Mechanical Metamaterials with Negative Extensibility: Nonlinear Analysis and Phase Diagram Calculation

Mechanical metamaterials are characterized by their unnatural elastic constants. The metamaterial property of negative extensibility is investigated. A five-member unit-cell structure is able to contract against the line of increasing applied tension. If stiffness were to be measured during this contraction it would appear to be negative. More accurately, it would be a negative `incremental’ stiffness since the slope of the force-response curve shifts from positive to negative after the load is applied. Negative extensibility is contrasted with other mechanical phenomena including negative compressibility, negative Poisson’s ratio, stretch-densification and the reversal of St.-Venant’s edge effects. The potential function of the unit-cell is highly nonlinear due to geometry. The unit-cell is bistable. For certain combinations of member stiffnesses and geometry, the force-response curve is hysteretic. The structure is characterized by five basic nonlinear mechanical responses: monostability (MS), superelasticity (SE), superplasticity (SP), negative extensibility of the superelastic type (NESE) and negative extensibility of the superplastic type (NESP). The negative extensibility response is defined by a `pinched’ hysteresis loop. Over the `pinched’ region the work done by the system is negative. Energy methods used in computational thermodynamics for the calculation of phase diagrams are re-purposed in order to map the mechanical responses of the unit-cell structure to regions in a phase diagram. The axes of the phase diagram are dimensionless system parameters that define the unit-cell geometry and element stiffnesses. The boundary lines represent the onset of a particular mechanical response. The process of computing boundary lines is informed by the principles of catastrophe theory. For instance, it is possible to compute with precision the onset of a hysteresis loop using a mathematical set of conditions. Similar to thermodynamics, on the phase diagram there are `triple points’ which simultaneously satisfy three conditions of mechanical equilibrium. The discussion ends by showing how the unit-cell structure can be arranged into a periodic array. The response of the periodic array is shown to be similar to that of its constituent unit-cell. Fabrication of a periodic structure that is able to contract against applied tension remains an open question. Viability of the phenomenon in practice is evaluated.