Reversible viscoelasticity and irreversible elastoplasticity in the power law creep and yielding of gels and fibre network materials under stress

Technical Summary: Reversible Viscoelasticity and Irreversible Elastoplasticity in Power Law Creep and Yielding of Gels and Fibre Network Materials

Problem Statement
Network materials, ranging from biological cytoskeletons and hydrogels to paper and colloidal gels, frequently exhibit power law creep under constant stress, where the strain rate decays as $\dot{\gamma} \sim t^n$ ($n < 0$) and strain grows sublinearly. While this behavior is ubiquitous in soft matter, the underlying physical mechanisms remain debated. Specifically, it is unclear whether power law creep arises from purely reversible viscoelastic deformations (non-affine rearrangements) or irreversible elastoplastic damage (bond breakage). Furthermore, the conditions under which these materials transition from slow creep to catastrophic yielding and failure, and the extent to which they recover their original shape upon unloading, require systematic characterization. This work addresses these questions using a minimal computational model of athermal fibre networks.

Methodology
The authors employ numerical simulations of a minimal athermal disordered network model in $d=2$ spatial dimensions.

• Model Construction: Networks are generated by randomly packing bidisperse disks, connecting overlapping centers with Hookean springs, and iteratively pruning to achieve a target average coordination number $Z$. Each spring is linear elastic up to a failure strain $b$, beyond which it breaks irreversibly.
• Rheological Protocol: The study utilizes a constant imposed shear stress $\Sigma$ protocol, which is less common in simulations than constant strain rate protocols.
• Numerical Algorithm: To maintain constant stress, the authors developed and validated two algorithms. Algorithm I iteratively adjusts the strain rate via a Newton-Raphson method to match the target stress. Algorithm II imposes a strain rate $\dot{\gamma} = (\Sigma - \sigma)/\eta$ using a fictitious Newtonian solvent of viscosity $\eta$. The authors demonstrate that both algorithms converge to the same results in the limit $\eta \to 0$, subsequently using the more efficient Algorithm II with $\eta = 0.16$.
• Parameters: Simulations are performed with $N=16,000$ nodes. Key variables include network connectivity $Z$, filament breakage threshold $b$, and applied stress $\Sigma$.

Key Contributions and Results

1. Purely Viscoelastic Regime (No Filament Breakage):

   • Stiff Networks ($Z > 4$): For networks above the Maxwell rigidity threshold ($Z > 2d = 4$), the material behaves as a Hookean solid at long times, reaching a steady-state strain $\gamma_{ss} \propto \Sigma$. The transient creep is purely viscoelastic, driven by non-affine internal deformations.
   • Floppy Networks ($Z < 4$): Networks below the rigidity threshold exhibit a "floppy" response where a vanishingly small stress induces a finite strain. The strain rate decays as $\dot{\gamma} \sim t^{-2}$, and the strain saturates to a constant.
   • Marginal Connectivity ($Z \approx 4$): A unique regime emerges for marginally connected networks ($Z \gtrsim 4$). Here, the system exhibits sustained power law creep with $\dot{\gamma} \sim t^{-0.5}$ and $\gamma \sim t^{0.5}$ over a prolonged duration before saturating. This behavior is purely viscoelastic and arises from non-affine deformations without any bond breaking.

2. Elastoplastic Regime (With Filament Breakage):

   • Power Law Creep in Sub-Rigid Gels: When filament breakage is allowed ($b < \infty$), the range of conditions supporting power law creep expands. Specifically, initially floppy networks ($Z < 4$) that would otherwise show rapid saturation in the purely viscoelastic case now exhibit a prolonged regime of power law creep ($\dot{\gamma} \sim t^n$ with $-1 < n < 0$) driven by the accumulation of plastic damage.
   • Yielding and Failure: For stresses exceeding a critical yield stress $\Sigma_y$, the network undergoes catastrophic failure after a delay time $\tau_y$. The yield stress scales roughly linearly with the breakage threshold $b$. The delay time $\tau_y$ increases dramatically as $\Sigma$ approaches $\Sigma_y$ from above.
   • Validation: The yield stress measured in stress-imposed simulations agrees with the limit of zero strain rate in strain-imposed startup simulations, confirming the robustness of the measured yield point.

3. Strain Recovery:

   • The study quantifies the recoverable vs. unrecoverable strain upon unloading.
   • Stiff Networks ($Z > 4$): Full shape recovery occurs if the stress is removed before significant plastic damage accumulates. Remarkably, even if a small fraction of springs break, the network can fully recover its shape provided the remaining connectivity stays above the rigidity threshold ($Z > 4$).
   • Floppy Networks ($Z < 4$): These networks exhibit partial or no recovery even at small strains, as they lack the elastic backbone required for recoil. Unrecoverable strain sets in immediately upon loading, even with negligible bond breakage.

4. Damage Localization:

   • Visual analysis of failure modes reveals distinct patterns. In initially stiff networks, plastic damage localizes into macroscopic bands spanning the system. In contrast, damage in initially floppy networks is distributed more diffusively across the material.

Significance and Claims
The paper claims to identify two distinct physical mechanisms capable of generating power law creep in network materials:

1. Viscoelastic Mechanism: Arising in marginally connected networks ($Z \gtrsim 4$) due to non-affine deformations without damage.
2. Elastoplastic Mechanism: Arising in sub-rigid networks ($Z < 4$) due to the progressive accumulation of bond breakage.

The authors conclude that while these two mechanisms explain power law creep in their minimal model, they do not claim these are the only mechanisms responsible for the phenomenon observed in the wide variety of experimental soft materials. They modestly suggest that other factors, such as Rouse-like chain motion, fractal network structures, or thermal linker dynamics, may also contribute to the ubiquity of power law creep in real-world systems. The work provides a computational framework for distinguishing between reversible viscoelasticity and irreversible elastoplasticity in the context of creep and yielding.