# Feasibility study of near surface residual stress field characterisation using the eigenstrain reconstruction method

Samantha Veck, Foroogh Hosseinzadeh, John Bouchard, H K Kim, Andrew Barrow

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Samantha Veck, Foroogh Hosseinzadeh, John Bouchard, H K Kim, Andrew Barrow. Feasibility study of near surface residual stress field characterisation using the eigenstrain reconstruction method. ICRS 11 - The 11th International Conference of Residual Stresses, SF2M; IJL, Mar 2022, Nancy, France. hal-03998011

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# FEASIBILITY STUDY OF NEAR SURFACE RESIDUAL STRESS FIELD CHARACTERISATION USING THE EIGENSTRAIN RECONSTRUCTION METHOD

Samantha Veckª,*, Foroogh Hosseinzadehª, P John Bouchardª, HK Kimª, Andrew Barrowᵇ

## ABSTRACT

Near surface (< 0.25 mm) characterisation of residual stresses is necessary to support improved life degradation predictions in high-risk sectors such as aerospace and nuclear. Experimental measurements at the near surface produce limited point-wise data and therefore incomplete characterisation of the stress state. Reconstructive methods aim to provide the complete residual stress field based on limited point-wise experimental measurements and elastic theory. The Eigenstrain Reconstruction Method (ERM) is an inverse technique which determines the eigenstrain (inelastic strain) distributions which give rise to elastic strain. Previous works investigating near surface reconstruction utilised synchrotron diffraction measurements and have yet to consider the effect of limited spatial resolution and measurement point availability at the near surface from laboratory-based measurement techniques.

This paper evaluates the feasibility of applying the ERM using laboratory-based measurement data for the characterisation of near surface residual stresses. Finite element analysis of a four-point bent beam is conducted to provide a comparative reference for residual stress reconstruction. Representative data sets for Incremental Centre Hole Drilling (ICHD) and the contour method are produced from the finite element model for use with the reconstruction method. Reconstruction is conducted with the Simple Triangle (SIMTRI) method using the synthetic data sets. The capability and limitations of SIMTRI for near surface residual stress characterisation are discussed. This paper finds that while near surface reconstruction is feasible, further work is required to optimise the basis function width at the near surface to enable reconstructions of residual stress with great detail in this region.

**Keywords:** Residual stress; Eigenstrain; Reconstruction; Finite element analysis; Limited measurements

## 1. Introduction

Near surface (< 0.25 mm) residual stress characterisation is challenging, with limited spatial resolution provided by conventional laboratory techniques such as Incremental Centre Hole Drilling (ICHD) [1]. Improvement in spatial resolution in this region is desirable to reduce the magnitude of safety margins applied to life degradation predictions for high-risk industries. Synchrotron diffraction is the preferred measurement technique for the characterisation of near surface residual stresses but requires additional resources beyond a conventional laboratory

<table><thead><tr><td>COUNTRY</td><td>EXPORT CLASSIFICATION</td><td>DATE</td></tr></thead><tbody><tr><td>UK</td><td>Not Listed</td><td>28 JAN 2022</td></tr></tbody></table>

<footnote>ªThe Open University, School of Engineering & Innovation, MK7 6AA, Milton Keynes, UK</footnote>

<footnote>ᵇRolls-Royce plc, PO Box 2000, DE21 7XX, Raynesway, Derby, UK</footnote>

<footnote>* Corresponding author. samantha.veck@open.ac.uk</footnote>
setting. To seek a solution to the challenge of limited spatial resolution without the use of non-laboratory techniques, the application of reconstruction theory is considered.

According to Withers et al. [2] residual stresses are those which remain in a body which is stationary and in equilibrium with its surroundings. Residual stresses develop from misfits between different regions or phases within a material [2] which often arise due to inelastic manufacturing processes [3], from misfitting parts in assembly, or chemical and thermal processes [4]. Misfits were defined as eigenstrains by Mura [5] in 1987 and they indicate permanent or non-elastic strain such as creep strain, transformation strain, thermal strain and other forms generated after non-elastic processes.

The eigenstrain distribution which gives rise to residual stress is often unknown and it must be found from incomplete elastic strain, stress or displacement measurements before the stress field can be reconstructed [6]. Inverse methods aim to find residual stress distributions which are compatible with such measurements and elasticity theory to reduce the solution space [7]. The Eigenstrain Reconstruction Method (ERM) [8] is an inverse method used for determining eigenstrain distributions. The ERM is a possible solution to the limited spatial resolution of near surface measurement data, as the complete stress state in the part is reconstructed from the limited measurement data, not simply interpolated between measurement points.

Despite the availability of literature describing the application of eigenstrain reconstruction to various manufacturing processes, performance of reconstructive methods at the near surface (< 0.25 mm) has had limited attention. Coratella et al. [9] used contour measurement data to reconstruct a shot peened profile and achieved good agreement with synchrotron and neutron diffraction. However, the reconstruction methodology used in this study is limited to shot peened samples, relying on a linear relationship between eigenstrain and laser peened induced residual stresses. Jun et al. [10] applied the ERM to a shot peened cone, reconstructing synchrotron diffraction results with a spatial resolution of approximately 0.02 mm for the first 0.5 mm from the surface. The reconstructions produced were excellent at the near surface but the level of spatial resolution for the reconstruction reference measurement data is not achievable using laboratory-based techniques.

The purpose of the study was therefore to investigate the feasibility of near surface eigenstrain reconstruction using laboratory techniques including ICHD and the contour method. Three variables were considered to evaluate their impacts on near surface reconstruction error: application of individual and combined near surface and bulk data sets; Triangle Pulse Function base width; and location of Triangle Pulse Function application on the part domain.

## 2. Reference and Synthetic Data

### 2.1 Reference Residual Stress Profile

Residual stresses in a four-point bent beam were predicted using Finite Element (FE) analysis and used as a reference stress profile by which results from the ERM were compared and evaluated. This residual stress distribution was selected due to prior use in reconstructive studies [11] to enable comparison of results, availability of an analytical model to describe the residual stress distribution [12] and a clearly defined eigenstrain region [13] which is a key requirement for the ERM.

Commercial FE software Abaqus was selected for generation of numerical models. A 2D beam FE model was produced (see Figure 1). The beam material is Alloy 625 ($E$ = 207 GPa, $v$ = 0.278, $\sigma_{0.2\gamma}$ = 894 MPa). The beam was constructed as a half model, with a symmetry Boundary Condition (BC) applied to the beams width at the centre to reduce computational time. For the ERM portion of analysis, a node at the centre of the midpoint of the symmetry BC was constrained in all degrees of freedom as recommended in [14]. The mesh size at the midpoint of the beam varied through width according to the measurement point spacing presented in Table 1. The mesh was biased along the beam length from 0.1 mm at the beam centre to 5 mm at the far end of the beam.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic drawing of the four-point bent beam showing the dimensions of the beam, the set-up for bending, the extent of plastically deformed region and location of boundary conditions in the FE model</figcaption></figure>

A plane stress elastic plastic four-point bending analysis with isotropic hardening was performed on the beam to produce a reference stress field for use during reconstruction. The dimensions of the plastic region (eigenstrain location) are presented in Figure 1. To validate the bent beam numerical model, an analytical model of the beam was calculated for elastic-perfectly plastic properties. The FE model dimensions and mesh were used for all reconstructions performed in this study.

## 2.2 Synthetic Reference Data Sets

In this paper, synthetic residual stress measurement data has been extracted from the FE bent beam simulation. ICHD and contour synthetic residual stress data were produced by reducing the FE simulated data set to a representative data set for each measurement type as reported in Table 1. No uncertainties were added to the synthetic data sets.

<table><caption>Table 1. Synthetic measurement data details</caption><thead><tr><th>Synthetic Data</th><th>Depth Range from Surface</th><th>Measurement Point Spacing</th><th>Measurement Points (M)</th></tr></thead><tbody><tr><td>ICHD [1]</td><td>0.016 mm - 1.024 mm</td><td>Strain gauge 062RE residual stress spacing [15]</td><td>28</td></tr><tr><td>Contour [16]</td><td>0.5 mm - through bulk</td><td>0.1 mm</td><td>134</td></tr><tr><td>ICHD and contour combined</td><td>0.016 mm - through bulk</td><td>As ICHD then 0.1 mm</td><td>152</td></tr></tbody></table>

## 3. Eigenstrain Reconstruction Method (ERM)

The ERM has been well documented in prior publications [8][17], therefore only a summary of the method will be presented here. The distribution of eigenstrain (inelastic strain) along the beam's width (x (Figure 1)) is described by Equation 1. The distribution of eigenstrain was initially assumed using basis functions $F_i(x)$ where i represents the basis function order. The order of approximation, or number of basis functions, N was less than or equal to the measurement points (M (Table 1)) in the synthetic data set being used. The eigenstrain distribution was subsequently refined with coefficients $c_i$ that were calculated using a least squares minimisation approach between reconstructed and synthetic residual stress data sets.

$$ \epsilon_{yy}^*(x) = \sum_{i=1}^{N} c_i F_i(x) \qquad (1) $$
## 3.1 Simple Triangle (SIMTRI) Method

A limitation of the eigenstrain approach for residual stress field reconstruction is the requirement for the user to have expertise in the selection of basis functions to adequately describe the eigenstrain distribution [8]. The Simple Triangle (SIMTRI) method, which provides a FE framework for ERM [8], offers a solution to this requirement by representing the eigenstrain distribution with Triangle Pulse Functions (TPFs).

The TPFs dimensions were defined by Equations 2 - 4. A denotes the rising edge position, B is the TPF peak, C is the falling edge. The TPF base width is equal to C - A. For each reconstruction, the number (N) of TPFs used was less than or equal to the number of synthetic measurement points (M) of the data set being used. Equation 5 describes the eigenstrain in the yy direction (Figure 1) for each individual basis function i. The coefficients ($c_i$) were given an arbitrary value of 0.001 to ensure residual stress values were of the correct magnitude [18] and later replaced with the coefficients for the final eigenstrain distribution run represented by Equation 1. FE nodal coordinate values (x) were requested by a Fortran user subroutine. Equations 2 – 5 ensured TPFs were equally spaced across the domain with a 50 % overlap of the TPF bases and that the peak position of the first and last TPF in the domain was centred over the domain edge (Figure 2).

$$A = \frac{\text{width of TPF application zone}}{N - 1} \times (i - 2) \qquad (2)$$

$$B = \frac{\text{width of TPF application zone}}{N - 1} \times (i - 1) \qquad (3)$$

$$C = \frac{\text{width of TPF application zone}}{N - 1} \times i \qquad (4)$$

$$\varepsilon_{yy,i}^{*}(x) = \begin{cases} ((x - A)/(B - A)) \times c_i & A \le x \le B \\ ((C - x)/(C - B)) \times c_i & B < x \le C \\ 0 & A > x > C \end{cases} \qquad (5)$$

The TPFs can either be applied to the eigenstrain region or across the entire measurement data set [19] to reconstruct the entire stress field. Figure 2 shows a visual representation of the application of TPFs to the part domain in these two configurations. Zonal and full field approaches were compared during this study.

<figure><img src="image_4.png" /><figcaption>Fig. 2. TPF application along beam width (a) placement in areas containing plastic deformation (zonal TPF placement); (b) placement across full beam width (full field TPF placement).</figcaption></figure>

Nine different test configurations of synthetic data sets were evaluated during the study, as displayed in Table 2. As there were two eigenstrain regions for the bent beam, column ‘Total TPFs Used’ in Table 2 also contains a bracketed number for how many TPFs were applied to each region if the zonal approach was used. The residual stress fields reconstructed using SIMTRI were extracted for analysis as described in section 4.
<table><caption>Table 2. Reconstruction configurations</caption><thead><tr><th>Test ID</th><th>Data Set</th><th>TPF<br>Placement</th><th>Total TPFs Used (N)</th><th>TPF Base Width<br>(mm) (2 d.p.)</th></tr></thead><tbody><tr><td>1</td><td>ICHD</td><td>Zonal</td><td>20 {10}</td><td>0.97</td></tr><tr><td>2</td><td>Contour</td><td>Zonal</td><td>20 {10}</td><td>0.97</td></tr><tr><td>3</td><td>ICHD and Contour</td><td>Zonal</td><td>20 {10}</td><td>0.97</td></tr><tr><td>4</td><td>ICHD</td><td>Full Field</td><td>24</td><td>1.04</td></tr><tr><td>5</td><td>Contour</td><td>Full Field</td><td>24</td><td>1.04</td></tr><tr><td>6</td><td>ICHD and Contour</td><td>Full Field</td><td>24</td><td>1.04</td></tr><tr><td>7</td><td>Contour</td><td>Zonal</td><td>42 {21}</td><td>0.47</td></tr><tr><td>8</td><td>ICHD and Contour</td><td>Zonal</td><td>42 {21}</td><td>0.47</td></tr><tr><td>9</td><td>ICHD and Contour</td><td>Zonal</td><td>100 {50}</td><td>0.20</td></tr></tbody></table>

## 4. Results

Test IDs 1, 4, 7 and 9 (Table 2) did not converge (coefficients of too high magnitude). Further discussion is provided in section 5. Reconstructions were successfully performed for all other experiment configurations. Results from successful reconstructions are presented in Figures 3 - 4.

<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Reconstructed residual stress field using test IDs 2,3 and 8; (b) near surface region.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Reconstructed residual stress fields using test IDs 5 and 6; (b) near surface region.</figcaption></figure>
The error of all successful reconstructions was evaluated at the near surface (0.25 mm) using Root Mean Square (RMS) error analysis [20]. Error analysis results described in the following paragraphs are displayed in Figure 5. For a TPF function base width of approximately 1 mm, the full field approach to TPF positioning worked well producing an RMS error of less than 6 MPa. For the full field method there was also a negligible difference between error values when reconstructions were performed with bulk data (ID 5) or a combination of near surface and bulk data sets (ID 6). For the zonal method, introduction of near surface and bulk data sets (ID 3) into the reconstruction significantly reduced near surface reconstruction error compared with bulk data alone (ID 2). Reducing the TPF base width from 0.97 mm to 0.47 mm (ID 8) for zonal reconstruction produced a slight decrease in error for at the near surface.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Root Mean Square (RMS) error analysis of experiment configurations with successful reconstructions (IDs 2, 3, 5, 6 and 8) at the near surface (< 0.25 mm)</figcaption></figure>

**5. Discussion**

The findings from this study suggest that the SIMTRI approach for eigenstrain reconstruction can be used for near surface reconstruction if limited laboratory-based reference data is used. The full field method appears more promising for near surface reconstruction based on error alone. However, this may only be due to the linear nature of the residual stress profile for the bent beam. Residual stress profiles which have sharp changes in stress < 0.5 mm from the surface may not be captured. The reduced error for full field TPF placement compared to zonal was not unexpected; [19] found improvement of agreement between reference and reconstructed profiles of a stir friction weld when TPFs spanned the whole range of measurement data compared to just the eigenstrain region. It is essential that this study is repeated with a different residual stress profile (e.g., shot peened) to further assess the advantages and disadvantages of full field and zonal approaches to reconstruction using laboratory-based reference data.

While not a limiting factor for the aim of the study, it should be emphasised that TPF placement across the full domain does not provide a realistic distribution of eigenstrain compared to the zonal approach and therefore has limited applicability for further analysis [21].

It was unsurprising that near surface data was unsuitable for through depth eigenstrain reconstruction without supporting bulk measurements due to an insufficient coverage of the eigenstrain region. ICHD synthetic data is taken over a depth of 1 mm from the surface (Table 1), resulting in only 20% coverage of the eigenstrain region (Figure 1). If destructive testing is not possible to provide laboratory-based bulk measurement data, expensive and difficult to acquire measurements at a synchrotron facility may be required.

The combined use of near surface and bulk residual stress data for zonal eigenstrain reconstruction provided a reduction in reconstruction error compared to bulk data alone. In contrast, increasing the number of basis functions used from 20 to 42 to further reduce error only produced a 0.1 MPa reduction which suggests that increasing the number of TPFs has minimal effect on error reduction at the near surface. However, while the TPF base width was
effectively halved from 0.97 mm to 0.47 mm, this only led to an increase of basis functions at the near surface from 1 to 2. The ratio of measurement points to TPFs is far lower for the bulk region in both analyses. Reducing the measurement point to TPF ratios at the near surface to match bulk ratios via decrease of TPF base width may lead to improved near surface error.

Reconstructions with TPF base width < 0.47 were not successful, furthering the concern that the TPF base width may be a limiting factor for near surface reconstruction. The FE reconstruction of eigenstrain for individual basis functions were investigated to determine a root cause for this limitation. It was found that when less than 5 nodes of the FE mesh were contained within the TPF base width, the TPF did not generate a triangular form and lead to linear summation issues. With a limitation of 5 nodal points required per TPF base width, for a 0.1 mm base width the FE mesh division must be a minimum of 0.02 mm. This suggests further work is required to refine the bent beam FE mesh to allow reduction of the TPF base width. However, even if refinement of the mesh is performed the maximum number of TPFs available is only equal to the number of measurement points which can also limit the minimum achievable base width.

To reduce computational time required for fine FE meshes and reduce the overall number of TPFs used in reconstruction to less than or equal the number of measurement points, the natural progression is modification of TPF width through depth, allowing the mesh size to increase away from the surface. The modification of base function width is briefly mentioned in [8] and has been performed where near surface synchrotron measurements of approximately 0.025 mm spatial resolution were available [10] which is considerably higher spatial resolution than available with laboratory-based measurement techniques. Less than 1.8 mm from the surface the TPF base width was 0.04 mm, increasing to 0.2 mm for the remainder of the domain. No guidance is provided for modification of TPF base width through depth nor is there published discussion into the effect this may have on reconstruction validity. Therefore, further work is required to determine if TPF base width is a severe limiting factor on the feasibility of near surface residual stress reconstruction with limited laboratory-based reference data for both zonal and full field SIMTRI reconstruction.

**6. Conclusions**

Near surface reconstruction using SIMTRI is feasible but heavily limited by FE mesh sizing and subsequent nodal density, measurement point data availability and the requirement for equal spacing of basis functions. Full field TPF placement produced a smaller RMS error than zonal TPF placement, however the capacity for capturing variations in residual stress at the near surface was inconclusive due to the linear residual stress distribution. If near surface reference measurement data does not cover a sufficient area of the eigenstrain region, it must be supported by destructive bulk measurements if reconstruction is to be performed without the use of synchrotron facilities.

Further research is required to determine if the base width of TPFs can be modified through depth with limited spatial resolution data to overcome limitations for near surface reconstruction identified in this study. A further reconstruction study should also be performed on an alternative residual stress distribution with zonal and full field TPF placement to allow for conclusive results on the advantages and disadvantages for each approach for near surface reconstruction.

**7. Acknowledgements**

This work is part funded by Rolls-Royce plc. The authors are grateful for communications with Associate Professor Tea-Sung Jun (Incheon National University) which enabled use of the SIMTRI method.

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