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Modulus of Resilience Calculator

Enter yield strength and Young's modulus to find the modulus of resilience.

How it is calculated

The tool applies Ur = σy² ÷ (2E), the area under the elastic part of the stress-strain curve up to yield. Yield strength σy is entered in megapascals (MPa) and Young's modulus E in gigapascals (GPa); the tool converts E to MPa by multiplying by 1,000. Because 1 MPa equals 1 MJ/m³, the result is first in MJ/m³ (four decimals) and also in kJ/m³ (two decimals). It also shows the strain at yield, εy = σy ÷ E, as a plain number with five decimals. Young's modulus must be greater than zero and yield strength cannot be negative. The formula assumes linear elastic behaviour all the way to yield, so it does not apply to materials with no clear yield point or to non-linear elastic ones such as rubber.

Worked example

Take a steel with yield strength 250 MPa and Young's modulus 200 GPa. Convert: E = 200 × 1,000 = 200,000 MPa. Square the yield strength: 250² = 62,500. Divide by 2E: 62,500 ÷ 400,000 = 0.15625, shown as 0.1563 MJ/m³ (156.25 kJ/m³). Strain at yield: 250 ÷ 200,000 = 0.00125. An alloy with σy = 275 MPa and E = 70 GPa gets 0.5402 MJ/m³, because the lower stiffness lets it stretch further elastically.

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Common mistakes

  • Entering Young's modulus in MPa instead of GPa. A steel at 200,000 would give a result a thousand times too small.
  • Entering yield strength in GPa or in psi. The field is in MPa.
  • Using ultimate tensile strength instead of yield strength. Resilience stops at yield, not at fracture, so using the larger number overstates the value.
  • Confusing the modulus of resilience with the modulus of toughness. Toughness is the area up to fracture and includes plastic deformation.
  • Assuming the stiffer or stronger material is always better. For energy absorption, a high σy with a low E gives a larger Ur, as the squared term dominates.

About this calculator

The modulus of resilience (Ur) is a materials-science property equal to the elastic strain energy a material can absorb per unit volume before it starts to permanently deform, calculated as Ur = σy²/(2E) — the yield strength squared divided by twice the Young's modulus. Graphically, it's the area under the stress-strain curve up to the yield point, representing energy the material returns fully once the load is removed.

Engineers use this property to select materials for applications that must absorb shock or store elastic energy without yielding — springs, bumpers, and impact-resistant components, for example, favor materials with a high modulus of resilience even if their overall strength or stiffness isn't the highest available. It's a standard calculation in mechanical and materials engineering coursework and in selecting alloys for fatigue-and-impact-sensitive parts.

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Comments (1)

  • Editorial team

    Because yield strength is squared, it dominates the result: at the same 200 GPa stiffness, a hypothetical spring steel at 1,400 MPa gives 4.9 MJ/m³ against 0.1563 for the default 250 MPa steel, about 31 times more. That is the logic behind picking high-yield alloys for springs. Remember the formula assumes linear behaviour right up to yield. Which materials would you like to compare side by side?

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