The Galois unsolvability of the sextic equation of anisotropic elasticity

Abstract

By an approximate numerical application of Galois theory it is proved that the sextic equation of anisotropic elasticity for cubic symmetry is in general unsolvable in radicals, elementary transcendental functions, or elliptic modular functions and that its group is the full symmetric group. This implies the same unsolvability for tetragonal, orthorhombic, monoclinic, and triclinic symmetry. A separate investigation proves the same unsolvability for trigonal symmetry. Special cases of cubic symmetry which might have solvable equations are examined. Directions restricted to {111} or {112} planes give unsolvable equations, in contrast to {100} and {110} planes. Three additional classes of elastic constants which give solvable equations are found but only two limiting cases are physically possible. An extensive survey suggests that any further special elastic constants are rather unlikely.