Cauchy never stopped being an engineer. In 1822 Claude-Louis Navier had presented to the Academy a theory of elastic solids based on forces between molecules, and Augustin Fresnel was explaining light as a transverse vibration of an elastic medium, the ether. Cauchy, a colleague of both at the Academy, followed this work closely. On 30 September 1822 he announced to the Academy a new theory, one that needed no molecules.

The imaginary cut

Think of a loaded beam, a bridge or a pillar. Pick a point inside and imagine a small flat cut through it, with a perpendicular direction $\mathbf{n}$. The material on one side of the cut pulls or pushes on the material on the other side. The force per unit area on that cut is a vector $\mathbf{t}$, which Cauchy called “pressure” and which is now called the stress vector.

This vector need not be perpendicular to the cut. The perpendicular part is the normal stress, which compresses or stretches; the parallel part is the shear stress, which tends to make the two faces slide over each other. And as the cut is turned about the same point, $\mathbf{t}$ changes.

Cauchy’s theorem

Cauchy’s discovery is that this dependence is the simplest possible: linear. There is an array of numbers $\sigma$ such that, for any direction $\mathbf{n}$,

$$\mathbf{t} = \sigma\,\mathbf{n}, \qquad \sigma = \begin{pmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} \end{pmatrix}.$$

Six numbers, because the array is symmetric, are enough to know the force on every possible cut through that point. This is Cauchy’s stress tensor.

The argument is the tetrahedron argument. Cut out a tiny tetrahedron with three faces parallel to the coordinate planes and the fourth perpendicular to $\mathbf{n}$. The forces on the faces are proportional to their areas, which shrink like the square of the size; weight and inertia are proportional to the volume, which shrinks like the cube. For a very small tetrahedron only the forces on the faces count, and they must balance. Since the area of each coordinate face is the area of the slanted face times one component of $\mathbf{n}$, the balance gives exactly $\mathbf{t} = \sigma\mathbf{n}$.

An example in the plane

In the plane the array has three numbers. Take, in megapascals,

$$\sigma = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}.$$

For a vertical cut, $\mathbf{n} = (1, 0)$, and $\mathbf{t} = (3, 1)$: normal stress 3 (tension) and shear 1. For a horizontal cut, $\mathbf{n} = (0, 1)$, and $\mathbf{t} = (1, 1)$: normal 1 and shear 1. Turn the cut in the figure and follow the two components:

The principal directions

Cauchy also showed that at each point there are three mutually perpendicular directions, the principal directions, in which the stress vector is perpendicular to the cut: pure tension or compression, with no shear at all. The corresponding values, the principal stresses, are the largest and smallest normal stresses possible at that point. They are what the engineer compares with the strength of the material. In today’s language they are the eigenvalues and eigenvectors of the tensor $\sigma$, ideas Cauchy developed in the same period while studying second-degree surfaces.

Exercise: the principal stresses

Find the principal stresses and principal directions of $\sigma = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}$.

We look for $\mathbf{n}$ with $\sigma \mathbf{n} = \lambda \mathbf{n}$. This requires $\det(\sigma - \lambda I) = 0$: $(3 - \lambda)(1 - \lambda) - 1 = \lambda^2 - 4\lambda + 2 = 0$, so $\lambda = 2 \pm \sqrt{2}$, that is, $3.414$ and $0.586$. For $\lambda = 2 + \sqrt{2}$, the first row gives $(1 - \sqrt{2})\,n_x + n_y = 0$, or $n_y = (\sqrt{2} - 1)\,n_x$, and $\sqrt{2} - 1 = \tan 22.5°$. The principal directions make angles of $22.5°$ and $112.5°$ with the $x$ axis.

Exercise: the prism

For an ordinary glass, $A \approx 1.5046$ and $B \approx 4\,200\ \text{nm}^2$. Compute the refractive index for violet light ($\lambda = 400$ nm) and for red light ($\lambda = 700$ nm).

Violet: $n = 1.5046 + \frac{4\,200}{160\,000} = 1.5046 + 0.0263 = 1.5309$. Red: $n = 1.5046 + \frac{4\,200}{490\,000} = 1.5046 + 0.0086 = 1.5132$. The difference is small, but it is what opens up the rainbow as light leaves a prism.

Sources: Augustin-Louis Cauchy, “Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques”, Bulletin de la Société philomatique (1823) · Augustin-Louis Cauchy, “De la pression ou tension dans un corps solide”, Exercices de mathématiques 2 (1827) · Augustin-Louis Cauchy, Mémoire sur la dispersion de la lumière (1836) · Isaac Todhunter and Karl Pearson, A History of the Theory of Elasticity, vol. 1 (1886) · Clifford Truesdell, Essays in the History of Mechanics (1968).

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