Nonlinear plate theory of single-layered MoS<sub>2</sub> with thermal effect

Abstract

The single-layered molybdenum disulfide (<inline-formula><tex-math id="M6">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M6.png"/></alternatives></inline-formula>) is a two-dimensional nanomaterial with wide potential applications due to its excellent electrical and frictional properties. However, there have been few investigations of its mechanical properties up to now, and researchers have not paid attention to its nonlinear mechanical properties under the multi-fields co-existing environment. The present paper proposed a nonlinear plate theory to model the effect of finite temperatures on the single-layered <inline-formula><tex-math id="M7">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M7.png"/></alternatives></inline-formula>. It is similar to the classical plate theory that both the in-plane stretching deformation and the out-of-plane bending deformation are taken into account in the new theory. However, the new theory consists of two independent in-plane mechanical parameters and two independent out-of-plane mechanical parameters. Neither of the two out-of-plane mechanical parameters in the new theory, which describe the resistance of <inline-formula><tex-math id="M8">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M8.png"/></alternatives></inline-formula> to the bending and the twisting, depends on the structure’s thickness. This reasonably avoids the Yakobson paradox: uncertainty stemming from the thickness of the single-layered two-dimensional structures will lead to the uncertainty of the structure’s out-of-plane stiffness. The new nonlinear plate equations are then solved approximately through the Galerkin method for the thermoelastic mechanical problems of the graphene and <inline-formula><tex-math id="M9">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M9.png"/></alternatives></inline-formula>. The approximate analytic solutions clearly reveal the effects of temperature and structure stiffness on the deformations. Through comparing the results of two materials under combined temperature and load, it is found, for the immovable boundaries, that (1) the thermal stress, which is induced by the finite temperature, reduces the stiffness of <inline-formula><tex-math id="M10">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M10.png"/></alternatives></inline-formula>, but increases the stiffness of graphene; (2) the significant difference between two materials is that the graphene’s in-plane stiffness is greater than the <inline-formula><tex-math id="M11">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M11.png"/></alternatives></inline-formula>’s, but the graphene’s out-of-plane stiffness is less than the <inline-formula><tex-math id="M12">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M12.png"/></alternatives></inline-formula>’s. Because the <inline-formula><tex-math id="M13">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M13.png"/></alternatives></inline-formula>’s bending stiffness is much greater than graphene’s, the graphene’s deformation is greater than MoS<sub>2</sub>’s with a small load. However, the graphene’s deformation is less than the <inline-formula><tex-math id="M14">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M14.png"/></alternatives></inline-formula>’s with a large load since the graphene’s in-plane stretching stiffness is greater than the MoS<sub>2</sub>’s. The present research shows that the applied axial force and ambient temperature can conveniently control the mechanical properties of single-layered two-dimensional nanostructures. The new theory provides the basis for the intensive research of the thermoelastic mechanical problems of <inline-formula><tex-math id="M15">\begin{document}${\rm{Mo}}{{\rm{S}}_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210160_M15.png"/></alternatives></inline-formula>, and one can easily apply the theory to other single-layered two-dimensional nanostructures.