Conical defects in growing sheets.

Area of Science:

• Physics and Materials Science: Focuses on the mechanics of growing or shrinking discs and their geometric transformations.

Background:

• Biological and synthetic discs can grow or shrink, altering their shape.
• Understanding the geometric and energetic consequences of disc deformation is crucial for various scientific fields.

Purpose of the Study:

• To investigate the conical shapes adopted by growing or shrinking discs.
• To characterize the discrete folding states of discs with a positive apex surplus angle.
• To determine the energies and stress distributions within these folded states.

Main Methods:

• Theoretical construction of conical states for discs with a positive intrinsic apex angle.
• Analysis of states in the regime where bending energy dominates.
• Identification of critical angles for self-contact and stability assessment.

Main Results:

• Discs with a positive apex surplus angle (phi(e) > 0) can fold into an infinite number of discrete states.
• All constructed states are stable below a critical apex angle where self-contact occurs.
• The ground state exhibits twofold symmetry.

Conclusions:

• The geometry of growing/shrinking discs leads to conical shapes.
• Bending-dominated mechanics dictate the folding into discrete, stable states.
• Apex angle critically influences disc self-contact and stability, with a twofold symmetric ground state observed.