Analysis and simulation on the bumpy-ridge structure of heavy equipment

Transcription

1 CASE STUDY 971 Analysis and simulation on the bumpy-ridge structure of heavy equipment H Zhang*, Y Yan, and R Zhang Department of Mechanical Engineering, Tsinghua University, Beijing, People s Republic of China The manuscript was received on 0 May 009 and was accepted after revision for publication on 9 October 009. DOI: / Abstract: The bumpy-ridge structure applied in the field of heavy equipment is analysed based on fracture mechanics and contact theory. At the same time, the elastic plastic contact of the ridges is simulated using the finite element method. Having been researched, the separated bumpy-ridge structure can be seen as a special integral structure that has a through crack. It was also found that, as long as there are adequate normal forces and treated contact surfaces, this kind of separated structure has the ability to resist detachment and parallel slide. The height of the notch and radius of the contact zone are the foremost indexes affecting the application of the bumpy-ridge technique, and these parameters are influenced by several factors, such as the shape of the ridges, the curvature of the contact point, the distance between neighbouring ridges, etc. The elastic contact of the ridge can be solved completely by the use of theoretical calculation and graphical method. Through comparison with the elastic contact, some principles of the elastic plastic contact can be obtained with a finite element analysis method. A bumpy-ridge structure has been successfully applied to a 350 MN die forging press and a 360 MN extrusion press in China. Keywords: bumpy-ridge, elastic contact, elastic plastic contact, simulation, heavy equipment 1 INTRODUCTION *Corresponding author: Department of Mechanical Engineering, Tsinghua University, Haidian District, Beijing , People s Republic of China. zhanghr80@163.com Typically, an integral structure is adopted for traditional heavy equipment. Taking the 5000 mm rolling mill of the Shanghai Baosteel Group, for instance, its frame weighs 400 tons [1], the roughcast weighs 500 tons, and molten iron reaches nearly 700 tons. Recently, a project involving a 400 MN aviation forging press, launched in Xian, China, faced a dilemma in terms of production, processing, and transportation because the frame weighs more than 3000 tons, and it is almost impossible to cast it in one piece. Even if the frame were divided into several beams and columns, the weight of each part would also be more than 800 tons. To solve this problem, Professor Yongnian Yan of Tsinghua University put forward a new method prestressing subdivision and the bumpy-ridge technique. The basic principle of this technique is further to partition the beams and columns of heavy equipment into many subparts, making the weight of each subpart about tons, and to coarsen the contact surfaces of the subparts through the special bumpy-ridge treatment. Finally, prestressing steel wires are used to wrap these subparts together, forming an integral structure. There are no metallurgical or mechanical connections between these subparts, such as welding, threaded connections, or rivets. In order to eliminate stress concentration, this technique mainly uses friction to prevent parallel slide along each contact surface. The keys located on the interfaces of the subparts are just used to orient each other. Therefore, the bumpyridge treatment is very important to this technique. The so-called bumpy-ridge treatment means artificially roughening the surfaces of the subparts, using methods such as knurling, linear cutting, milling, ball blast, etc., so that a multi-peak structure is formed on the contact surfaces of the subparts. Meanwhile, a layer of metal sheet is inserted into every contact interface of the subparts to increase the friction coefficient, as shown in Fig. 1. The different parameters of treatment have different effects on the bumpy-ridge structure. Research on this issue would

2 97 H Zhang, Y Yan, and R Zhang Fig. 1 Bumpy-ridge structure mainly involve multi-peak contact. Currently, the contact problems of single peaks and multiple peaks have been studied from different angles by many researchers, at home and abroad; for instance, Hertz initially solved the semi-infinite elastic body in contact with a single peak []; Greenwood researched the elastic contact model of rough surfaces [3]; the coupling effects between rigid multi-peak and flexible bodies were studied [4]; and the contact between elastic and perfectly plastic bodies was also analysed [5]. Non-linear contact has also been considered using a virtual load method [6]. Owing to the successful development of elastic contact theory, this paper first focuses on the elastic solution of parameters concerning bumpy-ridge treatment, and then uses a finite element analysis method to achieve some principles of elastic plastic contact. EXPLANATION OF BUMPY-RIDGE CONNECTION ON FRACTURE MECHANICS A bumpy-ridge structure could be seen as a special integral structure that has a through crack, and thus methods of fracture mechanics can be used to study it. According to the Griffith theory [7], if two separated pieces are imposed by axial force P (where P < 0; P can be looked upon as the pre-force produced by wires), these two objects will closely join together (as shown in Fig. (a)), and the total potential energy, W, of this system is equal to the summation of elastic strain energy, U, and surface energy, T. It can be expressed by equation (1), as follow W ¼ U þ T ð1þ U and T are, respectively, solved by equations () and (3) U ¼ P l AE T ¼ ga W ¼ P l ga AE ¼ P A g E ¼ P l A 3 E ðþ ð3þ ð6þ First, after the corresponding factors of equation (1) were substituted by equations () and (3), equation (4) would be achieved finally. Parameter l represents the height of the above two objects, A describes the contact area, E symbolizes the modulus of elasticity of the two objects (assuming that these two objects have identical E here), and g represents the density of the surface energy of the contact interface. Then, first-order and secondorder derivatives of W could be respectively solved, which are indicated in equations (5) and (6). It is not hard to find that the value is negative forever, W/@A is always positive. According to the potential energy principle, the system is in stable equilibrium. It is obviously impossible that objects 1 and would become detached along the direction of normal force P,

3 Analysis and simulation on the bumpy-ridge structure of heavy equipment 973 unless P changes into a tensile load. If these two objects were seen as subparts of heavy equipment, the bumpy-ridge structure would keep its integrality well, in the same manner. In addition, from the dynamics perspective, if there was a greater g, W could be less than zero, would become smaller. A system with less potential energy would promote the combination of subparts, which is the reason special approaches are applied to treat contact surfaces. In other words, methods such as knurling, linear cutting, milling, and ball blast would make these surfaces more active and improve the value of g. The parameter K I also could be used to analyse the trend of detachment along the normal direction: K I stands for the stress intensity factor of crack I; s represents the stress of the entire system; and a symbolizes the length of the crack. p K I ¼ s ffiffiffiffiffiffi pa ð7þ As long as P is adequately negative, then s is negative too, as shown in equation (7). The larger the absolute value of s, the lower K I is and then the normal crack is less likely to occur. Therefore, the above conclusion is proved once more from the new angle. With respect to parallel slide in a tangential direction, the composite stress intensity factors of cracks I and II need to be considered. In fact, not only are contact surfaces affected by tangential stress t, but also normal force P would be weakened by axial tension Q,as shown in Fig. (b). On this occasion, P, Q, and t can be regarded as the pre-force provided by the wires, the main load provided by the main cylinder of the press, and the accessional stress produced by the eccentric load. The approximate criterion of composite stress intensity is shown in equation (9) [7]. K II represents the stress intensity factor of crack II, and K IC, K IIC, respectively, stand for the critical values of the stress intensity factor of cracks I and II. Seen from equation (9), K II can be allowed to have a much larger value when K I is negative. It indicates that, as long as the axial tension Q can be fully balanced by normal force P, and the absolute value of the residual force is large enough, the subparts would not slide over each other in the tangential direction. Moreover, increasing the K IIC value can also improve the ability to resist parallel slide of the subparts. According to equation (8), enhancing the critical shear stress t C can improve K IIC t C is mainly associated with the compressive stress and coefficient of static friction. Therefore, if the contact surfaces were coarsened into a multi-peak shape, and an interlayer was inserted between the two subparts, the coefficient of static friction would be enhanced significantly, i.e. the K IIC value would rise Fig. Illustration of bumpy-ridge structure based on (a) Griffith theory and (b) stress intensity factor theory pffiffiffiffiffiffi K IIC ¼ t C pa K I K II þ K IC K IIC ¼ 1 K I þ K II ¼ K IC ð8þ ð9þ

4 974 H Zhang, Y Yan, and R Zhang 3 CALCULATION ON BUMPY-RIDGE CONNECTION WITH TRIBOLOGY PRINCIPLE AND ELASTIC PLASTIC CONTACT THEORY 3.1 Tribology principle of bumpy-ridge connection According to the basic principles of tribology, friction is mainly produced by mechanical engagement, adhesion, ploughing, molecular attraction, etc. The bumpy-ridge connection not only synthesizes the above-mentioned mechanisms of friction, but also artificially upgrades the effects of mechanical restriction and small-scale friction. As shown in Fig. 3, a notch on the interlayer would appear because of the compression of force P. When imposed by tangential force F, a space on the left side of the notch would occur owing to the elastic deformation of the notch wall, and on the right side the reaction force N and friction f would appear simultaneously. According to the static equilibrium, F should be equal to the composite force of N and f. If the height of the notch, h, was increased, the reaction force N provided by the notch wall would become correspondingly greater. Friction force f is associated with P and the friction coefficient. Generally, the friction coefficient is a constant. P mainly depends on A, the contact area, which is represented by the black arc in Fig. 3. Therefore, f is largely dependent on parameter A likewise. The above analysis shows that, under the combined allowable stress of the ridge and interlayer, the depth h of the notch and the contact area A are extremely important to resist parallel slide. Additionally, larger P can intensify adhesion (cold welding) between the rough ridge and interlayer. In fact, parameters h and A reflect two major mechanisms of the bumpy-ridge structure: h reflects the role of large-scale mechanical restriction, and A reflects the mechanism of small-scale friction. Experimental results show, however, that the largescale obstruction takes the dominant role, followed by small-scale friction [1]. h and A are related to a series of factors, such as the different materials (modulus of elasticity, yield strength) of the ridges and interlayer, different shapes of ridges, and different distances between neighbouring ridges. All these factors would affect the final application of the bumpy-ridge structure. Consequently, these factors would be analysed as follows through mechanics calculation and finite element method (FEM) simulation. On account of it being inconvenient to measure A (shown in Fig. 3), it has been replaced with a, indicating the half-width of the contact zone (shown in Fig. 5, later). 3. Elastic solutions on bumpy-ridge connection The contact surfaces are coarsened into multi-peak structures having various shapes. According to the initial dimension, the contact between a ridge and an interlayer can be divided into point contact, line contact, or surface contact. This paper will, respectively, discuss hemispherical point contact, triangular line contact, and cylindrical surface contact, as shown in Fig Calculation of parameters on hemispherical ridge contact The local amplification of the hemispherical ridge contact is shown in Fig. 5. A coordinate system is Fig. 4 Three types of ridge contact Fig. 3 Local analysis of bumpy-ridge connection Fig. 5 Model of hemispherical ridge contact

5 Analysis and simulation on the bumpy-ridge structure of heavy equipment 975 established, with the x-axis based on the initial tangent direction along the contact surface, the z-axis based on the common normal direction of the ridge and interlayer, and the y-axis vertical to the paper. Lines 1 0, 0,1, and, respectively, represent the outlines of the ridge and interlayer in both deformed and undeformed states. Under external load P, if the displacement of ridge d was smaller, it can be assumed that this model would conform to Hertz s hypothesis. According to the Hertz theory, the half-width of the contact zone, a, the displacement of the ridge, d, and the maximum value of the Hertz stress distribution, p 0, are, respectively, shown in the following equations (10) to (1) and Fig. 5 [1, 8]. Parameters R and E *, respectively, symbolize the radius of the ridge and the composite modulus of elasticity of the ridge and interlayer a ¼ 3PR 1=3 4E ð10þ d ¼ a R ¼ 9P 1=3 16RE ð11þ p 0 ¼ 3P 1=3 pa ¼ 6PE p 3 R ð1þ Under the action of distributing load p produced by P (note that p ¼ p 0 (1 r /a ) 1/, where r is the distance from the contact centre to any point on the surface of object ), u zi, standing for the z-displacements within the contact area (r 6 a) of the interlayer, have different expressions from the z-displacements of the outer contact area (r > a), u zo. They are, respectively, shown in equations (13) [8] and (14) [9]. Parameters y and E represent the Poisson ratio and modulus of elasticity of the object interlayer. The height of notch h is equal to the margin between the maximum absolute value of u zi and the minimum absolute value of u zo, as shown in equation (15) and Fig. 5. According to equation (16), for object, u zo can be regarded as a monotone decreasing function with r. Therefore, it is reasonable to assume that, when r reaches a critical value r 0, min(ju z0 j) would be approximately equal to zero. r 0 can be solved by enabling u zo ¼ 0, as shown in equation (17). Because r is obviously greater than a, sin (a/r) is assumed pffiffiffi to be 0, and then the critical value r 0 is about a u zi ¼ pp 0 1 n ða r Þ ðr 6 aþ ð13þ 4aE u zo ¼ p 0 1 n ða r Þ ae sinða=rþ þðr a a 4 Þ 1= ðr > aþ ð14þ h ¼ maxðju zi jþ minðju zo jþ ¼ maxðju zi jþ ¼ pp 0 1 n a E ju zo j ¼ 3p 0 1 a ða r Þ < 0 ð16þ E r 4 ½4a þ a sin ða=rþšr þ 4a 4 þ a 4 sin ða=rþ ¼0 ð17þ When multi-ridges come into contact with the interlayer, the distance between ridges, p l, should be at least bigger than r 0, that is, l > ffiffiffi a. Otherwise, the depth of notch h produced by one ridge would be weakened by the superposition of neighbouring ridges, as shown in Fig. 6. After substituting equation (10) into the expression r 0, the relationship between r 0 and R can be obtained, as shown in equation (18). Hence, a non-linear proportional relationship between r 0 and R can be obtained. When the radius of ridge R is relatively large, in order to improve the large-scale restriction of the bumpy-ridge structure, the distance l between neighbouring ridges should be increased accordingly. Parameter tan b can be used to reflect the blocking effect of the notch wall, as shown in equation (19). If R of the ridge becomes smaller, angle b will be greater, and the notch will have a deep, narrow feature. Substituting equations (10) and (1) into equation (16), the relationship between h and E can also be obtained, expressed by equation (0). Seen from equations (0) and (10), the depth h and half-width a would gradually increase with the enhancement of ratio E 1 /E (E 1 and y 1 represent the modulus of elasticity and Poisson ratio of the object 1 ridge). It means that the stiffness of the interlayer should be lower than that of the ridges. In addition, h would increase with the decrease in ridge radius R. However, R does not strictly abide by this rule: the smaller it is, the better it is. The maximum shear stress would increase with two to three times the power of R (seen from equation (1) [9]; t max symbolizes the maximum shear stress on the contact Fig. 6 Critical value of distance of neighbouring ridges for the condition of multi-ridge contact

6 976 H Zhang, Y Yan, and R Zhang zone). Too small R would magnify the stress concentration on the contact area. It should be noted that the above studies are based on ideally lubricated contact, ignoring friction, which makes parameters achieved in practice slightly different from the calculations r 0 ¼ 3 p ffiffiffi! 1=3 PR E ¼ 3 p ffiffiffi PR ð1 n 1 Þþð1 n ÞðE! 1=3 =E 1 Þ E tan b ¼ maxð u j zijþ ¼ 1 pffiffiffi n 3 PE r 0 E R! 1=3 ð18þ ð19þ h ¼ 1 n 9P 1=3 " # 1 16R ð1 n 1 ÞE þ 1 n E ðe =E 1 Þ ð0þ surface, h being the largest Du z would naturally be achieved. According to the Johnson equation, some curves Du z x concerning different elastic moduli of the interlayers were drawn by use of curve-fitting techniques, as shown in Fig. 8. An approximate value of h can be measured from this chart easily The calculation of parameters on rectangular ridge contact As shown in Fig. 9, the cross-section of the cylindrical ridge is rectangular (and so it is also termed a rectangular ridge in this paper); its h is illustrated in Fig. 9 and equation () []. The half-width a of the contact area is obviously the radius of the cylinder. It can be deduced from equation (3) [] that the point that enables its u zo to be zero is further away from the centre point of the cylindrical ridges. Parameter u zo has the same meanings, as mentioned above. Likewise, under interaction of cylindrical multi-ridges, as long as the distances l between the ridges remain 1=3 t max ¼ 0:31p 0 ¼ 0:18 PE R ð1þ 3.. The calculation of parameters on triangular ridge contact As stated above, the contact between the triangular ridge and interlayer should be considered as a line contact; namely, the dimension of the ridge along the y-direction is relatively larger. Therefore, it is reasonable to regard it as a plane strain model. However, the elastic calculation for the triangular ridge contact is extremely complicated, mainly because of the discontinuity of the triangular shape at its vertex. The stress on this point appears singular. Therefore, the depth of the notch produced by the triangular ridge, h, is difficult to solve through analytic methods. In fact, h includes two parts, h d and h u. The h d stands for the sunk depth of the contact area inside, and h u means the depth of the contact area outside. Despite its normal stress being infinite, the maximum shear stress is only a finite value. Therefore, on the basis of the Tresca yield criterion, when external load P is small, elastic theory can still be used to analyse it, as shown in Fig. 7. The expression of u z (it represents the z-displacements of every contact point), which was given by Johnson, is hardly used to calculate its specific value. Therefore, assuming that u z of the peak point of the triangular ridge was zero, and then solving the relative distance Du z (the absolute margin of other points u z and peak point u z ) on the interlayer Fig. 7 Model of elastic contact of triangular ridge Fig. 8 Du z x curves

7 Analysis and simulation on the bumpy-ridge structure of heavy equipment 977 large enough, there would be no effects on h. Although l is difficult to solve, it is certain that it should be much larger than the critical value of the triangular and hemispherical ridges h ¼ maxðju zi jþ ¼ ð1 n Þp 0a E u zo ¼ ð1 n Þp p 0 ðr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r E a Þ 3.3 FEM simulation of elastic plastic contact of bumpy ridge ðþ ð3þ For the majority of bumpy-ridge applications, elastic and plastic deformations have taken place on the contact surfaces. The analytic solution for elastic plastic contact is very complex. Hence, more practical issues need to use numerical methods, such as FEM, the boundary element method, the mathematical programming method [10], etc. This paper makes use of FEM to analyse elastic plastic contact, simulating hemispherical ridge contact, triangular ridge contact, and rectangular ridge contact, respectively. Assuming the half length of the bottom edge of each ridge w varies from 0.5 mm, 0.5 mm, to 1 mm, the distance l between ridges is taken as 0.5w, 0.8w, 1w, w, 4w, 6w. Parameter w is shown in Fig. 4. Elastic modulus E 1 of the ridge is still 00 GPa, and the Poisson ratio is 0.8. Then, elastic modulus E of the interlayer varies from 50 GPa, 110 GPa, 150 GPa, 00 GPa, 50 GPa, 300 GPa, 350 GPa, to 400 GPa, and the Poisson ratio is Assuming all the materials mentioned above were linear hardening materials, the relation between the stress and strain for the plastic stage is s s is the yield strength, H is the hardening coefficient, taken as 1/10 of E, and «p is the strain for the stage of plasticity. P changes from 100 N, 400 N, to 1600 N. Seen from Fig. 10, whether for elastic plastic contact or elastic contact, their h always decreases with an increase in the elastic modulus of the interlayer. When E of the interlayer reaches a certain degree, h will decline slowly and approach zero gradually. In fact, the conclusion is consistent with that of the preceding section, namely that the elastic modulus of the interlayer should be less than that of the ridge as much as possible. In addition, the simulation results are in accordance with Hertz s theoretical calculation on elastic contact basically, but h of the elastic plastic contact and the half-width a are both larger than those of the elastic contact. Figure 11 shows that the depths h of the hemispherical ridge and triangular ridge are much larger than that of the rectangular ridge. For the rectangular ridge, it is difficult to form a notch on the interlayer. The rectangular ridge tends to make the interlayer compressed as a whole. Generally, owing to the larger curvature, h of the triangular ridge should be greater than that of the hemispherical ridge. But, in this paper, it shows the opposite situation. It is mainly s ¼ s s þ H«p ð4þ Fig. 10 Comparison of theoretical calculations with simulation results Fig. 9 Model of rectangular ridge contact Fig. 11 The influence of ridge shape on h

8 978 H Zhang, Y Yan, and R Zhang because of the plane strain assumption for the triangular ridge. If the triangular ridge presented a conical shape, its h value would obviously be larger than that of the hemispherical ridge. But the cone is just an ideal geometry, without practical worth. From another perspective, the above analysis actually reflects this sequence: point contact has the largest h, followed by line contact, and surface contact has the smallest h. For the hemispherical ridge, the depth h also enlarges with an increase in the interval between ridges l at the beginning, as shown in Fig. 1. After l reaches a critical value l 0 (l 0 ¼ r 0 ), h remains unchanged. It is a fact that, the smaller the R of the hemispherical ridge, the larger the h of the notch. Without doubt, R should not be as small as possible. Once R was too small, it would magnify the stress concentration of the contact zone, which is adverse to the stability of the bumpy-ridge structure. In addition, the critical interval l 0 changes with the R of the hemispherical ridge. When R becomes greater, ratio l 0 /R gets smaller instead, but l 0 still increases basically. As can be seen from Fig. 1, l 0 is approximately equal to R when R ¼ 0.5 mm; that is, r 0 approaches R, which p is obviously greater than the above derivation ffiffiffi a based on elastic contact theory. In other words, the critical interval between ridges for elastic plastic contact needs to be greater than that for elastic contact. As shown in Fig. 13, the half-width a of the contact area is not greatly influenced by the yield strength of the interlayer. This shows that, for the stage of elastic plastic contact, a mainly depends on P, which is distinct from the elastic contact situation. Of the three ridges, the rectangular ridge has the largest half-width of the contact area, and its half-width is equivalent to w. The hemispherical ridge occupies second place, and the triangular ridge has the smallest half-width. 4 APPLICATIONS Applying the bumpy-ridge technique, a 350 MN die forging press has been designed by the Department of Mechanical Engineering, Tsinghua University. This press holds the second-biggest single-cylinder in the world. A 1/3.8 model of the press is illustrated in Fig. 14(a). In practice, the frame is capable of keeping its integrality well all the time. Figures 14(b) and (c) show the application of the bumpy-ridge structure on a 360 MN extrusion press, which was also designed by Tsinghua University. The arch beams of the press were dissected into several subparts, and contacting planes were coarsened to a multi-peak structure. Finally, all these subparts of the frame were wrapped together by steel wires. The eccentric-load experiment proved that the bumpy-ridge structure has a first-rate ability to resist parallel slide and detachment. At present, this press is in the debugging stage. 5 CONCLUSIONS According to the Griffith theory, the coarsening treatment on contact surfaces contributes to the connection of the bumpy-ridge structure; increasing the normal force and the tangential friction coefficient does help to improve the stress intensity factor of the bumpy-ridge structure, and it upgrades the capacity to resist parallel slide. Whether hemispherical, rectangular, or triangular, all ridges have a critical radius r 0 ; the analytic Fig. 1 The h R curves with different curvature radius of ridge Fig. 13 The a s s curves of multi-shape ridges

9 Analysis and simulation on the bumpy-ridge structure of heavy equipment 979 Fig. 14 Application of bumpy-ridge structure: (a) 1/3.8 model of 350 MN die forging press; (b) 360 MN extrusion press; (c) application of bumpy-ridge structure to 360 MN extrusion press calculation demonstrates that the smaller radius of ridge would increase depth h of the notch, but at the same time it also increases the maximum shear stress on the bumpy-ridge subparts. Therefore, R should be comprehensively considered; decreasing the elastic modulus of the interlayer would increase the half-width a of the contact area and depth h of the notch. FEM simulation shows that: the depth h of the elastic plastic contact and half-width a of the contact area are bigger than those of the elastic contact; h of the hemispherical and triangular ridges are much larger than that of the rectangular ridge; and the critical interval of the elastic plastic ridge contact is much larger than that of the elastic ridge contact. In general, the influence of the yield strength of the interlayer on a is not obvious. The bumpy-ridge structure has been successfully applied to a 350 MN die forging press and 360 MN extrusion press. Ó Authors 010 REFERENCES 1 Peng, J. B., Yan, Y. N., Zhang, R. J., and Lin, F. Principle and application of pre-stressed bumpy-ridge in field of heavy mechanism. Chin. J. Mech. Engng, 008, 44(6), Johnson, K. L. Contact mechanics, 1985 (Cambridge University Press, London). 3 Greenwood, J. A. and Williamson, J. B. Contact of nominally flat surfaces. Proc. R. Soc. London, 1966, 95, Komvopoulos, K. and Choi, D. H. Elastic finite element analysis of multi-asperity contacts. ASME J. Tribology, 199, 114, Yang, N., Chen, D. R., and Kong, X. M. Elastic Plastic finite element analysis of multi asperity contacts. Tribology, 000, 0(3), Zhao, H. The virtual contact loading method for contact problems considering material and geometric nonlinearities. Comput. Struct., 1996, 58, Li, Q. F. Fracture mechanics, 1998 (Haerbin Industrial University Press, Haerbin). 8 Timoshenko, S. and Goodier, J. N. Theory of elasticity, 1951 (McGraw-Hill Press, New York).

10 980 H Zhang, Y Yan, and R Zhang 9 Morton, W. B. and Close, L. J. Notes on Hertz theory of contact problems. Philosoph. Mag., 19, 43, Wang, X. C. and Kong, X. G. Contact mechanics and computational methods. J. South-West Jiaotong University, 1996, 31(3), APPENDIX Notation a half-width of contact area A contact area E modulus of elasticity of object 1 and E* composite modulus of elasticity E 1 modulus of elasticity of ridge E modulus of elasticity of interlayer f friction force F tangential force h depth of notch h u depth of notch on contact area, outside h d depth of notch on contact area, inside H hardening coefficient K I stress intensity factor of crack I K IC critical stress intensity factor of crack I K II stress intensity factor of crack II K IIC critical stress intensity factor of crack II l length of objects 1 and N supporting force of notch wall maximum contact stress p 0 p P Q r r 0 R T u z u zo u zi U y 1 y w W contact stress normal force tensile force distance from other points on interlayer to central point critical r radius of hemispherical ridge surface energy z-displacement z-displacement on interior of contact area z-displacement on exterior of contact area elastic strain energy Poisson ratio of ridge Poisson ratio of interlayer half-length of ridge bottom potential energy a angle of triangular ridge b angle of notch g density of surface energy d displacement of ridge Du z margin of z-displacement «p plastic strain l distance between neighbouring ridges l 0 critical distance of neighbouring ridges s normal stress s s yield strength t tangential stress t c critical shear stress maximum shear stress t max