DMT Thixotropic Models

This section documents the de Souza Mendes-Thompson (DMT) family of models for thixotropic yield-stress materials.

Overview

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

  • Yield stress behavior with structure-dependent yielding

  • Thixotropy (time-dependent structure buildup and breakdown)

  • Optional viscoelasticity (Maxwell backbone for stress overshoot and relaxation)

  • Shear banding (via nonlocal diffusion extension)

These models are particularly well-suited for:

  • Colloidal gels and suspensions

  • Structured emulsions and foams

  • Drilling fluids and muds

  • Waxy crude oils

  • Thixotropic pastes and slurries

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

Model Hierarchy

DMT Family
│
├── DMTLocal (Homogeneous)
│   ├── closure="exponential"
│   │   └── Smooth viscosity transition
│   │
│   └── closure="herschel_bulkley"
│       └── Explicit yield stress
│
└── DMTNonlocal (Spatial)
    └── Structure diffusion for shear banding
    └── Couette/channel flow profiles

When to Use Which Model

Behavior

DMTLocal

DMTNonlocal

Homogeneous flow

✓ Use this

Overkill

Shear banding

Cannot capture

✓ Use this

Stress overshoot

✓ (with elasticity)

✓ (with elasticity)

Delayed yielding

✓ Use this

✓ Use this

Few parameters

✓ Use this

More params

Key Features

Physical Foundation:

  • Structure parameter \(\lambda\) ∈ [0, 1] tracks microstructural organization

  • Competing buildup (aging) and breakdown (shear-induced) kinetics

  • Multiple viscosity closures: exponential or Herschel-Bulkley

  • Optional Maxwell backbone for viscoelastic effects

  • Fluidity interpretation with cooperativity length scale

Theoretical Extensions:

  • Fluidity-Maxwell formulation: Jeffreys/Oldroyd-B backbone with structure-dependent relaxation and retardation times for true stress relaxation and SAOS moduli

  • Nonlocal fluidity: Spatial diffusion for shear band regularization with cooperativity length \(\xi \sim \sqrt{D_{\lambda} \cdot t_{\text{eq}}}\)

  • Complete protocol equations: Full mathematical derivations for all rheological tests with closed-form solutions where available

Numerical Implementation:

  • JAX-accelerated kernels with jax.lax.scan integration

  • Papanastasiou regularization for smooth yield behavior

  • Full Bayesian inference support via NumPyro

Supported Protocols:

DMTLocal (all 6 protocols):

  • Flow curve (steady state) with viscosity bifurcation

  • Startup shear with stress overshoot mechanism

  • Stress relaxation after cessation (arrested by structure recovery)

  • Creep with delayed yielding and avalanche effect

  • Small amplitude oscillatory shear (SAOS) with Maxwell moduli

  • Large amplitude oscillatory shear (LAOS) with Fourier/Chebyshev analysis

DMTNonlocal (3 protocols):

  • Flow curve (steady state)

  • Startup shear

  • Creep

Quick Start

Exponential closure:

from rheojax.models import DMTLocal

model = DMTLocal(closure="exponential", include_elasticity=True)
model.fit(gamma_dot, stress, test_mode='flow_curve')

Herschel-Bulkley closure:

from rheojax.models import DMTLocal

model = DMTLocal(closure="herschel_bulkley", include_elasticity=True)
model.fit(gamma_dot, stress, test_mode='flow_curve')

Nonlocal for shear banding:

from rheojax.models import DMTNonlocal

model = DMTNonlocal(closure="exponential", n_points=51, gap_width=1e-3)
result = model.simulate_steady_shear(gamma_dot_avg=10.0, t_end=500.0)
banding = model.detect_banding(result, threshold=0.1)

Model Documentation

References

  1. de Souza Mendes, P. R. (2009). “Modeling the thixotropic behavior of structured fluids.” J. Non-Newtonian Fluid Mech., 164, 66-75.

  2. de Souza Mendes, P. R. & Thompson, R. L. (2012). “A critical overview of elasto-viscoplastic thixotropic modeling.” J. Non-Newtonian Fluid Mech., 187-188, 8-15.

  3. de Souza Mendes, P. R. & Thompson, R. L. (2013). “A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids.” Rheologica Acta, 52(7), 673-694. https://doi.org/10.1007/s00397-013-0699-1