Wrinkling of a compressible trilayer domain under large plane deformations

Instabilities that arise in layered systems have been a riveting course of study for the past few decades, having found utility in various fields, while also being frequently observed in biological systems. The trilayer structure, composed of a film, interphase and substrate, is employed in several applications where the structure undergoes large deformations and the materials used are far from incompressible. Due to their complex behavior and their potential applications, the instabilities of compressible tri-layered systems; as in how they are initiated and how they can be tuned, yet remain elusive and poorly understood. Hence, the main goal of this contribution is to shed light on the large deformation wrinkling behavior of a compressible, trilayer domain, wherein a theoretical solution which captures the instability behavior of a compressible trilayer system under plane deformations is developed. An excellent agreement is observed between the analytical solutions and numerical findings, obtained using FEM enhanced with eigenvalue analysis, for a wide range of geometrical and material parameters, including compressibility of the domains, stiffness ratios, and interphase thickness. The effect of compressibility is found to be particularly significant for the case of a more compliant interphase compared to the substrate. We rigorously establish a theoretical framework that yields a one-part solution for critical wavelength, which alone captures the different wrinkling modes that have been reported in trilayer structures but previously have been treated as a two-part problem. Finally, at the incompressibility limit, the solution here reduces to its counterparts established in literature.