Test Modes in Rheology¶

Learning Objectives

After completing this section, you will be able to:

Identify and describe four major rheological test modes (SAOS, relaxation, creep, flow)
Choose the appropriate test mode for a given material and research question
Interpret raw experimental data from each test mode
Understand advantages and limitations of each technique
Recognize which models are appropriate for each test mode

Prerequisites

What is Rheology? — Understanding of stress, strain, viscoelasticity
Material Classification — Material types (liquid, solid, gel)

Overview: Why Multiple Test Modes?¶

Different experimental techniques probe different aspects of material behavior:

Frequency domain vs. time domain: SAOS vs. relaxation/creep
Linear vs. nonlinear: Small strain (SAOS/relaxation/creep) vs. large strain/rate (flow)
Storage vs. dissipation: \(G'\) vs. \(G''\) vs. \(\eta\)
Timescale range: Fast (high \(\omega\)) vs. slow (low \(\omega\))

No single test mode provides complete characterization—each reveals complementary information.

Test Mode Summary Table¶

RheoJAX Test Modes Reference¶
TestModeEnum	Protocol	Description
`RELAXATION`	Stress relaxation \(G(t)\)	Step strain, measure stress decay over time
`CREEP`	Creep compliance \(J(t)\)	Step stress, measure strain growth over time
`OSCILLATION`	SAOS \(G^*(\omega)\)	Small-amplitude oscillatory shear, complex modulus
`FLOW_CURVE`	Steady-state \(\eta(\dot{\gamma})\)	Viscosity vs shear rate at equilibrium
`STARTUP`	Transient \(\sigma(t,\dot{\gamma})\)	Stress overshoot/undershoot at fixed shear rate
`ROTATION`	Legacy steady shear	Deprecated, use `FLOW_CURVE` instead

The Six Test Modes¶

1. Small-Amplitude Oscillatory Shear (SAOS)¶

What it is: Apply sinusoidal strain, measure sinusoidal stress response

Input: \(\gamma(t) = \gamma_0 \sin(\omega t)\)

Output: \(\sigma(t) = \gamma_0[G'(\omega) \sin(\omega t) + G''(\omega) \cos(\omega t)]\)

Measured quantities:

\(G'(\omega)\) — Storage modulus (elastic component, in-phase)
\(G''(\omega)\) — Loss modulus (viscous component, out-of-phase)
\(\tan(\delta) = G''/G'\) — Loss tangent (damping ratio)
\(\eta^*(\omega) = \sqrt{(G')^2 + (G'')^2} / \omega\) — Complex viscosity

Frequency sweep: Vary \(\omega\) from ~0.01 to ~100 rad/s

Why it’s powerful:

Probes linear viscoelasticity (non-destructive)
Direct access to \(G'\) and \(G''\) across timescales
Easiest to model mathematically (Fourier transform of relaxation)
Most common test in rheology

Limitations:

Limited to small strains (\(\gamma_0\) typically < 1%)
May miss nonlinear behavior
Requires stable oscillation (not all instruments can do low frequencies well)

When to use:

Characterizing material structure (crosslinking, gelation)
Model fitting for viscoelastic parameters
Quality control and formulation optimization
Frequency-dependent behavior (mastercurves)

Example applications:

Polymer melts: Molecular weight distribution from \(G'\) and \(G''\) curves
Gels: Gelation monitoring (crossover of \(G'\) and \(G''\))
Suspensions: Particle network structure

2. Stress Relaxation¶

What it is: Apply step strain, measure stress decay over time

Input: \(\gamma(t) = \gamma_0 H(t)\) — Step strain at \(t=0\)

Output: \(\sigma(t) = G(t) \gamma_0\) — Stress decays as material relaxes

Measured quantity:

\(G(t)\) — Relaxation modulus (stress/strain as function of time)

Time range: Typically 0.01 s to 1000 s

Why it’s powerful:

Direct measurement of relaxation spectrum
Time-domain data (easier to interpret physically)
Can access very long timescales
Simple experimental protocol

Limitations:

Requires fast strain application (instrument rise time < 0.01 s)
Inertial artifacts at short times
Sample slippage or edge fracture at long times
Limited to materials that don’t flow significantly

When to use:

Materials with long relaxation times (polymer melts, elastomers)
Studying molecular relaxation mechanisms
Validating viscoelastic models
Gelation studies (loss of relaxation as gel forms)

Example applications:

Polymers: Reptation dynamics, entanglement networks
Rubbers: Viscoelastic damping in elastomers
Biological tissues: Stress relaxation in cartilage, skin

3. Creep (Compliance)¶

What it is: Apply constant stress, measure strain increase over time

Input: \(\sigma(t) = \sigma_0 H(t)\) — Step stress at \(t=0\)

Output: \(\gamma(t) = J(t) \sigma_0\) — Strain increases as material creeps

Measured quantity:

\(J(t)\) — Creep compliance (strain/stress as function of time)

Time range: Typically 0.1 s to 10,000 s (hours)

Why it’s powerful:

Probes long-term deformation under constant load
Can distinguish viscous flow from elastic deformation
Sensitive to weak network structures
Physically intuitive (mimics real loading conditions)

Limitations:

Long experimental time
Difficult to apply truly constant stress (instrument drift)
Sample may flow out of geometry
Less common than SAOS or relaxation

When to use:

Studying long-term material stability (sagging, settling)
Weak gels and soft solids
Materials near yield stress
Validating linear viscoelastic models (creep and relaxation are related)

Example applications:

Asphalt: Long-term deformation under road load
Food products: Spreadability, flow under gravity
Soft tissues: Load-bearing capacity

4. Steady Shear Flow (Flow Curve)¶

What it is: Apply constant shear rate, measure viscosity at equilibrium

Input: \(\dot{\gamma}\) = constant — Shear rate (s \(^{-1}\))

Output: \(\sigma = \eta(\dot{\gamma}) \cdot \dot{\gamma}\) — Shear stress

Measured quantity:

\(\eta(\dot{\gamma})\) — Viscosity as a function of shear rate

Shear rate range: Typically 0.01 to 1000 s⁻¹

RheoJAX test mode: test_mode='flow_curve' (or legacy test_mode='rotation')

Why it’s powerful:

Directly measures flow behavior
Reveals nonlinear effects (shear thinning, shear thickening, yield stress)
Mimics processing conditions (pumping, mixing, extrusion)
Simple physical interpretation

Limitations:

Nonlinear regime (can’t predict from linear viscoelasticity)
Edge effects, wall slip, sample expulsion
May structurally damage sample
Not directly related to \(G'\) and \(G''\) (except at very small \(\dot{\gamma}\))

When to use:

Processing design (extrusion, coating, pumping)
Formulation optimization (pumpability, spreadability)
Quality control (viscosity specs)
Studying flow instabilities

Example applications:

Paints and coatings: Shear thinning for easy application
Food: Mouthfeel, pourability
Inks: Printing behavior
Blood: Cardiovascular fluid dynamics

5. Startup Shear¶

What it is: Apply constant shear rate, measure transient stress evolution

Input: \(\dot{\gamma}\) = constant (suddenly applied at \(t=0\))

Output: \(\sigma(t)\) — Stress as function of time at fixed shear rate

Measured quantity:

\(\sigma(t, \dot{\gamma})\) — Transient stress response (often shows overshoot/undershoot)

Time range: Typically 0.01 to 100 s

RheoJAX test mode: test_mode='startup'

Why it’s powerful:

Reveals thixotropic behavior (stress overshoot, undershoot)
Probes microstructural evolution during flow
Critical for understanding yielding dynamics
Used for elasto-plastic model validation (EPM, IKH)

Limitations:

Transient data harder to model than steady-state
Requires fast instrument response
Sample history-dependent (must control pre-shear)
May require multiple shear rates for complete characterization

When to use:

Studying thixotropy and shear rejuvenation
Validating constitutive models (EPM, IKH, STZ)
Understanding yield stress fluids and soft glasses
Investigating shear banding and flow instabilities

Example applications:

Colloidal gels: Stress overshoot indicates structural breakdown
Emulsions: Yielding dynamics in mayonnaise, cosmetics
Thixotropic fluids: Drilling muds, paints
Soft glassy materials: Foams, pastes

6. Large-Amplitude Oscillatory Shear (LAOS)¶

What it is: Apply sinusoidal strain at large amplitudes, analyze nonlinear stress response

Input: \(\gamma(t) = \gamma_0 \sin(\omega t)\) with \(\gamma_0 \gg\) linear limit

Output: \(\sigma(t)\) — Non-sinusoidal stress waveform

Measured quantities:

Higher harmonics: \(\sigma_3/\sigma_1\), \(\sigma_5/\sigma_1\)
Chebyshev coefficients (e₁, e₃, v₁, v₃)
Lissajous-Bowditch curves (\(\sigma\) vs \(\gamma\), \(\sigma\) vs \(\dot{\gamma}\))
SPP decomposition (sequence of physical processes)

RheoJAX test mode: test_mode='oscillation' with SPP models

Why it’s powerful:

Probes nonlinear viscoelasticity within single test
Fingerprints material microstructure
Distinguishes between similar linear rheology materials
Rich information in single experiment

Limitations:

Complex interpretation (multiple analysis frameworks)
Requires specialized instruments and software
Computationally intensive analysis
No universal standards for reporting

When to use:

Distinguishing materials with similar \(G'\), \(G''\)
Studying yielding and flow transitions
Material fingerprinting and quality control
Research on nonlinear constitutive behavior

Example applications:

Gels: Yield stress determination from Lissajous curves
Polymer melts: Strain-hardening/softening characterization
Complex fluids: Microstructural evolution during deformation

Visual Comparison of Test Modes¶

SAOS (Frequency Sweep)
───────────────────────
Input:  γ(t) = γ₀ sin(ωt)
        ┌───┐     ┌───┐
        │   │     │   │
────────┘   └─────┘   └────

Output: σ(t) (phase-shifted)
        ┌───┐     ┌───┐
      ┌─┘   └───┬─┘   └──
──────┘         └────────

Measure: G'(ω), G"(ω)


STRESS RELAXATION
─────────────────
Input:  γ(t) = γ₀ (step)
              ┌──────────
        γ₀ ───┤
              │
──────────────┘

Output: σ(t) (decays)
              ╱────
        σ₀ ──╱
            ╱
───────────╱

Measure: G(t)


CREEP
─────
Input:  σ(t) = σ₀ (step)
              ┌──────────
        σ₀ ───┤
              │
──────────────┘

Output: γ(t) (increases)
                    ╱─────
                  ╱
                ╱
───────────────╱

Measure: J(t)


STEADY SHEAR FLOW
─────────────────
Input:  γ̇ = const (various rates)
        │ ╱╱╱╱╱╱╱╱╱
        │╱╱╱╱╱╱╱╱╱
        ╱╱╱╱╱╱╱╱╱

Output: σ (steady state)
        │     ●
        │   ●
        │ ●
        └────────── γ̇

Measure: η(γ̇)

Relationships Between Test Modes¶

Linear Viscoelasticity: SAOS ↔ Relaxation ↔ Creep¶

In the linear regime, all three are related by Fourier transform:

SAOS to Relaxation:

\[ \begin{align}\begin{aligned}G'(\omega) = \omega \int_0^\infty G(t) \sin(\omega t) \, dt\\G"(\omega) = \omega \int_0^\infty G(t) \cos(\omega t) \, dt\end{aligned}\end{align} \]

Relaxation to SAOS (inverse transform):

\[G(t) = \frac{2}{\pi} \int_0^\infty \frac{G"(\omega)}{\omega} \cos(\omega t) \, d\omega\]

Creep and Relaxation (Laplace space):

\[\tilde{J}(s) \cdot \tilde{G}(s) = \frac{1}{s^2}\]

Practical implication: If you fit a model to SAOS data, you can predict relaxation and creep behavior (and vice versa).

Flow vs. Linear Viscoelasticity¶

Cox-Merz rule (empirical, often holds for polymers):

\[\eta(\dot{\gamma}) \approx \eta^*(\omega) \quad \text{at } \dot{\gamma} = \omega\]

Limitation: Only valid for some materials, breaks down for structured fluids (suspensions, gels).

Choosing the Right Test Mode¶

Decision Flowchart¶

[What do you want to know?]
   │
   ├─→ "Frequency-dependent viscoelasticity (G', G")"
   │      └─→ SAOS (frequency sweep)
   │
   ├─→ "Long-term deformation under load"
   │      └─→ CREEP
   │
   ├─→ "Relaxation timescales and spectrum"
   │      └─→ STRESS RELAXATION
   │
   ├─→ "Flow behavior, processing conditions"
   │      └─→ STEADY SHEAR FLOW
   │
   └─→ "Comprehensive characterization"
          └─→ Combine SAOS + Relaxation + Flow

Practical Guidelines¶

Use SAOS when:

You need \(G'\) and \(G''\) for modeling
Material is stable over long time
You want non-destructive testing
You’re monitoring gelation or curing

Use Stress Relaxation when:

You need time-domain data
Material has long relaxation times
You’re studying molecular mechanisms
You want to validate SAOS-derived models

Use Creep when:

You’re studying long-term stability (sagging, settling)
Material is near yield stress
You have very long experimental time available
You want to separate viscous flow from elastic deformation

Use Steady Shear Flow when:

You’re designing processing equipment
You need viscosity at specific shear rates
You’re studying shear thinning/thickening
You need to detect yield stress

Model Compatibility with Test Modes¶

Model Family	SAOS	Relax	Creep	Flow	Startup	LAOS
Classical (Maxwell, Zener)	✓	✓	✓	✗	✗	✗
Fractional Models	✓	✓	✓	✗	✗	✗
Flow (PowerLaw, Carreau, HB)	✗	✗	✗	✓	✗	✗
SGR (Soft Glassy Rheology)	✓	✓	✓	✗	✗	✗
Fluidity (Local, Nonlocal)	✓	✗	✗	✓	✗	✗
EPM (Lattice, Tensorial)	✗	✓	✓	✓	✓	✗
IKH (MIKH, MLIKH)	✓	✓	✓	✗	✗	✗
HL (Hébraud-Lequeux)	✓	✓	✗	✗	✗	✗
STZ (Shear Transformation)	✓	✓	✗	✓	✓	✗
SPP (LAOS Analysis)	✗	✗	✗	✗	✗	✓

7. DMTA / DMA (Tensile Oscillation)¶

What it is: Apply oscillatory tensile (or bending/compression) deformation and measure \(E^*(\omega)\)

Instruments: DMTA (TA Instruments RSA-G2, Netzsch DMA, Mettler Toledo DMA/SDTA, PerkinElmer DMA 8000)

Output: \(E'(\omega)\) (storage modulus, tensile) and \(E''(\omega)\) (loss modulus, tensile)

Relationship to shear: \(E = 2(1+\nu)G\) where \(\nu\) is the Poisson ratio

In RheoJAX: All oscillation-capable models (41 of 53) accept DMTA data directly:

from rheojax.models import FractionalZenerSolidSolid

model = FractionalZenerSolidSolid()
model.fit(omega, E_star,
          test_mode='oscillation',
          deformation_mode='tension',
          poisson_ratio=0.5)  # rubber

# predict() returns E* automatically
E_pred = model.predict(omega, test_mode='oscillation')

RheoJAX converts \(E^* \to G^*\) before fitting (model parameters stay in shear space) and converts back on predict(). See DMTA / DMA Analysis for theory and workflows.

Key distinctions:

Linear viscoelastic (Classical, Fractional, IKH): SAOS, relaxation, creep
Flow models (PowerLaw, Carreau, HB): Nonlinear steady shear only
Soft matter physics (SGR, HL, Fluidity): Statistical mechanics approaches
Elasto-plastic (EPM, STZ): Startup transients and flow curves
Nonlinear oscillatory (SPP): LAOS analysis and yield stress

Worked Example: Multi-Mode Characterization¶

Material: Polymer melt (polystyrene)

Goal: Complete rheological characterization

Experimental protocol:

SAOS frequency sweep (10⁻² to 10² rad/s)
- Result: \(G' \sim \omega^2\), \(G'' \sim \omega\) at low \(\omega\) (liquid-like)
- Crossover at \(\omega_c \approx 1\) rad/s \(\to \tau \approx 1\) s
- Fit: Fractional Maxwell Liquid (FML)
Stress relaxation (t = 0.01 to 100 s)
- Result: \(G(t)\) decays from \(10^5\) Pa to <100 Pa
- Confirms liquid-like behavior (no plateau)
- Validates FML fit from SAOS
Steady shear flow (\(\dot{\gamma} = 10^{-2}\) to \(10^3\) s \(^{-1}\))
- Result: Shear thinning (\(\eta\) decreases with \(\dot{\gamma}\))
- Zero-shear viscosity \(\eta_0 \approx 10^5\) Pa·s
- Fit: Carreau model for processing predictions

Outcome: Complete characterization for both linear viscoelasticity (SAOS/relaxation) and nonlinear flow (steady shear).

Key Concepts¶

Main Takeaways

SAOS: Frequency-dependent \(G'\) and \(G''\), most common test, linear regime
Stress Relaxation: Time-domain \(G(t)\), direct measurement of relaxation spectrum
Creep: Long-term deformation \(J(t)\), good for weak gels and stability
Steady Shear Flow: Nonlinear viscosity \(\eta(\dot{\gamma})\), for processing design
Linear tests are related via Fourier/Laplace transforms—fitting one predicts others

Self-Check Questions

You need to predict how a material flows during extrusion. Which test mode is most relevant?

Hint: Think about processing conditions (shear rate)
Why can’t you use a flow curve (\(\eta\) vs \(\dot{\gamma}\)) to predict SAOS behavior (\(G'\) vs \(\omega\)) ?

Hint: Linear vs. nonlinear regimes
A material has \(G' = G'' = 1000\) Pa at 1 rad/s. Can you predict \(G(t)\) at \(t = 1\) s exactly?

Hint: Need full frequency sweep, not single point
You observe stress relaxation \(G(t)\) from \(10^5\) Pa to \(10^3\) Pa over 100 s. Is this a liquid or solid?

Hint: Check if it plateaus or continues decaying
Why is SAOS preferred over creep for routine characterization?

Hint: Consider experimental time and frequency range

Summary¶

Four major test modes probe different aspects of rheology: SAOS (frequency-dependent \(G'\), \(G''\)), stress relaxation (time-domain \(G(t)\)), creep (long-term \(J(t)\)), and steady shear flow (nonlinear \(\eta(\dot{\gamma})\)). Linear viscoelastic tests (SAOS, relaxation, creep) are mathematically related, while flow tests probe nonlinear behavior. Choose test modes based on your material and research question.

Next Steps¶

Proceed to: Parameter Interpretation

Learn the physical meaning of rheological parameters like \(G'\), \(G''\), \(\tau\), and \(\alpha\).

Test Modes in Rheology¶

Overview: Why Multiple Test Modes?¶

Test Mode Summary Table¶

The Six Test Modes¶

1. Small-Amplitude Oscillatory Shear (SAOS)¶

2. Stress Relaxation¶

3. Creep (Compliance)¶

4. Steady Shear Flow (Flow Curve)¶

5. Startup Shear¶

6. Large-Amplitude Oscillatory Shear (LAOS)¶

Visual Comparison of Test Modes¶

Relationships Between Test Modes¶

Linear Viscoelasticity: SAOS ↔ Relaxation ↔ Creep¶

Flow vs. Linear Viscoelasticity¶

Choosing the Right Test Mode¶

Decision Flowchart¶

Practical Guidelines¶

Model Compatibility with Test Modes¶

7. DMTA / DMA (Tensile Oscillation)¶

Worked Example: Multi-Mode Characterization¶

Key Concepts¶

Further Reading¶

Summary¶

Next Steps¶