Sharing the load: a deal between two materials
By now you know the composite bargain from guides 1 and 2: a soft, humble matrix binds everything together, protects the surfaces, and passes force along, while a stiff, strong reinforcement actually carries most of that force — steel rebar in concrete, straw in mud, each covering the other's weakness. This guide answers the very next question a designer asks: put a known amount of stiff fibre into a known matrix, and how stiff, how strong is the result? Wonderfully, you can predict it with arithmetic a beginner can do — the rule of mixtures — as long as you respect where it holds and where it breaks.
The one number the whole calculation turns on is the volume fraction — what share of the composite's volume is fibre. Write it Vf for the fibre and Vm for the matrix, and since there is nothing else, Vf + Vm = 1. A 60 percent fibre lay-up (typical for aerospace carbon-fibre) means Vf = 0.6 and Vm = 0.4. Everything below is just those two fractions weighting the two materials' properties. But be careful which fraction you use: the rule of mixtures wants volume fraction, not weight fraction, and because carbon is lighter than epoxy the two numbers are not the same.
Springs in parallel: the isostrain upper bound
Pull a fibre composite along the fibres. Because fibre and matrix are bonded, they are forced to stretch by the same amount — they share one strain. This is the isostrain condition, and the picture to hold is a stiff steel cable and a soft rubber band clamped side by side and stretched together: two springs in parallel. Springs in parallel add their stiffnesses. Whoever is stiffer takes a bigger share of the force, but both feel the identical stretch.
Add up the forces and out drops the longitudinal rule of mixtures for stiffness: Ec = Vf times Ef + Vm times Em. The composite's Young's modulus is just the fibre and matrix moduli, each weighted by its volume fraction. Try it on that 60 percent carbon/epoxy, with Ef = 230 GPa and Em = 3 GPa: Ec = 0.6 times 230 + 0.4 times 3 = 138 + 1.2 = 139 GPa. The stiff fibres dominate so completely that the matrix's contribution (1.2 GPa) barely registers. In fact the fibres carry a load-share of (Ef times Vf) / (Em times Vm) = 138 / 1.2 = about 115 to 1 — the fibres shoulder roughly 99 percent of the force, the matrix almost none.
Springs in series: the isostress lower bound
Now turn the load 90 degrees and pull across the fibres. The force now has to pass through fibre and matrix in turn, like a chain of alternating stiff and soft links — springs in series. Series springs share the same force, so both phases feel the same stress (the isostress condition), but they stretch by wildly different amounts: the soft matrix strains a great deal, the stiff fibre almost not at all. And just as the weakest link governs a chain, the softest phase governs the compliance. The stiff fibre is now nearly useless.
Because compliances (the inverse of stiffness) add in series, the transverse rule of mixtures is a reciprocal sum: 1/Ec = Vf/Ef + Vm/Em. Feed in the same 60 percent carbon/epoxy: 1/Ec = 0.6/230 + 0.4/3 = 0.0026 + 0.133 = 0.136, so Ec = about 7.4 GPa. Stop and stare at that. The same material, at the same 60 percent fibre, is 139 GPa along the fibres and only 7.4 GPa across them — nearly 19 times stiffer one way than the other. That gap is not a flaw; it is anisotropy, and designing with it on purpose is the whole art of guide 5.
RULE OF MIXTURES -- CFRP: 60% carbon fibre (Ef = 230 GPa) in epoxy (Em = 3 GPa) loading model formula Ec -------------- ------------- ---------------------- ------- LONGITUDINAL springs in Ec = Vf x Ef + Vm x Em 139 GPa upper bound (along fibres) PARALLEL = 0.6(230) + 0.4(3) (isostrain) TRANSVERSE springs in 1/Ec = Vf/Ef + Vm/Em 7.4 GPa lower bound (across fibres) SERIES = 0.6/230 + 0.4/3 (isostress) Same material, same 60% fibre -> ~19x stiffer ALONG than ACROSS. The two formulas BRACKET reality: any other fibre arrangement lands between them.
Load transfer: how a fibre actually gets loaded
The isostrain story quietly assumed the fibres are continuous — running unbroken from grip to grip. But most real composites use chopped, finite fibres, and a short fibre has a problem: its ends carry no load at all. Force cannot leap onto a fibre by magic; it has to be handed over from the matrix through shear at the interface, like your hand gripping a rope. You can only pull the rope as hard as your grip allows, and a very short end just slips. Tension therefore builds up gradually from each fibre end inward, and only a long-enough fibre gets its middle stressed all the way up to the fibre's own breaking strength.
- Grip: the bonded matrix drags on the fibre's surface as the composite is pulled — no bond, no transfer.
- Shear at the interface: the softer matrix wants to stretch more than the stiff fibre, so it shears against the fibre and feeds tension into it.
- Build-up: fibre tension rises from zero at each tip, climbing inward over a transfer length.
- Plateau: if the fibre is long enough, the middle reaches the full fibre strength and does real work; if not, it is under-used everywhere.
The shortest fibre whose midpoint just reaches full fibre stress is the critical fibre length, lc = (fibre strength times diameter) / (2 times interface shear strength). Put in numbers: a fibre of strength 4000 MPa and diameter 0.010 mm, bonded with an interface that shears at 50 MPa, gives lc = (4000 times 0.010) / (2 times 50) = 40/100 = 0.4 mm — about 40 fibre diameters. Fibres much longer than lc (a rule of thumb is 15 times lc) behave essentially like continuous ones and the isostrain formula holds; fibres shorter than lc snap out and never reach full strength, so they reinforce far less. This one length is why chopped-fibre plastic is cheaper but weaker than continuous-fibre laminate.
From stiffness to strength — and the honest limits
Strength gets its own rule of mixtures, but with a subtle twist. Along aligned continuous fibres, the composite's strength is (composite strength) = Vf times (fibre strength) + Vm times (matrix stress at the strain where the fibres break). That last part matters: carbon fibre is stiff and snaps at a low strain, well before the ductile matrix reaches its own ultimate strength. So you must use the matrix's stress at the fibre's failure strain, not the matrix's peak — a smaller number. Get this wrong and you badly over-predict the strength. As with stiffness, the fibres dominate, and the matrix contributes little to the headline number while doing the vital background jobs.
Now the honest caveats, because a rule of mixtures is a clean average and reality is messier. First, it is a best case along the fibres only: pull across them and the composite is as weak as the matrix and interface — a fibre composite is strong along the fibres and weak across them, full stop, which is why fibre orientation is a life-or-death design choice. Second, fibres only pull well in tension; in compression they can buckle and kink, so compressive strength does not follow the same tidy line. Third, no single 'fibre strength' is even trustworthy — like any brittle solid, a fibre fails at its worst flaw, so strengths scatter and are really described by Weibull statistics, not one number.
Finally, the same two bounds work for particle-reinforced composites too, but you rarely sit on either line. Large-particle composites like concrete (stone aggregate in cement) or cemented carbide (hard WC grains in tough cobalt) have roughly equiaxed, randomly packed particles, so their real modulus lands between the isostrain and isostress bounds — the bounds bracket it rather than nail it. And beware one true exception: dispersion-strengthened composites, with a few percent of tiny hard particles, do not really work by load-sharing at all. Those particles are far too small and sparse to carry force; they strengthen by pinning dislocations just like precipitation strengthening, a mechanism from the strengthening rung, not a rule-of-mixtures effect.