Once fundamental scientific laws are understood, engineering technology depends on advances in materials – materials science – more than any other factor. Historically, ages have been named for the most advanced materials technology available: Stone Age, Bronze Age, Iron Age. If you look around the room you are in, you will recognize that our current age is the age of manufactured materials: plastics, ceramics, acrylics, polyester fibers, etc. Underlying the “Information Age” are the advances in electronic materials that have made possible the computers, displays, and communications equipment that enable information transfer and processing. Selecting materials with desirable properties and then processing to enhance these properties is now standard practice. The next generation of materials will include nanoscale manipulation and fabrication of materials unlike anything found in nature. These new nanomaterials can be predicted to have properties exceeding any currently available materials.

Materials are characterized by response functions that describe the materials’ response to applied forces or fields. For example, the electromagnetic properties of materials can be characterized by the electric permittivity (electric polarization of materials in response to an applied electric field), electrical conductivity (electrical current in response to an applied electric field), and magnetic permeability (magnetic polarization in response to an applied magnetic field). One of the fundamental mechanical properties is the elasticity, which describes the deformation of the material in response to applied forces. The elasticity, as with other response functions, depends on the physics and chemistry at the atomic scale and on the structure of the material at a micro- and mesoscale, which is larger than the atoms but smaller than the macroscopic sample. For example, crystal structure, grain boundaries, and dislocations are key determining factors for elastic, plastic, and fracture properties. There are three elastic regimes: tensile/compressive elasticity, shear elasticity, and bulk elasticity. These are all special cases of a generalized elastic tensor, but we will consider them separately to begin.