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How periodic surfaces bend
1Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.
Summary
Researchers explored deformation modes in periodic surfaces, finding membrane and bending modes are often inversely related. The total number of these modes is limited to three, impacting surface mechanics and origami structures.
Area of Science:
- Materials Science
- Mechanical Engineering
- Computational Geometry
Background:
- Periodic surfaces possess invariance under a 2D lattice of translations.
- Deformation modes are categorized as effective membrane modes (lattice stretching) or effective bending modes (lattice bending).
Purpose of the Study:
- To investigate the relationship between effective membrane and bending modes in periodic surfaces.
- To establish constraints on the total number of deformation modes for such surfaces.
Main Methods:
- Theoretical analysis of deformation modes for periodic piecewise smooth simply connected surfaces.
- Mathematical formulation to demonstrate orthogonality between membrane and bending modes.
- Illustrative examples derived from curved-crease origami tessellations.
Main Results:
- Effective membrane and bending modes exhibit a form of orthogonality for periodic surfaces.
- An increase in membrane modes corresponds to a decrease in bending modes, and vice versa.
- The combined total number of effective membrane and bending modes is constrained, never exceeding three.
Conclusions:
- The interplay between membrane and bending modes in periodic surfaces is fundamentally constrained.
- This constraint limits the total degrees of freedom for deformation, influencing structural behavior.
- Findings are relevant to the design and analysis of origami/kirigami-inspired structures.
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