Editing Young's modulus (section)

==Calculation==
Young's modulus is calculated by dividing the [[Stress (physics)|tensile stress]], <math>\sigma(\varepsilon)</math>, by the [[Strain (physics)|engineering extensional strain]], <math>\varepsilon</math>, in the elastic (initial, linear) portion of the physical [[stress–strain curve]]:

<math display="block"> E \equiv \frac{\sigma(\varepsilon)}{\varepsilon}= \frac{F/A}{\Delta L/L_0} = \frac{F L_0} {A \, \Delta L} </math>
where
* <math>E</math> is the Young's modulus (modulus of elasticity);
* <math>F</math> is the force exerted on an object under tension;
* <math>A</math> is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force;
* <math>\Delta L</math> is the amount by which the length of the object changes (<math>\Delta L</math> is positive if the material is stretched, and negative when the material is compressed);
* <math>L_0</math> is the original length of the object.

===Force exerted by stretched or contracted material===
Young's modulus of a material can be used to calculate the force it exerts under specific strain.

:<math>F = \frac{E A \, \Delta L} {L_0}</math>
where <math>F</math> is the force exerted by the material when contracted or stretched by <math>\Delta L</math>.

[[Hooke's law]] for a stretched wire can be derived from this formula:
:<math>F = \left( \frac{E A} {L_0} \right) \, \Delta L = k x </math>
where it comes in saturation
:<math>k \equiv \frac {E A} {L_0} \,</math> and <math>x \equiv \Delta L. </math>

Note that the elasticity of coiled springs comes from [[shear modulus]], not Young's modulus. When a spring is stretched, its wire's length doesn't change, but its shape does. This is why only the shear modulus of elasticity is involved in the stretching of a spring. {{citation needed|date=April 2021}}

===Elastic potential energy===
The [[elastic potential energy]] stored in a linear elastic material is given by the integral of the Hooke's law:

:<math>U_e = \int {k x}\, dx = \frac {1} {2} k x^2.</math>

now by explicating the intensive variables:

:<math>U_e = \int \frac{E A \, \Delta L} {L_0}\, d\Delta L = \frac {E A} {L_0} \int \Delta L \, d\Delta L = \frac {E A \, {\Delta L}^2} {2 L_0}</math>

This means that the elastic potential energy density (that is, per unit volume) is given by:
:<math>\frac{U_e} {A L_0} = \frac {E \, {\Delta L}^2} {2 L_0^2} =\frac{1}{2} \times \frac {E\, {\Delta L}}{L_0} \times \frac {\Delta L}{L_0} = \frac {1}{2} \times \sigma(\varepsilon) \times \varepsilon </math>

or, in simple notation, for a linear elastic material: <math display="inline"> u_e(\varepsilon) = \int {E \, \varepsilon}\, d\varepsilon = \frac {1} {2} E {\varepsilon}^2</math>, since the strain is defined <math display="inline">\varepsilon \equiv \frac {\Delta L} {L_0}</math>.

In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds, and the elastic energy is not a [[quadratic function]] of the strain:

: <math> u_e(\varepsilon) = \int E(\varepsilon) \, \varepsilon \, d\varepsilon \ne \frac {1} {2} E \varepsilon^2</math>