Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists

In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.


Introduction
Let us take a look back at the history of contact mechanics.In 1882, Hertz [1] solved the elastic normal contact problem of parabolic (not necessarily axisymmetric) bodies.This solution was expanded in 1941 by Föppl [2,3] and in 1942 by Schubert [4], who found an analytical solution for contacts of bodies with arbitrary dependence on the polar radius but restricted to axisymmetric shapes (this theory became widely known due to Ref. [5] by Sneddon).Despite the simplicity and elegance of this solution, the restriction to axisymmetric contacts was too strong to open the way for practical engineering applications.
An attempt to overcome the restriction of axial symmetry was undertaken in 1990 by Barber and Billings [6].Their approach is based on Betti's reciprocity theorem, as suggested by Shield [7] in 1967 and the extremal principle found by Barber [8].
However, Barber and Billings [6] merely illustrated their method by examples of contacts with "linear profiles" (i.e., pyramids with polygonal cross sections), since an analytical execution of their procedure is possible only for this case.In Ref. [9] (also in the correction (Ref.[10])), the extremal principle of Barber [8] was applied to contacts of profiles, which are not axially symmetrical, but slightly deviate from the axial symmetry.In Ref. [9], several examples were considered, which showed surprisingly good agreement with more rigorous numerical solutions.
The accuracy of the approximation is comparable, e.g., to the accuracy of the Cattaneo [11]-Mindlin [12] approximation for tangential contacts and Fabrikant's approximation [13] for the pressure distribution under a rigid flat punch of arbitrary shape, which are widely used in contact mechanical applications.Moreover, in Ref. [14], the analytical approximate solution in Ref. [9] has been generalized for the application to power-law graded materials.
www.Springer.com/journal/40544| Friction In the present paper, the solutions found in Refs.[9,14] are extensively validated by comparison with "numerically exact" solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM) for various convex profiles, that are, in part, deviating strongly from the axisymmetric shape.Thus, we suggest that the presented analytical solutions provide a simple and reasonably accurate method for solving general normal contact problems with compact contact areas in engineering design and tribology.
The remainder of the manuscript is organized as follows: In Section 2, the theoretical foundations of the approach used in Refs.[9,14] are summarized.In Section 3, the solution for non-axisymmetric contacts of homogeneous materials found in Ref. [9] is reproduced for the convenience of the reader.After that, in Section 4, analytical examples for the homogeneous solution are shown and compared to rigorous numerical solutions in Section 5.Sections 6-8 repeat the structures of Sections 3-5, for the case of power-law elastic grading, and a discussion of the advantages, drawbacks, possible extensions, and scope of application of the suggested method concludes the manuscript.

Fundamentals
The first basic idea behind the present approach is to apply Betti's reciprocity theorem to contact problems.This idea and its first applications have been described by Shield [7], who showed that the normal force F N (A) appearing due to the indentation of the profile f(x,y) to a depth d (where A is the contact area in this state) is given by Eq. ( 1): where the pressure distribution p * (x, y) is that under a flat punch with the cross section shape A, which is indented by a unit distance.Eq. ( 1) is an exact statement, which, however, does not allow any practical application, as neither the correct contact area nor the pressure distribution under a flat-ended punch with this cross section is known.An important step for the practical application of Eq. ( 1) was made by Barber [8] in 1974: He proved that the correct contact area fulfilling the usual contact conditions (the pressure inside the contact area is positive, and there is no interpenetration; outside the contact the pressure is zero, and the distance between surfaces is positive), corresponds to the maximum force at a given indentation depth.Thus, Barber [8] formulated an extremal principle, which can be used for finding the shape of the contact area, either by rigorous variation of the contact boundary or using an approximation in the sense of a Ritz ansatz.
Barber's proof [8] is based on a harmonic function representation of the elastic displacements.Let us briefly demonstrate an alternative approach (which is easier to generalize for non-homogeneous problems), based on Mossakovski's [15] ingenious idea of understanding a general indentation problem as a series of incremental flat punch indentations.
We can write Eq. ( 1) in the form of with the flat punch contact stiffness As the difference between two normal contact configurations with the contact regions A and A + dA, following Mossakovski [15], can be thought of as an incremental indentation dd by a flat punch with the planform A, the contact stiffness (Eq.( 3)) is universal.Hence, and therefore (the case dA = 0, once again, corresponds only to the flat punch contact which is the necessary condition for Barber's maximum principle [8].That "proof" ① applies to any normal contact problem, which can be thought of as a superposition of flat punch indentations, and especially to normal contacts of (locally isotropic) layered or functionally graded elastic materials.Nonetheless, to constructively apply Barber's principle [8], the pressure distribution p * under a flat-ended punch with an arbitrary cross section has to be known.Barber and Billings [6] proposed to use Fabrikant's approximation [13] for this pressure distribution.This is the last step, which closes the procedure and reduces it to the solution of a variational problem for the contact boundary line.Fabrikant's hypothesis [13] is that the stress distribution in polar coordinates (r,  ) is given in a good approximation by Eq. ( 7): where ( ) a  is the equation for the contact boundary in (r,  ) in the contact plane, and J 0 is the linear where E 1 and E 2 are the Young's moduli of the contacting bodies, and ν 1 and ν 2 are their Poisson's ratios.The motivation for the ansatz (Eq.( 7)) is not rigorous, but robust and convincing.Eq. ( 7) is known to be exact for elliptical punches with arbitrary eccentricity (as was already found by Hertz [1]), and it provides the correct asymptotic behavior in the vicinity of the contact boundary if the boundary line is smooth [15] (for punches with sharp corners, the corner singularity of the pressure distribution is of a higher order than the captured singularity by Fabrikant's approximation [13], as will be demonstrated in Section 5).
Note that Fabrikant's approximation [13] (Eq. ( 7)) is still not yet completely defined, as for its unambiguous definition the origin of the polar coordinate system has to be chosen.Both Fabrikant [13] and Barber [16] suggest to choose it at the centroid of the area A. However, there are no cogent theoretical reasons for this choice.The basic motivation for the use of Fabrikant's approximation [13] does not depend on the exact position of this point.This freedom can be used for choosing the position of the polar center in the most convenient way to simplify the calculations.
With the approximation (Eq.( 7)), Eq. (1) becomes The boundary of the contact area should now be found by maximizing this functional.We will omit the derivation of the solution of this variational problem, which was shown in Ref. [9].The main results will, nonetheless, be stated in Section 3.
Before, however, let us consider inhomogeneous materials with elastic grading of the form 0 ( ) i.e., the elastic modulus shall vary with depth z according to a power-law.The homogenous problem is always included as the special case (m = 0).The generalization of Eq. ( 7) for the indentation of a power-law graded elastic half-space by a rigid flat punch of arbitrary cross section has been given in Ref. [14] and reads (the index "m" corresponds to the inhomogeneous problem, while the index "0" shall always denote a solution for the homogenous problem) with the effective "modulus" [17]  where Γ denotes the Gamma function, and the nonlinear Note that for the contact of two elastic materials, a summation of the inverse effective "moduli"-in the spirit of Eq. ( 9)-is possible, if both materials have the same exponent m of elastic grading [18].
Hence, the generalization of Eq. ( 10) for power-law graded media is given by Eq. ( 15): Again, to obtain the approximate contact solution, the variational problem resulting from Barber's extremal principle [8] needs to be solved.The complete procedure is detailed in Ref. [14], and its most important results will be stated briefly in Section 6.

Approximate solution for normal contact of elastically homogeneous and slightly non-axisymmetric profiles
Let us consider an "arbitrary" profile ( , ) underlying the following restrictions: 1) The profile deviates only weakly from an axisymmetric one.
2) f(r,  ) is a monotonously increasing function of the polar radius r.
Under these conditions, the following simple analytical procedure can be applied, as shown in Ref. [9]: I.The origin of the polar coordinate system is placed at the lowest point of profile, so that (0, ) 0 II.In the second step, an "equivalent axisymmetric profile" is determined, which is simply the profile, averaged over the angles.
III.The profile (Eq.( 16)) is decomposed into an axisymmetric part and the deviation IV.With the equivalent axisymmetric profile (Eq.( 18)), the usual solution procedure of the method of dimensionality reduction (MDR) is applied [19].In particular, the transformed profile is determined as and V. The relation between the indentation depth and the effective contact radius a is determined by Eq. ( 22): VI.The normal force is given by Eq. ( 23): VII.The true non-axisymmetric contact area is given by Eq. ( 24): where ( ) a   is determined as with the prime denoting the first derivative with respect to the (first or only) functional argument, and and VIII.Finally, the pressure distribution can be calculated from the integral Friction 12(2): 340-355 (2024) where the effective contact radius must be understood as a function of ( ), a  as given implicitly by Eqs.(24) and (25).The upper star on the contact quantities under the integral in Eq. ( 28) denote the values during the indentation procedure, as the pressure distribution is determined by the integral over all incremental indentation steps, i.e., from an indentation depth d * = 0 until the final value d * = d [9].
VIIIa.A much more convenient way to determine the pressure distribution is scaling the axisymmetric solution for the contact pressure (axi)  p to the actual contact area, i.e., While this will exactly give the same result as the evaluation of the more complex Eq.(28) in the case of power-law indenters, as will be shown below, it will most probably always provide a satisfactory approximate solution for the real contact pressure distribution.
Eq. ( 30) is a generalization of Fabrikant's ansatz [13] for flat-ended punches.In the latter, the pressure distribution under an arbitrary flat punch is equal to the one under an axisymmetric punch, but "rescaled" to the true shape of the contact area.Similarly, Eq. (30) states that the contact pressure is equal to the one under an "equivalent axisymmetric indenter", but rescaled to the true contact area.
In the present paper, it will be shown that Eq. ( 30) is the exact approximation for power-law profiles (with arbitrary cross section), independently of the exponent of the power-law.Although it is not proved in a general case, the independence of the exponent gives hope that it will be a good approximation for arbitrary profiles.In that regard, one has to also bear in mind that the complete procedure is approximate (albeit yielding very good results, as will be demonstrated in Sections 5 and 8); therefore, in most cases, probably not much is gained by evaluating the seemingly more rigorous Eq. ( 28), compared to the scaling idea expressed in Eq. ( 30).
The whole procedure can be summarized as follows.First, an axisymmetric profile is produced by averaging the given profile over the polar angle.After that, the contact problem is solved for this equivalent axisymmetric profile.Finally, the true contact area is found by Eqs. ( 24) and ( 25), and the pressure distribution is calculated from either Eq. ( 28) or the more convenient Eq. (30).

Analytical examples for homogeneous materials 4.1 Solution for power-law shapes
Consider a profile having the form which means that all vertical cross sections are self-similar, differing only by a scaling factor.For all profiles, which can be written as a product of a radial and an angular function, the decomposition (Eq.( 19)) is very simple, as it affects only the angular factor.
( ) with the average value of ( ) For the MDR-transformed profile, we get Accordingly, for the deviations (Eqs.( 26) and ( 27)), and the boundary of the contact area is given by Eq. (36): where the effective contact radius is determined by Eq. (37): Note that Eq. ( 36) differs from the corresponding Eq. (67) in Ref. [9].The correct one is Eq.[36], while Eq.[67] in Ref. [9] contains an error, which was corrected in Ref. [10].
The normal force, according to Eq. ( 23), equals to Hence, for the average contact pressure p For the pressure distribution in the contact area normalized by the average pressure, we obtain Eq. (40) according to Eq. (28).
It is the same result as that for an axisymmetric contact (page 78, Ref. [18]), scaled to the non-axisymmetric contact area ( ) a  , as expressed in Eq. (30).However, this phenomenon is only possible due to the selfsimilarity property const.( ) ( ) Therefore, it is not generally correct, at least not in the rigorous (albeit asymptotic) sense, as shown above for power-law (and hence self-similar) profiles.

BEM results for elastically homogeneous bodies with complex shapes
Now, we shall compare the asymptotic analytical results obtained in the previous section with rigorous numerical simulations of the corresponding problems, done with the BEM for an elastic half-space [20], accelerated by the FFT.All problems are solved under displacement-controlled conditions, i.e., the indentation depth shall be prescribed.
Let us introduce two normalized measurements to quantify the error made by the analytical, but approximate solution, compared to the numerically  ( ) where the superscripts (th) and (sim) represent theoretical and simulated values. (th) a is the average value of (th) .
a N is the number of discretization points along the contact boundary, and the theoretical effective contact radius is determined from the indentation depth via Eq.(37).

Indenters with cross sections of regular polygons with sharp corners
First, we consider regular n-polygons given by Eq. ( 46): In Fig. 1, the theoretically predicted contact area (the red line) is compared to the area resulting from a BEM simulation (the grey filled shape), for the indentation by conical indenters (k = 1) with n = 3, 4, 5, and 6.In all cases, the theoretical and numerical results are normalized by the same value, specifically the theoretical effective contact radius, which can be determined immediately from the indentation depth, based on the first equation of Eq. (42).Apparently, the approximate solution works better for larger values of n.This is to be expected, as the limit n → ∞ corresponds to the indentation by a perfect cone, for which the analytical solution is, of course, exact.Moreover, for all values, except n = 3, the agreement between the analytical and numerical results is very satisfactory.
In Fig. 2, the error measurements introduced in Eqs. ( 44) and (45) are shown for conical (k = 1) and parabolic (k = 2) indenters, as well as for different values of n.It can be seen that the error of the approximate solution rapidly decreases with the increasing n.
In Figs. 3 and 4, the pressure distributions in contacts with pyramidal and parabolic indenters with square  cross sections are presented-both from the BEM simulation and the approximate analytical solution.The numerical results for the pressure distribution converge rapidly with the mesh size, except for the sharp edge (i.e., the cross section at 45°), which leads to the appearance of a weak (logarithmic) stress singularity.Accordingly, the theoretical asymptotic prediction is in good agreement with the BEM results almost everywhere, with the exception of the immediate vicinity of the sharp edges.

Indenters with cross sections of regular polygons with rounded corners
As shown in Section 5.1, the analytical approximate solution gives very good results, except for the vicinity of sharp edges of the indenter profile.To better understand the role of sharp edges, let us consider the same regular power-law polygonal indenters as before, but with rounded corners.Figure 5 shows the results for the same cross sections,   as shown in Fig. 1, but with rounded corners.Figures 6  and 7 show the corresponding results for the pressure distribution, along the same directions, as shown in Figs. 3 and 4, respectively, for polygon cross sections with sharp corners.The pressure distribution under an indenter with rounded corners has no singularity, and the simulation results rapidly converge everywhere with the mesh size.However, a significant maximum of pressure along the diagonal direction and correspondingly significant deviations between numerical and analytical results still remain.

Power-law indenters with irregular cross sections
One might argue that the quality of the asymptotic solution demonstrated in Sections 5.1 and 5.2 is only due to the fact that regular polygons are still relatively similar to axisymmetric indenter cross sections (at least, they exhibit discrete rotational symmetry).To check this hypothesis, we randomly generated several irregular horizontal indenter cross sections and  compared the corresponding normal contact solutions based on the approximate analytical solution and the BEM simulation.However, self-similarity was retained, as the radial dependence of the indenter profile was still chosen in the form of a power-law.In Fig. 8, the results are shown for the error measurements introduced in Eqs. ( 36) and (37), for conical (k = 1) and parabolic (k = 2) indenters, for 160 irregular cross sections that were generated randomly.The diagrams give the normalized error of the approximate solution as a function of the standard deviation of the cross section from perfect rotational symmetry.Naturally, the stronger the profiles deviate from axial symmetry, the larger the error of the asymptotic solution.However, no qualitative "misjudgements" of the asymptotic solution can be detected.
Interestingly, the approximate solution provides consistently better results for the parabolic indenters (k = 2) than for the conical ones (k = 1).
In Fig. 9, a detailed comparison of calculated contact areas is shown for six selected irregular shapes, marked with the red triangles 1-6 in Fig. 8.

Indenters with three-dimensional irregular shapes
For the application of the approximate solution in various tribological contexts, it is interesting to know whether the solution can also be used satisfactorily in the case of a three-dimensionally irregular indenter shape, e.g., to analyze local features of rough surfaces.So, let us turn our attention away from power-law shapes, and consider a general three-dimensional indenter profile.In Fig. 10, a comparison is shown between the approximate solution and the BEM simulation, for the relations between the macroscopic contact quantities (normal force, indentation depth, and contact area) in the indentation of an elastic half-space by the general indenter shape, as shown in the subplot of Fig. 10(a).All relations are in properly normalized variables, and it is obvious that the agreement between the approximate and the rigorous numerical solutions is very good.
Figure 11 gives the corresponding comparison for the pressure distributions at three instances of the indentation procedure, as marked in Fig. 10(b).

Approximate solution for normal contact of elastically inhomogeneous, slightly non-axisymmetric profiles
Let us now turn our attention to the mathematically slightly more general-yet, physically, probably more specific-problem of the indentation of a power-law graded elastic half-space by a slightly non-axisymmetric indenter.Compared to the algorithm described in Section 3, the steps I-III, i.e., the definition of the profile and its separation into axisymmetric and non-axisymmetric components, remain the same.However, the contact solution has to be generalized to account for the elastic grading, as shown in Eq. (11).Specifically, the generalized auxiliary profiles for the inhomogeneous problem are given by Eq. (48): The relations between the indentation depth, effective contact radius, and normal force are determined by the axisymmetric solution (Eq. ( 49)): and the deviation of the contact boundary from circular form can be calculated from Eq. ( 50). ( m G a a g a a mg a a g a Finally, the pressure distribution is found as where the upper star on variables under the integral denotes contact quantities during the indentation process from d * = 0 to the final value d * = d.
Once again, a more convenient way of determining the pressure distribution is to scale the respective axisymmetric result to the real asymmetric contact area, as expressed in Eq. (30).As in the homogeneous case, this will provide exactly the same solution as the evaluation of Eq. ( 51) www.Springer.com/journal/40544| Friction in the case of power-law indenters with arbitrary exponents and cross sections-as will be shown below-and will therefore probably be in close agreement with that of Eq. ( 51) in any even more general case.

General approximate inhomogeneous solution for power-law shapes
As an analytical example, we will consider the power-law indenter with arbitrary self-similar cross sections again, as shown in Eq. ( 31).The auxiliary MDR profiles are given by Eqs. ( 53) and (54): Hence, the boundary of the contact area equals to where the effective contact radius is determined by Eq. ( 56): Moreover, we obtain for the normal force and the pressure distribution in the contact area, normalized by the average pressure with the same definition of ρ, as shown in Eq. (40).Once again, Eq. ( 58) is the same result as that for an axisymmetric contact (page 265, Ref. [18]), scaled to the non-axisymmetric contour a ( )  .

BEM results for elastically inhomogeneous bodies with complex shapes
Let us, once more, compare the analytical approximate results obtained above to rigorous numerical simulations based on the BEM [21].We will not reproduce all the extensive numerical studies shown in Section 5 for homogeneous materials to account for the elastic grading, because all main findings are similar in the inhomogeneous case.To demonstrate the influence of material grading, some results will suffice.In Fig. 12, a comparison is shown for the predictions of the contact domain, based on the analytical approximate solution and BEM simulations, for pyramidal indenters (k = 1) with rounded regular n-polygon cross sections (n = 3, 4, and 6) and for two exponents of the power-law elastic grading, specifically m = 0.5 and −0.3.It is apparent that the approximate solution works significantly better for the case of positive elastic grading (m > 0, i.e., a soft surface with a harder material core).In fact, for m > 0, the approximate solution provides consistently better results than that in the homogeneous case (m = 0).This is to be expected, as it is known-and obvious from the pressure distribution in Eq. ( 12)-that positive elastic grading reduces the order of edge and corner singularities for the stress distributions, and therefore the quality of the approximate solution (which is only compromised at sharp edges or corners, as shown throughout this manuscript) will be improved by positive grading (and in turn reduced for negative elastic grading, i.e., hard surfaces with a softer material core).
This phenomenon is confirmed again in Fig. 13, showing the comparison for the predicted pressure distribution due to the indentation of a graded elastic half-space with m = 0.5 by a pyramidal indenter (k = 1) with rounded square cross section (n = 4).As can be seen, the prediction based on the analytical approximate solution is in almost perfect agreement with the rigorous numerical solution.

Discussion
The main milestones of contact mechanics, which had (and still have) a lasting effect on practical tribological applications in science and engineering, were the original work of Hertz [1] from 1882 for elliptical parabolic profiles, followed by the solution of Föppl [3] and Schubert from [4] in 1941 and 1942, respectively (which became widespread thanks to Ref. [5] by Sneddon in 1965) for axially symmetric but otherwise arbitrary indenter shapes.The solution presented in the paper at hand is an extension of this axisymmetric solution to non-axisymmetric profiles.
As the Boussineq problem (i.e., the frictionless normal contact problem) for arbitrary contact domains is too complicated to allow for an exact, general analytical solution, it has been believed for a very long time that the scope of analytical contact mechanics is restricted to very specific geometries, specifically, problems with either axial or plane symmetry, which were solved in a general form already in Ref. [4].However, in the present paper, we have laid out a general procedure for the solution of the Boussinesq problem for compact, but otherwise arbitrary, contact domains, which retains the analytical simplicity of the axisymmetric solution, but which-despite its approximate nature and asymptotic character of its derivation-has proved highly robust in its predictions even in the case of contacts that are far from axial symmetry.
Nonetheless, some restrictions must be kept in mind when applying the obtained procedure to real engineering contacts; restrictions, which, on the one hand, originate from the modelling abstractions of the Boussinesq problem (i.e., linear elasticity and the half-space approximation), and on the other hand, from the approximate nature of the solution, which, at least selectively, should be checked by more rigorous numerical simulations, e.g., based on the BEM or finite element method (FEM).
On the other hand, the procedure can be used to solve other classes of contact problems that reduce to the Boussinesq problem, e.g., the viscoelastic normal contact-via the elastic-viscoelastic correspondence principle [22,23]-or the tangential contact with friction, within the framework of the Cattaneo [11]-Mindlin [12] approximation.
The last point deserves a brief elaboration: It has been shown that the reduction of the tangential contact with friction to the frictionless normal contact problem via the principle of Jäger [24] and Ciavarella [25], in an approximate sense, is also possible for general three-dimensional [26] and even rough contacts [27].The quality of this approximation (i.e., of the reduction procedure) is of the same order as the quality of the approximate normal contact solution discussed in the present manuscript.In other words, this approximate contact solution can be applied straightforwardly to incorporate tangential (frictional) forces.
The approximate contact solution presented in this manuscript can be used for different tasks in tribology and engineering.On the one hand, it can be applied for the fast analysis of macroscopic contacts with complex shapes, e.g., within the framework of indentation testing.On the other hand, it may serve as a tribological tool for modelling single microcontacts ("asperities") with complex (or random) shapes in the contact of rough surfaces.In that regard, one, however, has to bear in mind that the solution is intended (in the first place) for the analysis of single contacts with simply connected (compact) contact domains.

Fig. 1
Fig. 1 Comparison of contact areas for pyramid indenters (k = 1) with regular n-polygon cross sections.The contact size is normalized by the theoretical value of the effective contact radius, as shown in the first equation of Eq. (42).

Fig. 2 Fig. 3
Fig. 2 Normalized errors of calculated normal force and average deviation of contact boundary between approximate solution and numerically exact BEM results for conical (k = 1) and parabolic (k = 2) indenters.

Fig. 4
Fig. 4 Pressure distributions of parabolic indenters with square cross section (k = 2 and n = 4).(a) Pressure at cross sections of a, b, c, and d corresponding to 0°, 30°, 40,° and 45°, respectively, as shown in subplots (b) and (c).Symbols are numerical results.(b) Contour line diagram of pressure distribution according to BEM simulation.(c) Contour line diagram of pressure distribution according to approximate analytical solution.

Fig. 5
Fig. 5 Comparison of contact areas for pyramid indenters (k = 1), whose cross sections are polygons with rounded corners.The contact boundary is normalized by the theoretical value of the effective contact radius (red: approximate solution, grey: BEM simulation).

Fig. 6
Fig. 6 Pressure distributions at different cross sections of pyramid with square cross section (k = 1 and n = 4).(a) Pressure at cross sections of a, b, and c corresponding to 0°, 30°, and 45°, respectively, as shown in subplot.The symbols are the numerical results.(b) Contour line diagram of pressure distribution according to BEM simulation.(c) Contour line diagram of pressure distribution according to approximate analytical solution.

Fig. 7
Fig. 7 Pressure distributions at three cross sections of parabolic indenters with square cross section (k = 2 and n = 4).(a) Pressure at cross sections of a, b, and c corresponding to 0°, 30°, and 45°, respectively, as shown in the subplot.The symbols are the numerical results.(b) Contour line diagram of pressure distribution according to BEM simulation.(c) Contour line diagram of pressure distribution according to approximate analytical solution.

Friction 12 ( 2 )Fig. 8
Fig. 8 Normalized errors of calculated normal force and average deviation of contact boundary between approximate solution and numerically exact BEM results, depending on standard deviation of irregular angular profile function ( )   .160 random profile cross sections were realized.The blue stars correspond to the pyramid indenters (k = 1), and the orange circles correspond to the parabolic indenters (k = 2).The examples marked by 1-6 are shown in Fig. 9.

Fig. 9
Fig. 9 Comparison of contact areas for six pyramid indenters (k = 1) with irregular cross sections.The orange curves are the approximate solution.The contact boundary is normalized by the theoretical value of the effective contact radius (red: approximate solution, grey: BEM simulation).

Friction 12 ( 2 )
Fig. 10 (a) Normal force-indentation depth and (b) contact area-indentation depth dependencies for indentation by general three-dimensional indenter.The subplot of (a) is the profile of the indenter.The red stars are the approximate solution, and the blue cycles are the BEM simulation with the 1024 × 1024 meshing grid.Comparisons for the shape of the contact area at three different indentation depths are shown in (b).

Fig. 12
Fig.12 Comparison of contact areas for pyramid indenters (k = 1) in contact with graded materials.The contact boundary is normalized by the theoretical value of the effective contact radius (red: approximate solution, grey: BEM simulation).