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Structural Analysis Fundamentals / Engineering materials

Elastic constants and how they connect

E, G, ν and K — why only two of the four are independent.

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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

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These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

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Why this matters

For isotropic linear elasticity, two independent elastic constants determine the others. Check this relationship when combining material data from different measurements or sources.

By the end of this lesson you should be able to

  • Define Young's modulus, shear modulus, Poisson's ratio and bulk modulus
  • Use G = E/2(1 + ν) and K = E/3(1 − 2ν)
  • Explain why ν cannot exceed 0.5
  • Calculate volumetric strain under hydrostatic pressure

What you should already know

  • Hooke's law and Young's modulus (Module 7)
  • Poisson's ratio and lateral strain (Module 7)
  • Shear stress and shear strain (Module 7)

Four constants describe how an isotropic elastic material responds, each to a different kind of loading:

  • Young's modulus E — resistance to stretching along one axis. σ = Eε.
  • Shear modulus G — resistance to changing shape at constant volume. τ = Gγ.
  • Poisson's ratio ν — how much the material contracts sideways when stretched. ν = −εlateral/εaxial.
  • Bulk modulus K — resistance to changing volume under all-round pressure. p = −K εvol.

The important structural fact is that these are not four independent pieces of information. An isotropic material looks the same in every direction, and that single symmetry statement is strong enough to tie them together:

Shear modulus from E and ν
G=E2(1+ν)G = \frac{E}{2(1+\nu)}

What it calculates: the shear modulus of an isotropic material

G
shear modulus (N/mm²)
E
Young's modulus (N/mm²)
ν
Poisson's ratio (dimensionless)

This assumes

  • The material is isotropic — the same in every direction
  • Behaviour is linearly elastic

In plain terms: A state of pure shear is the same thing as equal tension and compression on planes at 45°. Because that is true, the resistance to shear cannot be independent of the resistance to stretching.

Bulk modulus from E and ν
K=E3(1−2ν)K = \frac{E}{3(1-2\nu)}

What it calculates: resistance to volume change under hydrostatic pressure

K
bulk modulus (N/mm²)
E
Young's modulus (N/mm²)
ν
Poisson's ratio (dimensionless)

This assumes

  • Isotropic, linearly elastic material

In plain terms: Squeeze a cube equally on all six faces and it shrinks. K measures how hard that is. The (1 − 2ν) in the denominator is where the limit on Poisson's ratio comes from.

For a stable isotropic linear elastic material with finite positive E, G and K, Poisson’s ratio lies between −1 and 0.5. The upper limit represents incompressibility as K tends to infinity. This restriction does not apply unchanged to directional Poisson ratios in anisotropic materials.

Common approximate values are 0.30 for steel, 0.33 for aluminium and 0.20 for concrete. Cork can have a value near zero; negative values are possible in auxetic materials.

Predict first

A data sheet quotes a material as having E = 200 000 N/mm² and ν = 0.62. What should you conclude?

Worked example

Checking a set of material data for consistency

Given

  • A steel is quoted as E = 205 000 N/mm² and ν = 0.30
  • A separate test on the same steel gives G = 79 000 N/mm²

Find

Whether the three quoted values are mutually consistent, and the bulk modulus.

    Practice

    A steel has E = 205 000 N/mm² and ν = 0.30. What is its shear modulus, in N/mm²?

    Practice

    An aluminium alloy is measured as E = 70 000 N/mm² and G = 26 300 N/mm². What is its Poisson's ratio?

    Practice

    For the same steel (E = 205 000 N/mm², ν = 0.30), what is the bulk modulus, in N/mm²?

    Summary

    • E, G, ν and K describe stretching, shape change, lateral contraction and volume change
    • For an isotropic material only two are independent
    • G = E/2(1 + ν) and K = E/3(1 − 2ν)
    • ν ≤ 0.5 is a physical limit: at 0.5 the material is incompressible
    • Steel ν ≈ 0.30, aluminium ≈ 0.33, concrete ≈ 0.20