Zhao Yan, Cao Yanping, Hong Wei, Wadee M Khurram, Feng Xi-Qiao
AML and CNMM , Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University , Beijing 100084 , People's Republic of China.
Department of Aerospace Engineering , Iowa State University , Ames, IA 50011 , USA.
Proc Math Phys Eng Sci. 2015 Jan 8;471(2173):20140695. doi: 10.1098/rspa.2014.0695.
Compression of a stiff film on a soft substrate may lead to surface wrinkling when the compressive strain reaches a critical value. Further compression may cause a wrinkling-folding transition, and the sinusoidal wrinkling mode can then give way to a period-doubling bifurcation. The onset of the primary bifurcation has been well understood, but a quantitative understanding of the secondary bifurcation remains elusive. Our theoretical analysis of the branching of surface patterns reveals that the wrinkling-folding transition depends on the wrinkling strain and the prestrain in the substrate. A characteristic strain in the substrate is adopted to determine the correlation among the critical strain of the period-doubling mode, the wrinkling strain and the prestrain in an explicit form. A careful examination of the total potential energy of the system reveals that beyond the critical strain of period-doubling, the sinusoidal wrinkling mode has a higher potential energy in comparison with the period-doubling mode. The critical strain of the period-doubling mode strongly depends on the deformation state of the hyperelastic solid, indicating that the nonlinear deformation behaviour of the substrate plays a key role here. The results reported here on the one hand provide a quantitative understanding of the wrinkling-folding transition observed in natural and synthetic material systems and on the other hand pave the way to control the wrinkling mode transition by regulating the strain state in the substrate.
当压缩应变达到临界值时，在柔软基底上压缩坚硬薄膜可能会导致表面起皱。进一步压缩可能会引起起皱 - 折叠转变，然后正弦形起皱模式可能会让位于倍周期分岔。一级分岔的起始已得到很好的理解，但对二级分岔的定量理解仍然难以捉摸。我们对表面图案分支的理论分析表明，起皱 - 折叠转变取决于起皱应变和基底中的预应变。采用基底中的一个特征应变以明确形式确定倍周期模式的临界应变、起皱应变和预应变之间的相关性。对系统总势能的仔细研究表明，超过倍周期的临界应变后，与倍周期模式相比，正弦形起皱模式具有更高的势能。倍周期模式的临界应变强烈依赖于超弹性固体的变形状态，这表明基底的非线性变形行为在此起关键作用。这里报道的结果一方面提供了对在天然和合成材料系统中观察到的起皱 - 折叠转变的定量理解，另一方面为通过调节基底中的应变状态来控制起皱模式转变铺平了道路。
