The rules of macroscopic elastic response are derived in an exact way by first stating the time rate at which mechanical work is performed in deforming a collection of molecules, which is the time rate at which internal elastic energy is being reversibly stored in the molecular bonds. From this work rate, the definition of the average stress tensor is obtained as well as the exact statement of the strain rate. An additional time derivative of the average stress tensor then gives Hooke’s law in its most general nonlinear form. How the elastic stiffnesses in Hooke’s law change with changing strain is derived. Displacement is defined and the shape change and volume change of a sample are understood through how the displacements of the surface bounding the sample are related to the strain tensor. Elastodynamic plane body-wave response is obtained, as is reflection and refraction of plane body waves from an interface and evanescent surface waves. It is shown how sources of elastodynamic waves such as cracking and explosions are represented as equivalent body forces.

The general definition of elasticity is given, and as a special case the linear elasticity with Hooke’s law, is presented together with its derivation on the basis of the Cayley–Hamilton theorem. Some applications of elasticity theory in soil mechanics are presented.

The generalized Hooke's law is introduced, which represents six linear relations between the stress and strain components in the case of small elastic deformations. For isotropic materials, only two independent elastic constants appear in these stress–strain relations. Each longitudinal strain component depends linearly on the three orthogonal components of the normal stress; the relationship involves two constants: Young's modulus of elasticity and Poisson's coefficient of lateral contraction. Each shear strain component is proportional to the corresponding shear stress component; the shear modulus relates the two. The volumetric strain is proportional to the mean normal stress, with the elastic bulk modulus relating the two. The inverted form of the generalized Hooke's law is derived, which expresses the stress components as a linear combination of strain components. Lamé elastic constants appear in these relations. The Duhamel–Neumann law of linear thermoelasticity is formulated, which incorporates the effects of temperature on stresses and strains. The Beltrami–Michell compatibility equations with and without temperature effects are derived.