**9. Local plasticity limitation: description and possible correction**

The ASTM E837 standard reports that satisfactory results can be achieved when measured residual stresses do not exceed about 80% of the yield stress in the case of "thick" workpieces and 50% in the case of "thin" workpieces.

The previous parameters and FE results are used for the evaluation of the elastically evaluated plasticity factor *fel*, which is expressed through bivariate

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

Nobre et al. [21] provide a similar approach for the estimation of the plasticity

Beghini et al. [22] propose a special 4-grid strain gage rosette for the correction of the plasticity effect, which is available on the market (HBM). The correction is valid for the standard 3-grid rosettes only if the perpendicular grids are oriented

The evaluation of uncertainties associated with measurement of residual stresses by the hole-drilling method is a topic that has been little investigated. The evaluation, mainly in the case of non-uniform stress fields, involves a large number of parameters from different sources, including the properties of the materials under

Standard ASTM E837-13a [1] contains only some basic information about precision and bias associated with the hole-drilling measurement method, mainly in the case of uniform stress calculation. In fact, the standard states that the bias associated with a residual stress measurement by the hole-drilling method is less than �10% when dealing with uniform residual stresses. Based on the results of round-robin test programs, the precision (random error) is such as to give a standard deviation of �14 MPa for AISI 1018 carbon steels and a standard deviation of �12 MPa for type AISI 304 stainless steels. The standard also reports that the uncertainties in the case of non-uniform stress measurements are expected to be much larger than for

One of the first papers on the subject of evaluation of uncertainty was published by Oettel [25] (UNCERT Code of Practice 15). The work proposes an approach for the evaluation of hole-drilling uncertainty in the case of uniform stress fields and takes into account typical errors in the determination of material properties, errors in the measurement of acquired strains, the hole diameter and the influence of calculation coefficients. The code of practice can be applied only to uniform residual stress calculation equations based on ASTM E837-95 and cannot be used with the

Scafidi et al. [26] further developed this methodology by applying it to the recent version of the average uniform stress calculation and considering additional

Regarding evaluation of uncertainty in the case of non-uniform stresses, the first approach was provided by Schajer et al. [24] based on the Integral method. They consider a number of input estimates including the properties of materials (i.e. Young's modulus), strain readings, hole diameter and hole depths. The uncertainty components have statistical normal distributions with zero mean and are indepen-

The uncertainties of the measured strains are considered as an input, but fixed

factor f. The material characteristics are taken into account by measuring the variation of Vickers hardness, which estimates the material strain hardening due to

Plasticity generates a non-linearity on strain measurements.

testing, the strain readings and the hole execution methods.

*<sup>f</sup> el* <sup>¼</sup> *<sup>f</sup>* <sup>þ</sup> *Wf <sup>μ</sup>* (9)

polynomials, as a function of the parameters W and μ

the increase of plastic deformation.

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

in the directions of the principal strains.

**10. Evaluation of uncertainty**

uniform stress measurements.

for each step.

**73**

current version of the ASTM standard [1].

parameters, such as the step-by-step drilling depth.

dent of each other and each one is linearly combined.

The need for these stress limits is explained by the stress concentration generated by the drilled hole. When a hole is drilled on a loaded workpiece, it generates a stress concentration in the area around the hole. The magnitude of the stress concentration depends on several parameters including the diameter of the drilled hole, the load orientation and the distance of the strain gage grids from the hole. If the stress level is high, localized plastic deformation occurs around the hole, which generates larger overall surface strains (**Figure 11**).

The hole-drilling method requires that the strain gage grids be placed really close to the hole. For this reason, if local plasticity occurs, it may be that the strain measured by the gage is the arithmetical sum of the linear elastic strains and the plastic strains.

In any case, "thick" workpieces are less sensitive to the plasticity effect. This is due to the presence of material in the lower part of the blind hole determining a local reinforcement and reducing the stress concentration factor [3]. This explains the higher measurement limit in the case of a blind hole (80% of σY) compared to the case of a through hole (50% of σY).

Few research studies have been published on this topic to provide possible corrections for this error.

The work of Beghini et al. [20] provides a numerical procedure for correcting the effect of local plasticity in the case of a blind hole for uniform stress calculation. To carry out the stress correction, it is necessary that both the yield stress and the stress–strain curve in the plastic region are defined.

The equivalent stress, corrected to take into account the presence of plasticity *σeq*, is evaluated considering the elastic equivalent stress *σeq*,*<sup>i</sup>*, the yield stress *σ<sup>Y</sup>* of the material under testing and the plasticity factor *f* defined as following:

$$f = \frac{\sigma\_{eq} - \sigma\_{eq,i}}{\sigma\_Y - \sigma\_{eq,i}} \tag{8}$$

The correction algorithm obviously considers the geometry of the strain gage rosette and therefore the authors provide the calculation coefficients for several strain gage rosettes available on the market.

**Figure 11.** *Local plasticity areas with low applied loads (left side) or high loads (right side).*

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

The previous parameters and FE results are used for the evaluation of the elastically evaluated plasticity factor *fel*, which is expressed through bivariate polynomials, as a function of the parameters W and μ

$$f\_{el} = f + \mathcal{W}f^{\mu} \tag{9}$$

Nobre et al. [21] provide a similar approach for the estimation of the plasticity factor f. The material characteristics are taken into account by measuring the variation of Vickers hardness, which estimates the material strain hardening due to the increase of plastic deformation.

Plasticity generates a non-linearity on strain measurements.

Beghini et al. [22] propose a special 4-grid strain gage rosette for the correction of the plasticity effect, which is available on the market (HBM). The correction is valid for the standard 3-grid rosettes only if the perpendicular grids are oriented in the directions of the principal strains.
