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A chord modulus is the slope of the line connecting two specified points on a stress-strain curve:Instron's glossary defines the calculation by selecting a starting strain and an ending strain and taking the slope between their curve points.^[ref] Report the endpoints and strain definition with the value. A linear regression through every observation between them is generally a different estimator.

For an original branch $\sigma =E_0\epsilon +B{\epsilon }^2$, subtraction and factorization giveWith $E_0=2\mathrm{GPa}$, $B=100\mathrm{GPa}$, ${\epsilon }_1=0.01$ and ${\epsilon }_2=0.03$, the endpoint stresses are $30$ and $150\mathrm{MPa}$, so $E_{\mathrm{chord}}=6\mathrm{GPa}$. The secant modulus from the unloaded origin to the second point is instead $5\mathrm{GPa}$. For this quadratic branch, the chord happens to equal the tangent modulus at the midpoint strain; this midpoint equality does not hold for every nonlinear curve.

More generally, the fundamental theorem of calculus gives, for a smooth branch,Thus a chord samples the response across a finite interval. Shrinking that interval can approach a tangent modulus, but measurement noise becomes more consequential when divided by a small strain difference.

For engineering stress and local engineering strain measured with an extensometer,An original specimen with $A_0=10\mathrm{m}{\mathrm{m}}^2$ and gauge length $L_g=50\mathrm{mm}$ gives $6\mathrm{GPa}$ when force changes from $300$ to $1500\mathrm{N}$ and local extension changes from $0.5$ to $1.5\mathrm{mm}$. A constant readout offset cancels in each difference when those same physical states are used; scale errors and machine compliance do not generally cancel. Specify temperature, loading direction and history, since a chord across different branches need not describe one material response.

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]

The secant modulus at a specified point of a stress-strain curve is the slope of the line from the declared unloaded origin to that point:It depends on the chosen point when the curve is nonlinear. Mississippi State University's solid-mechanics laboratory material defines it geometrically in this way.^[ref] Specify both the strain measure and the origin when reporting a secant modulus.

For an original branch $\sigma =E_0\epsilon +B{\epsilon }^2$, with $E_0=2\mathrm{GPa}$ and $B=100\mathrm{GPa}$,At ${\epsilon }_{\ast }=0.02$, stress is $80\mathrm{MPa}$, the secant modulus is $4\mathrm{GPa}$ and the tangent modulus is $6\mathrm{GPa}$. The first slope relates accumulated stress and strain; the second predicts a small further increment. In linear elasticity through the unloaded origin they both equal Young's modulus.

The relation is especially clear for a smooth curve with $\sigma (0)=0$. The fundamental theorem of calculus givesThus the secant modulus averages the local tangent slopes over that interval. A chord modulus instead joins two specified states, with slope $({\sigma }_2-{\sigma }_1)/({\epsilon }_2-{\epsilon }_1)$; its first state need not be the unloaded origin.

For a tensile measurement using initial cross-sectional area $A_0$ and local gauge length $L_g$, engineering stress and engineering strain givewhere extension is measured from the appropriate unloaded state. An original $F=800\mathrm{N}$, $A_0=10\mathrm{m}{\mathrm{m}}^2$, $L_g=50\mathrm{mm}$ and $\Delta L_g=1\mathrm{mm}$ yields $E_{\sec }=4000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-2}=4\mathrm{GPa}$. A local extensometer avoids attributing all machine compliance to specimen extension. Retain the loading history, temperature and target stress or strain: a secant from an arbitrary preload and an unloading slope are different measurements, particularly after plastic deformation.

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.

The tangent modulus is the local slope of a specified uniaxial stress-strain curve:It converts a small additional strain near ${\epsilon }_0$ into the corresponding first-order change of stress. For nonlinear elastic deformation, this slope can vary with the operating point.^[ref] In uniaxial linear elasticity, $\sigma =E\epsilon$, so the tangent modulus equals the constant Young's modulus.^[ref]

For an original nonlinear branch, stipulateThen $E_{\tan }=E_0+2B\epsilon$. At ${\epsilon }_0=0.02$, the stress is $80\mathrm{MPa}$ and the tangent modulus is $6\mathrm{GPa}$. The secant modulus from the stipulated unloaded origin is instead $\sigma /{\epsilon }_0=4\mathrm{GPa}$. The two slopes answer different questions: the secant compares total stress and strain, while the tangent predicts a small further increment.

For $\Delta \epsilon =0.001$, the tangent predicts $\Delta \sigma \simeq 6\mathrm{MPa}$. The exact change in the chosen polynomial is $6.1\mathrm{MPa}$ because the omitted term is $B(\Delta \epsilon {)}^2=0.1\mathrm{MPa}$. This is a linear approximation whose error decreases quadratically with the increment, not a claim that the nonlinear material obeys Hooke's law globally.

To estimate a tangent modulus experimentally, specify the stress and strain measures, temperature, loading history and loading rate. For engineering stress $\sigma =F/A_0$ and engineering strain $\epsilon =\Delta L_g/L_g$ from a local extensometer,Fit a local window about the chosen operating point and check how the estimate changes with that window. For an original $L_g=50\mathrm{mm}$, $A_0=10\mathrm{m}{\mathrm{m}}^2$ and local slope $1200\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$, this gives $6000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-2}=6\mathrm{GPa}$. Differentiation amplifies measurement noise, and crosshead displacement can include machine compliance. Where plastic deformation or history-dependent response occurs, identify the loading branch: its local slope need not equal the unloading slope or an elastic material constant.