Mathematical Aspects of Materials Engineering

Materials Science and Engineering is a scientific discipline which is concerned with complex physical systems, vaguely defined boundary conditions and typically chemically ‚dirty‘ substances. Nevertheless, at the end of the day, a materials engineer has to come up with a prediction of material properties for given processing conditions with high reliability, for instance for the toughness of a turbine blade in a jet engine which has to perform reliably and reproducibly in extreme environments.

Unfortunately, contrary to common belief of materials engineers, the processing conditions do not constitute the state parameters of the properties of a material; rather the engineering properties of a processed product are determined by both the overall chemical composition and the internal distribution of elements and crystal defects, also referred to as microstructure. Since the overall chemical composition remains unchanged, the microstructure serves as state parameter of material properties. For predicting material properties with given processing history, the evolution of microstructure has to formulated, usually over many processing steps.

For select examples it will be demonstrated, how to set up constitutive equations, what microstructural variable to define, which mathematical tools to use and eventually, which quantitative predictions can be made.