Strain2Disp_FE

Finite-element reconstruction of displacement fields from measured strain or deformation-gradient maps

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Overview

Strain2Disp_FE reconstructs displacement fields from measured strain or deformation-gradient fields using a finite-element least-squares formulation. It is designed for cases where high-resolution experiments provide rich local strain information but not the full displacement field required for fracture-mechanics quantities such as the J-integral and stress intensity factors.

The method is especially useful for experimental mechanics workflows involving:

• high-angular-resolution electron backscatter diffraction (HR-EBSD),
• diffraction-based strain mapping,
• 2D and 3D crack-tip benchmark fields,
• indentation-induced cracks,
• displacement reconstruction before Abaqus-based fracture analysis,
• extraction of K_I, K_II, K_III and J from reconstructed fields.

The central idea is simple: measured strain or deformation-gradient components are treated as known quantities, while nodal displacements are treated as unknowns. The domain is discretised with finite elements, the displacement-gradient equations are assembled over the measurement points, and the resulting overdetermined system is solved in a least-squares sense.

2D validation using a synthetic crack-tip field. The finite-element reconstruction reproduces the displacement field more accurately than a finite-difference-style integration route.

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Why this tool exists

Modern full-field experiments can measure strain with excellent spatial resolution. However, strain fields alone do not always provide the individual displacement-gradient terms needed for fracture analysis. For example, the shear strain contains coupled displacement-gradient terms, but the J-integral requires specific displacement derivatives.

This toolbox fills that gap by converting measured local deformation information into a mechanically usable displacement field.

In practical terms:

measured strain / deformation-gradient field
        -> finite-element least-squares integration
        -> continuous displacement field
        -> fracture parameters, validation, visualisation, or Abaqus input

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What the method does

The workflow can be summarised as follows.

flowchart LR
    A[Measured field] --> B[Clean and regularise data]
    B --> C[Build FE mesh]
    C --> D[Choose element order and integration points]
    D --> E[Assemble A d = b]
    E --> F[Least-squares solution for nodal displacement]
    F --> G[Remove rigid-body motion]
    G --> H[Export Ux, Uy, Uz]
    H --> I[Fracture or field analysis]

Loading

The formulation supports two major input families:

Input family	Typical source	Reconstructed output
2D strain fields	HR-EBSD, DIC-derived strain, analytical benchmarks	Ux, Uy
3D strain or deformation-gradient fields	3D diffraction, DVC-derived gradients, synthetic benchmarks	Ux, Uy, Uz

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Features

• Finite-element integration of measured strain or deformation-gradient data.
• 2D and 3D formulations.
• Linear and quadratic element concepts.
• Least-squares solution of overdetermined displacement-gradient systems.
• Synthetic validation using crack-tip fields.
• HR-EBSD application to indentation-induced cracks.
• Optional post-processing for rigid-body translation and rotation removal.
• Reconstructed displacement fields suitable for downstream fracture analysis.
• Compatible with workflows where Abaqus is used to compute J, K_I, K_II and K_III.

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Example applications

1. 2D mode-I crack-tip validation

The 2D benchmark integrates a synthetic crack-tip strain field and compares the reconstructed displacement field with the analytical displacement solution.

Linear and quadratic elements with reduced and full integration-point configurations.

Typical use:

addpath(genpath(pwd));

cfg = struct();
cfg.problem        = "2D_mode_I_validation";
cfg.inputType      = "strain";
cfg.dimension      = 2;
cfg.elementType    = "linear";     % linear or quadratic, depending on the implementation branch
cfg.integration    = "full";       % reduced or full
cfg.units          = "mm";
cfg.removeRigidBody = true;

% Use the repository's top-level driver or input desk for the 2D validation case.
% Example wrapper name if your branch exposes one:
% results = Strain2Disp_FE(cfg);

Expected outputs include reconstructed U_x and U_y maps, error maps, and line-profile comparisons against the analytical solution.

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2. 3D mixed-mode crack-tip validation

The 3D benchmark reconstructs a displacement field from a synthetic mixed-mode crack field and checks whether the reconstructed field recovers the prescribed fracture parameters.

3D validation using a synthetic mixed-mode crack field and Abaqus-based fracture-parameter extraction.

Typical use:

addpath(genpath(pwd));

cfg = struct();
cfg.problem        = "3D_mixed_mode_validation";
cfg.inputType      = "strain";
cfg.dimension      = 3;
cfg.elementType    = "brick8";
cfg.integration    = "full";
cfg.units          = "mm";
cfg.exportAbaqus   = true;

% Use the repository's 3D validation driver or adapt the template above.
% results = Strain2Disp_FE(cfg);

Relevant reconstructed fields:

Ux, Uy, Uz
H11, H12, H13, H21, H22, H23, H31, H32, H33

These fields can then be used for J-integral and mixed-mode SIF extraction.

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3. HR-EBSD indentation-crack analysis

The experimental workflow reconstructs displacement fields from HR-EBSD elastic strain maps around Vickers-induced cracks in monocrystalline silicon.

HR-EBSD strain and lattice-rotation maps around an indentation-induced crack field.

The reconstructed displacement fields can be compared against AFM-measured topography and used as boundary conditions for fracture analysis.

Comparison between measured AFM topography and reconstructed displacement fields.

Typical use:

addpath(genpath(pwd));

cfg = struct();
cfg.problem         = "HR_EBSD_indentation";
cfg.inputType       = "elastic_strain";
cfg.dimension       = 3;
cfg.units           = "um";
cfg.material        = "anisotropic_silicon";
cfg.depthResolution = 700;       % nm, example value used for the EBSD information depth
cfg.removeCracks    = true;
cfg.removeRigidBody = true;
cfg.exportAbaqus    = true;

% Use the repository's HR-EBSD input desk or driver.
% results = Strain2Disp_FE(cfg);

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Installation

Clone the repository and add it to your MATLAB path:

git clone https://github.com/Shi2oon/Strain2Disp_FE.git
cd Strain2Disp_FE

Then in MATLAB:

addpath(genpath(pwd));
savepath;   % optional

Recommended environment:

• MATLAB R2021a or later.
• Optimisation or linear algebra routines available in base MATLAB.
• Abaqus, only if exporting reconstructed displacement fields for fracture analysis.

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Input data

The toolbox is intended for regular or regularised measurement grids. Typical inputs are:

2D field

Column / array	Meaning
X, Y	measurement coordinates
eps_xx, eps_yy, eps_xy	in-plane strain components
optional omega_xy	in-plane rotation if available

3D field

Column / array	Meaning
X, Y, Z	measurement coordinates
eps_xx, eps_yy, eps_zz	normal strain components
eps_xy, eps_xz, eps_yz	shear strain components
optional omega_xy, omega_xz, omega_yz	lattice rotations if available
optional F11 ... F33	deformation-gradient components

For diffraction-based elastic strain maps, be careful about missing data, cracks, grain boundaries and pattern-centre artefacts. Poorly resolved or physically inconsistent fields will reconstruct poor displacement fields.

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Outputs

Typical outputs include:

Output	Description
Ux, Uy, Uz	reconstructed displacement components
Hij	reconstructed displacement-gradient components
residual	least-squares residual or reconstruction error measure
mesh	nodal coordinates and element connectivity
figures	displacement maps, validation plots and line profiles
Abaqus files	optional boundary-condition or model files for fracture analysis

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Abaqus and fracture-parameter workflow

When the reconstructed displacement field is exported to Abaqus, the common downstream workflow is:

reconstructed displacement field
        -> apply as boundary conditions or field constraints
        -> elastic or anisotropic constitutive model
        -> interaction-integral / domain-integral analysis
        -> J, K<sub>I</sub>, K<sub>II</sub>, K<sub>III</sub>

This is particularly useful when the experimental boundary conditions, loads or crack geometry are difficult to idealise analytically.

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Practical guidance

Element choice

• Use linear elements for robust and efficient integration on dense regular grids.
• Use higher-order elements only where the measurement density and field smoothness justify the additional cost.
• In 3D, full integration is preferred when the data density is sufficient.

Data cleaning

• Remove cracks, voids and non-material regions before integration.
• Mask unreliable HR-EBSD points, grain boundaries, damaged patterns and severe discontinuities.
• Keep a sufficiently large field of view around the defect.
• Do not interpret near-tip singular pixels as reliable integration constraints unless the measurement quality supports it.

Rigid-body motion

Strain integration cannot determine absolute rigid-body translation or rotation. These should be corrected after reconstruction using a reference region or a physically motivated rigid-body removal step.

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Limitations

This is not a magic strain-to-truth converter. The quality of the reconstructed displacement field depends on the quality and physical consistency of the measured input field.

Important limitations:

• Diffraction-based strain mapping usually measures elastic strain, not total strain.
• Missing rotations or incomplete deformation-gradient information can limit uniqueness.
• Crack flanks, grain boundaries and discontinuities can introduce incompatibility.
• Out-of-plane displacement reconstruction depends on assumptions about measurement depth.
• Abaqus-derived fracture parameters remain sensitive to crack-tip placement, mesh construction and contour selection.

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Suggested repository structure

A clean project layout is:

Strain2Disp_FE/
├── README.md
├── Data/
│   └── Assets/
│       ├── Fig1_FE_elements.png
│       ├── Fig2_2D_modeI_validation.png
│       ├── Fig3_3D_mixed_mode_validation.png
│       ├── Fig5_HR_EBSD_strain_rotation_maps.png
│       └── ...
├── Examples/
│   ├── 2D_ModeI_Validation/
│   ├── 3D_MixedMode_Validation/
│   └── HR_EBSD_Indentation/
├── functions/
├── src/
└── outputs/

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Citation

If you use this repository, please cite:

@article{Koko2026Strain2DispFE,
  title   = {Computation of displacements from strain fields: Derivation, validation, and application},
  author  = {Koko, Abdalrhaman and Elmukashfi, Elsiddig and Karamched, Phani S. and Marrow, T. James},
  journal = {Acta Materialia},
  volume  = {304},
  pages   = {121818},
  year    = {2026},
  doi     = {10.1016/j.actamat.2025.121818}
}

The article states that the associated code is available through Zenodo DOI 10.5281/zenodo.6411572.

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Figure attribution

Figures in functions/Assets are extracted and cropped from:

Koko, A., Elmukashfi, E., Karamched, P. S., and Marrow, T. J. Computation of displacements from strain fields: Derivation, validation, and application. Acta Materialia 304, 121818, 2026.

The article is open access under the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Cropping and README formatting changes were made for display clarity.

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Related repositories

• DIC2ABAQUS - DIC-to-Abaqus workflows for fracture analysis.
• Defect_Descriptor - configurational-force and mixed-mode SIF extraction from measured fields.

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Contributing

Contributions are welcome, especially for:

• additional experimental examples,
• clearer input-desk templates,
• Abaqus export examples,
• uncertainty quantification,
• support for more diffraction data formats,
• documentation of 3D workflows.

Please open an issue or submit a pull request with a minimal reproducible example.