Pitting modeling of a spur gear in mesh.

Cananau, Sorin ; Stoica, Gina

1. INTRODUCTION

The principal mode of failure of gears systems is the surface pitting of gear teeth flanks. Pitting occurs within small cracks are initiated on the tooth surface or just below the surface (Johnson, 1989). The main problem which concern also this paper is to give the correct prediction of possible gear failure and is bounded by several requirements as high loads, high speed, minimal weight etc. The main cause of the initiation consists in metal-to-metal contact of asperities or defects due to low lubricant film thickness. The cracks may start at inclusion in the gear flank (or foreign particle contamination of lubricant), which act as stress concentrators and propagate due to normal and tangential loads (Glodez et al., 1998) This process is dependent of material and operating conditions. In any case the complete fatigue process of micropit formation consists of several stages: the crack initiation, the crack growth, the destruction of the surface layer at high rate and the destruction of the surface layer at low rate. In the paper we are going to study these stages using the tools of FEM analysis.

2. PHISICAL MODEL OF PITTING DESTRUCTION

Consequent with the physical model the process and the stages presented in introduction, the calculus model will be divided into two parts: the first model is associated with fatigue crack initiation under load and the second model is associated with fatigue crack development under contact loading.

The main criterion for separation of these two models is present also in classical standardized procedures DIN 3990 and AGMA 218.01 for prediction of service life in gears operation and consists in defining two numbers of stress cycles (Glodez et al., 1998): Ni (the number required for the appearance of the initial crack in the material), [N.sub.j] (the number required for growth of the crack in the material). The theoretical model presented in figure 2 represents the model with an inclusion (Glodez et al., 1998). The contact projection of a spherical inclusion is characterized by the diameter [d.sub.in] and the shear modulus [G.sub.in]. This inclusion is residing in a slip zone, an ellipse with semi-major axis L and semi-minor axis l.

The shear modulus of the material, G, is the shear modulus of the gear material (Glodez et al., 1999). The model of crack initiation due to a grain presence assumes that the dislocation stress. This model assumes that the dislocation stress (consequent tangential load) td is equal to (Tanaka & Mura, 1982):

[FIGURE 1 OMITTED]

[[tau].sub.d] = [[gamma].sub.1] G x [G.sub.in] / G + [G.sub.in] (1)

where [[gamma].sub.1] is the plastic strain in the grain due to one cycle loading with applied shearing stress [t.sub.max]:

[[gamma[.sub.1] = (l + L) / 1 x [[tau].sub.max] - [[tau].sub.f] / G (2)

and the meaning of [[tau].sub.f] is the tangential stress due to frictional forces in the meshing process.

We consider the process of grains dislocation is irreversible. Connected with this hypothesis we can say that the number of cycles [N.sub.i] will be a function of plastic strain amplitude (Sun et al., 1991). The constant k will take into account the irreversibility factor and this factor takes values:

k = [10.sup.-1] x [10.sub.-4] (3)

In this case, the number of cycles N is expressed in equation (4):

[N.sub.i] = k 8(G + [G.sub.in]) / [d.sub.in] x G x [[tau].sub.max] - [[tau].sub.f] / [([DELTA] [tau] - 2 [[tau].sub.f]).sup.2] (4)

Concerning the crack development we can assume that the cracks are defined as being short because their length are small and are associated with the process of surface pitting (Wang, 1998)

The theory of short crack growth is used to model the crack development associated with pitting phenomenon. We can use the theory of short crack growth only considering the characteristic dimension of material microstructure (the grain dimension) of the same seize with the length of the crack development.

But the dimension of the diameter of the hole is from units to thousands of grain characteristic dimension. That means the cracks extend from a dimension of a small grain to a much larger scale. The model of crack growth must taken into account the transfer of the phenomenon from the grain of the initial crack to tens (or thousands) neighboring grains. This implies the discontinuity of this process. We have also to consider the decrease of crack growth rate as the crack approaches, grain by grain, to a grain-boundary. We can express a relation between stress intensity range AK and plastic displacements [DELTA]u, using the geometric characteristics as are shown in fig.2:

[FIGURE 2 OMITTED]

3. FEM ANALISYS

Numerical analysis has the aim to establish the relationship [DELTA]K=f(a) utilizing a virtual crack extension method. In the first step we will study the possibility of the crack initiation as consequence of normal and tangential load applied.

The finite element analysis was conducted to asses with large approximation grains-polycrystalline distribution at the surface, considering a contact zone. The analysis is a 2D-plain strain analysis, with 4-nodes quadrilateral elements. The number of total elements is 2176. The present model taken into account the load applied corresponding to single pair teeth engagement and the maximum value for the load applied in inner point of this load distribution (Cananau, 2005).

In accordance with our hypotheses we consider in the same time a frictional load at the contact surface as tangent vectors of profile geometry. The values of these forces are determinate using Amontons-Coulomb friction laws and a coefficient of friction [mu]=0.06--an average value considered for lubricated gears. We suppose that the pitting phenomenon is the cause of the initiation of crack. In the FEM analysis we are going to simulate the presence of a short crack and we are going to investigate the effect of the crack in the gear function.

The diameter [d.sub.in] of the inclusion (grain) we assume to be as the same size as the finite element size in the meshing contact area. For the geometric model presented and for the chosen scale, we can assume the average value [d.sub.in] = 100[micro]m.

In the second step of this analysis we develop on the base of previous results, the study of crack propagation. The local meshing around the initial crack is sown in Fig.3. For this step we used special fracture finite elements. With this type of element, we can calculate the stress intensity factor.

[FIGURE 3 OMITTED]

In the next figure (Fig.4) is shown the torsional mesh stiffness result for a short crack presence (about 2 mm) at the root of the tooth of the driven gear.

[FIGURE 4 OMITTED]

4. DISCUSSION AND CONCLUSION

The paper presents a new model for the initiation and growth of the pits and cracks in the contact area of spur gears The FEA model showed that the tooth crack affects the stiffness of the tooth over the phase of engagement of the tooth. The stress intensity factor was investigated, as the crack propagates through material to the surface.

5. REFERENCES

Cananau, S., (2005) Pitting model in the contact area of spur gears, Tribology, The Annals of University "Dunarea de Jos" of Galati, Fasc.VIII, p. 89-92, ISSN 1221-4590

Glodez, S., Ren Z., Flasker, J., (1998) Simulation of surface pitting due to contact loading, Int. J. Num. Meth. In Eng., Vol. 43-23, p.33-50.

Glodez, S., Ren Z., Flasker, J., (1999), Surface fatigue of gear teeth flanks, J. Comp. and Struct., Vol. 73, p.475-483.

Johnson, K.L. (1989) The strenght of surfaces in rolling contact, J. Mech. Eng. Science, Vol. 203, p. 151-163.

Sun, Z., Rios, E.R., Miller, K.J., (1991), Short and long fatigue cracks interacting with grain boundaries, Fatigue and Eng. Mat., Vol. 14, p.277-291.

Tanaka, K., Mura, T., (1982), A theory of fatigue crack initiation at inclusion, Met. Trans., Vol. 13A-23, p.117-123.

Wang, C.H., (1998), Effects of Stress Ratio on Short Fatigue Crack Growth, ASME J. Eng. Mater. Techn, Vol. 118, p.362-366

Table 1. Geometrical characteristics
of spur gear model

 Pinion Gear

Module mm 5
Standard pressure angle 20 [degrees]
Number of teeth 15 21
Addendum coefficient +0.571 0
modification
Tip circle diameter mm 93.210 117.5
Center distance mm 90
Face width mm 18 16
Contact ratio 1.246