⚡ electrical engineering

Limit-state geometry of trailing-edge tensile cracks in soil slopes with a tensile strength threshold

This study presents a novel limit-state formulation that simultaneously determines the depth and inclination of trailing-edge tensile cracks in cemented soil slopes by treating tensile strength and shear resistance as independent failure mechanisms, thereby offering a more comprehensive analytical framework validated through theoretical benchmarks, field data, and discrete element simulations.

Original authors: Jijie Du, Liansheng Tang, Jianing Huang, Zihua Cheng

Published 2026-11-02

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CC BY 4.0

Original authors: Jijie Du, Liansheng Tang, Jianing Huang, Zihua Cheng

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

Slopes are not static monuments; they are living systems that slowly yield to gravity. When a hillside begins to fail, it rarely happens all at once. Instead, the ground often prepares for a slide by cracking at the very top, near the crest, before the main body of earth begins to slip downward. These initial cracks, known as trailing-edge tensile cracks, are more than just surface scars. They act as warning signs and, crucially, as pathways that allow rainwater to seep deep into the soil, accelerating the collapse. For engineers and geologists, understanding exactly how deep these cracks go and at what angle they tilt is vital for predicting when a slope might give way. Traditionally, models used to predict these cracks have relied on a simplifying assumption: that the cracks are perfectly vertical, like a wall cut straight down. This makes the math easier, but it ignores the reality that soil often breaks at an angle, and that the soil's ability to resist being pulled apart is just as important as its ability to resist being pushed together.

A team of researchers has now developed a new way to calculate the shape of these cracks that removes the assumption of verticality. Instead of forcing the crack to be straight up and down, their method treats both the depth and the angle of the crack as unknowns to be solved simultaneously. They built a model for slopes made of cemented soil—earth that has natural or artificial binders giving it a bit of strength—under conditions where water can drain away freely. The model balances the forces of gravity pulling the soil down against the soil's internal strength, which comes from two sources: the friction and stickiness that hold the soil together when squeezed, and a separate threshold that determines how much pulling force the soil can take before it snaps. By solving for the point where these forces reach a critical limit, the researchers can determine the most likely geometry of the crack without guessing its angle in advance.

To test their new approach, the team compared their calculations against established theories, real-world measurements from a loess slope in northwest China, and advanced computer simulations that mimic the behavior of individual soil particles. The results showed that their method aligns well with existing theories regarding how deep cracks should be under various conditions. When they applied their model to the real loess slope, where a crack had been measured at nearly 13 meters deep, their calculations produced a range of possible depths that included the actual measurement, provided a reasonable estimate for the soil's tensile strength was used. The computer simulations, which tracked the breaking of bonds between virtual soil particles, confirmed that the model correctly identified when a crack would form and when it would not. While the exact depth predicted by the model was often deeper than what the computer simulations showed, the angle of the crack predicted by the new method matched the simulated cracks with remarkable precision, differing by only a few degrees on average.

The study also revealed how different soil properties change the shape of the crack. The researchers found that as the soil's ability to resist being pulled apart increases, the cracks become shallower and tilt more steeply, sometimes becoming so resistant that no crack forms at all. Conversely, if the soil is stickier or more cohesive, the cracks tend to go deeper. The weight of the soil itself plays a role too; heavier soil leads to shallower cracks, though it has little effect on the angle. Interestingly, the angle of the soil's internal friction does not have a simple, one-way effect on the crack's shape; its influence depends heavily on how strong the soil is against being pulled apart. Through a detailed analysis of how these factors interact, the team determined that the soil's stickiness is the most dominant factor in deciding whether a crack will form and how deep it will be, followed closely by the soil's tensile strength.

This work offers a more complete picture of how slopes fail by treating the crack's geometry as a result of physical forces rather than a fixed rule. By acknowledging that cracks can be inclined and that the soil's resistance to pulling is a distinct property from its resistance to sliding, the new model provides a more nuanced tool for assessing slope stability. It does not claim to predict every future landslide, but it offers a clearer, mechanically grounded reference for understanding the initial stages of slope failure. The findings suggest that ignoring the tensile strength of soil or assuming cracks are always vertical may lead to inaccurate assessments of risk, particularly in cemented soils where the ability to resist tension is a key part of the material's character.

Technical Summary: Limit-state geometry of trailing-edge tensile cracks in soil slopes with a tensile strength threshold

Problem Statement
Trailing-edge tensile cracks are critical precursors to tensile–shear slope failures, altering stress transfer paths and facilitating water infiltration. While existing analytical models effectively estimate crack depth, they typically prescribe the crack inclination as a vertical plane and often neglect tensile strength ( $\sigma_t$ ) or treat it indirectly through modified shear parameters. This study addresses the limitation of treating crack geometry as a fixed input rather than a coupled outcome. The authors argue that the inclination of a trailing-edge tensile crack significantly influences the geometry of the potentially detached soil mass and the subsequent stability assessment, necessitating a formulation where both depth and inclination are determined simultaneously under mechanical constraints.

Methodology
The study develops a two-dimensional limit-state formulation for an initially intact, homogeneous cemented soil slope under drained conditions. The model represents a local tensile–shear wedge bounded by a trailing-edge tensile crack (BF) and a lower shear surface (EF).

Governing Principles: Shear resistance follows the Mohr–Coulomb criterion ( $c, \phi$ ), while tensile failure is governed independently by a macroscopic tensile strength threshold ( $\sigma_t$ ).
Coupled Unknowns: Unlike previous approaches, the crack depth ( $L$ ) and inclination ( $\delta$ ) are treated as coupled unknowns. The solution must satisfy force equilibrium, moment equilibrium, tensile failure conditions, and geometric admissibility ( $0 < \theta < \delta < \pi/2$ ).
Solution Procedure: Due to the lack of a closed-form solution for the coupled angular variables, the authors employ a hierarchical constrained search algorithm. This involves discretizing the shear plane angle ( $\theta$ ), solving for admissible crack inclinations ( $\delta$ ) that satisfy moment equilibrium, and selecting the minimum positive limit-state depth ( $L$ ) as the governing solution.
Validation: The formulation is evaluated through:
1. Theoretical Benchmarks: Comparison with established vertical crack depth solutions (e.g., Michalowski, Utili, Zhu et al.).
2. Field Comparison: A bounded consistency check against a loess slope case in Northwest China.
3. Numerical Simulation: Comparison with MatDEM (Discrete Element Method) simulations where crack geometry is extracted from bond-breakage and force-chain evolution.
4. Sensitivity Analysis: Local sensitivity analysis and global Sobol sensitivity analysis to quantify parameter influence.

Key Results

Benchmark Consistency: The model reproduces established trends in crack depth regarding cohesion, unit weight, and friction angle. Crucially, it introduces an explicit dependence on tensile strength, showing that higher tensile strength reduces the admissible crack depth.
Field Comparison: For a loess slope with a measured crack depth of 12.78 m, the model predicts a range of depths (9.90 m to 21.80 m) depending on the tensile-strength ratio ( $T = \sigma_t/c$ ). The field measurement falls within this calculated range, intersecting the model curve at $T \approx 0.7\text{--}0.8$ .
DEM Validation: Comparisons with MatDEM simulations show that the analytical model preserves the distinction between crack and no-crack states. While the analytical depth ( $L$ ) tends to be larger than DEM-extracted depths (Mean Absolute Error $\approx$ 1.66 m), the inclination angles show closer correspondence (Mean Absolute Error $\approx$ 4.22°). The authors attribute depth discrepancies to the difference between a theoretical limit-state reference and the specific stage of crack development captured in DEM.
Parametric Effects:
- Tensile Strength ( $\sigma_t$ ): Increasing $\sigma_t$ reduces both $L$ and $\delta$ . At high thresholds, it can suppress the formation of a finite positive-depth solution entirely.
- Cohesion ( $c$ ): Generally increases both $L$ and $\delta$ and expands the domain where a finite solution exists.
- Unit Weight ( $\gamma$ ): Primarily reduces $L$ with negligible effect on $\delta$ .
- Friction Angle ( $\phi$ ): Its effect on $L$ is non-monotonic and dependent on $\sigma_t$ ; at high tensile strengths, increasing $\phi$ may not increase $L$ due to the restrictive tensile threshold.
Sensitivity: Local sensitivity near the baseline identifies cohesion as the most influential parameter for depth. Global Sobol analysis ranks the parameters for the zero-including depth response as: Cohesion > Tensile Strength > Friction Angle > Unit Weight. Tensile strength is particularly significant for its role in the transition between finite-depth and crack-suppression states.

Significance and Claims
The authors claim that this study provides a mechanically constrained basis for interpreting trailing-edge tensile crack geometry by reformulating the problem from a prescribed geometric condition to a set of coupled unknowns. The primary contribution is the explicit integration of tensile strength as an independent failure criterion alongside shear strength, allowing for the simultaneous determination of crack depth and inclination.

The paper maintains a modest scope, clarifying that the formulation yields an idealized local limit-state geometry rather than a time-dependent prediction of crack propagation or a complete slope-failure process. It is intended as a reference for failure interpretation, back-analysis, or subsequent stability assessments, particularly for cemented soils where tensile resistance is non-negligible. The study does not claim to replace advanced numerical simulations but rather to offer an efficient, physically interpretable analytical tool that captures the coupled influence of tensile and shear failure mechanisms.

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