Abstract: A shoe comprises an upper portion, a sole, and a sheet of material. The upper portion is configured to receive a foot of a user. The sole is attached to the upper portion. The sheet of material is coupled to the sole. The material includes a substrate and one or more movable projections. The one or more movable projections are configured to extend from the substrate. The one or more movable projections are configured to move between a first orientation and a second orientation relative to the substrate, in response to the sole of the shoe moving between a generally flat configuration and a generally flexed configuration. The movable projections can have a triangular shape, a concave shape, a convex shape, a rectangular shape, or a barbed shape; and can be arranged in a one-direction pattern, a three-column pattern, a half pattern, a 16x2 pattern, or a checkerboard pattern.

Abstract: An auxetic metamaterial including a plurality of interconnected building blocks that are deformable between a collapsed position and an expanded position upon the application of a load. The building blocks are bistable, and define a stable state in both the collapsed position and the expanded position. In the stable state, the building blocks maintain the collapsed position or the expanded position, even after removal of the load applied on the building block.

Abstract: A kirigami structure including a thin flat sheet and a square-shaped array of perforations cut in the sheet. The array includes alternating rows and columns of adjacent orthogonal perforations interconnected via respective ligaments. The array forms a two-dimensional planar surface in a load-free state, and a three-dimensional kirigami surface in a tensile-load state in which a tensile load applied to the sheet causes out-of-plane buckling of the ligaments.

Abstract: An auxetic metamaterial including a plurality of interconnected building blocks that are deformable between a collapsed position and an expanded position upon the application of a load. The building blocks are bistable, and define a stable state in both the collapsed position and the expanded position. In the stable state, the building blocks maintain the collapsed position or the expanded position, even after removal of the load applied on the building block.