Programming bistability in geometrically perturbed mechanical metamaterials

This repository provides a versatile design framework for designing and optimizing bistable kirigami metamaterials with target shapes and mechanical properties.

The framework allows users to design custom kirigami patterns by prescribing shape-change lattice vectors in two stable states ($\boldsymbol{\ell}_1^R, \boldsymbol{\ell}_2^R$ in the first stable state and $\boldsymbol{\ell}_1^D, \boldsymbol{\ell}_2^D$ in the second), enabling the realization of homogeneous shape-morphing kirigami metamaterials.

Please use the code to design your own Kirigami pattern!

We highlight the richness of the design space by producing kirigami patterns that exhibit a variety of axial and shearing shape changes.

Paper

This repository contains the code developed for the paper below. Feel free to try this code to create your own planar Kirigami pattern.

Peng, Y., Niloy, I., Kam, M., Celli, P., & Plucinsky, P. (2024). "Programming bistability in geometrically perturbed mechanical metamaterials". Physical Review Applied, 22(1), 014073.

Getting Started

Prerequisites

• MATLAB (tested on R2023a and later versions)
• Optimization Toolbox (for FMINCON)

Usage

1. Main Script:

• Run Main.m to generate homogeneous bistable kirigami designs with prescribed Bravais lattice vectors.
  • Figure 1 & 2: Display the unit cell in the two stable states.
  • Figure 3 & 4: Illustrate the 2-by-2 designs in the reference and deformed states, respectively.

2. Post-Processing:

• Run Postresult.m to generate energy-related results.

  • Figure 5: Depicts the relationship between elastic energy and the characteristic stretch $\lambda$
  • Figure 6: Illustrates the relationship between the energy and the actuation angle $\xi$.

Key Functions

• Main.m: The main MATLAB code to implement optimization.
• Para.m: Parameterizes the kirigami geometry by the design variables and plots the figures in two states by setting p = 1 (with prescribed Bravais lattice vectors).
• Obj.m: Defines the objective function used for the simulation such as the target energy barrier,and target stiffnesses at two stable states.
• Constraint.m: Specifies the inequality constraints used for the optimization.
• Energy_spr.m and EnergyBarrier.m: Construct the functions for the elastic energy and energy barrier respectively.
• Rot.m: Represents the 2D rotation tensor.
• ccc.m: Is used for the clearance.

Customization and Optimization

• Feel free to experiment with different shape-change lattice vectors and energy barriers to customize your kirigami pattern.
• If the optimization does not yield a desirable pattern, consider the following:
  • Adjusting fmincon parameters (e.g., MaxFunctionEvaluations, MaxIterations, ConstraintTolerance) to refine the optimization process.
  • Modifying the prescribed Bravais lattice vectors to explore different configurations.
  • Tuning the objective function to better match your design goals.
  • Changing the initial point.
• Note: Some extreme shape changes, while mathematically valid, may not yield physically realizable kirigami patterns. Users are encouraged to explore a wide range of configurations.

How to cite

If you use this code in your work, please cite the following paper

@article{peng2024programming,
  title={Programming bistability in geometrically perturbed mechanical metamaterials},
  author={Peng, Yingchao and Niloy, Imtiar and Kam, Megan and Celli, Paolo and Plucinsky, Paul},
  journal={Physical Review Applied},
  volume={22},
  number={1},
  pages={014073},
  year={2024},
  publisher={APS}
}