Double restabilization and design of force–displacement response of the extensible elastica with movable constraints

A highly deformable rod, modelled as the extensible elastica, is connected to a movable clamp at one end and pin sliding along frictionless curved profile other. Bifurcation analysis shows that axial compliance provides stabilizing effect in compression, but unstabilizing tension. Moreover, with varying constraint's curvature origin vs bending rod's stiffness, addition possible buckling tension, structure displays none, two, or even four bifurcation loads, last two associated only first mode compression. Therefore, straight configuration may lose recover stability times, thus evidencing single double restabilization, feature never observed before. By means of closed-form solution for quasi-static behaviour analytically described under large rotations strain. The presented exploited, together an {\it ad hoc} developed optimization algorithm, design shape necessary obtain desired force-displacement curve, so realize force-limiter mechanical device capable delivering complex force response upon application continuous displacement both positive negative direction.