Anomalous size effects with fixed criticality in bistable flexible mechanical metamaterials

When the structure deformation is dominated by the low-energy deformation mode, the structure hardens with the increase in the size (number of units) at small sizes. This anomalous behavior will eventually disappear with the decay length of the finite structure converging to a size-independent characteristic quantity, but the specific critical point at which the anomalous behavior disappears still cannot be accurately and concisely described. Here, under two steady states of the bistable chain, we observed anomalous size effects with constant and oscillating criticality (the proportion of inhomogeneous deformation), two criticalities exactly separate the increasing and decreasing intervals of stiffness variation. They are interrelated due to the implied symmetries between the two steady states. On the other hand, they are distinguished because of the opposite superposition modes under the two steady states. Specifically, the constant criticality corresponds to the anomalous size effect achieved by the competition mechanism, while the oscillating criticality reveals an anomalous size effect achieved by the new mechanism (cancellation mechanism). In the anomalous size effect achieved by the cancellation mechanism, the singular characteristics generated by the completely cancelled deformation make it very robust. This robustness reflects in that the anomalous effect is no longer limited to linear small deformation, but it can still be observed stably in nonlinear large deformation. Our study reinterprets the anomalous size effect at a quantitative level, and the proposed cancellation mechanism expands the possible application range of this anomalous effect.