Visual walkthrough — Hooke's law in 3D — generalized stress-strain (tensor)

Step 1 — What is a "little cube of material"?

WHAT. Pick any point deep inside a solid — say the aluminium wall of a fuel tank. Zoom in until you can imagine a tiny cube of material sitting there. Everything we discuss happens on the six faces of this one cube.

WHY a cube? Because a cube gives us three clean directions to talk about — left/right, front/back, up/down. We name them $x$, $y$, $z$. Any push or slide on the material can be described by what happens to these six flat faces. A sphere would have no flat faces to push on cleanly; a cube is the simplest shape that hands us three perpendicular directions for free.

PICTURE. Below, the cube with its three axes. Each pair of opposite faces has an outward normal (the little arrow sticking straight out of the face). The normal is how we name a face: the "$x$-face" is the face whose normal points along $x$.

Step 2 — Stress: two arrows are needed, so two indices are needed

WHAT. Push or pull on a face. Force spread over an area is called stress. But here is the subtle part: to describe a stress you must say two things — which face you are acting on, and which way the force points on that face.

WHY two things? Look at the $x$-face. I can pull it straight outward (force along $x$) — that stretches the cube. Or I can drag it sideways (force along $y$) — that shears the cube, like sliding the top of a deck of cards. Same face, totally different effect. One arrow (the face) cannot capture both. So stress needs two labels: ${\sigma }_{ij}$ where $i$ = the face's normal direction and $j$ = the force's direction.

PICTURE. The $x$-face carrying a pull (${\sigma }_{xx}$, orange, straight out) and a drag (${\sigma }_{xy}$, teal, sideways). The first subscript is the face; the second is the arrow.

There are 3 faces $\times$ 3 arrow-directions $=9$ numbers. They live in a $3\times 3$ table — see Stress tensor and Cauchy's relation for why ${\sigma }_{ij}={\sigma }_{ji}$ (the table is symmetric, so only 6 of the 9 are independent).

Step 3 — Strain: measuring how the cube actually changed shape

WHAT. Stress is what we do to the cube. Strain is how the cube responds — how much it stretched or sheared, as a fraction. If a $10\text{mm}$ edge grows to $10.01\text{mm}$, the strain is $0.01/10=0.001$. It is a pure ratio, no units.

WHY a fraction, not a length? Because a big cube and a small cube of the same material stretch by the same fraction under the same pull. Fractions make the material property, not the sample size, the star. We track a point's tiny displacement $\mathbf{u}$ (how far it moved) and compare it to its neighbour.

PICTURE. Left: normal strain ${\epsilon }_{xx}$ — the cube gets longer in $x$, the fractional stretch. Right: shear strain — the right angle between two edges tips over by a small angle; that tilt is the shear. The two together describe every possible small change of shape.

Cloze check:

Step 4 — The naive guess and why one number $E$ fails

WHAT. In 1D, Hooke's law is one line: $\sigma =E\epsilon$ — "pull is proportional to stretch," with the stiffness $E$ as the constant. Could we just reuse one $E$ for the cube?

WHY it fails. Pull the cube in $x$. It gets longer in $x$ — fine, ${\epsilon }_{xx}$ grows. But it also gets thinner in $y$ and $z$! That sideways shrinking is the Poisson effect. A single number $E$ relating "pull in $x$" to "stretch in $x$" has no way to also predict the $y$ and $z$ shrinking. So we need a rule that lets stress in one direction cause strain in several directions.

PICTURE. Pull the cube along $x$ (orange arrows). Watch the dashed original outline: it stretches long-ways but the sides squeeze inward (plum arrows). One input, three outputs.

Step 5 — The map between tables: enter the 4-index tensor

WHAT. A rule that turns each of the 9 strains into each of the 9 stresses. To point at "the effect of strain ${\epsilon }_{kl}$ on stress ${\sigma }_{ij}$" we need to name four directions: $i,j$ (which stress) and $k,l$ (which strain). That four-labelled object is the stiffness tensor $C_{ijkl}$.

WHY four indices? Two for the stress we produce, two for the strain that produces it. We add up every strain's contribution — that summation is written by repeating the letters $k$ and $l$ (Einstein's shorthand: a repeated letter means "sum over $1,2,3$").

PICTURE. A wiring diagram: on the left the 6 strains, on the right the 6 stresses, and $C_{ijkl}$ as the bundle of wires connecting every strain to every stress. Each wire has a gain — that gain is one number inside $C$.

Step 6 — Symmetry as a shredder: 81 → 21 → 2

WHAT. We now delete redundant numbers using three facts already on the page.

WHY each cut works:

• Stress table is symmetric (${\sigma }_{ij}={\sigma }_{ji}$, Step 2) $\Rightarrow$ swapping $i,j$ can't change anything: $C_{ijkl}=C_{jikl}$.
• Strain table is symmetric (${\epsilon }_{kl}={\epsilon }_{lk}$, Step 3) $\Rightarrow$ $C_{ijkl}=C_{ijlk}$.
• Stored elastic energy is a single scalar, and mixed second derivatives of it commute, so the whole front pair swaps with the back pair: $C_{ijkl}=C_{klij}$ (see Elastic strain energy density).

These three cuts take $81\to 36\to 21$. A material with 21 independent numbers is fully anisotropic (different in every direction) — think carbon-fibre laminate, see Isotropic vs anisotropic materials (composites).

PICTURE. A funnel: 81 grains of sand pour in the top, symmetry sieves let only 21 through, and the final "isotropy" sieve (same in every direction) lets only 2 through — the two Lamé numbers $\lambda ,\mu$.

Step 7 — Assembling the isotropic law and reading every term

WHAT. With only two knobs $\lambda$ and $\mu$, the stiffness is forced to be $C_{ijkl}=\lambda {\delta }_{ij}{\delta }_{kl}+\mu ({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}).$ Feed it into ${\sigma }_{ij}=C_{ijkl}{\epsilon }_{kl}$ and it collapses to the workhorse formula.

WHY it splits into two clean pieces. The $\lambda {\delta }_{ij}{\delta }_{kl}$ block, when summed against ${\epsilon }_{kl}$, pulls out the trace ${\epsilon }_{kk}={\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33}$ — the fractional volume change — and stamps it only on the diagonal via ${\delta }_{ij}$. The $\mu$ block hits every component directly, governing shape change (shear). Volume vs shape: two effects, two knobs.

PICTURE. The cube's response split into two: a uniform "breathing" (volume, $\lambda$-term) plus a "twisting/squaring" (shape, $2\mu$-term). Any deformation is a sum of these two.

Step 8 — Edge & degenerate cases (never leave the reader stranded)

WHAT. Push the formula to its limits and check it still makes sense.

WHY. A law you trust only in the middle is no law. We test the corners: no shear, all-around pressure, and the incompressible limit.

PICTURE. Three mini-cubes: (a) pure uniaxial pull recovers $\sigma =E\epsilon$; (b) equal pressure on all sides — pure volume change, no shear; (c) $\nu \to \frac{1}{2}$, the cube refuses to change volume.

The one-picture summary

This single figure chains the whole walkthrough: cube → two-index stress → two-index strain → one-number guess fails (Poisson) → 4-index map $C$ → symmetry shredder (81→21→2) → the isotropic law split into volume ($\lambda$) and shape ($2\mu$) parts.

Downstream, this law feeds Thin-walled pressure vessels (tank walls), the Von Mises yield criterion (when does it break?), and the element blocks of the Finite Element Method — stiffness matrices.

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