Papers - The Lattice Parameters and Solubility Limits of Alpha Iron as Affected by Some Binary Transition-Element Additions

A general relalionship between the amount of glide shear (due to slip) and the macroscopic shape change has been developed. Since the deformation can be large, finite strain analysis is employed. In this treatnzent, the shape change is expressed by a deforlnation gradient matrix, F - [axi/aXj/ where X and x refer lo the initial and final positions of a particle. This matrix is readily evaluated in termas of the amounl of glide shear, a , and the direction cosines of the slip-plane normal and the slip direction for a single active slip systena. In the case of deforma tion from slip on several systems, the product F(a) of the several deformation gradient matrices is first calculated. Then, by assuming that the final configuration is reached by a long series of small shears of magnitude a, occurring more or less alternately in the several slip sys- IN problems of crystal plasticity, it is often necessary to relate the amount of glide in the operating slip systems to the macroscopic strain components. This relationship is relatively simple when only small strains are considered. In this case the separate strain contributions from several slip systems are additive. Such a procedure is incorrect in the case of large plastic deformation. To the authors' knowledge, no general treatment of the latter problem has appeared in the literature. In specific instances, Mark, Polanyi, and schmid3 have derived the relationship between glide and axial elongation during tensile pulling of a single crystal when only one slip system is active. A similar relationship for duplex slip was worked out by v. Göler and Sachs.4 Taylor and Elam1 have also studied in detail problems of large plastic deformation. Due to uncertainties as to the slip systems at the time, ! terns, the final deformation gradient matrix can be obtained. Mathematically, the problem is reduced to finding the limit of F(aN as N — m while a — 0 in such a way that the Product Na = a finite conslant designatzng the accumulated amount of shear. It turns out that this limit is simply eafF1, where F, is the matrix whose elements are the coefficients of a in the F(a) twatrzx. Application is made of the present treatment to a fcc crystal of Permalloy compressed on the (110) plane and contrained to elongate in the [112] directzon. In addition, it is shown that one rnay readily obtain from the general analyszs the well-known formulas relating elongation, amount of glide shear, and amount of lattice rotation for crystals deforming by single and double slip under tension. however, they were mainly concerned with proving that slip occurs on {111}(110) systems in fcc metals. The present general treatment arose out of recent investigations of magnetic anisotropy in cold-worked Fe-Ni alloys6 as well as some related problems in strength anisotropy of single crystals.' Detailed application of the general analysis will be provided in the case of a Permalloy single crystal compressed on the (110) plane and constrained to elongate only in the [112] direction. It will be seen that, as expected, the small strain approximation leads to significant errors after moderate straining. It will also be shown that the present general treatment yields the formulas of Mark et al. and of v. Goler and Sachs in those specific cases. GENERAL CONSIDERATIONS Consider a homogeneous deformation in which a material point initially at moves to (xl,xz,x3), both positions being referred to the same set of Cartesian axes. Such a deformation can be