**3. Advancements in residual stress calculation**

After acquisition of the relaxed strain values, the residual stresses need to be evaluated. First of all, it is necessary to choose between blind-hole and through-hole calculation and between uniform or non-uniform stress distribution along the depth. When the residual stress value is uniform along the depth, the ASTM E837-13a standard specifies that the formulas described in Section 8 of the standard be used. The standard suggests assuming the stress to be uniform only when prior information about the expected stress field is available or if a representative size of the magnitude of the residual stress is required.

Typical applications generally exhibit a non-uniform state of residual stresses. In this case, it is necessary to follow the instructions reported in Section 9 of ASTM E837-13a for the calculation of residual stresses along the depth. The standard provides the calibration matrices, derived by the integral method, that must be multiplied with the acquired strains to derive the stress values. The matrix coefficients, corresponding to each calculation depth, are dimensionless and almost independent of the material [1].

For the sake of simplicity, only the equation for the calculation of the combination stress P is shown below.

$$\left(\overline{a}^T \overline{a} + a\_P \overline{c}^T \overline{c}\right) \cdot P = \frac{E}{1+\nu} \overline{a}^T p \tag{1}$$

It is important to point out that dependence of the stresses on the Poisson's ratio, as shown in Eq. 1, is simplified [13].

All the coefficients reported in the calibration matrices A and B, for a nonuniform stress field, are strictly related to the nominal hole diameter (DN) of 2 mm. If the diameter of the drilled hole (D0) differs from the nominal value, each matrix coefficient reported in the standard needs to be corrected using Eq. 2 reported below:

$$
\overline{a}\_{j,kNEW} = \left(\frac{D\_0}{D\_N}\right)^2 \overline{a}\_{j,k} \tag{2}
$$

The dependency of the coefficients of the calibration matrix on the drilled hole diameter, as expressed in Eq. 2, is approximated as explained by Alegre et al. [7]. For example, using a rosette with a strain gage circle diameter (D) of 5.13 mm, the nominal hole diameter (DN) is equal to 2 mm and the allowed diameter of the drilled hole (D0) can vary from 1.88 to 2.12 mm. The ratio ð Þ *D*0*=DN* 2 ranges between 0.88 and 1.12.

Recently, some developments have been carried out to overcome the limits of the ASTM E837 standard previously described and to take into account other parameters affecting the results that are not considered in the standard.

Beghini et al. [8, 9] of the University of Pisa introduced a generalized integral method based on the analytical definition of influence functions. The method is substantially an evolution of the Integral Method and it also overcomes the limitation of the ASTM E837 standard regarding the maximum allowable value of

allows an accuracy of few microns (**Figure 1a**). An automatic system allows a higher number of drilling steps, a uniform feed rate and a fixed stabilization time. Moreover, it can be controlled remotely, minimizing operator presence near the drilling unit [2]; this is particularly important in the case of hole-drilling measure-

*Hole drilling measurements: typical automatic measuring device (MTS3000-Restan system—SINT*

*(a) LVDT sensor installed on the drilling device and (b) orbital drilling slide.*

*New Challenges in Residual Stress Measurements and Evaluation*

To obtain accurate measurements, it is very important to establish the point that corresponds to the "zero" cutter depth. The standard identifies it as the point at which the end mill begins to lightly scratch the surface of the workpiece, during slow advance drilling. It is clear that the quality of the results of this process greatly depends on the skill of the operator who carries out the measurements and may not

In the case of tests on conductive materials, it is possible to use the electrical contact technique that identifies the contact when the electrical connection occurs

ments on polymeric or composite materials.

be very accurate.

**62**

**Figure 2.**

**Figure 1.**

*Technology).*


#### **Table 1.**

*Comparison between the limitations of the ASTM E837-13a method and those of the generalized integral method based on the influence functions.*

Eqs. (3)–(5) are applicable only in the case of concentric holes, where it is

*(a) Two calibration matrices a and b for the ASTM E837 test method. (b) Calibration matrix A, composed of*

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

by separating stress and relieved strain in equibiaxial and shear components. Therefore, the problem will be solved using the Cartesian reference system of the

> *<sup>A</sup>* <sup>þ</sup> *<sup>α</sup>CTC* � *<sup>S</sup>* <sup>¼</sup> *EA<sup>T</sup>*

> > *<sup>x</sup>* , *σ*ð Þ*<sup>k</sup> <sup>y</sup>* , *τ* ð Þ*k*

*AT*

*xy <sup>T</sup>*

ð Þ*k* <sup>1</sup> , *ε* ð Þ*k* <sup>2</sup> , *ε* ð Þ*k* 3

*<sup>T</sup>*

Beghini et al. [8, 9] and, more recently, Barsanti et al. [13] extended the Integral Method by including a correction for the eccentricity of the hole with respect to the

For this general problem, no symmetry can be used and no advantage is obtained

The relationship between the strain and the stress can be re-written as reported

*S* and *e* have different sizes compared to the standard ASTM E837 approach. In this different formulation, the two vectors are defined using a 3 *k* � 1 arrangement

The matrix *A* has a size of 3 k � 3 k and implicitly includes the dependency of the calculated stress on the hole diameter, type and dimension of the strain gage rosette (including the gage circle diameter, length and width of the strain grids), Poisson's ratio, eccentricity and, if applicable, hole bottom chamfer and thickness of

The equations reported above include also the Tikhonov regularization, as in the

As for the strain and the stress components, also the new matrix of calibration

coefficients is defined for blocks of 3 � 3 elements as reported below:

*e* (6)

is the vector of the stress compo-

is the vector of strain reading; *A* **=**

possible to decouple the strain components.

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

*9 blocks, for the generalized integral method.*

strain-gage rosette.

where **<sup>S</sup>** <sup>¼</sup> *<sup>σ</sup>*ð Þ<sup>1</sup>

ð Þ1 <sup>1</sup> , *ε* ð Þ1 <sup>2</sup> , *ε* ð Þ1 <sup>3</sup> , … , *ε*

nents; *e* ¼ *ε*

the workpiece.

**65**

ASTM E837 standard.

*<sup>x</sup>* , *σ*ð Þ<sup>1</sup> *<sup>y</sup>* , *τ* ð Þ1 *xy* , … , *σ*ð Þ*<sup>k</sup>*

with a block structure of 3-elements.

generalized matrix of calibration coefficients.

rosette.

**Figure 3.**

below:

eccentricity. Using the Influence functions approach, it is also possible to include the real dependency of the Poisson's ratio and the diameter of the drilled hole on the calculated stress. In detail, the proposed methodology is based on analytical influence functions relating the measured relieved strains to the residual stress by means of integral equations. By processing the results of accurate finite element simulations, continuous analytical influence functions are produced.

The generalized integral method is more universal compared to the ASTM E837 standard and is currently the most suitable to include the influence parameters not considered in the standard and therefore to overcome its limitations. With this calculation method, it is possible to take into account other influence parameters, such as hole bottom chamfer and intermediate thickness.

**Table 1** provides a comparison of the limitations of the ASTM E837 standard and of the generalized integral method based on the influence functions.

### **4. Generalized integral method**

The ASTM E837 standard (Section 9.3) uses the integral method, including the Tikhonov regularization, to calculate non-uniform residual stresses. The residual stresses for each hole depth *j* are computed by solving the following matrix equations:

$$\left(\overline{\mathfrak{a}}^T \overline{\mathfrak{a}} + \alpha\_P \mathbf{c}^T \mathbf{c}\right) \cdot \mathbf{P} = \frac{E}{1+\nu} \mathbf{p} \tag{3}$$

$$\left(\overline{\mathbf{b}}^T \overline{\mathbf{b}} + a\_Q \mathbf{c}^T \mathbf{c}\right) \cdot \mathbf{Q} = E \mathbf{q} \tag{4}$$

$$\left(\overline{\mathbf{b}}^T \overline{\mathbf{b}} + \alpha\_T \mathbf{c}^T \mathbf{c}\right) \cdot \mathbf{T} = \mathbf{E}t \tag{5}$$

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

#### **Figure 3.**

eccentricity. Using the Influence functions approach, it is also possible to include the real dependency of the Poisson's ratio and the diameter of the drilled hole on the calculated stress. In detail, the proposed methodology is based on analytical influence functions relating the measured relieved strains to the residual stress by means of integral equations. By processing the results of accurate finite element simula-

*Comparison between the limitations of the ASTM E837-13a method and those of the generalized integral*

The generalized integral method is more universal compared to the ASTM E837 standard and is currently the most suitable to include the influence parameters not considered in the standard and therefore to overcome its limitations. With this calculation method, it is possible to take into account other influence parameters,

**Table 1** provides a comparison of the limitations of the ASTM E837 standard and

The ASTM E837 standard (Section 9.3) uses the integral method, including the Tikhonov regularization, to calculate non-uniform residual stresses. The residual stresses for each hole depth *j* are computed by solving the following matrix

*<sup>a</sup><sup>T</sup><sup>a</sup>* <sup>þ</sup> *<sup>α</sup>Pc<sup>T</sup><sup>c</sup>* � *<sup>P</sup>* <sup>¼</sup> *<sup>E</sup>*

*<sup>b</sup>* <sup>þ</sup> *<sup>α</sup><sup>Q</sup> <sup>c</sup><sup>T</sup><sup>c</sup>* 

*<sup>b</sup>* <sup>þ</sup> *<sup>α</sup>Tc<sup>T</sup><sup>c</sup>* 

*b T*

> *b T*

1 þ ν

*p* (3)

**Generalized integral method, based on the influence functions [8, 9]**

Not applicable if outside the range

Variable. Strain gage circle diameter (D), grid length (GL) and width (GW) can be used as input parameters

No limitation

0.25–0.45

No limitation

No limitation

geometry)

Not considered Considered (for a specific end mill

� *Q* ¼ *Eq* (4)

� *T* ¼ *Et* (5)

tions, continuous analytical influence functions are produced.

**ASTM E837 for non-uniform stress field**

differs from the nominal value used for the evaluation of the calibration coefficients

differs from the value used for the evaluation of the calculation coefficients

(A, B and C) reported in the standard.

Eccentricity correction is not included. Is considered acceptable if lower than

>1.0D thin (uniform/non-uniform stress)

<0.2D thick (uniform stress)

such as hole bottom chamfer and intermediate thickness.

**4. Generalized integral method**

**Parameters Limitations**

Hole eccentricity

radius

**Table 1.**

Workpiece thickness

Hole-bottom chamfer

**calculation [1]**

Hole diameter Approximated correction if hole diameter

*New Challenges in Residual Stress Measurements and Evaluation*

Poisson's ratio Approximated correction if Poisson's ratio

Rosette geometry Fixed only for the strain gage rosette

0.004D

*method based on the influence functions.*

equations:

**64**

of the generalized integral method based on the influence functions.

*(a) Two calibration matrices a and b for the ASTM E837 test method. (b) Calibration matrix A, composed of 9 blocks, for the generalized integral method.*

Eqs. (3)–(5) are applicable only in the case of concentric holes, where it is possible to decouple the strain components.

Beghini et al. [8, 9] and, more recently, Barsanti et al. [13] extended the Integral Method by including a correction for the eccentricity of the hole with respect to the strain-gage rosette.

For this general problem, no symmetry can be used and no advantage is obtained by separating stress and relieved strain in equibiaxial and shear components. Therefore, the problem will be solved using the Cartesian reference system of the rosette.

The relationship between the strain and the stress can be re-written as reported below:

$$\left(\overline{\mathbf{A}}^T \overline{\mathbf{A}} + a \mathbf{C}^T \mathbf{C}\right) \cdot \mathbf{S} = \mathbf{E} \overline{\mathbf{A}}^T \mathbf{e} \tag{6}$$

where **<sup>S</sup>** <sup>¼</sup> *<sup>σ</sup>*ð Þ<sup>1</sup> *<sup>x</sup>* , *σ*ð Þ<sup>1</sup> *<sup>y</sup>* , *τ* ð Þ1 *xy* , … , *σ*ð Þ*<sup>k</sup> <sup>x</sup>* , *σ*ð Þ*<sup>k</sup> <sup>y</sup>* , *τ* ð Þ*k xy <sup>T</sup>* is the vector of the stress components; *e* ¼ *ε* ð Þ1 <sup>1</sup> , *ε* ð Þ1 <sup>2</sup> , *ε* ð Þ1 <sup>3</sup> , … , *ε* ð Þ*k* <sup>1</sup> , *ε* ð Þ*k* <sup>2</sup> , *ε* ð Þ*k* 3 *<sup>T</sup>* is the vector of strain reading; *A* **=** generalized matrix of calibration coefficients.

*S* and *e* have different sizes compared to the standard ASTM E837 approach. In this different formulation, the two vectors are defined using a 3 *k* � 1 arrangement with a block structure of 3-elements.

The matrix *A* has a size of 3 k � 3 k and implicitly includes the dependency of the calculated stress on the hole diameter, type and dimension of the strain gage rosette (including the gage circle diameter, length and width of the strain grids), Poisson's ratio, eccentricity and, if applicable, hole bottom chamfer and thickness of the workpiece.

The equations reported above include also the Tikhonov regularization, as in the ASTM E837 standard.

As for the strain and the stress components, also the new matrix of calibration coefficients is defined for blocks of 3 � 3 elements as reported below:

A ¼ Að Þ <sup>11</sup> <sup>11</sup> <sup>A</sup>ð Þ <sup>11</sup> <sup>12</sup> <sup>A</sup>ð Þ <sup>11</sup> <sup>13</sup> 000 Að Þ <sup>11</sup> <sup>21</sup> <sup>A</sup>ð Þ <sup>11</sup> <sup>22</sup> <sup>A</sup>ð Þ <sup>11</sup> <sup>23</sup> ⋯ ⋯ 000 Að Þ <sup>11</sup> <sup>31</sup> <sup>A</sup>ð Þ <sup>11</sup> <sup>32</sup> <sup>A</sup>ð Þ <sup>11</sup> <sup>33</sup> 000 ⋮ ⋱ *:* ⋮ ⋮ *:* ⋱ ⋮ Að Þ k1 <sup>11</sup> <sup>A</sup>ð Þ k1 <sup>12</sup> <sup>A</sup>ð Þ k1 <sup>13</sup> <sup>A</sup>ð Þ kk <sup>11</sup> <sup>A</sup>ð Þ kk <sup>12</sup> <sup>A</sup>ð Þ kk 13 Að Þ k1 <sup>21</sup> <sup>A</sup>ð Þ k1 <sup>22</sup> <sup>A</sup>ð Þ k1 <sup>23</sup> ⋯ ⋯ <sup>A</sup>ð Þ kk <sup>21</sup> <sup>A</sup>ð Þ kk <sup>22</sup> <sup>A</sup>ð Þ kk 23 Að Þ k1 <sup>31</sup> <sup>A</sup>ð Þ k1 <sup>32</sup> <sup>A</sup>ð Þ k1 <sup>33</sup> <sup>A</sup>ð Þ kk <sup>31</sup> <sup>A</sup>ð Þ kk <sup>32</sup> <sup>A</sup>ð Þ kk 33 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (7)

**6. Eccentricity error: description and possible corrections**

rosette (approx. D = 2.56 mm).

the first and third quadrant (0°, 90°, 225°).

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

easily obtained by image analysis techniques.

configurations or using correction algorithms.

**Figure 4.**

**67**

*Eccentricity in hole-drilling measurements.*

The eccentricity between the drilled hole and the strain gage circle greatly influences the strain measurements. The ASTM standard requires a near perfect concentricity between the drilled hole and the rosette and prescribes an allowable eccentricity value that depends on the dimension of the strain gage rosette (0.004D). Using a standard rosette with a gage circle diameter D = 5.13 mm, the maximum allowable eccentricity is 0.02 mm. This limit increases (0.04 mm) or decreases (0.01 mm) using bigger (approx. D = 10.26 mm) or smaller types of

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

As shown in **Figure 4**, the eccentricity error is influenced by the eccentricity coordinates ex and ey and by the parameters of the strain gage rosette (D, GL and GW). The type B rosette generally shows a higher sensitivity to eccentricity errors compared to type A rosettes; this can be explained by the orientation of the gage grids, which are concentrated only in the first quadrant (0°, 45°, 90°), instead of in

For these reasons, the correction of eccentricity errors requires accurate determination of the position of the drilled hole in the reference system of the strain gage rosette; eccentricity can be measured by a special procedure using the drilling system microscope. Using a digital microscope, the eccentricity coordinates can be

As shown in **Figure 5**, the sensitivity of the grids is directly influenced by the hole eccentricity. When the eccentricity has the same direction as the grid, if the hole is closer to the grid the absolute value of the relaxed strain is greater. On the contrary, if the eccentricity has a transverse direction with respect to the grid, then a portion of the grid has a greater sensitivity, while the other portion has a lower sensitivity; this implies, by symmetry, that the error is almost compensated [13]. The eccentricity correction can be done using strain gage rosettes with special

The studies of Beghini et al. [10] and of Nau et al. [11] introduced the correction of eccentricity using a special six-grid rosette and an eight-grid rosette respectively. Both the rosettes are produced by HBM and make it possible to compensate the first-order of eccentricity error. However, the corrections based on special strain gage geometries do not correct the higher order eccentricity errors (higher than

**Figure 3** compares the visual interpretation of the calibration matrices of the ASTM standard and that of the new generalized calibration matrix.
