Intuition The ONE core ideaWhen you push or twist a solid, it deforms — and pushing it in one direction quietly changes its size in every direction at once. To capture "all pushes → all deformations" we need a bookkeeping device with enough slots for every direction-pair: that device is a tensor , and the whole of 3D Hooke's law is just one big linear rule connecting the "push tensor" to the "deform tensor".
This page assumes you have seen nothing . We build every letter, index, and picture the parent note leans on, in an order where each idea rests only on the one before it.
Before any physics, we need a way to say "this direction, not that one".
Definition The three axes and their labelsWe pick three mutually perpendicular directions and call them x , y , z 1 , 2 , 3 1 = x 2 = y 3 = z counter instead of spelling out x, y, z every time.
Why numbers instead of letters? Because we will soon have quantities that carry two directions at once (a face + a force). Writing σ 12
Definition What a subscript meansA subscript (the little number under a letter) is just an address . u i i u u 1 x u 2 y u 3 z i placeholder that stands for "whichever of 1, 2, 3 you like".
Intuition One index vs two indices — the picture
One index (v i x y z Two indices (M ij i j two directions to make sense — like "force in direction j i
Read M ij
A vector is an arrow with a length and a direction. In our axes it is stored as three numbers ( v 1 , v 2 , v 3 ) u
The bold u u i
Definition Tensor (rank = number of indices)A tensor is the general name for "a box of numbers, each labelled by directions".
Rank 0 = a plain number (a scalar ), no indices. Example: temperature.Rank 1 = a vector, one index, 3 numbers.Rank 2 = a grid, two indices, 3 × 3 = 9 Rank 4 = four indices, 3 × 3 × 3 × 3 = 81
Why does the topic need a rank-2 tensor? Because both stress and strain describe deformation states that require two directions to pin down. Why a rank-4 one? Because relating one 3 × 3 3 × 3 four address slots — that is the stiffness tensor C ij k l
Stress σ ij Pa = N / m 2 first index i = which face the force acts on (the face whose outward normal points along axis i second index j = which direction the force itself points.
σ 11 x x pull/push straight into the face (normal stress).σ 12 y x sideways drag along the face (shear stress, often written τ 12
Intuition Normal vs shear — the pictureIf the little arrow is perpendicular to its face, it stretches or squashes → normal stress (the diagonal entries σ 11 , σ 22 , σ 33 flat along its face, it slides layers past each other → shear stress (the off-diagonal entries).
The Greek letter is sigma σ τ tau ) is just a nickname for the off-diagonal shear entries.
Definition Sign convention (memorise this once)
Normal stress: tension is positive , compression is negative. So a rod being pulled apart has σ 11 > 0 σ 11 < 0 p σ 11 = σ 22 = σ 33 = − p Normal strain: stretch is positive , shrink is negative. Pull a rod and ε 11 > 0 ε 22 , ε 33 < 0 Shear sign: σ ij i = j + i + j ε ij + i + j decreases .
Every worked example in the parent note uses exactly this "tension +, compression −" convention.
Strain ε ij how much the material deformed, as a fraction (stretch ÷ original length). It has no units. Normal strain ε 11 x ε 12
Why fractional and not absolute? A 1 mm stretch means a lot for a 2 mm wire and nothing for a 2 m beam. Dividing by the original length makes strain a property of the deformation , independent of the object's size — exactly what a material law needs. The Greek letter is epsilon ε
Definition The two flavours of shear strain
Tensor shear ε 12 Engineering shear γ 12 = 2 ε 12 total angle change you'd measure with a protractor.
They differ by a factor of 2. The parent note's "factor-of-2 trap" is entirely about keeping these straight.
Common mistake "Small strain" is only the linearized measureThe definition ε ij = 2 1 ( ∂ j u i + ∂ i u j ) infinitesimal (small-strain) tensor : it keeps only terms linear in the displacement gradient and throws away the quadratic ones.
Why it feels safe: for stiff metals and typical spacecraft loads strains are tiny (∼ 1 0 − 3 Where it breaks: for large deformations (rubber seals, inflatables, buckled panels stretched by tens of percent) the discarded quadratic part matters and you must switch to a finite-strain measure (e.g. the Green–Lagrange strain). Everything on this page and in the parent note assumes small strain.
Intuition Why the grid mirrors across its diagonalIf σ 12 x y not equal σ 21 y x σ ij = σ j i
Because of this mirror, the 9 slots hold only 6 independent numbers : the 3 diagonal, plus 3 off-diagonal pairs. That "6" is why engineers can pack a tensor into a 6-slot list (Voigt notation).
The nine numbers σ ij any tilted plane. What force does that arbitrary plane feel?
Definition Traction vector and Cauchy's relationSlice the material with a plane whose outward normal is the unit arrow n = ( n 1 , n 2 , n 3 ) traction vector t t i = σ ij n j ( sum over j ) . Cauchy's stress theorem : the stress tensor is exactly the machine that turns "which way is the cut facing" (n t Stress tensor and Cauchy's relation .
Why this matters here: it tells you why stress needs two indices. One index (j i σ ij every imaginable slice through the point.
Definition Kronecker deltaδ ij = { 1 0 if i = j if i = j 1 0
Why the topic needs it: it is the only direction-blind building block available. When a material behaves the same in every direction (isotropic), the only tensors you're allowed to build its law from are made of δ
Definition Repeated index ⇒ add it upWhen an index appears twice in one term, you secretly sum over 1 , 2 , 3
C_{ijkl}\,\varepsilon_{kl} \;\equiv\; \sum_{k=1}^{3}\sum_{l=1}^{3} C_{ijkl}\,\varepsilon_{kl}.$$
Intuition The picture behindε k k x y z fractional change in volume . That's why ε k k volumetric strain : it's a swelling/shrinking meter.
Common mistake Free index vs summed indexSlip: thinking every repeated letter means "sum".
Fix: an index that appears once in a term (a free index, like i , j σ ij = C ij k l ε k l not summed — it labels which equation you're looking at. Only the index that appears twice (here k l
Definition Partial derivative and its shorthand∂ x j ∂ u i x j u i slope — rise (u i x j ∂ j u i ≡ ∂ x j ∂ u i . ∂ j j
Why this tool and not a plain fraction? A plain ratio Δ u /Δ x local stretch rate at a single point — which is exactly what strain needs, since strain can vary from point to point inside a bent structure. The nine slopes ∂ j u i displacement-gradient grid; its symmetric half is the strain tensor (see Strain tensor and displacement gradient ).
Intuition Symmetric vs antisymmetric half — stretch vs spinAny grid splits into a mirror-symmetric part and an anti-mirror part. The symmetric part is genuine stretching/shearing (stores energy). The antisymmetric part is pure rotation — the whole cube just turns without changing shape, storing no energy. Strain keeps only the symmetric half, which is why ε ij = 2 1 ( ∂ j u i + ∂ i u j )
Now that stress, strain, indices, and summation are all built, we can state the central rule of the parent note as its own definition .
C ij k l C ij k l 3 4 = 81
Minor symmetry (stress side): since σ ij = σ j i i ↔ j C ij k l = C j ik l Minor symmetry (strain side): since ε k l = ε l k k ↔ l C ij k l = C ij l k Major symmetry (energy): the stored elastic energy is a single number, so swapping the whole pair ij ↔ k l C C ij k l = C k l ij
The two minor symmetries drop 81 → 36 36 → 21 Isotropic vs anisotropic materials (composites) .
Definition Young's modulusE ν
==Young's modulus E E E
==Poisson's ratio ν x shrink is ν Poisson's ratio and material limits .
Hooke 3D sigma equals C epsilon
Test yourself — cover the right side and answer before revealing.
What does the subscript i u i The i u u 1 = x
How many independent numbers does a symmetric 3 × 3 6 — three diagonal plus three off-diagonal pairs, since the grid mirrors (σ ij = σ j i
In σ 12 Force pointing in direction 2 y 1 x
What is the sign convention for normal stress and strain? Tension/stretch positive, compression/shrink negative; a pressure p σ ii = − p
What does the traction relation t i = σ ij n j The stress tensor turns a plane's normal n t
What is δ ij i = j i = j 1 i = j 0 i = j
Write out ε k k ε 11 + ε 22 + ε 33
Which index in C ij k l ε k l k l i j
Why does strain keep only the symmetric half of ∂ j u i The antisymmetric half is pure rotation, which changes no shape and stores no elastic energy.
What does ∂ j Shorthand for ∂ / ∂ x j j
When does the small-strain formula ε ij = 2 1 ( ∂ j u i + ∂ i u j ) For large deformations — it drops quadratic gradient terms, so finite-strain measures are needed then.
How many independent constants for a fully anisotropic solid, and after isotropy? 21 (from 81 via minor + major symmetries), dropping to 2 for isotropy.
What is the relation between engineering shear γ 12 ε 12 γ 12 = 2 ε 12
Why do we use a partial derivative instead of a ratio Δ u /Δ x The derivative is the local, step-size-independent stretch rate at a point; a ratio depends on how big a step you take.
How many material constants does an isotropic solid need? Two (e.g. E ν λ μ
