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

(a) Along the fibres → fibre and matrix stretch by the same amount (iso-strain), like springs in parallel. Use the Rule of Mixtures, $E_1=E_f{V}_f+E_m{V}_m$. It is the upper bound. (b) Across the fibres → the load passes through both in turn, so they feel the same stress (iso-stress), like springs in series. Use the inverse (harmonic) rule, $\frac{1}{E_2}=\frac{V_f}{E_f}+\frac{V_m}{E_m}$. It is the lower bound.

CFRP = Carbon Fibre Reinforced Polymer. GFRP = Glass Fibre Reinforced Polymer. CFRP is stiffer per kilogram: carbon fibre reaches ~230 GPa at density ~1.6 g/cm³, while glass reaches ~72 GPa at ~2.5 g/cm³. See Specific strength and stiffness in aerospace materials.

$V_m=1-0.55=0.45$. Longitudinal (parallel, add): $E_1=230(0.55)+3(0.45)=126.5+1.35=127.85\text{GPa}$ Transverse (series, add reciprocals): $\frac{1}{E_2}=\frac{0.55}{230}+\frac{0.45}{3}=0.002391+0.15=0.152391$ $E_2=\frac{1}{0.152391}=6.562\text{GPa}$ Notice $E_1/E_2\approx 19.5$ — the ply is ~20× stiffer along fibres than across. This is why a single ply is useless off-axis and we must stack angles.

$V_m=0.40$. $E_1=72(0.60)+3.5(0.40)=43.2+1.4=44.6\text{GPa}$ Compare to carbon's ~128–139 GPa: glass fibre is intrinsically less stiff, so even a good GFRP ply is far softer than a CFRP ply — the reason wing skins go carbon and radomes go glass.

Same strain $\epsilon$ in both phases (iso-strain). Load in each phase $=$ stress $\times$ area, and area fraction $=$ volume fraction, so: $\frac{F_f}{F_{\text{total}}}=\frac{E_f{V}_f}{E_f{V}_f+E_m{V}_m}=\frac{230(0.55)}{127.85}=\frac{126.5}{127.85}=0.9894$ The fibres carry ≈98.9% of the load. The matrix, though it is 45% of the volume, carries barely 1%. This is the quantitative version of "fibres carry load, matrix holds them in place" — see Fibre–matrix interface and load transfer.

What $B$ is: the entry that links in-plane stretching to bending. If $B\ne 0$, pulling the panel makes it curl, and heating it makes it warp. Why it vanishes for symmetric stacks: for every ply at height $+z$ with stiffness $\bar{Q}$, symmetry guarantees an identical ply at $-z$. The term $\frac{1}{2}\bar{Q}(z_k^2-z_{k-1}^2)$ measures a signed first moment of stiffness about the mid-plane. Look at the figure: the two mirror plies sit at equal distances on opposite sides, so their first-moment contributions are equal in magnitude and opposite in sign and cancel term-by-term. Summed over all mirror pairs, $B=0$.

What $B=0$ removes: the bending–stretching coupling. A symmetric panel stretches when pulled and bends when a moment is applied, but the two never leak into each other — so it stays flat when you just pull it, and does not warp on cooling from the cure oven.

A standard answer: $[0/45/-45/90{]}_s$ — written out that is the 8-ply stack $0,45,-45,90,90,-45,45,0.$ Quasi-isotropic: the four distinct angles $0,45,-45,90$ are spread evenly at $45^{\circ }$ steps, so in-plane stiffness is the same in all directions ($A$ matrix isotropic). See Anisotropy and crystal directions for why even angular spread averages out directionality. Balanced: the $+45$ is paired with a $-45$, so tension–shear coupling in $A$ cancels. Symmetric: the subscript $s$ means the second half mirrors the first about the mid-plane, so $B=0$ and the panel will not warp on cool-down. ✔ all three properties met.

Rule: bending stiffness is $D_{ij}=\frac{1}{3}{\sum }_k{\bar{Q}}_{ij,k}(z_k^3-z_{k-1}^3)$. The $z^3$ means a ply's bending contribution scales with how far it is from the mid-plane, cubed. So put stiff plies at the surfaces. Estimate. Approximate each thin ply's contribution by $\bar{Q}\cdot t\cdot z^2$ (a thin slab at distance $z$ contributes $\approx \bar{Q}t{z}_{\text{mid}}^2$; this is the $\int z^2$ form of $D$). With equal $\bar{Q}$ and $t$: $\frac{D_{\text{surface}}}{D_{\text{centre}}}\approx \frac{z_{\text{surf}}^2}{z_{\text{cen}}^2}=\frac{(1.0{)}^2}{(0.2{)}^2}=\frac{1.0}{0.04}=25$ The surface ply is worth 25× the same ply at the centre for bending. Look at the figure: outer fibres see strain $z\kappa$, which is largest at the surface, so they do the most work resisting the bend.

(a) Specific modulus $E/\rho$ (units GPa·cm³/g): $\text{CFRP: }\frac{139.2}{1.6}=87.0,\text{Al 7075: }\frac{71}{2.80}=25.36$ (b) Ratio $=87.0/25.36=3.43$. CFRP delivers ~3.4× the stiffness for the same mass. (c) Bending: for a plate the bending stiffness scales as $E{t}^3$, and for a fixed mass a lower-density material lets you use a thicker panel ($t\propto 1/\rho$). Since $t^3$ enters, the density advantage is amplified cubically — CFRP's already-large specific-modulus edge grows even further for bending-dominated skins. This is the core reason the Boeing 787 is ~50% CFRP by weight.

Flaw 1 — not symmetric. With only two plies at $+z$ and $-z$ carrying different stiffnesses ($0^{\circ }$ vs $90^{\circ }$), the first-moment terms do not cancel: $B\ne 0$. The panel will warp as it cools from the cure temperature (thermal residual stress) and will bend when merely stretched. Flaw 2 — not balanced in shear. There is no $-\theta$ to pair with any $+\theta$ off-axis ply, so if any angled ply were added there'd be tension–shear coupling; even as-is the stack has no shear-coupling cancellation guarantee. Flaw 3 — not quasi-isotropic. Only $0^{\circ }$ and $90^{\circ }$ are present, so the panel is stiff along two axes but soft on the $\pm 45^{\circ }$ diagonals — it is not equally stiff in all directions. Minimum fix: make it symmetric by mirroring: $[0/90{]}_s=0,90,90,0$ (kills $B$, stops warp). For genuine all-direction stiffness, upgrade to a quasi-isotropic symmetric stack such as $[0/45/-45/90{]}_s$. This ties back to Fracture and fatigue in composites: a warping, off-axis-soft panel concentrates stress and fails early in fatigue.

L1 which model for transverse load? ::: inverse (series, same stress), a lower bound. L2 $E_2$ for $V_f=0.55$ carbon/epoxy? ::: ≈ 6.56 GPa. L3 fibre load fraction there? ::: ≈ 98.9%. L4 surface vs centre ply for bending, $|z|=1.0$ vs $0.2$ mm? ::: ~25× stronger at surface. L5 CFRP vs Al specific-modulus ratio? ::: ~3.4×, and bending amplifies it further.