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Surface normal deformation in elastic quarter-space
Tribology International 114 (2017) 358–364
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Tribology International journal homepage: www.elsevier.com/locate/triboint
Surface normal deformation in elastic quarter-space a
b
b,⁎
MARK
a
W. Wang , L. Guo , P.L. Wong , Z.M. Zhang a b
School of Mechatronics Engineering and Automation, Shanghai University, Shanghai, China Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Elastic quarter space Surface deformation Flexibility matrix
An efficient and explicit solution for the surface deformation of quarter-space under normal load is developed using the concept of flexibility matrix, which serves like springs in response to loads. Quarter-space is characterized by the unbounded side surface, such as in roller bearings and gears. The solution method is verified using a typical case. The edge effect on surface deformation under three load types namely, Hertzian point, flat cylindrical punch and Hertzian line, are evaluated. The effect can be considerable if the applied load is close to edge. The flexibility matrix is constant for a given case. Hence, the solution method is highly efficient, and particularly suitable for quarter-space problems which require iterative calculations, such as elastohydrodynamic lubrication analyses.
1. Introduction Acquiring the elastic deformation of contact surfaces is important in engineering. The solution process also needs to be fast and efficient for certain applications that require iterative calculations for the surface deformation, such as the analysis of tribo-pairs operating under the elasto-hydrodynamic lubrication (EHL) regime. Some common engineering components, such as roller bearings, gears and camfollowers, are characterized by the existence of free edge surfaces. Contact problems of these components are, in fact, more accurately modeled by elastic quarter-space (Fig. 1(a)). Nevertheless, the available solutions of elastic quarter-space are very complex, such that the elastic half-space model (semi-infinite body) is widely adopted for calculating contact stress and deformation in practical mechanical systems, such as those aforementioned applications, for their contact solutions are readily obtained with Bussinessq or Love formulae [1,2]. The assumption of semi-infinite body model is obviously not satisfied in these practical cases. For example, in the contact of gears and roller bearings, the length of the gear tooth and bearing roller are finite. Thus, the effect of free edge surfaces cannot be ignored and these components cannot be taken as semi-infinite bodies. The elastic quarter-space model is, indeed, more appropriate. Hetenyi [3] tackled the quarter-space problem with the concept of iteratively overlapping mutually orthogonal half-spaces with mirrored load pairs till fulfilling the boundary conditions of the quarter-space. Keer et al. [4] utilized Hetenyi's overlapping half-space idea and derived two integral equations to describe the quarter-space problem. They solved the equations with Fourier transform. Nevertheless, their ⁎
Corresponding author. E-mail address: [email protected] (P.L. Wong).
http://dx.doi.org/10.1016/j.triboint.2017.04.044 Received 23 February 2017; Received in revised form 20 April 2017; Accepted 25 April 2017 Available online 29 April 2017 0301-679X/ © 2017 Elsevier Ltd. All rights reserved.
method can only be applied to cases where the load can be Fouriertransformed. Later on, Hanson and Keer [5] overcame this limitation with a direct numerical solution of the quarter-space by solving two dimensional integral equations. Thus, any load type can be considered. The difference in the stress obtained with quarter-space and half-space models was studied. As pointed out in [5], the magnitude and position of the maximum stress calculated with quarter-space and half-space vary, especially when the load is located in the immediate neighborhood of the free end. Guilbault [6] made use of a correction factor which multiplies the Hetenyi's mirrored loads to simultaneously correct the influence of stresses on elastic deformation of a quarterspace, which provides a much faster solution than a complete Hetenyi process. This correction factor method was applied by Najjari and Guilbault [7] to investigate the edge effect in EHL analysis of roller bearings. Nevertheless, this method gives only approximate solutions. The present authors [8] have recently obtained the limit of Hetenyi's iteration with a matrix method and developed an explicit solution to the stress field of quarter-space problems. The aforementioned methods are all based on the overlapping half-space concept of Hetenyi. Apart from these, Bower et al. [9] adopted finite element method (FEM) to analyze the ratcheting limit of rail's plastic deformation. Hecker and Romanov [10] applied Mellin transform to solve the stress distribution of quarter-space. Ritz's method was also applied by Guenfoud et al. [11] to obtain the displacement solution of quarterspace. There is another perspective of the contact problem of quarterspace by considering its contact with a rigid body. Gerber [12] was the first to study a contact between a rigid body and an elastic quarter-
Tribology International 114 (2017) 358–364
W. Wang et al.
Fig. 1. Quarter-space solutions equivalent to overlapping half-spaces.
current load distribution. The theoretical derivation of the flexibility matrix for the elastic surface deformation with the quarter-space model is presented. The solution method is also validated through a special case study. The difference in the surface normal deformation calculated with quarter-space and half-space models are not yet investigated comprehensively. Therefore, the results of the surface normal deformation of quarter- and half-space models under different typical loads are also presented and discussed.
space. He obtained the stress distribution in a quarter-space which is pressed by a rectangular punch. Keer et al. [13] studied a quarter-space loaded with a rigid cylindrical indenter by integral transform techniques. Hanson and Keer [14] solved the contact stresses between a spherical indenter and a quarter-space. Wang et al. [15] studied the problem of a quarter-space in contact with a rigid sphere using equivalent inclusion method. Zhang et al. [16,17] analyzed the contact of a rigid roller and a finite-length elastic body. To include the effect of the free edge surfaces of the elastic body, they applied the method of Zhang et al. [8] in the study. However, the shear stresses on a free end surface as induced by the mirrored loads on the plane of the other end surface cannot be eliminated, i.e. it does not fulfill the zero stress boundary condition of the free surface. The elastic deformation of quarter-spaces is needed in the solution of many engineering problems, such as rail/wheel contacts [14] and EHL analyses of roller contacts [18–20]. The analyses of these application examples require iterative calculations. Thus, it requires not only accurate but also efficient solution for surface deformation of elastic quarter-space. In this paper, the solution of surface deformation of a quarter-space is developed resembling a matrix of springs. The deformation of a spring is obtained by simply dividing the load over its stiffness, or multiplying the load with its flexibility. If such a simple process can be implemented into the calculation loops of the above examples, the solution process would be significantly simplified and shortened. The complete calculation times can thus be much lower, especially if a great many times of iteration is needed. In order to realize this, a characteristic property of elastic quarter-space concerning its stiffness or flexibility must be known prior to entering the loop of calculation. This characteristic property must be independent of the load, and derived without the knowledge of the current load. The present paper achieves this aim by extending our recently proposed technique [8] for quarter-space solutions, such that an explicit form of the flexibility matrix of the elastic quarter-space is derived. This flexibility matrix is independent of the load, so that it can be used in every loop of the iteration process. The elastic surface deformation can be immediately obtained by simply multiplying this matrix with the
2. Solution of surface normal deformation with quarterspace model 2.1. Derivation of flexibility matrix A quarter-space problem with a distribution load P on the top surface as shown in Fig. 1(a) can be solved by making use of the solutions of two mutually orthogonal half-spaces as shown in Fig. 1(b), which is based on the overlapping half-space idea of Hetenyi [3]. To solve a quarter-space problem with matrix formulation [8], the horizontal and vertical surfaces of the quarter-space are discretized with rectangular meshes of different sizes. Fig. 2(a) shows schematically the mesh pattern on the top surface. The region near the free edge surface is discretized with finer meshes in order to enhance the accuracy of the deformation results close to the free edge. The solution of Fig. 1(a) is obtained by superimposing the half-space solutions of load-pairs: Ph , Ph and Pv , Pv ( Ph and Pv are mirror loads of Ph and Pv , respectively). Making use the stress boundary conditions of the original quarter-space (Fig. 1(a)): P on the top surface and zero stress on the vertical side surface, explicit solutions of the equivalent load Ph and Pv of the half-spaces (Fig. 1(b)) can be readily obtained by [8],
Ph = A ∙ P
(1)
Pv = B ∙ P
(2)
The Appendix shows how the two coefficient matrices A and B can be calculated. All pressure distributions are in matrix format as,
Fig. 2. (a) Mesh pattern and (b) the ith patch on top surface.
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Fig. 3. Example of an elastic block under Hertzian point load.
⎡ (p )1 ⎤ ⎡ (p )1 ⎤ ⎡ p1 ⎤ ⎢ v ⎥ ⎢ h ⎥ ⎢p ⎥ ( p ) 2 ⎥ ⎢ (pv )2 ⎥ ⎢ h 2⎥ ⎢ P= ; P = ; P = ⎢ ⋮ ⎥ h ⎢ ⋮ ⎥ v ⎢ ⋮ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢⎣ pkh ⎥⎦ ⎢⎣ (ph )kh ⎥⎦ ⎢⎣ (pv )k v ⎥⎦
D = C ∙ Ph Combining with Eq. (1) yields,
where F = C∙A . Matrix F links the originally applied load P to the normal deformation vector D. The normal deformation of the top surface of the elastic quarter-space can be readily obtained by multiplying the pressure P with F. Matrix F, which is related only to the contact material and surface meshing but remains constant for different loads, can be calculated using Eq. (3) with an unit pressure distribution for a given quarter-space problem. F is, thus, the flexibility matrix of the elastic quarter-space. This approach is very convenient for engineering calculation. 2.2. Verification of calculation for elastic deformation A Fortran program that calculates the elastic surface deformation of a quarter-space and a half-space was written. The solution of half-space is directly obtained with the applied load using the classical theory of Love [2]. The program was validated with an example of a Hertzian contact, as illustrated in Fig. 3. The radius of the Hertzian contact is a. The dimensions of the block are Ax =20a, Ay =40a and Az =20a. The vertical y-z plane is taken to be the targeted free side surface and its effect is studied. Ay is double in length of Ax. The number of mesh on the top surface (x-y plane) is 25,070 (230×109). The vertical side surface (y-z plane) has the same number of mesh (109×230). As illustrated in Fig. 2(a), finer mesh resolution is chosen in the regions close to the free side surface and along the x-axis where the applied load is located. A semi-elliptical pressure distribution according to the Hertzian point contact theory is assumed and applied at a distance of d from the free edge along the x-axis on the top surface, as shown in Fig. 3. The Hertz equation, which provides analytical solutions for a point contact on an elastic half-space, has sufficient precision to give deformation and stress if a is much smaller than the surface size and the load center is far from the free side surface. For this example, a is taken as 0.5 mm. The Hertzian pressure distribution for a point contact is expressed as,
⎞ ⎞ ⎛ Ph(xi , yi ) ⎧⎡ ⎛ ⎨⎢Bz⎜xj − xi + αi, yj − yi + βi⎟ δji⎜xj , yj ⎟ = ⎠ ⎠ ⎝ 4πG ⎩⎣ ⎝ ⎪ ⎪
⎞ ⎛ ⎞ ⎛ + Bz⎜xj − xi − αi, yj − yi − βi⎟ − Bz⎜xj − xi + αi, yj − yi − βi⎟ ⎠ ⎝ ⎠ ⎝ ⎞⎤ ⎡ ⎛ ⎛ ⎞ − Bz⎜xj − xi − αi, yj − yi + βi⎟⎥ + ⎢Bz⎜xj + xi + αi, yj − yi + βi⎟ ⎠⎦ ⎣ ⎝ ⎝ ⎠ ⎞ ⎛ ⎞ ⎛ + Bz⎜xj + xi − αi, yj − yi − βi⎟ − Bz⎜xj + xi + αi, yj − yi − βi⎟ ⎠ ⎝ ⎠ ⎝ ⎞⎤⎫ ⎛ − Bz⎜xj + xi − αi, yj − yi + βi⎟⎥⎬ = (cji + cji )Ph(xi , yi ) ⎠⎦⎭ ⎝ (3) ⎪ ⎪
and
ln(R + x ) + x
ln (R + y )]
(4)
where G is the shear modulus and ν is the Poisson's ratio; xi and yi are the coordinates of the center of ith pressure patch; xj and yj are the coordinates of deformation point; α and β are the half-length and half1/2 width of the pressure patch (Fig. 2(b)) and R = (x 2 + y 2 ) . The total deformation at point j is thus given as,
⎞ ⎛ δj ⎜xj , yj ⎟ = ⎠ ⎝
∑ (cji +
(7)
D=F ∙ P
where kh and k v are, respectively, the number of mesh on the top and vertical side surfaces. For the calculation of the surface normal (vertically downward) deformation of the quarter-space induced by the distributed load P as shown in Fig. 1(a), there is no effect generated by the load-pair, Pv and its mirror image Pv due to the symmetry of Pv and Pv with respect to the x-axis (the originally loaded surface of the quarter-space) in Fig. 1(b), Therefore, only the surface normal deformation caused by Ph and Ph is considered. The elastic deformation of a point on a plane due to a uniformly load patch on the same surface is given by the classical Love solution [2]. The normal deformation at point j on the top contact surface, which is caused by a uniform pressure, Ph(xi , yi ), on a patch i and its mirror image, Ph(−xi , yi ), on the top surface, can be expressed as [5],
Bz(x, y ) = 2(1 − ν)[y
(6)
P(r ) = Pmax (1−(r / a )2 ),
r≤a
(8)
The normal surface deformation is given as [1],
1−ν 2 π ∙Pmax (2a 2 − r 2 ), r ≤ a E 4a 1−ν 2 Pmax uz = {(2a 2 − r 2 )sin−1(a / r ) + ra 1−(a / r )2 }, E 2a uz =
cji )Ph(xi , yi )
(5)
The total normal deformation caused by the entire load vectors Ph and Ph can be expressed with a vector D. All coefficients cji + cji are also expressed in the form of a matrix C. Thus,
r>a
(9)
Table 1 lists the parameters of the Hertz contact. This example has the center of the Hertzian pressure distribution at the mid-span of the 360
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To evaluate the effect of a vertical free surface on the surface normal deformation, pressure distributions of three typical contacts, namely Hertzian point, flat cylindrical punch and Hertzian line, are chosen for case studies.
The quarter-space results coincide well with the ANSYS results, which substantiates the correctness and accurateness of the developed Fortran program. The minor difference between the two is due to the mesh difference. The meshing in ANSYS is 3-dimensional and it is difficult to divide the contact body into too many grids. The presented quarter-space treatment is semi-analytical and its results would converge to the exact solution if the resolution of the mesh is increased. The differences between the results of quarter-space (solid curve) and half-space (large dotted curve) are significant. The actual quarterspace deformation is bigger than that of half-space model due to the effect of the free side surface. One can consider that there exists no free surface in the half-space model and its structure is thus stronger than a quarter-space. Such a difference becomes increasingly obvious as the load center approaches the side edge. Furthermore, when the pressure center is shifted towards the free surface, as shown in Fig. 5(a), the maximum deformation point moves away from the pressure center towards the free end. This result contradicts the prediction of the Hertz theory. Fig. 6 shows the differences between the magnitude and the position of the maximum deformation with different locations of pressure center in the x-axis. The differences rise sharply when the pressure center is located close to the free surface. When the center of pressure is located at d = a from the edge, the error of the half-space deformation can be as large as 30% (Fig. 6(a)) and the position shift of the maximum deformation (Fig. 6(b)) reaches 28.4% of the load radius a. The effect of free edge surface fades gradually when the center of pressure shifts away from the free end. The location of maximum deformation almost coincides with the pressure center when d reaches 2a. The effect of free edge surface on the magnitude of deformation remains significant even when the center of pressure is relatively far (d =5a) from the free end, as shown in Fig. 6(a).
3.1. Hertzian point contact
3.2. Flat cylindrical punch contact
The surface normal deformations under a Hertzian point contact with the same parameters listed in Table 1 but different d values were calculated. The load distribution is expressed with Eq. (8). The load centers were, respectively, located at d = a, 2a, 3a, 4a and 5a (compared to the full length of the block, 20a). The effect of the free side surface at x=0 was studied. It was assumed that the free surface on the other side at x=20a produces no significant effect due to it being far from the load center. Hence, finer mesh resolution was adopted in the region close to the free edge at x=0 as depicted in Fig. 2(a). Fig. 5 shows deformations along the x-axis on the top surface (z=0), which were directly calculated using the derived flexibility matrix. To highlight the effect of free edge surface, the same set of results calculated using the half-space model is also illustrated. Finite element modeling (using ANSYS) was also complied and results are presented in Fig. 5 to demonstrate the accuracy of the matrix solution of quarter-space model. The insets are blow-ups of the load center region where the deformation curves acquire the maximum values.
The quarter-space loaded with a uniform pressure distribution which simulates the contact of a flat cylindrical punch is also investigated. The pressure is given as,
Table 1 Parameters of the Hertzian point contact. Max. Hertzian pressure: Pmax = 0.5 GPa Elastic modulus: E = 201 GPa, Poisson's ratio: ν = 0.3 Contact radius: a = 0.5 mm
block, i.e. d =10a, for the validation of the computer program. Since the load is away from the two free edges and the contact area of radius a is far less than the top surface area of the block, the half-space criterion for the use of the Hertz theory is valid. Therefore, the surface normal deformation (as the output of the developed computer program) calculated with the quarter-space and the half-space models should be the same or very much similar to the analytical solution of Hertz as expressed in Eqn. (9). The normal deformation of the top surface along the x-axis is depicted in Fig. 4. The good correlation of the three curves as shown in Fig. 4(b) proved the assumption and the developed program. Fig. 4(b) is an enlarged plot of the maximum deformation in the Hertzian contact region (from 9a to 11a), which also shows that the deformation calculated using the quarter-space model is slightly larger than that of the half-space model or Hertz theory. The differences are attributed to the effect of the two free end surfaces. 3. Effect of free side surface on surface normal deformation
P(r ) = 0. 5GPa, P(r ) = 0,r > a
r≤a (10)
Results were obtained with the pressure center located at d = a, 3a and 5a along the x-axis on the top surface (full length of the block: 20a). Fig. 7 illustrates the deformed top surfaces obtained with the quarter-space model as well as the half-space model. The results show that if the pressure center is located close to the free edge, the deformed profile deviates significantly from that predicted by the half-space model. The effect of free edge side can be considerable. For example, the maximum deformation that reaches to 0.00757a occurs at the free surface when cylindrical pressure is applied at d = a, which is much larger than the corresponding 0.00454a of half-space model. Fig. 7 depicts that the half-space model gives smaller deformation when
Fig. 4. Surface normal deformation: comparison of half-space model, quarter-space model and Hertz theoretical solution.
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Fig. 5. Results of deformation along x-axis on the top surface for different d values.
deformation for various distances between the center of pressure and free surface. More than 40% error in maximum deformation is found in the half-space model when the distance between the load center and the free surface is equal to the radius of the cylindrical load distribu-
compared with the quarter-space, and the maximum deformation locates right at the pressure center. These results indicate failure of the half-space model to provide accurate results when the load is applied near the free surface. Fig. 8 shows the deviations in maximum
Fig. 6. Difference in the values of maximum deformation and the positions of quarter- and half-space models. (a) Difference in magnitude. (b) Position shift.
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Fig. 9. Deformation of a block of width 20a under Hertz line contact in quarter- and half-space models (dashed line: half-space; solid line: quarter-space).
Fig. 7. Deformation results based on quarter- and half-space models under cylindrical load distribution (dashed line: half-space; solid line: quarter-space; full length: 20a).
quarter-space deformed profile is small. With increasing the loading length to 8a, i.e. the free side surfaces are closer to the load, the difference is increased. When the length of the line load is extended to 20a, as same as the length of the elastic block, the quarter-space deformation profile depicts the maximum deformation at the two ends whereas the half-space attains the maximum deformation at the midlength (the block center as well as the load center). Results show that when the load is close to the free side surface, the influence of the side surface is considerable and the use of half-space model may lead to significant errors.
4. Conclusions Fig. 8. Maximum deformation deviations for various locations of applied load.
An accurate and fast method for the solution of surface normal deformation in quarter-space is introduced in this paper. The surface normal deformation can be obtained directly by multiplying the original normal load on the top surface of the quarter-space with a flexibility matrix. The flexibility matrix, which is constant for a given discretized quarter-space, can be retrieved and used for different loads. This facilitates fast and accurate calculation for the quarter-space normal surface deformation. This paper has evaluated three different load types on a quarterspace, namely Hertzian point, flat cylindrical punch and Hertzian line. The maximum load is up to 0.5 GPa and the material of the quarterspace under consideration is taken as steel. It is found that if the load is close to a free edge surface, the effect of the free surface can be considerable. For the specified cases, the error in the maximum deformation can be up to 40% (for a load located very close to the edge) if the inappropriate half-space model is adopted. Furthermore, the effect of the free edge surface leads to the position shift of the maximum deformation point and asymmetric deformation profile.
tion. 3.3. 3.3 Hertzian line contact The third case is a Hertzian line contact, which simulates an elastic roller contact. The Hertz contact radius a is 0.5 mm in the y-direction. The pressure distribution is given as,
⎛ y ⎞2 P(y ) = 0. 5 1−⎜ ⎟ GPa, ⎝a⎠ P(y ) = 0,
y >a
y ≤a (11)
Deformation results were calculated with three different loading lengths of 4a, 8a and 20a. The center of these loads is fixed at the center of the block (mid-span, d =10a). These loading cases are axisymmetric about the mid-section at x=10a. Only the half-length of the distributed load was adopted in the calculation of the surface deformation of the block. The mirror image of that calculated surface deformation gives the deformation due to the other half of the load. Superimposing the two deformation results gives the surface deformation of the applied load. Fig. 9 shows the calculated deformation with the quarter-space and half-space models. When the line Hertzian load of 4a in length is applied at the center and its edges are far from the two free side surfaces at x=0 and 20a, the difference in the half-space and
Acknowledgement The study described in this paper was fully supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region, China (Project No. CityU11213914).
Appendix The stresses acting on the vertical (y-z plane) and the top planes (x-y plane) of the quarter-space as the results of the load-pairs: Ph , Ph and Pv , Pv , respectively, as shown in Fig. 1 can be expressed in vector matrix format as,
σyz = M ∙ Ph
(A1)
σxy = N ∙ Pv
(A2)
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where M and N are two-dimensional coefficient matrices, and they are directly determined by the Love theory [2]. Overlapping the solutions of the two half-spaces in Fig. 1(b) and considering the boundary conditions yields,
−Ph + σxy=−Ph + N ∙ Pv = −P
(A3)
−Pv + σyz=−Pv + M ∙ Ph = 0
(A4)
Combining (A3) and (A4) gives,
Ph = (1 − N ∙ M)−1 ∙ P
(A5)
Comparing with Eq. (1),
A = (1 − N ∙ M)−1 Similarly, from Eq. (2),
B = M ∙ (1 − N ∙ M)−1
boundaries and bonded wedges. Mater Sci Eng: A 1993;164:411–4. [11] Guenfoud S, Bosakov S, Laefer DF. A Ritz's method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space. Int J Solids Struct 2010;47:1822–9. [12] Gerber CE. Contact problems for the elastic quarter-plane and for the quarterspace. UMI Diss. Services; 1968. [13] Keer LM, Lee JC, Mura T. A contact problem for the elastic quarter space. Int J Solids Struct 1984;20:513–24. [14] Hanson M, Keer L. Analysis of edge effects on rail-wheel contact. Wear 1991;144:39–55. [15] Wang Z, Jin X, Keer LM, Wang Q. Numerical methods for contact between two joined quarter spaces and a rigid sphere. Int J Solids Struct 2012;49:2515–27. [16] Zhang H, Wang W, Zhang S, Zhao Z. Modeling of elastic finite-length space rollingsliding contact problem. Tribol Int 2016. [17] Zhang H, Wang W, Zhang S, Zhao Z. Modeling of finite-length line contact problem with consideration of two free-end surfaces. J Tribol 2016;138:021402. [18] Liu X, Yang P. Analysis of the thermal elastohydrodynamic lubrication of a finite line contact. Tribol Int 2002;35:137–44. [19] Liu XL, Yang PR. Numerical analysis of the oil-supply condition in isothermal elastohydrodynamic lubrication of finite line contacts. Tribol Lett 2010;38:115–24. [20] Sun H, Chen X . Thermal EHL analysis of cylindrical roller under heavy load. In: Proceedings of IUTAM symposium on elastohydrodynamics and micro-elastohydrodynamics. Springer; 2006. p. 107–120.
References [1] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985. [2] Love AEH. The stress produced in a semi-infinite solid by pressure on part of the boundary. Philos Trans R Soc Lond Ser A, Contain Pap A Math Or Phys Character 1929;228:377–420. [3] Hetenyi M. A general solution for the elastic quarter space. J Appl Mech 1970;37:70–6. [4] Keer L, Lee J, Mura T. Hetenyi's elastic quarter space problem revisited. Int J Solids Struct 1983;19:497–508. [5] Hanson M, Keer L. A simplified analysis for an elastic quarter-space. Q J Mech Appl Math 1990;43:561–87. [6] Guilbault R. A fast correction for elastic quarter-space applied to 3D modeling of edge contact problems. J Tribol 2011;133:031402. [7] Najjari M, Guilbault R. Edge contact effect on thermal elastohydrodynamic lubrication of finite contact lines. Tribol Int 2014;71:50–61. [8] Zhang Z, Wang W, Wong P. An explicit solution for the elastic quarter-space problem in matrix formulation. Int J Solids Struct 2013;50:976–80. [9] Bower AF, Johnson KL, Kalousek J. A ratchetting limit for plastic deformation of a quarter-space under rolling contact loads. In: Proceedings international symp on contact mechanics and wear of rail/wheel systems II. The Netherlands: University of Waterloo Press; 1987. p. 117–32. [10] Hecker M, Romanov A. The stress fields of edge dislocations near wedge-shaped
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Tribology International journal homepage: www.elsevier.com/locate/triboint
Surface normal deformation in elastic quarter-space a
b
b,⁎
MARK
a
W. Wang , L. Guo , P.L. Wong , Z.M. Zhang a b
School of Mechatronics Engineering and Automation, Shanghai University, Shanghai, China Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Elastic quarter space Surface deformation Flexibility matrix
An efficient and explicit solution for the surface deformation of quarter-space under normal load is developed using the concept of flexibility matrix, which serves like springs in response to loads. Quarter-space is characterized by the unbounded side surface, such as in roller bearings and gears. The solution method is verified using a typical case. The edge effect on surface deformation under three load types namely, Hertzian point, flat cylindrical punch and Hertzian line, are evaluated. The effect can be considerable if the applied load is close to edge. The flexibility matrix is constant for a given case. Hence, the solution method is highly efficient, and particularly suitable for quarter-space problems which require iterative calculations, such as elastohydrodynamic lubrication analyses.
1. Introduction Acquiring the elastic deformation of contact surfaces is important in engineering. The solution process also needs to be fast and efficient for certain applications that require iterative calculations for the surface deformation, such as the analysis of tribo-pairs operating under the elasto-hydrodynamic lubrication (EHL) regime. Some common engineering components, such as roller bearings, gears and camfollowers, are characterized by the existence of free edge surfaces. Contact problems of these components are, in fact, more accurately modeled by elastic quarter-space (Fig. 1(a)). Nevertheless, the available solutions of elastic quarter-space are very complex, such that the elastic half-space model (semi-infinite body) is widely adopted for calculating contact stress and deformation in practical mechanical systems, such as those aforementioned applications, for their contact solutions are readily obtained with Bussinessq or Love formulae [1,2]. The assumption of semi-infinite body model is obviously not satisfied in these practical cases. For example, in the contact of gears and roller bearings, the length of the gear tooth and bearing roller are finite. Thus, the effect of free edge surfaces cannot be ignored and these components cannot be taken as semi-infinite bodies. The elastic quarter-space model is, indeed, more appropriate. Hetenyi [3] tackled the quarter-space problem with the concept of iteratively overlapping mutually orthogonal half-spaces with mirrored load pairs till fulfilling the boundary conditions of the quarter-space. Keer et al. [4] utilized Hetenyi's overlapping half-space idea and derived two integral equations to describe the quarter-space problem. They solved the equations with Fourier transform. Nevertheless, their ⁎
Corresponding author. E-mail address: [email protected] (P.L. Wong).
http://dx.doi.org/10.1016/j.triboint.2017.04.044 Received 23 February 2017; Received in revised form 20 April 2017; Accepted 25 April 2017 Available online 29 April 2017 0301-679X/ © 2017 Elsevier Ltd. All rights reserved.
method can only be applied to cases where the load can be Fouriertransformed. Later on, Hanson and Keer [5] overcame this limitation with a direct numerical solution of the quarter-space by solving two dimensional integral equations. Thus, any load type can be considered. The difference in the stress obtained with quarter-space and half-space models was studied. As pointed out in [5], the magnitude and position of the maximum stress calculated with quarter-space and half-space vary, especially when the load is located in the immediate neighborhood of the free end. Guilbault [6] made use of a correction factor which multiplies the Hetenyi's mirrored loads to simultaneously correct the influence of stresses on elastic deformation of a quarterspace, which provides a much faster solution than a complete Hetenyi process. This correction factor method was applied by Najjari and Guilbault [7] to investigate the edge effect in EHL analysis of roller bearings. Nevertheless, this method gives only approximate solutions. The present authors [8] have recently obtained the limit of Hetenyi's iteration with a matrix method and developed an explicit solution to the stress field of quarter-space problems. The aforementioned methods are all based on the overlapping half-space concept of Hetenyi. Apart from these, Bower et al. [9] adopted finite element method (FEM) to analyze the ratcheting limit of rail's plastic deformation. Hecker and Romanov [10] applied Mellin transform to solve the stress distribution of quarter-space. Ritz's method was also applied by Guenfoud et al. [11] to obtain the displacement solution of quarterspace. There is another perspective of the contact problem of quarterspace by considering its contact with a rigid body. Gerber [12] was the first to study a contact between a rigid body and an elastic quarter-
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Fig. 1. Quarter-space solutions equivalent to overlapping half-spaces.
current load distribution. The theoretical derivation of the flexibility matrix for the elastic surface deformation with the quarter-space model is presented. The solution method is also validated through a special case study. The difference in the surface normal deformation calculated with quarter-space and half-space models are not yet investigated comprehensively. Therefore, the results of the surface normal deformation of quarter- and half-space models under different typical loads are also presented and discussed.
space. He obtained the stress distribution in a quarter-space which is pressed by a rectangular punch. Keer et al. [13] studied a quarter-space loaded with a rigid cylindrical indenter by integral transform techniques. Hanson and Keer [14] solved the contact stresses between a spherical indenter and a quarter-space. Wang et al. [15] studied the problem of a quarter-space in contact with a rigid sphere using equivalent inclusion method. Zhang et al. [16,17] analyzed the contact of a rigid roller and a finite-length elastic body. To include the effect of the free edge surfaces of the elastic body, they applied the method of Zhang et al. [8] in the study. However, the shear stresses on a free end surface as induced by the mirrored loads on the plane of the other end surface cannot be eliminated, i.e. it does not fulfill the zero stress boundary condition of the free surface. The elastic deformation of quarter-spaces is needed in the solution of many engineering problems, such as rail/wheel contacts [14] and EHL analyses of roller contacts [18–20]. The analyses of these application examples require iterative calculations. Thus, it requires not only accurate but also efficient solution for surface deformation of elastic quarter-space. In this paper, the solution of surface deformation of a quarter-space is developed resembling a matrix of springs. The deformation of a spring is obtained by simply dividing the load over its stiffness, or multiplying the load with its flexibility. If such a simple process can be implemented into the calculation loops of the above examples, the solution process would be significantly simplified and shortened. The complete calculation times can thus be much lower, especially if a great many times of iteration is needed. In order to realize this, a characteristic property of elastic quarter-space concerning its stiffness or flexibility must be known prior to entering the loop of calculation. This characteristic property must be independent of the load, and derived without the knowledge of the current load. The present paper achieves this aim by extending our recently proposed technique [8] for quarter-space solutions, such that an explicit form of the flexibility matrix of the elastic quarter-space is derived. This flexibility matrix is independent of the load, so that it can be used in every loop of the iteration process. The elastic surface deformation can be immediately obtained by simply multiplying this matrix with the
2. Solution of surface normal deformation with quarterspace model 2.1. Derivation of flexibility matrix A quarter-space problem with a distribution load P on the top surface as shown in Fig. 1(a) can be solved by making use of the solutions of two mutually orthogonal half-spaces as shown in Fig. 1(b), which is based on the overlapping half-space idea of Hetenyi [3]. To solve a quarter-space problem with matrix formulation [8], the horizontal and vertical surfaces of the quarter-space are discretized with rectangular meshes of different sizes. Fig. 2(a) shows schematically the mesh pattern on the top surface. The region near the free edge surface is discretized with finer meshes in order to enhance the accuracy of the deformation results close to the free edge. The solution of Fig. 1(a) is obtained by superimposing the half-space solutions of load-pairs: Ph , Ph and Pv , Pv ( Ph and Pv are mirror loads of Ph and Pv , respectively). Making use the stress boundary conditions of the original quarter-space (Fig. 1(a)): P on the top surface and zero stress on the vertical side surface, explicit solutions of the equivalent load Ph and Pv of the half-spaces (Fig. 1(b)) can be readily obtained by [8],
Ph = A ∙ P
(1)
Pv = B ∙ P
(2)
The Appendix shows how the two coefficient matrices A and B can be calculated. All pressure distributions are in matrix format as,
Fig. 2. (a) Mesh pattern and (b) the ith patch on top surface.
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Fig. 3. Example of an elastic block under Hertzian point load.
⎡ (p )1 ⎤ ⎡ (p )1 ⎤ ⎡ p1 ⎤ ⎢ v ⎥ ⎢ h ⎥ ⎢p ⎥ ( p ) 2 ⎥ ⎢ (pv )2 ⎥ ⎢ h 2⎥ ⎢ P= ; P = ; P = ⎢ ⋮ ⎥ h ⎢ ⋮ ⎥ v ⎢ ⋮ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢⎣ pkh ⎥⎦ ⎢⎣ (ph )kh ⎥⎦ ⎢⎣ (pv )k v ⎥⎦
D = C ∙ Ph Combining with Eq. (1) yields,
where F = C∙A . Matrix F links the originally applied load P to the normal deformation vector D. The normal deformation of the top surface of the elastic quarter-space can be readily obtained by multiplying the pressure P with F. Matrix F, which is related only to the contact material and surface meshing but remains constant for different loads, can be calculated using Eq. (3) with an unit pressure distribution for a given quarter-space problem. F is, thus, the flexibility matrix of the elastic quarter-space. This approach is very convenient for engineering calculation. 2.2. Verification of calculation for elastic deformation A Fortran program that calculates the elastic surface deformation of a quarter-space and a half-space was written. The solution of half-space is directly obtained with the applied load using the classical theory of Love [2]. The program was validated with an example of a Hertzian contact, as illustrated in Fig. 3. The radius of the Hertzian contact is a. The dimensions of the block are Ax =20a, Ay =40a and Az =20a. The vertical y-z plane is taken to be the targeted free side surface and its effect is studied. Ay is double in length of Ax. The number of mesh on the top surface (x-y plane) is 25,070 (230×109). The vertical side surface (y-z plane) has the same number of mesh (109×230). As illustrated in Fig. 2(a), finer mesh resolution is chosen in the regions close to the free side surface and along the x-axis where the applied load is located. A semi-elliptical pressure distribution according to the Hertzian point contact theory is assumed and applied at a distance of d from the free edge along the x-axis on the top surface, as shown in Fig. 3. The Hertz equation, which provides analytical solutions for a point contact on an elastic half-space, has sufficient precision to give deformation and stress if a is much smaller than the surface size and the load center is far from the free side surface. For this example, a is taken as 0.5 mm. The Hertzian pressure distribution for a point contact is expressed as,
⎞ ⎞ ⎛ Ph(xi , yi ) ⎧⎡ ⎛ ⎨⎢Bz⎜xj − xi + αi, yj − yi + βi⎟ δji⎜xj , yj ⎟ = ⎠ ⎠ ⎝ 4πG ⎩⎣ ⎝ ⎪ ⎪
⎞ ⎛ ⎞ ⎛ + Bz⎜xj − xi − αi, yj − yi − βi⎟ − Bz⎜xj − xi + αi, yj − yi − βi⎟ ⎠ ⎝ ⎠ ⎝ ⎞⎤ ⎡ ⎛ ⎛ ⎞ − Bz⎜xj − xi − αi, yj − yi + βi⎟⎥ + ⎢Bz⎜xj + xi + αi, yj − yi + βi⎟ ⎠⎦ ⎣ ⎝ ⎝ ⎠ ⎞ ⎛ ⎞ ⎛ + Bz⎜xj + xi − αi, yj − yi − βi⎟ − Bz⎜xj + xi + αi, yj − yi − βi⎟ ⎠ ⎝ ⎠ ⎝ ⎞⎤⎫ ⎛ − Bz⎜xj + xi − αi, yj − yi + βi⎟⎥⎬ = (cji + cji )Ph(xi , yi ) ⎠⎦⎭ ⎝ (3) ⎪ ⎪
and
ln(R + x ) + x
ln (R + y )]
(4)
where G is the shear modulus and ν is the Poisson's ratio; xi and yi are the coordinates of the center of ith pressure patch; xj and yj are the coordinates of deformation point; α and β are the half-length and half1/2 width of the pressure patch (Fig. 2(b)) and R = (x 2 + y 2 ) . The total deformation at point j is thus given as,
⎞ ⎛ δj ⎜xj , yj ⎟ = ⎠ ⎝
∑ (cji +
(7)
D=F ∙ P
where kh and k v are, respectively, the number of mesh on the top and vertical side surfaces. For the calculation of the surface normal (vertically downward) deformation of the quarter-space induced by the distributed load P as shown in Fig. 1(a), there is no effect generated by the load-pair, Pv and its mirror image Pv due to the symmetry of Pv and Pv with respect to the x-axis (the originally loaded surface of the quarter-space) in Fig. 1(b), Therefore, only the surface normal deformation caused by Ph and Ph is considered. The elastic deformation of a point on a plane due to a uniformly load patch on the same surface is given by the classical Love solution [2]. The normal deformation at point j on the top contact surface, which is caused by a uniform pressure, Ph(xi , yi ), on a patch i and its mirror image, Ph(−xi , yi ), on the top surface, can be expressed as [5],
Bz(x, y ) = 2(1 − ν)[y
(6)
P(r ) = Pmax (1−(r / a )2 ),
r≤a
(8)
The normal surface deformation is given as [1],
1−ν 2 π ∙Pmax (2a 2 − r 2 ), r ≤ a E 4a 1−ν 2 Pmax uz = {(2a 2 − r 2 )sin−1(a / r ) + ra 1−(a / r )2 }, E 2a uz =
cji )Ph(xi , yi )
(5)
The total normal deformation caused by the entire load vectors Ph and Ph can be expressed with a vector D. All coefficients cji + cji are also expressed in the form of a matrix C. Thus,
r>a
(9)
Table 1 lists the parameters of the Hertz contact. This example has the center of the Hertzian pressure distribution at the mid-span of the 360
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To evaluate the effect of a vertical free surface on the surface normal deformation, pressure distributions of three typical contacts, namely Hertzian point, flat cylindrical punch and Hertzian line, are chosen for case studies.
The quarter-space results coincide well with the ANSYS results, which substantiates the correctness and accurateness of the developed Fortran program. The minor difference between the two is due to the mesh difference. The meshing in ANSYS is 3-dimensional and it is difficult to divide the contact body into too many grids. The presented quarter-space treatment is semi-analytical and its results would converge to the exact solution if the resolution of the mesh is increased. The differences between the results of quarter-space (solid curve) and half-space (large dotted curve) are significant. The actual quarterspace deformation is bigger than that of half-space model due to the effect of the free side surface. One can consider that there exists no free surface in the half-space model and its structure is thus stronger than a quarter-space. Such a difference becomes increasingly obvious as the load center approaches the side edge. Furthermore, when the pressure center is shifted towards the free surface, as shown in Fig. 5(a), the maximum deformation point moves away from the pressure center towards the free end. This result contradicts the prediction of the Hertz theory. Fig. 6 shows the differences between the magnitude and the position of the maximum deformation with different locations of pressure center in the x-axis. The differences rise sharply when the pressure center is located close to the free surface. When the center of pressure is located at d = a from the edge, the error of the half-space deformation can be as large as 30% (Fig. 6(a)) and the position shift of the maximum deformation (Fig. 6(b)) reaches 28.4% of the load radius a. The effect of free edge surface fades gradually when the center of pressure shifts away from the free end. The location of maximum deformation almost coincides with the pressure center when d reaches 2a. The effect of free edge surface on the magnitude of deformation remains significant even when the center of pressure is relatively far (d =5a) from the free end, as shown in Fig. 6(a).
3.1. Hertzian point contact
3.2. Flat cylindrical punch contact
The surface normal deformations under a Hertzian point contact with the same parameters listed in Table 1 but different d values were calculated. The load distribution is expressed with Eq. (8). The load centers were, respectively, located at d = a, 2a, 3a, 4a and 5a (compared to the full length of the block, 20a). The effect of the free side surface at x=0 was studied. It was assumed that the free surface on the other side at x=20a produces no significant effect due to it being far from the load center. Hence, finer mesh resolution was adopted in the region close to the free edge at x=0 as depicted in Fig. 2(a). Fig. 5 shows deformations along the x-axis on the top surface (z=0), which were directly calculated using the derived flexibility matrix. To highlight the effect of free edge surface, the same set of results calculated using the half-space model is also illustrated. Finite element modeling (using ANSYS) was also complied and results are presented in Fig. 5 to demonstrate the accuracy of the matrix solution of quarter-space model. The insets are blow-ups of the load center region where the deformation curves acquire the maximum values.
The quarter-space loaded with a uniform pressure distribution which simulates the contact of a flat cylindrical punch is also investigated. The pressure is given as,
Table 1 Parameters of the Hertzian point contact. Max. Hertzian pressure: Pmax = 0.5 GPa Elastic modulus: E = 201 GPa, Poisson's ratio: ν = 0.3 Contact radius: a = 0.5 mm
block, i.e. d =10a, for the validation of the computer program. Since the load is away from the two free edges and the contact area of radius a is far less than the top surface area of the block, the half-space criterion for the use of the Hertz theory is valid. Therefore, the surface normal deformation (as the output of the developed computer program) calculated with the quarter-space and the half-space models should be the same or very much similar to the analytical solution of Hertz as expressed in Eqn. (9). The normal deformation of the top surface along the x-axis is depicted in Fig. 4. The good correlation of the three curves as shown in Fig. 4(b) proved the assumption and the developed program. Fig. 4(b) is an enlarged plot of the maximum deformation in the Hertzian contact region (from 9a to 11a), which also shows that the deformation calculated using the quarter-space model is slightly larger than that of the half-space model or Hertz theory. The differences are attributed to the effect of the two free end surfaces. 3. Effect of free side surface on surface normal deformation
P(r ) = 0. 5GPa, P(r ) = 0,r > a
r≤a (10)
Results were obtained with the pressure center located at d = a, 3a and 5a along the x-axis on the top surface (full length of the block: 20a). Fig. 7 illustrates the deformed top surfaces obtained with the quarter-space model as well as the half-space model. The results show that if the pressure center is located close to the free edge, the deformed profile deviates significantly from that predicted by the half-space model. The effect of free edge side can be considerable. For example, the maximum deformation that reaches to 0.00757a occurs at the free surface when cylindrical pressure is applied at d = a, which is much larger than the corresponding 0.00454a of half-space model. Fig. 7 depicts that the half-space model gives smaller deformation when
Fig. 4. Surface normal deformation: comparison of half-space model, quarter-space model and Hertz theoretical solution.
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Fig. 5. Results of deformation along x-axis on the top surface for different d values.
deformation for various distances between the center of pressure and free surface. More than 40% error in maximum deformation is found in the half-space model when the distance between the load center and the free surface is equal to the radius of the cylindrical load distribu-
compared with the quarter-space, and the maximum deformation locates right at the pressure center. These results indicate failure of the half-space model to provide accurate results when the load is applied near the free surface. Fig. 8 shows the deviations in maximum
Fig. 6. Difference in the values of maximum deformation and the positions of quarter- and half-space models. (a) Difference in magnitude. (b) Position shift.
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Fig. 9. Deformation of a block of width 20a under Hertz line contact in quarter- and half-space models (dashed line: half-space; solid line: quarter-space).
Fig. 7. Deformation results based on quarter- and half-space models under cylindrical load distribution (dashed line: half-space; solid line: quarter-space; full length: 20a).
quarter-space deformed profile is small. With increasing the loading length to 8a, i.e. the free side surfaces are closer to the load, the difference is increased. When the length of the line load is extended to 20a, as same as the length of the elastic block, the quarter-space deformation profile depicts the maximum deformation at the two ends whereas the half-space attains the maximum deformation at the midlength (the block center as well as the load center). Results show that when the load is close to the free side surface, the influence of the side surface is considerable and the use of half-space model may lead to significant errors.
4. Conclusions Fig. 8. Maximum deformation deviations for various locations of applied load.
An accurate and fast method for the solution of surface normal deformation in quarter-space is introduced in this paper. The surface normal deformation can be obtained directly by multiplying the original normal load on the top surface of the quarter-space with a flexibility matrix. The flexibility matrix, which is constant for a given discretized quarter-space, can be retrieved and used for different loads. This facilitates fast and accurate calculation for the quarter-space normal surface deformation. This paper has evaluated three different load types on a quarterspace, namely Hertzian point, flat cylindrical punch and Hertzian line. The maximum load is up to 0.5 GPa and the material of the quarterspace under consideration is taken as steel. It is found that if the load is close to a free edge surface, the effect of the free surface can be considerable. For the specified cases, the error in the maximum deformation can be up to 40% (for a load located very close to the edge) if the inappropriate half-space model is adopted. Furthermore, the effect of the free edge surface leads to the position shift of the maximum deformation point and asymmetric deformation profile.
tion. 3.3. 3.3 Hertzian line contact The third case is a Hertzian line contact, which simulates an elastic roller contact. The Hertz contact radius a is 0.5 mm in the y-direction. The pressure distribution is given as,
⎛ y ⎞2 P(y ) = 0. 5 1−⎜ ⎟ GPa, ⎝a⎠ P(y ) = 0,
y >a
y ≤a (11)
Deformation results were calculated with three different loading lengths of 4a, 8a and 20a. The center of these loads is fixed at the center of the block (mid-span, d =10a). These loading cases are axisymmetric about the mid-section at x=10a. Only the half-length of the distributed load was adopted in the calculation of the surface deformation of the block. The mirror image of that calculated surface deformation gives the deformation due to the other half of the load. Superimposing the two deformation results gives the surface deformation of the applied load. Fig. 9 shows the calculated deformation with the quarter-space and half-space models. When the line Hertzian load of 4a in length is applied at the center and its edges are far from the two free side surfaces at x=0 and 20a, the difference in the half-space and
Acknowledgement The study described in this paper was fully supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region, China (Project No. CityU11213914).
Appendix The stresses acting on the vertical (y-z plane) and the top planes (x-y plane) of the quarter-space as the results of the load-pairs: Ph , Ph and Pv , Pv , respectively, as shown in Fig. 1 can be expressed in vector matrix format as,
σyz = M ∙ Ph
(A1)
σxy = N ∙ Pv
(A2)
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where M and N are two-dimensional coefficient matrices, and they are directly determined by the Love theory [2]. Overlapping the solutions of the two half-spaces in Fig. 1(b) and considering the boundary conditions yields,
−Ph + σxy=−Ph + N ∙ Pv = −P
(A3)
−Pv + σyz=−Pv + M ∙ Ph = 0
(A4)
Combining (A3) and (A4) gives,
Ph = (1 − N ∙ M)−1 ∙ P
(A5)
Comparing with Eq. (1),
A = (1 − N ∙ M)−1 Similarly, from Eq. (2),
B = M ∙ (1 − N ∙ M)−1
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