Numerical simulation of the rheology of alumina abrasive particles applied to UHMWPE during the microabrasion test

The material used for 50 years in orthopedics for joint replacements is ultra-high molecular weight polyethylene (UHMWPE). Joint replacements are designed so that the metal is in surface contact with the polymeric material, and it is used in most degenerative joint disorders. A problem with joint replacements is the half-life of their UHMWPE components resulting from surface contact, leading to premature wear. This leads to the detachment of wear particles causing osteolysis. In the present investigation, a numerical simulation of the rheology of 5 μm size alumina abrasive particles applied to UHMWPE type GUR 1020 was performed by duplicating the microabrasion test. An analogy was proposed between the alumina particles observed in the scanning electron microscope (SEM) and the gravel stone obtaining a geometry similar to that of the alumina particles. A 3D scanner was used, obtaining different randomly selected gravel models. They were scanned and rendered to obtain the IGS files to be used in the finite element software. Using Archad’s abrasive wear theory, the volume displaced by the abrasive particles was obtained as a function of the geometry of the abrasive grains, influenced by the entry position of one or more of the different cutting edges contained in the geometry, the hardness of the softer material, the value of the normal load and the sliding distance. Individual trace profiles were obtained by 2D profilometry and wear traces were analyzed by SEM. Explicit dynamic analysis of particle motion was used to simulate the entry, transition and displacement during the abrasive wear test caused by detached particles. The experimental results of the microabrasion test were compared with finite element analysis, demonstrating that simulation can be used as a reliable tool to obtain information about the type of wear of a material.


Introduction
Abrasive particles in mechanical systems where there is surface contact and sliding, such as gears, bearings, motors and specifically joint replacements, are the cause of premature wear, imminent failures, production line stoppages and what is known as aseptic loosening in prostheses.The most important variable that influences the rate of wear by abrasion and erosion mechanisms is particle geometry.One of the characteristics of abrasive particles is that they are highly amorphous, and their abrasive potential depends on their orientation and angle of penetration with respect to the wear surface.
The main geometric shapes used by several authors are functional shapes such as spheres, cones, pyramids and/or recently combinations of these [1,2].
To perform the modeling of abrasive particles in three dimensions, the methodology used is through the generation of bodies of revolution from a generatrix axis, and they are used to represent average roughness shapes [3].
The challenge is the characterization, modeling and generation of abrasive particles in three dimensions.In most abrasion problems such particles can be treated as hard convex bodies.Bonifazi developed the use of boundary fractal techniques for the generation of random particles [4].In his research, Fang elaborates a threedimensional model of abrasion to improve the realism and stick to the prediction of the study of wear types [5].Williams and Hyncica analyzed the contacting surfaces of bearings contaminated with abrasive particle volumes of 10 and 50 μm, used a simple theoretical model with trapezoid shapes in X and Y axis direction to consider a three-body abrasion [6].Chen and Li simulated a two-dimensional abrasive model to simulate erosion wear [7].
Lukasz Rypina and Dariusz Lipinski analyzed the influence of the geometrical parameters of abrasive grains with respect to their different angles (attack, apex and opening), as well as the width and length of the particle.It indicated that the variation of the cutting edge angle of the abrasive particles in a wear process can have a significant impact on the damage caused to the surface.The analyses were carried out on the basis of the results of experiments, performing a simulation using the finite element method of a single abrasive grain.Also important part of his work was that he made a classification of the angles of incidence depending on their opening and damage to the contact surface, classifying them into: Acute angles of incidence (0°to 30°), oblique angles of incidence (30°to 60°), straight angles of incidence (90°) and angles of incidence greater than 90 degrees (greater than perpendicular).Acute angles are the ones that present greater penetration and therefore are the particles that generate more damage on the surface, generating detachment of material, microcracks and microfatigue [8].
These theories of particle elaboration make it possible for the characterization to be simpler but not very close to reality.
The objective of this research is to simulate the behavior of particles in three-body abrasive wear by modeling them using a 3D scanner and to compare them with experimental microabrasion tests to determine if they can be used as an auxiliary tool in tribology, since laboratory wear tests are long, expensive and require specialized equipment.

Method
The material used was a 30 mm diameter bar of UHMWPE type GUR 1020 provided by a prosthesis manufacturing company in Mexico.The specimens were machined to a thickness of 8 mm.A 25 mm diameter sphere of 52100 steel with a hardness of 860 HV, a surface roughness of 400 nμ and an abrasive with powder mixture with alumina particles with average grain size of 5 μm, 10 g of glucose, 1 g of sodium chloride and 85 ml of distilled water were used.The mixture was used as an abrasive liquid (slurry).
Alumina was selected as the abrasive particle to be used as a slurry to penetrate between the steel ball and the polyethylene test tube, causing considerable damage so that the traces left by the micro-abrasion tests are deep and thus to obtain information on the profile, depth by means of profilometry.
The tests were performed in an abrasive testing machine under NOM.ASTM G77.The specimens were prepared by grinding with 350, 500, 600, 600, 800 and 1200 μm sandpaper to achieve a surface finish with a roughness of Ra 0.177 ± 0.03 μm.During the test the sphere was set to a rotation with a speed of 41 cycles/ minute to allow the abrasive mixture to penetrate between the surfaces involved.The normal forces applied to the sphere were 5 N. Every 100 turns of the sphere, 10 drops of lubricant were added.One drop is equivalent to 1.228 ± 0.075 ml of the mixture (the value of each drop applied with a commercial plastic dropper was obtained with the graduated measure of a 50 ml test tube, reaching this value with 65 drops).The tests were carried out at 2000 cycles.
To obtain the amorphous geometry of alumina, an analogy was made with gravel stone.Figure 1 shows a 750 X micrograph of alumina with 5 μm size, figure 1(a) and (a) full-size photograph of a gravel stone mound, figure 1(b), the different anisotropic geometries of gravel are by far very similar to that of alumina.
The analogy of gravel stone with alumina particles was made because of their morphological similarity.A main characteristic of abrasive particles is their amorphous geometry with different edge angles, and as a consequence it is impossible to find two alike.This comparison can be extended to various materials that give off particles having the same configuration, causing the 3-body abrasive wear effect.
Therefore, the decision was made to use a 3D scanner to obtain their irregular geometries and replicate them.
These were scanned and digitized with the 3D scanner to obtain a file in STL format, which was imported into software where it was rendered and aligned to be saved as a solid object to be used in the simulation of the microabrasion test , shown in figure 2.
The models simulating the abrasive particles were imported into the numerical methods analysis software as discrete rigid, shown in figure 3, which means that the particles take on the characteristics of being stiffer than the rest of the parts in the model and their deformation is considered negligible.
The part that simulates the test specimen has mechano-plastic properties of UHMWPE added to it.The UHMWPE material specimens are in the elastic regime, behaving linearly according to data obtained from tensile tests.The Johnson-Cook (JC) plasticity model, which is an isotropic hardening model that includes the effects of strain rate loosening and hardening and temperature softening, was used to calculate the dynamic strain of the material under high strain rates.Cyclic effects and possible material anisotropy were not considered for the simulation.The JC model describes the yield stress as a function of strain rate and temperature that is   based on von Mises plasticity, with closed-form analytical equations specifying the hardening behavior and the strain rate dependence of the yield stress.The melting temperature of the material was taken at 130 degrees Celsius.Table 1 shows the data used in the simulation, of the elastic regime of UHMWPE from the Johnson-Cook plasticity model inserted into the software.
Figure 4 shows the calibrated JC Plastic and JC damage evolution model data for UHMWPE in the numerical simulation software, fed the plasticity and fracture model data.
From one of the profiles of the wear track, 3 representative measurements were obtained for the maximum depth left by the peaks of the alumina abrasive grains, 3D profilometry was used to obtain the real depth with a Tempor ® P15 instrument profilometer to obtain the surface graph corresponding to the microabrasion test, as shown in figure 5, Vision software was used to analyze the deepest point of the scan and sample of the profile obtained by 2D profilometry.
Figure 6(a) and (b), shows the amorphous geometrical characteristics of the alumina, observed in the SEM.figure 6(b) shows the abrasive particle embedded in the UHMWPE specimen during the microwear test, which causes the abrasive wear.figure 6(c) is an EDS analysis to ensure that the observed particle is alumina and not another material.
Given the geometrical characteristics of alumina, which presents an amorphous geometry observed in the micrograph in the Scanning Electron Microscope.An abrasive particle has different cutting angles, θ1, θ2, θ3... θn under a load W used during the test.The particle penetrates a depth of cut d at a distance traveled l as seen in figure 7(a).The particle bears half of the load at the front of the contact surface between the abrasive cone and the abraded material, due to the cutting direction, shown in figure 7(b).
The equation used for abrasive wear was developed considering the wear produced by a single grain, making it ideal.From the geometric description in figure 5(c), it is considered that the material to be abraded is in contact with only half of the surface of the abrasive particle, forming the triangle ACBO.From the cross section of the abrasive particle half is taken, obtaining a right triangle OAB, with a point angle θ for the vertical.Where r, is the radius of the circular segment AB, and AC is the height of the triangle or opposite side of the triangle, so the area of the triangle OAC is: Since this is half of the abrasive particle, equation (1) is multiplied by 2 to obtain the total area.Therefore, equation (1) is the cross-sectional area of the abrasive particle.Considering the hardness of the softer material H = 40 MPa, as the normal load P between the area A, and the cross-sectional area as a half-circle approximation.
The derivative of the volume is the area.The volume of wear produced at a distance dx Deriving equation (1) we have: Substituting the values of hardness and area and integrating equation (2).Equation (3) is the most representative expression of the abrasive wear produced by a single grain over a total sliding distance x.
Where tan θ is the average of the individual conical roughnesses, H is the hardness of the softest surface and P is the value of the normal load applied to the abrasive particle.Equation (3) has a similar structure to Archard's equation for the calculation of abrasive wear [11].
Substituting in equation (3) the values of the specimens of tip angle θ, the load P, the hardness of the softer material, which is H = 40 MPa, the displacement of the abrasive particle of x.The wear analysis of the present project was carried out to observe and analyze the wear mechanics of the material used in the fabrication of prosthesis for its possible application in improving its tribological properties.
The angle of attack of the peaks of the particles that meet the surface of the polyethylene was measured, image J was used to obtain the angles of the peaks shown in figure 8, indicating the front view of the particles, which are enclosed in a circle the peaks that are in contact with the UHMWPE specimen in the simulated test, obtaining the entry angles.Figure 8(b) shows the bottom view of the particles, which shows the contact peaks from another perspective.
Abrasive particles can have different random geometries, affecting the way in which they come into contact with the surface and the characteristics of the damage caused.In wear phenomena, abrasive particles impact the surface at different angles due to the nature of the process.This can result in irregular wear and the formation of specific wear patterns.However, for the present investigation we searched for gravel stone geometries that had similar angles found in the wear tracks analyzed by 2D and 3D profilometry, and that were compatible in geometry with the alumina particles.Therefore, gravel stones were scanned with characteristics of angles greater than 90 degrees indicated in table 2 and shown in figure 9.
The values in table 2. were obtained as a function of the inlet shear angle of the abrasive particle position, these angles will be used in the wear calculation equation.
Substituting values in equation (3) at load P = 5 N, Q, 1 Q 2 and Q 3 from table 2, and a hardness of UHMWPE H = 45 MPa, for Q 1 and a sliding distance of x = 0.15 mm.
Performing the operations for angles Q 2 and Q, 3 table 3 shows the obtained values of the displaced volumes at a distance used in the simulation.
The profiles of the tracks obtained by the microabrasion tests were also analyzed using profilometry, which are shown in figure 9, taking into consideration only the craters with similar angles to the scanned particles that leak as edge angles in contact with the polymeric surface.The angles were obtained with the image J tool.
The grooves of the microabrasion tests were analyzed by obtaining the angles in the profiles of each one of them.
Table 4 shows the angles measured with image J, angles close to the values of the cutting edge peaks of the scanned gravel stones were used.
A section cut, perpendicular to the particle motion, was obtained at the displacement distance used in the simulation (x = 0.15 mm). Figure 10 shows the profile of the abrasive particles interacting with the abraded material.

Results
The wear analysis of the present project was carried out to observe the wear mechanics of the material used in the fabrication of prosthesis for its better understanding of the wear particle behavior and its application in improving its tribological properties.
The results of the simulations and microwear tests are based directly on the comparison of the profiles of the peaks and their cutting-edge angles, as well as the von Mises stresses and the plastic deformation of the specimens.
Figure 11 shows a crater left by a microabrasion test where the folds generated by the displacement of the abrasive particle can be observed.These folds are due to the plastic deformation of the material.Figure 12 also shows the depth of the groove and the angle generated by the cutting tip of the specimen.These profiles are obtained by 2D profilometry.
Comparing the results of the displacement volumes indicated in table 3 for the virtual particles and table 4 for the profiles obtained from the wear tests, they are similar, because it is denoted that the equation of volume displaced by a single particle only takes as a variable derived from the geometry the edge angle that penetrates the material and displaces it, regardless of the actual morphology of an abrasive particle, which has 'n' numbers of peaks that can wear the material in contact.Because equation (3) was derived from a cone-shaped particle.This is observed in figures 9 and 10, where the profiles of the virtual and laboratory tests are presented.Figure 12 shows that as the alumina particle penetrates further into the abraded material, the angles of interaction can be   not just one, but several.If the angles of incidence interacting with the surface are not acute there will be no material detachment and more material displacement to the side edges.This is because angles of attack are greater than 90 degrees and tend to bounce or slide on the damaged surface.So it is observed in the cross sections of the particles in figures 12(a), (b) and c that there is little penetration with the characteristics of adhesive wear type.It is also observed that the effect of angles greater than 90 degrees gives a more noticeable surface roughness due to the action of particle dragging, coinciding in the analysis of surface damage with the author Lukasz Rypina [8].
Figure 12 circles the edges that generate the wear grooves.Profiles 1, 2, 3 and virtual particle profile 2 contain at least two edges that modify the geometry of the wear track.
In figure 13(a) von Mises value of 4.47 MPa was obtained, which is well below the ultimate yield stress of the material which is 28 MPa [12].Describing the images of this same figure, as a result of the simulation, the maximum von Mises flow stress is located in the stress concentration zone (figure 13(a), in direct contact with the particle edge face.It is in this zone that the largest amount of plastic deformation is also found with a value of PEEQ 5.021 (see figure 13(c).This value indicates the inelastic deformation of the material.If this variable is greater than zero it means that the material has yielded [13].At the beginning of the contact of the alumina particle with the polyethylene, the displacement of the material or burr moves towards the top, forming the material lip and the particle profile (see figure 13(b)).Figure 13(d) shows the plastic deformation without exceeding the yield stress of the material since it does not present detachment of the material, eliminating the abrasive particles from the scene for better detail.

Discussion
Several methodologies have been used by various authors for the generation of geometric shapes, the most common being the creation of spheres, cones, pyramids and polyhedra through the generation of solid bodies of revolution, and others use two-dimensional modeling.But the main characteristic of abrasive particles is their amorphous geometry, which is omitted in the aforementioned models.So we decided to scan gravel stone for three-dimensional modeling to have a better prediction and analysis of the type of wear during the microabrasion test.Equation (3), used to calculate the abrasive wear volume produced by a single grain along a displacement distance X, takes into consideration that the abrasive particle has only one peak or cutting edge, which interacts with the surface in contact and is therefore far from what happens in reality, including the geometry of the particle.In the wear traces left by the displacements of the virtual particles, we can observe that by having 'n' number of cutting edges and also depending on the entry position of this, the penetration depth depending on the load and the displacement distance, the number of cutting edges that interact with the worn surface will be more than one.On the other hand, the plastic deformations in the simulations present the same material displacement for the formation of folds and lips, leaving similar characteristics in terms of shape and volume displaced by the particles in laboratory tests.Therefore, the simulations of the particles obtained from the scanning of the gravel stone show a behavior seen in the laboratory tests.

Conclusions
The different geometric shapes of abrasive particles considered by several authors such as solids of revolution, cones, spheres and pyramids are far from being real, being very ideal for obtaining the displacement volume, since the formula used in equation (3) only takes into consideration the angle of a single edge of the particle, but disqualifies the amorphous geometries and their 'n' possible wear edge peaks that could be interacting with the surface in contact (figure 10).Therefore, it is concluded that the displacement volume considering only the tangent angle is not representative of a real displacement of the amorphous particle geometry.
Comparison and scanning of gravel stone, and its subsequent downscaling to micrometer levels, is an option to see the actual behavior of various particles during their sliding stage and generation of abrasive wear.
The stress concentration value of 4.47 MPa is below the creep limit of the material which is 28 MPa according to literature.Therefore, it is concluded that there is no immediate detachment of the polymeric material.
The value of the plastic deformation indicates that it presents large displacements of the material, this is shown in the micrographs taken from the wear tests and in the simulations of the replica of the test, emphasizing that the values of JC plastic and JC damage with a melting temperature of 130 degrees Celsius were taken as data of plasticity of the material obtained from the bibliography.
The behavior of the material by observing the 3D profilometry shows similarities in terms of lip formations, displaced material, folds, plastic deformations, and groove depths.Giving us an idea of the behavior of the particle in detail, concluding that the methodology used can be taken as a tool to determine the behavior of different abrasive particles at micrometric levels.
The frontal displacement of the material generates the formation of lips, and it is there where the highest concentration of stresses and trapped energy is located, which is released with plastic deformation and lateral folds without detachment of the material indicated by the von Mises value.This can generate residual stresses, causing cracks or, in critical situations, fatigue failures in the material.
The geometric profiles obtained by profilometry and finite element analysis do not present a geometric or dimensional similarity, since it depends on the plastic deformation of the material, speed and depth of the amorphous geometry of the particle.

Figure 1 .
Figure 1.Shows the geometrical similarity between alumina and gravel particles, (a) shows a micrograph of alumina at 750 X magnification, (b) shows a full-size photograph of the gravel stone, both parts have similarity in their amorphous geometry.

Figure 2 .
Figure 2. Gravel stones, in part (a) is shown Randomly selected particles of gravel stone to be scanned.(b) Gravel scanned and converted into models of virtual solid bodies.

Figure 3 .
Figure 3. 3D model with the characteristics of a discrete rigid body type.

Figure 4 .
Figure 4. Data in the program for the numerical simulation of the Jhonson-Cook Damage and Plastic hardening microabrasion test.

Figure 6 .
Figure 6.(a) Shows SEM of a single aluminum abrasive particle, (b) magnification of the particle at 20 mm and (c) EDS of the embedded particle.

Figure 7 .
Figure 7. Geometry of an abrasive particle (a) Cross section.(b) Isometric view of the carriage, and (c) Half-particle cross section.

Figure 8 .
Figure 8. Front and bottom perspective of the virtual particles, (a) front view (x, y), (b) bottom view (x, z).

Figure 9 .
Figure 9. α, β and γ angles of the profiles created in the craters of the microabrasion tests.

Figure 10 .
Figure 10.Cross section cut of abrasive particles and UHMWPE material (a) Particle profile 1, (b) Particle profile and (c) Particle profile 3.

Figure 11 .
Figure 11.Microabrasion test analysis showing the grooves generated by the alumina particles.

Figure 12 .
Figure 12.Abrasive particle profiles obtained by profilometry and transversal cutting using numerical methods software.

Table 2 .
Peak angles of attack of virtual alumina abrasive particles.