Rough Surface Contact

Transcription

1 Int. Jnl. of Multiphysics Volume Number 7 97 ABSAC ough Surface Contact Nguyen, B Alzahabi* Kettering University, USA his paper studies the contact of general rough curved surfaces having nearly identical geometries, assuming the contact at each differential area obeys the model proposed by Greenwood and Williamson. In order to account for the most general gross geometry, principles of differential geometry of surface are applied. his method while reuires more rigorous mathematical manipulations, the fact that it preserves the original surface geometries thus makes the modeling procedure much more intuitive. For subseuent use, differential geometry of axis-symmetric surface is considered instead of general surface (although this general case can be done as well) in Chapter.. he final formulas for contact area, load, and frictional torue are derived in Chapter... INODUCION he need of understanding rough surfaces contact has long been recognized. One primary focuses of the early studies is to predict real contact area as it varies with load. Since a rough surface is known to include layers of micro-asperities, the real area of contact can be extremely small comparing to the apparent area observed by our eyes and is very difficult to measure. his problem has been addressed and resolved for the first time by Archard, Greenwood and Williamson using novel fractal and statistical models to mathematically describe the microscopic surface structure. heir works have been the basis for various subseuent studies on contact mechanics, describing the surface geometry (asperities distribution, geometry) and material behavior (elastic, plastic flow) (Yastrebov et al. 4). ecently, a deterministic approach to model rough surface contact has grown rapidly with the advance of computational capability, providing further insights to the study of contact mechanics. he fact that only nominally flat rough surfaces case is focused has limited the scope of this model. One reason for this shortage was given by Greenwood and rip, as generally the curvatures difference of curved surfaces creates a cluster effect which makes asperities interaction becomes significant. hus an intensive analysis similar to those performed in the nominally flat rough surfaces contact is not freuently performed in the case of rough curved surfaces contact. ather, the latter in only loosely studied through the inspection of axial contact between two rough curved surfaces having constant curvatures, by replacing them with a nominally flat rough surface and a smooth curved surface having aneuivalentcurvature (Johnson 985). his method although gives a uick approximation of pressure distribution, it does not allow one to account for: More general analysis, such as the contact is non-axial or the surfaces have varying curvatures *Corresponding Author: balzahab@kettering.edu

2 98 ough Surface Contact More detailed analysis, such as the true distribution of contact pressure which is important to the calculation of cumulative frictional torue in rotating parts. his paper studies the contact of general rough curved surfaces having nearly identical geometries, assuming that the contact at each differential area obeys the model proposed by Greenwood and Williamson (GW model for short). In order to account for the most general geometry, principles of differential geometry of surface are applied. his method reuires more rigorous mathematical manipulations, as it preserves the original surface geometries (i.e. not reuire the original system to be replaced by any euivalent system) makes the modeling procedure much more intuitive. For subseuent use, differential geometry of axis-symmetric surface is considered instead of general surface (although this general case can be done as well) in Chapter.. he final formulas for contact area, load, and frictional torue are derived. One direct application of this study is the analysis of roughness-dependent frictional torue occuring rotating parts, whose geometry is often axis-symmetric. For flat surfaces contact (i.e. two flat surfaces slide across each other), effect of friction is generally uantified with the calculation of frictional force value. Similarly, for curved surface contact (e.g. in journal bearing), the value of frictional torue is freuently reuired. Unlike the former situation, where surface roughness does not affect the frictional force if one uses the Coulomb s friction model (since the total reaction force at the points of asperity contact always eual to the load), surface roughness changes the distribution of contact pressure across the curved surfaces (even when Coulomb s model holds), thus frictional torue value would be different. Furthermore, the frictional torue will not vary linearly with the load like when one models contacting surfaces smooth, rather it will also be dependent on the roughness. his topic is clarified through two specific examples. Lastly, additional analysis on the load contact area and frictional torue load relationship is presented.. MODELING OUGH SUFACE CONAC: Greenwood and Williamson (966) proposed a method to mathematically model the stochastic nature of surface s microscopic structure by using a probabilistic approach, which introduce the concept of asperity and consider their height to be normally distributed over the entire rough surface. In practice, such statement is valid for most high-end engineering surfaces (i.e. homogeneous, isotropic surface), yet not uite so for other lower-end ones (Bhushan ). For the latter situation, Kotwal and Bhushan (996) have developed an analytical method to generate probability density functions of non-gaussian distributions, but will not be considered in this study. One key assumption in the work of Greenwood and Williamson is that each individual contact does not affect the deformation of its neighbors and the asperity is spherical with curvature at its peak (Figure ). his conveniently allows the each individual contact to be modeled independently by implementing the Hertzian theory. As mentioned previously, this method limits the contact to only be between nominally flat rough surfaces, so that the above assumption holds.

3 Int. Jnl. of Multiphysics Volume Number 7 99 Consider two rough surfaces in contact which could be replaced by a system of two other surfaces with euivalent asperities curvature and MS roughness parameter. he first surface is perfectly smooth and is located at some distance l from the reference line h. he second surface is considered rough with the asperities height z varies randomly around the reference line which is described by the Gaussian distribution (Skewness and Kurtosis ) (Figure ): ( z h ) Φ( z) exp π () where is the MS roughness parameter. If the total number of asperities on this surface is N, the number of asperities having height in the interval [ z, z + dz] comes into contact is ν N Φ ( z) dz and thus the total number of asperities in contact is N x x l ν x N Φ( z) dz. Furthermore, according to the Hertzian contact theory, when a elastic sphere is indented to depth z l (from now refer as the indentation depth) in an elastic half-space: he contact area is 4 * F sin g E o ( z l). hus the cumulative contact area is x l A πa π ( z l ) and the reuired force is sin g x x A νa π ( z l ) N Φ( z) dz sin g x l x l and the cumulative reuired force is x x 4 * ν sin g o with x l x l ν ν * E E E F F E ( z l ) N Φ( z) dz modulus of elasticity and can be found using * E is the euivalent + ( E, E, ν, ν are the moduli of elasticity and Poisson s ratios of the two bodies) (Figure ). Noting that if a thin layer of coating is present, according to Liu et al (5) a different euivalent modulus of elasticity should be used.

4 ough Surface Contact Figure. GW model of a single asperity Figure. GW model of rough surfaces contact

5 Int. Jnl. of Multiphysics Volume Number 7 Figure.Contact between a elastic sphere and elastic half-space Very recently, the exact solutions for these integrals have been found in Jackson and Green (): If / z ( ) ( ) ( ) exp a π σs σs and Iep z a Φ z dz with Φ z I ( z a) Φ( z) dz ea a, then: I ep aexp( γ) ( α ) K ( γ) α K ( γ) a π σ + > s 5 Γ a 4 π 4 σ s π aexp( γ) ( + α ) I ( γ) + ( + α ) I ( γ) + α ( I ( γ) + I5 ( γ)) σ a < s () where: a α, α a α I ea exp( ) erfc () π σ s α γ σ s 4 I() and K() are the modified Bessel function of the first & second kind respectively erfc() is the complimentary error function Γ () is the gamma function

6 ough Surface Contact. NEALY-IDENICAL OUGH CUVED SUFACES CONAC: Since the two curved surfaces considered in this study are nearly-identical, contact at each infinitesimal area could be treated as the contact between two nominally flat rough surface, which can then be integrated to describe the overall contact behavior between gross geometries. his approach reuires the number of asperities and the indentation depth at each differential surface contact to be found. Furthermore, the magnitude, direction and location of application of contact pressure (i.e. define a bound vector) at each individual asperity as well as corresponding differential surface area should also be stated. What is known is the applied load, geometry and material properties of considered surfaces, and therefore any expression should be written in terms of these given parameters... Differential geometry of surfaces: i) Vector formalism of line in D space: It is very convenient to express a general bound vector in D space using vector formulation. A bound vector is completely defined if its initial point, magnitude and direction are specified. In this problem, three uantities need to be expressed vectorially are the asperity s direction, the reaction force and the friction force. Consider a straight line L is defined by two parameters ( xp, l ) (Figure 4). An arbitrary point Q on the line has position vector x Q is given by xq xp + λl where λ is an arbitrary scalar. We are also interested in he point where an asperity comes into contact: a point I is the intersection of two line L ( x, l ) and L ( x, l ) has position vector x I given by: x I (4) x + λ x + l λ l Figure 4. D line parameters

7 Int. Jnl. of Multiphysics Volume Number 7 Figure 5. D surface parameters ii) Differential geometry of surface of revolution: o understand the contact of any asperity on the surface, differential geometry is used to describe many of the surface s characteristics as function of surface s parameters (Figure 5). A general surface S in D space can be generated by two parameters ξ and ξ. Any p p( ξ, ξ ). In orthonormal coordinates the point on the surface has the position vector surface of revolution can always be expressed as (Gray 997): Let p ξ a and φ ( ξ ) cos( ξ) ( ξ, ξ ) φ( ξ ) sin( ξ) ψ ( ξ ) p a : ξ a a p (5) [ ϕsin( ξ ) ϕcos( ξ )] [ ϕ cos( ξ ) ϕ sin( ξ ) ψ ] [ ϕ( ψ cos( ξ ) ϕψ sin( ξ ) ϕϕ ] aa % aa % ϕ ψ + ϕ he unit normal vector n of the surface is:

8 4 ough Surface Contact ~ ψ cos( ξ) a a sgn( φ) ξ ψ, ξ ) sin( ξ ) a~ a φ + ψ φ n (6) ( a ~ a dξdξ sgn( φ) φ ψ + φ dξ ξ d da (7) For a surface of revolution, it is natural to pick ξ r, ξ φ (Figure 6). Figure 6. A general surface of revolution If a surface is obtained by rotating the curve z f() r from z z to z z around the, from E. (5) any point on this surface has the position vector: z axis with ϕ () r r and ψ () r f() r z. r cosϕ p( r, ϕ ) r sinϕ (8) f ( r) From E. (6) and E. (7), the corresponding unit normal vector n and the corresponding area of a differential surface element da is: z cos( ϕ) n ( r, ϕ) z sin( ϕ) (9) + ( z )

9 Int. Jnl. of Multiphysics Volume Number 7 5 da + ( z ) rdrdϕ () Furthermore, if is: gz ( ) f, the (apparent) area of this surface given by Anton (999) z Aapperent g( z) + z [ g ( z) ] dz π ().. Calculation of contact area, contact pressure and frictional torue: Consider a surface of revolution S with vertex O is at the origin of frame O is at the origin of frame ( i, i, i) and a surface of revolution S with vertex ( j, j, j ) (Figure 7). ( i, i, i ) can always be chosen such that the position vector of O with respect to e ex ez. Because the surface is axis-symmetric and the asperities are randomly distributed, the eccentricity vector F F F, which is a known vector. is the eccentricity vector [ ] is assumed to be parallel to the load [ ] r z Figure 7. Contacting surfaces of revolution According to the GW model, let S is a rough surface and S is a perfectly smooth surface. he geometries of S and S are generated by rotating the curves z f( r) and z f ( r ) around i and j respectively.

10 6 ough Surface Contact Consider a differential area at point I on the rough surface S defined by two parameters ( r, φ ) and associated with the normal vector n( r, φ ) (resolved in smooth surface S at point I. n( r, φ) I (resolved in ). From E. (9) and E. (): ). n intersects the is normal vector of the smooth surface S at point z cos( ϕ ) z sin( ϕ ) [ n n n ] n, x, y, z + ( z ) + ( z ) + ( z ) z cos( ϕ ) z sin( ϕ ) [ n n n ] n, x, y, z + ( z ) + ( z ) + ( z ) () () da + ( z) r dr dϕ (4) ( r, φ ) di (, I) and δ di (, P) with P is any point on the line II. Introduce Physically,δ, n, n represent the height, the direction of an individual asperity and the direction of reaction force respectively (Figure 8). Figure 8. epresentation of several surface parameters

11 Int. Jnl. of Multiphysics Volume Number 7 7 System of euations for the intersection at I can then be derived from E. (4) and E. (8): e x + r cos( ϕ ) ( r, ϕ) n, x + r cos( ϕ) r sin( ϕ ) ( r, ϕ) n, y + r sin( ϕ) e + z z ( r, ϕ) n, z + z (5) By solving this system, we obtain ( r, φ) as well as r( r, φ ), r φ (, φ ) into E. (), we could express indentation depth of an individual asperity can be found as: r ( r, φ ), φ( r, φ ). Substitute n in terms of r and φ. he I r, ϕ ) ( + δ )cos( ) (6) ( θ with cos( ) n n θ (Figure 9). In a differential area, the asperities can be assumed to be unidirectional (i.e. having the same direction vector). If the asperities density is ℵ, he number of asperities in that differential area is ℵ da ℵ + ( z ) r dr dφ. Since the asperities height is normally distributed described by the Gaussian distribution, the number of asperities on a differential δδ, + dδ is: area that height in the interval [ ] ν ℵ da Φ( δ ) dδ ℵΦ( δ ) + ( z ) r dr dϕdδ (7) In terms of contact pressure distribution: ν F 4 Π * E ℵΦ( ) I o δ dδ n da In terms of contact pressure distribution: ν F 4 Π * E ℵΦ( ) I o δ dδ n da Furthermore, the position vector of a single point of contact I and the moment arm (Figure ) respectively are:

12 8 ough Surface Contact p p g p + ( r, ϕ ) n sin sin g p sin g sin g ( p i ) i (8) ϖ i Assume the relative angular velocity is force at an asperity contact is:, the directional unit vector of the frictional n f sgn( ϖ )sgn( n ) sgn( )sgn( ), x n, y ϖ n, x n, x ( n f n n, x + n, y n, x + n, y ) (9) Finally, we attain the expressions for the number of asperities, real contact area, reaction force components and frictional torue in terms of an individual asperity, a differential surface area and the entire contacting surfaces: Figure 9. Contact of a single asperity

13 Int. Jnl. of Multiphysics Volume Number 7 9 able. Expressions for the number of asperities, real contact area, reaction force components and frictional torue Single asperity Number of asperities in contact eal contact area Differential surface area Entire surface Single asperity Differential surface area Entire surface ν ℵ Φ( δ ) + ( ) π dz o A A d sin g ν N π c I z r dr dϕ dδ ν Asin g π ℵΦ( δ ) I + ( z ) () r dr dϕ dδ diff π dz o d A A eaction force 4 * (x - component) Single asperity Fsin g, x E o I ( n i ) Differential 4 * F, x E o ℵΦ( δ ) I + ( z ) r dr dϕdδ surface area Entire surface diff real diff π dz o d F diff, x eaction force 4 * (y - component) Single asperity Fsin g, y E o I ( n i ) Differential 4 * F, y E o ℵΦ( δ ) I + ( z ) r dr dϕdδ surface area Entire surface F F x diff π dz o d diff, y eaction force 4 * (z - component) Single asperity Fsin g, z E o I ( n i) Differential 4 * F, z E o ℵΦ( δ ) I + ( z ) r dr dϕdδ surface area Entire surface diff π dz o d F diff, z Frictional 4 * torue Single asperity ( ~ sin g µ E o I psin g n ) Differential 4 * ( ) diff ν sin g µ E o ℵΦ δ I + ( z ) r dr d surface area Entire surface F z () ( n () ( n i ) () ( n (4) ϕ dδ ( ~ p π dz o d diff f i ) sin g (5) i ) n ) f

14 ough Surface Contact E. () shows the number N c of asperities that are in contact. E. () can be solved using E. () that approximates the real contact area. E. (), (), (4) yield the components of the cumulative reaction force which according to Newton s third law have to be eual to the components of the applied load. Finally, E. (5) gives the expression for the cumulative frictional torue. Figure. epresentation of moment arm 4. APPLICAION O SOME COMMON SUFACE GEOMEIES: 4.. Concentric spherical annulus: Consider two concentric spherical surfaces (Figure ) created by the revolution of: z r f( r ) r z (6) r z r f ( r ) r z (7) r with: sph [ F ] F (8) [ ε ]( < ) e ε (9)

15 Int. Jnl. of Multiphysics Volume Number 7 ( ε <<, ) ε () Figure. Schematic of a concentric spherical annulus Dropping the subscript of r,ϕ. It can be readily verified that, ϕ) ε (r and: r r n n cos( ) sin( ϕ) r ϕ () hus from E. (6) and E. (8): (r () I, ϕ ) δ + ε ε p r r g ε cos( ϕ ) sin( ϕ sin ) () ~ ε p g n f p r g sin sin (4) Substitute E. (), (6), (), (4) into E. (5), we attain the frictional torue expression:

16 ough Surface Contact 4 * µ E o ℵ Iδ I r I sph f ϕ (5) where: π Iϕ dϕ π I I δ r ( δ + ε ) ε d d r r / π dr δ exp arctan dδ r r I ( δ can be solved using E. ()) r r 4.. Eccentric cylindrical annulus: Consider two cylindrical surfaces (Figure ) defined by the revolution: d d r (6) z with: z r (7) cyl cyl [ F ϕ ) F sin( ϕ ) ] F cos( (8) [ ε ϕ ) ε sin( ) ] e cos( ϕ (9) da hdϕ (4) E. (5) then becomes: ε + cos( ϕ ) (, ϕ)cos( ϕ) + sin( ϕ ) (, ϕ)sin( ϕ) + h h cos( ϕ) ( a) sin( ϕ) ( b) ( c)

17 Int. Jnl. of Multiphysics Volume Number 7 Suaring (a) and (b) then take their sum: (.cos( ϕ ) + + [ + cosϕ ε ) ε cos( ϕ ) + ε cos( ϕ )] + [ + (.sin( ϕ ) + ε ε After solving for and dropping higher order terms of ε, we get: sin ϕ ) cos( ϕ ) + ε cos( ϕ ) + ε (4) + ε cos( ) ϕ Solve for ϕ ),sin( ) then substitute into E. () and E. (): cos( ϕ [ ϕ ) sin( ) ] n cos( ϕ (4) n ε ε cos( ) sin ( ϕ ) sin( ϕ) + cos( ϕ ) sin( ) ϕ (4) ϕ hus from E. (6), (8) and (9): n n I(, ϕ ) δ δ ( + ε cos( ϕ)) r (44) sin g [ + )cos( ϕ ) ( + )sin( ϕ ) ] p (45) ( sin g n f ( ( sin ( ϕ ) cos ( ϕ) + ε sin ( ϕ) cos ( )) ~ p ϕ (46) Substitute E. (), (5), (44), (46) into E. (), (), (5), we attain the contact area, applied load and cumulative frictional torue expressions: A cyl π δ π ℵ h exp ( δ ( π + (47)

18 4 ough Surface Contact F cyl x 4 * E o π ℵ π δ exp h ( δ ( + ε cos( ϕ ))) (cos( (48) cyl f 4 * µ E o π ℵ π δ exp h ( δ ( ( sin ( ϕ)cos ( ϕ) + ε ( sin ( ϕ)cos ( ϕ + ε cos( ϕ ))) ) s (49) 5. ANALYSIS: However, the eccentricity ( e ) is hard to measure without special instruments. hus it is more convenient to consider the frictional torue load relationship. By eliminating e using E. () to E. (5), we can find: τ F, µ,, G ) (5) f ( i where: F and µ are the applied load and the friction coefficient is MS roughness parameter G i s are the surfaces geometric parameters * f τ ( G ) µ F (5) i Which means the frictional torue is linearly dependent on the applied load value. It is expected that * f f and the two values become closer if more assumptions about the geometries are made. For example, in Section IV., if we assume ε ε <<, ), then from E.(4) and E. (5): (

19 Int. Jnl. of Multiphysics Volume Number 7 5 A cyl π δ π ℵ h exp ( δ ( π + (5) d sin( θ, d sin( θ ) After replacing ), E. (5) becomes: A cyl π δ π ℵ h exp ( δ ( π + (5) E. (5) is exactly the formula for the frictional torue applied load relationship if the contacting surfaces are modeled as smooth that is obtained in several studies such as the one from Grégory (4). Considering the example in Section 4. where no additional geometric assumption is made. First, the consistency between theories of contact mechanics should be taken into account. In the early days, Hertz set the foundation of contact mechanics by analytically predicting the compressive force reuired to indent a smooth sphere into a infinite smooth half-space, which was then broadened to account for other shapes. According to Sneddon 965, in the case of parallel-axis cylinders contact, the applied force as function of indentation depth ε + is: F smooth π (54) h( ε + ) / 4 hus E. (48) should be identical to E. (54) as since both describe contact in the smooth case. aking the limit of E. (48): ) lim F lim π 4 * E o ℵh cyl rough ( x F π δ exp ( δ ( + ε cos( ϕ))) ε (cos( ϕ) sin ( ϕ )) dδdϕ (55) Noting that the Gaussian distribution function becomes the Dirac delta function as, which has a special property: δ f ( x) f () α (55)

20 6 ough Surface Contact Using this property, E. (54) becomes: F 4 * ) E o ℵh rough ( Let π π ( ε cos( ϕ )) ε (cos( ϕ) sin ε Iϕ ( ε cos( ϕ)) (cos( ϕ) sin ( ϕ)) dδdϕ ( ϕ )) dδdϕ he analytical solution for this definite integral could be found using Wolfram Alpha: I ϕ a + b 5b ( 6a c a b + 58ab c 6b ) ( a b)( 6a c ab + b c) b b E π E a + b a + b b b F π F a + b a + b (56) where: a, b ε, c ε F ( ) and E ( ) are the incomplete elliptic integrals of first and second kind Since it is reuired that F ( ) F, we attain: rough smooth * πh( ε + E o ℵh 4I 4 ) (57) Next, using E. (), () we could analytically evaluate the first integrations in E. (47), (48), (49). Based on these expressions, a MALAB script is written to numerically evaluate E. (47), (48) and (49) at increments of ε and. he integration command integral uses global adaptive uadrature method integral with absolute error tolerance of e- (Shampine, 8). Graphs to Graph 6 show the relationships between contact area, applied force, frictional torue with eccentricity and roughness. Graph 7 indicates that even though the contact area applied load relationship might be linear at a particular roughness, the contact area increases faster with applied load as roughness goes up. We even observe this behavior more clearly in the case of frictional torue applied load relationship.

21 Int. Jnl. of Multiphysics Volume Number 7 7 Graph. Contact area as function of eccentricity and roughness Graph. Contact area as function of eccentricity at various roughness

22 8 ough Surface Contact Graph. Applied force as function of eccentricity and roughness Graph 4. Applied force as function of eccentricity at various roughness

23 Int. Jnl. of Multiphysics Volume Number 7 9 Graph 5. Frictional torue as function of eccentricity and roughness Graph 6. Frictional torue as function of eccentricity at various roughness

24 ough Surface Contact Graph 7. Contact area as function of applied force at various roughness Graph 8.Frictional torue as function of applied force at various roughness

25 Int. Jnl. of Multiphysics Volume Number 7 6. CONCLUSION he paper proposes a method to account for the contact of rough curved surface having nearly identical geometries. General formulas for the true contact area as well as the applied force have been deduced in terms of definite integrals. hese integrals could be simplified analytically. Numerical techniue and programming are then implemented in order to perform analysis of a special case: two cylindrical rough surfaces in contact. EFEENCES [] Anton, H., 999, Calculus: A New Horizon, sixth ed., Wiley, New York. [] Bhushan, B.,, Surface oughness Analysis and Measurement echniues, in: Bhushan, B. (Ed.), Modern ribology Handbook, CC Press, Boca aton. [] Blau, P.J.,, he significance and use of friction coefficient, ribology International, 4(9), [4] Gray, A., 997, Surfaces of evolution, Modern Differential Geometry of Curves and Surfaces with Mathematica, second ed., Boca aton, CC Press, Florida, [5] Greenwood, J.A., Williamson, J.B.P. (966) Contact of nominally flat surfaces, Proceedings of the oyal Society A, 95(44), 9. [6] Grégory, A., 4, On the Mechanical Friction Losses Occurring In Automotive Differential Gearboxes, he Scientific World Journal, 4. [7] Jackson,.L., et al.,, Contact mechanics, in: Menezes, P.L., Nosonovsky, S., Lovell, S., (Eds.), ribology for Scientists and Engineers, Springer, New York. [8] Jackson,. L., Green, I.,, On the Modeling of Elastic Contact between ough Surface, ribology ransactions, 54, -4. [9] Liu, S.B., 5, An extension of the Hertz theory for three-dimensional coated bodies, ribology Letters, 8(), 4. [] Lu, W., 4, Prediction of Surface opography at the End of Sliding unning-in Wear Based on Areal Surface Parameters, ribology ransactions, 57(), [] Menezes, P.L., et al.,, Fundamentals of Engineering Surfaces, in: Menezes, P.L., Nosonovsky, S., Lovell, S., (Eds.), ribology for Scienties and Engineer, Springer, New York. [] MuPAD, Symbolic Math in MALAB, [] Shampine, L.F., 8, MALAB Program for Quadrature in D, Applied Mathematics and Computation, (), [4] Sneddon, I. N., 965, he elation between Load and Penetration in the Axisymmetric Boussines Problem for a Punch of Arbitrary Profile, International Journal of Engineering Science, (), [5] Yastrebov, V.A., Anciaux, G., Molinari, J.F., From Infinitesimal to Full Contact between ough Surfaces: Evolution of the Contact Area, International Journal of Solids and Structures, 5, 8-.

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