Finite approximations to the second-order properties closure in single phase polycrystals

Extension of the first-order theory of microstructure design to considerations of morphological texture is addressed in this paper. The main challenges include the r-interdependence of the 2-point correlation functions of lattice orientation, construction of the corresponding microstructure hull, and its corresponding properties closure(s). It is shown that the correlation functions can be expressed in terms of an intermediate construct, called the texture function; the correlation functions have quadratic dependence in the texture functions. A complete (finite) texture hull is readily constructed for the texture functions in Fourier space, and is found to be a convex polytope. Eigen-texture functions occupy its corner (extreme) points. Microstructure design proceeds directly from homogenization relations evaluated at the corner points. This gives rise to (combined) properties closures, from which second-order microstructure design can proceed. This is demonstrated in a brief case study.