# Modulus of Resilience

## Table of Contents

The modulus of resilience is given by:

$$ U_r = \int_{0}^{\epsilon_Y}\sigma\mathop{d\epsilon} \approx \frac{\sigma_{YS}^2}{2E} $$

Where

- \( \small U_r \) is the modulus of resilience,
- \( \small \sigma \) is the stress,
- \( \small \epsilon \) is the strain,
- \( \small \epsilon_Y \) is the yield strain,
- \( \small \sigma_{YS} \) is the yield strength, and
- \( \small E \) is the elastic modulus (Young’s modulus).

Note that the first formula is th exact formula, equal to the area under the stress-strain curve up to the yield strain, and the second formula is an approximation for the modulus of resilience, which is valid for materials with a linear elastic region.