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Computation of the deformation of rhombi-slit kirigami

Frederic Marazzato

TL;DR

This work tackles the deformation of rhombi-slits kirigami by formulating a nonlinear, degenerate, sign-changing PDE $-\mathrm{div}(B(\xi)\nabla\xi)=0$ and regularizing it via a complex dissipation to enable analysis and numerics. A limiting absorption principle yields existence results for the continuous problem, while a complex-valued, piecewise-linear finite-element discretization coupled with a Schauder fixed-point framework provides a convergent numerical scheme. The authors demonstrate convergence and apply the method to auxetic, non-auxetic, and mixed kirigami patterns, comparing results with experiments and showing good qualitative agreement for carefully chosen $\varepsilon$, though small $\varepsilon$ can lead to boundary-condition loss and instability. The work advances the numerical treatment of sign-changing, degenerate PDEs in metamaterial modeling and offers practical tools and code for validating kirigami deformation predictions against experimental data.

Abstract

Kirigami are part of the larger class of mechanical metamaterials, which exhibit exotic properties. This article focuses on rhombi-slits, which is a specific type of kirigami. A nonlinear kinematic model was previously proposed as a second order divergence-form PDE with a possibly degenerate, and sign-changing coefficient matrix. We first propose to study the existence of solutions to a regularization of this equation by using the limiting absorption principle. Then, we propose a finite element method with complex polynomials to approximate the solutions to the nonlinear equation. Finally, simulations are compared with experimental results.

Computation of the deformation of rhombi-slit kirigami

TL;DR

This work tackles the deformation of rhombi-slits kirigami by formulating a nonlinear, degenerate, sign-changing PDE and regularizing it via a complex dissipation to enable analysis and numerics. A limiting absorption principle yields existence results for the continuous problem, while a complex-valued, piecewise-linear finite-element discretization coupled with a Schauder fixed-point framework provides a convergent numerical scheme. The authors demonstrate convergence and apply the method to auxetic, non-auxetic, and mixed kirigami patterns, comparing results with experiments and showing good qualitative agreement for carefully chosen , though small can lead to boundary-condition loss and instability. The work advances the numerical treatment of sign-changing, degenerate PDEs in metamaterial modeling and offers practical tools and code for validating kirigami deformation predictions against experimental data.

Abstract

Kirigami are part of the larger class of mechanical metamaterials, which exhibit exotic properties. This article focuses on rhombi-slits, which is a specific type of kirigami. A nonlinear kinematic model was previously proposed as a second order divergence-form PDE with a possibly degenerate, and sign-changing coefficient matrix. We first propose to study the existence of solutions to a regularization of this equation by using the limiting absorption principle. Then, we propose a finite element method with complex polynomials to approximate the solutions to the nonlinear equation. Finally, simulations are compared with experimental results.
Paper Structure (15 sections, 4 theorems, 54 equations, 11 figures)

This paper contains 15 sections, 4 theorems, 54 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Approximate continuous problem
  3. Rhombi-slits
  4. Linear problem
  5. Nonlinear problem
  6. Approximate Discrete problem
  7. Discrete setting
  8. Discrete problem
  9. Convergence
  10. Numerical tests
  11. Auxetic kirigami
  12. Non-auxetic kirigami
  13. Mixed type kirigami
  14. Discussion
  15. Conclusion

Key Result

Lemma 1

For all $\varepsilon > 0$, there exists a unique solution $\zeta_\varepsilon \in V_D$ to One has where $C_{tr} > 0$ is the constant from the trace theorem and $\alpha > 0$ is a coercivity constant described in the proof. Note that both constants are independent of $\xi$.

Figures (11)

  • Figure 1: Notations.
  • Figure 2: Unit cells zheng2022continuum.
  • Figure 3: Eigenvalues of $B(\xi)$.
  • Figure 4: Auxetic kirigami
  • Figure 5: Non-auxetic kirigami: $\Re(\xi_h)$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Remark 3
  • Proposition 4
  • proof
  • Theorem 5
  • proof