Buckling and postbuckling of stiff lamellae in a compliant matrix

Theoretical analysis and numerical simulations are performed to investigate the buckling and postbuckling behaviors of stiff lamellae embedded in a compliant matrix under uniaxial compression. First, the analytical solution is derived for the critical compressive strain and wrinkle wavelength of a stiff layer sandwiched between two different soft layers, in which the effects of interfacial shear stresses and matrix thicknesses have been taken into account. During postbuckling, the system may keep the sinusoidal buckling shape or bifurcate into period-doubling morphology. A phase diagram is established, which enables us to easily predict the morphological evolution from the geometric and material parameters of the system. Then the above analysis is extended to two or more parallel stiff lamellae within a compliant matrix. Different buckling modes are found in such multilayer systems, i.e., short-wave mode, long-wave mode, and hierarchical mode. This study not only sheds light on the stability and morphological evolution of lamella composites and structures but also helps understand the morphogenesis of some biological tissues.