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

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), [...]

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: It should read: On page 4 where it was written: (see subsection 5.

1). Thus, […]
On page 4, equations 20 where it was written: It should read: On page 4, equations 21 where it was written: It should read: ) f X ( ) On page 4, equations 22 where it was written: On page 4, equations 24 where it was written:

N
On page 5 where it was written: 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: On page 5, equation 30 where it was written: 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:

MATERIAL CHARACTERIZATION
It should read:

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].

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: a) 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 B and its uncertainty σ B (Eq. 25 and Eq. 26); b) Apply Eq. 2 in reverse order to estimate the expected value of the principal stress difference 'T T T 1 2 and its uncertainty; V 'T (Eq. 27and Eq. 28); c) 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].

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: a) 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 B and its uncertainty σ B (Eq. 25 and Eq. 26); b) Apply Eq. 2 in reverse order to estimate the expected value of the principal stress difference 'T T T 1 2 and its uncertainty; V 'T (Eq. 27and Eq. 28); c) 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 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 5, where it was written: It should read: 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].
Applying the procedure indicated in Section 2.2 for the continuous technique, [...]On page 8, Table2header, where it was written: 5.1.1 for the continuous technique, [...]It should read: