Plasticity of Crystals with Interacting Slip Systems
A new, non-conventional approach to the analysis of crystalline lattice rotations caused by large plastic strains of crystals is presented. Only rigid-ideally plastic crystals are considered. Contrary to the theory based on the Schmid law, the proposed model assumes that a yield initiation depends on stress states of all slip systems, i.e. interactions between slip systems are taken into account. In the paper an interaction rule, founded on microscopic observations, is governed by one integer constant n. For n = 1, the crystals satisfy the Mises criterion with a quadratic yield surface. For n → ∞ , the crystals are described by smooth (with rounded-off corners) yield surfaces tending to those generated by the Schmid law. For a fixed n, the interaction rule determines constitutive relations in terms of the strain rate tensor, the plastic spin tensor and the stress tensor. Three additional constitutive equations for the plastic spin components lead to an explicit descriptions of lattice reorientations. A generalized plastic potential for the strain rate and the plastic spin is introduced. A smooth yield condition generated by this potential enables to formulate a complete system of equations for the model, what considerably simplifies the numerical analysis. The f.c.c, crystals in tension and compression are examined in detail. The presented strain paths demonstrate a predominant influence of interactions between slip systems on the lattice reorientations.
E.SCHMID, Proc. Int. Congr. Appl. Mech., 324, Delft, 1924.
R.MISES, Mechanik der plastischen Formanderung von Kristallen, ZAMM, 8, 161, 1928.
G.I. TAYLOR, Plastic strain in metals, J. Inst. Metals, 62, 307, 1938.
J.F.W.BISHOP and R. HILL, A theory of plastic distortion of a polycrystalline aggregate under combined stress, Phil. Mag., 414, 1951.
J.DIEHL, Zugverformung von Kupfer-Einkristallen, Z. Metallk., 47, 331, 1956.
J.MANDEL, Generalization de la theorie de plasticite de W.T.Koiter, Int.J.Solids Struct., 1, 273, 1965.
J.J.GILMAN, Micromechanics of flow in solids, McGraw-Hill. Book Company, 1969.
R.HILL and J.R.RICE, Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20, 401, 1972.
D.HULL, lntroduction to dislocations, Pergamon Press, 1975.
R.J.ASARO, Geometrical effects in the inhomogeneous deformation of ductile single crystals, Acta Metall., 27, 445, 1979.
R.J.ASARO, Crystal plasticity, J. Appl. Mech., 50, 921, 1983.
W.GAMBIN, A model of rigid-ideally plastic crystals, J. Techn. Phys., 28, 309, 1987.
W.GAMBIN, Plastic behaviour of crystals, in: Proc. Int. Sym. on the lnelastic Behaviour of Solids: Models and Utilizations, Besancon 1988.
W.GAMBIN, A simplified model of rigid-ideally plastic crystal, J. Techn. Phys., 29, 155, 1988.
W.GAMBlN, Plasticity and lattice rotations in crystals [in Polish], IFTR Reports, 10, 1988.
Copyright © 2014 by Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland