Different forms of modulus of elasticity

We use stiffness to describe the resistance of a material against change in geometry when loaded. However, how clear are we all about what we are talking about? The modulus of elasticity is the term for describing stiffness within the regime of (linear) elasticity. It comes in many different forms, depending on how the material is loaded. A material can stretch, compress uniformly, shorten within rigid boundaries, or distort under shear. Each response is described by a different elastic modulus.

The fundamental difference between volume change and shape change

The clearest way to distinguish them is to separate volumetric behaviour —a change in the space a material occupies—from shear behaviour—a change in its proportions or angles. The illustration below shows these differences, with deformation exaggerated for clarity.

Shear modulus: changing shape without changing volume

Shear modulus, G, measures resistance to distortion caused by tangential loading (shear). The tangential loading can take the form of the difference between principal stresses in a tri-axial condition, which is what Mohr circle describes.

Imagine pushing the top of a block sideways while holding its base. Its square side profile becomes a parallelogram.

• Shape: changes because the angles change.

• Volume: remains unchanged in ideal linear-elastic shear.

Shear modulus therefore isolates shape-changing stiffness, rather than resistance to compression.

Shear modulus has high significance in geotechnical engineering, such as foundation design, earthworks design and tunnel design, where a shear failure plane can develop in the ground when subject to a tri-axial shear condition (non-uniform tri-axial load).

Bulk modulus: changing volume without changing shape

Bulk modulus, K, measures resistance to uniform pressure applied equally in every direction.

Imagine a cube compressed equally on all its faces. It becomes a smaller cube of the same aspect ratio, rather than a flattened or skewed one.

• Shape: unchanged—the proportions and angles stay the same.

• Volume: decreases under compression.

Bulk modulus isolates volumetric stiffness. A high bulk modulus means that substantial pressure produces relatively little volume change.

Bulk modulus has relatively higher significance in geotechnical engineering than structural engineering, as we often find a node of ground is tri-axially loaded, and sometimes uniformly tri-axially loaded, whereas a case of uni-axially loaded ground (especially soil) is rare.

Young’s modulus: stretching or compressing with free sides

Young’s modulus, E, measures resistance to axial deformation when the sides are free to expand or contract.

Imagine compressing a block vertically. It becomes shorter but wider – the two sides bulging out is due to Poisson’s ratio. The harder it is for the material to change volume, the higher the Poisson’s ratio is; at a Poisson’s ratio = 0.5, the material is theoretically not able to change volume.

• Shape: changes because the block’s proportions change.

• Volume: generally changes too. The lateral deformation does not necessarily compensate fully for the axial deformation. For an incompressible material, however, the volume remains unchanged.

Young’s modulus therefore describes a hybrid response involving both volumetric and shape deformation, not stretching alone.

It us the most commonly used and the primary form of modulus of elasticity in structural engineering, as most structural tests derives Young’s modulus directly.

Constrained modulus: axial compression with no sideways movement

Constrained modulus, M, measures axial stiffness when deformation in both lateral directions is prevented.

Imagine a block compressed vertically inside a rigid, close-fitting enclosure. Its height decreases, but its width and depth cannot change. The boundaries develop lateral reaction stresses.

• Shape: changes—the block becomes flatter, despite having no lateral movement.

• Volume: decreases because its height decreases while its cross-sectional area stays fixed.

This is the essential contrast with Young’s modulus: Young’s modulus involves free lateral deformation; constrained modulus involves zero lateral deformation. Preventing sideways movement makes the axial response stiffer.

The key comparison

Modulus	Loading or restraint	Volume change?	Shape change?
Young’s, E	Axial loading; sides free	(Normally) Yes	Yes
Bulk, K	Equal pressure in all directions	Yes	No
Constrained, M	Axial loading; lateral strain prevented	Yes	Yes
Shear, G	Tangential loading	No, in ideal elastic shear	Yes

Within the remit of elastic theory, these moduli are related and inter-convertible, based on Poisson’s ratio.