Truesdell Rate

In the study of how materials deform, from the stretching of a rubber band to the flow of tectonic plates, a fundamental challenge arises: how do we describe physical laws in a way that is independent of the observer? The laws of nature should not change whether we observe them from a stationary lab or a spinning spacecraft. This principle of material frame-indifference is a cornerstone of continuum mechanics, yet it presents a significant problem when we try to measure how stress changes over time. The most intuitive measure, the material time derivative of stress, fails this test, as its value is contaminated by the observer's own rotation.

This article addresses this critical knowledge gap by exploring the concept of objective stress rates, which are mathematical tools designed to provide a true, observer-independent measure of stress change. It focuses on one of the most powerful and physically insightful of these tools: the Truesdell rate. Across two chapters, you will gain a deep understanding of this essential concept.

• The ​​Principles and Mechanisms​​ chapter will unravel the "observer problem," introduce simpler objective rates like the Jaumann rate, and then derive the Truesdell rate, highlighting its unique structure and physical meaning.
• The ​​Applications and Interdisciplinary Connections​​ chapter will demonstrate the Truesdell rate's practical superiority in predictive modeling, its foundational role in modern computational simulations, and its relevance in fields beyond solid mechanics, such as the study of non-Newtonian fluids.

Imagine you’re on a fast-spinning merry-go-round, trying to describe the elegant swirl of cream you just poured into your coffee. To you, the cream seems to be flying outwards, pushed by some mysterious force, while its path curves in a baffling way. Someone standing on the ground, however, sees a much simpler picture: the cream is just moving with the rotating coffee, slowly mixing in. The fundamental physics governing the fluid—its viscosity, density, and how it resists being stirred—hasn't changed. The only thing that has changed is you, the observer. The laws of nature must be written in a way that they look the same for both you and the person on the ground. This powerful idea is called the ​​principle of material frame-indifference​​, or more simply, ​​objectivity​​. In the world of materials, it is not just a philosophical preference; it is a strict requirement for any physical law.

When we study how materials deform—a steel beam bending under a load, a polymer stretching, or a tectonic plate shifting—we need to write down constitutive laws that relate the forces within the material (​​stress​​) to its deformation rate (​​strain rate​​). We want to understand how the stress changes over time. The most obvious way to describe a change in stress, $\boldsymbol{\sigma}$, is to simply take its time derivative, which we write as $\dot{\boldsymbol{\sigma}}$. This is what we call the ​​material time derivative​​; it tells us how stress changes for a tiny piece of material as it moves and deforms.

But here we run into the merry-go-round problem. It turns out that this simple, intuitive derivative is not objective. If you, the observer, decide to rotate your frame of reference while a material is deforming, the $\dot{\boldsymbol{\sigma}}$ you measure will be different from what a stationary observer measures. Your own rotation "contaminates" the measurement. Mathematically, if a new observer is rotating with a spin $\boldsymbol{\Omega}$ relative to the old one, the new time derivative $\dot{\boldsymbol{\sigma}}^\ast$ is related to the old one by:

where $\boldsymbol{Q}$ is the rotation tensor relating the two frames. An objective quantity should transform simply as $\boldsymbol{A}^\ast = \boldsymbol{Q}\boldsymbol{A}\boldsymbol{Q}^{\mathsf{T}}$. The two extra terms, $\boldsymbol{\Omega}\boldsymbol{\sigma}^{\ast} - \boldsymbol{\sigma}^{\ast}\boldsymbol{\Omega}$, are the problem. They are artifacts of the observer's spin, not a real physical change in the material's stress state. A constitutive law built on $\dot{\boldsymbol{\sigma}}$ would predict different material behaviors depending on how the scientist chose to spin—a clear absurdity.

How do we fix this? The key is to define a "smarter" derivative that is immune to the observer's spin. The idea is to measure the rate of change of stress in a coordinate system that spins along with the material itself. This is called a ​​corotational frame​​. By doing this, we can cancel out the non-objective rotational effects.

One of the most famous ways to do this leads to the ​​Zaremba–Jaumann rate​​ (or simply Jaumann rate). We take the material time derivative $\dot{\boldsymbol{\sigma}}$ and add correction terms based on the material's own rate of rotation, which is described by the ​​spin tensor​​, $\boldsymbol{W}$. The spin tensor $\boldsymbol{W}$ is the skew-symmetric part of the velocity gradient $\boldsymbol{L}$. The definition is:

The magic here is that the spin tensor $\boldsymbol{W}$ transforms in just the right way to cancel out the pesky $\boldsymbol{\Omega}$ terms from the observer's rotation. The Jaumann rate $\overset{\mathrm{J}}{\boldsymbol{\sigma}}$ is thus ​​objective​​. We can build constitutive laws with it, confident that our physical predictions won't change if we happen to be on a merry-go-round. The Jaumann rate is just one of many possible objective rates; others, like the ​​Green–Naghdi rate​​, use different measures of the material's spin but achieve the same goal of objectivity.

So, are we done? We have objective rates. Why the need for anything else? This is where the profound insight of Clifford Truesdell, one of the founders of modern continuum mechanics, comes into play. He argued that deformation is more than just spinning. It also involves stretching and, importantly, changes in volume.

The ​​Cauchy stress​​, $\boldsymbol{\sigma}$, is defined as force per current area. When a material is compressed or stretched, its volume and density change. This means the very area over which the force is distributed is changing. Shouldn't a truly comprehensive stress rate account for this effect?

To build such a rate, we can use a clever conceptual device. Let's define a different stress measure, the ​​Kirchhoff stress​​, $\boldsymbol{\tau}$, as:

where $J$ is the ratio of the current volume of a piece of material to its original volume. You can think of the Kirchhoff stress as a "volume-weighted" version of the Cauchy stress.

Now, a very natural objective rate in continuum mechanics is the ​​Oldroyd upper-convected derivative​​, which is defined for any tensor $\boldsymbol{A}$ as $\dot{\boldsymbol{A}}^{\nabla}=\dot{\boldsymbol{A}}-\boldsymbol{L}\boldsymbol{A}-\boldsymbol{A}\boldsymbol{L}^{\mathsf{T}}$. It accounts for changes due to the full ​​velocity gradient​​, $\boldsymbol{L}$, which includes both stretching ($\boldsymbol{D}$) and spin ($\boldsymbol{W}$). Let's apply this robust rate to our Kirchhoff stress $\boldsymbol{\tau}$.

The ​​Truesdell rate​​ of the Cauchy stress, denoted $\overset{\triangle}{\boldsymbol{\sigma}}$, is then defined as the rate that is consistent with the Oldroyd rate of the Kirchhoff stress, scaled by the volume change $J$. That is, we demand:

This might look like just a mathematical definition, but it's a profound physical statement. We are defining the "correct" rate of Cauchy stress by linking it to the rate of a stress measure that is already connected to volume changes. Now, we can unveil the Truesdell rate. We substitute $\boldsymbol{\tau} = J\boldsymbol{\sigma}$ and use the kinematic identity $\dot{J} = J\,\mathrm{tr}(\boldsymbol{L})$ (which states that the rate of volume change is related to the trace of the velocity gradient). After some calculus, the $J$s cancel out, and we are left with a beautiful and revealing expression:

This is the Truesdell rate. The first three terms are the Oldroyd rate of $\boldsymbol{\sigma}$. The final term, $(\mathrm{tr}\,\boldsymbol{L})\boldsymbol{\sigma}$, is the crucial addition. It arises directly from the time derivative of the volume ratio $J$. Since $\mathrm{tr}\,\boldsymbol{L}$ represents the rate of volume expansion of the material, this term explicitly accounts for how the changing density of the material affects the stress. For a purely dilatational motion, where the material expands or contracts uniformly without changing shape, this term plays a central role.

We now have two objective rates, Jaumann and Truesdell. What is the fundamental difference between them? We can find out by simply subtracting one from the other. After using the decomposition $\boldsymbol{L} = \boldsymbol{D}+\boldsymbol{W}$ (where $\boldsymbol{D}$ is the symmetric rate-of-deformation, or stretching, tensor), the math reveals a wonderfully clear result:

Notice what's missing: the spin tensor $\boldsymbol{W}$. The entire difference between the two rates depends only on the ​​stretching​​ part of the motion, $\boldsymbol{D}$, and the stress state $\boldsymbol{\sigma}$.

This tells us everything. The Jaumann rate is a minimal correction; it only accounts for the material's spin. The Truesdell rate does more: it accounts for spin and stretching. The two rates are only identical if the material is not deforming ($\boldsymbol{D}=\boldsymbol{0}$) or if the stretching and stress satisfy the special condition $(\mathrm{tr}\,\boldsymbol{D})\boldsymbol{\sigma} = \boldsymbol{D}\boldsymbol{\sigma} + \boldsymbol{{sigma}}\boldsymbol{D}$. In all other cases, they give different answers, reflecting their different physical underpinnings.

The final piece of the puzzle reveals the deepest physical consequence. Any stress state can be split into two parts: a ​​spherical​​ (or hydrostatic) part, $p\boldsymbol{I}$, which represents pressure and relates to changes in volume (size), and a ​​deviatoric​​ part, $\boldsymbol{s}$, which relates to shear and changes in shape. So, $\boldsymbol{\sigma} = p\boldsymbol{I} + \boldsymbol{s}$.

When we apply the Jaumann rate to this decomposed stress, we find that it keeps the two parts neatly separate. The rate of change of the spherical part depends only on $\dot{p}$, and the rate of change of the deviatoric part depends only on $\dot{\boldsymbol{s}}$ and $\boldsymbol{s}$. There is no cross-talk.

The Truesdell rate, however, is different. Because its definition includes the stretching tensor $\boldsymbol{D}$, it ​​couples​​ the spherical and deviatoric responses. The rate of change of pressure can be influenced by the deviatoric stress $\boldsymbol{s}$, and the rate of change of the deviatoric stress is influenced by the pressure $p$. For instance, in a simple shear flow, which is a pure shape-changing motion, the difference between the Truesdell and Jaumann deviatoric rates can depend directly on the pressure $p$ in the material.

This coupling is not a mathematical artifact; it reflects a potential physical reality. For many materials, compressing them (increasing pressure) makes them stiffer and more resistant to shearing. The Truesdell rate provides a natural framework to capture this type of behavior. The choice between the Jaumann, Truesdell, or other objective rates is not just a matter of taste; it is a constitutive choice that embeds different assumptions about the fundamental behavior of the material itself. The journey from a spinning coffee cup to these subtle distinctions in tensor calculus reveals a core principle of physics: the search for a description of nature that is independent of the observer often leads us to a deeper understanding of the thing being observed.

In the previous chapter, we navigated the subtle and often confusing waters of kinematics to arrive at a definition for the rate of change of stress that an observer on a spinning asteroid could agree on. We constructed several "objective" rates, including the Truesdell rate, $\overset{\triangle}{\boldsymbol{\sigma}}$. But a definition, no matter how elegant, is merely a promise. The true test of any scientific concept lies in its power to describe the world, to make sense of experiments, and to build reliable tools. Is the Truesdell rate just a mathematical curiosity, or is it a truly useful piece of physics? As the saying goes, the proof of the pudding is in the eating. Let's put the Truesdell rate to work.

A brilliant way to test any theory in mechanics is to throw it into a situation that is simple to describe but surprisingly revealing: simple shear. Imagine spreading a deck of cards with your hand. Each card slides a little bit relative to the one below it. The deck as a whole is deformed, yet no single card is stretched. This motion is a combination of pure shearing and a continuous internal rotation, making it a perfect trap for a poorly constructed theory.

Let's conduct a thought experiment. Suppose we invent a simple, idealized material we'll call a "hypoelastic" solid. For this material, we'll postulate a very simple law: its objective stress rate is directly proportional to its rate of stretching. In mathematical terms, we write $\overset{\circ}{\boldsymbol{\sigma}} = 2G\,\boldsymbol{D}$, where $\boldsymbol{D}$ is the stretching tensor and $G$ is the shear modulus, a measure of stiffness. Now comes the crucial choice: which objective rate $\overset{\circ}{\boldsymbol{\sigma}}$ should we use?

A natural first guess might be the Jaumann rate, a popular choice which corrects for the average spin of the material. If we plug the Jaumann rate into our simple law and simulate the simple shear experiment, something bizarre happens. The predicted shear stress doesn't just increase as you'd expect; it oscillates, rising and falling as the shear continues. It's as if the material is having a crisis of conscience, periodically getting stronger, then weaker, then stronger again. This is not what we see in most real materials. The Jaumann rate, for all its intuitive appeal, seems to be misled by the continuous rotation inherent in simple shear.

Now, let's repeat the experiment, but this time we'll use the Truesdell rate. We set $\overset{\triangle}{\boldsymbol{\sigma}} = 2G\,\boldsymbol{D}$ and apply the same steady shear. The result is completely different. The predicted shear stress increases steadily and smoothly with the amount of shear. This is a much more physically plausible behavior for a solid being sheared. The same story unfolds if we look at the normal stresses—the stresses that press down on the "cards." The Truesdell rate predicts a sensible, growing normal stress, while the Jaumann rate again predicts a peculiar oscillation.

This test tells us something profound. It's not that the Jaumann rate is mathematically "wrong"—it's perfectly objective. But as a tool for building simple, predictive models of materials, it can fail spectacularly. The Truesdell rate, with its more sophisticated structure, proves to be a more robust and physically insightful choice. It's a powerful reminder that our mathematical choices have real, measurable physical consequences.

What gives the Truesdell rate its superior predictive power in these tests? A key part of the answer lies in a term that sets it apart from simpler co-rotational rates: the term $(\mathrm{tr}\,\boldsymbol{L})\boldsymbol{\sigma}$. The quantity $\mathrm{tr}\,\boldsymbol{L}$ measures the rate at which the material's volume is changing. This term, therefore, explicitly connects the stress rate to the material's expansion or compression.

Imagine a block of foam under some pressure. If we suddenly let it expand in all directions, the internal forces become spread out over a larger volume. The stress, which is force per current area, naturally decreases just because of this geometric "dilution," even before the material's internal structure has had time to relax. The Truesdell rate automatically and correctly captures this fundamental effect through its $(\mathrm{tr}\,\boldsymbol{L})\boldsymbol{\sigma}$ term. Simpler rates that lack this term are essentially blind to this important piece of the physics of compressible materials.

This might seem like just another correction, but it points to a deeper, more elegant truth. In large deformations, the Cauchy stress $\boldsymbol{\sigma}$ (force per current area) can be an awkward quantity precisely because the "current area" is a moving target. What if we instead defined a stress relative to the original, undeformed state? This idea leads us to the ​​Kirchhoff stress​​, denoted $\boldsymbol{\tau} = J\boldsymbol{\sigma}$, where $J$ is the ratio of the current volume to the original volume. The Kirchhoff stress is, in essence, a measure of force scaled to remove the effect of volume change.

And here lies a beautiful moment of unity. If we ask for the simplest objective rate of this more natural Kirchhoff stress, we arrive at the Oldroyd rate, given by $\boldsymbol{\tau}^{\nabla} = \dot{\boldsymbol{\tau}} - \boldsymbol{L}\boldsymbol{\tau} - \boldsymbol{\tau}\boldsymbol{L}^{\mathsf{T}}$. As it turns out, this clean and simple rate for $\boldsymbol{\tau}$ is directly related to the more complicated-looking Truesdell rate for $\boldsymbol{\sigma}$. The relationship is astonishingly simple:

The apparent complexity of the Truesdell rate is just a shadow cast by the simpler physics governing the Kirchhoff stress! The "messy" volumetric term in the Truesdell rate is simply what's needed to translate the clean physics of the reference configuration into the language of the deforming, current configuration.

This theoretical elegance is not just for show; it has enormous practical consequences. Today, engineers rarely build and break thousands of prototypes of cars or airplanes. Instead, they use powerful software based on the ​​Finite Element Method (FEM)​​ to simulate the behavior of complex systems under extreme conditions—from the response of a skyscraper in an earthquake to the inflation of an airbag in a car crash.

These simulations must accurately model materials undergoing immense stretching, twisting, and compression. To do this, they solve the fundamental equations of motion and the material's constitutive law step-by-step in time. The choice of objective stress rate is not an academic detail here; it is the very heart of the algorithm.

Because of its deep and clean connection to the undeformed state (via the Kirchhoff stress), the Truesdell rate forms the backbone of the most robust and widely used computational methods, especially for ​​hyperelastic​​ materials—those that store and release strain energy almost perfectly, like rubber. A constitutive law formulated with the Truesdell rate can be integrated in time in a way that remains consistent with the material's underlying stored energy function. This ensures that the simulation doesn't artificially create or destroy energy, which would lead to nonsensical results.

Furthermore, the form of the rate has direct implications for the stability of the simulation itself. When these equations are solved with an explicit time-stepping algorithm (which is like taking many small, quick snapshots of the evolution), each time step $\Delta t$ must be small enough to prevent the numerical solution from "exploding." The very structure of the Truesdell rate equation dictates the maximum allowable time step. By analyzing the mathematical properties of the operator associated with the Truesdell rate, engineers can precisely determine these stability limits. They find that a rapid expansion might require a much smaller, more careful time step than a slow shear, for instance. This analysis is crucial for creating simulations that are not just accurate, but can be run at all.

The principles we've discussed are so fundamental that they ripple out from solid mechanics into other fields. Consider the strange and fascinating world of ​​non-Newtonian fluids​​. These are substances like polymer melts, paints, ketchup, and biological fluids that defy the simple, linear behavior of water or air. When you stir them, they can climb the stirring rod; when you shear them, they can develop forces in a direction perpendicular to the flow.

To describe this complex behavior, rheologists build constitutive models that often borrow a page from the solid mechanics playbook. These fluids have "memory" and can exhibit "elastic-like" responses. Their behavior is often captured by rate-type equations that relate a stress tensor to the rate of deformation. And just as in solids, to be physically meaningful, these equations must be objective. It is no surprise, then, that objective rates like the Truesdell rate appear in advanced fluid models, helping to capture the bizarre and wonderful phenomena observed in these complex materials. This demonstrates the unifying power of continuum mechanics: the same deep principles forged to understand the bending of a steel beam can illuminate the flow of a polymer melt.

We return to our original question: is the Truesdell rate a good definition? The evidence speaks for itself. It is not just another entry in a catalog of mathematical possibilities. It is a carefully crafted tool that has been tested and has proven its worth. It provides physically sensible predictions where simpler rates can lead to paradoxes. It elegantly accounts for the physics of compressibility by revealing its connection to the more fundamental Kirchhoff stress. It provides a robust, stable, and consistent foundation for the modern computational simulations that are indispensable to science and engineering. Its conceptual framework even finds echoes in the study of complex fluids. The journey to define the "rate of change of stress" has led us not just to a correct answer, but to a tool of remarkable power and intellectual beauty—a testament to the quest for a description of nature that is at once objective, consistent, and deeply unifying.