Simulation of the surface and volume stress in the contact archwire/bracket for classical friction and critical contact angle

Mohammed Khalil El Kouifat; Houcine Zniker; Ikram Feddal; Bennaceur Ouaki

doi:10.1556/1848.2024.00764

International Review of Applied Sciences and Engineering

Volume/Issue:: Volume 15: Issue 3

Simulation of the surface and volume stress in the contact archwire/bracket for classical friction and critical contact angle

Authors:

Mohammed Khalil El Kouifat

Mohammed Khalil El Kouifat Rabat National School of Mines (ENSMR), BP: 753 Agdal-Rabat, Morocco

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elkouifatkhalil@gmail.com https://orcid.org/0000-0002-7130-6053

Houcine Zniker

Houcine Zniker Faculty of Sciences, Ibn Tofaïl University, Kenitra, Morocco

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Ikram Feddal

Ikram Feddal Department of Industrial and Civil Sciences and Technologies, Abdelmalek Essadi Tetouan, Morocco

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, and

Bennaceur Ouaki

Bennaceur Ouaki Rabat National School of Mines (ENSMR), BP: 753 Agdal-Rabat, Morocco

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Pages:: 424–434

Online Publication Date:: 02 Feb 2024

Publication Date:: 25 Oct 2024

Article Category:: Case Report

DOI:: https://doi.org/10.1556/1848.2024.00764

Keywords:: bracket arch wire interface; contact mechanics; orthodontic treatment; stress

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Abstract

Aim

The occlusion of the teeth is affected due to the appearance of micro-cracks resulting from maximum stresses in the contact zones between the wire and the bracket under normal and tangential loading. The objective of this study is to evaluate the surface and volume constraints in the presence of bonding and partial sliding zones on the contact surface between wires and supports. Knowledge of these stress fields will make it possible to better limit the surfaces where most of the micro-cracks occur. Indeed, the evaluation of the stress will facilitate the modelling or application of established micro crack models on this subject, because the initiation of micro-cracks often appears on the contact surface or just below it.

Materials and methods

In this study, the two most common situations of contact between the wire and the bracket were studied; the first corresponds to the situation where the wire is positioned in the center of the support (classic friction), and the second corresponds to the situation where the inclined wire touches the ends of the supports (critical contact angle).The MATHCAD software was used to simulate the damage zones for normal loading in the two cases studied (classic friction, critical contact angle). We proposed a Hertzian loading for the first case and a linear loading for the second case. Also, the effect of the additional load during wire tightening applied by the orthodontist was studied.

Results

The charge concentration is located above the contact zone, of the order of 0.3P0 (pressure per unit of arbitrary normal length), according to Mathcad simulation results. The adhesion zone/micro-slip zone contact generates the largest tangential load, which is directed towards the side experiencing the most stress. We also observed that the stick area shifts towards the recessed side when the additional load is applied. Additionally, comparing the configurations, the critical contact angle resulted in a higher maximum shear stress.

Abstract

Aim

Materials and methods

Results

1 Introduction

Malocclusion is the misalignment of teeth, or when the association between the upper and lower dental arches is incorrect. Orthodontics is the discipline that consists in promoting better dental occlusion by correcting the defects of the positioning of the teeth. For this purpose, orthodontics uses wires and orthodontic brackets which are fixed on the external face of the teeth. When the wire is engaged in the slot of the brackets it generates the necessary forces for orthodontic tooth movement [1–4].

During the metallurgical examination of the contact zone between two surfaces, it was often noticed that the morphology of these surfaces mainly depends on the loading mode imposed. Referring to the various (fatigue induced by small deflection) simulation tests [5], it has been noticed that under a given normal load P, the application of cyclic stress generates a certain asymmetry of the facies characterizing the micro-sliding generated on both sides. Thus, if the load is high enough that macro-sliding does not occur, the area located on the side of the eccentric will be more stressed than that located on the embedded side [6, 7]. Under these conditions, the alternating deformation will therefore be weaker if not zero on the side of the embedding, and consequently the micro-sliding can only occur on the side most stressed.

In this context, the roughness of the contact surface has an important effect on a variety of elements such as friction, wear, and other potential failures [8, 9]. For various dental materials, the coefficient of friction in various dental materials varies greatly depending on the surface roughness [10, 11]. A good understanding of this interaction is essential in dental applications because it affects how materials interact as well as their overall performance and durability provides an in-depth explanation of the complex interplay of forces and stresses that compose the behavior of contact zones by carefully examining the different loading directions given to the system.

Different numerical approaches exist in the literature to simulate the behavior of the archwire in contact with the bracket [12–15]. Similar geometries have been recreated in a variety of applications, including seal simulations [15] and coil springs [16]. As a result, Xinwen et al. created a finite element approach for calculating orthodontic force in the entire dentition utilizing a three-dimensional mandible, brackets, and archwire model. The simulation results showed higher accuracy with absolute force differences ranging from 0.5 to 22.7 cN and relative differences ranging from 2.5 to 11. 0%. To investigate support wire and support wire interactions, Jaeger et al. [17] used stainless steel and titanium-molybdenum alloy wires in their three support systems. The contact configurations are obtained by experimental and numerical investigations, they involve transitions that cause large variations in the effective wire stiffness (EWS). EWS is at its lowest when the wire touches one wing and increases by up to 300% when the wire touches two wings on the opposite side. This study improves the basic knowledge and has clinical implications for multi-support-wire contact arrangements.

As a result, in order to evaluate the volume stress and the surface stress in the presence or absence of an additional loading during contact between the cylindrical wire of diameter 0.4572 mm and the square slot bracket of 0.5588 mm, a simulation using Mathcad version 14 was carried out for two imposed normal loads, we have studied two of the most encountered cases of the contact between the wire and the bracket, the first case where the wire is located in the middle of the bracket, and the second case where the inclined wire touches the extremes of the brackets [18]. We have proposed a Hertzian loading for the first case and a linear loading for the second case. Also, we have studied the effect of the additional load during the tightening of wires applied by the orthodontist.

2 Materials and methods

2.1 Materials

We have modelled the contact between a wire with a diameter of 0.4572 mm made of 304L stainless steel in accordance with the ASTM A240 [19] standard and a support with a slot of 0.5588 mm for the two most frequent cases [20]. The simulation of the contact between the orthodontic wire and the bracket is performed using Mathcad 14 software (Fig. 1a and b).

Fig. 1.

Classical friction (a) and critical contact angle (b)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Since the wire has a circular section, while the slot is with a rectangular form, we can see a contact between a plane and a cylinder as shown in Fig. 2.

Fig. 2.

Contact between wire and bracket (cylinder/plane)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

2.2 Methods

2.2.1 Volume stress

The study consists in evaluating the surface stress, volume stress, and more particularly the maximum shear stress, as well as in the presence of an additional tangential load at the level of the contact between the wire and the brackets. For the case of classic friction, the critical contact angle is obtained in a partial slip condition, when the tangential load never exceeds the product of the normal force P by the coefficient of friction (Q < µP) [21]. Note that the preload applied by the orthodontist placing the bracket in contact with the wire is the source of the initial stress.

The contact modelling for the two configurations mentioned above are shown in Fig. 3.

Fig. 3.

Modelling of the contact for classical friction (a) and critical contact angle (b)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Equations ((1)–(3)) theoretically demonstrated provide a basis for determining the state of stress induced in a body under a given pressure p (x) and for a shear distribution q (x). For an infinitesimal element dξ located on the contact surface, xthe stresses

σ_{x x}, σ_{y y}, τ_{x y} a n d τ_{x y \max}

generated below the contact can be expressed by K.L. Johnson [22].

σ_{x x} = \frac{2}{π y} (\int_{- a}^{a} \frac{p (ξ) {(x - ξ)}^{2} d ξ y^{2}}{{[{(x - ξ)}^{2} + y^{2}]}^{2}} + \int_{- a}^{a} \frac{q (ξ) {(x - ξ)}^{3} d ξ y}{{[{(x - ξ)}^{2} + y^{2}]}^{2}})

(1)

σ_{y y} = \frac{2}{π y} (\int_{- a}^{a} \frac{p (ξ) d ξ y^{4}}{{[{(x - ξ)}^{2} + y^{2}]}^{2}} + \int_{- a}^{a} \frac{q (ξ) (x - ξ) d ξ y^{3}}{{[{(x - ξ)}^{2} + y^{2}]}^{2}})

(2)

τ_{x y} = \frac{2}{π y} (\int_{- a}^{a} \frac{p (ξ) (x - ξ) d ξ y^{3}}{{[{(x - ξ)}^{2} + y^{2}]}^{2}} + \int_{- a}^{a} \frac{q (ξ) {(x - ξ)}^{2} d ξ y^{2}}{{[{(x - ξ)}^{2} + y^{2}]}^{2}})

(3)

2.2.2 Surface stress without additional loading

Initially, we recall that in the presence of a micro-sliding zone, the pressure and stress distributions on the surface (Fig. 4) are given by:

Fig. 4.

Micro-slip zone (a) and stick zone (b)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

-For $| x |$ ≤ a:

p (x) Normal load distribution

(4)

-For − a ≤ x ≤ a1 and a1 ≤ x ≤ a:

q (x) = q^{'} (x) = μ p (x)

(5)

-For $| x |$ ≤ a1:

q (x) = q^{'} (x) + q^{″} (x) = μ p (x) - μ \frac{a_{1}}{a} p_{1} (x)

(6)

Where a1 represents the limit of the stick area and p1(x) is the normal load for the a1 such as:

a_{1} = a \sqrt{1 - \frac{Q}{µ P}}

(7)

Q and P respectively represent the tangential force and the normal force applied to the contact.

2.2.3 Formulations of the Surface stress with additional loading

Therefore, by taking as a starting point the model established by D. Nowell and DA Hills [23], one can consider the two-dimensional problem of a simple contact between a cylinder and a half-plane as schematized on Fig. 5a and b.

Fig. 5.

Cylinder-half-plane contact whether or not there is additional loading

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

If, under the effect of the stresses σ1 and σ2 applied on both sides of the contact (Fig. 5b), the limits c1 and c2 of the stick zone have the same value e, these will no longer be symmetrical with respect to the center of the contact and the additional tangential load is given by:

| σ_{1} - σ_{2} | = σ_{0}

(8)

c_{1} = a_{1} + e

(9)

c_{2} = - a_{1} + e

(10)

Where –a₁ and a₁ represent the limits of this zone in the absence of the additional loading as shown in Fig. 6).

Fig. 6.

Action of a normal and tangential load distributed on an infinite half-plane, case of the Hertzian load

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

The distribution of the contact pressure on the contact zone is given by:

q (x) For | x | \leq a :

(11)

Where p₀ is given by:

p_{0} = \frac{2 P}{µ a}

(12)

In this case, the distribution of the shear stresses generated on the contact surface is such that:

q (x) = q ’ (x) = µ q (x) For - a \leq x \leq c 1 and c 1 \leq x \leq a

(13)

q (x) = q ’ (x) + q^{″} (x) For c 2 \leq x \leq c 2

(14)

Where q’’(x) is to be determined in this particular case and μ represents the coefficient of friction.

The evaluation of the function q’’(x) is solved by N.I. Muskhelishvilli et al. [24] and is in the following form:

q^{″} (x) = - \frac{\sqrt{(x - c_{2}) (c_{1} - x)}}{π} \int_{c_{2}}^{c_{1}} \frac{(\frac{μ p_{0}}{a}) ξ - (\frac{σ_{0}}{4})}{\sqrt{(ξ - c_{2}) (c_{1} - ξ)} (ξ - x)} d ξ

(15)

And will require the consistency of the integral between c₁ and, c₂ namely:

\int_{c_{2}}^{c_{1}} \frac{(μ p_{0} t / a) - (σ_{0} / 4)}{\sqrt{(t - c_{2}) (c_{1} - t)}} dt = 0

(16)

On the other hand, the evaluation of equation (16) is reduced to:

c_{1} + c_{2} = \frac{σ_{0} a}{2 µ σ_{0}}

(17)

We are, thereafter, led to establish another equation taking into account the total tangential load applied to the contact imprint (-a, a). Thus, we have:

Q = \int_{- a}^{+ a} q (x) dx = μ P + \int_{C_{2}}^{C_{1}} q^{″} (x) dx

(18)

Q = μ P - μ P {(\frac{c_{1} - c_{2}}{2 a})}^{2}

(19)

Is : c_{1} - c_{2} = 2 a \sqrt{1 - \frac{Q}{μ P}}

(20)

From equations (15) and (17) we deduce the formulated result for this proposal by D. Nowell and D.A. Hills, namely:

c_{1} = \frac{σ_{0} a}{4 μ P_{0}} + a \sqrt{1 - \frac{Q}{μ P}}

(21)

c_{2} = \frac{σ_{0} a}{4 μ P_{0}} - a \sqrt{1 - \frac{Q}{μ P}}

(22)

The first case: classical friction

In this application we choose the load as a maximum in the middle and it tends towards zero in the edges of the contact, The possible load to model in this distribution is the normal Hertzian load expressed in the form [25]:

p (x) = p_{0} \sqrt{1 - {(\frac{x}{a})}^{2}}

(23)

Where a is the half-width of the contact and p0 the maximum load in the middle of the contact given by:

p_{0} = \frac{2 P}{π a}

(24)

An example of this Herz distribution is given in Fig. 6 [26].

The second case: Critical contact angle

For the case where the wire is inclined (critical contact angle), we choose the load in which its value is maximum at the edges of the half contact widths, and decreases when the contact area decreases while moving away. The normal load is in a form taking into account the low forces applied with no concavity and convexity. This type of distribution is given in Fig. 7.

p (x) = p_{0} (1 + \frac{x}{a})

(25)

Fig. 7.

Distribution of a linear load

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

3 Results

3.1 Experimental results

We performed fretting experiments regarding the interaction of dental wire and brackets in our previous experimental research [27, 28]. Using scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS), significant results were obtained. When we looked at these results, we found some interesting data regarding the contact area on the eccentric side of the beam. It was about 0.02 mm².This contact zone has diverse characteristics that are worth discussing. The results indicate the presence of two unique zones: a stick zone (zone 1) and a micro-slip zone (zone 2), as shown in Fig. 8.

Fig. 8.

The contact area between wire and bracket, stick zone (1), micro-slip (2). Outside the contact (3) [27 (CC BY-NC 4.0)]

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

3.2 Simulations results

We will give the analytical results for the proposed models for the two dental positions in this section. To validate the presence of the experimentally observed micro-slip and stick zone.

3.2.1 The first case: classical friction

Stresses of volume: Hertzien load

In this configuration we have plotted the contours of the volume stresses of $\frac{σ_{x x}}{p_{0}}$ , $\frac{σ_{y y}}{p_{0}}$ , $\frac{τ_{x y}}{p_{0}}$ and $\frac{τ_{x y \max}}{p_{0}}$ in the plane (x, y) for a coefficient of friction of 0.3, a half-width of 0.25 mm, with a normal load of 5 N mm⁻¹. The graphical representation of stress distribution is shown in Figs 9a, 9b, and 9c [17].

Fig. 9a.

Iso-stresses $\frac{σ_{xx}}{P_{0}}$ in the case of Hertzien loading for µ = 0.3

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Fig. 9b.

Iso-stresses $\frac{σ_{yy}}{P_{0}} in the case of$ Hertzien loading for µ = 0.3

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Fig. 9c.

Iso-stresses $\frac{τ_{xy}}{P_{0}} in the case of$ Hertzien loading for µ = 0.3

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Furthermore, we conducted comprehensive simulations using Mathcad to generate iso-stress plots for volumes $\frac{σ_{x x,}}{P_{0}}$ , $\frac{σ_{y y}}{P_{0}}$ and $\frac{τ_{x y}}{P_{0}}$ under Hertzian loading conditions. For a coefficient of friction of 0.3 and a width of 0.5 mm, our analysis revealed noteworthy observations. Notably, $σ_{x x,}$ closely approximated P0 and exhibited concentration at a depth of approximately 0.2 mm. In contrast, the maximal value of $\frac{σ_{y y,}}{P_{0}}$ , around 0.8 was situated at a depth of 0.18 mm. $\frac{τ_{x y}}{P_{0}}$ exhibited a peak value of 0.3, as illustrated in Fig. 10.

Fig. 10.

Maximum shear iso-stresses $\frac{τ_{xy \max}}{P_{0}} in the case of$ Hertzien loading for µ = 0 (a) and µ = 0.3 (b)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

\frac{τ_{x y m a x}}{P_{0}}

is a fundamental parameter that was introduced to evaluate the maximum shear stress (Fig. 11). This parameter is given:

τ_{xymax} = \sqrt{{(\frac{σ_{x x +} σ_{y y}}{2})}^{2} + {τ_{x y}}^{2}}

(26)

Fig. 11.

The distribution of the tangential load along the contact for the case of a coefficient of friction of µ = 0.3 for σ₀ = 0 MPa (a) and σ₀ = 7.316 MPa (b)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Significantly, the absence of tangential loading led to maximum shear iso-stresses symmetrically centered at a depth of 0.2 mm. On the other hand, in the presence of tangential loading, the maximum shear stress underwent a shift of 0.3 units towards the most stressed side.

Surface stress: Hertzien load

In this part of the paper, we provide the findings of our research into the deterioration modes caused by the interplay of a tangential load and stress on the wire in contact with the bracket. For this purpose, we used a Mathcad algorithm based on experimental data from wire-bracket contact. The specific parameters adopted for the analysis are as follows:

Half-width of the contact: a = 0.25 mm (Half of 0.4572 mm)
Load per unit of total tangential length: Q = 1.2 N
Pressure per unit of arbitrary normal length: P₀ = 12.7 N
Coefficient of friction: µ = 0.3
Additional tangential load: σ₀ = 7.316 MPa.

To be able to highlight these degradation modes generated under the effect of the tangential load in the presence of an additional load on the wire in contact with the bracket, we use the Mathcad program that has been developed with the experimental data adopted during the wire-bracket contact [29], namely:

Half-width of the contact: a = 0.25 mm (Half of 0.4572 mm)
Load per unit of total tangential length: Q = 1.2 N
Pressure per unit of arbitrary normal length: P0 = 12.7 N
Coefficient of friction: µ = 0.3
Additional tangential load: σ0 = 7.316 MPa.

3.2.2 The second case: critical contact angle

We have also plotted the contours of the volume stresses of $\frac{σ_{x x}}{p_{0}}$ , $\frac{σ_{y y}}{p_{0}}$ , $\frac{τ_{x y}}{p_{0}}$ and $\frac{τ_{x y m a x}}{p_{0}}$ for this linear normal load with the same conditions described above and with a critical contact angle $θ_{c} = 3 °$ . The results are shown in Figs 12a, 12b, 12c, and 13.

Fig. 12a.

Iso-stresses $\frac{σ_{xx}}{P_{0}} in the case of$ linear loading for µ = 0.3

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Fig. 12b.

Iso-stresses $\frac{σ_{yy}}{P_{0}} in the case of$ linear loading for µ = 0.3

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Fig. 12c.

Iso-stresses $\frac{τ_{xy}}{P_{0}} in the case of$ linear loading for µ = 0.3

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

Fig. 13.

Iso-stresses $\frac{τ_{x y m a x}}{P_{0}} in the case of$ linear loading for µ = 0 (a) and µ = 0.3 (b)

Citation: International Review of Applied Sciences and Engineering 15, 3; 10.1556/1848.2024.00764

We have simulated on Mathcad, the Iso-stresses of volumes $\frac{σ_{x x}}{p_{0}}$ , in the first case of a linear loading with a coefficient of friction of 0.3 for a width of 0.5 mm. Note that $\frac{σ_{x x}}{p_{0}}$ max reaches the value 1.6 P0 (pressure per unit of arbitrary normal length).

For $\frac{σ_{y y}}{P_{0}}$ the maximum value is 1.4 and located around 0.1 mm in depth. Shear stress $\frac{τ_{x y}}{P_{0}}$ reaches the maximum value of 0.6, located at the edges of the contact 0.25 mm.

On the maximum shear load in the absence of tangential loading, we note that the value of 0.5 is concentrated at the edge 0.25 mm. On the other hand, in the presence of friction, the value reaches 0.7. This explains the noticeable degradation that is often observed at the edge of the bracket as reported in many research [14].

4 Discussions

4.1 Experimental results

The stick zone (Zone 1) plays an important role in understanding the fretting behaviour of the dental wire and the bracket surface. This zone provides a significant interaction between the two materials as it maintains the stability of the system under the fretting circumstances. The ability of the stick zone to restrict relative motion between the wire and the bracket helps to prevent extreme wear and possible failure.

The micro-slip zone (Zone 2), which is close to the stick-zone, on the other hand, exhibits a different element of the fretting phenomena. The specific feature of oxides inside this zone shows that material deterioration is restricted. This degradation is most likely the result of cyclic loading and relative movement, which causes surface layer disintegration and probable chemical interactions between the materials involved.

The appearance of a stick zone and a micro-slip zone within the contact zone emphasizes the friction problems between the wire and the brackets. This interaction illustrates the stability between sticky and abrasive wear mechanisms. Although the bonding zone protects by reducing relative movement, the deterioration of the micro-slip zone demonstrates the potential difficulties in preserving the integrity of the material over time.

The observations of this study have an impact on the design and performance of dental brackets and wires. The results revealed a complex interplay that combines a stick zone and a micro-slip zone, as established by SEM and EDS analysis. Hence, it may be possible to mitigate fretting effects by improving the composition and surface treatment of materials in the contact area, especially in cyclic load conditions. Future research could examine the chemical and mechanical complexities of these zones, potentially leading to measures to improve the longevity and dependability of dental orthodontic systems.

4.2 Analytical prevention results

4.2.1 The first case: classical friction

•Stresses of volume: Hertzien load

The analysis of our results provides good knowledge of the stress distribution within the system studied, particularly under specific loading conditions. The contour plots and iso-stress simulations provide valuable visual insights into how distinct parameters influence the distribution of stresses. Valuable visual information was obtained through contour plots and iso-stress simulations on how different parameters affect the stress distribution.

The dominance of $σ_{xx}$ over $σ_{yy}$ highlights the important influence of the applied normal load on the stress distribution along the x-axis. The location of σ_xx at a shallow depth of approximately 0.2 mm highlights the importance of surface stresses and its possible influence on the behaviour of the material.

The presence of the micro-sliding zone, represented by $\frac{τ_{xy}}{p_{0}}$ , reflects the importance of the effect of frictional stresses. The displacement of peak shear stress due to tangential loading emphasizes the critical interaction between friction and stress models, particularly in scenarios involving relative motion.

Practical implications for design and engineering considerations were obtained from these identified models and stress variations. Information from this analysis can be used to control material selection, geometry, and loading conditions to reduce stress concentrations and improve overall system durability.

Surface stress: Hertzien load

Mathcad software results produced using the specified parameters provide insight into the different degradation modes that manifest due to the combined impacts of tangential loading and additional loading on the wire/bracket contact. We can further explore the implications of the chosen circumstances by applying these parameters to the analysis.

The half-width of the contact (a = 0.25 mm) is essential to establish the extent of contact between the wire and the support. This parameter affects the stress and deformation distribution in the contact region, which can affect wear patterns and material reactions.

Friction-induced stresses are produced when the load per unit total tangential length (Q = 1.2 N) interacts with the coefficient of friction (= 0.3). The importance of this load regarding the friction coefficient influences the friction forces at work, which in turn influence the wear and deformation mechanisms.

The normal forces experienced within the contact are influenced by the pressure per unit of arbitrary normal length (= 12.7 N). This parameter, associated with the additional tangential load (= 7.316 MPa), modifies the stress distribution, which can lead to localized degradation of the material.

Through the combination of these parameters simulated via the Mathcad program, we can observe and analyse the degradation modes that develop under the specified conditions. The interaction of tangential and normal loads, together with the coefficient of friction, shows the complex nature of wear, deformation and material behaviour upon contact with the support wire.

4.2.2 The second case: critical contact angle

The results of our investigation provide important insights into the stress distribution under linear normal loading. Volume stress plots provide a visual picture of how several parameters influence stress patterns.

Under linear loading, the significant increase in $\frac{σ_{x x}}{p_{0}}$ , reaches 1.6 P0 (pressure per unit of arbitrary normal length), indicating increased stress concentration along the x-axis. This concentration of stresses, particularly at the edge of the support, can have consequences on the deformation of the material and wear at this location.

The presence of a maximum value of 1.4 for $\frac{σ_{y y}}{p_{0}}$ , at a shallow depth of 0.1 mm shows the importance of normal loading to influence the stress distribution. This observation is relevant for understanding the behavior of the system at applied loads.

The apparent shear stress $\frac{τ_{x y}}{p_{0}}$ at the edges of the contact zone, which reaches 0.6, indicates the possibility of local material deterioration due to shear forces. This phenomenon can participate to wear and material degradation, especially in areas experiencing higher stress concentrations.

The difference between the maximum shear stresses in the absence and presence of transverse loading highlights the important effect of friction on the evolution of stress patterns. The increase in frictional shear stress helps explain the pronounced deterioration reported at the support edge in previous research. The increase in frictional shear stress helps explain the pronounced degradation reported by Khan et al. at the edge of the support [30] in previous research

5 Conclusion

In this study, we carried out a simulation on MATHCAD14 for a better understanding of the concentration of the maximum shear load for the case of a Hertzian load, which models the case of classical friction, and a linear load allowing us to model the real case of a critical contact angle.

The simulations showed that the concentration of the load is located above the contact; it is of the order of 0.3P₀ (pressure per unit of arbitrary normal length). Moreover, we have simulated the contact between the wire and bracket in Mathcad14 using the Johnson formulations for the case of a linear and Hertzian loading with an additional tangential load, we clearly see that the maximum tangential load is obtained at the interface between the stick-zone and the micro-slip zone and is oriented towards the most stressed side. We also note that the stick zone is shifted towards the recessed side when the additional load was applied and for comparison between the two configurations studied, we found that the maximum shear stress is greater in the case of the critical angle contact.

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International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
Address of the institute: Faculty of Engineering, University of Debrecen
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2023
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Scimago H-index	11
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International Review of Applied Sciences and Engineering
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Article Processing Charge	1100 EUR/article
Regional discounts on country of the funding agency	World Bank Lower-middle-income economies: 50% World Bank Low-income economies: 100%
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International Review of Applied Sciences and Engineering
Language	English
Size	A4
Year of Foundation	2010
Volumes per Year	1
Issues per Year	3
Founder	Debreceni Egyetem
Founder's Address	H-4032 Debrecen, Hungary Egyetem tér 1
Publisher	Akadémiai Kiadó
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	2062-0810 (Print)
ISSN	2063-4269 (Online)