Background and question

Cube oriented grains in face centered cubic materials are metastable under plane strain. Once loading begins, the lattice rotates, and the axis of that rotation, the normal direction ND, the rolling direction RD, or the transverse direction TD, has not been settled by experiment alone. Measurements on single crystals and on cube oriented specimens since the 1930s show that the micro texture that builds up inside a single cube grain depends sensitively on how the specimen is loaded and on how much initial misorientation the crystal carries before deformation. Early models attributed the splitting to rotation about ND and RD, while later analyses favored rotation about TD. A consistent picture requires a model that responds sharply to two small inputs: a slight initial orientation spread and a change in friction at the contact surfaces.

The study reported here provides such a picture. Raabe, Zhao, and Roters simulate the plane strain compression of a cube single crystal with a crystal plasticity finite element method in which the starting texture is not a single orientation but a quantitative distribution in orientation space. All simulations run to a 50 percent engineering thickness reduction, the regime of a typical channel die experiment.

Method: a texture component finite element model

The sample is a block of 500 three dimensional linear elements, each with eight integration points, bounded by four rigid surfaces. Linear Coulomb friction is applied separately to the top and bottom contact surfaces and, in a separate series, to all four side surfaces. The constitutive equations are integrated implicitly with the viscoplastic formulation of Kalidindi, Bronkhorst, and Anand, using {111}<110> slip systems and rate dependent hardening. The specific part of the model is the way the starting texture is fed into the mesh.

The cube orientation is described as one spherical Gaussian component in orientation space, defined by its height and its full width at half maximum. In a first step the ideal cube orientation, Euler triple 0°, 0°, 0° in Bunge notation, is assigned as a single rotation matrix to every integration point. In a second step the individual orientation portions are re oriented so that their ensemble reproduces the prescribed Gaussian, here with a full width at half maximum of 2.5°, distributed in random lateral order across the 500 elements. After this mapping the component concept is dropped: every portion evolves independently during the deformation. The procedure converts a distribution that originally exists only in orientation space into a distribution that exists simultaneously in real space and orientation space, which is the situation of an as grown single crystal that already contains small misorientations.