TNT Stretch-Creation (Enhanced Reformation) — Handbook

Quick Reference

Use when:

Networks where bond creation rate depends on chain stretch (strain-enhanced crosslinking, stretch-activated association)

Parameters:

4 parameters: \(G\) (Pa), \(\tau_b\) (s), \(\kappa\) (creation enhancement factor, dimensionless 0-5), \(\eta_s\) (Pa·s)

Key equation:

Creation rate: \(k_{on}(S) = \frac{1}{\tau_b} \left(1 + \kappa \left(\text{tr}(S) - 3\right)\right)\)

Test modes:

All 6: FLOW_CURVE, OSCILLATION, STARTUP, RELAXATION, CREEP, LAOS

Material examples:

Strain-crystallizing rubbers, mechanophore-activated networks, strain-induced gelation systems, adaptive polymers

Key characteristics:

Strain hardening through enhanced crosslink formation, positive feedback under extension

Notation Guide

Symbol

Units

Meaning

\(S\)

dimensionless

Dimensionless conformation tensor (second moment)

\(G\)

Pa

Plateau modulus (network elasticity)

\(\tau_b\)

s

Breakage timescale (bond lifetime at equilibrium)

\(\kappa\)

dimensionless

Creation enhancement factor (0-5 typical range)

\(k_{on}(S)\)

s⁻¹

Stretch-dependent bond creation rate

\(k_{off}\)

s⁻¹

Bond breakage rate (constant, \(1/\tau_b\))

\(\text{tr}(S)\)

dimensionless

Trace of conformation tensor (mean-square chain stretch)

\(\eta_s\)

Pa·s

Solvent viscosity (Newtonian background)

\(\kappa_{flow}\)

s⁻¹

Velocity gradient tensor

\(D\)

s⁻¹

Rate-of-strain tensor (symmetric part of \(\kappa_{flow}\))

\(\sigma\)

Pa

Cauchy stress tensor

\(I\)

dimensionless

Identity tensor

Overview

The Stretch-Creation variant of the Tanaka-Edwards transient network model introduces a positive feedback mechanism between chain stretch and bond formation rate. Unlike the base model where creation and breakage rates are constant, this variant recognizes that some polymer networks form new crosslinks more readily when chains are already extended.

Physical Motivation

Several physical systems exhibit strain-enhanced crosslinking:

  1. Strain crystallization in natural rubber: Stretched polymer chains align and crystallize, creating additional physical crosslinks. This is the classic mechanism responsible for the remarkable strength and toughness of natural rubber.

  2. Mechanophore-activated networks: Chemical groups that become reactive when subjected to mechanical force. Chain extension activates binding sites that were previously inaccessible.

  3. Stretch-induced gelation: Some polymer solutions gel more rapidly under extension as chain stretching promotes intermolecular contacts and association.

  4. Adaptive hydrogels: Biomimetic networks where mechanical loading triggers crosslink formation, similar to biological tissue remodeling.

Distinction from Elastic Stiffening

The stretch-creation mechanism is fundamentally different from FENE-like elastic stiffening:

  • FENE models: Chain force increases nonlinearly with extension due to finite extensibility (elastic stiffening)

  • Stretch-creation: Number of crosslinks increases with extension (kinetic stiffening)

Both lead to strain hardening, but through different mechanisms. The stretch-creation variant can be combined with FENE to capture both effects simultaneously.

Positive Feedback and Stability

The coupling \(k_{on} \propto (1 + \kappa(\text{tr}(S) - 3))\) creates positive feedback:

  1. External force stretches chains → \(\text{tr}(S)\) increases

  2. Enhanced creation rate → more crosslinks form

  3. More crosslinks → higher stress under same strain

  4. Can lead to further stretching if load-controlled

This positive feedback is stabilized by:

  • Constant breakage rate \(k_{off} = 1/\tau_b\)

  • Flow-induced relaxation (advection, rotation)

  • Finite \(\kappa\) values prevent runaway

The balance between enhanced creation and constant breakage sets the steady-state network structure.

Physical Foundations

Stretch-Activated Association

The core physical idea is that bond formation probability increases with chain extension. Several microscopic mechanisms can lead to this behavior:

Entropy-driven exposure: Coiled chains hide binding sites; stretching exposes them.

Alignment-induced association: Extended chains align parallel to each other, promoting intermolecular contacts and hydrogen bonding or van der Waals attraction.

Force-activated chemistry: Mechanical force lowers activation barriers for certain chemical reactions (mechanophore activation).

The mean-field coupling \(k_{on} \propto \text{tr}(S)\) assumes that bond creation rate scales with the average mean-square chain stretch. More sophisticated treatments could use the full distribution of chain stretches, but the mean-field approximation captures the essential physics.

Strain Crystallization Physics

Natural rubber’s exceptional mechanical properties arise from strain-induced crystallization. At rest, polymer chains are amorphous. Under extension:

  1. Chains align along the stretching direction

  2. Aligned segments pack into crystalline lamellae

  3. Crystallites act as additional physical crosslinks

  4. Network modulus increases dramatically

Flory (1947) first recognized this mechanism. The stretch-creation variant phenomenologically captures this effect through \(\kappa > 0\).

Mechanophore Networks

Modern mechanochemistry enables design of polymers with force-sensitive chemical groups (mechanophores). Examples include:

  • Spiropyran that isomerizes under tension

  • Cyclobutane rings that open to form reactive radicals

  • Hidden thiols that become exposed for disulfide exchange

These systems can be engineered so that chain extension activates crosslinking chemistry, directly realizing the stretch-creation coupling.

Mean-Field Coupling Assumption

The model assumes all chains see the same average stretch \(\text{tr}(S)/3\). In reality:

  • Chain length polydispersity creates distribution of stretches

  • Network heterogeneity (defects, entanglements) causes local variations

  • Flow history affects different chain populations differently

Despite these simplifications, the mean-field coupling \(\kappa(\text{tr}(S) - 3)\) provides a tractable framework that captures qualitative behavior and can guide experimental design.

Stability and Bounds

The linear coupling \(k_{on} = (1/\tau_b)(1 + \kappa(\text{tr}(S) - 3))\) has no built-in saturation. For very large \(\kappa\) or \(\text{tr}(S)\), the creation rate can become arbitrarily large, potentially leading to numerical instability or unphysical predictions.

Practical considerations:

  • \(\kappa \leq 2\) for most fits (mild to moderate enhancement)

  • \(\kappa > 5\) may require additional damping or saturation terms

  • Ensure steady-state solutions exist (balance with breakage)

  • For extreme extensions, combine with FENE to bound \(\text{tr}(S)\)

Governing Equations

Modified Conformation Tensor Evolution

The key modification is the stretch-dependent creation rate:

\[k_{on}(S) = \frac{1}{\tau_b} \left(1 + \kappa \left(\text{tr}(S) - 3\right)\right)\]

The conformation tensor \(S\) evolves according to:

\[\frac{dS}{dt} = \kappa_{flow} \cdot S + S \cdot \kappa_{flow}^T - k_{off}(S - S_{eq})\]

where \(k_{off} = 1/\tau_b\) and the equilibrium tensor \(S_{eq}\) is determined by the balance of creation and destruction.

At equilibrium (rest), \(dS/dt = 0\) implies:

\[k_{on}(S_{eq}) S_{eq} = k_{off} I\]

For the base model (\(\kappa = 0\)), \(k_{on} = k_{off}\), so \(S_{eq} = I\).

For stretch-creation (\(\kappa > 0\)), the equilibrium is modified:

\[\left(1 + \kappa(\text{tr}(S_{eq}) - 3)\right) S_{eq} = I\]

This is a nonlinear equation for \(S_{eq}\). For small \(\kappa\), \(S_{eq} \approx I\) (perturbation). For large \(\kappa\), \(S_{eq}\) can deviate significantly.

However, in the current implementation, the creation term is written as:

\[\frac{dS}{dt} = \kappa_{flow} \cdot S + S \cdot \kappa_{flow}^T - \frac{1}{\tau_b}(S - I) - \frac{\kappa}{\tau_b}(\text{tr}(S) - 3) I\]

This form separates the base creation-destruction term \(-(1/\tau_b)(S - I)\) from the stretch-enhancement correction \(-(\kappa/\tau_b)(\text{tr}(S) - 3) I\).

Constitutive Equation (Stress)

The stress tensor is the sum of network and solvent contributions:

\[\sigma = G(S - I) + 2 \eta_s D\]

where:

  • \(G(S - I)\) is the elastic network stress (deformation from equilibrium)

  • \(2\eta_s D\) is the Newtonian solvent stress

The modulus \(G\) is constant (linear elasticity of Gaussian chains). Nonlinearity enters through the evolution of \(S\).

Steady Shear Flow

For steady simple shear \(\dot{\gamma}\), the velocity gradient is:

\[\begin{split}\kappa_{flow} = \begin{pmatrix} 0 & \dot{\gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\end{split}\]

At steady state, \(dS/dt = 0\). The conformation tensor components \(S_{xx}\), \(S_{xy}\), \(S_{yy}\) satisfy:

\[\dot{\gamma} S_{xy} &= \frac{1}{\tau_b}(S_{xx} - 1) + \frac{\kappa}{\tau_b}(\text{tr}(S) - 3)\]
\[\dot{\gamma}(S_{yy} - S_{xx}) &= \frac{1}{\tau_b} S_{xy}\]
\[0 &= \frac{1}{\tau_b}(S_{yy} - 1) + \frac{\kappa}{\tau_b}(\text{tr}(S) - 3)\]

where \(\text{tr}(S) = S_{xx} + S_{yy} + S_{zz}\) and \(S_{zz} = 1 + (\kappa/\tau_b)(\text{tr}(S) - 3)\).

The shear stress is:

\[\sigma_{xy} = G S_{xy} + \eta_s \dot{\gamma}\]

This is a nonlinear system coupling \(S_{xx}, S_{xy}, S_{yy}\) through \(\text{tr}(S)\). For \(\kappa = 0\), the system decouples and reduces to the base model’s analytical solution.

For \(\kappa > 0\), numerical root-finding is required. The solution exhibits strain hardening: \(\sigma_{xy}\) increases more steeply with \(\dot{\gamma}\) compared to \(\kappa = 0\).

Small Amplitude Oscillatory Shear (SAOS)

For linearized perturbations around equilibrium \(S = I + \epsilon e^{i\omega t}\), the stretch-creation correction \(\kappa(\text{tr}(S) - 3)\) is second-order in \(\epsilon\) and does not affect the linear viscoelastic response.

Thus, the complex modulus for SAOS is identical to the base model:

\[G^* = G' + iG'' = G \frac{i\omega\tau_b}{1 + i\omega\tau_b}\]
\[G' = G \frac{(\omega\tau_b)^2}{1 + (\omega\tau_b)^2}\]
\[G'' = G \frac{\omega\tau_b}{1 + (\omega\tau_b)^2}\]

This is a single Maxwell element. The stretch-creation mechanism is invisible in the linear regime.

Startup of Steady Shear

Starting from rest (\(S(0) = I\)), apply \(\dot{\gamma}\) and integrate:

\[\frac{dS}{dt} = \kappa_{flow} \cdot S + S \cdot \kappa_{flow}^T - \frac{1}{\tau_b}(S - I) - \frac{\kappa}{\tau_b}(\text{tr}(S) - 3) I\]

The stress \(\sigma_{xy}(t) = G S_{xy}(t) + \eta_s \dot{\gamma}\) initially grows as chains stretch, potentially overshoots if \(\kappa\) enhances the transient buildup, and then decays to the steady-state value.

The magnitude and timing of the overshoot depend on \(\kappa\):

  • \(\kappa = 0\): Standard Tanaka-Edwards overshoot (moderate)

  • \(\kappa > 0\): Enhanced overshoot due to faster creation during transient stretching

  • \(\kappa \gg 1\): Pronounced overshoot, potentially delayed relaxation

Stress Relaxation After Cessation

After stopping flow from steady state, the conformation tensor relaxes:

\[\frac{dS}{dt} = -\frac{1}{\tau_b}(S - I) - \frac{\kappa}{\tau_b}(\text{tr}(S) - 3) I\]

The relaxation is no longer single-exponential due to the coupling between \(S\) components through \(\text{tr}(S)\).

For small \(\kappa\), the relaxation is approximately exponential with timescale \(\tau_b\). For larger \(\kappa\), the decay is slower initially (creation resists relaxation) then faster as \(\text{tr}(S) \to 3\).

Creep and Recovery

Under constant stress \(\sigma_0\), the conformation tensor evolves with \(\kappa_{flow} = \dot{\gamma}(t) e_x \otimes e_y\), where \(\dot{\gamma}(t)\) is the instantaneous shear rate determined by:

\[\sigma_0 = G S_{xy}(t) + \eta_s \dot{\gamma}(t)\]

This is a differential-algebraic system. The stretch-creation coupling modifies the creep compliance curve, especially at long times where steady-state network structure affects terminal flow rate.

Large Amplitude Oscillatory Shear (LAOS)

Under \(\gamma(t) = \gamma_0 \sin(\omega t)\), the conformation tensor and stress evolve according to the full nonlinear ODE. The stretch-creation mechanism generates higher harmonics in the stress response.

The third harmonic \(I_3/I_1\) is enhanced by \(\kappa\) since strain-induced creation amplifies the stress at large instantaneous strain.

Parameter Table

Parameter

Symbol

Default

Bounds

Description

Modulus

\(G\)

1000 Pa

(1, 10⁸) Pa

Plateau modulus (network elasticity)

Breakage time

\(\tau_b\)

1.0 s

(10⁻⁶, 10⁴) s

Characteristic bond lifetime at equilibrium

Creation enhancement

\(\kappa\)

0.5

(0.0, 5.0)

Stretch-creation coupling strength (dimensionless)

Solvent viscosity

\(\eta_s\)

0.0 Pa·s

(0.0, 10⁴) Pa·s

Newtonian background viscosity

Parameter Interpretation

Plateau Modulus \(G\)

  • Directly measurable from \(G'\) at high frequencies (above \(1/\tau_b\))

  • Related to network strand density: \(G = \nu k_B T\) where \(\nu\) is number density of elastically active strands

  • Typical range: 100 Pa (weak gels) to 1 MPa (elastomers)

Breakage Timescale \(\tau_b\)

  • Sets the characteristic relaxation time in the linear regime

  • Inverse of the peak in \(G''(\omega)\)

  • For physical networks: microseconds to hours depending on bond energy

  • For chemical networks with dynamic covalent bonds: seconds to days

Creation Enhancement \(\kappa\)

This is the key parameter that distinguishes the stretch-creation variant.

Physical interpretation: Fractional increase in creation rate per unit excess chain stretch.

  • \(\kappa = 0\): No stretch-creation coupling (base Tanaka-Edwards)

  • \(\kappa = 0.1 - 0.5\): Mild enhancement, subtle strain hardening

  • \(\kappa = 1 - 2\): Moderate enhancement, noticeable strain stiffening

  • \(\kappa > 2\): Strong coupling, pronounced strain hardening and flow instability

Typical values:

  • Strain-crystallizing rubber: \(\kappa \sim 0.5 - 1.5\)

  • Mechanophore networks: \(\kappa \sim 0.2 - 1.0\) (depends on activation force)

  • Stretch-induced gelation: \(\kappa \sim 1 - 3\)

Fitting strategy: Start with \(\kappa = 0\) (base model), fit SAOS to get \(G, \tau_b\). Then fit nonlinear startup or flow curve with \(\kappa > 0\) as the only free parameter.

Solvent Viscosity \(\eta_s\)

  • Often negligible for polymer melts (\(\eta_s = 0\))

  • Important for solutions and gels (can dominate at high shear rates)

  • Sets Newtonian plateau at high \(\dot{\gamma}\) (above network relaxation)

Validity and Assumptions

Assumptions

  1. Gaussian chain statistics: No finite extensibility (chains can stretch indefinitely). Combine with FENE variant if chains approach full extension.

  2. Mean-field stretch coupling: All chains see average \(\text{tr}(S)/3\). Ignores distribution of chain extensions.

  3. Linear coupling: \(k_{on} \propto (1 + \kappa(\text{tr}(S) - 3))\) is the simplest functional form. Real systems may saturate at large stretch.

  4. Affine deformation: Chains deform with the continuum (no slip, no reptation). Valid for well-crosslinked networks.

  5. Single relaxation time: All bonds have the same lifetime \(\tau_b\). Polydispersity would require spectrum of \(\tau_b\).

  6. Incompressibility: Trace of stress is determined by pressure (not modeled). Only deviatoric stresses matter.

Validity Regime

The stretch-creation model is physically justified when:

  • Bond formation rate increases measurably with chain extension (mechanophore activation, strain crystallization)

  • Network is well-connected (percolated, no dangling ends)

  • Deformations are not so large that chains reach full extension (\(\text{tr}(S) \ll b^2\) where \(b\) is Kuhn length)

  • Timescales are slow enough that local equilibration is faster than bond dynamics

Breakdown Scenarios

The model breaks down when:

  1. Extreme extension: \(\text{tr}(S) \to b^2\) (Kuhn length). Need FENE correction.

  2. Runaway instability: If \(\kappa\) is too large, positive feedback can cause numerical blow-up. Watch for \(\text{tr}(S) \to \infty\).

  3. High frequency: Entanglements or glassy modes faster than \(1/\tau_b\) are not captured.

  4. Nonaffine deformation: Loosely crosslinked gels or near gelation point may exhibit nonaffine rearrangements.

  5. Saturation neglected: Real \(k_{on}(S)\) likely saturates at large stretch. Linear coupling is first-order approximation.

Regimes and Behavior

Linear Viscoelastic Regime

For \(\gamma_0 \ll 1\) or \(\omega\tau_b \gg 1\):

  • \(G', G''\) identical to base model (single Maxwell element)

  • \(\kappa\) is invisible (second-order in strain)

  • Crossover frequency \(\omega_c = 1/\tau_b\)

Key point: SAOS alone cannot determine \(\kappa\). Need nonlinear tests.

Moderate Strain Regime

For \(\gamma_0 \sim 0.1 - 1\) or \(\text{Wi} = \dot{\gamma}\tau_b \sim 1\):

  • Stress begins to exceed linear prediction

  • Strain hardening becomes measurable

  • \(\kappa\) controls magnitude of enhancement

  • Startup overshoot is amplified

Signature: Flow curve \(\sigma(\dot{\gamma})\) curves upward relative to base model.

Large Strain Regime

For \(\gamma_0 > 1\) or \(\text{Wi} > 1\):

  • Significant additional stress from enhanced crosslinking

  • \(\text{tr}(S)\) can be much larger than 3

  • Positive feedback becomes strong (creation accelerates)

  • Risk of numerical issues if \(\kappa\) too large

Behavior: Stress can increase superlinearly with strain, resembling strain hardening in filled rubbers.

Startup Transients

Upon imposing \(\dot{\gamma}\):

  1. Initial loading (\(t \ll \tau_b\)): Elastic response, \(S\) grows affinely

  2. Overshoot (\(t \sim \tau_b\)): Competition between stretching and breakage, enhanced by \(\kappa > 0\)

  3. Decay to steady state (\(t \gg \tau_b\)): Network reaches new equilibrium structure

Effect of \(\kappa\):

  • \(\kappa = 0\): Moderate overshoot

  • \(\kappa \sim 1\): Enhanced overshoot (more chains created during loading)

  • \(\kappa > 2\): Pronounced overshoot, delayed relaxation

Steady-State Flow

At \(t \to \infty\), the network reaches a balance between stretch-enhanced creation and constant breakage.

  • Low \(\text{Wi}\): Newtonian-like (\(\sigma \propto \dot{\gamma}\))

  • High \(\text{Wi}\): Strain hardening (\(\sigma \propto \dot{\gamma}^{1+\alpha}\) with \(\alpha > 0\) depending on \(\kappa\))

The flow curve \(\sigma_{xy}(\dot{\gamma})\) is steeper than the base model, indicating enhanced resistance to flow due to more crosslinks forming under deformation.

What You Can Learn

From SAOS Data

  • Plateau modulus \(G\) from high-frequency \(G'\)

  • Relaxation time \(\tau_b\) from \(G''\) peak or crossover

  • Solvent viscosity \(\eta_s\) from high-frequency \(G''\) tail

Cannot determine \(\kappa\) from SAOS alone (linear regime insensitive to stretch-creation coupling).

From Startup Shear

  • Overshoot magnitude sensitive to \(\kappa\)

  • Time to peak modified by creation enhancement

  • Compare to base model (\(\kappa = 0\)) to isolate effect

Strategy: Fit base model to SAOS, then fit startup with \(\kappa\) as single free parameter.

From Flow Curves

  • Strain hardening exponent reflects \(\kappa\) magnitude

  • High-rate plateau (if present) from \(\eta_s\)

  • Curvature in log-log plot indicates nonlinear creation kinetics

Diagnosis: Upward curvature in \(\sigma(\dot{\gamma})\) suggests stretch-creation coupling.

From Creep/Recovery

  • Steady-state compliance affected by \(\kappa\) through modified network structure

  • Recovery shape nonexponential due to stretch-dependent relaxation

  • Permanent strain (if any) indicates irreversible bond rearrangement

From LAOS

  • Higher harmonics \(I_3/I_1, I_5/I_1\) enhanced by \(\kappa\)

  • Pipkin diagram shows expanded nonlinear region compared to base model

  • Lissajous curves more elliptical (strain stiffening)

Experimental Design

Sample Preparation

  • Ensure network is fully formed and equilibrated before testing

  • Avoid pre-shear that might change network structure (unless studying thixotropy)

  • Temperature control critical (affects \(\tau_b\) exponentially via Arrhenius)

Control Samples

To isolate the stretch-creation effect:

  • Chemically crosslinked network (no bond dynamics): Should show only elastic response

  • Base transient network (no mechanophore or strain-crystallization): Compare \(\kappa = 0\) prediction

Avoiding Artifacts

  • Wall slip: Use serrated geometries or small gaps

  • Edge fracture: Stay below critical strain (~100-300% for elastomers)

  • Strain crystallization melting: Keep temperature above crystalline melting point unless studying that effect

  • Shear heating: Use small gaps and low frequencies for viscous samples

Data Quality Checks

  • SAOS: \(G', G''\) must satisfy Kramers-Kronig relations

  • Startup: Repeatability across cycles (network should be reversible)

  • Flow curve: No hysteresis between up and down sweeps (unless thixotropic)

  • LAOS: Fourier spectrum should decay monotonically with harmonic number

Computational Implementation

Numerical Considerations

The stretch-creation variant requires ODE integration for all test modes except SAOS (which is analytical). Key challenges:

  1. Nonlinear coupling: \(\text{tr}(S)\) couples all components

  2. Positive feedback: Large \(\kappa\) can cause stiffness in ODE

  3. Steady-state root-finding: Implicit equations for flow curve

  4. JIT compilation: JAX automates differentiation for gradients

JIT Compilation with JAX

The ODE right-hand side \(dS/dt\) is JIT-compiled for efficiency:

@jax.jit
def ode_rhs(S, kappa_flow, kappa):
    trace_S = jnp.trace(S)
    dSdt = (jnp.dot(kappa_flow, S) + jnp.dot(S, kappa_flow.T)
            - (1/tau_b) * (S - jnp.eye(3))
            - (kappa/tau_b) * (trace_S - 3) * jnp.eye(3))
    return dSdt

JAX automatically differentiates this for use in NLSQ fitting (Jacobian-based optimization).

Steady-State Root Finding

For flow curve prediction, solve \(dS/dt = 0\):

from jax.scipy.optimize import root

def residual(S_flat, gamma_dot, kappa):
    S = S_flat.reshape(3, 3)
    dSdt = ode_rhs(S, kappa_flow(gamma_dot), kappa)
    return dSdt.flatten()

S_ss = root(residual, S_init.flatten(), gamma_dot, kappa).reshape(3, 3)
sigma = G * S_ss[0, 1] + eta_s * gamma_dot

For \(\kappa > 0\), need good initial guess (e.g., base model solution or previous \(\dot{\gamma}\) value).

Vectorization for Efficiency

When predicting over arrays of \(\dot{\gamma}\) or \(\omega\), use jax.vmap:

predict_single = lambda gamma_dot: solve_steady_state(gamma_dot, kappa)
predict_vectorized = jax.vmap(predict_single)
sigma_array = predict_vectorized(gamma_dot_array)

This compiles to parallel execution on GPU/TPU.

Handling Numerical Instability

If \(\text{tr}(S) \to \infty\) (runaway creation):

  1. Reduce :math:`kappa`: Likely unphysical value

  2. Add saturation: Modify \(k_{on}\) to plateau at large stretch

  3. Combine with FENE: Bound chain extension

  4. Check initial conditions: Ensure \(S(0) = I\) for rest

Typical symptom: Solver fails with “maximum iterations exceeded” or NaN in stress.

Fitting Guidance

Hierarchical Fitting Strategy

Step 1: Linear viscoelasticity (SAOS)

Fit base model (\(\kappa = 0\)) to determine:

  • \(G\) from plateau \(G'\)

  • \(\tau_b\) from crossover frequency

  • \(\eta_s\) from high-frequency \(G''\)

Step 2: Nonlinear startup (validate base)

Predict startup with \(\kappa = 0\), check agreement with data. If overshoot is under-predicted, proceed to Step 3.

Step 3: Fit :math:`kappa` from startup or flow curve

With \(G, \tau_b, \eta_s\) fixed, optimize \(\kappa\) to match:

  • Startup overshoot magnitude

  • Flow curve curvature at high \(\text{Wi}\)

Step 4: Joint refinement (optional)

Re-optimize all 4 parameters simultaneously on combined SAOS + startup + flow data.

Parameter Bounds

Parameter

Typical Range

Fitting Bounds

\(G\)

100 - 10⁶ Pa

(1, 10⁸) Pa

\(\tau_b\)

0.001 - 1000 s

(10⁻⁶, 10⁴) s

\(\kappa\)

0.0 - 2.0

(0.0, 5.0)

\(\eta_s\)

0 - 1000 Pa·s

(0, 10⁴) Pa·s

Common Pitfalls

  1. Overfitting :math:`kappa`: Without SAOS constraint, can fit noise. Always anchor with linear data first.

  2. Confusing :math:`kappa` with FENE: Both cause strain hardening. Check if hardening appears in extension (FENE) vs creation rate (stretch-creation).

  3. Ignoring :math:`eta_s`: High-rate plateau mistaken for creation effect. Fit solvent viscosity separately.

  4. Too large :math:`kappa`: Values above 3-5 often unphysical, may cause numerical issues.

  5. Wrong initial guess: Root-finding for steady state needs reasonable \(S\) initial guess, especially large \(\kappa\).

Optimization Settings (NLSQ)

RheoJAX uses NLSQ (JAX-accelerated Levenberg-Marquardt):

result = model.fit(
    data,
    test_mode='startup',
    max_iter=5000,
    ftol=1e-8,
    xtol=1e-8,
)

For stretch-creation variant:

  • Increase max_iter to 5000-10000 if \(\kappa > 1\)

  • Set ftol=1e-8 for tight convergence

  • Monitor residuals: should decrease monotonically

If fit fails to converge:

  • Check data quality (noise, outliers)

  • Reduce number of free parameters (fix \(\eta_s = 0\) if applicable)

  • Try different initial guess for \(\kappa\) (e.g., 0.1, 0.5, 1.0)

Diagnostics

After fitting, check:

  1. Residual plot: Should be random scatter, no systematic trends

  2. Predicted vs observed: \(R^2 > 0.95\) for good fit

  3. Parameter uncertainties: Bootstrap or Bayesian inference

  4. Cross-validation: Predict held-out test (e.g., different \(\dot{\gamma}\))

Physical sanity:

  • \(G\) within expected range for material class

  • \(\tau_b\) consistent with bond energy (Arrhenius check)

  • \(\kappa\) positive (creation enhances with stretch)

Usage Examples

Basic Usage

from rheojax.models import TNTSingleMode
from rheojax.core import RheoData
import numpy as np

# Create model with stretch-creation breakage
model = TNTSingleMode(breakage="stretch_creation")

# Load experimental data (startup shear)
t = np.linspace(0, 10, 200)  # seconds
sigma_exp = load_experimental_stress(t)  # Pa

data = RheoData(x=t, y=sigma_exp, test_mode='startup')

# Fit model
result = model.fit(data, gamma_dot=1.0)  # shear rate in s^-1

# Inspect fitted parameters
print(f"G = {model.G.value:.2e} Pa")
print(f"tau_b = {model.tau_b.value:.3f} s")
print(f"kappa = {model.kappa.value:.3f}")
print(f"eta_s = {model.eta_s.value:.2e} Pa·s")

# Predict and plot
sigma_pred = model.predict(t, test_mode='startup', gamma_dot=1.0)

import matplotlib.pyplot as plt
plt.plot(t, sigma_exp, 'o', label='Data')
plt.plot(t, sigma_pred, '-', label='Fit')
plt.xlabel('Time (s)')
plt.ylabel('Stress (Pa)')
plt.legend()
plt.show()

SAOS Prediction

# Small amplitude oscillatory shear
omega = np.logspace(-2, 2, 50)  # rad/s

# Set parameters manually or from previous fit
model.G.value = 1000.0        # Pa
model.tau_b.value = 1.0       # s
model.kappa.value = 0.5       # dimensionless
model.eta_s.value = 0.0       # Pa·s

# Predict complex modulus (kappa has no effect in linear regime)
G_complex = model.predict(omega, test_mode='oscillation')

G_prime = G_complex.real
G_double_prime = G_complex.imag

plt.loglog(omega, G_prime, label="G'")
plt.loglog(omega, G_double_prime, label='G"')
plt.xlabel('Frequency (rad/s)')
plt.ylabel('Modulus (Pa)')
plt.legend()
plt.show()

Flow Curve Prediction

# Steady shear flow
gamma_dot = np.logspace(-2, 2, 50)  # s^-1

# Predict steady-state stress
sigma = model.predict(gamma_dot, test_mode='flow_curve')

plt.loglog(gamma_dot, sigma)
plt.xlabel('Shear rate (1/s)')
plt.ylabel('Stress (Pa)')
plt.title(f'Flow curve (kappa = {model.kappa.value:.2f})')
plt.show()

# Compare with base model (kappa = 0)
model_base = TNTSingleMode(breakage="base")
model_base.G.value = model.G.value
model_base.tau_b.value = model.tau_b.value
model_base.eta_s.value = model.eta_s.value

sigma_base = model_base.predict(gamma_dot, test_mode='flow_curve')

plt.loglog(gamma_dot, sigma, label=f'Stretch-creation (κ={model.kappa.value:.2f})')
plt.loglog(gamma_dot, sigma_base, '--', label='Base (κ=0)')
plt.xlabel('Shear rate (1/s)')
plt.ylabel('Stress (Pa)')
plt.legend()
plt.show()

Comparing Variants

# Create three models: base, Bell, stretch-creation
model_base = TNTSingleMode(breakage="base")
model_bell = TNTSingleMode(breakage="bell")
model_stretch = TNTSingleMode(breakage="stretch_creation")

# Set common parameters
for m in [model_base, model_bell, model_stretch]:
    m.G.value = 1000.0
    m.tau_b.value = 1.0
    m.eta_s.value = 0.0

# Variant-specific parameters
model_bell.F_0.value = 50.0       # kT
model_stretch.kappa.value = 1.0   # dimensionless

# Predict startup
t = np.linspace(0, 5, 200)
gamma_dot = 1.0

sigma_base = model_base.predict(t, test_mode='startup', gamma_dot=gamma_dot)
sigma_bell = model_bell.predict(t, test_mode='startup', gamma_dot=gamma_dot)
sigma_stretch = model_stretch.predict(t, test_mode='startup', gamma_dot=gamma_dot)

plt.plot(t, sigma_base, label='Base')
plt.plot(t, sigma_bell, label='Bell (force-sensitive)')
plt.plot(t, sigma_stretch, label='Stretch-creation')
plt.xlabel('Time (s)')
plt.ylabel('Stress (Pa)')
plt.title('Startup comparison')
plt.legend()
plt.show()

Composition with FENE

# Combine stretch-creation with FENE (bounded extensibility)
from rheojax.models import TNTSingleMode

model = TNTSingleMode(breakage="stretch_creation", elasticity="fene")

# Set FENE parameter (finite extensibility)
model.L.value = 10.0  # Maximum stretch ratio

# Set stretch-creation parameter
model.kappa.value = 1.5

# Other parameters
model.G.value = 5000.0
model.tau_b.value = 0.5
model.eta_s.value = 0.0

# Predict large-strain startup
t = np.linspace(0, 3, 300)
gamma_dot = 5.0  # high rate

sigma = model.predict(t, test_mode='startup', gamma_dot=gamma_dot)

plt.plot(t, sigma)
plt.xlabel('Time (s)')
plt.ylabel('Stress (Pa)')
plt.title('Stretch-creation + FENE')
plt.show()

# The FENE term prevents S from diverging at large strain
# The kappa term enhances stress buildup during extension

Bayesian Inference

from rheojax.pipeline import BayesianPipeline

# Load data
data = RheoData(x=t, y=sigma_exp, test_mode='startup')

# Create pipeline
pipeline = BayesianPipeline()

# Fit with NLSQ first (warm-start)
pipeline.set_model(model)
pipeline.set_data(data)
nlsq_result = pipeline.fit_nlsq(gamma_dot=1.0)

# Run Bayesian inference (NUTS sampler)
bayes_result = pipeline.fit_bayesian(
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,
)

# Plot posterior distributions
pipeline.plot_pair()      # Pairwise correlations
pipeline.plot_forest()    # Credible intervals
pipeline.plot_trace()     # MCMC chains

# Extract credible intervals
intervals = model.get_credible_intervals(
    bayes_result.posterior_samples,
    credibility=0.95
)

print("95% Credible Intervals:")
for param, (low, high) in intervals.items():
    print(f"  {param}: [{low:.3e}, {high:.3e}]")

Composition with Other Variants

The stretch-creation variant can be combined with other TNT variants to capture multiple physical effects simultaneously.

With Bell Breakage (Force-Activated Breakage)

Combining stretch-creation and Bell breakage models a network where both creation and destruction are stress-dependent:

# Not directly supported in single breakage parameter
# Would require custom model combining both mechanisms
# Physically: stress-activated breakage + stretch-activated creation

This combination is relevant for dual-responsive networks (e.g., mechanophore activation + mechano-sensitive degradation).

With FENE Elasticity (Finite Extensibility)

As shown above, this is the recommended combination for large strains:

model = TNTSingleMode(breakage="stretch_creation", elasticity="fene")

  • FENE bounds \(\text{tr}(S) < 3L^2\), preventing divergence

  • Stretch-creation enhances stress within that bound

  • Captures both kinetic stiffening (creation) and elastic stiffening (FENE)

Multi-Mode Extension

The stretch-creation mechanism can be extended to multi-mode models:

\[\sigma = \sum_{i=1}^N G_i (S_i - I) + 2\eta_s D\]

Each mode \(i\) has its own \(\tau_{b,i}\) and could have its own \(\kappa_i\) if creation enhancement varies with timescale.

Typically, a single \(\kappa\) is used (same mechanism across all modes).

Rheopecty: Anti-Thixotropic Behavior

Rheopecty is the opposite of thixotropy: viscosity increases with time under shear, rather than decreasing. While thixotropic materials (modeled by DMT or fluidity kinetics) lose structure under flow, rheopectic materials build structure under deformation.

In the stretch-creation model, rheopecty arises naturally from the coupling between chain stretch and bond creation:

  1. Flowing chains stretch, increasing \(\text{tr}(\mathbf{S})\)

  2. Enhanced trace increases the creation rate \(k_{on}(\mathbf{S})\)

  3. More bonds form, strengthening the network

  4. The strengthened network resists flow more strongly

This constitutes a positive feedback loop: more flow leads to more stretch, which leads to more bonds, which leads to more resistance to flow. The material stiffens under sustained deformation rather than softening.

This is fundamentally different from thixotropic models (such as the DMT or fluidity frameworks) where shear breaks structure. In the stretch-creation framework, shear builds structure. The two behaviors correspond to opposite signs of the feedback between deformation and microstructural evolution:

  • Thixotropy (negative feedback): shear destroys bonds, reducing viscosity, permitting more shear

  • Rheopecty (positive feedback): shear stretches chains, creating more bonds, increasing viscosity, resisting further shear

Experimental systems that exhibit rheopectic or strain-building behavior include:

  • Strain-crystallizing rubbers (polyisoprene under extension): Chain alignment promotes crystallite nucleation, creating additional physical crosslinks that stiffen the network.

  • Mechanophore-containing networks (spiropyran-functionalized gels): Mechanical force activates latent reactive groups, forming new covalent or non-covalent bonds under deformation.

  • Peptide hydrogels with strain-activated hydrogen bonding: Mechanical alignment of peptide fibers promotes inter-fiber hydrogen bonding, strengthening the gel under shear.

Physical Mechanisms for Stretch-Dependent Creation

Several distinct physical mechanisms can lead to stretch-dependent bond creation, each corresponding to a different molecular-level process:

  1. Strain-induced crystallization. Chain alignment promotes crystallite nucleation, and crystallites act as additional crosslinks. The crystal nucleation rate increases with chain orientation, which scales with \(\text{tr}(\mathbf{S}) - 3\). This is the dominant mechanism in natural rubber and other strain-crystallizing elastomers.

  2. Mechanochemical activation. Mechanophore groups along the chain backbone activate under tension, exposing reactive sites that form new bonds. Well-known examples include spiropyran (which isomerizes to the reactive merocyanine form under force) and dioxetane (which cleaves to aldehyde fragments that can recombine). The activation rate depends on the force in the chain, which correlates with stretch.

  3. Proximity-enhanced association. Chain stretch brings reactive end-groups or side-groups into closer proximity with potential binding partners, increasing the effective encounter rate. In a coiled conformation, reactive groups may be sterically shielded; extension exposes them and reduces the mean distance to neighboring chains.

  4. Alignment-enhanced bonding. Pre-aligned chains have reduced conformational entropy cost for association, lowering the activation barrier for bond formation. Two parallel, extended chains can zip together more easily than two randomly coiled chains, because the entropic penalty for forming a contact is smaller when the chains are already aligned.

The parameter \(\kappa\) captures the net effect of these mechanisms: \(\kappa > 0\) means that stretch enhances creation, while \(\kappa = 0\) recovers the base Tanaka-Edwards model with constant creation rate. The linear form \(k_{on} = (1/\tau_b)(1 + \kappa(\text{tr}(\mathbf{S}) - 3))\) is the simplest phenomenological coupling; more detailed models could include saturation, activation thresholds, or distinct \(\kappa\) values for different mechanisms.

Shear Thickening Transient

During startup flow at constant shear rate \(\dot{\gamma}\), the stretch-creation model predicts that the stress rises faster than the linear elastic prediction. This shear thickening transient is a direct consequence of the positive feedback between stretch and bond creation.

The physical sequence during startup is:

  1. Initial elastic loading: At early times \(t \ll \tau_b\), the network deforms affinely and the stress grows linearly with strain: \(\sigma(t) \approx G \dot{\gamma} t\).

  2. Stretch enhancement activates: As chains extend, \(\text{tr}(\mathbf{S})\) exceeds 3, and the creation rate \(k_{on}\) begins to increase above its equilibrium value \(1/\tau_b\).

  3. Network densification: The enhanced creation rate produces more elastically active chains than the base model, so the effective modulus increases with time: \(G_{\text{eff}}(t) > G_0\).

  4. Super-linear stress growth: The stress grows faster than the elastic prediction: \(\sigma(t) > G \dot{\gamma} t\), reflecting the increasing number of load-bearing chains.

  5. Enhanced steady state: Eventually the system reaches a steady state where the enhanced creation rate balances the constant breakage rate, yielding a higher effective viscosity than the base Tanaka-Edwards model.

The steady-state stress satisfies:

\[\sigma_{\text{steady}}(\dot{\gamma}) > \sigma_{\text{TE}}(\dot{\gamma})\]

where \(\sigma_{\text{TE}}\) is the base Tanaka-Edwards steady-state stress (with \(\kappa = 0\)). The ratio \(\sigma_{\text{steady}} / \sigma_{\text{TE}}\) increases with both \(\kappa\) and the Weissenberg number \(\text{Wi} = \dot{\gamma} \tau_b\).

The transient shear thickening ratio \(\sigma_{\text{peak}} / \sigma_{\text{TE,peak}}\) can significantly exceed 1 for large \(\kappa\), because the peak stress occurs at a time when \(\text{tr}(\mathbf{S})\) is near its maximum transient value.

This startup behavior contrasts sharply with the Bell model, where startup shows a stress overshoot: the stress rises above the steady-state value and then relaxes back down. In the stretch-creation model, the stress rises monotonically to a steady state that is higher than the base model prediction, without necessarily overshooting.

Creep Ringing and Hardening

Under applied constant stress (creep), the stretch-creation model can exhibit a damped oscillatory approach to steady state, known as creep ringing. This phenomenon arises from the coupling between deformation, chain stretch, and network creation.

The mechanism proceeds as follows:

  1. Stress application: Applying a constant stress \(\sigma_0\) causes chains to stretch in order to carry the load. The trace \(\text{tr}(\mathbf{S})\) increases.

  2. Enhanced creation: The increased trace raises the creation rate \(k_{on}\), forming additional bonds. The network stiffens, and the strain rate \(\dot{\gamma}(t)\) decreases.

  3. Reduced stretch: The lower strain rate reduces chain stretch, causing \(\text{tr}(\mathbf{S})\) to decrease. The creation rate drops, and the network softens slightly.

  4. Oscillatory approach: This back-and-forth between stiffening and softening creates damped oscillations in \(\dot{\gamma}(t)\), analogous to creep ringing in inertial rheometry but arising here from a purely constitutive mechanism.

At long times, the strain rate \(\dot{\gamma}(t)\) may approach a lower steady value than the Tanaka-Edwards prediction, reflecting the hardened network state. For strong coupling (large \(\kappa\)), the material may effectively arrest: \(\dot{\gamma} \to 0\) as the network continuously builds under sustained deformation, analogous to a sol-gel transition driven by the applied stress.

This long-time behavior is the opposite of creep rupture seen in force-dependent breakage models (Bell), where the applied stress accelerates bond breakage and eventually leads to catastrophic flow. In the stretch-creation model, the applied stress strengthens the network rather than weakening it.

Possible Gelation Under Flow

For sufficiently large \(\kappa\), the positive feedback between stretch and creation can drive a sol-gel transition under flow. This is one of the most distinctive predictions of the stretch-creation model.

The mechanism is as follows:

  • At rest: The network has an equilibrium chain density \(n_0 = k_{on}(I) / k_{off} = 1\), and the conformation tensor is \(\mathbf{S} = \mathbf{I}\).

  • Under flow: Chain stretch increases \(\text{tr}(\mathbf{S})\), which enhances the creation rate. If the enhanced creation rate exceeds the destruction rate by a sufficient margin, the network density grows without bound.

  • Critical condition: Flow-induced gelation occurs when the stretch-enhanced creation overwhelms the constant breakage. The critical condition can be expressed as:

\[\kappa \left(\text{tr}(\mathbf{S}_{\text{ss}}) - 3\right) > \kappa_{\text{gel}}\]

where \(\kappa_{\text{gel}}\) is a critical threshold that depends on the flow type and the Weissenberg number.

Flow-induced gelation is observed experimentally in several systems:

  • Wormlike micelle solutions at high shear rates: Shear-induced structures (SIS) form above a critical shear rate, dramatically increasing viscosity and sometimes leading to flow instabilities.

  • Polymer solutions near the theta-point: Flow-induced crystallization in polyethylene melts and solutions, where extensional flow promotes chain alignment and nucleation.

  • Associative polymer solutions under extensional flow: Hydrophobically modified polymers can gel under extension as chain stretch exposes associating groups and promotes intermolecular bridging.

The gelation threshold depends on both \(\kappa\) and the flow type. Extensional flows are more effective at driving gelation than shear flows because they produce larger chain stretch for the same deformation rate. In simple shear, the rotational component partially mitigates the stretching effect, raising the critical \(\kappa\) required for gelation.

Failure Mode: Gelation

Failure Mode

Mechanism: Positive feedback between chain stretch and bond creation leads to runaway network formation. The material solidifies under deformation as the creation rate overwhelms the constant breakage rate.

Signature: Stress that grows without bound during startup; strain rate \(\dot{\gamma} \to 0\) in creep (complete arrest); diverging viscosity with increasing shear rate in the flow curve.

Physical origin: When \(\kappa \cdot (\text{tr}(\mathbf{S}) - 3)\) exceeds a critical threshold, the creation rate overwhelms destruction, and the network density grows without limit. The system crosses a flow-induced sol-gel transition.

Implication: The model predicts a flow-induced sol-gel transition. Above the critical deformation rate, the material cannot reach a steady-state flow; instead, the stress and network density grow indefinitely.

Numerical warning: Large \(\kappa\) values (typically \(\kappa > 3\)) can cause ODE stiffness due to the rapidly increasing creation rate. This may require adaptive time stepping, implicit integrators (BDF methods), or reduced step sizes (\(\Delta t < 0.01 \tau_b\)).

Design insight: This mechanism is desirable in self-healing materials and injectable hydrogels. During injection, the material flows; at rest (plus under residual stress), enhanced gelation restores and even exceeds the original network strength. The stretch-creation coupling provides a built-in self-reinforcement mechanism.

See Also

TNT Shared Reference:

TNT Base Model:

Opposite Physics — Stress-Activated Destruction:

Related TNT Variants:

External Thixotropic/Rheopectic Models:

API Reference

Model Class

class TNTSingleMode(BaseModel):
    """
    Single-mode Tanaka-Edwards transient network model
    with stretch-creation variant.

    Parameters
    ----------
    breakage : str
        Breakage mechanism ("base", "bell", "stretch_creation")
    elasticity : str
        Elasticity type ("gaussian", "fene")

    Attributes
    ----------
    G : Parameter
        Plateau modulus (Pa)
    tau_b : Parameter
        Breakage timescale (s)
    kappa : Parameter
        Stretch-creation enhancement factor (dimensionless, 0-5)
    eta_s : Parameter
        Solvent viscosity (Pa·s)
    """

Key Methods

def fit(data: RheoData, **kwargs) -> FitResult:
    """
    Fit model to experimental data.

    Parameters
    ----------
    data : RheoData
        Experimental data with test_mode set
    **kwargs
        Protocol-specific parameters (gamma_dot, omega, etc.)

    Returns
    -------
    FitResult
        Fitted parameters, residuals, R^2, convergence info
    """

def predict(x: ArrayLike, test_mode: str, **kwargs) -> Array:
    """
    Predict rheological response.

    Parameters
    ----------
    x : array_like
        Independent variable (time, frequency, shear rate)
    test_mode : str
        Protocol ("startup", "oscillation", "flow_curve", etc.)
    **kwargs
        Protocol parameters (gamma_dot, gamma_0, omega, etc.)

    Returns
    -------
    y : Array
        Predicted response (stress, modulus, etc.)
    """

def fit_bayesian(data: RheoData, num_warmup=1000, num_samples=2000,
                 num_chains=4, **kwargs) -> BayesianResult:
    """
    Bayesian parameter inference using NUTS.

    Parameters
    ----------
    data : RheoData
        Experimental data
    num_warmup : int
        NUTS warmup iterations
    num_samples : int
        Posterior samples per chain
    num_chains : int
        Number of MCMC chains

    Returns
    -------
    BayesianResult
        Posterior samples, diagnostics (R-hat, ESS)
    """

Supported Test Modes

Test Mode

Description

"oscillation"

Small amplitude oscillatory shear (SAOS): \(G^*(\omega)\)

"flow_curve"

Steady shear flow: \(\sigma(\dot{\gamma})\)

"startup"

Startup of steady shear: \(\sigma(t)\) at constant \(\dot{\gamma}\)

"relaxation"

Stress relaxation after cessation: \(\sigma(t)\)

"creep"

Creep under constant stress: \(\gamma(t)\) at constant \(\sigma_0\)

"laos"

Large amplitude oscillatory shear: \(\sigma(t)\) at \(\gamma(t) = \gamma_0 \sin(\omega t)\)

Parameter Access

# Get parameter value
G_value = model.G.value  # Pa

# Set parameter value
model.kappa.value = 1.0

# Get parameter bounds
lower, upper = model.kappa.bounds

# Get all parameters as dict
params = model.get_parameter_dict()

References

Foundational Theory

  1. Tanaka, F., & Edwards, S. F. (1992). Viscoelastic properties of physically crosslinked networks. 1. Transient network theory. Macromolecules, 25(5), 1516-1523. https://doi.org/10.1021/ma00031a024

    • Original transient network theory (base model)

  2. Flory, P. J. (1947). Thermodynamics of crystallization in high polymers. I. Crystallization induced by stretching. Journal of Chemical Physics, 15(6), 397-408. https://doi.org/10.1063/1.1746537

    • Classical theory of strain-induced crystallization

  3. James, H. M., & Guth, E. (1943). Theory of the elastic properties of rubber. Journal of Chemical Physics, 11(10), 455-481. https://doi.org/10.1063/1.1723785

    • Non-Gaussian network theory (precursor to FENE)

Mechanophore and Adaptive Networks

  1. Vernerey, F. J., Long, R., & Brighenti, R. (2017). A statistically-based continuum theory for polymers with transient networks. Journal of the Mechanics and Physics of Solids, 107, 1-20. https://doi.org/10.1016/j.jmps.2017.05.016

    • Modern continuum framework for adaptive networks

  2. Vernerey, F.J. (2018). Transient response of nonlinear polymer networks: A kinetic theory. Journal of the Mechanics and Physics of Solids, 115, 230–247. DOI: 10.1016/j.jmps.2018.02.018 PDF

    • Kinetic theory for transient networks with nonlinear chain response

  3. Wagner, R.J., Hobbs, E., & Vernerey, F.J. (2021). A network model of transient polymers: exploring the micromechanics of nonlinear viscoelasticity. Soft Matter, 17, 8742–8757. DOI: 10.1039/d1sm00753j PDF

    • Discrete network model validating continuum TNT predictions

  4. Wang, Q., Gossweiler, G. R., Craig, S. L., & Zhao, X. (2014). Cephalopod-inspired design of electro-mechano-chemically responsive elastomers for on-demand fluorescent patterning. Nature Communications, 5, 4899. https://doi.org/10.1038/ncomms5899

    • Mechanophore-crosslinked elastomers (experimental)

  5. Creton, C. (2017). 50th Anniversary Perspective: Networks and Gels: Soft but Dynamic and Tough. Macromolecules, 50(21), 8297-8316. https://doi.org/10.1021/acs.macromol.7b01698

    • Review of dynamic networks and toughening mechanisms

Strain Crystallization

  1. Candau, N., Laghmach, R., Chazeau, L., Chenal, J.-M., Gauthier, C., Biben, T., & Munch, E. (2014). Strain-induced crystallization of natural rubber and cross-link densities heterogeneities. Macromolecules, 47(16), 5815-5824. https://doi.org/10.1021/ma5006843

    • Modern experimental study of strain crystallization

  2. Tosaka, M. (2007). Strain-induced crystallization of crosslinked natural rubber as revealed by X-ray diffraction using synchrotron radiation. Polymer Journal, 39(12), 1207-1220. https://doi.org/10.1295/polymj.PJ2007059

    • In-situ X-ray studies of crystallization kinetics

Numerical Methods

  1. Hulsen, M. A., Fattal, R., & Kupferman, R. (2005). Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms. Journal of Non-Newtonian Fluid Mechanics, 127(1), 27-39. https://doi.org/10.1016/j.jnnfm.2005.01.002

    • Numerical techniques for viscoelastic constitutive equations

  2. Owens, R. G., & Phillips, T. N. (2002). Computational Rheology. Imperial College Press. ISBN: 978-1860941863

    • Comprehensive reference for solving rheological ODEs/PDEs