Model Families Overview

Learning Objectives

After completing this section, you will be able to:

  1. Distinguish between classical, fractional, and flow model families

  2. Identify which family applies to your experimental data

  3. Understand the advantages and limitations of each family

  4. Select candidate models within each family

  5. Recognize when to upgrade from simple to complex models

Prerequisites

The Model Families

RheoJAX provides 53 rheological models organized into families based on their theoretical foundations:

Model Families Overview

Category

Model Family

Physical Basis

classical

Maxwell, Zener, SpringPot

Linear viscoelastic elements

fractional_maxwell

FractionalMaxwellGel, etc.

Fractional calculus extensions (Maxwell-based)

fractional_zener

FractionalZenerSS, etc.

Fractional calculus extensions (Zener-based)

fractional_advanced

Burgers, Jeffreys, Poynting-Thomson

Multi-element fractional models

flow

PowerLaw, Carreau, Bingham, HB

Viscosity vs. shear rate

multi_mode

GeneralizedMaxwell

Discrete relaxation spectrum

sgr

SGRConventional, SGRGeneric

Trap model for amorphous materials

fluidity

FluidityLocal, FluidityNonlocal

Cooperative flow models

fluidity_saramito

FluiditySaramitoLocal, FluiditySaramitoNonlocal

Tensorial EVP with thixotropic fluidity

epm

LatticeEPM, TensorialEPM

Lattice/Tensorial elasto-plasticity

ikh

MIKH, MLIKH

Isotropic kinematic hardening (thixotropy)

hl

HebraudLequeux

Mean-field soft matter model

stz

STZConventional

Shear transformation zone theory

spp_laos

SPPYieldStress

Sequence of Physical Processes (LAOS)

giesekus

GiesekusSingleMode, GiesekusMultiMode

Nonlinear viscoelastic polymer (tensor ODE)

fikh

FIKHLocal, FMLIKHLocal

Fractional isotropic-kinematic hardening

dmt

DMTLocal, DMTNonlocal

Structure-parameter thixotropy (de Souza Mendes)

itt_mct

ITTMCTSchematic, ITTMCTIsotropic

Mode-coupling theory (dense suspensions)

tnt

SingleMode, Cates, LoopBridge, MultiSpecies, StickyRouse

Transient network theory (5 variants)

vlb

VLBLocal, MultiNetwork, Variant, Nonlocal

Distribution tensor network (4 variants)

hvm

HVMLocal

Hybrid Vitrimer Model (3 subnetworks)

hvnm

HVNMLocal

Vitrimer Nanocomposite (4 subnetworks)

Quick Reference Table

Family

Test Modes

Material Types

Key Feature

Classical

SAOS, Relaxation, Creep

Simple liquids/solids

Exponential decay

Fractional

SAOS, Relaxation, Creep

Complex materials

Power-law relaxation

Flow

Steady Shear

Nonlinear fluids

Shear thinning/thickening

SGR

SAOS, Relaxation

Soft glasses

Noise temperature \(x\)

Fluidity

SAOS, Flow

Cooperative systems

Spatial correlations

EPM

Relaxation, Creep, Startup

Elasto-plastic solids

Plastic rearrangements

IKH

SAOS, Relaxation, Creep

Thixotropic materials

Kinematic hardening

HL

SAOS, Relaxation

Soft matter

Mean-field dynamics

STZ

SAOS, Relaxation, Flow

Amorphous solids

Shear transformation

SPP

LAOS

Yield stress fluids

Physical process decomposition

Giesekus

All 6 protocols

Polymer solutions/melts

Nonlinear shear thinning + \(N_1\)

Saramito

All 6 protocols

EVP fluids

Tensorial yield + thixotropy

FIKH

All 6 protocols

Complex thixotropic

Fractional memory + hardening

DMT

All 6 protocols

Structured fluids

Structure parameter \(\lambda\)

ITT-MCT

All 6 protocols

Dense colloids, glasses

Memory kernel, glass transition

TNT

All 6 protocols

Associative polymers

Network attachment/detachment

VLB

All 6 protocols

Transient networks

Distribution tensor \(\boldsymbol{\mu}\)

HVM

All 6 protocols

Vitrimers

Bond exchange reactions

HVNM

All 6 protocols

Filled vitrimers

Interphase + Guth-Gold

Family 1: Classical Viscoelastic Models

Who they’re for: Simple polymers, dilute solutions, basic characterization

Test modes: SAOS, stress relaxation, creep

Key characteristic: Exponential relaxation (single or discrete timescales)

The Models

  1. Maxwell (2 parameters)

    • Type: Viscoelastic liquid

    • Equation: \(G(t) = G_0 \exp(-t/\tau)\)

    • Use for: Single relaxation time, simple liquids

    • Example: Low molecular weight polymer melts

  2. Zener (3 parameters)

    • Type: Viscoelastic solid (Standard Linear Solid)

    • Equation: \(G(t) = G_e + G_m \exp(-t/\tau)\)

    • Use for: Materials with equilibrium modulus (crosslinked)

    • Example: Lightly crosslinked rubbers

  3. SpringPot (2 parameters)

    • Type: Pure fractional element (bridges solid and liquid)

    • Equation: \(G(t) \sim t^{-\alpha}\)

    • Use for: Power-law behavior, conceptual studies

    • Example: Critical gels

When to Use Classical Models

✅ Use classical models when:

  • Material is simple (monodisperse polymer, dilute solution)

  • Relaxation appears exponential in log-log plot

  • You need quick characterization

  • Material is well-described by single timescale

❌ Avoid classical models when:

  • Relaxation curve is not straight in semi-log plot (exponential)

  • Material has broad molecular weight distribution

  • Gel-like or power-law behavior observed

  • Fitting fails or requires many Maxwell elements in parallel

Example: Fitting Maxwell Model

from rheojax.models import Maxwell

# Stress relaxation data
model = Maxwell()
model.fit(time, G_t, test_mode='relaxation')

# Parameters
G0 = model.parameters.get_value('G0')    # Modulus (Pa)
eta = model.parameters.get_value('eta')  # Viscosity (Pa·s)
tau = eta / G0                            # Relaxation time (s)

For detailed equations: Classical Viscoelastic Models

Family 2: Fractional Viscoelastic Models

Who they’re for: Complex materials with broad relaxation spectra

Test modes: SAOS, stress relaxation, creep

Key characteristic: Power-law relaxation via fractional derivatives

Why Fractional Models?

Real materials often have distributions of relaxation times due to:

  • Polydispersity (molecular weight distribution)

  • Branching and entanglements

  • Fractal or heterogeneous structure

Fractional calculus captures these distributions with a single parameter \(\alpha\) (fractional order).

Power-law signatures:

  • Relaxation: \(G(t) \sim t^{-\alpha}\)

  • Oscillation: \(G', G'' \sim \omega^\alpha\) (parallel scaling)

The 11 Fractional Models

Liquids (zero equilibrium modulus):

  1. Fractional Maxwell Liquid (FML) — Most common for polymer melts

  2. Fractional Maxwell Model (FMM) — Generalized Maxwell

  3. Fractional Jeffreys — Retardation + relaxation

Solids (finite equilibrium modulus):

  1. Fractional Zener Solid-Solid (FZSS) — Two solid elements

  2. Fractional Kelvin-Voigt (FKV) — Solid with fractional damping

  3. Fractional KV-Zener — Extended Kelvin-Voigt

  4. Fractional Zener Liquid-Liquid (FZLL) — Two liquid elements

  5. Fractional Zener Solid-Liquid (FZSL) — Hybrid

Gels (power-law across all frequencies):

  1. Fractional Maxwell Gel (FMG) — Critical gel

  2. Fractional Burgers — Complex gel behavior

  3. Fractional Poynting-Thomson — Extended gel model

When to Use Fractional Models

✅ Use fractional models when:

  • Log-log plot shows parallel \(G'\) and \(G''\) lines (power-law)

  • Material has broad molecular weight distribution

  • Relaxation is NOT purely exponential

  • Classical models fail to capture curvature

  • You observe gel-like behavior

❌ Avoid fractional models when:

  • Material is simple (use classical first)

  • You have very few data points (< 10)

  • Single exponential fits perfectly well

The Fractional Order Parameter (\(\alpha\))

\(\alpha\) characterizes the breadth of the relaxation spectrum:

  • \(\alpha \to 1\): Narrow spectrum (nearly liquid-like, approaches Maxwell)

  • \(\alpha \approx 0.5\): Broad spectrum (gel-like, power-law)

  • \(\alpha \to 0\): Narrow spectrum (nearly solid-like, approaches elastic)

Typical values:

  • Monodisperse polymers: \(\alpha = 0.8 - 0.95\)

  • Polydisperse polymers: \(\alpha = 0.5 - 0.7\)

  • Gels: \(\alpha = 0.3 - 0.6\)

  • Soft glassy materials: \(\alpha = 0.1 - 0.3\)

Example: Fitting Fractional Zener Solid-Solid

from rheojax.models.fractional_zener_ss import FractionalZenerSolidSolid

# SAOS frequency sweep (omega, [G', G"])
model = FractionalZenerSolidSolid()
model.fit(omega, G_star, test_mode='oscillation')

# Parameters (automatic smart initialization in v0.2.0)
Ge = model.parameters.get_value('Ge')           # Equilibrium modulus
alpha = model.parameters.get_value('alpha')     # Fractional order
tau = model.parameters.get_value('tau_alpha')   # Characteristic time

print(f"Gel-like character (α): {alpha:.3f}")

For detailed equations: Fractional Viscoelastic Models

For fractional calculus background: Fractional Viscoelasticity: Mathematical Reference

Family 3: Flow Models

Who they’re for: Processing, formulation, quality control

Test modes: Steady shear flow (rotation)

Key characteristic: Nonlinear viscosity \(\eta(\dot{\gamma})\)

Why Flow Models?

Most complex fluids exhibit shear-dependent viscosity:

  • Shear thinning: \(\eta\) decreases with shear rate (polymers, suspensions)

  • Shear thickening: \(\eta\) increases with shear rate (dense suspensions)

  • Yield stress: Material doesn’t flow below critical stress (pastes, gels)

These are nonlinear effects not captured by linear viscoelastic models.

The 6 Flow Models

Newtonian and Shear Thinning:

  1. PowerLaw (2 parameters) — Simple shear thinning

    • \(\eta(\dot{\gamma}) = K \dot{\gamma}^{n-1}\)

    • \(n < 1\): Shear thinning

    • \(n = 1\): Newtonian

  2. Carreau (4 parameters) — Shear thinning with Newtonian plateaus

    • \(\eta(\dot{\gamma}) = \eta_\infty + (\eta_0 - \eta_\infty) [1 + (\lambda \dot{\gamma})^2]^{(n-1)/2}\)

    • Most flexible for polymers

  3. Carreau-Yasuda (5 parameters) — Extended Carreau

  4. Cross (4 parameters) — Alternative to Carreau

Yield Stress:

  1. Bingham (2 parameters) — Newtonian above yield stress

    • \(\sigma = \sigma_y + \eta_p \dot{\gamma}\) (for \(\sigma > \sigma_y\))

    • Use for: Drilling fluids, simple pastes

  2. Herschel-Bulkley (3 parameters) — Power-law above yield stress

    • \(\sigma = \sigma_y + K \dot{\gamma}^n\)

    • Use for: Soft solids, yield-stress fluids

When to Use Flow Models

✅ Use flow models when:

  • You have steady shear data (shear rate vs. viscosity or stress)

  • Material exhibits shear thinning or shear thickening

  • You need to predict processing behavior (extrusion, pumping)

  • Material has yield stress

❌ Avoid flow models when:

  • You have oscillation or relaxation data (use viscoelastic models)

  • Material is in linear regime

  • You need to predict elastic behavior (\(G'\))

Example: Fitting PowerLaw Model

from rheojax.models import PowerLaw

# Steady shear data (shear rate, viscosity)
model = PowerLaw()
model.fit(shear_rate, viscosity, test_mode='rotation')

# Parameters
K = model.parameters.get_value('K')     # Consistency (Pa·s^n)
n = model.parameters.get_value('n')     # Flow index

if n < 1:
    print(f"Shear thinning: n = {n:.3f}")
elif n > 1:
    print(f"Shear thickening: n = {n:.3f}")
else:
    print("Newtonian fluid")

For detailed equations: Flow Curve Models

Family 4: Soft Glassy Rheology (SGR) Models

Who they’re for: Soft glassy materials (foams, emulsions, pastes, colloidal suspensions)

Test modes: SAOS, stress relaxation

Key characteristic: Noise temperature \(x\) parameterizes material state

The SGR models are based on Sollich’s trap model (1998), treating soft glassy materials as ensembles of mesoscopic elements that hop between energy traps.

The 2 SGR Models

  1. SGR Conventional (3 parameters: \(x, G_0, \tau_0\))

    • Original Sollich formulation

    • \(x < 1\): Glass (aging), \(x > 1\): Ergodic (flows)

    • Use for: Foams, emulsions, pastes

  2. SGR GENERIC (3 parameters)

    • Thermodynamically consistent (Fuereder & Ilg 2013)

    • Better stability near \(x \to 1\)

    • Use for: Systems near glass transition

For detailed equations: SGR Conventional (Soft Glassy Rheology) — Handbook

Family 5: Fluidity Models

Who they’re for: Materials exhibiting cooperative flow

Test modes: SAOS, steady shear flow

Key characteristic: Fluidity field describes local flow propensity

The 2 Fluidity Models

  1. Fluidity Local (2-3 parameters)

    • Local fluidity dynamics

    • Use for: Simple shear-thinning materials

  2. Fluidity Nonlocal (3-4 parameters)

    • Includes spatial correlations via cooperativity length

    • Use for: Systems with cooperative rearrangements

For detailed equations: Fluidity Local (Homogeneous Fluidity Model) — Handbook

Family 6: Elasto-Plastic Models (EPM)

Who they’re for: Yield stress fluids, amorphous solids, metallic glasses

Test modes: Stress relaxation, creep, startup shear, flow curves

Key characteristic: Plastic rearrangements via local yielding events

The 2 EPM Models

  1. Lattice EPM (multiple parameters)

    • Discrete lattice of mesoscopic blocks

    • Tracks stress redistribution after plastic events

    • Use for: Granular materials, dense suspensions

  2. Tensorial EPM (multiple parameters)

    • Full tensorial stress formulation

    • Use for: Complex loading conditions

For detailed equations: Elasto-Plastic Models (EPM)

Family 7: Isotropic Kinematic Hardening (IKH)

Who they’re for: Thixotropic materials with structural recovery

Test modes: SAOS, relaxation, creep

Key characteristic: Structure parameter evolves with deformation

The 2 IKH Models

  1. MIKH (Modified IKH) — Standard thixotropic model

    • Structure builds up at rest, breaks down under shear

    • Use for: Drilling muds, waxy crude oils

  2. MLIKH (ML-enhanced IKH) — Machine learning augmented

    • Neural network-enhanced constitutive relations

    • Use for: Complex thixotropic behavior

For detailed equations: Maxwell-Isotropic-Kinematic Hardening (MIKH)

Family 8: Hébraud-Lequeux (HL) Model

Who they’re for: Soft glassy materials, dense suspensions

Test modes: SAOS, relaxation

Key characteristic: Mean-field dynamics for plastic flow

The HL Model (3-4 parameters)

  • Mean-field approach to soft matter flow

  • Describes stress distribution evolution

  • Use for: Dense colloidal suspensions, soft glasses

For detailed equations: Hébraud–Lequeux (HL) Model — Handbook

Family 9: Shear Transformation Zone (STZ) Models

Who they’re for: Metallic glasses, amorphous solids

Test modes: SAOS, relaxation, flow

Key characteristic: STZ density evolves with deformation

The STZ Model (4+ parameters)

  • Based on Falk-Langer STZ theory

  • Tracks shear transformation zone population

  • Use for: Metallic glasses, bulk amorphous materials

For detailed equations: Shear Transformation Zone (STZ)

Family 10: SPP Models (LAOS Analysis)

Who they’re for: Yield stress fluids characterized via LAOS

Test modes: Large-amplitude oscillatory shear (LAOS)

Key characteristic: Sequence of Physical Processes decomposition

The SPP Model

  1. SPP Yield Stress — LAOS-based yield stress analysis

    • Decomposes stress into elastic storage + plastic dissipation

    • Extracts yield stress from oscillatory data

    • Use for: Gels, pastes, structured fluids

For detailed equations: SPP Yield Stress Model

Family 11: Giesekus Models

Who they’re for: Polymer solutions and melts exhibiting shear thinning and normal stresses

Test modes: All 6 (flow curve, SAOS, startup, relaxation, creep, LAOS)

Key characteristic: Nonlinear tensor ODE with mobility parameter \(\alpha\)

The 2 Giesekus Models

  1. Giesekus Single-Mode (3 parameters: \(G, \lambda_1, \alpha\))

    • Upper-convected Maxwell + anisotropic drag (\(\alpha\) controls nonlinearity)

    • \(\alpha = 0\): recovers upper-convected Maxwell; \(\alpha = 0.5\): maximum shear thinning

    • Predicts first and second normal stress differences (\(N_1, N_2\))

    • Use for: Polymer solutions, dilute/semi-dilute systems

  2. Giesekus Multi-Mode (\(3N\) parameters)

    • N parallel Giesekus elements with solvent viscosity

    • Use for: Polydisperse polymers, entangled melts

For detailed equations: Giesekus Nonlinear Viscoelastic Models

For tutorial: ODE-Based Constitutive Models

Family 12: Fluidity-Saramito EVP Models

Who they’re for: Elastoviscoplastic materials with thixotropy

Test modes: All 6 protocols

Key characteristic: Tensorial stress with Von Mises yield + thixotropic fluidity evolution

The 2 Saramito-Fluidity Models

  1. Fluidity-Saramito Local (7-8 parameters)

    • Tensorial stress \([\tau_{xx}, \tau_{yy}, \tau_{xy}]\) with Von Mises yielding \(\alpha = \max(0, 1 - \tau_y/|\tau|)\)

    • Coupling modes: “minimal” (\(\lambda = 1/f\)) or “full” (\(\lambda + \tau_y(f)\) aging yield)

    • Predicts normal stresses \(N_1\), stress overshoot, creep bifurcation

    • Use for: Yield stress fluids, carbopol gels, soft solids

  2. Fluidity-Saramito Nonlocal (8-9 parameters + n_points)

    • Includes spatial cooperativity for shear banding

    • Use for: Systems showing heterogeneous flow

For detailed equations: Fluidity-Saramito EVP Model

For tutorial: ODE-Based Constitutive Models

Family 13: FIKH (Fractional IKH) Models

Who they’re for: Complex thixotropic materials with long-term memory

Test modes: All 6 protocols

Key characteristic: Fractional derivatives + kinematic hardening for power-law memory

The 2 FIKH Models

  1. FIKH Local — Fractional IKH

    • Combines structure-parameter thixotropy with fractional Caputo derivatives

    • Power-law memory captures long-time relaxation tails

    • Use for: Waxy crude oils, drilling muds with complex aging

  2. FMLIKH Local — Fractional Modified Leonov IKH

    • Extended fractional variant with Leonov constitutive equation

    • Use for: Highly nonlinear thixotropic materials

For detailed equations: Fractional Multi-Lambda IKH (FMLIKH)

For tutorial: ODE-Based Constitutive Models

Family 14: DMT (de Souza Mendes-Thompson) Models

Who they’re for: Thixotropic materials with structure buildup and breakdown

Test modes: All 6 protocols

Key characteristic: Structure parameter \(\lambda \in [0, 1]\) with viscosity closure

The 2 DMT Models

  1. DMT Local (5-7 parameters)

    • Structure kinetics: \(d\lambda/dt = (1-\lambda)/t_{eq} - a\lambda|\dot{\gamma}|^c/t_{eq}\)

    • Two closures: “exponential” (smooth) or “herschel_bulkley” (explicit yield)

    • Optional Maxwell elasticity for stress overshoot and SAOS

    • Use for: Drilling muds, waxy crude oils, food products

  2. DMT Nonlocal (6-8 parameters + n_points)

    • Structure diffusion \(D_\lambda \nabla^2 \lambda\) for spatial heterogeneity

    • Use for: Shear-banding thixotropic materials

For detailed equations: DMT Thixotropic Models

For tutorial: Thixotropy and Yield Stress Analysis

Family 15: ITT-MCT (Mode-Coupling Theory) Models

Who they’re for: Dense colloidal suspensions and glass-forming systems

Test modes: All 6 protocols

Key characteristic: Memory kernel from microscopic pair correlations; glass transition

The 2 ITT-MCT Models

  1. ITT-MCT Schematic (\(F_{12}\)) (5-6 parameters)

    • Memory kernel \(m(\Phi) = v_1 \Phi + v_2 \Phi^2\) with strain decorrelation

    • Glass transition at \(v_2 = 4\): \(\epsilon = (v_2 - 4)/4\)

    • Semi-quantitative; first JIT compile takes 30-90s

    • Use for: Generic glass-forming systems, fast exploration

  2. ITT-MCT Isotropic (4-5 parameters)

    • Uses Percus-Yevick structure factor \(S(k)\) for quantitative predictions

    • Volume fraction \(\varphi\) as primary control parameter

    • Use for: Hard-sphere colloids, quantitative comparison with experiments

For detailed equations: ITT-MCT Models

For tutorial: Dense Suspensions and Glassy Materials

Family 16: TNT (Transient Network Theory) Models

Who they’re for: Associative polymers, wormlike micelles, telechelic polymers, biological gels

Test modes: All 6 protocols

Key characteristic: Dynamic crosslinks with attachment/detachment kinetics

The 5 TNT Models

  1. TNT SingleMode (3 params) — Simplest transient network

  2. TNT Cates (3-4 params) — Living polymers (wormlike micelles)

  3. TNT LoopBridge (4-5 params) — Telechelic loop↔bridge dynamics

  4. TNT MultiSpecies (\(2N+1\) params) — Multiple chain populations

  5. TNT StickyRouse (4-5 params) — Rouse chains with sticky associations

For detailed equations: Transient Network Theory (TNT)

For tutorial: Transient Polymer Network Models (TNT + VLB)

Family 17: VLB (Vernerey-Long-Brighenti) Models

Who they’re for: Transient polymer networks with distribution tensor formulation

Test modes: All 6 protocols

Key characteristic: Chain distribution tensor \(\boldsymbol{\mu}\) tracks end-to-end vector statistics

The 4 VLB Models

  1. VLB Local (3-4 params) — Basic distribution tensor network

  2. VLB MultiNetwork (\(3N\) params) — Multiple interacting networks

  3. VLB Variant (5-6 params) — Bell force sensitivity + FENE extensibility

  4. VLB Nonlocal (4-5 params + n_points) — PDE for shear banding

For detailed equations: VLB Transient Network Models

For tutorial: Transient Polymer Network Models (TNT + VLB)

Family 18: HVM (Hybrid Vitrimer Model)

Who they’re for: Vitrimers — polymers with covalent + exchangeable crosslinks

Test modes: All 6 protocols

Key characteristic: 3 subnetworks (Permanent + Exchangeable + Dissociative) with TST kinetics

  • Bond Exchange Reactions (BER) via transition state theory

  • Factor-of-2 relaxation: \(\tau_E = 1/(2k_{BER})\) — both \(\boldsymbol{\mu}\) and \(\boldsymbol{\mu}_{nat}\) relax

  • \(\sigma_E \to 0\) at steady state (natural state tracks deformation)

  • 5 factory methods for limiting cases (neo-Hookean, Maxwell, Zener, etc.)

For detailed equations: HVM (Hybrid Vitrimer Model)

For tutorial: Vitrimer and Nanocomposite Models

Family 19: HVNM (Hybrid Vitrimer Nanocomposite Model)

Who they’re for: Nanoparticle-filled vitrimers with interphase reinforcement

Test modes: All 6 protocols

Key characteristic: 4 subnetworks (P + E + D + Interphase) with Guth-Gold amplification

  • Extends HVM with a 4th interphase subnetwork at NP-polymer interface

  • \(X(\varphi) = 1 + 2.5\varphi + 14.1\varphi^2\) modulus amplification

  • Dual TST kinetics: independent matrix and interphase exchange rates

  • \(\varphi = 0\) recovers HVM exactly (machine precision verified)

  • 5 factory methods for limiting cases

For detailed equations: HVNM (Hybrid Vitrimer Nanocomposite Model)

For tutorial: Vitrimer and Nanocomposite Models

Model Selection Flowchart

START: What type of data do you have?
   │
   ├─→ SAOS / Relaxation / Creep (linear viscoelasticity)
   │     │
   │     ├─→ Simple exponential decay?
   │     │      └─→ YES: Classical (Maxwell, Zener)
   │     │
   │     ├─→ Power-law / gel-like / complex?
   │     │      └─→ YES: Fractional (FML, FZSS, FMG)
   │     │
   │     └─→ Multiple relaxation processes?
   │            └─→ YES: TNT MultiSpecies, VLB MultiNetwork, GeneralizedMaxwell
   │
   ├─→ Steady Shear Flow (nonlinear)
   │     │
   │     ├─→ Shear thinning without yield stress?
   │     │      └─→ YES: PowerLaw, Carreau, Giesekus
   │     │
   │     ├─→ Yield stress present?
   │     │      └─→ YES: Bingham, Herschel-Bulkley, Saramito
   │     │
   │     └─→ Thixotropic (time-dependent)?
   │            └─→ YES: DMT, Fluidity, IKH
   │
   ├─→ Transient (Startup / Stress Overshoot)
   │     │
   │     ├─→ Polymer solution/melt?
   │     │      └─→ YES: Giesekus, TNT
   │     │
   │     ├─→ Thixotropic fluid?
   │     │      └─→ YES: DMT, IKH, Fluidity-Saramito
   │     │
   │     └─→ Vitrimer / adaptive material?
   │            └─→ YES: HVM, HVNM
   │
   └─→ Dense suspension / Glass?
         │
         ├─→ Colloidal glass?
         │      └─→ YES: ITT-MCT, SGR
         │
         ├─→ Metallic glass / amorphous solid?
         │      └─→ YES: STZ, EPM
         │
         └─→ Soft glass (foam, emulsion)?
                └─→ YES: SGR, HL

Complexity Ladder: When to Upgrade Models

Start simple, add complexity only if needed:

Level 1: Classical (2-3 parameters)

  • Try first for all linear viscoelastic data

  • If fit is poor, move to Level 2

Level 2: Fractional (3-4 parameters)

  • Better for complex materials

  • If still poor, check data quality or try Level 3

Level 3: Multi-mode (6+ parameters)

  • Discrete spectrum (multiple Maxwell/Zener elements)

  • Only if you have high-quality data over wide range

Red flags for over-fitting:

  • More parameters than data points / 3

  • Parameters vary wildly with small data changes

  • Excellent fit but unphysical parameter values

Key Concepts

Main Takeaways

  1. Classical models (Maxwell, Zener): Exponential relaxation, simple materials

  2. Fractional models (FML, FZSS, FMG): Power-law relaxation, complex materials

  3. Flow models (PowerLaw, Carreau, HB): Nonlinear viscosity, steady shear

  4. Nonlinear viscoelastic (Giesekus, Saramito): Tensor ODE, normal stresses, yielding

  5. Thixotropic (DMT, IKH/FIKH, Fluidity): Structure parameter, time-dependent viscosity

  6. Soft matter physics (SGR, HL, ITT-MCT): Statistical mechanics, glass transition

  7. Elasto-plastic (EPM, STZ): Amorphous solid yielding dynamics

  8. Transient networks (TNT, VLB): Dynamic crosslink attachment/detachment

  9. Vitrimers (HVM, HVNM): Bond exchange reactions, nanocomposite reinforcement

  10. LAOS analysis (SPP): Nonlinear oscillatory characterization

  11. Start simple: Try classical first, upgrade only if needed

  12. Physics determines family: Linear (classical/fractional) → nonlinear (flow/Giesekus) → thixotropic (DMT/IKH) → network (TNT/VLB) → vitrimer (HVM/HVNM)

Self-Check Questions

  1. You observe \(G'\) and \(G''\) scaling as \(\omega^{0.4}\) (parallel lines). Which family?

    Hint: Power-law scaling indicates fractional

  2. Your material shows exponential stress relaxation. Start with classical or fractional?

    Hint: Exponential indicates classical (simpler)

  3. Can you use a Maxwell model for steady shear flow data?

    Hint: Maxwell is linear viscoelastic, not for flow

  4. What does \(\alpha = 0.9\) tell you about the material?

    Hint: Close to 1 means narrow relaxation spectrum, nearly liquid-like

  5. You fit PowerLaw and get \(n = 0.6\). What does this mean?

    Hint: \(n < 1\) indicates shear thinning

Further Reading

Within this documentation:

Model handbooks (full equations and theory):

Advanced tutorials:

Summary

RheoJAX provides 53 rheological models across 22 families (20 listed below; Fractional Maxwell, Fractional Zener, and Fractional Advanced are sub-families counted separately in some contexts):

  • Classical (3): Maxwell, Zener, SpringPot — exponential relaxation, simple materials

  • Fractional (11): FML, FZSS, FMG, Burgers, Jeffreys, etc. — power-law relaxation

  • Flow (6): PowerLaw, Carreau, HB, Bingham, Cross — nonlinear viscosity

  • Multi-Mode (1): Generalized Maxwell — discrete relaxation spectrum

  • Giesekus (2): Single/multi-mode — nonlinear viscoelastic tensor ODE

  • SGR (2): Conventional, GENERIC — soft glassy materials

  • Fluidity (2): Local, Nonlocal — cooperative flow

  • Saramito (2): Local, Nonlocal — tensorial EVP + thixotropy

  • EPM (2): Lattice, Tensorial — elasto-plastic yielding

  • IKH (2): MIKH, MLIKH — isotropic-kinematic hardening

  • FIKH (2): FIKH, FMLIKH — fractional kinematic hardening

  • DMT (2): Local, Nonlocal — structure parameter thixotropy

  • HL (1): Hébraud-Lequeux — mean-field soft matter

  • STZ (1): Shear transformation zones — amorphous solids

  • SPP (1): LAOS yield stress analysis

  • ITT-MCT (2): Schematic, Isotropic — mode-coupling theory

  • TNT (5): SingleMode, Cates, LoopBridge, MultiSpecies, StickyRouse

  • VLB (4): Local, MultiNetwork, Variant, Nonlocal

  • HVM (1): Hybrid vitrimer — 3 subnetworks + TST kinetics

  • HVNM (1): Vitrimer nanocomposite — 4 subnetworks + Guth-Gold

DMTA/DMA Support: All 41+ oscillation-capable models also support tensile modulus (E*) via automatic E* ↔ G* conversion when deformation_mode='tension' is specified. See DMTA / DMA Analysis for details.

Always start simple and add complexity only when necessary.

Next Steps

Proceed to: Model Selection Guide

Learn to use decision flowcharts and compatibility checking to choose the right model for your data.