= Young's modulus
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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> $\varepsilon=\Delta L_g/L_g$. In the proportional range,
$$
E=\frac{d\sigma}{d\varepsilon}=\frac{L_g}{A_0}\frac{dF}{d(\Delta L_g)}.
$$
The ratio $\sigma/\varepsilon$ 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>: $1\ \mathrm{Pa}=1\ \mathrm{N\,m^{-2}}$. By contrast, the <spring constant> of a uniform tensile specimen of length $L$ is $k_s=EA_0/L$, with units $\mathrm{N\,m^{-1}}$. 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: $\delta A_0/A_0\simeq2\delta d/d$. 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.<Young; Thomas (1773–1829); physician, physicist and Egyptologist by Royal Society>[Royal Society person record NA7947, Activity: formulation of a measure of elasticity called Young's Modulus]{ref}
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