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Computation of stresses in jammed packings modeled with Tresca friction

Frédéric Marazzato, Shankar Venkataramani

TL;DR

This work addresses computing stresses in jammed packings of rigid polygonal cells governed by a Tresca friction law. It develops a constrained energy minimization for interfacial friction with nonpenetration constraints and derives a dual problem whose solution yields interfacial normal stresses, followed by a consistent $H(\mathrm{div})$-conforming stress reconstruction using lowest-order Raviart–Thomas elements $\mathbb{RT}_0$. The approach is demonstrated on Voronoi meshes, with numerical results showing consistency with boundary loads and robustness across mesh types, including shear and compression tests. The method provides a practical framework to predict internal stress distributions in masonry-like jammed systems without elastic deformation, highlighting the role of friction in stress transfer and the possibility of local tension under compression.

Abstract

This paper is interested in the computation of stresses within jammed packings of rigid polygonal cells. The cells are considered to follow a Tresca friction law. First, a constrained minimization problem is introduced where the friction energy is minimized while enforcing the non-interpenetration of neighboring cells as inequality constraint. The corresponding dual maximization problem is then deduced and its solution provides normal stresses at the interface between cells. Finally, lowest order Raviart-Thomas finite elements are used to reconstruct a consistent stress field by solving local problems. Numerical results are presented to showcase the consistency and robustness of the proposed methodology.

Computation of stresses in jammed packings modeled with Tresca friction

TL;DR

This work addresses computing stresses in jammed packings of rigid polygonal cells governed by a Tresca friction law. It develops a constrained energy minimization for interfacial friction with nonpenetration constraints and derives a dual problem whose solution yields interfacial normal stresses, followed by a consistent -conforming stress reconstruction using lowest-order Raviart–Thomas elements . The approach is demonstrated on Voronoi meshes, with numerical results showing consistency with boundary loads and robustness across mesh types, including shear and compression tests. The method provides a practical framework to predict internal stress distributions in masonry-like jammed systems without elastic deformation, highlighting the role of friction in stress transfer and the possibility of local tension under compression.

Abstract

This paper is interested in the computation of stresses within jammed packings of rigid polygonal cells. The cells are considered to follow a Tresca friction law. First, a constrained minimization problem is introduced where the friction energy is minimized while enforcing the non-interpenetration of neighboring cells as inequality constraint. The corresponding dual maximization problem is then deduced and its solution provides normal stresses at the interface between cells. Finally, lowest order Raviart-Thomas finite elements are used to reconstruct a consistent stress field by solving local problems. Numerical results are presented to showcase the consistency and robustness of the proposed methodology.
Paper Structure (16 sections, 3 theorems, 35 equations, 11 figures)

This paper contains 16 sections, 3 theorems, 35 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Tresca Friction
  3. Discrete Model
  4. Setup
  5. Facet values
  6. Primal problem
  7. Dual problem
  8. Stress reconstruction
  9. Geometric setup
  10. Discrete problem
  11. Numerical results
  12. Linear program
  13. Implementation
  14. Verification tests
  15. Shear test
  16. ...and 1 more sections

Key Result

Proposition 1

Assume that $\bm g$ is such that $E(\bm u) \to +\infty$ when $\| \bm u\|_V \to +\infty$. Then, the functional $\mathtt E$ defined in eq:primal energy convex admits a minimizer.

Figures (11)

  • Figure 1: Examples of jammed packings
  • Figure 2: Voronoi mesh $\mathcal{M}$ of $\Omega$.
  • Figure 3: Sketch of the contact between a rigid domain and $\Omega$.
  • Figure 4: Notation for edge quantities
  • Figure 5: Stress reconstruction in an internal cell.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Proposition 1
  • Remark 2
  • proof : Proof of Proposition \ref{['th:existence']}
  • Proposition 3: Weak Duality
  • Remark 4
  • Remark 5
  • Proposition 6
  • Remark 7
  • proof : Proof of Proposition \ref{['th:recon']}