= Single-fibre tensile test
A <single-fibre tensile test> pulls one <material fibre> between grips while measuring axial <force> and extension over a defined <gauge length>. A <load cell> supplies force information; a direct specimen-extension measurement or a corrected <crosshead displacement> supplies extension. With initial area $A_0$ and initial gauge length $L_g$, the conventional <engineering stress> and <engineering strain> are
$$
\sigma=\frac{F}{A_0},\qquad \varepsilon=\frac{\Delta L}{L_g}.
$$
The initial proportional slope estimates <Young's modulus>; a changing local slope is a <tangent modulus>. The area and displacement conventions must accompany that result.
The glass- and carbon-fibre study by <Rajnish Kumar (materials scientist)>, <Lars Pilgaard Mikkelsen>, <Hans Lilholt> and <Bo Madsen (materials scientist)> used a semi-automated single-fibre machine and compared stress-strain-based modulus determinations.<Understanding the mechanical response of glass and carbon fibres: stress-strain analysis and modulus determination by Rajnish Kumar and others>[Public primary abstract, apparatus and modulus analysis]{ref} Its dataset contains individual tensile curves, gauge-length groups and test-setup compliance calculations.<Single fiber tensile test - Glass and Carbon fibers by Rajnish Kumar and others>[Description, Data/Results files and Compliance spreadsheets]{ref}
For an original ideal calculation, take a circular strand of diameter $10\,\mathrm{\mu m}$ and $L_g=20\,\mathrm{mm}$. Its area is $A_0=7.854\times10^{-5}\,\mathrm{mm^2}$. A force increment $0.010\,\mathrm N$ produces measured crosshead travel $0.040\,\mathrm{mm}$. Suppose independently established <machine compliance> is $C_m=0.50\,\mathrm{mm\,N^{-1}}$, with no grip slip and the same deforming specimen length as $L_g$. Then
$$
\Delta L=\Delta x-C_m\Delta F=0.035\,\mathrm{mm},\qquad
E=\frac{L_g}{A_0}\frac{\Delta F}{\Delta L}\approx72.8\,\mathrm{GPa}.
$$
Using the uncorrected travel would instead give $63.7\,\mathrm{GPa}$. These chosen increments represent a proportional segment, not measured dataset values or a complete uncertainty analysis.
Check grip slip, specimen alignment and the force zero before interpreting the slope. An irregular area cannot automatically be replaced by a circular area from one apparent diameter. A <vibroscope> offers another route to fibre-area estimation through vibration under <tension>.<Vibroscope method for determination of cross-sectional area of glass and carbon fibres – Experiments and analyses by Bo Madsen and others>[Public abstract, area estimation from tensioned-fibre frequency measurements]{ref} Preserve the original force and displacement records so corrections and area assumptions remain inspectable.
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