Validation of Hole-Drilling Residual Stress Measurements in Workpieces of Various Thickness

This study uses a thick, aluminum alloy plate with one shot peened surface and samples removed from that plate. The plate geometry is cut from a large 25.4 mm thick AA 7050-T7451 rolled plate. The 7050 aluminum alloy is commonly used in aerospace structural applications and has been used in previous studies related to residual stress measurement [7, 11, 12]. The mechanical properties of this material in the T7451 temper are shown in Table 1 [13]. From the 25.4 mm thick plate, a smaller plate was machined to dimensions shown in Fig. 1, measuring 356 mm along the rolling direction (L) by 406 mm (long transverse, LT) by 20 mm (short transverse, ST). The 20 mm thickness was obtained by milling the original stock from only one 356 by 406 mm surface; the milled surface was designated as the top surface. Residual stress was induced in the plate by shot peening the top surface uniformly using SAE 170 cast steel shot at an Almen intensity of 6-10A and 100% coverage.

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From the shot-peened plate, six smaller samples are cut as shown in Figs. 1 and 2, each measuring 102 mm (L) by 32 mm (LT) by 20 mm (ST). These samples are designated as samples A, B, C, D, E, and F and permanently inscribed with their letter designator in a corner of the shot-peened surface. Samples A through D are further cut to a specific, reduced thickness by wire electric discharge machining (EDM) on a plane parallel to the sample top (shot peened) surface, the plane being offset by the target workpiece thickness (see Table 2) plus half the wire EDM cut width. Samples E and F are set aside for further work. The four different final sample thicknesses fall into the range of “thin” or “intermediate” thickness for residual stress measurement by hole-drilling, as established by ASTM E837-20 [2] (and described further below).

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A series of residual stress measurements are performed in the shot peened plate and the removed samples using the hole-drilling method. Measurements follow ASTM E837-20 [2], which comprises the following steps. A strain gage rosette, having three individual strain gages positioned around a gage circle, is affixed to the workpiece surface. An end mill is used to mill a blind hole at the center of the gage circle in a series of prescribed depth steps. After reaching each depth step, strains measured by each of the three gages are recorded. Finally, after all depth steps are complete, the depth profiles of three in-plane components of residual stress are calculated from the strain versus hole depth data using the stress calculation procedure in ASTM E837-20 [2].

Each hole-drilling measurement incorporates the same materials and procedures, with key details as follows. Strain gage rosettes are Type A as defined in ASTM E837-20 [2] (Micro Measurements CEA-13-062UL-120), having a gage circle diameter of D = 5.13 mm. Rosettes are bonded to the workpiece using cyanoacrylate adhesive (Vishay M-Bond 200) and oriented such that Gage 1 of the rosette lays along the longitudinal (rolling) direction of the workpiece. The schedule of depth steps is shown in Table 3, which follows the procedure listed in ASTM E837-20, Sect. 8 [2] but adds smaller steps near the workpiece surface (depth < 0.3 mm). Holes are produced with a specialized milling station. Each measurement makes use of a new, 1.59 mm diameter, carbide end mill. The hole depth increments are cut such that the hole diameter is 2.0 mm using an orbital path with a spindle speed of 30,000 RPM, a plunge speed of 0.005 mm/s, and a travel speed of 1.0 mm/s. Measurements in the thick plate are performed with the plate clamped to the milling station table. Measurements in samples A, B, C, and D are performed with the samples clamped in a cantilevered configuration to a block on the milling station table. The cantilever clamping is used to avoid restraining deformation caused by hole-drilling, which is recommended for making useful measurements (see Schajer [1], Sect. 4.6). Figure 3 shows the cantilever clamping configuration; shims (not shown) are placed between the two clamps to avoid flattening existing sample curvature.

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Residual stress versus depth is calculated using the most recent revision to the hole-drilling standard, ASTM E837-20, and elastic properties in Table 1. Workpiece thickness ranges are designated in the standard using the ratio of workpiece thickness W to gage circle diameter D. Workpieces with W/D > 0.6 are designated as “thick”; this range of thickness is considered large enough that the strain versus depth response is independent of W. Workpieces with W/D < 0.25 are designated as “thin”; this range of thickness is considered small enough that the strain versus depth response is proportional to W. Workpieces with W/D between 0.25 and 0.6 are designated as “intermediate”, where strain versus depth response depends on W in the manner described by Schajer [10]. Table 2 lists the thickness (W) and W/D values for the shot-peened plate and samples A, B, C, and D. The plate is a thick workpiece, sample A is a thin workpiece, and all other samples are intermediate workpieces.

A total of 17 residual stress measurements are performed: 5 measurements in the shot peened plate prior to cutting, and 3 measurements each on samples A, B, C, and D. Measurement locations for the plate are shown in Fig. 1 with coordinates listed in Table 4 (data reported for 5 of the 6 locations; one location was spare). Measurement locations on the smaller samples are shown in Fig. 2 with coordinates listed in Table 5 (data reported for 3 of the 5 locations; two locations were spare). The final hole depth is measured using a micrometer after each measurement is completed and the strain gage (and adhesive) removed. A depth offset is added to all depth steps so that the total of all depth steps matches the measured final depth [5].

Given the repeated hole-drilling measurements for each sample configuration, the average residual stress versus depth profile and its standard deviation are computed for further assessment. The residual stress versus depth data for each measurement is interpolated to mid-step depths of Table 3, the interpolation being necessary because the required depth corrections vary slightly among measurements. The average residual stress versus depth profile, and its standard deviation versus depth, is computed from the multiple measurements and reported. The data are also used for further assessment.

Although all samples are removed from a single plate that had been uniformly shot peened, the residual stress is expected to depend on the sample thickness. We expect that the uniformly shot-peened thick plate will have a thin layer of compressive stress at the peened surface that is balanced by low stresses away from the surface; therefore, the thick plate should not experience noticeable deformation after shot peening. However, we expect a thin sample removed from the surface of the same plate would deform, as the minimally stressed bulk of the plate is no longer present to resist the larger stresses near the peened surface. In deforming, a thin sample releases some of the near-surface compressive residual stress induced by the peening process; a thinner sample should have lower residual stress than a thicker sample, the thinner sample taking on a curvature indicative of the stress release. To quantify the deformation of each sample after removal from the plate by EDM cutting, we measure the EDM cut surface form using an area-scanning profilometer.

Because of the expected differences in residual stress among samples, a method is needed to compare residual stresses in the different thicknesses. One way to do this is to perform an elastic stress analysis [7] using a finite element (FE) model. Using a commercial FE code [14], a model is created for each sample thickness. Each model is rectangular with dimensions matching the sample, elastic material properties as given in Table 1, and the FE mesh composed of hexahedral elements. The mesh has 50 elements along each of the longitudinal (X) and long transverse (Y) directions. Element thickness in the short transverse (Z) direction is 0.0125 mm, resulting in 100 elements through thickness for sample A, 120 elements for sample B, 160 elements for sample C, and 240 elements for sample D.

For each sample thickness, the FE model takes an input stress versus depth profile and provides as an output relaxed residual stresses and a deformed sample shape. The input comprises an average residual stress versus depth profile for each in-plane stress component (longitudinal, transverse, and shear), developed by taking the average of the five measurements in the plate at each mid-step depth (where each depth step is defined in Table 3). The average stress versus depth profile of each stress component is input by setting a stress initial condition at the centroid of each element, using linear interpolation between depths as needed. For element centroids beyond the final mid-step depth in the plate, it is assumed that stress remains constant below the final mid-step depth; for centroids before the first mid-step depth, it is assumed that stress remains constant until the first mid-step depth. No other forces are imposed on the models. Constraints are applied only at the node at the center of the top surface: this node is fully constrained. Each model is allowed to reach equilibrium under these conditions. The output from each model is the longitudinal, transverse, and shear residual stress through the thickness at the center of the sample. This output residual stress versus depth profile is called the expected residual stress profile. The expected residual stress profile from FE is compared to the measured residual stress profile by plotting them together, allowing a graphical comparison for each sample thickness. A quantitative comparison is performed by computing the root mean-square (RMS) difference between the expected and measured stress profiles.

Residual stress versus depth profiles measured in the thick plate are shown in Fig. 4. For all measurements, residual stress is negative (compressive) near the surface of the plate and increases in magnitude until reaching a minimum at a depth of roughly 0.1 mm; residual stress then decreases in magnitude until it approaches zero a depth of roughly 0.25 mm. A residual stress versus depth profile of this shape is typical for shot peened materials [15]. The maximum compressive stress in the longitudinal direction (σ_xx) is -300 ± 40 MPa; the maximum compressive stress in the transverse direction (σ_yy) is -315 ± 25 MPa. For all measurements in the thick plate, shear stress (σ_xy) was small (magnitude < 20 MPa) at all depth steps and is not therefore reported. The standard deviation of the measured residual stress in the thick plate is shown in Fig. 5. The standard deviation of the measurements is maximum (45 MPa for longitudinal stress and 60 MPa for transverse stress) at the first depth increment (0.0127 mm); decreases rapidly until a depth of roughly 0.1 mm; increases again to a local maximum at 0.14 mm (30 MPa for both longitudinal and transverse stress); and finally decreases rapidly to values of 10 MPa or less in the subsurface region (depths past 0.3 mm).

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Figure 6(a) shows a comparison of longitudinal strain versus hole depth for one representative measurement in the thick plate and each removed sample (data in Fig. 6(b) are described later). For shallow depths (depth < 0.2 mm), the strain versus hole depth behavior is similar for all removed samples, rising to roughly 100 με at a depth of roughly 0.18 mm, with those strains being lower than strains in the thick plate. Past this depth, the strain versus hole depth behavior varies significantly with workpiece thickness. For samples A, B, and C, the strain begins to fall at hole depths past 0.2 mm, with the thinner samples approaching zero strain more rapidly than in the thicker samples. For sample D, the strain continues to increase slowly until reaching a maximum (of roughly 125 με) at a hole depth of 0.4 mm, then slowly decreasing with greater depth. The strain versus hole depth behavior in the thick plate differs completely from that in the removed samples, with the strain being larger and then continuing to increase for all depths.

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Figure 7 shows residual stress for sample D; Fig. 8 for sample C; Fig. 9 for sample B; and Fig. 10 for sample A. A comparison of average residual stress versus depth profiles for all samples, as well as the thick plate, can be found in Fig. 11(a) and (b). For all measurements, the normal components of residual stress are negative (compressive) near the surface of the sample and increase in magnitude until reaching a minimum at a depth of roughly 0.09 mm; residual stress then increases until reaching a maximum at a depth of roughly 0.25 mm, and finally decreases slowly over greater depths. The average maximum compressive stress in the longitudinal direction (σ_xx) is -218 MPa for sample D, -205 MPa for sample C, -171 MPa for sample B, and -141 MPa for sample A. The average maximum compressive stress in the transverse direction (σ_yy) is consistently larger in magnitude than in the longitudinal, being -234 MPa for sample D, -215 MPa for sample C, -184 MPa for sample B, and -157 MPa for sample A. For all samples, shear stress (σ_xy) was small (magnitude < 20 MPa) for all depth increments and is not therefore reported. The residual stress in the removed samples (Fig. 11) matches typical behavior for shot peened material, but the magnitude of the compressive stress increases with thickness, and the slope of the subsurface stress (beyond 0.3 mm) decreases with thickness. This matches our expectation that thin samples experience a stress release when removed from the thicker plate, with thinner samples experiencing a larger stress release.

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The standard deviation of the replicate measurements in each sample are shown in Fig. 12 and included as error bars in Fig. 11. For all samples and for both longitudinal (σ_xx) and transverse (σ_yy) directions, the standard deviation is maximum at the initial depth step; reaches a local minimum at a depth of roughly 0.09 mm (approximately the depth at which the residual stress profile reaches its minimum value); reaches another local maximum at a depth of roughly 0.15 mm; and finally decreases to relatively small (0–10 MPa) values by a depth of roughly 0.03 mm and remains relatively small for the rest of the depth steps. For samples A and B, the maximum standard deviation is 30.0 ± 5.0 MPa for both longitudinal and transverse stresses; for sample C, the maximum standard deviation is 47.5 ± 1.0 MPa for both longitudinal and transverse stresses; for sample D, the maximum standard deviation is 40.0 ± 2.0 MPa for both longitudinal and transverse stresses. It is notable that thinner samples did not exhibit larger standard deviation (more dispersion) as compared to thicker samples.

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Figure 13 shows the form of the EDM surface of each sample after cutting from the thick plate by wire EDM. A radius of curvature is computed from each surface form by using circular fits to surface height data along two lines, one along the X-direction and the other along the Y-direction, both lines being near the middle of the samples. Table 6 compares radii of curvature for the samples. The radius of curvature is smallest for the thinnest sample (A) and largest for the thickest sample (D). This suggests that thinner samples experienced a greater release of stress during EDM cutting, which is consistent with the residual stress of Fig. 11.

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Expected residual stress (based on FE modelling) is shown for all sample thicknesses in Fig. 11(c) and (d). The trends in expected residual stress with thickness generally agree with those for the measured residual stress (Fig. 11(a) and (b)). The magnitude of the peak compressive stress increases with sample thickness while the subsurface (depth > 0.3 mm) slope decreases with sample thickness.

While Fig. 11 shows general agreement between measured and expected residual stress, the measured stresses are more significantly affected by thickness than are the expected stresses, especially for small thickness. For the smallest thickness (sample A), the measured longitudinal peak compression is -140 MPa while the expected peak is -190 MPa, which is a significant difference. For the intermediate thickness (sample D), the measured and expected longitudinal peak compression are similar (about -220 MPa). The measurement data also suggest that thickness affects the depth at which the peak compression occurs, being shallower in the smallest thickness (sample A). The subsurface (depth > 0.3 mm) stress gradient is similarly affected, the measured and expected longitudinal stress gradient being in reasonable agreement for the intermediate thickness (sample D) and quite different for the smallest thickness (sample A). The RMS differences between the average measured residual stress profile and the expected residual stress profile are given in Table 7 for longitudinal and transverse stress in each sample. The table shows that RMS difference is largest for the smallest thickness and decreases steadily with increasing thickness; the RMS difference is mostly independent of the stress direction.

It is useful to compare the thick plate residual stress versus depth data (Fig. 4) to prior work. Numerous previous studies [4, 11, 16‐18] provide residual stress measurement data for AA 7050-T7451 workpieces treated with similar (although not identical) shot peening parameters, with residual stress measurements by hole-drilling, slitting, and x-ray diffraction with layer removal. Figure 14 compares the average longitudinal residual stress versus depth profile in the present thick plate to data from the prior studies. The data show similar trends, with surface stress of -100 to -150 MPa giving way to maximum compressive stress of -250 to -300 MPa at a depth of 0.07 to 0.10 mm, and a depth of compression (zero crossing) of about 0.30 mm. Overall, we conclude that the present thick plate residual stresses are typical of shot peened aluminum plate.

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To quantify the value of the new residual stress calculation procedure for intermediate and thin workpieces in ASTM E837-20, we perform additional residual stress calculations for one measurement in each sample using the thick workpiece calculation procedure. It is expected that the thick workpiece calculation procedure, when inappropriately used for intermediate and thin samples, will produce erroneous residual stress results relative to those from the thickness-dependent calculation procedure. Figures 15 and 16 compare the two normal residual stress components calculated using the two procedures. For sample D, with W/D (0.585) just below the threshold for a thick workpiece (W/D = 0.6), the two procedures give similar results. For smaller W/D, the thick workpiece procedure exaggerates the minimum and maximum stress values, the exaggeration being larger for smaller thickness. For the smallest thickness (sample A with W/D = 0.244), the thick workpiece calculation exaggerates the minimum stress by 60 to 90 MPa and the maximum stress by 60 MPa. Figures 15 and 16 also show that stresses from the thick workpiece procedure have roughly the same maximum compressive stress for every sample thickness; constant maximum compressive stress with decreasing thickness is erroneous and arises from not using the thickness-dependent calculation procedure. We conclude that the thickness-dependent residual stress calculation procedure provides value, as it produces the expected trends in residual stress for thin and intermediate samples that the thick workpiece calculation fails to capture.

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To provide an additional comparison for the FE-calculated expected residual stress, two additional hole-drilling measurements are performed on the back (EDM) surface (opposite the shot-peened top surface) of two samples, one on sample A (1.25 mm thickness), and one on sample D (3.0 mm thickness). Figure 17 displays measured residual stress data for sample A, with the average of measurements on the top surface and the single measurement on the back surface, along with the expected residual stress for sample A. Figure 18 displays a similar comparison for sample D. Inspection of Figs. 17 and 18 shows that, for both samples A and D, the residual stress measured near the back surface falls close to the expected stress (notwithstanding a sharp increase in measured residual stress within roughly 0.1 mm of the back surface that is consistent with localized tensile stresses created by wire EDM). The similarity is encouraging.

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Figures 17 and 18 include an estimate of the bending stress in the two samples, which is computed from the measured radii of curvature (Table 6) using a strength of materials (SoM) analysis. SoM relates the radius of curvature in beam bending to a linear distribution of stress through the beam thickness. The dashed lines in Figs. 17 and 18 reflect this trend, computed by combining the moment–curvature relation with the typical equation for bending stress, which provideswhere E is the elastic modulus (Table 1), ρ is radius of curvature (Table 6), and y' is distance from the beam mid-thickness. For both samples the slope of the bending stress from SoM is in excellent agreement with the slope of hole-drilling data at the back surface and the expected stress (from FE); this trend holds for both longitudinal and transverse stresses. Figure 17 shows that the slope of the front surface measurement data, at depths from 0.3 to 0.6 mm, does not match the slopes of the other trends (SoM bending stress, expected stress, or the back surface stress). A similar outlying trend of front surface measured stress is not evident for sample D in Fig. 18. We conclude that there may be some systematic error in the measurement technique used for sample A or perhaps in the thin-workpiece residual stress calculation procedure.

The goals of clamping for hole-drilling residual stress measurements are: first, to secure the sample to achieve good precision in cutting; second, to introduce near-zero strain (not flatten existing sample curvature); and third, to allow the sample to deform after cutting each hole depth step. These goals are easily met for the thick plate by applying clamps at locations relatively far (greater than 100 mm) from the site of the strain gage rosette. However, proper clamping of the thin and intermediate thickness samples is more difficult, as their non-flat geometry (Fig. 13) causes conflict between the goals listed above. The cantilever configuration (Fig. 3) has clamps applied within 25 mm of the strain gages, due to the small sample size, and shims are used to reduce clamping strain. This allows clamping with a strain of no more than 7 με.

To evaluate the effect of clamping on hole-drilling measurements in sample A, two additional measurements are performed with the sample firmly clamped, as shown in Fig. 19. Measurements are located either at the main sites shown in Fig. 2 or mid-way between those sites. Figure 20 compares residual stress versus depth data for measurements with the normal cantilever clamping (normal, Fig. 3) and with firm clamping. For shallow depth (< 0.2 mm), the firmly-clamped data agree closely with the expected stress, including the magnitude and depth of maximum compressive stress. At larger depths (0.2 to 0.4 mm) the two measurements diverge from one another.

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For purposes of comparison, it is useful to add the residual stresses from the sample in the normal cantilever clamping configuration to the additional stress introduced to the sample by firm clamping. This “corrected” residual stress (from the normal clamping measurements) can be compared to the residual stress from the firmly-clamped measurements to determine whether the agreement of the firmly-clamped measurements with the expected residual stress is solely due to the addition of clamping stress. The clamping stresses are estimated as bending stresses, with the surface stress computed from strains measured after clamping. Clamping strains in the second firm-clamping measurement are as follows: -966 με in gage 1 (along the longitudinal direction), -578 με in gage 2 (45°), and -232 με in gage 3 (90°). Surface stresses calculated from these strains using planar stress–strain relations give σ_xx of -83 MPa and σ_yy of -44 MPa. The corrected normal cantilever clamping data (Fig. 21) show similar maximum compressive stress to the firmly-clamped data, and the two datasets behave similarly past the depth of maximum compressive stress. However, the firmly-clamped data agree more closely with the expected stress than the corrected normal cantilever clamping data at shallow depths. This suggests that the difference between the residual stress profiles measured with normal cantilever clamping and firm clamping is not solely due to the introduction of clamping strains. While there is improved agreement between the firmly-clamped measurements and the expected stress, the firm clamping conflicts with good practices described by Schajer [1] and NPL [3]. It would be useful to assess the effects of clamping for thin samples in further work.

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To further probe the relationship between sample thickness and measured residual stress, an additional set of hole-drilling experiments is performed. The main set of experiments address the influence of W/D by varying W (thickness) while keeping D (gage circle diameter) constant at 5.13 mm. The set of additional measurements varies the gage circle diameter, with measurements in two samples. In sample A (1.25 mm thickness), one additional measurement is performed using a strain gage rosette with a gage circle diameter of 2.57 mm. In sample D (3.0 mm thickness), two additional measurements are performed, one each with gage circle diameters of 2.57 mm and 10.26 mm. These complement the prior measurements with a gage circle diameter of 5.13 mm. The schedule of depth steps used for the additional measurements are in Table 8 (2.57 mm diameter) and Table 9 (10.26 mm). The hole diameter is 1 mm and 4 mm for the small and large size gages respectively. Varying the gage circle diameter on a single sample thickness changes the value of W/D and therefore changes the residual stress calculation procedure and, possibly, the sample thickness designation. For example, the original measurement on sample A has W/D = 0.244, which is designated as “thin”; however, the follow-on measurement with gage circle diameter 2.57 mm has W/D = 0.486, which is designated as “intermediate”.

Data from the additional set of measurements underscore how W/D affects strain release and the residual stress calculation. Figure 6(b) compares longitudinal strain versus hole depth for measurements on sample D with gage circle diameters D = 2.57 mm (W/D = 1.17), D = 5.13 mm (W/D = 0.585), and D = 10.26 mm (W/D = 0.292). The strains depend strongly on gage circle diameter, with smaller gages (larger W/D) experiencing larger strains. Figure 22 compares residual stress data for measurements in sample A (for two values of W/D) while Fig. 23 compares the data for sample D (for three values of W/D). For both samples (A and D), the residual stress data are comparable at all depths regardless of gage circle diameter, despite the obvious differences in measured strain (Fig. 6(b)). Some differences are observed in measured residual stress values using various gage circle diameters at the first depth step (0.0127 mm or 0.0254 mm, depending on the gage circle diameter used). However, we note that even for measurements using the same sample and gage circle diameter, the standard deviation of the measured residual stress is largest at the first depth step, as shown by the error bars in Figs. 22 and 23. Shortly after the first depth step (at a depth of approximately 0.05 mm), the trend of measured residual stress data for all gage circle diameters begin to match more closely, and the trend remains similar through the full depth of the measurement. This suggests that the new thickness-dependent residual stress calculation procedure produces results that are independent of gage circle diameter, thereby validating the new thickness-dependent residual stress calculation procedure.

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