Mohr-Coulomb criterion

From the ground we build on to the mountains that shape our landscapes, a simple question lies at the foundation of civil engineering and earth sciences: when will this material break? While some materials have well-understood behaviors, frictional materials like soil, rock, and concrete present a unique challenge. Their strength is not fixed but depends intricately on the pressure squeezing them together. This article delves into the Mohr-Coulomb criterion, the elegant and powerful model that first captured this fundamental behavior. It addresses the central problem of predicting failure in these ubiquitous materials by defining strength as a simple interplay between 'glue' (cohesion) and 'grip' (friction). This exploration is divided into two parts. In the first chapter, 'Principles and Mechanisms,' we will dissect the core equation, uncover the genius of effective stress in fluid-filled materials, and visualize the criterion's majestic 3D geometry. Following this, the 'Applications and Interdisciplinary Connections' chapter will reveal the astonishing reach of this single idea, showing how it governs phenomena from dam stability and induced earthquakes to the evolution of burrowing animals and the precision of 3D printing.

Imagine trying to slide a heavy book across a tabletop. The heavier the book, the harder you have to push sideways to get it to move. This resistance, which increases with the force pressing the surfaces together, is what we call ​​friction​​. Now, what if the book was slightly stuck to the table with a bit of weak glue? Even before you press down on it, you’d need to apply a certain initial sideways force just to break that bond. This intrinsic, pressure-independent stickiness is what we can call ​​cohesion​​.

This simple thought experiment contains the entire soul of the Mohr-Coulomb criterion. It’s a beautifully simple model that describes when a vast range of materials—from the soil under your feet and the concrete in our buildings to the rock deep within the Earth’s crust—will fail. Instead of a book and a table, we speak of stresses. The "sideways push" is the ​​shear stress​​, denoted by $\tau$, which tries to slide one layer of material past another. The "downward press" is the ​​normal stress​​, $\sigma_n$, which squeezes the material together.

The great insight of Charles-Augustin de Coulomb and later refined by Christian Otto Mohr was to propose that the total shear stress a material can withstand before failing ($\tau_f$) is a simple linear combination of its "glue" and its "friction":

This is the celebrated ​​Mohr-Coulomb failure criterion​​. Let’s look at the two star players here:

• ​​Cohesion ($c$)​​: This is the material's inherent shear strength when there's no normal stress at all ($\sigma_n = 0$). It's the "y-intercept" of our failure line. For a pile of dry sand, the cohesion is essentially zero—there's no glue holding the grains together. But for a cemented sandstone or a sticky clay, the mineral cement or electrochemical forces provide a significant intrinsic bond, giving them a positive cohesion.

• ​​Angle of Internal Friction ($\phi$)​​: This parameter governs how much additional shear strength the material gains as you squeeze it. The term $\tan\phi$ is effectively the coefficient of friction. Why an angle? Because it represents the slope of the failure line in the $\tau$-$\sigma_n$ plane. A higher $\phi$ means a steeper line, indicating that the material's strength is highly sensitive to confining pressure. This friction arises from the microscopic reality of grains grinding and sliding past one another, a process that is often accompanied by ​​dilatancy​​—the tendency of a dense granular material to expand in volume as it shears.

This simple linear relationship is the bedrock of our understanding, a first principle from which a remarkably rich and complex picture of material failure emerges.

Our simple model works beautifully for dry materials. But what happens when the pores and cracks within a material are filled with a fluid, like water in soil? This is where Karl von Terzaghi, the father of soil mechanics, had a moment of profound genius. He realized that it's not the total normal stress that governs a material's strength, but the ​​effective stress​​—the stress actually carried by the solid skeleton of the material.

Imagine the grains of soil as being squeezed together. If there is water in the pores between them, and that water is under pressure, it pushes the grains apart. This pore water pressure, $u$, counteracts the total normal stress, $\sigma_n$. The effective normal stress, $\sigma_n'$, is therefore:

The material's strength responds only to $\sigma_n'$. This single idea explains a host of geological phenomena, from landslides triggered by heavy rainfall (which increases pore pressure and reduces effective stress) to the dangers of quicksand.

The story gets even more fascinating when we consider unsaturated soils, which contain both air and water in their pores. Here, the water doesn't just push; it can also pull. Due to ​​capillarity​​ (the same effect that makes water climb up a thin straw), the water menisci between soil grains can create a negative pressure, or ​​suction​​. This suction, also known as matric suction ($s = u_a - u_w$, where $u_a$ is air pressure and $u_w$ is water pressure), pulls the soil grains together, effectively increasing the normal stress between them. The result? The soil becomes stronger. This is why a sandcastle holds its shape when it's damp, but collapses when it's completely dry or completely saturated.

This effect can be captured by extending the effective stress principle, for instance with Bishop's effective stress concept. By incorporating a parameter, $\chi$, that depends on how saturated the soil is, the effective stress becomes $\sigma_n' = (\sigma_n - u_a) + \chi(u_a - u_w)$. When we plug this into our Mohr-Coulomb equation, we find that the suction contributes a term that adds directly to the cohesion. This new, larger intercept is called the ​​apparent cohesion​​. A soil that has very little "true" cohesion when saturated can exhibit enormous strength when partially saturated due to this suction effect. This showcases the power and flexibility of the core Mohr-Coulomb idea: the fundamental physics of friction and cohesion remains, but we must be clever about identifying the true stress that governs it.

The simple failure line is a 2D picture. But a real material lives and is stressed in three dimensions. What does the Mohr-Coulomb criterion look like in the full 3D space of principal stresses ($\sigma_1, \sigma_2, \sigma_3$)? The answer is not a line, but a surface: a ​​yield surface​​. For any stress state inside this surface, the material behaves elastically. If the stress state reaches the surface, the material yields.

Let’s compare this to other famous yield criteria. For many metals, which are largely unaffected by hydrostatic pressure, the yield surfaces (like those of Tresca or von Mises) are infinite cylinders in principal stress space. You can squeeze a piece of steel from all sides, increasing its pressure to immense values, and it won't yield. Its yielding depends only on the differences in stress, not the average pressure.

The Mohr-Coulomb criterion, however, is fundamentally different. Because its strength depends on friction, it is inherently ​​pressure-dependent​​. In 3D stress space, its yield surface is not a cylinder but an infinitely long ​​pyramid​​ with its axis aligned with the hydrostatic line ($\sigma_1=\sigma_2=\sigma_3$). The pyramid gets wider as the compressive pressure increases, graphically showing that the material becomes stronger under confinement.

The cross-section of this pyramid, viewed in the "deviatoric plane" (a slice perpendicular to the hydrostatic axis), is an ​​irregular hexagon​​. Why a hexagon and not a smooth circle? The answer lies in the criterion's core postulate: failure is governed by the most critical plane, which corresponds to the largest of the three possible Mohr's circles. The six straight sides of the hexagon represent the six possible orderings of the principal stresses ($\sigma_1, \sigma_2, \sigma_3$). Each side corresponds to a regime where a specific pair of principal stresses (e.g., $\sigma_1$ and $\sigma_3$) defines the largest Mohr's circle and thus dictates failure. The corners of the hexagon are special states, like ​​triaxial compression​​ ($\sigma_2 = \sigma_3$) or ​​triaxial extension​​ ($\sigma_1 = \sigma_2$), where the system is on the cusp of switching which plane is the most critical. This hexagonal shape, a direct consequence of the physics, reveals that the intermediate principal stress, $\sigma_2$, plays a crucial role in determining failure—a feature ignored by simpler, circular criteria.

This pyramid also has a distinct orientation. Its apex points toward the tensile region of stress space. This tells us that materials like soil, rock, and concrete are profoundly asymmetric: they are much, much stronger in compression than they are in tension. You can build a mountain out of rock, but you can't hang a rope made of it.

Staring at a 3D hexagonal pyramid can be daunting. To make sense of it, engineers and scientists often slice it open and look at a 2D cross-section called the ​​meridian plane​​. In this view, we plot a measure of distortional or "shear-like" stress, typically the invariant $q$, against a measure of average pressure, $p$.

In this specially chosen plane, the complex geometry of the pyramid collapses into something wonderfully simple: the boundaries of the yield surface become straight lines again!. The slope of these lines is determined solely by the friction angle $\phi$, while their intercept is a function of the cohesion $c$. This provides an incredibly powerful tool. Experimentalists can perform tests, like the triaxial compression tests on rock or soil samples, measure the principal stresses at failure, calculate the corresponding ($p, q$) points, and plot them. The data points will fall on a straight line, and from its slope and intercept, they can directly determine the fundamental material properties, $c$ and $\phi$. This is how we take the abstract theory and connect it directly to tangible, measurable properties of the world around us.

We end our journey with a final, beautiful insight. What happens to our theory in the limit where friction disappears? Let's consider a material where the friction angle $\phi$ is zero.

Physically, this means the material's strength no longer depends on the confining pressure. Sliding the book on the table now takes the same force regardless of how hard you press down. Mathematically, setting $\phi=0$ makes the term $\tan\phi$ zero. The Mohr-Coulomb criterion simplifies to:

The failure line in the $\tau$-$\sigma_n$ plane becomes horizontal. The mighty hexagonal pyramid in 3D stress space becomes a hexagonal ​​prism​​. The criterion becomes pressure-insensitive.

And in this form, it is none other than the ​​Tresca yield criterion​​, one of the classical models used to describe the yielding of metals! This is a profound unification. A theory developed to understand the behavior of soil and rock contains, as a special case, a cornerstone theory of metal plasticity. It reveals that the crucial feature separating these two great classes of materials is, at its heart, the presence or absence of internal friction. In discovering the simple rule of friction and glue, we find a principle that not only explains the stability of the ground we stand on but also connects it to the behavior of the materials we use to build our industrial world. The apparent complexity of nature often hides an underlying, unifying simplicity.

In the previous chapter, we dissected the elegant rule that governs when a pile of sand, a mountain slope, or a block of concrete will hold its shape or give way to flow. That rule, the Mohr-Coulomb criterion, given by the simple relation $\tau_f = c' + \sigma'_n \tan\phi'$, is far more than a dusty equation in an engineering handbook. It is a key that unlocks a staggering variety of phenomena, a common thread weaving through disciplines that might otherwise seem worlds apart. Now that we understand the 'what' and 'how' of this principle, we shall embark on a journey to discover the 'so what?'—to witness its power in action, from the grand scale of civil engineering and planetary geology to the intimate mechanics of a worm burrowing in the mud and the high-tech world of 3D printing. Prepare to be surprised by the unity of it all.

Perhaps the most direct and crucial application of the Mohr-Coulomb criterion lies in geomechanics, the science of making sure the ground beneath our feet stays put. When engineers design a dam, carve a tunnel, or lay the foundation for a skyscraper, they are locked in a constant negotiation with this rule.

Consider the immense earthen dams that hold back vast reservoirs. You might naturally think that having a full reservoir pushing against the dam is its most dangerous state. But what happens when the water level is dropped quickly? Let's say, for emergency repairs or after a flood. The water that has seeped into the soil of the dam's slope can't drain out immediately. This trapped water continues to exert an outward pore pressure, effectively trying to push the soil grains apart, while the stabilizing counter-pressure from the reservoir water is suddenly gone. As we know from the principle of effective stress, this reduces the effective normal stress $\sigma'_n$ holding the grains together, which in turn drastically lowers the soil's shear strength. In this precarious moment, the dam slope is much more likely to fail and slump, all because the water was taken away. It is this kind of counter-intuitive, yet vital, insight that the Mohr-Coulomb criterion provides, allowing engineers to design for the worst-case scenario.

The same principles allow us to ask fundamental questions about what is possible. How high can we make a vertical cut into a bank of clay before it collapses under its own weight? Using the tools of plasticity theory, which are built upon yield criteria like Mohr-Coulomb's, engineers can construct simple, theoretical stress fields. If they can find even one such field that satisfies equilibrium and doesn't violate the yield condition, they can prove, with mathematical certainty, a safe height for the excavation. This "lower bound" approach gives a guaranteed-safe design—a testament to the power of theoretical mechanics in practical safety engineering.

The criterion does not just describe static stability; it also governs the earth in motion. In recent years, you may have heard reports of earthquakes occurring in regions not typically known for them, often linked to human activities like geothermal energy extraction or the disposal of wastewater from oil and gas operations. This phenomenon, known as induced seismicity, is a direct and unsettling demonstration of Mohr-Coulomb at a geological scale.

Deep underground, ancient faults exist in a state of equilibrium. The immense compressive stress from the surrounding rock, $\sigma_n$, clamps the fault shut, and the friction between the rock faces prevents slip. But these faults are not dry; their pores are filled with water at some background pressure. When we inject more fluid into the ground, we dramatically increase this pore pressure. This pressure pushes back against the clamping stress, reducing the effective normal stress $\sigma'_n$ on the fault plane. The shear strength, which depends directly on $\sigma'_n$, plummets. Suddenly, the existing shear stress, which the fault had been resisting for millennia, is enough to overcome the weakened friction, and the fault slips—an earthquake is born. We are not creating new stresses, but merely reducing the strength that was holding ancient stresses in check.

A similar story of a sudden loss of strength plays out on snow-covered mountains. A deep layer of compacted snow can be stable for months, its strength described by its own cohesion and internal friction. But imagine a warm spell melts the surface, creating a layer of wet, heavy slush. This slush flows like a very slow, thick fluid, and as it creeps downslope, it exerts a shear stress on the stable snowpack below. At a certain point, the shear drag from the flowing slush, combined with the normal weight of the slush, becomes just enough to meet the Mohr-Coulomb failure condition of the underlying snow layer. The solid base gives way, and a catastrophic avalanche is triggered. This example wonderfully illustrates the interplay between a material behaving like a viscous fluid and one behaving like a frictional solid, with the Mohr-Coulomb criterion acting as the judge that determines the moment of failure. In more extreme cases, for granular materials flowing rapidly, the material can be thought of as being constantly at the point of yielding, allowing us to model its complex motion using the same fundamental principles.

Here, our journey takes a surprising turn. The same law that governs mountainslides and dam failures also dictates the shape and form of the living world in its interaction with the ground. Physics, it turns out, is a harsh master for biology, and a successful organism is one that has evolved a way to work with its laws.

Walk along a riverbank. You will see that the banks lined with trees, shrubs, and grasses are much steeper and more stable than those of bare soil. This is not a coincidence. The intricate network of plant roots acts like a natural form of rebar, weaving through the soil matrix. This network provides a "root cohesion," $c_r$, that adds directly to the soil's inherent cohesion, $c'$. The total shear strength becomes $\tau_f = (c' + c_r) + \sigma'_n \tan\phi'$. This extra biological contribution to strength can be the deciding factor that allows a bank to withstand the erosive forces of flowing water and the destabilizing effects of saturation, keeping an entire ecosystem intact.

Now, for one of the most beautiful connections of all: consider an earthworm tunneling through the soil. To us, it's just a worm digging. To a physicist, it is a soft-bodied hydraulic machine performing a controlled act of mechanical failure. To expand its burrow, the worm must generate an internal pressure, $p$, that pushes radially outwards on the burrow wall. This pressure must be great enough to overcome the sediment's inherent strength. What is that strength? For a cylindrical burrow being pushed open, the resistance is provided by the material's unconfined compressive strength—which, under the Mohr-Coulomb model, can be shown to depend only on its cohesion $c$ and friction angle $\phi$. An animal's ability to burrow is therefore not a matter of pure will, but a question of whether its physiology can generate the pressure required by the laws of soil mechanics.

This simple idea has profound implications for the history of life. The "Cambrian Explosion," some 540 million years ago, saw the rapid diversification of animal body plans. This was concurrent with the "agronomic revolution," when animals first began to burrow deeply into the seafloor, churning sediments on a massive scale. The ability to do so was a powerful evolutionary advantage. But this ability was constrained by physics. Animals burrowing quickly in fine-grained mud found an "undrained" material whose strength increased with depth; animals burrowing slowly in sand found a "drained" material with different mechanical rules. The evolution of strategies to cope with these constraints—such as secreting a mucus lining that acts as an additional cohesive glue—was likely a critical innovation. In this way, the Mohr-Coulomb criterion helped shape the ecological landscape of the ancient oceans, acting as an evolutionary filter that favored certain body plans and bio-mechanical strategies over others, with consequences we still see today in the fossil record.

Lest we think this principle is confined to the great outdoors or ancient history, it shows up in our everyday lives and at the forefront of technology.

Anyone who has built a sandcastle at the beach knows that dry sand is useless, and soaking wet sand is a soupy mess. The magic happens with just a little bit of water. The tiny menisci of water held in the pores between sand grains create a surface tension effect, pulling the grains together. This effect acts just like cohesion. We can model this "apparent cohesion" as a parameter that first increases with water content and then decreases as the pores become flooded. The Mohr-Coulomb criterion, armed with a moisture-dependent cohesion term, can then perfectly predict the Angle of Repose—the steepest stable slope—of our sand pile, explaining why a pinch of water lets us build magnificent towers.

Finally, let us leap to the cutting-edge of manufacturing. In powder bed additive manufacturing (a form of 3D printing), a thin layer of fine metal or ceramic powder is spread by a recoater blade, and a laser then selectively melts a pattern. The quality of the final part depends critically on the uniformity of the powder bed. But what happens if an oversized powder particle gets in the way? As the blade pushes this particle through the powder bed, it's mechanically identical to pushing a foundation into soil. The particle plastically deforms the powder around it, creating a wake of lower-density material—a "comet tail" defect. The pressure needed to initiate this plastic flow, and thus create the defect, can be calculated precisely using a model based on the Mohr-Coulomb criterion, applied to the tiny world of the powder bed. Understanding this helps engineers design better processes to create flawless parts for aerospace, medicine, and beyond.

From the stability of a dam to the quaking of the earth, from the engineering of a riverbank by tree roots to the evolutionary pressures on a Cambrian worm, from the construction of a child's sandcastle to the precision of a 3D printer—the Mohr-Coulomb criterion emerges again and again. It is a stunning example of the power of a single, simple physical law to describe, predict, and unify a vast and complex world. It reminds us that the fundamental rules of nature are written not just in the stars, but in every grain of sand.