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Comparison of LMP and MHS+MG Models for Oxidation-Corrected Creep Rupture Life Prediction of Grade 23 Steel

This study demonstrates that the Modified Hyperbolic Sine plus Monkman–Grant (MHS+MG) model outperforms both constrained and unconstrained quadratic Larson–Miller Parameter (LMP) models in predicting the oxidation-corrected creep rupture life of Grade 23 steel by offering superior accuracy, stability, and physically consistent stress–life trends across a wide temperature range.

Original authors: Qiang Xu, Abdul Wahab RAO

Published 2026-08-11

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CC BY 4.0

Original authors: Qiang Xu, Abdul Wahab RAO

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an engineer building a bridge that has to withstand the scorching heat of a summer sun for decades. You need to know exactly when the metal will finally give way and snap. This is the world of creep, a slow, silent stretching of metal under constant stress and high heat. It's the reason why old power plants need careful monitoring; if you guess wrong about when a pipe will burst, the consequences are disastrous. To predict this, scientists use mathematical "crystal balls" called models. Two famous ones are the Larson-Miller Parameter (LMP), which is like a simple, old-school recipe that mixes temperature and time, and the Monkman-Grant relation, which looks at how fast the metal stretches before it breaks. But there's a catch: when these metal pipes sit in hot air for years, they rust (oxidize) on the outside. This rust eats away a tiny bit of the metal's thickness, making the pipe feel more stressed than it actually is, which tricks the models into thinking the metal is weaker than it really is. To get the true picture of the metal's strength, scientists have to mathematically "peel off" this rust effect first.

Now, enter a team of researchers from the University of Huddersfield who decided to put these different crystal balls to the test. They focused on a special type of steel called Grade 23 (or T23), which is a superhero version of older steel used in boilers. They gathered a massive collection of 135 real-world test results, all carefully cleaned of that "rust" effect, and asked a simple question: Which mathematical model can best predict when this steel will break without getting confused by the data? They compared the classic, flexible Larson-Miller method against a newer, more complex approach called MHS+MG (a mix of a Modified Hyperbolic Sine law and the Monkman-Grant relation). The goal wasn't just to find the model with the lowest number of errors, but to find the one that behaves like a real, honest metal—always getting weaker as stress increases, never doing something weird and impossible.

The researchers ran their numbers using two different ways of fitting the data: a standard "best fit" method and a "robust" method that tries to ignore outliers. Here is what they found. The classic Larson-Miller model, when allowed to wiggle freely, did a decent job at guessing the numbers, but it had a secret flaw. At the highest temperature tested (650 °C), the model's curve did a strange U-turn, suggesting that if you pulled the steel harder, it would actually last longer. That is physically impossible; it's like saying if you push a car harder, it will drive further without running out of gas. When the researchers forced the model to behave and never turn back, it stopped making that impossible prediction, but its accuracy plummeted, becoming much worse at guessing the actual time.

On the other hand, the MHS+MG model was the clear winner. It didn't just guess the numbers better; it did so while staying physically sensible. It never made that weird U-turn. In fact, the standard "best fit" version of MHS+MG was the most accurate of all, with a prediction error (RMSE) of 0.2614 in the logarithmic time scale and a success rate (R²) of 0.9363. Even the "robust" version, which tried to be extra careful with the data, performed very closely behind it with an error of 0.2719 and a success rate of 0.9311. The researchers were careful to point out that the MHS+MG model's success wasn't just because they used a fancy "robust" math trick; even when they used the simple, standard math, it still beat the Larson-Miller model by a wide margin.

However, the team wasn't ready to declare this a "solved problem" for all time. They noted that the MHS+MG model is currently just a "rupture-life" predictor, meaning it guesses when the break happens, but it hasn't been fully tested against the speed of the stretching because that specific data wasn't available for this steel. Also, the math behind the model had some "wobbly" parts where the numbers could shift around depending on how the calculation was done. So, while the MHS+MG model is currently the best balance of accuracy and common sense for this specific steel, the researchers suggest we should keep refining it. They also noted that for the highest temperature tests, they used 39 data points out of 51 available, and future work needs to be very clear about why some points were left out. Ultimately, this study shows that for Grade 23 steel, the MHS+MG model is the most reliable guide we have right now, provided we remember to strip away the rust first.

Technical Summary: Comparison of LMP and MHS+MG Models for Oxidation-Corrected Creep Rupture Life Prediction of Grade 23 Steel

Problem Statement
Reliable prediction of creep rupture life is critical for the design and integrity assessment of high-temperature power plant components. Grade 23 steel (T23), a low-alloy ferritic steel, offers superior creep strength compared to conventional 2.25Cr–1Mo steels. However, interpreting long-term rupture data is complicated by oxidation. During prolonged exposure in air, oxide growth reduces the specimen's cross-sectional area, increasing effective stress and artificially shortening the apparent rupture life. To accurately model the material's intrinsic creep resistance rather than environmental section loss, oxidation-corrected datasets are required. Furthermore, existing parametric models like the Larson–Miller Parameter (LMP) may produce physically inadmissible results (e.g., non-monotonic stress-life trends) when unconstrained, while alternative formulations require rigorous validation against the same datasets to isolate model formulation effects from calibration strategies.

Methodology
This study conducts a controlled comparison of three modeling approaches using a unified, oxidation-corrected dataset of 135 rupture observations for Grade 23 steel across four temperatures (500, 550, 600, and 650 °C). The data were digitized from previous work by Igarashi et al.

The study evaluates:

Unconstrained Quadratic LMP: An empirical baseline using a quadratic polynomial to relate the Larson–Miller parameter to log-stress.
Monotonic Quadratic LMP: The same quadratic form but constrained to ensure the derivative of the predicted parameter with respect to log-stress remains non-positive ( $dP/dx \leq 0$ ), enforcing a physically admissible trend where life decreases as stress increases.
Modified Hyperbolic Sine + Monkman–Grant (MHS+MG): A stress-function model combining a modulus-normalized hyperbolic sine term with a Monkman–Grant rate-life relationship. This model was calibrated using two distinct strategies:
- Ordinary Least Squares (OLS): Equal weighting of all observations.
- Robust Residual-Weighted: Iterative weighting to minimize the influence of outliers.

All models were evaluated using unweighted Root Mean Square Error (RMSE) in log-life space, $R^2$ , and Mean Absolute Percentage Error (MAPE) in the natural time domain. The comparison specifically isolates the impact of the mathematical formulation (LMP vs. MHS+MG) and the calibration strategy (OLS vs. Robust) by applying them to the identical dataset.

Key Results

Superiority of MHS+MG: The MHS+MG formulation demonstrated the strongest overall performance. The OLS-calibrated MHS+MG model achieved the lowest error metrics (RMSE = 0.2614, $R^2$ = 0.9363, MAPE = 0.5082), slightly outperforming the robust-weighted MHS+MG variant (RMSE = 0.2719, $R^2$ = 0.9311, MAPE = 0.5303).
Limitations of Quadratic LMP: While the unconstrained quadratic LMP provided reasonable numerical accuracy (RMSE = 0.3774), it developed an internal non-monotonic turning point (vertex) within the fitted stress range, most notably at 650 °C. This resulted in a physically questionable prediction where rupture life appeared to increase with stress in a localized region.
Cost of Physical Constraints: Enforcing monotonicity on the quadratic LMP removed the turning point but caused a substantial degradation in predictive accuracy (RMSE increased to 0.5499, MAPE to 2.0832).
Calibration Strategy Impact: The fact that the OLS MHS+MG model performed slightly better than the robust-weighted version indicates that the MHS+MG advantage stems primarily from its stress-function formulation rather than the use of a sophisticated weighting scheme.

Significance and Claims
The paper claims that within the context of this oxidation-corrected dataset, the MHS+MG stress-function formulation provides the best balance of predictive accuracy, numerical stability, and physically consistent stress–life behavior.

The study emphasizes that the improved performance of MHS+MG is intrinsic to its formulation, which accommodates curvature without requiring an internal reversal of the stress-life trend, unlike the unconstrained polynomial LMP. The authors explicitly state that these results should not be interpreted as a universal constitutive validation. Because the model was fitted as a rupture-life regression without matched oxidation-corrected minimum creep-rate data, and because the $q$ parameter showed weak identifiability at several temperatures, the fitted parameter values should be treated with caution. The primary contribution is the demonstration that the MHS+MG route offers a more reliable practical balance between numerical accuracy and physical admissibility for Grade 23 steel creep life prediction compared to standard quadratic LMP approaches.

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