Friction-Dependent Crossover Strain

• Friction-dependent crossover strain is defined as the strain threshold where frictional dissipation transitions, validated by shear experiments in granular media.
• The topic examines competing mechanisms like frictional healing and anelasticity, quantified through constitutive laws and inertial number scaling.
• It governs mechanical transitions such as strain-stiffening, yielding, and fluidization, offering insights for designing materials with tailored friction properties.

Friction-dependent crossover strain describes the strain or deformation threshold at which the dominant mechanisms of frictional energy dissipation, mechanical rigidity, and flow characteristics of a system fundamentally change. In frictional materials—ranging from granular assemblies to molecular fluids, soft condensed matter, polymeric chains, and engineered surfaces—this crossover is governed by the interplay between microscopic contact mechanics, dissipative processes such as frictional healing and anelasticity, and collective phenomena including fluidization, aging, yielding, and phase transition. The concept is rigorously quantified in terms of constitutive laws, critical nondimensional parameters, and scaling relations that define the transition point between regimes characterized by different friction–strain relationships.

1. Fundamental Constitutive Laws and Critical Parameters

The prototypical expression for friction-dependent crossover strain emerges from shear experiments on granular media (Kuwano et al., 2011), where the macroscopic friction coefficient $\mu$ transitions from a regime of velocity weakening (negative shear-rate dependence) to velocity strengthening (positive shear-rate dependence) at a critical inertial number $I_c$. The inertial number is defined as

$I = \dot{\gamma}\sqrt{\frac{m}{P d}},$

where $\dot{\gamma}$ is the shear rate, $m$ is grain mass, $P$ the confining pressure, and $d$ the grain diameter. The central constitutive law is

$\mu = c_1 I - \alpha\ln(\dot{\gamma}\tau) + \mu_0,$

with $c_1 \approx 0.6$ (anelastic strengthening), $\alpha$ (healing-weakening), $\tau$ the healing time, and $\mu_0$ a reference friction. The crossover value $I_c = \alpha/c_1$ marks where frictional healing and anelasticity contribute equally—a precise, friction-controlled strain threshold separating quasistatic from inertial flow. Scaling relations in the inertial regime, such as $\Delta\mu \propto c_1 I$ and $\Delta H^* \propto c_2 I$ (with $c_2 \approx 0.2$), link frictional strengthening directly to dilatancy.

In dense granular flows and compressed granular media, the friction-controlled crossover strain is identified with a characteristic macroscopic strain window $\delta\varepsilon_p$ associated with nucleation of sliding events (Gimbert et al., 2012). As the applied stress increases, both stress and strain correlation lengths diverge—no unique shear-band thickness emerges; rather, strain localization depends continuously on the deformation window, with the friction threshold setting the scale for event nucleation but not for fixed spatial features.

2. Microscopic Physics: Healing, Anelasticity, and Statistical Contact Laws

The crossover between frictional regimes is rooted in the competition of distinct physical mechanisms:

• Frictional healing: Dominates at low strain rates (below $I_c$), where slow interparticle motion allows the formation and strengthening of microscopic contacts, producing time- and history-dependent friction.
• Anelasticity: Becomes dominant for fast, inertial flows (above $I_c$), where energy is dissipated not only in interparticle friction but also through internal viscoelastic deformation.

In statistical models of frictional interfaces, the crossover strain is a population effect that emerges from the distribution of micro-junction states ("pinned" vs. "slipping") (Thøgersen et al., 2014). The effective strain required to trigger macroscopic sliding, and thus change in friction, depends on the current distribution of junction stretches, the evolution of these populations during transient sliding or deceleration, and the microscopic laws dictating pinning and repinning. The result is strong history dependence, with the macroscopic static friction coefficient set by the preparation of the interface (e.g., deceleration dynamics, slow slip phenomena) and the underlying statistical distributions.

3. Strain-Stiffening, Yielding, and Nonmonotonic Dependence on Friction

Friction-dependent crossover strains govern the transition between mechanical regimes such as strain-stiffening, yielding, and fluidization in complex assemblies:

• In granular chains and entangled packings, the emergent strain-stiffening and yielding response is nonmonotonic with respect to friction (Dumont et al., 2017, Shaebani et al., 2022). The maximum stress and shear penetration depth reach extrema at intermediate friction values, reflecting an optimal network connectivity for force transmission.
• Models describe resistance force scaling as

$$\log(F/F_0) \sim \mu \sqrt{\mathcal{N} \Phi^{11/8} z/b},$$

where $\mathcal{N}$ is beads per chain, $\Phi$ the volume fraction, $z$ indentation, and $b$ bead size. The penetration depth of shear into granular chains

$$\ell \propto \xi\text{ (function of chain topology, bond gaps, and directional persistence)},$$

is modulated by packing-generated connectivity and local friction, with stiffening and yield transitions optimized at crossover friction.

Experimental and simulation analyses show that in cyclically sheared granular materials (Yuan et al., 17 Oct 2025), friction controls the crossover from intermittent, aging-like rearrangements to continuous, fluidized motion. The dynamic state diagram in $(\Gamma, \mu)$ space (strain amplitude, friction) identifies a friction-dependent crossover strain $\Gamma_C$, at which aging gives way to fluidization. The behavior is reentrant: increasing friction initially suppresses fluidization by stabilizing contacts, but further increase enhances continuous rearrangement due to a larger density of metastable configurations, leading to fluidization through creep. The Van Hove probability distribution and mean squared displacement provide markers for the transition, with exponential-tailed displacement distributions in the aging regime and Gaussian for the fluidized regime.

4. Rheology and Generalized Constitutive Models

Rheological models systematically incorporate friction into the crossover between solid-like and flowing regimes. In discrete element simulations of dry granular systems (Man et al., 2022), the effective friction coefficient and solid fraction obey

$$\mu_{\textrm{eff}} = \mu_s + \frac{\mu_1 - \mu_s}{1 + I_0/I},$$

with $\mu_s$ (static friction), $\mu_1$ (inertial friction), and $I_0$ (transitional inertial number) all rising with increased interparticle friction, shifting the crossover to larger strain rates. The transition point is further unified via a dimensionless "frictional number" $\mathcal{M}$,

$\mathcal{M} = I/\sqrt{\mu_{\rm fl} + \beta \mu_p},$

and, incorporating granular temperature $\Theta$,

$$\left(\frac{\mu_{\textrm{eff}}}{\mu_s}\right)\Theta^{0.5}$$

collapses data across regimes onto a single curve. This formalism establishes that the crossover strain is a function of both microscopic friction and inertial effects, universalizing the behavior in frictional rheology.

In spherocylinder packings (Heussinger, 2021), the shear modulus transitions from large (friction-dependent $g_{\infty}$) to small (frictionless $g_0$) at a crossover strain

$\gamma_c \sim \mu p / (k_t z \phi),$

where $p$ is pressure, $k_t$ tangential stiffness, $z$ contact number, $\phi$ packing fraction. Above $\gamma_c$, sliding friction induces a yield stress $\sigma_y = g_{\infty}\gamma_c$ and shear-thinning behavior ensues.

5. Frictional Crack Propagation, Plasticity, and Phase Transitions

At fracture interfaces and in high-pressure plasticity, friction-dependent crossover strain determines rupture dynamics and transformation thresholds. In stress-controlled frictional cracks (Brener et al., 27 Oct 2024), the amplitude of the singular elastic field is set by the slip velocity rise $\Delta v$ rather than a finite stress drop. The scaling

$K \sim \mu (\Delta v / c_r) \sqrt{H}$

(with $\mu$ shear modulus, $c_r$ crack velocity, $H$ strip height) replaces classical fracture mechanics, requiring a nonmonotonic steady-state friction law to permit coexistence of stick and slip regions across the interface.

In high-pressure plastic strain-induced phase transformations (Levitas et al., 2022), friction at the contact boundaries modifies stress–plastic strain fields, influencing the spatial heterogeneity and magnitude of the crossover strain required to trigger transformations such as the $\alpha$–$\omega$ phase change in zirconium. Coupled experimental–analytical–computational methods quantify the correction to the transformation threshold introduced by friction, demonstrating that while friction affects local distributions of plastic strain, the minimum pressure required for transformation is independent of the specific shear–compression path taken.

6. Strain-dependent Friction in Functional Interfaces and Fluids

Friction-dependent crossover strain appears in the tuning of nanoscale systems and complex fluids. In texture-induced strained graphene (Mescola et al., 2021), the local strain field drives transitions between high and ultra–low friction states (superlubricity), with the friction force exhibiting anisotropy controlled by the direction and magnitude of induced strain, quantified by

$\varepsilon_x = (l - \lambda)/\lambda,$

and experimental fits using $f \propto (L_{\textrm{ext}} + L_0)^{2/3}$ (load-dependent friction).

For molecular fluids (Post et al., 2022), targeted molecular dynamics combined with cumulant expansions of Jarzynski’s identity compute the friction profile as a function of imposed strain and velocity. The generalized Langevin equation

$$m \frac{d^2x}{dt^2} = -\frac{\partial G}{\partial x} - \int K(t, \tau) \dot x(\tau) d\tau + \eta(t) + f(t)$$

encapsulates non-Markovian memory effects; the friction coefficient $\Gamma(v)$ displays a characteristic "crossover velocity" $v_c = (x - x_0)/\tau_C$ beyond which dissipation channels become suppressed and friction decreases. This reflects a crossover strain rate beyond which structural rearrangement cannot fully compensate for applied driving, producing non-Newtonian behavior.

7. Broader Implications and Applications

Friction-dependent crossover strains are critical in determining the mechanical response, aging, phase stability, and transport properties in a wide array of natural and engineered systems. They underlie the nonequilibrium relaxation and compaction in athermal granular assemblies (Yuan et al., 17 Oct 2025), the mechanics of functional interfaces in nanoelectromechanical systems, the rheology of industrial and geological materials, and the control of phase transitions in electronic materials such as nickelates (Cui et al., 2023). The existence of well-defined, friction-controlled crossover points enables predictive modeling, guided experiment, and rational design of materials and devices where precise control of deformation and energy dissipation are mandatory for optimized performance. Furthermore, the mathematical framework and experimental markers developed across this literature establish the friction-dependent crossover strain as a unifying principle in the physics of disordered, driven, and frictional matter.