## 2 Materials and Methods  
### Characterization of the Stress Field Around Volumetric Flaws  
Metallic structures subjected to mechanical stresses may present geometric imperfections, which in turn constitute mechanical stress concentrators. A factor, \(K_{c}\), representing the mechanical stress concentrator, can be defined as the ratio between the maximum value of the mechanical stress at the defect tip, \(\sigma_{max}\), and the value of the nominal stress, \(\sigma\), from the net section of the studied component. The nominal stress, \(\sigma\), is the mechanical tension in areas far from the defect area.  
Mathematically, this ratio is defined as follows:  
\[K_{c}=\frac{\sigma_{max}}{\sigma} \tag{1}\]  
The severity of the mechanical stress concentrator depends on the geometrical configuration of the analyzed flaw [17]. For the special case of a circular gap with a radius of \(a\), the solutions for the radial and circumferential stresses can be obtained depending on the distance \(r\) and the angle \(\theta\)[18].  
The normalized mechanical stresses for this situation are described by the following equations:  
\[\widetilde{\sigma}_{r}=\frac{\sigma_{r}}{\sigma_{\infty}}=\frac{1}{2}\bigg{(} 1-\frac{a^{2}}{r^{2}}\bigg{)}-\frac{cos2\theta}{2}\bigg{(}\frac{3a^{4}}{r^{4}} -\frac{4a^{2}}{r^{2}}+1\bigg{)} \tag{2}\]  
\[\widetilde{\sigma}_{\theta}=\frac{\sigma_{\theta}}{\sigma_{\infty}}=\frac{1}{2 }\bigg{(}1-\frac{a^{2}}{r^{2}}\bigg{)}-\frac{cos2\theta}{2}\bigg{(}\frac{3a^{ 4}}{r^{4}}+1\bigg{)} \tag{3}\]  
where \(\sigma_{r}\) represents radial stress, and \(\sigma_{\theta}\) represents hoop stress.  
Figure 1 shows the distribution of the normalized stress, \(\widetilde{\sigma}_{\theta}\), as a function of the normalized radius \(r/a\) along the x axis (\(\theta=0\)). The normalized stress drops quite quickly from value 3 to value 1 with an increase in the \(r/a\) ratio. We consider this kind of mechanical stress distribution around a defect in the analysis of the morphology of the hydrides located at the tips of the volumetric defects on the Zr-2.5%Nb alloy CANDU pressure tube.  
Also, the stress field generated at the flaw tip, considering planar dimensions, could be characterized by the fracture mechanics parameter \(K_{I}\), which is the stress intensity factor for _Mode I_ (opening mode) of the crack faces. The samples tested in this work were of the SENT (single-edge notched tension) type and were mechanically subjected to tensile stress, as schematically shown in Figure 2.  
The stress intensity factor for _Model_ (opening mode) of the crack faces, \(K_{I}\), is given by the following relationship [20]:  
\[K_{I}=\frac{P}{B\sqrt{W}}f\left(\frac{a}{W}\right) \tag{4}\]  
Figure 1: Distribution of normalized stress (\(\widetilde{\sigma_{\theta}}\)) as a function of distance.  
with:  
\[f\Big{(}\frac{a}{W}\Big{)}=\frac{\sqrt{2 tan\big{(}\frac{\pi a}{ZW}\big{)}}}{cos( \frac{\pi a}{ZW})}\Big{[}0.752+2.02\Big{(}\frac{a}{W}\Big{)}+0.37\Big{(}1-sin \Big{(}\frac{\pi a}{2W}\Big{)}\Big{)}^{3}\Big{]} \tag{5}\]  
Relationships (4) and (5) were used to calculate the stress intensity parameter, \(K_{I}\), for the experimental samples made of the Zr-2.5%Nb alloy CANDU pressure tubes used in the present study.  
To analyze the stress field influence around the stress concentrator, that is, the flaw tip, on the zirconium hydride morphology, both parameters, \(K_{c}\) and \(K_{I}\), were used.