= Fibre
{disambiguate=material}
{wiki=Fiber}
= Material fibre
{synonym}
A \b[material fibre] is a long, slender physical strand: its <length> is large compared with its transverse dimensions. The geometry is shared by materials with different roles. <Carbon fibres> and <glass fibres> can reinforce a <composite material>, while an <optical fibre> guides <light>. The material's properties, the strand dimensions and the way strands are assembled all affect its response.
A <single-fibre tensile test> grips one strand, measures its axial <force> and extension over a specified <gauge length>, and uses its <cross-sectional area> to obtain <stress> and <strain>. A 2020 study used a semi-automated instrument to characterize <glass fibres> and <carbon fibres> through their <stress-strain curves> and inferred moduli.<Understanding the mechanical response of glass and carbon fibres: stress-strain analysis and modulus determination by Rajnish Kumar and others>[Public primary abstract: single-fibre apparatus and modulus analysis]{ref} Its public dataset supplies individual tensile curves at several <gauge lengths>, summary sheets and calculations of test-setup <machine compliance>.<Single fiber tensile test - Glass and Carbon fibers by Rajnish Kumar and others>[Dataset description, gauge-length groups and compliance files]{ref} This matters because <crosshead displacement> can include deformation of the apparatus as well as extension of the <material fibre>; its slope alone need not give the specimen's <Young's modulus>.
An original ideal model shows why making thinner strands changes some responses more strongly than others. Keep total material <cross-sectional area> $A$, <length> $L$ and <Young's modulus> $E$ fixed. Compare one solid circular rod with $N$ identical circular fibres, each of <radius>
$$
r_N=\sqrt{\frac{A}{N\pi}}.
$$
For axial loading, grip every strand so that all undergo the same extension $\Delta L$. Each supplies <force> $E(A/N)\Delta L/L$, and adding the <forces> gives the <axial stiffness>
$$
\boxed{K_{\rm ax}=\frac{EA}{L}},
$$
independent of $N$ under these assumptions. This is the uniform-bar <linear elasticity> relation applied to parallel strands.
Now make a separate pure-bending comparison of an <unbonded material-fibre bundle>. Allow longitudinal sliding, neglect <friction>, and let each strand bend about its own <neutral axis> with the same <curvature> $\kappa$. A circular strand has <second moment of area> $I_N=\pi r_N^4/4$ and <filament bending modulus> $EI_N$. Its <bending moment> is $EI_N\kappa$ in the small-deformation model.<Stresses in Beams by David Roylance>[Normal Stresses, equations3–6, pp.2–4: linear-elastic moment-curvature relation]{ref} Adding those moments gives the <bending rigidity of an unbonded fibre bundle>:
$$
\boxed{B_{\rm bundle}=NEI_N=\frac{EA^2}{4\pi N}=\frac{B_{\rm solid}}{N}}.
$$
Each <material fibre> has its own <neutral axis>, which is crucial to this calculation. Bonding strands together transfers <shear stress> and couples their axial deformation; the independent-strand sum then ceases to describe the assembly. In a reinforced <composite material>, that coupling is part of <fibre-matrix load transfer>.
For chosen $A=1\,\mathrm{mm}^2$, $L=100\,\mathrm{mm}$ and $E=70\,\mathrm{GPa}$, both axial specimens have $K_{\rm ax}=700\,\mathrm{N\,mm^{-1}}$. At $N=100$, each <radius> is approximately $0.0564\,\mathrm{mm}$, and the independent-bending value is $55.7\,\mathrm{N\,mm^2}$ compared with $5570.4\,\mathrm{N\,mm^2}$ for the solid rod. The parameters are stipulated for this model; $A$ counts material and excludes spaces between strands.
\Image[/unbonded-fibre-bundle-scaling.png]
{title=Equal material area with different axial and independent-fibre bending responses}
{description=Original linear-elastic model with fixed material area, length and modulus. Axial loading imposes equal extension; the separate bending comparison permits sliding and independent neutral axes. The curves are calculated rather than measured fibre data.}
A useful mechanical comparison therefore records which <material fibres> are gripped, whether they can slide, their arrangement and the loading direction. Equal material quantity alone does not determine an assembly's <stiffness>, and small <filament bending modulus> does not imply small axial <Young's modulus>.
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