The value of the strain-energy density obtained by setting ε1 = εR in Eq1 from the lesson Strain-Energy Density, where εR is the strain at rupture, is known as the modulus of toughness of a material. It is equal to the area under the entire stress-strain diagram as shown:
and represents the energy per unit volume required to cause the material to rupture. It is clear that the toughness of a material is related to its ductility as well as to its ultimate strength, and that the capacity of a structure to withstand an impact load depends upon the toughness of the material used. For example, railroad couplers are made of ductile steel which has a large modulus of toughness.