AI 中文总结
针对随机异质膜-基底系统，建立均质化不稳定性理论，通过三阶近似推导色散关系，揭示刚度对比度对起皱临界载荷和波数的影响规律，为异质膜-基底系统的设计与评估提供力学工具。
AI 中文摘要
柔性基底上的刚性膜的起皱不稳定性会受到膜刚度空间波动的强烈影响。我们针对具有随机弯曲刚度的一维膜-基底系统建立了均质化不稳定性理论，将异质稳定性方程重新表述为曲率场的Lippmann-Schwinger方程，并利用局域-非局域核分解和腔场公式推导了强对比度展开。在三阶截断时，得到了用于预测临界载荷和波数的有效极化率与Dyson型色散关系。将刚度建模为指数映射的高斯随机场，可解析获取所需的两点和三点连通统计量。该理论通过广义本征值计算和傅里叶谱模拟得到验证：刚度对比度增大时，临界载荷降低，不稳定性向更高波数偏移，产生更短的皱纹；弱对比度下，阈值遵循通用标度律N_c^(0)-N_c~ε²，中强对比度下三阶近似比二阶理论更准确。波长选择由主导材料波长λ*与调和平均参考波长λ_H的比值控制，当λ*/λ_H<1时，调和平均模型可准确预测皱纹波长。该框架为统计异质膜-基底系统的可靠性评估与设计提供了基于力学的工具。
英文摘要
Wrinkling instabilities in stiff films on compliant substrates are strongly affected by spatial fluctuations in film stiffness. We develop a homogenized instability theory for one-dimensional film--substrate systems with random bending stiffness. The heterogeneous stability equation is reformulated as a Lippmann--Schwinger equation for the curvature field, and a strong-contrast expansion is derived using a local--nonlocal kernel decomposition and a cavity-field formulation. Truncation at third order yields an effective polarizability and a Dyson-type dispersion relation for predicting the critical load and wavenumber. The stiffness is modeled as an exponentially mapped Gaussian random field, allowing the required two- and three-point connected statistics to be obtained analytically. The theory is validated against generalized eigenvalue calculations and Fourier spectral simulations. Increasing stiffness contrast lowers the critical load and shifts the instability toward higher wavenumbers, producing shorter wrinkles. At weak contrast, the threshold follows the universal scaling $N_c^{(0)}-N_c\sim\varepsilon^2$, whereas at moderate and strong contrast the third-order approximation is more accurate than the second-order theory. Wavelength selection is controlled by the ratio of the dominant material wavelength $λ^\ast$ to the harmonic-mean reference wavelength $λ_H$. For $λ^\ast/λ_H<1$, the harmonic-mean model accurately predicts the wrinkle wavelength. The framework provides a mechanics-based tool for reliability assessment and design of statistically heterogeneous film--substrate systems.
