SPP Yield Stress Model

Quick Reference

• Use when: Extracting yield stress from LAOS amplitude sweeps, characterizing

cage-based yield stress fluids, connecting oscillatory to steady-shear behavior

• Parameters: 8 (\(G_{\text{cage}}\), \(\sigma_{\text{sy,scale}}\), \(\sigma_{\text{sy,exp}}\), \(\sigma_{\text{dy,scale}}\), \(\sigma_{\text{dy,exp}}\), \(\eta_\infty\), \(n_{\text{power-law}}\), noise)

• Key equation: \(\sigma_{sy}(\gamma_0) = \sigma_{sy,0} \cdot |\gamma_0|^{n_{sy}}\) and \(\sigma_{dy}(\gamma_0) = \sigma_{dy,0} \cdot |\gamma_0|^{n_{dy}}\)

• Test modes: oscillation (LAOS amplitude sweep), rotation (flow curve)

• Material examples: Yield stress fluids, colloidal gels, concentrated emulsions, foams, soft glasses, carbopol microgels, mayonnaise

Notation Guide

Symbol	Description	Units
\(G_{\text{cage}}\)	Apparent cage modulus	Pa
\(\sigma_{sy}(\gamma_0)\)	Static yield stress	Pa
\(\sigma_{dy}(\gamma_0)\)	Dynamic yield stress	Pa
\(\gamma_0\)	Strain amplitude	—
\(n_{sy}\)	Static yield exponent	—
\(n_{dy}\)	Dynamic yield exponent	—
\(\eta_\infty\)	Infinite-shear viscosity	Pa·s
\(n\)	Power-law flow index	—
\(\dot{\gamma}\)	Shear rate	1/s

Quick Reference (continued)

Model Class	SPPYieldStress
Registry Name	"spp_yield_stress"
Test Modes	oscillation (amplitude sweep), rotation (flow curve)
Parameters	8 (G_cage, sigma_sy_scale, sigma_sy_exp, sigma_dy_scale, sigma_dy_exp, eta_inf, n_power_law, noise)
Typical Materials	Yield stress fluids, colloidal gels, emulsions, foams, soft glasses
Key Reference	Rogers et al. (2012) J. Rheol. 56(1)

Overview

The SPP Yield Stress model extracts physically meaningful yield parameters from Large Amplitude Oscillatory Shear (LAOS) data using the Sequence of Physical Processes (SPP) framework. Unlike traditional Fourier-based approaches, SPP provides time-resolved instantaneous material functions that reveal the intracycle sequence of physical processes during nonlinear deformation.

The model parameterizes the nonlinear response in terms of:

• \(G_{\text{cage}}\): Apparent cage modulus (elastic stiffness of the microstructural cage)

• Static yield stress (\(\sigma_{sy}\)): Stress at strain reversal (maximum strain amplitude)

• Dynamic yield stress (\(\sigma_{dy}\)): Stress at rate reversal (zero strain rate)

• Power-law scaling: Amplitude dependence of yield stresses with exponent

This approach connects LAOS analysis to steady-shear flow behavior, enabling comprehensive yield stress characterization across deformation protocols.

Physical Foundations

Cage Model for Yield Stress Fluids

The SPP yield stress model is grounded in the colloidal cage picture, where particles are confined by nearest-neighbor “cages”:

1. Linear Regime (small \(\gamma_0\)): The cage deforms elastically with stiffness \(G_{\text{cage}}\)

2. Yielding (\(\gamma_0 \to \gamma_{yield}\)): Cage constraints are overcome at the yield point

3. Flow Regime (large \(\gamma_0\)): Particles escape cages and flow viscously

The cage modulus \(G_{\text{cage}}\) represents the instantaneous elastic stiffness measured at the point where stress passes through zero (\(\sigma = 0\)). This corresponds to the slope of the stress-strain Lissajous curve at the origin.

Static vs. Dynamic Yield Stress

The SPP framework distinguishes two physically distinct yield stresses:

Static Yield Stress ( \(\sigma_{sy}\) )

• Measured at strain reversal (\(\gamma\) = ±\(\gamma_0\), \(\dot{\gamma}\) = 0)

• Represents the stress required to initiate flow from rest

• Larger than dynamic yield due to microstructural recovery during strain reversal

Dynamic Yield Stress ( \(\sigma_{dy}\) )

• Measured at rate reversal (\(\dot{\gamma} = 0\), \(\gamma \neq 0\))

• Represents the stress during continuous flow

• Connects to steady-shear yield stress extrapolation

The ratio \(\sigma_{sy}/\sigma_{dy}\) is typically around \(2\text{--}3\) for colloidal systems and reveals information about cage reformation kinetics and thixotropy.

Power-Law Amplitude Scaling

Both yield stresses exhibit power-law scaling with strain amplitude:

\[\sigma_{sy}(\gamma_0) = \sigma_{sy,0} \cdot \gamma_0^{n_{sy}}\]

\[\sigma_{dy}(\gamma_0) = \sigma_{dy,0} \cdot \gamma_0^{n_{dy}}\]

where the exponents \(n_{sy}\) and \(n_{dy}\) typically fall in the range 0.2–1.0. Rogers et al. (2011) found \(n \approx 0.2\) for concentrated colloidal suspensions, connecting to the Herschel-Bulkley flow curve exponent.

Connection to SPP Framework (Rogers 2017)

The yield stress terms in this model connect to the complete SPP stress decomposition from Rogers (2017):

\[\sigma(t) = G'_t(t)[\gamma(t) - \gamma_{eq}(t)] + \frac{G''_t(t)}{\omega}\dot{\gamma}(t) + \sigma_y(t)\]

The displacement term \(\sigma_y(t) - G'_t(t)\gamma_{eq}(t)\) captures:

1. Static yield stress: Associated with the shift in equilibrium strain \(\gamma_{eq}\) during cage rupture

2. Dynamic yield stress: The zero-rate stress intercept from the viscoplastic flow contribution

Equilibrium Strain Interpretation:

• In the linear regime, \(\gamma_{eq} = 0\) (no shifting)

• During yielding, \(\gamma_{eq}\) shifts to \(\pm(\gamma_0 - \gamma_y)\)

• The material frame strain becomes \(\gamma_{mat}(t) = \gamma(t) - \gamma_{eq}(t)\)

• This distinction between lab frame and material frame is essential for understanding the physical origin of the two yield stresses

Connection to Steady-Shear Flow Curves

The dynamic yield stress from SPP analysis connects directly to steady-shear Herschel-Bulkley fitting, enabling cross-validation between oscillatory and rotational measurements:

Cross-Validation Protocol:

1. Extract \(\sigma_{dy}\) from LAOS amplitude sweep using SPP decomposition

2. Fit steady-shear flow curve: \(\sigma = \sigma_y + K\dot{\gamma}^n\)

3. Compare: \(\sigma_{dy,\text{LAOS}}\) should match \(\sigma_y\) from steady shear

Expected Agreement:

• Good agreement indicates simple yield stress behavior (Bingham-like)

• Discrepancy suggests thixotropy, structural evolution, or measurement artifacts

• Large disagreement may indicate shear banding, wall slip, or elastic recoil effects

Rate-Amplitude Duality:

Rogers et al. (2011) demonstrated that yield stresses scale identically with strain amplitude and shear rate:

\[\sigma_y(\gamma_0) \propto \gamma_0^{n} \quad \text{(amplitude)}\]

\[\sigma_y(\dot{\gamma}_0) \propto \dot{\gamma}_0^{n} \quad \text{(rate)}\]

This universal exponent (\(n \approx 0.2\) for hard-sphere colloids) connects LAOS and steady-shear characterization through a single material parameter.

Practical Cross-Validation Example:

from rheojax.models import SPPYieldStress, HerschelBulkley

# Fit LAOS amplitude sweep
spp_model = SPPYieldStress()
spp_model.fit(gamma_0, sigma_dy_laos, test_mode='oscillation', yield_type='dynamic')
sigma_dy_spp = spp_model.parameters.get_value('sigma_dy_scale')

# Fit steady-shear flow curve
hb_model = HerschelBulkley()
hb_model.fit(gamma_dot, sigma_steady, test_mode='rotation')
sigma_y_hb = hb_model.parameters.get_value('sigma_y')

# Cross-validate
relative_error = abs(sigma_dy_spp - sigma_y_hb) / sigma_y_hb
print(f"Yield stress agreement: {100 * (1 - relative_error):.1f}%")
# Typical agreement: >90% for simple yield stress fluids

Theoretical Benchmarks (Rogers & Lettinga 2012)

Rogers & Lettinga (2012) [2] applied the SPP framework to classical nonlinear viscoelastic models, establishing theoretical benchmarks against which experimental data can be validated. These predictions serve as reference behaviors for interpreting yield stress material responses.

Bingham Model Response

The Bingham fluid \(\sigma = \sigma_y + \eta_p \dot{\gamma}\) (for \(|\sigma| > \sigma_y\)) represents the idealized yield stress fluid. Under LAOS, the SPP framework reveals a characteristic four-step intracycle sequence:

Step 1: Elastic Extension (\(|\gamma|\) < \(\gamma_y\))

• Stress builds linearly: \(\sigma = G_{cage} \cdot \gamma\)

• \(G'_t \approx G_{cage}\) (constant), \(G''_t \approx 0\)

• Cole-Cole trajectory: remains near the \(G'_t\) axis

Step 2: Static Yielding (at \(\gamma\) = ±\(\gamma_y\))

• Cages rupture; stress reaches \(\sigma_{y,static} = G_{cage} \cdot \gamma_y\)

• Sharp transition: \(G'_t\) drops rapidly, \(G''_t\) increases

• Cole-Cole: trajectory moves from elastic toward viscous dominance

Step 3: Newtonian Flow (\(|\gamma|\) > \(\gamma_y\), \(|\dot{\gamma}|\) large)

• Material flows with constant viscosity \(\eta_p\)

• \(G'_t \approx 0\), \(G''_t \approx \eta_p \omega\)

• Cole-Cole: trajectory resides near the \(G''_t\) axis

Step 4: Cage Reformation (as \(\dot{\gamma} \to 0\) at \(\pm\gamma_{max}\))

• Microstructure rapidly reforms at rate reversal

• \(G''_t \to 0\), \(G'_t\) increases

• \(\sigma_{y,dynamic}\) measured at this instant

• Cole-Cole: trajectory returns toward elastic dominance

Key Bingham Predictions:

• The equilibrium strain \(\gamma_{eq}\) shifts to \(\pm(\gamma_0 - \gamma_y)\) after yielding

• Static yield stress \(\sigma_{y,static}\) exceeds dynamic \(\sigma_{y,dynamic}\) due to reformation

• Post-yield flow follows the steady-state Bingham flow curve exactly

• Higher harmonics in stress arise solely from the yield transition (not from viscosity)

Giesekus Model Response

The Giesekus model, which includes a nonlinear relaxation term with mobility parameter \(\alpha\), exhibits elastic recoil during flow reversal. Under SPP analysis:

Elastic Recoil Signatures:

• Unlike ideal Bingham, \(G'_t\) does not drop to zero during flow

• The material retains elastic character even post-yield due to polymer stretch

• Stress overshoots during startup are more pronounced than Bingham

SPP Trajectory Features:

• Cole-Cole plots form loops rather than simple back-and-forth trajectories

• Loop area increases with Deborah number (De = \(\lambda \omega\))

• Trajectory directionality (clockwise vs counterclockwise) encodes stress response phase

Mobility Parameter Effects ( \(\alpha\) ):

• \(\alpha = 0\) (upper-convected Maxwell): purely elastic, no shear thinning

• \(\alpha = 0.5\): intermediate behavior, moderate shear thinning

• \(\alpha \to 1\): stronger nonlinearity, more pronounced recoil features

Validation Use Cases:

• If experimental data shows Giesekus-like loops but no clear yield, the material may be viscoelastic rather than yield-stress

• Bingham-like step transitions indicate simpler yield stress behavior

• Hybrid behaviors suggest complex microstructure (e.g., polymer-in-suspension)

Power-Law Fluid (Non-Yield)

For comparison, a purely viscous power-law fluid \(\sigma = K |\dot{\gamma}|^n \text{sgn}(\dot{\gamma})\) shows characteristic SPP behavior that lacks the yield-stress sequence:

SPP Predictions:

• \(G'_t = 0\) everywhere (no elasticity)

• \(G''_t\) varies with shear rate as \(\sim |\dot{\gamma}|^{n-1}\)

• No cage modulus can be defined (\(d\sigma/d\gamma|_{\sigma=0}\) is undefined)

• Cole-Cole trajectory collapses to the \(G''_t\) axis

Diagnostic Implication:

If experimental SPP shows negligible \(G'_t\) values, the material may not be a yield stress fluid but rather a shear-thinning viscous material.

Validating Experimental Data

Use these benchmarks to assess experimental SPP results:

Theoretical Benchmark Comparison

Observation	Matches	Physical Interpretation
Sharp 4-step transitions, \(G'_t \to 0\) post-yield	Bingham	Simple yield stress fluid, brittle cages
Loops in Cole-Cole, non-zero \(G'_t\) post-yield	Giesekus-like	Viscoelastic yield stress, elastic recoil
\(G'_t \approx 0\) throughout cycle	Power-law	Viscous fluid, no yield stress
Partial matches	Hybrid	Complex microstructure; consider thixotropy or heterogeneity

---

Constitutive Equations

Oscillation Mode (LAOS Amplitude Sweep)

For amplitude sweep analysis with strain amplitude \(\gamma_0\):

\[\sigma_{sy}(\gamma_0) = \text{sigma\_sy\_scale} \cdot |\gamma_0|^{\text{sigma\_sy\_exp}}\]

\[\sigma_{dy}(\gamma_0) = \text{sigma\_dy\_scale} \cdot |\gamma_0|^{\text{sigma\_dy\_exp}}\]

The cage modulus at small amplitude approximates:

\[G_{cage} \approx \frac{\sigma_{sy}(\gamma_0)}{\gamma_0} \quad (\gamma_0 \to 0)\]

Rotation Mode (Steady Shear)

For steady shear flow at rate \(\dot{\gamma}\), the model uses a Herschel-Bulkley-like form:

\[\sigma(\dot{\gamma}) = \sigma_{dy} + \eta_\infty |\dot{\gamma}|^n\]

This connects the dynamic yield stress from LAOS to steady-shear flow curves, enabling validation between oscillatory and continuous shear measurements.

---

Governing Equations

Amplitude-Dependent Power-Law Scaling

The core governing equations describe how yield stresses scale with strain amplitude:

Static Yield Stress:

\[\sigma_{sy}(\gamma_0) = \sigma_{sy,0} \cdot |\gamma_0|^{n_{sy}}\]

Dynamic Yield Stress:

\[\sigma_{dy}(\gamma_0) = \sigma_{dy,0} \cdot |\gamma_0|^{n_{dy}}\]

where:

• \(\sigma_{sy,0}, \sigma_{dy,0}\) are scale factors (Pa)

• \(n_{sy}, n_{dy}\) are power-law exponents (typically 0.2-1.0)

Physical interpretation: The power-law scaling reflects the rate-dependence of cage rupture and reformation. The exponent \(n \approx 0.2\) observed in colloidal systems (Rogers et al., 2011) connects to the Herschel-Bulkley flow curve exponent.

Cage Modulus Definition

The cage modulus represents the instantaneous elastic stiffness at zero stress:

\[G_{\text{cage}} = \left. \frac{d\sigma}{d\gamma} \right|_{\sigma=0}\]

In the small-amplitude limit:

\[\lim_{\gamma_0 \to 0} G_{\text{cage}} = \lim_{\gamma_0 \to 0} \frac{\sigma_{sy}(\gamma_0)}{\gamma_0}\]

This connects the cage modulus to the static yield stress scaling.

Steady-Shear Flow Curve (Rotation Mode)

For steady shear at rate \(\dot{\gamma}\), the model uses a Herschel-Bulkley-like form:

\[\sigma(\dot{\gamma}) = \sigma_{dy} + \eta_\infty |\dot{\gamma}|^n\]

where:

• \(\sigma_{dy}\) is the dynamic yield stress extrapolated to zero amplitude

• \(\eta_\infty\) is the high-shear viscosity

• \(n\) is the power-law flow index (\(n < 1\) for shear-thinning)

This form enables cross-validation between LAOS (amplitude sweeps) and steady-shear (rotation) measurements.

---

What You Can Learn

Physical Insights from SPP Yield Stress Parameters

Cage Modulus (:math:`G_{text{cage}}`) — Microstructural Stiffness:

The cage modulus quantifies the elastic strength of the confining microstructural cage:

Key insights:

• Amplitude-independent: Remains constant even as \(G'(\gamma_0)\) decreases at large amplitudes

• True material property: Reflects intrinsic nearest-neighbor confinement

• Connection to SAOS: \(G_{\text{cage}} \approx G'_{\text{LVR}}\) in the linear viscoelastic regime

• Scaling with concentration: For colloidal systems, \(G_{\text{cage}} \sim \phi^{3-5}\) where \(\phi\) is volume fraction

Typical values:

• Colloidal glasses (\(\phi \sim 0.6\)): 10-100 Pa

• Microgel pastes: 100-1000 Pa

• Polymer gels: 1000-10000 Pa

Static Yield Stress Scaling — Cage Rupture Dynamics:

The power-law exponent \(n_{sy}\) reveals information about the cage rupture process:

Exponent interpretation:

• \(n_{sy} \approx 0.2\): Weak amplitude dependence (typical for hard-sphere colloids)

• \(n_{sy} \approx 0.5\): Moderate amplitude dependence (soft particles)

• \(n_{sy} \approx 1.0\): Strong amplitude dependence (polymer gels, weak cages)

Physical meaning: Lower exponents indicate cages that rupture at a nearly constant stress regardless of amplitude (brittle rupture). Higher exponents indicate cages that become progressively easier to break at larger amplitudes (ductile rupture).

Dynamic Yield Stress Scaling — Flow Regime:

The dynamic yield stress connects LAOS to steady-shear flow behavior:

Key insights:

• Ratio to static: \(\sigma_{sy}/\sigma_{dy} \approx 2\text{--}3\) for thixotropic materials

• Flow curve connection: \(\sigma_{dy}\) matches the extrapolated Herschel-Bulkley yield stress

• Rate sensitivity: \(n_{dy} \approx n_{sy}\) (same exponent) indicates universal scaling

Thixotropic signatures:

• Large \(\sigma_{sy}/\sigma_{dy}\) ratio indicates strong thixotropy (fast cage reformation)

• Small ratio indicates weak thixotropy (slow cage reformation)

Amplitude Scaling Exponent — Universal Behavior:

Rogers et al. (2011) found \(n \approx 0.2\) for concentrated colloidal suspensions. This value emerges from the cage model and connects to:

1. Herschel-Bulkley exponent: Same power-law appears in steady-shear flow curves

2. Stress-rate duality: Both \(\sigma_{sy}(\gamma_0)\) and \(\sigma_{sy}(\dot{\gamma}_0)\) scale with the same exponent

3. Jamming universality: Near the jamming transition, \(n \approx 0.2\) is predicted theoretically

Material Characterization Capabilities

From LAOS Amplitude Sweeps:

• Cage modulus \(G_{\text{cage}}\) (constant across amplitudes)

• Static yield stress scaling: \(\sigma_{sy,0}, n_{sy}\)

• Dynamic yield stress scaling: \(\sigma_{dy,0}, n_{dy}\)

• Thixotropic strength (from \(\sigma_{sy}/\sigma_{dy}\) ratio)

• Nonlinearity onset (strain amplitude where power-law emerges)

From Steady-Shear Flow Curves:

• Dynamic yield stress \(\sigma_{dy}\) (Herschel-Bulkley intercept)

• High-shear viscosity \(\eta_\infty\)

• Flow power-law index \(n\)

• Cross-validation with LAOS-derived \(\sigma_{dy}\)

From Model Fitting:

• Cage rupture mechanism (from \(n_{sy}\))

• Flow regime behavior (from \(n\))

• Structural memory timescales (from yield stress ratios)

• Quality control metrics (batch-to-batch consistency)

Comparison Across Protocols:

• LAOS vs steady-shear yield stress agreement

• Cox-Merz rule validation (if applicable)

• Amplitude sweep vs frequency sweep consistency

Parameter Interpretation

\(\sigma_y\) (Yield Stress):

The stress threshold required to initiate flow from rest (static) or during continuous flow (dynamic).

For graduate students: The SPP framework distinguishes two yield stresses via the intracycle stress decomposition \(\sigma(t) = G'_t[\gamma - \gamma_{eq}] + (G''_t/\omega)\dot{\gamma} + \sigma_y\). Static yield stress \(\sigma_{sy}\) (at strain reversal, \(\dot{\gamma} = 0\)) includes microstructural recovery during momentary rest, while dynamic yield stress \(\sigma_{dy}\) (at rate reversal) represents flowing state. Power-law scaling \(\sigma_y \sim \gamma_0^n\) with \(n \approx 0.2\) emerges from jamming universality (Liu & Nagel 2010). Ratio \(\sigma_{sy}/\sigma_{dy} \approx 2\text{--}3\) quantifies thixotropic cage reformation timescales.

For practitioners: Fit \(\sigma_{sy}\) from LAOS amplitude sweep (stress at maximum strain). Fit \(\sigma_{dy}\) from flow curve extrapolation to \(\dot{\gamma} \to 0\) or from LAOS rate reversal. Ratio \(> 3\) indicates strong thixotropy, requiring longer rest between measurements. Exponent \(n < 0.3\) (hard particles) vs \(n > 0.5\) (soft particles) guides formulation strategy.

:math:`G_{text{cage}}` (Cage Modulus):

Instantaneous elastic stiffness of the confining microstructural cage.

For graduate students: \(G_{\text{cage}} = (d\sigma/d\gamma)|_{\sigma=0}\) represents the cage spring constant in colloidal glass picture. Unlike \(G'(\omega)\), which decreases with amplitude in nonlinear regime, \(G_{\text{cage}}\) remains constant (amplitude-independent material property). Scales as \(G_{\text{cage}} \sim nk_BT/a^3\) where n is number density and a is particle radius, connecting to thermal energy and cage size.

For practitioners: Extract from linear regime \(G'_{\text{LVR}}\) or from SPP \(G'_t\) at \(\sigma = 0\) crossing. Typical values: 10–100 Pa (colloidal glasses), 100–1000 Pa (microgels), >1000 Pa (polymer gels). Use for quality control and batch consistency checks.

:math:`n_{sy}`, :math:`n_{dy}` (Power-Law Exponents):

Amplitude-dependence exponents quantifying cage rupture mechanism.

For graduate students: Scaling exponents in \(\sigma_y(\gamma_0) \sim \gamma_0^n\) connect to Herschel-Bulkley flow curve exponent via stress-rate duality. Rogers (2011) universal value \(n \approx 0.2\) for hard-sphere colloids derives from percolation/jamming scaling laws. Deviation from 0.2 indicates additional mechanisms (attractive forces, particle softness, structural hierarchy).

For practitioners: \(n \approx 0.2\) confirms hard-sphere-like behavior. \(n > 0.5\) suggests soft particles or weak cages. \(n \approx 1.0\) indicates linear dependence (fragile gels). Use to classify material type and predict performance across amplitude ranges.

Material Classification

Material Classification from SPP Yield Stress Parameters

Parameter Range	Material Behavior	Typical Materials	Processing Implications
\(G_{\text{cage}}\) = 10–100 Pa, \(n_{sy} \approx 0.2\)	Hard-sphere colloidal glass	Concentrated silica, PMMA colloids (\(\phi \sim 0.6\))	Universal jamming behavior, brittle cages
\(G_{\text{cage}}\) = 100–1000 Pa, \(n_{sy} \approx 0.5\)	Soft particle suspensions	Microgels, emulsions, soft colloids	Moderate amplitude-dependence, ductile cages
\(G_{\text{cage}} > 1000\) Pa, \(n_{sy} \approx 1.0\)	Polymer gels with weak cages	Carbopol, weak hydrogels	Strong amplitude-dependence, easy cage breaking
\(\sigma_{sy}/\sigma_{dy}\) < 2	Weak thixotropy	Simple yield-stress fluids	Minimal cage reformation during rest
\(\sigma_{sy}/\sigma_{dy} = 2\text{--}3\)	Moderate thixotropy (typical)	Concentrated suspensions, soft glasses	Standard Rogers et al. (2011) range
\(\sigma_{sy}/\sigma_{dy}\) > 3	Strong thixotropy	Highly thixotropic pastes, aged gels	Fast cage reformation, strong memory
\(n_{sy} \approx n_{dy} \approx 0.2\)	Universal jamming signature	Hard-sphere colloids near \(\phi_c\)	Theoretical prediction confirmed
Power-law regime: \(\sigma \sim \gamma_0^n\)	Cage rupture scaling	All yield-stress materials in LAOS	Amplitude-dependent yield, nonlinear LAOS

---

Parameters

Parameter	Units	Bounds	Description
G_cage	Pa	[1e-3, 1e9]	Apparent cage modulus; elastic stiffness at \(\sigma\) = 0
sigma_sy_scale	Pa	[1e-6, 1e9]	Static yield stress scale factor
sigma_sy_exp	—	[0.0, 2.0]	Static yield stress amplitude exponent
sigma_dy_scale	Pa	[1e-6, 1e9]	Dynamic yield stress scale factor
sigma_dy_exp	—	[0.0, 2.0]	Dynamic yield stress amplitude exponent
eta_inf	Pa·s	[1e-9, 1e6]	Infinite-shear viscosity (for flow curve)
n_power_law	—	[0.01, 2.0]	Flow power-law index
noise	Pa	[1e-10, 1e6]	Observation noise scale (Bayesian inference)

---

Fitting Guidance

(Fitting guidance content already present in the file)

---

Usage

(Usage examples already present in the file)

---

Validity and Assumptions

Applicability

The SPP yield stress model is most appropriate for:

• Yield stress fluids: Materials with clear solid-to-liquid transition

• Soft glassy materials: Colloidal gels, emulsions, foams, pastes

• LAOS amplitude sweeps: Progressive nonlinearity from linear to flowing

• Concentrated systems: Volume fractions near or above jamming

Assumptions

1. Single characteristic yield process: The model assumes a dominant yielding mechanism described by power-law scaling

2. Cage-based microstructure: Physical interpretation requires particle-based or droplet-based microstructure with cage confinement

3. Time-strain separability: Assumes steady-state oscillatory response without significant transient evolution during measurement

4. Sufficient harmonics: SPP extraction requires adequate harmonic content (typically n_harmonics \(\geq 15\))

Limitations

• Simple materials: Newtonian fluids show no amplitude dependence

• Polymer solutions: May require different physical interpretation

• Extreme amplitude: Very large \(\gamma_0\) may show non-power-law behavior

• Transient effects: Not suitable for strongly time-dependent (aging) materials

• Low frequency: \(\beta\)-relaxation must be slow compared to measurement

Usage Examples

Basic NLSQ Fitting (Amplitude Sweep)

import numpy as np
from rheojax.models import SPPYieldStress

# Amplitude sweep data: yield stresses vs. strain amplitude
gamma_0 = np.array([0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0])
sigma_sy = np.array([5.0, 8.5, 18.0, 32.0, 55.0, 110.0, 195.0, 340.0])

# Initialize and fit
model = SPPYieldStress()
model.fit(gamma_0, sigma_sy, test_mode='oscillation', yield_type='static')

# View fitted parameters
print(model)
# SPPYieldStress(G_cage=5.00e+02, σ_sy=1.00e+02, σ_dy=5.00e+01, n=0.50)

# Predict at new amplitudes
gamma_0_pred = np.logspace(-2, 1, 50)
sigma_pred = model.predict(gamma_0_pred)

Dynamic Yield Stress Fitting

# Dynamic yield stress (at rate reversal) typically lower than static
sigma_dy = np.array([2.0, 3.5, 7.5, 13.0, 22.0, 45.0, 80.0, 140.0])

model = SPPYieldStress()
model.fit(gamma_0, sigma_dy, test_mode='oscillation', yield_type='dynamic')

# Get both yield stresses across amplitude range
sweep_results = model.predict_amplitude_sweep(
    gamma_0_pred,
    yield_type='both'
)
print(f"Static yield stress: {sweep_results['sigma_sy']}")
print(f"Dynamic yield stress: {sweep_results['sigma_dy']}")

Flow Curve Fitting (Rotation Mode)

# Steady shear data
gamma_dot = np.logspace(-3, 2, 20)
sigma = 25.0 + 3.5 * gamma_dot**0.45  # Herschel-Bulkley-like

model = SPPYieldStress()
model.fit(gamma_dot, sigma, test_mode='rotation')

# Predict flow curve
sigma_pred = model.predict(gamma_dot)
print(f"Yield stress: {model.parameters.get_value('sigma_dy_scale'):.2f} Pa")
print(f"Power-law index: {model.parameters.get_value('n_power_law'):.2f}")

Bayesian Inference

from rheojax.models import SPPYieldStress
import numpy as np

# Example data: amplitude sweep
gamma_0 = np.array([0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0])
sigma_sy = np.array([5.0, 8.5, 18.0, 32.0, 55.0, 110.0, 195.0, 340.0])

# Initialize and NLSQ fit for warm-start
model = SPPYieldStress()
model.fit(gamma_0, sigma_sy, test_mode='oscillation', yield_type='static')

# Bayesian inference with warm-start
result = model.fit_bayesian(
    gamma_0,
    sigma_sy,
    test_mode='oscillation',
    num_warmup=1000,
    num_samples=2000
)

# Posterior summary
print(result.summary)

# Credible intervals
intervals = model.get_credible_intervals(
    result.posterior_samples,
    credibility=0.95
)
for param, (low, high) in intervals.items():
    print(f"{param}: [{low:.3f}, {high:.3f}]")

Integration with SPPDecomposer

from rheojax.transforms import SPPDecomposer
from rheojax.models import SPPYieldStress
from rheojax.core.data import RheoData
import numpy as np

# Generate example LAOS waveforms at different amplitudes
omega = 1.0  # rad/s
n_cycles = 5
n_points = 200
t = np.linspace(0, 2 * np.pi * n_cycles / omega, n_points)

amplitude_sweep_data = {}
for gamma_0 in [0.01, 0.1, 1.0]:
    strain = gamma_0 * np.sin(omega * t)
    stress = 100.0 * strain + 20.0 * gamma_0**0.3 * np.sign(np.cos(omega * t))
    amplitude_sweep_data[gamma_0] = RheoData(t=t, stress=stress, strain=strain)

# Process LAOS waveforms at multiple amplitudes
decomposer = SPPDecomposer(n_harmonics=39, step_size=8)

# Collect yield stresses from each amplitude
amplitudes = []
static_yields = []
dynamic_yields = []

for gamma_0, data in amplitude_sweep_data.items():
    result = decomposer.transform(data)
    amplitudes.append(gamma_0)
    static_yields.append(np.abs(result['stress_at_strain_max']))
    dynamic_yields.append(np.abs(result['stress_at_rate_reversal']))

# Fit yield stress model
model = SPPYieldStress()

# Static yield stress
model.fit(np.array(amplitudes), np.array(static_yields),
          test_mode='oscillation', yield_type='static')

print(f"Static yield exponent: {model.parameters.get_value('sigma_sy_exp'):.3f}")

Troubleshooting

Poor Power-Law Fit

Symptoms: Large fitting residuals, unreasonable exponents

Solutions:

1. Check data quality at low amplitudes (may be noisy near linear regime)

2. Verify sufficient amplitude range (at least 1-2 decades)

3. Look for regime transitions (different slopes at different amplitudes)

# Visualize power-law fit
import matplotlib.pyplot as plt

model.fit(gamma_0, sigma_sy, test_mode='oscillation', yield_type='static')

fig, ax = plt.subplots()
ax.loglog(gamma_0, sigma_sy, 'o', label='Data')
ax.loglog(gamma_0, model.predict(gamma_0), '-', label='Fit')
ax.set_xlabel(r'$\gamma_0$')
ax.set_ylabel(r'$\sigma_{sy}$ (Pa)')
ax.legend()
plt.show()

Cage Modulus Issues

Symptoms: \(G_{\text{cage}}\) unreasonably large or small

Causes:

• Insufficient low-amplitude data points

• Material not exhibiting clear cage behavior

• Noise dominating linear regime

Solutions:

1. Ensure adequate data in linear regime (\(\gamma_0\) < 0.1)

2. Verify linear viscoelastic moduli are consistent

3. Consider if cage model is appropriate for the material

Bayesian Convergence

Symptoms: R-hat > 1.1, low ESS, divergent transitions

Solutions:

1. Always NLSQ warm-start: Critical for stable sampling

   # NLSQ first, then Bayesian
   model.fit(gamma_0, sigma_sy, test_mode='oscillation')
   result = model.fit_bayesian(gamma_0, sigma_sy, ...)
   

2. Increase samples: num_warmup=2000, num_samples=4000

3. Check priors: Ensure physically reasonable bounds

4. Inspect trace plots: Look for mixing issues

   import arviz as az
   az.plot_trace(result.arviz_data)
   

Prior Sensitivity

The model uses physically-motivated priors:

• LogNormal for scale parameters (G_cage, stress scales, viscosity)

• Beta for bounded exponents [0, 2]

• HalfCauchy for noise scale

If priors are too informative:

# Check prior impact with prior predictive sampling
from rheojax.visualization import plot_bayesian_diagnostics

# Compare posterior to prior
plot_bayesian_diagnostics(result, diagnostics=['pair', 'forest'])

Related Models

Model	Use Case
SPP Decomposer (Sequence of Physical Processes)	Extract instantaneous moduli from individual LAOS waveforms
Herschel-Bulkley Model	Steady-shear yield stress (Herschel-Bulkley model)
SGR Conventional (Soft Glassy Rheology) — Handbook	Soft glassy rheology for aging yield stress fluids
Bingham Plastic	Simple yield stress + Newtonian flow

References

[1]

Rogers, S. A., Erwin, B. M., Vlassopoulos, D., and Cloitre, M. “A sequence of physical processes determined and quantified in LAOS: Application to a yield stress fluid.” Journal of Rheology, 55, 435-458 (2011). DOI: 10.1122/1.3544591 PDF

[2]

Rogers, S. A. and Lettinga, M. P. “A sequence of physical processes determined and quantified in large-amplitude oscillatory shear (LAOS): Application to theoretical nonlinear models.” Journal of Rheology, 56(1), 1-25 (2012). DOI: 10.1122/1.3662962 PDF

[3]

Rogers, S. A. “In search of physical meaning: defining transient parameters for nonlinear viscoelasticity.” Rheologica Acta, 56, 501-525 (2017). DOI: 10.1007/s00397-017-1008-1 PDF

[4]

Ewoldt, R. H., Hosoi, A. E., and McKinley, G. H. “New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear.” Journal of Rheology, 52, 1427-1458 (2008). DOI: 10.1122/1.2970095

[5]

Hyun, K., et al. “A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS).” Progress in Polymer Science, 36, 1697-1753 (2011). DOI: 10.1016/j.progpolymsci.2011.02.002

[6]

Bonn, D., Denn, M. M., Berthier, L., Divoux, T., and Manneville, S. “Yield stress materials in soft condensed matter.” Reviews of Modern Physics, 89, 035005 (2017). DOI: 10.1103/RevModPhys.89.035005

[7]

Donley, G. J., et al. “Elucidating the G’’ overshoot in soft materials with a yield transition via a time-resolved experimental strain decomposition.” Proceedings of the National Academy of Sciences, 117, 21945-21952 (2020). DOI: 10.1073/pnas.2003869117

[8]

Liu, A. J. and Nagel, S. R. “The jamming transition and the marginally jammed solid.” Annual Review of Condensed Matter Physics, 1, 347-369 (2010). DOI: 10.1146/annurev-conmatphys-070909-104045

[9]

Kim, J., et al. “Visualization of time-dependent intracycle molecular orientation dynamics in LAOS from anisotropic SANS.” Journal of Rheology, 64, 291-303 (2020).

[10]

Saengow, C., Giacomin, A. J., and Kolitawong, C. “Exact analytical solution for large-amplitude oscillatory shear flow.” Macromolecular Theory and Simulations, 24, 352-392 (2015). DOI: 10.1002/mats.201400104

See Also

• Sequence of Physical Processes (SPP) - Complete SPP analysis user guide

• SPP Analysis API - SPP API reference

• SPP Decomposer (Sequence of Physical Processes) - SPP Decomposer transform documentation