Non-Associative Flow Rule

The behavior of materials like soil and rock under load is governed by the principles of plasticity, the science of irreversible deformation. At the core of this theory are two key concepts: a ​​yield surface​​, which defines the stress limits a material can withstand before failing, and a ​​plastic potential​​, which dictates the direction of material flow once that limit is reached. The simplest assumption, known as the ​​associated flow rule​​, is that these two concepts are one and the same. While this elegant model works remarkably well for metals, it falls short when describing the complex behavior of granular materials. These materials often exhibit volume changes, such as expansion (dilatancy) or compaction, that are not correctly predicted by their frictional strength alone.

This article addresses this critical gap by exploring the ​​non-associative flow rule​​, a more powerful framework that decouples strength from flow behavior. By adopting this approach, we gain the flexibility needed to create realistic models for a wide range of geomaterials. The following chapters will guide you through this essential concept. "Principles and Mechanisms" delves into the theoretical foundations, contrasting the associated and non-associative rules and outlining the physical reasoning and computational consequences. Subsequently, "Applications and Interdisciplinary Connections" demonstrates the indispensable role of non-associativity in solving real-world problems in soil mechanics, geology, and engineering.

To understand how materials like soil, sand, and rock behave under extreme loads—why a sandcastle holds its shape and how a mountain stands—we must venture beyond the simple world of elastic springs. We enter the realm of ​​plasticity​​, the study of permanent, irreversible deformation. At the heart of this field lies a beautiful geometric idea, but one that nature, in its complexity, often chooses to modify. Our journey is to understand this idea, its elegant simplicity, its limitations, and the more powerful, nuanced concept that arises from them.

Imagine you are standing on a vast frozen lake. As you walk, the ice beneath your feet is in a state of stress. As long as the stress is small, the ice behaves elastically; if you retrace your steps, the ice returns to its original state. However, there is a limit. If the stress becomes too great, the ice will crack. The collection of all possible stress states that the ice can withstand just before it cracks forms a boundary in the abstract "space of stresses." This boundary is called the ​​yield surface​​, and we describe it with a mathematical function, let's call it $f$. For any stress state inside this surface ($f 0$), the material is elastic. For any state on the surface ($f = 0$), plastic deformation, or "yielding," is possible.

This answers the question of when a material fails. But it doesn't answer the question of how it fails. When the ice cracks, in which direction do the cracks propagate? When a metal bar is stretched until it permanently deforms, how does the material inside flow? This direction of plastic deformation is a separate physical question. The governing principle, as it turns out, is also defined by a surface, which we call the ​​plastic potential​​, denoted by the function $g$. A central tenet of plasticity theory, known as the ​​normality rule​​, states that the "direction" of the plastic strain rate is perpendicular (or normal) to this potential surface $g$ in stress space.

So, we have two fundamental concepts, embodied by two surfaces: the yield surface $f$, which acts as a trigger for plasticity, and the plastic potential $g$, which dictates the geometry of the subsequent flow.

What is the simplest, most elegant relationship we can propose between these two surfaces? We could suppose they are one and the same: $f = g$. This beautifully simple assumption is known as an ​​associated flow rule​​. In this case, the plastic strain rate is normal to the yield surface itself. The trigger for failure and the direction of failure are coupled, governed by a single, unified law.

This idea is not just mathematically convenient; it works astonishingly well for many materials, most notably metals. For metals, the yield strength is largely unaffected by hydrostatic pressure. A common model is the ​​von Mises yield criterion​​, which states that yielding occurs when the shear energy reaches a critical value. Its yield surface in stress space is a simple cylinder aligned with the hydrostatic axis. Now, consider the normal vector to this cylindrical surface. It points purely in the shear (or ​​deviatoric​​) directions, with no component along the pressure axis. According to the associated flow rule, the plastic strain rate must also point in this direction. This implies that the plastic flow of metals involves only a change in shape, with no change in volume. The trace of the plastic strain rate tensor, which represents the volumetric change, is zero: $\mathrm{tr}(d\boldsymbol{\varepsilon}^p) = 0$. This prediction of incompressible plastic flow is in excellent agreement with experimental observations for metals. The associated flow rule seems like a triumph of theoretical elegance and predictive power.

The world, however, is not made only of metal. Consider granular materials like sand, soil, and rock. Anyone who has walked on a beach has noticed that the dense, wet sand underfoot seems to dry out and turn pale just before your foot sinks in. This is a manifestation of a phenomenon called ​​dilatancy​​: the sand is actually expanding in volume as it is being sheared.

To model such materials, we need a yield criterion that accounts for their pressure-sensitive nature; unlike metals, their strength increases with confining pressure. A classic model for this is the ​​Drucker-Prager criterion​​, which can be visualized as a cone in stress space, contrasted with the von Mises cylinder. The yield function can be written as $f = q + \alpha p - k = 0$, where $q$ is a measure of shear stress, $p$ is the mean stress (pressure), and $\alpha$ is a parameter related to the material's internal friction.

Let's apply our elegant associated flow rule to this conical yield surface. The normal vector to a cone is not perpendicular to the pressure axis; it points outwards and upwards. The associated flow rule thus predicts that when a frictional material yields, it should both deform in shear and expand in volume (dilate). This is qualitatively correct! We see dilatancy in dense sand.

But here lies the rub. The associated flow rule intrinsically links the amount of friction ($\alpha$) to the amount of dilatancy. A higher friction angle implies a greater tendency to dilate. When we measure these properties in the laboratory, we find a stark disagreement. For most soils and rocks, the measured volume expansion is significantly less than what the associated flow rule predicts based on their measured frictional strength. For a typical sand with a friction angle of $\phi = 30^{\circ}$, an associated model would predict a certain rate of volume expansion. If we measure the actual volume expansion, we might find it corresponds to a much smaller "dilation angle" of $\psi = 10^{\circ}$. The associated model could be overpredicting the dilatancy by a factor of three or more, a massive error in any engineering calculation.

Nature is telling us that for these materials, the rule for "when to yield" is different from the rule for "how to flow." The solution is to break the elegant coupling of the associated model and embrace a ​​non-associative flow rule​​, where the plastic potential $g$ is different from the yield function $f$.

We maintain the yield function $f(\boldsymbol{\sigma}) = q + \alpha p - k$ to accurately capture the material's strength, which depends on friction. But for the flow rule, we introduce a separate plastic potential, $g(\boldsymbol{\sigma}) = q + \beta p$, where the parameter $\beta$ is chosen independently to match the observed dilatancy. The plastic strain rate is now normal to the surface of $g$, not $f$.

This freedom is incredibly powerful.

• We can now accurately model a dense sand that has high frictional strength (a large $\alpha$) but only modest dilatancy (a smaller $\beta$).
• We can capture the concept of a ​​critical state​​, a key idea in soil mechanics, where after extensive shearing, a material deforms at constant volume. This corresponds to a non-dilatant flow, which we can model perfectly by setting $\beta = 0$, while keeping the frictional strength $\alpha > 0$. An associated model could never achieve this, as setting $\alpha=0$ would mean the material has no friction.
• We can even model ​​compaction​​, where a loose material becomes denser under shear, by choosing an appropriate potential function (e.g., by allowing $\beta$ to be negative, depending on the sign convention).

By allowing the two surfaces, $f$ and $g$, to be distinct, we sacrifice the initial simplicity of a single governing law, but we gain the flexibility to describe a much richer and more realistic range of material behaviors.

This newfound freedom is not without its own set of rules and consequences. We can't just invent any plastic potential $g$ that we like.

First, our models must obey the laws of physics, specifically the Second Law of Thermodynamics. This requires that any dissipative process, such as plastic deformation, cannot create energy from nothing. The rate of plastic work, or ​​plastic dissipation​​ $D = \boldsymbol{\sigma} : \dot{\boldsymbol{\varepsilon}}^p$, must always be non-negative. For a non-associative model, this condition translates into a direct constraint on the mathematical form of the plastic potential $g$. It turns out that to guarantee non-negative dissipation for all possible stress states, the plastic potential $g$ must be a ​​convex function​​ that has its global minimum at the origin of stress space. For our Drucker-Prager potential, $g = \sqrt{J_2} + \beta p$, a detailed analysis shows that this thermodynamic requirement enforces the condition that $\beta \ge 0$ (assuming compression is positive). This means that while we can model non-dilatant flow ($\beta=0$), we cannot model plastic compaction with this specific form of the potential without violating a fundamental law of physics. Physics provides a boundary for our modeling creativity.

Second, there is a computational price to pay for realism. In modern engineering, we use computers and the ​​Finite Element Method (FEM)​​ to simulate the behavior of complex structures. This involves solving vast systems of equations. The speed and stability of this process depend critically on the properties of the system's "stiffness matrix." For an associated plasticity model, the resulting material stiffness matrix is beautifully ​​symmetric​​. This allows engineers to use extremely efficient and robust numerical solvers.

However, when we introduce a non-associative flow rule where $f \neq g$, this elegant symmetry is broken. The material stiffness matrix becomes ​​non-symmetric​​. This non-symmetry is not a mathematical artifact; it is a direct consequence of the physics of decoupling strength from flow. It forces engineers to use more complex, slower, and more memory-intensive solvers to handle the non-symmetric systems of equations. This is the computational cost of realism—a practical trade-off that engineers and scientists must make when choosing a model that is both accurate and computationally feasible. The journey from the simple associated rule to the complex non-associated one is a perfect example of how science progresses: a beautiful idea confronts inconvenient facts, and a more nuanced, powerful, and ultimately more truthful picture of the world emerges.

Now that we have grappled with the principles and mechanisms of non-associative plasticity, you might be asking a very fair question: "This is all very clever mathematics, but what is it for? Where does this seemingly abstract idea meet the real world?" This is where our journey becomes truly exciting. The non-associative flow rule isn't just a theoretical refinement; it is the key that unlocks our ability to understand and predict the behavior of a vast range of materials that shape our world, from the soil beneath our feet to the rocks deep within the Earth's crust. It is the difference between a crude sketch and a faithful portrait of nature's mechanical artistry.

The previous chapter showed that the essence of non-associativity is the separation of two distinct physical phenomena: the condition for failure (the yield surface, $f$) and the rule for deformation during that failure (the plastic potential, $g$). For many materials in our daily experience, like a steel beam, these two rules are one and the same—they are "associative." But for a vast and important class of materials known as geomaterials—soil, rock, concrete, and sand—this is simply not the case. Their story is written in the language of non-associativity.

Let us start with something you can scoop up in your hands: sand. If you take a bag of densely packed sand and shear it, something remarkable happens. In order for the grains to move past one another, the entire volume must expand. The grains have to "climb" over their neighbors. This phenomenon is called ​​dilatancy​​. Conversely, if you take a bag of very loose sand and shear it, the grains will tend to fall into the gaps, and the volume will decrease or ​​compact​​.

An associated flow rule, where $g=f$, would rigidly link the material's shear strength (its friction) to a specific amount of dilation. But experiments tell us this is not what happens. The amount a soil dilates depends on its density and confining pressure, and it is a property independent of its shear strength. The non-associative flow rule is precisely the tool we need to capture this independence. We can define a friction angle, $\phi$, that governs the yield surface $f$, and a separate dilation angle, $\psi$, that governs the plastic potential $g$. By doing so, we can accurately derive the relationship between plastic shear strain and plastic volume change, a cornerstone of modern soil mechanics.

This isn't just an academic exercise. Imagine you are a geotechnical engineer designing a foundation for a skyscraper or a retaining wall for a highway. You need to predict how the soil will deform under load. Will it settle? By how much? To answer these questions, you go to the lab. You perform tests, like the triaxial compression test, on soil samples taken from the site. From these tests, you extract data on how strain develops with stress. These data are then used to calibrate the parameters of your constitutive model—including the crucial parameters that define the non-associative plastic potential. This process transforms an abstract theory into a powerful predictive tool for safe and efficient engineering design.

The story gets even more dramatic when we add water. Most soil in the ground is saturated. When a saturated, contractive soil is sheared quickly—so quickly that the water cannot escape (an "undrained" condition)—the tendency of the soil grains to rearrange and compact squeezes the water. This increases the pore water pressure. According to the principle of effective stress, this rise in water pressure pushes the soil grains apart, reducing their contact forces and drastically weakening the soil. An associated flow rule would often over-predict this plastic compaction, leading to an overly conservative—and incorrect—prediction of pore pressure rise. A non-associative rule, which allows us to tune the amount of plastic compaction, is essential for accurately modeling the pore pressure response and predicting the true stability of the soil. This is the physics behind soil liquefaction during earthquakes, a phenomenon that can cause buildings to topple and the ground to flow like a liquid.

Many of the most challenging problems in engineering involve not a single, steady load, but thousands or millions of cycles of loading: the shaking of an earthquake, the pounding of waves on an offshore platform's foundation, the daily passage of traffic over a bridge abutment. Here, the non-associative nature of geomaterials reveals one of its most striking consequences: ​​ratcheting​​.

To understand this, let's contrast a soil with a simple metal. If you take a metal bar and bend it back and forth symmetrically, it will deform plastically, but after the cyclic loading stops, it will be more or less the same length. The plastic strains in one direction are cancelled out by the plastic strains in the other. This is because the metal's plastic flow is largely incompressible and associative.

A soil is different. As we've seen, shearing a dilative soil causes it to expand. Because this plastic expansion is governed by the plastic potential $g$, it often occurs regardless of the direction of the shear. Shearing it one way causes it to expand, and shearing it back the other way can cause it to expand again. The result is that even under perfectly symmetric cyclic shearing, the soil accumulates volumetric strain, cycle after cycle. This, in turn, drives an accumulation of strain in one direction—a relentless drift known as ratcheting. This phenomenon is simply impossible to capture with an associative model, but it is a natural outcome of a non-associative bounding surface model for soils. It is this very ratcheting behavior, governed by a non-associativity parameter, that determines whether a loose sand will compact toward liquefaction or a dense sand will dilate and stiffen under cyclic loading.

The importance of the non-associative flow rule extends far beyond the traditional boundaries of civil engineering. It is a fundamental concept in the Earth sciences.

In petroleum geology, engineers are concerned with how fluids flow through porous reservoir rocks like sandstone. Under the immense pressures deep underground, these rocks can fail not by dilating, but by collapsing in localized zones called ​​compaction bands​​. These bands act as barriers to fluid flow, completely changing the dynamics of a reservoir. Modeling the formation of these bands requires a plastic potential $g$ that, under high mean stress $p$, predicts plastic volume decrease, or compaction. The condition for this is that the derivative of the potential with respect to pressure, $\frac{\partial g}{\partial p}$, must be negative. This is a classic application of non-associative plasticity to a geological-scale problem.

In rock mechanics and mining, engineers must predict the stability of tunnels and slopes in jointed rock masses. The behavior of these masses is dominated by sliding along the joints. A simple Coulomb friction law, which states that slip occurs when the shear stress reaches a fraction of the normal stress, can be elegantly framed within the theory of plasticity. A non-dilatant friction model—one where sliding does not cause the joint to open—is precisely a non-associative model where the plastic potential $g$ depends only on the shear stress, not the normal stress. This principle is built into powerful numerical tools like Discontinuous Deformation Analysis (DDA), which simulate the complex interaction of large rock blocks.

Perhaps the most profound interdisciplinary connection is in the field of ​​poro-mechanics​​, which studies the full coupling between fluid flow and the deformation of a porous solid. When a rock shears and dilates plastically near a fault, it creates new pore volume. In an undrained environment, this expansion of the solid matrix forces the fluid to expand to fill the new space, causing a drop in fluid pressure—a phenomenon called dilatancy-induced suction. In a drained environment, this local pressure drop creates a gradient that sucks fluid into the dilating zone. This plastic volume change, governed by the non-associative flow rule, acts as a fundamental source/sink term in the fluid mass conservation equations. It is essential for modeling everything from earthquake aftershocks and volcanic activity to induced seismicity from hydraulic fracturing.

Finally, it is worth appreciating the computational implications of this physical reality. In the world of associative plasticity, the governing equations possess a beautiful mathematical symmetry. The problem of finding the equilibrium state can be cast as finding the minimum of a single energy potential. The resulting stiffness matrices in a Finite Element simulation are symmetric, which allows us to use very efficient and robust numerical solvers.

The non-associative flow rule breaks this symmetry. Because $g \neq f$, there is no single underlying potential to be minimized. The problem can no longer be solved by simply rolling downhill on an energy landscape. The resulting algorithmic tangent matrices are, in general, ​​non-symmetric​​. This presents a significant computational challenge, demanding more sophisticated and expensive solution algorithms.

But we should not see this as a flaw. Instead, it is a beautiful reflection of the true complexity of nature. The fact that materials like soil and rock refuse to be described by the simplest symmetric laws forces us to be more clever. The quest to build robust computational tools that can handle this inherent non-associativity is where physics, numerical analysis, and engineering converge. It is the engine that powers our ability to turn these profound physical insights into predictions that keep structures safe, resources managed, and our understanding of the planet ever-deepening.