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Direction of crack growth initiation in roller contact: strain energy density criterion
Theoretical and Applied Fracture Mechanics 15 (1991) 191-198 Elsevier
191
Direction of crack growth initiation in roller contact: strain energy density criterion R. Wojcik Institute of Aeronautical Engineering and Applied Mechanics, Warsaw University of Technology, Warsaw, Poland
Small defects or cracks near the surface of roller contact could spread and lead to failure at large. Their growth behavior depends on the rolling load, size and orientation of the initial defects, and material property in addition to friction at the contacting surfaces. Stress intensity factors K1 and K2 are obtained for three different crack types near the surface between the roller and contacting solid. Various possible directions of crack growth initiation are obtained as the different roller loads are moved relative to the crack. The results are indicative of railway failure observed in service and are helpful to future studies on subcritical and/or critical crack growth.
1. Introduction A basic problem in determining the strength degradation of machine elements is rolling contact where cracks could be initiated near the surface on account of inhomogeneities arising from the material microstructure. The orientation and size of these cracks are dependent on the load condition and material. Once cracks are nucleated, the local stresses are amplified and could result in additional damage. Linear elastic fracture mechanics theory has advocated the use of the stress intensity factor quantity for determining the onset of rapid fracture when it reaches a critical value, say Ka¢, that is considered to be characteristic of the material. However, the onset of crack growth does not always correspond to rapid fracture; subcritical crack growth is generally the rule rather than the exception. Such a consideration should be kept in mind when selecting failure criterion. Another practical aspect of crack initiation analysis is the allowance for the crack to alter its direction of growth because the loads may not be aligned symmetrically about the crack at all times. This situation prevails in roller contact where l o a d / c r a c k symmetry is never preserved. For inplane extension, at least two stress intensity factors would prevail which could be identified separately as contributions arising from symmetrical and skew-symmetrical loading if the problem is linear; they would be written as K t and K 2. Hence, the condition for crack initiation would 0167-8442/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
depend on both K a and K 2 and the crack may turn and grow in a non-self-similar manner. The classical energy release rate concept that invokes self-similar crack growth cannot be readily applied. To this end, Sih first proposed to use the strain energy density factor S for predicting the direction of crack initiation. The critical value of S or S¢ is also indicative of the material fracture toughness as in the case of K1¢. It, however, possesses the additional feature of being able to account for subcritical crack growth by assuming that the quotient Sj/rj remains constant during crack growth where j = 1, 2, etc., represent the number of crack growth segment with rj being the j t h segment of crack extension. Furthermore, since S / r is identified with the energy density function d W / d V , the criterion can be applied even in the presence of plastic deformation. It is for the aforementioned reasons that the energy density concept is chosen for finding the crack initiation direction in rolling contact. Solutions for three different crack geometries are obtained by application of the finite element method. Assuming linear elastic behavior, it suffices to use S that is expressible as a quadratic function of K 1 and K 2. Results for possible crack growth directions are found and displayed graphically.
2. Roller contact of body with cracks Depicted in Fig. 1 is a schematic of the roller pressed on a block of elastic material with Young's
192
R. Wojcik / Direction of crack growth initiation in roller contact
Yl x 0
"~ ~
Crack
Fig. 1. Schematic of roller contact on elastic medium.
l'
il
3. Method of solution
(a) Vertical crack
/.."2,~,~.'=oOl a../ -< Ib) S l a n t e d c r a c k
material is given in Fig. 2(a). Figure 2(b) corresponds to an edge crack of length a inclined at an angle a = 20 o with respect to the contact surface while Fig. 2(c) shows an embedded crack parallel to the contact surface and has the same length as the contact. The situations in Figs. 2(b) and 2(c) have been commonly observed in railway failure and their crack initiation a n d / o r growth behavior are not completely understood. In what follows, the strain energy density criterion will be applied to predict the crack initiation behavior of these cracks for ratios of a/b = 0.7, 1.0 and 1.2 while the magnitude of the normal load R will also be altered.
I
Use is made of the two-dimensional finite element procedure for linear elastic deformation, the method of which can be found in [1]. Consideration is first given to the application of unit forces on the contact nodes between the roller and elastic body. The stiffness matrices in the x- and y-direction are then constructed. Solved are the stresses and displacements arising from roller contact.
3.1. Basic equations Let F and R denote, respectively, the force applied to the roller in the x- and y-direction such that F=[ll{T},
~c) Horizontal crack Fig. 2. Configurations of subsurface crack near roller contact.
modulus E and Poisson's ratio v. The vertical load has a magnitude R while the length of contact is 2b. An inclined crack of length a extends from the contact surface and is oriented at an angle a. A plane strain state of stress is assumed to occur in the xy-plane. Coulomb's friction 1 is assumed for determining the friction forces between the roller and the elastic body. Three types of initial crack configurations are assumed to exist under the roller; they are shown in Figs. 2(a) to 2(c) inclusive. A surface crack of length a penetrates at a right angle into the 1 The same applies to the crack surfaces.
R=[ll{N}
(1)
where T and N represent the tangential and normal contact force between the roller and elastic medium. Only R prevails in the work to follow. The governing equations for the contact problem with friction can be expressed in terms of relative displacements of the roller with reference to the contacting body and of the crack surfaces. If x ° and y0 stand for the initial distances between a particular point on the roller and a corresponding point on the body that would come in contact, then displacement compatibility in the y- and x-direction as indicated in Fig. 1 would require that [A1]{N ) + [ A z ] { T }
+[A3]{P}
+ [ A 4 ] ( Q } + { A y } ~---{ y O } [A,]{N}
+ [ A 6 ] { T } "Jr" [ A 7 ] { P )
+ [ A s ] { Q } + { A x } = {x °}
(2)
193
R Wojcik / Direction of crack growth initiation in roller contact
In eqs. (2), Ax and Ay correspond to the displacement of the roller relative to the contacting body in the x- and y-directions, respectively. The normal and tangential forces resulting from the contact of the crack surfaces are, respectively, P and Q. Similarly, two additional expressions can be written for the displacements u,0 and u 0t of the crack surfaces in the normal and tangential direction; they take the forms
Fig. 3. Vector representation of strain energy density factor.
[A9](N } + [A10]{T} + [ A n ] { P )
at- ["4121{Q)~---{ uO} [A,3I(N} + [A,4]{T} + [ & d ( P }
+ [A16]{Q} = {U 0 }
(3)
The quantities [A1], [A2] .... , [A16] are the stiffness matrices for the roller/body contact system. The foregoing equations are solved numerically for determining the contact area between the roller and elastic body and portion of the crack surfaces that are closed. Their corresponding normal and tangential forces at the contacts are also found. Once the stresses and displacements are known, the stress-intensity factors K 1 and K 2 can be extracted without difficulty.
Here, r and 9 are local polar coordinates where 0 is measured from the plane bisecting the crack surfaces. A plane strain condition is assumed in eq. (6).
3.3. Strain energy density factor According to the strain energy density criterion [2,3], crack would initiate in a radial direction along which the strain energy density function d W / d V attains a relative minimum. For a fixed distance, say to, from the crack front, it suffices to consider the variation of the factor S with the angle O as in Fig. 3 since
3. 2. Stress-intensity factors
dW S d--V = r
If u~ and uy denote, respectively, the displacement components in the x- and y-direction, then the K 1 and K 2 factors can be determined as
Under plane strain, the energy stored in a unit element can be written as
K2 = [ D ]
[Uy]
(4)
in which the matrix [D] gives the near field r- and 9-dependence: 1 +v r [dll [DI=--V-2f~-~ [d~
du] d=]
(7)
dW l+v[ dV = 2E t°x2+°2-v(°x+°Y)/+2"rx2y]
in which the stresses ox, %, %y and % are given by %= ~
(51
K1
+~
9(
. 9 . 39'~
cos ~ . 1 + sin ~ sm T J
K2 [ . 9 9 39"~ [sln~ cos-~ cos-~- )
In eq. (5), the quantities d~j (i, j -- 1, 2) stand for
O'y
d.=sm . -914(1_v)
~
Sln -~ sin ~ )
- 2 cos 2°] K:
d,2= - c o s O [2(1 - 2 v ) - 2 s i n 20 ] (6) d21 : cosO [2(1 - 2 v ) + 2 sin2-~ ] ' 9 1 4 ( 1 - u ) + 2 cos2-~ ] d22 = sln-~
(8)
Txy ~" ~
. 9[2+ 39~ sin / cos 0 cosT)
K1 [ 9. 9 39 [cos~sln$cos-~-)
K2 9/ + \~/2~r ~c°s~/1
. 9 . 39~ - sm~sm-~--)
R. Wojcik / Direction of crack growth initiation in roller contact
194 K1
C
0
oz=2~ 2-~-7~cos ~ K2 . 0 - 2p~sm~-
6(~
(9)
,¢
'K2
Substituting eqs. (9) into (8) and eliminating d W / d V between eqs. (7) and (8), it is found that
S = a n K 2 + 2a12KIK 2 + a22K~
1 all = 16qTG (3 - 4v - cos 0)(1 + cos 0) a12
=
1 16vG 2 sin O[cos O -
a22 = ~
~ 40
-~ 26 K° (smooth)--~. K1 (fri~t ~/#" - 3
(1
-2
with G being the shear modulus of elasticity. M i n i m u m of S can be found from the conditions that and
02S -~>0
(12)
from which the direction of crack growth 00 can be found. Equation (10) m a y be inserted into the first of eqs. (12) to yield 0 . 8 ( K 2 - K 2) sin 0 + ( K 2 - 3K~) sin 20
+4K1K 2cos20-1.6KaK 2=0,
X/b
-2
[4(1 - v)(1 - cos 0) + (1 + cos 0 ) ( 3 cos 0 - 1)]
0S 0---0=0
0 1 2 Distance
-1
(11)
- 2v)]
o (smooth)
c
(10)
The coefficients a,j (i, j = 1, 2) stand for [2,3]:
(.friction)
o
for0=
:L00 (13)
in which v = 0.3 has been taken. Once the values of K 1 and K 2 are known, eq. (13) can be applied to solve for 8o .
4. Discussion of results The finite element program in [1] is applied to obtain numerical results of the stresses and displacements for a coefficient of friction /~ = 0.4 between the roller and elastic b o d y with u = 0.3. Effect of sliding can be seen f r o m the results for a smooth contact.
4.1. Crack tip stress intensification The intensification of the local stress field for the three different crack configurations in Figs.
/
~,q
,,
V Fig. 4. Variations of stress intensity factors with normalized distance to contact load R = 130 kN for edge crack normal to surface with a / b = 0.7.
2(a) to 2(c) can be best reflected by the stress intensity factors K 1 and K 2. Displayed in Fig. 4 are variations of Kj ( j = 1, 2) with the normalized distance x / b as a n o r m a l rolling load of R = 130 k N moves f r o m the left h a n d side of the crack corresponding to negative x to the right h a n d side of the crack corresponding to positive x. The crack with a = 0.7b in Fig. 2(a) terminates on the surface at x = 0. N o t e that the in-plane shear m o d e K 2 dominates 2 whereas K1 is relatively small. The curves for contact with and without friction are nearly the same. W h e n the crack with a = 1.2b is tilted at an angle a = 20 ° as in Fig. 2(b), the idealization of a s m o o t h contact would have led to an unrealistically estimate of K2, Fig. 5. As the load R = 260 k N passes over the crack, the shear action increases as evidenced by a rise in K 2 while K1 diminishes as a result of normal compression. Referring to Fig. 2(c) where the crack of length 2b is e m b e d d e d at a depth 0.5b under the surface, b o t h K 1 and K 2 fluctuated as the load R = 130 k N traverses across the crack. Fig-
2 Such an influence has been observed experimentally [4].
R. Wojcik / Direction of crack growth initiation in roller contact
ure 6 shows that the friction at the contact has a significant influence on Kt and K 2. The assumption of a smooth contact between the crack surfaces tends to overestimate the shear mode stress intensity factor K 2. Because of the presence of both K~ and K 2, the critical crack initiation load must involve a combination of K~ and K 2 such as the expression in eq. (10). The minimum S or Smin can be obtained by inserting the solution of eq. (13) for 0 = 00 into eq. (10), i.e., Smin = S ( O o )
195
E
Ko2 (smooth)-,
g. ~ 30~-
20 /--" KO(smooth) K2(friot ion)-/--7 \ ~ | / / [ \ 'c~ | l r K' (friction} /~J . . . . _~.~ 10I ~ N N ~
(14)
The initiation of subcritical crack growth corresponds to dW Sm,n = r0 ( --d--~-)c
(15)
Fig. 6. Variations of stress intensity factors with normalized distance to contact load R = 130 kN for embedded crack under surface with a / b = 1.0.
J where r0 is the distance of the nearest neighboring element next to the crack front and (dW/dV)c is the area under the true stress and true strain curve [5]. Onset of rapid fracture occurs when
1201 /'~ Ke (smooth) 100
Smin ~ Sc =
E 80
g.
(1 + ~,)(1 - 2v)K12c 2~E
(16)
I
4.2. Direction of crack initiation
°/
-I'-,~ K2 (friction) x I ! I I
o
t
-3
-2
~l
-1 -20 -40
Distan
K~ (smooth) K 1 (friction)
.~
Fig. 5. Variations of stress intensity factors with normalized distance to contact load R = 260 kN for crack inclined to surface at a= 20 ° with a/b =1.2.
Equation (13) may be solved for the direction of crack growth from the known values of K 1 and K 2 in Figs. 4 to 6 inclusive. Three sets of solutions are obtained; they correspond to the crack configurations in Figs. 2(a) to 2(c). Crack growth angle 00 is measured from the plane bisecting the upper and lower crack surface such that anti-clockwise is positive and clockwise is negative. Edge crack ( a / b = 0.7). Plotted in Fig. 7 are the variations of the angle O0 with x/b. The limiting values of 0o are _+82.3 °. When the crack tip first experiences the roller load, only K 2 prevails and 0o = 82.3 ° for ~, = 0.3. The fracture angle 00 decreases due to the presence of K t until it vanishes altogether when the roller is directly on top of the edge crack. This corresponds to a pure opening mode with K 2 = 0 and K~ is maximum. As the roller travels over the positive range of x, the opposite situation occurs where the fracture angle 00 now becomes negative and reaches a
R. Wojcik / Direction of crack growth initiation in roller contact
196
_ ___
~
Inclined crack (a/b = 1.2). The crack initiation behavior for the inclined surface crack in Fig. 2(b) is quite different. Figure 8 shows that the fracture angle 00 is negative at all time. That is, the crack always grows into the solid away from the free surface. A slight fluctuation in 00 is detected in the range - 3 0 ° to - 6 0 ° as the roller first approaches the crack. After that, the crack tends to grow in a direction almost normal to the original plane at 00 = - 8 2 . 3 ° where K l vanishes while K 2 attains a m a x i m u m for 0.50 <~x/b<~ 2.33 with friction taken into account. Again, the fracture angle is not affected appreciably by friction. Embedded crack (a/b = 1.0). The case of an embedded crack with its length equal to that of the contact region as shown in Fig. 2(c) is also analyzed. Since the results at the two crack tips are not the same, they will be presented separately in Figs. 9 and 10. As the roller approaches the left crack tip, Fig. 9 shows that the crack tends to initiate towards the roller, the direction of which turned almost normal to the free surface for a small advancement of the roller. Friction effect became significant when the roller passed beyond the left crack tip where crack growth direction starts to deviate away from the contact surface until 00 approached zero at x / b = 7/3. The intensification of the stresses, however, is small be-
R
.80 °
Crack
~I "40° .20 °
.X
Distance
x/b
-2o0I -4d
/-- S moot h
-60°I~ I \< 80°I"
=/_Friction
"~"~
- - "
Fig. 7. F r a c t u r e initiation angle versus n o r m a l i z e d d i s t a n c e to c o n t a c t load R = 130 k N for e d g e c r a c k n o r m a l to s u r f a c e with
a / b = 0.7.
maximum value of - 8 2 . 3 ° when K] ~ 0 and g 2 reaches maximum. Friction has no significance on the direction of crack growth.
R
CD ° 4(]
rack
2d I I
-2
I
-1
0
-x
I
I
I
1
2
3
Distance
xib
-24
Friction t
-8
. _
J
Smooth
_..___
Fig. 8. F r a c t u r e initiation angle versus n o r m a l i z e d d i s t a n c e to c o n t a c t load R = 260 k N for crack inclined to surface at a = 20 o with a / b =1.2.
197
R. Wojcik / Direction of crack growth initiation in roller contact R
~
Crack
(I~ 4C (9
,~ 2c; I
I
-2
f
-1
0
~
2 i\ I
I
4
IL
I
Distance x/b
-2(
'ric'°ni / - 1 i
-4(
L
I
3
-6t
', ~Smooth
./
_
-8d
Fig. 9. Fracture initiation angle versus normalized distance to contact load R =130 kN for left tip of subsurface crack with a / b = 1.0.
cause the l o a d has m o v e d f a r t h e r a w a y f r o m the crack. V a r i a t i o n s of O0 with x / b at the right c r a c k tip are given in Fig. 10. A f t e r the initial oscillation, 00 stabilized at - 82.3 ° for 1 ~< x / b ~< 3.833
w i t h / t = 0.4. T h e f r a c t u r e angle O0 returns to zero as the roller m o v e s a w a y f r o m the crack a n d the m a g n i t u d e o f the stress i n t e n s i t y factors decrease accordingly.
R
:ra:k
o 4( ¢P O~
~ 2d I I
I
-2
-1
0
-26
,~x
I
I
!
1
2
3
I
I
Distance x/b
I I
mooth I
I
Frict ion---~l
~j
I
-6d -8d
Fig. 10. Fracture initiation angle versus normalized distance to contact load R =130 kN for fight tip of subsurface crack with a / b = 1.0.
198
R. Wojcik / Direction of crack growth initiation in roller contact
5. Conclusions The strain energy density criterion has been applied to predict the direction of crack initiation for three different orientations of cracks as the load exerted by roller contact is varied relative to the crack location. Various possible fracture angles are obtained as the load intensity and crack size a n d / o r orientation are changed. Prediction on the critical roller load and location that would initiate crack growth is a more complicated problem as it would require information on determining the critical energy state in relation to the fracture toughness of the material. This m a y be determined by a plot of Smin in eq. (14) with x / b . If the m a x i m u m of Smin or Sn~ max exceeds ro(dW/dV)c in eq. (15), then subcritical crack growth initiation could be assumed to occur. The crack can thus grow in segments until Smmax reaches Sc in eq. (16) which corresponds to global instability. The foregoing scheme could be applied to explain the cracking of railway. Residual stress effects due to repeated application of rolling contact loads can also be included using the incremental
theory of plasticity where the strain energy density remains valid. A brief discussion on how this is applied to the contact problem can be found in [6]. Such an effort, however, is left for future research.
References [1] M. Olzak, J. Stupnicki and R. Wojcik, Crack tip propagation due to the contact stresses, Proc. of 5th Int. Congress on Tribology, EUROTRIB'89, Hensinki (1989) 289-294. [2] G.C. Sih, Strain-energy-density factor applied to mixed mode crack problems, Int. J. Fract. 9 (1974) 305-323. [3] G.C. Sih, "Introductory Chapters--Mechanics of Fracture", Vol. I to Vol. VII, ed. G.C. Sih (Martinus Nijhoff Publishers: The Netherlands, 1972-1982). [4] R. Wojcik, "The determination of stress intensity factors using speckle photography method", Proc. of 4th Autumn Seminar for Holographic Interferometry, GDR (1985) 210218. [5] L.F. Gillemot, Criterion of crack initiation and spreading, J. Eng. Fract. Mech. 8 (1976) 239-253. [6} G.C. Sih and D.Y. Tzou, Discussion on subsurface crack propagation due to surface traction in sliding wear, Z Appl. Mech. 52 (1985) 237-239.
191
Direction of crack growth initiation in roller contact: strain energy density criterion R. Wojcik Institute of Aeronautical Engineering and Applied Mechanics, Warsaw University of Technology, Warsaw, Poland
Small defects or cracks near the surface of roller contact could spread and lead to failure at large. Their growth behavior depends on the rolling load, size and orientation of the initial defects, and material property in addition to friction at the contacting surfaces. Stress intensity factors K1 and K2 are obtained for three different crack types near the surface between the roller and contacting solid. Various possible directions of crack growth initiation are obtained as the different roller loads are moved relative to the crack. The results are indicative of railway failure observed in service and are helpful to future studies on subcritical and/or critical crack growth.
1. Introduction A basic problem in determining the strength degradation of machine elements is rolling contact where cracks could be initiated near the surface on account of inhomogeneities arising from the material microstructure. The orientation and size of these cracks are dependent on the load condition and material. Once cracks are nucleated, the local stresses are amplified and could result in additional damage. Linear elastic fracture mechanics theory has advocated the use of the stress intensity factor quantity for determining the onset of rapid fracture when it reaches a critical value, say Ka¢, that is considered to be characteristic of the material. However, the onset of crack growth does not always correspond to rapid fracture; subcritical crack growth is generally the rule rather than the exception. Such a consideration should be kept in mind when selecting failure criterion. Another practical aspect of crack initiation analysis is the allowance for the crack to alter its direction of growth because the loads may not be aligned symmetrically about the crack at all times. This situation prevails in roller contact where l o a d / c r a c k symmetry is never preserved. For inplane extension, at least two stress intensity factors would prevail which could be identified separately as contributions arising from symmetrical and skew-symmetrical loading if the problem is linear; they would be written as K t and K 2. Hence, the condition for crack initiation would 0167-8442/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
depend on both K a and K 2 and the crack may turn and grow in a non-self-similar manner. The classical energy release rate concept that invokes self-similar crack growth cannot be readily applied. To this end, Sih first proposed to use the strain energy density factor S for predicting the direction of crack initiation. The critical value of S or S¢ is also indicative of the material fracture toughness as in the case of K1¢. It, however, possesses the additional feature of being able to account for subcritical crack growth by assuming that the quotient Sj/rj remains constant during crack growth where j = 1, 2, etc., represent the number of crack growth segment with rj being the j t h segment of crack extension. Furthermore, since S / r is identified with the energy density function d W / d V , the criterion can be applied even in the presence of plastic deformation. It is for the aforementioned reasons that the energy density concept is chosen for finding the crack initiation direction in rolling contact. Solutions for three different crack geometries are obtained by application of the finite element method. Assuming linear elastic behavior, it suffices to use S that is expressible as a quadratic function of K 1 and K 2. Results for possible crack growth directions are found and displayed graphically.
2. Roller contact of body with cracks Depicted in Fig. 1 is a schematic of the roller pressed on a block of elastic material with Young's
192
R. Wojcik / Direction of crack growth initiation in roller contact
Yl x 0
"~ ~
Crack
Fig. 1. Schematic of roller contact on elastic medium.
l'
il
3. Method of solution
(a) Vertical crack
/.."2,~,~.'=oOl a../ -< Ib) S l a n t e d c r a c k
material is given in Fig. 2(a). Figure 2(b) corresponds to an edge crack of length a inclined at an angle a = 20 o with respect to the contact surface while Fig. 2(c) shows an embedded crack parallel to the contact surface and has the same length as the contact. The situations in Figs. 2(b) and 2(c) have been commonly observed in railway failure and their crack initiation a n d / o r growth behavior are not completely understood. In what follows, the strain energy density criterion will be applied to predict the crack initiation behavior of these cracks for ratios of a/b = 0.7, 1.0 and 1.2 while the magnitude of the normal load R will also be altered.
I
Use is made of the two-dimensional finite element procedure for linear elastic deformation, the method of which can be found in [1]. Consideration is first given to the application of unit forces on the contact nodes between the roller and elastic body. The stiffness matrices in the x- and y-direction are then constructed. Solved are the stresses and displacements arising from roller contact.
3.1. Basic equations Let F and R denote, respectively, the force applied to the roller in the x- and y-direction such that F=[ll{T},
~c) Horizontal crack Fig. 2. Configurations of subsurface crack near roller contact.
modulus E and Poisson's ratio v. The vertical load has a magnitude R while the length of contact is 2b. An inclined crack of length a extends from the contact surface and is oriented at an angle a. A plane strain state of stress is assumed to occur in the xy-plane. Coulomb's friction 1 is assumed for determining the friction forces between the roller and the elastic body. Three types of initial crack configurations are assumed to exist under the roller; they are shown in Figs. 2(a) to 2(c) inclusive. A surface crack of length a penetrates at a right angle into the 1 The same applies to the crack surfaces.
R=[ll{N}
(1)
where T and N represent the tangential and normal contact force between the roller and elastic medium. Only R prevails in the work to follow. The governing equations for the contact problem with friction can be expressed in terms of relative displacements of the roller with reference to the contacting body and of the crack surfaces. If x ° and y0 stand for the initial distances between a particular point on the roller and a corresponding point on the body that would come in contact, then displacement compatibility in the y- and x-direction as indicated in Fig. 1 would require that [A1]{N ) + [ A z ] { T }
+[A3]{P}
+ [ A 4 ] ( Q } + { A y } ~---{ y O } [A,]{N}
+ [ A 6 ] { T } "Jr" [ A 7 ] { P )
+ [ A s ] { Q } + { A x } = {x °}
(2)
193
R Wojcik / Direction of crack growth initiation in roller contact
In eqs. (2), Ax and Ay correspond to the displacement of the roller relative to the contacting body in the x- and y-directions, respectively. The normal and tangential forces resulting from the contact of the crack surfaces are, respectively, P and Q. Similarly, two additional expressions can be written for the displacements u,0 and u 0t of the crack surfaces in the normal and tangential direction; they take the forms
Fig. 3. Vector representation of strain energy density factor.
[A9](N } + [A10]{T} + [ A n ] { P )
at- ["4121{Q)~---{ uO} [A,3I(N} + [A,4]{T} + [ & d ( P }
+ [A16]{Q} = {U 0 }
(3)
The quantities [A1], [A2] .... , [A16] are the stiffness matrices for the roller/body contact system. The foregoing equations are solved numerically for determining the contact area between the roller and elastic body and portion of the crack surfaces that are closed. Their corresponding normal and tangential forces at the contacts are also found. Once the stresses and displacements are known, the stress-intensity factors K 1 and K 2 can be extracted without difficulty.
Here, r and 9 are local polar coordinates where 0 is measured from the plane bisecting the crack surfaces. A plane strain condition is assumed in eq. (6).
3.3. Strain energy density factor According to the strain energy density criterion [2,3], crack would initiate in a radial direction along which the strain energy density function d W / d V attains a relative minimum. For a fixed distance, say to, from the crack front, it suffices to consider the variation of the factor S with the angle O as in Fig. 3 since
3. 2. Stress-intensity factors
dW S d--V = r
If u~ and uy denote, respectively, the displacement components in the x- and y-direction, then the K 1 and K 2 factors can be determined as
Under plane strain, the energy stored in a unit element can be written as
K2 = [ D ]
[Uy]
(4)
in which the matrix [D] gives the near field r- and 9-dependence: 1 +v r [dll [DI=--V-2f~-~ [d~
du] d=]
(7)
dW l+v[ dV = 2E t°x2+°2-v(°x+°Y)/+2"rx2y]
in which the stresses ox, %, %y and % are given by %= ~
(51
K1
+~
9(
. 9 . 39'~
cos ~ . 1 + sin ~ sm T J
K2 [ . 9 9 39"~ [sln~ cos-~ cos-~- )
In eq. (5), the quantities d~j (i, j -- 1, 2) stand for
O'y
d.=sm . -914(1_v)
~
Sln -~ sin ~ )
- 2 cos 2°] K:
d,2= - c o s O [2(1 - 2 v ) - 2 s i n 20 ] (6) d21 : cosO [2(1 - 2 v ) + 2 sin2-~ ] ' 9 1 4 ( 1 - u ) + 2 cos2-~ ] d22 = sln-~
(8)
Txy ~" ~
. 9[2+ 39~ sin / cos 0 cosT)
K1 [ 9. 9 39 [cos~sln$cos-~-)
K2 9/ + \~/2~r ~c°s~/1
. 9 . 39~ - sm~sm-~--)
R. Wojcik / Direction of crack growth initiation in roller contact
194 K1
C
0
oz=2~ 2-~-7~cos ~ K2 . 0 - 2p~sm~-
6(~
(9)
,¢
'K2
Substituting eqs. (9) into (8) and eliminating d W / d V between eqs. (7) and (8), it is found that
S = a n K 2 + 2a12KIK 2 + a22K~
1 all = 16qTG (3 - 4v - cos 0)(1 + cos 0) a12
=
1 16vG 2 sin O[cos O -
a22 = ~
~ 40
-~ 26 K° (smooth)--~. K1 (fri~t ~/#" - 3
(1
-2
with G being the shear modulus of elasticity. M i n i m u m of S can be found from the conditions that and
02S -~>0
(12)
from which the direction of crack growth 00 can be found. Equation (10) m a y be inserted into the first of eqs. (12) to yield 0 . 8 ( K 2 - K 2) sin 0 + ( K 2 - 3K~) sin 20
+4K1K 2cos20-1.6KaK 2=0,
X/b
-2
[4(1 - v)(1 - cos 0) + (1 + cos 0 ) ( 3 cos 0 - 1)]
0S 0---0=0
0 1 2 Distance
-1
(11)
- 2v)]
o (smooth)
c
(10)
The coefficients a,j (i, j = 1, 2) stand for [2,3]:
(.friction)
o
for0=
:L00 (13)
in which v = 0.3 has been taken. Once the values of K 1 and K 2 are known, eq. (13) can be applied to solve for 8o .
4. Discussion of results The finite element program in [1] is applied to obtain numerical results of the stresses and displacements for a coefficient of friction /~ = 0.4 between the roller and elastic b o d y with u = 0.3. Effect of sliding can be seen f r o m the results for a smooth contact.
4.1. Crack tip stress intensification The intensification of the local stress field for the three different crack configurations in Figs.
/
~,q
,,
V Fig. 4. Variations of stress intensity factors with normalized distance to contact load R = 130 kN for edge crack normal to surface with a / b = 0.7.
2(a) to 2(c) can be best reflected by the stress intensity factors K 1 and K 2. Displayed in Fig. 4 are variations of Kj ( j = 1, 2) with the normalized distance x / b as a n o r m a l rolling load of R = 130 k N moves f r o m the left h a n d side of the crack corresponding to negative x to the right h a n d side of the crack corresponding to positive x. The crack with a = 0.7b in Fig. 2(a) terminates on the surface at x = 0. N o t e that the in-plane shear m o d e K 2 dominates 2 whereas K1 is relatively small. The curves for contact with and without friction are nearly the same. W h e n the crack with a = 1.2b is tilted at an angle a = 20 ° as in Fig. 2(b), the idealization of a s m o o t h contact would have led to an unrealistically estimate of K2, Fig. 5. As the load R = 260 k N passes over the crack, the shear action increases as evidenced by a rise in K 2 while K1 diminishes as a result of normal compression. Referring to Fig. 2(c) where the crack of length 2b is e m b e d d e d at a depth 0.5b under the surface, b o t h K 1 and K 2 fluctuated as the load R = 130 k N traverses across the crack. Fig-
2 Such an influence has been observed experimentally [4].
R. Wojcik / Direction of crack growth initiation in roller contact
ure 6 shows that the friction at the contact has a significant influence on Kt and K 2. The assumption of a smooth contact between the crack surfaces tends to overestimate the shear mode stress intensity factor K 2. Because of the presence of both K~ and K 2, the critical crack initiation load must involve a combination of K~ and K 2 such as the expression in eq. (10). The minimum S or Smin can be obtained by inserting the solution of eq. (13) for 0 = 00 into eq. (10), i.e., Smin = S ( O o )
195
E
Ko2 (smooth)-,
g. ~ 30~-
20 /--" KO(smooth) K2(friot ion)-/--7 \ ~ | / / [ \ 'c~ | l r K' (friction} /~J . . . . _~.~ 10I ~ N N ~
(14)
The initiation of subcritical crack growth corresponds to dW Sm,n = r0 ( --d--~-)c
(15)
Fig. 6. Variations of stress intensity factors with normalized distance to contact load R = 130 kN for embedded crack under surface with a / b = 1.0.
J where r0 is the distance of the nearest neighboring element next to the crack front and (dW/dV)c is the area under the true stress and true strain curve [5]. Onset of rapid fracture occurs when
1201 /'~ Ke (smooth) 100
Smin ~ Sc =
E 80
g.
(1 + ~,)(1 - 2v)K12c 2~E
(16)
I
4.2. Direction of crack initiation
°/
-I'-,~ K2 (friction) x I ! I I
o
t
-3
-2
~l
-1 -20 -40
Distan
K~ (smooth) K 1 (friction)
.~
Fig. 5. Variations of stress intensity factors with normalized distance to contact load R = 260 kN for crack inclined to surface at a= 20 ° with a/b =1.2.
Equation (13) may be solved for the direction of crack growth from the known values of K 1 and K 2 in Figs. 4 to 6 inclusive. Three sets of solutions are obtained; they correspond to the crack configurations in Figs. 2(a) to 2(c). Crack growth angle 00 is measured from the plane bisecting the upper and lower crack surface such that anti-clockwise is positive and clockwise is negative. Edge crack ( a / b = 0.7). Plotted in Fig. 7 are the variations of the angle O0 with x/b. The limiting values of 0o are _+82.3 °. When the crack tip first experiences the roller load, only K 2 prevails and 0o = 82.3 ° for ~, = 0.3. The fracture angle 00 decreases due to the presence of K t until it vanishes altogether when the roller is directly on top of the edge crack. This corresponds to a pure opening mode with K 2 = 0 and K~ is maximum. As the roller travels over the positive range of x, the opposite situation occurs where the fracture angle 00 now becomes negative and reaches a
R. Wojcik / Direction of crack growth initiation in roller contact
196
_ ___
~
Inclined crack (a/b = 1.2). The crack initiation behavior for the inclined surface crack in Fig. 2(b) is quite different. Figure 8 shows that the fracture angle 00 is negative at all time. That is, the crack always grows into the solid away from the free surface. A slight fluctuation in 00 is detected in the range - 3 0 ° to - 6 0 ° as the roller first approaches the crack. After that, the crack tends to grow in a direction almost normal to the original plane at 00 = - 8 2 . 3 ° where K l vanishes while K 2 attains a m a x i m u m for 0.50 <~x/b<~ 2.33 with friction taken into account. Again, the fracture angle is not affected appreciably by friction. Embedded crack (a/b = 1.0). The case of an embedded crack with its length equal to that of the contact region as shown in Fig. 2(c) is also analyzed. Since the results at the two crack tips are not the same, they will be presented separately in Figs. 9 and 10. As the roller approaches the left crack tip, Fig. 9 shows that the crack tends to initiate towards the roller, the direction of which turned almost normal to the free surface for a small advancement of the roller. Friction effect became significant when the roller passed beyond the left crack tip where crack growth direction starts to deviate away from the contact surface until 00 approached zero at x / b = 7/3. The intensification of the stresses, however, is small be-
R
.80 °
Crack
~I "40° .20 °
.X
Distance
x/b
-2o0I -4d
/-- S moot h
-60°I~ I \< 80°I"
=/_Friction
"~"~
- - "
Fig. 7. F r a c t u r e initiation angle versus n o r m a l i z e d d i s t a n c e to c o n t a c t load R = 130 k N for e d g e c r a c k n o r m a l to s u r f a c e with
a / b = 0.7.
maximum value of - 8 2 . 3 ° when K] ~ 0 and g 2 reaches maximum. Friction has no significance on the direction of crack growth.
R
CD ° 4(]
rack
2d I I
-2
I
-1
0
-x
I
I
I
1
2
3
Distance
xib
-24
Friction t
-8
. _
J
Smooth
_..___
Fig. 8. F r a c t u r e initiation angle versus n o r m a l i z e d d i s t a n c e to c o n t a c t load R = 260 k N for crack inclined to surface at a = 20 o with a / b =1.2.
197
R. Wojcik / Direction of crack growth initiation in roller contact R
~
Crack
(I~ 4C (9
,~ 2c; I
I
-2
f
-1
0
~
2 i\ I
I
4
IL
I
Distance x/b
-2(
'ric'°ni / - 1 i
-4(
L
I
3
-6t
', ~Smooth
./
_
-8d
Fig. 9. Fracture initiation angle versus normalized distance to contact load R =130 kN for left tip of subsurface crack with a / b = 1.0.
cause the l o a d has m o v e d f a r t h e r a w a y f r o m the crack. V a r i a t i o n s of O0 with x / b at the right c r a c k tip are given in Fig. 10. A f t e r the initial oscillation, 00 stabilized at - 82.3 ° for 1 ~< x / b ~< 3.833
w i t h / t = 0.4. T h e f r a c t u r e angle O0 returns to zero as the roller m o v e s a w a y f r o m the crack a n d the m a g n i t u d e o f the stress i n t e n s i t y factors decrease accordingly.
R
:ra:k
o 4( ¢P O~
~ 2d I I
I
-2
-1
0
-26
,~x
I
I
!
1
2
3
I
I
Distance x/b
I I
mooth I
I
Frict ion---~l
~j
I
-6d -8d
Fig. 10. Fracture initiation angle versus normalized distance to contact load R =130 kN for fight tip of subsurface crack with a / b = 1.0.
198
R. Wojcik / Direction of crack growth initiation in roller contact
5. Conclusions The strain energy density criterion has been applied to predict the direction of crack initiation for three different orientations of cracks as the load exerted by roller contact is varied relative to the crack location. Various possible fracture angles are obtained as the load intensity and crack size a n d / o r orientation are changed. Prediction on the critical roller load and location that would initiate crack growth is a more complicated problem as it would require information on determining the critical energy state in relation to the fracture toughness of the material. This m a y be determined by a plot of Smin in eq. (14) with x / b . If the m a x i m u m of Smin or Sn~ max exceeds ro(dW/dV)c in eq. (15), then subcritical crack growth initiation could be assumed to occur. The crack can thus grow in segments until Smmax reaches Sc in eq. (16) which corresponds to global instability. The foregoing scheme could be applied to explain the cracking of railway. Residual stress effects due to repeated application of rolling contact loads can also be included using the incremental
theory of plasticity where the strain energy density remains valid. A brief discussion on how this is applied to the contact problem can be found in [6]. Such an effort, however, is left for future research.
References [1] M. Olzak, J. Stupnicki and R. Wojcik, Crack tip propagation due to the contact stresses, Proc. of 5th Int. Congress on Tribology, EUROTRIB'89, Hensinki (1989) 289-294. [2] G.C. Sih, Strain-energy-density factor applied to mixed mode crack problems, Int. J. Fract. 9 (1974) 305-323. [3] G.C. Sih, "Introductory Chapters--Mechanics of Fracture", Vol. I to Vol. VII, ed. G.C. Sih (Martinus Nijhoff Publishers: The Netherlands, 1972-1982). [4] R. Wojcik, "The determination of stress intensity factors using speckle photography method", Proc. of 4th Autumn Seminar for Holographic Interferometry, GDR (1985) 210218. [5] L.F. Gillemot, Criterion of crack initiation and spreading, J. Eng. Fract. Mech. 8 (1976) 239-253. [6} G.C. Sih and D.Y. Tzou, Discussion on subsurface crack propagation due to surface traction in sliding wear, Z Appl. Mech. 52 (1985) 237-239.