**11. Experimental validation of the method on a four-point bending rig**

The entire measurement chain and the testing parameters (rotational speed, type of end mill, feed rate and delay time) were verified using a special apparatus, developed by SINT Technology, which applies a known bending stress on a specially designed specimen.

The specimen is a flat rectangular cross section cantilever beam, fixed at one end, and loaded at the other end by means of a pneumatic actuator (**Figure 12**).

The material used for the specimen is aluminum alloy AW7075 T651. The maximum applied bending stress was approximately 25 MPa.

An approximately 1.90 mm hole diameter was drilled with 130 incremental drilling depths of 10 μm up to a 1.30 mm total depth. The rotational speed was approx. 400,000 rpm and the feed rate 0.2 mm/min. To prevent any interaction between the tip and the specimen, the cutter was fully raised for each drilling step. The diameter was accurately measured after drilling, for each test, with the microscope installed on the system and two dial gages, also to determine the residual eccentricity.

The (uniaxial) stress due to bending was easily obtained from the beam theory, **Figure 12b**, Eq. 15:

$$
\sigma\_{\text{Be}} = 6 \frac{Fb}{wh^2} \tag{15}
$$

**Figure 12.**

*(a) Bending test bench to simulate a known reference residual stress and (b) shows the linear distribution of the bending stress and orientation of the strain gage rosette grids and hole eccentricity definition.*

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses DOI: http://dx.doi.org/10.5772/intechopen.90392*

where *b* is the distance between the load axis and the rosette strain gage center, *wh2* is the width and height of the beam cross section, and *F* is the load imposed by the pneumatic actuator.

A known load was used for determining the properties of the material. In fact, the elasticity parameters (Young's modulus E and Poisson's ratio ν) of the material were measured before drilling by applying a preliminary bending load before drilling.

Grid 1 of each strain gage should be aligned with the beam axis. The manual strain gage installation unavoidably introduces a misalignment. However, the angle between grid 1 and the beam axis can be found from Eq. 16 (accurate approximation for small values of γ):

$$\gamma = \frac{1}{2} \cdot \frac{\varepsilon\_1^F(\mathbf{0}) - 2\varepsilon\_2^F(\mathbf{0}) + \varepsilon\_3^F(\mathbf{0})}{\varepsilon\_1^F(\mathbf{0}) - \varepsilon\_3^F(\mathbf{0})} \tag{16}$$

The measured strains need to be decoupled in order to deduce the relaxed strain due to the bending stress. The relaxed strains due to the residual stresses and the relaxed strains due to the bending stresses are obtained as:

$$\begin{aligned} \boldsymbol{\varepsilon}\_i^{\rm RS}(\mathbf{z}\_j) &= \boldsymbol{\varepsilon}\_i(\mathbf{z}\_j) \\ \boldsymbol{\varepsilon}\_i^{\rm B\epsilon}(\mathbf{z}\_j) &= \boldsymbol{\varepsilon}\_i^F(\mathbf{z}\_j) - \boldsymbol{\varepsilon}\_i(\mathbf{z}\_j) - \boldsymbol{\varepsilon}\_1^F(\mathbf{0}) \end{aligned} \tag{17}$$

Strain *ε<sup>F</sup> <sup>i</sup>* ð Þ 0 needs to be subtracted in the second member of Eq. 17 since the relaxed strains are defined as the effect of introduction of the drilled hole, therefore they need to be zero at zero depth. Finally, the experimental data are the bending relaxed strains as a function of hole depth increments *εBe <sup>i</sup> zj* .

### **12. Test results and analysis**

The following testing conditions were adopted during the measurements.


The following parameters were then used for the stress calculation and for the uncertainty evaluation:


relation *f* (sensitivity coefficients) which, in the formula (12), multiply the standard

This calculation procedure, which is implemented in the EVAL 7 software, requires the execution of a high number of stress calculations for the uncertainty

In particular, considering a measurement carried out according to the ASTM standard using 20 calculation depths, the uncertainty evaluation requires the repetition and therefore the combination of the results obtained with 206 different

**11. Experimental validation of the method on a four-point bending rig**

The entire measurement chain and the testing parameters (rotational speed, type of end mill, feed rate and delay time) were verified using a special apparatus, developed by SINT Technology, which applies a known bending stress on a spe-

The specimen is a flat rectangular cross section cantilever beam, fixed at one end, and loaded at the other end by means of a pneumatic actuator (**Figure 12**). The material used for the specimen is aluminum alloy AW7075 T651. The max-

An approximately 1.90 mm hole diameter was drilled with 130 incremental drilling depths of 10 μm up to a 1.30 mm total depth. The rotational speed was approx. 400,000 rpm and the feed rate 0.2 mm/min. To prevent any interaction between the tip and the specimen, the cutter was fully raised for each drilling step. The diameter was accurately measured after drilling, for each test, with the microscope installed on the system and two dial gages, also to determine the residual

The (uniaxial) stress due to bending was easily obtained from the beam theory,

*<sup>σ</sup>Be* <sup>¼</sup> <sup>6</sup> *Fb*

*(a) Bending test bench to simulate a known reference residual stress and (b) shows the linear distribution of the*

*bending stress and orientation of the strain gage rosette grids and hole eccentricity definition.*

*wh*<sup>2</sup> (15)

uncertainty square *u x*ð Þ<sup>i</sup> of each input estimate *x*i.

*New Challenges in Residual Stress Measurements and Evaluation*

evaluation related to hole drilling measurements.

imum applied bending stress was approximately 25 MPa.

stress calculations.

cially designed specimen.

eccentricity.

**Figure 12.**

**76**

**Figure 12b**, Eq. 15:


After performing the drilling tests, the relaxed strains were imported into the EVAL 7 calculation software developed by SINT Technology. The calculation of non-uniform stresses was carried out according to the following two methods: the original ASTM 837-13a standard and the generalized integral method, based on the Influence Functions, by applying the algorithms described in the previous sections and correcting some systematic errors.

The extended features are shown in **Table 4**.

The stresses were calculated considering a distribution of 20 constant steps within 1 mm of depth.

Next, the bending stress distribution was calculated by the generalized integral method and then compared with the expected bending stress distribution.

Finally, based on the calculated stress curves, the uncertainty of measurement was evaluated considering the input quantities reported above (Section 12). The measurement uncertainties are expressed as standard uncertainties multiplied by a coverage factor equal to 2 (which in the case of normal distribution corresponds to a confidence level of about 95%).

**Figure 13** compares the expected bending stresses with the stress components σx, σ<sup>y</sup> and τxy, calculated from the interpolated relaxed strains with their associated uncertainty.

The purpose of the authors is to highlight the importance of the correction of each source of error, which is not contemplated in the ASTM E837 calculation. This has been achieved by showing the effects on the calculated stresses in the event that those corrections are not considered. For this reason, one by one, all the corrections have been deselected from the generalized integral method with all the active corrections.

**Figure 14** shows the percentage error on σBe when the generalized integral method is not applied, due to the hole eccentricity, the combination of Poisson's ratio and the hole diameter, and the geometry of the strain gage rosette.

Regarding eccentricity, a maximum error of approximately 1.5% is committed, in the area closest to the surface. It is necessary to highlight that the eccentricity radius error, that affects this data, is similar to the maximum value tolerated by the ASTM 837-13a standard. In some real cases, due to the inexperience of the operator


or to non-standard test conditions, the eccentricity radius can be higher than the limitation reported by the standard and, therefore, correction of eccentricity is

Regarding the influence of Poisson's ratio and the hole diameter, the maximum deviation is around 8%. Indeed, the calibration constants are not

essential for an accurate evaluation of residual stresses.

*Percentage error on σBe when the generalized integral method is not applied.*

*Comparison between expected bending stresses and the calculated stress components.*

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses*

*DOI: http://dx.doi.org/10.5772/intechopen.90392*

**Figure 13.**

**Figure 14.**

**79**

#### **Table 4.**

*Comparison between the ASTM E837 & generalized integral method, based on the influence functions.*

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses DOI: http://dx.doi.org/10.5772/intechopen.90392*

**Figure 13.** *Comparison between expected bending stresses and the calculated stress components.*

**Figure 14.** *Percentage error on σBe when the generalized integral method is not applied.*

or to non-standard test conditions, the eccentricity radius can be higher than the limitation reported by the standard and, therefore, correction of eccentricity is essential for an accurate evaluation of residual stresses.

Regarding the influence of Poisson's ratio and the hole diameter, the maximum deviation is around 8%. Indeed, the calibration constants are not

• Zero-depth uncertainty (*z0*): 0.005 mm

and correcting some systematic errors.

within 1 mm of depth.

uncertainty.

corrections.

**Table 4.**

**78**

confidence level of about 95%).

• Depth measurement uncertainty (*z*): 0.01 mm

*New Challenges in Residual Stress Measurements and Evaluation*

The extended features are shown in **Table 4**.

After performing the drilling tests, the relaxed strains were imported into the EVAL 7 calculation software developed by SINT Technology. The calculation of non-uniform stresses was carried out according to the following two methods: the original ASTM 837-13a standard and the generalized integral method, based on the Influence Functions, by applying the algorithms described in the previous sections

The stresses were calculated considering a distribution of 20 constant steps

method and then compared with the expected bending stress distribution.

Next, the bending stress distribution was calculated by the generalized integral

Finally, based on the calculated stress curves, the uncertainty of measurement was evaluated considering the input quantities reported above (Section 12). The measurement uncertainties are expressed as standard uncertainties multiplied by a coverage factor equal to 2 (which in the case of normal distribution corresponds to a

**Figure 13** compares the expected bending stresses with the stress components σx, σ<sup>y</sup> and τxy, calculated from the interpolated relaxed strains with their associated

The purpose of the authors is to highlight the importance of the correction of each source of error, which is not contemplated in the ASTM E837 calculation. This has been achieved by showing the effects on the calculated stresses in the event that those corrections are not considered. For this reason, one by one, all the corrections have been deselected from the generalized integral method with all the active

**Figure 14** shows the percentage error on σBe when the generalized integral method is not applied, due to the hole eccentricity, the combination of Poisson's

Regarding eccentricity, a maximum error of approximately 1.5% is committed, in the area closest to the surface. It is necessary to highlight that the eccentricity radius error, that affects this data, is similar to the maximum value tolerated by the ASTM 837-13a standard. In some real cases, due to the inexperience of the operator

listed in the standard

*Comparison between the ASTM E837 & generalized integral method, based on the influence functions.*

**ASTM E837-13a Generalized integral method,**

market

**based on the influence functions**

To any rosette available in the

ratio and the hole diameter, and the geometry of the strain gage rosette.

Eccentricity correction Not available Available

Poisson's ratio correction Approximate Complete Hole diameter correction Approximate Complete Hole-bottom chamfer correction Not available Available Intermediate thickness extension Not available Available Tikhonov regularization Available Available

**Features Comparison**

Applicability to strain gage rosettes Only to rosettes

expressed as a function of Poisson's ratio and the diameter of the measured hole: only the approximate correction is provided. In this case, both the measured diameter (D = 1.88 mm) and the Poisson's ratio considered (n = 0.33 mm) are far from those used to generate the calibration matrices reported in the standard.

All the features reported above have been introduced in dedicated software for the evaluation of residual stresses and related uncertainty. Finally an experimental

*Recent Advancements in the Hole-Drilling Strain-Gage Method for Determining Residual Stresses*

σ<sup>x</sup> residual stress normal component in the x direction [MPa] σ<sup>y</sup> residual stress normal component in the y direction [MPa] τxy residual stress shear component in the xy plane [MPa]

test, performed on a 4-point bending test rig, is described.

σMIN minimum principal residual stress [MPa] σMAX maximum principal residual stress [MPa]

σ<sup>Y</sup> yield stress of the testing material [MPa] D diameter of the strain gage circle [mm] D0 diameter of the drilled hole [mm]

DN ASTM E837 nominal hole diameter [mm]

ε1, ε2, ε<sup>3</sup> strains acquired from the strain gage rosette [μm/m]

a, b calibration constants used in the calculation of uniform stress

**a**, **b** calibration matrix constants used in the calculation of non-uniform

σeq,i equivalent residual stress producing the onset of plasticity in the 2D

ð Þ *ΔV=V <sup>j</sup>*,*<sup>x</sup>* electrical output reading for each grid x (x = 1,2,3) and for each

*uC*ð Þ*y* combined standard uncertainty for measurement result

β principal angle [rad]

*DOI: http://dx.doi.org/10.5772/intechopen.90392*

GL grid length [mm] GW grid width [mm]

ν Poisson's ratio

p, q, t combination strain [μm/m] P, Q, T combination stress [MPa] E Young's modulus [MPa]

s workpiece thickness [mm] n number of acquisition steps j number of hole depth steps

stress

case

zj depth increment j

**81**

k sequence number for hole depth steps

aj,k, bj,k calibration matrix for isotropic and shear stresses

f,f el plasticity factor calculated in plastic and elastic field

z0 depth error during the zero-depth determination Kx gage factors of the strain gage for each grid x (x = 1,2,3)

*A* generalized matrix of calibration coefficients **C** generalized Tikhonov regularization matrix ex, ey eccentricity component of the x and y directions σeq,el equivalent stress [MPa] corrected for plasticity effect

W, μ coefficients for the plasticity correction

depth increment j y measurement result (output estimate)

**c** Tikhonov regularization matrix **αP**, **αQ**,**α<sup>T</sup>** Tikhonov regularization factors

**S** vector of the stress components **e** vector of the strain components

**List of symbols**

Finally, regarding the influence of the geometry of the strain gage rosette, the maximum deviation is approximately 2%. It represents the error due to use of a rosette that is different from the geometry envisaged in the standard. In this case, the rosette that was used is very similar to type B of the standard. In other cases, the errors may be higher.
