长程力诱导软弹性固体表面不稳定模式的预拉伸控制
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摘要
本文的研究对象是一类以聚二甲基硅氧烷(PDMS)为代表的,弹性范围内可以承受较大变形的软弹性固体材料。这类材料普遍具有优良的材料性能,广泛应用于日常生活和科学研究各个领域。
我们所讨论的问题是由这类软弹性固体材料制成的衬底在外部作用下表面形貌的稳定性问题。外部作用可以分为两类:一类是外部接触体与软弹性固体表面的范德华相互作用;另一类是对软弹性固体本身施加的均匀预拉伸变形。前者能够促进表面小扰动的发展,而弹性变形引起的应变能却能抑制表面失稳。这两类外部作用相互竞争可以产生稳定有序的表面形貌。我们通过线形稳定性分析,研究了有限变形软弹性固体在范德华力作用下的表面稳定性问题,并探讨了通过施加不同类型的有限变形调控软弹性固体表面失稳的具体形貌。
应用非线形弹性力学理论,选择有限变形后的构形为参考构形,我们推导了在此构形下扰动变形场的平衡方程与边界条件。详细讨论了这类材料的应力应变关系,并进一步建立了计入一般有限变形效应,应力张量与扰动位移的关系式。
首先,考虑了一个较为简单的二维平面应变问题,预先施加等比双向拉伸的软弹性固体薄膜在其表面与一刚性接触体存在范德华力作用下的稳定性问题。经过线性稳定性分析,我们揭示了薄膜失稳的临界载荷以及临界波长与预拉伸程度之间的关系。临界载荷与临界波数随着拉伸的主伸长比增加而变大,因而我们可以通过对软弹性薄膜进行等比双向预拉伸来调控表面失稳的形貌。
接着,我们研究了经过单向预拉伸的软弹性薄膜,在范德华力作用下的形貌失稳的临界条件以及失稳的具体形貌。经过单向拉伸的薄膜在参考构形中表现出面内各向异性,通过进行线性稳定性分析,我们发现单向预拉伸可以引发薄膜表面形成平行于拉伸方向的条纹状失稳图样。同时,我们还可以通过增大预拉伸的主伸长比来减小条纹状图样的间隔
接着,我们对前面所讨论的问题做进一步的拓展,考虑更加一般的情况,分析经过任意均匀有限变形的薄膜的形貌稳定性。对各向同性不可压缩软弹性薄膜,预先在面内两个垂直方向上分别均匀变形。通过线性稳定性分析,我们得到载荷参数的表达式,并进一步讨论薄膜形貌失稳的临界载荷以及临界波长,揭示了描述有限变形的参数,即三个方向的主拉伸比,对形貌失稳的临界条件以及失稳具体形貌的影响。并且我们利用一般有限变形的结果,分析了一个面内剪切变形的实例,验证已有的结论。
最后,我们对处于有限单向预拉伸软弹性衬底上的液体薄膜由于范德华力驱动而产生的失稳进行了线性稳定性分析。通过采用润滑理论近似,我们推导出液体表面扰动生长指数的具体表达式。相应地,我们还得到了失稳的判据,也详细讨论了扰动的最不稳定的模式。通过具体的数值运算,我们发现对衬底进行预拉伸可以对液体薄膜在拉伸方向起到抑制失稳的作用,而在其他方向却使薄膜更容易失稳。液体膜会在垂直于拉伸方向发生周期性失稳,而失稳的波长与衬底的拉伸程度相关。因而对衬底的预拉伸致使液体薄膜发生各向异性的去浸润,而各向异性的程度可以通过改变主拉伸来调控,这一现象可以用来制备某些细长而有序的微结构。
We consider a kind of soft elastomers with extreme flexibility, taken Polydimethylsiloxane (PDMS) as a representative. Such soft elastomers are wildly used in both daily life and scientific research, due to their excellent material properties.
Here we focus on the surface stability of the soft elastomeric substrate with external interactions, which can be sorted into the van der Waals interaction between the surface of elastomer and a rigid contactor, and the uniformly pre-deformation of the elastomer itself. The former facilitates the perturbation on the surface, while the latter inhibits. As a result, it provides a promising way to fabricate well-ordered pattern on the elastomer surface. By a linear analysis, we show how the finite pre-deformation affects the surface morphology of the soft elastomers, and discuss the regulating of the pattern.
By applying the theory of nonlinear elasticity, we obtain the equilibrium equations and the boundary conditions in terms of the additional fluctuations in the finitely deformed state taken as the reference configuration. The stress-strain relations of such soft elastomers are discussed in detail, and the stress tensor with respect to the general finite deformation and additional disturbance is derived explicitly.
First of all, we study a simple two-dimensional problem under plane strain condition, viz. elastic buckling of an equi-biaxially pre-tensioned soft elastic film due to van der Waals interaction between the surface of film and a rigid contactor. By a linear stability analysis, we reveal the dependence of critical buckling pattern on the pre-tension. And with increasing the magnitude of pre-tension, the critical wave number and corresponding interaction parameter increase. Accordingly the buckling of elastic film can be regulated by equi-biaxial pre-tension.
Secondly, we consider the buckling of a uni-axially pre-tensioned soft elastic film due to van der Waals interaction. The uni-axial pre-tension leads to that the film material behaves anisotropically with respect to the reference configuration. It is shown that the film will buckle into stripe parallel to the tensile direction by a linear stability analysis. Moreover the spacing of the stripe pattern decreases with increasing the magnitude of the pre-tension.
Then we develop the foregoing discussion into a general case, which is the stability of an elastic film pre-deformed homogeneously due to van der Waals interaction. Consider the surface buckling of an isotropic incompressible soft film, which is deformed uniformly in two perpendicular directions respectively. The expression of the interaction parameter is deduced through a linear stability analysis. Furthermore, we investigate the criterion and pattern of the surface buckling, which are dependent on three principal extension ratios related to the finite pre-deformation. Besides, we take a soft elastomeric film with shear pre-deformation for instance, which is coincident with the general conclusion.
Finally, we study the instability of a thin liquid film resting on a uniformly stretched soft elastomeric substrate driven by van der Waals forces by a linear analysis. By adopting the lubrication approximation, the growth rate of a fluctuation of the liquid surface is derived. The corresponding instability criterion has been obtained, and the most unstable mode of the fluctuation has been discussed. It is shown that the pre-stretch stabilize the liquid film in the direction parallel to the tension while destabilize it in other directions. The liquid film is most unstable against the periodic fluctuation normal to the tensile direction with a special wavelength depend on the magnitude of the principal stretch. Therefore, the pre-stretch of the substrate leads to anisotropic dewetting of the thin liquid film, and the degree of anisotropy can be tuned by varying the principal stretch. This phenomenon may be used to create microstructures with elongated features.
我们所讨论的问题是由这类软弹性固体材料制成的衬底在外部作用下表面形貌的稳定性问题。外部作用可以分为两类:一类是外部接触体与软弹性固体表面的范德华相互作用;另一类是对软弹性固体本身施加的均匀预拉伸变形。前者能够促进表面小扰动的发展,而弹性变形引起的应变能却能抑制表面失稳。这两类外部作用相互竞争可以产生稳定有序的表面形貌。我们通过线形稳定性分析,研究了有限变形软弹性固体在范德华力作用下的表面稳定性问题,并探讨了通过施加不同类型的有限变形调控软弹性固体表面失稳的具体形貌。
应用非线形弹性力学理论,选择有限变形后的构形为参考构形,我们推导了在此构形下扰动变形场的平衡方程与边界条件。详细讨论了这类材料的应力应变关系,并进一步建立了计入一般有限变形效应,应力张量与扰动位移的关系式。
首先,考虑了一个较为简单的二维平面应变问题,预先施加等比双向拉伸的软弹性固体薄膜在其表面与一刚性接触体存在范德华力作用下的稳定性问题。经过线性稳定性分析,我们揭示了薄膜失稳的临界载荷以及临界波长与预拉伸程度之间的关系。临界载荷与临界波数随着拉伸的主伸长比增加而变大,因而我们可以通过对软弹性薄膜进行等比双向预拉伸来调控表面失稳的形貌。
接着,我们研究了经过单向预拉伸的软弹性薄膜,在范德华力作用下的形貌失稳的临界条件以及失稳的具体形貌。经过单向拉伸的薄膜在参考构形中表现出面内各向异性,通过进行线性稳定性分析,我们发现单向预拉伸可以引发薄膜表面形成平行于拉伸方向的条纹状失稳图样。同时,我们还可以通过增大预拉伸的主伸长比来减小条纹状图样的间隔
接着,我们对前面所讨论的问题做进一步的拓展,考虑更加一般的情况,分析经过任意均匀有限变形的薄膜的形貌稳定性。对各向同性不可压缩软弹性薄膜,预先在面内两个垂直方向上分别均匀变形。通过线性稳定性分析,我们得到载荷参数的表达式,并进一步讨论薄膜形貌失稳的临界载荷以及临界波长,揭示了描述有限变形的参数,即三个方向的主拉伸比,对形貌失稳的临界条件以及失稳具体形貌的影响。并且我们利用一般有限变形的结果,分析了一个面内剪切变形的实例,验证已有的结论。
最后,我们对处于有限单向预拉伸软弹性衬底上的液体薄膜由于范德华力驱动而产生的失稳进行了线性稳定性分析。通过采用润滑理论近似,我们推导出液体表面扰动生长指数的具体表达式。相应地,我们还得到了失稳的判据,也详细讨论了扰动的最不稳定的模式。通过具体的数值运算,我们发现对衬底进行预拉伸可以对液体薄膜在拉伸方向起到抑制失稳的作用,而在其他方向却使薄膜更容易失稳。液体膜会在垂直于拉伸方向发生周期性失稳,而失稳的波长与衬底的拉伸程度相关。因而对衬底的预拉伸致使液体薄膜发生各向异性的去浸润,而各向异性的程度可以通过改变主拉伸来调控,这一现象可以用来制备某些细长而有序的微结构。
We consider a kind of soft elastomers with extreme flexibility, taken Polydimethylsiloxane (PDMS) as a representative. Such soft elastomers are wildly used in both daily life and scientific research, due to their excellent material properties.
Here we focus on the surface stability of the soft elastomeric substrate with external interactions, which can be sorted into the van der Waals interaction between the surface of elastomer and a rigid contactor, and the uniformly pre-deformation of the elastomer itself. The former facilitates the perturbation on the surface, while the latter inhibits. As a result, it provides a promising way to fabricate well-ordered pattern on the elastomer surface. By a linear analysis, we show how the finite pre-deformation affects the surface morphology of the soft elastomers, and discuss the regulating of the pattern.
By applying the theory of nonlinear elasticity, we obtain the equilibrium equations and the boundary conditions in terms of the additional fluctuations in the finitely deformed state taken as the reference configuration. The stress-strain relations of such soft elastomers are discussed in detail, and the stress tensor with respect to the general finite deformation and additional disturbance is derived explicitly.
First of all, we study a simple two-dimensional problem under plane strain condition, viz. elastic buckling of an equi-biaxially pre-tensioned soft elastic film due to van der Waals interaction between the surface of film and a rigid contactor. By a linear stability analysis, we reveal the dependence of critical buckling pattern on the pre-tension. And with increasing the magnitude of pre-tension, the critical wave number and corresponding interaction parameter increase. Accordingly the buckling of elastic film can be regulated by equi-biaxial pre-tension.
Secondly, we consider the buckling of a uni-axially pre-tensioned soft elastic film due to van der Waals interaction. The uni-axial pre-tension leads to that the film material behaves anisotropically with respect to the reference configuration. It is shown that the film will buckle into stripe parallel to the tensile direction by a linear stability analysis. Moreover the spacing of the stripe pattern decreases with increasing the magnitude of the pre-tension.
Then we develop the foregoing discussion into a general case, which is the stability of an elastic film pre-deformed homogeneously due to van der Waals interaction. Consider the surface buckling of an isotropic incompressible soft film, which is deformed uniformly in two perpendicular directions respectively. The expression of the interaction parameter is deduced through a linear stability analysis. Furthermore, we investigate the criterion and pattern of the surface buckling, which are dependent on three principal extension ratios related to the finite pre-deformation. Besides, we take a soft elastomeric film with shear pre-deformation for instance, which is coincident with the general conclusion.
Finally, we study the instability of a thin liquid film resting on a uniformly stretched soft elastomeric substrate driven by van der Waals forces by a linear analysis. By adopting the lubrication approximation, the growth rate of a fluctuation of the liquid surface is derived. The corresponding instability criterion has been obtained, and the most unstable mode of the fluctuation has been discussed. It is shown that the pre-stretch stabilize the liquid film in the direction parallel to the tension while destabilize it in other directions. The liquid film is most unstable against the periodic fluctuation normal to the tensile direction with a special wavelength depend on the magnitude of the principal stretch. Therefore, the pre-stretch of the substrate leads to anisotropic dewetting of the thin liquid film, and the degree of anisotropy can be tuned by varying the principal stretch. This phenomenon may be used to create microstructures with elongated features.
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