Erratum: On the precision and accuracy of the acoustic birefringence technique for stress evaluation

In the article “On the precision and accuracy of the acoustic birefringence technique for stress evaluation”, with the DOI code number: https://doi.org/10.1590/1517-7076-RMAT-2022-0146, published at Matéria (Rio de Janeiro) 27(3):e20220146:

On page 2 where it was written:

These statistical concepts and their use here are briefly reviewed in the following subsections [4].

2.1.1. Statistical errors: basic concepts

Consider a sample of N direct measurements of a variable x {x1, x2, …, xN }, the standard estimate for the expected result of the measurement is […]

If the number of measurements N is not sufficiently large (which is usually the case at the experiments carried out at IEN’s ultrasonic laboratory), […]

It should read:

These statistical concepts and their use here are briefly reviewed in the following subsections [4, 5].

2.1.1. Statistical errors: basic concepts

The result of the measurement of a quantity should be given by the best estimate of its expected value and the uncertainty associated with this estimate. The best estimate for the expected value of a quantity from a sample of N direct measurements of a variable x {x1, x2, …, xN }, is the mean or average value of these measurements, and the uncertainty associated with this estimate is the standard deviation of this mean or average value.

Thus, the result of the measurement of a quantity is defined by, […]

If the number of direct measurements N is not sufficiently large for its standard deviation to be an adequate estimate of the standard deviation of the totality of the possible measurements (which is usually the case at the experiments carried out at IEN’s ultrasonic laboratory), […]

On page 3, equation 8 where it was written:

$${\sigma }_{\overline{x}}=\frac{s_x}{\sqrt{N}}=\sqrt{{\displaystyle {\sum }_{i=1}^N\frac{{\left(x_i\rightleftarrows \overline{x}\right)}^2}{N\left(N-1\right)}}}$$

It should read:

$${\sigma }_{\overline{x}}=\frac{s_x}{\sqrt{N}}=\sqrt{{\displaystyle {\sum }_{i=1}^N\frac{{(x_i-\text{}\overline{x})}^2}{N\left(N-1\right)}}}$$

On page 4 where it was written:

(see subsection 5.1). Thus, […]

It should read:

(see subsection 2.3.1). Thus, […]

On page 4, equations 20 where it was written:

$$\varphi =f(X)=f(\overline{X})+{\displaystyle \sum _{k=1}^M{\left(\frac{\partial f}{\partial x_k}\right)}_{\overline{X}}(x_k-}{\overline{x}}_k)$$

It should read:

$$\text{Φ}=f(X)=f(\overline{X})+{\displaystyle \sum _{k=1}^M{\left(\frac{\partial f}{\partial x_k}\right)}_{\overline{X}}(x_k-}{\overline{x}}_k)$$

On page 4, equations 21 where it was written:

$$\overline{\varphi }=f(\overline{X})$$

It should read:

$$\overline{\text{Φ}}=f(\overline{X})$$

On page 4, equations 22 where it was written:

$${\sigma }_{\varphi }=\sqrt{{\displaystyle \sum _{k,l}^M{\left(\frac{\partial f}{\partial x_k}\right)}_{\overline{X}}{V}_{kl}{\left(\frac{\partial f}{\partial x_l}\right)}_{\overline{X}}}}$$

It should read:

$${\sigma }_{\text{Φ}}=\sqrt{{\displaystyle \sum _{k,l}^M{\left(\frac{\partial f}{\partial x_k}\right)}_{\overline{X}}{V}_{kl}{\left(\frac{\partial f}{\partial x_l}\right)}_{\overline{X}}}}$$

On page 4, equations 24 where it was written:

$${\sigma }_{\overline{\varphi }}=\frac{{\sigma }_{\varphi }}{\sqrt{N}}$$

It should read:

$${\sigma }_{\overline{\text{Φ}}}=\frac{{\sigma }_{\text{Φ}}}{\sqrt{N}}$$

On page 5 where it was written:

$$\overline{B}=f({\overline{t}}_{31},{\overline{t}}_{32})=2\left(\frac{{\overline{t}}_{31}-{\overline{t}}_{32}}{{\overline{t}}_{31}+{\overline{t}}_{32}}\right)$$

It should read:

$$\overline{B}=f({\overline{t}}_{31},{\overline{t}}_{32})=-2\left(\frac{{\overline{t}}_{31}-{\overline{t}}_{32}}{{\overline{t}}_{31}+{\overline{t}}_{32}}\right)$$

On page 5 where it was written:

The experimental precision is characterized by relative error given by the ratio between the standard deviation of the mean (Eq. 8) and the absolute value of the mean (Eq. 4), whereas the experimental accuracy is defined by the ratio between the standard deviation of the mean (Eq. 8) and the […]

It should read:

The experimental precision is characterized by relative error given by the ratio between the standard deviation of the mean (Eq. 8) and the absolute value of the mean (Eq. 4), whereas the experimental accuracy is the relative error obtained by dividing the absolute value of the difference between the principal stress difference (Eq. 27) and the reference value Xref by the […]

On page 5 where it was written:

In the continuous technique, an initial direction aligned with one of the symmetry axes is chosen (say direction 1) and kept fixed while a sequence of N (usually five to ten) ultrasonic shear waves are generated and polarized […]

I should read:

In the continuous technique, an initial direction aligned with one of the symmetry axes is chosen (say direction 1) and kept fixed while a sequence of N (usually five to ten) ultrasonic shear waves is generated and polarized […]

On page 5, equation 29 where it was written:

$$\mathit{PRECISION}=\frac{{\sigma }_x}{\left|\overline{x}\right|}$$

It should read:

$$\mathit{PRECISION}=\frac{{\sigma }_{\overline{x}}}{\left|\overline{x}\right|}$$

On page 5, equation 30 where it was written:

$$\mathit{ACCURACY}=\frac{\left|\text{Δ}\overline{T}\right|}{\left|x_{\mathit{REF}}\right|}$$

It should read:

$$\mathit{ACCURACY}=\frac{|\text{Δ}\overline{T}-X_{\mathit{REF}}|}{\left|X_{\mathit{REF}}\right|}$$

On page 5 where it was written:

The waves’ time-of-flight is determined from their electronic signals using the mathematical techniques of cross correlation and data interpolation [5].

It should read:

The waves’ time-of-flight is determined from their electronic signals using the mathematical techniques of cross correlation and data interpolation [6].

On page 6 where it was written:

For stress states in which the principal stresses are not aligned with the material symmetry axes, a modified version of Eq. (2) must be employed [1,6].

It should read:

For stress states in which the principal stresses are not aligned with the material symmetry axes, a modified version of Eq. (2) must be employed [1,7].

On page 6 where it was written:

2.4. MATERIAL CHARACTERIZATION

It should read:

2.3.1. Material characterization

On page 6 where it was written:

The direction of applied loading should coincide with one of the material symmetry axes 1 or 2, but the specific choice may affect the parameters’ results and deserves further study [7]:

It should read:

The direction of applied loading should coincide with one of the material symmetry axes 1 or 2, but the specific choice may affect the parameters’ results and deserves further study [8]:

On page 6 where it was written:

The number of significant figures to be retained in the results is based on the number of significant digits and relative error (precision) of the input data and on the standard deviation of the mean of the computed variables [5].

2.5. Estimation of the principal stress difference

With the material parameters characterized, Eq. (3) can be applied in reverse order to estimate the principal stress difference in selected points of a structure under loading:

1. First select the technique (continuous in this work) for propagating the ultrasonic shear waves and acquire and treat the data (the waves’ time-of-flight) accordingly (Section 4) to obtain the expected value of the birefringence $\overline{B}$ and its uncertainty ${\sigma }_{\overline{B}}$ (Eq. 25 and Eq. 26);

2. Apply Eq. 2 in reverse order to estimate the expected value of the principal stress difference Δ$\overline{T}$ = (T ₁ – T ₂) and its uncertainty; σ_ΔT (Eq. 27 and Eq. 28);

3. Determine the experimental precision and accuracy (if a reference solution is available) of the result according to the Section 3.4.

It should read:

The number of significant figures to be retained in the results is based on the number of significant digits and relative error (precision) of the input data and on the standard deviation of the mean of the computed variables [6].

2.3.2. Estimation of the principal stress difference

With the material parameters characterized, Eq. (2) can be applied in reverse order to estimate the principal stress difference in selected points of a structure under loading:

• First select the technique (continuous in this work) for propagating the ultrasonic shear waves and acquire and treat the data (the waves’ time-of-flight) accordingly to obtain the expected value of the birefringence $\overline{B}$ and its uncertainty ${\sigma }_{\overline{B}}$ (Eq. 25 and Eq. 26);

• Apply Eq. 2 in reverse order to estimate the expected value of the principal stress difference Δ$\overline{T}$ = (T ₁ – T ₂) and its uncertainty; σ_ΔT (Eq. 27 and Eq. 28);

• Determine the experimental precision and accuracy (if a reference solution is available) of the result using Eq. 29 and Eq. 30.

On page 7 where it was written:

Applying the procedure indicated in Section 5.1.1 for the continuous technique, […]

It should read:

Applying the procedure indicated in Section 2.2 for the continuous technique, […]

On page 8, Table 2 header, where it was written:

5000 Kgf

10000 Kgf

15000 Kgf

20000 Kgf

25000 Kgf

30000 Kgf

It should read:

5,000 Kgf

10,000 Kgf

15,000 Kgf

20,000 Kgf

25,000 Kgf

30,000 Kgf

On page 8, section 3.2., where it was written:

Applying the continuous technique for the stress estimation (Section 5.2), …

… were used instead of the average value indicated in Table 5.

… and Eq. (28) as

It should read:

Applying the continuous technique for the stress estimation (Section 2.2), …

… were used instead of the average value indicated in Table 3.

… and Eq. (28) as indicated in Table 5.

On page 8, section 3.3., where it was written:

… are given by the elementary theory of the strength of materials [8] as: …

It should read:

… are given by the elementary theory of the strength of materials [9] as: …

On page 9, Table 4 header, where it was written:

42000 Kgf

It should read:

42,000 Kgf

On page 9, Table 4, line “Mean (birefringence)”, where it was written:

-0,0023

0,0018

0,0008

-0,0021

It should read:

-0.0023

0.0018

0.0008

-0.0021

On page 9, Table 4, line “Uncertainty^a”, where it was written:

0,0001

0,0001

0,0001

0,0002

It should read:

0.0001

0.0001

0.0001

0.0002

On page 9, Table 5, where it was written: Table 5.Principal stress difference at points C1 e C5 (birefringence technique).

POINT	m (mm2/Kgf)	B0	$\overline{B}$	ΔT(= T1 – T2) (Kgf/mm2)	RELATIVE ERROR (%)
C1	–0.000230 ± 0.000002	–0.0023 ± 0.0001	0.0018 ± 0.0001	–17,8 ± 0.6	3,37
C5	–0.000230 ± 0.000002	0.0008 ± 0.0001	–0.0021 ± 0.0002	12,6 ± 1.0	7,94

It should read:

Table 5. Principal stress difference at points C1 e C5 (birefringence technique).

POINT	m (mm2/Kgf)	B 0	$\overline{B}$	ΔT(= T1 – T2) (Kgf/mm2)	RELATIVE ERROR (%)
C1	–0.000230 ± 0.000002	–0.0023 ± 0.0001	0.0018 ± 0.0001	–17.8 ± 0.6	3.37
C5	–0.000230 ± 0.000002	0.0008 ± 0.0001	–0.0021 ± 0.0002	12.6 ± 1.0	7.94

On page 9, Table 6, column “APPLIED LOAD (Kgf)” where it was written:

42000

It should read:

42,000

On page 10, section 4, where it was written:

This will require the modification of the equation relating the principal stress difference and the birefringence as discussed by THOMPSON et al. [6].

It should read:

This will require the modification of the equation relating the principal stress difference and the birefringence as discussed by THOMPSON et al. [7].

On page 10, section “Bibliography” where it was written:

…

[5] BITTENCOURT, M.S.Q., “Desenvolvimento de um sistema de medida do tempo decorrido da onda ultrasônica e análise do estado de tensões em materiais metálicos pela técnica da birrefringência acústica”. Tese (Doutorado em Ciências), Universidade Federal do Rio de Janeiro, Rio de Janeiro, 2000.

[6] THOMPSON, R.B., LEE, S.S., SMITH, J.F., “Angular dependence of ultrasonic wave propagation in a stressed, orthorhombic continuum: theory and application o the measurement of stress and texture”, The Journal of the Acoustical Society of America, v. 80, n. 3, pp. 921–931, 1986. http://dx.doi.org/10.1121/1.393915.

[7] DUTRA, M.A.M., “Avaliação acustoelástica do aço 20 MnMoNi 55, material estrutural do vaso de pressão dos reatores nucleares de Angra II e Angra III ”. Tese (Mestre em Ciências), Instituto de Engenharia Nuclear, Rio de Janeiro, RJ, 2009.

[8] TIMOSHENKO, S.P., Strengthen of materials. 2nd ed., New York: D. Van Nostrand Company, Inc., 1949.

It should read:

…

[5] TAYLOR, J. R., Error analysis: the study of uncertainties in physical measurements. 2nd ed., University Science Books, 1997.

[6] BITTENCOURT, M.S.Q., “Desenvolvimento de um sistema de medida do tempo decorrido da onda ultrasônica e análise do estado de tensões em materiais metálicos pela técnica da birrefringência acústica”. Tese (Doutorado em Ciências), Universidade Federal do Rio de Janeiro, Rio de Janeiro, 2000.

[7] THOMPSON, R.B., LEE, S.S., SMITH, J.F., “Angular dependence of ultrasonic wave propagation in a stressed, orthorhombic continuum: theory and application of the measurement of stress and texture”, The Journal of the Acoustical Society of America, v. 80, n. 3, pp. 921–931, 1986. http://dx.doi.org/10.1121/1.393915.

[8] DUTRA, M.A.M., “Avaliação acustoelástica do aço 20 MnMoNi 55, material estrutural do vaso de pressão dos reatores nucleares de Angra II e Angra III ”. Dissertação (Mestre em Ciências), Instituto de Engenharia Nuclear, Rio de Janeiro, RJ, 2009.

[9] TIMOSHENKO, S.P., Strength of materials. 2nd ed., New York: D. Van Nostrand Company, Inc., 1949.

Publication Dates

• Publication in this collection
  14 Apr 2023
• Date of issue
  2023