Modulus of toughness is defined as the amount of strain energy density (strain on a unit volume of material) that a given material can absorb before it fractures. Modulus of toughness is measured in units of PSI or Pascals. It can be calculated in a test by calculating the total area under the stress-strain curve up until the fracture point of the specimen.
To put this into perspective, two materials that have the same yield strength could have entirely different modulus of toughness if one is more ductile than the other. A material that is more ductile will stretch further and have a greater area under its stress-strain curve.
The most common application of modulus of toughness in engineering is in the designing of new structures. If a structure being designed is more susceptible to being overloaded, a more ductile material with a higher modulus of toughness is best, even if other materials have the same yield strength. This can sometimes lead to a higher cost material being used but will also increase the safety of the structure.