Theory of lattice strain in a material undergoing plastic deformation: Basic formulation and applications to a cubic crystal

Theory of lattice strain in a polycrystalline aggregate under deviatoric stress is extended to include the influence of ongoing plastic deformation. When deviatoric stress is applied to a polycrystalline material at high temperatures (or above the yield stress), applied macroscopic stress is redistributed to individual grains by plastic deformation according to their orientations with respect to the macroscopic stress and plastic anisotropy of a given crystal. This microstress causes elastic deformation of individual grains that can be measured by x-ray diffraction. Consequently, the observed lattice strain depends on two material properties, viscosity (plasticity) and elastic compliance as well as the applied macroscopic stress and the stress-strain distribution among various grains. The influence of plastic deformation on lattice strain is analyzed using an anisotropic and nonlinear power-law constitutive relationship. In this model, the dependence of inferred macroscopic stress on the crystallographic orientation of diffraction plane $(hkl)$ comes from elastic and plastic anisotropy of a crystal. In many materials, plastic anisotropy dominates over elastic anisotropy. This explains the observed large dependence of inferred stress on the diffraction plane and means that the determination of elastic anisotropy is difficult when plastic deformation occurs with anisotropic plasticity. When elastic anisotropy is known, plastic anisotropy of single crystal and/or stress-strain distribution in a deformed polycrystal can be determined from radial x-ray diffraction using the present model. Some examples are presented using the data on MgO.