Geometry and mechanics of thin growing bilayers

Area of Science:

• Materials Science
• Solid Mechanics
• Geometric Mechanics

Background:

• Thin sheets undergo complex shape changes (morphing) due to various stimuli like thermal expansion or differential growth.
• Understanding the relationship between a sheet's initial shape and its final morphed state is crucial for predicting material behavior.

Purpose of the Study:

• To develop an analytical model explaining how arbitrary thin sheets morph under isotropic in-plane expansion.
• To introduce a novel metric for sheet slenderness incorporating thickness and shape.
• To investigate the relationship between initial shape, curvature, and the final isometric state.

Main Methods:

• Development of a geometry-inspired analytical model.
• Introduction of a new slenderness measure.
• Numerical simulations to analyze morphing behavior.
• Experimental verification of model predictions.

Main Results:

• The analytical model rationalizes how disk shape influences morphing from spherical bending to an isometric limit.
• A new slenderness measure effectively characterizes sheets by thickness and plate shape.
• The mean curvature of the isometric state is found to be 3/4 of the natural curvature, confirmed numerically and experimentally.
• Numerical analyses reveal a preferred bending direction in the isometric state.

Conclusions:

• The presented analytical model accurately describes the morphing of thin sheets under expansion.
• The model's scalability makes it applicable to sheets across a wide range of sizes.
• The findings provide fundamental insights into the mechanics of thin sheet deformation and shape evolution.