⚡ electrical engineering

Interpretable hybrid XGBoost-MH models for predicting crack coalescence stress from direct shear testing of rock-like specimens containing non-persistent joints

This study develops an interpretable hybrid XGBoost-Butterfly Optimization model, validated on 450 direct shear tests and integrated into a user-friendly GUI, to accurately predict crack coalescence stress in jointed rock masses while identifying key mechanical drivers through SHAP analysis.

Original authors: Fariborz Matinpour, Shadman Mohammadi Bolbanabad, Vahab Sarfarazi, Mohammad Rezaei, Matin Eghdami, Mahdi Hasanipanah

Published 2026-07-10

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CC BY 4.0

Original authors: Fariborz Matinpour, Shadman Mohammadi Bolbanabad, Vahab Sarfarazi, Mohammad Rezaei, Matin Eghdami, Mahdi Hasanipanah

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 have a giant block of rock, but it's not a solid, perfect cube. It's more like a puzzle made of broken pieces glued together, with cracks running through it that don't quite go all the way across. These are called "non-persistent joints." Now, imagine you try to slide the top half of this block sideways while pushing down on it. What happens? The rock doesn't just slide; the cracks start to grow, they meet up, and suddenly the whole thing snaps. The exact moment the cracks join forces to break the rock is called the Crack Coalescence Stress (CCS).

For a long time, figuring out exactly when this snap happens has been like trying to guess the ending of a mystery novel by reading only the first page. Scientists have done hundreds of experiments, but the math behind it is so twisty and complicated that it's hard to predict without doing the messy lab work every single time.

That's where this study comes in. The researchers decided to teach a super-smart computer brain to be the detective. They didn't just use one brain; they built a team of four different "metaheuristic" algorithms—think of them as four different types of treasure hunters, each with their own unique way of searching for the best answer. These hunters were:

The Honey Badger (HBA): A tough, relentless digger.
The Butterfly (BOA): A sensitive searcher that follows scent trails.
The Crow (CSA): A clever bird that remembers where it hid food and watches others.
The Grey Wolf (GWO): A pack leader that coordinates a hunt.

Their job was to tune a powerful machine learning tool called XGBoost. Think of XGBoost as a very fast, very accurate calculator that can learn from patterns. But like a video game character, it needs its settings tweaked perfectly to work its best. The four "hunters" (algorithms) raced to find the perfect settings for the calculator.

The Big Discovery
The team ran 450 direct shear tests on rock-like specimens made of gypsum and cement. They crunched all that data and let the four hunters do their thing. Here is the result: All four hunters were good, but the Butterfly (BOA) was the clear winner.

The XGBoost–BOA model didn't just guess; it nailed the prediction. When tested on new, unseen data, it got it right 99.62% of the time (a score called VAF). Its error was tiny, with a Root Mean Square Error (RMSE) of just 1.2680 and a Mean Absolute Error (MAE) of 0.8159. In the world of rock physics, that's like hitting a bullseye from across the room. The other hunters (GWO, CSA, and HBA) were close, but the Butterfly's predictions were the most stable and reliable, with the least amount of "wobble" or uncertainty.

What Actually Matters?
The researchers didn't just want a black box that gives a number; they wanted to know why the computer made its choice. Using a special "X-ray" tool called SHAP analysis, they looked inside the model's brain. They found that three things were the heavy hitters controlling when the rock breaks:

Normal Stress (σₙ): How hard you are pushing down on the rock.
Uniaxial Compressive Strength (σc): How strong the rock material itself is.
Jointing Coefficient (JC): A number that describes how much of the rock's surface is actually a crack versus solid rock.

The model showed that if you push down harder (higher normal stress) or if the rock is stronger (higher compressive strength), the cracks need more force to join up. Interestingly, other factors like the rock's elasticity or Poisson's ratio played a much smaller role in this specific scenario. The model learned that the "big three" are the ones calling the shots.

What the Paper Rules Out
The study explicitly argues against the idea that you can easily predict this behavior by looking at just one factor in isolation, like just the joint length or just the angle. The paper suggests that the interaction between the normal stress, the material strength, and the jointing coefficient is too complex to untangle with simple experiments alone. The old way of trying to guess these relationships qualitatively (just by looking and guessing) isn't enough; you need this kind of data-driven, hybrid approach to see the full picture.

How Sure Are They?
The authors are very confident in their numbers, but they are careful not to claim they have solved the universe. They proved their model works incredibly well on the 450 specific tests they ran in the lab. They used a technique called 10-fold cross-validation, which is like testing a student on ten different practice exams to make sure they aren't just memorizing answers but actually learning the lesson. The results held up every time.

However, they are honest about the limits. This model was trained on rock-like specimens made of gypsum and cement mixtures in a lab setting. They don't claim this model works perfectly on every single type of rock in the real world (like a mountain made of granite or sandstone) without more testing. They suggest that while the tool is powerful, future work needs to test it on a wider variety of materials and stress conditions to be truly universal.

The Cool Bonus: A Video Game for Engineers
To make this useful for real people, the team didn't just leave the math in a notebook. They built a Graphical User Interface (GUI). Imagine a simple app where an engineer can type in the rock's strength, the pressure, and the crack details, click a "Predict" button, and instantly get the answer. It turns a complex, high-tech math problem into a simple tool anyone can use to design safer tunnels, dams, and slopes.

In short, the paper shows that by letting a "Butterfly" hunt for the best settings in a super-computer, we can predict exactly when a cracked rock will snap under pressure with amazing accuracy, giving us a clearer map of how the ground beneath our feet might behave.

Technical Summary: Interpretable Hybrid XGBoost-MH Models for Predicting Crack Coalescence Stress

Problem Statement
Crack coalescence stress (CCS) is a critical mechanical parameter governing shear-induced failure in jointed rock masses, playing a vital role in the stability analysis of underground and surface engineering projects. While laboratory research has extensively examined crack initiation and propagation under various stress states, the simultaneous and quantitative influence of normal stress ( $\sigma_n$ ), intrinsic mechanical properties of the intact material, and the jointing coefficient (JC) on CCS remains insufficiently understood. Traditional experimental approaches often struggle to isolate these coupled nonlinear effects, particularly when large datasets are available but the underlying relationships are obscured. This study addresses this gap by developing a data-driven framework to predict CCS for rock-like specimens containing non-persistent joints subjected to direct shear loading.

Methodology
The study utilizes a comprehensive database derived from 450 direct shear tests conducted on cubic rock-like specimens (gypsum–cement mixtures) with varying non-persistent joint configurations. The experimental program systematically varied the jointing coefficient (JC), normal stress ( $\sigma_n$ ), and material mechanical properties (uniaxial compressive strength $\sigma_c$ , tensile strength $\sigma_t$ , elastic modulus $E$ , Poisson's ratio $\nu$ , cohesion $C$ , and internal friction angle $\phi$ ).

To predict CCS, the authors developed a hybrid modeling framework integrating the Extreme Gradient Boosting (XGBoost) algorithm with four population-based metaheuristic optimization techniques:

Honey Badger Algorithm (HBA)
Butterfly Optimization Algorithm (BOA)
Crow Search Algorithm (CSA)
Grey Wolf Optimizer (GWO)

The metaheuristic algorithms were employed to systematically optimize the hyperparameters of the XGBoost model (e.g., number of estimators, learning rate, max depth). The modeling process involved:

Data Partitioning: The dataset was split into 80% training and 20% testing sets.
Optimization: A 10-fold cross-validation strategy was implemented on the training set to minimize overfitting and select optimal hyperparameters by minimizing the Root Mean Square Error (RMSE).
Evaluation: Model performance was assessed using statistical indices ( $R^2$ , VAF, RMSE, MAE), information-theoretic criteria (AIC, BIC), graphical diagnostics (Taylor diagrams, Regression Error Characteristic curves), residual analysis, and uncertainty assessment.
Interpretability: The best-performing model was analyzed using SHapley Additive exPlanations (SHAP) and Partial Dependence Plots (PDP) with Individual Conditional Expectation (ICE) to elucidate feature importance and non-linear effects.
Application: A user-friendly Graphical User Interface (GUI) was developed to facilitate practical engineering application.

Key Results
All four hybrid models demonstrated excellent predictive capabilities. However, the XGBoost–BOA model consistently outperformed the others, achieving the best balance between accuracy, generalization, and robustness.

Performance Metrics: The XGBoost–BOA model achieved a test-stage $R^2$ of 0.9960 and a Variance Accounted For (VAF) of 99.62%. It recorded the lowest prediction errors with an RMSE of 1.2680 and MAE of 0.8159.
Generalization: Unlike the XGBoost–CSA model, which showed the highest training accuracy but signs of overfitting (higher test error), the XGBoost–BOA model maintained superior performance on the independent test set.
Uncertainty: Uncertainty analysis confirmed the XGBoost–BOA model had the narrowest confidence bound width (WCB = 0.4056), indicating the highest reliability for unseen data.
Feature Importance: SHAP analysis identified normal stress ( $\sigma_n$ ) as the dominant factor influencing CCS, followed by uniaxial compressive strength ( $\sigma_c$ ) and the jointing coefficient (JC). Other mechanical parameters ( $\sigma_t, \nu, E, C, \phi$ ) were found to have secondary influence within the scope of the dataset.
Non-linear Effects: PDP–ICE plots revealed distinct stepwise behaviors and non-linear interactions, particularly a sharp drop in predicted CCS at a JC of 0.65, and a strong, nearly monotonic increase in CCS with increasing normal stress.

Significance and Contributions
The paper claims that the primary contribution of this work is the development of a practical and transparent tool for evaluating crack coalescence stress in jointed rock systems under direct shear conditions. By coupling XGBoost with metaheuristic optimization, the study successfully captures the complex, coupled effects of geometric and mechanical parameters that are difficult to isolate through laboratory testing alone.

The significance of the work lies in three main areas:

Predictive Accuracy: The hybrid XGBoost–BOA model provides a highly accurate and robust method for predicting CCS, outperforming other metaheuristic-tuned configurations.
Interpretability: Unlike "black-box" models, the integration of SHAP and PDP–ICE analyses ensures the model's predictions are physically consistent with established fracture mechanics concepts, validating the dominance of normal stress and joint geometry.
Engineering Utility: The integration of the predictive model into a GUI allows engineers to rapidly explore parametric effects and estimate CCS for design and experimental assessment, bridging the gap between advanced computational modeling and practical rock engineering applications.

The authors modestly acknowledge that the study is based on laboratory-scale specimens with specific material compositions and joint configurations. They suggest that while the database is extensive, future expansion to include broader geometries, stress paths, and material types would further strengthen the general applicability of the proposed framework.

Technical Summary: Interpretable Hybrid XGBoost-MH Models for Predicting Crack Coalescence Stress

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