Intuition The One Core IdeaA composite is two materials glued into one: thin stiff fibers that carry load along their length, held inside a soft matrix that binds them and passes load between them. Everything in this topic is bookkeeping for one question — if I pull, push, or twist this layered material in some direction, how much does it stretch and how much stress builds up inside?
Before you can read a single stiffness matrix on the parent page, you must own every letter it uses. This page introduces each symbol in an order where nothing is used before it is built. Read top to bottom once and the parent note becomes plain English.
Everything begins with pulling on a block of material. Two questions arise: how hard am I pulling per unit of surface? and how much did the block stretch compared to its original length?
Figure 1 below shows all three ideas side by side: a block being stretched (normal stress), and a block being skewed (shear stress). Look at the cyan outline (original shape) versus the amber outline (deformed shape) — the whole page rests on seeing that difference.
σ τ Stress is force divided by the area it acts on: σ = F / A A F Pa = N/m 2 1 GPa = 1 0 9 Pa
Normal stress σ : force acts perpendicular to the face (pulling apart or pushing together).Shear stress τ : force acts along the face, sliding one layer over the next — like pushing the top of a deck of cards sideways.
Definition Sign convention — tension positive, compression negativeStress and strain carry a sign , and everyone must agree on it or the matrices give wrong answers:
Tension (pulling apart, block gets longer ) → σ > 0 ϵ > 0 Figure 1 the amber block is longer than cyan, so both are positive.Compression (pushing together, block gets shorter ) → σ < 0 ϵ < 0 σ = F / A negative force because it points inward.Direction matters : each stress lives on a named face and axis. σ 1 σ 2 which face, which way, and what sign .
ϵ γ Strain is the fractional stretch: ϵ = Δ L / L ϵ = 0.01 negative ϵ as a fraction of itself .
Engineering shear strain γ : the total change in a corner's right angle, measured in radians. Precisely: take a corner that started at exactly 9 0 ∘ 9 0 ∘ − γ γ decrease in the right angle , not a "lean" — the whole shrinkage of the angle, counting both edges that tilt. In Figure 1 it is the amber corner's departure from square.This is the engineering definition (the one the parent's matrices use); a related "tensorial" version equals γ /2 γ γ ≈ tan γ
Intuition Why split into normal and shear at all?Any way you can deform a flat piece of material is a mixture of just three moves: stretch left-right, stretch up-down, and skew (shear). That is why every matrix on the parent page has exactly three rows. Master these three and you have covered all cases of in-plane loading.
Pull harder, stretch more. The ratio between them is a property of the material, not of the block's size. That ratio is stiffness .
Definition Young's modulusE Young's modulus is stress divided by strain: E = σ / ϵ E E
Picture two springs: a stiff one barely moves under your hand, a floppy one moves a lot. E
Units: same as stress, GPa. Carbon fiber E f ≈ 230 E m ≈ 3
G G 12 Shear modulus G G = τ / γ E
For an isotropic material one number G directional ply we must say which plane is being sheared, so it gains subscripts: ==G 12 G 12 G
Intuition Why we need bothE G E G
Stretch a rubber band and it gets thinner . Pull a block one way and it shrinks the other way. That coupling has its own symbol.
Figure 2 shows this directly: the cyan block (before) becomes the amber block (after) — taller and narrower. The two cyan inward arrows are the sideways shrink that Poisson's ratio measures.
Definition Poisson's ratioν Poisson's ratio ν ν = − lengthwise strain sideways strain ν 0.2 0.35
Picture the block getting longer and narrower at once (Figure 2 ). ν
ν = 0 ν = 0.5
Because a composite is not the same in every direction, it gets two Poisson's ratios:
Definition Major and minor Poisson's ratiosν 12 ν 21 ν 12 ν 21 not equal, but they are tied by the symmetry rule the parent uses:
ν 21 = ν 12 E 1 E 2
Intuition WHY the two Poisson ratios must obeyν 21 = ν 12 E 2 / E 1 the work you do stretching a material does not depend on the order you apply the loads. Here is the argument in three visible steps.
Step 1 — write the two cross-couplings. From the parent's compliance relations, pulling only in direction 1 with stress σ 1 ϵ 2 = − E 1 ν 12 σ 1 only in direction 2 with stress σ 2 ϵ 1 = − E 2 ν 21 σ 2 E 1 ν 12 E 2 ν 21
Step 2 — energy cannot care about order. Load the block to a final state ( σ 1 , σ 2 ) σ 1 σ 2 σ 2 σ 1 same number either way (it is a property of the final squeezed state, like height on a hill — path-independent). Writing out the work for each path, the only way the two totals can match for every load is if the two cross-terms are equal:
E 1 ν 12 = E 2 ν 21 symmetric (mirror-image across its diagonal). This is exactly why the parent can write Q 12
Step 3 — rearrange. Multiply both sides by E 2 ν 21 = ν 12 E 1 E 2 E 2 ≪ E 1 ν 21 ν 12 0.3 → 0.0214
A single material like aluminium behaves the same whichever way you pull — it is isotropic (from Greek: "equal in all directions"). A composite does not: fibers run one way, so that way is stiff and across is soft. It is anisotropic . To keep track, we name two axes glued to the fibers.
Figure 3 shows a tilted ply: the amber lines are fibers, the white arrow labelled 1 runs along them, the cyan arrow labelled 2 runs across, and the amber arc is the angle θ x
Definition Material axes: direction 1 and direction 2
Direction 1 : along the fibers. This is the strong, stiff direction — modulus E 1 Direction 2 : across the fibers (perpendicular, in the plane of the ply). This is weak — modulus E 2 ≪ E 1 The little subscripts on E 1 , E 2 , G 12 , ν 12 G 12
Definition Fiber orientation angleθ not line up with the load. ==θ x 0 ∘ x 9 0 ∘ 4 5 ∘ x Figure 3 ).
Intuition Two coordinate systems, one materialThere is the material system (1, 2 — glued to the fibers) and the laminate system (x , y x y θ [ Q ˉ ] x y
To predict composite stiffness you need the recipe: what fraction of the material is fiber versus matrix.
Definition Volume fractionsV f V m V f is the fraction of the total volume that is fiber; V m V f + V m = 1 V f = 0.6 V m = 0.4 V f ≈ 0.55 0.65
Subscript f m
Mnemonic Reading the subscriptsf f iber, m m atrix, numbers 1/2 x / y which piece and which direction before you worry about which quantity .
The parent stacks three stresses and three strains into columns and links them with a grid of numbers. A grid that turns one list of numbers into another is a matrix .
Definition What a matrix does hereA matrix is a rectangular table of numbers that maps one column of quantities to another. Feeding in the strains ( ϵ 1 , ϵ 2 , γ 12 ) ( σ 1 , σ 2 , τ 12 ) this strain contribute to that stress."
[ Q ] stiffness matrix: strains in → stresses out. Big numbers = stiff.[ S ] compliance matrix: stresses in → strains out. It is the exact reverse, [ S ] = [ Q ] − 1 [ T ] transformation matrix: rebuilds the numbers when you rotate from axes 1–2 to axes x y θ cos θ sin θ
Intuition Why invert, why theQ 66 = G 12 1/ E strains and want stresses , so they need the inverse, [ Q ] Q 66 = G 12 symmetry of §3 pays off: it makes [ Q ] Q 12
The diagram below renders as a flowchart — read it bottom-up: the whole parent topic is these building blocks stacked.
Force over area = stress sigma tau
Fractional stretch = strain eps gamma
Poisson ratio nu couples directions
Angle theta and transform T
Transformed stiffness Qbar in x y
The whole parent topic (Laminate Theory ) is these building blocks stacked. This foundation connects onward to material selection , feeds the finite-element models , and its stiffness numbers ultimately drive launch-load and mass-budget trades. Anisotropy also matters for thermal design and layers stack into sandwich structures .
Test yourself — you are ready for the parent page only if each answer comes instantly.
What does stress σ Force per unit area, F / A
What sign does a compressive stress carry, and why? Negative — the force points inward, so plugging it into σ = F / A σ < 0 ϵ < 0
How is strain ϵ Strain is the fractional stretch Δ L / L
Precisely, what is engineering shear strain γ The decrease (in radians) of an originally right angle when the block is sheared: the corner goes from 9 0 ∘ 9 0 ∘ − γ
Give the defining ratio for Young's modulus E E = σ / ϵ E
What physical move does shear modulus G G 12 Skewing/sliding — the change of a right angle (shear strain γ
Which direction is axis 1 versus axis 2 in a ply? 1 = along fibers (stiff, E 1 E 2
What does the angle θ Angle from the structure's x
Why do V f V m They are the only two ingredients; together they fill the whole volume.
What is the difference between [ Q ] [ S ] [ Q ] [ S ] [ S ] = [ Q ] − 1
Why must ν 21 = ν 12 E 2 / E 1 Elastic energy is path-independent, forcing the compliance table to be symmetric (ν 12 / E 1 = ν 21 / E 2
