Rate of Strain

Motion is more than just moving from one point to another. For any object that can change shape—from a flowing river to a piece of metal being forged—the simple concept of velocity is insufficient. How do we precisely describe the stretching, shearing, and twisting that occurs within a deforming body? The answer lies in a powerful mathematical concept that captures the very essence of shape change: the rate of strain. This article provides a journey into this fundamental quantity. We will begin in "Principles and Mechanisms" by deconstructing motion to uncover the rate of strain tensor, exploring its mathematical properties and profound physical meanings, from volumetric change to its connection with energy and power. Subsequently, in "Applications and Interdisciplinary Connections," we will see how this single concept serves as a unifying language across diverse fields, explaining the behavior of non-Newtonian fluids, the creep of solids under high temperature, and even the mechanics of biological growth.

Imagine you are floating in a river. If the entire river moves at the same speed, you simply drift downstream. This is pure translation. But what if the water near the right bank moves faster than the water near the left bank? You would start to spin. What if the water ahead of you moves faster than the water behind you? You would be stretched. The simple concept of velocity is not enough to describe the rich motion of a continuous body like a river, a piece of metal being forged, or the air flowing over a wing. We need a way to capture how the velocity changes from point to point.

The tool for this job is a mathematical object called the ​​velocity gradient​​, denoted by $\mathbf{L} = \nabla\mathbf{v}$. It's a tensor that acts as a complete local map of the flow's character. If you tell it the relative position of a nearby water particle, it tells you the difference in your velocities. The remarkable thing about the velocity gradient is that it can always be split into two distinct parts: a symmetric part and a skew-symmetric part.

The skew-symmetric part, $\mathbf{W} = \frac{1}{2}(\mathbf{L} - \mathbf{L}^T)$, is called the ​​spin tensor​​. It describes the local rate of rigid-body rotation—the spinning part of our river analogy. If you have a body undergoing a pure rigid rotation, like a spinning top, its velocity gradient will be purely skew-symmetric, and this tensor $\mathbf{W}$ will capture its angular velocity perfectly.

The symmetric part, $\mathbf{d} = \frac{1}{2}(\mathbf{L} + \mathbf{L}^T)$, is the hero of our story. It is called the ​​rate of deformation tensor​​, or more simply, the ​​rate of strain​​. This tensor describes all the motion that actually changes the shape of a body—the stretching, squashing, and shearing. A motion is defined as "rigid" if and only if its rate of deformation is zero everywhere. For anything to bend, stretch, or flow, its rate of strain tensor must be non-zero. This is the quantity that separates a rigid stone from flowing water or a deforming metal.

Let's dissect this tensor $\mathbf{d}$ and see what its components tell us. It’s a 3x3 matrix, and each entry has a direct, physical meaning.

The elements on the main diagonal, $d_{11}$, $d_{22}$, and $d_{33}$, represent the ​​rates of elongation​​ (or contraction, if negative) along the coordinate axes. Imagine a block of material whose velocity field is given by $\mathbf{v} = (k_1 x_1, k_2 x_2, k_3 x_3)$. Points farther from the origin move faster along each axis. A quick calculation shows that the rate of strain tensor for this flow is perfectly diagonal:

This represents a pure stretch or compression along the $x_1$, $x_2$, and $x_3$ axes, with the rates of stretching given by the constants $k_1$, $k_2$, and $k_3$.

The sum of these diagonal elements is the ​​trace​​ of the tensor, $\text{tr}(\mathbf{d}) = d_{11} + d_{22} + d_{33}$. This scalar quantity has a profound physical meaning: it is the ​​volumetric strain rate​​, or the rate of change of volume per unit volume. If you want to know how quickly a small parcel of fluid is expanding or being compressed, you just need to calculate the trace of its rate of strain tensor. If $\text{tr}(\mathbf{d}) = 0$, the material is undergoing ​​incompressible​​ flow; its volume is not changing, even as its shape might be distorting wildly. This is an excellent approximation for most liquids and is a cornerstone assumption in modeling the plastic flow of metals.

The off-diagonal elements, like $d_{12}$, describe the ​​rate of shearing​​. They measure how quickly the angles between material lines are changing. For example, the component $d_{12} = \frac{1}{2}(\frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1})$ quantifies half the rate of decrease of the angle between two line segments that were initially parallel to the $x_1$ and $x_2$ axes. A non-zero shear rate means that an imaginary square drawn in the material is being deformed into a rhombus.

Even in a complex flow with both stretching and shearing, we can ask: are there any special directions in which a line element is only stretched, with no rotation? The answer is yes. These directions are called the ​​principal directions of strain​​, and the corresponding rates of stretching are the ​​principal strain rates​​. Mathematically, they are the eigenvectors and eigenvalues of the rate of strain tensor $\mathbf{d}$. This is a beautiful piece of physics: the tensor's abstract mathematical properties (its eigenvalues and eigenvectors) correspond directly to the most physically intuitive aspects of deformation (the directions and magnitudes of maximum stretch).

We've called $\mathbf{d}$ the "rate of strain," which strongly implies it is the time derivative of some measure of strain. Let's see if this is true. In the world of ​​small deformations​​, where displacements are tiny, we define the infinitesimal strain tensor as $\boldsymbol{\varepsilon} = \frac{1}{2}(\nabla \mathbf{u} + (\nabla \mathbf{u})^T)$, where $\mathbf{u}$ is the displacement field. If we take the material time derivative of this strain tensor (i.e., we follow a particle and ask how its strain is changing), we find a wonderfully simple result. Under the small-strain approximation, this time derivative is exactly equal to the rate of deformation tensor:

This result is a crucial check on our intuition and confirms the name "rate of strain."

But what about ​​large deformations​​, where things can stretch and twist dramatically? Here, the infinitesimal strain tensor is no longer adequate. We need a more robust measure, like the ​​Right Cauchy-Green deformation tensor​​, $\mathbf{C} = \mathbf{F}^T \mathbf{F}$, where $\mathbf{F}$ is the full deformation gradient. This tensor accurately measures the squared change in length of material fibers. What is its rate of change? In a truly profound result, it can be shown that the material time derivative of the Cauchy-Green tensor is directly related to the rate of deformation tensor $\mathbf{D}$ (the large-deformation counterpart of $\mathbf{d}$):

And if we consider the strain relative to the current state, the relationship becomes even more stark. The time derivative of the relative Cauchy-Green tensor, at the instant it is being calculated, is simply twice the rate of deformation tensor. This solidifies the role of $\mathbf{D}$ as the true, fundamental "speedometer" for strain, valid for any deformation, no matter how large.

So, the rate of strain tensor tells us how a material's shape is changing. Why is this so monumentally important in physics and engineering? Because it is inextricably linked to forces and energy.

When you deform a material, you do work on it. The rate at which you do this internal work—the power—is what heats the material, stores elastic energy, or drives chemical reactions. It turns out that this internal power per unit volume, $\mathcal{P}_{int}$, is given by an elegantly simple expression:

Here, $\boldsymbol{\sigma}$ is the ​​Cauchy stress tensor​​ (the true, physical measure of internal forces) and the ":" symbol represents a double dot product, which is a way of "multiplying" two tensors to get a scalar. This relationship means that stress and the rate of strain are ​​work-conjugate​​. They form a pair. Stress is the "force" of the continuum, and rate of strain is the "velocity" of deformation. Their product is power. This central relationship, which arises from the Principle of Virtual Work, is the foundation for almost every constitutive law that describes how a material behaves. It is the bridge connecting the kinematics of motion ($\mathbf{d}$) to the kinetics of forces ($\boldsymbol{\sigma}$).

This work-conjugate relationship gives us the power to model incredibly complex material behaviors. Take a metal paperclip. If you bend it slightly, it springs back. This is ​​elastic​​ deformation. If you bend it too far, it stays bent. This is ​​plastic​​ deformation. The rate of strain tensor allows us to build a beautiful framework to describe this.

The key idea is that the total deformation is a combination of a permanent, plastic part and a recoverable, elastic part. In finite deformation theory, this is expressed with the multiplicative decomposition of the deformation gradient, $\mathbf{F} = \mathbf{F}^e \mathbf{F}^p$. The magic happens when we look at the rates. The total rate of deformation, $\mathbf{D}$, can be additively split into an elastic part and a plastic part:

Now, let's look at the stress power using this decomposition:

(Here we use the Kirchhoff stress $\boldsymbol{\tau}$, a close relative of the Cauchy stress, which is naturally work-conjugate to $\mathbf{D}$ per unit reference volume.)

Each term has a clear physical meaning. The term $\boldsymbol{\tau} : \mathbf{D}^e$ represents the rate of work that is stored as recoverable elastic potential energy, like the energy in a compressed spring. The second term, $\mathcal{D} = \boldsymbol{\tau} : \mathbf{D}^p$, is the ​​plastic dissipation​​. This is the power that is irretrievably lost as heat. It's why the paperclip gets warm when you bend it back and forth! The second law of thermodynamics demands that this dissipation can never be negative, $\mathcal{D} \ge 0$.

From the simple idea of a velocity gradient, we have journeyed to the heart of thermodynamics and material science. The rate of strain tensor is more than just a mathematical definition; it is a fundamental quantity that measures the very essence of shape change, governs the flow of energy, and allows us to distinguish between a springing elastic solid and a flowing plastic metal. It's a testament to the unifying power of physical principles, showing how the geometry of motion is profoundly and beautifully connected to the mechanics of force and the laws of energy.

Now that we have grappled with the mathematical machinery of the rate of strain tensor, you might be tempted to put it back in its box, a tidy but abstract tool for the specialist. But to do so would be to miss the entire point! This tensor is not some isolated piece of intellectual machinery; it is a golden key that unlocks a breathtaking variety of phenomena in the world around us. It is the common language spoken by flowing honey, creeping glaciers, growing trees, and the turbine blades in a jet engine. Let us now take a journey through these diverse realms and see how this single concept brings a beautiful and unexpected unity to them all.

We all have an intuitive feel for how water flows. If you stir it, the resistance you feel is proportional to how fast you stir. This simple, linear relationship is the essence of a Newtonian fluid. The stress—the internal force within the fluid—is directly proportional to the rate of strain tensor, $D$. A faster deformation rate means a proportionally higher stress. It’s a clean, simple picture.

But the world is full of "weird stuff" whose behavior defies this simple rule. Think of ketchup: it sits stubbornly in the bottle until you shake it vigorously, at which point it suddenly flows freely. This is called shear thinning. How do we begin to describe such behavior? We must allow for a more sophisticated conversation between stress and strain rate. The Reiner-Rivlin fluid model is a beautiful step in this direction. It proposes that for an isotropic fluid, the stress doesn't have to depend just on $D$, but can also depend on its square, $D^2$. The constitutive law takes the general form:

Here, $\alpha_1$ and $\alpha_2$ are not necessarily constants; they can be functions of the invariants of the strain rate, meaning they can change depending on how intense the deformation is. This simple-looking addition of the $D^2$ term opens the door to a rich world of non-Newtonian behaviors.

To understand the origin of these behaviors, we must look deeper, into the microscopic structure of the fluid. Consider a polymer solution, which is like a tangled mess of long, spaghetti-like molecules. At rest, these molecules are randomly coiled. When the fluid is sheared slowly, the molecules have time to wriggle and relax back to their coiled state. The situation is not so different from a Newtonian fluid. But what happens if we shear it quickly? The story changes completely. The external deformation rate, characterized by the shear rate $\dot{\gamma}$, starts to compete with the polymer's intrinsic relaxation rate, $1/\tau$, where $\tau$ is the time it takes for a stretched molecule to relax back to a coil. The crucial parameter is the dimensionless ratio of these two rates, the Weissenberg number, $Wi = \dot{\gamma}\tau$.

When $Wi \ll 1$, the molecules relax much faster than they are deformed. They remain happily coiled, and the viscosity is constant. When $Wi > 1$, the flow is deforming the molecules faster than they can relax. They are forced to stretch out and align with the flow, like logs floating down a river. This alignment makes it much easier for the fluid layers to slide past one another, and the apparent viscosity drops. This is shear thinning! The same principle explains why you need to stir paint before using it, or why silly putty can stretch slowly but snap if you pull it fast.

We tend to think of solids as rigid and unyielding. But under the right conditions—high stress or high temperature—solids flow. This plastic deformation, a permanent change in shape, is at its heart a rate process, governed by the same fundamental ideas we've been discussing.

On a macroscopic level, we can measure a plastic strain rate. But where does it come from? The answer lies in the microscopic world of the crystal lattice. Real crystals are not perfect; they are riddled with line defects called dislocations. Plastic deformation occurs when these dislocations glide through the crystal, like a ruck in a carpet being pushed from one end to the other. The macroscopic plastic shear strain rate, $\dot{\gamma}$, is the collective result of this microscopic motion. The famous Orowan equation provides the fundamental link:

Here, $\rho_m$ is the density of mobile dislocations, $b$ is their characteristic size (the Burgers vector), and $\bar{v}$ is their average velocity. The continuum concept of strain rate is thereby demystified; it is a direct measure of the flux of these tiny defects.

This perspective is crucial for understanding one of the most important failure modes in engineering: creep. Creep is the slow, time-dependent deformation of a material under a constant stress, especially at high temperatures. Imagine a turbine blade inside a jet engine. It is subjected to immense stress at temperatures that would make steel glow red. Over thousands of hours, it slowly stretches. If it stretches too much, the engine fails. The lifetime of that engine depends entirely on understanding and predicting the creep strain rate.

A typical creep test reveals three stages. In the first, primary creep, the strain rate decreases over time. This happens because as dislocations move and multiply, they get tangled up, creating a "traffic jam" that hinders further motion. This process is called strain hardening. At the same time, the high temperature allows for recovery processes, like dislocation climb, where dislocations find ways to get around obstacles. In the primary stage, hardening wins, and the flow slows down.

Eventually, a dynamic equilibrium is reached where the rate of hardening is perfectly balanced by the rate of recovery. This marks the beginning of the second stage, secondary or steady-state creep, where the strain rate is constant. This minimum creep rate is the most critical parameter for engineering design. Decades of research have shown that it can be described remarkably well by the Norton Power Law:

Every part of this equation tells a story. The strain rate increases with stress $\sigma$ raised to a power $n$, where $n$ reveals the specific microscopic mechanism at play (e.g., dislocation climb). The Arrhenius term, $\exp(-Q/RT)$, tells us that creep is a thermally activated process, a race against an energy barrier $Q$, which becomes easier to win at higher temperatures $T$.

The rate-dependence of plastic flow also explains a common materials test: hardness. In a hardness test, a sharp indenter is pressed into a material. The "hardness" is related to the force required to make a permanent dent of a certain size. If you perform this test on a polymer by applying the load very quickly, you will measure a higher hardness value than if you apply it slowly. Why? A rapid load corresponds to a very high rate of strain under the indenter. This high strain rate increases the material's intrinsic resistance to plastic flow—a phenomenon known as strain rate hardening—so a smaller dent is formed for the same force, leading to a higher calculated hardness.

The rate of strain tensor is not confined to mechanics; its influence extends across physics, engineering, and even biology.

When a material deforms viscously or plastically, energy is dissipated, usually as heat. Bend a paperclip back and forth, and you'll feel it get warm. The rate at which this energy is dissipated per unit volume is called the stress power, and it is given by a beautifully simple expression: the double contraction of the stress tensor and the rate of strain tensor, $\mathcal{P} = \sigma_{ij} D_{ij}$. This term connects the kinematics of deformation ($D$) directly to the first law of thermodynamics. It is the price, in energy, that must be paid to make a material flow.

The rich, time-dependent behavior of materials presents a challenge: how can we experimentally pick apart the different contributions to deformation? Clever experimental designs, centered on the concept of strain rate, provide the answer. Consider a material that exhibits both instantaneous elastic (spring-like) response and time-dependent viscoplastic (creep) flow. By applying a sudden jump in stress at a constant temperature, we can measure the instantaneous jump in strain. Since viscous flow takes time, this instantaneous strain jump must be purely elastic, allowing us to measure the material's stiffness. Then, by applying a sudden jump in temperature at a constant stress, we can observe the resulting change in the creep rate. Since creep is a thermally activated process, this jump in rate allows us to directly calculate the activation energy $Q$ for the underlying atomic mechanisms. These "jump tests" are a powerful form of physical interrogation, allowing us to extract a material's deepest secrets.

Perhaps the most exciting application of these ideas lies at the frontier of biology. How does a plant grow? How does an embryo develop its shape? These are not just questions of biochemistry; they are questions of mechanics. A growing tissue can be viewed as a continuum that is actively changing its own reference state. A powerful approach models the total deformation, $F$, as a product of elastic deformation, $F_e$, and a growth tensor, $F_g$, which describes the local creation of new mass ($F = F_e F_g$). From this, we can derive the velocity gradient and the rate of strain tensor, just as we did for inert matter. The trace of the strain rate tensor, $\nabla \cdot \mathbf{v}$, represents the local volumetric growth rate. By applying the principles of continuum mechanics, scientists can now build predictive models of morphogenesis, linking gene expression to the physical forces and deformations that shape a living organism.

Finally, let's not forget the simple, elegant geometry that underlies it all. The rate of strain tensor $\mathbf{d}$ is a symmetric tensor, which means it has real eigenvalues (principal strain rates) and orthogonal eigenvectors (principal axes). These axes represent the directions in which a tiny sphere of material is being stretched or compressed most rapidly. They give us a local, instantaneous picture of the flow's geometry—a map of pure deformation, stripped of any rigid body rotation.

From the kitchen to the jet engine, from a steel beam to a living cell, the rate of strain tensor is the unifying thread. It is a testament to the power of physics to find simple, profound principles that govern the complex and beautiful motion of the world.