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When the structure deformation is dominated by the low-energy deformationmode, the structure hardens with the increase in the size (number of units) atsmall sizes. This anomalous behavior will eventually disappear with the decaylength of the finite structure converging to a size-independent characteristicquantity, but the specific critical point at which the anomalous behaviordisappears still cannot be accurately and concisely described. Here, under twosteady states of the bistable chain, we observe anomalous size effects withconstant and oscillating criticality (the proportion of inhomogeneousdeformation), two criticalities exactly separate the increasing and decreasingintervals of stiffness variation. They are interrelated because of the impliedsymmetries between the two steady states, on the other hand, they aredistinguished because of the opposite superposition mode under the two steadystates. Specifically, the constant criticality corresponds to the anomaloussize effect achieved by the competition mechanism, while the oscillatingcriticality reveals an anomalous size effect achieved by the new mechanism(cancellation mechanism). In the anomalous size effect achieved by thecancellation mechanism, the singular characteristics generated by thecompletely cancelled deformation make it very robust. This robustness reflectsin that the anomalous effects are no longer limited to linear smalldeformation, but it can still be observed stably in nonlinear largedeformation. Our study reinterprets the anomalous size effect at a quantitativelevel, and the proposed cancellation mechanism expands the possible applicationrange of this anomalous effect.

Anomalous size effects with fixed criticality in bistable flexible mechanical metamaterials