# Comparison of Stress Concentration Factors Obtained by Different Methods
## 3 Description of the Notch Effect  
The so-called dimensionless stress concentration factor \(\alpha\), also called form factor, is used to describe the notching effect. Two kinds of stress concentration factors can be distinguished. In the technical literature, it is most commonly defined as the ratio of the local maximum stress \(\sigma_{\text{max}}\), e.g., at the root of the notch and the nominal (mean, average) stress value \(\sigma_{\text{nom}}\) in the weakened cross-section, according to the equation  
\[\alpha=\frac{\sigma_{\text{max}}}{\sigma_{\text{nom}}}. \tag{1}\]  
However, there is some inconsistency in practice when applying the value of the nominal stress to consider its value in the unweakened cross-section. However, this difference may disappear if the dimensions of the notch are small compared to the overall cross-sectional dimensions. The stress concentration factor takes into account the stress increase induced by the change in the local geometry of the body as well as the effect of detailed changes in the force flux in the cross-section. In general terms, however, the factor is related to the stress component that gives it the largest value. It is therefore sometimes referred to by indices of principal stress, e.g.,  
\[\alpha_{1}=\frac{(\sigma_{\text{max}})_{1}}{\sigma_{\text{nom}}}. \tag{2}\]  
In the case of uniaxial tension, e.g., in the case of circular cross-sections of diameter \(d\) subjected to a tensile force \(F\), the nominal tension itself will be expressed by the relation  
\[\sigma_{\text{nom}}=(\sigma_{\text{nom}})_{1}=\frac{4F}{\pi\cdot d^{2}}. \tag{3}\]  
In the case of complex loading of the component where multi-axial stresses are assumed, it is appropriate to consider transforming the stress components into an equivalent uniaxial stress,  
\[(\sigma_{\text{max}})_{\text{eq}}=\
u\cdot(\sigma_{\text{max}})_{1}, \tag{4}\]  
Figure 1: The 6 configurations of shape-discontinuity-dimension-load stress concentrators identifiable as (**a–f**).  
which introduces the equivalent stress and the associated equivalent stress concentration factor into the calculation  
\[\alpha_{\text{eq}}=\frac{\left(\sigma_{\text{max}}\right)_{\text{eq}}}{\sigma_{ \text{nom}}}=\
u\cdot\alpha_{1}. \tag{5}\]  
\(\
u\) is a dimensionless stress coefficient (factor) dependent on the ratio of the stress components at the most stressed location and on the strength hypothesis considered. For example, according to the [PERSON] hypothesis, which can be used in the case of minimal plastic deformation occurring, with a simple type of loading, not considering relaxation processes in a triaxial tension with a triad of principal stresses \(\sigma_{1}>\sigma_{2}>\sigma_{3}\), the coefficient \(\
u\) can be expressed according to the relation  
\[\
u^{\text{VM}}=\frac{\left(\sigma_{\text{max}}\right)_{\text{eq}}^{\text{VM} }}{\sigma_{1}}=\frac{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}-\sigma _{1}\sigma_{2}-\sigma_{2}\sigma_{3}-\sigma_{3}\sigma_{1}}}{\sigma_{1}}. \tag{6}\]  
In the case of this hypothesis and considering the biaxial tension \(\left(\sigma_{3}=0\right)\), this coefficient is determined by  
\[\
u^{\text{VM}}=\sqrt{1-\frac{\sigma_{2}}{\sigma_{1}}+\left(\frac{\sigma_{2} }{\sigma_{1}}\right)^{2}}. \tag{7}\]  
The theoretical stress concentration factor related to an ideally elastic material generally corresponds to a higher stress concentration than actually occurs in the element due to the initiation of local plastic deformations. For this reason, the so-called effective stress concentration factor, which is equivalent to the effective notch factor, is introduced. With more significant plastic deformations, the shape of the notch already changes and the stress reduction is already significant. In practice, the resulting stress-strain state can be expressed analytically by the so-called Neuber rule, which, among other things, relates the stress concentration factor to the deformation. However, this rule loses accuracy with increasing degree of plasticization [10; 11].