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Published January 2022 | Accepted Version
Journal Article Open

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


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 L²-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 experimental 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.

Additional Information

© 2021 Elsevier Ltd. Received 19 April 2021, Revised 2 October 2021, Accepted 7 October 2021, Available online 24 October 2021. The core part of this work was conducted during HZ's visit to Los Alamos National Laboratory (LANL) to work with RAL under LANL's LDRD program, USA, whose support is acknowledged. RAL's work is also supported by LANL's Science Campaign 2 program, USA. KB and HZ were also partially supported by the Air Force Office of Scientific Research, USA through the MURI grant FA9550-16-1-0566. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Hao Zhou, Ricardo A. Lebensohn, Péter Reischig, Wolfgang Ludwig, Kaushik Bhattacharya, Corrigendum to 'Imposing equilibrium on measured 3-D stress fields using Hodge decomposition and FFT-based optimization' [Mech. Mater., 164 (2022) 104109], Mechanics of Materials, Volume 174, 2022, 104448, ISSN 0167-6636, https://doi.org/10.1016/j.mechmat.2022.104448. (https://www.sciencedirect.com/science/article/pii/S0167663622002150)

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August 22, 2023
October 23, 2023