During the past two decades, mathematical modeling has been gaining acceptance as a legitimate part of materials science and engineering. However, as common to all relatively new disciplines, we still lack a realistic perspective regarding the uses, limitations, and even the optimal methodologies of mathematical modeling techniques.

The term “mathematical modeling” covers a broad range of activities, including molecular dynamics, other atomistic scale systems, continuum fluid and solid mechanics, deformation processing, systems analysis, input-output models, and lifecycle analyses. The common point is that we use algebraic expressions or differential equations to represent physical systems to varying degrees of approximation and then manipulate these equations, using computers, to obtain graphical output.

While it is becoming an accepted fact that some kind of mathematical modeling will be needed to make most research programs complete, there is still considerable ambiguity as to what form this should take and what might be the actual usefulness of such an effort.

Among the more seasoned and successful practitioners of this art, clear guidelines have emerged regarding the uses and limitations of the mathematical modeling approach. We seek to illustrate these uses through the successful modeling examples presented by some leading practitioners. Some general principles may be worth repeating as an introduction to this interesting collection of articles.