Fractional IKH (FIKH) Models

This section documents the Fractional Isotropic-Kinematic Hardening (FIKH) family of models for thixotropic elasto-viscoplastic (TEvp) materials with power-law memory.

Overview

The FIKH family extends the classical Isotropic-Kinematic Hardening (IKH) Models framework by replacing the integer-order structure kinetics with a Caputo fractional derivative. This captures the power-law memory observed in many complex fluids:

• Standard IKH ( \(\alpha\) = 1): Exponential recovery \(\lambda \sim \exp(-t/\tau)\)

• Fractional FIKH ( \(\alpha\) < 1): Power-law recovery \(\lambda \sim t^{\alpha-1}\) at long times

Fractional derivatives introduce a fading memory where recent deformation history affects the current structure more than distant past. This single parameter \(\alpha\) captures a broad distribution of restructuring timescales without requiring multiple modes.

Thermokinematic coupling adds:

• Temperature-dependent yield stress: \(\sigma_y(\lambda, T)\)

• Arrhenius viscosity: \(\eta(T) = \eta_0 \cdot \exp(E_a/RT)\)

• Thermal evolution from plastic dissipation

These models are particularly suited for:

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

• Colloidal gels with hierarchical structure

• Materials exhibiting stretched-exponential recovery

• Systems with thermal feedback (shear heating)

Model Hierarchy

FIKH Family
│
├── FIKH (Single Mode)
│   ├── 12 parameters (base)
│   ├── 20 parameters (with thermal coupling)
│   ├── 22 parameters (full: thermal + isotropic hardening)
│   └── Single fractional structure variable
│
└── FMLIKH (Multi-Mode)
    ├── Per-mode: G_i, η_i, C_i, γ_dyn_i, τ_thix_i, Γ_i
    ├── Shared or per-mode fractional order α
    └── Global yield with distributed kinetics

When to Use FIKH

Experimental Signatures

Use FIKH when you observe:

1. Power-law stress relaxation at long times: \(G(t) \sim t^{-\alpha}\), not exp(-t/\(\tau\))

2. Stretched exponential recovery after shear cessation

3. Broad relaxation spectrum in frequency sweep (Cole-Cole depression)

4. Delayed yielding in creep tests below apparent yield stress

5. Temperature-dependent flow with Arrhenius-like behavior

6. Stress overshoot with long tail in startup (not sharp exponential decay)

Decision Tree

Is recovery exponential (single timescale)?
├── YES → Use MIKH (simpler, faster)
└── NO → Is recovery power-law?
    ├── YES → Use FIKH (single α captures spectrum)
    └── NO → Is there hierarchical structure?
        ├── YES → Use FMLIKH (multiple modes)
        └── NO → Consider SGR or DMT models

Model Comparison

Behavior	Single Mode (FIKH)	Multi-Mode (FMLIKH)
Power-law recovery	✓ Use this	Also works
Single structural population	✓ Use this	Overkill
Broad relaxation spectrum	Limited	✓ Use this
Few parameters needed	✓ Use this	More params
Hierarchical structure	Limited	✓ Use this
When \(\alpha \to 1\) (exponential)	Consider MIKH	Consider ML-IKH

Material-Specific Recommendations

Material	Recommended Model	Typical \(\alpha\)	Key Protocol
Waxy crude oils	FIKH (thermal)	0.5-0.7	Startup at different T
Colloidal gels	FMLIKH	0.3-0.6	Frequency sweep
Food gels	FIKH	0.6-0.8	Creep recovery
Drilling muds	FIKH (thermal)	0.4-0.6	Flow curve + relaxation
Greases	FIKH	0.5-0.7	LAOS + startup
Cement pastes	FMLIKH	0.4-0.6	Multiple rest times

Key Features

Fractional Structure Evolution:

• Caputo derivative captures power-law fading memory

• Single \(\alpha\) parameter spans exponential (\(\alpha=1\)) to strong memory (\(\alpha \to 0\))

• Mittag-Leffler relaxation generalizes the exponential

Armstrong-Frederick Kinematic Hardening:

• Back-stress A tracks deformation history

• Captures Bauschinger effect in cyclic loading

• Dynamic recovery prevents unbounded hardening

Full Thermokinematic Coupling:

• Arrhenius temperature dependence for viscosity

• Structure-temperature yield stress coupling

• Plastic dissipation heating with heat loss

Supported Protocols:

• Flow curve (steady state)

• Startup shear (stress overshoot)

• Stress relaxation (Mittag-Leffler decay)

• Creep (delayed yielding, thermal runaway)

• SAOS (fractional Maxwell moduli)

• LAOS (harmonic generation, Lissajous figures)

Quick Start

Basic FIKH with thermal coupling:

from rheojax.models.fikh import FIKH

# Create model with fractional order α = 0.7
model = FIKH(include_thermal=True, alpha_structure=0.7)

# Set key parameters
model.parameters.set_value("G", 1000.0)
model.parameters.set_value("sigma_y0", 10.0)
model.parameters.set_value("delta_sigma_y", 50.0)
model.parameters.set_value("tau_thix", 100.0)

# Fit to startup data
model.fit(t, stress, test_mode='startup', strain=strain)

# Predict flow curve
sigma = model.predict(gamma_dot, test_mode='flow_curve')

Multi-mode FMLIKH:

from rheojax.models.fikh import FMLIKH

# Create 3-mode model with shared fractional order
model = FMLIKH(n_modes=3, include_thermal=False, shared_alpha=True)

# Set per-mode parameters
for i, tau in enumerate([1.0, 10.0, 100.0], 1):
    model.parameters.set_value(f"tau_thix_{i}", tau)

# Fit to oscillation data
model.fit(omega, G_star, test_mode='oscillation')

Model Documentation

• Fractional Isotropic-Kinematic Hardening (FIKH)
• Fractional Multi-Lambda IKH (FMLIKH)

References

Fractional Calculus:

1. Podlubny, I. (1999). Fractional Differential Equations. Academic Press.

2. Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press.

3. Diethelm, K. (2010). The Analysis of Fractional Differential Equations. Springer.

Fractional Rheology:

4. Jaishankar, A. & McKinley, G.H. (2014). “A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids.” J. Rheol., 58, 1751-1788.

IKH Foundation:

For foundational IKH references (Dimitriou 2014, Geri 2017, Wei 2018), see Isotropic-Kinematic Hardening (IKH) Models.