Volumetric strain is the change in volume of a body divided by its original volume: ev = ΔV / V. It is a pure number with no unit. For a body strained in three directions it equals the sum of the three linear strains, ev = ex + ey + ez. The stress that causes a change in volume, usually a uniform pressure, is called volumetric stress, and the two are linked by the bulk modulus K.
Volumetric strain formulas
| Case | Volumetric strain ev |
|---|---|
| General definition | ΔV / V |
| Rectangular block, stresses in x, y, z | ex + ey + ez = (σx + σy + σz)(1 − 2ν) / E |
| Bar under a single axial stress σ | (σ / E)(1 − 2ν) = e(1 − 2ν) |
| Cube under equal pressure p on all faces | 3p(1 − 2ν) / E = p / K |
| Cylinder (thin or solid) | eL + 2 ed (longitudinal + twice diametral strain) |
| Sphere | 3 × diametral strain |
Here E is Young’s modulus and ν is Poisson’s ratio. A positive ev means the volume increased; a negative value means it decreased.
Volumetric stress and bulk modulus
When a body is squeezed equally from all sides, for example a solid deep under water, the stress on every face is the same pressure p. This is called volumetric (hydrostatic) stress. The ratio of volumetric stress to volumetric strain is the bulk modulus:
K = volumetric stress ÷ volumetric strain = p ÷ (ΔV / V)
K is measured in pascals, the same as stress. Steel has K ≈ 160 GPa, aluminium about 76 GPa and water about 2.2 GPa, which is why water is often treated as incompressible in simple problems.
The elastic constants are related by E = 3K(1 − 2ν). This also shows why Poisson’s ratio cannot exceed 0.5: at ν = 0.5, 1 − 2ν = 0 and the material would not change volume at all, like rubber, which is nearly incompressible.
Derivation for a rectangular bar
Take a block of length L, width B and depth D. Its volume is V = LBD. After straining, each side changes slightly: L(1 + ex), B(1 + ey), D(1 + ez). The new volume is LBD(1 + ex)(1 + ey)(1 + ez). Since the strains are tiny, their products are negligible, so V’ ≈ V(1 + ex + ey + ez). Therefore ΔV / V = ex + ey + ez.
For a bar pulled along x only, ex = σ/E and the lateral strains are ey = ez = −νσ/E, giving ev = (σ/E)(1 − 2ν).
Solved examples
Example 1: steel bar in tension. A steel bar 100 mm × 50 mm × 20 mm carries an axial tensile stress of 100 MPa. E = 200 GPa and ν = 0.3. Find the change in volume.
- Axial strain e = 100 / 200,000 = 0.0005
- ev = 0.0005 × (1 − 2 × 0.3) = 0.0005 × 0.4 = 0.0002
- V = 100 × 50 × 20 = 100,000 mm³
- ΔV = 0.0002 × 100,000 = 20 mm³ (an increase)
Example 2: steel cube under pressure. A steel cube is subjected to a uniform pressure of 50 MPa. K = 160 GPa. Volumetric strain = −p/K = −50 / 160,000 = −3.1 × 10⁻⁴, a decrease in volume of about 0.03%.
Example 3: sphere. A sphere’s diameter shrinks by 0.01% under pressure. Its volumetric strain is 3 × (−0.0001) = −0.0003.
Where volumetric strain matters
- Design of thick cylinders, pressure vessels and hydraulic systems.
- Soil mechanics, where volume change under load causes settlement.
- Deep-sea equipment and submarines under large hydrostatic pressure.
- Hydraulic fluids, where compressibility affects the stiffness of the system.
Frequently asked questions
What is the formula for volumetric strain?
Volumetric strain ev = ΔV / V. For three-dimensional loading, ev = ex + ey + ez.
What is the unit of volumetric strain?
It has no unit because it is a ratio of two volumes.
What is volumetric stress?
Volumetric stress is the uniform normal stress (pressure) acting on all faces of a body that changes its volume without changing its shape. Volumetric stress = K × volumetric strain.
What is the volumetric strain of a cylinder?
For a cylinder, volumetric strain = longitudinal strain + 2 × diametral (circumferential) strain.
References
- R. K. Bansal, Strength of Materials, chapter on elastic constants.
- S. Ramamrutham, Strength of Materials, volumetric strain and bulk modulus.