Higher-order displacement, strain, and stress analyses of origami graphene auxetic metamaterial-reinforced cylindrical shell

Abstract

In this new work, we will investigate the application of a higher-order shear deformable model for elastostatic higher-order displacement, strain, and stress analyses of a shear deformable cylindrical shell. The composite structure is assumed to be composed of a Cu-based matrix reinforced with foldable nanographene, in a thermal environment. The cylindrical shell is reinforced with graphene origami auxetic metamaterial subjected to mechanical and thermal loads. The constitutive relations are extended in thermal environment using three-dimensional Hooke's law where using the mathematical and statistical approach, the effective materials characteristics are experimentally derived as some modification coefficients of foldability, content of reinforcement, and thermal load. The principle of virtual work is used to derive governing equations taking thermal loading into consideration. One can arrive at a parametric analysis and a numerical result investigation through employing an analytical approach to explore elastostatic displacement/strain/stress along the radial coordinate with changes of thermal loads, folding, content characteristics, and other parameters.

Appendix

$$\left\{{\mathcal{F}}_{1},{\mathcal{F}}_{2},{\mathcal{F}}_{3},{\mathcal{F}}_{4}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \left(1-\nu \right)\left(R+z\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{5},{\mathcal{F}}_{6},{\mathcal{F}}_{7},{\mathcal{F}}_{8}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu \left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{9},{\mathcal{F}}_{10},{\mathcal{F}}_{11}\right\}={\int }_{-h/2}^{h/2}\lambda \nu \left(R+z\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{12},{\mathcal{F}}_{13},{\mathcal{F}}_{14}\right\}={\int }_{-h/2}^{h/2}\lambda \left(1-\nu \right)\left(R+z\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{15},{\mathcal{F}}_{16},{\mathcal{F}}_{17},{\mathcal{F}}_{18}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu \left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{19},{\mathcal{F}}_{20},{\mathcal{F}}_{21},{\mathcal{F}}_{22}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu \left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{23},{\mathcal{F}}_{24},{\mathcal{F}}_{25}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{26},{\mathcal{F}}_{27},{\mathcal{F}}_{28},{\mathcal{F}}_{29}\right\}={\int }_{-h/2}^{h/2}\lambda \frac{\left(1-2\nu \right)}{2}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{30},{\mathcal{F}}_{31},{\mathcal{F}}_{32},{\mathcal{F}}_{33}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \left(1-\nu \right){{z}}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{34},{\mathcal{F}}_{35},{\mathcal{F}}_{36},{\mathcal{F}}_{37}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{z}}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{38},{\mathcal{F}}_{39},{\mathcal{F}}_{40}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{z}}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{41},{\mathcal{F}}_{42},{\mathcal{F}}_{43}\right\}={\int }_{-h/2}^{h/2}\lambda \left(1-\nu \right){{z}}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{44},{\mathcal{F}}_{45},{\mathcal{F}}_{46},{\mathcal{F}}_{47}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{z}}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{48},{\mathcal{F}}_{49},{\mathcal{F}}_{50},{\mathcal{F}}_{51}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu z\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{52},{\mathcal{F}}_{53},{\mathcal{F}}_{54}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}{{z}}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{55},{\mathcal{F}}_{56},{\mathcal{F}}_{57},{\mathcal{F}}_{58}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}{{z}}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{59},{\mathcal{F}}_{60},{\mathcal{F}}_{61},{\mathcal{F}}_{62}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \left(1-\nu \right){{{z}}}^{2}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{63},{\mathcal{F}}_{64},{\mathcal{F}}_{65},{\mathcal{F}}_{66}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {z}^{2}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{67},{\mathcal{F}}_{68},{\mathcal{F}}_{69}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{2}\left(R+{{z}}\right)\left\{\mathrm{1,2}z,3{z}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{70},{\mathcal{F}}_{71},{\mathcal{F}}_{72}\right\}={\int }_{-h/2}^{h/2}\lambda \left(1-\nu \right){z}^{2}\left(R+z\right)\left\{\mathrm{1,2}z,3{z}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{73},{\mathcal{F}}_{74},{\mathcal{F}}_{75},{\mathcal{F}}_{76}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{2}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{77},{\mathcal{F}}_{78},{\mathcal{F}}_{79},{\mathcal{F}}_{80}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{2}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{81},{\mathcal{F}}_{82},{\mathcal{F}}_{83}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}{{{z}}}^{2}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{84},{\mathcal{F}}_{85},{\mathcal{F}}_{86},{\mathcal{F}}_{87}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}{{{z}}}^{2}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{88},{\mathcal{F}}_{89},{\mathcal{F}}_{90},{\mathcal{F}}_{91}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \left(1-\nu \right){{{z}}}^{3}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{92},{\mathcal{F}}_{93},{\mathcal{F}}_{94},{\mathcal{F}}_{95}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{3}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{96},{\mathcal{F}}_{97},{\mathcal{F}}_{98}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{3}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{99},{\mathcal{F}}_{100},{\mathcal{F}}_{101}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}{{{z}}}^{3}\left(R+{{z}}\right)\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{102},{\mathcal{F}}_{103},{\mathcal{F}}_{104},{\mathcal{F}}_{105}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \frac{\left(1-2\nu \right)}{2}{{{z}}}^{3}\left(R+{{z}}\right)\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{106},{\mathcal{F}}_{107},{\mathcal{F}}_{108},{\mathcal{F}}_{109}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\frac{\lambda \left(1-\nu \right)}{\left(R+{{z}}\right)}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{110},{\mathcal{F}}_{111},{\mathcal{F}}_{112},{\mathcal{F}}_{113}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu \left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{114},{\mathcal{F}}_{115},{\mathcal{F}}_{116}\right\}={\int }_{-h/2}^{h/2}\lambda \nu \left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{117},{\mathcal{F}}_{118},{\mathcal{F}}_{119},{\mathcal{F}}_{120}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\frac{\lambda \left(1-\nu \right)}{\left(R+{{z}}\right)}{{z}}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{121},{\mathcal{F}}_{122},{\mathcal{F}}_{123},{\mathcal{F}}_{124}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{z}}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{125},{\mathcal{F}}_{126},{\mathcal{F}}_{127}\right\}={\int }_{-h/2}^{h/2}\lambda \nu {{z}}\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{128},{\mathcal{F}}_{129},{\mathcal{F}}_{130},{\mathcal{F}}_{131}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\frac{\lambda \left(1-\nu \right)}{\left(R+{{z}}\right)}{{{z}}}^{2}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{132},{\mathcal{F}}_{133},{\mathcal{F}}_{134},{\mathcal{F}}_{135}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{2}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{136},{\mathcal{F}}_{137},{\mathcal{F}}_{138}\right\}={\int }_{-h/2}^{h/2}\lambda \nu {{{z}}}^{2}\left\{\mathrm{1,2z},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{139},{\mathcal{F}}_{140},{\mathcal{F}}_{141},{\mathcal{F}}_{142}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\frac{\lambda \left(1-\nu \right)}{\left(R+{{z}}\right)}{{{z}}}^{2}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{143},{\mathcal{F}}_{144},{\mathcal{F}}_{145},{\mathcal{F}}_{146}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \nu {{{z}}}^{2}\left\{1,{{z}},{{{z}}}^{2},{{{z}}}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathcal{F}}_{147},{\mathcal{F}}_{148},{\mathcal{F}}_{149}\right\}={\int }_{-h/2}^{h/2}\lambda \nu {{{z}}}^{2}\left\{1,2{{z}},3{{{z}}}^{2}\right\}{\text{d}}z.$$

$$\left\{{\mathbb{C}}_{1},{\mathbb{C}}_{2},{\mathbb{C}}_{5},{\mathbb{C}}_{6}\right\}={\int }_{-\frac{h}{2}}^\frac{h}{2}\lambda \alpha \left(R+z\right)\left(1+\vartheta \right)\left\{1,z,{z}^{2},{z}^{3}\right\}{\text{d}}z.$$

$$\left\{{\mathbb{C}}_{3},{\mathbb{C}}_{4},{\mathbb{C}}_{7},{\mathbb{C}}_{8}\right\}={\int }_{-h/2}^{h/2}\lambda \alpha \left(1+\vartheta \right)\left\{1,z,{z}^{2},{z}^{3}\right\}{\text{d}}z.$$