The Jaumann Rate: Objectivity in Continuum Mechanics

In the study of deforming materials, a fundamental question arises: how does internal stress change over time? While a simple time derivative seems like an obvious answer, it harbors a deep-seated flaw—it fails the principle of material frame-indifference, meaning its value depends on the observer's own motion. This article addresses this critical problem in continuum mechanics by introducing the concept of an objective stress rate. First, in the "Principles and Mechanisms" chapter, we will unravel why the simple derivative is insufficient and derive the Jaumann rate, an elegant solution that measures stress from a co-rotating perspective. Following this, the "Applications and Interdisciplinary Connections" chapter will demonstrate the Jaumann rate’s vital role in explaining complex fluid behaviors and enabling modern engineering simulations, while also critically examining its celebrated shortcomings that have paved the way for more sophisticated models.

How does the stress in a material change over time? An intuitive first approach, a beautifully simple one, is to just take the time derivative. If we have a stress tensor, which we'll call $\boldsymbol{\sigma}$, we might propose that its rate of change is simply $\frac{d\boldsymbol{\sigma}}{dt}$, or $\dot{\boldsymbol{\sigma}}$ for short. This seems as natural as saying the rate of change of position is velocity. But as is so often the case in science, a simple question leads us down a rabbit hole into a much deeper and more beautiful reality.

The trouble begins with a fundamental principle that underpins all of physics: the ​​principle of material frame-indifference​​, or ​​objectivity​​. This is a grand way of saying that the laws of nature must not depend on the observer. Whether you are standing on the ground or spinning on a merry-go-round, the physical laws governing a piece of stretched rubber in your hand should be the same. Your motion should not alter the material's internal physics.

Let's see if our simple time derivative, $\dot{\boldsymbol{\sigma}}$, respects this principle. Imagine two observers. One is "fixed", and the other is spinning on a merry-go-round. The spinning observer sees the world through a rotating reference frame, described by a rotation tensor $\mathbf{Q}(t)$. If the fixed observer measures a stress $\boldsymbol{\sigma}$, the spinning observer will measure a rotated version of it, $\boldsymbol{\sigma}^{\ast} = \mathbf{Q}\boldsymbol{\sigma}\mathbf{Q}^{\mathsf{T}}$. This is perfectly fine; it just accounts for their different perspectives.

But what happens when they measure the rate of change? The spinning observer calculates the time derivative of what they see, $\dot{\boldsymbol{\sigma}}^{\ast}$. A bit of calculus (using the product rule on $\mathbf{Q}\boldsymbol{\sigma}\mathbf{Q}^{\mathsf{T}}$) reveals something startling:

Here, $\boldsymbol{\Omega}$ is the spin of the merry-go-round itself. Look at that equation! The rate measured by the spinning observer, $\dot{\boldsymbol{\sigma}}^{\ast}$, is not just the rotated version of the fixed observer's rate, $\mathbf{Q}\dot{\boldsymbol{\sigma}}\mathbf{Q}^{\mathsf{T}}$. There are extra terms! These extra terms, $\boldsymbol{\Omega}\boldsymbol{\sigma}^{\ast} - \boldsymbol{\sigma}^{\ast}\boldsymbol{\Omega}$, depend on the observer's own spin.

This is a disaster for physics. It means that two observers will disagree on the rate of stress change even for a completely static, unchanging block of material. If the material stress $\boldsymbol{\sigma}$ is constant, the fixed observer measures $\dot{\boldsymbol{\sigma}}=\mathbf{0}$. But the spinning observer will measure a non-zero rate of change! Their own spinning motion creates a "phantom" rate of stress change. We cannot build a universal law of material behavior—a ​​constitutive law​​—on a quantity that is contaminated by the observer's motion. The simple time derivative is not ​​objective​​.

The solution is as elegant as the problem is profound. If we want a rate of change that is independent of any external observer's motion, we must measure it from a special reference frame: one that lives inside the material and tumbles and spins along with it. Think of it as attaching a tiny set of coordinate axes to a grain of dust within a deforming piece of clay. As the clay is sheared and twisted, our little axes ride along, spinning with the local neighborhood of material. A rate of change measured from the perspective of this spinning frame would be a true measure of the material's intrinsic changes, cleansed of any rigid rotational effects. This is the idea behind a ​​corotational stress rate​​.

But how do we know how the material is locally spinning? The answer lies in the ​​velocity gradient tensor​​, $\mathbf{L}$. This tensor describes how the velocity of the material changes from one point to a neighboring point. It is the key to understanding the local deformation. Any matrix can be split into a symmetric part and a skew-symmetric part. For the velocity gradient, this split reveals two distinct physical actions:

1. The ​​rate of deformation tensor​​, $\mathbf{D} = \frac{1}{2}(\mathbf{L}+\mathbf{L}^{\mathsf{T}})$, which is symmetric. This tells us how the material is being stretched or compressed.

2. The ​​spin tensor​​, $\mathbf{W} = \frac{1}{2}(\mathbf{L}-\mathbf{L}^{\mathsf{T}})$, which is skew-symmetric. This tells us the average angular velocity of the material at that point—its local rate of spin.

The spin tensor $\mathbf{W}$ seems like a perfect candidate to describe the spinning of our internal coordinate system! Let's choose this as the spin we want to "subtract out."

By subtracting the apparent rate of change due to the local spin $\mathbf{W}$ from the "naive" material time derivative $\dot{\boldsymbol{\sigma}}$, we arrive at the celebrated ​​Jaumann rate​​, often denoted $\boldsymbol{\sigma}^{\nabla}$:

The term we are subtracting, the commutator $[\mathbf{W}, \boldsymbol{\sigma}] = \mathbf{W}\boldsymbol{\sigma} - \boldsymbol{\sigma}\mathbf{W}$, is precisely the rate of change of the stress tensor that would be observed if it were just passively rotating with the local material spin $\mathbf{W}$. By removing it, we are left with the part of the stress rate that is due to actual material response—to stretching and straining, not just spinning.

Amazingly, this mathematical construction works perfectly. If we check how the Jaumann rate transforms between our fixed observer and our friend on the merry-go-round, we find that the non-objective terms from $\dot{\boldsymbol{\sigma}}$ are exactly cancelled by the non-objective terms that arise from the transformation of $\mathbf{W}$. The result is a clean, objective transformation: $(\boldsymbol{\sigma}^{\nabla})^{\ast} = \mathbf{Q}\boldsymbol{\sigma}^{\nabla}\mathbf{Q}^{\mathsf{T}}$. We have found an objective rate!

A crucial sanity check for any objective rate is what it predicts for a body that is already stressed and then undergoes a pure rigid body rotation (no stretching, so $\mathbf{D}=\mathbf{0}$). Physically, the stress should just be carried along with the rotation; there should be no new stress generated. The material stress state, seen from its own rotating frame, is constant. Therefore, a true objective rate should be zero. The Jaumann rate passes this test with flying colors. For a rigid rotation, it turns out that $\dot{\boldsymbol{\sigma}}$ is exactly equal to $\mathbf{W}\boldsymbol{\sigma} - \boldsymbol{\sigma}\mathbf{W}$. When plugged into the Jaumann rate formula, the terms cancel perfectly, and we get $\boldsymbol{\sigma}^{\nabla} = \mathbf{0}$. This gives us confidence we are on the right track.

You might wonder why you survived your first mechanics courses without ever hearing about this. The answer lies in approximation. For the ​​infinitesimal strains​​ and small rotations common in introductory problems (like a slightly sagging beam), the spin $\mathbf{W}$ is very small, and the stress $\boldsymbol{\sigma}$ is also small. The correction term, which is a product of the two, is a "higher-order" term that can be safely neglected. For small motions, the simple time derivative is good enough.

However, when we enter the world of ​​finite deformations​​—the world of metal forging, car crashes, and lava flows—rotations can be enormous. Here, the correction terms are not just significant; they are essential. A constitutive law written using the non-objective $\dot{\boldsymbol{\sigma}}$ would make bizarre and unphysical predictions, such as generating enormous stresses in a fluid that is simply stirred without any stretching. The Jaumann rate, by providing an objective framework, allows us to write sensible laws for these complex scenarios.

So, is the Jaumann rate the final answer? The ultimate way to measure stress change? Nature, as always, is more subtle. The Jaumann rate is a brilliant and indispensable concept, but it's not perfect. Its choice of the spin tensor $\mathbf{W}$ is very convenient, but it doesn't always capture the true "material" rotation.

A classic example that reveals this flaw is ​​simple shear​​—think of shearing a deck of cards. Let's take a block of a simple elastic material and shear it. Our intuition, and experiments, tell us that the shear stress should increase smoothly as the shear increases. But if we use a simple elastic model based on the Jaumann rate, we get a bizarre prediction: the shear stress oscillates, and even stranger, normal stresses appear and disappear on the top and bottom faces of the block as it is sheared. For instance, a Jaumann-based model predicts a normal stress that varies with the shear amount $\gamma$ as $\sigma_{11}(\gamma) = \mu(1 - \cos(\gamma))$, where $\mu$ is the shear modulus. These oscillating, "spurious" stresses are not what real materials do.

This tells us that the local spin $\mathbf{W}$ is not always the best proxy for the material's intrinsic rotational state. The search for a better corotational frame has led to a whole family of objective rates. For example, the ​​Green-Naghdi rate​​ uses a different spin, one derived from the rotation tensor $\mathbf{R}$ in the polar decomposition of the deformation gradient ($\mathbf{F}=\mathbf{R}\mathbf{U}$). This spin is thought to be more "materially" based, and indeed, it performs better in simple shear, avoiding some of the spurious effects of the Jaumann rate. Other famous rates, like the ​​Truesdell rate​​ and ​​Oldroyd rate​​, also exist, each with different properties and theoretical underpinnings.

The journey of the Jaumann rate is a wonderful lesson in science. It starts with a simple question and a nagging inconsistency, leads to an elegant mathematical solution that fixes the problem, and then, upon closer inspection, reveals even deeper subtleties about the nature of rotation and deformation. It shows us that the very act of defining "rate of change" for a tensor is not a trivial mathematical step, but a profound physical choice—a constitutive assumption about how a material's properties should be measured as it tumbles through space and time.

Having unraveled the beautiful, almost geometric, logic behind the Jaumann rate, you might be tempted to think of it as a clever piece of mathematical formalism, a curiosity for the theoretician. Nothing could be further from the truth. The necessity of observing the world from a spinning perspective, of separating true change from mere rotation, is not an abstract conceit. It is a fundamental challenge that appears everywhere, from the swirling of paint in a can to the intricate dance of atoms in a deforming crystal, and its consequences are profoundly practical. In this chapter, we will embark on a journey to see how this one idea—the Jaumann rate—serves as a crucial key, unlocking puzzles and building bridges between seemingly disparate fields of science and engineering.

Let's begin with something familiar: stirring a thick fluid, like honey or a polymer melt. A simple experiment reveals a bizarre phenomenon that Newtonian physics cannot explain. If you rotate a rod in a vat of such a fluid, you would expect the fluid to be dragged around the rod. But something else happens: the fluid also tries to climb up the rod. This is known as the Weissenberg effect. Where does this upward force come from? The motion is purely rotational shear, yet the fluid pushes back in a direction normal to the shear plane.

This is precisely the kind of puzzle where the Jaumann rate shines. Imagine modeling such a a "hypoelastic" fluid, where the rate of stress is related to the rate of deformation. If we apply a simple shear flow, our intuition suggests only a shear stress should develop. But when we write our constitutive law using the Jaumann rate to ensure objectivity—that is, to get the physics right from the perspective of a spinning fluid element—a startling prediction emerges. The model reveals that normal stresses, $\sigma_{11}$ and $\sigma_{22}$, must arise to keep the equations balanced. Even for a steady, unchanging flow, the Jaumann rate of shear stress turns out to be proportional to the difference in these very normal stresses. The upward climb of the fluid is the macroscopic manifestation of these hidden normal stress differences. The Jaumann rate provides a mathematical porthole into this non-intuitive world, forming a cornerstone of rheology, the science of how complex materials like plastics, foods, and biological fluids flow.

Let's now leap from the fluid dynamics lab to the world of supercomputers, where engineers design everything from cars to airplanes. When simulating a car crash or the forging of a metal beam, the materials involved undergo enormous deformations and rotations. An engineer writing a Finite Element Method (FEM) code faces a critical question: how does the stress inside a piece of metal evolve as it is crushed and sent tumbling through space?

If the simulation code were to use a simple time derivative, it would predict that stress is generated just because the object is rotating—a physical absurdity. A free-falling steel beam doesn't become more stressed just because it's spinning. To prevent such "spurious stresses," the code must be objective. It must use a rate like the Jaumann rate.

This choice has deep, practical consequences for the software itself. Implementing the Jaumann rate introduces extra terms into the governing equations of the simulation. These terms, which depend on the current stress state and the rate of rotation, have a specific mathematical character: they typically make the overall system of equations non-symmetric. For the computational scientist, this is a crucial piece of information. It dictates that they must use more complex and often slower numerical solvers designed for non-symmetric systems. The abstract principle of objectivity, brought to life by the Jaumann rate, reaches right down into the algorithmic heart of the most advanced engineering software on the planet.

So far, the Jaumann rate appears to be a heroic scientific tool. It explains bizarre fluid phenomena and makes billion-dollar simulations possible. But the story of science is never so simple, and great ideas are great not because they are perfect, but because they are clear enough to reveal their own flaws.

Let's return to our simple shear experiment. The Jaumann-based model correctly predicts the emergence of normal stresses. But if we continue shearing the material indefinitely, it predicts something truly strange: the shear stress doesn't just increase and level off; it oscillates, like a sine wave. It's as if by continuously pushing a box across the floor, the friction force were to periodically increase, decrease, and even change direction. This is, for most materials, physically incorrect. This infamous "spurious oscillation" is a well-known pathology of the Jaumann rate when coupled with simple elastic models under large shear. It’s an artifact of the model, a ghost in the machine.

There are other, more subtle cracks in the foundation. The fundamental balance of angular momentum dictates that, in the absence of internal body couples, the Cauchy stress tensor must be symmetric. It turns out that a hypoelastic law using the Jaumann rate does not strictly guarantee that an initially symmetric stress tensor will remain symmetric during a general deformation. In a numerical simulation, this can lead to an imbalance of internal moments, creating "spurious torques" that can corrupt the solution. These flaws do not invalidate the principle of objectivity, but they tell us that the Jaumann rate, while objective, might not be the perfect way to achieve it. It is a good answer, but perhaps not the final answer.

The discovery of these flaws did not lead scientists to abandon objectivity. Instead, it spurred a deeper search for a more physically-grounded way of looking at rotation. The breakthrough came from the field of materials science, specifically in the study of single crystals.

What, after all, is stress in a metal? It is a measure of the elastic stretching of the atomic lattice. Thus, the most physically meaningful reference frame to measure stress change is one that rotates with the crystal lattice itself. The generic "material spin" $\mathbf{W}$ used in the Jaumann rate is a kinematic average that doesn't always coincide with the spin of the underlying atomic structure. In crystal plasticity, this lattice spin, let's call it $\mathbf{W}^e$, can be distinguished from the spin due to plastic slip.

This is a profound insight. By constructing a new objective rate, one that is co-rotational with the lattice spin $\mathbf{W}^e$ instead of the material spin $\mathbf{W}$, scientists created a more sophisticated model. And the result? When applied to the simple shear problem, the unphysical stress oscillations vanish!. The predicted shear stress now grows monotonically, just as one would expect. This is a spectacular example of scientific progress: a problem identified in a mathematical model (the oscillations) is solved by appealing to a deeper physical picture (the motion of the atomic lattice).

The Jaumann rate, then, is not just a formula. It is a character in a great scientific story. It is the first and most intuitive solution to the problem of describing nature in a way that doesn't depend on how we, the observers, are spinning. It gives us profound insights into the behavior of complex fluids and provides the essential scaffolding for modern computational engineering.

But its story is also a beautiful cautionary tale. It teaches us that even our most elegant theories must be relentlessly tested against physical reality. Its shortcomings—the spurious oscillations and symmetry issues—were not failures but signposts, pointing the way toward a deeper understanding of motion and matter. The journey from the simple material spin of Jaumann to the refined lattice spin of crystal plasticity encapsulates the very nature of science: a perpetual, iterative quest for a more perfect, more objective, and more beautiful description of our world.