Isotropic-Kinematic Hardening (IKH) Models

This section documents the Isotropic-Kinematic Hardening (IKH) family of models for thixotropic elasto-viscoplastic (TEvp) materials.

Overview

The IKH family provides comprehensive constitutive equations for complex fluids that exhibit:

  • Yield stress behavior with structure-dependent yielding

  • Thixotropy (time-dependent structure buildup and breakdown)

  • Viscoelasticity (stress relaxation, creep)

  • Kinematic hardening (Bauschinger effect, directional memory)

These models are particularly well-suited for:

  • Waxy crude oils (pipeline restart, cold flow assurance)

  • Drilling fluids and muds (borehole stability, pump circulation)

  • Greases and lubricants (NLGI grades, bearing applications)

  • Colloidal gels (bidisperse systems, hierarchical structure)

  • Structured emulsions (dense emulsions, foams)

  • Thixotropic cements and pastes (self-leveling, 3D printing)

Thixotropy Fundamentals

Thixotropy is the reversible, time-dependent decrease in viscosity under constant shear rate, with subsequent recovery at rest. It arises from competition between microstructural breakdown (shear) and buildup (aging).

Physical Mechanisms:

  • Breakdown: Shear disrupts network bonds, aggregates, or particle structures

  • Buildup (aging): Brownian motion, attractive forces, or reaction kinetics rebuild structure

  • Structure parameter (\(\lambda\)): Dimensionless variable tracking microstructural state (0-1)

Characteristic Experimental Signatures:

  1. Hysteresis loops: Different stress-strain rate curves for increasing vs decreasing shear

  2. Stress overshoot: Peak stress in startup flow before steady-state

  3. Delayed yielding: Time-dependent creep response, viscosity bifurcation

  4. Recovery kinetics: Gradual viscosity increase after shear cessation

Common Kinetic Equation:

\[\frac{d\lambda}{dt} = \underbrace{\frac{1-\lambda}{t_{eq}}}_{\text{aging}} - \underbrace{a\lambda|\dot{\gamma}|^c/t_{eq}}_{\text{rejuvenation}}\]

where \(t_{eq}\) is equilibration time, \(a\) is breakdown rate, and \(c\) is shear-rate exponent.

Model Selection Guide:

Model Family

Best For

Key Features

DMT Thixotropic Models

Industrial fluids

Simple kinetics, exponential/HB closures

Isotropic-Kinematic Hardening (IKH) Models

Metal plasticity

Hardening/softening, yield surface evolution

Fluidity Models

Yield stress fluids

Fluidity evolution, Saramito viscoelasticity

Experimental Protocols for Thixotropic Materials:

  • Three-interval test: Low rate → high rate → low rate to measure breakdown/recovery

  • Step-rate tests: Instantaneous rate changes to probe kinetics

  • Startup flow: Constant rate from rest to observe overshoot

  • Creep: Constant stress to observe delayed yielding

Both models include comprehensive Industrial Applications sections with typical parameter ranges from field studies, and Parameter Estimation Methods covering sequential fitting, multi-start optimization, Bayesian inference, and regularization techniques for ill-conditioned problems

Model Hierarchy

IKH Family
│
├── MIKH (Single Mode)
│   └── 11 parameters
│   └── Single structural timescale
│   └── Exponential recovery
│
└── ML-IKH (Multi-Mode)
    ├── Per-Mode Yield: 7N+1 parameters
    │   └── N independent yield surfaces
    │   └── Parallel mechanical connection
    │
    └── Weighted-Sum Yield: 6+3N parameters
        └── Single global yield surface
        └── Distributed kinetics

When to Use Which Model

Behavior

Single Mode (MIKH)

Multi-Mode (ML-IKH)

Exponential recovery

✓ Use this

Overkill

Stretched-exponential recovery

Poor fit

✓ Use this

Single structural population

✓ Use this

Overkill

Hierarchical structure

Poor fit

✓ Use this

Few parameters needed

✓ Use this

More params

Complex aging behavior

Limited

✓ Use this

Key Features

Physical Foundation:

  • Built on classical plasticity theory (Armstrong-Frederick kinematic hardening)

  • Incorporates structural kinetics for thixotropy (Goodeve-Moore framework)

  • Maxwell viscoelasticity for liquid-like long-time behavior

  • Perzyna regularization for smooth yield transitions

Industrial Applications:

  • Quantitative parameter ranges from field studies and laboratory characterization

  • Application-specific guidance for pipeline operations, drilling, lubrication

  • Mode selection rules for multi-timescale materials (\(\beta\) to \(N\) mapping)

Parameter Estimation:

  • Sequential fitting strategies exploiting timescale separation

  • Multi-start global optimization for complex parameter landscapes

  • Bayesian inference with NLSQ warm-start and prior selection guidance

  • Regularization methods for correlated parameters

Numerical Implementation:

  • Two formulations: ODE (for creep/relaxation) and return mapping (for startup/LAOS)

  • JAX-accelerated kernels for efficient computation

  • Full Bayesian inference support via NumPyro

Supported Protocols:

  • Flow curve (steady state)

  • Startup shear

  • Stress relaxation

  • Creep

  • Small amplitude oscillatory shear (SAOS)

  • Large amplitude oscillatory shear (LAOS)

Quick Start

Single-mode model:

from rheojax.models import MIKH

model = MIKH()
model.parameters.set_value("G", 1000.0)
model.parameters.set_value("sigma_y0", 20.0)
model.parameters.set_value("tau_thix", 10.0)

# Predict flow curve
sigma = model.predict_flow_curve(gamma_dot)

Multi-mode model:

from rheojax.models import MLIKH

model = MLIKH(n_modes=3, yield_mode='weighted_sum')
model.parameters.set_value("G", 1000.0)
model.parameters.set_value("sigma_y0", 20.0)

# Set distributed timescales
for i, tau in enumerate([0.1, 1.0, 10.0], 1):
    model.parameters.set_value(f"tau_thix_{i}", tau)

Model Documentation

References

  1. Dimitriou, C. J. and McKinley, G. H. (2014). “A comprehensive constitutive law for waxy crude oil: a thixotropic yield stress fluid.” Soft Matter, 10(35), 6619-6644. DOI: 10.1039/C4SM00578C PDF

  2. Geri, M., Venkatesan, R., Sambath, K., and McKinley, G. H. (2017). “Thermokinematic memory and the thixotropic elasto-viscoplasticity of waxy crude oils.” J. Rheol., 61(3), 427-454. DOI: 10.1122/1.4978259 PDF

  3. Wei, Y., Solomon, M. J., and Larson, R. G. (2018). “A multimode structural kinetics constitutive equation for the transient rheology of thixotropic elasto-viscoplastic fluids.” J. Rheol., 62(1), 321-342. DOI: 10.1122/1.4996752 PDF