Anisotropic material; Metal plasticity; Microstructures; Optimization; Polycrystalline material
The support of this research by the Army Research Office is gratefully acknowledged. M. Lyon is supported by a National Science Foundation Graduate Research Fellowship. Fourier analysis is implemented on the orientation distribution of a polycrystalline microstructure. The linearity and convexity of the Fourier space, with respect to orientation, allows one to consider all possible distributions by considering all linear combinations of single-grain orientations. The limits of the Fourier space are therefore defined by the solutions to a set of linear programming problems. A unique approach to the linear programming, similar to the Krylov subspace methods for obtaining solutions to linear systems, is presented. The method is particularly efficient for this application where a large number of independent variables is often required. These solutions are then used as the constraints in the gradient-based optimization of non-linear functions within the Fourier space. In the example, Taylor yield theory and an anisotropic solution for the stress concentration around a hole in a plate of cubic-orthotropic polycrystalline material are expressed as non-linear functions within the Fourier space. The maximum obtainable ratio of Taylor factor to stress concentration for any polycrystalline orientation distribution in copper is found to be 1.22, more than double the minimum value.