Shape-Programming in Hyperelasticity Through Differential Growth

Area of Science:

• Solid Mechanics
• Materials Science
• Computational Mechanics

Background:

• The growth-driven shape-programming problem seeks to determine material growth to achieve desired deformations.
• Existing methods often rely on simplifying assumptions like stress-free conditions.

Purpose of the Study:

• To develop and analyze an optimal control framework for shape programming in hyperelastic bodies.
• To investigate both compatible and incompatible growth scenarios.
• To extend shape programming to include boundary conditions and external loads.

Main Methods:

• Formulation within optimal control theory in hyperelasticity.
• Utilizing Hausdorff distance for shape comparison and incorporating actuation complexity in cost functionals.
• Mathematical analysis for well-posedness and gradient-based optimization algorithms for numerical approximation.
• Application of inverse techniques for broader problem applicability.

Main Results:

• The study proves the well-posedness of the formulated optimal control problem.
• Gradient-based optimization algorithms are successfully applied for numerical approximation.
• Inverse techniques are demonstrated to handle more generic situations than analytical methods.
• Numerical experiments on beam-like and shell geometries validate the proposed scheme.

Conclusions:

• The proposed optimal control framework effectively addresses the growth-driven shape-programming problem in hyperelasticity.
• The inclusion of boundary conditions and external loads enhances the applicability of shape programming.
• Numerical methods, particularly inverse techniques, provide a powerful tool for approximating solutions in complex scenarios.