ON THE PRECISION AND ACCURACY OF THE ACOUSTIC BIREFRINGENCE TECHNIQUE FOR STRESS EVALUATION

This paper presents a numerical procedure for estimation of the precision and accuracy of the acoustic birefringence technique as used in the Instituto de Engenharia Nuclear (IEN) for evaluation of residual and applied stresses in structures and components, mainly of the nuclear power industry. This procedure shall be incorporated to the signal processing module of the ultrasonic system used at IEN’s Ultrasonic Laboratory to account in an automatic and systematic way for the uncertainties in the input data and their propagation throughout the calculations. The acoustic birefringence is generally defined from the speeds of two mutually orthogonal volumetric waves of normal incidence, but when the use of a pulse-echo measurement system is feasible, the birefringence can be defined directly from the time-of-flight of the waves, since they travel the same physical space. The times-of-flight of the waves can thus be regarded as the primary variables of interest. They are estimated by coupling the mathematical techniques of cross correlation and data interpolation, whereas the material’s acoustoelastic constant is determined via a weighted linear regression. An Excel spreadsheet performs all calculations taking into account the uncertainties and the number of significant digits in the results. As an example of the procedure developed, the estimation of the precision and accuracy in the evaluation of the stresses acting in a beam under bending is presented. The analytical solution derived from the strength of materials theory was used as the reference value for accuracy estimation purpose.


INTRODUCTION
This paper proposes a numerical procedure for estimation of the precision and accuracy of the acoustic birefringence technique as used in the Instituto de Engenharia Nuclear (IEN) for evaluation of residual and applied stresses in structures and components, mainly of the nuclear power industry.This numerical procedure shall be incorporated to the signal processing module of the ultrasonic system used at IEN's Ultrasonic Laboratory in order to account in an automatic and systematic way for the uncertainties in the input data and their propagation throughout the calculations.
The acoustic birefringence is generally defined from the speeds of two mutually orthogonal volumetric waves of normal incidence, but when the use of an ultrasonic pulse-echo measurement system is feasible, the birefringence can be defined directly from the time-offlight of the waves since they travel the same physical space.In the pulse-echo mode, the time-of-flight of the ultrasonic wave can thus be regarded as the primary variable for stress measurement by the acoustic birefringence technique.The theoretical fundaments of ultrasonic wave propagation in elastic, homogeneous, isotropic and anisotropic solids are reviewed in ref. [1].In the approach followed at IEN, the times-of-flight are determined by coupling the mathematical techniques of cross correlation and data interpolation [2].This paper starts with the basic concepts and equations of the acoustic birefringence technique for stress measurement and the fundamentals of experimental error analysis focusing on the statistics concepts employed in the work.This is followed by a description of the alternative techniques available at IEN's Ultrasonic Laboratory for generating the ultrasonic waves, and for estimating their times-of-flight and the acoustic birrefringence at a material point.The numerical procedure proposed to improve and systematize the estimation of the precision and accuracy of the birefringence technique for stress evaluation is then presented.It relies on consistently accounting for the uncertainties in the input data and their propagation throughout the calculations up to the final results.A weighted linear regression is employed for determination of the material acoustoelastic constant, which accounts for the uncertainties in the times-of-flight and in the applied loads during the loading test.Finally, the mathematical treatment of the equation relating the birefringence and the stresses and their experimental uncertainties is discussed.The experimental precision is characterized by the relative error given by the ratio between the standard deviation of the individual measures and their average value, and the experimental accuracy, by the ratio between the average value of the individual measures and a reference value.An Excel spreadsheet performs all calculations taking into account in a consistent way the number of significant digits and the uncertainties in the data field.
As an example of the procedure developed, the estimation of the experimental precision and accuracy in the evaluation of the stresses acting in a beam under simple bending is presented.The experimental input data (waves' time-of-flight) was taken from a M.Sc.dissertation developed at IEN's Ultrasonic Laboratory.The analytical solution derived from the strength of materials theory was used as the reference value for accuracy estimation purpose.

DETERMINATION OF THE PRINCIPAL STRESSES DIFFERENCE BY THE ACOUSTIC BIREFRINGENCE TECHNIQUE
The acoustical birefringence B is the normalized difference of the speeds V 31 and V 32 of two shear waves polarized orthogonally along the material symmetry axes x 1 and x 2 and propagating through the thickness (x 3 axis) of a component with flat and parallel surfaces [1,3,4].B is evaluated according to equation (1)

( ) ( )
where the times-of-flight t 31 and t 32 are used to replace the speeds since, in principle, waves start exactly at the same point and travel exactly the same distance.B is the sum of the anisotropy from the material texture B 0 and from the internal and applied stresses.For an homogeneous material, the last contribution depends linearly by means of the acoustoelastic constant m from the difference of the principal stresses 1 T and 2 T aligned with material symmetry axes x 1 and x 2 , ( ) 0 B and m can be characterized through a tensile test by linearly fitting the values of the birefringence at increasing loading.Equations ( 1) and (2) show that the birefringence technique can provide a thickness-averaged and relative stress magnitude only.

FUNDAMENTALS OF EXPERIMENTAL ERROR ANALYSIS
Every physical measurement is subject to a certain degree of uncertainty.
) is estimated by solving Eq. (2) in reverse order assuming B, B 0 and m as independent variables (input data).These statistical concepts and their use here are briefly reviewed in the following subsections [5].The numerical procedure proposed is described in more detail in Sections 4 and 5.

Statistical errors: Basic Concepts
Consider a sample of N direct measurements of a variable x {x 1 , x 2 , ... , x N }, the standard estimate for the expected result of the measurement is and 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), the standard deviation of individual measurements x σ is replaced by s x , termed the experimental standard deviation of individual repeated measurements, ( ) so that the standard deviation of the mean becomes, ( ) The covariance of two sets of random variables x and y extracted from two samples of N direct measures, { x 1 , x 2 , ... , x N } and { y 1 , y 2 , ... , y N } is given by and Pearson correlation coefficient (r) of the sets x and y by The two sets of random variables x and y are said to be strongly correlated when the value of r is close to unity, and to be uncorrelated when r is null.

Data Fitting: Weighted Linear Regression
Assume that there is a strong correlation among the measurements associated with a pair of variables (x, y) and that a relationship of cause and effect exist between them described by a functional relationship ) (x f y = . The determination of this relationship is known as curve fitting (or regression).The simplest functional relationship is when the Pearson correlation coefficient (r) among the pairs (x i , y i ) of N measurements of the variables x and y is close to unity.In this case, the variation of y as a function of x can be expressed as a linear relationship, where the angular (a) and linear (b) coefficients of the straight line can be estimated by the method least squares by minimizing the functional expression, ) , ( The identification of the material acoustoelastic constant m and initial birefringence B 0 (and their uncertainties) is done by applying the weighted linear regression described above to the linear function given by equation ( 2) relating the birefringence B with the principal stress difference T ).The input data for this procedure are the values of the i B determined at N loading levels i T ∆ and their associated uncertainties ( ) and

Error Propagation
Error propagation is determined using the formulas provided by differential calculus based on the Taylor expansion of a multivariate function.Thus, if a variable φ depends of M other variables ) , , , ( , according to some function f(X), and if the N measurements of each one of the independent variables are distributed around the mean value ) , , , ( , such that around this neighborhood f(X) can be approximated by the first terms of a Taylor expansion, i.e., The estimate for the expected value of φ is given by the value of f(X) at the mean value point X , and the uncertainty associated with each indirect measure of φ by where and the uncertainty in the mean value of φ is The multivariate functions considered in this work are given by equation ( 1), for the birefringence B, and equation ( 2), for the difference of the principal stress ( ) The former is a two-variable function of the times-of-flight t 31 and t 32 .Applying the above procedure to account for the propagation of errors in the calculations, the expected value of the birefringence is Equation ( 2) for the principal stress difference should be treated as a subtraction and division of numbers with independent uncertainties, i.e. ( ). Applying the general procedure to account for the propagation of errors, the expected value of the principal stress difference is given by with uncertainty ( )

Precision and Accuracy
The experimental precision is characterized by relative error given by the ratio between the standard deviation of the mean (equation 8) and the absolute value of the mean (equation 4), whereas the experimental accuracy is defined by the ratio between the standard deviation of the mean (equation 8) and the absolute value of the reference value REF x , that is, and The number of significant figures used to report the results are determined considering the number of significant figures of the input data and the standard deviation of the mean of the variable under consideration (times-of-flight, birefringence, acoustoelastic constant and stresses) constrained by the precision of the input data.An Excel spreadsheet performs all the required calculations.

THE EXPERIMENTAL PROCEDURE USED AT IEN
The generation of the ultrasonic volumetric waves and the evaluation of their times-of-flight used in the birrefringence technique for the stress determination in a particular specimen are done at IEN's Laboratory of Ultrasonic using the equipment system schematically depicted in Figure 1.The core of the ultrasonic system is comprised of a generator of ultrasonic waves, an oscilloscope for signal acquisition, and a personal computer for signal processing.The wave time-of-flight is determined by coupling the mathematical techniques of cross correlation and data interpolation [2].Transducers of different frequencies are available for specific applications.In the continuous technique, an initial direction aligned with one of the material 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 in this direction, their signals captured and their times-of-flight (t 31 ) i (i =1, N) determined using the mathematical techniques of cross correlation and data interpolation.Only after the series of signals acquisition is completed, the transducer is rotated to the direction orthogonal to the first one (direction 2) and a similar series of measurements are made leading to the set (t 32 ) i (i =1, N).These results are then used to determinate the waves' average time-of-flight in each orthogonal direction ) , ( In the pair-to-pair technique, the transducer is aligned with a chosen material symmetry axis (say direction 1) and the time-of-flight t 31 of the shear wave generated and polarized in this direction is determined.The transducer is next rotated to the material symmetry axis 2 (orthogonal to direction 1) and the procedure repeated to calculate the wave's time-of-flight t 32 .Using the orthogonal waves' time-of-flight t 31 and t 31 , the birefringence B is determined in the sequel according to equation ( 1).This procedure is repeated a pre-determined number of times N (again five to ten measurements).Differently from the previous technique, only after the series of N measurements are completed resulting in the set B i (i = 1, N), the average value of the birrefringence B and its corresponding uncertainty B σ are calculated.

THE NUMERICAL PROCEDURE PROPOSED
Before starting this Section, it is recalled that the birefringence technique (Section 2), can only provide a thickness-averaged and relative stress magnitude.The thickness-averaged constraint is due to the application of volumetric shear waves, and the relative stress constraint relates to the fact that only the principal stress difference and not their nominal values can be generally estimated.An additional limitation comes from the fact that equation 2 refers to stress states in which the principal stresses are aligned with the material symmetry axes.In some particular applications, however, as in a beam under bending to be presented, it is possible to obtain the stress distribution along a beam's cross-section normal to the deformation plane by using shear waves propagating on the cross-section's plane and orthogonally polarized along the material symmetry axes on the deformation plane.For this example, the nominal value of the principal stress can also be estimated for the most external fibers on the deformation plane, since one the principal stress acting there has actually a null value.For stress states in which the principal stresses are not aligned with the material symmetry axes, a modified version of equation 2 has to be employed [1,6].
The numerical procedure proposed for the systematic evaluation of the experimental precision and accuracy of the birefringence technique as employed at IEN' Ultrasonic Laboratory for evaluation of residual and applied stresses is now summarized.The procedure is divided in two main steps, material characterization (i.e., estimation of m and B 0 parameters) and stress estimation (i.e., estimation of the principal stress difference 2 1 ).

Material Characterization
Material characterization is done by a uniaxial loading test (T 1 or T 2 is null) at different stress levels below the material yielding stress.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].

Continuous technique:
a) For each load level

Pair-to-pair technique:
a) For each load level The number of significant figures to be retained in the final results is based on the number of significant digits and relative error (precision) of the input data and on the standard deviation of the of the mean of the computed variables [5].

Estimation of the Principal Stress Difference
With the material parameters characterized, equation 3 can be applied in reverse order to estimate the principal stress difference in selected points of a structure under loading: The number of significant figures to be retained in the final results is based, as in the previous case, on the number of significant digits and relative error (precision) of the input data and on the standard deviation of the of the mean of the computed variables.

EXPERIMENTAL APPLICATION
To illustrate the numerical procedure proposed for estimation of the precision and accuracy of the acoustic birefringence ultrasonic technique as used at IEN's Ultrasonic Laboratory for the evaluation of stresses in structures, some experimental results reported by M. A. Monteiro Dutra [7] in his M.Sc.dissertation for the behavior of a beam under bending are exploited here.An analytical solution derived from the strength of materials theory was used as the reference value for accuracy estimation purpose.The beam was simply supported in two points 827 mm apart, and then loaded up to 42,000 Kgf (96% of the material yield limit) at its central region at half of the beam length.The ultrasonic signals were acquired first for the beam in the unloaded condition, and then loaded.
Only points located along the beam height and located at half of its length were selected for the measurements (Fig. 2).The continuous technique was used to acquire the ultrasonic signals.Shear waves were propagated along the beam thickness and polarized along the material symmetry axes x 1 (longitudinal direction: length) and x 2 (transversal direction: height).In each point, 5 pairs of signals were acquired to determine the wave average time of flight, making a total of 10 signals acquisition per point.The first 5 signals were acquired with the shear wave polarized along the beam's longitudinal direction and the remaining 5 ones with the shear wave polarized along its transversal direction (the material symmetry directions).A data acquisition system using a dual element transducer of 0,5 MHz was mounted to generate, receive and treat the shear wave echoes in order to determine the waves' time-of-flight.

Material characterization: Determination of the Acoustoelastic Constant and of the Initial Birefringence
To obtain the acoustoelastic constant, which is the angular coefficient of the straight line that approximates the relationship between the wave velocity and the applied load, a sample of the beam material with a length of 60 mm and a cross section of 40x40 mm 2 was subjected to a loading (compression) program consisting of 6 load increments of 5,000 Kgf each at its central point.Shear waves were propagated along the beam thickness and polarized along the material symmetry axes x 1 and x 2 , and their times of flight were acquired at each load level.Monteiro Dutra [7] considered two alternative load testing in which the direction of applied compression load is alternatively aligned with the material symmetry axes x 1 and x 2 , finding two different values for the acoustoelastic constant.Here, the acoustoelastic constant for the load aligned with the material direction 2 (transversal direction) was used to evaluate the principal stress difference.
Applying the procedure indicated in Section 5.1.1 for the continuous technique, and taking into account the time-of-flight of the longitudinal and transversal waves indicated in Tables 1  and 2, the following expected values (and uncertainties) where obtained for the acoustoelastic constant and the initial birefringence:

Principal Stress Difference Estimation by the Birefringence Technique
Applying the continuous technique for the stress estimation (Section 5.2), the following times-of-flight and birefringence were determined at points C1 and C5 at the center crosssection of the beam (Figure 1).The values of the initial birefringence B 0 at points C1 and C5 (Table 4) indicate that the material of the beam is acoustically heterogeneous.In order to account for this material characteristic, as a first approximation, the local values of B 0 were used instead of the average value indicated in Table 3.With this consideration, the expected principal stress difference (with uncertainties) at points C1 and C5 were then estimated from equations 27 and 28 as

Principal Stresses Determination by the Strength of Materials Theory
For the case of simply supported, rectangular beam subjected to a uniformly distributed load acting on a small area of the beam midway between the supports, the magnitudes of the principal stresses T 1 (direction longitudinal, 1 x ) and T 2 (direction transversal, 2 x ) are given by the elementary theory of the strength of materials [8] as: a) For points at the central section 2 For points at the central section 2 where, q is the distributed load acting on length d , l is the distance between the supports, h is the height of the beam, b is the thickness of the beam (along the x 3 axis).The geometric and loading data are compiled in Table 6, while the nominal stresses determined at points C1 and C5 are shown I Table 7.

Evaluation of the Precision and Accuracy of the Experimental Procedure
Considering equations 29 and 30, and the results in Tables 5 and 7, the precision and accuracy in the estimation of principal stress difference at points C1 and C5 are finally summarized in Table 8.

CONCLUSIONS
In this work, a numerical procedure has been proposed for estimation of the precision and accuracy of the acoustic birefringence technique as used in the Instituto de Engenharia Nuclear (IEN) for evaluation of residual and applied stresses in structures and components, mainly of the nuclear power industry.This procedure shall be incorporated to the signal processing module of the ultrasonic system used at IEN's Ultrasonic Laboratory to account in an automatic and systematic way for the uncertainties in the input data and their propagation throughout the calculations.
For the case showed here, the acoustic birefringence technique provided reasonably precise and accurate results.When assessing these results, however, it should be kept in mind that the material of the beam showed an acoustically heterogeneous behaviour and because of that some additional approximation had to be introduced in the analysis.A better solution would be to develop the original birefringence equations ( 1) and ( 2) directly for this kind of material behaviour.
Future work shall be directed to the analysis of structures in which the principal stresses are not aligned with the material axes of symmetry and a more appropriate treatment of acoustically heterogeneous materials.This will require the modification of the equation relating the principal stress difference and the birefringence as discussed by Thompson et al. [6].
deviation of the mean) the values of the angular (a) and linear (b) coefficients are given by

Figure 1 .
Figure 1.Schematic drawing of the ultrasonic system finally used to calculate the birefringence value B and its uncertainty B σ .

1 ,
P (P = nº of stress levels at the loading test)], estimate the waves' average time-of-flight in each orthogonal direction ) 8, and then estimate the birefringence mean value B and its uncertainty B σ using equations 4 and 8 once again; b) For the set of values k B and T k and associated uncertainties k B σ and k T σ , estimate the parameters m and B 0 and their uncertainties m σ and 0 B σ applying equations 17 to 19.

1 ,m and B 0 and their uncertainties m σ and 0 Bσ
P (P = nº of stress levels at the loading test)], estimate the wave's time-of-flight for the pair of orthogonal directions ) the birefringence value B k .From the set of B k values estimate the birefringence mean value B and its corresponding uncertainty B σ using equations 4 and 8. b) For the set of values k B and T k and associated uncertainties applying equations 17 to 19.

a)
First select the technique (continuous or pair-to-pair) 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 B σ (equations 25 and 26); b) Apply equation 2 in reverse order to estimate the expected value of the principal stress difference uncertainty; T ∆ σ (equations 27 and 28); c) Determine the experimental precision and accuracy (if a reference solution is available) of the result according the Section 3.4.

A
sample of 20 MnMoNi 55 steel supplied by NUCLEP -Nuclebrás Equipamentos Pesados S.A. was used to manufacture the beam specimen.The beam dimensions were 107 mm height x 95 mm thickness x 895 mm span length (827 mm support length) (see Fig 2).
There are fundamentally two different types of experimental errors, termed Statistical and Systematic.Statistical errors are random in nature: repeated measurements will differ from each other and from the true value by amounts which are not individually predictable, although the average behaviour over many repetitions can be predicted.Systematic errors arise from problems in the design of the experiment.They are not random, and affect all measurements in some well-defined way.The Total error is the sum of the two types of errors.To avoid the limitations of classification of the uncertainties in statistical or systematic and to unify the procedures to report the errors in measurements, the Bureau International des Poids et Mesures recommends that the uncertainties be classified in accordance with the method of assessment as uncertainties of type A, which are those evaluated by statistical methods over repeated measurements, and uncertainties of type B, which are evaluated by other methods.
Including the resolution of the equipment and the effects of data interpolation.

Table 2 . Waves' time-of-flight (under loading)
Including the resolution of the equipment and the effects of data interpolation.

Table 4 . Waves' time-of-flight and birefringence at selected points of the beam
Including the resolution of the equipment and the effects of data interpolation.