Editing Young's modulus (section)

===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>