Revisiting the compressive buckling of periodic multi-layers reinforced composites: Theoretical and numerical analysis

Multi-layered composites consisting of stiff inclusions and compliant matrices are widely found in biological tissues, geological features, waveguides, and flexible electronics. Capturing their buckling instabilities is crucial for understanding structural morphogenesis and informing the design of functional devices. However, a unified theoretical framework capable of accurately identifying buckling modes under compressive loading is still lacking. In this work, using the energy method, we develop a mechanics model for periodically reinforced composites with stiff layers by rigorously accounting for in-plane displacements and interfacial shear stress. The proposed model provides accurate predictions of the critical compressive strain and wavelength for the local-wrinkling mode, and effectively distinguishes between local-wrinkling and long-wave modes over a wide range of geometric and material parameters. A general phase diagram is constructed to reveal how geometric and material parameters govern the selection between the two buckling modes. Moreover, an explicit expression for the critical condition of the mode transition is analytically derived, offering direct guidance for the controllable design of buckling patterns. These findings provide new insights into the buckling instability of multi-layered composites.