410 S。 Sen, B。 Aksakal / Materials and Design 25 (2004)   407–417

Fig。 2。 (a) Dimensions of the solution zone, (b) FEM mesh and the boundary conditions (D ¼ 2R and d ¼ 2r)。

Fig。 3。 Variation of stresses in the shaft–hub with r=R along OC line (l=d ¼ 0:5, h ¼ 42 lm)。

for three different interference values, e。g。 h ¼ 42, 64 and 74 lm。

3。Finite element model

Because of the geometric symmetry, the problem was analysed as 2D axisymmetric model using ANSYS 5。6 software [29–32]。 As given in Figs。 1 and 2 the solutions took place for only the hatched areas。 By considering

the symmetry in r axis, the model used in the analysis and boundary condition was given in Fig。 2。 In this figure AB is the contact line which is the most critical zone in the whole system。 As thermal boundary condi- tions of the model the heat transfers between CD, DB, BE, and EF surfaces and free atmosphere as convection, along AB surface between two mating contact surfaces as conduction heat transfer and along y and r axis, along the central of model。 It was assumed that the points along the r axis in the y direction and the points along

S。 Sen, B。 Aksakal / Materials and Design 25 (2004)   407–417 411

l/d=0。5 , h=42 mic。

Fig。 4。 Variation of the radial displacements in the hub and shaft along interference region, AB line (l=d ¼ 1, h ¼ 64 lm)。

the y axis in the r direction have zero displacement。 In order to get better results at the interface of the both,

and—1hub and shaft were the main concern, a finer mesh was used in this region。 In the outer part, hub, 255 element and in the inner part, shaft, 525 element with quadri-

fTn  þ 1g¼ ½C]ffRhgDt þ ½½C]— ½½Kc]þ ½Kh]]Dt]fTnggð4Þ

lateral four node axisymmetric isoparametric element was taken and with the effect of heat transfer between these two parts and for force transformation in the shaft–hub surface region connection 306 elements, 3 node axisymmetric contact element with elastic coulomb friction was used。 To provide these kinds of interfer- ences low temperatures are needed and the trends of these results are in good agreement with analytical re- sults in [4,5,8]。 The validity of the solutions can be seen in both Figs。 3 and 4。 The energy and other related equations are also given in references of [29–32]。

4。Finite  element formulation

In the present study, a thermo-elastoplastic problem     is

assumed to possess a bilinear elastic–plastic  behaviour in which the material is kinematically hardening which obeys to the von Mises yield criterion, and details are given in [30]。 In the elastic region the r–e relation can be written as

dr ¼ Ceðde — d~eT Þ: ð5Þ

In the plastic  region

dr ¼ Cepðde — d~eT Þþ dr~T; ð6Þwhere

The solution domain X bounded by surface C is pided into n elements at each node (m)。 The temperature and its

gradients within each element can be expressed as 。 。

Applying the method of weighted residuals and inte- grating by Gauss’s theorem, which introduces surface integrals of the heat flow across the element boundary, and after doing some manipulations, the resulting ele- ment equation becomes a non-linear transient form [30,31] as follows: