Analytical displacement-controlled solutions to bifurcation buckling and post-buckling of soft functional beams with constrained ends

The soft functional beams in many modern devices usually have elastic bifurcation buckling under the end-displacement control, which is essential to their consequent functions. The concise and accurate analytical solutions for the buckling and post-buckling analysis are needed to fast design these beams. Here we derive some closed-form displacement-controlled solutions for the bifurcation buckling and post-buckling of such end-constrained beams via the precise consideration of the deformed configuration of the beams. The displacement-controlled solutions to the potential energy, structural deformation, internal forces and their critical results are obtained in concise form for the beams with six typical boundary conditions. We find that these beams have only one unique but universal normalized potential energy surface depending on only two dimensionless quantities. The valley bottom pathways on the potential energy surface show that the critical buckling state is not only a bifurcation point but also a valley-ridge inflection point, and the energy increases quadratically before the point and increases linearly with a slope of 2 beyond the point. The axial forces are gradually increasing during post-buckling, greater than the traditional prediction. Our theoretical expressions provide a direct description on the end-displacement-controlled bifurcation buckling and post-buckling of the soft beams with finite deformation, which would inspire the derivation of the analytical displacement-controlled solutions for some other elastic bifurcation buckling problems.