Objective Stress Rate

In the field of continuum mechanics, one of the most fundamental tasks is to describe how the state of stress within a material evolves as it deforms. While this may seem straightforward, a profound challenge arises when the material undergoes large rotations in addition to being stretched or compressed. How can we formulate physical laws for stress that are universal, independent of how we, the observers, might be spinning through space?

The core problem, which this article addresses, is that the simple, intuitive time derivative of the stress tensor is fundamentally flawed. It mixes the true material response with spurious effects caused by rigid body rotation, leading to laws that are not objective. To solve this, the concept of the ​​objective stress rate​​ was developed—a "smarter" derivative that provides a clean, rotation-free measure of how stress is truly changing within the material's own reference frame.

This article will guide you through this essential concept. In the first chapter, ​​Principles and Mechanisms​​, we will explore the theoretical foundation of objectivity, use analogies to understand why the naive rate fails, and examine the construction and consequences of different objective rates like the Jaumann and Truesdell rates. Subsequently, in ​​Applications and Interdisciplinary Connections​​, we will see how these theoretical ideas have critical, real-world consequences in computer simulations, structural engineering, and even the study of complex fluids.

Imagine you are trying to describe the wind. You have a weather vane, a magnificent golden rooster on a spire. As the wind blows, the rooster turns. You, a diligent scientist, note down the rate at which the pointer is changing its direction. But then a mischievous friend comes along and starts spinning the entire spire. Now, the pointer is changing direction much faster! If you naively write down this new, faster rate, you would wrongly conclude the wind has changed dramatically. Your measurement is contaminated by the spin of your instrument. To find the true change in the wind, you must be clever enough to subtract the spire's spin from your measurement.

This is precisely the dilemma we face in continuum mechanics when describing how stress in a material changes. The simple material time derivative of stress, $\dot{\boldsymbol{\sigma}}$, is like the naive measurement of the spinning rooster. It sees both the change in stress due to the material actually being squished and stretched (deformation) and the "change" in stress components that happens just because the material itself is rotating in space. A physical law, however, should not depend on how fast we decide to spin our laboratory. It must be independent of the observer, a principle we call ​​material frame indifference​​ or ​​objectivity​​. Our naive rate, $\dot{\boldsymbol{\sigma}}$, fails this test spectacularly.

Let's see why this "contamination" happens. Suppose an observer in one reference frame sees a stress tensor $\boldsymbol{\sigma}$. A second observer, whose frame is rotating with respect to the first by a rotation $\boldsymbol{Q}(t)$, will see a different stress tensor, $\boldsymbol{\sigma}^* = \boldsymbol{Q}\boldsymbol{\sigma}\boldsymbol{Q}^{\mathsf{T}}$. This is just the standard rule for how the components of a tensor change when you rotate your coordinate system.

The trouble begins when we ask how the rate of stress changes. Using the product rule for differentiation, the rate seen by the second observer is: $\dot{\boldsymbol{\sigma}}^* = \dot{\boldsymbol{Q}}\boldsymbol{\sigma}\boldsymbol{Q}^{\mathsf{T}} + \boldsymbol{Q}\dot{\boldsymbol{\sigma}}\boldsymbol{Q}^{\mathsf{T}} + \boldsymbol{Q}\boldsymbol{\sigma}\dot{\boldsymbol{Q}}^{\mathsf{T}}$

If we define the spin of the observer's frame as the skew-symmetric tensor $\boldsymbol{\Omega} = \dot{\boldsymbol{Q}}\boldsymbol{Q}^{\mathsf{T}}$, a little algebra shows that this becomes: $\dot{\boldsymbol{\sigma}}^* = \boldsymbol{Q}\dot{\boldsymbol{\sigma}}\boldsymbol{Q}^{\mathsf{T}} + \boldsymbol{\Omega}\boldsymbol{\sigma}^* - \boldsymbol{\sigma}^*\boldsymbol{\Omega}$

Look at that! The transformed rate $\dot{\boldsymbol{\sigma}}^*$ is not simply the rotated version of the original rate, $\boldsymbol{Q}\dot{\boldsymbol{\sigma}}\boldsymbol{Q}^{\mathsf{T}}$. There are two extra terms, $\boldsymbol{\Omega}\boldsymbol{\sigma}^* - \boldsymbol{\sigma}^*\boldsymbol{\Omega}$, that depend on the observer's spin $\boldsymbol{\Omega}$. This is the mathematical signature of the "spinning rooster" problem. A constitutive law like $\dot{\boldsymbol{\sigma}} = \mathbb{C}:\boldsymbol{D}$ (where $\mathbb{C}$ is an elasticity tensor and $\boldsymbol{D}$ is the rate of deformation) cannot be a fundamental law of nature, because its very form depends on the observer.

How do we fix this? We must invent a "smarter" stress rate, one that is purified of these rotational effects. We need to define a new kind of derivative, let's call it $\stackrel{\circ}{\boldsymbol{\sigma}}$, that is truly ​​objective​​, meaning it transforms cleanly like a proper tensor should: $\stackrel{\circ}{\boldsymbol{\sigma}}^* = \boldsymbol{Q}\stackrel{\circ}{\boldsymbol{\sigma}}\boldsymbol{Q}^{\mathsf{T}}$.

The most intuitive way to do this is to measure the rate of stress in a coordinate system that is spinning along with the material. This is the essence of a ​​corotational rate​​. We take our naive rate $\dot{\boldsymbol{\sigma}}$ and add correction terms that are designed to exactly cancel the spurious rotational parts. The general form looks like this: $\stackrel{\circ}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{\omega}\boldsymbol{\sigma} + \boldsymbol{\sigma}\boldsymbol{\omega}$ Here, $\boldsymbol{\omega}$ is a carefully chosen skew-symmetric tensor that represents our best guess for the material's spin rate.

A very common and natural choice is to use the spin tensor $\boldsymbol{W}$, which is simply the skew-symmetric part of the velocity gradient $\boldsymbol{L}$. This gives birth to the famous ​​Zaremba-Jaumann rate​​ (or simply ​​Jaumann rate​​): $\stackrel{\nabla}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{W}\boldsymbol{\sigma} + \boldsymbol{\sigma}\boldsymbol{W}$ This new rate is, by construction, objective. A beautiful test of its validity is to consider a body that is undergoing a pure rigid rotation without any deformation. In this case, there is no stretching or squeezing, so $\boldsymbol{D} = \mathbf{0}$. We would physically expect that the "real" rate of stress change is zero. For a pure rigid rotation, it turns out that $\dot{\boldsymbol{\sigma}} = \boldsymbol{W}\boldsymbol{\sigma} - \boldsymbol{\sigma}\boldsymbol{W}$. Plugging this into the Jaumann rate definition gives $\stackrel{\nabla}{\boldsymbol{\sigma}} = (\boldsymbol{W}\boldsymbol{\sigma} - \boldsymbol{\sigma}\boldsymbol{W}) - \boldsymbol{W}\boldsymbol{\sigma} + \boldsymbol{\sigma}\boldsymbol{W} = \mathbf{0}$. It works perfectly! Our smarter rate correctly reports that nothing interesting is happening to the stress in the material's own co-rotating frame. This allows us to formulate an objective constitutive law: $\stackrel{\nabla}{\boldsymbol{\sigma}} = \mathbb{C}:\boldsymbol{D}$.

But wait. Is the spin tensor $\boldsymbol{W}$ the only possible choice for $\boldsymbol{\omega}$? Of course not! We've opened a Pandora's box of possibilities. Physicists and engineers have defined a whole menagerie of objective rates, each based on a different definition of the material's spin.

For instance, the ​​Green-Naghdi rate​​ uses the spin of the rotation tensor $\boldsymbol{R}$ from the polar decomposition of the deformation gradient ($\boldsymbol{F} = \boldsymbol{R}\boldsymbol{U}$), which represents the rotation of the material's "fibers" themselves. Another important one is the ​​Truesdell rate​​, which is not strictly corotational but is derived by considering the rate of change of stress in the undeformed reference configuration and mapping it forward. It has a more complicated form: $\stackrel{\triangledown T}{\boldsymbol{\sigma}} = \dot{\boldsymbol{\sigma}} - \boldsymbol{L}\boldsymbol{\sigma} - \boldsymbol{\sigma}\boldsymbol{L}^{\mathsf{T}} + (\text{tr }\boldsymbol{D})\boldsymbol{\sigma}$.

All of these rates (and many others!) are perfectly objective. They all correctly handle superposed rigid motions. But they are not the same. If we subject a material to a simple shear, the spin tensor $\boldsymbol{W}$ is different from the spin of the material fibers used in the Green-Naghdi rate. This leads to a fascinating and deeply important question: if we build a constitutive model using different objective rates, do we get different physical predictions?

The answer is a resounding yes. The choice of objective rate is not a mere mathematical subtlety; it has profound physical consequences. To see this, let's consider a famous thought experiment: an elastic block undergoing a large, continuous simple shear, like a deck of cards being pushed from the top.

We use a simple rate-based constitutive model called a ​​hypoelastic​​ model, which states that the objective stress rate is proportional to the rate of deformation: $\stackrel{\circ}{\boldsymbol{\sigma}} = 2G\,\boldsymbol{D}$, where $G$ is the shear modulus.

If we use the ​​Truesdell rate​​, we find that the shear stress $\sigma_{12}$ grows linearly with the amount of shear $\gamma$: $\sigma_{12}^{T}(\gamma) = G\gamma$. This seems plausible; more shear, more stress.

But if we use the ​​Jaumann rate​​, something utterly bizarre happens. The shear stress does not grow indefinitely. Instead, it oscillates: $\sigma_{12}^{J}(\gamma) = G\sin(\gamma)$! The model predicts that as you keep shearing the material, the stress will rise, then fall, then become negative, and so on. This is highly counter-intuitive and doesn't match the behavior of most real materials in large shear. The difference between the two predictions, $\Delta \sigma_{12}(\gamma) = G(\gamma - \sin(\gamma))$, becomes enormous as the shear grows large.

This result is a red flag. It tells us that these simple hypoelastic models, while objective, are not capturing the full picture of elasticity. The model built with the Jaumann rate exhibits a strange ​​path dependence​​. If you shear it by an amount $2\pi$ and then unshear it back to the start, you end up with zero stress, but the work you've done is not zero. You've somehow generated or lost energy in a closed loop, which a truly elastic material should never do.

This brings us to the deepest idea of all. The gold standard for describing elastic behavior is ​​hyperelasticity​​. A hyperelastic material is one whose stress state is derived from a ​​stored energy potential​​, a function $\psi$ that depends only on the current state of deformation. Think of it like a perfect spring, where the potential energy is $\frac{1}{2}kx^2$. The force is the derivative of the potential, $F = kx$, and it depends only on the current position $x$, not on how it got there. The work done in moving from $x_1$ to $x_2$ and back to $x_1$ is always zero. This is a conservative system.

The work done in a deformation process can be thought of as a journey on a landscape. For a hyperelastic material, this landscape is like a real mountain range under gravity. The work done to climb from one point to another only depends on the change in altitude (the potential), not the specific path taken.

The hypoelastic model with the Jaumann rate, however, is like a "magic mountain". You can walk in a closed loop and end up back where you started, but find that you've gained or lost energy. The work done is path-dependent. Mathematically, we say the "stress power one-form is not exact". This means that for a general hypoelastic model, there is no underlying stored energy function $\psi$. This is why the Jaumann-rate model is called hypoelastic ("lesser" elasticity), not hyperelastic (true elasticity).

Is there any hypoelastic model that is also hyperelastic? Yes, but only in very special cases. For example, a model using a specific rate called the "logarithmic rate" is mathematically equivalent to a hyperelastic model whose energy is a function of the logarithmic (or Hencky) strain.

This gives us a crucial insight. If a material is truly elastic, we shouldn't be using a rate-based model in the first place. We should be defining a stored energy function, for example as a function of the right Cauchy-Green tensor, $W(\boldsymbol{C})$. This approach, used in ​​hyperelasticity​​, builds objectivity in from the very beginning because the tensor $\boldsymbol{C} = \boldsymbol{F}^{\mathsf{T}}\boldsymbol{F}$ is automatically invariant to superposed rotations. Since stress is calculated directly from the current deformation, not by integrating a rate, the entire issue of objective stress rates becomes unnecessary. Objective rates are tools for the world of inelasticity—plasticity and viscoplasticity—where the material's state inherently depends on the path taken.

After all this, you might be wondering: if the naive rate $\dot{\boldsymbol{\sigma}}$ is so wrong, why is it used in every introductory engineering textbook? The answer lies in a beautiful scaling argument.

Remember that the naive rate is contaminated by a spurious rotational term of the form $\boldsymbol{\omega}\boldsymbol{\sigma} - \boldsymbol{\sigma}\boldsymbol{\omega}$. Let's compare the size of this term to the size of the "real" stress rate coming from physical straining.

In the ​​infinitesimal-strain regime​​, where all strains and rotations are very small (say, of order $\epsilon \ll 1$), the stress is also small ($\|\boldsymbol{\sigma}\| \sim \epsilon$), and the spin rate is small ($\|\boldsymbol{\omega}\| \sim \dot{\epsilon}$). The spurious rotational rate is therefore of order $\sim \dot{\epsilon} \cdot \epsilon$. The physical rate from straining is of order $\sim \dot{\epsilon}$. The ratio of the spurious term to the physical term is of order $\epsilon$. Since $\epsilon$ is very small, the rotational error is a negligible, higher-order term! For small deformations, our naive rate is a perfectly good approximation of the true, objective rate.

However, in the ​​finite-strain regime​​, rotations can be large, even if the strains are small. The spin rate $\|\boldsymbol{\omega}\|$ can be of order 1. Now, the spurious rotational rate is of order $\sim 1 \cdot \epsilon = \epsilon$. This is the same order of magnitude as the physical stress rate! The error is no longer negligible; it's a dominant effect.

This is the final piece of the puzzle. For a slightly bending beam or a bridge under traffic, the rotations are tiny, and we can safely use the simplified small-strain theory where the distinction between stress rates vanishes. But for a spinning car tire, a piece of sheet metal being stamped into a complex shape, or the turbulent flow of a polymer, the rotations are large. In this world, the concept of the objective stress rate is not an academic curiosity—it is the essential key to writing physical laws that make any sense at all.

Now that we’ve grappled with the abstract principle of objectivity, you might be left with a nagging question: does this mathematical formalism truly matter? Is there a tangible difference between one objective rate and another, or is it all just academic bookkeeping? The answer is a resounding yes. Getting the rate right is often the difference between a computer simulation that faithfully predicts the behavior of the real world and one that produces utter nonsense.

Let us now embark on a journey to see this principle in action. We will venture into the heart of digital crash tests, witness the strange behavior of materials under extreme stress, and uncover a surprising connection to the world of flowing polymers. We will discover that this single, elegant idea is a thread that weaves through vast and varied fields of science and engineering, ensuring that our virtual worlds speak the same language as physical reality.

Imagine the immense computational power used to design everything from the lightweight frame of a modern aircraft to the crash-worthiness of a family car. At the core of these simulations are numerical methods like the Finite Element Method (FEM) and the Material Point Method (MPM), which break a complex object down into a mosaic of simpler, interconnected pieces. The simulation's job is to figure out how this entire assembly deforms, stresses, and potentially fails under load.

The global behavior is governed by grand physical laws, like the principle of virtual work, which balances internal forces against external loads. But the real soul of the simulation lives inside each tiny element: the constitutive model. This is the set of rules—the material's "DNA"—that dictates how stress responds to deformation. And it is precisely here, in the moment-to-moment update of the stress state, that our objective rates come to life.

As we've learned, we can't just use the simple time derivative of the Cauchy stress, $\dot{\boldsymbol{\sigma}}$. We need an objective rate. But which one? The world of continuum mechanics offers a whole menu of choices. The two most famous are the Zaremba-Jaumann rate, a historical workhorse, and the Green-Naghdi rate, a more sophisticated choice. The essential difference lies in what they choose to "co-rotate" with. The Jaumann rate spins with the local fluid-like vorticity of the material, $\boldsymbol{W}$, while the Green-Naghdi rate spins with the true material rotation, $\boldsymbol{R}$, extracted from the deformation itself.

Does this choice matter? Consider a cautionary tale: the case of large-amplitude cyclic shear. Imagine taking a block of metal and shearing it back and forth, over and over again, always returning to the starting point. Intuitively, if we cycle the strain symmetrically around zero, the stress response should also stabilize into a symmetric loop. However, if we run this simulation using the Jaumann rate, something bizarre happens. With each cycle, the mean stress begins to drift, or "ratchet," to higher and higher values, as if the material were being pushed progressively in one direction. It’s like rocking a car back and forth on level ground and watching it mysteriously creep forward on its own. This is physically wrong.

The Green-Naghdi rate, by correctly aligning its reference frame with the material's true rotation, predicts no such drift. It shows a stable, symmetric stress-strain loop, just as experiments do. This dramatic failure of the simpler model reveals a profound truth: the Jaumann rate, while objective, is not "integrable" to a true elastic potential. This means that for a purely elastic material, it can predict that energy is created from nothing over a closed deformation cycle—another physical impossibility! More advanced formulations, such as those based on logarithmic strain, are built upon a solid foundation of hyperelastic energy potentials, elegantly sidestepping these issues from the start. The choice of rate is not merely a matter of taste; it is a choice about the physical integrity of your simulation.

The principle of objectivity extends far beyond simple elastic-plastic models. The world is filled with materials whose behavior is far more intricate.

Consider materials that exhibit "memory." When you bend a paperclip, it not only deforms but also becomes harder to bend further. If you bend it back, it remembers the direction of the initial loading. In plasticity theory, this is modeled using a concept called kinematic hardening, which introduces an internal "backstress" tensor, often denoted $\boldsymbol{\alpha}$, that tracks the center of the yield surface in stress space. Just like the Cauchy stress $\boldsymbol{\sigma}$, this backstress tensor is a physical quantity that lives in the deforming material. Therefore, to correctly model the material's memory under large rotations, the backstress itself must be updated using an objective rate. Forgetting to do so would be like trying to navigate a spinning ship with a compass that doesn't account for the ship's rotation.

What about extreme events, like a high-speed car crash or a projectile impacting armor? Here, the material's response depends not just on the amount of strain, but also on the rate of strain and the temperature. Models like the Johnson-Cook law are designed for these scenarios. They capture this complexity in a scalar flow stress function. Because the physics of hardening is packed into scalar invariants (quantities that are naturally independent of rotation), these models are inherently compatible with any valid objective stress rate. The fundamental requirement of objectivity remains, but the specific formulation for hardening doesn't prefer one objective rate over another.

Perhaps the most dramatic consequences of choosing the right (or wrong) rate appear in the field of structural stability. An engineer must be able to predict not just how a bridge will bend under traffic, but at what point it might disastrously buckle or begin to flutter uncontrollably, like the infamous Tacoma Narrows Bridge.

These life-or-death questions are answered by analyzing the structure's tangent stiffness matrix, $K_T$, which represents its instantaneous resistance to deformation. The stability of the structure is encoded in the eigenvalues of this matrix. A crucial property of $K_T$ is its symmetry. As it turns out, the choice of objective stress rate in a simulation has a direct and profound impact on this symmetry.

Formulations based on a true energy potential, like hyperelasticity, always produce a symmetric $K_T$. However, hypoelastic models based on the Jaumann rate generally produce a non-symmetric $K_T$. The spin terms in the Jaumann rate introduce non-symmetric components into the very mathematical fabric of the stiffness matrix.

Why is a non-symmetric matrix so troubling? A symmetric system can only buckle (a static instability). A non-symmetric system, however, can also "flutter"—a dynamic instability where vibrations grow exponentially in time. A standard stability analysis that assumes a symmetric matrix would be completely blind to this possibility. An engineer using a Jaumann-rate-based model without the proper non-symmetric analysis tools might certify a design as stable, while in reality, it's a flutter catastrophe waiting to happen. Once again, a seemingly subtle choice in the constitutive model has far-reaching consequences for predicting the safety and reliability of large-scale engineering systems.

So far, our journey has remained in the realm of solids. But the beauty of a fundamental principle lies in its universality. Does the concept of objectivity appear elsewhere?

Let's venture into the world of non-Newtonian fluid mechanics—the study of strange fluids like molten polymers, paints, and biological fluids. When these complex fluids flow, they stretch and rotate, generating internal stresses, much like solids. A fluid dynamicist trying to model the flow of plastic into a mold faces the exact same problem as a solid mechanician modeling a steel beam: the constitutive equation relating stress to the rate of deformation must be frame-indifferent.

And so, fluid dynamicists, working independently, developed their own family of objective derivatives! You will encounter names like the ​​upper- and lower-convected Oldroyd derivatives​​ and the ​​Gordon-Schowalter derivative​​. While the names and physical contexts are different, these are the fluid mechanical cousins of the objective rates we've studied. In fact, they can be shown to be mathematically related, forming a continuous spectrum of possible objective rates.

This discovery is a wonderful example of the unity of physics. The same deep principle—that the laws of nature do not depend on the observer—imposes the same mathematical challenge on scientists modeling wildly different physical systems. Whether you are simulating the crash of a car, the buckling of a bridge, or the extrusion of a polymer, the need for an objective rate is a common, unifying thread. It is the essential mathematical tool that ensures our computational models are not just solving equations, but are truly speaking the language of the physical universe.