Modeling and Simulation of the Defect Form Effect on Mechanical Behaviour of Shrink-Fit Assemblies.
Hamid Boutoutaou
Department of Mechanic University of Boumerdes
Algeria
hamid_boutoutaou@yahoo.fr
Jean-François Fontaine
IUT of Auxerre University of Bourgogne
French
Jean-francois.Fontaine@iut-dijon.u-bourgogne.fr
Abstract—The calculation methods of shrink fit assemblies remain traditional and have hardly changed for several years.
They consider that the contact surfaces are perfectly smooth and do not take into account their geometrical defects (shape, roughness ....). The models created impose too high manufacturing costs to make consistent assumptions of calculations with the operational conditions of realization. To reduce manufacturing costs, the study of the influence of defect form on assemblies’ resistance characteristics is essential. The objective of this work is to study the influence of the defect form on the strength characteristics of shrink fit assemblies. It is shown that the defect form is not prejudicial to the strength of the assembly: the mean pressures are almost equivalent to the conventional case of surfaces without defects. Various finite element simulations were performed. The influence of the amplitude and of the default period was studied for different types of tightening.
Keywords—shrink-fit; finite element; modeling of contact;
machining; behavior.
I. INTRODUCTION
The shrink fit is an assembly technique by tightening between two generally cylindrical parts. This tightening causes a pressure at the contact interface which maintains them in solidarity with the friction. The calculation methods of shrink fit assemblies remain traditional and have hardly changed for several years. They are based on the thick-walled tube of theory with internal pressures developed by Timoshenko [1]. This model is restricted to the elastic behavior (low tightening) and simple cylindrical parts whose contact surfaces are assumed perfectly smooth [2]. The models created and impose too high manufacturing costs to make consistent assumptions of calculations with the operational conditions of realization. To reduce manufacturing costs, the study of the influence of the shape of default on assembly strength characteristics is essential. Some recent studies show the importance of having finer models to better understand and develop the shrink fit assemblies.
Zhang Y, B Mc Clain, Fang XD. [3] studied the distribution of stresses in the interface a ball bearing particularly on the edges with the finite element method. They established a strength criterion based on two safety factors λS(safety factor ensuring)
the resistance of each element and λp (safety factor ensuring the connection between the two parts). Truman, J.D. Booker [4] have analyzed the effect of the loading of the tightening on the micro-shifts for the fretting of a gear phase having a non- constant radial stiffness over an axis to predict rupture. Adnan O, Emsettin T, D Murat A, S. Sadri [5] noted the need to simulate the assembly process for example in the case of flexible joint fitting to better design this type of assembly. By studying the strengthening of local resistance by laser heat treatment, Sniezek L, Zimmerman A. [6] showed that 50% of the resistance can be increased by performing a geometric dephasing of the treatment on the axis and bore. Croccolo D, De Agostinis M, N. Vincenzi [7] concentrate their work on the behavior of assemblies, including resistance to fatigue.
However, the assumptions of previous studies are very restrictive since the surfaces are considered geometrically perfect and smooth. This does not correspond to reality, it is indeed not possible to produce perfect surfaces, these have actually defects in processes for the production. Do not consider this defect form the disadvantage of having to make the surfaces of the parts assembled with great precision to be consistent with the calculations. This increases production costs and requires the use of methods super finish as rectification. But there are also advantages the fact of considering the presence of the interface defect. The assembly strength increases in the presence of defects form. JF Fontaine, Siala IE. [8] Showed that the defect form had a significant influence on the local stress at the contact interface.
Hosseinzadeh [9] 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. To take into account the effects of stress gradient properly, Lanoue et al. [10] 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. As indicated by equations (1) and (2), when the parts are perfectly cylindrical, the main constraints in the interface in the part having the bore (hub, disc etc.) are composed of a radial compression σr and circumferential strain σθ. The axial stresses are negligible except at the edges.
2 1 2 2
22
2 1
r r pr
r
(1)
2 1
2 2
22
2 1
r r pr
(2)And the Von Mises stress :
2 2
2
( ) ( )
) 2 (
1
VM
r
r
(3)Which give :
2
1 2 2
22
2 2
r r pr
VM
(4)With p interface pressure, r1and r2inner and outer radii of the hub and ρ radius shrink. This state gives an equivalent Von Mises stress can be greater than twice the interface pressure (see equation (4)). Defect form has the effect of changing the local stress because the radial stressr(pressure) is maximum when the tightening is greatest. Instead, the orthoradial stress
is lower, which has the effect of limiting the equivalent Von Mises stress. It is so possible for the same Von Mises stress increase the contact pressure therefore resistance of the assembly.
The objective of this work is to study the influence of the defect form on the strength characteristics extraction of shrink fit assembly. Axisymmetric modeling phases shrink and extraction are presented. Several defect form parameters representing the conditions for obtaining the bore are then considered. Finally, the results will be compared with those calculated by the methods proposed by the standards [11].
II. MODELING AND SIMULATION
The assembly studied is composed of a steel shaft and a hub duralumin. Nominal diameters d = 20 mm and the contact length L = 12 mm. The elastic properties of steel are: Young's modulus E = 210 103MPa, Poisson ν = 0.29, yield σe = 350 MPa and those of duralumin hub: E = 74,103MPa, ν = 0.33 and σe= 270 MPa. Hub outer diameter D = 40 mm. The law of plasticity is used classical Prandtl-Reuss law with isotropic hardening. The contact surfaces have different defects form of representative process to obtain (turning). Thus the cylindrical profiles have lobes in the direction of the generator “fig.1”.
Several hypotheses have been considered in the models:
- The thermal dilatation phase is not taken into account. It does not affect the material properties and the geometry of the interface.
- The behavior is elastic-plastic; contact with small slip is
chosen to model the shrinking phase. While a contact with important shift is selected for the extraction phase.
- The coefficient of friction between steel and duralumin is chosen at a conventional value of 0.15. The models are made with the finite element software ABAQUS.
Fig.1. Profiles form of the shaft and hub
“Fig.2” shows the mesh used for areas near the interface.
Along the contact surface, the mesh size is (0.05x0.15) mm to the axis and (0.05x0.05) mm for the hub. The choice of fine mesh and regular on the contact interface facilitates the analysis and exploitation of results.
Fig. 2. Meshing interface shaft – hub
“Fig.3” shows the boundary conditions used during the various simulations. For shrinking phase, we blocked the axis of the shaft in both the radial and axial directions and the generatrix of the hub in the axial direction. In the extraction phase, we imposed axial translation to the axis of a few microns.
Fig. 3. Boundary conditions for the extraction phase
9,99 9,995 10 10,005 10,01
0 2 4 6 8 10 12
Radius values (mm)
profile lengh (mm)
Shaft with two lobes Hub with three lobes
Shaft Hub
II. RESULTS SIMULATIONS
Defect form changes the traditional concept of tightening since the radius then varies at each point. This creates zero tightening and pressure zones and the areas where the tightness and pressure reach their maximum value. As shown in “fig. 4”, at the interface, the pressure follows the shape of the defect and is no longer constant as in the case without default. It is up to the top of the lobes and null in the hollow where there is no contact.
Fig. 4. Distribution of pressure at the interface of contact for cases with and without defect form
“Fig. 5” shows the distribution of Von Mises stress at the contact interface. Unlike the pressure, this stress has an important value in areas where the pressure is zero. Its maximum value is reached at the top of the lobes. A geometric phase difference between the radial and circumferential stress explains why the maximum Von Mises stress is lower than the pressure and does not reach twice as in the perfect case “fig.
6”
Fig. 5. Distribution of Von Mises stresses at the interface of contact for cases with and without defect form
Fig. 6. Distribution of principal stresses at the interface of contact
Fig. 7. Influence of the period defect on mechanical characteristics
“Fig. 8” shows that the magnitude of the defect greatly influences the values of stress, pressure and extraction force at the interface. Higher the value of the amplitude is greater, higher the value of the pressure increases as a function of the tightening and its difference with the maximum stress of Von Mises. For large values of tightening, the lobes of the hub are crushed by plastication, a loss of clamping occurs affecting the contact pressure and especially Von Mises who is stabilized on a quasi constant value.
Fig. 8. Influence of the amplitude defect on mechanical characteristics The intensity of the tightening differently influences the behavior of assemblies where the parameters of defect are not identical. For the same amplitude of defects and two different periods “fig. 9 (a)”, the variation of the resistance to extraction is almost identical. Against by, in the case of the same period of defects and two different amplitudes “fig. 9” (b), the behavior of the assembly changes. A growing gap between the resistances values of both cases is observed. When the amplitude of defect is important, the assembly becomes more resistant. This difference is explained by the influence of the tightening which increases as the value of the amplitude increases. Figures (a) and (b) also show an assembly with a
0 50 100 150 200 250 300 350
0 2 4 6 8 10 12
Contact pressure distribution (MPa)
Contact lenght (mm)
hub with decfect hub without defect
10 60 110 160 210
0 2 4 6 8 10 12
Von Mises stress distribution (MPa)
Contact lenght (mm)
hub with defect shaft with defect hub and shaft without defect
-490 -390 -290 -190 -90 10 110
0 2 4 6 8 10 12
Principal stresses (MPa)
Lenght contact (mm)
Radial stress S11 Circonferntial stress S22
0 50 100 150 200 250 300 350 400
contact pressure(Mpa) hub V.Mises stress(Mpa) shaft V.Mises stress(Mpa) extraction force x 0,1 (N)
100 200 300 400 500 600 700 800
5 10 15 20 25 30
Amplitude of the defect forme (µm)
hub V.Mises stress(Mpa) shaft V.Mises stress(Mpa) contact pressure(Mpa)
extraction force with defect x0,1 (N) extraction force without defect x0,1 (N)
0 2π π 2π/3 π/2
Periodicity of defect form (rad.)
defect form is more resistant than a traditional assembly (without defect).
Fig. 9. Extraction force variation with maximum tightening ∆M
(a) different periods; (b) different amplitudes.
III. CONCLUSION
In this article, we studied the influence of geometric parameters of defect form, such as period, amplitude and tightening, on the behavior and mechanical strength of the shrink fit assemblies. We have shown that it is not only possible but essential to integrate defect form in Modeling of these assemblies. While conventional calculation models do not allow because of their simplifying assumptions, it is quite possible today thanks to the finite element method, to integrate them into a numerical modeling. The stakes are significant in economic terms. Indeed, a first view, defects are inherent in the process of obtaining mechanical surfaces, not to consider in modeling forced to be consistent in terms of practical use very expensive manufacturing methods. Then from another point of view, defect form contributes to the strength of the assembly. It may require a resistance equal or higher than without the presence of the defect by decreasing the pressure- Von Mises stress report.
References
[1] Timoshenko SP. Strength of materials part II: “advanced theory and problems”. 3rd ed. Krieger Pub. Co.; 1956. p. 205–13.
[2] NF E22-620 et E22-622.”Assemblages frettés sur portée cylindrique”.
AFNOR, Paris la Défense; January 1984.
[3] Zhang Y, Mc Clain B, Fang XD.”Design of interference fits via finite element method”.Int J Mech Sci; 42:1835–50, (2000).
[4] Truman CE, Booker JD. “Analysis of a shrink-fitEng. Fail Anal; 14:557–
72, (2007).
[5] Adnan O, Emsettin T, Murat D A, Sadri S. “Stress analysis of shrink- fitted joints for various fit forms via finite element method”.Mater Des;26:281–9, (2005).
[6] Sniezek L, Zimmerman A.“The carrying capacity of conical Interference Fit joints with laser reinforcement zones”. J Mater Proc Technol;
210:914–25, (2010).
[7] Croccolo D, DeAgostinis M, Vincenzi N. “Static and dynamic strength evaluation of interference fit and adhesively bonded cylindrical joints”.
Int J. Adhes 2010; 30:359–66.
[8] Fontaine JF, Siala IE. “Three dimensional modelling of a shrinkage fit taking into account The form defects”.Eur J Mech A/Solids;
17/1:107–19, (1998).
[9] Hosseinzadeh F.Residual stresses in shrink fits and quenched components.PhD thesis. UK: University of Bristol; 2009.
[10] Lanoue F, Vadean A, Sanschagrin B.Finite element analysis and contact, modeling considerations of interference fits for fretting fatigue strength calculations.Simul.Model Pract Theory 2009; 17:1587–602.
[11] NF E22-620. ‘’Assemblage frettés sur portée cylindrique : fonction, réalisation, calcul’’. AFNOR, Paris la Défense, Jan.1984.
1400 3600 5800 8000 10200
5 10 15 20 25 30
Extraction force (N)
Amplitude of maximum tightening ∆M(µm)
T=2Pi/3 T=Pi/2 T= 0
1400 3600 5800 8000 10200 12400
5 10 15 20 25 30
Extraction force N
Amplitude of maximum tightening ∆M(µm)
Df= 10µm Df= 20µm Df= 0 µm
T= π/2 (b)
Df= 20 µm (a)
