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Multiscale Calculation of Equivalent Elastic Modulus for Rock Mass Based on Perturbation Theory

This paper proposes a multiscale calculation approach based on perturbation theory that incorporates initial damage and meso-heterogeneity to accurately determine the equivalent elastic modulus of rock masses, demonstrating superior accuracy compared to conventional methods through validation with laboratory and field test results.

Original authors: Lei Wen, Wu Yang, Huayun Yang

Published 2026-07-24

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CC BY 4.0

Original authors: Lei Wen, Wu Yang, Huayun Yang

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 trying to predict how a giant, crumbling castle made of stone will hold up under pressure. But here's the twist: the castle isn't a solid block of marble. It's a messy pile of rocks, some smooth, some jagged, held together by patches of cement, with hidden cracks running through them like spiderwebs. This is the real world of "rock mass" engineering. When engineers build tunnels, mines, or dams, they can't just test a perfect, tiny cube of rock in a lab and assume the whole mountain will behave the same way. That's like testing a single brick to see if a whole skyscraper will survive an earthquake. The problem is that rocks are messy, full of tiny flaws and cracks that change how they bend and break. Scientists have long tried to bridge the gap between the tiny world of microscopic cracks and the huge world of massive rock formations, but their old maps often led to wrong turns. They needed a new way to look at the problem that could see both the forest and the trees at the same time.

This paper, titled "Multiscale Calculation of Equivalent Elastic Modulus for Rock Mass Based on Perturbation Theory," is like a new, super-smart pair of glasses that lets engineers see exactly how a bumpy, cracked rock will squish and stretch. The authors, Lei Wen, Wu Yang, and Huayun Yang, developed a clever math trick called "multiscale calculation" that treats the rock's initial damage (those tiny cracks that are already there) as a starting point. Instead of pretending the rock is perfect, they use a method called "perturbation theory"—which is basically a fancy way of saying "let's nudge a perfect system slightly to see how it wobbles"—to figure out how the tiny cracks mess up the rock's stiffness. They call the strain (the squishing) caused by these tiny cracks the "bridge" that connects the microscopic world to the macroscopic world.

The team didn't just sit at a computer; they went into a deep metal mine in southern China to grab real rocks. They deliberately damaged these rocks by squeezing them with different amounts of force (specifically at 0.25, 0.4, 0.5, 0.65, and 0.75 times the strength of a perfect rock) to create "initial damage." Then, they put these damaged rocks through two tests: a "Brazilian split" (where they squeeze a rock from the sides until it splits like an orange) and a standard "uniaxial compression" (squishing it from top to bottom). They also used sound waves (P-wave velocity) and CT scans to see the invisible cracks inside the rocks.

What they found is that their new math method is much better at predicting reality than the old, standard computer simulations. When they compared their results to the actual lab tests, their "multiscale" approach drew curves that matched the real rocks almost perfectly. The old methods were okay at guessing the peak strength (how hard you can push before it breaks), but they got the shape of the curve wrong. The new method showed that the tensile load-displacement curves (from the splitting test) bend in a specific, "strictly lower convex" way that matches real life, whereas the old simulations missed this nuance. In contrast, the compression stress-strain curves (from the squishing test) follow a different "convex" shape. They even took their method to a real-world construction site where a lead-zinc mine had collapsed. They used it to calculate the strength of "grouting solidified bodies"—basically, broken rock that had been glued back together with cement slurry. The math predicted the behavior of this glued-up mess accurately before it reached its breaking point.

However, the authors are careful to note that their method isn't magic. While it nailed the behavior before the rock broke, it struggled to predict exactly what happened after the peak stress, when the rock started to crumble and soften. This is because their model doesn't yet account for the messy "strain-softening" that happens when rock fails completely. But for the most important part—figuring out how stiff and strong a damaged rock mass is before it gives way—their new "bridge" between the micro and macro worlds works better than the conventional tools. It suggests that by acknowledging the rock's initial scars and using this perturbation math, engineers can get a much clearer picture of whether a rock mass will hold up or collapse, making our tunnels and mines safer.

Technical Summary: Multiscale Calculation of Equivalent Elastic Modulus for Rock Mass Based on Perturbation Theory

Problem Statement
The accurate determination of the equivalent elastic modulus is critical for the safety and stability assessment of rock mass engineering. Current methodologies face significant limitations: laboratory tests on intact rock specimens fail to account for the macro and micro-joints and cracks inherent in engineering rock masses, while field tests are often unrepresentative due to their limited spatial scope. Furthermore, traditional reduction methods used to scale laboratory results to field conditions rely on unreliable reduction factors and cannot capture the mechanical response process during rock mass failure. There is a recognized need for a systematic, multiscale coupling approach that bridges the gap between intact rock mechanics and the heterogeneous behavior of damaged rock masses.

Methodology
The authors propose a multiscale calculation approach grounded in perturbation theory, specifically designed to account for initial damage in rock masses. The methodology treats the rock mass with initial damage as a perturbation system where a small parameter ( $\delta$ ) reflects the multiscale expansion process of internal cracks.

Key methodological steps include:

Theoretical Framework: The strain tensor caused by meso-heterogeneity is utilized as a "bridge" connecting macro and micro scales. The displacement field is subjected to asymptotic expansion, separating variables into macroscopic coordinates ( $x$ ) and microscopic neighborhoods ( $y$ ).
Governing Equations: The study derives governing equations by coupling the equilibrium differential equation with a constitutive equation that incorporates a micro-elastic damage factor ( $D^\delta$ ). The initial damage is represented by an initial elastic damage tensor.
Homogenization: By solving the motion equations under different physical dimension scales, the authors derive a homogenized macro equivalent elasticity modulus ( $E_{mnkl}^\delta$ ). This involves solving for tensor eigenfunctions ( $\chi$ ) using the finite element method based on variational principles.
Numerical Implementation: The approach was implemented using MATLAB for iterative calculations of the damage coefficient and COMSOL for finite element modeling. The model updates load ( $Q$ ) and displacement ( $u$ ) at each step to reflect the evolving damage state.
Validation: The approach was validated through two primary cases:
1. Laboratory Validation: Uniaxial compression and Brazil split tensile tests were conducted on limestone specimens collected from a deep metal mine in southern China. Specimens were pre-damaged to varying degrees (0.25 $\sigma_{cs}$ to 0.75 $\sigma_{cs}$ ) and characterized via P-wave velocity and CT scanning.
2. Field Validation: The method was applied to grouting solidified bodies in a lead–zinc mine stope, where the grouted material was treated as a rock mass with irregular joint fissures.

Key Contributions

Multiscale Perturbation Model: The paper establishes a theoretical framework where the strain tensor arising from meso-heterogeneity serves as the connecting mechanism between micro and macro scales, allowing for the calculation of equivalent elastic modulus considering initial damage.
Dynamic Damage Evolution: Unlike conventional models that often treat elastic modulus as a constant, this approach reflects the dynamic development of the equivalent elastic modulus and the evolution of both macro and micro damage during the loading process.
Improved Accuracy: The study demonstrates that the proposed multiscale calculation yields results closer to actual laboratory and field test data compared to conventional finite element simulation methods.

Results

Laboratory Findings:
- P-wave velocity ( $V_p$ ) was found to decrease as initial damage increased, confirming its utility as a damage indicator.
- CT scans revealed that higher initial damage correlated with increased porosity and crack density.
- Brazil Split Tensile Test: The load–displacement curves exhibited a strictly lower convex function. The multiscale calculation results aligned more closely with experimental curves than conventional simulations, particularly in the pre-peak region.
- Uniaxial Compression Test: Stress–strain curves were monotonically increasing concave functions. The multiscale approach accurately predicted the equivalent elastic modulus, with values closer to test results than conventional simulations. Damage evolution showed a gradual increase initially, followed by rapid development and stabilization.
Field Findings:
- In the field compression test of grouting solidified bodies, the multiscale approach accurately predicted the stress–strain curve up to the peak uniaxial compressive stress.
- The calculated equivalent elastic modulus for the grouting solidified bodies matched field test data, validating the method's applicability to complex engineering scenarios involving broken ore and cement slurry aggregates.
Limitations Observed: The study noted that while the pre-peak results were accurate, the post-peak stress–strain curves for the grouting solidified bodies diverged from field results. This discrepancy is attributed to the model's lack of a strain-softening parameter, which is necessary to capture the post-failure behavior of materials that differ from deep brittle rock.

Significance and Claims
The paper claims that the proposed multiscale calculation approach provides a robust theoretical method for solving complex problems in rock engineering. By considering initial damage and utilizing perturbation theory, the method successfully bridges the gap between microstructural heterogeneity and macroscopic mechanical behavior. The authors assert that this approach offers a more accurate alternative to conventional simulation methods for determining the equivalent elastic modulus of rock masses. They suggest that if a suitable bridge between macro and micro scales can be identified, this methodology could be extended to determine mechanical parameters for other heterogeneous materials. The work is presented as a reference for improving the assessment of rock mass stability in engineering projects, particularly where initial damage and scale effects are critical factors.

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