Elastic constants

An isotropic elastic material, one that behaves the same in every direction, is fully described by just two independent constants. The four commonly used are Young's modulus E (axial stress over axial strain), the shear modulus G (shear stress over shear strain), the bulk modulus K (pressure over volumetric strain) and Poisson's ratio ν (lateral strain over axial strain, taken as a positive number). Because only two are independent, the others follow from relationships between them.

Poisson's ratio describes how a bar gets thinner when it is stretched. For structural steel it is about 0.3, for concrete roughly 0.15 to 0.2, and for rubber close to 0.5, which means rubber hardly changes volume. A value of 0.5 is the theoretical upper limit for a stable isotropic material; above it the bulk modulus would be negative, meaning the volume would shrink under tension. The theoretical lower limit is −1; a few engineered foams and lattices have negative Poisson's ratios, but ordinary construction materials lie between about 0.1 and 0.5.

The relations E = 2G(1 + ν) and E = 3K(1 − 2ν) are used to get G for torsion and shear deflection when only E is known, and to understand confined behaviour, such as a bearing pad squeezed between steel plates or soil under a wide footing. Volumetric strain of a bar under axial stress is ε(1 − 2ν).

Real materials are not perfectly isotropic. Timber is far stiffer along the grain than across it; rolled steel plates and fibre composites show directional properties, and concrete's modulus changes with age, stress level and moisture. Codes therefore give design values for E and, where needed, G, rather than expecting designers to measure them.