Elastic Constants Calculator: E, G, K and Poisson's Ratio

Find the other elastic constants of a material from any two, and the strains for a given tensile stress.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

For an isotropic, linear elastic material only two of the elastic constants are independent. Young's modulus E relates stress to strain in tension, the shear modulus G relates shear stress to shear strain, the bulk modulus K relates pressure to volume change, and Poisson's ratio is the ratio of the lateral to the axial strain.

Give any two and the calculator finds the other two. Give an axial stress as well and it finds the axial strain, the lateral strain, which is opposite in sign, and the change in volume.

Formulas used

  • G = \frac{E}{2(1 + \nu)}
  • K = \frac{E}{3(1 - 2\nu)}
  • \varepsilon_{lat} = -\nu\,\varepsilon
  • \frac{\Delta V}{V} = \varepsilon(1 - 2\nu)

Worked example

With the default numbers in the calculator above:

  1. Young's modulus: 200 GPa
  2. Poisson's ratio: 0.3
  3. Shear modulus: 76.92 GPa
  4. Bulk modulus: 166.7 GPa
  5. Lamé constant: 115.4 GPa
  6. Axial strain under that tensile stress: 500 microstrain
  7. Lateral strain (the section gets thinner): -150 microstrain
  8. Volume strain: 200 microstrain
  9. Extension per metre of length: 0.5 mm per metre

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is Poisson's ratio for steel?

About 0.30. Aluminium is about 0.33, copper 0.34 and rubber close to 0.5.

Why must Poisson's ratio be less than 0.5?

A value of 0.5 means no change of volume at all (an incompressible material, so the bulk modulus is infinite). Above 0.5 the material would grow in volume when it is squeezed, which is not possible.

Does it work for timber or composites?

No. Those are not isotropic: they have different constants in different directions.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.

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