This chapter provides a brief summary of the physics of elastic constants that has important applications for the inference of the constitution of Earth's interior (Chapter 17) as well as for understanding the origin of lateral heterogeneity in seismic wave velocities (Chapter 20) and seismic anisotropy (Chapter 21). Elasticity is one of the key properties of materials that control seismic wave propagation. To the extent that elastic constants reflect the nature of chemical bonding and crystal structure, the elasticity of a given material also influences its plastic properties. Starting from the definition of elastic constants and their symmetry properties, a brief summary of the experimental techniques of measurements of elastic constants and some materials science (solid-state physics) fundamentals of elastic constants are discussed including the origin of the variation of the elastic constant with pressure and temperature (anharmonicity) and the influence of chemical composition and phase transformations on elastic constants. Birch's law on the relationship between density and elastic wave velocities and the Debye model of lattice vibration provide a unified framework to understand the relationship among various anharmonic parameters although these relations (or models) are only approximately satisfied in real materials.

Key words Hooke's law, elastic constants, bulk modulus, shear modulus, Lamé constants, Poisson's ratio, Cauchy relation, X-ray diffraction, Brillouin scattering, ultrasonic wave techniques, Birch's law, Grüneisen parameter, Anderson–Grüneisen parameter, Debye model, soft mode, polyhedral model, Pauling's rule.