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单层二硫化钼(
${\rm{Mo}}{{\rm{S}}_2}$ )是有广泛应用前景的二维纳米材料, 但其力学性质还没有被深入研究, 特别是其热弹耦合力学行为迄今还没有被关注到. 本文首次提出了考虑热应力影响的单层${\rm{Mo}}{{\rm{S}}_2}$ 的非线性板理论, 并对比研究了其与石墨烯的热弹耦合力学性质. 对于不可移动边界, 结果显示: 1)有限温度产生的热应力降低了${\rm{Mo}}{{\rm{S}}_2}$ 的刚度, 但提高了石墨烯的刚度; 2)在相同几何尺寸和温度条件下, 变形较小时${\rm{Mo}}{{\rm{S}}_2}$ 的刚度大于石墨烯, 但伴随变形的增大,${\rm{Mo}}{{\rm{S}}_2}$ 的刚度将小于石墨烯. 研究结果表明, 边界预加轴向外力和环境温度可以调节单层二维纳米结构力学性质. 本文建立的热弹耦合板模型, 可以推广至其他单层二维纳米结构.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
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-
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Fig. 1. 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 2
$a = b = 6\;{\rm{nm}}$ ,$N_{xx}^0 = N_{yy}^0 = 0$ 时, 结构在两个不同温度下的载荷变形幅值曲线Fig. 2. Loads-response curves with two temperatures for
$a = b = 6\;{\rm{nm}}$ and$N_{xx}^0 = N_{yy}^0 = 0$ .
图 6 给定边界轴向力和温度条件下的载荷、几何尺寸及变形幅值曲面
Fig. 6. Loads- dimensions-response surfaces with the given stretching stresses and the temperature.
图 3
$a = b = 6\;{\rm{nm}}$ ,$T = 0\;\rm K$ 时, 在两个不同边界拉力下的载荷变形幅值曲线Fig. 3. Loads-response curves with two edge stretching stresses for
$a = b = 6\;{\rm{nm}}$ and$T = 0\; \rm K$ .
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Fig. 4. Loads-response curves with two edge stresses and two temperatures for
$a = b = 6\;{\rm{nm}}$ . -
[1] Tan C, Cao X, Wu X J, He Q, Yang J, Zhang X, Chen J, Zhao W, Han S, Nam G, Sindoro M, Zhang H 2017 Chem. Rev. 117 6225
Google Scholar
[2] Pumera M, Sofer Z 2017 Adv. Mater. 29 1605299
Google Scholar
[3] 王靖慧, 焦丽颖 2017 科学通报 62 2158
Google Scholar
Wang J H, Jiao L Y 2017 Chin. Sci. Bull. 62 2158
Google Scholar
[4] 王慧, 徐萌, 郑仁奎 2020 物理学报 69 017301
Google Scholar
Wang H, Xu M, Zheng R K 2020 Acta Phys. Sin. 69 017301
Google Scholar
[5] Song X, Hu J, Zeng H 2013 J. Mater. Chem. C 1 2952
Google Scholar
[6] Zhao J, Liu H, Yu Z, Quhe R, Zhou S, Wang Y, Liu C C, Zhong H, Han N, Lu J, Yao Y, Wu K 2016 Prog. Mater. Sci. 83 24
Google Scholar
[7] 顾品超, 张楷亮, 冯玉林, 王芳, 苗银萍, 韩叶梅, 张韩霞 2016 物理学报 65 018102
Google Scholar
Gu P C, Zhang K L, Feng Y L, Wang F, Miao Y P, Han Y M, Zhang H X 2016 Acta Phys. Sin. 65 018102
Google Scholar
[8] 魏争, 王琴琴, 郭玉拓, 李佳蔚, 时东霞, 张广宇 2018 物理学报 67 128103
Google Scholar
Wei Z, Wang Q Q, Guo Y T, Li J W, Shi D X, Zhang G Y 2018 Acta Phys. Sin. 67 128103
Google Scholar
[9] Hong Y, Liu Z, Wang L, Zhou T, Ma W, Xu C, Feng S, Chen L, Chen M, Sun D, Sun D, Chen X, Chen H, Ren W 2020 Science 369 670
Google Scholar
[10] 黄坤, 殷雅俊, 吴继业 2014 物理学报 63 156201
Google Scholar
Huang K, Yin Y J, Wu J Y 2014 Acta Phys. Sin. 63 156201
Google Scholar
[11] 黄坤, 殷雅俊, 屈本宁, 吴继业 2014 力学学报 46 905
Google Scholar
Huang K, Yin Y J, Qu B N, Wu J Y 2014 Chin. J. Theoret. Appl. Mechan. 46 905
Google Scholar
[12] Cao K, Feng S, Han Y, Gao L, Lu Y 2020 Nat. Commun. 11 284
Google Scholar
[13] Li X, Zhu H 2015 J. Materiomics 1 33
Google Scholar
[14] Xiong S, Cao G 2016 Nanotechnology 27 105701
Google Scholar
[15] Jiang J, Qi Z, Park H, Rabczuk T 2013 Nanotechnology 24 435705
Google Scholar
[16] Late D, Shirodkar S, Waghmare U, Dravid V, Rao C 2014 Chemphyschem 15 1592
Google Scholar
[17] Hu X, Yasaei P, Jokisaari J, Öğüt S, Salehi-Khojin A, Robert F, Klie R 2018 Phys. Rev. Lett. 120 055902
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Google Scholar
[19] Akinwande D, Brennan C, Bunch J, Egberts P, Felts J, Gao H, Huang R, Kim J, Li T, Li Y 2017 Extreme Mech. Lett. 23 42
[20] Wei Y, Yang R 2018 Natl. Sci. Rev. 6 324
[21] Chen S, Chrzan D C 2011 Phys. Rev. B 84 5409
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Google Scholar
[23] Zhou L, Cao G 2016 Phys. Chem. Chem. Phys. 18 1657
Google Scholar
[24] Gao E, Xu Z 2015 J. Appl. Mech. 82 121012
Google Scholar
[25] Audoly B, Pomeau Y 2010 Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells (New York: Oxford University Press) pp157-213
[26] Kudin K, Scuseria G, Yakobson B 2001 Phys. Rev. B 64 235406
Google Scholar
[27] Landau L, Lifshitz E 1997 Theory of Elasticity 3rd (Oxford: Butterworth Heinemann) pp38−50
[28] O'NEILL B 2006 Elementary Differential Geometry (Singapore: Elsevier) pp364−376
[29] Eduard E, Krauthammer T 2001 Thin Plates and Shells: Theory, Analysis, and Applications (New York: Marcel Dekker) pp191−240
[30] 胡海昌 1981 弹性力学的变分原理及其应用 (北京: 科学出版社)pp322−342
Hu H 1981 Variational Principles of Theory of Elasticity with Applications (Beijing: Science Press) pp322−342 (in Chinese)
[31] Liu K, Yan Q, Chen M, Fan W, Sun Y, Suh J, Fu D, Lee S, Zhou J, Tongay S, Ji J, Neaton J, Wu J 2014 Nano Lett. 14 5097
Google Scholar
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Google Scholar
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Google Scholar
[34] Luongo A, Egidio A 2005 Nonlinear. Dynam. 41 171
Google Scholar
[35] Luongo A, D'Annibale F 2013 Int. J. Nonlin. Mech. 55 128
Google Scholar
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