INVESTIGATION ON THE INTERFACIAL ADHESION OF A CYLINDRICAL FIBRILLAR ON A SUBSTRATE UNDER THE COUPLING EFFECT OF TENSION AND TORQUE
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摘要: 受壁虎刚毛可逆黏附性能的启发, 本文建立了单根弹性圆柱纤维与刚性基底黏附接触的理论和数值模型, 同时考虑了拉伸和扭转载荷的耦合作用及纤维半径对界面黏附性能的影响. 研究发现耦合载荷作用下柱形纤维同样存在一个临界半径, 当纤维半径小于该临界尺寸时, 界面应力达到均匀的理论强度分布, 接触边界应力集中消失, 出现缺陷不敏感现象; 当纤维半径大于该临界尺寸时, 界面以裂纹扩展而失效. 在耦合载荷作用下纤维的临界半径小于纯拉伸而大于纯扭转时的临界尺寸, 且该临界半径随着施加扭转载荷的增大而减小. 表明在纯拉伸载荷下使界面黏附强度达到最优的柱形纤维, 在拉伸和扭转载荷耦合作用下, 由于界面失效形式的转变使界面易发生脱黏, 并且界面脱黏时的拉脱力随着扭转载荷的增大而减小, 理论和数值结果一致. 本文结果进一步应用揭示了壁虎可以通过调控施加在其最小黏附单元上的载荷形式实现纯拉伸载荷下强黏附及耦合载荷下易脱黏的力学机制.Abstract: Inspired by the reversible adhesion of gecko seta, theoretical and numerical models of an elastic cylindrical fibrillar on a rigid substrate are established in the present paper, in which the coupling effect of tension and torque as well as the radius of the fibrillar on the interfacial adhesion is considered. It is found that when the fibrillar is subjected to the coupling effect of tension and torque, there also exists a critical radius of the fibrillar, below which the interfacial stress can reach the theoretical strength uniformly and the stress concentration at the contact edge vanishes, leading to the phenomenon of flaw insensitivity. When the radius of the fibrillar is larger than the critical value, the detachment of the interface between the fibrillar and substrate occurs in the form of crack propagation. The critical radius of the fibrillar under the coupling effect of tension and torque decreases with increasing the applied torque, but it is smaller than the case of pure tension and is larger than the case of pure torque. It can be concluded that if a fibrillar can reach the optimal theoretical adhesion strength under the pure tensile load, it could be easily detached under the coupling effect of tension and torque due to the different interface failure modes under different loading forms. The pull-off force decreases with the increase of the applied torsion load. The theoretical results are well consistent with the numerical ones. The obtained results in the present paper can be further applied to disclose the mechanical mechanism of the reversible adhesion of gecko seta through applying pure tension to achieve strong attachment and applying coupling effect of tension and torque to achieve easy detachment.
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引 言
自然界生物经过亿万年的进化, 形成各种复杂的生命系统, 获得众多特异的功能以适应它们的生存环境[1-3]. 从自然界汲取经验和灵感, 是现代科技创新和工程难题解决的重要方法[4-5], 目前军民领域应用的很多科技均是通过仿生获得. 在众多的仿生研究中, 壁虎等一类生物超强的黏附爬行能力吸引了研究人员的广泛兴趣. 壁虎不仅具有超强的黏附能力, 而且能从表面随意脱黏, 且其黏附性能几乎不受任何环境和材料性质的影响. 对壁虎脚掌的强黏附和易脱黏性能进行仿生研究, 不仅对设计新型攀爬装备及发展新型黏附材料具有重要意义, 而且在生物医学、可穿戴柔性器件、太空碎片抓捕等领域具有重要的应用价值, 同时能发展表/界面黏附接触力学.
壁虎脚掌优异的黏附性能很早就受到了研究人员的关注, 直到最近20年, 随着观测和测量设备及实验技术的发展, 壁虎黏附系统的宏微观结构及黏附机理才得以揭示. 2000年, 美国科学家Autumn等[6-8]利用双悬臂微机电系统首次实验测量了壁虎单根刚毛的切向和法向黏附力, 并否定了之前人们假想的吸盘、机械啮合、黏液等为壁虎黏附机制的假说, 侧面验证了壁虎黏附能力主要来源为范德华力[6]. 随后, 他们进一步开展了壁虎脚掌和单根刚毛在不同材料表面黏附力的测量实验, 给出了壁虎黏附机理为范德华力的直接实验证据. 范德华力是普遍存在于任何接触物体界面间非常微弱的中性分子力, 它如何能支撑壁虎体重？这主要得益于壁虎脚掌精细的微观黏附组织. 壁虎脚掌的黏附系统是一种多分级、多纤维状结构, 每个脚趾生有数百万根倾斜排列微米尺度的刚毛, 每根刚毛末端进一步分叉为成千上百根纳米尺度的绒毛[8]. 壁虎脚掌共数十亿根纳米绒毛足以积累大量范德华力产生惊人的黏附力. Arzt 等[9]实验观察比较了一系列具有强黏附性能昆虫和动物黏附系统的微观结构, 发现不同生物最小黏附纤维尺寸从微米到纳米不等, 并且生物体重越大, 其黏附系统结构越精细. 壁虎由于体重最大, 黏附系统结构最为精细, 黏附能力最强, 因而成为人们主要的仿生研究对象. Stewart等[10]实验比较了死壁虎和活体壁虎的黏附力, 发现二者基本一致, 更进一步证实了壁虎黏附力主要源自与生物生命特征无关的范德华力. 另有实验发现, 环境湿度产生的毛细力[11-12]及接触界面产生的静电力[13-14]对壁虎黏附亦有重要影响. 为了澄清壁虎如何适应其复杂多变的生存环境, 研究人员分别实验研究了基底粗糙度[15-16]、环境相对湿度[12, 17]、环境温度[18-19]、表面灰尘[20-21]等因素对壁虎黏附行为的影响.
基于壁虎黏附组织的宏微观特征, 目前在微结构阵列黏附表面的仿生制备方面同样取得了显著进展. Geim等[22]仿壁虎黏附组织的刚毛结构, 首次利用电子光束刻蚀和氧离子处理制备了微米尺度的高弹聚酰亚胺柱状阵列表面, 每平方厘米面积可负重3 N, 远高于相同材料的光滑表面黏附强度. del Campo等[23]利用不同实验技术制备出末端形状不同的PDMS柱状纤维阵列, 研究了接触纤维末端形状对界面黏附的影响. Murphy等[24]仿生制备了多级柱状纤维阵列表面, 发现多级结构能显著提高界面黏附性能. Hassan等[25]发现将一个大尺寸柱状阵列表面分成多个阵列表面同时与基底黏附, 总黏附力得到提高, 并且考虑了细分数目对界面黏附性能的影响. 还有学者制备了微柱阵列仿生黏附表面, 开展了该表面的摩擦和黏附性能[26]、自清洁性能[27]等方面的研究[28]. 研究人员注意到将柱状纤维末端修饰为蘑菇形能显著提高界面黏附性能[29-31], 并基于蘑菇形纤维阵列表面设计了多种黏附抓手[30-31]. 为了实现界面黏附性能的调控, 目前主要通过制备非对称纤维阵列[32-33]或引入外场(例如温度场[34]、磁场[35]、电场[36]、光场[37]等)通过改变实际接触面积实现界面黏附性能的调控.
以黏附接触力学和表/界面力学为基础, 目前已发展了不同尺度的理论模型来揭示壁虎等生物的黏附机制及主要影响因素. 宏观方面, 研究人员注意到壁虎脚掌的黏附组织整体表现出材料各向异性及梯度变化特征, 建立了各向异性和梯度材料的黏附接触力学模型, 理论揭示了壁虎利用材料各向异性实现宏观可逆黏附的力学机制[38], 给出了材料梯度特征对界面黏附的影响机制[39]. 微观方面, 研究人员首先从壁虎单根黏附纤维出发分析其界面黏附行为, Gao等[40]建立了拉伸载荷下单根弹性圆柱与刚性基底接触的界面黏附模型, 得到了界面缺陷不敏感的临界圆柱半径, 解释了壁虎单根黏附纤维的尺寸优化特性. Chen等[41]理论分析了扭转载荷下单根圆柱纤维界面缺陷不敏感问题. Spolenak等[42]研究了柱状纤维末端形状对界面黏附性能的影响. Arzt等[8]提出了接触细化模型, 发现将一个大尺度纤维末端细化为多个小尺度纤维与基底接触, 能显著提高界面黏附力. Yao等[43]建立了自相似多分级纤维结构的界面黏附模型, 发现总黏附力随纤维级数增加成指数增长. Hui等[44-45]对多纤维柱状阵列表面的黏附行为开展了相关研究, 揭示了纤维尺寸对界面黏附的影响. Chen等[46]理论分析柱形纤维阵列表面黏附性能的优化问题. Balijepalli等[47]和Luo等[48]数值研究了由软硬性质不同的双材料组成的柱状复合材料纤维(靠近界面材料较软, 远离界面材料较硬)的界面黏附性能, 发现复合材料纤维界面黏附强度高于单一材料纤维. He等[49]分析了倾斜柱状纤维与基底间的黏附行为, 主要考虑了界面黏附性能的方向依赖性. 考虑到壁虎最小黏附单元的真实形状类似于有限尺寸纳米薄膜, Chen等[50-51]基于经典Kendall撕脱模型, 理论研究了多级薄膜结构及预应力对壁虎黏附性能的影响. Tian等[8]提出了一种摩擦黏附模型, 得到了仿生薄膜界面黏附力与撕脱角的定量关系. Peng等[52-55]以壁虎最小黏附纤维为仿生对象, 建立了一系列有限尺寸仿生纳米薄膜与基底黏附接触的理论模型, 揭示了壁虎可逆黏附的微观力学机理及主要影响因素.
由以上研究可以看出, 目前对壁虎脚掌黏附系统的微观结构、黏附的宏微观力学机理方面已开展系统的实验和理论研究. 在仿壁虎黏附功能表面制备方面, 多采用柱状纤维阵列表面仿生壁虎黏附系统结构[56]. 纤维阵列表面的黏附行为与单根纤维的黏附性能密切相关. 目前, 对单根柱形纤维黏附性能研究相对较少, 且仅考虑了柱形纤维在单一拉伸或扭转载荷下界面黏附的尺寸效应[40-41]. 实验观察到壁虎在脱黏过程中脚掌伴随着扭转动作, 表明壁虎刚毛同时受到拉伸和扭转载荷的作用. Kang等[57]实验制备了末端为蘑菇型的仿生纤维阵列表面, 发现扭转载荷有利于界面脱黏. 然而, 目前对仿生黏附纤维在拉伸和扭转载荷耦合作用下的界面黏附机理仍不清楚. 在耦合载荷作用下, 柱形纤维是否仍存在使界面达到缺陷不敏感的临界尺寸？扭转载荷如何影响界面脱黏时的拉脱力？
针对以上存在的关键问题, 本文首先建立了单根圆柱形弹性纤维与刚性基底黏附接触的理论模型, 同时考虑拉伸和扭转载荷的耦合作用及纤维尺寸对界面黏附性能的影响. 进一步建立了与理论对应的数值模型, 并将理论结果和数值结果进行了对比. 最后, 应用本文结果揭示了壁虎脚掌最小黏附单元实现可逆黏附的力学机理.
1. 弹性圆柱纤维与刚性基底黏附接触的理论模型
弹性圆柱纤维与半无限大刚性基底黏附接触的力学模型如图1所示, 柱形纤维同时受拉伸载荷P和扭转载荷T的作用, 纤维半径为R. 考虑到实际接触过程中, 表面灰尘和表面粗糙度将引起界面缺陷, 假设真实界面接触半径为
$a = \alpha R$ ,$0 < \alpha < 1$ .$\alpha R < r < R$ 表示界面缺陷区. 因此, 该接触问题可以看作为圆柱纤维在拉伸和扭转载荷共同作用下接触区存在边界裂纹的界面断裂力学问题. Gao等[40]和Chen等[41]分别研究了圆柱纤维在单一拉伸和扭转载荷作用下界面黏附问题, 并分别用到了拉伸和扭转载荷作用下对应的I型裂纹和III型裂纹尖端应力强度因子![]()
图 1 弹性圆柱纤维与半无限大刚性基底黏附接触示意图. 纤维同时受到拉伸载荷P和扭转载荷TFigure 1. Schematic diagram of an elastic cylindrical fibrillar in contact with a rigid half-space substrate. The fibirllar is subjected to both tension P and torque T$$ {K_{\rm{I}}} = \frac{P}{{\sqrt {{\text{π}}{a^3}} }}{F_1}\left( \alpha \right) $$ (1) $$ {K_{{\rm{III}}}} = \frac{{2T}}{{\sqrt {{\text{π}}{a^5}} }}{F_3}\left( \alpha \right) $$ (2) 其中
$$ {F_1}\left( \alpha \right) = \sqrt {1 - \alpha } {G_1}\left( \alpha \right) $$ (3) $$ {F_3}\left( \alpha \right) = \sqrt {1 - \alpha } {G_3}\left( \alpha \right) $$ (4) $$ {G_1}\left( \alpha \right) = \frac{1}{2}\left( {1 + \frac{1}{2}\alpha + \frac{3}{8}{\alpha ^2} - 0.363{\alpha ^3} + 0.731{\alpha ^4}} \right) $$ (5) $$ {G_3}\left( \alpha \right) = \frac{3}{8}\left( {1 + \frac{1}{2}\alpha + \frac{3}{8}{\alpha ^2} + \frac{5}{{16}}{\alpha ^3} + \frac{{35}}{{128}}{\alpha ^4} + 0.208{\alpha ^5}} \right) $$ (6) 当圆柱纤维同时受到拉伸和扭转载荷时, 该界面裂纹属于I型和III型混合裂纹. 将式(1)和式(2)带入Griffith能量准则, 可得
$$ \frac{{K_{\rm{I}}^2}}{{2{E^*}}} + \frac{{K_{{\rm{III}}}^2}}{{4{\mu ^*}}} = \Delta \gamma $$ (7) 即
$$ \frac{{{P^2}F_1^2\left( \alpha \right)}}{{2{\text{π}}{\alpha ^3}{R^3}{E^*}}} + \frac{{{T^2}F_3^2\left( \alpha \right)}}{{{\text{π}}{\alpha ^5}{R^5}{\mu ^*}}} = \Delta \gamma $$ (8) 其中, 系数2和4是由刚性基底引入,
$\Delta \gamma = {\gamma _1} + {\gamma _2} - {\gamma _{12}}$ 为界面黏附能,${\gamma _1}$ 和${\gamma _2}$ 分别为纤维和基底的表面能,${\gamma _{12}}$ 为界面能,${E^*} = {E \mathord{\left/ {\vphantom {E {(1 - {\nu ^2})}}} \right. } {(1 - {\nu ^2})}}$ ,${\mu ^*} = {E \mathord{\left/ {\vphantom {E {[2(1 + \nu )}}} \right. } {[2(1 + \nu )}}]$ ,$E$ 和$\nu $ 分别是弹性圆柱纤维的弹性模量和泊松比.由式(8)可得界面失效时的拉脱力为
$$ P = \sqrt {\frac{{2{\text{π}}{\alpha ^3}{R^3}{E^*}\Delta \gamma }}{{F_1^2\left( \alpha \right)}} - \frac{{2{E^*}{T^2}F_3^2\left( \alpha \right)}}{{{\mu ^*}{\alpha ^2}{R^2}F_1^2\left( \alpha \right)}}} $$ (9) 根据断裂力学知识, 圆柱纤维在拉伸和扭转载荷作用下, 界面正应力和切应力在裂纹尖端处均存在应力奇异性[58]. 实际上, 界面失效时裂纹尖端应力并不会无穷大, 而是由其理论强度控制. 假设在耦合载荷作用下界面应力达到理论强度时正应力
$\sigma $ 和切应力$\tau $ 满足$$ \lambda {\text{ = }}{\left( {\frac{\sigma }{{{\sigma _0}}}} \right)^2} + {\left( {\frac{\tau }{{{\tau _0}}}} \right)^2} = 1 $$ (10) 其中,
$ {\sigma _0} $ 和$ {\tau _0} $ 分别为正应力和切应力的理论强度.类似于圆柱纤维受单一拉伸或扭转载荷情况, 当纤维半径小于某一临界尺寸时, 界面各点应力均匀达到其理论强度分布[40-41]. 假设在耦合载荷作用下, 同样存在一个临界纤维半径
${R_{{\text{cr}}}}$ , 当纤维半径小于该临界值时, 界面脱黏时刻各点应力能达到均匀理论强度分布, 即满足式(10). 此时可以分成以下两种情况讨论.(1) 当扭转载荷较小时, 界面各点的切应力与到圆心的距离成正比, 并可表示为
$$ \tau = \frac{r}{{\alpha R}}{\left. \tau \right|_{r = \alpha R}} $$ (11) 其中,
${\left. \tau \right|_{r = \alpha R}} = {{2 T} \mathord{\left/ {\vphantom {{2 T} {({\text{π}}{\alpha ^3}{R^3}) \leqslant {\tau _0}}}} \right. } {({\text{π}}{\alpha ^3}{R^3}) \leqslant {\tau _0}}}$ 为接触界面边缘的切应力. 由式(10)可得此时界面正应力的分布为$$ \sigma {\text{ = }}{\sigma _0}\sqrt {1 - {{\left( {\frac{{r{{\left. \tau \right|}_{r = \alpha R}}}}{{\alpha R{\tau _0}}}} \right)}^2}} $$ (12) 界面切应力和正应力的分布规律如图2(a)所示.
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图 2 当扭转载荷不同时, 接触界面各点应力达到理论强度时切应力和正应力的分布规律, 其中$ \alpha R $ 为接触界面半径, R为纤维半径Figure 2. The distribution of shear and normal stress at contact interface under different torsion loads when the interfacial stress reaches the theoretical strength uniformly,$ \alpha R $ is the radius of the contact interface and R is the radius of the fiber由式(12)可得界面各点均以理论强度失效时的拉脱力为
$$ \begin{split} & {P_c} = \int_0^{\alpha R} {2{\text{π}}r{\sigma _0}\sqrt {1 - {{\left( {\frac{\tau }{{{\tau _0}}}} \right)}^2}} {{\rm{d}}} r}= \\ &\qquad \frac{{{{\text{π}}^3}{\alpha ^8}{R^8}{\sigma _0}}}{{6{T^2}{\tau _0}}}\left[ {\tau _0^3 - {{\left( {\tau _0^2 - \frac{{4{T^2}}}{{{{\text{π}}^2}{\alpha ^6}{R^6}}}} \right)}^{\frac{3}{2}}}} \right],\;\;{\left. \tau \right|_{r = \alpha R}} \leqslant {\tau _0} \end{split} $$ (13) 根据式(9)和式(13), 当扭转载荷较小时, 界面各点应力达到均匀理论强度分布时的临界纤维半径可由下式得到
$$ \begin{split} & \frac{{{{\text{π}}^3}{\alpha ^8}{R^8}{\sigma _0}}}{{6{T^2}{\tau _0}}}\left[ {\tau _0^3 - {{\left( {\tau _0^2 - \frac{{4{T^2}}}{{{{\text{π}}^2}{\alpha ^6}{R^6}}}} \right)}^{\frac{3}{2}}}} \right]= \\ &\qquad \sqrt {\frac{{2{\text{π}}{\alpha ^3}{R^3}{E^*}\Delta \gamma }}{{F_1^2\left( \alpha \right)}} - \frac{{2{E^*}{T^2}F_3^2\left( \alpha \right)}}{{{\mu ^*}{\alpha ^2}{R^2}F_1^2\left( \alpha \right)}}} \end{split} $$ (14) (2) 当施加的扭转载荷增大到
$T = {{{\text{π}}{\alpha ^3}{R^3}{\tau _0}} \mathord{\left/ {\vphantom {{{\text{π}}{\alpha ^3}{R^3}{\tau _0}} 2}} \right. } 2}$ 时,$\;{\left. \tau \right|_{r = \alpha R}} = {\tau _0}$ . 随着扭转载荷的进一步增大, 接触界面边缘附近的切应力均达到其理论强度, 正应力为零, 如图2(b)所示. 假设在${a_1} \leqslant r \leqslant \alpha R$ 范围内,$ \tau = {\tau _0} $ , 在$0 < r < {a_1}$ 范围内,$ \tau = {{r{\tau _0}} \mathord{\left/ {\vphantom {{r{\tau _0}} {{a_1}}}} \right. } {{a_1}}} $ . 根据力矩平衡有$$ T = \int_0^{{a_1}} {2{\text{π}}r\left( {\frac{r}{{{a_1}}}{\tau _0}} \right)r{{\rm{d}}} r} + \int_{{a_1}}^{\alpha R} {2{\text{π}}r{\tau _0}r{{\rm{d}}} r} $$ (15) 由上式可得
$$ {a_1} = \sqrt[3]{{4{\alpha ^3}{R^3} - \frac{{6T}}{{{\text{π}}{\sigma _0}}}}} $$ (16) 根据式(12)可得此时界面以理论强度脱黏时的拉脱力为
$$\begin{split} & {P_c} = \int_0^{{a_1}} {2{\text{π}}r{\sigma _0}\sqrt {1 - {{\left( {\frac{\tau }{{{\tau _0}}}} \right)}^2}} {{\rm{d}}} r} {\text{ = }}\frac{{2{\text{π}}{\sigma _0}}}{3}{\left( {4{\alpha ^3}{R^3} - \frac{{6T}}{{{\text{π}}{\tau _0}}}} \right)^{\frac{2}{3}}}, \\ &\qquad\qquad\quad {\left. \tau \right|_{{a_1} \leqslant r \leqslant \alpha R}} = {\tau _0} \end{split} $$ (17) 根据式(9)和式(17), 当扭转载荷较大时, 界面各点应力达到均匀理论强度分布时的临界纤维半径可由下式得到
$$ \begin{split} &\frac{{2{\text{π}}{\sigma _0}}}{3}{\left( {4{\alpha ^3}{R^3} - \frac{{6T}}{{{\text{π}}{\tau _0}}}} \right)^{\frac{2}{3}}} = \\ &\qquad \sqrt {\frac{{2{\text{π}}{\alpha ^3}{R^3}{E^*}\Delta \gamma }}{{F_1^2\left( \alpha \right)}} - \frac{{2{E^*}{T^2}F_3^2\left( \alpha \right)}}{{{\mu ^*}{\alpha ^2}{R^2}F_1^2\left( \alpha \right)}}}\end{split} $$ (18) 根据公式(9)、式(13)和式(17)可得随着圆柱纤维半径的变化界面脱黏时的拉脱力为
$$ P = \left\{ \begin{split} & \sqrt {\frac{{2{\text{π}}{\alpha ^3}{R^3}{E^*}\Delta \gamma }}{{F_1^2\left( \alpha \right)}} - \frac{{2{E^*}{T^2}F_3^2\left( \alpha \right)}}{{{\mu ^*}{\alpha ^2}{R^2}F_1^2\left( \alpha \right)}}} ,{\text{ }}R > {R_{{\text{cr}}}} \\ & \frac{{{{\text{π}}^3}{\alpha ^8}{R^8}{\sigma _0}}}{{6{T^2}{\tau _0}}}\left[ {\tau _0^3 - {{\left( {\tau _0^2 - \frac{{4{T^2}}}{{{{\text{π}}^2}{\alpha ^6}{R^6}}}} \right)}^{\frac{3}{2}}}} \right],\\ &\qquad{\text{ }}R \leqslant {R_{{\text{cr}}}}{\text{ and }}{\left. \tau \right|_{r = \alpha R}} \leqslant {\tau _0} \\ & \frac{{2{\text{π}}{\sigma _0}}}{3}{\left( {4{\alpha ^3}{R^3} - \frac{{6T}}{{{\text{π}}{\tau _0}}}} \right)^{\frac{2}{3}}},{\text{ }}R \leqslant {R_{{\text{cr}}}}{\text{ and }}{\left. \tau \right|_{{a_1} \leqslant r \leqslant \alpha R}} = {\tau _0} \end{split} \right. $$ (19) 对上式进一步进行无量纲化可得
$$ \frac{P}{{{\text{π}}{R^2}{\sigma _0}}} = \left\{ \begin{gathered} \sqrt {\frac{{2{\alpha ^3}}}{{{\text{π}}F_1^2\left( \alpha \right)}} \cdot \frac{{{E^*}\Delta \gamma }}{{R\sigma _0^2}} - \frac{{8F_3^2\left( \alpha \right)}}{{9{\alpha ^2}F_1^2\left( \alpha \right)}} \cdot {{\left( {\frac{T}{{{T_0}}}} \right)}^2} \cdot \frac{{{E^*}}}{{{\mu ^*}}} \cdot {{\left( {\frac{{{\tau _0}}}{{{\sigma _0}}}} \right)}^2}} ,{\text{ }}R > {R_{{\text{cr}}}} \\ \frac{{3{\alpha ^8}}}{8} \cdot {\left( {\frac{{{T_0}}}{T}} \right)^2}\left\{ {1 - {{\left[ {1 - \frac{{16}}{{9{\alpha ^6}}} \cdot {{\left( {\frac{T}{{{T_0}}}} \right)}^2}} \right]}^{\frac{3}{2}}}} \right\},{\text{ }}R \leqslant {R_{{\text{cr}}}}{\text{ and }}{\left. \tau \right|_{r = \alpha R}} \leqslant {\tau _0} \\ \frac{2}{3}{\left( {4{\alpha ^3} - 4\frac{T}{{{T_0}}}} \right)^{\frac{2}{3}}},{\text{ }}R \leqslant {R_{{\text{cr}}}}{\text{ and }}{\left. \tau \right|_{{a_1} \leqslant r \leqslant \alpha R}} = {\tau _0} \\ \end{gathered} \right. $$ (20) 其中,
${T_0} = \dfrac{{2{\text{π}}{R^3}}}{3}{\tau _0}$ .2. 结果与讨论
图3表示在不同
$\alpha $ 和不同扭矩$ {T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} $ 时无量纲拉脱力随无量纲纤维半径的变化规律, 其他参数取值为$ {{{\tau _0}} \mathord{\left/ {\vphantom {{{\tau _0}} {{\sigma _0}}}} \right. } {{\sigma _0}}} = 1 $ ,$ {{{E^*}} \mathord{\left/ {\vphantom {{{E^*}} {{\mu ^*}}}} \right. } {{\mu ^*}}} = 2.67 $ . 由图3可知, 当纤维半径减小到某临界尺寸${R_{{\rm{cr}}}}$ 时, 无量纲拉脱力保持不变且与纤维半径无关, 表明当圆柱纤维半径小于该临界尺寸时, 接触界面各点应力均达到其理论强度分布, 即出现缺陷不敏感现象. 当纤维半径大于该临界尺寸时, 界面脱黏以I型和III型混合裂纹扩展而逐渐失效, 无量纲拉脱力随着纤维半径的增大而减小. 在一定的扭转载荷$ {T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} $ 或$\alpha $ 下, 当纤维半径增大到另一临界半径${R'_{{\rm{cr}}}}$ 时, 无量纲拉脱力减小为零, 说明此时扭转载荷已经使界面失效, 界面脱黏时无需拉伸载荷的作用. 当纤维半径不变时, 界面脱黏时的拉脱力随着扭转载荷的增大而减小, 当扭转载荷增大到某临界值(${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.34$ )时, 无论纤维半径多大, 界面脱黏时的拉脱力基本为零, 如图3(b). 由图3可知扭转载荷有利于界面脱黏, 当柱形纤维同时受到拉伸和扭转载荷时, 界面脱黏时的拉脱力总是小于单一拉伸载荷($ {T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0 $ )下的拉脱力, 本文理论结果与Kang等[57]的实验结果一致. 在一定的扭转载荷下, 当纤维半径$R \geqslant {R'_{{\rm{cr}}}}$ 时, 界面脱黏以III型裂纹扩展而失效, 说明此时无需拉伸载荷, 扭转载荷已经使界面脱黏; 当${R_{{\rm{cr}}}} < R < {R'_{{\rm{cr}}}}$ 时, 界面脱黏以I型和III型混合裂纹扩展而失效; 当$R \leqslant {R_{{\rm{cr}}}}$ 时, 裂纹尖端应力集中消失, 界面各点应力均达到其理论强度, 此时界面黏附性能最优.![]()
图 3 (a)不同$\alpha $ 和(b)不同扭矩${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} $ 时无量纲拉脱力随无量纲纤维半径的变化规律Figure 3. The relationship between the dimensionless pull-off force and the dimensionless fiber radius with (a) different$\alpha $ and (b) different torque${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} $ 柱形纤维在耦合载荷作用下产生缺陷不敏感的临界半径随施加扭矩大小的变化规律如图4所示. 可以发现, 扭转载荷对柱形纤维缺陷不敏感的临界半径具有重要影响. 在耦合载荷作用下, 柱形纤维的临界半径随施加扭矩的增大而减小, 该临界半径小于纤维受单一拉伸载荷的临界尺寸而大于受单一扭转载荷下的临界尺寸. 由图4可知, 对于某一特定尺寸的柱形纤维, 如果纤维半径刚好满足在纯拉伸载荷下缺陷不敏感尺寸, 在单一拉伸载荷下, 界面各点应力达到均匀的理论强度分布, 界面黏附性能最优; 当同时施加拉伸和扭转载荷时, 由于柱形纤维在耦合载荷下的缺陷不敏感临界半径小于单一拉伸载荷下临界尺寸, 在耦合载荷作用下, 界面脱黏转变为以裂纹扩展而失效, 此时界面脱黏时的拉脱力小于在单一拉伸载荷下拉脱力, 并且该拉脱力随着扭转载荷的增大而减小. 因此, 我们可以通过改变施加在柱形纤维上的载荷形式, 实现单一拉伸载荷下强黏附和耦合载荷下易脱黏的界面黏附性能的调控.
为了验证上述理论, 我们进一步采用商业有限元软件(ABAQUS 2020)建立了与理论模型对应的数值模型. 圆柱纤维采用线弹性材料, 弹性模量
$ {\text{ }}E = {\text{2 GPa}} $ , 泊松比$ \nu = 0.25 $ . 弹性纤维与刚性基底间的相互作用通过内聚力模型表征. 界面内聚力模型的牵引力−分离位移关系采用如图5所示的梯形形式表示, 其中,$ \lambda = {\left( {{\sigma \mathord{\left/ {\vphantom {\sigma {{\sigma _0}}}} \right. } {{\sigma _0}}}} \right)^2} + {\left( {{\tau \mathord{\left/ {\vphantom {\tau {{\tau _0}}}} \right. } {{\tau _0}}}} \right)^2} $ 表示界面混合应力,$ \sigma $ 和$ \tau $ 分别为界面正应力和切应力,$ {\sigma _0} $ 和$ {\tau _0} $ 分别为正应力和切应力的理论强度. 当$ \lambda {\text{ = }}1 $ 时界面应力达到其理论强度. 假设界面相互作用为分子间的范德华力, 因此内聚力模型的主要参数取值为:${\sigma _0} = {\tau _0} = 20{\text{ MPa}}$ , 界面黏附能$\Delta \gamma = 0.009\;9{\text{ J/}}{{\text{m}}^2}$ (梯形区域的面积),$ {\delta _1} = 0.005{\text{ nm}} $ ,$ {\delta _2} = 0.495{\text{ nm}} $ , 最大分离位移${\delta _{\text{c}}} = 0.5{\text{ nm}}$ . 柱形纤维的顶端同时受到拉伸和扭转载荷, 弹性纤维采用C3D8R单元, 界面层选用COH3D8内聚力单元. 假设界面实际接触面积约为纤维横截面的50%, 即$ \alpha {\text{ = }}0.7 $ .![]()
图 4 耦合载荷下柱形纤维缺陷不敏感的临界半径随扭矩的变化规律Figure 4. Variation of the critical radius of the fiber at flaw insensitivity with the torque under the coupling load图6表示当扭转载荷
${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.15$ 时, 半径分别为20 nm, 140 nm和200 nm的柱形纤维在拉伸和扭转载荷共同作用下界面脱黏时的应力分布云图. 可以看出, 随着纤维半径的减小, 接触区的界面应力逐渐趋于均匀分布. 当纤维半径R = 200 nm时, 界面失效时接触区边界处应力最大, 存在明显的应力集中; 当纤维半径减小到R = 140 nm时, 界面应力达到理论强度的区域增大, 接触区中心的应力提高, 整个接触区各点界面应力差异减小; 当纤维半径R = 20 nm时, 界面各点应力达到均匀的理论强度分布, 应力集中消失, 产生缺陷不敏感现象.图7为拉脱力仿真值与理论值的对比结果, 其中图7(a)表示柱形纤维在纯拉伸载荷
$({T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0) $ 及拉伸和扭转耦合载荷$({T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.15) $ 作用下的无量纲拉脱力与无量纲纤维半径关系的数值和理论结果对比. 可以看出, 无论是理论分析还是数值模拟均发现柱形纤维存在一个临界半径, 当纤维半径小于该临界尺寸时, 界面各点应力均能达到理论强度分布, 此时无量纲拉脱力保持不变且与纤维半径无关; 当纤维半径大于该临界尺寸时, 裂纹尖端存在显著的应力集中, 界面脱黏以裂纹扩展而失效. 理论和数值结果均表明当纤维受到拉伸和扭转耦合载荷时, 界面脱黏时的拉脱力明显小于纯拉伸时的界面拉脱力, 表明扭转有利于界面脱黏. 当纤维半径一定时, 耦合载荷下界面脱黏时的拉脱力随施加扭矩的变化关系如图7(b) 所示, 数值结果和理论结果一致. 由图7(a)可以看出, 理论和有限元模型预测的临界纤维半径在数值上存在一定的差异, 这主要因为在理论分析中采用宏观条件下的格里菲斯断裂准则, 而数值模拟采用的界面内聚力模型, 两者采取界面失效准则不同. 另外, 有限尺寸的柱形纤维与宏观裂纹也存在一定的差别.上述分析结果可以进一步应用揭示壁虎脚掌最小黏附单元实现可逆黏附的力学机制. 将壁虎的最小黏附单元看成弹性圆柱形纤维[40], 取壁虎刚毛及界面范德华力相互作用的参数:
$ {\sigma _0} = {\tau _0}{\text{ = 20 MPa}} $ ,$ \Delta \gamma {\text{ = 0}}{\text{.01 J/}}{{\text{m}}^2} $ ,${\delta _{\text{c}}} = 0.5{\text{ nm}}$ ,$ E = {\text{2 GPa}} $ ,$ \nu = 0.25 $ . 由式(13)可得当圆柱纤维受纯拉伸载荷(T = 0)时出现界面缺陷不敏感的临界半径为${R_{{\rm{cr}}}} \approx 257{\text{ nm}}$ , 该尺寸与壁虎最小黏附单元的真实尺寸(200 ~ 300 nm)[6]非常接近, 说明在纯拉伸载荷下壁虎最小黏附单元脱黏时界面各点应力均能达到理论强度分布, 产生界面缺陷不敏感现象, 此时界面黏附强度最优. 由于柱形纤维在耦合载荷下缺陷不敏感临界半径明显小于单一拉伸载荷下的临界尺寸(图4), 当壁虎最小黏附单元同时受到拉伸和扭转载荷时, 不再满足界面缺陷不敏感条件, 界面裂纹尖端处存在应力集中, 界面脱黏将以裂纹扩展而失效, 并且界面拉脱力随着扭转载荷的增大而减小(图7), 说明扭转载荷有利于壁虎脱黏. 因此, 壁虎可以通过调节施加在其最小黏附单元上的载荷形式实现单一拉伸载荷下强黏附及耦合载荷下易脱黏的可逆黏附行为.![]()
图 6 半径分别为20 nm, 140 nm和200 nm的柱形纤维在耦合载荷下拉脱时刻的界面应力$\lambda $ 分布云图, 此时施加的扭转载荷不变${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.15 $ Figure 6. The distribution of interfacial stress$\lambda $ at the moment of interface detachment under the coupling load with the radius of 20 nm, 140 nm and 200 nm, respectively, where the applied torque is constant${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.15 $ ![]()
图 7 数值结果与理论结果的对比. (a) 纯拉伸和耦合载荷下界面拉脱力与纤维半径关系; (b) 纤维半径一定时, 界面拉脱力随扭矩的变化关系Figure 7. Comparison of the numerical and theoretical results. (a) Variation of the pull-off force with the fiber radius under pure tension and coupling load of tension and torque; (b) Variation of the pull-off force with the torque as the fiber radius fixed3. 结论
本文首先建立了弹性圆柱纤维与刚性基底黏附接触的理论模型, 同时考虑了拉伸和扭转载荷的耦合作用及纤维尺寸对界面黏附性能的影响. 结果表明, 在耦合载荷作用下, 柱形纤维同样存在一临界半径, 当纤维半径小于该临界尺寸时, 界面应力达到均匀的理论强度分布, 出现界面缺陷不敏感现象; 当纤维半径大于该临界尺寸时, 界面脱黏以裂纹扩展而失效. 耦合载荷下纤维的临界半径小于单一拉伸载荷下的临界尺寸而大于单一扭转载荷下的临界尺寸, 并且纤维在耦合载荷下缺陷不敏感的临界半径随着扭矩的增大而减小. 当纤维半径不变时, 界面脱黏时的拉脱力随扭矩的增大而减小, 表明扭转有利于界面脱黏. 进一步建立了与理论对应的数值模型, 理论结果与数值结果一致. 通过对比不同载荷形式下的纤维临界半径, 揭示了壁虎可以通过调控施加在最小黏附单元上的载荷形式, 实现单一拉伸载荷下强黏附和耦合载荷下易脱黏的力学机制. 本文结果不仅能加深理解壁虎可逆黏附性能的微观力学机理, 揭示了壁虎最小黏附单元的尺寸优化特性, 而且能对设计新型黏附功能表面提供理论依据.
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图 1 弹性圆柱纤维与半无限大刚性基底黏附接触示意图. 纤维同时受到拉伸载荷P和扭转载荷T
Figure 1. Schematic diagram of an elastic cylindrical fibrillar in contact with a rigid half-space substrate. The fibirllar is subjected to both tension P and torque T
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图 2 当扭转载荷不同时, 接触界面各点应力达到理论强度时切应力和正应力的分布规律, 其中
$ \alpha R $ 为接触界面半径, R为纤维半径Figure 2. The distribution of shear and normal stress at contact interface under different torsion loads when the interfacial stress reaches the theoretical strength uniformly,
$ \alpha R $ is the radius of the contact interface and R is the radius of the fiber![]()
图 3 (a)不同
$\alpha $ 和(b)不同扭矩${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} $ 时无量纲拉脱力随无量纲纤维半径的变化规律Figure 3. The relationship between the dimensionless pull-off force and the dimensionless fiber radius with (a) different
$\alpha $ and (b) different torque${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} $ ![]()
图 4 耦合载荷下柱形纤维缺陷不敏感的临界半径随扭矩的变化规律
Figure 4. Variation of the critical radius of the fiber at flaw insensitivity with the torque under the coupling load
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图 6 半径分别为20 nm, 140 nm和200 nm的柱形纤维在耦合载荷下拉脱时刻的界面应力
$\lambda $ 分布云图, 此时施加的扭转载荷不变${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.15 $ Figure 6. The distribution of interfacial stress
$\lambda $ at the moment of interface detachment under the coupling load with the radius of 20 nm, 140 nm and 200 nm, respectively, where the applied torque is constant${T \mathord{\left/ {\vphantom {T {{T_0}}}} \right. } {{T_0}}} = 0.15 $ ![]()
图 7 数值结果与理论结果的对比. (a) 纯拉伸和耦合载荷下界面拉脱力与纤维半径关系; (b) 纤维半径一定时, 界面拉脱力随扭矩的变化关系
Figure 7. Comparison of the numerical and theoretical results. (a) Variation of the pull-off force with the fiber radius under pure tension and coupling load of tension and torque; (b) Variation of the pull-off force with the torque as the fiber radius fixed
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