We examine the problem of an elastic film bonded to an elastic substrate and subjected to compressive stress (induced by growth of the film or lateral compression of the whole system). Following the linear analysis previously carried out to determine the critical growth/compression ratio required to induce wrinkling in the film, we carry out a weakly-nonlinear analysis to derive an amplitude equation that describes the evolution of the wrinkling amplitude beyond the bifurcation point. We carry out a comprehensive numerical bifurcation analysis of the problem using the finite element method and show excellent agreements between the weakly-nonlinear analysis and the numerical experiments. We are also able to solve directly for the bifurcation point in our discretized system and characterize the effect of implementation details such as the aspect ratio of the computational domain on the observed bifurcation point. Finally, we explore solutions of the amplitude equation in the case that the wrinkling amplitude is allowed to vary over long spatial and/or temporal scales.

我们研究了弹性膜粘结到弹性基材上并受到压缩应力（由膜的生长或整个系统的横向压缩引起）的问题。在先前进行线性分析以确定在薄膜中引起皱纹所需的临界生长/压缩比之后，我们进行了弱非线性分析以得出一个振幅方程，该方程描述了超过分叉点的皱纹振幅的演变。我们使用有限元方法对问题进行了全面的数值分叉分析，并证明了弱非线性分析与数值实验之间的优异一致性。我们还能够直接解决离散系统中的分叉点，并表征实现细节的影响，例如计算域的纵横比对观察到的分叉点的影响。最后，在允许起皱幅度在较长的空间和/或时间尺度上变化的情况下，我们探索幅度方程的解。