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

Huang Kun; Wang Teng-Fei; Yao Ji

doi:10.7498/aps.70.20210160

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:

PACS:

62.20.D-	(Elasticity)
62.25.-g	(Mechanical properties of nanoscale systems)
81.40.Jj	(Elasticity and anelasticity, stress-strain relations)
81.40.Lm	(Deformation, plasticity, and creep)

Authors and contacts

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

Corresponding author: Huang Kun, kunhuang2008@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12050001, 11562009)

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1. 引　言

自机械剥离获得单层石墨烯以来, 多种单层二维纳米结构不断被发现^[1-4]. 比如, 与石墨烯一样具有单原子厚度的六方氮化硼(h-BN)^[5]; 原子不在同一面内的准二维纳米单层, 例如硅烯^[6]; 以及不同原子构成的“三片单层结构”(three-sheet single layer), 例如二硫化钼( ${\rm{Mo}}{{\rm{S}}_2}$ )^[7,8]等. 最近还发现了“五片单层结构”的 ${\rm{MoS}}{{\rm{i}}_2}{{\rm{N}}_4}$ ^[9]. 石墨烯因优异的物理化学性质, 具有广泛的潜在应用价值. 其力学性质引起了广泛的兴趣, 并得了大量的研究成果^[10-12]. 但针对 ${\rm{Mo}}{{\rm{S}}_2}$ 单层的研究还主要集中在电学、热学和摩擦性质上, 对其力学性质的研究还较少^[13]. 特别是, ${\rm{Mo}}{{\rm{S}}_2}$ 面外抗弯能力的研究还很少被关注^[13-15]. 此外对于二维纳米结构, 实际应用中常常需要考虑环境温度变化的影响. 虽然 ${\rm{Mo}}{{\rm{S}}_2}$ 的热学性质已经引起了研究者的关注^[16-18]. 但迄今为止, 还没有针对单层 ${\rm{Mo}}{{\rm{S}}_2}$ 热弹耦合(热-力耦合)的系统研究. 任何材料在工程应用前, 都需要对其力学性质有基本的认识, ${\rm{Mo}}{{\rm{S}}_2}$ 也不会例外. 纳米结构通常对温度变化比较敏感, 因此在力学模型中考虑温度的影响是必要的.

需要指出的是, 迄今还没有统一的理论来描述单层二维结构的力学性质^[19-20]. 这是因为, 如何建立化学键理论和二维材料力学性质的关系, 是具有高度挑战性的难题. 考虑到以量子化学为基础, 来建立“多片”单层二维结构力学理论的巨大困难, 研究者通常退而求其次, 使用唯像的宏观连续介质理论来描述其力学性质^[19]. 连续介质理论中的力学参数, 则使用量子计算和实验来拟合. 其中密度泛函紧束缚(FDTB)计算和分子动力学(MD)是两种被广泛使用的方法^[19-24]. 迄今的文献中, 多数 ${\rm{Mo}}{{\rm{S}}_2}$ 的力学参数是在绝对零度下得到的, 忽略了有限温度的影响^[13-15]. 但温度对二维材料力学性质的影响, 在理论和实践中都是需要考虑的. 对于石墨烯, 温度对弹性模量, Poisson比、以及面外弯曲和扭转刚度的影响不大^[21]. 对于 ${\rm{Mo}}{{\rm{S}}_2}$ , 弹性参数和温度的关系还缺乏系统的研究. 近期的分子动力学计算(MD)显示, ${\rm{Mo}}{{\rm{S}}_2}$ 在温度较低情况下可以忽略温度对弹性参数的影响^[18]. 虽然温度对 ${\rm{Mo}}{{\rm{S}}_2}$ 的弹性参数影响较小, 但温度会产生显著的热膨胀效应. 一个有趣的现象是, 在有限温度下(1000 K以下), ${\rm{Mo}}{{\rm{S}}_2}$ 热胀冷缩, 而同为单层的石墨烯却是热缩冷胀^[16,17]. 造成这种截然相反结果的原因, 是石墨烯和 ${\rm{Mo}}{{\rm{S}}_2}$ 有着不同的热变形机制^[22]: 由于面外弯曲刚度很小, 石墨烯在温度场中波浪式起伏, 并造成负的热膨胀系数; 但 ${\rm{Mo}}{{\rm{S}}_2}$ 的面外弯曲刚度远远大于石墨烯, 可有效抑制温度造成的起伏; 这使得 ${\rm{Mo}}{{\rm{S}}_2}$ 在高温下的体积膨胀, 导致正的热膨胀系数. 但无论是正或负的热膨胀系数, 在边界受限时都将产生热应力. 二维材料的抗弯能力通常较弱, 热应力可能导致热屈曲, 并使结构丧失功能. 因此澄清二维材料的热弹耦合力学性质, 在工程应用中具有重要的意义. 本文将基于唯像的宏观Föppl-von Karman 板理论, 来建立具有独立面内和面外力学参数的单层 ${\rm{Mo}}{{\rm{S}}_2}$ 的热弹耦合力学理论. 该理论可方便的推广到其他单层二维纳米材料.

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Föppl-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Föppl-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\r\nFig. 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 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

下载: 全尺寸图片幻灯片

$\begin{split} \;&U = {U_{\rm{e}}} + {U_{\rm{s}}} =\\ & \int\nolimits_S \left[ \frac{1}{2}{k_{\rm{B}}}{{\left( {2H} \right)}^2} + {k_{\rm{G}}}K + \frac{1}{2}{k_{\rm{b}}}{{\left( {2J} \right)}^2} - {k_{\rm{g}}}Q \right]{\rm{d}}S . \end{split}$

(1)

方程(1)中, $H$ , $K$ 为平均曲率和Gauss曲率, k_B和k_G分别为弯曲刚度和扭转刚度; $Q = {\rm{tr}}\left( {{\varepsilon _0}} \right)$ , $J = \det \left( {{\varepsilon _0}} \right)$ 是板中面的二维应变张量 ${\varepsilon _0}$ 的两个不变量, ${k_{\rm{b}}}$ 和 ${k_{\rm{g}}}$ 是面内刚度. 在宏观Föppl-von Karman板理论中, 面内和面外刚度间有如下关系: ${k_{\rm{B}}}/{k_{\rm{b}}} = - {k_{\rm{G}}}/{k_{\rm{g}}} = {h^2}/12$ , $h$ 为板厚. 和经典的板理论不同, 二维材料的面内和面外刚度间没有此关系^[19]. 即不能通过面内力学参数, 直接通过二维材料的厚度得到面外的弯曲和扭转刚度. 这种不一致性称为Yakobson悖论(Yakobson paradox)^[19]. 迄今为止, 对Yakobson悖论还没有得到一致认可的物理解释. 文献[23]指出, 石墨烯的在有限变形时的各向异性导致了Yakobson悖论. 使用板的等效厚度概念, 可以方便地使用连续介质力学理论, 为设计提供方便^[24]. 本文将放弃引入等效厚度, 直接把面内刚度和面外刚度作为独立的力学参数. 这样处理既回避了Yakobson悖论产生的不确定性, 在理论上也没有逻辑矛盾. 对比方程(1)和经典板壳的变形能, 有:

${k_{\rm{b}}} = \frac{Y}{{1 - {\nu ^2}}},\quad {k_{\rm{g}}} = \frac{Y}{{1 + \nu }}.$

(2)

其中, $Y$ , $\nu$ 分别为二维弹性模量和Poisson比. k_B和k_G 需要通过原子计算或实验确定. 在直角坐标系中, 令 $u, v, w$ 分别为 $x, y, z$ 方向的位移, 如图1所示. 使用von Karman非线性应变, 则面内应变张量的分量为^[25,27]

$\begin{split} &\varepsilon _{xx}^0 = \frac{{\partial u}}{{\partial x}} + \frac{1}{2}{\left( {\frac{{\partial w}}{{\partial x}}} \right)^2}, \\ & \varepsilon _{yy}^0 = \frac{{\partial v}}{{\partial y}} + \frac{1}{2}{\left( {\frac{{\partial w}}{{\partial y}}} \right)^2}, \\ &\varepsilon _{xy}^0 = \frac{1}{2}\left( {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}} + \frac{{\partial w}}{{\partial x}}\frac{{\partial w}}{{\partial y}}} \right). \end{split}$

(3)

面内应变张量的两个不变量为^[25]

$\begin{split} & 2J = \frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{1}{2}\left[ {{{\left( {\frac{{\partial w}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial w}}{{\partial y}}} \right)}^2}} \right], \\ & Q = \left[ {\frac{{\partial u}}{{\partial x}} + \frac{1}{2}{{\left( {\frac{{\partial w}}{{\partial x}}} \right)}^2}} \right]\left[ {\frac{{\partial v}}{{\partial y}} + \frac{1}{2}{{\left( {\frac{{\partial w}}{{\partial y}}} \right)}^2}} \right]\\ &\;\;\;\;\;\;- \frac{1}{4}{\left( {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}} + \frac{{\partial w}}{{\partial x}}\frac{{\partial w}}{{\partial y}}} \right)^2}. \end{split}$

(4)

平均曲率和Gauss曲率为^[28]

$\begin{split} &H \approx \frac{1}{2}\left( {\frac{{{\partial ^2}w}}{{\partial {x^2}}} + \frac{{{\partial ^2}w}}{{\partial {y^2}}}} \right),\quad \\ &K \approx \frac{{{\partial ^2}w}}{{\partial {x^2}}}\frac{{{\partial ^2}w}}{{\partial {y^2}}} - {\left( {\frac{{{\partial ^2}w}}{{\partial x\partial y}}} \right)^2}. \end{split}$

(5)

在此需要指出的是, 方程(5)是经典Föppl-von Karman板理论中对曲率的近似表达. 对于宏观板而言, 在小变形和小截面转动条件下, 可以忽略近似的曲率表达和精确表达的差别^[25]. 若定义应力Airy函数为 $F$ , 可仿照经典板壳理论, 定义面内的二维应力为^[25,29]

${N_{xx}} = \frac{{{\partial ^2}F}}{{\partial {y^2}}},\quad {N_{xy}} = - \frac{{{\partial ^2}F}}{{\partial x\partial y}},\quad {N_{yy}} = \frac{{{\partial ^2}F}}{{\partial {x^2}}}.$

(6)

从变形能(1), 可得:

$\begin{split} &{N_{xx}} = {k_{\rm{b}}}\varepsilon _{xx}^0 + \left( {{k_{\rm{b}}} - {k_{\rm{g}}}} \right)\varepsilon _{yy}^0,\quad {N_{xy}} = {k_{\rm{g}}}\varepsilon _{xy}^0, \\ & {N_{yy}} = \left( {{k_{\rm{b}}} - {k_{\rm{g}}}} \right)\varepsilon _{xx}^0 + {k_{\rm{b}}}\varepsilon _{yy}^0, \end{split}$

(7)

进一步可把面内应变表示为应力的形式:

$\begin{split} &\varepsilon _{xx}^0 = \frac{1}{\chi }{N_{xx}} - \frac{1}{\lambda }{N_{yy}},\quad \varepsilon _{yy}^0 = \frac{1}{\lambda }{N_{xx}} - \frac{1}{\chi }{N_{yy}},\\ &\varepsilon _{xy}^0 = \frac{{{N_{xy}}}}{{{k_g}}}. \end{split}$

(8)

其中 $\chi = {k_{\rm{g}}}\left( {2{k_{\rm{b}}} - {k_{\rm{g}}}} \right)k_{\rm{b}}^{{\rm{ - 1}}}$ , $\lambda = {k_{\rm{g}}}\left( {2{k_{\rm{b}}} - {k_{\rm{g}}}} \right)( {k_{\rm{b}}} -$ ${k_{\rm{g}}} )^{ - 1}$ . 此时面内变形能密度可写为

$\begin{split} &\frac{1}{2}{k_{\rm{b}}}{\left( {2J} \right)^2} - {k_{\rm{g}}}Q = \frac{1}{{2\chi }}{\left( {{N_{xx}} + {N_{yy}}} \right)^2} \\ &- \frac{1}{{{k_{\rm{g}}}}}\left[ {{N_{xx}}{N_{yy}} - {{\left( {{N_{xy}}} \right)}^2}} \right]. \end{split}$

(9)

把方程(6)代入 (9) 式, 有:

$\begin{split} {U_s} =\;& \int\nolimits_S \left[ \frac{1}{{2\chi }}{{\left( {\frac{{{\partial ^2}F}}{{\partial {y^2}}} + \frac{{{\partial ^2}F}}{{\partial {x^2}}}} \right)}^2}\right.\\ &\left.- \frac{1}{{{k_{\rm{g}}}}}\left( {\frac{{{\partial ^2}F}}{{\partial {y^2}}}\frac{{{\partial ^2}F}}{{\partial {x^2}}} - {{\left( {\frac{{{\partial ^2}F}}{{\partial x\partial y}}} \right)}^2}} \right) \right]{\rm{d}}S . \end{split}$

(10)

为保证位移场的连续单值性, 应变场满足如下的完备性条件^[25]: $2\dfrac{{{\partial ^2}\varepsilon _{xy}^0}}{{\partial x\partial y}} - \dfrac{{{\partial ^2}\varepsilon _{xx}^0}}{{\partial {y^2}}} - \dfrac{{{\partial ^2}\varepsilon _{yy}^0}}{{\partial {x^2}}} = K$ , 并可等价的表示为应力函数的形式, ${\Delta ^2}F = - \lambda K$ . 应力函数引入时, 需要在势能泛函中引入Lagrange乘子 $l\left( {x, y} \right)$ , 则(1)式可写为

$\begin{split} U =\;& \int\nolimits_S {\Bigg\{ {\frac{1}{2}{k_{\rm{B}}}{{\left( {2H} \right)}^2} + {k_{\rm{G}}}K + \frac{1}{{2\chi }}{{\left( {\frac{{{\partial ^2}F}}{{\partial {x^2}}} + \frac{{{\partial ^2}F}}{{\partial {y^2}}}} \right)}^2}} } \\ & - \frac{1}{{{k_{\rm{g}}}}}\left[ {\frac{{{\partial ^2}F}}{{\partial {x^2}}}\frac{{{\partial ^2}F}}{{\partial {y^2}}}\left. { - {{\left( {\frac{{{\partial ^2}F}}{{\partial x\partial y}}} \right)}^2}} \right]} \right.\\ &+ l\left( {x,y} \right)\left( {{\Delta ^2}F + \chi K} \right) \Bigg\}{\rm{d}}S.\\[-15pt] \end{split}$

(11)

对(11)式进行复杂但直接的计算, 并识别Lagrange乘子后(具体计算方法可参考文献[25]), (11)式可化为

$\begin{split} U = \;&\int\nolimits_S {\Bigg\{ {\frac{1}{2}{k_{\rm{B}}}{{\left( {2H} \right)}^2} + {k_{\rm{G}}}K} } + \frac{1}{{2\chi }}\left( {\frac{{{\partial ^2}F}}{{\partial {x^2}}} + \frac{{{\partial ^2}F}}{{\partial {y^2}}}} \right)^2 \\ &- \frac{1}{{{k_{\rm{g}}}}}\left[ {\frac{{{\partial ^2}F}}{{\partial {x^2}}}\frac{{{\partial ^2}F}}{{\partial {y^2}}}} - {{\left( {\frac{{{\partial ^2}F}}{{\partial x\partial y}}} \right)}^2} \right] \\ &+ \frac{1}{2}\left[ \frac{{{\partial ^2}F}}{{\partial {y^2}}}{{\left( {\frac{{\partial w}}{{\partial x}}} \right)}^2} + \frac{{{\partial ^2}F}}{{\partial {x^2}}}{{\left( {\frac{{\partial w}}{{\partial y}}} \right)}^2}\right.\\ &\left.- 2\frac{{{\partial ^2}F}}{{\partial x\partial y}}\frac{{\partial w}}{{\partial x}}\frac{{\partial w}}{{\partial y}} \right] \Bigg\}{\rm{d}}S.\\[-15pt] \end{split}$

(12)

为了考虑温度对结构的影响, 假设边界轴向外力和热应力的势能为

$\begin{split} W =\;& \int\nolimits_S {q\left( {x,y,t} \right){\rm{d}}S} + \frac{1}{2}\int\nolimits_S {\left[ {\left( {N_{xx}^0 - N_{xx}^{\rm{T}}} \right){{\left( {\frac{{\partial w}}{{\partial x}}} \right)}^2}} \right.} \\ & \left. { + 2N_{xy}^0\frac{{\partial w}}{{\partial x}}\frac{{\partial w}}{{\partial y}} + \left( {N_{yy}^0 - N_{yy}^{\rm{T}}} \right){{\left( {\frac{{\partial w}}{{\partial y}}} \right)}^2}} \right]{\rm{d}}S. \end{split}$

(13)

在此, $q\left( {x, y, t} \right)$ 为面内载荷; $N_{xx}^0, N_{yy}^0$ 为边界 $x, y$ 轴方向的初始二维应力, $N_{xx}^{\rm{T}}, N_{yy}^{\rm{T}}$ 为边界 $x, y$ 轴方向的二维热应力. 边界上的轴向应力, 压为负, 拉为正. 当不考虑温度对弹性参数的影响时, 均匀温度场下的热应力为^[29]

$N_{xx}^{\rm{T}} = {k_b}\varepsilon _{xx}^{\rm{T}} = {k_{\rm{b}}}\alpha T,\quad N_{yy}^{\rm{T}} = {k_b}\varepsilon _{yy}^{\rm{T}} = {k_b}\alpha T.$

(14)

在此, $\varepsilon _{xx}^{\rm{T}}$ , $\varepsilon _{yy}^{\rm{T}}$ 为热应变, $\alpha$ 为热膨胀系数. 把 $w, F$ 作为独立变量, 构造Lagrange函数, $L = {U_{\rm{e}}} + {U_{\rm{s}}} -$ $W$ , 并对其变分, 令 $\delta L = 0$ ^[30], 可得考虑均匀温度场产生的热应力影响的受力平衡方程为

$\begin{split} &{k_{\rm{B}}}{\nabla ^4}w = - \left( {\frac{{{\partial ^2}F}}{{\partial {y^2}}} + N_{xx}^0 - N_{xx}^{\rm{T}}} \right)\frac{{{\partial ^2}w}}{{\partial {x^2}}} - \bigg( {\frac{{{\partial ^2}F}}{{\partial {x^2}}}} \\ & { + N_{yy}^0 - N_{yy}^{\rm{T}}} \bigg)\frac{{{\partial ^2}w}}{{\partial {y^2}}} + 2\left( {\frac{{{\partial ^2}F}}{{\partial x\partial y}} + N_{xy}^0} \right)\frac{{{\partial ^2}w}}{{\partial x\partial y}} + q, \\ &{\nabla ^4}F = - \chi \left[ {\frac{{{\partial ^2}w}}{{\partial {x^2}}}\frac{{{\partial ^2}w}}{{\partial {y^2}}} - {{\left( {\frac{{{\partial ^2}w}}{{\partial x\partial y}}} \right)}^2}} \right]. \\[-15pt] \end{split}$

(15)

由于变形能(12)和经典板的势能形式相似, $F, w$ 的边界条件和经典板理论中的相同. 对于四边铰支的板, 有:

$\begin{split} &w\left( {x = 0,a} \right) = 0,\quad \frac{{{\partial ^2}w}}{{\partial {x^2}}}\left( {x = 0,a} \right) = 0, \\ & w\left( {y = 0,b} \right) = 0,\quad \frac{{{\partial ^2}w}}{{\partial {y^2}}}\left( {x = 0,b} \right) = 0, \\ & \int\nolimits_0^a {\left[ {\frac{1}{\chi }\frac{{{\partial ^2}F}}{{\partial {y^2}}} - \frac{1}{\lambda }\frac{{{\partial ^2}F}}{{\partial {x^2}}} - \frac{1}{2}{{\left( {\frac{{\partial w}}{{\partial x}}} \right)}^2}} \right]} {\rm{d}}x = 0, \\ &\int\nolimits_0^a {\left[ {\frac{1}{\lambda }\frac{{{\partial ^2}F}}{{\partial {y^2}}} - \frac{1}{\chi }\frac{{{\partial ^2}F}}{{\partial {x^2}}} - \frac{1}{2}{{\left( {\frac{{\partial w}}{{\partial y}}} \right)}^2}} \right]} {\rm{d}}x = 0. \end{split}$

(16)

在此需要注意的是, 当前的理论具有4个独立的力学参数: 面内和面外的力学参数是独立的！该理论没有人为的引入二维结构的厚度, 很好的解决Yakobson悖论引发的不确定性. 这是和已有的单层二维纳米结构力学模型最明显的区别, 例如文献[10, 11, 26]中的板壳理论. 事实上, 本文的二维结构热弹耦合非线性板理论具有4个独立的力学参数和一个热学参数(热膨胀系数). 对于不同的单层二维结构, 只要通过原子计算(例如FDTB或MD), 得到5个参数, 就可以得到相应的力学模型. 因此, 该理论可以用来描述其他单层原子结构的力学性质, 例如石墨烯、h-BN等. 下节将通过平衡方程(15)来对比研究石墨烯和 ${\rm{Mo}}{{\rm{S}}_2}$ 的力学性质.

3. 算例及讨论

本节通过方程组(15), 来讨论单层 ${\rm{Mo}}{{\rm{S}}_2}$ 和石墨烯的热弹耦合力学性质. 假设有四边铰支, 边长为分别为 $a$ 和 $b$ 的单层片(如图1所示), 此时的边界条件为(16)式. 由于方程组(15)是非线性的, 除了个别的情况, 精确的解析解是很难得到的^[25,27,29]. 故本文使用Galerkin方法^[30]近似求解方程(15). 方程(15)和(16)的形式和经典板壳理论相似(但力学参数和板厚无关), 可以引入和Föppl-von Karman板相似的近似解析解^[29]:

$\begin{split} & w = \sum\limits_{m,n = 1}^\infty {{\eta _{mn}}\sin \frac{{n{\rm{\pi }}x}}{a}} \sin \frac{{m{\rm{\pi }}y}}{b}, \\ & F = \sum\limits_{j,k = 1}^\infty {{\xi _{jk}}\sin \frac{{j{\rm{\pi }}x}}{a}} \sin \frac{{k{\rm{\pi }}y}}{b}. \end{split}$

(17)

为了简化讨论, 仅取一项, 并令 ${\eta _{11}} = \eta, \quad {\xi _{11}} = \xi$ . 把(17)式代入(15)式, 等号两边同时乘以 $\sin \left( {{\text{π}}{a^{ - 1}}x} \right)$ $\sin \left( {{\text{π}}{b^{ - 1}}y} \right)$ , 并在矩形期间积分(Galerkin积分^[30]), 整理得到:

${k_1}\eta - {k_2}\eta \xi - {k_0}q = 0,\quad \xi = - {k_3}{\eta ^2},$

(18)

在此

$\begin{split} & {k_1} = \frac{{{k_{\rm{B}}}{{\rm{\pi }}^4}{{\left( {{b^2} + {a^2}} \right)}^2}}}{{{a^4}{b^4}}} + \frac{{{{\rm{\pi }}^2}\left( {N_{xx}^0 - {k_{\rm{b}}}\alpha T} \right)}}{{{a^2}}} \\ &\qquad+ \frac{{{{\rm{\pi }}^2}\left( {N_{yy}^0 - {k_{\rm{b}}}\alpha T} \right)}}{{{b^2}}}, \\ & k_2= \frac{{32{{\rm{\pi }}^2}}}{{3{a^2}{b^2}}},\quad {k_3} = \frac{{16{a^2}{b^2}\chi }}{{3{{\rm{\pi }}^2}{{\left( {{a^2} + {b^2}} \right)}^2}}},\quad {k_0} = \frac{{16}}{{{{\rm{\pi }}^2}}}. \end{split}$

(19)

方程组(18)可简化为一个方程:

${k_1}\eta + {k_2}{k_3}{\eta ^3} = {k_0}{\kern 1pt} q.$

(20)

(20)式即确定结构中心点的变形幅值和载荷关系的非线性代数方程.

这里需要指出的是, 对于 ${\rm{Mo}}{{\rm{S}}_2}$ 的面内力学参数, 二维杨氏模量和Poisson比, 已经有比较系统的研究^[13,31-33]. 面外的弯曲刚度 ${k_{\rm{B}}}$ , 近期已经受到了关注^[14-15]. 但是, 扭转刚度(Gauss 刚度) ${k_{\rm{G}}}$ , 迄今还没有相关报到. 如何确定Gauss刚度是一个需要进一步研究的问题. 但根据微分几何中的经典理论, Gauss-Bonnet积分公式^[28], 有 $\displaystyle \iint\nolimits_S {K{\rm{d}}S }=$ $2{\text{π}} - \displaystyle \sum\nolimits_{i = 1}^n {{a_i}} - \displaystyle \oint\nolimits_C {{k}{\rm{d}}s}$ . 其中 ${k}$ 为边界曲线的测地曲率, ${a_i}$ 为角点的外角. 若石墨烯片的边界在变形前后处于相同的平面内, 则 $\displaystyle \oint\nolimits_C {{k}{\rm{d}}s} = 0$ , 此时矩形的石墨烯片有 $2{\text{π}} - \displaystyle \sum\nolimits_{i = 1}^n {{a_i}} = 0$ , 故 $\displaystyle \iint\nolimits_S {K{\rm{d}}S = }0$ . 因此Gauss曲率不出现在平衡方程和边界条件中, 不影响本文的结果. 但当结构出现自由边界时, Gauss曲率刚度将出现在边界条件中, 并影响结构的受力变形, 我们将在后续研究中处理该问题. 在此取 ${\rm{Mo}}{{\rm{S}}_2}$ 的力学参数为^[14-15,33]: $Y = 120\, {\rm{N}}/{\rm{m}}$ , $\nu =$ $0.23$ , ${k_{\rm{{\rm B}}}} = 9.61\, {\rm{eV}}$ . 为直观了解 ${\rm{Mo}}{{\rm{S}}_2}$ 的力学性质, 用石墨烯来对比研究. 石墨烯的力学参数为^[19]: $Y = 340\, {\rm{N}}/{\rm{m}}$ , $\nu = 0.165$ , ${k_{\rm{{\rm B}}}} = 1.6\, {\rm{eV}}$ . 热膨胀系数^[17]: ${\rm{Mo}}{{\rm{S}}_2}$ 为 $\alpha = 6.49 \times 1{0^{ - 5}}~{{\rm K}^{ - 1}}$ , 石墨烯为 $\alpha =$ $- 2.14 \times 1{0^{ - 5}}\, {{\rm K}^{ - 1}}$ . 根据方程(2), ${\rm{Mo}}{{\rm{S}}_2}$ 有, ${k_{\rm{b}}} =$ $792\, {\rm{eV}}/{\rm{n}}{{\rm{m}}^2}$ , ${k_{\rm{g}}} = 610\, {\rm{eV}}/{\rm{n}}{{\rm{m}}^2}$ ; 石墨烯有, ${k_{\rm{b}}} =$ $2814\, {\rm{eV}}/{\rm{n}}{{\rm{m}}^2}$ , ${k_{\rm{g}}} = 1822\, {\rm{eV}}/{\rm{n}}{{\rm{m}}^2}$ . 为便于讨论, 几何尺寸取为正方形, 即 $a = b$ . 根据方程(20)可得 ${\rm{Mo}}{{\rm{S}}_2}$ 和石墨烯, 在不同情况下的中心点变形和参数的关系, 如图2—图6所示.

$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\r\nFig. 2. Loads-response curves with two temperatures for $a = b = 6\;{\rm{nm}}$ and $N_{xx}^0 = N_{yy}^0 = 0$.$

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

$N_{xx}^0 = N_{yy}^0 = 0$

下载: 全尺寸图片幻灯片

$图 6 给定边界轴向力和温度条件下的载荷、几何尺寸及变形幅值曲面\r\nFig. 6. Loads- dimensions-response surfaces with the given stretching stresses and the temperature.$

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.

下载: 全尺寸图片幻灯片

先讨论温度对结构的影响. 令 $N_{xx}^0 = N_{yy}^0 = 0$ , 从方程(20)可得石墨烯和 ${\rm{Mo}}{{\rm{S}}_2}$ 在相同载荷下的变形幅值, 如图2所示. 从图2可以看出: 对于0 K条件下的正方形片, 载荷较小时石墨烯的变形大于 ${\rm{Mo}}{{\rm{S}}_2}$ ; 但伴随载荷的增大, 石墨烯的变形将小于 ${\rm{Mo}}{{\rm{S}}_2}$ 的变形. 这说明, 小变形时石墨烯的刚度小于 ${\rm{Mo}}{{\rm{S}}_2}$ , 而大变形时石墨烯的刚度大于 ${\rm{Mo}}{{\rm{S}}_2}$ . 出现此反常现象的原因是: 1) ${\rm{Mo}}{{\rm{S}}_2}$ 的弯曲刚度远远大于石墨烯, 而小变形时面外的抗弯刚度是抵抗变形的主要来源, 因此 ${\rm{Mo}}{{\rm{S}}_2}$ 的变形小于石墨烯; 2)伴随变形的增大, 面内的刚度逐渐成为抵抗变形的主要因素, 而石墨烯的面内弹性模量远远大于 ${\rm{Mo}}{{\rm{S}}_2}$ , 这使得大变形时石墨烯的变形小于 ${\rm{Mo}}{{\rm{S}}_2}$ . 但是, 当温度出现小幅升高时, 例如仅升高 $1\;\rm K$ (如图2所示), 石墨烯的变形就一致地小于 ${\rm{Mo}}{{\rm{S}}_2}$ . 这表明两种材料的力学性质对温度很敏感.

来看轴向力的影响, 此时令 $T = 0$ . 从图3可知, 虽然轴向的拉力可显著强化石墨烯和 ${\rm{Mo}}{{\rm{S}}_2}$ 的刚度, 但变形趋势和图2一致. 相对于轴向力的影响, 温度对石墨烯和 ${\rm{Mo}}{{\rm{S}}_2}$ 力学性质的影响更复杂和有趣. 在有限温度条件下(T < 1000 K), 石墨烯的热膨胀系数 $\alpha < 0$ , 是热缩冷胀材料; 而 ${\rm{Mo}}{{\rm{S}}_2}$ 的热膨胀系数 $\alpha > 0$ , 是热胀冷缩材料^[16-18]. 这使得温度强化了石墨烯的刚度, 但降低了 ${\rm{Mo}}{{\rm{S}}_2}$ 的刚度. 图2和图4显示, 相同条件下轻微的温度升高就使得石墨烯的变形幅值小于 ${\rm{Mo}}{{\rm{S}}_2}$ . 但是, 当同时考虑温度和轴向力的影响时, 情况会变得复杂. 如图5所示, 温度较低时, 在 $N_{xx}^0 = N_{yy}^0 = 7.0\;{\rm{nN}}/{\rm{nm}}$ 的轴向拉力作用下 ${\rm{Mo}}{{\rm{S}}_2}$ 的变形, 小于在 $N_{xx}^0 = N_{yy}^0 =$ $0.1\;{\rm{nN}}/{\rm{nm}}$ 的轴向拉力作用下石墨烯的变形; 但当温度超过临界值时, 变形幅值大小的顺序将颠倒. 从方程(20)的系数表达式(19)还可以看出, 变形幅值和结构的几何尺寸紧密相关, 如图6所示. 以上结果表明, 温度和边界轴向力对两种二维材料的力学性质均有显著影响. 在此需要指出的是, 本文仅考虑轴向拉力对两种材料力学性质的影响. 当边界出现轴向压力时, 结构可能屈曲. 此时温度的提高将增加石墨烯的抗屈曲能力, 但会降低 ${\rm{Mo}}{{\rm{S}}_2}$ 的抗屈曲能力. 屈曲将使得结构的力学行为, 特别是动力学行为变得异常复杂^[34-35]. 我们将另文讨论该问题. 在此还需要指出的是, 本文根据唯像的连续介质力学建立了单层 ${\rm{Mo}}{{\rm{S}}_2}$ 的热-弹耦合模型, 并依据原子计算结果取定力学和热膨胀系数. 尽管新的模型建立在严谨的逻辑推理之上, 但该理论的正确性最终需要实验和量子计算的检验. 相对于被广泛深入研究的石墨烯而言, ${\rm{Mo}}{{\rm{S}}_2}$ 还缺乏系统量子计算和实验研究^[1,22,23]. 特别是多场耦合条件下的力学问题还需要深入研究.

$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\r\nFig. 3. Loads-response curves with two edge stretching stresses for $a = b = 6\;{\rm{nm}}$ and $T = 0\; \rm K$.$

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

$T = 0\; \rm K$

下载: 全尺寸图片幻灯片

$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\r\nFig. 4. Loads-response curves with two edge stresses and two temperatures for $a = b = 6\;{\rm{nm}}$.$

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

$a = b = 6\;{\rm{nm}}$

下载: 全尺寸图片幻灯片

$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\r\nFig. 5. Loads-temperatures-response surfaces with the given stretching stresses 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

Fig. 5. Loads-temperatures-response surfaces with the given stretching stresses for

$a = b = 6\;{\rm{nm}}$

下载: 全尺寸图片幻灯片

4. 结　论

假设面内张拉和剪切变形能, 与面外弯曲和扭转变形能相互独立, 本文建立了单层 ${\rm{Mo}}{{\rm{S}}_2}$ 的热弹耦合非线性板理论, 并通过该理论对比研究了单层 ${\rm{Mo}}{{\rm{S}}_2}$ 和石墨烯的力学性质. 结果显示, 在边界不移动条件下, 边界的预加轴向力和有限温度对结构的力学性质具有显著影响. 因此可以通过边界预加轴向载荷或环境温度来改变二维纳米结构的力学性质. 例如, 通过提高环境温度来弱化 ${\rm{Mo}}{{\rm{S}}_2}$ 的刚度, 或强化石墨烯的刚度. 此外, 本文提出的热弹耦合非线性板理论不依赖单层二维结构的厚度, 可以方便的应用于描述其他单层二维纳米结构的力学性质.

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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

Figure 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

DownLoad: Full-Size Img PowerPoint

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

Figure 2. Loads-response curves with two temperatures for $a = b = 6\;{\rm{nm}}$ and $N_{xx}^0 = N_{yy}^0 = 0$ .

DownLoad: Full-Size Img PowerPoint

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

Figure 6. Loads- dimensions-response surfaces with the given stretching stresses and the temperature.

DownLoad: Full-Size Img PowerPoint

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

Figure 3. Loads-response curves with two edge stretching stresses for $a = b = 6\;{\rm{nm}}$ and $T = 0\; \rm K$ .

DownLoad: Full-Size Img PowerPoint

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

Figure 4. Loads-response curves with two edge stresses and two temperatures for $a = b = 6\;{\rm{nm}}$ .

DownLoad: Full-Size Img PowerPoint

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

Figure 5. Loads-temperatures-response surfaces with the given stretching stresses for $a = b = 6\;{\rm{nm}}$ .

DownLoad: Full-Size Img PowerPoint

[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
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Vol. 70, Issue 13 - 2021-07-05

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Nonlinear plate theory of single-layered MoS₂ with thermal effect

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Corresponding author: Huang Kun, kunhuang2008@163.com

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1. 引　言

2. 单层 ${\rm{Mo}}{{\rm{S}}_2}$ 的热弹耦合非线性板理论

3. 算例及讨论

4. 结　论

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Article

留言板

Nonlinear plate theory of single-layered MoS2 with thermal effect

Abstract

Authors and contacts

Corresponding author: Huang Kun, kunhuang2008@163.com

FullText HTML

1. 引 言

2. 单层MoS2{\rm{Mo}}{{\rm{S}}_2}的热弹耦合非线性板理论

3. 算例及讨论

4. 结 论

References

Catalog

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Authors

Referees

About Journal

About

Nonlinear plate theory of single-layered MoS₂ with thermal effect

1. 引　言

2. 单层 ${\rm{Mo}}{{\rm{S}}_2}$ 的热弹耦合非线性板理论

4. 结　论