Diameter ratio and friction coefficient effect on equivalent plastic strain (PEEQ) during contact between two brass solids

Abstract The study of contact mechanics is rapidly expanding in scope and importance. However, most studies of spherical do not spend significant time comprehensively analysing PEEQ, especially in the effect of friction coefficient and diameter ratio. This study aims to introduce comprehensive research examining diameter ratio and friction coefficient influencing equivalent plastic strain (PEEQ) during contact between two brass solids. In this study, the finite element method analysed an elastic-perfectly plastic brass material. The finite element model was two hemispheres with diameter ratios ranging from 1 to 5. In addition, the diameter in the upper hemisphere is 17.5 mm in all variations, while the bottom hemisphere follows the diameter ratio. The current study confirmed that the finite element results agreed well with the previous results’ analytical contact models. The findings also revealed that differences in ratio diameter and friction coefficient were significant to PEEQ. Therefore an increase in the coefficient of friction causes an expansion of the maximum PEEQ area for a diameter ratio of 1 and a reduction in the PEEQ maximum area for diameter ratios of 2 to 5. Expansion and contraction on the PEEQ area indicated that the contact radius widens and contracts as the coefficient of friction and diameter ratio change. Further research is required to investigate the effect of other parameters in PEEQ analysis, such as material properties and multiple cycle loading conditions. Furthermore, the practical implications of these findings may contribute to our understanding of engineering design and failure analysis.


PUBLIC INTEREST STATEMENT
The scope and relevance of contact mechanics study are constantly rising; nevertheless, most spherical studies do not devote enough effort to adequately analyse equivalent plastic strain (PEEQ), notably the effect of friction coefficient and diameter ratio.This work presented a detailed investigation of the impacts of diameter ratio and friction coefficient on PEEQ during contact between two brass solids.The finite element method used in the study was to analyse an elastic-perfectly plastic brass material.The finite element model consisted of two hemispheres with diameter ratios ranging from one to five.The data showed that changes in ratio diameter and friction coefficient were essential to PEEQ.

Introduction
The study of contact mechanics is rapidly expanding in scope and importance.Surface roughness plays a critical role in the development of contact mechanics.Contact mechanics is a fundamental concept in the study of mechanics and materials science, encompassing adhesions (Achanta et al., 2009;Upadhyay & Kumar, 2019), friction (Xie & Zhu, 2021) - (Taylor, 2022), and deformation (Ghaednia et al., 2017;Johnson, 1995).On a microscopic scale, perfectly smooth surfaces become rough due to irregular peaks, referred to as asperities.During contact between the two engineering surfaces, the actual contact area is a small fraction of the nominal contact surface area because the asperities model on the two machining planes presents the surface contact points (Ghaednia et al., 2017).In applications such as gears, the surface contact of asperities may occur when the teeth of the gears interlock.Despite appearing smooth and flat, principally, the teeth may have imperfect surface contact of asperities with numerous unevenness.When both teeth of gears come into contact, only a portion surface contact of asperities on each tooth makes contact with each other.As a result, surface contact between the two surface gears occurs only at tooth locations where the asperities form of one tooth fill the space between the asperities form of the other gear.Figure 1 depicts the contact procedure.To decrease computing time for the asperities model in a rough surface is a simplification by single asperity analysis (L.Wang & Xiang, 2013), where a single asperity model can be a hemisphere.
Several investigations conducted on surface contact between asperities included experimental (Greenwood & Williamson, 1966) - (Johnson & Shercliff, 1992), finite element (Shankar & Mayuram, 2008) - (Jackson & Green, 2005), and analytical studies (Brake, 2012;Zhao et al., 2000).Subsequently, surface contact models for experimental, finite element, and analytical studies are to simplify and accelerate the results for the analysis.The development of the surface contact model started with the research from Hertz (Öner, 2021a), which discussed the behavior of two spheres in contact.This research is then called the Herzian contact model.However, Hertzian contact only discusses elastic regimes (Fischer-Cripps, 2006).It means that concepts of the Herzian contact are not thoroughly applied to metal materials since they can undergo contact elastic, elastic-plastic, and fully plastic phases.
The analysis of contact models elastic-plastic has been widely studied by Chang et al. (CEB model) (Chang et al., 1987), Kogut and Etsion (KE model) (Kogut & Etsion, 2002), and Jackson and Green (JG model) (Jackson & Green, 2005), in which a model is a sphere against a rigid flat.In this case, most of the study assumes contact to be frictionless.Friction and wear may occur in every surface contact, especially in real-world conditions.Friction is a phenomenon that occurs in the area of surface contact between two moving objects.In the engineering system, understanding the friction law is essential (Zhang et al., 2019).Several studies have analyzed the frictional contact problem between a rigid cylindrical punch and an elastic half-plane.The frictionless contact between a functionally graded material (FGM) coating and an orthotropic substrate was investigated by Oner and Erdal (Öner, 2021a) using a semi-analytical method.Comez and Yilmaz (Çömez & Yilmaz, 2019) considered a monoclinic half-plane in the framework of linear elasticity theory.They reported that low friction, small indentation load, and large punch radius could decrease tensile peaks in in-plane stress to prevent surface damage and crack initiation near the trailing edge.Oner et al (Oner et al., 2015).studied the contact problem of two elastic layers loaded by a rigid circular punch and resting on a semi-infinite plane using analytical and finite element methods but assumed frictionless contact.Comez, Isa (Çömez, 2021) investigated the steady-state thermoelastic receding double contact problem between a rigid cylindrical punch and a homogeneous layer on a half-plane.Birinci et al (Birinci et al., 2015).conducted an analytical and finite element study of continuous and discontinuous contact cases.Oner and Birinci (Öner & Birinci, 2014) also investigated a surface contact problem involving two elastic layers sitting against a half-infinite plane loaded with a rigidity stamp to determine initial separation loads and distances between the layers and a lower-layer elastic half-plane.According to Comez and Guler (Comez & Guler, 2017), increasing the friction coefficient can increase the contact area between two contacting layers, minimizing the chance of damage on the contact surface.Oner et al (Öner et al., 2014).utilized analytical and finite element methods to solve a receding contact problem that involves two elastic layers supported by a Winkler foundation.Polat et al (Polat et al., 2018).solved the continuous surface contact problem for a functionally graded layer on an elastically semi-infinite plane.Oner and Birinci (Öner & Birinci, 2020) tackled the discontinuous contact problem between a functionally graded layer and a homogeneous half-space loaded symmetrically with point load P through a rigid block.Oner, Erdal (Öner, 2021b) proposed an analytical method for solving the two-dimensional frictionless continuous contact problem of an orthotropic layer pressed by the rigidity stamp, and Yaylaci et al (Yaylaci et al., 2021).examined both continuous and discontinuous contact problems for a functionally graded layer and a homogeneous half-plane using analytical and finite element methods.Ozsahin and Talat Sukru (Ozsahin, 2007) investigated the contact problem between an elastic layer and an elastic halfplane under external loads, providing numerical solutions for continuous and discontinuous contact cases.Despite these studies, research on the effects of plastic deformation and damage development during friction remains limited (Moshkovich et al., 2019).
Fully plastic contact is strongly avoided in machinery because it causes geometric changes even though the load has been removed (Jamari & Schipper, 2006).In the artificial hip joint, plastic deformations and strains can increase wear and clinical failure (Jamari et al., 2014).Sheet metal forming is adversely affected by friction, an unfavourable phenomenon that increases the forming force, making the quality reduction of the draw pieces' surface and reducing forming limitation (Trzepieciński et al., 2022).Nevertheless, the complicated cross-sectional shape can be produced directly by plastic deformation instead of machining, which guarantees higher mechanical properties and results in significant cost savings in the metal forming process (Xiaotao & Fan, 2012).The extreme load allows the model to be subjected to fracture (Latypov et al., 2012).Thus, this assertion creates an urgency favouring strain analysis for a crucial process to help understand when the fracture occurs.
Recently, deformation ratio studies against the diameter ratio have been performed by Johnson & Shercliff (Johnson & Shercliff, 1992), Ismail et al (Ismail et al., 2015), and Lamura et al (Lamura et al., 2023).Johnson & Shercliff reported the diameter ratio did not affect plastic deformations in geometries with the same hardness.In contrast, different findings indicated that the diameter ratio at the same material hardness significantly influenced plastic deformation (Ismail et al., 2015;Lamura et al., 2023).Lamura et al (Lamura et al., 2023).reported an insignificant friction coefficient against plastic deformation at high loads.However, there was an influence on the distribution of von Mises stress marked by reducing the area of highly localized or green ellipses.
Equivalent plastic strain (PEEQ) is a superior parameter for characterizing plastic deformations under the influence of the loading process (Zheng et al., 2018).For instance, the metal-forming process has been investigated by analysis of PEEQ distribution and strain pathways in the most critical areas (Zheng et al., 2018) - (Kriflou et al., 2021).Son et al (Son et al., 2011).reported differences were quantitatively analyzed by comparing PEEQ in the wire resulting from the interaction between wire and inclusion.Here, there is a correlation between the drawing stress increase and the PEEQ distribution value for AI203 and SUS304 inclusions.Liu et al (Liu et al., 2017).reported plastic deformation history in the infeed rotary swaging process.During the reduction zone, the central exhibits more deformation (higher PEEQ) than the upper layers; this is less evident for the higher friction coefficient.
Furthermore, most studies of spherical contacts did not thoroughly examine the PEEQ parameters.There have been very few previous studies that have mentioned PEEQ in the conformal contact model.There are quite a few past researches that have mentioned PEEQ in the conformal contact model.For example, Hu et al. analysed an elastic-plastic contact model in-cylinder (Hu et al., 2020).They reported PEEQ displayed an evolution of plastic zone.Also, loading configuration may affect partial slip contact between an elastic-plastic sphere and a rigid flat surface (Shi et al., 2013).They reported the plastic strain is significantly higher in the trailing portion of the contact area than that in the leading region.Also, continuous accumulation in the plastic straining occurs in both parts under periodic tangential displacement loading, indicating a plastic shakedown and potential fatigue failure.However, the previous investigations by Hu et al. and Shi et al. did not comprehensively analyse the effect of friction coefficient and diameter ratio on PEEQ during contact.
Therefore, this study attempted to evaluate the influence of friction coefficient and diameter ratio on PEEQ value in the contact between two hemispheres.This research draws an extension from Lamura et al (Lamura et al., 2023).studied the effect of diameter ratio and friction coefficient on plastic deformation without PEEQ analysis.Moreover, the PEEQ analysis is of particular concern as a significant parameter for characterizing plastic deformation.Through PEEQ analysis, this study would provide information on the location of material deformation due to the applied load.

Finite element modelling
The surface contact between the two hemispheres models is in Figure 2. The two-dimensional axisymmetric finite element (FE) model is simplified by the three-dimensional using a rotational symmetry axis.This selected approach previously used by Mars et al. addressed a similar problem (Mars et al., 2015).The FE model in this research used 4-node bilinear elements and reduced integration with hourglass control (CAX4R elements in ABAQUS) for the upper and the bottom hemispheres.Reduction integration is more effective than the default total stiffness formulation for dealing with the nonlinear material response at high strain levels.In addition, using a scale factor will help improve accuracy for course mesh with only a slightly higher computational cost.The mesh size and sensitivity studies used in this paper are according to our previous study by Lamura et al (Lamura et al., 2023).
The influence of the diameter ratio on PEEQ was studied using five different diameter models.The diameter ratio varied from 1 to 5 in each model.The lower hemisphere has a smaller diameter value than the upper hemisphere.The diameter of the upper hemisphere is then adjusted to be the same in each ratio, which is 17.5 mm.The detail of the model configuration is in Table 1.In the boundary condition, loading at 8000 N applied above the hemisphere is in Figure 2. The load was distributed across the surface using a coupling constraint, with the model constrained to move only along the y-axis.Additionally, the base of the lower hemisphere was fixed in place to prevent movement in any direction.
The material of the hemisphere in this study is brass.The material properties are in Table 2. Main model assumptions regarding asperity contact are according to an elastic-perfectly plastic material behavior of homogeneous and isotropic.Brass material selected in the study is because it has high mechanical strength and higher corrosion resistance for many instances and scenarios (Gaurav et al., 2022).Brass material is also widely used in machine applications such as cartilage (Yaqoob et al., 2022), bearing (Ünlü, 2009), and gears (Phokane et al., 2018).

Friction coefficient
The surface roughness is one of the most significant characteristics of the metal-forming process.
Here the rough surface has a static friction coefficient at the contact interface between two hemispheres in this model.The variation of the friction coefficient in this study was 0, 0.05, 0.1, and 0.4.Several parameters determine the friction coefficient, such as the material, friction conditions, temperature, and topography of the tool and workpiece (Trzepieciński et al., 2022).The value of this friction coefficient is selected based on the range of friction coefficients that occur in the metal-forming process from previous studies (Fratini et al., 2006) - (Bailey, 1975).
Further static friction coefficient is an essential parameter for analyzing surfaces with asperities or smooth surfaces since it accurately represents the frictional behavior of a micro slip contact, an important consideration when simulating contact mechanics (X.Wang et al., 2018).Additionally, the value of the static friction coefficient can be determined from force measurements in experiments, which simplifies the computational model and reduces the computational time required for the simulations (Papangelo et al., 2015;X. Wang et al., 2018).The static friction coefficient follows the friction law because the value is the average of friction calculation in the experimental result.And then, the hemisphere is modeled with a smooth surface and static contact in general.This study also explored the static friction coefficient using the Lagrangian multiplier formula.The reason is that the Lagrangian multiplier is more suitable for analysing high-deformation frictional contacts, as reported by Sheng et al (Phokane et al., 2018).(Sheng et al., 2005).

Normal contact in Spherical contact models
Researchers (Chang et al., 1987;Jackson & Green, 2005;Kogut & Etsion, 2002) have made significant efforts to model the elastic-plastic properties of two contacting bodies.Chang et al (Chang et al., 1987).developed a contact model (CEB model) based on volume conservation of the plastically deforming asperity.The critical interference value (ω c ) is defined in Equation 1 and the critical load value (P c ) is defined in Equation 2. (2) Where, And, the Initial yield is defined by K = 0.4 as the maximum contact pressure factor.Load (P) of CEB models can be expressed as follows: In the 2002, Kogut and Etsion (model KE) (Kogut & Etsion, 2002) used the value of K.In addition, the critical interference value of KE is shown on Equation 6 Where, The relationship between dimensionless load (P/P c ) and dimensionless interference is define in the Equation 9 and (11).
In 2005, Jackson and Green (JG model) (Jackson & Green, 2005) acquired the critical interference value based on the value of von Mises stress, as shown in Equation 11. Where, Equations ( 15) and ( 16) are defined the correlation between the dimensionless load and dimensionless interference. Where, where the ω c is same with ω c in this paper and ω* is ω/ω c .The analytical approach from CEB, KE and JG models will be compared with present study in the diameter ratio 1.The two hemispheres with same hardness equal to deformable sphere against rigid flat.However, in the higher diameter ratio variation will be compared with experimental results from Lamura et al (Lamura et al., 2023).and simulation results from Ismail et al (Ismail et al., 2015).This is because KE and JG formula is defined with dimensionless interference maximum at 110.

Comparison of current study and previous study analysis
In this study, we conducted mesh sensitivity analysis as a follow-up to our previous research (Lamura et al., 2023).The present study results are in close agreement with those of the analytical contact model using the JG, KE, and CEB model (Figure 3a).The increasing load yielded higher deformation because the loading presses on the hemisphere surface.The present results agreed closely with the KE and JG models.Instead, the comparative data in Figure 3b confirmed that the results of the present study agreed very well with those from previous experimental results.The deformation ratio determined here is the difference in deformation between the upper and lower hemispheres.The deformation ratio increased as the diameter ratio increased because the diameter and volume of the material changed to obtain a similar level of deformation.This trend shows a good agreement and an error of below 10% for the experiments and below 3% with experimental results from Ismail et al (Ismail et al., 2015).Accordingly, the simulation is valid.
Figure 4 also depicts the influence of the diameter ratio on the PEEQ and deformation unloading in the frictionless contact under the same load at different diameter ratios.As a result of the increased diameter ratio, the deformation unloading increases, and the same is noticeable in the max PEEQ value because when the diameter of the hemisphere gets bigger, the volume of material that undergoes deformation also increases, meaning that the elastic potential energy stored in the material will increase as well.As a result, when the material is under unloading, it will deform more as the diameter of the hemisphere increases.

The effect of friction coefficient on the PEEQ
Figure 5 shows the relationship of the maximum PEEQ value to the friction coefficient in each model, which is crucial to understanding the physics of the problem.In the case of M1, the increasing friction coefficient increases the maximum value of PEEQ.However, for M2-M5, an increasing friction coefficient resulted in a decrease in the maximum PEEQ value.This feature is because existing friction induces energy dissipation, which reduces the system's deformation and strain energy.The different responses observed in M1 and M2-M5 highlighted the importance of considering the effects of friction on the contact mechanics of hemispherical bodies.
In Figure 6, the distribution of PEEQ using different colors represents the maximum, minimum, and median values.The results show that M1 exhibits a relatively uniform distribution of PEEQ in the upper and lower hemispheres, while M2-M5 has significant differences in PEEQ distribution between the two hemispheres.The maximum PEEQ area in the lower hemisphere expands as the  diameter ratio increases, while the maximum ones in the upper hemisphere decrease.This observation can relate to the fact that a smaller diameter hemispherical body suffers more deformation and strain energy under the same load, resulting in a higher PEEQ value.Furthermore, the increasing friction coefficient on M1 increases the maximum PEEQ area, while the decreasing PEEQ area could be on M2-M5 models.These findings highlight the significance of friction in determining the distribution of PEEQ and the overall contact mechanics of hemispherical bodies.

The effect of friction coefficient on the contact radius
Figure 7 shows how the friction coefficient affects the contact radius in each model.In all models, the increasing friction coefficient reduced the contact radius.This decrease may relate to an increase in tangential force between the contacting surfaces, which leads to a reduction in the normal force and, as a result, a decrease in the contact area.The reduced contact area is directly proportional to the rise in contact pressure, which can result at the beginning of plastic deformation in the contacting surfaces.When analysing the mechanical behaviour of contacting surfaces, it is critical to carefully consider the effect of the friction coefficient on the contact area and the resulting contact pressure.

Discussion
Previous research has included a convergence study.The M1 model with load variations agrees well with JG, KE, and CEB contact model formulations.The current result in the M1 model is midway between JG and KE.This result of the JG and KE models only analyses up to Ltot/c = 110.As a result, we compared the diameter fluctuation to the experimental and FE data from Ismail et al (Ismail et al., 2015).There is also good agreement between the experiment and FE results from Ismail et al (Ismail et al., 2015).Accordingly, the model presented in this work is correct.
PEEQ is the accumulation of plastic deformations during the loading process (Zheng et al., 2018) and has main characteristics for measuring plastic strain cumulative after cycle loading-unloading (Li et al., 2019).According to Sulaiman et al (Sulaiman et al., 2012), the material will yield a PEEQ value of more than zero.In this study, PEEQ is sensitive to the friction coefficient.Good lubrication is beneficial for achieving a more homogeneous distribution of plastic strain, while contact pairs with a high friction coefficient are beneficial for achieving cold hardening on surfaces (Liu et al., 2017).One of the purposes of lubrication is to avoid plastic deformations.
The higher diameter ratio will increase the plastic deformation ratio, which has been reported by previous studies (Ismail et al., 2015;Jamari et al., 2015;Lamura et al., 2023).The distribution of PEEQ appeared to expand at the bottom hemisphere.Otherwise, the upper hemisphere is to shrink.This feature indicates that loading has a high impact on the lower hemisphere.Instead, the increasing friction coefficient to M1 causes the maximum PEEQ value and area to expand.In contrast, raising the friction coefficients in M2-M5 models reduced the highest PEEQ values.Growing and shrinking occurring in the PEEQ area may relate to the contact radius also widening and shrinking with the increase in the coefficient of friction for each diameter ratio.This result is due to the difference in the contact regime values between M1 and M2-M5 models.According to Kogut and Etsion (KE) analytics solutions (Kogut & Etsion, 2002), fully plastic contacts start at dLtot/dc-KE more than 110 because dLtot/dc-KE on M1 is 97.425.The surface contact has not yet reached the real fully Where dLtot is the total interference loading (the amount of interference from the upper and lower hemispheres), dc-KE is the critical interference KE.This increase and decrease in the maximum value of the PEEQ area reported by Chari et al (Kriflou et al., 2021).conducted PEEQ analysis in equal-channel angular rolling using a response surface.The maximum value of PEEQ decreases at low thickness ratios and increases the friction coefficient in strip-to-die at high thickness ratios.The thickness ratio obtained was by comparing the thickness of the intake channel strip to the thickness of the outlet channel strip.The diameter ratio in the current investigation is similar to the effect of friction on the thickness ratio reported by Chari et al (Kriflou et al., 2021).and Li et al (Li et al., 2019), who suggested a decrease in the maximum value of PEEQ in plastic deformation observed the angle of internal friction.
According to Jamari et al (Jamari et al., 2014), the PEEQ improves wear and failure.The theory explains that smaller PEEQ values are better for avoiding material failure during the metal-forming process.According to Liu et al (Liu et al., 2017), good lubrication may promote homogeneous plastic strain distribution, while a high friction coefficient value is suitable for cold hardening.The reduction of PEEQ value in high friction may relate to the material experiencing hardening.Nevertheless, the reduced maximum PEEQ value is too small to detect the plastic deformation ratio with variation friction coefficient (Lamura et al., 2023) because the material is elasticperfectly plastic.
This study has some limitations.First, a 2D axisymmetric model was employed in this investigation using the adiabatic principle.In plastic materials theory, hardening behaviour depends on temperature and material annealing behaviour, which describes the loss of accumulated plastic strain that signals metal softening at high temperatures (Muránsky et al., 2015).The temperature distribution over the contact zone has a substantial link with PEEQ, according to the research on bonding mechanisms in cold spray technology for metals (Viscusi et al., 2020).The steady increase in strain may be due to particle aggregation and deposition behaviour under the low-pressure cold spray, particle speed, and temperature (Z.Wang et al., 2020).Second, to represent surface roughness, a static friction coefficient is modelled.Third, loading and unloading does not repeat for several cycles.After the first loading, the contact behaviour becomes elastic when there are many cycles or repetitive stationary contact for a given load (Jamari & Schipper, 2008).Fourth, this study did not consider the hardening factor.The high load generates a suitably high pressure, promoting the plastic strain and the hardening of the working material (Juettner et al., 2022).According to Burbank et al (Burbank & Woydt, 2016), strain hardening occurs when ductile materials undergo plastic deformation.This constraint becomes the direction of our future research.

Conclusions
The study examined the effect of friction coefficient and diameter ratio on PEEQ value when two hemispheres came into contact.The finite element results in this study accord well with the analytical models from the KE and JG models.Ismail et al.'s experimental (Ismail et al., 2015) and FE results correlate well with the current work.In conclusion, the results of this investigation show that both the diameter ratio and the friction coefficient have a substantial impact on the PEEQ.The maximum area distribution in the lower hemisphere expands as the diameter ratio increases.As the friction coefficient increases, the maximum PEEQ value for M1 increases and expands, while the maximum PEEQ value for M2-M5 decreases and shrinks.The PEEQ area extends and contracts, showing that the contact radius similarly expands and contracts as the coefficient of friction increases for each diameter ratio.More research is needed to investigate the impact of other parameters on PEEQ, such as material properties and loading conditions.Furthermore, future studies must address the practical implications of the findings in engineering design and failure analysis.

Figure 1 .
Figure 1.Illustration of contact asperity in gear application.

Figure
Figure 3. Comparisons include: a) the current study with prior researcher models, b) the current study with experimental results and past researcher models.

Figure 5 .
Figure 5.The effect of Friction Coefficient on the PEEQ.

Figure
Figure 7.The relationship between contact radius and friction coefficient.