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A carbon fibre is a slender reinforcement made predominantly of carbon, used to carry tension along its length. Its axial Young's modulus, tensile strength and mass density are especially useful when comparing how much elastic load a lightweight component can carry. A fibre's measured tensile strength also depends on defects and the length tested, so one strength value does not characterize every specimen.

Embedding carbon fibres in a polymer produces carbon-fibre-reinforced polymer. The composite matrix supports the fibre arrangement and permits fibre-matrix load transfer; the axial performance of an aligned assembly does not imply equal transverse performance. Tensile testing of single fibres measures the reinforcement response, while testing composite strips at different orientations measures how that response survives assembly into a composite material.

Carbon-fibre-reinforced polymer is a composite material in which carbon fibres reinforce a polymer composite matrix. The fibres carry axial tension efficiently, while the composite matrix maintains their arrangement and transfers load through interfacial shear stress. A component's response therefore depends on fibre direction, fibre volume fraction and bonding as well as the constituent properties.

For continuous aligned reinforcement, the longitudinal rule of mixtures predicts the initial axial Young's modulus. A stack with fibres in several directions trades some axial performance for reinforcement in those directions. Such an arrangement can reduce in-plane anisotropy, but does not automatically make the through-thickness response identical. Comparing tensile testing along the fibres, across them and through the assembled layers reveals different possible failure mechanisms, including fibre fracture and loss of fibre-matrix load transfer.

A composite material contains distinguishable constituent phases whose arrangement contributes to the bulk properties. In a fibre-reinforced example, strong fibres carry much of the axial load while a composite matrix holds them together and transmits shear stress between them. Carbon-fibre-reinforced polymer combines carbon fibres with a polymer composite matrix; glass fibres provide another reinforcement.

The constituent proportions alone do not determine the result. Aligned fibres produce anisotropy, whereas changing their orientations changes which directions are reinforced.^[ref] In an ideal long-fibre composite, the longitudinal rule of mixtures estimates the axial Young's modulus. A comparison of tensile testing along and across the fibres exposes the importance of geometry and fibre-matrix load transfer. The same idea of combining phase properties with geometry appears in effective conductivity, although the relevant constitutive equations differ.

The composite matrix in a composite material is its continuous surrounding phase, distinguished from embedded fibres or particles. It holds the reinforcement together and transfers load through interfacial stresses; it can also protect the reinforcement from damage or the environment.

For a fibre-reinforced polymer, the polymer is the composite matrix even if the fibres carry much of the axial load. Matrix stiffness, interfacial adhesion and failure therefore affect the whole specimen. This materials meaning differs from a rectangular array of numbers in linear algebra.

Fibre-matrix load transfer describes how interfacial shear stress transfers axial force between a reinforcing fibre and its surrounding composite matrix. For a cylindrical fibre of diameter $d$, axial force balance gives $d{\sigma }_f/dx=4\tau /d$ when $\tau$ is the appropriately signed interfacial shear stress.

The fibre stress is therefore small near a free end and builds over a transfer length. Short fibres may never reach their tensile strength before the interface slips or the matrix yields. The critical fibre length quantifies this competition in a simplified constant-shear model.

The fibre volume fraction is the fraction of a composite material's total volume occupied by reinforcing fibres:For a specimen with only fibres and a composite matrix, their fractions sum to one. Voids require a third fraction; they must not be silently counted as composite matrix. The fibre volume fraction is different from the fibre mass fraction because the constituents can have different mass densities.

For a void-free material, a measured fibre mass fraction $w_f$ giveswhere ${\rho }_f$ and ${\rho }_m$ are the constituent mass densities. For example, masses in the ratio $3:2$ and mass densities in the ratio $3:1$ give $V_f=1/3$, not $3/5$. Inserting $V_f$ into the longitudinal rule of mixtures estimates axial Young's modulus; orientation and fibre-matrix load transfer remain separate requirements.

A glass fibre is a thin strand of glass used as a reinforcement. When embedded in a composite matrix, many glass fibres can carry axial tension while the surrounding material maintains their positions and transfers load between them. The reinforcement is effective only if the interface transmits the required shear stress; an unbonded fibre can slide without carrying the expected load.

Continuous aligned glass fibres reinforce one direction strongly, whereas short fibres with varied orientations distribute reinforcement over more directions.^[ref] Even if the glass itself is approximately isotropic, a composite material containing aligned glass fibres is generally anisotropic. Comparing specimens with equal fibre volume fraction but different orientations separates the effects of constituent quantity and arrangement.

A hybrid fibre composite contains two or more types of reinforcing fibre in a common composite matrix. Combining carbon fibres and glass fibres, for example, provides another design choice besides changing the amount or orientation of a single reinforcement. The word hybrid specifies the combination of reinforcements, not a guarantee that every mechanical property improves.

For continuous, parallel fibres, perfect bonding and elastic axial loading, equal strain gives the extension of the longitudinal rule of mixturesHere the $V_i$ are constituent fractions by volume and the $E_i$ are axial Young's moduli. This estimate describes the initial elastic response. It does not by itself predict tensile strength: when one fibre population breaks, its load is redistributed through the composite matrix, changing the stress carried by the surviving population.

A long-fibre composite contains reinforcing fibres whose lengths are large enough that end regions do not dominate their load-carrying response. In a well-bonded continuous aligned specimen pulled along the fibres, the fibres and composite matrix undergo approximately the same axial strain. Their different Young's moduli then produce different axial stresses, and adding their forces gives the longitudinal rule of mixtures.

Load enters a fibre through shear stress at its interface with the composite matrix. Near a free fibre end the axial stress must rise from zero, so a short fibre can spend much of its length below its useful load. The critical fibre length expresses the associated scale in a simple strength-transfer model. Whether a fibre is mechanically long therefore depends on its diameter, the interface and the required stress, rather than on an absolute length in millimetres. Changing fibre direction still changes the response: long reinforcement does not remove anisotropy.

The longitudinal rule of mixtures estimates the axial Young's modulus of continuous aligned fibres in a composite matrix:The model assumes perfect bonding, small elastic strain, no voids and loading parallel to the fibres. Equal extension over a common initial length gives equal axial strain $\epsilon$ in both phases. Adding their forces gives the average axial stress $\sigma =V_f{E}_f\epsilon +(1-V_f)E_m\epsilon$, which proves the formula. It is the axial Voigt elastic approximation.

For example, $V_f=0.6$, $E_f=70\mathrm{GPa}$ and $E_m=3\mathrm{GPa}$ give $E_L=43.2\mathrm{GPa}$. These are illustrative inputs, not universal constituent values. The formula makes reinforcement efficient because the stiffer constituent carries the larger stress at the common strain. Loading across the fibres changes the compatibility and traction conditions, so the same formula is not a general transverse-modulus law. Fibre waviness or poor fibre-matrix load transfer can also invalidate the ideal axial estimate.