Small/large amplitude vibration, snap-through and nonlinear thermo-mechanical instability of temperature-dependent FG porous circular nanoplates

Abstract

The current paper investigates the nonlinear thermo-mechanical instability and small/large amplitude vibration of thermally pre/post-buckled functionally graded (FG) porous circular nanoplates in bifurcation/limit load buckling. The thermo-mechanical properties of the graded nanoplate under uniform heating are considered to be functions of temperature according to the Touloukian model. To describe the functionally graded porous materials, two different patterns of porosity distribution are adopted in which the first pattern has a uniform distribution, but the second pattern has an uneven one. The novelty of the present study and its significance can be summarized into: (i) investigating the effect of the porosity distribution and geometrical imperfection on the nonlinear thermo-mechanical bending and small/large amplitude vibration of the temperature-dependent FG circular nanoplates during bifurcation buckling. (ii) Studying the snap-through instability and small amplitude vibration of the thermally post-buckled FG porous circular nanoplates during limit load buckling. To this aim, the nonlocal elasticity theory alongside the von-Kármán nonlinear assumption is imposed to derive the nonlinear motion equations of the geometrically imperfect FG porous circular nanoplates in the framework of Hamilton’s principle. By employing the Ritz approach together with the Chebyshev polynomial as the basic functions, the coupled nonlinear equations are discretised for both clamped and simply supported edge conditions. Next, depending on the nonlinear problem at hand, three different numerical algorithms, including the Newton-Raphson iterative method, the direct displacement control strategy, and the cylindrical arc-length technique, are implemented to assess the static and dynamic behaviour of the porous nanosystem. After validating the developed mathematical model, a comprehensive examination is performed to determine the influence of the temperature dependence of materials, porosity distribution patterns, nonlocal parameter, material gradient index, imperfection sensitivity, and edge conditions on the nonlinear bending, snap-through instability, and vibration responses of the graded circular nanoplates in the pre- and post-buckling domains of bifurcation/limit load buckling.

Appendices

Appendix A

The entries of matrices given by Eq. 23 are computed as follows

$$\begin{aligned} M_{nm}^{uu}&=\int _{0}^{a}{\left[ {{I}_{0}}rN_{m}^{u}N_{n}^{u}+\mu {{I}_{0}}r\frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{u}}{\text {d}r} \right] }\,\text {d}r \\ M_{nm}^{uw}&=\int _{0}^{a}{\left[ -{{I}_{1}}r\frac{\text {d}N_{m}^{w}}{\text {d}r}N_{n}^{u}-\mu {{I}_{1}}r\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{u}}{\text {d}r} \right] }\,\text {d}r \\ M_{nm}^{wu}&=\int _{0}^{a}\left[ -{{I}_{1}}rN_{m}^{u} \frac{\text {d}N_{n}^{w}}{\text {d}r} -\mu {{I}_{1}}r\frac{\text {d}N_{m}^{u}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\right. \\&\quad -\mu {{I}_{1}}N_{m}^{u}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}-\mu {{I}_{1}}\frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} -\mu \frac{{{I}_{1}}}{r}N_{m}^{u}\frac{\text {d} N_{n}^{w}}{\text {d}r} \\&\quad +\mu {{I}_{0}}r\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) N_{m}^{u} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} +\mu {{I}_{0}}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \\&\quad N_{m}^{u}\frac{\text {d}N_{n}^{w}}{\text {d}r} +{{\mu }^{2}}{{I}_{0}}r\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\left. {{\mu }^{2}}{{I}_{0}} \left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} +\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \right] \,\text {d}r \\ M_{nm}^{ww}&=\int _{0}^{a}\left[ {{I}_{0}}rN_{m}^{w}N_{n}^{w} -\mu {{I}_{0}}rN_{m}^{w}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}-\mu {{I}_{0}}N_{m}^{w}\frac{\text {d} N_{n}^{w}}{\text {d}r}\right. \\&\quad +{{I}_{2}}r\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{I}_{2}}r\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\mu {{I}_{2}}\frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} +\mu {{I}_{2}}\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu \frac{{{I}_{2}}}{r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} -\mu {{I}_{1}}r\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}} N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu {{I}_{1}}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}-{{\mu }^{2}}{{I}_{1}}r \left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} +\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}}\right) \\&\quad \left. \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} -{{\mu }^{2}}{{I}_{1}}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r}\right] \,\text {d}r \\ \end{aligned}$$

$$\begin{aligned} K_{nm}^{uu}&=\int _{0}^{a}\left[ {{A}_{11}}r \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{u}}{\text {d}r}+{{A}_{12}}N_{m}^{u}\frac{\text {d}N_{n}^{u}}{\text {d}r}\right. \\&\qquad \left. +{{A}_{12}}\frac{\text {d}N_{m}^{u}}{\text {d}r} N_{n}^{u}+\frac{{{A}_{22}}}{r}N_{m}^{u}N_{n}^{u} \right] \,\text {d}r \\ K_{nm}^{uw}&=\int _{0}^{a}\left[ \frac{{{A}_{11}}}{2}r \frac{\partial w}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{u}}{\text {d}r} +{{A}_{11}}r\frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{u}}{\text {d}r}\right. \\&\quad -{{B}_{11}}r\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{u}}{\text {d}r} -{{B}_{12}}\frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{u}}{\text {d}r}\\&\quad +\frac{{{A}_{12}}}{2}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}N_{n}^{u}+{{A}_{12}} \frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}N_{n}^{u}\\&\quad \left. -{{B}_{12}}\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}N_{n}^{u}-\frac{{{B}_{22}}}{r} \frac{\text {d}N_{m}^{w}}{\text {d}r}N_{n}^{u} \right] \,\text {d}r\\ K_{nm}^{wu}&=\int _{0}^{a}\left[ {{A}_{11}}r\left( \frac{\partial w}{\partial r}+\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}\right. \\&\quad +\mu {{A}_{11}}r\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\mu {{A}_{11}} \left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} +\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}}\right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +{{A}_{12}}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) N_{m}^{u} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu {{A}_{12}}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) N_{m}^{u} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{12}}}{r}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) N_{m}^{u} \frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu {{A}_{12}}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{12}}}{r}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu \frac{{{A}_{22}}}{r}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) N_{m}^{u} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\mu \frac{{{A}_{22}}}{{{r}^{2}}} \left( \frac{\partial w}{\partial r}+\frac{\partial {{w}^{*}}}{\partial r} \right) N_{m}^{u}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -{{B}_{11}}r\frac{\text {d}N_{m}^{u}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} -{{B}_{12}}N_{m}^{u}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad \left. -{{B}_{12}}\frac{\text {d}N_{m}^{u}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} -\frac{{{B}_{22}}}{r}N_{m}^{u}\frac{\text {d}N_{n}^{w}}{\text {d}r} \right] \,\text {d}r\\ \end{aligned}$$

$$\begin{aligned} K_{nm}^{ww}&=\int _{0}^{a}\left[ \frac{{{A}_{11}}}{2}r \left( {{\left( \frac{\partial w}{\partial r} \right) }^{2}} +\frac{\partial w}{\partial r}\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\right. \\&\quad +\mu \frac{{{A}_{11}}}{2}r \frac{\partial w}{\partial r}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{11}}}{2}\frac{\partial w}{\partial r} \left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} +\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}}\right) \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +{{A}_{11}}r\left( \frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} +{{\left( \frac{\partial {{w}^{*}}}{\partial r}\right) }^{2}}\right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu {{A}_{11}}r\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}\frac{\partial {{w}^{*}}}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}} N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu {{A}_{11}}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}\frac{\partial {{w}^{*}}}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}}\right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad -{{B}_{11}}r\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -\mu {{B}_{11}}r \left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{{{\text {d}}^{2}} N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{{{\text {d}}^{2}} N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu {{B}_{11}}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} +\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad -{{B}_{12}}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -\mu {{B}_{12}}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu \frac{{{B}_{12}}}{r}\left( \frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}+\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu \frac{{{A}_{12}}}{2}\left( {{\left( \frac{\partial w}{\partial r} \right) }^{2}}+\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{12}}}{2r}\left( {{\left( \frac{\partial w}{\partial r} \right) }^{2}}+\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu {{A}_{12}}\left( \frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} +{{\left( \frac{\partial {{w}^{*}}}{\partial r} \right) }^{2}} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{12}}}{r}\left( \frac{\partial w}{\partial r}\frac{\partial {{w}^{*}}}{\partial r} +{{\left( \frac{\partial {{w}^{*}}}{\partial r}\right) }^{2}}\right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad -\mu {{B}_{12}}\left( \frac{\partial w}{\partial r} +\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu \frac{{{B}_{12}}}{r}\left( \frac{\partial w}{\partial r}+\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -\mu \frac{{{B}_{22}}}{r}\left( \frac{\partial w}{\partial r}+\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}} N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad -\mu \frac{{{B}_{22}}}{{{r}^{2}}} \left( \frac{\partial w}{\partial r}+\frac{\partial {{w}^{*}}}{\partial r} \right) \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -\frac{{{B}_{11}}}{2}r \frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}} N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -{{B}_{11}}r \frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} +{{D}_{11}}r\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +{{D}_{12}}\frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} -\frac{{{B}_{12}}}{2}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -{{B}_{12}}\frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}+{{D}_{12}}\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad \left. +\frac{{{D}_{22}}}{r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \right] \,\text {d}r \\ \end{aligned}$$

$$\begin{aligned} \tilde{K}_{nm}^{uu}& =0,\quad \tilde{K}_{nm}^{uw}=0, \quad \tilde{K}_{nm}^{wu}=0 \\ \tilde{K}_{nm}^{ww}&=\int _{0}^{a}\left[ {{N}^{T}}r \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{N}^{T}}r\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\right. \\&\quad +\mu {{N}^{T}} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r}+\mu {{N}^{T}} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad \left. +\mu \frac{{{N}^{T}}}{r}\frac{\text {d} N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \right] \,\text {d}r \\ F_{n}^{u}&=\int _{0}^{a}\left[ {{N}^{T}}r \frac{\text {d}N_{n}^{u}}{\text {d}r}+{{N}^{T}}N_{n}^{u} \right] \,\text {d}r \\ F_{n}^{w}&=\int _{0}^{a}\left[ {{N}^{T}}r\frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{N}^{T}}r\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\right. \\&\quad +\mu {{N}^{T}}\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{N}^{T}}\frac{\partial {{w}^{*}}}{\partial r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\mu \frac{{{N}^{T}}}{r}\frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{n}^{w}}{\text {d}r} -{{M}^{T}}r\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} -{{M}^{T}}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad \left. +qrN_{n}^{w}-\mu qr\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}-\mu q\frac{\text {d}N_{n}^{w}}{\text {d}r}\right] \,\text {d}r \end{aligned}$$

$$\begin{aligned} \tilde{K}_{nm}^{uu}& =0,\quad \tilde{K}_{nm}^{uw}=0, \quad \tilde{K}_{nm}^{wu}=0 \\ \tilde{K}_{nm}^{ww}&=\int _{0}^{a}\left[ {{N}^{T}}r \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{N}^{T}}r\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\right. \\&\quad +\mu {{N}^{T}} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r}+\mu {{N}^{T}} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad \left. +\mu \frac{{{N}^{T}}}{r}\frac{\text {d} N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \right] \,\text {d}r \\ F_{n}^{u}&=\int _{0}^{a}\left[ {{N}^{T}}r \frac{\text {d}N_{n}^{u}}{\text {d}r}+{{N}^{T}}N_{n}^{u} \right] \,\text {d}r \\ F_{n}^{w}&=\int _{0}^{a}\left[ {{N}^{T}}r\frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{N}^{T}}r\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\right. \\&\quad +\mu {{N}^{T}}\frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{N}^{T}}\frac{\partial {{w}^{*}}}{\partial r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\mu \frac{{{N}^{T}}}{r}\frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{n}^{w}}{\text {d}r} -{{M}^{T}}r\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} -{{M}^{T}}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad \left. +qrN_{n}^{w}-\mu qr\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}-\mu q\frac{\text {d}N_{n}^{w}}{\text {d}r}\right] \,\text {d}r \end{aligned}$$

Appendix B

The entries of the tangent stiffness matrix introduced by Eq. 29 are determined as follows

$$\begin{aligned} T_{nm}^{uu}&=K_{nm}^{uu} \\ T_{nm}^{uw}&=K_{nm}^{uw}+\int _{0}^{a}\\&\quad \left[ \frac{{{A}_{11}}}{2}r\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{u}}{\text {d}r} +\frac{{{A}_{12}}}{2}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}N_{n}^{u} \right] \,\text {d}r \\ T_{nm}^{wu}&=K_{nm}^{wu} \\ \end{aligned}$$

$$\begin{aligned} T_{nm}^{ww}&=K_{nm}^{ww}-\tilde{K}_{nm}^{ww}+\int _{0}^{a}\\&\quad \left[ {{A}_{11}}r\frac{\partial u}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}+\mu {{A}_{11}}r\frac{\partial u}{\partial r} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \right. \\&\quad +\mu {{A}_{11}}\frac{\partial u}{\partial r} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r} +{{A}_{12}}u\frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu {{A}_{12}}u \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}+\mu \frac{{{A}_{12}}}{r}u \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu {{A}_{12}} \frac{\partial u}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} +\mu \frac{{{A}_{12}}}{r}\frac{\partial u}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu \frac{{{A}_{22}}}{r}u\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} +\mu \frac{{{A}_{22}}}{{{r}^{2}}}u\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +{{A}_{11}} r{{\left( \frac{\partial w}{\partial r} \right) }^{2}} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\frac{{{A}_{11}}}{2}r\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu \frac{{{A}_{11}}}{2}r\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}} N_{n}^{w}}{\text {d}{{r}^{2}}} +\mu \frac{{{A}_{11}}}{2}r{{\left( \frac{\partial w}{\partial r} \right) }^{2}}\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{11}}}{2}r\frac{\partial w}{\partial r} \frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{11}}}{2}\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu \frac{{{A}_{11}}}{2}{{\left( \frac{\partial w}{\partial r} \right) }^{2}}\frac{{{\text {d}}^{2}} N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu \frac{{{A}_{11}}}{2}\frac{\partial w}{\partial r} \frac{{{\partial }^{2}}{{w}^{*}}}{\partial {{r}^{2}}} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r}+{{A}_{11}}r\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad +\mu {{A}_{11}}r\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} \\&\quad +\mu {{A}_{11}}\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r}\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{\text {d}N_{n}^{w}}{\text {d}r} -{{B}_{11}}r\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -\mu {{B}_{11}}r\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}\frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} -\mu {{B}_{11}}\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r}\\&\quad -{{B}_{12}}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r} -\mu {{B}_{12}}\frac{\partial w}{\partial r} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu \frac{{{B}_{12}}}{r}\frac{\partial w}{\partial r} \frac{{{\text {d}}^{2}}N_{m}^{w}}{\text {d}{{r}^{2}}} \frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{A}_{12}}{{\left( \frac{\partial w}{\partial r} \right) }^{2}} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{12}}}{2}\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}} +\mu \frac{{{A}_{12}}}{r}{{\left( \frac{\partial w}{\partial r} \right) }^{2}}\frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r} \\&\quad +\mu \frac{{{A}_{12}}}{2r}\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} +\mu {{A}_{12}}\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad +\mu \frac{{{A}_{12}}}{r}\frac{\partial w}{\partial r} \frac{\partial {{w}^{*}}}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} -\mu {{B}_{12}}\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu \frac{{{B}_{12}}}{r}\frac{{{\partial }^{2}}w}{\partial {{r}^{2}}}\frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}-\mu \frac{{{B}_{22}}}{r} \frac{\partial w}{\partial r}\frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad -\mu \frac{{{B}_{22}}}{{{r}^{2}}}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r}\frac{\text {d}N_{n}^{w}}{\text {d}r} -\frac{{{B}_{11}}}{2}r\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{{{\text {d}}^{2}}N_{n}^{w}}{\text {d}{{r}^{2}}}\\&\quad \left. -\frac{{{B}_{12}}}{2}\frac{\partial w}{\partial r} \frac{\text {d}N_{m}^{w}}{\text {d}r} \frac{\text {d}N_{n}^{w}}{\text {d}r}\right] \,\text {d}r \end{aligned}$$