Analysis of dynamic shear bands in an FCC single crystal

We study plane strain dynamic thermomechanical deformations of an fcc single crys- tal compressed along the crystallographic direction (010) at an average strain rate of 1000 sec -~ . Two cases are studied; one in which the plane of deformation is parallel to the plane (001) of the single crystal, and another one with deformation occurring in the plane (10i) of the single crystal. In each case, the 12 slip systems are aligned symmetrically about the two centroidal axes. We assume that the elastic and plastic deformations of the crystal are symmetrical about these two axes. The crystal material is presumed to exhibit strain hardening, strain-rate hardening, and thermal softening. A simple combined isotropic-kinematic hardening expression for the crit- ical resolved shear stress, proposed by Weng, is modified to account for the affine thermal soft- ening of the material. When the deformation is in the plane (001) of the single crystal, four slip systems (111)(1 i0), (11 i)(l i0), (1 i i)(1101, and (111)(110) are active in the sense that significant plastic deformations occur along these slip systems. However, when the plane of deformation is parallel to the plane (10i) of the single crystal, slip systems (1 ll)(110), (111)(011), (111)(110), and (11 l)(0i 1) are more active than the other eight slip systems. At an average strain of 0.108, the maximum angle of rotation of a slip system within a shear band, about an axis perpendic- ular to the plane of deformation, is found to be 20.3 ° in the former case, and 22.9 ° in the latter. One way to understand the micromechanics of shear band formation in polycrystalline materials is to study their initiation and growth in a single crystal. Several investigators, e.g. SAWKILL and HONEYCOMBE (1954), PRICE and KELLY (1964), SA~OTO et al. (1965), and Ct-LAr~C and ASARO (1981), have observed regions of localized shearing in fcc sin- gle crystals deformed quasistatically. ZmRY and NEMAT-NASSER (1990) have recently studied numerically the phenomenon of shear banding in an fcc single crystal undergo- ing plane-strain tensile deformations at high strain rates. We refer the reader to their article for a list of references and a brief outline of the historical development of the subject. They used the double cross-slip model proposed by KOEHLER (1952) and later by OROWAN (1954) during the entire loading history. Here we study a similar problem with the crystal deformed in compression rather then tension, assume that all 12 slip sys- tems are potentially active at any instant of loading, use constitutive relation for the crit- ical shear stress that is different from the one employed by ZIKRY and NEMAT-NASSER (1990), employ a different technique to integrate the system of equations, and consider two loadings. With the axis of compression aligned along the crystallographic direction (010), the plane of deformation is taken to be either parallel to the plane (001) or (10i) of the single crystal. When the plane of deformation is parallel to the plane (001) of the single crystal, a single shear band making an angle of 45 ° with the horizontal line ensues from the cen- troid of the cross-section and is reflected back from the top loading surface, the angle of reflection being essentially equal to the angle of incidence. The slip strains on the slip systems (111)(1 i0), (111)(110), (1H)(110), and (111)(110) are high, and these constitute