Materials and Design 32 (2011) 3692–3701 Contents lists available at ScienceDirect Materials and Design journal homepage: www.elsevier.com/locate/matdes Modeling of interference fits taking form defects of the surfaces in contact into account H. Boutoutaou a, M. Bouaziz b, J.F. Fontaine c,⇑ a Laboratoire de Dynamique des Moteurs et Vibroacoustique, Université M’hamed BOUGARA, faculté des sciences de l’ingénieur, 35000 Boumerdes, Algeria Laboratoire de mécanique et développement, Ecole nationale Polytechnique, Département de Génie Mécanique, 16200 El Harrach, BP 182, Algeria c Laboratoire d’électronique, informatique et Image, Université de Bourgogne, route des plaines de l’Yonne, 89 000 Auxerre, France b a r t i c l e i n f o Article history: Received 22 November 2010 Accepted 25 March 2011 Available online 31 March 2011 Keywords: C. Machining D. Mechanical fastening E. Shrinkage a b s t r a c t The technique of assembly by shrink fit is increasingly used today. However, the methodology of parts sizing has not changed in 50 years. Assembled parts are assumed to have accurate dimensions and very low form defects. This has the disadvantage of increasing the cost of parts production. To reduce manufacturing costs, the study of the influence of form defects on the characteristics of assembly strength is essential. Taking default form into account assumes that the tightening (difference between the diameters of the shaft and the bore) is defined. In the case under consideration, the tightening depends locally on the radius. Two definitions of the tightening are proposed: maximum tightening and mean tightening. It is shown that the form defect is not detrimental to the assembly strength: the mean pressures are nearly equivalent to the classical case of surfaces without defects. Various finite element simulations were performed. The influence of the value and the type of defect have been studied for conventional tightening (elastic materials) and more intensive tightening (elasto-plastic behavior) in the case of axisymmetrical and non-axisymmetrical parts. The theoretical results correlate well with those obtained through experiments. However, for intensive tightening, the behavior of the roughness is not negligible. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Shrunk parts assemblies have been performed for a long time, for example to give wooden wheels a more resistant rolling surface. These hoops were put in place by force or by thermal expansion. No accuracy was necessary because of the high elasticity of the assembled parts. Machining methods having advanced, it was then possible to assemble metal parts that required greater accuracy to guarantee a given strength and geometry, as in the case of wrapped guns, or to transmit higher power with sprockets keyed on their shafts. Actually, this mode of assembly is widely used and has almost replaced other methods such as securing parts by inserting a pin in a groove. For requirements of optimizing characteristics, mechanical progress requiring use of materials as close as possible to their limits has led to the development of the study of this assembly method using theoretical studies, computer generated calculations and experimental trials. However, in the literature, few studies have been done and the calculation rules contained in the standard have not changed in ⇑ Corresponding author. E-mail address: jffont@u-bourgogne.fr (J.F. Fontaine). 0261-3069/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2011.03.059 50 years [1]. They are based on the classical solution of the thickwalled tube with internal pressures developed by Timoshenko [2]. This model is still limited to simple cylindrical parts which do not allow us to simulate the behavior of most industrial cases. Some recent studies show the advantage of higher resolution models to understand interference-fits joints better. Zhang et al. [3] have studied the stress distribution at the interface of a ball bearing in particular on the edges with finite element method. They established a strength criterion based on two safety factors ks (factor of safety to ensure component strength) and kp (factor of safety to ensure no slippage on mating surfaces). Eyercioglu et al. [4] used finite elements modeling to design a tool for forging shrunk parts to ensure the final dimensions of the finished product. Truman and Booker [5] have analyzed the effect of loading clamps on micro-slidings during the shrinking phase of a gear having a non-constant radial stiffness on an axis to predict fracture. By studying the strengthening of local resistance by heat treatment with laser, Sniezek et al. [6] have shown that it is possible to increase the resistance by 50% by producing a geometric dephasing of treatment on the shaft and on the bore. Adnan et al. [7] note the need to simulate the assembly process, for example in the case of insertion of a flexible joint, to improve design of this kind of assembly. Sun et al. [8] used simulation to validate the deformations due to heating of a crankshaft during 3693 H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 insertion of frets. Croccolo et al. [9] focus on the joint behavior, including resistance to fatigue. However the assumptions of these previous studies are very restrictive because the surfaces are considered to be smooth and geometrically perfect. This does not correspond to reality. Indeed, it is not possible to achieve perfect surfaces or eliminate defects due to manufacturing processes completely. Failing to take form defects into account presents the drawback of having to manufacture surfaces with greater accuracy to be consistent with calculations. This increases manufacturing costs and requires a finishing process such as grinding. But there are others benefits in considering the presence of interface form defects. The strength of the joint is increased when there are form defects. Fontaine and Siala [10] showed that form defects have a significant influence on the local stress state at the contact area. As shown in Eqs. (1) and (2), when the parts are perfectly cylindrical, the main stresses at the interface in the part with the bore (hub, disc, etc.) consist of a radial compression rr and a hoop tension rh. Axial stress remains negligible except at the edges. rr   2  r2 r ¼ p 2 1 2 1  22 ðr2  r 1 Þ r1 ð1Þ   2  r21 r2 1 þ ðr22  r 21 Þ r 21 ð2Þ rh ¼ p rmm rmm 1 ¼ pffiffiffi 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrr  rh Þ2 þ ðrr Þ2 þ ðrh Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffi r2 r2 2 þ 22 ¼p 2 1 2 ðr 2  r 1 Þ r1  1 2p Z 2p 0 ðr a  r h Þdh  ð3Þ ð4Þ The maximum interference DM which is defined by the following equation: DM ¼ 2: max½ðra  rh Þ rh ∆M ra ∆m ∆ m ∆M A - perfect axis and hub B - perfect axis and hub with form defect Fig. 1. Definition of the mean tightening and the maximum tightening: (A) without form defect Dm = DM, and (B) with form defect Dm – DM. 2. Modeling of the extracting strength 2.1. Shrink process modeling with p interface pressure, r1 inner radius and r2 outer radius of the hub. This state gives an allowable equivalent stress, i.e., Von Mises stress rmm , which may be greater than twice the interface pressure p (see Eq. (3)). The form defect results in a change of the local stress state because the radial stress (pressure) is maximal when the tightening is maximal. On the contrary, the hoop stress is lower at this location. That has the effect of limiting the difference between both stresses. So, the geometrical dephasing between the radial and axial stresses tends to smooth the variation of the Von Mises stress. This property can be exploited to increase the resistance of the assembly. The objective of this study is to analyze the influence of defect shape on the assembly resistance to extraction for non-axisymmetrical parts. Initially, the modeling of the shrink step will be presented. Several parameters of the form defect representing conditions of obtaining the bore will be considered. Then, the extraction strength modeling will be detailed and theoretical results will be compared to experimental results. The introduction of the form defect changes the traditional notion of tightening since the radii vary in every point. Therefore, two definitions of the interference (tightening) are introduced (see Fig. 1). The mean interference Dm which is defined by the following equation: Dm ¼ 2 rh ra ð5Þ with ra, local outer radius of the shaft and rh, local inner radius of the hub 2.1.1. General assumptions A perfect solid steel cylinder is considered with a radius varying from 7.985 mm to 7.995 mm. Its length is equal to 20 mm, with the following elastic characteristics: Young modulus E = 210  103 MPa, Poisson’s ratio m = 0.3 and Yield stress re = 350 MPa. The axis is assembled with a duralumin hub with two different external shapes (circular with outer diameter Dm = 39 mm and rectangular with outer length L = 40 mm and width W = 30 mm). The mechanical characteristics of the duralumin are: E = 71  103 MPa, Poisson’s ratio m = 0.28 and Yield stress re = 250 MPa. The plastic law used is the classical Prandtl–Reuss law with isotropic hardening. The bore presents different form defects resulting from the machining process (turning process). So, the cylindrical profile consists of lobes in the directrix direction. One lobe exhibits a classic form defect due to the geometry of the machine (form defect periodicity P = p. Three lobes exhibit the effect of a defect due to a clamping chuck with three jaws (form defect periodicity P = 2p/3. Four lobes exhibit a more complex defect (form defect periodicity P = p/2. The tightening is considered standard, i.e. lower than the conventional value of a relative tightening of 0.26%. Several assumptions have been adopted in the modeling: – the axis is considered to be perfect geometrically before the hub bore defects; – the thermal dilatation step is not taken into account. It is considered not to affect the properties of materials or the interfacial micro-geometry; – the behavior is elastoplastic, the formulation is expressed in large deformations and a contact with little sliding is chosen for the fitting step. A contact with large sliding is taken into account with geometrical non linearity for the extracting step; – the coefficient of friction is depending on many parameters (i.e. interference, surfaces, working stresses, stress concentration, fretting, assembly,. . .). It is known that the coefficient of friction at shrink-fit pressures is much lower than that quoted for low normal loads [11]. Booker and Truman [12] have shown that the coefficient of friction is area independent. It is the same for full and partial radial contact of a hub on a shaft. Here in 3694 H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 the present study, the form defects affect only the local state of stress. So, the coefficient of friction is considered identical for each test. It is chosen at a conventional value of 0.15 between steel and duralumin; – the modeling is performed with the finite element method using ABAQUSÒ software. 2.1.2. Modeling of shrink step To take into account the effects of stress gradient properly, Lanoue et al. [13] indicate that the mesh must be refined near the interface of the shaft and hub. A convergence study has been performed to reveal this influence and different contact constraints have been tested to achieve a compromise between accuracy and computation time. The interference fit option available on ABAQUSÒ can easily model this step. Fig. 2 shows the mesh used. At the interface, following the contact area, the mesh size is 0.2 mm  0.2 mm for the axis and 0.05 mm  0.05 mm for the hub. In the case of a form defect, the pressure at the interface is not constant as shown in Fig. 3. Axis Mesh Fig. 4 shows the cartographies of Von Mises stresses. At the interface, the distribution of Von Mises stress does not follow the distribution of pressure. It reaches a level equivalent to the maximum pressure, while in the case of the hub surface being perfect, it is equal to twice the pressure (see Fig. 5). In Fig. 6, a geometrical dephasing between the maximum radial and hoop stresses can be seen. This explains why the maximum Von Mises stress is less than twice the maximum pressure. However, the Von Mises stress is not maximum at the interface but rather inside the hub in an area close to the contact surface. (see Fig. 7). The periodicity of the defect tends to increase the pressure and the maximum Von Mises stress (see chart on left in Fig. 8.). In the context of a high form defect, because the contact pressure may be low or even zero if the local clamping is negative (no local contact), the mean pressure does not increase in the same proportion (see chart on right in Fig. 8). Reduced to the maximum Von Mises stress, this tends to decrease. Fig. 9 shows that the form defect value and the mean tightening influence directly the level of stress at the interface. The variation Hub Mesh Fig. 2. Shaft and hub meshes used for modeling the shrink step. Fig. 3. Distribution of contact pressures along the interface for the case of free defect assembly, circular and rectangular hub with four lobes defect (P = p/2). H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 3695 Fig. 4. Von Mises stresses fields for circular and rectangular hub with four lobes (P = p/2). Fig. 5. Von Mises stress distribution in the Hub at the interface. is roughly linear. The higher the form defect value is, the more the maximum pressure rises according to tightening increases as well as differences with the maximum Von Mises stress. However the mean pressure does not increase in the same proportions. This is due to a smaller real contact area for high form defect values. Fig. 10 indicates the evolution of the ratio between the real contact area AR and the nominal area AN corresponding to the case without defect. When the real contact area tends to the nominal area, the maximum Von Mises stress reaches the maximum pressure value (ef = 0.01 mm and mean tightening Dm = 0.03 mm). When the hub area near the interface becomes plastic, the maximum pressure continues to increase but to a lesser extent because of work hardening. Fig. 11 shows that for a form defect with four lobes (P = p/2), the extracting strength does not decrease because of the plasticity. This can be explained by the increase of the real contact area due to the plastic strain. On the contrary, the maximum pressure decreases due to strain hardening. The form defect influence also edge effects (see Fig. 12). Özel et al. [14] studied the influence of the shape of the shaft and the hub on the pressure distribution for a shrink fit. They have shown that the edge effects strongly depend on the geometry both of the interface and external shapes of the hub. Zhang et al. [15] have modeling a turbocharger compressor used in a diesel locomotive. The assembly presents an axis and a sleeve in steel and an impeller in aluminum. They show that the edge effect depends strongly of the outer shape of the impeller. It is so necessary to computer with accuracy the contact pressure between the different elements. Fontaine and Siala [16] showed that it was theoretically possible to remove the stress concentration due to edge effects by defining a suitable chamfer. Practically, this chamfer is difficult to achieve because it would require very precise processes of manufacturing and the presence of form defects would further complicate the task. 2.2. Extracting strength modeling To model the extracting strength, conditions with the following boundaries were applied: a displacement of the upper end of the axis was imposed. The plane face of the hub and the outer cylindrical surface were immobile following the Z-axis; a small area near the hub of the axis was free to take into account the gap between the axis and the support device (see Fig. 13). The friction coefficient was set at an arbitrary value f = 0.15. An example of evolution extraction load is given in Fig. 14. The periodicity of form defect influences the extraction force slightly as shown in Table 1 for samples with or without defects (the periodicity of the defect taking the values P = 2p, 3p/2 and p/2). Fig. 15 shows that, depending on the definition of the tightening, the effect of form defect on the extraction load will vary. If the mean tightening is considered, as compared to a perfect bore, when the hub presents form defects, the extraction load is always 3696 H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 Position on the circumference of the interface (mm) Position on the circumference of the interface (mm) Fig. 6. Radial stress (above) and hoop stress (below) in the hub at the interface. Fig. 7. Von Mises stress distribution along the radial direction. greater. This means that the defects increase the mechanical strength. This evolution according to the value of the form defects is nonlinear. On the contrary, if the maximum tightening is considered, it is the opposite and the higher the tightening, the wider the gap between the results. The results are close around the regression line and the evolution of the strength will be more or less linear depending on the form. To conclude, both definitions of tightening are possible, the first giving results close to the calculations based on parts without defects, and the second better correlated to the calculation of extraction strength. The edge effects have little influence on the extracting strength. Due to high localized pressures, the form variation generated by the deformation of the axis tends to block the extraction. The increase of extraction force is about 5 to 10%. It tends to be higher when the hub exhibits significant form defects. 3. Experimental study The aim of the study is to predict the strength of a shrunk assembly in the presence of geometrical defects. Although in this H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 3697 Fig. 8. Maximum pressure, maximum Von Mises stress and mean pressure versus periodicity of the form defect for Dm = 0,02 mm (left chart). Ratios mean contact pressure and maximum contact pressure on Von Mises versus periodicity of the form defect for Dm = 0,02 mm (right chart). Fig. 9. Maximum pressure, maximum Von Mises stress and mean pressure versus mean tightening and form defect value for a defect with four lobes. They have been shrunk to steel calibrated shafts with a diameter equal to 7.996 mm. The presented results concern only the rectangular hubs. The samples were machined on a Direct Numerical Control machine (DNC). The form defect was obtained with a small round cutter. A finishing pass on a lathe yielded different roughness. The samples were measured precisely before assembly using the following procedure: – measurement of the mean diameter on a Coordinate Measurement Machine (50 points taken on two sections with a maximum distance between them); – form defect measurement on a cylindrical geometry measurement machine (digitizing of the profile every 5°) on the sections corresponding to these for the mean diameter measurement; – roughness measurement with a traditional contact stylus device (four measurements using a generatrix at different locations on the surface of the bore). Fig. 10. Ratio between real and nominal contact areas versus the mean tightening. paper, the modeling concerns only the shape defects but it seemed useful to have samples for different surface textures. Thus, different samples were made: they have a form defect in 1, 3 or 4 lobes (P = 2p, 2p/3 and p /2), the same average diameter equal to 7.98 mm and a form defect amplitude ranging from 0.02 mm to 0.06 mm. The roughness pitch varies from 0.14 mm to 0.5 mm and the average roughness RZ from 2.74 lm to 22 lm. Then, each sample has been modeled using the geometric measured data. The profile of the bore has been reconstructed by superimposing the profile of the form defect to the average diameter measured on two different sections. The comparison of the experimental results with the modeling results at the interface remains a difficult task because it is impossible to place strain gauges on the interface without damaging them. Several techniques have been the subject of previous work. 3698 H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 Fig. 11. Stress state in the hub with four lobes defect at the interface, real contact area and extracting strength versus mean tightening (plasticity occurring for a tightening of 0,035 mm). Fig. 12. Edge effect for a hub with defect (four lobes) and without defect. Lewis et al. [17] have attempted to measure interface pressures with ultrasound. Unfortunately, the accuracy of the calibration depends on the similarity of the roughness of the specimen contacting surfaces to that of the interference faces. As yet, they note that Fig. 13. Limits conditions for the extracting step. there is no method to obtain directly the contact pressure or real area of contact, analytically from measurements of reflection. Recently, Lewis et al. [18] have used the technique of neutron diffraction to measure the stresses generated during a shrink fit and then subjected to torsional stress. For a conventional relative tightening of 0.09%, the measurement results obtained overlap those given by the F.E. model. But, because of measurement uncertainties, they cannot highlight the edge effects. The determination of the stress state depends on the identification of micro-deformation of the structure. So, the knowledge of the behavior law is necessary. Under an elasto-plastic law, the measurement is more complex due to the nonlinearity of the behavior. This technique requires substantial equipment and does not exactly measure stresses at the interface but only near it. Hosseinzadeh [19] has adapted the technique of deep hole drilling (DHD) to measure residual stresses in a shrink fit to take into account the elastoplastic state of the material. The incremental method (ICHD) developed permits to measure the stresses in the three directions [20]. The authors show that the residual stresses due to the surfaces machining materially alter the effective stresses at the interface. This technique gives a good agreement between theory and experience but it is difficult to implement. Finally, stress measurement at the interface is an operation difficult despite the current progress of experimental techniques. Therefore, the extraction force, easily measurable, is used here to compare modeling with experiments. H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 3699 Fig. 14. Example of the evolution of the extracting load versus the relative displacement between the shaft and the hub for different tightening for four lobes form defect. Table 1 Theoretical extracting strength for a mean tightening Dm = 0.01 mm. Number of lobes Without defect One lobe (P = 2p) Three lobes (P = 3p/2) Four lobes (P = p/2) Extracting load 279 N 288 N 289.5 N 312 N Table 2 shows the different attributes for each shrunk sample, and the experimental and calculated extraction forces. Fig. 16 shows the evolution of the experimental extraction load compared to the theoretical load according to the maximum relative tightening DM% defined by the following conventional relation: Dm% = Dm/dA (with Dm maximum tightening and et dA axis diameter). The influence of the form defect period is notable: for example, the last points of the diagram have a same tightening (Dm% = 0.56) but the extraction load is greater for a sample corresponding to a period P = p/2. It can be noticed that for conventional tightening lower than 0.26%, the difference between modeling and experimentation is weak. When the tightening exceeds 0.3%, differences between theoretical and experimental results are more important. This is due to the behavior of the asperities that becomes quickly plastic. This effect is known as clamping loss or smoothing. It depends on the value of roughness. Fontaine et al. [21] have shown that the tightening loss can reach a non negligible value in the order of several hundred Newton. So, at this level, it is necessary to take the behavior of the roughness into account. 4. Method to calculate bore dimensions taking form defects into account Taking form defect into account requires an alternative definition of a priori specifications of the elements assembled. It is essential to define specifications that take into account geometric constraints as well as the strength of the joint. The mean or maximum tightening may be associated with a measureand ensuring the mechanical strength. However, as indicated in the conclusion, this work remains to be done. In a first approach, the conventional method for determining the strength characteristics of an assembly can be adapted to the case of a shrink fit of a bore exhibiting form defects. Assuming Fig. 15. Influence of the tightening definition on the extracting load for four lobes form defect. 3700 H. Boutoutaou et al. / Materials and Design 32 (2011) 3692–3701 Table 2 Characteristic of the experimental samples and modeling and experimentation results. Nb sample Form defect periodicity Form defect amplitude ef (mm) Max. tightening DM (mm) Min. tightening Dm (mm) Max. rel. tightening. DM% Min. rel. tightening Dm (%) Theoretical load (N) Experimental load (N) 1LB05 3LB05 1LB50 4LB10 4LB15 1LB30 3LB30 4LB30 2p 3p/2 2p p/2 p/2 2p 3p/2 p/2 0.022 0.026 0.033 0.02 0.030 0.029 0.06 0.06 0.025 0.026 0.044 0.05 0.06 0.066 0.09 0.09 0.012 0.014 0.027 0.03 0.03 0.051 0.0546 0.0538 0.16 0.16 0.28 0.31 0.38 0.41 0.56 0.56 0.08 0.09 0.17 0.19 0.19 0.32 0.34 0.34 2231 1847 4393 5209 5785 6587 6768 7828 2200 2000 4500 4900 5500 6000 6400 7200 (2) Similarly, a bore with an average radius of 7.98 mm and a form defect amplitude of ±0.02 mm guarantees that the limit of elasticity will not be exceeded. This radius is equal to 7.969 mm for a geometrically perfect bore. (3) An average radius between both mini and maxi radii values satisfies the conditions of strength. Table 3 summarizes these elements and shows that taking form defect into account can reduce production costs. 5. Conclusion Fig. 16. Experimental and theoretical extracting load versus maximum relative tightening. the diameter of the shaft is free of defects and knowing the type of form defect of the hole obtained on a given machine, the proposed methodology is the following: (1) Determination by numerical calculation of the maximum radius or diameter to obtain the specified minimum strength. (2) Determination by numerical simulation of the average minimum radius or diameter taking into account a range of controlled defects so that the Von Mises stress does not exceed the elastic limit. For example: To determine the radius of a hole with p/4 defect period and a minimum desired extracting strength equal to 650 N. (1) Using numerical calculation to simulate this defect, a bore having an average radius of 7.985 mm and a form defect amplitude of ±0.01 mm can satisfy the conditions of resistance. This radius is equal to 7.978 mm for a geometrically perfect bore. Table 3 Determination of dimensions of bore for the without defect and form defect cases. Without defect case Form defect case Mean radius (mm) Tolerance (mm) Manufacturing process dm = 7.9974 mm ±0.0045 Grinding 7.98 mm < dm < 7.985 mm ±0.02 Turning Taking the form defect of the interface of an interference fit into account allows for the use of a less expensive manufacturing process than the process indicated by the classical standard. In this case, the tightening is not constant and it is necessary to define it. In this paper, two definitions are proposed: the mean tightening that is closer to the classical result and the maximum tightening that is more correlated to the strength. To define a standard specification, it would be interesting to base further work on the Geospelling concept developed by Dantan et al. [22]. This concept allows for the unambiguous expression of a specification with geometrical and mechanical conditions. This is the case of shrink fits where both types of conditions are necessary. Actually, this paper shows that finite element modeling can be used to validate the geometrical specification of the assembled parts taking into account form defects. For higher relative tightenings, it is necessary to take roughness behavior into account. Several approaches can be taken: the first consists of intervening directly on the contact law by acting on the damping coefficient of the penalty method. Introduced for computational reasons, the justification of this coefficient is the asperity behavior. However, this ultra-realistic approach requires an adjustment of the coefficient with the microgeometrical surface texture. 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