The Corotational Rate: A Principle of Objectivity in Continuum Mechanics

Describing how a material deforms and resists forces is a central goal of continuum mechanics. However, a significant challenge arises when a body is both deforming and rotating: how can we separate the true change in internal stress from the apparent change caused by the body simply tumbling through space? This "observer's dilemma" means that simple, intuitive physical laws often fail, predicting stress from pure rotation, which is a physical contradiction. To formulate laws that are valid regardless of the observer's motion—a concept known as objectivity—we need a more sophisticated tool.

This article introduces the ​​corotational rate​​, an elegant solution that provides an objective measure of change in deforming materials. First, in the "Principles and Mechanisms" chapter, we will explore the fundamental problem of objectivity, see how the corotational rate is constructed to solve it, and examine different types of rates like the Jaumann and Green-Naghdi rates. Subsequently, the "Applications and Interdisciplinary Connections" chapter will demonstrate the vital role of this concept in practical fields, showing how it enables realistic simulations in computational engineering and how the same idea unifies our understanding of materials as diverse as metals and liquid crystals.

Imagine you are a tiny physicist, and your laboratory is a small speck of dust embedded in a piece of taffy. As a giant pulls and twists the candy, your world is thrown into chaos. It stretches, it shears, it tumbles end over end. Your mission, as a good physicist, is to write down the laws that describe how the taffy deforms—how it resists being stretched. But how can you distinguish the pure stretching from the fact that your entire lab is spinning wildly through space? If you measure the velocity of a nearby dust speck, is it moving away because the taffy is stretching, or just because you're both on a spinning merry-go-round? This is the central problem of describing change in continuum mechanics, and its solution is one of the most elegant ideas in the field: the ​​corotational rate​​.

Let's make our thought experiment more precise. The state of stress in a material—a measure of the internal forces—is described by the ​​Cauchy stress tensor​​, which we can call $\boldsymbol{\sigma}$. The most natural way to describe how this stress changes in time is to use the ​​material time derivative​​, $\dot{\boldsymbol{\sigma}}$, which tracks the rate of change as we follow a specific material particle. A simple, intuitive constitutive law might say that the rate of change of stress is proportional to the rate of deformation.

But this intuitive approach leads to a spectacular failure. Consider a solid body that is simply spinning like a top at a constant angular velocity, with no change in its shape or size. Since the body is not deforming, our intuition screams that the stress in the material should not be changing. Yet, if we use a naive law like $\dot{\boldsymbol{\sigma}} = \lambda\,\mathrm{tr}(\mathbf{L})\,\mathbf{I} + 2\,\mu\,\mathbf{L}$ (where $\mathbf{L}$ is the velocity gradient), we find that it predicts a non-zero $\dot{\boldsymbol{\sigma}}$! It seems to suggest that stress is being created from nothing, just by the act of rotation.

This is a physical contradiction. The problem is that our simple time derivative, $\dot{\boldsymbol{\sigma}}$, is not ​​objective​​. It doesn't just measure the change in the material; it also picks up the change due to the rigid rotation of the material itself. It mixes the physics of deformation with the kinematics of rotation. A valid physical law must be independent of such rigid motions; it must give the same result whether we observe the spinning top from a fixed laboratory or from a frame that is spinning along with it. This is the ​​principle of material frame indifference​​, and it demands that we find an ​​objective stress rate​​.

The solution is as simple as it is brilliant: if the spinning is the problem, let's stop observing from the fixed "ground" and instead build our laboratory on a conceptual merry-go-round that spins exactly with the material. This is the ​​corotational frame​​. By describing the physics in this rotating frame, the confounding effects of the body's overall rotation vanish, allowing us to see the pure deformational changes.

Mathematically, this means we "correct" the naive material time derivative. The instantaneous rotation of the material is described by the ​​spin tensor​​, $\mathbf{W}$, which is the skew-symmetric part of the velocity gradient $\mathbf{L}$. The simplest corotational rate, known as the ​​Zaremba–Jaumann rate​​ (or simply Jaumann rate), is defined as:

The correction terms, $-\mathbf{W}\boldsymbol{\sigma} + \boldsymbol{\sigma}\mathbf{W}$, precisely cancel out the stress change generated by pure rotation. If we go back to our spinning, undeforming body, we find that the Jaumann rate is exactly zero, $\overset{\circ}{\boldsymbol{\sigma}} = \mathbf{0}$, restoring physical consistency. This process of defining a rate that is insensitive to rigid rotation is the essence of objectivity. The Jaumann rate measures the rate of change of stress as seen by an observer spinning with the local neighborhood of the material point.

Now, a fascinating question arises: is the spin tensor $\mathbf{W}$ the only choice for our corotating merry-go-round? The answer is no, and this leads to a whole "zoo" of different objective rates, each with its own physical interpretation and mathematical justification.

1. ​​The Green–Naghdi Rate​​: The total deformation of a body is captured by the ​​deformation gradient​​ $\mathbf{F}$. A fundamental theorem in mechanics, the ​​polar decomposition​​, tells us that any deformation can be uniquely split into a pure stretch followed by a pure rotation: $\mathbf{F} = \mathbf{R}\mathbf{U}$. Here, $\mathbf{R}$ represents the rotation of the material fibers themselves. We could argue that this rotation, $\mathbf{R}$, defines a more physically intrinsic rotating frame than the one associated with the spin tensor $\mathbf{W}$. The spin of this "material" frame is given by $\boldsymbol{\Omega} = \dot{\mathbf{R}}\mathbf{R}^{\top}$. In general, this spin $\boldsymbol{\Omega}$ is not the same as the continuum spin $\mathbf{W}$. Using $\boldsymbol{\Omega}$ in our correction formula gives us the ​​Green–Naghdi rate​​. For many practical problems, like the large shearing of a material, the Green–Naghdi rate gives more physically plausible results, avoiding unphysical stress oscillations that can sometimes appear when using the Jaumann rate.

2. ​​The Truesdell Rate​​: We can approach the problem from another direction entirely. The Cauchy stress $\boldsymbol{\sigma}$ represents force per unit current area. As the material deforms, this area stretches, rotates, and changes size. A more fundamental approach might be to define a rate that accounts for this convection and "dilution" of the stress tensor. This line of reasoning leads to the ​​Truesdell rate​​. Its definition, $\dot{\boldsymbol{\sigma}} - \mathbf{L}\boldsymbol{\sigma} - \boldsymbol{\sigma}\mathbf{L}^{\top} + \mathrm{tr}(\mathbf{L})\boldsymbol{\sigma}$, is more complex. Unlike the Jaumann rate, which only depends on the spin part of the velocity gradient, the Truesdell rate depends on the full velocity gradient $\mathbf{L}$, accounting for stretching ($\mathbf{D}$) and volume change ($\mathrm{tr}(\mathbf{L})$) in its very definition. It is the rate that naturally arises when pulling back a stress tensor to a reference configuration and then pushing it forward again.

The choice of which rate to use is not arbitrary; it has real consequences. In computational mechanics, for instance, the choice of objective rate affects the ​​consistent tangent modulus​​—the stiffness matrix used in numerical simulations to solve for the material's response—which can alter the convergence and accuracy of a simulation.

With all these different rates, one might worry that the physics has become ambiguous. But here, nature reveals a beautiful, unifying simplicity. Let's return to the original problem that started our quest: a body undergoing a pure rigid rotation, with no deformation ($\mathbf{D} = \mathbf{0}$). What do these different rates predict for the evolution of stress in this simple case?

It turns out that for this specific motion, the Jaumann, Green-Naghdi, and Truesdell rates all become mathematically identical! They all simplify to the same differential equation: $\dot{\boldsymbol{\sigma}} = \mathbf{W}\boldsymbol{\sigma} - \boldsymbol{\sigma}\mathbf{W}$. The solution to this equation is simply $\boldsymbol{\sigma}(t) = \mathbf{R}(t)\boldsymbol{\sigma}(0)\mathbf{R}^{\top}(t)$, which states that the stress tensor simply rotates rigidly with the body. This is a profound result. Although these rates are derived from different physical arguments and behave differently in complex deformations, they all agree on the fundamental principle: they all correctly handle pure rotation, which was the very problem they were invented to solve. This demonstrates the profound consistency underlying the theory.

The corotational framework is a powerful tool for analyzing materials that undergo large rotations but only small or moderate strains. By filtering out the large rigid body rotations, it allows us to use simpler, small-strain constitutive laws in the local, rotating frame.

However, this magic has its limits. The entire framework hinges on the assumption that the local deformation is small. What happens if our piece of taffy is not only rotating but also being stretched to twice its original length? In this case, the strain is no longer small (a 100% strain!). The small-strain measures and additively-decomposed plasticity models used in the corotated frame break down completely. They are no longer accurate kinematic measures, they violate thermodynamic principles of energy conjugacy, and they cannot capture the complex physics of finite plastic deformation.

At this point, the corotational small-strain assumption must be abandoned. We are forced to move to a fully ​​finite-strain​​ theory, employing more sophisticated strain measures (like the Green-Lagrange or logarithmic strain) and kinematically-correct plasticity models based on the multiplicative decomposition of the deformation gradient ($\mathbf{F} = \mathbf{F}_{\mathrm{e}}\mathbf{F}_{\mathrm{p}}$). The concept of objectivity remains absolutely critical, but it becomes embedded within a more powerful and general mathematical structure. The journey to understand the humble piece of twisting taffy takes us ever deeper into the beautiful and intricate world of continuum mechanics.

Now that we have grappled with the principles and mechanisms of the corotational rate, we might find ourselves asking a very fair question: so what? What is this elegant mathematical machinery for? Is it a beautiful but abstract concept, confined to the blackboards of theoreticians? The answer, you might be happy to hear, is a resounding no. The idea of an objective rate is not just a theoretical nicety; it is a vital and powerful tool that unlocks our ability to describe and predict the behavior of the physical world in a staggering range of applications, from the crash of a car to the shimmering display of a smartphone. It is one of those wonderfully unifying concepts that, once understood, reveals a hidden connection between seemingly disparate corners of science and engineering.

Let’s begin our journey by imagining ourselves as a tiny observer, a speck of dust embedded in a block of clay. The clay is being squashed, stretched, and twisted all at once. From our vantage point, everything is a dizzying swirl of motion. How could we possibly figure out how much our immediate neighborhood is being truly deformed—the part that causes internal resistance—versus how much we are just being passively tumbled and spun around? This is the essential problem that the corotational rate solves. It gives our observer a "non-spinning" reference frame, a way to measure the pure rates of stretching and shearing, untainted by the rigid rotation of the material. It allows us to write down physical laws that are objective, that do not depend on the arbitrary spin of the observer.

In the world of engineering, especially in the design of cars, airplanes, and buildings, we often rely on computer simulations to test how materials and structures will behave under extreme stress. For very small deformations—a floor joist bending slightly under your weight—the math is relatively simple. But what about the large, violent deformations of a car crash, the intricate folding of sheet metal into a complex part, or the slow, immense creep of a glacier? Here, the material not only stretches but also rotates dramatically.

If we naively try to build a simulation by relating the simple time derivative of stress to the rate of stretching, we will get disastrously wrong answers. The simulation would predict stresses appearing even if we just spin an object without deforming it at all! This is because the simple time derivative, as we've seen, mixes the real change in stress from deformation with an "apparent" change that comes purely from the object's rotation. A material's constitutive law—its rulebook for how it resists deformation—must be objective. It cannot depend on the observer's spinning perspective.

This is precisely where the corotational rate becomes the hero of the story. In modern computational tools like the Finite Element Method (FEM) or the Material Point Method (MPM), the stress in each little piece of the simulated object is updated incrementally in time. To do this correctly, the simulation must use an objective stress rate, such as the Jaumann rate. The algorithm essentially says: "To find the new stress, take the old stress, and add a change. But that change must be calculated based only on the part of the motion that is pure stretching, not the rotational part".

In practice, this is accomplished through elegant numerical schemes. A common approach is to computationally "un-rotate" the changing material element, apply the stretching according to the material's physical laws in this non-rotating frame, and then "re-rotate" the result back into the global frame. This process often involves mathematical objects like the matrix exponential, which precisely handle the integration of these rotational dynamics over a small time step. This isn't just an academic exercise; it is the core machinery inside the software that ensures a simulated car crumples realistically, rather than behaving in some physically nonsensical way. Furthermore, this objective formulation is critical for deriving the material's effective stiffness during the deformation, a quantity known as the algorithmic tangent, which is essential for the stability and efficiency of the entire simulation.

Once we accept that we must "subtract out" a spin to be objective, a more subtle and profound question emerges: which spin should we subtract? The total, observable spin of the material element? Or is there a more physically motivated choice? This question leads us into the heart of modern material science, particularly the theory of plasticity, which describes permanent deformation.

Imagine a material that can deform both elastically (like a spring) and plastically (like a sticky slider). This is a good model for most metals. The total deformation can be thought of as a product of these two parts: an elastic distortion of the atomic crystal lattice, and a plastic distortion from planes of atoms slipping past one another. The total spin of the material element is therefore a sum of the spin from the elastic lattice rotation ($W_e$) and the spin from the plastic slip ($W_p$).

Now, the stress in the material arises from the stretching and straining of the elastic atomic lattice. It seems physically natural that the constitutive law for the material's "springiness" should only care about what's happening to the lattice itself. It shouldn't be directly affected by the complex tumbling and vorticity generated by the plastic slipping process. This insight suggests that instead of using the total spin to define our corotational rate, we should use only the elastic spin $W_e$.

This choice gives rise to objective rates like the Green-Naghdi rate. By using the elastic spin, we formulate a "cleaner" constitutive law where the objective stress rate depends only on the elastic part of the deformation rate. This effectively decouples the elastic response from the intricacies of the plastic spin. This is a beautiful example of how deeper physical reasoning leads to a refinement of our mathematical tools, creating models that are not only predictive but also more faithful to the underlying physics.

Perhaps the most striking illustration of the corotational rate's power is that it appears in a completely different field of physics: soft matter. Let's leave the world of steel beams and car crashes and consider the material inside your laptop or smartphone screen—a liquid crystal.

A nematic liquid crystal can be pictured as a fluid composed of microscopic, rod-like molecules. While the material can flow like a liquid, the rods tend to align with their neighbors along a common direction. This average orientation at any point is described by a vector field called the "director," denoted by $\mathbf{n}$. The magic of a liquid crystal display (LCD) comes from our ability to control this director field with electric fields, which in turn changes the material's optical properties.

But what happens when a liquid crystal flows? The fluid motion, described by a velocity field $\mathbf{v}$, carries the director rods along. In a region of non-uniform flow, the fluid will have both a rate-of-strain (stretching) and a vorticity (spin). Both of these affect the director. The strain tends to align the rods in a particular way, while the vorticity simply tumbles them around. To describe the physics of how the director orientation evolves, we need an equation that separates these effects.

The celebrated Ericksen-Leslie theory provides just such an equation. At its heart lies a balance of torques on the director. The viscous torque, exerted by the flowing fluid on the director, depends on the director's rate of change. But just as with stress in a solid, the simple time derivative of the director is not the right physical quantity. We need a rate of change that is independent of the local fluid's rigid-body rotation. The theory naturally introduces a co-rotational derivative of the director, $\mathbf{N} = \dot{\mathbf{n}} - \mathbf{W}\cdot \mathbf{n}$, where $\mathbf{W}$ is the vorticity, or spin tensor, of the fluid flow.

This is exactly the same mathematical structure we saw in solid mechanics! The co-rotational rate $\mathbf{N}$ represents the rate of change of the director's orientation as seen by an observer tumbling with the local fluid element. The viscous torque is then found to be proportional to this objective rate. The full theory connects the director dynamics, the fluid flow, and the stress tensor through a set of viscosity coefficients, embodying a rich and complex coupling.

It is a remarkable testament to the profound unity of physics that the very same mathematical idea allows us to formulate objective laws for materials at opposite ends of the mechanical spectrum. The corotational rate is a universal tool for disentangling true physical change from the trivial effects of rotation, whether we are describing the lattice of a deforming metal or the tumbling dance of molecules in a flowing liquid crystal. It offers us, in every case, a truly objective perspective.