Toughness - Mathematical Definition

Mathematical Definition

Toughness can be determined by measuring the area (i.e., by taking the integral) underneath the stress-strain curve and its energy of mechanical deformation per unit volume prior to fracture. The explicit mathematical description is:

Where

• is strain
• is the strain upon failure
• is stress

Another definition is the ability to absorb mechanical (or kinetic) energy up to failure. The area covered under stress strain curve is called toughness.

If the upper limit of integration up to the yield point is restricted, then the energy absorbed per unit volume is known as the modulus of resilience. Mathematically, the modulus of resilience can be expressed by the product of the square of the yield stress divided by two times the Young's modulus.