Mohr-Coulomb Failure Criterion

Why do some materials crumble under pressure while others hold firm? From the stability of a mountain range to the foundation beneath a skyscraper, understanding the point of failure is a central challenge in engineering and earth sciences. Materials like soil, rock, and concrete don't simply break at a fixed stress; their strength is a complex interplay of internal friction and inherent 'stickiness'. The Mohr-Coulomb failure criterion provides an elegant and powerful framework to predict this critical threshold, offering a lens through which we can ensure the safety of our structures and comprehend the forces shaping our natural world.

This article delves into this cornerstone of mechanics, providing a comprehensive overview for students and professionals alike. In the following chapters, we will first dissect the fundamental ​​Principles and Mechanisms​​ that underpin the theory. We will explore how the concepts of cohesion and friction are unified, visualized through the genius of Mohr's circle, and modified by the crucial effect of pore pressure. Following this, we will journey through the criterion's diverse ​​Applications and Interdisciplinary Connections​​, revealing how this single model provides insights into everything from geotechnical design and energy exploration to the mechanics of climate change and even the behavior of living organisms.

To truly understand how a mountain stands, how the ground beneath our feet can suddenly give way, or how we can safely tunnel through rock, we must first ask a very simple question: what makes something break? The answer, like so many profound truths in physics, is both beautifully simple and elegantly complex. It is a story of friction, stickiness, and a brilliant geometric insight that allows us to predict the point of failure.

Imagine two piles of material: one of dry sand, the other of solid granite. What holds them together?

For the sand, the answer is ​​friction​​. If you try to slide one layer of sand over another, the grains rub against each other. If you squeeze the pile, this friction increases, making it much harder to cause a slide. The strength of the sand pile depends directly on how much it is confined or squeezed.

Now, consider the granite. It also has internal friction between its mineral grains, but it has something more: an intrinsic "stickiness" that holds it together even when it's not being squeezed. This property is called ​​cohesion​​. It’s the result of chemical bonds and interlocking crystals.

The Mohr-Coulomb criterion is, at its heart, a magnificent unification of these two ideas. It proposes that the strength of a material against sliding on any internal plane is simply the sum of its inherent cohesion and a frictional component that depends on how much that plane is being pressed together.

The French physicist Charles-Augustin de Coulomb was the first to formalize this. Imagine trying to slide a heavy book across a table. The resistance you feel is due to friction, and it's proportional to the book's weight. If you press down harder on the book, the resistance increases. Now, imagine there's a bit of weak glue under the book—that's cohesion.

Coulomb's genius was to write this down in a simple, powerful equation. He stated that the maximum shear stress, $\tau$ (the stress trying to cause a slide), that a plane can withstand is:

Here, $c$ is the ​​cohesion​​, the material's inherent shear strength, like the glue. The term $\sigma_n$ is the normal stress, which is the stress acting perpendicular to the plane, squeezing it together—like you pressing down on the book. The quantity $\tan\phi$ is the coefficient of internal friction, where $\phi$ is called the ​​angle of internal friction​​. Just as with the book, the more you squeeze (higher $\sigma_n$), the more frictional resistance you generate, and the higher the total shear strength $\tau$ becomes.

Coulomb's law is perfect for a single, known plane. But inside a solid block of rock under pressure, there are an infinite number of potential failure planes, all oriented in different directions. On which one will it fail? It will fail on the plane that is most "critically stressed"—the one where the applied shear stress comes closest to reaching the shear strength defined by Coulomb's law.

Finding this plane was a puzzle that stymied engineers until the German engineer Otto Mohr devised a tool of pure graphical genius: the ​​Mohr's circle​​.

For any state of stress at a point, which we can describe by the principal stresses (the maximum, intermediate, and minimum squeezing, denoted $\sigma_1$, $\sigma_2$, and $\sigma_3$), Mohr's circle allows us to find the normal stress ($\sigma_n$) and shear stress ($\tau$) on any plane passing through that point. Let's consider the 2D case for simplicity, looking at the plane containing the largest and smallest principal stresses, $\sigma_1$ and $\sigma_3$. The Mohr's circle is drawn on a graph with normal stress on the horizontal axis and shear stress on the vertical axis. The circle has its center at $\frac{\sigma_1 + \sigma_3}{2}$ and a radius of $\frac{\sigma_1 - \sigma_3}{2}$. Every single point on this circle's circumference represents the $(\sigma_n, \tau)$ pair for a unique plane orientation. It’s a complete map of the stress world at that point.

Now, let's combine these two powerful ideas. We can plot Coulomb's law, $\tau = c + \sigma_n \tan\phi$, on the same graph as Mohr's circle. It forms a straight line, known as the ​​failure envelope​​.

This sets up a dramatic and clear condition for failure. As long as the Mohr's circle for a stress state lies entirely below this failure envelope, the material is stable. No plane within the material has a combination of shear and normal stress that is sufficient to cause it to slide. But as the external loads increase, the principal stresses $\sigma_1$ and $\sigma_3$ change, and the Mohr's circle grows larger.

Failure occurs at the precise moment the Mohr's circle expands just enough to touch the failure envelope. This point of tangency is the moment of truth. It tells us the exact normal and shear stress on the failure plane, and its angle reveals the physical orientation of the plane on which the material will break. This beautiful geometric condition—the tangency of the circle and the line—is the essence of the Mohr-Coulomb failure criterion.

In the real world, especially in geology, rocks and soils are not always dry. They are often porous and saturated with fluids like water, oil, or, in the case of carbon sequestration, injected CO₂. This fluid exists under pressure—the ​​pore pressure​​, $p$. This pressure acts outwards in all directions, pushing the solid grains of the material apart.

Crucially, this pore pressure counteracts the external squeezing stress, $\sigma_n$, that holds the material together. Think of an air hockey table: the puck glides with almost no friction because the upward pressure of the air counteracts its weight. In the same way, high pore pressure reduces the "clamping" force between grains, thereby lowering the frictional component of the material's strength.

This fundamental insight was formalized by Karl von Terzaghi in his ​​principle of effective stress​​. He stated that a material's strength and deformation are governed not by the total stress, but by the ​​effective stress​​, $\sigma'$, which is the total stress minus the pore pressure: $\sigma'_n = \sigma_n - \alpha p$, where $\alpha$ is a material property called the Biot coefficient, typically close to 1.

The failure criterion thus becomes a condition on the effective stress:

This explains a host of critical geological phenomena. For example, injecting fluids into the ground for geothermal energy or CO₂ storage increases the local pore pressure. This can reduce the effective normal stress on a pre-existing fault to the point where the existing shear stress is enough to cause it to slip, potentially triggering earthquakes.

Moving from a 2D plane to a 3D solid, we must consider all three principal stresses, $\sigma_1 \ge \sigma_2 \ge \sigma_3$. Failure is governed by the largest of the three possible Mohr's circles. If we map out all the combinations of $(\sigma_1, \sigma_2, \sigma_3)$ that satisfy the Mohr-Coulomb criterion, what geometric shape do we get in the 3D space of principal stresses?

The result is a magnificent and somewhat troublesome object: an irregular ​​hexagonal pyramid​​. The central axis of this pyramid is the hydrostatic line, where all stresses are equal ($\sigma_1 = \sigma_2 = \sigma_3$) and there is no shear. The farther a stress state is from this axis, the more shear it involves. The surface of the pyramid represents the boundary of failure.

If we slice this pyramid with a plane of constant pressure (known as the ​​$\pi$-plane​​ or deviatoric plane), we see the source of its geometric complexity: the cross-section is an irregular hexagon. This is fundamentally different from simpler models like the von Mises criterion, which is used for metals and whose cross-section is a perfect circle.

This hexagonal shape tells us something profound: the strength of a Mohr-Coulomb material depends not just on the overall amount of shear stress, but also on the type of shear. The distance from the center of the hexagon to its corners is greater than the distance to the middle of its flat sides. The corners correspond to specific stress states, such as ​​triaxial compression​​ (squeezing a cylinder, where $\sigma_1 > \sigma_2 = \sigma_3$) and ​​triaxial extension​​ (stretching a cylinder, where $\sigma_1 = \sigma_2 > \sigma_3$). The fact that the hexagon is not a circle means the material is predicted to be stronger in triaxial compression than in triaxial extension, a feature observed in many geological materials. This dependence on the type of shear state is captured mathematically by the ​​Lode angle​​.

The sharp corners and flat edges of the Mohr-Coulomb hexagon, while physically meaningful, are a nightmare for computer simulations. In computational plasticity, the direction of plastic flow (how the material deforms once it yields) is determined by the normal (the perpendicular direction) to the yield surface. On a smooth surface like a circle, the normal is uniquely defined everywhere. But at a corner or an edge of a hexagon, what is the normal? It’s not unique; any direction between the normals of the two adjoining faces is technically valid. This non-uniqueness can cause numerical algorithms to oscillate and fail to find a solution.

To get around this, engineers often use a simplified model called the ​​Drucker-Prager criterion​​. This is essentially a smooth, circular cone that approximates the pointy Mohr-Coulomb pyramid. This model is computationally friendly because its circular cross-section has a unique normal everywhere (except at the very tip of the cone).

However, this simplification comes at a cost. By replacing the hexagon with a circle, the model loses all sensitivity to the Lode angle. It predicts the same strength in compression and extension, which is often inaccurate. There is also no single "best" way to fit the circle to the hexagon. One can circumscribe it (matching the strength at the compression corners), inscribe it, or match the area, with each choice representing a different compromise between accuracy and safety.

To formalize these models for computation, scientists often switch from the language of principal stresses ($\sigma_1, \sigma_2, \sigma_3$) to the language of ​​stress invariants​​. The two most important are the mean effective stress, $p'$, which measures the average "squeezing" pressure, and the deviatoric stress invariant, $q$, which measures the magnitude of the total shear or distortion.

In this $(p', q)$ language, the complex geometry of the Mohr-Coulomb criterion simplifies beautifully. For a specific loading path, like triaxial compression, the yield condition becomes a simple straight line:

The slope $M$ is directly related to the friction angle $\phi$, and the intercept $k$ is related to the cohesion $c$. For instance, in triaxial compression, a rigorous derivation shows that $M = \frac{6 \sin\phi}{3 - \sin\phi}$. This formulation is the bridge between the intuitive physical picture of Mohr's circles and the algebraic expressions used in modern finite element software. It also helps us clearly distinguish between a bulk material model, defined by invariants, and a simple friction law on a specific interface, defined by local tractions.

Finally, we must acknowledge one last layer of sophistication. The Mohr-Coulomb criterion tells us when a material yields. But it doesn't necessarily have to tell us how it deforms. In the simplest models ("associated plasticity"), the plastic flow is assumed to be perpendicular to the yield surface. For Mohr-Coulomb, this predicts a large increase in volume, known as ​​dilatancy​​, as the grains are forced to ride up and over each other.

For many soils, this predicted dilatancy is too large. To fix this, modelers can define a separate function, called a ​​plastic potential​​ ($g$), to govern the direction of flow. This potential often has a form similar to the yield function, but uses a ​​dilatancy angle​​, $\psi$, which is typically smaller than the friction angle $\phi$. This "non-associated" framework decouples the condition for yielding from the rule for flow, offering greater flexibility to match the complex reality of material behavior.

From a simple idea of friction and stickiness, we have journeyed through a landscape of elegant geometry, real-world complications, and computational pragmatism. The Mohr-Coulomb criterion remains a cornerstone of mechanics because it captures a fundamental truth about our physical world in a framework that is at once intuitive, versatile, and profoundly insightful.

Having journeyed through the principles and mechanisms of the Mohr-Coulomb criterion, we might be tempted to view it as a specialized tool for a narrow field of engineering. But to do so would be to miss the forest for the trees. The true beauty of a fundamental physical law is not in its complexity, but in its universality. The simple relationship between cohesion, friction, and failure is one of these profound unifiers, a secret handshake between seemingly disparate parts of our world. It governs not just the stability of a colossal dam, but the brief existence of a child's sandcastle; not only the ground beneath our cities, but the soil churned by a humble earthworm.

Let us now explore this expansive landscape, to see how this single idea provides a lens through which we can understand an astonishing variety of phenomena, bridging disciplines from civil engineering to ecology, planetary science, and even biology.

The most natural home for the Mohr-Coulomb criterion is in geotechnical engineering, the science of building on, in, or with the Earth. Here, the theory is not an academic curiosity; it is the bedrock of safety and design.

Slopes, Hills, and Sandcastles

Why does a pile of dry sand form a cone with a specific, characteristic angle? And why, with just a little water, can you build vertical walls for a sandcastle that would instantly collapse if dry? The answer to both questions is the Mohr-Coulomb criterion. For dry sand, the cohesion $c$ is virtually zero, and the slope's stability is a direct contest between gravity pulling material downslope and the internal friction angle $\phi$ holding it back. The maximum stable angle, the famous "angle of repose," is simply equal to $\phi$.

When we add a bit of water, something magical happens. Surface tension in the water menisci between sand grains creates a suction effect, pulling the grains together. This effect acts like a "glue," giving the sand an apparent cohesion. As we saw in our discussion of principles, cohesion allows a material to resist shear stress even when the normal stress is low. This newfound cohesion is what allows the moist sand to stand in vertical cuts, at least for a while. Of course, too much water saturates the sand, eliminates the suction, and turns the castle back into a soupy pile.

This simple beach-day observation scales up to the level of entire landscapes. The stability of natural hillsides and riverbanks is governed by the same interplay of friction, cohesion, and the weight of the material. In ecology and environmental engineering, we find a beautiful interdisciplinary connection: the roots of plants and trees act as a natural reinforcement for the soil. This "root cohesion" can be quantified and added directly to the soil's own cohesion in the Mohr-Coulomb equation. A vegetated riverbank is stronger and more resistant to erosion and failure precisely because the living root network provides this extra shear strength, a wonderful example of nature's own engineering.

When we want to analyze the stability of a complex slope, perhaps for a highway cutting or a dam, modern engineers turn to powerful computational tools like the Finite Element Method (FEM). In a technique called the Shear Strength Reduction Method, engineers create a virtual model of the slope and systematically reduce the soil's cohesion $c$ and friction $\tan\phi$ by a common factor, $F_s$, until the computer model "collapses." The value of $F_s$ at which failure occurs is the factor of safety of the real slope. It's a testament to the power of the criterion that we can use it to "break" a virtual mountain in a computer to ensure the real one never does.

Holding Back the Earth

Whenever we dig a hole or build a structure to hold back soil, like a basement wall or a highway retaining wall, we are fighting a battle against the forces described by the Mohr-Coulomb criterion. The soil behind the wall wants to fail and move, exerting a powerful lateral pressure. Rankine's theory, derived directly from the Mohr-Coulomb condition, allows us to calculate this "active earth pressure" that the wall must be designed to resist.

A fascinating phenomenon occurs when the soil has cohesion, such as in clay. The cohesion provides an inherent "tensile" strength (in a manner of speaking) to the soil mass. Near the top of the retaining wall, the pressure from the soil's weight is low, and the cohesive strength can actually be greater than the push from the soil. This results in a theoretical region of negative pressure, or tension, between the soil and the wall. Since soil can't physically pull on a wall, what happens in reality is that a "tension crack" forms, a gap between the soil and the structure. If you've ever seen a vertical crack in the ground running parallel to the edge of a deep excavation in clayey soil, you have witnessed a life-sized manifestation of the cohesive term in the Mohr-Coulomb equation. For a wall shorter than this tension crack depth, the soil could theoretically stand on its own, exerting no pressure on the wall at all!

The same principles apply in reverse when we consider the ground supporting a structure. The ultimate bearing capacity of a foundation—the maximum pressure it can exert on the ground before the soil fails in shear—is calculated using the Mohr-Coulomb parameters. A crucial interdisciplinary twist, connecting to fluid mechanics and poro-mechanics, is the role of time and water. In a low-permeability clay, a rapidly applied load (short-term) doesn't give the water in the soil's pores time to escape. The incompressible water carries much of the load, generating high pore pressure and changing the failure mechanics. The analysis in this "undrained" condition uses a total stress approach with a single parameter, the undrained shear strength $s_u$. If the load is applied slowly, or after a long time (long-term), the excess pore pressure dissipates, and the soil skeleton carries the full load. This "drained" condition is analyzed using the familiar effective stress parameters $c'$ and $\phi'$. A soil's strength is not a single number; it depends on how quickly you ask it to respond.

The power of the Mohr-Coulomb model extends far beyond traditional civil engineering, appearing in any domain where granular or brittle materials are put under stress.

Energy, Resources, and a Trip to the Underworld

In the quest for geothermal energy or hydrocarbon resources, engineers drill boreholes thousands of meters into the Earth's crust. Down there, the rock is under immense stress from the weight of the overlying formations. Drilling a hole is a violent act that drastically alters this stress field. The rock at the wall of the borehole wants to fail and collapse inwards. To prevent this, a drilling fluid, or "mud," is used to exert an outward pressure on the borehole wall.

Wellbore stability analysis is a high-stakes application of the Mohr-Coulomb criterion. The hoop stress (tangential stress) around the borehole is typically compressive and is greatest at the points aligned with the minimum far-field horizontal stress. If this hoop stress exceeds the rock's strength, defined by its cohesion and friction, the borehole wall will crush and spall off, a phenomenon known as a "breakout." The analysis is a beautiful synthesis of mechanics, incorporating the far-field earth stresses, the mud pressure, the rock's own pore pressure, and even thermal stresses caused by the drilling fluid being colder than the rock formation. That cooling can cause failure is a wonderful piece of physics: the cold mud makes the rock at the borehole wall want to shrink, but it is constrained by the warmer rock around it. This generates a tensile thermal stress, which effectively reduces the rock's compressive strength and brings it closer to failure.

A Changing Planet: The Mechanics of Permafrost

Vast regions of our planet's high latitudes are underlain by permafrost—soil that has remained frozen for two or more consecutive years. The strength of this frozen ground is not just due to the soil particles themselves. The ice that fills the pores and binds the particles together acts as a powerful cohesive agent. This "ice cohesion" is a dominant component of the soil's strength and can be modeled within the Mohr-Coulomb framework as a temperature-dependent cohesion, $c(T)$, which increases dramatically as the temperature drops further below freezing.

Here, the Mohr-Coulomb criterion connects directly to climate science. As global temperatures rise, permafrost begins to thaw. This thawing is a mechanical catastrophe: the ice cohesion is lost. The ground loses its strength, leading to massive ground subsidence, landslides, and the catastrophic failure of buildings, pipelines, and roads built upon it. Understanding the mechanics of frozen soil through this lens is critical for predicting the impacts of climate change and designing resilient infrastructure in the Arctic.

The Living World: How to Dig a Burrow

Perhaps the most surprising application takes us into the realm of biology. Consider an annelid—an earthworm—tunneling through the ground. It has no claws, no teeth, no shovel. How does it do it? The worm is a master of hydrostatic pressure. By contracting its muscles, it pressurizes the coelomic fluid within its body segments, pushing outward on the surrounding soil. To extend its burrow, the pressure it generates must be sufficient to cause the soil at the burrow wall to fail in shear and deform plastically.

The minimum pressure the worm must generate is nothing other than the unconfined compressive strength of the soil, a value determined directly from the soil's Mohr-Coulomb parameters, $c$ and $\phi$. In essence, the worm is solving a geotechnical engineering problem with its own body. This field, known as bioturbation, shows how the physical constraints of the environment shape the evolution of biological form and function. Life does not exist in a vacuum; it must constantly negotiate the physical laws of its medium, and for organisms that live in soil, that law is often Mohr-Coulomb.

Throughout this tour, we have spoken of cohesion $c$ and friction angle $\phi$ as if they were known, god-given constants for a given material. But they are not. They are parameters in a model, and their values must be inferred from noisy, imperfect experimental data. This brings us to our final, and perhaps most profound, interdisciplinary connection: to the world of statistics and data science.

How do we determine $c'$ and $\phi'$ from a set of triaxial tests in the lab? Traditionally, one might plot the failure points and draw a "best-fit" line. But what is the "best" fit? And how certain are we of the resulting parameters?

Modern approaches frame this as a Bayesian inference problem. We start with a prior belief about the parameters (perhaps from experience with similar soils). We then perform experiments, collecting data like peak stress values. The Mohr-Coulomb model, combined with a model for measurement error, allows us to define a likelihood: the probability of observing our data given a particular set of parameters. Bayes' theorem then provides a mathematically rigorous way to combine our prior beliefs with the likelihood from our new data to arrive at a posterior probability distribution for the parameters. This posterior is the complete expression of our updated knowledge: it gives us the most likely values for $c'$ and $\phi'$, but also quantifies our uncertainty in them. This is science in action—a continuous cycle of proposing models, testing them against reality, and formally updating our state of knowledge.

From the sandcastle to the stars (or at least, the deep earth), the Mohr-Coulomb criterion is more than just a formula. It is a testament to the unifying power of physical principles, revealing the hidden mechanical logic that connects the inanimate and the living, the engineered and the natural, the known and the uncertain. It is a simple idea, but one that has helped to shape our understanding of the world.