Indentation mechanics of soft bilayers across multiple length scales

Indentation is a powerful method to probe the mechanical properties of soft materials, yet conventional theories fail to accurately describe multilayered systems, such as soft tissues, fruits, and thin flexible electronics. We present a theoretical, computational, and experimental study on the indentation response of elastic bilayers. By constructing an asymptotic model that recovers known solutions in limiting cases, we obtain a physically intuitive and simple equation that predicts the force–displacement behavior of those bilayers under spherical indentation. Finite element simulations confirm the validity of the model across a wide range of modulus and thickness ratios between the two layers. Experiments using PDMS bilayers further demonstrate that out theory accurately captures indentation response of bi-layered polymers. Our model bridges the gap between complex numerical analyses and oversimplified single-layer assumptions, offering a general framework applicable across a broad range of indentation scales – from microscale measurement using Atomic Force Microscopy to macroscale mechanical indentation.