Two approaches for estimation and expression of uncertainty in stress strain tests.

In the present age of globalization, where mutual recognition agreement is the key word for intercontinental business, test certificates from an accredited laboratory gain in importance. As per accreditation criteria of ISO/IEC 17025:1999 standard, all testing laboratories have to provide an expression and estimation of uncertainty involved in each test parameter.

In this article, an attempt has been made to express and estimate uncertainty involved in a stress-strain test, measured by a universal testing system, for the test parameters including modulus at 300% extension and tensile strength for a vulcanized rubber compound. Tests were performed as per ISO37:1994, similar to ASTM D412.

Calibration laboratories have been estimating and quoting the calibration uncertainty for a long time as a part of the calibration process and generally using the concept of sensitivity coefficient. While calculating the uncertainty contribution of each parameter, this method involves more mathematics and is easy to use by calibration engineers. In testing laboratories, estimating and quoting the measurement uncertainty is in the primary stage and a lot of work is yet to be done. In this sector, whatever work is being done is mostly in analytical testing and is mainly based on the Eurachem Guide. Thus scientists/ analysts engaged in this area are familiar with the relative uncertainty concept while estimating the uncertainty contribution of each parameter.

In this article, we have given the uncertainty estimation by a first method in detail, and then for modulus at 300% we have given the calculation as per a second method too. Later, the correlation between these two methods is shown mathematically so the user from both sectors can find things of their interest. Definitions used:

* Modulus at 300% elongation (M300) is the tensile stress at the test length required to produce 300% elongation.

* Tensile strength (TS) is the maximum tensile stress recorded in extending the test piece to breaking point.

First method--using the concept of sensitivity coefficient

The general information for calculation of uncertainty of parameter M300 and TS:

* Instrument used: Zwick Universal Test System 1445 with non-contact type optical extensometer;

* range: 0-10 kN;

* sample tested: vulcanized rubber compound; and

* laboratory temp: 25[degrees]C.

(1) Mathematical model for modulus at 300% elongation M[300.sub.x] = M[300.sub.a]+e

Where: M[300.sub.x] = estimated value of modulus at 300% elongation

M[300.sub.a] = average value of modulus at 300% elongation e = uncertainty involved in the test

(2) M[300.sub.x] = [[L/(W x T)].sub.at 300%elongation]+ e

Where: L is the load in Newtons (N); W is the width of the test specimen, normally considered as 6.00 mm as of die dimension of ISO37 Type 1, similar to ASTM D412 Die C; and T is the thickness of the test specimen.

Equation 2 can be represented as:

(3) M[300.sub.x] = f (L, W, T, EXT, e)

Hence the contributory variances are as follows:

(4) [u.sub.c.sup.2] (M[300.sub.x]) = [C.sub.LC.sup.2] [u.sup.2]([delta]LC) + [C.sub.DIE.sup.2] [u.sup.2]([delta]DIE) + [C.sub.GM.sup.2] [u.sup.2]([delta]GM) + [C.sub.EXT.sup.2] [u.sup.2]([delta]EXT) + [C.sub.e.sup.2] [u.sup.2]([delta]e)

Where: [delta]LC is the uncertainty in the load cell calibration; [delta]DIE -uncertainty in the cutting die calibration; [delta]GM is the uncertainty in the gauge meter calibration; [delta]EXT is the uncertainty in the optical extensometer calibration; [delta]e is the uncertainty in the M300 measurement (repeatability); and [C.sub.LC], [C.sub.DIE], [C.sub.GM], [C.sub.EXT] and [C.sub.e] are the sensitivity coefficients respectively for load cell, cutting die, gauge meter, optical extensometer and M300 measurement (repeatability). Note 1: Temperature effect on the load cell is assumed negligible because the testing temperature, as well as the load cell calibration temperature, were the same.

Sensitivity coefficients can be calculated as follows:

(5) [C.sub.LC] = ([differential]f/[differential]L) = [[1/(WxT)].sub.at 300% elongation]

(6) [C.sub.DIE] = ([differential]f/[differential]W) = [[-L/([W.sup.2]xT)].sub.at 300% elongation]

(7) [C.sub.GM] = ([differential]f/[differential]T) = [[-L/(Wx[T.sup.2])].sub.at 300% elongation]

(8) [C.sub.EXT] = ([differential]f/[differential]EXT) = [-(change in load by [+ or -]1% change in extension)/(WxT)]

(9) [C.sub.e] = ([differential]f/[differential]e) = 1

Type A uncertainty evaluation for modulus at 300% elongation

The uncertainty due to repeatability of the test is evaluated by a statistical method using the normal probability distribution for n = 10 measured data (table 1).

Number of observations = n = 10

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Standard deviation of the mean = s/[square root of (n)] = 0.151/[square root of (10)] = 0.048

Uncertainty contribution u([delta]e) = 0.048 MPa

Sensitivity coefficient = 1 (by equation 9)

Degree of freedom = n-1 = 9

Type B uncertainty evaluation for M300

As per the calibration certificate, load cell calibration uncertainty is [+ or -] 1% without a confidence level. So, assuming rectangular probability distribution, the standard uncertainty u([delta]LC) = 1.3797/[square root of (3)] = 0.7966 N. The value 1.3797 is one percent of the average value of the measured load. Degree of freedom = [infinity].

By our last ten-year experience, we know that from a calibrated die, width of the narrow portion of the dumbbell is always coming within [+ or -] 1% of 6 mm, the width of the die. So assuming rectangular probability distribution, the standard uncertainty u([delta]DIE) = 0.06/[square root of (3)] = 0.0346 mm. Degree of freedom = [infinity].

From the supplier certificate, the gauge meter accuracy is [+ or -] 0.01 mm, having least count of 0.001 mm. So assuming rectangular probability distribution, the standard uncertainty u([delta]GM) = 0.01/[square root of (3)] = 0.0058 mm. Degree of freedom = [infinity].

As per calibration certificate, extensometer calibration uncertainty is [+ or -] 1% without a confidence level. So assuming rectangular probability distribution, the standard uncertainty u([delta]LC) = 1/[square root of (3)] = 0.5774%. Degree of freedom = [infinity].

Combined standard uncertainty evaluation for M300

The sensitivity coefficients in equation 4 are computed as follows:

[C.sub.LC] = ([differential]f/[differential]L) = [1/(6.00 x 2.028)] = 0.0822 [mm.sup.-2] (see equation 5);

[C.sub.DIE] = ([differential]f/[differential]W) = [-137.97/([6.00.sup.2] x 2.028)] = -1.8898 N[mm.sup.-3] (see equation 6);

[C.sub.GM] = ([differential]f/[differential]T) = [-137.97/(6.00 x [2.028.sup.2])] = -5.5911 N[mm.sup.-3] (see equation 7);

[C.sub.EXT] = ([differential]f/[differential]EXT) = [-0.6920/(6.00 x 2.028)] = 0.0569 N[mm.sup.-2] (see equation 8).

Note 2: To get data of change in load by change in extension 1% from either side of 300%, we assume that the stress-strain curve is linear from 300% elongation to the break point. The mean load value at break (572.2%) was observed at 326.34 N. So the change in load by per unit, (here %), change in extension = (326.34-137.97)/(572.2-300.0) = 0.6920 N.

Substituting the individual terms discussed previously in equation 4 yields the combined standard uncertainty, [u.sub.c](M[300.sub.x]).

[u.sub.c.sup.2] (M[300.sub.x]) = [(0.0822).sup.2] x [(0.7966).sup.2] + [(-1.8898).sup.2] x [(0.0346).sup.2] + [(-5.5911).sup.2] x [(0.0058).sup.2] + [(0.0569).sup.2] x [(0.5774).sup.2] + [(1).sup.2] x [(0.048).sup.2] = 0.0130 [u.sub.c] (M[300.sub.x]) = [square root of (0.0130)] = 0.1140 MPa

Effective degree of freedom evaluation for M300

To obtain the effective degree of freedom, the degree of freedom for each standard uncertainty component is required. For a component obtained from Type A evaluation, the degree of freedom is obtained from a number of independent repeated observations. For a component obtained from Type B evaluation, the degree of freedom is obtained from the judged reliability of the value of the component, which is often the case in practice [v.sub.i] [right arrow] [infinity] Hence:

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As the effective degree of freedom more than 100 is treated as [infinity], so [v.sub.eff](M[300.sub.x]) = [infinity].

Expanded uncertainty evaluation for M300

Since the effective degree of freedom is [infinity], from student t-distribution table for 95% confidence level, the coverage factor k = 1.960.

(10) Expanded uncertainty, U(M300) = k x [[u.sub.c](M[300.sub.x])]
                                   = 1.960 x 0.1140
                                   = 0.2235 MPa
                                   = ~0.2 MPa

Reporting of results, M300

For the test parameter modulus at 300% elongation, M300, the uncertainty U(M300) is [+ or -] 0.2 MPa determined from a combined standard uncertainty [u.sub.c](M[300.sub.x]) = 0.114 MPa and a coverage factor k = 1.96 based on the student t-distribution for [v.sub.eff](M[300.sub.x]) = [infinity] degree of freedom and estimated to have a level of confidence of 95%. Hence:

M300 = 11.3 [+ or -] 0.2 MPa

Uncertainty budget

The uncertainity budget for modulus at 300% elongation, is shown in table 2.

Mathematical model for tensile strength

(11) T[S.sub.x] = T[S.sub.a]+e T[S.sub.x] = estimated value of tensile strength T[S.sub.a] = average value of tensile strength e = uncertainty involved in the test

(12) T[S.sub.x] = [L/[(W x T)].sub.at break] + e

Where: L is the load in Newtons (N); W is the width of the test specimen, normally considered as 6.00 mm as of die dimension of ISO37 Type 1, similar to ASTM D412 Die C; and T is the thickness of the test specimen.

Equation (12) can be represented as:

(13) T[S.sub.x] = f(L, W, T, e)

Hence the contributory variances are as follows:

(14) [u.sub.c.sup.2] (T[S.sub.x]) = [C.sub.LC.sup.2] [u.sup.2]([delta]LC) + [C.sub.DIE.sup.2] [u.sup.2]([delta]DIE) + [C.sub.GM.sup.2] [u.sup.2]([delta]GM) + [C.sub.e.sup.2][u.sup.2]([delta]e)

Where: [delta]LC is the uncertainty in the load cell calibration; [delta]DIE--uncertainty in the cutting die calibration; [delta]GM is the uncertainty in the gauge meter calibration; [delta]e is the uncertainty in the TS measurement (repeatability); [C.sub.LC], [C.sub.DIE], [C.sub.GM] and [C.sub.e], are the sensitivity coefficients, respectively, for load cell, cutting die, gauge meter, TS measurement (repeatability).

Note 3: Temperature effect on the load cell is assumed negligible because the testing temperature, as well as the load cell calibration temperature, were the same.

Sensitivity coefficients can be calculated as follows:

(15) [C.sub.LC] = ([differential]f/[differential]L) = [[1/(W x T)].sub.at break]

(16) [C.sub.DIE] = ([differential]f/[differential]W) = [[-L/([W.sup.2] x T)].sub.at break]

(17) [C.sub.GM] = ([differential]f/[differential]T) = [[-L/(W x [T.sup.2])].sub.at break]

(18) [C.sub.e] = ([differential]f/[differential]e) = 1

Type A uncertainty evaluation for tensile strength

The uncertainty due to repeatability of the test is evaluated by a statistical method using the normal probability distribution for n = 10 measured data (table 3).

Number of observations = n = 10

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Standard deviation of the mean = s/[square root of (n)] = 0.3974//[square root of (10)] = 0.1257

Uncertainty contribution u([delta]e) = 0.1257 MPa

Sensitivity coefficient = 1 (by equation 18)

Degree of freedom = n-1 = 9

Type B uncertainty evaluation for tensile strength

As discussed earlier, type-B uncertainties as well as the sensitivity coefficient will be the same.

Combined standard uncertainty evaluation for TS

By substituting the individual terms discussed previously in equation (14) yields the combined standard uncertainty, [u.sub.c](T[S.sub.x]).

[u.sub.c.sup.2] (T[S.sub.x]) = [(0.0822).sup.2] x [(0.7966).sup.2] + [(-1.8898).sup.2] x [(0.0346).sup.2] + [(-5.5911).sup.2] x [(0.0058).sup.2] + [(1).sup.2] x [(0.1257).sup.2] =0.0254 [u.sub.c](T[S.sub.x]) = [square root of ((0.0254))] = 0.1594 MPa

Effective degree of freedom evaluation for TS

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Expanded uncertainty evaluation for TS

Since the effective degree of freedom is 25, from the student t-distribution table lot 95% confidence level, the coverage factor k = 2.06.

Expanded uncertainty, U(TS) = k x [[u.sub.c](T[S.sub.x])]
                            = 2.06 x 0.1594
                            = 0.3284 MPa
                            = -0.3 MPa

Reporting of results, tensile strength

For the test parameter tensile strength, the uncertainty U(TS) is [+ or -] 0.3 MPa determined from a combined standard uncertainty [u.sub.c](T[S.sub.x]) = 0.1594 MPa and a coverage factor k = 2.06 based on the student t-distribution for [v.sub.eff](T[S.sub.x]) ~25 degree of freedom and estimated to have a level of confidence of 95%. Hence:

TS = 26.8 [+ or -] 0.3 MPa

The uncertainty budget for tensile strength is shown in table 4.

Second method--using the concept of relative uncertainty

Mathematical model for M300 can be written as:

Modulus at 300% = load (Newtons) x repeatability/ width (mm) x thickness (mm)

= 137.97/6 x 2.028 x 1 = 11.34 MPa

The equation shows that uncertainty in modulus arises from four factors--load, width, thickness and repeatability. Here we have combined the uncertainty contribution of load cell calibration and the effect of an extensometer to detect the exact 300% elongation, i.e., indirect effect on load measurement at 300%. (This can be done in the first method also, but the mathematical contribution will be the same).

The value of each will be calculated as follows:

* Uncertainty due to repeatability--for the mean value of 11.34 MPa, the standard deviation of the mean is 0.048 MPa (calculated in the first method, type B uncertainty), corresponding to the relative standard deviation of the mean as 0.048/11.34 = 0.00423.

* Uncertainty due to load--the uncertainty of the load arises from: a) uncertainty of load cell calibration, i.e., 1% corresponding to uncertainty in the load of 137.97 x 1/100 = 1.3797 N. As the confidence level has not been provided, the standard uncertainty will be 1.3797/[square root of (3)] = 0.79657 N: b) uncertainty in the reading of extensometer. As the uncertainty of the extensometer is 1%, the uncertainty of this reading will cause a variation in load. The change in load as calculated per unit is 0.6920 N (calculated in first method, combined standard uncertainty evaluation). As the confidence level has not been provided, the distribution may be taken as rectangular. The standard uncertainty is thus 0.692/[square root of (3)] = 0.399526.

u (load) = [square root of ([(0.79657).sup.2] + [(0.399526).sup.2])] = 0.89115 N

* Uncertainty in width--the uncertainty as per experience has been 1%, which is equivalent to 0.06 mm for a 6 mm die. As the confidence level has not been provided, the uncertainty may be 0.06/[square root of (3)] = 0.0346.

* Uncertainty in thickness--the thickness has been measured with gauge meter having an uncertainty of 0.01 mm. There will also be type A uncertainty in the measurement, but as the repeatability of the whole method has been taken into calculations, the same need not be accounted again. Therefore, the uncertainty in thickness measurement is 0.01/ [square root of (3)] = 0.0058 mm.

Calculation of total uncertainty

The total uncertainty is due to the four factors as listed above. The same may be tabulated (table 5) to find out the relative uncertainty contribution of each factor. Then these relative uncertainties are combined as per the following rule to calculate the total combined uncertainty.

For models involving only a product or quotient, e.g., y = k(pqr ...), the combined standard uncertainty is given by:

[u.sub.c] (y) = y.k. [square root of (u[((p)/p).sup.2] + u[((q)/q).sup.2] + .........)]

where (u(p)/p), etc., are the uncertainties in the parameter expressed as relative standard deviation.

[u.sub.c] (M300) = 11.34 [square root of ([(0.00423).sup.2] + [(0.00646).sup.2] + [(0.00577).sup.2] + [(0.00286).sup.2])] = 0.114N

We can observe that the combined uncertainty worked out by this method is the same as calculated by the first method, and thus the expanded uncertainty, at 95% confidence level, remains the same as in the first method.

Correlation between methods

In the second (relative uncertainty) method, we can write the square of combined uncertainty as:

[u.sub.c.sup.2] (M[300.sub.x) = [(Mean).sup.2] x ([delta][L.sup.2]/[L.sup.2] + [delta][W.sup.2]/[W.sup.2] + [delta][T.sup.2]/[T.sup.2] + [delta][e.sub.2]/[Mean.sup.2])

This equation can be rearranged as shown in figure 1.

Figure 1

[u.sub.c.sup.2](M[300.sub.x] = [(Mean).sup.2] x [delta][L.sup.2]/[L.sup.2] + [(Mean).sup.2] [delta][W.sup.2]/[W.sup.2] + [(Mean).sup.2] [delta][T.sup.2]/[T.sup.2] + [delta][e.sub.2]

By replacing the value of the mean, M300 by L/(W x T) in the above equation:

[u.sub.c.sup.2](M[300.sub.x] = [(L).sup.2]/[(W x T).sup.2] x [delta][L.sup.2]/[L.sup.2] + [(L).sup.2]/[(W x T).sup.2] x [delta][W.sup.2]/[W.sup.2] + [(L).sup.2]/[(W x T).sup.2] x [delta][T.sup.2]/[T.sup.2] + [delta][e.sup.2]

The above equation can be rearranged as:

[u.sub.c.sup.2](M[300.sub.x] = (1)/[(W x T).sup.2] x [delta][L.sup.2]/1 + [(L).sup.2]/[([W.sup.2] x T).sup.2] x [delta][W.sup.2]/1 + [(L).sup.2]/[(W x [T.sup.2]).sup.2] x [delta][T.sup.2]/1 + [delta][e.sup.2]

In determining load at 300% elongation, two uncertainty contributing factors are the load cell itself and the extensometer, which indicate the exactness of 300%, thus the [delta]L here is:

[delta][L.sup.2] = [(uncertainty contribution from load cell).sup.2] + [(uncertainty contribution from extensometer).sup.2]

From this fact and equations 5 to 9:

[u.sub.c].sup.2] (M[300.sub.x]) = [C.sub.LC.sup.2] [u.sup.2]([delta].sub.LC]) + [C.sub.DIE.sup.2] [u.sup.2]([delta].sub.DIE]) + [C.sub.GM.sup.2] [u.sup.2]([[delta].sub.GM]) + [C.sub.EXT.sup.2] [u.sup.2]([[delta].sub.EXT]) + [C.sub.e.sup.2] [u.sub.2]([delta]e)

This is equivalent to equation 4. This shows the equivalence of the second method with the fast one.

Table 1--M300 results from the UTS and
deviation from the mean value

S. no.     M300   Gauge       Load      M300i-   SQ (M300i-
          (MPa)    (mm)        (n)   M300 avg.   M300 avg.)

     1     11.5    2.03     140.07        0.16       0.0256
     2     11.6    2.03     141.29        0.26       0.0676
     3     11.3    2.11     143.06       -0.04       0.0016
     4     11.2    2.07     139.10       -0.14       0.0196
     5     11.1    2.04     135.86       -0.24       0.0576
     6     11.4     1.9     129.96        0.06       0.0036
     7     11.3    2.03     137.63       -0.04       0.0016
     8     11.4    2.18     149.11        0.06       0.0036
     9     11.4    1.87     127.91        0.06       0.0036
    10     11.2    2.02     135.74       -0.14       0.0196
   Sum   113.40   20.08   1,379.74        0.00       0.2040
  Mean    11.34   2.028     137.97

Table 2--uncertainty budget for modulus at 300% elongation

Source of          Estimates       Limits   Probability distribution/
uncertainty (Xi)        (xi)    ([+ or -]       uncertainty type-A or
                               [DELTA]xi)                   B/divisor
Repeated
  observations      11.3 MPa          --               Normal, Type A
Load cell
  calibration       137.97 N    1.3797 N         Rectangular, Type-B,
Cutting die                                      [square root of (3)]
  calibration        6.00 mm     0.06 mm         Rectangular, Type-B,
Gauge meter                                      [square root of (3)]
  calibration       2.028 mm     0.01 mm         Rectangular, Type-B,
Optical                                          [square root of (3)]
  extensometer
  calibration           300%          1%         Rectangular, Type-B,
Combined                                         [square root of (3)]
  uncertainty             --          --                           --
Expanded
                    11.3 MPa          --    t-distribution, k = 1.960

Source of             Standard            Sensitivity
uncertainty (Xi)   uncertainty            coefficient
                         u(xi)                     Ci
Repeated
  observations       0.048 MPa                      1
Load cell
  calibration         0.7966 N     0.0822 [mm.sup.-2]
Cutting die
  calibration        0.0346 mm   -1.8898 N[mm.sup.-3]
Gauge meter
  calibration        0.0058 mm   -5.5911 N[mm.sup.-3]
Optical
  extensometer                    0.0596 N[mm.sup.-2]
  calibration          0.5774%
Combined
  uncertainty               --                     --
Expanded
                                                   --

Source of           Uncertainty    Degree of
uncertainty (Xi)   contribution      freedom
                      u(y), MPa           vi
Repeated
  observations           0.0480            9
Load cell
  calibration            0.0655   [infinity]
Cutting die
  calibration            0.0654   [infinity]
Gauge meter
  calibration            0.0324   [infinity]
Optical
  extensometer           0.0329   [infinity]
  calibration
Combined
  uncertainty            0.1140   [infinity]
Expanded
                     0.2235 MPa   [infinity]

Table 3--TS results from the UTS and deviation
from the mean value

S. no.       TS   Gauge       Load   TSi-TS   SQ (TSi-
          (MPa)    (mm)        (n)     avg.   TS avg.)

     1     26.2    2.03     319.12    -0.63     0.3969
     2     27.2    2.03     331.30     0.37     0.1369
     3     26.8    2.11     339.29    -0.03     0.0009
     4     26.5    2.07     329.13    -0.33     0.1089
     5     26.7    2.04     326.81    -0.13     0.0169
     6       27     1.9     307.80     0.17     0.0289
     7     26.8    2.03     326.42    -0.03     0.0009
     8     26.5    2.18     346.62    -0.33     0.1089
     9     27.6    1.87     309.67     0.77     0.5929
    10       27    2.02     327.24     0.17     0.0289
   Sum   268.30   20.28   3,263.39     0.00     1.4210
  Mean            2.028

Table 4--uncertainty budget for tensile strength

Source of un-    Estimates       Limits   Probability distribution/
certainty (Xi)        (xi)    ([+ or -]       uncertainty type-A or
                             [DELTA]xi)                   B/divisor

Repeated
  observations    26.8 MPa           --              Normal, type A
Load cell
  calibration     137.97 N     1.3797 N        Rectangular, Type-B,
Cutting die                                    [square root of (3)]
  calibration      6.00 mm      0.06 mm        Rectangular, Type-B,
Gauge meter                                    [square root of (3)]
  calibration     2.028 mm      0.01 mm        Rectangular, Type-B,
Combined                                       [square root of (3)]
  uncertainity          --           --                          --
Expanded
  uncertainty     26.8 MPa           --    t-distribution, k = 2.06

Source of un-       Standard            Sensitivity
certainty (XJ)   uncertainty            coefficient
                       u(xi)                     Ci

Repeated
  observations    0.1257 MPa                      1
Load cell
  calibration       0.7966 N     0.0822 [mm.sup.-2]
Cutting die
  calibration      0.0346 mm   -1.8898 N[mm.sup.-3]
Gauge meter
  calibration      0.0058 mm    -5.5911 N[mm.sup-3]
Combined
  uncertainity            --                     --
Expanded
  uncertainty     0.1594 MPa                     --

Source of un-     Uncertainty    Degree of
certainty (XJ)   contribution      freedom
                    u(y), MPa           vi

Repeated
  observations         0.1257            9
Load cell
  calibration          0.0655   [infinity]
Cutting die
  calibration          0.0654   [infinity]
Gauge meter
  calibration          0.0324   [infinity]
Combined
  uncertainity         0.1594   [infinity]
Expanded
  uncertainty      0.3284 MPa   [infinity]

Table 5--calculations

Parameter        Value      Standard      Relative
                         uncertainty   uncertainty

Repeatability     1.00       0.00423       0.00423
Load, N         137.97       0.89115       0.00646
Width, mm         6.00        0.0346       0.00577
Thickness, mm    2.028        0.0058       0.00286