A Computational Method for the Determination of the Elastic Displacement Field using Measured Elastic Deformation Field

A novel approach was derived to compute the elastic displacement field from a measured elastic deformation field (i.e., deformation gradient or strain). The method is based on integrating the deformation field using Finite Element discretisation. Space and displacement fields are approximated using piece-wise interpolation functions. Hence, the full elastic deformation field can be expressed as nodal displacements, the unknowns. The nodal displacements are then obtained using a least square method. The proposed method was applied to the symmetrical (residual) elastic deformation field measured using high (angular) resolution electron backscatter diffraction around a Vickers micro-indenting impression on a (001) mono-Si crystal sample with the integrated out-of-plane surface displacements matched with the impression topography measured using. The (residual) displacement field was used as the boundary conditions to calculate the three-dimensional stress intensity factors (K_(I,II,III)) at the cracks emanating from the indentation.

💡 Research Summary

The paper introduces a novel computational framework for reconstructing a three‑dimensional elastic displacement field directly from experimentally measured deformation data, such as deformation gradients or strain tensors obtained by high‑angular‑resolution electron backscatter diffraction (HR‑EBSD). The core of the method is a finite‑element (FE) discretisation of the measured deformation field combined with a least‑squares (LSE) solution for the nodal displacements.

First, the measured deformation gradient F (or strain ε) is expressed at any point inside an FE element as the gradient of the interpolated displacement field u(x) = Σ N_i(x) u_i, where N_i are standard shape functions and u_i are the unknown nodal displacement vectors. By equating the analytical expression of the displacement gradient ∇u with the measured deformation gradient, a set of linear equations K U = F̂ is assembled, where K contains integrals of shape‑function derivatives and F̂ is a “virtual load” vector constructed from the experimental deformation data. Because the number of measured data points (pixels) far exceeds the number of nodal degrees of freedom, the system is over‑determined. The authors therefore minimise the squared residual ‖K U − F̂‖², leading to the normal equations KᵀK U = KᵀF̂. The solution is obtained either by direct inversion (when KᵀK is full rank) or by a Moore‑Penrose pseudoinverse when the matrix is ill‑conditioned.

The method is implemented using eight‑node hexahedral elements. HR‑EBSD provides the local lattice rotations and deviatoric strains through cross‑correlation of electron backscatter patterns; these are decomposed into symmetric (strain) and antisymmetric (rotation) parts. The anisotropic elastic constants of single‑crystal silicon are used to convert strain to stress, ensuring that the material’s crystallographic anisotropy is fully accounted for.

Experimental validation is performed on a (001) silicon wafer indented with a Vickers diamond pyramid under a 50 g load for 1 s, producing half‑penny cracks. The EBSD step size is 0.25 µm, yielding an 800 × 600 pixel strain map. Monte‑Carlo simulations of electron trajectories (Casino) estimate the effective interaction depth of the EBSD signal; the mean depth is 173 nm (50 % probability). The measured strain field is integrated for several assumed depths (40 nm to 850 nm). The out‑of‑plane component of the reconstructed displacement field is compared with high‑resolution atomic force microscopy (AFM) topography of the indentation imprint. The best agreement occurs for the 173 nm depth, confirming the method’s ability to recover realistic surface displacements.

The reconstructed 3‑D displacement field is then used as a boundary condition for a linear elastic finite‑element model of the cracks. By applying the displacement field on the crack faces, the authors compute the three stress‑intensity factors K_I, K_II, and K_III for each crack tip, thus providing a full mixed‑mode fracture‑mechanics characterization without any a‑priori assumptions about crack length or geometry. This contrasts with traditional indentation‑fracture methods, which rely on empirical relations and suffer from large scatter, especially for anisotropic brittle materials.

Supplementary material details synthetic benchmark tests, regularisation strategies, and the implementation workflow for exporting the reconstructed displacement field to commercial FE software (ABAQUS).

In conclusion, the paper delivers (1) a rigorous FE‑based integration scheme that converts measured deformation gradients into nodal displacements via least‑squares optimisation, (2) experimental verification using HR‑EBSD, AFM, and focused‑ion‑beam (FIB) tomography, and (3) a direct route from measured micro‑scale deformation to quantitative mixed‑mode stress‑intensity factors. The approach bridges the gap between high‑resolution full‑field deformation measurements and fracture‑mechanics analysis, offering a powerful tool for micro‑mechanical testing, materials design, and reliability assessment of brittle, anisotropic crystals.