A material point method based ploughing model to study the effect of asperity geometry on the ploughing behaviour of an elliptical asperity

A material point method (MPM)-based numerical model has been used to study the effect of asperity size and orientation relative to sliding direction on the ploughing behaviour of a rigid, ellipsoidal asperity. Based on the simulated ploughing behaviour, an analytical model has been extended to calculate the ploughing depths over the wear track and compute the forces acting over the contacting surface of an ellipsoidal asperity sliding through a rigid-plastic substrate. The analytical model results have been compared with the MPM model results. The MPM model results are also validated to be in good agreement with the friction forces and ploughing depths measured from the ploughing experiments on lubricated steel sheets with ellipsoidal indenters up to certain sizes and orientations.


Introduction
The geometry of an asperity plays a significant role in determining the forces acting on it while sliding through a substrate. Thus, a wide range of asperity geometries ranging from two dimensional wedges and cylinders [1,2] to three dimensional pyramids (with square and hexagonal bases) and spheres have been analysed in terms of frictional forces while ploughing through a substrate. Mathematical models have provided a great deal of support in understanding the effects of asperity geometry on friction during sliding. Both analytical and numerical models have been used to simulate the ploughing behaviour of a rigid-asperity sliding through a substrate [3][4][5]. The friction in a system of two surfaces sliding relative to each other has been attributed to the plastic deformation of the asperities on the surface of the contacting bodies and shearing of the contact interface [6]. In sliding of an asperity through a substrate, ploughing is defined as displacing material from the sliding path of the asperity, without involving any actual material removal.
Some of the initial work on the effect of asperity geometry on the friction force was done by Bowden and Tabor [6], by analysing a spade, a cylinder and a sphere-shaped steel tool when ploughing through a metallic surface. Challen and Oxley [1] computed steady state solutions for the coefficient of friction for a two dimensional wedge shaped asperity sliding against a soft substrate based on Green's slip-line field theory [7]. Hokkirigawa and Kato [8] mapped the friction and wear in sliding of a spherical asperity as a function of 'degree of penetration', defined as the ratio of penetration depth and contact length and 'interfacial friction factor' defined as the ratio of interfacial shear strength and shear strength of the substrate. They modified the expressions for coefficient of friction by correcting the degree of penetration for three-dimensional, spherical asperities using an experimental fitting factor. The spherical asperity geometry was also assumed by most statistical contact models in modelling contacting surfaces [9]. Such an assumption is useful only in describing isotropic surfaces but cannot be easily extended to model anisotropic surfaces with variable asperity geometries. According to [10,11], anisotropic rough surfaces can be best described by elliptical asperities, with the contact being mapped as a set of elliptic patches. In order to compute friction using a multi-asperity model, it is important to model contacting asperities as ellipsoids and elliptic paraboloids based on the asperity height distribution and contact mapping [10,12,13].
Initial work to build analytical ploughing models [14] for calculating forces acting on an elliptical asperity while sliding through a substrate was done by van der Linde [4,15], where a hexagonal pyramid was approximated by an elliptic-paraboloid shaped asperity. The net force acting on the contacting faces of the hexagonal-pyramid shaped asperity was calculated from the force due to the contact pressure, acting along the normal direction into the asperity's surface and the force due to the interfacial shear, acting along the tangent to the asperity's surface in the direction of plastic flow. The total force, calculated using the material properties and the unit vectors, was resolved in three dimensions. This approach was extended and used to calculate the forces acting on ellipsoidal and elliptic paraboloid shaped asperities, ploughing through a rigid-plastic substrate (having negligible elastic deformation) [16]. The forces acting on the asperity were shown to be a function of its axes length, ellipticity ratio and orientation with respect to the sliding direction. The ploughing forces calculated from the elliptical asperities with reduced geometrical parameters were compared with simpler analytical models for spherical [3] and hexagonal-pyramidal [5] asperities and were shown to be in close agreement.
Analytical models, in spite of being simple and fast cannot be applied to real materials due to their complex deformation and shear behaviour. Typically, numerical simulations of single-asperity sliding have been done using finite element (FE) method [17,18] and molecular dynamics (MD) [19][20][21]. However, FE simulations have found modelling of large scale local plastic deformation in ploughing challenging [22,23]. In the available commercial FE codes, element deletion and adaptive re-meshing techniques are commonly used to model ploughing using simpler bilinear elastic-plastic material behaviour [24]. Typically, Coulomb friction is used to model interfacial friction in these FE models [25,26]. These in overall hinder the accuracy of the FE models and particularly increase the computational time. Variations to the standard FE such as crystal plasticity FE methods [27] and MD [19] have been used to model scratch at a nano-scale on single crystal substrate. Hence, particle methods such as MD have dealt with challenges of scaling and selection of interatomic potentials for modelling ploughing at small length scales [28][29][30]. Particle based methods such as smooth particle hydrodynamics (SPH) have also been used to model ploughing, although without proper experimental validation [31]. Recently, the material point method (MPM) has been successfully implemented to develop a ploughing model which has been validated for the coefficient of friction and deformation results using ploughing experiments [32].
From the available literature it can be seen that neither analytical nor numerical models to compute the deformation and friction in an elliptical asperity sliding through a metallic substrate are available. Further, physical validation of the model and an extended study on the effect of asperity geometry on the ploughing behaviour has not been done using elliptical (ellipsoids and elliptic paraboloids with an elliptical central cross-section/base in the sliding plane) indenters.

Nomenclature of symbols
In this regard, the current work extends the analytical model, introduced in [16], to compute the ploughed profile and ploughing forces on an elliptical asperity sliding through a rigid-plastic substrate (with negligible elastic deformation). The present work accounts for the asymmetry in plastic flow and interfacial shear with varying size and orientation of the asperity to compute the ploughing depth and the total force over the contacting region of an ellipsoidal asperity. Both the ploughing depth and the coefficient of friction obtained from the extended analytical model are compared with the MPM-based model [32], for ploughing of a rigid-plastic substrate by ellipsoidal asperities of varying size, orientation and applied load. Furthermore, ploughing experiments have also been performed using elliptical pins with varying size, orientation and applied load on a lubricated steel sheet. Also the ploughing depths and coefficient of friction from the MPM-based ploughing model are compared and validated with the results of the ploughing experiments.

Calculating ploughed profile and ploughing forces
Previously an analytical model to compute the forces in ploughing of a rigid-plastic substrate by an elliptical asperity was introduced in Mishra et al. [16]. The model decomposed the net force acting on the asperity into the force due to contact pressure and interfacial shear stress (see Fig. 1a). As shown in Fig. 1a, the contact pressure due to plastic deformation acted along the normal direction into the surface while the shear force acted tangential to the surface along the direction of plastic flow. The surface was divided into infinitesimal small (tetrahedral) elements and their normal vector and the tangential flow vector were calculated. The expressions for the projected area of the surface of the tetrahedral element on the Cartesian coordinate planes were derived and also expressed in the spherical coordinates. The boundaries of the asperity-substrate contact and separation of plastic flow into positive and negative components were obtained from the points on the elliptic contact patch with zero and infinite slopes m respectively, (see Fig. 1b). As shown in Fig. 1b, for the contact patch centred at C on the sliding plane, points N and S mark the end of the asperity-substrate contact while point M indicates the point of separation of plastic flow into þy and -y components. The x and y coordinates of N,S and M are given using the contact lengths CE ¼ a x , CF ¼ a y and angle of orientation β in equation (1). The component of force in the x, y and z axes, acting on a surface-element is expressed as the product of the elemental area and the (deformation and interfacial shear) stress along the x, y and z component of the unit vector corresponding to the stress. The total force due to deformation and interfacial shear in x, y and z axes are obtained by integrating the elemental forces over the contact and flow boundaries. The ploughing forces in [23] were calculated by assuming constant ploughing depth over the elliptic contact plane. ; The current model builds on the findings of the analytical model in [16] and extends it to compute the depth profile and the forces in ploughing by an ellipsoidal asperity. It uses insights of the MPM-based ploughing simulations of ellipsoidal asperities to study and develop theoretical understanding of the effect of asperity geometry on the ploughed profile. Using the applied load, the model computes the initial ploughing depth of the substrate. (The ploughing depth is then calculated over the ploughed profile using fitting terms obtained from the ratios of the projected contact lengths, such as p, q and w which are further explained in section 2.1.1 and 2.1.2. These factors account for the additional pile-up or sink-in due to the asymmetric distribution of the plastic flow separated around the (arc L \ M) asperity in the þy and -y directions, conservation of the distributed plastic flow and resistance or assistance to the plastic flow/deformation due to interfacial shear.)

Calculation of ploughed profile
The ploughing depth for a load F applied by a rigid asperity on a plastically deforming substrate is given by balancing the applied load on contact area A with the stress underneath the asperity, i.e. hardness of the deforming substrate H as shown in equation (2.1). In this analysis, an ellipsoidal asperity with its axes along the x,y and z axis, slides in the xy plane with axis length r along the z direction. Thus its other axes lengths are e x r and e y r, where e x and e y are the ellipticity ratios along the x and y direction respectively. The reference contact length is taken as a z , where the contact length along the x and y axis is given as a x and a y . For a rigidplastic substrate, the frontal half of the asperity is only in contact with the substrate during ploughing, and hence the contact area at a fixed depth is halved as shown in equation (2.1). The penetration depth d is then computed as the difference between the height of the asperity c ¼ r, and the separation δ of the centre of the asperity from the surface of the un-deformed substrate δ ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi r 2 a 2 z p , as shown in equation (2.2). However, the ploughing depth for an ellipsoidal asperity oriented at an angle to the sliding direction cannot be calculated directly from equation (2.2). Due to the skewness and asymmetry in the ellipsoidal asperity oriented along the sliding direction, the plastic flow due to deformation and interfacial shear is altered. Hence, the ploughing depth over the ellipsoidal asperity-substrate contact region is modified and calculated using fitting factors accounting for variations in the plastic flow. Ploughing is considered as a dynamic, visco-plastic event where the initial deformation of the substrate is followed by subsequent shearing and plastic flow of the deformed substrate. Here, the ploughing depth d is calculated as the sum of the pile-up height h pu and the groove depth d g over the ploughed profile, as shown in Fig. 2b and c. The pile-up height h pu0 and the groove depth d g0 in measuring the total ploughing depth d for the spherical asperity (e x ¼ e y ¼ 1) are taken as the reference values in modifying the ploughing depth (pile-up height and groove depth) due to plastic deformation for an ellipsoidal asperity. The ratio of h pu 0 and d is taken as l.

Change in ploughing depth due to plastic deformation
Piling-up of deformed substrate in front of the asperity significantly affects the ploughing depth and forces. Here, the effect of asperity geometry on pile-up height will be discussed in terms of slope of asperity in the sliding (xy and xz) planes and projected area in the (yz) plane perpendicular to sliding. As an ellipsoidal asperity, with its major axis in the sliding direction (x-axis), is rotated in the sliding plane (see Fig. 2a), the projected length of its major axis in x-direction, L x decreases while that in y-axis, L y increases. Firstly, the contact area projected in the yzplane increases with increase in L y resulting in increasing the resistance to plastic flow in the x-direction (see Fig. 2c and d). This results in increased deformation of the substrate and its piling up in front of the asperity. Secondly, the slope of the asperity-substrate contact in the xzplane decreases with increase in L x . This results in decreasing the pile-up height due to the ease of plastic flow of the piled up substrate under the asperity, which follows its slope in the xz plane (see in Fig. 2b). Finally, the sharpness of the asperity tip in the sliding direction, i.e. the change of slope of the elliptic contact profile in the xy-plane increases with increase in L x and decreases with increase in L y (see in Fig. 2b). A sharp asperity tip assists the deformation of the substrate and the distribution of plastic flow of the piled up substrate around the asperity. A sharp asperity tip thus, reduces pile-up of the substrate. Thus, the total pile-up height h pu is taken proportional to L y /L x i.e. the ratio of y and x-coordinates of points N and M respectively. In equation (3.1) h pu is given by fitting power n to y N =x M .
As the asperity continues to plough through the substrate, the piled up material in front of the asperity shares the applied load with the material in the groove. To maintain load balance following equation (2.1), higher pile-up height results in lower groove depth d g (see Fig. 2c and d). Hence, the groove depth is proportional to L x =L y (see Fig. 2c and d). Similarly, in equation (3.2), d g is given by fitting power o to x M =y N . The height of the piled up substrate varies across the ploughed profile from the tip of the asperity to its edges (see Fig. 2c and d). Unlike, the groove depth, the distribution of pile-up material does not follow the shape of the asperity. Hence, the average of the pile-up height of all contacting points is taken here as h pu .
The mean ploughing depth d' for contacting points on the xy-plane, is taken as the sum of groove depth d g and average pile-up height h pu , and is given in equation (3.3) by fitting power p to y N =x M . The ploughed profile consists of the pile-up both in front of the asperity and in its periphery and the ploughed groove. The piled-up substrate in front of the asperity dominates the ploughed profile during ploughing. Hence, p is taken as a positive fraction in equation (3.3) in calculating the forces acting on the asperity during ploughing. The deformed substrate is subsequently distributed over the contacting surface as the asperity slides over a given section of the substrate, therefore, the pile-up subsides. Now the groove depth and pile-up on side of the asperity are only accounted in measuring the final ploughing depth where, p is taken as a positive or negative fraction in equation (3.3) depending on the measurement point (asperity periphery/front).
The asymmetry in a rotated ellipsoidal asperity results in asymmetric plastic flow. This also results variations in ploughing depths on either sides of the asperity (see Fig. 3a). The difference in the plastic flow in þy and -y axis (see Fig. 1) is proportional to the difference in resistance to plastic flow in þy and -y axis in the xz plane. The difference in resistance to plastic flow is expressed as difference in the projected contact lengths of arcs MS and MN in the x-axis as shown in equation (4.1) (see Fig. 3b). Thus the change in ploughing depth d* at the periphery of the asperity, along the y-axis is taken proportional to the ratio of the projected contact length NS in x-axis 2x N to the un-rotated contact length along x axis 2a x .

Change in ploughing depth due to interfacial shear
The applied normal force in equation (2.1) is taken as the force responsible for deformation of the substrate before sliding starts. As sliding begins, an additional force F sh z acts on the asperity along the zaxis due to the shearing of the interface. The force on the asperity due to interfacial shear acts in the direction opposite to the relative velocity of the asperity with respect to the deforming substrate at the interface. The corresponding force on the substrate due to interfacial shear acts in the opposite direction, F sh z . Thus a component of the force acts on the substrate in the þz axis due to interfacial shear and a component of force acts on the substrate in the -z axis due to plastic compressive pressure. On one hand, the force due to interfacial shear by its nature, restricts plastic flow around the contacting asperity which then decreases the ploughing depth. On the other hand, the forces acting on the asperitysubstrate contact due to interfacial shear and plastic deformation result in a bi-axial tension on the deforming substrate elements. Such a stress-state caused faster yielding and increases the ploughing depth to maintain load balance. These ploughing depth derived in equation (2.2) is modified using F sh z by multiplying factors u and v which take into account factors for reducing or increasing plastic deformation. Combining both the factors u and v to a single factor w for including the effect of interfacial shear force, the final ploughing depth d '' is given in equation (5).

Calculation of ploughing forces
The forces acting on the ellipsoidal asperity ploughing through the rigid-plastic substrate is calculated based on the method developed in [16]. The surface of the asperity in contact is divided into infinitesimal elements whose projected areas on the 3 Cartesian planes are calculated and expressed in spherical coordinate axes. The coordinates of the boundaries of contact of the asperity with the substrate is calculated based on the modified ploughing depths derived in section 2.1. The components of forces in the 3 axis due to plastic deformation and interfacial shear are now integrated over the modified contact boundaries to obtain the corresponding components of the total force.

Calculation of the projected areas
The projected contact area between the ellipsoidal asperity and the substrate is corrected due to the correction in the ploughing depths. The projected area along the xy, yzand yz planes are calculated by taking an small elemental area on the contact surface and resolving it in x, yand zdirection. The elemental projected contact areas are then transformed from the Cartesian coordinate system C(x,y,z) into the spherical coordinate system S(θ, φ, r) using the Jacobian determinant ¼ detj∂C =∂Sj. This gives us the expressions for the elemental projected areas dAxy, dAxz and dAyz in terms of variables θ and φ as spherical coordinates (azimuthal angle and polar angles) and constants r and β (angle of orientation).
Integrating the expressions 6.1-6.3, we obtain the projections of the total contact area between the asperity and the substrate in the xy, yzand zx planes. Now the limits of the integration are found by obtaining the boundaries of contact. For an ellipsoid ploughing through the substrate at an angle β with respect to sliding direction x, the contact is divided into two regions with positive plastic flow and negative plastic flow, where the component of relative plastic flow velocity is along þy and -y directions respectively. The separation of plastic flow occurs at the point on the asperity surface where the slope is infinite in the sliding plane. The termination of plastic flow occurs on either side of the asperity where the slope is zero in the sliding plane as shown in Fig. 1b. The flow separation curve is shown as arc LM, while the arcs consisting the boundaries of plastic flow are shown by curves SN, LN and LS in the Fig. 1a.
The coordinates of the point N,Mand S change as the ploughing depths are modified. However, in spherical coordinates this change is reflected only as a change in the polar angles φ of the points. The azimuthal angle θ and the radial distance of the points remain the same as they lie on the surface and maintain a slope of either zero or infinity. Hence the coordinates of the points S, M, N and L are given in set of equations (7.1)-(7.3) in spherical coordinates. The detailed derivation can be found in [16].

Calculation of the total force components
Now the components of the forces acting on the asperity along Cartesian coordinate axes are calculated from the expressions of the unit normal and unit tangent. The unit vector expressions are functions of integrand of the corrected projected areas, nðφÞ, lðθ; φÞ and mðθ; φÞ given in equation (6.1)-(6.3) as derived in [16]. The limits of integration are taken from equations (7.1)-(7.3). Integrating the total projected area responsible for net positive plastic flow the total components of force due to plastic deformation are obtained in equation (8.1) The asperity-substrate contacting region can be given as the sum of the ellipsoidal cap with horizontal base at N and the ellipsoidal segment between S and N. The area of such an ellipsoidal segment is approximated as half the area of the ellipsoidal band, which is the surface of the asperity bounded by horizontal planes intersecting at S and N. In calculating the total force due to ploughing and shearing in the xand z direction the integration of elemental forces is done over the ellipsoidal cap with base at N and the ellipsoidal segment between Sand N as shown in Fig. 4a T. Mishra et al.

Numerical model and experimental setup
Both numerical calculations and experimental tests were conducted up for ploughing ellipsoidal pins using the same geometrical parameters. Simulations were also done using the numerical model to compare the results obtained from the modified analytical model using the same geometrical parameters for the ellipsoidal asperities. The axis size r of the ellipsoidal asperity is taken as 0.2 mm for comparison with the analytical model and 0.5 mm for comparison with experiments. The size of the ellipsoid are changed for both analytical and experimental studies involving cases where only one axis size is varied either along or perpendicular to the sliding direction or both axes sizes are varied such that e x e y ¼ 1. For experimental studies, as listed also in Table 4 in the first case, ellipticity ratios e x are 1/2, 2/3, 5/6, 1, 6/5, 3/2 and 2 and e y ¼ 1 and vice versa, while in the second case e x values are 1/2, 3/5, 3/4, 1, 4/3, 5/3 and 2 and e y ¼ 1=e x . The orientation of the ellipsoid with respect to x-axis β is varied at 15 0 interval between 0 0 90 0 . For analytical studies, the axes sizes are varied with ellipticity ratios e x as 1/ 4, 2/7, 1/3, 2/5, 1/2, 2/3, 4/5, 1, 5/4, 3/2, 2, 5/2, 3, 7/2, 4 and the orientation β varied at 7:5 � interval between 0 0 90 0 for the same cases.

Computational method
The MPM based ploughing model is used to simulate ploughing using an ellipsoidal asperity. The ellipsoidal asperity is made up of a triangulated mesh with mesh size varying from 5 to 20 μm. These triangles have no self-interaction which makes the asperity perfectly rigid. The substrate is made up of particles which interact with each other following the 'linear-MPM pair-wise' interaction algorithm. The interaction between the ellipsoidal asperity and the substrate follows from the 'triangle-MPM pair-wise' interaction algorithm as mentioned in [32]. The substrate is modelled as a half cylinder in order to optimize the number of particles and hence the computational time. The size of the MPM particles are varied from 5 to 20 μm for studying the convergence of the model-results with particle resolution. The parameters for the MPM based ploughing model are listed in Table 1. The MPM-based ploughing model has been shown in Fig. 7.
The deformation in the substrate is modelled by using the material model composed of the linear elastic equation of state and the Bergstr€ om van Liempt material model [33] as given in equations (9.1) and (9.2) respectively. The total stress is computed from the hydrostatic and deviatoric components. The hydrostatic stress σ hyd is computed using the equation of state model from the bulk modulus K and volumetric strain αI where Iis the identity matrix. The deviatoric stress is computed from the deviatoric strain using a 'radial return' plasticity algorithm [32]. For    validating the MPM-based ploughing model with the analytical model, a perfect-plastic material behaviour is chosen for the substrate whose parameters are listed in Table 2. In order to compare the results of MPM-based ploughing model with the experiments, the Bergstr€ om van Liempt material model (yield stress: σ BL y ) for DX56 steel is chosen with its parameters as listed in Table 3.
Further the interfacial shear stress is modelled using either of the two models that have been developed using experimental fitting and characterization [32]. (1) A theoretical model is used for validation of the MPM-based ploughing model with the analytical model. The first interfacial friction model takes the interfacial shear stress τ sh as a fraction, f(interfacial friction factor), of the bulk shear stress of the substrate (equation (10.1)). The bulk shear stress κis taken as σ y =√3 based on von Mises yield criterion where σ y is the yield stress. (2) An empirical model is used to compare the results from the MPM based ploughing model with the experimental results. The second interfacial friction model takes the interfacial shear stress as a power-law function of the contact pressure between the triangles of the indenter in contact with the MPM particles of the substrate (equation (10.2)). Based on the experimental characterization of the interfacial shear strength of DX56 steel sheet lubricated with 'Quaker Ferrocoat N136' lubricant under varying loads (contact pressure P c ) and fitting of results, coefficient C p ¼ 1:34 and exponent n p ¼ 0:88 are obtained [32,34].

Experimental procedure
The experimental procedure consisted of ploughing experiments using indenters with ellipsoidal tips on a DX-56 steel sheet lubricated using 'Quaker Ferrocoat N136' forming lubricant. The section elaborates on the design of the ellipsoidal pin specimens with different ellipticity ratios, preparation of sheet specimens and the test set-up used for the ploughing experiments. The experiments were done 3 times for repeatability.

Material
The material used for making indenters is D2 tool steel DIN 1.2379,  obtained by heat treatment in vacuum. The pin is heat treated to a hardness of 62 � 2 HRC (746HV or 7.316 GPa). The elastic modulus of the pin is 210 GPa and its Poisson's ratio is 0.3. After heat treating the D2 tool steel cylinders, the outside base diameter is ground followed by high precision milling and polishing to obtain the ellipsoidal shape at the tip to validate the numerical model. Ten different ellipticity ratios were chosen. Seven of the designs, B1 to B7 have ellipticity ratios changing along one of the ellipsoids axis awhile the size of the other axis b remained constant. Three of the pins A1-A3 had the size of the axis such that the ellipticity ratio of one axis was reciprocal of the other axis. The reference radius of the ellipsoidal pins was kept at 0.5 mm.
The designed pins were marked with slots with 15 0 intervals for aligning the pins at the desired orientation with respect to the sliding direction as shown in Fig. 5a. The surface of the pin was measured using a confocal microscope to verify for the axes size and surface roughness as shown in Fig. 5b. The axes sizes a and b were within the design limits a d � a m and b d � b m as shown in Table 4. The mean roughness R a of the pin tips were about 0.5-1 μm as polishing the tips was challenging (Fig. 5d). The root mean squared roughness R q and average surface roughness R a of the pins are listed in Table 4. The sheet, made up of DX 56 steel, was hot mounted on 50 mm diameter bakelite resin disc and polished using a lapping machine. Initially sandpaper grit P220 was used to grind out the unevenness and 1, 3 and 9 μm size diamond suspensions were used for mirror polishing. A final R a of 5 nm was obtained on the DX56 steel sheets prior to the ploughing experiments, see Fig. 5c. The polished sheet was then lubricated with 2g/m 2 of 'Quaker Ferrocoat N136' lubricant.

Method
The ploughing experiments were carried out using the designed ellipsoidal pins in the Bruker's UMT-2 tribometer. The tribometer has been adapted to be a scratch test set-up as shown in Fig. 6a. The UMT-2 scratch set-up consisted of three stages for motion in all three directions. The z-carriage was used to adjust the height of the ellipsoidal pin while also applying the given load on the contact. The pin was slid along the xaxis using the x-slider. The y-stage was used to mount the specimen to be tested and was also for any sliding required for the purpose of ploughing or offsetting. The z-carriage, x-slider and y-stage were moved using a stepper motor drive by translating rotational into linear motion using a lead screw and guide rails. The y-stage consisted of an eccentric screw which along with two other screws was used to clamp the disc specimen onto the stage. The y-stage was connected to the motor using a lead screw with 2 mm pitch.
The load applied on the pin and the friction in both xand ydirection were measured using the ATI F/T mini 40 (3D) load sensor with a load range of 0-60 N and 0.01 N resolution in the z-axis and a load range of 0-20 N and 0.05 N resolution in the x-and y-axis. In this way, all the force components involved in ploughing with an elliptical indenter can be measured. The 3D load sensor was connected to the pin holder using a mount and to the upper drive stage using a suspension block as shown in Fig. 6b. The suspension block with its spring plates helped in adjusting for possible shock loads. The pin holder consisted of a hole with inner diameter same as that of the base of the pin and a marking along the xaxis, i.e. direction of x-slider to adjust the orientation of the ellipsoidal pin. Load controlled tests were performed at 7 N and 16 N, normal loads for the different pin sizes and orientation along the sliding direction as shown in Fig. 6a.

Results and discussion
MPM-based ploughing simulations have been performed on a rigidplastic substrate and DX56 steel sheet in order to compare and validate the results with the analytical model and the ploughing experiments as elaborated in section 4.1 and 4.2 respectively. An ellipsoidal indenter has been loaded in the z-axis and slid along the x-axis as seen in Fig. 7 based on the ploughing model parameters listed in Tables 1-3.

Comparison of analytical results with numerical results
Fitting factors p and q (section 2.1) are used to calculate the change in ploughing depth in front and at the periphery of the asperity-substrate contact respectively, due to the change in asperity size and orientation relative to sliding. Values of p and q change with the asperity shape (ellipticity ratio) and applied load. The change in ploughing depth due to interfacial shear is measured using w ¼ 1in equation (5) for all cases. Incorporating the depth corrections due to the plastic flow and interfacial shear, the forces acting on an asperity were recalculated using equation (8.1)-(8.6). The results were compared with the results obtained from the MPM-based ploughing simulation of ellipsoidal asperity on a rigid-plastic substrate. The modelling parameters are listed in Tables 1 and 2 The superscripts of 's', 'p' represent interfacial shear and plastic deformation respectively. The subscripts 'ex', 'ey' and 'exy' represent the changing ellipticity ratio along only x-direction, only ydirection, both x-and y-direction such that e x e y ¼ 1. The subscript '1 N', '2 N' represent the applied load. The subscript 'x', 'y', 'z' represent the corresponding axis.

Comparison of friction results
The coefficient of friction plot was resolved into the components in the xand yaxis due to plastic deformation and interfacial shear and plotted for different axis size and load in Fig. 8. The coefficient of friction in x-direction due to plastic deformation μ p x increased with β from 0 0 to 90 0 due to increase in projected area along the plane perpendicular to the x-axis as shown in equation (8.1). The coefficient of friction in the yaxis due to plastic deformation μ p y increases to its maximum at 30 0 due to the difference in the net projected area perpendicular to the y-axis for    Fig. 8b. The decrease in ellipticity ratio (a/b)decreases the difference in the projected contact area in the yz plane at 0 0 and 90 0 orientation and also the difference in the projected contact area in the xzplane in þy and -ydirection with orientation β. Hence the increase in μ p x and the change in μ p y decreases with β for a decrease in the ellipticity ratio a/b (asymmetry) of the asperity (compared to Fig. 8a) as shown in Fig. 8c. For an ellipsoidal asperity with major axis perpendicular to the sliding direction, the trends in the coefficient of friction with β are reversed as shown in Fig. 8d. The results obtained from the numerical ploughing simulation for a rigid-plastic substrate are in good agreement with that obtained from the analytical model, taking into account the modified ploughed profile including e.g. pile up as discussed in section 2.1.
The effect of asperity size and applied load on the coefficient of friction due to plastic deformation and interfacial shear has been shown for three different cases in Fig. 9. For the case where the ellipticity ratio e x is increased while keeping e y as 1, the coefficient of friction due to ploughing decreases. This is because the ploughing depth decreases with increase in asperity size to maintain the same contact area for a given load. However if the ellipticity ratio e x is increased as the reciprocal of e y , the coefficient of friction due to plastic deformation decreases due to the decrease in projected area perpendicular to the sliding direction. The depth also increases with e x as explained in 4.1.2., which combined with the change in projected area results in the friction plots in Fig. 9a. On the third case, as the ellipticity ratio e y increases keeping e x as 1, the ploughing depth decreases while the projected area perpendicular to the sliding direction increases. This decreases the coefficient of friction due to plastic deformation with increase in e y , although at a lower rate compared to the previous two cases. The coefficient of friction due to plastic deformation increases with load while that due to interfacial shear remains mostly constant as shown in Fig. 9b. However for low values of e y , μ s increases with e y due to the faster increase in projected area perpendicular to the x-axis due to an increase in ploughing depth.  The analytical model shows good agreement with the numerical model except for small e y , where large deformation due to cutting increases the numerical friction force.

Comparison of ploughed profile
In order to study the ploughed profile, and compare the numerically simulated ploughing depths, a section of the substrate material surface was chosen as shown in Fig. 10. The section comprised of different regions with size of a unit particle volume and the average position of each of the region was plotted as a function of the sliding distance. Fig. 10a shows the average position of the particle in the region under the central axis of the asperity in the sliding direction over the whole sliding distance, shown as black mark in Fig. 10b. It can be said that as the orientation of the asperity changes towards 90 0 the pile up increase rapidly initially with a small decrease in ploughing depth as further discussed in the previous and upcoming sections. The ploughing depth d p is calculated as the sum of the pile-up height and groove depth (see Fig. 2) in the ploughed profile. The ploughed profile is obtained from the final average position of the particles in the ploughed cross section as shown in Fig. 10a and b.
Following equation (2), the ploughing depth should remain constant irrespective of the orientation of the asperity. However, the developed model has accounted for the plastic and shear flow behaviour to develop fitting factors to compute the ploughing depth as a function of asperity orientation. Fig. 11 shows the ploughing depth obtained from the analytical and numerical model for various orientation, size and ellipticity ratio of the ellipsoidal asperity. It can be seen from Fig. 11a that the total ploughing depth in front (at point C shown in Fig. 3a) of an ellipsoidal asperity (a ¼ 100μm; b ¼ c ¼ 200μm) during ploughing decreases with increase in β. This is due to the decrease in A yz and the resistance to plastic flow, which increases the pile-up height in front of the sliding asperity (see section 2.1.1). The change in ploughing depth in Fig. 11a corresponds to the change in friction force due to plastic deformation in the Fig. 8d. The interfacial shear stress aids to the yielding of the substrate and marginally increases the ploughing depth as shown for f ¼ 0.45in Fig. 11a (see section 2.1.2). The decrease in (pileup height) ploughing depth in front of the asperity with β in Fig. 11a corresponds to an increase in the groove depth of the ploughed profile to balance the load shared deformed substrate (see section 2.1). Hence the depth of the ploughed track (sum of groove depth and pile-up height of the peripherical ridges of the ploughed track), given as the average of d þy and d y in Fig. 11b, increases with increase in β for the same asperity.
The asymmetric separation of flow due to asymmetry in the ploughing asperity results in a variable ploughing depth on either ends of the ploughed track (points S and N shown in Fig. 3), as given in Fig. 11b and c. The change in asperity orientation from 0 0 to 90 0 results in an rapid divergence in ploughing depths d þy and d y on either side of ploughed wear track followed by their steady convergence resulting as shown in Fig. 11b and c. This behaviour is explained by the variation in distribution of piled-up substrate material in front of the asperity to its periphery with β, as shown in section 2.1 using equations (4.2) and (4.3). The depth profiles are reversed as the major axis of the asperity is perpendicular to sliding direction as shown in Fig. 11c. The ploughing depths for ellipsoidal asperity increases with decrease in e y only following load balance and plastic flow correction as shown in Fig. 11d. However the ploughing depth increases with e y , which increases as reciprocal of e x , due to increase in the resistance to plastic flow. The numerical ploughing depth does agree fairly well with the analytical ploughing depths.

Experimental validation of numerical results
The overall coefficient of friction along the sliding direction and perpendicular to the sliding direction was measured from the experiments and compared with the MPM-based ploughing simulations. The  ploughed profile of the DX-56 sheet was obtained from the observed cross-section of the wear track after the ploughing experiments. These cross-sections will be compared to the MPM-based simulation results. The effect of ellipsoid size, orientation and applied load on the coefficient of friction and deformation are explained as a part of the effect of asperity geometry on the ploughing behaviour.

Validation of friction results
In order to obtain the coefficient of friction from the ploughing experiments and ploughing simulations, the force components acting on the indenter were plotted over the sliding distance as shown in Fig. 12. The sliding distance of the numerical simulations was kept small to reduce computational cost. The average friction force was taken in the steady state of friction in the last one-third of the sliding distance. The   Fig. 15. Confocal image at 50x magnification of the ploughed track showing surface height distribution of a substrate ploughed by a load of 7N using an ellipsoidal asperity of axes size a ¼ 1000μm, b ¼ 250μm orientated at angle with respect to sliding direction (a)     ¼ 1000 μm, b ¼ 250 μm, (b) a ¼ 833 μm, b ¼  300 μm, (c) a ¼ 667 μm, b ¼ 375 μm and (d) a ¼ 1000 μm, b ¼ 250 μm at 7Nload. friction force in the xand ydirection was divided by the applied load to obtain the coefficient of friction in the xand y direction as μ x and μ y . The coefficient of friction values obtained from the numerical simulation were converged for particle and mesh sizes of 5,10 and 20 μm for the substrate and indenter respectively to obtain the final result (see [32] for convergence study background). It can be seen in Fig. 12 that the orientation of the asperity results in friction force acting on the indenter in both x and y direction.
Loads of 7 N and 16 N were applied to the indenters B1-B7 with their reference axis either along or perpendicular to the sliding direction leading to the case with varying ellipticity ratio e x and e y . The coefficient of friction was plotted against the asperity size increasing from 250 to 2000 μm both along the sliding direction and perpendicular to the sliding direction as shown in Fig. 13. It can be seen that the coefficient of friction is only measured along the sliding direction due to the axis symmetry in the sliding ellipsoidal indenter. As it can be seen from Fig. 13a that for the case μ vs e x of axis size the agreement between the experiment and the numerical results is shown for ellipsoidal indenters which are less skewed and have ellipticity ratio closer to unity. The results obtained from ploughing experiments with smaller ellipsoidal tips with e x of 0.5 and 0.33 deviate largely from numerical results at high loads as cutting wear sets instead of ploughing and friction becomes unstable. For large ellipsoidal tips with e x of 0.75 and 1 the friction results obtained from the ploughing experiments exceed those obtained from the numerical model. The measured friction for indenters with large axis size are highly sensitive to alignment with respect to both loading and sliding direction. The regions in the μ vs e x and μ vs e y plots with the aforementioned effects of cutting wear and misalignment, are marked as CW and MA respectively in Fig. 13.
The coefficient of friction was studied as a function of the angle of orientation with respect to the sliding direction β for ellipsoidal pins with varying ellipticity ratio and applied load. The ellipticity ratio of the axes e x =e y in the sliding plane was varied from 1.8, 2.8 to 4 such that the product e x e y was 1 and the coefficient of friction was plotted in Fig. 14a, b and c respectively. A load of 7 N was applied to avoid any cutting effects and maintain ploughing wear. The load was changed to 16 N for ellipticity ratio of 1.8 and the coefficient of friction was plotted in Fig. 14d. For lower ellipticity ratio of 1.8 the coefficient of friction plots both in the xand yaxis 7 N and 16 N as shown in Fig. 14a and d. However, as the ellipticity ratio of the pins were increased to 2.8 and 4, the coefficient of friction plots both in xand yaxis obtained from the ploughing experiments exceeded than that obtained from the numerical simulations for lower β values. The spread in values of the measured coefficient of friction can be attributed to the high sensitivity of the friction force with respect to misalignment for highly skewed and long indenters. A small forward tilt in the indenter in the x and y axis could lead to an increase in the friction force experienced by the indenter. Although theoretically the coefficient of friction for angle β ¼ 0 0 should reduce with increasing ellipticity ratio due to decrease in projected area in the sliding x direction, the observed increase in friction with increasing ellipticity ratio could be explained due to a minor misalignment and subsequent increase in contact area and friction due to interfacial shear.

Validation of ploughed profile
The ploughing depth d was obtained as the sum of the maximum groove depth, d g and the maximum pile-up height h pu on either side of the ploughed wear track for both the experiments as well as the simulations. The ploughing depth on the þy axis was given as d þy and on the -yaxis was termed as d y . The ploughed track that was observed under the confocal microscope for experiments done at different orientations for a ellipsoidal asperity with ellipticity ratio e xy of 4 are shown in Fig. 15. It can be clearly seen particularly in Fig. 15a and somewhat in Fig. 15d that for orientation of 0 0 and 90 0 the ploughing depths on either side of the wear track are more or less similar. Any possible difference in the ploughing depth can be due to the sliding misalignment in clamping the ellipsoidal pin mark with the pin holder marking to set the require orientations of 0 0 and 90 0 . As the ellipsoid is oriented at an angle of 30 0 or 60 0 there is a difference in the ploughing height on either side of the ploughed wear track as seen in Fig. 15b and c. It can also be seen that the ploughing depth reduces while the contact width increases from Fig. 15a-d which follows the asperity geometry.
The total ploughing depths obtained from ploughing experiments and simulation has been plotted as a function of the ellipsoid size in xand ydirection for applied loads of 7 N and 16 N as shown in Fig. 17. It can be seen in Fig. 17a and b that the ploughing depth decreases with increase in size of the ellipsoidal pin tip as also follows from the load balance given in equations (2.1) and (2.2)., The ploughing depths for pins with e y between 0.5 and 1 is higher as compared to the pins with e x between 0.5 to 1 as shown in Fig. 17. This follows from equations (3.1) and (3.2) where a smaller axes size in the y-direction results in a lower resistance to plastic flow, and hence a higher penetration of the pin into the substrate. However, both the numerical and experimental ploughing depths decreases as e x and e y increase from 1 to 2, the experimental ploughing depth in fact marginally increases as the ellipticity ratio e x increases from 1 to 2. The increase in ploughing depth could be due to misalignment of the ellipsoidal pins in the vertical plane which results in deeper penetration of the pin into the substrate. As discussed previously, an ellipsoidal pin with its major axis in the sliding direction is sensitive to misalignment in the x or y axis. The increase in ploughing depth for larger pins in Fig. 17a also explains the increase in their corresponding coefficient of friction as shown in Fig. 13a.
It can also be seen that the ploughing depths for the ellipsoidal pins in experiments are smaller than those obtained from the simulations. The difference in penetration can be attributed to the high surface roughness of the pins as listed in Table 4. To compare the effect of roughness on penetration, when a highly polished reference spherical ball of 0.5 mm radius was used instead of pin 'B4' of radius 0.5 mm, the ploughing depth obtained was about 1.5-2 times higher as seen in Fig. 16a. The effect of roughness is mostly prominent for ellipsoids with low ellipticity ratio and large size where the geometry of the ellipsoid doesn't allow for deeper penetration and the surface roughness affects the ploughing depths. However, for high ellipticity ratio the numerical and the experimental wear profile compare very well as shown in Fig. 16b. The ploughing depth obtained from experiments and simulations for either side of the ploughed track are plotted for applied loads of 7 N using ellipsoidal pins of ellipticity ratios 4, 2.8 and 1.8 and applied load of 16 N for ellipticity ratio 4 as shown in Fig. 18. It can be seen from Fig. 18a and d that for high ellipticity ratio of 4, the ploughing depths obtained on the þy and -y axis of the ploughed tracks are in good agreement for both experiments and simulation for both applied loads of 7 N and 16 N. However, as the ellipticity ratio decreases and the ellipsoidal pin is closer to a spherical shape, the roughness effects sets in and the experimental ploughing depths decreases on either side of the ploughed track (see Fig. 16a). This results in experimental ploughing depth being lower as compared to the simulated ploughing depth for ellipsoidal pins with ellipticity ratio of 1.8 and 2.8 at 16 N load as shown in Fig. 18b and c.
The models have been successfully validated for forces and penetration depths in ploughing of an elliptical asperity through a lubricated steel sheet. In the transition to cutting wear, in ploughing experiments for smaller ellipsoidal pin sizes, unstable and high friction is observed along with the formation of wear debris (chips). Modelling of friction and material removal in cutting wear requires inclusion of robust damage models in the numerical model, which is beyond the scope of the current paper. However, the effect of the size, orientation relative to the sliding direction, of the elliptical asperity and the applied load on the ploughing forces and the ploughed profile have been studied using both the analytical and the numerical models. The forces for each asperity size and orientation can be combined to calculate the ploughing forces in a multi-asperity sliding contact. Hence, by describing the contact between a (anisotropic) rough, hard tool surface and a soft, smooth sheet as a set of elliptic contact patches [11], the forces in ploughing of the sheet by the tool asperities and the ploughed profile of the sheet can be computed by the developed models.

Conclusion
An analytical model to compute the ploughed profile and forces in ploughing by ellipsoidal asperities of varying size, ellipticity ratio and orientation has been extended and compared to the numerical model for a rigid-plastic substrate. The effect of asymmetry in asperity geometry on the ploughing depths and ploughing friction has been discussed through results of the numerical model. The MPM-based ploughing model has further been validated with ploughing experiments using ellipsoidal pins on lubricated steel sheets. The results are in good agreement for most asperity geometries. The comparison between the analytical model, simulations and experiments have been done for both the forces involved in ploughing as well as the profile of the ploughed track. A good agreement has been found between the approaches. The deviation in the experimental and numerical results are described through roughness and alignment effects.