From beams to bilayers: A unifying approach towards instabilities of compressible domains under plane deformations

Instabilities that form when a domain of compliant elastic material goes under compressive forces are prevalent in nature and have found many applications. Even though instabilities are observed in a myriad of fields and materials, the large deformation bifurcation analysis of compressible domains, may it be beams, half-spaces, or bilayers, remains understudied compared to the incompressible case. In this work, we present a unifying approach for the instability analysis of a compressible elastic domain under plane deformations, wherein the unifying approach is then particularized for beams, half-spaces, and bilayers. First, the large-deformation incremental analysis for a rectangular, compressible, hyperelastic domain under plane deformations is developed, which serves as a generic and all-encompassing framework for other geometries. Subsequently, this generic framework is applied to the specific domains of beam, half-space, and lastly as the superimposition of the two; bilayer. Obtained analytical results for the onset of wrinkling in the beam, half-space and bilayer geometries are explored in the full range of compressibility and for various geometrical parameters, including their comparison with computational simulations using the finite element method, cultivating excellent agreements between analytical and numerical results all across the material and geometrical parameter spectrum. The analytical framework presented here provides grounds for further works on other modes of instabilities and more complex geometries.