Theory of the response of cubic crystals to [111] loading

Abstract

The mechanical behavior of cubic crystals in unconstrained [111] (i.e., body diagonal) uniaxial loading is studied theoretically. A general axisymmetric loading path, of necessity, passes successively through three zeros where the lattice takes on the bcc, sc, and fcc configurations; based on this behavior, a proof is given of the intrinsic instability of the sc lattice in general. Elastic moduli (taken as any fourth-rank tensor of moduli) are characterized according to symmetry along the path; for the three unstressed cubic states, the connections are made with the usual bulk and shear moduli. As noted previously, actual elastic moduli of a crystal under load depend upon choice of strain measure. Three sets of moduli (two of which transform tensorially) are discussed, both for general crystals and for lattice models employing pairwise interactions, and formulas are presented for transforming from one set to another or from one reference state to another along the path. Detailed computations of the theoretical response of a particular lattice model to the axisymmetric path of uniaxial loading are presented. The computations (including internal energy, load, stress, lattice parameters, and elastic moduli) span the three unstressed cubic states. The Born or notional concept of stability is particularized to the path of [111] uniaxial loading, and numerical computations of the domains of notional stability are made within the framework of the lattice model.