Physical Modeling of Grinding Forces

Abstract

In order to address the increasing demands on precision in manufacturing, the prediction of various processes by model-based methods is increasingly becoming a key technology. With respect to this, the grinding process still reveals a lot of potential in terms of reliable predictions. In order to exploit this potential and to improve the understanding of the process itself, a physical force model is developed. Here, process-typical influencing factors, as well as commonly used cooling lubricants, are considered. In addition to the simulative effort for the actual model, basic experimental investigations have to be carried out. In single scratch tests, it has been found that process and deformation mechanisms such as rubbing, ploughing, and cutting of the material and also the pile-up of this material on both sides of the cutting grain are significantly involved in the development of forces. It also turned out that the resulting forces are greater when cooling lubricants are used and that the topographic characteristics of a scratch are also affected by them. For a realistic mapping of these effects within the force model, the deformation model, according to Johnson and Cook, and a discretization, according to Arbitrary Lagrangian-Eulerian, proved most suitable. For integrating the cooling lubricants, the Reynolds equation using a subroutine proves to be a suitable instrument. The challenge to complete the force model is combining the scratch and the Reynolds equation simulation.

1 Introduction

For the development of new technologies or the optimization of existing systems, an essential guarantor for this is high-precision manufactured components and tools. A key element for a precision-manufactured component is choosing the right manufacturing process and how well it is understood. The grinding process is one of the most important surface-finishing processes in the industry [1]. This is also due to the fact that grinding is indispensable for precision-manufactured high-performance components for major industries such as aerospace, the automotive industry, and the energy sector [2]. Grinding itself is a material-removing process in which geometrically undefined abrasive grains are used as tools [3]. Due to these irregularly shaped abrasive grains and the very high process speeds, grinding is a complex manufacturing technology [4]. While this technology has been used by humankind since ancient times, its complexity leads to the fact that the grinding process is still not adequately understood and researched to this day [5, 6]. Due to the complex abrasion mechanisms between abrasive grain and material, experimental investigations are extremely costly or hardly possible when considering an entire grinding wheel [7, 8]. Existing approaches to determining characteristics of the grinding process by experimental investigations are usually associated with very high costs and therefore are not economically lucrative [9, 10].

In order to avoid these expensive investigations, there exist already several approaches. For example, there are considerations to reduce the required experiments to individual aspects and to measure only the total forces during grinding [11, 12]. In addition to examining the entire grinding wheel, some considerations relate to a single abrasive grain. For instance, Nie et al. [13] have mathematically mapped an abrasive grain statistically and used it to describe the influence of cutting speed and cutting depth on the process forces. In addition, several further considerations attempt to describe the process using numerical or analytical approaches [11, 15]. Notably, the cooling lubricants commonly used in manufacturing are not included or play a minor part in most considerations [13, 14]. The latter influence the process itself and are essential to produce the required fine surface finish [15].

The simulation of a manufacturing process in a computer model has proven to be a suitable method to optimize such processes [16, 17]. Therefore, experiments are mainly needed to validate the model and do not have to be carried out for each constellation of test parameters. An additional advantage of such a model is the possibility of observing processes that are taking place inside the material, such as the stresses that occur. Depending on the complexity of the manufacturing process, it is easier or more difficult to represent these effects within a model. The grinding process reveals a high potential with respect to modeling approaches. This is also due to the fact that special removal mechanisms such as rubbing, ploughing and cutting are not yet sufficiently considered in models for grinding. The model developed here for the grinding process is intended to close the gap by considering and modeling deformation processes and process-required additives such as cooling lubricants. In this context, it is important to map the influence of the cooling lubricants and the mechanical removal mechanisms in detail. In particular, processes in the very small gaps between the tool and the material are of special complexity. For this purpose, the behavior in these areas is to be approached by employing the Reynolds equation.

2 Experimental Investigation

This section discusses the procedure and results of the experiments performed. The experiments carried out here are single scratch tests. In these tests, a diamond tip, which represents a grit of the grinding tool, is driven through the surface of a sample, which represents the material of the workpiece. The scratches created in this way are used for further investigations. Real experiments are important for developing a physical force model for several reasons.

On the one hand, they serve to compare the data from the grinding model with reality. The focus here is on the forces that occur, which in this case, are divided into normal and tangential forces. On the other hand, the material behavior of the samples used is investigated. Special attention is paid to process-typical effects such as rubbing, ploughing, and cutting. But also the effects, such as chip formation and the pile-up of the material. For the development of the physical force model, it is necessary to take all these effects into account and reproduce them since they also contribute significantly to force development.

2.1 Requirements for Performing Experiments

In order to simulate the pile-up effect, an appropriately suitable test rig is required. Therefore, this project uses a scratch test rig with longitudinally guided grains. This makes it possible to generate individual scratches in a reproducible manner. Consequently, the pile-up effect can also be reproduced and investigated. By using an entire grinding wheel, on the other hand, it is not possible to examine the pile-up effect for each scratch individually. In addition, it is possible to determine the force distributions individually for each abrasive grain using the single scratch test rig.

2.1.1 Test Rig for Scratch Tests

In order to generate reproducible and utilizable scratches on a sample surface, the test rig displayed in Fig. 1 shows its most important components. It is important to underline that the abrasive grain in the test rig used here is moved through the sample exclusively in a translational movement. This is also the major difference to test rigs with a rotating grinding wheel. To realize this translational movement, an elementary component of this test rig is the linear unit. The sample is clamped on this unit via a corresponding device and moves translationally through the indenter tip during the scratch test. The linear unit can also be used to set and perform the different scratch speeds (0 mm/s to 800 mm/s) for the tests. Here, too, stepless adjustment is possible, with the restriction that the drive used only works reliably up to a speed of 800 mm/s and is no longer precisely controlled beyond this threshold.

Fig. 1.

Test rig with the main modules and functional elements to perform scratch tests in dry and lubricated conditions

In order to ensure ideal fixation, it is necessary that the indenter does not twist during clamping (important when using pyramidal indenters) or does not move up or down (occurs while using a screw to press the indenter against a wall). A three-jaw chuck has proven to be a reliable clamping with low susceptibility to mistakes.

A confocal distance laser (CL-3000 series from Keyence) is used to set the required penetration depth of the indenter into the corresponding sample. The test rig is equipped with a dynamometer (type 9109AA from Kistler) to record the forces during a scratch test. This dynamometer records the tangential and normal forces.

In order to be capable of carrying out the scratch tests either dry, i.e., unlubricated, or lubricated with cooling lubricants, a dispensing unit (2000 series from Vieweg) for various liquids is integrated into the test rig. With the aid of this dispensing unit, the reference oil, here FVA2 and FVA3, is applied onto the sample directly in front of the indenter. As a sample material, aluminum alloy A2024-T351 (see Table 1) is used to represent ductile material.

Table 1. Material parameters of aluminum alloy A2024-T351 used as samples

2.1.2 Indenter

In order to get reproducible scratches, it is advisable to use indenters instead of real abrasive grains. The problem of low reproducibility results primarily from the complexity of placing the abrasive grains always in the same position or classifying their adjusted position. Indenters used for the experiments can be seen schematically in Fig. 2. The first two indenters (from left to right) are geometrically standardized, whereas the last indenter has an undefined geometry and therefore comes closest to the real abrasive grain, but again with the previously described problems in the usage.

Fig. 2.

Conical indenter (left); pyramidical indenter (center); indenter with real grit (right) used in single grit scratch experiments to measure grinding forces

Both the conical and the pyramidal indenters are available in different versions with regard to the angle of their tip. For the experiments carried out here, mainly conical indenters of 90° to 150° are used. The advantage of using conical indenters is their simple and rotationally symmetrical geometry, which makes the alignment of the indenter considerably easier than with the pyramidal indenter. When aligning the pyramidal indenter, paying attention to the orientation of the pyramid faces is always necessary. In Fig. 3, three different examples of conical indenters are shown. The first picture shows a microscope image of an indenter in a three-dimensional perspective, and the two following pictures show two indenters with different factory-specified angles of their tip. The angles were measured manually to check the production precision. The diamond tip and the carrier material of the indenter are also clearly visible based on the three-dimensional image. The three-dimensional scan is required for the development of the force model. This scan enables the remodeling of the indenter in the finite element software used. The examination of the angle of the indenter tips, on the one hand, is necessary for quality control. On the other hand, they are required for the later documentation and simulation of the wear of the indenter tips.

Fig. 3.

3D magnification of a conical Indenter (left); conical indenter with a factory-specified angle of 105° (center); conical indenter with a factory-specified angle of 120° (right)

2.2 Preparations for the Scratch Tests

Since the samples are manufactured in a metal processing facility, it is important to ensure they are free of production residues, such as lubricating oils and greases, before they are used in the test rig. Hence, they are cleansed of possible impurities with acetone in an ultrasonic bath. The procedure is the same for the respective indenter to be used. The cleaned samples are then fixed in the clamping device of the test rig, and the indenter is inserted into the jaw chuck. Due to the weight of the indenter (approximately 5 g), the tip of the indenter now lightly contacts the surface of the sample, and the chuck is then firmly tightened. The position of the sample to the indenter tip is set as scratch depth “zero” in the used LabView program. After the sample has been moved away from the indenter tip, the confocal distance laser is used to set the specified scratch depth for a specific test run. Moving the linear unit to the start position completes the setup for a scratch test.

2.3 Performing Scratch Tests in Dry Conditions

In order to exclusively investigate the effect of different scratching speeds, scratching depths, and indenter angles on the samples in relation to the normal and tangential force and the sample topography, scratching tests are first carried out in a dry environment. This approach ensures that only the pure interaction between the diamond tip of the indenter and the aluminum surface of the sample is investigated. The distribution of the forces and their changes under different test parameters allow essential conclusions to be drawn about the material parameters. These parameters are important for the grinding model to predict forces with this model in its final state. For this reason, a range of parameter constellations is tested for these scratch tests. Scratch depths from 50 μm to 250 μm, scratch speeds from 50 mm/s to 1000 mm/s, and indenter angles from 90° to 120° have turned out to be practicable constellations.

Regarding the selection of the scratch depth, it is important to consider that the deeper the scratch depth, the more reliable the results are and the less they scatter. This is because the indenter tip is slightly rounded. However, it is important not to scratch too deep since from a scratch depth of approximately 300 μm, the substrate material of the indenter can be partially involved in the formation of the scratch. The selected scratch depths are also subject to the production-related properties of the indenters used. Due to the rounded tip, a certain penetration depth is required to obtain error-free data for the simulation and its validation. Although smaller scratch depths are common in grinding, characterizing processes are scalable in most cases. After overhauling the test rig, it was decided not to use the maximum speed of the device due to technical control reasons and to use it only up to 800 mm/s.

Following the setting of the corresponding test parameters, the test is started via the implemented software of the test rig. After a pre-defined acceleration phase of the sample, which is mounted on the linear unit, the sample moves with a constant speed under the indenter in the scratching area and is scratched. Figure 4 shows a sample used for the scratch tests. On this sample three sets each with nine scratches can be seen. The same test parameters apply within each set. Therefore, all scratches within one area are repetition tests. The three sets differ in their scratch speed. The upper set with nine scratches generated at 50 mm/s feed rate, the middle set with nine scratches were generated at 400 mm/s feed rate and the lower set with nine scratches were generated at 750 mm/s feed rate.

Fig. 4.

Three sets of scratches with nine repetitions each on an aluminum sample, the sets differ in scratch speed. Test parameters: scratch speed 50, 400 and 750 mm/s; scratch depth 0.08 mm; conical indenter of 105°; lubricated with FVA2

The topographic properties of the scratched samples are then examined and evaluated with suitable optical equipment. For this purpose, 3D-capable microscopes such as the confocal microscope µSurf-explorer from NanoFocus or the digital microscope VHX of the 7000 series from Keyence are generally used. Based on the topography obtained in this way, conclusions can be drawn about the deformation behavior of the sample material. These are important to realistically simulate effects such as the pile-up of the displaced material.

The dynamometer integrated with the test rig directly records the force signals and saves them corresponding to the test parameters. Figure 5 shows a typical normal and tangential force distribution during a scratch test.

Fig. 5.

Typical distribution of normal and tangential force during a scratch test. For the evaluation, only the forces without the edge areas are used to calculate the mean values. Here illustrated exemplary by the area within the dashed lines.

To avoid interfering edge effects where the indenter enters and leaves the material, 10% of the force signal after entering and before leaving are each ignored in the evaluation. Figure 5 illustrates which section of the force signal is used to calculate the mean values for further evaluation. Figure 5 also clearly shows that the normal forces in a scratch test are higher than the corresponding tangential forces. This property is also unaffected by the selected indenter angle and the set scratch speed, as shown in Fig. 6.

Fig. 6.

Mean values of normal and tangential forces for different scratch speeds and two indenter angles with a scratch depth of 50 μm: cone angle of 105° (top); cone angle of 120° (bottom)

Furthermore, it can be seen in Fig. 6 that with increasing scratching speed, both the normal and tangential forces decrease. This trend has already been observed in previous publications [18, 19]. One reason for this behavior could be a temporary temperature increase in the cutting area, which reduces the flow stress in the area.

2.4 Performing Scratch Tests in Wet Conditions

In industrial production, grinding processes are almost exclusively done in combination with cooling lubricants, and it is also necessary to perform scratch tests with cooling lubricants. This is also important to determine whether, for example, the pile-up effect is increased or decreased by the influence of cooling lubricants. Another important point is to investigate the influence of cooling lubricants on the tangential and normal forces. This also raises the question of whether these forces increase or decrease.

The attached dispensing unit on the test rig enables the application of selected cooling lubricants under which the scratch test will be carried out. Since industrially used cooling lubricants generally contain various additives to adjust their properties, they are not suitable for basic research with regard to the scratch test. One of the reasons for this is that the added substances cannot always be determined qualitatively and quantitatively due to confidentiality. In addition, evaluating the tests is difficult when such additives are present since many processes occur in part at the molecular level. For this reason, reference oils (FVA2 and FVA3 from Weber Reference Oils) are used instead of real cooling lubricants for the tests carried out here. The two reference oils essentially differ in their viscosity. Here the reference oil FVA2 with 85 mm²/s at 20 ℃ has a significantly lower viscosity than the reference oil FVA3 with 300 mm²/s at 20 ℃. The tests with reference oils are carried out the same way as those in a dry environment.

Based on the results obtained from dry and wet tests, it is then possible to detect and evaluate both analogies and differences. The first effect that attracts attention is that it can be confirmed that the normal forces are still larger than the tangential forces when using cooling lubricants. This can be seen in Fig. 7. Figure 7 also displays another important influence of the cooling lubricants. It can be seen that the normal and tangential forces under dry test conditions are smaller than those under wet test conditions. One possible reason for this may be the additional liquid phase that must be displaced by the indenter, which increases the force.

Fig. 7.

Normal and tangential force for scratch tests under dry and wet conditions. Test parameters: scratch speed 50 mm/s; scratch depth 0.08 mm; conical indenter of 105°

Another remarkable aspect is that the deviation in the forces caused by the reference oil FVA2 (low viscosity) is slightly wider than that of the reference oil FVA3 (high viscosity). The different viscosities of the two reference oils can be used as an important clue to explain this phenomenon. The reference oil FVA3, with a higher viscosity than the reference oil FVA2, might absorb possible vibrations of the indenter due to its higher viscosity; as a result, a small deviation in the forces is recorded.

Another essential part of the realization of a physical force model is the understanding of the surface characteristics of the workpiece after a scratch test. Here, however, the focus of the investigation is less on the quality of the surface itself but more on the topography and characteristics of a single scratch. In exclusively dry tests, the topographical characteristics of a scratch are mainly influenced by the use of different indenter angles. Since a further component is contributed when cooling lubricants are used, it is important to find out how this additional component affects the scratch characteristics.

To examine the respective scratches, all scratches of a sample are measured optical. With the help of the digital microscope VHX 7000 from Keyence, the surface is recorded and converted into a three-dimensional image. These surface profiles are then exported as scatter plots and further processed in Matlab. During further processing, the data volume of the scatter plot is reduced from the data export. This is necessary for performance reasons only. Thus, a surface profile can be generated, as shown in Fig. 8.

Fig. 8.

Topography of a scratch magnified by using a digital microscope (left); recreated relief of such a scratch by using the exported csv data (right)

In a subsequent step, various profiles are extracted from such a relief (cf. Fig. 9), describing the scratch and its cross-section.

Fig. 9.

Extracted profile from the relief of a scratch with the key values scratch width and scratch depth to evaluate the scratch topography. The extracted profile depicts the cross section of the scratch from Fig. 8 at the border line between the colored and grey areas. Values are displayed in µm.

The scratch width $d$ and the scratch depth $t$ have proven to be suitable values for comparing the topography of different scratches. Instead of using the values individually, the ratio $d/t$ is used. The reason for this is that, for example, the scratch depth $t$ changes depending on the scratch depth set before the start of the test. However, since the scratch always represents the negative image of the indenter used and the indenter always has the same height and corresponding width ratio, this ratio must also be present in the imprint. Detectable deviations of this ratio within a scratch profile are, therefore, due to process- or parameter-related influences. The values d and t used are determined by Matlab from the measured values of the digital microscope. Figure 10 shows the ratio $d/$ t depending on the three environmental conditions dry, FVA3, and FVA2.

Fig. 10.

Ratio of scratch width $d$ to scratch depth $t$ for velocities 50 mm/s and 400 mm/s and the scratch depth of 80 μm and 250 μm, dry and wet condition generated by a conical indenter with an angle of 105°

Figure 10 shows that the ratio $d/t$ becomes smaller as soon as the tests are carried out with the reference oils. It is also notable that the reference oil FVA2 (low viscosity) differs from the reference oil FVA3 (high viscosity) with regard to this ratio. FVA2 always shows the larger ratio $d/t$. However, in wet condition the ratio is always smaller than in dry conditions, which indicates that the scratches become smaller when performed with cooling lubricants. A material buildup in front of the indenter without reference oils could cause this phenomenon. This buildup of material could cause an increase in material removal at the sides of the indenter. However, we cannot explain this phenomenon with absolute certainty at present. To identify the exact causes, further experimental setups designed for this purpose must be developed. For example, it is necessary to optically examine the cutting front during the scratch test.

3 Development of the Grinding Model

For the development of the grinding model, the realistic behavior between abrasive grain and sample material has to be considered and implemented in the model. To develop a reliable force model, first of all, it is essential to simulate the material behavior of the sample in a physically precise manner. To accomplish this, it is necessary to take a more detailed consideration of simulation approaches and techniques. In the first instance, it is advisable to consider the indenter as rigid to focus on the sample material and its behavior. With this assumption transferred to the scratching problem, the represented abrasive grain is treated as wear-free and fracture-resistant.

3.1 Selection of the Suitable Material Model

To simulate the behavior of the material realistically, a suitable approach must be selected and implemented. The deformation model, according to Johnson and Cook (JC), is a candidate for this purpose. Using the JC model, the strain hardening of the corresponding material can be described analytically. Furthermore, the strain rate and temperature dependence of a material are also described. The material behavior, according to JC, is integrated as standard in most finite element method (FEM) programs. The von Mises stress $\sigma$ can be calculated according to the JC model by using the equation

$$\sigma =\left[A+B{\left(\varepsilon \right)}^{n}\right]\left[1+C\mathrm{ln}\left(\frac{\dot{\varepsilon }}{{\dot{\varepsilon }}_{0}}\right)\right]\left[{\left(\frac{T-{T}_{room}}{{T}_{melt}-{T}_{room}}\right)}^{m}\right].$$

(1)

Here, $A$ is the quasi-static yield stress, $B$ is the modulus of strain hardening, $n$ is the work hardening exponent, $C$ is the strain rate sensitivity, and $m$ defines the temperature sensitivity. Significant temperature development in the process of a single scratch is not expected. Due to the high thermal conductivity of the aluminum selected here, it is assumed that any process heat is immediately transported off the scratch area. Hence, temperature-relevant effects in the JC model are of negligible importance for the current state (Table 2).

Table 2. Johnson–Cook material parameters used for aluminum [21]

The Crystal Plasticity Finite Element Simulation Method (CPFEM) has been considered an alternative method for describing material behavior. However, the comparison between the JC model and CPFEM carried out in this context does not indicate any significant advantage with respect to CPFEM. This is also due to the fact that the data used here for the CPFEM by [20] is still from its initial phase.

3.2 Discretization Approaches

Various approaches exist in continuum mechanics and are implemented in many finite element programs to simulate the motion of material points. In Abaqus, the used FE program for this study, the mesh-based approaches according to Lagrangian (LAG) and Arbitrary Lagrangian-Eulerian (ALE), and the mesh-free smooth particle hydrodynamics (SPH) approach are to be mentioned. Each approach is particularly well suited for certain problems, so comparing the three approaches was first carried out on a simplified 2D scratching process.

3.2.1 2D Discretization Benchmark

The computational effort differs significantly depending on the type of discretization and how fine the mesh is set. For the estimation of basic aspects of the individual approaches, the problem is therefore first considered on a two-dimensional level. For the preliminary study conducted here, we started with a tool rake angle with a positive value of $\upgamma =20^\circ$ in dry conditions, similar to a turning process, for which experimental and simulated data are available to compare from the literature [21]. For the sample, aluminum alloy A2024 T351 was used. Figure 11 shows the FE output models used to evaluate the different discretization approaches. For the Lagrangian (LAG) and Arbitrary Lagrangian-Eulerian (ALE) approach, the sample is divided into three layers, with a sacrificial layer L2 separating the chip area L1 from the base material L3. When an element in the sacrificial layer L2 reaches a critical damage value, the elements are deleted and separated. The element size of the mesh-based approach is similar to the particle size of the mesh-free smooth particle hydrodynamics approach. No sacrificial layer is required here since separation can occur between any pair of particles when the cohesive bonds are no longer sufficiently large.

Fig. 11.

2D simulative output model for benchmarking of the LAG/ALE models (left) and SPH model (right), according to [22]

Figure 12 shows the results from the simulation and the experimental values. Based on this figure, it can be seen that, with respect to the tangential forces, all approaches are close to the experimental values with a deviation from the experimental value by a maximum of 1%. In contrast, if the normal force is observed, it is noticeable that all approaches have a deviation of about 40%. A possible reason for this large deviation could be the active element deletion in the mesh-based approaches. When the affected elements are deleted, they can no longer cause any force to be exerted on the tool. In practice, however, no material elements are deleted, and the remaining material builds up in front of the tool and leads to an increase in force. In the mesh-free approaches, a weighting function controls when a cohesive material bond is dissolved. The parameters used here may be the reason why the composite is dissolved earlier.

In contrast to the measured forces shown in Fig. 5 and Fig. 6, the tangential forces are much larger than the normal forces in the 2D simulation. Besides the already discussed effects of element deletion and weighting function on the forces, an important aspect is the effects of material deformation when rubbing, ploughing, and cutting occurs. Since these material deformations run in all directions, these effects can only be displayed reasonably in a 3D simulation.

Fig. 12.

Comparison of the tangential forces (left) and normal forces (right) of the discretizational approaches with the experimental results from [22] for the 2D orthogonal cutting model with tool rake angle γ = 20°, according to ref. [22]

3.2.2 Single Grit Scratch Model as 3D Approach

Even if investigations in two-dimensional form can provide values for approximate predictions, a three-dimensional approach is necessary for an overall analysis since only in this way boundary effects such as the pile-up of the material and the general material deformation effects like rubbing, ploughing, and cutting can be represented in a useful way. A simplified three-dimensional model of a scratch test is used to investigate the discretization approaches. The model height is 0.2 mm, the model depth is 0.5 mm, the model length is 1 mm and the spacing between the elements or particles are 0.003 mm. Figure 13 shows the stress distribution for the ALE and SPH approaches.

Fig. 13.

Simulative output model for benchmarking LAG, ALE, and SPH in 3D. Already with an example of the distribution of the von Mises stress in the ALE and SPH models. Test parameters: model height 0.2 mm; model depth 0.5 mm; model length 1 mm; indenter geometry cone 105°; feed rate 200 mm/s; scratch depth 0.03 mm; element spacing for the LAG/ALE and particle spacing for the SPH 0.003 mm, respectively [22].

Figure 14 shows the simulated and experimentally determined values for the normal and tangential forces of the three approaches.

Fig. 14.

Process forces of measurement and discretization approaches for a 3D single grit scratch (cutting speed ${v}_{c}$ = 200 mm/s, depth of cut ${a}_{p}$ = 50 μm that corresponds to ${a}_{p,\mathrm{sim}}$ = 30 μm, cone angle of $\upgamma$= 105°), according to [22]

As can be seen from these results, the simulated tangential forces now show the measured tendency with smaller values than the normal forces. Additionally, the ALE approach agrees best with the values from the experiments. Therefore, the ALE approach will be used for the more detailed development of the physical force model.

3.3 Simulative Integration of the Cooling Lubricants

As experimental investigations have already shown that cooling lubricants have a detectable influence on the normal and tangential forces during scratching and that the topographical nature of a scratch also changes under their influence, the cooling lubricants must also be integrated into the grinding model. The inclusion of an additional material turns out to be non-trivial. In addition to the reproducibility of the influence of cooling lubricants during scratch tests, the realizability via discretization approaches must also be discussed. A standard discretization for a liquid film in combination with the very small gap height between the indenter tip and the material is computationally almost impossible or even difficult to perform. Figure 15 shows the interspace that can be rated as problematic by classical FEM discretization.

Fig. 15.

Schematic illustration of the small gap height between the indenter tip and the material in which the lubricant is located

3.3.1 Basic Principle According to Reynolds Equation

After various solution methods for this problem have been considered, the approach, according to Reynold, is the most suitable solution. The Reynold equation describes and calculates pressure distribution problems of thin viscous fluid films in lubrication theory. The Reynolds equation can be described by

$$\frac{\partial }{\partial x}\left(\frac{{h}^{3}}{12\eta }\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial z}\left(\frac{{h}^{3}}{12\eta }\frac{\partial p}{\partial z}\right)=\frac{1}{2}\frac{\partial \left({U}_{2}-{U}_{1}\right)h}{\partial x}+\left({V}_{1}-{V}_{2}\right)+\frac{1}{2}\frac{\partial \left({W}_{2}-{W}_{1}\right)h}{\partial z}.$$

(2)

Here, $h$ is the gap height between two plates, $U$ and $W$ are the respective velocities of the plates in the $x$ and $z$ directions, $V$ is the velocity in the $y$ direction, $p$ is the pressure between the plates, and η is the dynamic viscosity of the fluid. In principle, the Reynolds equation converts a three-dimensional problem into a two-dimensional one. Applied to the problem of Fig. 15, it is no longer necessary to describe the interspace with a very fine mesh.

3.3.2 Implementation of the Reynold Equation by a User Element

With Abaqus, a direct implementation of the Reynolds equation is not possible without further effort. However, special subroutines can be integrated into Abaqus by scripts. Such subroutines are called user elements (UEL) and are used to apply the Reynolds equation to the problem under consideration. [23] has already programmed such a UEL to investigate and simulate plain bearings in Abaqus. With the help of this UEL, it is sufficient to discretize the liquid within the gap with only one element of thickness. The UEL converts this three-dimensional mesh to a two-dimensional layer, solves the Reynolds equation in it, and returns the results to the nodes of the three-dimensional sample. Therefore, the pressure distribution of a fluid within a very small gap can be calculated and simulated. Figure 16 shows an exemplary simulation of oil between two plates. In the simulation, the upper gray body represents a rigid planar plate, which is loaded initially with a pressure field in the negative $y$-direction. The lower element is designed as a user-defined deformable material. In this example, the material properties and the corresponding deformation behavior are assumed for the aluminum alloy A2024-T351. The element height is 0.04 mm, the element depth is 0.1 mm, the model length is 0.3 mm and the spacing between the elements are 0.0006 mm. The meshing in this element was done manually.

A parameterizable fluid is located between the two bodies. The force from the rigid body causes a pressure field in the fluid, which transmits a resulting force to the deformable body. The liquid can flow in the $x$-direction, but cannot flow in the $z$-direction due to the infinite expansion of the elements in this direction.

Fig. 16.

Example of a simulation with the Reynolds equation integrated as user element (UEL) in Abaqus. Test parameters: deformable model height 0.04 mm; model depth 0.1 mm; model length 0.3 mm; gap height 0.005 mm; applied pressure 1 MPa; element spacing in height 0.0133 mm; element spacing in length 0.006 mm.

Figure 17 shows the numerical solution by the UEL compared with the analytical solution of the gap height over time, which slowly reduces due to the force applied by the upper body. The reduction in gap height over time is considered here as a reference point. The expression

$$h\left(t\right)=\sqrt{\frac{{h}_{0}^{2}{B}^{3}L\eta }{2{p}_{p}BL{h}_{0}^{2}t+{B}^{3}L\eta }}(3)$$

is derived by transforming the Reynolds equation according to the gap height h(t). Here, $h$ is the gap height, $B$ and $L$ are the dimensions of the element, ${p}_{p}$ is the applied pressure, and $\eta$ is the dynamic viscosity. Based on the two solutions, it can be seen that they are identical. This proves that the UEL in Abaqus reliably computes the Reynolds equation.

Fig. 17.

Comparison of numerical and analytical solutions related to the change in gap height

A highly simplified initial model is considered first to adapt the UEL to the scratch test problem. Figure 18 displays this initial model and the stress development due to the pressure field of the oil film after solving the Reynolds equation. The gray body represents an infinitely extended indenter in $z$-direction. The lower part represents the aluminum sample as a deformable material. Between these two bodies is the liquid, which can flow off in $x$-direction.

Fig. 18.

Initial model for adapting the UEL to the scratch test problem

Even though the results of the additional pressure field due to the fluid film are promising, the model must include some adaptations. Momentarily the sample (blue body) and indenter (gray body) need to be infinite in $z$-direction. For the further course of the project, the quasi-two-dimensional simplification will be removed step by step. In this way, the Reynolds equation will also be rendered implantable for a complete three-dimensional scratch test. This requires modifications to both the UEL and the modeling of the indenter tip and sample in Abaqus. Additionally, further considerations are required to simulate the flow of the reference oils between the indenter tip and the sample in the most efficient way. The Reynolds equation does not provide for flow through the gap. To solve this problem, a further UEL may be necessary.

4 Conclusion

For the development of the grinding model presented here, two major aspects have been addressed in this report. On the one hand, the approach to develop the actual model and, on the other hand, the necessary basic experimental investigation. The experimental data are important for reproducing the material behavior and mapping the forces correctly in the simulation. Single scratch tests were considered here as the basis for the experimental investigation. These were carried out either dry, for the basic behavior, or wet, to consider cooling lubricants. The normal forces in each test parameter constellation are shown to be higher than the corresponding tangential forces. Moreover, it turned out that the resulting forces in scratch tests also depend on the deformation behavior of the sample material used. Thus, effects such as the pile-up of the material have a non-negligible influence on the normal and tangential forces. With the addition of cooling lubricants, the trend of higher normal force remains. However, it has been shown that both normal and tangential forces are generally higher when cooling lubricants are used. In addition to the influence of the forces during scratching, the topography of a scratch itself is also affected by the cooling lubricants. Thus, with the help of the scratch width and depth ratio, it can be seen that scratches produced under the influence of cooling lubricants are less wide.

For the development of the grinding model, the experiments revealed that, on the one hand, the consideration of cooling lubricants is important to obtain a realistic force model and, on the other hand, how different test parameters influence the normal and tangential forces. It became clear that preliminary observations in a two-dimensional simulation are only of limited value. In particular, the pile-up effect of the material and its influence on the forces can only be mapped realistically in a three-dimensional simulation. The influence of the cooling lubricants and the associated narrower scratches can only be correctly reproduced in a three-dimensional simulation. With regard to the cooling lubricants to be simulated, the employment of the Reynolds equation using a user element (UEL) has crystallized as a promising option. Here, problems in discretizing the very small gap between the indenter and sample can be avoided. The challenge is to couple the three-dimensional scratch test, with all its deformation aspects, with the Reynolds equation.