic potentials are externally applied to them. The whole structure is also rested on Pasternak elastic foundation, and the thermal environment influence is evaluated on its behavior. The trigonometric type of higher-order shear deformation theory is employed, and the governing motion equations are derived to obtain more accurate results. Then, they are solved utilizing the generalized differential quadrature method for various boundary conditions. After ensuring the validity of the results by comparing them with known data in the available literature, the effect of the most vital parameters on the frequencies of the plate is discussed. For instance, it is seen that the impact of the porosity coefficient on the natural frequencies is utterly dependent on the pores’ distribution patterns.

Appendix

Rewriting the governing motion equations of (32)–(36) in terms of displacement and axisymmetric state leads to the below equations:

$$\begin{gathered} \delta u: \hfill \\ - A_{1} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}u + \left( {A_{13} {\mkern 1mu} - A_{7} {\mkern 1mu} - A_{1} {\mkern 1mu} } \right)\frac{1}{r}\frac{\partial }{\partial r}u + \frac{{A_{{19{\mkern 1mu} }} }}{{r^{2} }}u + A_{2} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial r^{3} }}w + \left( {A_{2} {\mkern 1mu} + A_{8} {\mkern 1mu} - A_{14} {\mkern 1mu} } \right)\frac{1}{r}\frac{{\partial^{2} }}{{\partial r^{2} }}w \hfill \\ - \frac{{A_{20} {\mkern 1mu} }}{{r^{2} }}\frac{\partial }{\partial r}w - A_{4} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\psi_{r} + \left( {A_{16} {\mkern 1mu} - A_{10} {\mkern 1mu} - A_{4} {\mkern 1mu} } \right)\frac{1}{r}\frac{\partial }{\partial r}\psi_{r} + \frac{{A_{22} {\mkern 1mu} }}{{r^{2} }}\psi_{r} - P_{1} {\mkern 1mu} \frac{\partial }{\partial r}\phi + \left( {P_{4} {\mkern 1mu} - P_{1} {\mkern 1mu} } \right)\frac{1}{r}\phi \hfill \\ - M_{1} {\mkern 1mu} \frac{\partial }{\partial r}\chi + \left( {M_{4} {\mkern 1mu} - M_{1} {\mkern 1mu} } \right)\frac{1}{r}\chi - I_{0} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial t^{2} }}u + I_{1} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial t^{2} \partial r}}w - I_{3} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial t^{2} }}\psi_{r} = 0, \hfill \\ \end{gathered}$$

(49)

$$\begin{gathered} \delta w: \hfill \\ - A_{2} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial r^{3} }}u + \left( {A_{14} {\mkern 1mu} - A_{8} {\mkern 1mu} - 2{\mkern 1mu} A_{2} {\mkern 1mu} } \right)\frac{1}{r}\frac{{\partial^{2} }}{{\partial r^{2} }}u + \frac{{A_{20} {\mkern 1mu} }}{{r^{2} }}\frac{\partial }{\partial r}u - \frac{{A_{20} {\mkern 1mu} }}{{r^{3} }}u + A_{3} {\mkern 1mu} \frac{{\partial^{4} }}{{\partial r^{4} }}w + \left( {A_{9} {\mkern 1mu} + 2{\mkern 1mu} A_{3} {\mkern 1mu} - A_{15} {\mkern 1mu} } \right)\frac{1}{r}\frac{{\partial^{3} }}{{\partial r^{3} }}w \hfill \\ - \left( {\frac{{A_{21} {\mkern 1mu} }}{{r^{2} }} + K_{G} - N_{r}^{ext} } \right)\frac{{\partial^{2} }}{{\partial r^{2} }}w + \left( {\frac{{A_{21} {\mkern 1mu} }}{{r^{3} }} - K_{G} + N_{r}^{ext} } \right)\frac{\partial }{\partial r}w + K_{W} w - A_{5} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial r^{3} }}\psi_{r} \hfill \\ + \left( {A_{17} {\mkern 1mu} - A_{11} {\mkern 1mu} - 2{\mkern 1mu} A_{5} {\mkern 1mu} } \right)\frac{1}{r}\frac{{\partial^{2} }}{{\partial r^{2} }}\psi_{r} + \frac{{A_{23} {\mkern 1mu} }}{{r^{2} }}\frac{\partial }{\partial r}\psi_{r} - \frac{{A_{23} {\mkern 1mu} }}{{r^{3} }}\psi_{r} - P_{2} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\phi + \left( {P_{{5{\mkern 1mu} }} - 2{\mkern 1mu} P_{2} } \right)\frac{1}{r}\frac{\partial }{\partial r}\phi - M_{2} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\chi \hfill \\ + \left( {M_{5} {\mkern 1mu} - 2{\mkern 1mu} M_{2} {\mkern 1mu} } \right)\frac{1}{r}\frac{\partial }{\partial r}\chi - \frac{{I_{1} {\mkern 1mu} }}{r}\frac{{\partial^{2} }}{{\partial t^{2} }}u - I_{1} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial t^{2} \partial r}}u - I_{0} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial t^{2} }}w + I_{2} {\mkern 1mu} \frac{{\partial^{4} }}{{\partial t^{2} \partial r^{2} }}w + \frac{{I_{2} {\mkern 1mu} }}{r}\frac{{\partial^{3} }}{{\partial t^{2} \partial r}}w - \frac{{I_{4} {\mkern 1mu} }}{r}\frac{{\partial^{2} }}{{\partial t^{2} }}\psi_{r} \hfill \\ - I_{4} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial t^{2} \partial r}}\psi_{r} = 0, \hfill \\ \end{gathered}$$

(50)

$$\begin{gathered} \delta \psi_{r} : \hfill \\ - A_{4} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}u + \left( {A_{16} {\mkern 1mu} - A_{4} {\mkern 1mu} - A_{{10{\mkern 1mu} }} } \right)\frac{1}{r}\frac{\partial }{\partial r}u + \frac{{A_{22} {\mkern 1mu} }}{{r^{2} }}u + A_{{5{\mkern 1mu} }} \frac{{\partial^{3} }}{{\partial r^{3} }}w + \left( {A_{11} {\mkern 1mu} + A_{5} {\mkern 1mu} - A_{17} {\mkern 1mu} } \right)\frac{1}{r}\frac{{\partial^{2} }}{{\partial r^{2} }}w \hfill \\ - \frac{{A_{23} {\mkern 1mu} }}{{r^{2} }}\frac{\partial }{\partial r}w - A_{6} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\psi_{r} + \left( {A_{18} {\mkern 1mu} - A_{12} {\mkern 1mu} - A_{6} {\mkern 1mu} } \right)\frac{1}{r}\frac{\partial }{\partial r}\psi_{r} + \left( {A_{31} {\mkern 1mu} + \frac{{A_{24} {\mkern 1mu} }}{{r^{2} }}} \right)\psi_{r} - \left( {P_{3} {\mkern 1mu} + P_{7} {\mkern 1mu} } \right)\frac{\partial }{\partial r}\phi \hfill \\ + \left( {P_{6} {\mkern 1mu} - P_{{3{\mkern 1mu} }} } \right)\frac{1}{r}\phi - \left( {M_{3} {\mkern 1mu} + M_{7} } \right){\mkern 1mu} \frac{\partial }{\partial r}\chi + \left( {M_{6} {\mkern 1mu} - M_{3} {\mkern 1mu} } \right)\frac{1}{r}\chi - I_{3} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial t^{2} }}u + I_{4} {\mkern 1mu} \frac{{\partial^{3} }}{{\partial t^{2} \partial r}}w - I_{{5{\mkern 1mu} }} \frac{{\partial^{2} }}{{\partial t^{2} }}\psi_{r} = 0, \hfill \\ \end{gathered}$$

(51)

$$\begin{gathered} \delta \phi : \hfill \\ P_{1} {\mkern 1mu} \frac{\partial }{\partial r}u + \frac{{P_{4} {\mkern 1mu} }}{r}u - P_{2} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}w - \frac{{P_{5} {\mkern 1mu} }}{r}\frac{\partial }{\partial r}w + P_{11} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\phi + \frac{{P_{11} {\mkern 1mu} }}{r}\frac{\partial }{\partial r}\phi - P_{13} {\mkern 1mu} \phi + \left( {P_{9} {\mkern 1mu} + P_{3} } \right){\mkern 1mu} \frac{\partial }{\partial r}\psi_{r} \hfill \\ + \left( {P_{9} {\mkern 1mu} + P_{6} {\mkern 1mu} } \right)\frac{1}{r}\psi_{r} + P_{14} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\chi + \frac{{P_{14} {\mkern 1mu} }}{r}\frac{\partial }{\partial r}\chi - P_{16} {\mkern 1mu} \chi = 0, \hfill \\ \end{gathered}$$

(52)

$$\begin{gathered} \delta \chi : \hfill \\ M_{1} {\mkern 1mu} \frac{\partial }{\partial r}u + \frac{{M_{4} {\mkern 1mu} }}{r}u - M_{2} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}w - \frac{{M_{5} {\mkern 1mu} }}{r}\frac{\partial }{\partial r}w + \left( {M_{9} {\mkern 1mu} + M_{3} {\mkern 1mu} } \right)\frac{\partial }{\partial r}\psi_{r} + \left( {M_{9} {\mkern 1mu} + M_{6} {\mkern 1mu} } \right)\frac{1}{r}\psi_{r} \hfill \\ + P_{14} {\mkern 1mu} \frac{{\partial^{2} }}{{\partial r^{2} }}\phi + \frac{{P_{14} {\mkern 1mu} }}{r}\frac{\partial }{\partial r}\phi - P_{16} {\mkern 1mu} \phi + M_{{11{\mkern 1mu} }} \frac{{\partial^{2} }}{{\partial r^{2} }}\chi + \frac{{M_{{11{\mkern 1mu} }} }}{r}\frac{\partial }{\partial r}\chi - M_{13} {\mkern 1mu} \chi = 0 \hfill \\ \end{gathered}$$

(53)

The used integral coefficients are introduced as:

$$\left[ {\begin{array}{*{20}c} {A_{1} ,} & {A_{2} ,} & {A_{3} ,} & {A_{4} ,} & {A_{5} ,} & {A_{6} } \\ \end{array} } \right] = \int\nolimits_{z} {Q_{11} (z)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} ,} & {f(z),} & {zf(z),} & {f(z)^{2} } \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {A_{7} ,} & {A_{8} ,} & {A_{9} ,} & {A_{10} ,} & {A_{11} ,} & {A_{12} } \\ \end{array} } \right] = \int\nolimits_{z} {Q_{12} (z)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} ,} & {f(z),} & {zf(z),} & {f(z)^{2} } \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {A_{13} ,} & {A_{14} ,} & {A_{15} ,} & {A_{16} ,} & {A_{17} ,} & {A_{18} } \\ \end{array} } \right] = \int\nolimits_{z} {Q_{21} (z)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} ,} & {f(z),} & {zf(z),} & {f(z)^{2} } \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {A_{19} ,} & {A_{20} ,} & {A_{21} ,} & {A_{22} ,} & {A_{23} ,} & {A_{24} } \\ \end{array} } \right] = \int\nolimits_{z} {Q_{22} (z)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} ,} & {f(z),} & {zf(z),} & {f(z)^{2} } \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {A_{25} ,} & {A_{26} ,} & {A_{27} ,} & {A_{28} ,} & {A_{29} ,} & {A_{30} } \\ \end{array} } \right] = \int\nolimits_{z} {Q_{66} (z)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} ,} & {f(z),} & {zf(z),} & {f(z)^{2} } \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {A_{31} ,} & {A_{32} } \\ \end{array} } \right] = \int\nolimits_{z} {\left[ {\begin{array}{*{20}c} {Q_{55} (z),} & {Q_{44} (z)} \\ \end{array} } \right]\left( {\frac{df(z)}{{dz}}} \right)^{2} } \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {P_{1} ,} & {P_{2} ,} & {P_{3} } \\ \end{array} } \right] = \int\nolimits_{z} {e_{31} \frac{\pi }{{h_{f} }}\sin \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {P_{4} ,} & {P_{5} ,} & {P_{6} } \\ \end{array} } \right] = \int\nolimits_{z} {e_{32} \frac{\pi }{{h_{f} }}\sin \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {P_{7} ,} & {P_{9} } \\ \end{array} } \right] = \int\nolimits_{z} {e_{15} \cos \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {\frac{df(z)}{{dz}}} \\ \end{array} } \right]} \,{\text{d}}z,\;\left[ {\begin{array}{*{20}c} {P_{8} ,} & {P_{10} } \\ \end{array} } \right] = \int\nolimits_{z} {e_{24} \cos \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {\frac{df(z)}{{dz}}} \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {P_{11} ,} & {P_{12} } \\ \end{array} } \right] = \int\nolimits_{z} {\left[ {\begin{array}{*{20}c} {s_{11} ,} & {s_{22} } \\ \end{array} } \right]\cos^{2} \left( {\frac{\pi z}{{h_{f} }}} \right)} \,{\text{d}}z,\;P_{13} = \int\nolimits_{z} {s_{33} \frac{{\pi^{2} }}{{h_{f}^{2} }}\sin^{2} \left( {\frac{\pi z}{{h_{f} }}} \right)} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {P_{14} ,} & {P_{15} } \\ \end{array} } \right] = \int\nolimits_{z} {\left[ {\begin{array}{*{20}c} {d_{11} ,} & {d_{22} } \\ \end{array} } \right]\cos^{2} \left( {\frac{\pi z}{{h_{f} }}} \right)} \,{\text{d}}z,\;P_{16} = \int\nolimits_{z} {d_{33} \frac{{\pi^{2} }}{{h_{f}^{2} }}\sin^{2} \left( {\frac{\pi z}{{h_{f} }}} \right)} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {M_{1} ,} & {M_{2} ,} & {M_{3} } \\ \end{array} } \right] = \int\nolimits_{z} {q_{31} \frac{\pi }{{h_{f} }}\sin \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {M_{4} ,} & {M_{5} ,} & {M_{6} } \\ \end{array} } \right] = \int\nolimits_{z} {q_{32} \frac{\pi }{{h_{f} }}\sin \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right]} \,{\text{d}}z,$$

$$\left[ {\begin{array}{*{20}c} {M_{7} ,} & {M_{9} } \\ \end{array} } \right] = \int\nolimits_{z} {q_{15} \cos \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {\frac{df(z)}{{dz}}} \\ \end{array} } \right]} \,{\text{d}}z,\;\left[ {\begin{array}{*{20}c} {M_{8} ,} & {M_{10} } \\ \end{array} } \right] = \int\nolimits_{z} {q_{24} \cos \left( {\frac{\pi z}{{h_{f} }}} \right)\left[ {\begin{array}{*{20}c} {1,} & {\frac{df(z)}{{dz}}} \\ \end{array} } \right]} \,{\text{d}}z,$$

$$M_{13} = \int\nolimits_{z} {\mu_{33} \frac{{\pi^{2} }}{{h_{f}^{2} }}\sin^{2} \left( {\frac{\pi z}{{h_{f} }}} \right)} \,{\text{d}}z$$