Module 8 · Lesson 8.2
Elastic constants and how they connect
E, G, ν and K — why only two of the four are independent.
Why this matters
You will meet four elastic constants and it is easy to assume they are four separate facts you have to look up. They are not. For an isotropic material, any two of them fix the other two, and knowing that relationship stops you accepting a set of material data that cannot possibly be self-consistent.
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:
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.
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.
That denominator is worth staring at. As ν approaches 0.5, the term (1 − 2ν) approaches zero and K goes to infinity: the material becomes incompressible. Its volume cannot change at all, no matter how hard you squeeze it. Rubber is close to this, at about ν = 0.4999.
A value of ν greater than 0.5 would make K negative, meaning the material would expand when you compressed it and release energy while doing so. That is thermodynamically impossible, so ν ≤ 0.5 is a hard physical ceiling, not a convention.
At the other end, ν = 0 means no lateral contraction at all — cork is close, which is exactly why a cork can be pushed into a bottle neck without bulging sideways.
Typical values worth carrying in your head: steel ν ≈ 0.30, aluminium ≈ 0.33, concrete ≈ 0.20, cork ≈ 0.
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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint