# Modulus of Toughness

## Table of Contents

The modulus of toughness is given by:

$$ U_t = \int_{0}^{\epsilon_f}\sigma\mathop{d\epsilon} $$

Where

- \( \small U_t \) is the modulus of toughness,
- \( \small \sigma \) is the stress,
- \( \small \epsilon \) is the strain, and
- \( \small \epsilon_f \) is the strain at fracture.

Note that this is the area under the stress-strain curve up to the failure/fracture strain. See the sources for some geometric approximations for the modulus of toughness.