立式銑床主軸及銑頭結(jié)構(gòu)設(shè)計(jì),立式銑床主軸及銑頭結(jié)構(gòu)設(shè)計(jì),立式,銑床,主軸,結(jié)構(gòu)設(shè)計(jì) at , D.A. g University Univers contact substrate has been investigated. The coecient of friction at the edge of contact, which characterizes the asymptotic stress field, is considered as the controlling parameter in the analysis. The predicted results are in agreement with the neering problems. They may include splint joints between shafts, bolted connections, and certain analysisbecausecrack nucleation andinitiationare more complex as the problem may involve a knowledge of the combined interaction of load, geometry and material in addition to the use of a components are reduced by cyclic fretting fatigue involving contact. Cyclic fretting loads tend to consider the presence of cracks and makes use of stress and strain in certain planes along the contact surface that are regarded as critical 58. The stress and strain, however, are highly elevated near the contact edge similar to that of a crack tip. A slight change in the fretting conditions, can cause Mechanics * geometries of gas turbine fan blade dovetails etc. These components are prone to failure by fatigue. Fretting fatigue failure can be divided into two stages: crack initiation and propagation. Life esti- mate of cracked structures may be solved by many existingmodels.Buttheeectoffrettingcontacton crack initiation is likely to be far less amenable to activate flaw at the contact surface which will lead to the development of cracks. However, the ana- lysis of fretting fatigue is dicult because the damage process may involve a multitude of cracks 1,2. In many test, multiple cracks are often found near the edge of contact or near the ship-stick boundary 3,4. Conventional approach does not experimental observations. The information gained may lend insight into the dierent stages of damage associated with the complex process of fretting fatigue. C211 2003 Elsevier Ltd. All rights reserved. Keywords: Fracture; Fretting fatigue; Mechanics; Crack nucleation 1. Introduction Contact problem covers a wide range of engi- valid failure criterion that preferably could handle crack nucleation, initiation and propagation. Fatigue life and endurance limit of mechanical Crack initiation Y.J. Xie a, * a Department of Mechanical Engineering, Liaonin Fushun, LN 113001, b Department of Engineering Science, Oxford Abstract By using the S-theory, the crack initiation angle from the Theoretical and Applied Fracture Corresponding author. E-mail address: (Y.J. Xie). 0167-8442/$ - see front matter C211 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2003.09.003 contact surface Hills b of Petroleum x , and the polar Fracture Mechanics 40 (2003) 279283 distance from the crack tip to locate the site of failure. This distance is r which coincides with the singular character of the strain energy density field. Moreover, r can also be used as the distance use contact and fracture mechanics for predicting the direction of crack initiation from the contact surface into the contacting body. 2. Strain energy density criterion The S-criterion can be used to describe the fracture behavior of stable and unstable crack growth, critical loads and crack initiation angle in mixed mode. It makes use of the strainenergy density factor S which is related to the strain en- ergy density function dW =dV by the relation dW =dV S=r where r is the distance from the crack tip. The critical value of dW =dV can be de- termined from the area under the uniaxial stress and strain curve 911. For the asymptotic ana- lysis S depends only on the angle h. In general, S can depend on r. In what follows, it suces to use the asymptotic form of S: S a 11 K 2 I a 12 K I K II a 22 K 2 II a 33 K 2 III 1 where a 11 1 l 8pE 3 C0 4l C0 cosh1 cosh a 12 1 l 8pE 2sinhcosh C0 1 2l a 22 1 l 8pE 41 C0 l1 C0 cosh 1 cosh3cosh C0 1C138 a 33 1 l 2pE 2 with l being the PoissonC213s ratio and E the elastic modulus. For the past several decades, S-theory has been widely used in analysis of crack problems by using the asymptotic stress field on r C01=2 . However, the theory remains valid when using the complete stress field. In this case, S would depend on r and h. It should be pointed out that the relation 280 Y.J. Xie, D.A. Hills / Theoretical and Applied from the contact edge in the absence of a pre- Fig. 1. Contact between a two-dimensional rectangular punch 1 2 coordinates r;h, both with the origin at left edge of contact, are selected. Normal force P and tan- gential force Q act on the punch. The normal and shear tractions along interface have been solved in closed form 2,12, which are px 1 C0 P sinkp p 2 C16 C0 x 1 a C17 kC01 x 1 a C16C17 C0k 3 and qx 1 fpx 1 4 where f is the coecient of friction; the k can be determined by tankp 21 C0 l f1 C0 2l ; 0 k 1 5 with l being the PoissonC213s ratio of the substrate. Eq. (3) shows that the stress state near the punch corners varies as r ij 0r kC01 ; x 1 2a 0r C0k ; x 1 0 C26 as r ! 0 6 When l 0:5 or with f 0, it yields k 0:5. This gives the same order of stress singularity as and a substrate. fore, the physical meaning of K I when used as a stress fields. This stress fields is equal to the Eq. (9) adding Eq. (11). Then the local (eective) k and new crack will form at a certain critical load. The Fracture Mechanics 40 (2003) 279283 281 The singular stress fields at the sharp edge of the contact between the rectangular rigid punch and substrate are known from the asymptotic contact analyses 12. Using the polar coordinates r;h, Fig. 1, the stresses at the left corner are found to vary as r I rr r I hh r I rh 0 B 1 C A C0 K I 2pr p cos h 2 1 sin 2 h 2 C18C19 cos 3 h 2 sin h 2 cos 2 h 2 0 B B B B B B 1 C C C C C C A 9 This expression is identical in form with the Mode I compressed stress field, but where K I P pa p 10 3.3. Mode II singular stress field From the solution in 12, the asymptotic stress failure criterion is not clear. The strain energy density theory does not have this limitation since it focuses attention on the failure of an element ref- erenced from the vicinity of a site where failure is likely to occur. In passing, it may be emphasized that failure is always assumed to initiate from a finite distance away from the crack tip whose exact location can never be found in the material. When l 0:5, the asymptotic stress boundary conditions of substrate in contact area next to left corner are r 22 j h0 C0 P p 2ar p 7 and r 21 j h0 C0 fP p 2ar p 8 3.2. Mode I singular stress field that for a crack from Eqs. (5) and (6). However, the stress intensity factor is not related to the en- ergy release rate as in the crack problem. There- Y.J. Xie, D.A. Hills / Theoretical and Applied field of Mode II is given by crack initiation angles can be obtained from the strain energy density factor criterion 911. It is assumed that the crack tends to run in the h 0 di- rection for which S min prevails. That is oS=oh 0 0. This leads to a 0 11 a 0 12 f a 0 22 f 2 0 15 where a 0 11 sinh 0 2l C0 1 cosh 0 a 0 12 2cos 2 h 0 C01 C0 2lcosh 0 C0 1 16 I k II , are expressed in terms of h as 13 k I hr hh h 2pr p 13 k II hr rh h 2pr p 14 where r hh hr I hh r II hh and r rh hr I rh r II rh . Fig. 2 shows the typical angular variations of Eqs. (13) and (14) with angle h and f. It indicates that k I h will be positive at certain h and f, which means that the contact boundary possesses typical fracture characteristics that can be investigated by using the S-theory. 4. Boundary cracking in complete fretting contact For singular stress fields next to the corners, a r II rr r II hh r II rh 0 B 1 C A C0 K II 2pr p sin h 2 1 C0 3sin 2 h 2 C18C19 C03sin h 2 cos 2 h 2 cos h 2 1 C0 3sin 2 h 2 C18C19 0 B B B B B B B 1 C C C C C C C A 11 where K II fP pa p 12 3.4. Characters of the stress fields The initiation of a crack depends on singular a 0 22 sinh 0 1 C0 2l C0 3cosh 0 282 Y.J. Xie, D.A. Hills / Theoretical and Applied Fig. 3 gives schematically the relationship of normalized cracking angle h 0 =p and the coecient Fig. 2. Variation of eective stress intensity factors 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 20.0 40.0 60.0 80.0 100.0 120.0 Cracking angle o deg Friction coefficient f Fig. 3. Cracking angle h 0 versus friction coecient f. Fracture Mechanics 40 (2003) 279283 of friction. Experimental observations 13 indicate that the crack initiation angle in fretting fatigue tests ranges widely from 25C176 to 80C176. In these fret- ting tests, f varies with cycling from initial value 0.20.4 to final stable value 0.71.2. Fig. 3 indi- cates that the cracking angles predicted by S- theory for crack initiating from the contact boundary are in agreement with the experimental results. 5. Conclusions The problem of contact between a rectangular rigid punch and flat-surface substrate has been in- vestigated by using the S-theory. Cracking angles in boundary of the substrate are theoretically pre- dicted. The present findings provide a method for determining crack initiation from a site that had no k I , k II with angle h, for dierent f. existingcrack.Keepinmindthatapre-requisitefor using fracture mechanics is that the material must be assumed to have an existing crack. Additionally, the work in 14 studied the stress state near the corner of complete contact subject to fretting action using an asymptotic analysis. Their results indicated that there are infinite combina- tions of friction coecient f and wedge angle u that would result for a )0.5 singularity next to the corner. Even though the order of the stress sin- gularity from the contact boundary without a crack is the same as that for a re-existing crack, the physical interpretation of the stress intensity fac- tors for the contact problem is dierent, the meaning of which is not clear. The use of the strain energy density theory does not have this problem. Its physical meaning in terms of predicting failure by fracture remains the same. This distinction should be recognized. Equipment, ASTM STP 1159, American Society for Testing and Materials, Philadelphia, 1992, pp. 6066. 2 D.A. Hills, D. Nowell, Mechanics of Fretting Fatigue, Kluwer Academic Publishers, Netherlands, 1996. 3 V. Lamacq, M.-C. Dubourg, L. Vincent, Crack path prediction under fretting fatiguea theoretical and experi- mental approach, ASME J. Tribol. 118 (1996) 711720. 4 M.H. Wharton, D.E. Taylor, R.B. Waterhouse, Metallur- gical factors in the fretting-fatigue behavior of 70/30 Bras and 0.7% carbon steel, Wear 23 (1973) 251260. 5 H.K. Kim, S.B. Lee, Crack initiation growth behavior of Al 2024-T4 under fretting fatigue, Int. J. Fatigue 19 (1997) 243251. 6 W. Cheng, H.S. Cheng, T. Mura, L.M. Keer, Microme- chanics modeling of crack initiation under contact fatigue, ASME J. Tribol. 116 (1994) 28. 7 M.P. Szolwinski, T.N. Frris, Mechanics of fretting fatigue crack formulation, Wear 198 (1996) 93107. 8 L.J. Fellows, D. Nowell, D.A. Hills, On the initiation of fretting fatigue cracks, Wear 205 (1997) 120129. 9 G.C. Sih, A special theory of crack propagation: methods of analysis and solutions of crack problems, in: G.C. Sih (Ed.), Mechanics of Fracture I, Noordho, Leyden, 1973, pp. 2145. 10 G.C. 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