Maxwell-Isotropic-Kinematic Hardening (MIKH)

Quick Reference

• Use when: Thixotropic elasto-viscoplastic materials with stress overshoot, Bauschinger effect, thixotropic hysteresis

• Parameters: 11 (G, \(\eta\), C, \(\gamma_{\text{dyn}}\), m, \(\sigma_{y0}\), \(\Delta\sigma_y\), \(\tau_{\text{thix}}\), \(\Gamma\), \(\eta_\infty\), \(\mu_p\))

• Key equation: \(\frac{d\sigma}{dt} = G(\dot{\gamma} - \dot{\gamma}^p) - \frac{G}{\eta}\sigma\) (Maxwell viscoelasticity with plasticity)

• Test modes: flow_curve, startup, relaxation, creep, oscillation, laos

• Material examples: Drilling fluids, greases, waxy crude oil, thixotropic cements, structured emulsions

class rheojax.models.ikh.mikh.MIKH

Bases: IKHBase

Maxwell-Isotropic-Kinematic Hardening (MIKH) Model.

A thixotropic elasto-viscoplastic model combining: 1. Armstrong-Frederick kinematic hardening (backstress evolution). 2. Isotropic hardening/softening via structural parameter lambda (thixotropy). 3. Maxwell viscoelastic element for proper relaxation behavior. 4. Viscous background solvent.

Two Formulations:

• Maxwell ODE (via Diffrax): For creep/relaxation protocols

• Return Mapping: For startup/LAOS protocols (incremental)

Governing Equations:

σ_total = σ + η_inf * γ̇

Stress Evolution (ODE formulation): dσ/dt = G(γ̇ - γ̇ᵖ) - (G/η)σ

Yield Surface: |σ - α| ≤ σ_y(λ) σ_y(λ) = σ_y0 + Δσ_y * λ

Structure Evolution: dλ/dt = (1-λ)/τ_thix - Γ*λ*|γ̇ᵖ|

Backstress Evolution (Armstrong-Frederick): dα = C*dγ_p - γ_dyn*|α|^(m-1)*α*|dγ_p|

Parameters:

• G – Shear modulus [Pa]

• eta – Maxwell viscosity [Pa·s] (controls relaxation time τ = η/G)

• C – Kinematic hardening modulus [Pa]

• gamma_dyn – Dynamic recovery parameter for backstress [-]

• m – AF recovery exponent [-] (typically 1.0)

• sigma_y0 – Minimal (destructured) yield stress [Pa]

• delta_sigma_y – Yield stress increment (structured - destructured) [Pa]

• tau_thix – Thixotropic rebuilding time scale [s]

• Gamma – Structural breakdown coefficient [-]

• eta_inf – High-shear viscosity [Pa·s]

• mu_p – Plastic viscosity for Perzyna regularization [Pa·s]

__init__()

Initialize base model.

predict_flow_curve(gamma_dot)

Predict steady-state flow curve.

Return type:

ndarray

predict_startup(t, gamma_dot=1.0)

Predict startup shear response.

Parameters:

• t (ndarray) – Time array

• gamma_dot (float) – Constant shear rate

Return type:

ndarray

Returns:

Stress vs time

predict_relaxation(t, sigma_0=100.0)

Predict stress relaxation.

Parameters:

• t (ndarray) – Time array

• sigma_0 (float) – Initial stress

Return type:

ndarray

Returns:

Stress vs time

predict_creep(t, sigma_applied=50.0)

Predict creep response.

Parameters:

• t (ndarray) – Time array

• sigma_applied (float) – Applied constant stress

Return type:

ndarray

Returns:

Strain vs time

predict_laos(t, gamma_0=1.0, omega=1.0)

Predict LAOS response.

Parameters:

• t (ndarray) – Time array

• gamma_0 (float) – Strain amplitude

• omega (float) – Angular frequency

Return type:

ndarray

Returns:

Stress vs time

model_function(X, params, test_mode=None, **kwargs)

NumPyro model function for Bayesian inference.

Accepts protocol-specific kwargs (gamma_dot, sigma_applied, sigma_0). Falls back to values cached during _fit() if not provided.

Notation Guide

Symbol	Units	Description
\(\sigma\)	Pa	Deviatoric stress (elasto-plastic component)
\(\alpha\)	Pa	Backstress (kinematic hardening variable)
\(\lambda\)	–	Structural parameter (0 = destructured, 1 = structured)
\(\dot{\gamma}\)	1/s	Total shear rate
\(\dot{\gamma}^p\)	1/s	Plastic shear rate
\(\sigma_y\)	Pa	Current yield stress (depends on \(\lambda\))
\(\xi\)	Pa	Relative stress (\(\xi = \sigma - \alpha\))

Overview

The MIKH (Maxwell-Isotropic-Kinematic Hardening) model is a comprehensive thixotropic elasto-viscoplastic constitutive equation developed by Dimitriou & McKinley (2014) for complex fluids like waxy crude oil. It combines:

1. Maxwell viscoelasticity: Stress relaxation via \(\eta\) (Maxwell viscosity)

2. Kinematic hardening: Backstress evolution (Armstrong-Frederick type)

3. Isotropic hardening: Yield stress evolution via structural parameter \(\lambda\)

4. Viscous background: High-shear Newtonian contribution (\(\eta_\infty\))

The model captures:

• Stress overshoot in startup flow

• Bauschinger effect (easier reverse flow after forward loading)

• Thixotropic loops (history-dependent stress-strain curves)

• Yield stress aging (rest-time dependence)

• Flow curve with shear-thinning and yield stress

Theoretical Background

Historical Context

The MIKH model emerges from the synthesis of three traditionally separate fields:

1. Classical Plasticity: The theory of plastic deformation in metals, particularly the work of Prager and Armstrong-Frederick on kinematic hardening to capture the Bauschinger effect.

2. Thixotropy: The time-dependent rheology first systematically studied by Freundlich and colleagues in the 1920s-30s, formalized through structural kinetics approaches (Goodeve 1939, Moore 1959).

3. Yield Stress Materials: The Herschel-Bulkley framework and its extensions to time-dependent yield stress materials (de Souza Mendes & Thompson 2019).

The unification of these frameworks by Dimitriou & McKinley provides a thermodynamically consistent model capable of describing the complex behavior of materials like waxy crude oil, drilling muds, and structured colloidal suspensions.

Material Class: Thixotropic Elasto-Viscoplastic Fluids (TEvp)

TEvp materials exhibit several characteristic behaviors:

1. Yield Stress: Below a critical stress \(\sigma_y\), the material responds elastically (reversibly). Above \(\sigma_y\), plastic flow occurs irreversibly.

2. Thixotropy: The material’s structure—and hence its properties—depend on mechanical history. Under shear, microstructure breaks down (destructuring); at rest, it recovers (restructuring). The structural parameter \(\lambda \in [0,\, 1]\) tracks this state:

• \(\lambda = 1\): Fully structured (maximum yield stress, maximum elasticity)

• \(\lambda = 0\): Fully destructured (minimum yield stress)

3. Viscoelasticity: Even in the elastic regime, the material exhibits stress relaxation over time due to microstructural rearrangements.

4. Kinematic Hardening: Under cyclic loading, the material exhibits directional memory—the Bauschinger effect. This is captured through the backstress \(\alpha\), which shifts the yield surface in stress space.

Physical Interpretation of the Microstructure

In waxy crude oils, the microstructure consists of:

• Wax crystals that form a space-spanning network below the gelation temperature

• Inter-crystalline bonds (van der Waals forces, crystal interlocking) that provide mechanical integrity

• Continuous oil phase that acts as the suspending medium

The structural parameter \(\lambda\) represents the fraction of intact inter-crystalline bonds. When sheared, bonds break (destructuring); at rest, thermal fluctuations allow bonds to reform (restructuring). This microscopic picture motivates the kinetic equations for the evolution of \(\lambda\).

For other TEvp materials:

• Drilling fluids: The parameter \(\lambda\) represents the organization of clay platelets and polymer chains

• Colloidal gels: The parameter \(\lambda\) represents the fraction of intact colloidal bonds

• Greases: The parameter \(\lambda\) represents the organization of thickener fibers

Thermokinematic Memory (FIKH Framework)

The Fractal IKH (FIKH) framework (Geri et al. 2017) extends the MIKH model to account for temperature-dependent microstructure. This is critical for materials like waxy crude oils where thermal history determines the precipitated wax content.

Effective Volume Fraction:

The wax volume fraction depends on temperature history:

\[\phi(T) = \phi_{\max} \cdot f(\Delta T)\]

where \(\Delta T = T - T_{wax}\) is the subcooling below the wax appearance temperature. The function \(f(\Delta T)\) captures the precipitation kinetics.

Thermokinematic Memory:

The material “remembers” its thermal history through:

1. The precipitated wax morphology (cooling rate affects crystal size/shape)

2. The equilibrium connectivity \(\xi_{eq}(T)\) at each temperature

3. The effective volume fraction \(\phi(T)\) determining available wax content

This leads to a modified structure evolution equation:

\[\frac{d\xi}{dt} = k_1(\xi_{eq} - \xi) - k_2 \xi |\dot{\gamma}^p|\]

where the equilibrium connectivity \(\xi_{eq}(T)\) replaces the constant target value of 1.

Parameter Scaling with Microstructure:

In the FIKH framework, macroscopic parameters depend on both temperature (via \(\phi\)) and structure (via \(\xi\)):

\[\begin{split}\mu_p(T, \xi) &= \mu_{p,\infty}(T) \cdot \left[1 - \frac{\xi}{\xi_c}\right]^{-[\eta]\phi} \\ \sigma_y(\xi) &= \sigma_{y,0} + \Delta\sigma_y \cdot \xi^n \\ C(\xi) &= C_0 \cdot \xi^p\end{split}\]

where \(\xi_c\) is the critical connectivity for jamming, \([\eta]\) is the intrinsic viscosity, and \(n, p\) are scaling exponents related to fractal dimension.

Fractal Microstructure Interpretation

The structure parameter \(\lambda\) can be interpreted as the normalized fractal connectivity \(\xi \in [0,\, 1]\) of the microstructural network.

Fractal Connectivity:

In a fractal network (e.g., wax crystal aggregates), the connectivity \(\xi\) represents the fraction of intact bonds relative to a fully percolated network:

\[\xi = \frac{N_{bonds}}{N_{bonds,max}}\]

The fractal nature of colloidal aggregates leads to power-law scaling of mechanical properties with connectivity:

• Yield stress: \(\sigma_y \propto \xi^n\) with \(n \approx 2-3\)

• Elastic modulus: \(G \propto \xi^{d_f/(d-d_f)}\) where \(d_f\) is fractal dimension

The Avalanche Effect:

The nonlinear coupling between structure (\(\xi\)) and yield stress (\(\sigma_y\)) leads to an important dynamic phenomenon—delayed yielding or the “avalanche effect”:

1. Under stress \(\sigma < \sigma_y(\xi_0)\), the material creeps slowly

2. Slow creep causes gradual structure breakdown: \(\xi \downarrow\)

3. As \(\xi\) decreases, \(\sigma_y(\xi)\) drops, accelerating breakdown

4. Eventually, catastrophic yielding occurs when \(\sigma > \sigma_y(\xi)\)

This positive feedback loop explains why thixotropic materials can appear solid for extended periods before suddenly flowing.

Relationship to Percolation:

Near the percolation threshold, the network connectivity follows:

\[\xi \propto (p - p_c)^\beta\]

where \(p\) is the bond occupation probability and \(p_c \approx 0.5\) for 3D networks. The critical exponent \(\beta\) depends on network topology.

Thermodynamic Consistency

The MIKH model can be derived within the Gurtin-Fried-Anand thermomechanical framework, ensuring:

1. Frame invariance: Constitutive equations are objective (independent of observer)

2. Second law compliance: Dissipation inequality is satisfied

3. Energy balance: Clear separation of stored (elastic) and dissipated energy

The key thermodynamic quantities are:

• Free energy: \(\psi(\gamma^e, \alpha, \lambda)\) storing elastic energy

• Dissipation: \(\mathcal{D} = \sigma \dot{\gamma}^p + X \dot{\alpha} + Y \dot{\lambda} \geq 0\)

where X and Y are thermodynamic forces conjugate to the internal variables. This framework guarantees that the model respects fundamental physics while allowing complex phenomenology.

Physical Foundations

Maxwell-Like Framework

The MIKH model uses a Maxwell-like viscoelastic element as its foundation. The Maxwell element consists of a spring (modulus G) in series with a dashpot (viscosity \(\eta\)), giving a relaxation time \(\tau = \eta / G\):

\[\frac{d\sigma}{dt} = G(\dot{\gamma} - \dot{\gamma}^p) - \frac{G}{\eta}\sigma\]

The first term represents elastic loading minus plastic flow. The second term represents viscoelastic relaxation with characteristic time \(\tau = \eta/G\).

Physical interpretation:

• At short times (\(t \ll \tau\)): Elastic response dominates, \(\sigma \approx G \cdot \gamma\)

• At long times (\(t \gg \tau\)): Viscous flow, \(\sigma \to 0\) under constant strain

• The Maxwell element captures the liquid-like long-time behavior of structured fluids

Kinematic Hardening (Armstrong-Frederick)

Kinematic hardening is a plasticity concept that accounts for the Bauschinger effect: when a material is deformed plastically in one direction, subsequent yield in the opposite direction occurs at a lower stress than the initial yield stress.

The backstress \(\alpha\) represents the “center” of the yield surface in stress space. As plastic deformation accumulates, \(\alpha\) evolves according to the Armstrong-Frederick (AF) law:

\[d\alpha = C \cdot d\gamma^p - \gamma_{dyn} |\alpha|^{m-1} \alpha |d\gamma^p|\]

Term 1 (Hardening): \(C \cdot d\gamma^p\)

• The backstress increases proportionally to plastic strain increment

• C is the kinematic hardening modulus [Pa]

• This creates a “memory” of the plastic deformation direction

Term 2 (Dynamic Recovery): \(-\gamma_{\text{dyn}} \cdot |\alpha|^{m-1} \cdot \alpha \cdot |d\gamma^p|\)

• Limits backstress saturation (prevents unbounded growth)

• \(\gamma_{\text{dyn}}\) controls recovery rate

• \(m\) controls nonlinearity (\(m = 1\) is linear, \(m > 1\) accelerates recovery at high \(\alpha\))

• Recovery is proportional to \(|d\gamma^p|\), so it only occurs during plastic flow

Steady-state backstress: At steady plastic flow:

\[\alpha_{ss} = \frac{C}{\gamma_{dyn}} \cdot \text{sign}(\dot{\gamma}^p)\]

The ratio \(C / \gamma_{\text{dyn}}\) determines the maximum backstress magnitude.

Note

Reparameterization for \(m \neq 1\): The original Dimitriou & McKinley (2014) paper defines backstress recovery through a back-strain \(A\) with \(\sigma_{back} = C \cdot A\) and recovery function \(f(A) = (q|A|)^m \operatorname{sign}(A)\). RheoJAX works directly with the backstress \(\alpha\) and uses \(\gamma_{dyn} \cdot |\alpha|^{m-1} \cdot \alpha\). For \(m = 1\), \(\gamma_{dyn} = q\) exactly. For \(m \neq 1\), the mapping is \(\gamma_{dyn} = q^m \cdot C^{1-m}\). Both parameterizations are mathematically equivalent; fitted parameter values should be interpreted accordingly.

Isotropic Hardening (Thixotropy)

The yield stress evolves with a structural parameter \(\lambda \in [0,\, 1]\):

\[\sigma_y(\lambda) = \sigma_{y,0} + \Delta\sigma_y \cdot \lambda\]

• \(\sigma_{y,0}\): Minimal yield stress when fully destructured (\(\lambda = 0\))

• \(\Delta\sigma_y\): Additional yield stress from structure

• \(\sigma_{y,\max} = \sigma_{y,0} + \Delta\sigma_y\): Maximum yield stress when fully structured (\(\lambda = 1\))

The structure evolves according to a first-order kinetic equation:

\[\frac{d\lambda}{dt} = \frac{1-\lambda}{\tau_{thix}} - \Gamma \lambda |\dot{\gamma}^p|\]

Term 1 (Buildup): \((1 - \lambda) / \tau_{\text{thix}}\)

• Structure recovers toward \(\lambda = 1\) with characteristic time \(\tau_{\text{thix}}\)

• At rest (\(\dot{\gamma}^p = 0\)): \(\lambda(t) = 1 - (1 - \lambda_0) \exp(-t / \tau_{\text{thix}})\)

• Physical origin: Brownian motion, thermal fluctuations allow bond reformation

Term 2 (Breakdown): \(\Gamma \cdot \lambda \cdot |\dot{\gamma}^p|\)

• Structure breaks down proportionally to plastic strain rate

• \(\Gamma\) is the breakdown efficiency coefficient

• Physical origin: Mechanical work breaks inter-particle bonds

Steady-state structure: At constant shear rate:

\[\lambda_{ss} = \frac{1/\tau_{thix}}{1/\tau_{thix} + \Gamma|\dot{\gamma}|}\]

Mathematical Formulation

Core Equations

Stress decomposition:

\[\sigma_{total} = \sigma + \eta_{\infty} \dot{\gamma}\]

The total stress consists of the elasto-plastic contribution \(\sigma\) and a purely viscous background \(\eta_{\infty} \dot{\gamma}\). The latter represents the suspending fluid’s viscosity.

Yield condition:

\[f = |\xi| - \sigma_y(\lambda) \leq 0 \quad \text{where} \quad \xi = \sigma - \alpha\]

The material yields when the relative stress \(|\xi| = |\sigma - \alpha|\) exceeds the current yield stress \(\sigma_y(\lambda)\). The backstress \(\alpha\) shifts the yield surface in stress space.

Plastic flow rule (Perzyna regularization):

\[\dot{\gamma}^p = \frac{\langle f \rangle}{\mu_p} \cdot \text{sign}(\xi)\]

where \(\langle \cdot \rangle\) denotes Macaulay brackets (positive part function):

\[\begin{split}\langle f \rangle = \max(f, 0) = \begin{cases} f & \text{if } f > 0 \\ 0 & \text{if } f \leq 0 \end{cases}\end{split}\]

The Perzyna regularization parameter \(\mu_p\) [Pa·s] controls how sharply the material transitions from elastic to plastic behavior. Small \(\mu_p\) gives rate-independent plasticity; larger \(\mu_p\) smooths the transition.

Complete System of ODEs

The MIKH model comprises three coupled ordinary differential equations:

\[\begin{split}\frac{d\sigma}{dt} &= G(\dot{\gamma} - \dot{\gamma}^p) - \frac{G}{\eta}\sigma \\ \frac{d\alpha}{dt} &= C\dot{\gamma}^p - \gamma_{dyn}|\alpha|^{m-1}\alpha|\dot{\gamma}^p| \\ \frac{d\lambda}{dt} &= \frac{1-\lambda}{\tau_{thix}} - \Gamma\lambda|\dot{\gamma}^p|\end{split}\]

With the plastic flow rate determined by:

\[\dot{\gamma}^p = \frac{\langle |\sigma - \alpha| - \sigma_y(\lambda) \rangle}{\mu_p} \cdot \text{sign}(\sigma - \alpha)\]

Two Formulations

The MIKH model uses two numerical formulations depending on the experimental protocol:

1. Maxwell ODE Formulation (for creep/relaxation)

Suitable for stress-controlled or strain-relaxation experiments where the full viscoelastic response is needed:

# State: [sigma, alpha, lambda]
d(sigma)/dt = G(gamma_dot - gamma_dot_p) - (G/eta)*sigma
d(alpha)/dt = C*gamma_dot_p - gamma_dyn*|alpha|^(m-1)*alpha*|gamma_dot_p|
d(lambda)/dt = (1-lambda)/tau_thix - Gamma*lambda*|gamma_dot_p|

This formulation uses adaptive ODE integration (Diffrax) for accurate time-stepping of the coupled system.

2. Return Mapping Formulation (for startup/LAOS)

Suitable for strain-driven experiments with incremental time stepping:

# Given strain increment Δγ:
1. Elastic predictor: σ_trial = σ_n + G·Δγ
2. Check yield: f = |σ_trial - α_n| - σ_y(λ_n)
3. If f > 0: Radial return with AF correction
4. Update λ AFTER stress (timing-consistent)

The return mapping algorithm provides:

• Exact stress update (radial return to yield surface)

• Implicit treatment of the plastic corrector

• Efficient JAX scan implementation for long time series

Critical timing fix: The structure parameter \(\lambda\) is updated AFTER the stress calculation, using the plastic strain rate from the current step. This ensures consistency with the physical picture where structure responds to the applied deformation.

Steady-State Analysis

At steady state (d/dt = 0), the flow curve follows from the equilibrium conditions.

Structure balance:

\[\lambda_{ss} = \frac{k_1}{k_1 + k_2|\dot{\gamma}|}\]

where \(k_1 = 1/\tau_{\text{thix}}\) and \(k_2 = \Gamma\).

Backstress saturation (from \(d\alpha/dt = 0\) at steady state, \(m = 1\)):

\[\alpha_{sat} = \frac{C}{\gamma_{dyn}}\]

Steady-state stress (Dimitriou & McKinley 2014, Eq. 28):

\[\sigma_{ss} = \frac{C}{\gamma_{dyn}} + \sigma_{y,0} + \Delta\sigma_y \cdot \lambda_{ss} + \eta_{\infty}|\dot{\gamma}|\]

Substituting the structure balance:

\[\sigma_{ss}(\dot{\gamma}) = \frac{C}{\gamma_{dyn}} + \sigma_{y,0} + \frac{\Delta\sigma_y}{1 + \Gamma\tau_{thix}|\dot{\gamma}|} + \eta_{\infty}|\dot{\gamma}|\]

This produces the characteristic shear-thinning flow curve:

• Low shear rate ( \(\dot{\gamma} \to 0\) ): \(\sigma \to C/\gamma_{dyn} + \sigma_{y,0} + \Delta\sigma_y\) (backstress + structured yield stress)

• High shear rate ( \(\dot{\gamma} \to \infty\) ): \(\sigma \to C/\gamma_{dyn} + \sigma_{y,0} + \eta_\infty \dot{\gamma}\) (backstress + linear viscous)

Governing Equations

The complete MIKH system comprises three coupled ODEs (see Mathematical Formulation for details):

\[\begin{split}\frac{d\sigma}{dt} &= G(\dot{\gamma} - \dot{\gamma}^p) - \frac{G}{\eta}\sigma \\ \frac{d\alpha}{dt} &= C\dot{\gamma}^p - \gamma_{dyn}|\alpha|^{m-1}\alpha|\dot{\gamma}^p| \\ \frac{d\lambda}{dt} &= \frac{1-\lambda}{\tau_{thix}} - \Gamma\lambda|\dot{\gamma}^p|\end{split}\]

With plastic flow rate:

\[\dot{\gamma}^p = \frac{\langle |\sigma - \alpha| - \sigma_y(\lambda) \rangle}{\mu_p} \cdot \text{sign}(\sigma - \alpha)\]

Validity and Assumptions

Valid for:

• Thixotropic elasto-viscoplastic fluids: Waxy crude oils, drilling muds, greases, structured emulsions

• Materials with Bauschinger effect: Easier reverse flow after forward loading

• Yield stress evolution: Rest-time dependent yield stress

• Moderate shear rates: Below onset of turbulence or flow instabilities

Assumptions:

• Single structural parameter \(\lambda\): One-dimensional structure kinetics (no multi-scale structure)

• Isotropic yielding: von Mises-like yield criterion (no anisotropy)

• Affine deformation: No spatial gradients (homogeneous flow)

• Incompressible: No density changes

• Isothermal: No temperature effects

Not appropriate for:

• Multi-timescale thixotropy: Use Multi-Lambda Isotropic-Kinematic Hardening (ML-IKH) instead

• Shear banding: Requires spatial extension (1D or 2D)

• Viscoelastic effects dominating over plasticity: Use Maxwell or Oldroyd-B models

• High-frequency oscillations: Limited by quasi-static assumption

What You Can Learn

From fitting Modified IKH to experimental data, you can extract insights about thixotropy, kinematic hardening (Bauschinger effect), and structural evolution in elasto-viscoplastic materials.

Parameter Interpretation

\(\lambda\) (Structure Parameter):

Dimensionless internal variable (\(0 \leq \lambda \leq 1\)) quantifying microstructural integrity. For graduate students: \(\lambda\) represents fraction of intact bonds/aggregates. Evolution: \(d\lambda/dt = (1 - \lambda)/\tau_{\text{thix}} - \Gamma \lambda |\dot{\gamma}^p|\). At steady state: \(\lambda_{\text{ss}} = 1/(1 + \Gamma \tau_{\text{thix}} |\dot{\gamma}|)\). Couples to yield stress via \(\sigma_y(\lambda) = \sigma_{y,0} + \Delta\sigma_y \cdot \lambda\), capturing aging-induced hardening. For practitioners: \(\lambda = 1\) (fully aged, maximum strength) vs \(\lambda = 0\) (fully broken down, minimum strength). Measure indirectly via yield stress recovery tests. Materials with long \(\tau_{\text{thix}}\) retain flow history.

\(\alpha\) (Kinematic Backstress):

Internal stress representing directional anisotropy from flow-induced microstructure. For graduate students: Armstrong-Frederick kinematic hardening: \(d\alpha/dt = C \cdot \dot{\gamma}^p - \gamma_{\text{dyn}} |\alpha|^{m-1} \alpha |\dot{\gamma}^p|\). Produces Bauschinger effect (easier reverse yielding). At steady state: \(\alpha_{\text{ss}} = C / \gamma_{\text{dyn}}\). Ratio \((\sigma_y - 2\alpha_{\text{ss}}) / \sigma_y\) quantifies asymmetry. For practitioners: Measure via reverse flow tests. High \(C\) leads to strong directional memory, pronounced Bauschinger effect. Typical for waxy crude oils, fiber suspensions.

\(\tau_{\text{thix}}\) (Thixotropic Rebuilding Time):

Timescale for structural recovery at rest. For graduate students: First-order kinetics for aging: \(\lambda \to 1\) with time constant \(\tau_{\text{thix}}\). Sets width of hysteresis loops in up-down flow ramps. For thermally-activated processes, \(\tau_{\text{thix}} \sim \tau_0 \exp(\Delta E_{\text{build}} / k_B T)\). For practitioners: Extract from rest-time dependent startup tests or step-strain recovery. Fast aging (\(\tau_{\text{thix}} < 10\) s) vs slow aging (\(\tau_{\text{thix}} > 100\) s). Critical for pumping restart protocols.

\(\Gamma\) (Breakdown Coefficient):

Efficiency of shear-induced destructuring (units: inverse shear rate). For graduate students: Controls shear-thinning: \(\lambda_{\text{ss}} = 1/(1 + \Gamma \tau_{\text{thix}} |\dot{\gamma}|)\). High \(\Gamma\) means rapid breakdown, low \(\Gamma\) means persistent structure. Connects to flow curve via \(\sigma_{\text{ss}}(\dot{\gamma}) = \sigma_{y,0} + \Delta\sigma_y / (1 + \Gamma \tau_{\text{thix}} |\dot{\gamma}|) + \eta_\infty |\dot{\gamma}|\). For practitioners: Fit from flow curve curvature. \(\Gamma \tau_{\text{thix}} \sim 1\) at characteristic shear rate where structure is half-broken.

C, \(\gamma_{\text{dyn}}\) , m (Kinematic Hardening Parameters):

Control backstress evolution and Bauschinger effect magnitude. For graduate students: \(C\) is hardening modulus, \(\gamma_{\text{dyn}}\) is dynamic recovery rate, \(m\) is recovery exponent (\(m = 1\) linear, \(m > 1\) nonlinear). Armstrong-Frederick model with \(m = 1\) widely used. Steady \(\alpha_{\text{ss}} = C / \gamma_{\text{dyn}}\) independent of \(m\). For practitioners: Identify from cyclic loading or reverse flow tests. \(C / \gamma_{\text{dyn}}\) sets saturation backstress (typ. 10-50% of \(\sigma_y\)).

Material Classification

Material Classification from Modified IKH Parameters

Parameter Range	Material Behavior	Typical Materials	Processing Implications
\(\tau_{\text{thix}} < 10\) s, \(\Gamma \tau_{\text{thix}} < 1\)	Fast aging, weak shear-thinning	Soft gels, cosmetics, paints	Rapid recovery, moderate thixotropy
\(\tau_{\text{thix}} = 10\text{--}100\) s, \(\Gamma \tau_{\text{thix}} = 1\text{--}10\)	Moderate aging, strong shear-thinning	Drilling muds, greases, emulsions	Pronounced thixotropy, history-dependent
\(\tau_{\text{thix}} > 100\) s, \(\Gamma \tau_{\text{thix}} > 10\)	Slow aging, extreme shear-thinning	Waxy crude oils, cement pastes	Long memory, pumping challenges
\(C / \gamma_{\text{dyn}} < 0.1 \sigma_y\)	Weak Bauschinger effect	Isotropic gels, simple colloids	Symmetric yielding
\(C / \gamma_{\text{dyn}} > 0.3 \sigma_y\)	Strong Bauschinger effect	Waxy crude oils, fiber suspensions	Directional flow history, asymmetric yielding

• Connection to SAOS: \(G \approx G'\) (storage modulus) at high frequency

5. Stress Overshoot Magnitude

• Overshoot ratio: \((\sigma_{\max} - \sigma_{\text{ss}}) / \sigma_{\text{ss}}\)

• Controlled by interplay of \(G\), \(C\), and \(\lambda_0\) (initial structure)

• Physical signature: Thixotropic materials show overshoot; purely viscoplastic do not

6. Yield Stress Aging

• Time dependence: \(\sigma_y(t_{\text{rest}}) = \sigma_{y,0} + \Delta\sigma_y \cdot (1 - \exp(-t_{\text{rest}} / \tau_{\text{thix}}))\)

• Aging rate: \(1 / \tau_{\text{thix}}\)

• Maximum recoverable yield stress: \(\sigma_{y,0} + \Delta\sigma_y\)

Dimensionless Groups

The model behavior can be characterized by several dimensionless numbers:

Weissenberg Number (Wi):

\[Wi = \dot{\gamma} \tau_{thix}\]

Ratio of shear rate to structure buildup rate. \(\text{Wi} \gg 1\) means structure breaks down faster than it recovers (destructured regime).

Deborah Number (De):

\[De = \frac{\tau}{\tau_{exp}} = \frac{\eta/G}{t_{exp}}\]

Ratio of relaxation time to experimental time scale. \(\text{De} \gg 1\) means elastic response dominates; \(\text{De} \ll 1\) means viscous response dominates.

Bingham Number (Bi):

\[Bi = \frac{\sigma_y}{\eta_{\infty}\dot{\gamma}}\]

Ratio of yield stress to viscous stress. \(\text{Bi} \gg 1\) means yield-dominated; \(\text{Bi} \ll 1\) means viscous-dominated.

Structure Number (Sn):

\[Sn = \Gamma \tau_{thix}\]

Relative efficiency of breakdown versus buildup. \(\text{Sn} \gg 1\) means structure breaks down efficiently under shear.

Industrial Applications

The MIKH model was developed for and validated against industrial thixotropic materials. This section provides application-specific guidance with typical parameter ranges from field studies and laboratory characterization.

Waxy Crude Oil Pipeline Operations

The MIKH model was originally developed for waxy crude oils (Dimitriou & McKinley 2014), making it the reference model for pipeline flow assurance applications.

Pipeline Restart After Shutdown:

When a pipeline shuts down, wax precipitates and forms a gel network. Key parameter ranges from field applications:

• \(\tau_{\text{thix}}\) = 100–10,000 s: Long aging times for gelled pipelines

• \(\sigma_{y,0} + \Delta\sigma_y\) = 50–500 Pa: Gel strength depends on cooling rate and rest time

• \(\Gamma \cdot \tau_{\text{thix}}\) > 10: Extreme shear-thinning for pipeline restart

Engineering implications:

• Restart pressure scales with \(\sigma_y(t_{\text{rest}})\) where \(t_{\text{rest}}\) can span hours to days

• Monitor thermokinematic memory (FIKH framework) for temperature-cycled systems

• Stress overshoot during restart indicates incomplete gel breakdown

Cold Flow Assurance:

For subsea pipelines below wax appearance temperature (WAT):

• Continuous low-shear flow prevents complete gelation

• Target operating shear rate: \(\dot{\gamma} > 1/(\Gamma \cdot \tau_{\text{thix}})\) to maintain destructured state

• Thermal cycling protocols require FIKH framework with temperature-dependent \(\phi\)

Drilling Fluids and Muds

Water-based drilling fluids exhibit pronounced IKH behavior due to clay platelet aggregation and polymer interactions.

Typical parameter ranges:

• \(\tau_{\text{thix}}\) = 1–100 s: Faster recovery than crude oils due to smaller particles

• \(\sigma_{y,0}\) = 5–15 Pa: API barite suspension requirements for cutting transport

• \(C/\gamma_{\text{dyn}} \approx 0.1\text{--}0.3\,\sigma_y\): Moderate Bauschinger effect from clay orientation

Borehole Stability:

• Gel strength must exceed cutting particle buoyancy: \(\sigma_y > \Delta\rho \cdot g \cdot d_{\text{particle}}\)

• Thixotropic recovery prevents fluid loss into formation during connections

• API 6rpm/300rpm readings map to MIKH parameters via flow curve fitting

Pump Circulation Restart:

After pipe connections or trips:

• Initial startup pressure \(\propto \sigma_y(t_{\text{connection}})\) where \(t_{\text{connection}} \sim 30\text{--}300\) s

• Stress overshoot magnitude indicates gel breakdown efficiency

• Design pumping rate to achieve \(\dot{\gamma} > 1/(\Gamma \cdot \tau_{\text{thix}})\) throughout annulus

Greases and Lubricants

Grease consistency (NLGI grades) correlates with MIKH parameters through the yield stress and thixotropic timescales.

NLGI Grade Correlation:

NLGI Grade	Application	\(\sigma_y\) (Pa)	\(\tau_thix\) (s)
000-00	Centralized systems	50-150	1-10
0-1	Enclosed gears	100-300	5-30
2	General purpose	200-500	10-100
3-6	High-consistency	400-2000	50-500

Bearing Startup Applications:

• Stress overshoot magnitude indicates grease breakdown risk under initial loading

• Kinematic hardening (C parameter) critical for reversing loads in oscillating bearings

• Channeling behavior: permanent structure breakdown when \(\dot{\gamma}\) peak > critical value

Kinematic Hardening in Reversing Loads:

The Bauschinger effect (controlled by \(C / \gamma_{\text{dyn}}\) ratio) is particularly important for greases in oscillating applications:

# Reverse flow simulation for oscillating bearing
model = MIKH()
model.parameters.set_value("C", 100.0)      # Kinematic hardening
model.parameters.set_value("gamma_dyn", 5.0) # Recovery rate
# Backstress saturation: α_max = C/γ_dyn = 20 Pa

# Simulate LAOS to observe Bauschinger effect
sigma_laos = model.predict_laos(t, gamma_0=0.5, omega=1.0)

Thixotropic Cements and Pastes

Cementitious materials exhibit structure evolution from early hydration and particle flocculation.

Pumping and Placement:

• \(\tau_{\text{thix}}\) = 10–1000 s: Depending on formulation and admixtures

• Structure recovery must match placement window for self-leveling vs. vertical stability

• High \(\Gamma\) values enable rapid breakdown for pumping, but may compromise build-up

Self-Leveling vs. Non-Sag Behavior:

Application	Parameter Requirement	Physical Interpretation
Self-leveling floors	Low \(\tau_{\text{thix}}\), high \(\Gamma\)	Fast breakdown, moderate recovery
Vertical surfaces	High \(\tau_{\text{thix}}\), moderate \(\Gamma\)	Slow breakdown, strong recovery
3D printing	Very high \(\sigma_{y,0} + \Delta\sigma_y\)	Immediate yield on deposition

Yield Stress Aging for Formwork Removal:

The time-dependent yield stress evolution determines safe formwork removal:

\[\sigma_y(t_{cure}) = \sigma_{y,0} + \Delta\sigma_y \cdot (1 - e^{-t_{cure}/\tau_{thix}})\]

For critical structural applications, \(\tau_{\text{thix}}\) must be characterized at the curing temperature to predict strength development.

Parameters

Parameter	Symbol	Units	Description
G	G	Pa	Elastic shear modulus. Controls initial stiffness and stress overshoot amplitude. Typical range: \(10^2 - 10^6\) Pa for structured fluids.
eta	\(\eta\)	Pa·s	Maxwell viscosity. Relaxation time \(\tau = \eta / G\). Large values = elastic solid. Setting \(\eta \to \infty\) recovers rate-independent plasticity.
C	C	Pa	Kinematic hardening modulus. Controls backstress buildup rate. Larger C = stronger Bauschinger effect.
gamma_dyn	\(\gamma_{\text{dyn}}\)	–	Dynamic recovery parameter. Limits backstress saturation. Saturation: \(\alpha_{\max} = C / \gamma_{\text{dyn}}\).
m	\(m\)	–	AF exponent (typically 1.0). Controls nonlinearity of recovery. \(m = 1\): linear AF; \(m > 1\): accelerated recovery at high \(\alpha\).
sigma_y0	\(\sigma_{y0}\)	Pa	Minimal yield stress (fully destructured state, \(\lambda = 0\)). This is the “static” yield stress after prolonged shearing.
delta_sigma_y	\(\Delta\sigma_y\)	Pa	Structural yield stress contribution. \(\sigma_{y,\max} = \sigma_{y0} + \Delta\sigma_y\) when \(\lambda = 1\). Controls strength of aging effect.
tau_thix	\(\tau_{\text{thix}}\)	s	Thixotropic rebuilding time. Time for structure recovery at rest. Typical: \(10^{-1}\) – \(10^4\) s depending on material.
Gamma	\(\Gamma\)	–	Breakdown coefficient. Efficiency of shear-induced destructuring. Higher \(\Gamma\) = faster breakdown under shear.
eta_inf	\(\eta_{\infty}\)	Pa·s	High-shear viscosity (solvent contribution). Dominates at high shear rates where structure is destroyed.
mu_p	\(\mu_p\)	Pa·s	Plastic viscosity (Perzyna regularization parameter). Small \(\mu_p\) = sharp yield; large \(\mu_p\) = smoothed transition.

Fitting Guidance

Initialization Strategy

1. Flow curve first: Fit \(\sigma_{y0}\), \(\Delta\sigma_y\), \(\tau_{\text{thix}}\), \(\Gamma\), \(\eta_\infty\) from steady-state data

2. Startup second: Fix flow curve params, fit \(G\), \(C\), \(\gamma_{\text{dyn}}\) from transient

3. Relaxation/creep: Fine-tune \(\eta\) (Maxwell viscosity)

Protocol Selection

Protocol	Best for
flow_curve	Steady-state parameters (\(\sigma_{y0}\), \(\Delta\sigma_y\), \(\eta_\infty\))
startup	Elasticity (\(G\)) and hardening (\(C\), \(\gamma_{\text{dyn}}\))
relaxation	Maxwell viscosity (\(\eta\))
creep	Combined viscoelastic-plastic response
laos	Full nonlinear characterization

Troubleshooting

Issue	Solution
Poor flow curve fit	Check \(\sigma_{y0}\) initialization; ensure \(\dot{\gamma}\) range spans structure transition
No stress overshoot	Increase \(G\) or decrease \(\Gamma\) (maintain structure during startup)
Overshoot too sharp	Increase \(\mu_p\) (plastic viscosity regularization)
No Bauschinger effect	Increase \(C\) (hardening) or decrease \(\gamma_{\text{dyn}}\) (less recovery)
Stress doesn’t relax	Decrease \(\eta\) (Maxwell viscosity); check \(\tau = \eta / G\) vs experiment time

Parameter Estimation Methods

The MIKH model’s 11 parameters span different experimental timescales and phenomena. Advanced estimation methods improve identifiability and uncertainty quantification beyond basic curve fitting.

Sequential Fitting Strategy

A sequential approach exploits the separation of timescales in the MIKH model to improve parameter identifiability:

Stage 1: Flow Curve (Steady State)

From flow curve data \(\sigma(\dot{\gamma})\), fit the steady-state parameters:

• \(\sigma_{y,0}\), \(\Delta\sigma_y\) (yield stress bounds)

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

• \(\Gamma \cdot \tau_{\text{thix}}\) product (controls shear-thinning curvature)

from rheojax.models import MIKH

model = MIKH()

# Fix elastic/hardening params, fit thixotropic
model.parameters.freeze(['G', 'C', 'gamma_dyn', 'eta', 'mu_p'])
model.fit(gamma_dot, sigma_ss, test_mode='flow_curve')

# Extract fitted values
sigma_y0_fit = model.parameters.get_value('sigma_y0')
delta_sigma_y_fit = model.parameters.get_value('delta_sigma_y')

Stage 2: Startup Transients

From startup stress overshoot \(\sigma(t; \dot{\gamma}_0)\), fit:

• \(G\) (controls initial slope and overshoot magnitude)

• \(C\), \(\gamma_{\text{dyn}}\) (kinematic hardening, Bauschinger effect)

• \(\tau_{\text{thix}}\) (recovery timescale, now separated from \(\Gamma\))

# Unfreeze elastic/hardening parameters
model.parameters.unfreeze(['G', 'C', 'gamma_dyn'])

# Fit startup data with flow curve params fixed
model.fit(t_startup, sigma_startup, test_mode='startup')

Stage 3: Relaxation/Creep

From stress relaxation \(\sigma(t)|_{\gamma=\text{const}}\), fit:

• \(\eta\) (Maxwell viscosity, determines \(\tau_{\text{relax}} = \eta / G\))

• \(\mu_p\) (Perzyna regularization, yield transition sharpness)

model.parameters.unfreeze(['eta', 'mu_p'])
model.fit(t_relax, sigma_relax, test_mode='relaxation')

Multi-Start Global Optimization

For datasets spanning wide parameter ranges or with multiple local minima, use multi-start optimization:

# Multi-start with parallel execution
model.fit(
    X, y,
    use_multi_start=True,
    n_starts=5,           # Number of random initializations
    parallel=True         # ThreadPoolExecutor for 3-5x speedup
)

When to use multi-start:

• Flow curves spanning >3 decades of shear rate

• Combined protocol fitting (flow + startup + relaxation)

• Initial fits show residual structure (systematic over/under-prediction)

• Materials with unusual parameter combinations (e.g., very high \(\tau_{\text{thix}}\))

Global optimization for multi-modal problems:

from rheojax.utils.optimization import nlsq_optimize_global

# Global search for challenging parameter landscapes
result = nlsq_optimize_global(
    objective_fn,
    initial_params,
    bounds=param_bounds
)

Bayesian Inference with MCMC

For uncertainty quantification, use NumPyro NUTS with NLSQ warm-start:

# Stage 1: Point estimate (fast, provides good initialization)
model.fit(X, y, test_mode='startup')

# Stage 2: Bayesian inference (4 chains for reliable R-hat)
result = model.fit_bayesian(
    X, y,
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,         # Production: 4 chains for R-hat diagnostics
    seed=42               # Reproducibility
)

# Check convergence diagnostics
print(f"R-hat: {result.r_hat}")   # Target: <1.01
print(f"ESS: {result.ess}")       # Target: >400

Prior Selection Guidance:

The choice of priors significantly affects Bayesian inference for the MIKH model:

Parameter	Recommended Prior	Rationale
\(\tau_{\text{thix}}\)	LogNormal(\(\mu = \log(10)\), \(\sigma = 1\))	Spans 1–100 s; positive, heavy-tailed
\(\Gamma\)	HalfNormal(\(\sigma = 10\))	Positive breakdown coefficient
\(\sigma_{y,0}\), \(\Delta\sigma_y\)	TruncatedNormal or Uniform	Material-dependent bounds
\(G\)	LogNormal(\(\mu = \log(1000)\), \(\sigma = 1\))	Typical modulus range for soft materials
\(C / \gamma_{\text{dyn}}\) ratio	LogNormal(\(\mu = \log(10)\), \(\sigma = 0.5\))	Backstress saturation constraint

Regularization for Ill-Conditioned Problems

When parameters are correlated (common for \(G\)-\(C\), \(\tau_{\text{thix}}\)-\(\Gamma\) pairs), use:

1. Tikhonov Regularization:

Add penalty \(\lambda \|\theta\|^2\) to objective function to stabilize optimization:

from rheojax.utils.optimization import nlsq_curve_fit

result = nlsq_curve_fit(
    model_fn, x, y, params,
    regularization='tikhonov',
    lambda_reg=1e-4  # Regularization strength
)

2. Bounds Tightening:

Physically constrain parameter ranges based on material knowledge:

# Constrain based on material class
model.parameters.set_bounds('tau_thix', (1.0, 1000.0))  # Drilling fluid
model.parameters.set_bounds('sigma_y0', (5.0, 50.0))   # API spec range

3. Combined Protocol Fitting:

Fitting multiple test modes simultaneously reduces parameter correlation by providing orthogonal constraints:

# Combined protocol fitting (pseudo-code pattern)
# Concatenate datasets with appropriate weighting
X_combined = combine_protocols(flow_data, startup_data)
weights = [1.0, 2.0]  # Emphasize transient data

model.fit(X_combined, y_combined, weights=weights)

Sensitivity Analysis

Identify which parameters most influence predictions to guide experimental design:

Local Sensitivity (Jacobian-based):

import jax
import jax.numpy as jnp

def compute_sensitivity(model, X, param_names):
    """Compute local parameter sensitivities."""

    def prediction_fn(param_values):
        for name, val in zip(param_names, param_values):
            model.parameters.set_value(name, val)
        return model.predict(X)

    # Get current parameter values
    param_values = jnp.array([
        model.parameters.get_value(name) for name in param_names
    ])

    # Compute Jacobian: ∂σ/∂θ
    jacobian = jax.jacobian(prediction_fn)(param_values)
    return jacobian

Sensitivity interpretation:

• High sensitivity: Parameter strongly influences predictions (well-constrained by data)

• Low sensitivity: Parameter weakly influences predictions (may be poorly identifiable)

• Correlated sensitivities: Parameters are coupled (consider reparameterization)

Practical recommendations:

1. Compute sensitivities at the fitted parameter values

2. Focus experimental design on regimes where target parameters have high sensitivity

3. For \(\tau_{\text{thix}}\): use startup/recovery data at \(t \sim \tau_{\text{thix}}\)

4. For \(G\): use early-time startup data (\(t \ll \tau_{\text{thix}}\))

5. For \(\Gamma\): use flow curve data near \(\dot{\gamma} \sim 1/(\Gamma \cdot \tau_{\text{thix}})\)

Usage

The MIKH model is available via:

from rheojax.models import MIKH

Common workflows:

1. Flow curve fitting: Determine \(\sigma_{y0}\), \(\Delta\sigma_y\), \(\eta_\infty\) from steady-state data

2. Startup fitting: Extract \(G\), \(C\), \(\gamma_{\text{dyn}}\) from transient stress overshoot

3. Creep/relaxation: Constrain \(\eta\) (Maxwell viscosity) and \(\mu_p\) (plastic viscosity)

4. Bayesian inference: Quantify uncertainty in thixotropic timescales

Integration with Pipeline:

from rheojax.pipeline import Pipeline

# Fluent API for complete workflow
(Pipeline()
 .load('startup_data.csv', x_col='time', y_col='stress')
 .fit_nlsq('mikh', test_mode='startup')
 .fit_bayesian(num_samples=2000)
 .plot_trace()
 .save('mikh_results.hdf5'))

Usage Examples

Basic Usage

import numpy as np
from rheojax.models import MIKH

# Initialize model
model = MIKH()

# Set parameters
model.parameters.set_value("G", 1000.0)
model.parameters.set_value("sigma_y0", 20.0)
model.parameters.set_value("eta_inf", 0.1)

Flow Curve

# Predict steady-state flow curve
gamma_dot = np.logspace(-2, 2, 50)
sigma = model.predict_flow_curve(gamma_dot)

Startup Shear

# Predict startup response
t = np.linspace(0, 10, 200)
sigma_startup = model.predict_startup(t, gamma_dot=1.0)

Stress Relaxation

# Predict relaxation from initial stress
t = np.linspace(0, 100, 200)
sigma_relax = model.predict_relaxation(t, sigma_0=100.0)

Creep

# Predict creep under constant stress
t = np.linspace(0, 100, 200)
strain = model.predict_creep(t, sigma_applied=50.0)

LAOS

# Large amplitude oscillatory shear
t = np.linspace(0, 20, 500)
sigma_laos = model.predict_laos(t, gamma_0=1.0, omega=1.0)

Fitting

# Fit to experimental data
model.fit(gamma_dot, sigma_data, test_mode='flow_curve')

# Bayesian inference with NLSQ warm-start
result = model.fit_bayesian(
    X_data, sigma_data,
    num_warmup=1000, num_samples=2000,
    test_mode='startup'
)

Relation to Other Models

Model	Relationship to MIKH
Herschel-Bulkley	MIKH reduces to HB at steady state without kinematic hardening (\(C = 0\)) and with power-law \(\eta_\infty\)
Saramito EVP	Similar framework but without kinematic hardening; uses Oldroyd-B instead of Maxwell for viscoelasticity
de Souza Mendes TEvp	Different structure kinetics formulation; uses viscosity-based approach rather than yield stress
Multi-Lambda Isotropic-Kinematic Hardening (ML-IKH)	Multi-mode extension with N parallel structural elements

References

[1]

Dimitriou, C. J. and McKinley, G. H. “A comprehensive constitutive law for waxy crude oil: a thixotropic yield stress fluid.” Soft Matter, 10(35), 6619-6644 (2014). DOI: 10.1039/C4SM00578C PDF

[2]

Geri, M., Venkatesan, R., Sambath, K., and McKinley, G. H. “Thermokinematic memory and the thixotropic elasto-viscoplasticity of waxy crude oils.” Journal of Rheology, 61(3), 427-454 (2017). DOI: 10.1122/1.4978259 PDF

[3]

Saramito, P. “A new elastoviscoplastic model based on the Herschel-Bulkley viscoplastic model.” Journal of Non-Newtonian Fluid Mechanics, 158, 154-161 (2009). https://doi.org/10.1016/j.jnnfm.2008.12.001

[4]

de Souza Mendes, P. R. and Thompson, R. L. “Time-dependent yield stress materials.” Annual Review of Fluid Mechanics, 51, 421-449 (2019). https://doi.org/10.1146/annurev-fluid-010518-040305

[5]

Armstrong, P. J. and Frederick, C. O. “A mathematical representation of the multiaxial Bauschinger effect.” CEGB Report RD/B/N731 (1966). https://doi.org/10.3184/096034007X207589

[6]

Mewis, J. and Wagner, N. J. “Thixotropy.” Advances in Colloid and Interface Science, 147-148, 214-227 (2009). https://doi.org/10.1016/j.cis.2008.09.005

[7]

Chaboche, J. L. “Constitutive equations for cyclic plasticity and cyclic viscoplasticity.” International Journal of Plasticity, 5(3), 247-302 (1989). https://doi.org/10.1016/0749-6419(89)90015-6

[8]

Larson, R. G. and Wei, Y. “A review of thixotropy and its rheological modeling.” Journal of Rheology, 63(3), 477-501 (2019). https://doi.org/10.1122/1.5055031

[9]

Dullaert, K. and Mewis, J. “A structural kinetics model for thixotropy.” Journal of Non-Newtonian Fluid Mechanics, 139(1-2), 21-30 (2006). https://doi.org/10.1016/j.jnnfm.2006.06.002

[10]

Prager, W. “A new method of analyzing stresses and strains in work-hardening plastic solids.” Journal of Applied Mechanics, 23, 493-496 (1956). https://doi.org/10.1115/1.4011389

See Also

• Multi-Lambda Isotropic-Kinematic Hardening (ML-IKH) — Multi-mode extension for distributed thixotropic timescales

• /models/dmt/dmt_local — Alternative thixotropic formulation (de Souza Mendes-Thompson)

• /models/fluidity/saramito — Fluidity-Saramito EVP models with thixotropy

• Section 3: Advanced Topics (Weeks 7-12) — Advanced thixotropic modeling workflows