Wear, 146 (1991) 229-240 229

Investigation of crack propagation during contact by a finite element method

M. Olzak, J. Stupnicki and R. Wojcik Institute of Aircraj? Engineering and Applied Mechanics, Warsaw University of Technology, Nowowiejska 22/24, 00-665 Warsaw (Poland)

(Received April 9, 1990; revised November 9, 1990; accepted December 4, 1990)

Abstract

This paper presents a numerical method, baaed on the finite element approach, for the analysis of stress states in cracks during contact. The problem is considered to be two- dimensional, but the method also allows three-dimensional cases to be solved. The investigations concern the contact between a steel roller and a rectangular steel block with cracks of known geometry, i.e. surface-breaking cracks perpendicular to the contact surface and oblique to it, and cracks parallel to the surface located at various depths. DiEerent configurations of load relative to the crack are considered which correspond to quasi-static rolling of the roller over the surface of the block. Normal and tangential loads are applied. As results, the changes in crack shape during rolling are presented. Diagrams of the tangential forces during contact in the following stages of rolling are shown. Finally, stress-intensity factors KI and K,, are calculated and presented on plots VS. the position of the load axis relative to the crack. In conclusion, existing hypotheses concerning crack propagation during contact are confirmed, and new hypotheses are presented.

1. Introduction

The phenomenon of crack initiation and propagation during contact (for example on railway tracks) has become an important problem in engineering and an object of intensive research in many scientific centers. Many theories have been developed, but the mechanism of the effect is still not fully explained and understood. In this paper, a numerical method for determination of the stress state in an area near the front of a crack existing during contact is presented. The method is used to investigate two-dimensional contact problems between objects of simplified geometry, i.e. a cylindrical roller and a rectangular block with a crack. Calculations were performed for different geometrical contact configurations. Normal and tangential loading were ap- plied. As a result, diagrams of normal and tangential interactions at the contact area and diagrams of crack shapes, as well as the stress intensity factors K1 and Kn were obtained.

2. Existing theories of crack propagation during contact and the aim of research

A review of publications concerning crack during contact, especially “shelling” type, is provided in ref. 1. Existing theories of crack propagation

0043-1648/91/$3.50 0 Elsevier Sequoia/Printed in The Netherlands

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during contact were reviewed by Bower [2-41; his conclusions can be formulated as follows.

(a) Initiation and propagation of a crack according to mode I fracture can be caused by residual stresses principally in the direction perpendicular to the crack edges.

(b) Propagation according to mode II fracture can be a result of shear stresses existing in the region at the front of the crack.

(c) Propagation according to mode I can be caused by fluid under pressure trapped in the crack. This mechanism can occur in two ways: fluid is forced into the crack by a rolling wheel, the so-called “mechanism of hydraulic pressure”, and presses on the faces of the crack causing tensile stresses on its front or fluid penetrates the crack before the contact and a moment later the rolling wheel closes the mouth of the crack and the trapped fluid presses on the faces of crack as in the previous case.

In both cases, stressing of the material near the crack tip can be so great that crack propagation is possible.

The other general conclusion of many authors is that the existence of cracks during contact does not affect the pressure distribution on the contact surface. For this assumption they usually take the hertzian (ellipsoidal) pressure distribution for further considerations. However, previous experi- mental results [5] and the present numerical results indicate that this assumption should be modified. The list of possible hypotheses to explain crack propagation should be lengthened.

The aim of the present research is an analysis of the strain and stress state around the front of a crack during contact. Finally, it is expected that the mechanism of crack propagation in this area can be explained.

3. Investigated model

The contact between a roller and a rectangular block with a crack of known geometry was considered (Fig. 1).

The material of the bodies in contact is elastic and isotropic, and the area around the crack front satisfies the assumptions of linear elastic fracture mechanics. The objects are three-dimensional, but at this stage only the two- dimensional contact problem is solved. The method allows the case of quasi- static rolling of a cylinder on an elastic plane to be analyzed. So far the following problems have been solved:

(a) the frictional problem of contact between the roller and block with a crack perpendicular to the running surface (Fig. 2), taking into consideration the influence of the rolling process on the distribution of tangential forces on the contact area;

(b) the normal problem of contact between roller and block with a crack inclined to the surface at an angle a=BO” (Fig. 3);

(c) the frictional problem of contact between the roller and the block with a crack situated at a depth h =0.5b parallel to the surface (b is half

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X -3

Fig. 1. Investigated model of contact, a roller and block with a crack of known geometry: a=5.7 mm, 8 mm; 20”<ag90”; 3.89bG5.4 mm; d=lOO mm; t=lO mm; h=16 mm; w-48 mm; 130<R< 260 kN; modulus of elasticity, E= 210 GPa.

Fig. 2. Finite element mesh for a surface-breaking crack perpendicular to the surface.

Fig. 3. Finite element mesh for surface-breaking crack oblique to the surface.

Fig. 4. Finite element mesh for a sub-surface crack parallel to the surface.

of the contact width). In case (c), when the boundaries of the loaded area are located above the crack tip, they become so-called Palmgren-Lundberg points (Fig. 4).

4. Method of solution

The method used is based on the finite element approach. Since the elements used for problems of fracture mechanics must be of a special type, the standard procedure of the finite element method was developed with

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such elements, which allow the real singularity type (r)- ‘I2 ahead of the crack tip to be represented. The shape of the special elements used to model the area near the crack tip, and the shape of the standard elements used for the remaining part of the structure are shown in F’ig. 5. The exact description of the elements used in this procedure can be found in ref. 6. The c~culations were done in three main stages.

4.1. State I of the calculations

This stage concerns the determination of the matrix of so-called “influence coefficients”, i.e. the flexibilities of couples of nodes which come into contact (nodes of the roller and block and nodes of the crack faces). The final matrix is a result of the solution of a certain number of problems in which the nodes of the roller, block and crack faces, which can contact each other, are loaded with unit forces in the y and x directions since normal and frictional problems are solved. This is shown in Fig. 6 for a block sector of the roller.

4.2. Stage II of the calculations

This stage concerns the full solution of the contact problem,

and a

which means that the areas of contact and the ~~bu~o~ of normal and tangential forces on these areas are determined. Originally, the solution runs “step by step”, i.e. for an assumed contact between following nodes of block and roller, or crack faces (depending on which comes into contact first), the increments of forces in preceding nodes are determined. The algorithm itself selects the order of nodes coming into contact. The iteration procedures are also applied to the solution of frictional problems where it is necessary to determine the state of load for which a given node on the contact surface or on the crack face starts sliding. For this the iteration process is conducted until the tangential force in the node reaches the limit value (according to

8 0 1

(al @I

Fii. 5. (a) Standard and (b) special elements used for modeling the roller and block.

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Fig. 6. Roller and block subjected to load by unit forces for determination of the matrices of flexibility.

the Coulomb law of friction). The full procedure is given in detail in Appendix A.

4.3. Stage III of the cak-ulations For the calculated nodal normal and tangential forces, the state of

displacements, and strains and stresses are determined. Finally, to describe the stressing of material directly at the crack tip, the stress intensity factors Kl and K,, are calculated. This is done using the analytical solution of Westergaard for the state of displacements and stresses in the area around the crack tip.

5. Results

The most complete solution obtained so far is for case (A) in Section 3; i.e. the contact of roller and block taking into consideration the friction between them and between the crack faces, and rolling which affects the distribution of tangent forces. The coefficient of friction was assumed to be ,LL= 0.4, although the procedure allows different values of CL to be used for the surface of the contact and the crack faces. The value of the total normal force acting on the roller was R= 136 kN, which corresponds to a maximum pressure on the contact area between the roller and the block of I’,,,, = 3400 MPa.

The best way to present the results seems to be to illustrate the changes in crack shape during subsequent stages of rolling. To obtain clear drawings, different scales for the geometry of the element mesh and for the displacement components in the x and @ directions were used. This sequence of changes is shown in Fig. 7. for an assumed direction of motion from left to right. Figure 7(b) shows the symmetrical position of the load with respect to the crack and confnms the result presented earlier [5] of existing tensile stresses in the crack tip area in such a configuration. At the same time, relative vertical displacements of nodes in the crack are visible, which is a result

cc> (4 Fig. 7. Deformation of a perpendicular crack during subsequent stages of rolling (direction of motion from left to right).

P[ kN/ rn]

Fii. 8. Shape of a perpendicular crack under symmetrical load (case without friction).

Fig. 9. Plot of contact normal forces corresponding to Fig. 7(a).

of friction (the components of the vector {p}). Comparison with Fig. 8 which presents analogous results but for the case without friction confirms this conclusion. The other important result is the division of the contact area into two separate surfaces at the moment when the opposite (right) face of the crack comes into contact (Fig. 7(a)). It is obvious that it must affect the pressure distribution during contact. The diagram of normal forces during contact is shown in Fig. 9. It can be seen that it differs from the hertzian distribution.

The diagrams presented in Figs. 10(a) and (b) show the distributions of tangential forces during contact between roller and block and between crack faces. The solid line represents the tangential forces in the nodes occurring just after pressing the roller to the block at an initial point situated

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o---- at ter press in +.----- aiter rolling of 5b surf ace

tsnvsnt Ial forces in crack

(a) @>

Fig. 10. (a) Plot of tangent&I forces in contact and their change as a resuft of rolliig; (b) plot of tangential forces in contact of crack faces, and their change as a result of rolling.

K [ M’a*n~“~] 80

I

K,,(rlth friction)

R/------

R=

Fig. 11. Distributions of stress intensity factors ICI and Ka for a perpendicular crack.

5b/3 left of the crack (where 2b is the width of the contact). The dashed line represents the distribution which was obtained at the same points after rolling from an initial point situated 5b to the left.

The influence of the rolling process is distinctly visible. Finally, the stress intensity factors Kr and ICI1 were determined as a function of the distance between he axis of the load and the axis of the crack. ltivo curves shown on Fig. 11 correspond to solutions of the frictionless normal problem and the case with friction respectively. As expected, the friction slightly reduces the maximum value of & caused by shear stresses in the crack tip. F’igure 12 shows results from the solution of case (B) which is a normal

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(a)

Fig. 12. Deformation of an oblique crack during subsequent stages of rolling (dire&ion of motion from left to right).

K , [ bPa* m ““’ ]

x/b

1 2 3

Fig. 13. Distributions of stress intensity Kl and KU for an oblique crack.

problem with a crack oblique to the surface of the block. It should be noted that the crack mouth is open when the load is relatively far from it. Then, as the roller approaches the crack its mouth starts to close, and fInally the

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Fig. 14. Deformation of a sub-surface crack under load.

R = 13000 kN/m _2 w

\ crack

Fig. 15. Distributions of surface intensity factor K, and K,, for a sub-surface crack.

mouth is closed while the faces near the tip have opened. Figure 13 presents the distributions of stress intensity factors for this case. Interesting results were expected for case (C) with the sub-surface crack parallel to the surface. This, however, was not confirmed by the results. However, Fig. 14 shows the effect of the opening of the crack faces near the right tip when the axis of the load approaches the left tip. Corresponding diagrams of the stress intensity factors Ki and K,, are shown in Fig. 15. Their values are, however, relatively small.

6. Conclusions

(a) The method applied appears to be very useful for the solution of contact problems with cracks.

(b) The earlier published hypothesis 5 concerning the existence of tensile stresses at the crack tip for a symmetrical load has been confirmed.

(c) The pressure distribution for a crack oblique to the surface and perpendicular to it differs from the hertzian distribution as a result of the division of the contact area into two separate parts.

(d) For a sub-surface crack the simultaneous action of the stress intensity factors Ki and K,, was confirmed.

(e) The maximum values of & for perpendicular and oblique cracks exceed the critical stress intensities &, which for rail steel is in the range

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27-55 MPa ml’*. If the fatigue problem is taken into consideration, then the threshold value of the stress intensity is exceeded in all investigated cases, even for mode I of crack extension, and propagation of cracks will occur.

(f) The results obtained for frictional probIems are a step forward in the analysis of real contact. They illustrate the capabilities of the method and expiain the effect of rolling more adequately.

(g) The effect of fluid pressure acting on crack faces described in refs. 4 and 7 can be a significant factor in analysis of the stress state around the crack tip. This should be examined in further investigations.

References

1 R. K. Steele, Recent North American experience with shelling in railroad rails, Report ORE 0173 (Department of Transportation, U.S.A.).

2 A. F. Bower, Stresses in a damaged body subjected to contact loads, University of Camlvridge Technical Report, 1986 (University of Cambridge, Cambridge, U.K.).

3 A. F. Bower, The strength of surface in rolliig contact, ~n~v~~t~ of Curve Technical Report, 1986 (University of Cambridge, Cambridge, U.K.).

4 A. F. Bower, The effects of crack face friction and trapped fluid on surface initiated rolling contact fatigue cracks, University of Cambridge Technical Report CUED/C-Mech/TR. 40, 1987 [University of Cambridge, Cambridge, U.K.).

5 R. W6jcik, The analysis of strain and stress state near the crack tip in contact of deformable bodies, Doctorate Thesis, Warsaw, 1981 (in Polish).

6 M. OIzak, Investigation for stress intensity factors for oblique crack in thick-walled cylinder under internal pressure, Doctomte Thesis, Warsaw, 1986 (in Polish).

7 S, Way, Pitting due to robing contact, J. Appl. Mech., 2 (1935) A49-A58.

Appendix A

For solution of the contact problem with friction it is necessary to know the whole process of loading, beginning with the initiaI state (usually zero loading) and carrying on to the final state for which the problem is being solved.

The algorithm presented below concerns the progression from the solution for a state corresponding to the normal load R’ and tangential load K’, to the state determined by forces 22” and X’, assuming that the increments of both forces are small in between the two states considered. Let us suppose now that for the two-dimensional contact problem between the roller and the block with a crack, the solution of the set of eqns. (A), for a certain state co~espon~g to forces R’ and E, is known. This means practically that the vectors of the normaI forces {P’) and (S}, the vectors of the tangential forces (2”) and {F’}, as well as the vectors of sliding {x03 and {pq are known.

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‘141 111 l&l 101 f-491

PI

where [A,] to [A,,] are flexibility matrices of contact nodes of the roller and block, (P} is the vector of normal forces in contact between roller and block, {T) is the vector of tangential forces in contact between roller and block, {s) is the vector of normal forces in contact between crack faces, (fl is the vector of tangential forces in contact between crack faces, R is the total force pressing the roller to the block in the y direction, K is the total force acting on the roller in the x direction, i.e. tangential to the surface of the block (if applied), M is the total normal reaction between crack faces, Nis the total tangential reaction between crack faces, y is the total displacement of the center of the roller with respect to the fixed base of the block in the normal direction, 2 is the total displacement of the center of the roller with respect to the fixed base of the prism in the tangential direction, {go} is the vector of prehminary distances between nodes of roller and block in the y direction, {x0) is the vector of displacement of the nodes of the roller with respect to the nodes of the block in the x direction as a result of sliding, (8’) is the vector of preliminary distances between nodes of crack faces in the normal direction, and {p} is the vector of displacements of nodes of one crack face with respect to the other in the tangential direction as a result of sliding.

The iteration procedure for determination of the vectors {x0’} and {p’) will be presented below.

The next stage is the solution for the state corresponding to the contact forces R” and 32’. For this we assume at the beginning that the nodes in contact are the same as for the loads R’ and K’. This assumption allows cases in which the contact appears at nodes at which it has not existed before, or in which there is loss of contact at nodes which were in contact before, or both these cases together, can be eliminated. The first step of the iteration procedure is a repeated solution of set (Al) for R’ and E substituted by R” and K”.

As a result, the new vectors of forces at the contact nodes {P”}, {S”}, {T”}, {F”} are calculated.

At the same time the program veties the conditions for frictional forces during contact according to Coulomb’s law, for all nodes in contact.

P”$ #u > Iq

for couples of nodes on the surface of the block,

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for couples of nodes on the crack faces, where p is the coefficient of friction. In couples of nodes on the surface of the block where condition (AZ)

is not satisfied, the quantity x°F1 is substituted by x”‘~ and after the solution of set (Al) (where (~“‘1 was substituted by {x”‘}) the value of IT,/ is reduced. An analogous procedure is performed for contact between crack faces. The procedure described of changes in the values of zoi and pi is carried on until condition (AZ) is satisEed at all contact nodes. Obviously, the changes in xoi and pi must be small enough to satisfy the conditions (A3).

Sip- IFi1 d E (A3)

where E is the value of assumed accuracy of the tangential forces determined. If during the assumed increment in the load (from R’ to 22” and from

K’ to K”), new nodes of contact appear (with index k), than forces in between R' , R” and K’, x’, R* and P, have to be found such that the distance gk between the nodes of this couple, in the direction normal to the contact surface, is close to zero.

Then, the new equation for this couple of nodes is added to the set of eqns. (Al), and xok (or Pk) is substituted by 71( which is the distance between the nodes measured in the direction tangential to the contact surface, and then the set eqns. (Al) is solved with respect to R” and K”.

The procedure is similar when contact is lost between any couple of nodes (for exampie with index 1). In this case we are looking for quantities R* and r” for which PL= 0), and the corresponding equations are removed from set (Al).

It is necessary to emphasize that this procedure of contact solution takes into account the simultaneous action of normal and tangential forces during contact, This means that the friction beCween crack faces which affects the distribution of normal forces is also taken into consideration (especially for the quasi-static rolling process). However, the friction between the block and roller affects the ~~bu~on of normal forces between crack faces (especially when a tangential load K is applied).