## 3 The Models
### Determining the Normal Force of Action through the Hardness Method  
Knowing, according to Brinell method [35], the hardness of the component in which the ball of diameter \"\(D\)\" left its mark and having the diameter \"\(d\)\" of the spherical indentation, the force with which the component part was acted upon can be determined, using a reasoning similar to the one applied to hardness determination by the Brinell method. According to the diagram depicted in Figure 3, the value of the normal force, that actsin static conditions, is given by relation (7).  
The hardness formula, according to [40], with normal force in kgf, and \"\(D\)\" and \"\(d\)\" in millimeters, is as follows:  
\[HB=\frac{2\cdot N}{\pi\cdot D\cdot\left(D-\sqrt{D^{2}-d^{2}}\right)} \tag{6}\]  
Figure 6: Diagram of forces for the mathematical model proposed.  
wherein  
\[N=\frac{\pi\cdot D\cdot HB\cdot\left(D-\sqrt{D^{2}-d^{2}}\right)}{2} \tag{7}\]  
The angle of the spherical cap that penetrated the material can be obtained from the triangle \(AOB\), as shown in Figure 3.  
\[\sin\left(\frac{\varphi}{2}\right)=\frac{\frac{d}{2}}{\frac{D}{2}} \tag{8}\]  
Wherein the depth with which the ball penetrates the material is determined from the triangle \(AOB\) and is described by the following relationship:  
\[h=\frac{D}{2}-\frac{D}{2}\cdot\cos\left(\frac{\varphi}{2}\right) \tag{9}\]  
The value of the normal force, which deforms the tube, described by relation (7), is compared with that of the proposed model given in relations (4) and (5).  
Determining the Normal Force of Action through Elastic-Plastic Contact Regime with Spherical Indentation  
During contact, the material usually begins yielding at very small interferences, especially in the case of metals. Therefore, the elastic regime only covers a small range of interferences. On the other hand, the fully plastic regime occurs at very large interferences. For the majority of contacts, part of the material is in the elastic regime, and part of it undergoes plastic deformations. This is called the elastic-plastic regime. So far, there are no closed-form solutions for elastic-plastic contacts that arederived from fundamental principles [30].  
Determining the normal force of action by the method described in [27] requires knowledge of the material properties, namely \(E_{1,2}\)--[PERSON]'s modulus for the tube and ball, respectively, in MPa, \(S_{y}\)--yield strength in MPa, and [PERSON]'s ratio for the tube with the role of the part where the deformation takes place and the ball with the role of the spherical indenter.  
\[\frac{1}{E}=\frac{1-\
u_{1}^{2}}{E_{1}}+\frac{1-\
u_{2}^{2}}{E_{2}} \tag{10}\]  
The equivalent radius, depending on the cylinder, \(R_{1}\), and ball, \(R_{2}\), in millimeter is as follows:  
\[\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \tag{11}\]  
A dimensionless factor is defined, using the notation from Figure 3  
\[Q=\frac{E}{S_{y}}\cdot\frac{h}{\frac{d}{2}} \tag{12}\]  
where, based on thenotation from Figure 3, the radius of truncated contact areas is determined:  
\[\frac{d}{2}=\sqrt{\left(\frac{D}{2}\right)^{2}-\left(\frac{D}{2}-h\right)^{2}} \tag{13}\]  
\[At=\pi\cdot\frac{d^{2}}{4} \tag{14}\]  
\[\frac{A\prime}{A}=0.05\cdot\left(\ln Q\right)^{2}-0.57\cdot\left(\ln Q\right) +2.41 \tag{15}\]  
where \(A\) and \(A^{\prime}\) are the real and truncated contact areas.  
When \(Q>1.78\), the elastic-plastic small deformation phase persists until \(Q\leq 21\). For \(21<Q\leq 400\), contact occurs in the elastic-plastic medium deformation phase, and it has been considered to be consistent with the conventional hardness relation (note that this is the value assumed in their model). When \(Q>400\), the contact is in elastic-plastic large deformation phase [27].  
Depending on the value of Q, for the cases analyzed, according to [27]  
\[\frac{N}{A}=2.9\cdot S_{y} \tag{16}\]  
\[N=A\cdot 2.9\cdot S_{y} \tag{17}\]  
Hence, the value of the normal force \"\(N\)\", described by (17), is compared with the one obtained by solving the system given by relations (4) and (5).