Discrete dislocation analysis of a tensile crack under fatigue

Materials Science and Engineering, 44 (1980) 63 - 72

63

© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

Discrete Dislocation Analysis of a Tensile Crack Under Fatigue

K. JAGANNADHAM and M. J. MARCINKOWSKI

Department of Mechanical Engineering and Engineering Materials Group, University of Maryland, College Park, Md. 20742 (U.S.A.) (Received May 26, 1 9 7 9 ; i n revised form August 20, 1979)

SUMMARY

The discrete dislocation m e t h o d is emp l o y e d to determine the effect o f pre-existing crack size on the behavior o f a plastic tensile crack under static loading. The behavior o f a tensile crack under cyclic and reverse loading is also determined by the same method. Three important steps in the fatigue crack growth during each cycle are identified. It is concluded that the details o f the fatigue crack growth during each cycle o f loading cannot be inferred from the fatigue striations observed on the crack surfaces after fracture. The statistical nature o f fatigue is explained by the range o f pre-existing crack sizes present in a material which may or may n o t contain a crack o f critical size that can propagate under cyclic or reverse loading.

1. INTRODUCTION

It is n o w well established that a nucleated crack at a slip band or a twin band in a material undergoes two stages of growth under cyclic stresses [1]. Stage I, which is characterized by cracking along slip planes and is difficult to observe at high stress amplitudes, has been determined to be slower than stage II. In stage II crack growth is found to be the result of reversed plastic shearing with plastic blunting at the tip and takes place on a plane normal to the direction of maximum tensile stress. The crack propagation mechanism involving the repetition of plastic blunting and resharpening is the characteristic aspect of stage II growth [1 - 4] by which the occurrence of fatigue striations is explained. This microscopic process and the energy considerations are inferred from experimental observations; however, there is no p r o o f y e t

of the exact mechanism of crack growth [1 4]. The discrete dislocation analysis of fatigue crack growth presented n o w illustrates the exact mechanism of crack growth and establishes the conditions for fatigue crack growth in terms of the energy of the crack configuration. The discrete dislocation analysis of the behavior of cracks in materials has been employed in the past to determine the various terms contributing to the total energy of the configuration. The advantage of this m e t h o d over analytical methods has been demonstrated in earlier studies of the behavior of plastic cracks [ 5 - 11]. The computation procedure used to determine the dislocation configuration of any crack consisted of minimizing the total energy of the configuration with respect to the position of each dislocation. The elastic constants for iron and its alloys were chosen for the determination of the energy versus size curves. The total energy of the configuration is given by E w = E s c + ESL + E I c + EICL + EIL +

+ E L + E f - - E w +E~

(1)

where Esc a n d ESL are the self-energy of crack and lattice dislocations respectively, E ~ is the interaction energy of crack dislocations, EIc L the interaction energy of crack and lattice dislocations, E~L the interaction energy of lattice dislocations, EL the energy of the ledge surface formed, Et the frictional energy expended in moving the lattice dislocations against a frictional stress rf, Ew the work done by the applied stress TA and E7 the surface energy of the crack. It should be noted that when the crack is allowed to grow continuously, as shown in Fig. 1, the irreversible frictional energy is additive whenever a lattice dislocation moves against the

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frictional stress. The details of the calculations used in determining the ET versus R curves have been given in earlier analyses of continuous plastic cracks [8 - 11]. Figure 1 shows the behavior of cracks of different preexisting sizes when rA = 600 X l 0 s dyn cm -2 and r~ = 300 X 10 s d y n cm -2 . The plastic zone is assumed to form on a slip plane perpendicular to the crack plane. The Burgers vector of the crack dislocations is 2.5 A and t h a t of the lattice dislocations is 1.25 A. It is seen that the E~ versus R curve depends on the pre-existing size of the crack. When the initial crack is below a certain size it cannot emit any lattice dislocations. Above this m i n i m u m size the crack emits lattice dislocations so as to reduce its energy and it is allowed to grow continuously. When the preexisting crack size R i is greater than 110 A the total energy of the crack decreases continuously with increasing size and therefore the crack is unstable. However, for 120 A > R i 80 A the total energy of the crack initially

decreases with increasing size, reaches a minimum, then increases to a m a x i m u m and finally decreases again. This m a x i m u m gives the activation energy required for the crack to grow continuously with decrease in energy. Thus the elastic energy stored is initially dissipated in generating the plastic zone in addition to the new crack surfaces but the plastic deformation around the tip causes blunting of the crack tip and a corresponding increase in energy. The energy m a x i m u m decreases with increasing size of the pre-existing crack until finally crack growth takes place with a continuous decrease in energy. The plastic blunting is n o t sufficient to arrest a large crack since the decrease in energy due to the release of strain is large compared with the increase in energy due to the formation of the plastic zone.

2. B E H A V I O R O F A C R A C K U N D E R CYCLIC LOADING

The behavior of a crack under cyclic loading where the loading cycle consists of the spontaneous application of a constant load followed by its removal will now be considered. The crack configuration is assumed to reach equilibrium instantaneously. The configurations under cyclic loading were determined as indicated before using the discrete dislocation analysis. In order to reduce the computer time required to determine the complete

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crack growth process, the applied stress rA was assumed to be large, i.e. 700 × l 0 s d y n cm -2 . The Griffith configuration of the elastic crack at this applied stress is found at a crack size Rc = 75 A, as shown in Fig. 2. The plastic zone is assumed to form by emitting lattice dislocations on a slip plane perpendicular to the crack plane. The crack tip stress field is in fact relaxed as much as possible on a slip plane perpendicular to the crack plane, although the c o m p o n e n t of the applied stress is zero [6 - 8]. This choice of slip plane is suitable for determining the effect of the crack tip stress field on the plastic zone under cyclic loading. The effect of cyclic loading on the behavior of the crack at rA = 700 × 10 s d y n cm -2 and Tf = 200 × l 0 s d y n c m - 2 w a s studied for two sizes of the preexisting crack, Ri = 50 A and R i = 100 £ . The points S on the elastic crack curve in Fig. 2 indicate the starting points of the loading cycle. The changes in the total energy of the crack and the dislocation configuration are illustrated in Figs. 3 and 4 for R i = 5 0 ~,. In Fig. 3, S indicates the total energy of the elastic crack when it is spontaneously loaded; point 1 shows when the crack generates the plastic zone, point 2 when it is unloaded, point 3 when the crack is spontaneously loaded a second time and point 4 when the crack is unloaded. Figure 4 illustrates the dislocation configuration of the crack at various stages indicated by the same number as in Fig. 3. The loading cycle was stopped

after two cycles since the crack did not grow during loading by decreasing its total energy. It is seen from Fig. 4 that a lattice dislocation moves into the crack on the first unloading but that on loading a second time the configuration obtained earlier is reached w i t h o u t any increase in crack size. Such cracks are very stable under cyclic loading and do not grow because there is not sufficient elastic energy stored in the crack to extend it. The crack does not close completely on unloading since the lattice dislocation present at the tip provides a ledge step on the crack surface that prevents the crack surfaces closing together. The cyclic loading of the fatigue crack of initial size Ri = 100 A is shown in Figs. 5 and 6. Figure 5 represents eight cycles of loading and unloading. The energy change in each cycle is shown separately. These could not be shown together on one plot owing to the large difference in the total energy of the configuration between each cycle. Figure 5 shows the starting point S, the minimum in the energy reached at R = 120 A at point 1 and the unloaded configuration with energy at point 2. The energy drops initially on spontaneously loading and decreases with increasing crack size, indicating t h a t the stored elastic energy propagates the crack, but increases again owing to the blunting of the crack tip by the plastic zone formed. The increase in energy due to the formation of the plastic zone arrests further crack growth. The crack is then unloaded to configuration 2. The initial decrease in energy is considered by other workers [ 1 - 4] to result from the instability of the crack configuration, while the subsequent increase is associated with plastic blunting. The instability arises from the release of the stored elastic energy to propagate the crack.

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In this region the work done by the applied stress is more than the increase in energy due to the formation of new surfaces of the crack, the frictional energy spent in generating the lattice dislocations and the strain energy arising from the self-energies and interaction energies of the crack and lattice dislocations. T h e dislocation configuration obtained during the first loading cycle is shown in Fig. 6 where configuration 1 corresponds to point 1 in the ET versus R curve during the first cycle. Configuration 2 corresponds to point 2 when the crack is unloaded. It is apparent that the crack does not close on unloading. In fact it consists of one anticrack dislocation required to screen

the stress field of the lattice dislocations so that the stresses vanish on the crack surfaces. The significance of the term "anticrack dislocation" is discussed at length in an earlier work [11]. The crack does not close completely in the unloaded state; the crack surfaces remain separated. The lattice dislocations present in the plastic zone generate stress on the crack surfaces. However, the freesurface boundary conditions on the crack surfaces can be satisfied by distributing dislocations of opposite sign to the crack dislocations on them. Since these surface dislocations in the crack region are opposite in sign to the crack dislocations present in the loaded

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state, they are called anticrack dislocations. The physical significance of the anticrack dislocations lies in their tendency to make the crack close. Thus, if the total Burgers vector of all lattice dislocations is equal to the total Burgers vector of the anticrack dislocations, by definition the crack is closed. When the lattice dislocations move towards the crack and are annihilated with the anticrack dislocations, there are no ledge steps available on the crack surface to keep the crack surfaces separated. The increase in energy of the configuration on unloading indicates t h a t the crack is stabilized by lattice dislocations, i.e. the frictional stress prevents the lattice dislocations from moving into the crack. The energy con-

tributions of various terms in the total energy of the crack at various points of crack growth are n o t presented here but the results can be obtained from the authors. The second loading cycle indicated in Fig. 6 shows that the crack size is 10 A greater than that of the previous unloaded configuration. The energy reaches a minimum at point 3 after passing through the instability and increases with further increase in the crack size. The unloaded configuration corresponds to point 4. The corresponding dislocation configurations are shown in Fig. 6. The second cycle of loading is not significantly different from the first, in the sense that the crack size again increases by 10 A and does n o t decrease on unloading. The crack size increases on loading to

68

point 5 in the third cycle b u t closes to its previous size on unloading. The energy of the unloaded configuration could n o t be plotted in the figure and hence is shown in parentheses. Figure 6 shows that the crack contains many lattice dislocations at point 5 b u t that many of these lattice dislocations move into the crack and are annihilated with the anticrack dislocations. The total energy of the configuration in each cycle increases as a result of the irreversible frictional energy extended in moving the lattice dislocations back and forth. The irreversible frictional energy is part of the energy expended in forming the plastic zone. Further loading cycles are illustrated in Fig. 5 and the dislocation configuration of the crack at various points is shown in Fig. 6. The energy of the configuration is shown in parentheses in cases for which it could not be plotted. It was found that the crack propagates with a continuous decrease in energy leading to complete instability after seven loading cycles. At this stage the stored elastic energy of the crack is sufficient to make the crack propagate without its arrest due to plastic blunting. The total energy of the configuration at the point of minimum energy during each loading cycle is plotted in Fig. 7 as a function of crack size to show the increase in energy with each cycle. The points indicated by the same numbers in Figs. 5 and 6 have the same significance. The shape of the crack surface at point 14 in Fig. 6 is shown schematically in Fig. 8. The relative positions of the lattice dislocations are n o t drawn to scale b u t their n u m b e r is indicated to show the height of the ledge steps. The

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presence of an anticrack dislocation causes the crack surfaces to close, as shown in Fig. 8. The fracture surface of the crack contains ridges produced by the ledge steps. The separation of the ledge steps depends on the position of the lattice dislocations around the crack in the final stage. Although each cycle of loading and unloading generates lattice dislocations at the crack tip, n o t all the lattice dislocations generated remain in the lattice. The ledge steps are eliminated whenever the lattice dislocations move into the crack and are annihilated and there is no specific identification on the crack surface to show the history of the growth of the crack configuration. Therefore the striations observed at the end of fatigue crack growth represent ledge

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steps in the final configuration. It cannot be inferred t h a t these striations formed during each cycle, i.e. there is no one-to-one correspondence between the n u m b e r of loading cycles and the number of striations observed. Thus the conclusions related by Forsyth and Ryder [4] seem to be invalid except in a special situation. However, these striations certainly indicate the plastic blunting for fatigue crack behavior and represent the ledge steps on the crack surface in the final configuration. The effect of frictional stress on the behavior of the crack under cyclic loading is studied next by decreasing the frictional stress to r~ = 100 × l 0 s d y n a m - 2 and assuming the same pre-existing crack size. The energy of the configuration of the crack during each loading cycle is shown in Fig. 9. The first cycle consists of the starting point S, the growth of the crack to point I and the unloaded configuration to point 2. There is an increase in the crack size to point 3 on the second loading and partial closure of the crack to its previous size on unloading to point 4. The third cycle is similar to the second and the fourth is similar to the first. The crack reaches complete instability after four cycles of loading and unloading as shown by points 9, 10 and 11. The dislocation configurations obtained at various stages of growth of the crack are shown in Fig. 10, the numbers corresponding

to those in Fig. 9. A decrease in the frictional stress on the movement of lattice dislocations leads to complete instability of the crack after a smaller number of cycles. The effect of frictional stress on the behavior of the crack is indirect. The growth of the crack during loading is altered by the distance to which the lattice dislocations are moved into the lattice. The frictional stress opposes the movement of lattice dislocations. Similarly, the number of lattice dislocations that move into the crack and are annihilated with the anticrack dislocations on unloading also depends on the frictional stress. The total energy of the crack at the minimum energy points of loading in each cycle are plotted as a function of crack size in Fig. 11. The points connected by full lines correspond to cyclic loading. The present study of the behavior of the crack at two frictional stress levels demonstrates that the effect of frictional stress is indirect and that a direct correlation of the behavior of the crack with frictional stress is n o t possible.

3. B E H A V I O R O F A T E N S I L E C R A C K U N D E R REVERSED LOADING

The behavior of a tensile crack under reversed loading is studied next, with the same pre-existing crack size as earlier and with the same frictional stress, i . e . R i = 100 A and vt =

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Fig. 11. T h e E T vs. R p l o t o f t h e m i n i m u m e n e r g y p o i n t s o f l o a d i n g f r o m Figs. 9 a n d 12. T h e points j o i n e d b y full lines r e p r e s e n t cyclic l o a d i n g a n d t h o s e j o i n e d b y b r o k e n lines r e p r e s e n t reversed l o a d i n g cycles. T h e n u m b e r e d p o i n t s c o r r e s p o n d t o t h o s e in Figs. 9 a n d 12.

100 X l 0 s d y n c m - 2 . The loading cycle consists of spontaneously loading the crack to TA = 700 X l 0 s d y n cm -2 , unloading it after the minimum energy is reached and loading the crack in the reverse direction. Figure 12 shows the total energy of the crack at various stages of the loading cycle. Point S indicates the total energy of the elastic crack configuration; point 1 shows when the energy reaches a

minimum, point 2 shows when the crack is unloaded and point 3 shows when the crack is loaded in the reverse direction. Figure 13 shows the dislocation configurations at these points. Configurations 1 and 2 are the same as those indicated in Fig. 10 with the same numbers. There are five lattice dislocations around the crack after unloading with the crack surfaces separated. Two lattice dislocations have moved into the crack after reverse loading. The presence of three lattice dislocations with three anticrack dislocations indicates t h a t the crack surfaces are closed, as shown earlier. When the crack is now loaded spontaneously t h e configuration reaches point 4 with a 10 A increment. The dislocation configuration is shown in Fig. 13. The third cycle of loading consists of an increase in crack size by 20 A on loading and closure of the crack by 10 A on unloading. The fourth cycle leads to complete instability of the crack. It is observed from Fig. 12 that the configuration on reverse loading in each cycle has lower energy than the configuration obtained on unloading. The points 2 and 3, 5 and 6, and 8 and 9 which correspond to the unloaded and reverse loaded configurations respectively show the same result. When the crack is loaded in the reverse direction, the work done by the applied stress on the crack dislocations is larger than the frictional energy due to the movement of lattice dislocations and the energy increase due to other terms. Therefore the total energy of the crack decreases from

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t h a t in t h e u n l o a d e d c o n f i g u r a t i o n . A c o m parison of dislocation configurations obtained d u r i n g cyclic l o a d i n g w i t h t h o s e o b t a i n e d d u r i n g reverse l o a d i n g cycles clearly s h o w s t h a t t h e r e are f e w e r l a t t i c e d i s l o c a t i o n s a r o u n d t h e c r a c k in reverse l o a d i n g cycles. T h e r e is t h e r e f o r e less r e s i s t a n c e t o t h e m o v e m e n t o f t h e c r a c k o n f u r t h e r loading. T h e reverse l o a d i n g r e d u c e s plastic b l u n t i n g a n d allows t h e c r a c k t o b e h a v e m o r e elastically.

T h e irreversible f r i c t i o n a l e n e r g y e x p e n d e d in m o v i n g t h e lattice d i s l o c a t i o n s i n t o t h e c r a c k on reverse l o a d i n g is additive in e a c h c y c l e a n d t h u s t h e t o t a l e n e r g y o f t h e c r a c k conf i g u r a t i o n increases f a s t e r d u r i n g e a c h cycle. The total energy of the crack configuration at minimum energy during the loading cycle is s h o w n in Fig. 11 b y t h e p o i n t s c o n n e c t e d b y b r o k e n lines. I t is seen t h a t t h e m i n i m u m e n e r g y o f t h e c r a c k o n l o a d i n g at a n y size

72 during reverse loading cycles is higher than that during cyclic loading. The difference is attributed to additional frictional energy expended in moving the lattice dislocations back and forth on reverse loading.

4. SIGNIFICANCE OF THE RESULTS The analysis of the behavior of a crack during either cyclic loading or reverse loading cycles is performed with very large crack sizes in order to reach complete instability within a few cycles. The c o m p u t i n g time required to reach the final stage of crack propagation is also shortened by the procedure. Cracks below a certain size do not propagate during cyclic loading because there is insufficient stored elastic energy. The fatigue crack growth depends on three factors. Firstly, an increase in the size of the crack due to initial instability when the stored elastic energy of the crack is dissipated in creating n e w crack surfaces and generating the plastic zone is important. Secondly, the formation of the plastic zone with increase in energy leads to plastic blunting and crack arrest. Thirdly, subsequent unloading with or without reverse loading must maintain the separation of the crack surfaces so that the crack does n o t close up; thus the frictional stress must retain the lattice dislocations at the tip. The pre-existing size of the crack has a large effect on the fatigue crack growth. When a material containing a range of crack sizes is subjected to fatigue, n o t all sizes of crack can propagate. When the stress amplitude is small, large cracks can only respond to cyclic loading. Since such large cracks m a y n o t be present in the material the fatigue life can be very long at low stress amplitudes. The statistical nature of the fatigue properties can be explained by the range of crack sizes present in a material. Present results also indicate that the striations on the crack surfaces arise as a result of the creation of ledge steps during crack growth. It cannot be inferred that the ledge steps found on the crack surfaces after fatigue crack growth are formed in each cycle, particularly since some ledge steps are eliminated. Therefore there is no one-to-one correspondence between the cycles of loading and the fatigue striations observed. The cracks some-

times close on unloading with or without reverse loading and thus the number of striations and the distances between them cannot be used to determine the details of the crack growth process. It m a y also be pointed o u t that the stress field of an elastic crack is very different from the stress field of a plastic crack undergoing cyclic loading or reverse loading. This difference is due to the plastic zone present around the crack [8]. There is therefore no justification for treating the crack as elastic in order to determine the crack tip stress field during each cycle of loading. The stress field of a plastic crack during cyclic loading is very different from that of an elastic crack of the same size.

ACKNOWLEDGMENTS The computing facilities for this project were provided in full by the Computer Science Center of the University of Maryland. Financial support for the present study was provided by the U.S. Department of Energy under Contract AT-(40-1)-3935. Discussions with Professor R. W. Armstrong of the Department of Mechanical Engineering, University of Maryland, are gratefully acknowledged.

REFERENCES 1 A. S. Tetelman and A. J. McEvily, Jr., Fracture o f Structural Materials, Wiley, New York, 1967. 2 A. J. McEvily and T. L. Johnston, Int. J. Fract. Mech., 3 (1967) 45. 3 C. Laird, Fatigue crack propagation, Am. Soc. Test. Mater., Spec. Tech. Publ. 415, 1967, p. 131. 4 P. J. E. Forsyth and D. A. Ryder, Metallurgia, 63 (1961) 117. 5 M.J. Marcinkowski,J. Appl. Phys., 46 (1975) 496. 6 K. Jagannadham and M. J. Marcinkowski, Phys. Status Solidi, 42 (1977) 439. 7 K. Jagannadham and M. J. Marcinkowski, Mater. Sci. Eng., 33 (1978) 21.

8 K. Sadananda, K. Jagannadham and M. J. Marcinkowski, Phys. Status Solidi, 44 (1977) 633. 9 K. Jagannadham and M. J. Marcinkowski, J. Mater. Sci., 13 (1978) 1725. 10 K. Jagannadham and M. J. Marcinkowski, Int. J. Fract., 15 (1979) 119. 11 K. Jagannadham and M. J. Marcinkowski, Int. J. Fract., 16 (1980) 193.