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Abstract
Significant changes in the mechanical state of polycrystalline metals accompany the large deformations associated with forming operations. Crystallographic texture is one important source of the developing anisotropy in the flow stress and the macroscopic response observed in these materials. Polycrystalline models provide descriptions of macroscopic material behavior by averaging the response of an aggregate of crystals underlying a material point. For such averaged quantities to be representative at the continuum level, it is necessary to average over several hundred single crystals. Here a means of obtaining a macroscopic constitutive model rich in microstructural detail is provided, without the necessity of retaining large amounts of information at the level of the individual grains.
In order to construct a continuum description of anisotropy, the geometric hardening of an idealized planar assembly of two-dimensional grains is examined. A Taylor assumption is used to link the macroscopic and microscopic length scales. Consequently, the macroscopic deformation rate is imposed on each crystal. Equations of evolution for the grain orientation are derived and expressed in terms of the macroscopic deformation rate and a single microstructural parameter determined solely by the slip system geometry. An analytic expression for the plastic spin and a stability criterion for the developing texture of the aggregate are determined in terms of this parameter.
The anisotropy of the aggregate is represented by a continuous grain orientation distribution function expanded as a Fourier series. The set of even order moment tensors for the distribution are identified and equations of evolution for the moment tensors are derived. Lower moments provide descriptions of the distribution function that, near the onset of texturing, compare well with discrete polycrystalline simulations and analytic solutions in uniaxial tension and simple shear. Such comparisons identify quantitative levels of induced anisotropy beyond which higher moments are necessary to capture the material response.
Constitutive relations describing viscous slip at the grain level relate the slip system shearing rate to the resolved shear stress on the slip planes. Using the microscopic response at the crystal level, an average stress and an appropriate tangent stiffness for the aggregate are obtained using an orientation average.
