Diagnosing Shear Conventions and Loading Constraints in a Crystal Elasticity Laboratory

Abstract

A computational laboratory is developed in which students diagnose errors in elastic-tensor notation, distinguish transverse boundary conditions, and decide whether an averaged modulus adequately describes a specified crystal direction. Complete stiffness matrices for inorganic structures are examined under infinitesimal uniaxial loading. An exact isotropic check shows that doubling the shear products in a directional-compliance calculation changes a prescribed 100 GPa response into apparent values between 28.571 and 100 GPa. Omitting those products instead produces values between 100 and 600 GPa. A separate comparison of constrained extension with free transverse contraction identifies discrepancies above twenty percent in investigated axial cases. For the correctly evaluated crystal responses, the Hill-derived modulus differs by more than twenty percent from at least one tested direction in 863 structures. The count is unchanged when angular sampling increases from 1,024 to 16,384 points. A CaO--BN exercise connects nearly identical averaged moduli to markedly different strain intervals, while a TiIr example separates a limiting direction from the fraction of orientations exceeding a tolerance. Eight independent visualizations display atomic positions, shear errors, loading constraints, directional strain, and angular resolution. The resulting three-hour exercise contains complete reference answers and individual transfer questions. Its numerical results are verified, but its effect on student understanding has not been measured.