Collective buckling of an elastic beam array on an elastic substrate for applications in soft lithography

Abstract

We analyze the collective buckling of an array of vertical elastic beams with their lower ends built into an elastic substrate. The beams interact between themselves through the deformation of the elastic substrate. The present analysis is more sophisticated than previous ones on rigid beams on an elastic substrate in that the beams are regarded as elastic rather than rigid. From the linear theory for elastic beam buckling and the linear theory of elasticity, an eigenvalue problem is formulated and solved. Calculations show that the deformability of the beams lowers the critical height of the beams, but it does not affect the buckling pattern much. Our work also suggests that the collective buckling is dominated by the interaction of neighboring beams through the deformation of the substrate rather than whether the beams are rigid or elastic. The results are useful for the better understanding, design and application of the nanostructures produced by soft lithography.