Elastic Constants Notes

Last reviewed 16 Sept 2026 · 6 min read

The four elastic constants

For a homogeneous, isotropic material loaded within the elastic range, four constants describe how it deforms:

Constant	Definition	Symbol
Young's modulus	normal stress ÷ linear strain	$E = σ / ε$
Modulus of rigidity (shear modulus)	shear stress ÷ shear strain	$G$ or $C$ or $N$ $= τ / γ$
Bulk modulus	direct stress (equal in all directions) ÷ volumetric strain	$K = σ / ε_{v}$
Poisson's ratio	lateral strain ÷ linear strain (magnitude)	$μ$ or $ν$ $= - ε_{l a t} / ε$

Only two of them are independent for an isotropic material. Knowing any two, the other two follow from the relations derived below.

RememberNumber of independent elastic constants

Isotropic material: 2
Orthotropic material (e.g. timber, laminated composites): 9
General anisotropic material: 21

Poisson's ratio

When a bar is stretched it becomes longer and thinner. The ratio of the lateral contraction strain to the axial extension strain is constant within the elastic range:

μ = \frac{lateral strain}{longitudinal strain} = \frac{δ d / d}{δ L / L}

Material	Typical Poisson's ratio
Steel	0.25 – 0.30 (0.3 commonly used)
Aluminium	about 0.33
Cast iron	0.21 – 0.26
Concrete	0.15 – 0.20 (IS 456 uses 0.2 for elastic analysis)
Rubber	nearly 0.5
Cork	nearly 0
Perfectly plastic material (volume constant)	0.5

Exam TipWhy cork is used as a bottle stopper

Cork has $μ \approx 0$ : pushing it into the neck does not make it bulge sideways, so it goes in and stays sealed. Rubber, with $μ \approx 0.5$ , bulges and grips.

Limits of Poisson's ratio

From $E = 3 K (1 - 2 μ)$ , since $E$ and $K$ are positive, $1 - 2 μ > 0$ → $μ < 0.5$ . From $E = 2 G (1 + μ)$ , $μ > - 1$ . So theoretically

- 1 < μ < 0.5

For ordinary engineering materials $0 \leq μ \leq 0.5$ . At $μ = 0.5$ the material is incompressible ( $K \to \infty$ , no volume change).

Strains in three dimensions (generalised Hooke's law)

When normal stresses $σ_{x}$ , $σ_{y}$ , $σ_{z}$ act together (tension positive), each produces its own strain along its direction and lateral contraction in the other two:

ε_{x} = \frac{σ _{x}}{E} - μ \frac{σ _{y}}{E} - μ \frac{σ _{z}}{E}

ε_{y} = \frac{σ _{y}}{E} - μ \frac{σ _{z}}{E} - μ \frac{σ _{x}}{E}

ε_{z} = \frac{σ _{z}}{E} - μ \frac{σ _{x}}{E} - μ \frac{σ _{y}}{E}

Volumetric strain

Volumetric strain is the sum of the three linear strains (for small strains):

ε_{v} = \frac{δ V}{V} = ε_{x} + ε_{y} + ε_{z} = \frac{( σ _{x} + σ _{y} + σ _{z} ) ( 1 - 2 μ )}{E}

FormulaVolumetric strain — standard cases

Bar under axial stress $σ$ : $ε_{v} = \frac{σ}{E} (1 - 2 μ)$
Rectangular block: $ε_{v} = \frac{δ L}{L} + \frac{δ b}{b} + \frac{δ t}{t}$
Cylindrical rod: $ε_{v} = \frac{δ L}{L} + 2 \frac{δ d}{d}$
Sphere: $ε_{v} = 3 \frac{δ d}{d}$
Equal stress $σ$ in all three directions: $ε_{v} = \frac{3 σ ( 1 - 2 μ )}{E}$

Relation between E, K and μ

Apply equal tensile stress $σ$ on all faces of a cube. Each linear strain is

ε = \frac{σ}{E} (1 - 2 μ)

so $ε_{v} = 3 ε = \frac{3 σ ( 1 - 2 μ )}{E}$ . By definition $K = σ / ε_{v}$ , therefore

E = 3 K (1 - 2 μ)

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Elastic Constants

The four elastic constants

Poisson's ratio

Limits of Poisson's ratio

Strains in three dimensions (generalised Hooke's law)

Volumetric strain

Relation between E, K and μ

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