COMPUTER SOFTWARE FOR INTERACTIVE MANIPULATION OF REPRESENTATION SURFACES FOR HIGHER-RANK TENSOR PROPERTIES Andrew F. Stark¹, James S. Miller¹ and Bernhardt J. Wuensch², ¹ Artificial Intelligence Laboratory^{, 2}Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

We have found it useful, in teaching crystallography to students of materials science, to couple the development of symmetry theory with demonstration of the consequences of symmetry on the physical behavior of materials. An appropriate set of topics is tensor description of crystal properties, derivation of the restrictions that symmetry imposes on tensor elements, and the nature of the resulting anisotropy in physical properties. No single representation surface is adequate for third and higher-rank tensor properties, but the dependence on crystallographic direction (specified by direction cosines l_i) of important scalar moduli can be specified. Examples are Young's modulus E = s_11/e₁₁ = 1/(l_ml_nl_pl_qS_mnpq) that relates tensile stress s₁₁ to tensile strain e₁₁ in the same direction, and the longitudinal piezoelectric modulus d = P₁₁/s₁₁ = l_ml_nl_pd_mnp that provides the component of the polarization vector along the direction in which a uniaxial stress is applied. These two moduli are given by very complex polynomials that, for triclinic crystals, include 21 and 18 terms, respectively. A fast, interactive program with high-quality graphics has been developed to display the associated representation surfaces. The viewing perspective may be varied to convey appreciation of both the shape of the surface and that it conforms to the point group of the crystal. The magnitude of any tensor element in the polynomial may be "zoomed" up or down to show its contribution to the shape of the surface.

The software and accompanying documentation are publicly available over the World Wide Web at http://www-swiss.ai.mit.edu/~astark/crystal.html. Users need a UNIX workstation and a copy of the mathematical software MAPLE for Xwindows.