Computation of stresses in jammed packings modeled with Tresca friction

Technical Summary: Computation of Stresses in Jammed Packings Modeled with Tresca Friction

Problem Statement
This paper addresses the computation of stress fields within jammed packings of rigid polygonal cells, such as masonry structures or dense granular materials. Unlike traditional Discrete Element Methods (DEM) that often rely on elastic deformations to penalize interpenetration, this work focuses on systems where cells are fully rigid and do not deform elastically. In such "jammed" configurations, mechanical equilibrium is maintained exclusively through contact forces and friction, without significant elastic strain. The specific challenge lies in determining the internal stress distribution in these statically indeterminate (hyperstatic) systems, where the number of force variables exceeds the number of equilibrium constraints, leading to a non-unique solution space. The authors model the contact physics using a Tresca friction law, which posits a fixed threshold for tangential slip, distinct from the stress-dependent Coulomb friction.

Methodology
The proposed methodology follows a three-stage computational framework:

1. Primal Energy Minimization:
   The authors formulate a constrained minimization problem where the total frictional energy is minimized. The energy functional consists of the sum of Tresca friction work over all internal interfaces, defined as $s_T |[u]_e \cdot t_e|$, where $s_T$ is the friction coefficient, $[u]_e$ is the displacement jump, and $t_e$ is the tangent vector. This minimization is subject to:

   • Non-interpenetration constraints: The normal component of the displacement jump must be non-negative ($[u]_e \cdot n_e \geq 0$).
   • Confinement: A Neumann boundary condition ($g$) is applied to prevent cells from slipping to infinity, as the energy lacks elastic restoring forces.
   • Translation invariance: A global constraint is added to fix the center of mass.
     The cells are modeled as rigid bodies with translational degrees of freedom only; rotational degrees of freedom are explicitly neglected. This simplification is motivated by the observation that in jammed configurations, resistance to rotation (requiring dilatancy) is orders of magnitude higher than resistance to sliding.

2. Dual Maximization:
   Since the primal problem seeks displacements (which are effectively zero in a stable jammed state), the authors deduce a dual maximization problem. The dual variables correspond to the forces acting on the internal edges. Solving this dual problem yields the normal and tangential stresses at the interfaces. The existence of minimizers for the primal energy and solutions for the dual problem is established under appropriate hypotheses regarding the confinement term. The dual formulation effectively selects a specific solution from the hyperstatic manifold of possible stress states.

3. Stress Reconstruction:
   The output of the dual problem provides discrete stress values at the edges. To reconstruct a continuous stress field $\sigma$ within each cell, the authors employ lowest-order Raviart–Thomas ($RT_0$) finite elements. These elements are $H(\text{div})$-conforming, meaning their degrees of freedom correspond to the normal stresses at the edge barycenters. The reconstruction involves solving a local mixed problem in each cell to find a stress tensor $\sigma_c$ such that:

   • $\text{div}(\sigma_c) = 0$ (equilibrium in the absence of volume loads).
   • The normal stress on the boundary matches the values obtained from the dual solution.
   • The symmetry of the stress tensor ($\sigma = \sigma^T$) is imposed weakly via a Lagrange multiplier. This is necessary because the underlying discrete model neglects moment balance equations, meaning the reconstructed stress may possess a skew-symmetric component representing residual torques.

Key Contributions

• Rigid Friction Model: The paper introduces a model for jammed packings that relies solely on rigid friction (Tresca criterion) without elastic parameters, distinguishing it from standard DEM approaches.
• Hyperstaticity Handling: The work explicitly addresses the static indeterminacy of jammed packings. It demonstrates that the optimization approach selects a unique solution from the space of self-stress states by minimizing the frictional energy functional.
• Consistent Stress Reconstruction: A novel reconstruction technique is presented using $RT_0$ elements to generate a consistent stress field from discrete interface forces, ensuring local force equilibrium within each cell.
• Neglect of Rotation: The authors propose a simplified model that ignores rotational degrees of freedom, arguing that this is physically justified for compressed packings where translational sliding dominates, while acknowledging that this leads to non-symmetric stress tensors.

Numerical Results
The methodology was tested on various 2D polygonal meshes (specifically Voronoi tessellations) with the following findings:

• Verification: In tests with uniform boundary conditions, the reconstructed stress field matched the expected uniform stress state, confirming the consistency of the method.
• Shear Test: In a shear loading scenario, the reconstructed stress tensor exhibited asymmetry ($\sigma_{12} \neq \sigma_{21}$). The authors attribute this to the lack of enforced moment balance in the discrete model, noting that the skew-symmetric part accounts for residual torques.
• Homogeneous Compression: Despite uniform compressive boundary conditions, the solver produced a non-homogeneous stress field. The authors explain this as a consequence of the system's hyperstaticity; the optimization selects a specific energy-minimizing state from the null space, which naturally leads to the formation of "force chains" (sparse networks of high stress) and regions of low or zero stress.
• Tension in Compression: The results showed cells in tension (positive principal stresses) even under global compression. The authors identify this as a physical phenomenon arising from frictional interlocking and bridging effects, rather than a numerical artifact.

Significance and Claims
The paper claims that its proposed methodology offers a robust and consistent way to compute stresses in jammed packings by leveraging the physics of rigid friction and energy minimization. The authors argue that their approach better captures the underlying physics of jammed systems compared to elastic-based DEMs, particularly in regimes where elastic deformation is negligible.

The significance of the results lies in the demonstration that minimizing frictional energy (an $L_1$-type objective) naturally leads to sparse force networks, aligning with experimental observations of force chains in granular materials. The paper modestly acknowledges its limitations: the neglect of rotational equilibrium results in non-symmetric stress tensors, and the use of Tresca friction (fixed threshold) simplifies the problem compared to the more complex, non-convex Quasi-Variational Inequalities required for Coulomb friction. The authors suggest that future work could extend the model to include rotational degrees of freedom and more realistic friction laws to achieve full mechanical consistency.