Cusp singularity-based bistability criterion for geometrically nonlinear structures

Abstract

This letter introduces a method of analyzing multi-stable truss structures and unit cells of engineered materials using the third order derivative of the formulated system's potential energy function. The method can determine systematically all the cusp point singularities arising at the onset of bistability, and therefore identify the regions of hysteretic superelasticity and superplasticity from the usual geometrical nonlinearity in the solution space. The ability to design these behaviors is a great advantage that is highlighted with the example analysis of a bistable plane truss and a tetrahedral unit cell of a 3D lattice structure.

Negative extensibility metamaterials: Occurrence and design-space topology

“…When the system parameters a and b are varied in a design process, a stable equilibrium solution {x,y} of the equilibrium equations (17) may lose uniqueness at a point of supercritical pitchfork bifurcation [23] and split into two stable and one unstable solutions. For the systems with one degree of freedom, this occurred at the cusp points (8) within limit set (10) of the potential (6), and these points satisfied the additional condition: 0, see [19] for more details. We suggest that such a condition should generally apply to an internal degree of freedom responsible for the destabilization.…”

“…The condition (19) replaces 0 used in (10) for the analysis of structures with one independent degree of freedom. In an immediate vicinity of a limit point, the structure is in equilibrium, and therefore the criterion for an inflection point (structural destabilization) is the following:…”

“…Note that Fig.2a structure has only one independent design parameter, since the value a cannot vary in (7). Discussion of Fig.2c structure and other examples leading to the potential energy form (6) can be found in [19].…”

“…One of them is monostable and two are bistable -when two stable equilibrium configurations are possible for a same load. The bistable (hysteretic) response can be either structural superelasticity or structural superplasticity [19] depending on whether the initial configuration is recovered upon load removal. The term superplasticity implies that a full recovery of the overall (structural) plastic deformation is still possible upon load reversal, see Fig.3b.…”

“…The bottom line in the phase diagram, Fig.3a, represents the onset of bistability because all designs above this line are bistable and all those below are monostable. This line is a locus of points , that represent simultaneously a mechanical equilibrium, destabilization (zero stiffness) and an undulation point or cusp singularity [19][20][21][22] of the potential:…”