Shear Jamming Transition in Disordered Systems

• Shear jamming is a nonequilibrium rigidity transition in which disordered particle assemblies become mechanically stable under applied shear, forming a percolating force network.
• The transition exhibits critical scaling behavior, evidenced by abrupt jumps in contact number and elastic moduli, and is influenced by particle friction and preparation history.
• It plays a pivotal role in understanding rheology and nonlinear flow in granular materials, dense suspensions, foams, and glasses under shear deformation.

The shear jamming transition is a nonequilibrium rigidity transition in which disordered packings of particles, initially unjammed under isotropic conditions, develop a finite yield stress and elastic moduli in response to applied shear. This phenomenon is central to the mechanics and flow of granular materials, dense suspensions, foams, emulsions, and glasses. Unlike isotropic jamming—which occurs at a protocol-dependent critical density under compressive loading—shear jamming can be induced at densities below the isotropic jamming point, typically through the application of steady or cyclic shear deformation. The nature of the transition, its associated critical behavior, and the microscopic mechanisms involved depend sensitively on particle properties (specifically friction), system history, and driving protocol.

1. Definitions and Phenomenology

Shear jamming refers to the emergence of rigidity—i.e., the ability of the system to support static stress or respond elastically—when an initially unjammed, disordered assembly of particles is sheared. Under controlled strain or stress, this transition is marked by the sudden appearance of a percolating force network, nonzero pressure, and a system-spanning rigid cluster. In frictionless systems, the transition manifests when the average packing fraction is below the isotropic jamming point (φ_J), but above a threshold set by the system's preparation (e.g., annealing or mechanical training). In frictional systems, the scenario is more intricate, involving hysteresis, re-entrant flow regimes, and a true first-order rigidity transition at finite stress.

Experiments (e.g., two-dimensional photoelastic disk packings) show a sharp increase in pressure and force chain connectivity at a critical shear strain γ_j, even when the initial packing fraction is below φ_J (Pan et al., 2023). Simulations confirm that configurations prepared by cyclic or slow compression can be shear jammed at densities φ with φ_J < φ < φ_j, where φ_j marks the protocol-dependent jammed limit (Kawasaki et al., 2020, Pan et al., 2023).

In frictionless athermal spheres, the shear jamming transition is associated with the system attaining isostaticity—where the number of contacts per particle Z equals 2d (with d the spatial dimension)—and a discontinuous jump in Z is observed at the transition (Deng et al., 4 Mar 2024, Babu et al., 2023).

2. Rigidity, Isostaticity, and Microstructure

At the core of shear jamming is the formation of a system-spanning rigid cluster, whose emergence is marked by isostaticity. For frictionless spheres in d dimensions, isostaticity requires Z_J = 2d; for frictional systems, the isostatic condition is generalized to account for rotational degrees of freedom and mobilized contacts, leading to Z_μ^{iso} = D + 1 after correcting for the fraction of fully mobilized contacts (Babu et al., 2023).

The rigidity transition at shear jamming is discontinuous in the quasi-static limit—measured by a sudden, system-wide jump in the size of the rigid cluster and the mechanical contact number—both in frictionless and fully relaxed frictional granular packings (Babu et al., 2023, Deng et al., 4 Mar 2024). This discrete onset of rigidity connects the shear jamming transition to the universal features of the isotropic jamming transition.

The microstructure of shear-jammed states, as revealed by the bond and contact network, is generically anisotropic. Under shear, force chains and contacts preferentially align along the principal compressive axis imposed by deformation (Pan et al., 2023, Pan et al., 2022). The fabric tensor (measuring orientational order) retains a high degree of anisotropy after jamming, and this anisotropy is fundamental for the emergence of finite residual shear stresses and distinct elastic moduli in the jammed state (Baity-Jesi et al., 2016, Zheng et al., 2018).

3. Criticality, Scaling, and Universality

Despite the first-order nature of the rigidity transition (discontinuity in Z), the shear jamming point exhibits power-law scaling in various quantities, analogous to the criticality observed in isotropic jamming. Near the transition:

• The excess contact number δZ = Z – Z_c scales as δγ^{1/2}, with δγ = γ – γ_j, where γ_j is the critical shear strain (Babu et al., 2022).
• Pressure (P) and shear stress (σ) scale linearly with distance from the critical point: P, σ ∼ δγ.
• The vibrational density of states D(ω) exhibits a low-frequency plateau, and the crossover frequency ω* scales as δγ^{1/2} (Babu et al., 2022).
• The shear modulus G, in isotropic jamming, vanishes with δZ: G ∼ δZ. By contrast, in shear-jammed states, the modulus acquires a nonvanishing, discontinuous component due to anisotropy (Baity-Jesi et al., 2016, Pan et al., 2022, Pan et al., 2023).

The distributions of interparticle forces and nearest-neighbor gaps near jamming are characterized by exponents (θ_e, θ_l, γ) that satisfy mean field marginal stability relations; empirical values in shear-jammed and isotropic-jammed states are in close agreement (Babu et al., 2022). The fact that both transitions (isotropic and shear-induced) conform to the same universal scaling and marginal stability has been confirmed numerically and theoretically (Babu et al., 2022, Goodrich et al., 2015, Pan et al., 2023).

A key result is that, under cyclic shear, the transition is best described as a first-order phase transition with quenched disorder; sample-to-sample fluctuations in φ_J dominate the finite-size scaling of critical quantities. The relationship between the disconnected and connected susceptibilities, χ_dis ∼ (χ_con)^2, marks the influence of disorder and athermal dynamics, in analogy with the athermal random-field Ising model (Deng et al., 4 Mar 2024).

4. Rheology, Nonlinear Response, and Dissipation

The rheological signature of the shear jamming transition includes the emergence of a finite yield stress at the critical point and a diverging viscosity as the jammed state is approached under shear (Otsuki et al., 2012, Zaccone, 28 Oct 2024). In frictional systems, the flow curves (stress versus strain rate) display hysteresis and a re-entrant regime in the phase diagram: as shear stress is ramped, inertial flow gives way to a jammed solid, then to plastic flow at higher stress—a hallmark of a nonequilibrium first-order transition (Grob et al., 2013).

In frictionless systems, the viscosity diverges as η ∼ |z_c – z|^{–α} (with α ≈ 1.3 in 2D), while the shear modulus vanishes linearly with the excess coordination number G ∼ (z – z_c), indicating that the rigid phase is stabilized by the increase in mechanical contacts (Zaccone, 28 Oct 2024).

Below the transition, dissipation is dominated by a shrinking fraction of rapidly moving particles; the velocity distribution develops a heavy algebraic tail (P(v) ∼ v^{–3}), and different moments of velocity diverge with distinct exponents, underlining the localization of plastic dissipation near jamming (Olsson, 2015).

Nonlinear shear response is characterized by strain softening and then hardening as the contact network develops and anisotropy increases (Coulais et al., 2014, Kawasaki et al., 2020, Pan et al., 2022). For frictionless, deeply annealed systems, shear hardening displays robust critical scaling: G ~ σ^{2/5}, ΔZ ~ σ^{2/5}, μ ~ σ^{1/4}, with the shear modulus expressed as a sum of isotropic (∝ ΔZ) and anisotropic (∝ μ^2) contributions (Pan et al., 2022, Pan et al., 2023). This scaling is preserved across different preparation protocols.

5. Role of Friction, Protocol, and System Preparation

Friction plays a pivotal role in the nature of the shear jamming transition. In frictional assemblies, the jamming transition is discontinuous, with hysteresis and re-entrant regions in the stress–density phase diagram; the frictionless limit sees a merging of critical densities and a continuous transition at zero stress (Grob et al., 2013). The threshold density for stabilization by friction is set by the lowest density where shear-induced geometry allows enough contacts for mechanical stability; below this threshold, even high friction cannot stabilize the structure (Vinutha et al., 2015).

Protocol dependence is prominent: the jamming density attained depends on history (e.g., mechanical training, annealing), with deeply annealed or “trained” configurations displaying shear jamming at densities above the minimal φ_J (Kawasaki et al., 2020, Pan et al., 2023). Under cyclic shear, the memory of preparation is erased, and the system converges to a unique minimal φ_J (Deng et al., 4 Mar 2024), in contrast to the broad “J-line” of densities accessible under different preparation protocols.

Experimental and simulation protocols that rotate the direction of shear (“alternating shear rotation”) demonstrate clear differences between frictional and frictionless suspensions: the jamming point for frictionless systems remains stable near random close packing, while for frictional systems the jamming point increases with the tacking angle (Acharya et al., 16 Mar 2025). The contributions to viscosity from contact and hydrodynamic stresses can be mapped to the average contact number for both particle types, reinforcing the structural–rheological coupling.

6. Theoretical Frameworks and Mathematical Scaling

The scaling ansatz for elastic energy near the transition invokes the excess contact number ΔZ, packing fraction shift Δφ, and the applied strain, leading to scaling laws for energy, pressure, and shear stress that possess “emergent scale invariance” (Goodrich et al., 2015). In the thermodynamic limit, the difference between shear and isotropic jamming lies in the orientation of the dominant elastic eigenmode within the six-dimensional space of elastic moduli; rotational symmetry (SO(3)) is preserved in the five “soft” modes, establishing universality for both transitions (Baity-Jesi et al., 2016).

The mathematical description establishes key scaling relations:

• Shear modulus (solid phase): G ∝ z – 2d (above jamming)
• Viscosity (fluid phase): η ∼ |z_c – z|^{–α} (below jamming)
• Connected/disconnected susceptibilities: χ_dis ∼ (χ_con)^2 (finite size, first-order with disorder) (Deng et al., 4 Mar 2024)
• Stress–strain scaling at the transition: σ ∼ Δγ, ΔZ ∼ (Δγ)^{1/2} (Babu et al., 2022)

Critical exponents and force/gap distribution properties (e.g., θ_e, θ_l, γ) are found to saturate inequalities expected from marginal stability and mean-field theory, reinforcing analogies with hard-sphere glasses in infinite dimensions (Babu et al., 2022).

7. Broader Implications and Open Directions

The shear jamming transition serves as a "bridge" linking the rheology of thermal hard spheres (glasses) and athermal soft spheres, providing a generalized phase diagram that unifies frictionless and frictional systems (Pan et al., 2023). Many practical phenomena—including shear hardening, dilatancy, fragility, and discrete shear thickening—are closely connected to or governed by the underlying shear jamming transition.

Future research directions include:

• Exploring the critical behavior in three-dimensional frictional systems and extending universality claims (Otsuki et al., 2012).
• Disentangling the influence of mechanical noise, particle shape, and interaction range on the shear jamming threshold (Marschall et al., 2014, Acharya et al., 16 Mar 2025).
• Clarifying the dynamic yielding and flow properties near the merging point of shear jamming and yielding lines (Urbani et al., 2016).
• Developing a renormalization-group framework for the jamming transition that captures the role of disorder, finite-size scaling, and nonaffine dynamics (Goodrich et al., 2015, Zaccone, 28 Oct 2024).

The shear jamming transition, thus, constitutes a robust, quantitative paradigm for the understanding of rigidity and flow in amorphous matter across a wide spectrum of materials and length scales.