Bifurcation and bistability in pneumatically actuated periodically porous elastomers

Pneumatic actuation offers a convenient way to tune pattern formation in periodically porous elastomers. In this paper, we study deflation-induced buckling in periodically porous elastomers using both an analytical approach and the finite element method. By applying the Euler-Bernoulli beam model, an analytical bifurcation condition is derived using the energy approach, from which the critical buckling load can be identified analytically. Furthermore, the theoretical results are qualitatively validated by the corresponding finite element ones. It is noted that a unit cell will be assigned in finite element simulations, and usually, such a choice can be varied. In order to ensure the consistency of different unit cells in finite element simulations, we assign two representative cases and identify the proper boundary conditions that should be applied. By applying the same solution procedure, we carry out a stability analysis and analytically reveal the effect of an initial geometrical imperfection on the buckling behavior. Finally, the variation of band gaps in pneumatically activated periodic elastomers is studied by means of the finite-element method. It is hoped that the present analysis could provide useful insight not only into pattern formation in periodically porous elastomers under deflation but also into the choice of the unit cell in numerical simulations.