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A variationally consistent membrane wrinkling model based on spectral decomposition of the stress tensor

Daobo Zhang, Josef Kiendl

TL;DR

The paper addresses wrinkling in thin membranes by proposing a variationally consistent framework based on spectral decomposition of the stress tensor, unifying taut, wrinkled, and slack states within a single approach and aligning with mixed-wrinkling criteria. The method decomposes stresses and energies into positive/negative contributions to derive stress and constitutive tensors, includes a degradation factor for residual compressive stiffness, and adopts a ν* modification to realize mixed criteria, all within a variational formulation solved via isogeometric discretization. Key contributions include unifying strain-, stress-, and mixed-based criteria, ensuring the uniaxial tension condition, and demonstrating superior convergence and robustness across benchmark problems. The approach provides a general, efficient tool for predicting wrinkling in membranes under complex loading, with potential extensions to hyperelastic and anisotropic materials.

Abstract

We present a variationally consistent wrinkling model based on spectral decomposition of the stress tensor, providing a unified formulation that captures the three distinct membrane states. Compared to the previous strain-based spectral decomposition approach, the proposed model improves accuracy by satisfying the uniaxial tension condition from tension field theory and aligning with the mixed wrinkling criterion. It also demonstrates excellent performance under various loading conditions and offers enhanced generality by unifying strain-based, stress-based, and mixed criteria within a single framework. Beyond these improvements, the model retains the superior convergence properties of the previous approach, including the framework for the flexible inclusion or omission of residual compressive stiffness. This mitigates nonconvergence or singularities in slackening states. With these adjustments, new expressions for stress and constitutive tensors are consistently derived. Finally, extensive validation through analytical, numerical, and experimental benchmark tests highlights the robustness of the model. The results confirm its accuracy in capturing the mechanical response of wrinkled thin membranes, strong convergence properties, and value for advanced membrane wrinkling analysis.

A variationally consistent membrane wrinkling model based on spectral decomposition of the stress tensor

TL;DR

The paper addresses wrinkling in thin membranes by proposing a variationally consistent framework based on spectral decomposition of the stress tensor, unifying taut, wrinkled, and slack states within a single approach and aligning with mixed-wrinkling criteria. The method decomposes stresses and energies into positive/negative contributions to derive stress and constitutive tensors, includes a degradation factor for residual compressive stiffness, and adopts a ν* modification to realize mixed criteria, all within a variational formulation solved via isogeometric discretization. Key contributions include unifying strain-, stress-, and mixed-based criteria, ensuring the uniaxial tension condition, and demonstrating superior convergence and robustness across benchmark problems. The approach provides a general, efficient tool for predicting wrinkling in membranes under complex loading, with potential extensions to hyperelastic and anisotropic materials.

Abstract

We present a variationally consistent wrinkling model based on spectral decomposition of the stress tensor, providing a unified formulation that captures the three distinct membrane states. Compared to the previous strain-based spectral decomposition approach, the proposed model improves accuracy by satisfying the uniaxial tension condition from tension field theory and aligning with the mixed wrinkling criterion. It also demonstrates excellent performance under various loading conditions and offers enhanced generality by unifying strain-based, stress-based, and mixed criteria within a single framework. Beyond these improvements, the model retains the superior convergence properties of the previous approach, including the framework for the flexible inclusion or omission of residual compressive stiffness. This mitigates nonconvergence or singularities in slackening states. With these adjustments, new expressions for stress and constitutive tensors are consistently derived. Finally, extensive validation through analytical, numerical, and experimental benchmark tests highlights the robustness of the model. The results confirm its accuracy in capturing the mechanical response of wrinkled thin membranes, strong convergence properties, and value for advanced membrane wrinkling analysis.

Paper Structure

This paper contains 12 sections, 53 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Membrane formulation and theoretical background of wrinkling models
  3. A new variationally consistent wrinkling model based on spectral decomposition of the stress tensor
  4. Discretization
  5. Numerical examples
  6. A stretched rectangular membrane wrinkled under in-plane pure bending
  7. A rectangular membrane wrinkled by simple shear
  8. A flat square membrane wrinkled by biaxial corner stretching
  9. A square isotropic airbag wrinkled due to inflation
  10. A square blanket wrinkled by free fall under self-weight
  11. Conclusion
  12. Wrinkling model based on spectral decomposition of the strain tensor

Figures (17)

  • Figure 1: Possible states of a membrane: (a) taut, (b) wrinkled, (c) slack.
  • Figure 2: Schematic view of a membrane zhang2024variationally. The vectors $\mathbf{X}$ and $\mathbf{x}$ denote the positions on the midsurface in the reference and deformed configurations, respectively, with $\mathbf{u}$ as the displacement.
  • Figure 3: Comparison of membrane state classifications by wrinkling criteria.
  • Figure 4: Transformations among the various configurations of a membrane of length $L$.
  • Figure 5: (a) In-plane pure bending of a pre-tensioned rectangular membrane and (b) numerical simulation settings for the right half of the rectangular membrane under in-plane bending.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Remark 1