Non-Isothermal JMAK Phase Transformation Kinetics
State of the art in JMAK extensions for non-isothermal conditions in welding and additive manufacturing, with Koistinen-Marburger coupling and CCT diagram integration.
The core problem
Standard JMAK assumes isothermal conditions. AM/WAAM thermal cycles involve heating rates of 10^2 to 10^4 K/s and cooling rates of 10 to 10^3 K/s, violating the isokinetic assumption. The additivity rule (Scheil’s principle) breaks down when nucleation and growth have different activation energies.
Non-isothermal JMAK extensions
Temperature-dependent Avrami exponents
McNamara et al. (2022) developed a generalized non-isothermal JMAK framework using temperature-dependent Avrami exponents. Validated against in-situ synchrotron XRD for both IN718 and Ti-6Al-4V. Key finding: standard JMAK overpredicts transformation fractions by 20—40% under AM thermal cycles.
Citation: McNamara et al. (2022). “Predicting phase transformation kinetics during metal additive manufacturing using non-isothermal Johnson-Mehl-Avrami models.” Additive Manufacturing, Elsevier.
Simplified transformation kinetics for cyclic reheating
Lu, Li & Yang (2021) derived a simplified transformation kinetics (STK) model from JMAK for cyclic reheating in WAAM of IN718. Captured gamma-double-prime and gamma-prime precipitation evolution across multiple thermal cycles.
Citation: Lu, Li & Yang (2021). “Simulation of precipitates evolution driven by non-isothermal cyclic thermal history during wire and arc additive manufacturing of IN718 superalloy.” Journal of Manufacturing Processes.
Nonlinear cooling corrections
Wu et al. (2020) addressed non-isokinetic behavior under nonlinear cooling in TA15 titanium alloy. Showed JMAK fails when cooling rate changes mid-transformation. Proposed a segmented approach where each cooling regime uses locally calibrated parameters.
Citation: Wu et al. (2020). “Diffusion transformation model in TA15 titanium alloy: The case of nonlinear cooling.” Materials & Design.
Three-stage transformation with distinct exponents
Kaushik, Korayem & Hadadzadeh (2022) identified three-stage non-isothermal transformation with distinct Avrami exponents per stage in LPBF Ti-6Al-2Sn-4Zr-2Mo-0.08Si and Ti-6Al-4V.
Citation: Kaushik, Korayem & Hadadzadeh (2022). “Determination of alpha to beta phase transformation kinetics in laser-powder bed fused titanium alloys.” Materials Science and Engineering A.
Ultrafast laser quenching
Zhu et al. (2026) extended JMAK to ultrafast laser quenching (>10^4 K/s) where non-equilibrium effects dominate.
Citation: Zhu et al. (2026). “A Predictive Method for Hardness Distribution in Laser-Quenched Medium-Carbon Steel.” Journal of Materials Processing Technology.
DSC-calibrated kinetics for LPBF
Song et al. (2025) extended non-isothermal JMAK using DSC-calibrated kinetics for alpha/beta transformations in LPBF Ti-6Al-4V. Demonstrated that post-build heat treatment optimization requires non-isothermal corrections.
Citation: Song et al. (2025). “Advanced phenomenological models guided heat treating processes for LPBF Ti-6Al-4V alloy.” Materials Today Communications.
Koistinen-Marburger model for martensite
The KM equation remains the standard for diffusionless (athermal) martensitic transformation in welding FEM codes:
f_m = 1 - exp[-beta * (M_s - T)]
Implemented in Sysweld, Abaqus (UMAT/VUMAT), and most commercial welding simulation packages.
JMAK + KM coupling approach
- Diffusional products (ferrite, pearlite, bainite): JMAK with CCT-derived parameters
- Martensite: KM with M_s and beta calibrated from dilatometry
- Phase fractions must sum to unity: f_F + f_P + f_B + f_M = 1
- TRIP (Greenwood-Johnson effect) included via Leblond model
Key coupling papers
Murthy, Akyel, Reisgen & Olschok (2022) coupled JMAK with KM in a unified FEM framework for laser beam welding. Demonstrated that ignoring solid-state phase transformation effects underestimates residual stresses by 15—30%.
Citation: Murthy et al. (2022). “Simulation of transient heat transfer and phase transformation in laser beam welding.” Journal of Advanced Joining Processes.
Xia & Jin (2018) provided detailed implementation of JMAK+KM coupling in Abaqus user subroutines. Validated against Gleeble dilatometry.
Citation: Xia & Jin (2018). “Numerical modeling of coupling thermal-metallurgical transformation phenomena of structural steel in the welding process.” Advances in Engineering Software.
Ghafouri et al. (2020) used Machnienko model for diffusional transformations + KM for martensite in UHSS welding. Showed SSPT-induced volume expansion (up to 4%) significantly reduces tensile residual stress in HAZ.
Citation: Ghafouri et al. (2020). “Finite element simulation of welding distortions in ultra-high strength steel S960 MC.” Engineering Structures.
CCT diagram integration with FEM
Implementation approaches
- Sysweld (ESI Group): Native CCT/TTT diagram support. Metallurgical database with Leblond-based solver. Most widely used in industry.
- Abaqus user subroutines: UMAT/HETVAL for latent heat, USDFLD for phase-dependent properties. CCT data digitized and interpolated.
- ANSYS: User-programmable features for custom metallurgical models.
Integration workflow
- Experimental CCT diagrams (dilatometry) digitized into lookup tables
- FEM thermal solver provides T(t) history at each integration point
- Cooling rate dT/dt computed and interpolated on CCT diagram
- Phase fractions, transformation start/end temperatures extracted
- Latent heat fed back to thermal solver (weak or strong coupling)
- Phase-dependent mechanical properties assigned (mixture rule)
Key papers
Morawiec et al. (2022): Sysweld-based simulation with CCT/TTT diagrams for laser welding of multiphase steel. Validated predicted phase fractions against metallography.
Citation: Morawiec et al. (2022). “Numerical simulation and experimental analysis of thermal cycles and phase transformation behavior of laser-welded advanced multiphase steel.” Symmetry.
Jiao & Jin (2025): Abaqus-based thermal-metallurgical-mechanical coupling using SH-CCT diagrams. Multi-physics field coupling with phase-dependent material properties.
Citation: Jiao & Jin (2025). “Finite element and experimental analysis of residual stresses in G20Mn5 welded joints considering solid-state phase transformation.” Int. J. Pressure Vessels and Piping.
Multi-phase field approaches
Phase-field models resolve grain morphology, dendrite arm spacing, and solute segregation. Computational cost is 3—6 orders of magnitude higher than JMAK.
Computational cost comparison
| Method | Domain | Time | Resolution |
|---|---|---|---|
| JMAK (empirical) | Full part | Seconds to minutes | Volume fractions only |
| Cellular Automata | mm to cm scale | Minutes to hours | Grain morphology |
| Phase-field (direct) | um to mm scale | Hours to weeks | Dendrite/sub-grain |
| Phase-field + ML | mm to cm scale | Minutes to hours | Grain morphology |
ML-accelerated phase-field
Choi et al. (2024): ML-accelerated phase-field for LPBF. Composable ML predictions at small scale stitched together for full-part microstructure. Reduced computation time by ~100x.
Citation: Choi, Xue, Liao & Cao (2024). “Accelerating phase-field simulation of 3D microstructure evolution in LPBF with composable ML predictions.” Additive Manufacturing.
Xue et al. (2022): Graph neural network surrogate for phase-field. Trained on small-domain PF simulations, generalized to large domains.
Citation: Xue, Gan, Liao & Cao (2022). “Physics-embedded graph network for accelerating phase-field simulation of microstructure evolution in AM.” npj Computational Materials.
Lach (2026): Review comparing CA vs. phase-field for AM. Notes ML-accelerated PF for WAAM showing promise.
Citation: Lach (2026). “Cellular Automata and Phase-Field Modeling of Microstructure Evolution in Metal AM.” Metals.
Research gaps
Cyclic reheating kinetics
JMAK/KM are calibrated for single thermal cycles. WAAM involves 10—100+ reheating events per location. No validated model exists for cumulative transformation kinetics under repeated austenitization/cooling cycles with varying peak temperatures (partial vs. full re-austenitization).
Non-equilibrium phases at AM cooling rates
At cooling rates >100 K/s, bainite/martensite boundaries blur. Standard CCT diagrams (measured at ⇐50 K/s) do not cover the relevant regime. High-rate dilatometry data is sparse.
Multi-pass interaction effects
Previous layer microstructure serves as initial condition for next layer. No standard framework tracks inherited grain structure, texture, and precipitate state through dozens of thermal cycles.
Composition-dependent kinetics
JMAK parameters are alloy-specific. For WAAM with filler wire compositional variations (dilution gradients), spatially varying kinetics parameters are needed but rarely implemented.
ML surrogate opportunities
- Replace CCT lookup tables: Train NNs on dilatometry data to predict phase fractions as function of arbitrary T(t) histories, bypassing the additivity assumption entirely.
- Accelerate phase-field: GNN/CNN surrogates trained on small-domain PF predict microstructure at part-scale with 100—1000x speedup.
- Bridge scales: ML maps JMAK volume fractions to phase-field morphologies, creating hybrid models with both kinetics and morphology at reasonable cost.
- Inverse design: VAEs and generative models invert process to microstructure mappings for parameter optimization.
- Real-time control: ML surrogates of metallurgical models run at process speed for closed-loop microstructure control during WAAM.
Connection to TMM surrogate project
See TMM Surrogates for the overall architecture. The JMAK/KM kinetics form the metallurgical stage (stage 3) of the recommended neural operator chain. Since JMAK/KM equations are ODEs at each material point (decoupled spatially once T and epsilon are known), they can be solved analytically or with a small MLP approximating ODE integration.
The key challenge is handling cyclic reheating in WAAM, where standard JMAK fails. An ML surrogate trained on experimental data could bypass the additivity assumption and directly predict phase fractions from arbitrary thermal histories.