**2. Methods**

### *2.1. Bond Strength Tests of Adhesives*

The tests were carried out on 15 different polymer-cement adhesives, four samples for each adhesive. The adhesives compositions are cement mortars modified with various polymers [28]. In all cases, same-property foamed polystyrene was used as the substrate. The tests were carried out by three different research teams, which used various equipment (reproducibility conditions). They used the pull-off method according to ÖNORM B 6100:1998 10 01 [29]. A layer of polymer-cement adhesive was applied onto the substrate. Once cured in standardized conditions, 50 mm diameter round steel stamps weregluedon.Aforceperpendiculartothesurfacewasappliedtothestampsuntiltheypulled<sup>o</sup>ff.

Additional tests were also carried out in accordance with the ETAG 004 method [30]—squareshaped stamps of 50 mm were used in this case. The stamp shape impact on the test results is discussed in Section 4.

This can be a basic equation which describes the test model:

$$
\sigma = \frac{4F}{\pi D^2} \tag{1}
$$

where:

> σ—strength result, MPa

*F*—breaking force, N

*D*—diameter of the stamp in mm.

It should be noted that the pull-off test is commonly used in different variants, according to different standards and procedures. Configuration of tests used in this article is presented in Figure 3.In addition to adhesives, a pull-off test is used for a wide spectrum of construction materials, including all types of coatings, varnishes, materials for the protection and repair of concrete structures, etc.

**Figure 3.** Pull-off test configuration. (**a**) Test scheme; (**b**) photographic documentation of an example test.

### *2.2. Tensile Strength Tests of Plastics*

The object of the tensile strength tests were two plastics: ABS-GF and PC. The plastics tensile strength tests example uses the results obtained in laboratory proficiency tests carried out in 69 laboratories. This exercise was organized by the Deutsches Referenzbüro für Lebensmittel-Ringversuche und Referenzmaterialien GmbH (Kempten, Germany). The tests were carried out in accordance with the method described in ISO 527-1/-2 [31] standard. Reproducibility values were published in the organizer's report [32].

This can be a basic equation, which describes the test model:

$$
\sigma = \frac{F}{ab} \tag{2}
$$

where:

σ—strength result, MPa

*F*—force, N *a*—sample thickness, in mm *b*—sample width, in mm.

### **3. Estimation of Uncertainty**

### *3.1. Bond Strength Tests and Their Influence on the Results*

The general adhesion testing principle is based on stress determination, which results in the tested material detachment from the substrate—the substrate is a laboratory equivalent of the substrate on which the material is actually used. Although the test is intended to simulate the actual conditions of use, it differs in a number of aspects, including:


Therefore, it is difficult to translate the test result directly into the material's behavior when installed in the actual building; nevertheless, a test is a uniformed material assessment method and should reflect the differences between particular materials.

The most important factor affecting adhesion results is the adhesive composition. However, in conformity laboratory tests (like in our case) the laboratory only knows the use of the material and the resulting assessment criterion. The criterion must be met regardless of the composition.

The factors which can affect the test results (their uncertainty) can be divided into three groups:


glue chemical and physical influence on the adhesive, repeatability, and reproducibility of sample preparation (the sample preparation process for testing is multi-stage and involves a number of interactions, such as layer thickness, clamping force, conditioning variability, and others that may affect the test result), the material and sample heterogeneity.

It is noteworthy that in the case of all tensile strength tests there are usually some non-perpendicular components of stress. In some cases, first of all, related to scientific oriented research, emerging non-perpendicular components are the subject of analysis aimed at understanding material behavior and enriching knowledge about the material, such as in rock-engineering performance where visible discontinuities created by nature and adjacent areas (on a mm-scale) di ffering significantly in mechanical properties in tested material cause variability of tangent modules [33]. In the case of bond strength conformity tests, however, they will constitute an undesirable factor resulting from the imperfections of the measurement system and preparation of the sample. Ultimately, it reveals itself in a dispersion of results along with other uncontrolled factors. In this example, one can see a significant di fference between compliance and scientific tests. The same influencing factor is, in one case, a source of information, and in the other, an obstacle to the precise assessment.

The bond strength test, like most tests used to assess construction materials, is of a destructive nature; thus, it is not possible to repeat the action sequence on the same sample. This situation makes it impossible to isolate the impact of individual components from groups 2 and 3 on the result. Thus, from a practical point of view, a laboratory which performs a specific test obtains a limited test result pool (3 to 5 results) and the knowledge of the lab encompasses the device measurement characteristics and the final dispersion of the results.

If the tolerance limits are given in the description of the test method, it is assumed by default that observance of these limits ensures that group 2 interactions have a negligible impact on the test results. The variations related to material heterogeneity, sample preparation, and the test itself are manifested in the final dispersion of the results. Given the fact that a laboratory normally tests 3–5 samples, the listed components must be considered in conjunction.

Thus, in this situation, we can determine the components determined by measurement of force and diameter. These components can be included in the model equation. The rest of the components are contained in the dispersion of results; however, their impact remains unknown. It would be necessary to perform numerous inter-laboratory experiments in order to separate components related to the test from the components which result from the tested material or samples heterogeneity.

Such experiments are justified during the method validation period but few specifications indicate the method precision (standard deviation of repeatability and reproducibility), which could be treated as the uncertainty component.

### 3.1.1. Precision of Bond Strength Tests

Documents [29,30], which describe and recommend the use of bond strength tests, include no data on test precision. Numerous outcomes based on both repeatability and reproducibility tests indicate that a very large dispersion of results is characteristic for bond strength tests.

Table 1 shows the bond strength test results (average values of four lab tests for each adhesive in each laboratory) in the three research teams. The tests used 15 di fferent polymer-cement adhesive samples of 0.02 to 0.11 MPa adhesion range. The table also includes the result variability for individual adhesives in individual laboratories, as well as variations resulting from repeatability and reproducibility calculated in accordance with the ISO 5725 [34] based on all laboratories' results.

All variations in Table 1 are expressed as a *v* coe fficient of variation for better presentation:

$$v = \frac{s}{\overline{\sigma}} \, 100\% \tag{3}$$

where:

> *s*—standard deviation of repeatability or reproducibility

σ—mean value of bond strength in the result set with standard deviation calculated.



1 Outlier variance (outlier variance significant at the 99% level of confidence in the Cochran test); 2 Questionable variance (outlier variance significant at the 95% level of confidence in the Cochran test).

The data was subjected to the Grubbs test for outliers within the result set for a given adhesive. The test showed no outliers. The individual standard deviations for the three laboratories and each adhesive were subjected to the Cochran test. A questionable standard deviation value of 0.024 MPa (ν = 55.6%) was obtained for the 'e' sample and Laboratory 2. One standard deviation outlier of 0.021 MPa (ν = 33.4%) was obtained for sample 'n' and Laboratory 1. In each case, an increased standard deviation was recorded in a di fferent research team; thus, the questionable and outlier variance results were not rejected in the overall variance of repeatability and reproducibility calculations.

There is no statistically significant correlation between the standard deviation of repeatability and reproducibility values and the adhesion force value (respectively *r* = 0.13 and *r* = 0.38 for the critical value with the α = 0.05 coe fficient *r*cr = 0.514). There is no statistically significant correlation between intra-laboratory and inter-laboratory standard deviation either (*r* = 0.16) or between standard deviations in particular laboratories (*r* lies between 0.18–0.28).

The dispersion values are random and do not indicate that any of the laboratories involved in the experiment has demonstrated standard deviations, which were too high or too low. Minimum and maximum v values for repeatability are 9% and 42%, and for reproducibility 11% and 45% respectively. These values are very high and, thus, indicate that the test method is of low precision. Given that the adhesive assessment criterion is: σ > 0.08 MPa, the result variability may sugges<sup>t</sup> that this adhesive assessment test method involves a high risk.

### 3.1.2. Estimation of Bond Strength Results Uncertainty

The uncertainty estimation was carried out with the use of methods recommended in JCGM documents [20,21] and EA [25].

Considering the interaction impact assessment on the test results, the model equation, which makes it possible to determine the individual components influence on the total uncertainty of the adhesion test results, is

$$
\sigma = \frac{4F}{\pi D^2} + A\_\sigma \tag{4}
$$

where:

*A*σ—unknown value interaction impacts, which contribute to the result dispersion (includes material heterogeneity). The value *A*σ = 0—the error value which could be corrected is unknown; however, *A*σ uncertainty contributes to σ uncertainty. This uncertainty is revealed in a random dispersion of results expressed as the standard deviation of the σ value.

Other elements as in Equation (1).

The total value of standard uncertainty estimation σ is usually based on equation:

$$u\_c^2(y) = \sum \left(\frac{\partial f}{\partial \mathbf{x}\_i}\right)^2 u^2(\mathbf{x}\_i) \tag{5}$$

hence

$$
\mu\_{\sigma}^{2} = \left(\frac{\partial \sigma}{\partial D}\right)^{2} \mu\_{D}^{2} + \left(\frac{\partial \sigma}{\partial F}\right)^{2} \mu\_{F}^{2} + \left(\frac{\partial \sigma}{\partial A\_{\sigma}}\right)^{2} \mu\_{A\_{\sigma}}^{2} \tag{6}
$$

where:

> *uD*—standard uncertainty of *D uF*—standard uncertainty of *F*

*uA*σ—dispersion of σ results.

Expanded uncertainty is

$$M\_{\sigma} = k u\_{\sigma}^2 \tag{7}$$

where *k* is the coverage factor corresponding to 95 percent coverage interval.

Owing to the fact that there are no uniform uncertainty estimation principles and that, according to recommendations, uncertainty estimation should be based on the state of knowledge, laboratories may make di fferent assumptions about the value of *uD*, *uF*, *a*, and *k*, although they use the same equations as the starting point. For example, when estimating type B uncertainty related to force values, either measurement tolerances or specific uncertainty from the calibration certificate might be taken into consideration. Although *uA*σ laboratories mostly rely on historical data of the repeatability test on a larger number of samples in a given laboratory, the current series standard deviation might be used in many instances. Reproducibility standard deviation obtained from inter-laboratory experiments should be used, as recommended [25]. This, however, is a rare case.

*k* = 2 is sometimes taken as the coverage factor, although this assumption is not always justified (approximate normal distribution value corresponding with 95 percent coverage interval). *k* obtained from Student's *t*-distribution is also used, based on the e ffective number of degrees of freedom calculated from the Welch—Satterthwaite equation. The current approach outlined in the JCGM document [21] consists in the probability density distribution (PDF) propagation rather than the uncertainty propagation—this may also produce di fferent uncertainty results.

Table 2 presents some examples of how uncertainty components can be estimated and how can be used with the model equation presented herein. All of them have been created by the authors in accordance with the possibilities presented in the JCGM documents [20,21].


**Table 2.** Four example-approaches to uncertainty estimation.

> 1 Probability density function.

Figure 4a,b show the test results and coverage intervals resulting from the use of di fferent approaches towards uncertainty estimation (I, II, and IV). The uncertainties estimated with the use of di fferent approaches di ffer significantly, which seems particularly important. The detailed results are shown in Table 3.

**Figure 4.** Bond strength results with U value for three laboratories (1–3) and two materials (**a**) adhesive 'n'; (**b**) adhesive 'o'. Black error bars are based on actual standard deviation (approach II). Points min 1 and max 1 are 95% coverage intervals obtained in approach I (only B-uncertainty of *F* and *D v*alues), points min 2 and max 2 are 95% coverage intervals obtained by the Monte Carlo method in approach IV, where reproducibility standard deviation obtained from inter-laboratory comparisons was used.



The result differences in particular laboratories when testing the same adhesive are significant enough to affect the final conformity assessment against the requirements. The uncertainty value differences obtained by different laboratories using different approaches, and uncertainty values which exceed the test result value (e.g., approaches II and III for Lab 1, adhesive 'n') weaken the significance of such "uncertain" uncertainty when assessing the risk of incorrect assessment.

ISO 527 [31] based tests examine the tensile stress of plastics. Unlike bond strength testing, application conditions simulation is not the test objective; rather, the test results describe the material properties. A designer outlines the material conditions and, thus, determines material suitability for specific construction use.

#### *3.2. Tests of Tensile Strength and Their Influence on the Results*

The factors which can affect the test results can be divided into three groups, as with bond strength:


perpendicular stress, however, the design of the equipment and inaccurate clamping of the sample may imply the formation of tangential and torsional components that may affect the result.

### 3.2.1. Precision of Tensile Strength Tests

Tensile strength test precision is described in the ISO 527 standard applicable for various types of plastics. For PC repeatability and reproducibility standard deviations: sr = 0.18 MPa, sR = 0.89 MPa, for ABS sr = 0.18 MPa, sR = 1.93 MPa respectively.

As part of the laboratory proficiency testing organized by the Deutsches Referenzbüro für Lebensmittel-Ringversuche und Referenzmaterialien GmbH (DRRR), the standard reproducibility deviation values were determined: for PC sr=0.26, sR=0.73MPa, for ABS-GF sr=0.51MPa, sR=1.35MPa. DRRR inter-laboratory studies involved 69 laboratories, thus statistical data is very extensive.

The experiment results make it possible to conclude that, in contrast to the bond strength tests, according to ISO 527-1,2 [31] a random dispersion of results in the tensile strength test expressed as the reproducibility standard deviation is relatively small.

Variation coefficient *v* defined by Equation (3)—PC and ABS-GF respectively—repeatability: 0.4% and 0.9%, and reproducibility: 1.3 and 2.3%. Compared to the v value determined in bond strength tests, these values are significantly lower.

### 3.2.2. Estimation of Tensile Strength Results Uncertainty

The model equation which makes it possible to determine the individual components influence on the total uncertainty of tensile strength test results can be presented in the same way as was the case with bond strength tests (Equation (4)):

$$
\sigma = \frac{F}{ab} + A\_{\sigma} \tag{8}
$$

where:

*A*σ—unknown value interaction impacts which contribute to the results dispersion (includes material heterogeneity). Other elements as in Equation (2).

Standard uncertainty of σ is:

$$
\mu\_{\sigma}^{2} = \left(\frac{\partial \sigma}{\partial a}\right)^{2} \mu\_{a}^{2} + \left(\frac{\partial \sigma}{\partial b}\right)^{2} \mu\_{b}^{2} + \left(\frac{\partial \sigma}{\partial F}\right)^{2} \mu\_{F}^{2} + \left(\frac{\partial \sigma}{\partial A\_{\sigma}}\right)^{2} \mu\_{A\_{\sigma}}^{2} \tag{9}
$$

where:

> *ua,b,F*—standard uncertainty of *a*, *b*, *F*

*uA*σ—dispersion of σ results.

Uncertainties were estimated based on ISO 527 standard assumptions, i.e., force measurement with the use of a class 1 device, requirements for the sample shape: a = 10.0 ± 0.2 mm, b = 4.0 ± 0.2 mm.

Figure 5a,b show the test results and coverage intervals which result from the use of different approaches towards uncertainty estimation (I, II, and IV). The different approaches to uncertainty estimation differ slightly, especially when compared to the differences in the uncertainty values in the bond strength test. Therefore, the estimated uncertainties appear to provide the basis to claim that the risk of incorrect assessment can be determined in a credible manner. The data referring to the method precision given in the standard may be the basis for a uniform uncertainty estimation.

**Figure 5.** Tensile strength results with U value for three laboratories (X, Y, Z) and two materials (**a**) PC, (**b**) ABS. Black error bars are based on actual standard deviation (approach II). Points min 1 and max 1 are 95% coverage intervals obtained in approach I (only B-uncertainty of *F* and *D* values), points min 2 and max 2 are 95% coverage intervals obtained by the Monte Carlo method in approach IV, where reproducibility standard deviation is obtained from ISO 527-2 standard.
