Inverted cones and their elastic creases.

Area of Science:

• Solid Mechanics
• Materials Science
• Geometric Mechanics

Background:

• Creases in conical structures are common in natural and engineered systems.
• Understanding crease formation is crucial for predicting material behavior under stress.
• Previous studies often lack precise analytical models for controlled crease formation.

Purpose of the Study:

• To develop an analytical model for the elastic inversion of a right circular cone.
• To precisely define the characteristic width and shape of the boundary layer deformation (crease).
• To calculate the strain energy and applied force during cone inversion.

Main Methods:

• Utilizing a cylindrical shell analogy to simplify the deformation analysis.
• Solving the governing equation of deformation with carefully defined assumptions and boundary conditions.
• Deriving closed-form expressions for crease shape and inversion force.
• Comparing analytical predictions with geometrically nonlinear, large displacement finite element analysis (FEA).

Main Results:

• An exact expression for the crease shape was derived.
• The characteristic crease width was accurately predicted for both shallow and steep cones.
• Strain energy and applied inversion force were computed using an energy method.
• Analytical results showed excellent agreement with FEA data.

Conclusions:

• The cylindrical shell analogy and derived analytical model provide accurate predictions for cone inversion.
• The methodology offers a reliable approach for studying controlled crease formation in conical shells.
• This work complements existing research on creases formed during less controlled processes like sheet crumpling.