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62
UDC 539.22.003.12:62-233.28:539.4.001.24
CONSIDERATION OF ANISOTROPY AND CONTACT OF CRACKS EDGE AT STRESS CALCULATIONS OF ROLLING BEARINGS
Maksimovich O.V., Doctor of Technical Sciences Il'yushin A.V., Ivashhuk A.D., Post-graduate students Lutsk National Technical University, Lutsk, Ukraine E-mail: olesyamax@meta.ua
ABSTRACT
Investigation of influence of anisotropy on stress-deformed state of base (roller bearings track) considering the appearance of cracks with contacting edges in it is done in the work. The boundary integral equation method is used to determine the stresses. Solution of the Integral equation is done numerically by the mechanical quadrature method. At the task solution it is considered that cracks can be located in the compressive stresses areas, wherefore the cracks edges can contact. The unknown contact stresses on the cracks edges and contact area were determined correspondingly from the set quadratic programming problem. The stresses under the bearings were determined according to the Hertz formula. On the ground of the carried out investigations the following conclusions can be done: mechanical anisotropy of bearings at the track edge allows increasing resistance to brittle crack initiation or decreasing growth rate of fatigue cracks. At that, it is necessary to choose such thermo-mechanical methods of processing at which the module of elasticity in circular direction will be larger than in radial direction (perpendicular to the edge); additional compressive stresses decrease the stress intensity factor (SIF) and tension stresses increase. In this regard in the track neighborhood it is reasonable to do thermal processing to create residual compressive stresses in circular direction. One of the ways to create mechanical anisotropy and compression stresses is insertion of the previously heated thin-walled anisotropic cylinders; the calculated maximum values of the stress intensity factor (SIF) give the possibility to calculate the growth rate of fatigue cracks for different lengths and crack angular inclinations. At that, in the early stages the calculations process is simplified, because here there is the crack edges contact and hence K=0. Under such conditions a crack will grow along the strait line where it is located. At long lengths of cracks when the crack edges contact is absent, these SIFs allow calculating critical values of crack lengths. Analogical results are obtained for other values of the relations of elasticity modules w, friction factors k, values of parameter m, characterizing the level of additional stresses. These results (which are not provided here) confirm the stated above conclusions on influence of these parameters on the SIF values.
KEY WORDS
Stresses, crack, contact durability, roller bearing.
In roller bearings loading is transferred via roller system. It is known that high stresses appear in the contact area of the rollers and the track. Here, as a rule, edge fractures appear and generate at different angles relatively the edge. Further, repeated motion of the rollers along the track with the fractures which had appeared earlier, will result in the repeated loadings. That is why the initiated fractures in the early stage will generate according to fatigue mechanisms. In calculations it is necessary to take into consideration friction, that can sufficiently influence on stress of dies (rollers) and the track material during the manufacturing process and thermo-mechanical processing can obtain mechanical anisotropy (particularly, the areas where plastic deformation has taken place, become mechanically anisotropic areas). Also it is known that to increase the track durability the reinforcing thermo-mechanical methods are used, which can result in residual stresses. At the preset task consideration it is necessary to take into account the fracture edge contact, because compressive stresses take place around the roller.
Goal of Research: to investigate the influence of anisotropy on stress-deformed state of base (roller bearings track) considering the appearance of fractures with contacting edges in it.
CONDITIONS, MATERIALS AND METHODS
Let's present stresses in the die area as a sum of two constituents: stresses, corresponding to the die operation (they will be determined); the additional stresses, which change slowly, and which are conditioned with forces applied to the bearing. Let's assume that the known forces applied to each roller during the process of roller exploitation and the additional stresses in the track edge area (the last ones can be determined by the experimental methods).
The track radius is by an order greater than the roller radius and the contact area in its turn is small in comparison with the roller radius. Hence, we use known for these cases
simplification, having substituted the base with the semi-plane •y<0. Let's take account of
f E
mechanical anisotropy, wherefore we admit that elasticity modules E and r in the directions - parallel or perpendicular to the edge - can be different.
To determine the stresses conditioned by the roller (die) operation we will consider the
jr_y <" /7
pressure as the known pressure which according to Hertz has the form at ' c|
p{x)~ ) /a , where x° - is the center of the contact area, P - is the pressure in
the center (maximum). We can emphasize that at the known total force Q, influencing the
p
die, and at the radius of roller R, values P and a - are determined by the well-known formulas [3] for the isotropic material [4] - for the anisotropic one. Particularly, for the isotropic materials we have:
P0=4GalR, a = jQR0(jc + \)lJlnG j j
where + 1)'^ = 3_4,/] G- ¡s the shearing module, v-is the Poisson
coefficient.
With an allowance for friction forces we consider analogically as in [1, 4, 14], that the
t — —kp
common and tangential stresses under the die are connected by dependence ^ r , where k - is a friction coefficient. We investigate fracture generation at the stage of fracture initiation, which dimensions are proportionate with the contact area dimensions. In connection with this, let us assume that the additional stresses in the die area will be
constant and equal to °x ,cry where m - is the known constant,
determined experimentally.
To determine the stresses in the semi-plane with oblique edge fracture we apply the algorithm developed in the works [5-7] and based on the method of the boundary integral equation method.
To realize it is necessary to determine the stresses in the integral semi-plane, that are determined by the pressure applied to the edge. At first we examine the auxiliary problem for
semi-infinite plate to which edge the distributed loading is applied
ay - Txy ~ Y0q(x) ^ pQfgkiy hgpg ^g functjon q(x) is different from zero in the interval
(c,d) X0,Y0_ are ^g Specjfjeci constants. Let us denote complex potentials of Lekhnitsky
for this problem by means of ^p^^where Zl-2 + y} Sl-2 are the performance equation roots. For their determination first we deduce the potentials ^'x°^^o ^>xo) _
which correspond to the concentrated force effect (X,Y), being applied in an arbitrary point( x°'0). On the ground of [4], we have:
1 X + s2Y 1 m . . 1 X + sJ 1
O0(zl5x0) = -—-?--,W0(z2,x0) = - 1
ri ■ 3 - u V 2' U / O-
1 ^2 X0 ^2 ^2 ^^ 0
Then for the case of forces effect q(x) hence by integration we find:
} = lX^ + s^ ^ = +
2ni sl-s2 2m Sj -s2
(1).
where
d
F{z)=\
q(Qdl
z-t (2)
Let us apply this solution to the examined case of pressure from the Hertz problem, for
_ q(x) = 4a2-(x-x)2, Xo= 0,= -kP
0
j. \ y y \ c s 7 u 7" u T
which L L . Then integrating into (2), we obtain:
F(z) = n yj(z-xc f- a2 -(z-xc)
By means of this algorithm let us estimate the stresses near the oblique cracks depending on their distance to the die, angularities, and material mechanical characteristics.
RESULTS AND DISCUSSION
Consideration of Mechanical Anisotropy of Material. Let us consider semi-plane with
straight-line edge crack with the length L _ oblique at an angle of P to the straight-line boundary. Let us assume, that the material of the semi-plane has the modules of
EE v v G
elasticity r, the Poisson coefficients x>" and the shearing module ^. Let's consider
next the case of the weak anisotropy of material, when the modules of elasticity and the
average coefficient of Poisson v are known. Then the Poisson coefficients V*.>"VV were determined in the way that the average coefficient of Poisson was equal to the specified v
(further at the calculations we assumed v = 0>25). Therefore we obtained equation system
G - E
[4] ^ >* ~ t xy y ~ yx x ^ The shearing module was assumed as z\i + v) _ where E- the average Young module.
The calculated relation coefficients of stresses intensity for ^
case P = -№°,Lla = \ ^ xc=0t depending on the relative distance x = xJa are
x —
presented in Fig.la, where 0 the coordinate of crack ingress at the semi-plane edge.
K
Here IJI - are the stresses intensity factors (SIF). The relation of elasticity modules in the
w — E / E
coordinate axis direction x y \s indicated near the curves.
Analogical results for the relative lengths of cracks L/a = 0,5,0,25,2 gre presented in Fig. 1.b of Table 2.
c
The results of the calculations at the crack angle P ^ and the relative length L!a = 1 are presented in Fig. 3.
Figurel - Distribution of relative SIF 11 depending on the distance of crack to the die center at
P - -145" ^ LI a- \ (a) i LI a- 0,5
(b)
Figure 2 - Distribution of relative SIF 11 depending on the distance of crack to the die center at
/3 = -145" LI a - 0,25
(a) and L / a = 2 (b)
0.2 0.15 0.1 0.05 0
-0.05 -0.1 -0.15 -0.2
2
W\l,4 ~~W=1
-2 -1 F„
o
X
Figure 3 - Distribution of relative SIF 11 depending on the distance of crack to the die center at
£ = -160° L!a = 1
1
2
3
Consideration of Additional Stresses. The peculiarity of the specified problem is that the stresses at the crack can not be found separately from the die and the additional stresses with their further summing, as far as the crack edges contact, and that is why this problem is nonlinear. Let us estimate the effect of the additional stresses on SIF. For it let's examine the
oblique crack at an angle ^— —145 gt ^g re|atjve length LI a-1 we admit that the additional stresses in the crack area are constant and at examining the semi-plane we
assume 17^ where P^ - the pressure at the die center, which is maximum. The
calculated relative SIF are presented in Fig.4.a at m = (value m are pointed near
curves) for the isotropic material and in Fig. 4.b - for the mechanically anisotropic material at w = 1,2. We stress that practically in all examined cases the cracks edges contact take place, for which reason the relative SIFs are equal to zero for isotropic material and are small for anisotropic materials.
0.1
0.05 0
-0.05 -0.1 -0.15 -0.2 -0.25
-0.3
-o.K ITS'
/11
0.1
-3 -2
a
-0.1
-0.2
-0.3
\\ '
.......... \\\ i -0.1
Fn at/? = -145° L/a = 1 w = w = l,2,
Figure 4 - Distributions of relative SIF 11 at ^ " ^ ' a = 1, w =1 (a) and " ~ ^ ^ (b) taking
into account the additional stresses
Judging by Figure 4 it is possible to conclude that the additional stresses of compression reduce the SIF value (in 1,33 times in comparison with the case of the additional stresses absence). The additional stresses of tension increase the SIF value significantly. It correspondingly increases the rate of fatigue cracks growth. The material mechanical anisotropy gives the opportunity to reduce SIF additionally. The compressive stresses and the mechanical anisotropy allow reducing the maximum SIF in 1,55 times.
Maximum values of SIF. Let us calculate the maximum values of SIF at different angles depending on the crack length. Just the maximum values of SIF are the basic values in fatigue calculations and strength calculations, at that these SIFs correspond to the most unfavorable rollers distribution relatively the crack. In this regard we do calculations of SIF at different cracks lengths, construct the corresponding graphs, analogical to the ones that are presented in the figures above and further using them we calculate the maximum values of SIF.
The calculation results for cracks angles 90°, 120°, 145°, 160° are presented in Fig.5.a, where these angles are pointed out. Fig. 5.b presents the analogical results for the anisotropic material at w = 1,2.
Apparently the cracks with an angle 1200 to the axis Ox are the most dangerous, except for the short cracks (at L / a <0,4), when the perpendicular cracks grow with the maximum rate.
0.25
c2
/
0.25 0.2 □ .15 0.1 0.05
90°
É-—-12CP
"Í45
160°
i
04 0.6 08 ' 12 1.4 1 3 18 2 2.2 2/ 0.4 0.6 0.8 1 1.2 1 4 1.6 1 8 2 2.2 2.4
a Ua b Ua
F
Figure 5 - Dependence of maximum values of relative SIF 11 from the cracks lengths with an angle
900,1200,1450,1600
, at w=1 (a) and w=1,2 (b)
Friction Forces Consideration. The calculation results are analogical to the data presented in Fig. 5 above, at the consideration of friction forces at the friction coefficient k=0,1 are presented in Fig. 6.a at w=1 and in Fig. 6.a at w=1,2.
120
90°
V.........«5°.................--^
/
i i
o
0 0.5 1 1.5 2 2.Í 0.5 1 1.5 2 2.5
a L7a fa Ua
F
Figure 6 - Dependence of maximum values of relative SIF 11 from the cracks lengths, with an angle
90°, 120°, 145°, 160° at friction consideration
0.25 0.2 0.15 0.1 0.05
12[P i
s/ /
ièîF^ 145°
CONCLUSIONS
The roller mechanical anisotropy at the track edge allows increasing resistance to brittle crack initiation or decreasing growth rate of fatigue cracks. At that, it is necessary to choose such thermo-mechanical methods of processing at which the module of elasticity in circular direction will be larger than in radial direction (perpendicular to the edge).
Additional compressive stresses decrease the stress intensity factor (SIF) and tension stresses increase. In this regard in the track neighborhood it is reasonable to do thermal processing to create residual compressive stresses in circular direction. One of the ways to create mechanical anisotropy and compression stresses is insertion of the previously heated thin-walled anisotropic cylinders.
The calculated maximum values of the stress intensity factor (SIF) give the possibility to calculate the growth rate of fatigue cracks for different lengths and crack angular inclinations. At that, in the early stages the calculations process is simplified, because here there is the crack edges contact and hence KI. Under such conditions a crack will grow along the strait line where it is located (refer to [3]). At long lengths of cracks when the crack edges contact is absent, these SIFs allow calculating critical values of crack lengths.
Analogical results are obtained for other values of the relations of elasticity modules w,
friction factors k, values of parameter m, characterizing the level of additional stresses.
These results (which are not provided here) confirm the stated above conclusions on
influence of these parameters on the SIF values.
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