We study the surface wrinkling of a stiff thin elastic film bonded to a compliant graded elastic substrate subject to compressive stress generated either by compression or growth of the bilayer. Our aim is to clarify the influence of the modulus gradient on the onset and surface pattern in this bilayer. Within the framework of finite elasticity, an exact bifurcation condition is obtained using the Stroh formulation and the surface impedance matrix method. Further analytical progress is made by focusing on the case of short wavelength limit for which the Wentzel–Kramers–Brillouin method can be used to resolve the eigenvalue problem of ordinary differential equations with variable coefficients. An explicit bifurcation condition is obtained from which the critical buckling load and the critical wavelength are derived asymptotically. In particular, we consider two distinct situations depending on the ratio of the shear modulus at the substrate surface to that at infinity. If is of or small, the parameters related to modulus gradient all appear in the higher-order terms and play an insignificant role in the bifurcation. In that case, it is the modulus ratio between the film and substrate surface that governs the onset of surface wrinkling. If, however, , the modulus gradient affects the critical condition through leading-order terms. Through our analysis we unravel the influence of different material and geometric parameters, including the modulus gradient, on the bifurcation threshold and the associated wavelength which can be of importance in many biological and technological settings.

中文翻译：

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涂覆到分级基材上的薄膜的表面起皱

我们研究了粘合到柔顺梯度弹性基材上的硬弹性薄膜的表面起皱，该薄膜受到双层压缩或生长产生的压缩应力。我们的目的是阐明模量梯度对该双层的起始和表面图案的影响。在有限弹性框架内，使用Stroh公式和表面阻抗矩阵方法获得了精确的分叉条件。通过关注短波长极限的情况，取得了进一步的分析进展，在这种情况下，可以使用Wentzel-Kramers-Brillouin方法来解决变系数常微分方程的特征值问题。获得了显式分岔条件，由此渐近推导出临界屈曲载荷和临界波长。特别是，我们根据基材表面的剪切模量与无穷大处的剪切模量之比考虑两种不同的情况。如果 为或很小，则与模梯度相关的参数全部出现在高阶项中，对分岔的作用不显着。在这种情况下，薄膜和基材表面之间的模量比决定了表面起皱的发生。然而，如果 ，则模量梯度通过前导项影响临界条件。通过我们的分析，我们揭示了不同材料和几何参数（包括模量梯度）对分叉阈值和相关波长的影响，这在许多生物和技术环境中可能很重要。