Emergent Rigidity Transitions

• Emergent rigidity transitions are the onset of mechanical stability in disordered systems achieved via collective geometric, topological, and constraint-driven effects rather than traditional symmetry breaking.
• They are exemplified by phenomena such as force network organization, states of self-stress, and rigid cluster percolation observed in granular media, biological tissues, and metallic glasses.
• Analysis methods including rigidity matrices and dual force-space mapping provide actionable insights for designing tunable metamaterials and understanding complex biological mechanics.

Emergent rigidity transitions denote the onset of mechanical stability—shear, bulk, or generalized resistance to deformation—in disordered or underconstrained systems through collective effects, often without changes in canonical order parameters such as spatial density modulations. Unlike traditional solids that stiffen through energetically or entropically favored real-space ordering, many amorphous, granular, or biological materials develop rigidity via geometric, topological, or constraint-driven mechanisms. These transitions can arise as bifurcations in the structure of force networks, the spatial percolation of rigid clusters, or through the development of states of self-stress, frequently in the absence of conventional broken spatial symmetry. Emergent rigidity transitions unify phenomena across granular media, biological tissues, network glasses, confluent cell layers, and mechanical metamaterials.

1. Rigidity as an Emergent, Constraint-Driven Phenomenon

Traditionally, rigidity transitions were understood as symmetry-breaking events associated with the emergence of non-uniform spatial patterns: crystalline density waves, amorphous glassy frozen structure, or nematic orientation. In contrast, emergent rigidity is often decoupled from such real-space ordering. Notable cases include:

• Dry granular solids exhibit shear-induced jamming, described via broken translational symmetry not in particle positions but in a dual “force space” (a convex tiling in so-called height or force-vector space) (Sarkar et al., 2013, Sarkar et al., 2015). Force and torque balance, positivity constraints on contacts, and frictional inequalities collectively pin the configuration of forces, even while grain positions remain nearly static throughout the transition.
• Disordered spring and fiber networks (e.g., mikado or biopolymer models) can undergo rigidity transitions purely due to geometry: application of shear or extension transforms underconstrained (Maxwell-deficient) networks from floppy to rigid as geometric incompatibility accumulates and a system-spanning state of self-stress emerges (Vermeulen et al., 2017, Rens et al., 2019, Gandikota et al., 2022).
• Biological tissues (vertex/Voronoi models) and confluent 3D cell aggregates show rigidity controlled by purely geometrical order parameters—such as preferred cell shape index or dimensionless surface area—not by contact number or real-space order (Merkel et al., 2017, Thomas et al., 2022, Arzash et al., 2023).
• Metallic glasses and other network-forming liquids exhibit transitions where percolating aggregates with specific local motifs (e.g., five-fold symmetry) form a rigid cluster at the glass transition, showing that local structure can induce a global transition in mechanical response (Chu et al., 3 Jun 2025).

The key insight is that in these systems, mechanical stability emerges through collective constraints—whether geometric, topological, or force-balance—that prohibit low-energy deformations, even when standard order parameters remain featureless.

2. Universal Features and Mechanisms of Emergence

Multiple universalities have been identified in emergent rigidity transitions:

Rigidity Percolation

In network models, the appearance of a system-spanning rigid cluster underlies the transition. Rigidity percolation is sometimes realized as a first-order transition (abrupt appearance, e.g., in frictionless jamming) or as a second-order transition (continuous growth with power-law distributed cluster sizes, e.g., frictional jamming, bond-bending, and tensegrity networks) (Henkes et al., 2015, Stephenson et al., 2022).

System Type	Criticality	Transition Mechanism
Frictionless packings	First-order	Contact counting (isostaticity)
Frictional grains, bond-bending	Second-order	Rigid cluster percolation
Vertex/Voronoi tissue models	Geometric	Shape incompatibility, residual stress

Geometric and Topological Criteria

• States of Self-Stress: Rigidity is conferred by the appearance of nontrivial solutions to the equilibrium matrix (SVD singularity), manifesting as states of self-stress that allow the storage of mechanical load (Vermeulen et al., 2017).
• Convexity and Cyclic Conditions: In polygons or network loops, convexity is necessary and cyclicity is sufficient for a state of self-stress and hence rigidity (Gandikota et al., 2022).
• Geometric Incompatibility: In underconstrained networks and fiber/tissue models, incompatibility between preferred lengths (target areas, perimeters) and global constraints drives rigidity (Merkel et al., 2017, Aspinwall et al., 25 Aug 2025).

Force Network Organization

• Broken Symmetry in Force Space: In frictional granular matter, force-tiling vertices in height space pin into a non-uniform, persistent pattern across the transition; this broken translational invariance in force space, rather than position space, marks the true symmetry-breaking event (Sarkar et al., 2013, Sarkar et al., 2015).
• Load-Bearing Backbones: Percolating rigid clusters or backbones (e.g., in sheared granular packings) carry higher pressure and are identified by both force-based (dynamical matrix) and topological (pebble game) methods, exhibiting sponge-like structures below mean-field coordination thresholds (Liu et al., 2020).

3. Criticality, Scaling, and Multicriticality

Emergent rigidity often displays universal scaling near criticality:

• Power-Law Distributions: Rigid cluster size distributions follow power laws at the transition (e.g., p(n) ∝ n^–2.5 in frictional packings) (Henkes et al., 2015).
• Critical Exponents and Scaling Functions: Universal scaling forms describe elastic moduli, density correlations, and viscoelastic response, with distinct exponents for jamming and rigidity percolation (e.g., ν = 1 and ν = 1/2, respectively, in effective-medium theory) (Liarte et al., 2021).
• Multicriticality: Anisotropic networks exhibit multiple, direction-dependent transitions: rigidity may percolate first along one set of bonds, followed by others, leading to multicritical phase diagrams and crossover scaling (Wang et al., 13 Sep 2024).
• Scaling of Mechanical Response: In 3D Voronoi tissue models, the shear modulus scales linearly with distance from the critical shape index (g ∼ s₀* – s₀), reflecting a geometric origin differing from the square-root scaling of particulate jamming (Merkel et al., 2017).

4. Biological and Materials Implications

Biological Networks

• Cells and tissues tune rigidity both developmentally and dynamically by varying properties such as adhesion, cortical tension, or shape indices, control the percolation of rigid regions, and exploit fluid–solid transitions for morphogenetic processes (Thomas et al., 2022, Arzash et al., 2023, Aspinwall et al., 25 Aug 2025).
• Cytoskeletal and ECM networks utilize strain-stiffening, prestress, and geometric incompatibility to dynamically regulate rigidity while remaining undercoordinated by constraint counting (Aspinwall et al., 25 Aug 2025).
• Active remodeling and heterogeneities in target properties can shift the rigidity transition, suggesting a role for “double optimization” in biological control (Arzash et al., 2023).

Engineered and Natural Materials

• Metallic and network glasses: The percolation of specific short-range order (e.g. five-fold icosahedral clusters) provides not only a diverging correlation length at the glass transition but also a structural origin for macroscopic stiffness and toughness (Chu et al., 3 Jun 2025).
• Colloidal gels and metamaterials: Tunability between bending and stretching constraints allows for the design of materials with sharp stiffening or auxetic responses (Rens et al., 2019, Stephenson et al., 2022).
• Highly compressible granular media: The interplay of friction and geometric contact area between deformable grains introduces novel transition criteria based on the extent of interfacial contact, rather than solely on contact counting (Poincloux et al., 15 Apr 2024).

5. Mathematical Formalisms, Order Parameters, and Dual Spaces

Key mathematical frameworks structure the analysis of emergent rigidity:

• Rigidity matrix R and Maxwell-Calladine count: Quantifies first-order constraint counting and distinguishes between linear zero modes (LZMs) and states of self-stress.
• Prestress stability and higher-order rigidity: In undercoordinated/disordered systems, geometric incompatibility and prestress (nonzero constraint forces) stabilize otherwise floppy modes via higher-order expansion of the constraint functions (Aspinwall et al., 25 Aug 2025).
• Dual or height spaces: The mapping from real-space contact networks to dual force-space tilings (e.g., Maxwell–Cremona tiling, height vectors h) allows the identification of broken symmetry and persistent order in force networks (Sarkar et al., 2013, Sarkar et al., 2015).
• SVD of compatibility/equilibrium matrices: The vanishing of singular values corresponds to the emergence of system-spanning self-stress, marking the geometric rigidity transition (Vermeulen et al., 2017).
• Variational and Γ-convergence methods: Discrete-to-continuum limits of frustrated spin systems show that geometric constraints enforce “rigid” interfacial energies for domain transitions (Cicalese et al., 2019).

Key order parameters often derive from the persistence of force-space structure (e.g., overlap of force-tiling density fields) or from the scaling of the shear modulus and the correlation length of the percolating rigid cluster.

6. Open Problems and Future Directions

Several open avenues and implications arise from the paper of emergent rigidity transitions:

• Extension to three dimensions: Generalizing dual-space or force-tiling schemes and identifying appropriate topological constructs in higher-dimensional amorphous systems.
• Nonlinear, anisotropic, and active systems: Understanding transitions in active biological matter, networks with spatial/molecular heterogeneity, and systems with dynamic, feedback-driven remodeling.
• Rigidity-induced criticality in solids: Tensorial (as opposed to scalar) order parameters create new topologies of phase diagrams, new critical points, non-additive mixing energies, and microstructure sensitivity, particularly in geometrically nonlinear or networked solids (Grabovsky et al., 2023).
• Avalanche phenomena and mechanical learning: In tensegrities and certain disordered systems, the addition of a single constraint can collectively eliminate many floppy modes, leading to avalanches and self-organization at the transition (Stephenson et al., 2022).
• Quantitative design and control: The understanding of how local motifs, geometry, prestress, or connectivity dictate mechanical response opens pathways for the rational engineering of tunable matter from soft robots and metamaterials to synthetic tissues (Chu et al., 3 Jun 2025, Arzash et al., 2023).

7. Summary Table: Representative Mechanisms in Emergent Rigidity Transitions

System Type	Transition Driver	Order Parameter	Distinctive Feature
Dry granular solids	Shear, force/torque balance	Force-tiling density	Broken symmetry in force space (not real space)
Frictional packings	Cluster percolation	Largest rigid cluster	Second-order transition, critical cluster sizes
Spring/fiber networks	Affine/nonaffine strain, geometry	SSS via SVD of Q	Geometry-controlled, not just counting
Vertex/Voronoi tissues	Target shape/surface index	Shear modulus, G	Geometric incompatibility, residual stress
Metallic glasses	Percol. of five-fold clusters	System-spanning cluster	Maxwell isostatic point, diverging length scale
Tensegrity/metamaterial	Cables/rods percolation	Strongly conn. clusters	Nonlinear constraint avalanches
Biological tissues	Contractility/adhesion/shape	p₀*, G, force chains	Dynamic/adaptive tuning, multicriticality

• (Sarkar et al., 2013) Origin of Rigidity in Dry Granular Solids
• (Henkes et al., 2015) Rigid cluster decomposition reveals criticality in frictional jamming
• (Sarkar et al., 2015) Shear-induced rigidity of frictional particles: Analysis of emergent order in stress space
• (Merkel et al., 2017) A geometrically controlled rigidity transition in a model for confluent 3D tissues
• (Vermeulen et al., 2017) Geometry and the onset of rigidity in a disordered network
• (Rens et al., 2019) Rigidity and auxeticity transitions in networks with strong bond-bending interactions
• (Liu et al., 2020) Sponge-like rigid structures in frictional granular packings
• (Liarte et al., 2021) Universal scaling for disordered viscoelastic matter near the onset of rigidity
• (Gandikota et al., 2022) Rigidity transitions in zero-temperature polygons
• (Stephenson et al., 2022) Rigidity percolation in a random tensegrity via analytic graph theory
• (Grabovsky et al., 2023) Rigidity-induced critical points
• (Arzash et al., 2023) Rigidity of Epithelial Tissues as a Double Optimization Problem
• (Poincloux et al., 15 Apr 2024) Rigidity transition of a highly compressible granular medium
• (Wang et al., 13 Sep 2024) Rigidity transitions in anisotropic networks happen in multiple steps
• (Chu et al., 3 Jun 2025) Emergent rigidity percolation of five-fold aggregates enables controllable glass properties
• (Aspinwall et al., 25 Aug 2025) Rigidity and mechanical response in biological structures

These results point to a unifying principle: rigidity transitions in disordered, complex, or underconstrained systems can arise without conventional order, governed instead by the global organization of constraints, emergence of self-stress, or collective geometric incompatibility. Understanding the structure and scaling of these transitions is central to a predictive science of amorphous and living matter.