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Bo Madsen studies the physical and mechanical behaviour of composite materials. His DTU profile relates experiments and analytical modelling to microstructure, volumetric composition and manufacturing conditions.^[ref] He is also a named contributor to the shared glass- and carbon-fibre tensile dataset.^[ref]

In 2024 he was first and corresponding author of an experimental vibroscope study with Rajnish Kumar and Hans Lilholt. The apparatus uses frequency measurements on a fibre under tension to infer its cross-sectional area. The paper compared simple, direct and iterative analysis procedures, and reported distributions of individual fibre areas rather than treating all strands as geometrically identical.^[ref]

This method addresses a practical difficulty in a single-fibre tensile test: the area needed to convert force into engineering stress may be difficult to determine from an image of one apparent diameter. Comparing independently inferred areas with the resulting stress-strain curves tests more than a force sensor alone. It also keeps geometric variation between material fibres separate from differences in the material's constitutive response. The work is a collaboration; the byline does not imply that Madsen alone performed every measurement.

Hans Lilholt is a materials scientist whose DTU profile identifies research on composite materials, their preparation and their physical and mechanical properties.^[ref] He is a named contributor to the glass- and carbon-fibre tensile dataset, alongside Rajnish Kumar, Bo Madsen and Lars Pilgaard Mikkelsen.^[ref]

With Bo Madsen and Rajnish Kumar, he coauthored a 2024 experimental study of the vibroscope method for estimating fibre cross-sectional area from vibration under tension. It compared three analysis procedures and used tensile stress-strain curves in assessing the area estimates.^[ref]

An original calculation shows why this metrology affects a strength claim. At fixed measured force, inferring an area $\hat{A}=0.98A$ produces inferred engineering stressA chosen two-percent area underestimate therefore gives about a 2.04-percent stress overestimate. This is an illustrative calculation, not a new measurement from that study. The area measurement and its analysis belong alongside the reported tensile result; a force record alone does not determine material stress.

Lars Pilgaard Mikkelsen is a researcher in the mechanics of composite materials. His DTU profile describes combining the finite element method with experimental characterization of polymer matrix composites, particularly their mechanical fatigue and compression response.^[ref]

He coauthored the 2020 study of glass fibres and carbon fibres with Rajnish Kumar, Hans Lilholt and Bo Madsen. The study compared modulus determinations from single-strand stress-strain measurements.^[ref] The associated dataset identifies him by the full name Lars Pilgaard Mikkelsen and lists him as a contributor; it supplies individual tensile records and separate machine compliance calculations.^[ref]

The measurement distinction is important for mechanics models: crosshead displacement can include movement of the test apparatus as well as specimen extension. A simulation supplied with a modulus inferred from that combined displacement can inherit an apparatus-dependent response. The single-fibre tensile test explains a numerical correction example; retaining both the individual records and the correction assumptions makes that inference inspectable. This is joint experimental work, and the cited records do not assign every instrument or analysis step to one coauthor.

Rajnish Kumar is a researcher in mechanical testing of composite materials. His institutional profile identifies him within DTU's composites manufacturing and testing group and gives ORCID 0000-0002-1243-9585, distinguishing this materials researcher from namesakes.^[ref]

He was first and corresponding author of the 2020 glass- and carbon-fibre study with Lars Pilgaard Mikkelsen, Hans Lilholt and Bo Madsen. Their semi-automated single-fibre tensile test examined stress-strain-based modulus determinations. The public abstract reports an approximately strain-independent true-stress-based tangent modulus for the tested glass fibres, while the tested carbon fibres showed a tendency for that modulus to increase with strain.^[ref]

That distinction explains why one fitted Young's modulus need not summarize an entire stress-strain curve. A local slope and a slope fitted over a finite strain interval answer different questions; their definitions must accompany a comparison. The result concerns the tested specimens and the stated analysis, rather than every glass or carbon fibre. The associated dataset names Kumar as a contributor and preserves individual records and compliance calculations, allowing those measurement assumptions to be examined.^[ref]

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 areThe 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, Lars Pilgaard Mikkelsen, Hans Lilholt and Bo Madsen used a semi-automated single-fibre machine and compared stress-strain-based modulus determinations.^[ref] Its dataset contains individual tensile curves, gauge-length groups and test-setup compliance calculations.^[ref]

For an original ideal calculation, take a circular strand of diameter $10\mu \mathrm{m}$ and $L_g=20\mathrm{mm}$. Its area is $A_0=7.854\times 10^{-5}\mathrm{m}{\mathrm{m}}^2$. A force increment $0.010\mathrm{N}$ produces measured crosshead travel $0.040\mathrm{mm}$. Suppose independently established machine compliance is $C_m=0.50\mathrm{mm}{\mathrm{N}}^{-1}$, with no grip slip and the same deforming specimen length as $L_g$. ThenUsing 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.^[ref] Preserve the original force and displacement records so corrections and area assumptions remain inspectable.

A vibroscope measures the vibration of a tensioned material fibre. In fibre testing, the measured fundamental frequency, suspended length and applied tension can be used to estimate linear mass density; known mass density then permits an area estimate. The physical measurement complements the single-fibre tensile test, whose conversion from force to engineering stress requires cross-sectional area.

The 2024 study by Bo Madsen, Rajnish Kumar and Hans Lilholt compared area-analysis procedures using measured frequencies of tensioned glass and carbon strands.^[ref] A practical arrangement holds a known strand length, applies a known tension, excites small transverse oscillations and detects the frequency of its response. The supports and identification of the vibration mode enter the interpretation.

For an original idealization, treat the strand as a uniform flexible string with fixed endpoints, length $L$, tension $T$ and linear mass density $\mu$. Its wave speed is $\sqrt{T/\mu }$ and its fundamental wavelength is $2L$, giving the vibroscope area estimatorThe string wave-speed and fixed-end standing-wave relations supply this model.^[ref]

Choose $L=20\mathrm{mm}$, $T=1.00\mathrm{mN}$, $f_1=2000\mathrm{Hz}$ and $\rho =2500\mathrm{kg}{\mathrm{m}}^{-3}$. The model gives $\hat{\mu }=1.5625\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-1}$ and $\hat{A}=62.5\mu {\mathrm{m}}^2$. Only if a circular area is assumed does this correspond to diameter $\sqrt{4\hat{A}/\pi }\approx 8.92\mu \mathrm{m}$.

These are stipulated model values, not published fibre data. Checking several tensions tests the predicted proportionality of $f_1^2$ to $T$. Finite bending resistance, uncertain effective length and changes in strand dimensions under tension can invalidate the flexible-string estimate. A resonance peak alone does not establish that all of its assumptions hold.

The vibroscope area estimator converts the fundamental frequency of a tensioned material fibre into a cross-sectional area through a mechanical model. For a uniform flexible strand of length $L$, positive tension $T$, linear mass density $\mu$ and mass density $\rho$, the fixed-end string relation givesThe second equation uses $\mu =\rho A$: the frequency first identifies mass per length under the stated tension, and a chosen mass density then converts it into area. Circular geometry is needed only to infer a diameter from that area.^[ref]

The experimental study by Bo Madsen, Rajnish Kumar and Hans Lilholt used a Favimat+ fibre testing machine. Acoustic waves excited the strand, and a frequency sweep plus a polynomial fit located the resonance peak. A load cell measured tension, while grip separation supplied $L$. The reported frequency range was 500–4000 Hz, with an estimated resolution of 0.1 Hz; the stated force and length readout resolutions were $1\mu \mathrm{N}$ and $0.1\mu \mathrm{m}$. Those readout resolutions do not themselves specify the estimator's full measurement uncertainty.^[ref]

The area and mass density must refer to the same loaded or reference configuration. The companion theory derives volume and area corrections for recovering an untensioned reference area from the tensioned measurement.^[ref]

The vibroscope area uncertainty follows locally by logarithmic differentiation:Treat the small input errors as independent random variables, and let each $u$ denote the corresponding standard uncertainty. First-order uncertainty propagation then givesCorrelated inputs require their covariances as well. Four independent 1% relative standard uncertainties give $\sqrt{10}\%\approx 3.16\%$ for the area. A systematic 1% overestimate of frequency alone instead multiplies the area estimate by $(1.01{)}^{-2}\approx 0.9803$: it produces about a 1.97% underestimate.

Bending resistance can also increase the observed frequency. The companion theory paper models fully clamped supports and derives the corresponding frequency equation and deformation corrections.^[ref] An independently derived, simply supported example isolates the bending-stiffness bias in the vibroscope area estimator. Let $B=EI$ be the filament bending modulus and impose zero deflection and zero bending moment at each endpoint. Substituting $y=\sin (n\pi x/L)\cos (2\pi f_{n}t)$ into the tensioned Euler-Bernoulli beam equation $\mu y_{tt}+B{y}_{xxxx}-T{y}_{xx}=0$ givesWith $r=b/a={\pi }^{2}B/(T{L}^2)$, the uncorrected flexible-string estimate satisfies $\hat{A}/A=1/(1+r)$. At $r=0.10$, it is 9.09% too small. The same model yields two-mode separation of tension and bending stiffness:For $T>0$ and $B\ge 0$, the predicted mode ratio is $f_2/f_1=2\sqrt{(1+4r)/(1+r)}$, ranging from 2 toward 4. These identities belong to the simply supported model; clamped grips require their own boundary conditions. Neither a single resonance peak nor an assumed diameter establishes the appropriate support, density or bending correction.