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