A Solid-Based Approach for Modeling Simple Yield-Stress Fluids: Rheological Transitions, Overshoot and Relaxation

Abstract: Yield-stress fluids are ubiquitous and encountered in diverse fields ranging from natural muddy flows to industrial applications such as secondary battery electrode slurries and direct ink writing. Despite the proposal of various constitutive equations, few models have been shown to successfully predict both steady and transient rheological behaviors in yield-stress fluids. In this study, a constitutive equation is hereby proposed, offering a comprehensive description of the rheological characteristics observed in simple yield-stress fluids, excluding thixotropy, such as the Carbopol dispersion. The constitutive equation is derived from a Zener-type viscoelastic solid element combined with an additional linear dashpot connected in parallel, together with a nonlinear viscosity model, a flow rule, an evolution equation for the back stress, and the Kroner-Lee decomposition. This combination satisfies the principle of material frame invariance. The proposed model successfully reproduces the rheological characteristics qualitatively in a manner consistent with experimental observations conducted during start-up shear, creep, and stress relaxation tests. In particular, the present viscoelastic solid-based constitutive equation is shown to accurately predict stress overshoot during start-up shear. Importantly, the overshoot is found to originate from a homogeneous mechanism in which normal stress difference enhances the stress invariant and thereby accelerates the plastic response, rather than from isotropic hardening or spatially heterogeneous microstructural evolution. This study is expected to facilitate a deeper understanding of the intricate dynamics governing the flow of yield-stress fluids.

• The paper introduces a solid-based constitutive model that unifies steady-state and transient behaviors in yield-stress fluids.
• It employs a Zener-type viscoelastic solid with a parallel dashpot to capture phenomena like stress overshoot, residual stress, and creep transitions.
• The 3D tensorial extension aligns with experimental data, offering significant insights for industrial applications and computational simulations.

Solid-Based Constitutive Modeling of Yield-Stress Fluids: Framework and Rheological Predictions

Introduction

This paper introduces a constitutive equation founded on a solid-based framework for modeling simple yield-stress fluids, specifically microgel dispersions such as Carbopol. Standard rheological models (e.g., Herschel-Bulkley) capture steady-state phenomena, but fail to predict the full suite of transient behaviors observed experimentally, including stress overshoot during start-up shear, finite residual stress after stress relaxation, and creep transitions. The presented approach integrates viscoelastic solid mechanics with explicit solvent-induced dissipation, capturing both steady and transient rheological characteristics without invoking thixotropy or spatially heterogeneous microstructural dynamics.

Microstructural Motivation and Model Architecture

The microstructure of Carbopol dispersions is conceived as jammed aggregates of swollen microgel particles immersed in interstitial solvent. The proposed model employs a Zener-type viscoelastic solid element to represent the gel phase, enhanced by a linear dashpot in parallel to account for solvent dissipation. This configuration enables decomposition of total strain into elastic (intra-microgel) and plastic (inter-microgel rearrangement) components. The stress evolution is governed by a flow rule prescribing the plastic strain rate, an Armstrong-Frederick type evolution equation for back stress, and a nonlinear viscosity law (Eyring or Carreau-Yasuda models) to account for shear-thinning dynamics.

Mathematical Derivation: 1D and 3D Constitutive Equations

1D Formulation

The 1D model is analytically tractable and serves as a precursor for the tensorial (3D) extension. The additive strain decomposition yields ordinary differential equations for elastic and plastic strains, total stress, and back stress. The viscosity function, central to yielding behavior, is implemented both in stress- and strain-rate-dependent forms (Eyring and Carreau-Yasuda).

Key results from 1D analysis:

• Steady-state stresses as functions of shear rate reproduce the Herschel-Bulkley relation at low shear rates.
• Nonzero fully relaxed stress is predicted for stress relaxation after cessation of shear, matching experimental findings for Carbopol.
• The threshold for solid-to-fluid transition in creep is directly linked to the yield stress extracted from steady-state fits.

However, the 1D model is deficient in capturing stress overshoot during start-up shear, due to monotonic dependence of viscosity on stress difference.

3D Extension

Recognizing the limitations of the 1D representation, the model is generalized to three dimensions by adopting the Kroner-Lee decomposition for finite strains, thereby ensuring material frame invariance and applicability to complex flow kinematics. The tensorial formulation includes a flow rule for the plastic strain-rate tensor, a back-stress evolution equation consistent with continuum mechanics, and nonlinear hyperelasticity for the gel stress. The viscosity function $\nu(\|T_{\text{gel}} - T_B\|)$ is expressed in terms of the second invariant of the stress difference.

Distinct outcomes in the 3D model:

• Stress overshoot in start-up shear emerges naturally due to tensorial coupling, with the normal stress differences amplifying transient plastic response.
• The magnitude and location of stress overshoot are rate-dependent and align with empirical rheological data.
• The fully relaxed stress in stress relaxation experiments is modulated by the pre-shearing rate, in contrast to the fixed stress predicted by the 1D model, matching recent experimental observations.
• Creep transitions remain governed by the yield stress from steady-state fits, maintaining consistency with classical rheological criteria.

Rheological Phenomena: Detailed Numerical and Analytical Results

The solid-based constitutive model accurately reproduces four principal rheological features of simple yield-stress fluids:

1. Start-up Shear Overshoot: The 3D tensorial model predicts stress overshoots observed empirically. This result contradicts earlier models attributing overshoots to thixotropy or isotropic hardening. The overshoot mechanism arises from homogeneous, tensorial stress invariant coupling.
2. Steady-State Behavior: At low shear rates, the model fit is congruent with Herschel-Bulkley dynamics. At higher rates, the emergence of a second Newtonian plateau arises from the parallel dashpot, consistent with experimental observations.
3. Stress Relaxation Plateaus: The model reliably predicts non-zero, shear-rate-dependent residual stress after relaxation, matching the memory effects found in recent measurements.
4. Creep Transition: The transition from solid-like (strain saturation) to fluid-like (continuous strain growth) is governed by yield stress, confirming the material's viscoelastic solid-fluid duality in practical rheometry.

Noteworthy numerical results include the rate-dependent modulation of fully relaxed stress and precise reproduction of start-up overshoots, with the Carreau-Yasuda viscosity providing superior fit flexibility for intermediate shear-rate regimes.

Theoretical and Practical Implications

The study advances the theoretical foundation of yield-stress fluid modeling by:

• Demonstrating that viscoelastic solid-based constitutive equations, grounded in microstructural physical interpretation, suffice to capture all key features of simple yield-stress fluids without recourse to thixotropy or microstructural heterogeneity.
• Clarifying the origin of stress overshoot in start-up shear as a tensorial effect, decoupling it from isotropic hardening phenomena.
• Establishing a general constitutive framework capable of extension to finite element simulations and practical engineering applications, such as battery electrode slurries and direct ink writing, where yield-stress and viscoelasticity interplay is crucial.

The 3D model's ability to capture both steady and transient rheological transitions positions it as a robust tool for quantitative prediction and optimization in industrial processing and complex flow scenarios.

Future developments may involve incorporation of thixotropic effects for more complex fluids, parameter identification strategies for diverse material systems, and integration into large-scale computational rheology platforms.

Conclusion

The proposed solid-based constitutive model provides a unified, microstructurally informed framework for simple yield-stress fluids, reconciling steady-state and transient behaviors within a viscoelastic solid paradigm. The analytical and numerical results confirm alignment with experimental observations for Carbopol dispersions, including overshoot, relaxation plateaus, and creep transitions. The 3D tensorial formulation, satisfying material frame invariance, enables application to general deformation modes and paves the way for advanced rheological modeling in both basic and applied contexts (2604.03467).