Polymer-matrix composites — CFRP, GFRP; ply lay-up, laminate theory: Foundations

Before you can read the parent note, you need every letter, ratio, and picture it silently assumes. We build them from nothing, each one leaning on the one before.

1. Force, area, and the idea of "how hard is this material squeezed?"

Why do we need area at all? A 100 N pull on a thick bar barely stresses it; the same 100 N on a hair snaps it. The same force feels different depending on how much material it is shared over. That sharing is captured by the next symbol.

Figure — Polymer-matrix composites — CFRP, GFRP; ply lay-up, laminate theory

Figure 1 — Same pull $F$ , two different cross-sections. On the thick bar the arrows are spread over a large face $A$ (low stress); on the thin bar the same arrows crowd a tiny face (high stress). Alt text: two rods pulled by equal force, the thin one shaded to show higher stress, with the formula sigma equals F over A written below.

2. Stretch, and the idea of "how much did it give?"

Why fractional and not the raw stretch $Δ L$ ? A long rod and a short rod under the same pull stretch by different absolute amounts, but a good material description shouldn't depend on how long the sample happens to be. Dividing by $L$ removes that, leaving a property of the material, not the sample.

$Δ L :::$ the change in length (final minus original), in metres.

Why is

ε

unitless?

because it is length divided by length — the metres cancel.

See Stress, strain and Young's modulus for the fuller picture.

3. Stiffness: linking stress and strain

Figure 2 — Stress on the vertical axis versus strain on the horizontal axis. Two straight lines through the origin: a steep magenta one (stiff, big $E$ ) and a shallow orange one (soft, small $E$ ). The slope of each line IS its Young's modulus. Alt text: stress–strain graph with two lines of different steepness, slope labelled E.

4. The two players: fibre and matrix

The subscript is just a name-tag: $σ_{f}$ = the stress in the fibre, $ε_{m}$ = the strain in the matrix, and so on. Whenever you see a letter with $f$ or $m$ , read it as "…of the fibre" or "…of the matrix." Deeper on why we glue them: Fibre–matrix interface and load transfer and Thermoset vs thermoplastic polymers.

5. How much of each? Volume fractions

Why do fractions appear in the stiffness formulas? Because a phase that fills more of the material contributes more to the total. The formula weights each phase's stiffness by its share — exactly like a weighted average.

V_{f} + V_{m}

equals what, and why?

1

, because fibre and matrix together fill the whole material — no gaps in the ideal model.

Why does

V_{f} = A_{f} / A

for a ply?

the fibres run the full length

L

, so that common length cancels top and bottom — volume fraction becomes area fraction.

6. Direction matters: the ply and its 1-, 2-axes

Figure 3 — A single ply drawn as parallel horizontal threads. A magenta arrow along the threads marks axis 1 (stiff, $E_{1}$ ); an orange arrow across them marks axis 2 (floppy, $E_{2}$ ). Alt text: sheet of parallel fibres with a long-axis arrow labelled 1 and a cross-axis arrow labelled 2.

Definition The other three elastic constants a ply needs

Two moduli are not enough to fully describe a ply — a full stiffness picture needs four numbers:

$E_{1}$ — stiffness along the fibres (defined above).
$E_{2}$ — stiffness across the fibres (defined above).
$ν_{12}$ (the Greek "nu", Poisson's ratio) — when you pull along axis 1, the ply also gets thinner across axis 2. $ν_{12}$ = how much sideways shrink you get per unit lengthwise stretch: $ν_{12} = - ε_{2} / ε_{1}$ . Picture squeezing a stretched rubber band thinner as it lengthens. Its partner $ν_{21}$ is fixed by the tidy rule $ν_{12} / E_{1} = ν_{21} / E_{2}$ , so it is not an independent fifth number.
$G_{12}$ (shear modulus) — stiffness against skewing the ply (sliding the top edge sideways over the bottom, turning a rectangle into a parallelogram) rather than stretching it.

Why the parent note needs all four: to rotate a ply to a new angle $θ$ you must know how it responds to stretch AND to shear. Those four numbers ( $E_{1}, E_{2}, ν_{12}, G_{12}$ ) are the ingredients of each ply's stiffness $\overset{ˉ}{Q}$ , and $\overset{ˉ}{Q}$ is what gets summed into the $A$ , $B$ , $D$ matrices. Without $ν_{12}$ and $G_{12}$ the ABD bookkeeping simply cannot be filled in.

7. Parallel vs series: DERIVING the two rules

The parent note quotes two stiffness formulas. Here we build both from the raw stress/strain partitioning so no step is a black box.

Formula Parallel (iso-strain) →

$E_{1} = E_{f} V_{f} + E_{m} V_{m}$ Set-up: pull the ply along the fibres. Fibre and matrix are bonded end-to-end, so they must stretch by the same fraction — same strain $ε$ in both (this is the "iso-strain" assumption).

Step 1 — split the force. The total pull $F$ is shared: part goes through the fibres, part through the matrix. $F = F_{f} + F_{m} .$ Step 2 — write each force as stress × its own area (using $σ = F / A \Rightarrow F = σ A$ ): $σ_{1} A = σ_{f} A_{f} + σ_{m} A_{m} .$ Step 3 — divide through by the total area $A$ : $σ_{1} = σ_{f} \frac{A _{f}}{A} + σ_{m} \frac{A _{m}}{A} .$ Step 4 — swap area fractions for volume fractions using $A_{f} / A = V_{f}$ and $A_{m} / A = V_{m}$ (justified in §5, because the fibres are aligned and continuous): $σ_{1} = σ_{f} V_{f} + σ_{m} V_{m} .$ Step 5 — turn every stress into stiffness × strain ( $σ = E ε$ ), remembering the strain is common ( $ε_{f} = ε_{m} = ε$ ): $E_{1} ε = E_{f} ε V_{f} + E_{m} ε V_{m} .$ Step 6 — cancel the common $ε$ : $E_{1} = E_{f} V_{f} + E_{m} V_{m} .$ This is a plain volume-weighted average of the two stiffnesses — the stiff fibres dominate.

Formula Series (iso-stress) →

$\frac{1}{E _{2}} = \frac{V _{f}}{E _{f}} + \frac{V _{m}}{E _{m}}$ Set-up: pull across the fibres. Now the load passes through the fibre and then the matrix, one after the other, so they carry the same stress $σ$ but stretch by different amounts (the "iso-stress" assumption).

Step 1 — total stretch is the sum of the two stretches. As a fraction of the whole, each phase contributes its own strain weighted by its share of the thickness: $ε_{2} = V_{f} ε_{f} + V_{m} ε_{m} .$ Step 2 — turn each strain into stress ÷ stiffness ( $ε = σ / E$ ), with the common stress $σ$ : $ε_{2} = V_{f} \frac{σ}{E _{f}} + V_{m} \frac{σ}{E _{m}} .$ Step 3 — write the overall strain as $ε_{2} = σ / E_{2}$ and substitute: $\frac{σ}{E _{2}} = σ (\frac{V _{f}}{E _{f}} + \frac{V _{m}}{E _{m}}) .$ Step 4 — cancel the common $σ$ : $\frac{1}{E _{2}} = \frac{V _{f}}{E _{f}} + \frac{V _{m}}{E _{m}} .$ Because you add reciprocals, the softest phase dominates — that is why $E_{2}$ comes out small.

Figure 4 — Left: fibre-spring and matrix-spring side-by-side (parallel), sharing the same stretch, stiffnesses add to give big $E_{1}$ . Right: the two springs end-to-end (series), sharing the same force, reciprocals add to give small $E_{2}$ . Alt text: spring diagram contrasting parallel and series arrangements with labels iso-strain and iso-stress.

Check with the parent's numbers ( $E_{f} = 230$ , $E_{m} = 3$ GPa, $V_{f} = 0.6$ ):

Parallel: $E_{1} = 230 (0.6) + 3 (0.4) = 139.2$ GPa.
Series: $E_{2} = (\frac{0.6}{230} + \frac{0.4}{3})^{- 1} = 7.36$ GPa.

Same materials, 19× difference — purely because of parallel vs series. This is Fracture and fatigue in composites' starting point too: cracks love the weak direction.

Common mistake Forgetting the edge cases

$V_{f} \to 0$ and $V_{f} \to 1$ Both formulas must survive the extremes — check them:

$V_{f} = 1$ (all fibre, no matrix): $E_{1} = E_{f}$ and $E_{2} = E_{f}$ . Sensible — a block of pure fibre. But the series model also happily lets $V_{m} \to 1$ with $V_{f} \to 0$ , giving $E_{2} = E_{m}$ (pure matrix) — also fine.
The trap: the series formula $\frac{1}{E _{2}} = \frac{V _{f}}{E _{f}} + \frac{V _{m}}{E _{m}}$ never gives infinity for real inputs, because neither $E_{f}$ nor $E_{m}$ is zero. But if a phase's modulus were taken as zero (a void, a gap of pure air), that term $\frac{V}{0}$ blows up → $E_{2} \to 0$ . That is the model correctly screaming "a series chain is only as stiff as its softest link — a gap makes it floppy."
Beyond the model: near $V_{f} \to 1$ there is too little matrix to actually wet and bond the fibres, so the neat elastic straight-line assumption (§3) breaks down and the real ply is full of voids. The formulas stay finite but stop being true. That is why real lay-ups keep $V_{f} \approx 0.55$ – $0.65$ .

8. Stacking symbols: angle $θ$ , position $z$ , lay-up brackets

Figure 5 — The global $x$ -axis (main load) drawn horizontally. Three plies overlaid: $0^{\circ}$ along $x$ , $+ 4 5^{\circ}$ rotated anticlockwise, $- 4 5^{\circ}$ rotated clockwise, with the anticlockwise-positive arrow marked. Alt text: reference x-axis with plies at 0, plus 45 and minus 45 degrees showing the anticlockwise-positive sign convention.

Why do we need $z$ ? Because when a laminate bends, the outer plies (large $∣ z ∣$ ) stretch most and do the most work. Any formula about bending will therefore weight plies by their $z$ . That is the whole reason the parent's $D$ matrix carries a $z^{3}$ inside it.

What does the

s

[0/90]_{s}

mean?

the stack is symmetric — mirror the listed plies about the mid-plane.

Which plies matter most for bending stiffness, and why?

the outer ones (large

∣ z ∣

), because in bending they stretch the most.

For

+ 4 5^{\circ}

, which way are the fibres turned?

anticlockwise from the global

x

-axis (positive = anticlockwise convention).

How these foundations feed the topic

Equipment checklist

Cover the right-hand side and test yourself — you are ready for the parent note only if each reveals cleanly.

Stress

σ

force divided by area,

σ = F / A

, measured in Pa (here GPa) — the "crowding" of force.

The prefix G in GPa

giga = a billion;

1 GPa = 1 0^{9} Pa

Strain

ε

fractional stretch

Δ L / L

, a unitless ratio.

Young's modulus

E

stiffness, the slope of the stress–strain line,

E = σ / ε

— valid only on the straight (elastic) part.

Difference: stiffness vs strength

stiffness

E

resists stretching; strength is the stress that breaks it — different things.

Subscripts

f

and

m

"of the fibre" and "of the matrix" (e.g.

E_{f}

σ_{m}

Volume fractions

V_{f}, V_{m}

shares of fibre and matrix; they sum to 1.

Why

V_{f} = A_{f} / A

fibres run the full length, so that common length cancels — volume fraction equals cross-section area fraction.

Ply axes 1 and 2

1 = along fibres (stiff

E_{1}

); 2 = across fibres (floppy

E_{2}

Poisson's ratio

ν_{12}

sideways shrink per unit lengthwise stretch,

ν_{12} = - ε_{2} / ε_{1}

Shear modulus

G_{12}

stiffness against skewing (parallelogram distortion), not stretching.

Anisotropy

stiffness depends on direction — the reason one ply isn't enough.

Iso-strain (parallel) derivation

split force,

σ_{1} A = σ_{f} A_{f} + σ_{m} A_{m}

, divide by

A

, use

σ = E ε

, cancel

ε

→

E_{1} = E_{f} V_{f} + E_{m} V_{m}

Iso-stress (series) derivation

add strains,

ε_{2} = V_{f} ε_{f} + V_{m} ε_{m}

, use

ε = σ / E

, cancel

σ

→

1/ E_{2} = V_{f} / E_{f} + V_{m} / E_{m}

Edge cases

V_{f} \to 0, 1

formulas give pure matrix or pure fibre; a zero-modulus (void) term makes

E_{2} \to 0

, and near

V_{f} \to 1

the elastic assumption itself fails.

Ply angle

θ

and its sign

rotation from the global

x

-axis, positive = anticlockwise (right-hand convention).

Position

z

distance from the mid-plane; outer plies (large

∣ z ∣

) dominate bending.

Lay-up

[\dots]_{s}

list of ply angles;

s

means symmetric (mirrored about mid-plane).