⇦Chinese Physics Letters, 2017, Vol. 34, No. 9, Article code 096201 New Insights on the Deflection and Internal Forces of a Bending Nanobeam * De-Min Zhao(赵德敏), Jian-Lin Liu(刘建林)** Affiliations Department of Engineering Mechanics, College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao 266580 Received 16 May 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 11672334, 11672335 and 11611530541, and the Fundamental Research Funds for the Central Universities under Grant No 15CX08004A.
**Corresponding author. Email: liujianlin@upc.edu.cn Citation Text: Zhao D M and Liu J L 2017 Chin. Phys. Lett. 34 096201 Abstract Nanowires, nanofibers and nanotubes have been widely used as the building blocks in micro/nano-electromechanical systems, energy harvesting or storage devices, and small-scaled measurement equipment. We report that the surface effects of these nanobeams have a great impact on their deflection and internal forces. A simply supported nanobeam is taken as an example. For the displacement and shear force of the nanobeam, its dangerous sections are different from those predicted by the conventional beam theory, but for the bending moment, the dangerous section is the same. Moreover, the values of these three quantities for the nanobeam are all distinct from those calculated from the conventional beam model. These analyses shed new light on the stiffness and strength check of nanobeams, which are beneficial to engineer new-types of nano-materials and nano-devices. DOI:10.1088/0256-307X/34/9/096201 PACS:62.23.Hj, 62.25.-g, 68.35.Gy © 2017 Chinese Physics Society Article Text With the rapid development on miniaturization of devices, especially to the nanometer scale, naofibers, naowires and nanotubes have been widely used in micro/nano-electromechanical systems (MEMS/NEMS). These one-dimensional structures, i.e., nanobeams, have become the building blocks in micro-sensors, micro-actuators, micro-transistors, and energy harvesting or storage devices. Moreover, they are fundamental elements in the measurement of the mechanical properties of nano-materials, via atomic force microscopic (AFM) and the scanning tunneling microscope (STM).[1,2] So far, the exploration of the mechanical response of nanobeams remains an open problem if they have special performance due to the surface effects. Commonly speaking, the surface effects of nanomaterials are mainly caused by the surface energy and residual surface stress,[3] which make them demonstrate strong size-dependent behaviors.[4-6] A great deal of effort has been made to investigate the static and dynamic responses of nanobeams. For example, the bending deflections of nanobeams under different boundary conditions in infinitesimal and finite deformation have both been systematically studied.[7] In the analysis, the Euler–Bernoulli beam[8,9] and the Timoshenko beam[10] are often adopted as the models, where the shear force is ignored or considered. For the elastic stability of nanorods, Wang et al. gave the critical load of a slender nanobeam in compression, and computed its postbuckling morphology.[11,12] Furthermore, it has even been reported that the residual surface stress can induce the self-buckling of a nanobeam, and its critical value can be quantitatively calculated.[13] Moreover, Liu et al. discussed the adhesion of a nanobeam to a solid surface, and found that its critical adhesion length is affected by its surface effects.[14] For the dynamic behavior, Wang et al.[12,15] and He et al.[16] investigated the surface effects on the natural frequency of the simply supported, cantilever and fixed-fixed beam, using the linear vibration model. Recently, Zhao et al.[17] gave a comprehensive study on the nonlinear behaviors of a nanobeam with complex nonlinear vibration, including the quasi-periodic motion and chaotic motion. It should be mentioned that the above analyses are mainly based upon continuum mechanics, and another crucial issue is the molecular simulations. Actually a great deal of effort has been spent on this area.[18-20] Although much work has been carried out on all kinds of mechanical behaviors of nanobeams, attention is rarely paid to checking their strength and stiffness. As is well known, calculating the deflection and internal forces of a beam is a fundamental task in mechanics of materials, which is the preliminary step in designing structures made of beams. In this Letter, it is emphasized that at the nanoscale, the criteria of stiffness and strength check of nanobeams should be modified, as their surface effects play an important role in this process. Firstly, the governing equation and deflection curve of a nanobeam with surface effects are presented. Then the shear force and bending moment diagrams of the nanobeam are given. It shows that the dangerous section and maximum values of displacement and internal forces should be modulated when nanobeams are considered. We believe that this exploration can provide a comprehensive understanding of the physical mechanism of nano-materials, which is beneficial to the design of nano-devices. Without loss of generality, we take a simply supported beam $AB$ as a typical example. We refer to the Cartesian coordinate system $A-xy$, where the origin is located at point $A$. As schematized in Fig. 1, there is a concentrated force $P$ applied at point $C$ on the beam, the distance of $AC$ is $L_{1}$, and the total length of the beam is $L$. A position parameter $k$ is introduced, where $k=L_{1}/L$, and $0 < k < 1$. The cross section of the beam is a rectangle, with its width $b$ and height $h$, then the moment of inertia $I=bh^{3}/12$. In consideration of the surface effects, the model can be abstracted as a composite beam with a core-shell structure, where the surface is assumed with zero thickness. Young's modulus of the bulk material is $E$, and the surface elastic modulus is denoted by $E^{\rm s}$. For the nanobeam, the bending stiffness should be recast as $(EI)^{\ast}=\frac{1}{12}Ebh^3+\frac{1}{2}E^{\rm s}bh^2+\frac{1}{6}E^{\rm s}h^3$.[10,11] In the light of the surface elasticity theory developed by Cammarata and Murdoch[21] and GurtinMurdoch,[22] the residual surface stress $\tau _0$ in the surface layer can lead to a transversely distributed pressure $q_{\rm s}(x)$ along the longitudinal direction, which is termed as the Young–Laplace equation[8,9,15,22] $$ q_{\rm s} (x)=H{w''},~~ \tag {1} $$ where the parameter $H=2\tau _0 b$ is a constant parameter, $w$ is the deflection, and the symbol of prime denotes the derivative with respect to $x$. Thus the governing equation of the deflection $w(x)$ on the beam can be given as $$ w^{(4)}=q_{\rm s}(x),~~ \tag {2} $$ i.e., $w^{(4)}=r^2{w''}$, where the superscript (4) means the 4th derivative with respect to $x$, and the parameter $r^{2}=H/{(EI)^\ast}$, which can be positive or negative, depending on the value of the residual surface stress. In practice, numerous experimental and numerical results show that the sign of the residual surface stress can be positive or negative.[8,9,15]
cpl-34-9-096201-fig1.png
Fig. 1. Schematic diagram of a simply supported nanobeam with surface effects subjected to a concentrated force.
The general solution of Eq. (2) has different formats corresponding to the sign of $\tau _0$. When the residual surface stress $\tau _0 >0$, its solution can be written as $$ w(x)=\begin{cases} C_1 e^{rx}+C_2 e^{-rx}+C_3 x+C_4,~({0\leqslant x\leqslant L_1}), \\ C_5 e^{rx}+C_6 e^{-rx}+C_7 x+C_8,~({L_1 \leqslant x\leqslant L}), \\ \end{cases}~~ \tag {3} $$ where $C_{i}$ ($i=1, 2, {\ldots}8$) are integration constants. The boundary conditions of the beam are prescribed as $$ \begin{cases} w(0)=0,~{w''}(0)=0,~w(L)=0,~{w''}(L)=0, \\ w^-({L_1})=w^+({L_1}),~{w'}^-({L_1})={w'}^+({L_1})=0, \\ {w''}^-({L_1})={w''}^+({L_1})=0, \\ ({EI})^\ast [{{w'''}^+({L_1})-{w'''}^-({L_1})}]=-P, \end{cases}~~ \tag {4} $$ where the symbols $^+({L_1})$ and $^-({L_1})$ represent the cross sections, which approach point $C$ from the right and left, respectively. Substituting Eq. (3) into Eq. (4), the constants can be determined as $C_{2}=-C_{1}=\frac{P}{({EI})^\ast}\frac{({e^{2kLr}-e^{2Lr}})} {2r^3e^{kLr}({e^{2Lr}-1})}$, $C_{3}=\frac{P}{({EI})^\ast}\frac{({k-1})}{r^2}$, $C_{4}=0$, $C_{6}=-e^{2Lr}C_5 =\frac{P}{({EI})^\ast}\frac{e^{2Lr}({e^{2aLr}-1})}{2r^3e^{kLr}({e^{2Lr}-1})}$, $C_8 =-C_7 L=-\frac{P}{({EI})^\ast}\frac{kL}{r^2}$. When $\tau _0 < 0$, the solution to Eq. (2) should be expressed as $$ w(x)=\begin{cases} D_1 \sin ({rx})+D_2 \cos ({rx})+D_3 x+D_4,~({0\leqslant x\leqslant L_1}), \\ D_5 \sin ({rx})+D_6 \cos ({rx})+D_7 x+D_8,~({L_1 \leqslant x\leqslant L}), \\ \end{cases}~~ \tag {5} $$ where the constants $D_{i} $ ($i=1, 2, {\ldots}8$) are $D_1 =\frac{P}{({EI})^\ast}\frac{\sec ({kLr})[{\cot ({Lr})\tan ({kLr})-1}]}{r^3[ {1+\tan ( {kLr})^2}]}$, $D_2 =D_4 =0$, $D_3 =\frac{P}{({EI})^\ast}\frac{1-k}{r^2}$, $D_5 =\frac{P}{({EI})^\ast}\frac{\cot ({Lr})\sin ( {kLr})}{r^3}$, $D_8 =-D_7 L=\frac{P}{({EI})^\ast}\frac{kL}{r^2}$, and $D_6 =-\frac{P}{({EI})^\ast}\frac{\sin ({kLr})}{r^3}$. Although the solutions have different formats when $\tau _0>0$ and $\tau _0 < 0$, they can both degenerate to the same expression when $\tau _0$ approaches zero and $({EI})\ast=EI$. Using the L'Hopital rule, the limit of the deflection is derived as $$ \mathop {\lim}\limits_{r\to 0} w(x)=\begin{cases} -\frac{P({1-k})x}{6EI}({2kL^2-x^2-k^2L^2}),~({0\leqslant x\leqslant L_1}) \\ -\frac{P({1-k})}{6EI}[\frac{1}{1-k}({x-kL})^3+k(2-k)L^2x-x^3],~~({L_1 \leqslant x\leqslant L}) \end{cases}~~ \tag {6} $$ which is just the result presented in the textbook of mechanics of materials, representing the deflection of a macroscopic beam. This consistence validates the correctness of the above model, indicating that the surface elasticity theory is the generalization of the conventional continuum mechanics theory. Based on the expression of the beam deflection, the shear force $Q$ and bending moment $M$ of the Euler–Bernoulli beam can be deduced as $$\begin{align} Q=\,&({EI})^\ast {w'''},~~ \tag {7} \end{align} $$ $$\begin{align} M=\,&({EI})^\ast {w''}.~~ \tag {8} \end{align} $$ Consequently, the deflection curve and the internal force diagrams are be depicted in Fig. 2. In calculation, the physical parameters of the nanobeam are chosen as follows:[8,12] Young's modulus of bulk materials $E=76$ GPa, and Possion's ratio $\upsilon =0.3$. The residual surface stress $\tau _0$ is selected as $\tau _0 =0.89$, 0, and $-$0.89 N/m, respectively, and surface elastic modulus $E^{\rm s}=1.22$ N/m. The cross section is rectangular, with the height $h=50$ nm, width $b=100$ nm and length $L=1000$ nm. The concentrated force $P=2$ nN and the location parameter $k$ are set as 0.2 or 0.7.
cpl-34-9-096201-fig2.png
Fig. 2. Deflection and internal forces of a simply supported beam. (a) Deflection curve with $k=0.2$, (b) deflection curve with $k=0.7$, (c) shear force diagram with $k=0.2$, (d) shear force diagram with $k=0.7$, (e) bending moment with $k=0.2$, and (f) bending moment with $k=0.7$.
As shown in Figs. 2(a) and 2(b), at the same position of the beam $k=0.2$ or 0.7, the deflection has different values corresponding to different values of residual surface stress. For example, when $k=0.2$, deflections at the middle point of the beam are $-$0.2421, $-$0.2980 and $-$0.3876 nm, corresponding to $\tau _0 =0.89$, 0, and $-$0.89 N/m, respectively. It is clearly shown that the positive residual surface stress can cause less deformation of the beam, and the negative one can make the beam deform stronger, indicating that the nanobeam sounds 'stiffer' when it has a positive residual surface stress, and 'softer' when the residual surface stress is negative. This conclusion is in agreement with the previous results.[8,15] Let $x_{\rm d}$ represent the location where the deflection has the extreme value, i.e., ${w'}(x_{\rm d})=0$. For example, when $\tau _0>0$ and $k>0.5$, the extreme value condition leads to $x_{\rm d} =\frac{1}{r}{\rm \cosh}^{-1}\frac{({1-k})e^{kLr}( {e^{2Lr}-1})}{e^{2Lr}-e^{2kLr}}$. Then the relationship between $x_{\rm d}$ and the parameter $k$ can be demonstrated in Fig. 3(a). It is shown in the figure that at the same loading position, the value of $x_{\rm d}$ is different when the residual surface stress takes different values. This means that at the nanoscale, the dangerous section in stiffness check predicted by the surface elasticity theory is different from that by the conventional theory. Furthermore, the value of the maximum deflection $w_{\max}$ becomes different when the residual surface stress takes different values. Since we know that the criterion of stiffness check is $|{w_{\max}/L}|\leqslant[\delta]$, where $[\delta]$ is the tolerant relative displacement. When $k=0.7$, the maximum deflection $w_{\max}=-0.5435$, $-$0.4221 and $-$0.3442 nm, corresponding to $\tau _0=-0.89$, 0, and 0.89 N/m, respectively. If $[\delta]$ is taken as 4$\times10^{-4}$, when $\tau _0 =0.89$ N/m, the beam is safe. However, when $\tau _0 =-0.89$ N/m and 0, the beam is dangerous. This conclusion reemphasizes the necessity of adopting the surface elasticity theory when conducting the stiff check of nanobeams.
cpl-34-9-096201-fig3.png
Fig. 3. Location of the section position with maximum defection (a) or shear force (b) versus the position parameter $k$.
For the shear force diagram, each segment calculated by the conventional beam theory is a horizontal line. However, for the nanobeam, the segment predicted by the surface elasticity theory is normally nonlinear, as shown in Figs. 2(c) and 2(d). The position of the maximum value of the shear force, i.e., the dangerous section is different corresponding to different values of the residual surface stress. Let $x_{\rm s}$ represent the location where the shear force has an extreme value, i.e., $w^{(4)}(x_{\rm s})=0$. In the range of $0 < x < L_{1}$, the extreme value condition yields $\sinh (rx)=0$, and its solution is $x=0$. The further results show that $w^{(5)} < 0$ when $\tau_0>0$, and $w^{(5)}>0$ when $\tau _0 < 0$, indicating that the shear force curve in this segment is concave and convex, respectively. As a consequence, the maximum points of the shear force curve corresponding to different values of the residual surface stress are different. The dependence relationship between the maximum value position $x_{\rm s}$ and the parameter $k$ is shown in Fig. 3(b). It can be seen that in the range of $AC$ segment, when $\tau _0>0$, the dangerous section is located at the loading point, thus the curve is linear; when $\tau _0 < 0$, the dangerous section is at point $A$, and the function curve includes two horizontal lines; when $\tau _0=0$, the function is enclosed by the above-mentioned two curves. It is also shown that the maximum shear forces $Q_{\max}$ for different values of the residual surface stress are different. This indicates that when performing the strength check, the dangerous section and dangerous point at nanoscale should be reconsidered. For the bending moment diagram in Figs. 2(e) and 2(f), although the increasing trends of the three curves are the same, the case is more complicated for a nanobeam than a macroscopic beam. For the nanobeam, each segment of the curve is nonlinear, and not a straight line anymore. Similar to the previous results,[8,15] the negative residual surface stress can enhance the deformation of the beam, making it become softer, vice versa for the positive residual surface stress. For the first segment of the bending moment diagram, the extreme value condition yields $\cosh(rx_{m})=0$, where $x_{m}$ is the coordinate of the extreme point. However, this equation has no solution, meaning that the curve is monotonic, and the dangerous section is just located at the loading point. That is to say, the position of the dangerous section on the bending moment does not depend on the residual surface stress. Although the dangerous section does not alter no matter whether the residual surface stress is considered or not, the maximum values of the bending moment on this section for different values of the residual surface stress are different. However, it should be mentioned that we only concentrate on the present model, i.e., the simply supported beam with a concentrated force. In fact, for more examples, such as a beam with different configurations, different boundary conditions and experiencing complex loads, maybe we cannot draw the same conclusion. This again stresses the idea that at nanoscale, the strength check on elastic structures should be carefully noted. In summary, we have extensively studied the surface effect on the deflection and internal forces of the beam at nanoscale. A simply supported beam subjected to a concentrated force is taken as an example, and the distributed force induced by the residual surface stress is also applied on the beam. The solution of the beam deflection indicates that our model can naturally reduce to the conventional result, meaning that it is consistent with the classical theory. From the curves of deflection, shear force and bending moment, it can be seen that the nanobeam with a positive residual surface stress looks stiffer, and the one with a negative residual surface stress sounds softer; this conclusion matches well with the previous results. For the shear force and bending moment diagrams of the nanobeam, the curves are mostly nonlinear, which differs from the straight lines on the macroscopic beam. We also affirm that when performing the stiffness and strength check of the nanobeam, the dangerous sections on the deflection curve and shear force curve are different from those calculated by the macroscopic beam model. For the simply supported nanobeam, the dangerous section of the bending moment is not dependent upon the residual surface stress. For the nanobeam, all of the maximum values of the deflection, shear force and bending moment are different from those predicted by the conventional beam theory. These analyses provide a full understanding of stiffness and strength check of the nanobeam, which may be helpful to engineer better new-types of materials and devices at the micro and nanoscale. The obtained results can also give some inspirations on the education innovation in mechanics of materials. References High Performance Silicon Nanowire Field Effect TransistorsA review on applications of carbon nanotubes and graphenes as nano-resonator sensorsSurface Energy of Nanostructural Materials with Negative Curvature and Related Size EffectsSurface effects on elastic properties of silver nanowires: Contact atomic-force microscopySize Dependence of Young’s Modulus in ZnO NanowiresSurface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopyLarge displacement of a static bending nanowire with surface effectsSurface Effect on the Elastic Behavior of Static Bending NanowiresDeflections of Nanowires with Consideration of Surface EffectsTimoshenko beam model for static bending of nanowires with surface effectsSurface effects on buckling of nanowires under uniaxial compressionTimoshenko beam model for buckling and vibration of nanowires with surface effectsTowards Understanding Why the Thin Membrane Transducer Deforms: Surface Stress-Induced BucklingStiction of a Nano-Beam with Surface EffectEffects of surface elasticity and residual surface tension on the natural frequency of microbeamsSurface stress effect on bending resonance of nanowires with different boundary conditionsNonlinear free vibration of a cantilever nanobeam with surface effects: Semi-analytical solutionsMolecular Dynamics Simulations about Adsorption and Displacement of Methane in Carbon NanochannelsSurface and interface stress effects in thin films
[1] Cui Y, Zhong Z H, Wang D L, Wang W U and Lieber C M 2003 Nano Lett. 3 149
[2] Wang Q and Arash B 2014 Comput. Mater. Sci. 82 350
[3] Ouyang G, Wang C X and Yang G W 2009 Chem. Rev. 109 4221
[4] Jing G Y, Duan H L, Sun X M, Zhang Z S, Xu J, Li Y D, Wang J X and Yu D P 2006 Phys. Rev. B 73 235409
[5] Chen C Q, Shi Y, Zhang Y S, Zhu J and Yan Y J 2006 Phys. Rev. Lett. 96 075505
[6] Cuenot S, Fretigny C, Demoustier-Champagne S and Nysten B 2004 Phys. Rev. B 69 165410
[7] Liu J L, Mei Y, Xia R and Zhu W L 2012 Physica E 44 2050
[8] He J and Lilley C M 2008 Nano Lett. 8 1798
[9] Li H, Yang Z, Zhang Y M and Wen B C 2010 Chin. Phys. Lett. 27 126201
[10] Jiang L Y and Yan Z 2010 Physica E 42 2274
[11] Wang G F and Feng X Q 2009 Appl. Phys. Lett. 94 141913
[12] Wang G F and Feng X Q 2009 J. Phys. D 42 155411
[13] Liu J L, Sun J and Zuo P C 2016 Acta Mech. Solida Sin. 29 192
[14] Liu J L, Xia R and Zhou Y T 2011 Chin. Phys. Lett. 28 116201
[15] Wang G F and Feng X Q 2007 Appl. Phys. Lett. 90 231904
[16] He J and Lilley C M 2008 Appl. Phys. Lett. 93 263108
[17] Zhao D M, Liu J L and Wang L 2016 Int. J. Mech. Sci. 113 184
[18]Quan J and Yang X Y 2015 Acta Phys. Sin. 64 116201 (in Chinese)
[19]Lin Y, Peng J, Liu Y H, Xu Z H, Shan D B and Guo B 2014 Acta Phys. Sin. 63 016201 (in Chinese)
[20] Wu H A, Chen J and Liu H 2015 J. Phys. Chem. C 119 13652
[21]Gurtin M E and Murdoch A I 1975 Arch. Ration. Mechan. Anal. 57 291
[22] Cammarata R C 1994 Prog. Surf. Sci. 46 1