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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.

Engineering strain is the change in a measured segment's length divided by that segment's initial length:It is dimensionless. A segment initially $50\mathrm{mm}$ long that extends by $0.5\mathrm{mm}$ has $e=0.01$, or one percent. The initial gauge length must match the instrument's measured extension; actuator motion can include deformation of grips and the test frame.

For a uniform stretch ratio $\lambda =L/L_0$, engineering strain is $\lambda -1$. It agrees with the small axial strain to leading order, while the logarithmic length change is $\log \lambda$. Engineering strains over successive finite stretches do not simply add: two extensions of ten percent multiply length by $1.1^2$, giving a total of twenty-one percent. This makes the choice of reference length essential when comparing measurements taken over different loading stages.

An extensometer measures the change in distance between two specimen points. Its gauge length defines the reference length used to calculate engineering strain.

A 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.^[ref] Its public dataset supplies individual tensile curves at several gauge lengths, summary sheets and calculations of test-setup machine compliance.^[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 radiusFor 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 stiffnessindependent 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 $E{I}_N$. Its bending moment is $E{I}_N\kappa$ in the small-deformation model.^[ref] Adding those moments gives the bending rigidity of an unbonded fibre bundle: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 $K_{\mathrm{ax}}=700\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$. At $N=100$, each radius is approximately $0.0564\mathrm{mm}$, and the independent-bending value is $55.7\mathrm{N}\mathrm{m}{\mathrm{m}}^2$ compared with $5570.4\mathrm{N}\mathrm{m}{\mathrm{m}}^2$ for the solid rod. The parameters are stipulated for this model; $A$ counts material and excludes spaces between strands.

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.

The gauge length is the initial separation of the points whose extension defines a reported engineering strain. It need not equal the full deforming length between grips.

Machine compliance is the extra load-path displacement per applied force attributed to the apparatus in a specified measurement model. Its units are $\mathrm{m}{\mathrm{N}}^{-1}$, the reciprocal units of a spring constant. Here $C_m$ includes the frame, load cell, adapters and grips insofar as their deformation contributes to the recorded crosshead displacement $x$. It is a property of that configuration and readout, rather than a universal constant of the testing machine.

Assume a uniform specimen with initial length $L$, constant $A_0$ and Young's modulus $E$, small axial elastic deformation, quasistatic loading, no slip, and a linear, length-independent apparatus response over the chosen force window. Equilibrium makes the same force act through specimen and apparatus, while their extensions add. The two springs in series therefore giveThe offset $x_0$ describes the displacement origin after seating; the slope $q$ describes total compliance. Fitting an intercept avoids forcing an arbitrary zero through the origin, but cannot repair slip or a curved response.

If all of $x-x_0$ is incorrectly attributed to specimen extension, the inferred modulus isFor positive $C_m$, this is too small. The fractional bias is most serious for a short, thick, stiff specimen because its own compliance $L/(E{A}_0)$ is small. A local extensometer instead has $d(\Delta L_g)/dF=L_g/(E{A}_0)$ in this ideal model, excluding deformation outside its contacts.

A shaped specimen requires more care. With varying initial cross-sectional area $A(s)$ and uniform $E$, its axial compliance is $\int ds/[EA(s)]$ in the one-dimensional approximation. Shoulder and end deformation must be included explicitly or remain identical between tests and be absorbed into an effective intercept. Changing only the extensometer gauge length on an unchanged specimen does not change its total compliance and cannot perform a two-length machine compliance separation.

The reference configuration is a chosen arrangement used to label a body's material particles and measure their subsequent deformation. A particle at $\mathbf{X}$ in the chosen reference has position $\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$ in the current configuration. The reference coordinates are Lagrangian coordinates and remain attached to the particles.

The initial shape of a specimen is a convenient reference, but a reference configuration need not be stress-free. A preloaded or residually stressed specimen can also be chosen as the reference. For example, engineering strain compares length with the chosen original gauge length, while the deformation gradient compares infinitesimal material line elements. Current and reference mass densities satisfy $\rho J={\rho }_0$ for conserved mass, with $J$ the local volume ratio. Keeping this choice explicit prevents mixing stress or strain measures based on different areas and lengths.

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.

Strain measures local changes of length and angle relative to a chosen reference configuration. For a small displacement field $\mathbf{u}(\mathbf{X})$, the infinitesimal strain tensor isIts diagonal entries are fractional length changes along the coordinate axes. Off-diagonal entries describe changes of angle, with engineering shear strain equal to twice the corresponding tensor component.

A uniform translation has zero strain because its displacement gradient vanishes. An infinitesimal rigid rotation also has zero strain because its displacement gradient is antisymmetric. Finite rotations require finite strain theory, since the linear formula is then insufficient. In tensile testing, the simple ratio of extension to original gauge length is engineering strain; it coincides with the small axial component in the appropriate limit but is not a complete three-dimensional description.

Strain rate measures how rapidly a material deforms. In a one-dimensional small-strain test, $\dot{\epsilon }$ is extension rate divided by initial gauge length, with units ${\mathrm{s}}^{-1}$. Stretching to the same final strain quickly or slowly can produce different stresses in a rate-dependent material.

For a general flow, the instantaneous local length-change rate is described by the rate-of-strain tensorA material line of current unit direction $\mathbf{n}$ has relative length-change rate $\mathbf{n}\cdot D\mathbf{n}$. This uses the velocity gradient and excludes rigid-body spin. In finite strain theory, $D$ is not simply the componentwise time derivative of every possible finite strain measure. Newtonian fluid stress depends on $D$, while measurements at several strain rates test whether an elastic or plastic model needs explicit rate dependence.

MIT's Spring 2004 laboratory assignment Properties of Heat Treated Steel gives a concrete tensile testing protocol. Record transducer channels and calibration factors; measure specimen diameter and length, and the extensometer gauge length with its locking pin inserted. Zero the load cell, pin the threaded grips into their adapters, and attach the extensometer centrally. Apply the specified seating load, record actuator position, then acquire readings every second while moving at about $0.05\mathrm{cm}\mathrm{mi}{\mathrm{n}}^{-1}$. The handout derives engineering stress from load cell force, engineering strain from extensometer deformation, and Young's modulus by linear regression below the proportional limit; it warns that actuator-derived strain includes apparatus deformation.^[ref]

For a modulus-focused measurement, select a load window after seating, hold temperature and loading rate consistent, and record force, local extension and crosshead displacement together. Fit $F=F_0+k_g\Delta L_g$ within that window and calculate $E=k_g{L}_g/A_0$. Inspect the regression residuals for curvature; repeat at a lower maximum load and include unloading to check elastic deformation. These are diagnostic additions to the cited protocol, whose full test continues through failure. The handout's seating load is specific to its specimens and apparatus, rather than a universal preload.

An extensometer measures the change between its contact points; crosshead displacement measures travel of the machine's moving member. The manufacturer Instron explicitly distinguishes local strain from travel that includes machine deformation and machine compliance.^[ref] The gauge length $L_g$ can be shorter than the uniform length $L$ between grips. Thus even a perfectly rigid machine generally has $\Delta L_g\ne x$: the local instrument sees only its own segment. Do not divide one channel's extension by the other channel's reference length.

Grips must transmit axial force without slip. Misalignment adds bending moment and makes opposite sides experience different axial strains; an instrument on just one side may then report a biased modulus. A changing extension offset, hysteresis between loading and unloading, or load-dependent fitted slope is evidence to investigate seating, slip or material response. A straight line over a short interval alone does not identify which is responsible.

Young's modulus is the ratio of axial stress to axial strain in uniaxial linear elasticity. For an isotropic plate, the bending rigidity is $E{t}^3/[12(1-{\nu }^2)]$, with thickness $t$ and Poisson's ratio $\nu$.

A tensile modulus experiment asks how much a specimen stretches under an axial force, while allowing its sides to contract freely. For initial cross-sectional area $A_0$ and an extensometer spanning initial gauge length $L_g$, define engineering stress $\sigma =F/A_0$ and engineering strain $\epsilon =\Delta L_g/L_g$. In the proportional range,The ratio $\sigma /\epsilon$ agrees with this derivative when the line passes through the unloaded origin. A fitted offset should not be mistaken for a material property. Young's modulus has units of pressure: $1\mathrm{Pa}=1\mathrm{N}{\mathrm{m}}^{-2}$. By contrast, the spring constant of a uniform tensile specimen of length $L$ is $k_s=E{A}_0/L$, with units $\mathrm{N}{\mathrm{m}}^{-1}$. Two rods made of the same material can therefore have very different spring constants while sharing Young's modulus.

The normalization removes a useful geometrical ambiguity: doubling $L$ doubles the extension at a given force, whereas doubling $A_0$ halves it. For a round specimen $A_0=\pi d^2/4$; for a rectangular one $A_0=wt$, using initial diameter $d$, width $w$ and thickness $t$. Diameter error matters twice at first order: $\delta A_0/A_0\simeq 2\delta d/d$. Specify material direction, temperature and loading rate when comparing Young's modulus. Lateral restraint changes the stress state and can prevent this experiment from measuring the uniaxial modulus.

The name commemorates Thomas Young, whose formulation of a measure of elasticity is recorded by the Royal Society. This attribution alone does not establish priority for every modern definition or experimental method.^[ref]