The Cauchy stress tensor

The Cauchy stress tensor

It feels appropriate to mention here my journey in my quest to understand the Cauchy stress tensor.

The mysterious stress tensor has eluded me for a long time but I've finally come around to understanding it. The story begins surprisingly with Hooke's law! The first time I've encountered the stress tensor was when I tried to study Einstein's field equations. I thought that I knew enough Riemannian geometry to be able to understand the equations, but the formidable stress-energy tensor stumped me and I gave up for years. I came back to it at some point from the angle of Continuum mechanics; but the meaning of body forces and surface forces again felt cryptic and once more I quit. Recently I've been studying Zohdi's Finite Element Primer for Beginners and his Appendix B renewed my interest in understanding body and surface forces. Equipped with renewed determination, I decided to research material that would explain these forces and found a great ally in the notes of Michael E. McIntyre. With their aid, I derived Euler's equation I realized I was very close to the stress tensor AND Navier-Stokes equations. I was stoked! The final blow to ignorance came from humbling myself by reading the chapter on stress from the undergraduate book by Giancoli and watching a youtube video on the stress tensor.

If you've come across this article in hopes of understanding the stress tensor, I hope that you will leave satisfied!

Hooke's law

From Doug Giancoli's Physics, §12 Static Equilibrium; Elasticity and Fracture, I learned that Hooke's law,

\begin{align} F = -kx, \end{align}

is not merely about springs but applies generally to materials to tell us how they deform when forces push them or pull them (approximately and as long as $x \ll 1$.)

For example, for a solid cylinder of length $l$ and cross-sectional area $A$ if we apply a force $F$ on its ends we have

\begin{align} \frac{\Delta l}{l} = \frac{1}{E}\frac{F}{A}. \end{align}

The factor $\frac{\Delta l}{l}$ is called the strain while $\frac{F}{A}$ is called the stress. For example, a strain of $1$ means that the material is now twice as long. The constant $E$ is Young's modulus, it depends on the material and orientation (for instance, pulling perpendicularly to the grain in wooden material stretches differently than pulling parallel to it.)

Hooke's law implies that the conditions $\sum F = 0$ and $\sum \tau = 0$ (net linear and torsion forces) are not equivalent to no forces being present. This is simple but important to keep in mind; it tells us that internally, something is happening to the material.

Another type of deformation is shear deformation that occurs from shear stress. This occurs when the forces are tangent to the surfaces they are applied. A simple example is when we push with our hand the top cover of a book on a table; a shear deformation will occur. Again it follows Hooke's law but with a different constant called the shear modulus.

This was a real aha moment for me; I'm surprised that in my high school or undergrad education I was not told of this fact about Hooke's law. Hooke's law seems a lot more important than some spring-and-load exercises.

All in all, a cube may be deformed by forces in 9 different ways: 3 by pulling opposite sides in the normal direction and 6 by tangent forces. This is all collected in a 2-tensor $\sigma$ called the Cauchy stress tensor. For any normal vector $\vec{n}$, the quantity $\vec{n}\cdot\sigma$ is the stress force on the infinitesimal surface $\delta S$ corresponding to $\vec{n}$, and for any surface $S$, the quantity $\int_S \vec{n}\cdot\sigma dS$ is the net force on the surface, which by Stoke's theorem equals $\int_V \nabla\cdot\sigma dV$ for the enclosed volume $V$, thus $\nabla\cdot\sigma$ is the net force per volume.

I recommend chapter 12 from Giancoli's book and this youtube video if something hasn't made sense so far.

Example equations using the stress tensor

Cauchy's momentum equation

From Michael McIntyre's IB lecture notes on fluid dynamics (2nd section), I've learned how to understand the terms appearing in the momentum equation.

Viscocity is the internal friction in a fluid. For an inviscid fluid, where $m$ is the mass distribution, $\vec{v}$ is the velocity field, $\vec{F}$ are the body forces per volume and $p$ is the pressure, the following equation holds.

\begin{align} \frac{d}{dt} \int_V m \vec{v} dV = \int_V m\vec{F} dV - \int_S p\vec{n} dS - \int_S m\vec{v}(\vec{v}\cdot\vec{n})dS \end{align}

Numbering these terms as 1, 2, 3, 4, we have:

1. $m\vec{v}$ is the momentum field and the integral over $V$ is the total momentum in $V$. (check units: the time derivative of momentum is force from Newton's second law $F = \dot{p}$.)
2. A body force is really acceleration per volume; thus $\vec{F}dV$ is the acceleration of the particles in the fluid due to the body forces. (An example body force is $g$, the gravity of Earth.)
3. Pressure is the net momentum transfer between inner and outer regions of a surface (force per unit area.) Pressure in fluids is a scalar quantity, see Pascal's law and How Pressure Became a Scalar, Not a Vector. The basic idea is this: fluid is defined as not being able to hold its shape in the presence of a force. If fluid is in equilibrium, it means that the pressure in all directions is the same.
4. We're trying to determine how much fluid (momentum) has exited or entered $V$. If $m$ fluid moves with velocity $v$ through an opening $dS$ of the surface for $dt$ time, it traverses through a cylinder of base $dS$ and height $\vec{v}\cdot\vec{n}dt$. By multiplying the volume of the cylinder with the momentum field $m\vec{v}$ we obtain the net change in momentum from $dS$ in an instant $dt$.

The corresponding pointwise equation is Euler's equation. It follows by Stoke's theorem on the surface integrals and by using the conservation of mass equation $\partial_t m = -\nabla\cdot m\vec{v}$ (this is the continuity equation for mass transport). It is given by:

\begin{align} \label{eq:euler} m\frac{D\vec{v}}{Dt} = -\nabla p - m\vec{F}. \end{align}

Here $\frac{D\vec{v}}{Dt}$ denotes the material derivative of $\vec{v}$, that is $\partial_t\vec{v} + (\vec{v}\cdot\nabla)\vec{v}$.

By making some simplifications, we can arrive at Bernoulli's principle. Let $\vec{\omega} := \nabla\times\vec{v}$ and $H := u^2/2 + p/m + \Phi$, and assume steady flow ($\partial_t\vec{v} = 0$), constant pressure, and a potential for the body forces ($\vec{F} = -\nabla\Phi$), and write Euler's equation ($\ref{eq:euler}$) as:

\begin{align} \vec{\omega}\times\vec{v} + \nabla H = 0. \end{align}

It implies $\vec{\omega}\cdot\nabla H = 0$ and $\vec{v}\cdot\nabla H = 0$, and so $H$ is constant along vortex and stream lines. $H$ constant means that high pressure iff low velocity and low pressure iff high velocity. This perhaps unintuitive result is explained in the particle scale in this youtube video. The simple explanation is that high velocities are more likely to be tangential to the pipe, while low velocities are more likely to be random, thus having a larger normal component (pressing against the pipe surface) on average.

We should also note that the integrals we wrote down were under the assumption that the fluid is inviscid. If we had allowed for viscosity, we would have derived the Navier-Stokes equations instead. In this case, stress becomes a 2-tensor $\sigma$. With the intuition that $\nabla\cdot\sigma$ is the net force per volume, we can replace the gradient of pressure in ($\ref{eq:euler}$) with $\nabla\cdot\sigma$ to obtain Cauchy's momentum equation:

\begin{align} m\frac{D\vec{v}}{Dt} = \nabla\cdot\sigma - m\vec{F}. \end{align}

Euler's equation is a specific instance of this equation with $\sigma = -pI$ where $p$ is pressure (thus all surfaces feel the same pressure.) The Navier-Stokes equation makes a different choice; I will not go further into this topic since I don't know a lot about fluid dynamics.

Einstein's field equations

In a Riemannian manifold with metric $g$, the equations can be written in tensor form as

\begin{align} G + \Lambda g = \kappa T. \end{align}

The quantities $\Lambda$ and $\kappa$ are constants and $G$, the Einstein tensor, is a function of the metric $g$. Thus the entire left-hand side of the equation is a function of the metric, while in the right sits alone the stress-energy tensor $T$.

Classical electromagnetism

In classical electromagnetism, the stress tensor $\Theta_{ij}$ is given by

\begin{align} \Theta_{ij} = \epsilon_0E_iE_j + \frac{1}{\mu_0}B_iB_j - \frac 12 \delta_{ij}\left(\epsilon_0|E|^2 + \frac{1}{\mu_0}|B|^2\right). \end{align}

This equation cannot be explained from some more fundamental notions; rather, it is viewed as fundamental, motivated by experiments.