Modulus of Resilience Calculator

The modulus of resilience (U_r) is the maximum strain energy per unit volume a material can absorb and still recover elastically. It equals the area under the linear portion of the stress-strain curve up to the yield point: U_r = sigma_y^2 / (2E). Units are J/m^3 or MJ/m^3.

High-strength steels and spring steels have high resilience because they combine high yield strength with a moderate elastic modulus. Spring steel (sigma_y = 1,200 MPa, E = 200 GPa): U_r = 1200^2 / (2 * 200,000) = 3.6 MJ/m^3. Mild steel (sigma_y = 250 MPa): U_r = 0.16 MJ/m^3. Rubber has very high resilience from a different mechanism.

Resilience is the elastic energy absorption capacity (reversible). Toughness is the total energy absorption before fracture, including plastic deformation (irreversible). A spring must have high resilience. A crash-absorbing component needs high toughness. These are often competing properties.

Modulus of resilience is used to select materials for springs, snap-fit connectors, bow limbs, and impact-absorbing structures where elastic energy storage and release is the primary function. Engineers compare U_r per unit mass (specific resilience = U_r / density) for weight-critical applications.

Because U_r = sigma_y^2 / (2E), a lower Young's modulus increases resilience for the same yield strength. This is why titanium alloys and fiber composites often have better specific resilience than steel despite lower yield strengths: their lower E more than compensates. However, the stiffness penalty must also be considered.