The ability of a material to absorb energy when deformed elastically and to return it when unloaded is called resilience. This is usually measured by the modulus of resilience, which is the strain energy per unit volume required to stress the material from, zero stress to the yield stress s. The strain energy per unit volume for uniaxial tension is

(1)

From the above definition the modulus of resilience is

(2)

This equation indicates that the ideal material for resisting energy loads in applications where the material must not undergo permanent distortion, such as mechanical springs, is having a high yield stress and a low modulus of elasticity. Table 1 gives some values of modulus of resilience for different materials.

Table 1. Modulus of resilience for various materials

Material	E, psi	s0, psi	Modulus of resilience, Ur
Medium-carbon steel	30×106	45000	33,7
High-carbon spring steel	30×106	140000	320
Duraluminium	10,5×106	18000	17,0
Cooper	16×106	4000	5,3
Rubber	150	300	300
Acrylic polymer	0,5×106	2000	4,0

Toughness

The toughness of a material is its ability to absorb energy in the plastic range. The ability to withstand occasional, stresses above the yield stress without fracturing is particularly desirable in parts such as freight-car couplings, gears, chains, and crane hooks. Toughness is a commonly used concept, which is difficult to pin down and define. One way of looking at toughness is to consider that it is the total area under the stress-strain curve. This area is an indication of the amount of work per unit volume, which can be done, on the material without causing it to rupture. Figure 1.2 shows the stress-strain curves for high- and low-toughness materials. The high-carbon spring steel has a higher yield strength and tensile strength than the medium-carbon structural steel. However, the structural steel is more ductile and has a greater total elongation. The total area under the stresstrain curve is greater for the structural steel, and therefore it is a tougher material. This illustrates that toughness is a parameter that comprises both strength and ductility. The crosshatched regions in Fig. 1 indicate the modulus of resilience for each steel. Because of its higher yield strength, the spring steel has the greater resilience.
Several mathematical approximations for the area under the stress-strain curve have been suggested. For ductile metals that have a stress-strain curve like that of the structural steel, the area under the curve can be approximated by either of the following equations:

(3)

(4)

For brittle materials the stress-strain curve is sometimes assumed to be a parabola, and the area under the curve is given by

(5)

Figure 1. Comparison of stress-strain curves

All these relations are only approximations to the area under the stress-strain curves. Further, the curves do not represent the true behavior in the plastic range, since they are all based on the original area of the specimen.