Comparisons of crystal hardening laws in multiple slip (Q1097080)

This paper brings together and concisely reviews results from recent analytical investigations on single crystals (variously done alone or with students) in which predictions from different theoretical hardening laws are contrasted and compared with experimental studies. Finitely deforming f.c.c. crystals in both constrained and unconstrained multiple- slip configurations are considered. Four crystal hardening laws are given prominence. Two of these belong to a class of theories in which the physical hardening moduli relating rates-of-change of critical strengths (in the 24 crystallographically equivalent slip systems) to slip-rates are taken as symmetric. These are G. I. Taylor's classic isotropic hardening rule (proposed in 1923), which is almost universally adopted in the metallurgical literature for various approximate analyses of single and poly-crystal deformation, and a 2- parameter modification of Taylor's rule that has an empirical basis in the qualitative features of experimentally determined latent hardening in single slip. The other two hardening laws featured here belong to a class of theories that were introduced by the author in Acta Mech. 28, 139-151 (1977; Zbl 0365.73004). This class requires the above moduli to be nonsymmetric and explicitly dependent upon the current stress state in such a manner that the following consequences are assured. (1) The deformation-dependent hardening of latent slip systems necessarily develops anisotropically if there is relative rotation of gross material and underlying crystal lattice. (2) The theories admit self-adjoint boundary value problems for crystalline aggregates, hence a variational formulation. Theoretical results from the various hardening rules are contrasted and compared with finite strain experiments in the metallurgical literature. Both tensil-loaded crystals in 4, 6 and 8-fold symmetry orientations and compressively loaded crystals under conditions of channel die constraint are treated. A postulate of minimum plastic work introduced by the author in Proc. R. Soc. Lond., Ser. 17 378, 329-349 (1981; Zbl 0493.73033) plays a prominent role in the theoretical analyses, in many cases providing a unique solution to the slip system inequalities and deformation constraints (where applicable). The rather remarkable ability of the simple theory to reconcile diverse qualitative features of both constrained and unconstrained finite deformation of f.c.c. crystals is demonstrated. Finally, conditions for total loading (all systems active) in 6-fold symmetry are investigated, and certain concepts regarding the selection of active systems under prescribed straining are critically assessed.