1.Department of Engineering Mechanics, Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming 650500, China 2.Yunnan Key Laboratory of Disaster Reduction in Civil Engineering, Kunming University of Science and Technology, Kunming 650500, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 12050001, 11562009)
Received Date:24 January 2021
Accepted Date:24 February 2021
Available Online:30 June 2021
Published Online:05 July 2021
Abstract:The single-layered molybdenum disulfide (${\rm{Mo}}{{\rm{S}}_2}$) 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 ${\rm{Mo}}{{\rm{S}}_2}$. 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 ${\rm{Mo}}{{\rm{S}}_2}$ 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 ${\rm{Mo}}{{\rm{S}}_2}$. 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 ${\rm{Mo}}{{\rm{S}}_2}$, 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 ${\rm{Mo}}{{\rm{S}}_2}$’s, but the graphene’s out-of-plane stiffness is less than the ${\rm{Mo}}{{\rm{S}}_2}$’s. Because the ${\rm{Mo}}{{\rm{S}}_2}$’s bending stiffness is much greater than graphene’s, the graphene’s deformation is greater than MoS2’s with a small load. However, the graphene’s deformation is less than the ${\rm{Mo}}{{\rm{S}}_2}$’s with a large load since the graphene’s in-plane stretching stiffness is greater than the MoS2’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 ${\rm{Mo}}{{\rm{S}}_2}$, and one can easily apply the theory to other single-layered two-dimensional nanostructures. Keywords:single-layered MoS2/ graphene/ thermal stress/ nonlinear plate theory
2.单层${\rm{Mo}}{{\rm{S}}_2}$的热弹耦合非线性板理论对于二维材料, 被广泛采用的宏观连续介质力学模型是F?ppl-von Karman板理论[19,22-24].本文将建立 图1所示单层${\rm{Mo}}{{\rm{S}}_2}$的热弹耦合非线性板理论. 虽然还没有通过量子力学(或化学键)理论证明, 三原子层的二维结构的变形能(自由能)可写为经典的板理论形式. 但该唯像理论可以得到和量子计算一致的结构变形 [14-15]. 经典的F?ppl-von Karman板壳理论中, 绝度零度时的变形能包含面内的张拉和剪切变形能, 以及面外的弯曲和扭转变形能[25], 如方程(1)所示. 出人意料的是, 该变形能很好地描述了单层石墨烯的力学性质[25-27]: 图 1 单层${\rm{Mo}}{{\rm{S}}_2}$计算简图: (a) 顶视图; (b)侧视图; (c)等效板立体图; (d)边界载荷 Figure1. Computational model of single-layer ${\rm{Mo}}{{\rm{S}}_2}$: (a) Top view of the structure; (b) Side view of the structure; (c) Stereo plate model of the structure; (d) Applied edge loads
图 4$a = b = 6\;{\rm{nm}}$时, 在两个不同温度和边界载荷下的载荷变形幅值曲线 Figure4. Loads-response curves with two edge stresses and two temperatures for $a = b = 6\;{\rm{nm}}$.
图 5$a = b = 6\;{\rm{nm}}$时, 给定边界轴向力条件下的载荷、温度及变形幅值曲面 Figure5. Loads-temperatures-response surfaces with the given stretching stresses for $a = b = 6\;{\rm{nm}}$.