Shear Thickening and Dilatancy: From Oobleck to Earthquakes

Some fluids defy common sense. Instead of getting thinner when stirred, they thicken, resisting motion with increasing force. This counter-intuitive behavior, known as ​​shear thickening​​, is most famously demonstrated by a simple mix of cornstarch and water, which can feel like a liquid one moment and a solid the next. While seemingly a novelty, this phenomenon points to a profound principle in physics: the collective behavior of crowded particles under stress. This article addresses the knowledge gap between the popular understanding of shear thickening and its deep, unifying role across various scientific and engineering disciplines. It explores how the same fundamental mechanism of particles jamming and needing more space to move governs not only strange fluids but also the stability of the ground we stand on and the failure of advanced materials.

The following chapters will guide you through this fascinating landscape. First, in ​​"Principles and Mechanisms"​​, we will delve into the microscopic origins of shear thickening and its solid-state cousin, dilatancy, introducing the sophisticated mathematical language of plasticity theory used to describe them. Then, in ​​"Applications and Interdisciplinary Connections"​​, we will reveal how these concepts are harnessed in engineering design, from shock absorbers to pipelines, and how they provide a common thread linking the behavior of soils, rocks, porous metals, and even polymers.

Imagine a swimming pool filled not with water, but with a mixture of cornstarch and water, colloquially known as "oobleck". If you were to gently wade into this pool, your feet would sink in as if it were a thick liquid. But what if you tried to run across its surface? An amazing thing happens: the fluid seems to instantly turn solid under your feet, supporting your weight. The harder you strike it, the more it resists. This remarkable behavior is called ​​shear thickening​​. It's a property of certain fluids—known as ​​non-Newtonian fluids​​—whose viscosity is not constant. Instead, their apparent viscosity increases as the rate of shear or strain increases.

While running on oobleck is a dramatic demonstration, we can see the consequences of shear thickening in more controlled settings. Consider a fluid being pushed by pressure through a long, straight pipe. A normal, or ​​Newtonian​​, fluid like water develops a smooth, parabolic velocity profile, flowing fastest at the center and slowest at the walls. But a shear-thickening fluid behaves differently. Where the fluid is forced to move quickly relative to its neighbors (i.e., where the shear rate is high), its viscosity skyrockets. This happens most near the pipe walls. The high viscosity near the walls slows the fluid down, forcing the rest of the flow to move more uniformly. The result is a velocity profile that is much flatter or more "blunted" in the center, almost like a solid plug being pushed through the pipe. This "bluntness" is a direct macroscopic signature of the fluid fighting back against the imposed shear.

Why does this happen? What is going on at the microscopic level? The secret lies in organization and disorder. A typical shear-thickening fluid is a dense suspension, meaning it consists of a high concentration of solid particles (like the cornstarch grains) suspended in a liquid (like water).

At rest or under very gentle shear, the particles are happily separated, lubricated by a thin layer of the surrounding liquid. They can slide past one another with relative ease. The system is disordered but mobile. As you increase the shear rate—by stirring it faster or striking it—you try to force these particles past each other more quickly. There comes a point where the particles can no longer get out of each other's way by following the fluid flow. Instead, they are driven into direct, frictional contact.

This is the critical moment. The lubricating layers of fluid are squeezed out, and the solid particles begin to grind against one another. They rapidly organize into transient, sprawling, log-jam-like structures that span large regions of the fluid. These structures, often called ​​hydroclusters​​, are force-bearing chains of particles that are temporarily jammed together. This jammed state drastically resists further deformation, causing the huge, nearly instantaneous increase in viscosity that we observe. It's as if you tried to empty a jar of marbles by shaking it gently versus trying to force a plunger through it; in the second case, the marbles jam up and resist. When the high shear is removed, thermal motion and fluid pressure help break up these clusters, and the fluid returns to its low-viscosity state. The phenomenon is entirely reversible.

This idea of particles jamming and resisting shear is not unique to fluids. In fact, it is a cornerstone of the mechanics of granular materials like sand, soil, and powders. Imagine a tightly packed box of marbles. If you want to shear the top layer of marbles relative to the bottom, the marbles in the middle can't just slide past each other in a straight line; they are interlocked. To move, they must roll up and over their neighbors. This "rolling over" motion forces the entire pack of marbles to expand in volume.

This phenomenon—the increase in volume of a granular material when it is subjected to shear—is called ​​dilatancy​​. In a way, dilatancy in solids is the cousin of shear thickening in fluids. Both arise from the geometric constraints of densely packed particles. The material must expand, or dilate, to allow for deformation. A dense sand dune, a bag of rice, or a heap of gravel all exhibit this behavior. When you step on dense, wet sand at the beach, you may notice the sand around your foot momentarily dries and whitens. This is dilatancy in action: the sand grains under your foot are shearing, causing the sand to expand, which in turn sucks the surface water into the newly created pore spaces.

To understand and predict the behavior of these materials, physicists and engineers needed a more formal mathematical language. In the world of solid mechanics, this led to the elegant theory of ​​plasticity​​. The theory posits that to describe how a material deforms permanently (plastically), we need not one, but two, distinct mathematical rules, or functions.

First, we need a rule that tells us when the material will start to deform plastically. This is the ​​yield function​​, which we can call $f$. You can think of it as a boundary or a fence in the space of all possible stresses. As long as the stress state is inside this fence ($f \le 0$), the material behaves elastically, like a spring. But once the stress touches the fence ($f=0$), plastic deformation begins. For materials like sand, this fence is not a simple line; its position depends on both the shear stress (trying to make grains slide) and the confining pressure (squeezing them together).

Second, once the stress is on the fence, we need a rule that tells us in what direction the material will deform. Will it mostly shear? Will it mostly change volume? Or some combination? This rule is provided by the ​​plastic potential​​, which we can call $g$. The genius of the theory is that the direction of the plastic strain increment (a vector representing the change in shape and volume) is given by the gradient of this potential function, $\dot{\boldsymbol{\varepsilon}}^{p} = \dot{\lambda} \frac{\partial g}{\partial \boldsymbol{\sigma}}$. This is like a topographical map where the direction of steepest ascent points the way for the flow.

The most natural, simple, and aesthetically pleasing assumption one could make is that the rule for when to flow and the rule for where to flow are one and the same. In other words, let's assume the plastic potential is the same as the yield function, $g=f$. This is called an ​​associated flow rule​​. It has a beautiful geometric interpretation: the direction of plastic flow is always perfectly perpendicular (or "normal") to the yield surface.

This elegant theory makes a very strong prediction. For a frictional material whose strength is described by a friction angle $\phi$, the associated flow rule predicts that its dilatancy during shear is also governed by $\phi$. In fact, it predicts that the ​​dilatancy angle​​, $\psi$, which measures the ratio of volume expansion to shear deformation, must be equal to the friction angle, $\psi = \phi$.

Herein lies the rub. This beautiful prediction, for all its mathematical elegance, is often wrong. When we go to the lab and carefully measure the properties of real sand, we find a consistent and inconvenient fact: the sand dilates, but not nearly as much as the associated flow rule predicts. We might measure a friction angle of $\phi = 35^{\circ}$ but a dilatancy angle of only $\psi=10^{\circ}$. An associated model, in this case, would over-predict the volume expansion by a huge margin, rendering it useless for accurate engineering predictions.

Reality forces our hand. We must abandon the simple idea that $g=f$. We must allow the two functions to be different. This is the essence of a ​​non-associated flow rule​​. We "divorce" the plastic potential from the yield function.

This gives us incredible freedom. We can now design the yield function $f$ to accurately capture the material's strength, using the experimentally measured friction angle $\phi$. Simultaneously, we can design a separate plastic potential $g$ to accurately capture the material's flow behavior, using the experimentally measured dilatancy angle $\psi$. For instance, by choosing a potential of the form $g = q - M_p p$, we can show that the ratio of plastic volume change to plastic shear strain is directly and solely controlled by the dilatancy parameter $M_p$, which is related to $\psi$. If we want to model a material that shears without any volume change at all (like in the critical state of soil), we simply set its dilatancy parameter to zero ($\psi=0$), without having to alter our description of its frictional strength. This decoupling is the key to building realistic models of soils, rocks, concrete, and other complex materials.

But as always in physics, there is no free lunch. The elegant associated flow rule ($g=f$) is deeply connected to principles of material stability. When we adopt a non-associated rule ($g \neq f$), we are treading on dangerous ground. We may have a model that accurately describes the material's behavior under "normal" conditions, but we have also introduced the possibility of instability.

This instability is not just a mathematical quirk. It is the seed of catastrophic failure. It manifests as ​​localization​​, a phenomenon where deformation, instead of remaining uniform throughout the material, spontaneously concentrates into very narrow bands of intense shear. These ​​shear bands​​ are the precursors to faults in rock, shear planes in soil slopes, and fracture patterns in concrete.

The non-associated framework is so powerful that it not only accommodates this possibility but allows us to predict it. By analyzing the governing equations of the material model, we can find the precise conditions under which this instability, known as a loss of ellipticity, will occur. We can derive a ​​critical dilatancy parameter​​, $\beta_{\mathrm{crit}}$. If a material's actual dilatancy $\beta$ exceeds this critical value—a value which depends on the friction $\alpha$ and the material's elastic stiffness—the equations predict the formation of shear bands. What begins as an observation about a kitchen slurry (oobleck) leads us through a path of jamming particles, granular expansion, and the sophisticated mathematics of plasticity, only to arrive at a profound insight into how and when materials break. This is the inherent beauty and unity of physics: simple principles, when pursued with rigor and honesty, can reveal the deepest secrets of the world around us.

In our previous discussion, we unraveled the curious nature of shear-thickening fluids—materials that defy our everyday intuition by becoming more viscous, more resistant, the harder we try to stir them. This behavior, moving from a liquid-like to a more solid-like state under stress, might at first seem like a mere laboratory curiosity, a fun trick to play with cornstarch and water. But what good is a fluid that fights back? As it turns out, this very resistance is the key to a vast and fascinating array of applications, forging surprising connections between fluid dynamics, materials science, geophysics, and even the molecular world of polymers. This principle, in its various guises, helps us design futuristic armor, understand the ground beneath our feet, and predict how materials fail.

Let's begin in the world of engineering, where controlling fluids is paramount. Imagine trying to pump a shear-thickening slurry through a pipe. Unlike water, which flows fastest at the center and slows smoothly to a stop at the walls, a shear-thickening fluid behaves rather differently. Because it resists high shear rates, the fluid near the pipe wall, where shear is greatest, becomes extremely viscous. This forces most of the flow to occur more uniformly in the center, creating a "blunted" or "plug-like" velocity profile. The precise shape of this profile, of course, depends on the fluid's "flow behavior index" $n$; for a shear-thickening fluid with $n>1$, the velocity profile becomes progressively flatter than the classic parabolic profile of a Newtonian fluid as $n$ increases. This concentration of shear and pressure drop near the walls has profound implications for the design of pipelines and the energy required to pump such materials in industries from food processing to mining.

This ability to "push back" makes these fluids ideal candidates for energy absorption. Consider a hydraulic damper in a vehicle's suspension or a shock absorber in a landing gear. We want a device that offers little resistance to slow movements but provides immense opposition to sudden, violent shocks. A shear-thickening fluid does precisely this. Under high shear rates—like those during a major jolt—its effective viscosity skyrockets, dissipating a large amount of energy. But there is no free lunch in physics. This rapid energy dissipation manifests as heat. The rate of viscous heat generation is proportional to the shear stress multiplied by the shear rate. For a power-law fluid, this means the heating scales with the shear rate to the power of $n+1$. Since $n>1$ for a shear-thickening fluid, this leads to an explosive increase in heat generation at high speeds. An engineer designing a high-performance damper must therefore not only harness the fluid's desirable mechanical properties but also contend with the formidable thermal management challenge it creates.

The stabilizing nature of this viscous resistance can be applied in more subtle ways. In the field of fluid-structure interaction, engineers sometimes face the challenge of hydroelastic instabilities, where a flexible structure immersed in a flow begins to flutter or "flap" uncontrollably. By choosing a shear-thickening fluid, it is conceptually possible to introduce a powerful damping mechanism. As the structure begins to oscillate at high frequency, it induces high shear rates in the surrounding fluid. The fluid responds by becoming more viscous, creating a damping pressure that opposes the motion and can potentially quell the instability, turning the fluid's stubbornness into a source of stability.

Even a seemingly simple process like stirring is profoundly affected. When you stir a Newtonian fluid like water in a tank, the familiar vortex forms as a balance between centrifugal force and gravity. If you were to stir a shear-thickening fluid under the same conditions, the picture changes. Its tendency to behave more like a rigid body under rotation results in a completely different distribution of velocities throughout the tank, which in turn alters the shape and depth of the surface vortex. This principle is vital in chemical engineering, where ensuring proper mixing of complex, non-Newtonian mixtures is often the difference between a success and failure. The fluid's rheology also impacts heat transfer; the way a fluid's shear-thickening nature alters the relationship between the momentum boundary layer and the thermal boundary layer is a critical factor in designing efficient heat exchangers for non-Newtonian fluids.

Now, let us take a step back and see if this principle appears elsewhere. We will find it, wearing a different hat, in the world of "solids." Geologists and civil engineers have long known that dense, granular materials like sand, soil, and rock exhibit a peculiar property: when sheared, they expand. Imagine a tightly packed box of marbles. To slide one layer over another, the marbles in that layer must ride up and over the ones below, causing the entire pack to increase in volume. This phenomenon is called ​​dilatancy​​. It is the solid-state cousin of shear thickening. Its effect is everywhere, from the stability of building foundations to the mechanics of earthquakes.

In the sophisticated world of geomechanics, this behavior is captured using the mathematical framework of plasticity. A material's resistance to permanent deformation is described by a "yield surface," often modeled by criteria like the Mohr-Coulomb condition, which depends on a material's internal friction angle, $\phi$. However, the direction of plastic flow once yielding occurs—specifically, how much the material expands in volume as it deforms—is governed by a separate property, the ​​dilatancy angle​​, $\psi$. For many real materials like soil and rock, the friction that governs its strength is not the same as the dilatancy that governs its volume change ($\phi \neq \psi$). This leads to what is called a "non-associated" flow rule, a subtle but essential feature for accurately predicting the behavior of the ground under load. The idea of dilatancy is not just for static loads, either. In viscoplastic models used to simulate soil response during an earthquake or the behavior of concrete under high-speed impact, the rate of volumetric expansion is directly tied to a dilatancy parameter in the material's constitutive law.

Why is it that some materials, like wet sand, expand when sheared, while others, like a solid piece of steel, seem to deform at constant volume? The answer reveals a deep and beautiful unity. The tendency to dilate is intimately linked to whether the material's strength depends on hydrostatic pressure.

Classical plasticity models for metals, such as the von Mises or $J_2$ theory, assume that a metal's strength is independent of pressure. A wonderful consequence of this assumption is that plastic deformation must occur at constant volume—that is, it must be incompressible. In contrast, the strength of granular materials like soil is highly dependent on pressure—the more you squeeze it, the stronger it gets. And as we've seen, these are precisely the materials that exhibit dilatancy. The rule is simple and profound: pressure-insensitivity implies plastic incompressibility; pressure-sensitivity allows for dilatancy.

This principle even explains how a solid metal can become dilatant. If the metal contains microscopic pores or voids, its behavior changes. When this porous material is stretched, the voids grow. This growth of internal voids constitutes an overall increase in the material's volume—it is a form of dilatancy! The plastic potential in models for porous metals, like the Gurson-Tvergaard-Needleman (GTN) model, explicitly depends on hydrostatic pressure, linking the void growth directly to the stress state and provide a mechanism for plastic volume change that is essential for understanding ductile fracture.

The story reaches its most intimate scale in the world of polymers. Consider a glassy polymer, a tangled mess of long-chain molecules frozen in place. If you deform this glass rapidly, you can mechanically force the chains apart, creating tiny pockets of empty space—what physicists call "free volume." This strain-induced increase in free volume is a microscopic form of dilatancy. This extra "elbow room" has a dramatic effect: it allows the polymer chains to move and rearrange more easily. The material's structural relaxation time—a measure of how long it takes to respond to a change—can drop precipitously. A small, transient volumetric strain can make the glass behave as if it were "younger" and more mobile.

From the gritty reality of pumping industrial slurries to the subtle dance of polymer chains, we see the same fundamental principle at play. Shear thickening and dilatancy are the macroscopic expression of a microscopic struggle for space. Whether it's solid grains of sand jostling past each other, microscopic voids in a metal growing under tension, or polymer chains being pried apart, the story is the same: when you force a crowded system to rearrange, it often needs to expand. Recognizing this simple, unifying idea across such a vast landscape of scientific and engineering disciplines is a perfect example of the inherent beauty and interconnectedness of the physical world.