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We present a methodology to impose micromechanical constraints, i.e. stressequilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated)voxelized stress field obtained, for example, by means of synchrotron X-raydiffraction techniques. The method consists in finding the equilibrated stressfield closest (in $L^2$-norm sense) to the measured non-equilibrated stressfield, via the solution of an optimization problem. The extraction of thedivergence-free (equilibrated) part of a general (non-equilibrated) field isperformed using the Hodge decomposition of a symmetric matrix field, which isthe generalization of the Helmholtz decomposition of a vector field into thesum of an irrotational field and a solenoidal field. The combination of: a) theEuler-Lagrange equations that solve the optimization problem, and b) the Hodgedecomposition, gives a differential expression that contains the bi-harmonicoperator and two times the curl operator acting on the measured stress field.These high-order derivatives can be efficiently performed in Fourier space. Themethod is applied to filter the non-equilibrated parts of a synthetic piecewiseconstant stress fields with a known ground truth, and stress fields in GumMetal, a beta-Ti-based alloy measured in-situ using Diffraction ContrastTomography (DCT). In both cases, the largest corrections were obtained neargrain boundaries.

Imposing equilibrium on measured 3-D stress fields using Hodge decomposition and FFT-based optimization