= Chord modulus
{title2=$E_{\rm chord}=\Delta\sigma/\Delta\varepsilon$}
A \b[chord modulus] is the slope of the line connecting two specified points on a <stress-strain curve>:
$$
E_{\rm chord}=\frac{\sigma_2-\sigma_1}{\varepsilon_2-\varepsilon_1},\qquad \varepsilon_2\ne\varepsilon_1.
$$
Instron's glossary defines the calculation by selecting a starting strain and an ending strain and taking the slope between their curve points.<Modulus of Elasticity in the Instron glossary>[Calculating Different Types of Modulus, Chord Modulus]{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 $\sigma=E_0\varepsilon+B\varepsilon^2$, subtraction and factorization give
$$
E_{\rm chord}=E_0+B(\varepsilon_1+\varepsilon_2).
$$
With $E_0=2\,\mathrm{GPa}$, $B=100\,\mathrm{GPa}$, $\varepsilon_1=0.01$ and $\varepsilon_2=0.03$, the endpoint stresses are $30$ and $150\,\mathrm{MPa}$, so $E_{\rm chord}=6\,\mathrm{GPa}$. 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,
$$
E_{\rm chord}=\frac1{\varepsilon_2-\varepsilon_1}
\int_{\varepsilon_1}^{\varepsilon_2}E_{\rm tan}(u)\,du.
$$
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>,
$$
E_{\rm chord}=\frac{L_g}{A_0}\frac{F_2-F_1}{\Delta L_{g,2}-\Delta L_{g,1}}.
$$
An original specimen with $A_0=10\,\mathrm{mm^2}$ 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.
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