MECHANISMS

OF

FATIGUE

CRACK

PROPAGATION

IN METALS A. Ya.

Krasovskii

The most common service are fatigue and fatigue of the material. ferentiated, the origin accurately determined

UDC 539.4

reasons for the occurrence and subcritical growth of cracks in machine parts during corrosion under stress and a combination of them revealed in the form of corrosion Two stages of the "life" of a sample or part under the action of a cyclic load are difand the propagation of a fatigue crack. The boundary between these stages has not been and its position depends upon what size of microcrack is adopted as forming the nucleus.

In practice such a determination normally involves the resolving capacity of the instrument used for detecting a microcrack and therefore different methods lead to dissimilar conclusions on the ratio between the number of cycles until the origin of a microcrack N O and the full life of the sample or part Nf. According to the data of [i], depending upon the sample geometry, the level of stress amplitude, the material structure, the observed length of the microcrack, and other factors the No/Nf ratio may vary from 0.5 to 88%. The general tendency is such that the value of this ratio decreases with an increase in stress amplitude and an increase in the sharpness of the stress raiser. A reduction in temperature, as a rule, increases the value of N O /Nf. An increase in ductility of the material normally leads to an increase in crack growth rate. These data show that the fatigue crack propagation stage is not rarely the main portion of the life of a sample, as a result of which a study of this process is of value in itself. According to established concepts, the crack propagation stage may be divided into two stages. Stage I involves growth of the embryonic crack in a slip plane and stage II involves advance of the crack in a plane close to or coinciding in orientation with the plane of action of the maximum tensile stress. Below are considered possible mechanisms of fatigue crack propagation in stage II. At present more than 60 equations are known describing fatigue crack growth rate [2] (see also the review of P. Romvar' and his coauthors on this question, the publication of which is planned in the next issue of this journal), a fact which by itself is an indication of the great scientific value and practical importance of this problem. Normally in deriving such equations (if they are not empirical) one or another model of crack propagation reflecting in concentrated form the basic process responsible for advance of a crack during a load cycle is used. As a rule, the model considers a single mechanism of crack growth and therefore has limited value and is true only in that range of loads and for those materials for which the mechanism occurs. In view of the above, studying the specific mechanisms of fatigue crack propagation and establishing the limits of their applicability have important scientific and practical value. One of the basic methods of experimental study of mechanisms of crack propagation is electron fractography. In studying processes of fatigue crack propagation by this method, the greatest attention is devoted to formations on the failure surface which are specific for cyclic loading and are called fatigue striations. The majority of investigators have a tendency to assume that striations are the result of occurrence of local cyclic plastic deformations at the crack tip and are the most characteristic sign of fatigue fractures. HOWever, the method of crack propagation by the formation of fatigue striations is not the only one. Mechanisms of Fatigue Crack Growth in Steels. As a result of an analysis of many investigations made on different steels over a wide range of structural conditions with a change in loading parameters, the authors of [3] came to the conclusion that in this class of materials there are basically four types of crack growth mechanisms: striations, microspalling, merging of voids, and failure in grain boundaries. Propagation of a crack according to the striation mechanism is insensitive to the average cycle stress (with the exception of small loads) and to sample thickness (with the exception of large loads). Depending upon the level of load and the material structure the striation mechanism is accompanied or displaced by one of the above mechanisms, which leads to acceleration of crack growth. Another important observation noted in [3]

Institute of Strength P r o b l e m s , A c a d e m y of Sciences of the Ukrainian SSR, Kiev. T r a n s l a t e d f r o m P r o b l e m y P r o c h n o s t i , No. 10, pp. 65-72, 78, October, 1980. Original a r t i c l e submitted A p r i l 12, 1980.

0039-2316/80/1210-1255507.50

9 1981 Plenum

Publishing C o r p o r a t i o n

1255

is that l o w - s t r e n g t h steelw and alloys a r e m o r e prone to fail by the s t r i a t i o n m e c h a n i s m . With l o w - c y c l i c loads t e m p e r e d m a r t e n s i t i c s t e e l s have a tendency toward i n t e r g r a n u l a r failure and with high-cyclic loads toward the m e c h a n i s m of m e r g i n g of voids. As a rule s t e e l s with r e d u c e d f r a c t u r e toughness have a tendency toward a change f r o m failure a c c o r d i n g to the s t r i a t i o n m e c h a n i s m . Of all of the studied m a t e r i a l s , s t e e l s which failed a c c o r d i n g to the s t r i a t i o n m e c h a n i s m had the lowest r a t e of c r a c k growth with c o m p a r a b l e loads. T h e r e f o r e , the authors of [3] spoke p e s s i m i s t i c a l l y about the p r o s p e c t s of c r e a t i n g s t e e l s capable of r e s i s t i n g fatigue c r a c k propagation. With r e g a r d to r e c o m m e n d a t i o n s on the use of s t e e l s , in their use under conditions of cyclic loading those m a t e r i a l s which fail a c c o r d i n g to the s t r i a t i o n m e c h a n i s m under working loads m u s t be p r e f e r r e d . All of this i m p a r t s s p e c i a l value on a detailed investigation of this m e c h a n i s m . It should be noted that a t t e m p t s to e s t a b l i s h a c o r r e l a t i o n between the a v e r a g e spacing of the s t r i a t i o n s and the m a c r o s c o p i c c r a c k growth r a t e [4-8] have frequently led to a s a t i s f a c t o r y r e l a t i o n s h i p between these values if the a n a l y s i s was for the r a n g e of loads for which the P a r i s equation is c o r r e c t (the c e n t e r portion of the fatigue f a i l u r e curve): dl _ C h K n, dN

(1)

w h e r e d l / d N is the r a t e of c r a c k growth; AK, r a n g e of the s t r e s s intensity factor; and C and n, constants. In g e n e r a l , it may be e x p e r i m e n t a l I y c o n f i r m e d (in addition to the d i r e c t proofs of [9]) that each s t r i a t i o n is f o r m e d during a single Ioad cycle. A detailed a n a l y s i s of the r u l e s and m i e r o m e c h a n i s m s of fatigue c r a c k propagation in 08kp s t e e l at low t e m p e r a t u r e s is p r e s e n t e d in [7, 8]. Depending upon the t e m p e r a t u r e and load, five basic c r a c k growth m e c h a n i s m s w e r e o b s e r v e d on the f r a c t u r e s u r f a c e s : fatigue s t r i a t i o n s , spelling, pits, "stitching," and i n t e r g r a n u l a r failure. For e a c h a r e a of the fatigue failure c u r v e and e a c h t e m p e r a t u r e , a failure s u r f a c e r e l i e f typical for t h e m and making a basic contribution to advance of the c r a c k p r e d o m i n a t e s in the f r a c t u r e . F o r a c r a c k growth r a t e of d l / d N < 5 9 10 .6 m m / e y e I e a ' s t i t e h l i k e " s t r u c t u r e p r e d o m i n a t e s at alI t e m p e r a t u r e s and the m a e r o s t r u e t u r e is flat with a s m o o t h dull s u r f a c e . In such a f r a c t u r e in c o n t r a s t to fatigue s t r i a t i o n s the specific elongated portions a r e oriented p a r a l l e l to the d i r e c t i o n of c r a c k growth. T h e r e is a n opinion that this failure m e c h a n i s m is c a u s e d by p r o c e s s e s of s h e a r f o r m a t i o n and failure a c c o r d i n g to the antiplane def o r m a t i o n type (longitudinal s h e a r or type III crack). Since in the a r e a of s mall loads this f o r m of failure a l m o s t c o m p l e t e l y d i s p I a e e s ali other m e c h a n i s m s , it m u s t d e t e r m i n e the leveI of threshoId load Kth below which the c r a c k r e m a i n s s t a t i o n a r y . This i m p o r t a n t c h a r a c t e r i s t i c has fundamental value since it d e t e r m i n e s the l i m i t of the capability of a m a t e r i a l to r e s i s t fatigue failure and probably had a d i r e c t relationship to the fatigue l i m i t of a m a t e r i a l . In the r a n g e of c r a c k growth r a t e s of 5 910 .6 s d l / d N s 10 .3 m m / e y e I e at 20~ the p r i m a r y m e c h a n i s m of failure is the m e c h a n i s m of s t r i a t i o n s and at - 1 6 0 ~ spelling with a s m a l l s h a r e of ductile failure pits. As noted above, in the c e n t e r portion of the fatigue f a i l u r e c u r v e t h e r e is good a g r e e m e n t between the actual r a t e of c r a c k growth and the spacing of the s t r i a t i o n s (Fig. 1). F o r many m e t a l s it is c h a r a c t e r i s t i c that in the p r e s e n c e of active media, e s p e c i a l l y in the a r e a of low loads, they r e v e a l a tendency toward failure in g r a i n boundaries. F r e q u e n t I y , such failure o c c u r s in ecrubination with spailing and a l s o with the p a r t i c i p a t i o n of the q u a s i s t r i a t i o n m e c h a n i s m . An i n c r e a s e in loading frequency n o r m a l l y c a u s e s a d e c r e a s e in the r a t e of c r a c k growth p e r load cycle. T h e r e f o r e , p r o p a g a t i o n of a fatigue c r a c k may be d e t e r m i n e d by different m e c h a n i s m s . Of t h e m perhaps only the s t r i a t i o n and " t r a c k - m a r k " f o r m a t i o n [6, 10] m e c h a n i s m s may be c o n s i d e r e d typieaiIy fatigue, i.e., those caused by a l t e r n a t i n g sign plastic d e f o r m a t i o n at the c r a c k tip. The other m e c h a n i s m s , such as spalling, ductile pitted f a i l u r e , i n t e r g r a n u l a r f a i l u r e , and s e c o n d a r y c r a c k i n g , a r e the r e s u l t of the action of the m a x i m u m tensile load in the cycle. They a r e m o r e typical of static loading. The one additional a b o v e - d e s c r i b e d m e c h a n i s m , which leads to a s o - c a l l e d "stitchlike" s t r u c t u r e in the f r a c t u r e , has not r e c e i v e d a full explanation until now and it c a n hardly be stated definitely w h e t h e r it is typically fatigue. T h e r e f o r e , r e f e r r i n g the r e a d e r to s p e c i a l l i t e r a t u r e [6, 10, 11-18] on the question of nontypicai fatigue m e c h a n i s m s , let us t u r n in s o m e detail to models which explain the origin of fatigue s t r i a t i o n and s e r v e as a b a s i s for an a n a l y t i c a l d e s c r i p t i o n of the r a t e of c r a c k growth. M e c h a n i s m of Fatigue S t r i a t i o n F o r m a t i o n . F o r its i n t e r p r e t a t i o n , the L a i r d model is typical [19]. This p r e s e n t s c r a c k p r o p a g a t i o n as a s u c c e s s i o n of p r o c e s s e s of plastic blunting of a c r a c k tip in the loading halfcycle and the p r o c e s s of s h a r p e n i n g of a c r a c k with the f o r m a t i o n at its tip of two g r o o v e s in the unloading h a l f - c y c l e (Fig. 2). 1256

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Fig. 1. Relationship of the distance between fatigue striations to the rate of c r a c k growth d l / d N (white points) at various test t e m p e r a t u r e s and also the percent content in the a r e a of the f r a c t u r e of a r e a s o c cupied by s t r i a t i o n (dark points): 1) t = 20~ 2) t = - 1 0 ~ 3) t = - 4 0 ~ 4) t = - 7 0 ~ 5) t = - 1 0 0 ~ Fig. 2. P r o c e s s of plastic blunting of the tip of a fatigue c r a c k propagating in stage II: a) loading absent; b) s m a l l tensile load; c) large tensile load; d) small degree of unloading; e) m a x i m u m c o m p r e s s i v e load (or full unloading with a positive coefficient of cycle a s y m m e t r y ) ; f) small tensile load. Axis of loading is vertical. Double a r r o w s indicate the b r o a d e r a r e a of slip hand development. Fig. 3. P l a n of f o r m a t i o n of fatigue striations according to [9, 22]. The two most characteristic traits of this model should be noted. The first is that the basic process providing advance of the crack is plastic blunting of the tip with its characteristic linear dimension, the crack opening. Calculation or assumption of the configuration of the blunted tip makes it possible to obtain an equation for the growth of a crack during a loading cycle. There are many indications of the fact that the process of plastic blunting of the tip plays an important role in propagation of a fatigue crack according to the striation mechanism but there is not complete certainity that it alone provides for advance of the crack. The second characteristic trait of the Laird model is the fact that the groove of a fatigue striation is formed in the unloading half-cycle. This sign has been taken as the basis of many experimental confirmations of the model [19-21]. In the last of these works it was possible to obtain quite convincing proofs of the existence of the m e c h a n i s m of plastic blunting of the tip. Similar p r e m i s e s w e r e used as the basis of another model also accenting attention on the m e c h a n i s m of plastic blunting of a c r a c k tip [9, 22] (Fig. 3). The main difference between this model and the preceding is the fact that according to it a fatigue striation is formed at the s t a r t of the loading h a l f - c y c l e and not in unloading and, as a consequence, the groove of each s t r i a t i o n is located not at its s t a r t (corresponding to the position of the front of the c r a c k blunted in the given loading half-cycle) but at the end. This model permitted the authors of [9] to logically explain the f r a c t o g r a p h i c pictures r e v e a l e d in p r o g r a m m e d loading of aluminum alloys but only in that portion which did not c o n c e r n the unloading half cycle. A c o m p a r a t i v e analysis of the models presented in Figs. 2 and 3 was given in [20]. Of the other models, let us mention the Tomkins and Biggs model [23], which introduces into the cons i d e r a t i o n the p r o c e s s of s e p a r a t i o n of the m a t e r i a l by rupture along the line of c r a c k propagation in addition to the p r o c e s s of plastic blunting. Similarly to the model shown in Fig. 3, this model predicts the f o r m a t i o n of the whole fatigue s t r i a t i o n and its groove in the loading half cycle. A critique of the Tomldns and Biggs model is contained in [20, 24]. To explain the differences in morphology of fatigue failure s u r f a c e s of s a m p l e s tested in air and in vacuum, Pelloux [25] and B r o e k and Bowles [26] developed another model (Fig. 4). A c c o r d i n g to this model, the basic r e a s o n causing the appearance of fatigue s t r i a t i o n in air is the o c c u r r e n c e of oxide films on the newly formed c r a c k surface. With unloading the oxides r e t a r d the normal o c c u r r e n c e of slip of the opposite sign. The absence of the film in vacuum p e r m i t s portions of the newly f o r m e d c r a c k s u r f a c e s to again "stick together," which leads to a substantially lower c r a c k growth rate. This model has obtained detailed c r y s t a l l o g r a p h i c int e r p r e t a t i o n using as an example fcc metals in [26]. Attempts to give it an experimental basis a r e given in the detailed review of Broek [6]. A detailed investigation of the s t r i a t i o n m e c h a n i s m by the s t e r e o f r a c t o g r a p h i c method is given in [27-29]. The use in them of high purity nickel as the m a t e r i a l of the investigation made it possible to study the s t r i a t i o n

1257

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Fig. 4. Schematic representation of the changes in fatigue crack tip geometry occurring in air and in vacuum: a, b, e) opening of the crack in the process of "alternative" shear. Positions d-h (air) and i-m (vacuum) correspond to the positions on the load vs time curve shown at the right. Fig. 5. Succession of stages of operation of the mechanism the results of investigations made in high purity nickel.

of fatigue crack propagation from

Fig. 6. Schematic presentation of the mechanism of fatigue crack advance and the tip geometry related to it. On the right are shown approximately the point of the load cycle corresponding to each crack profile. mechanism in pure form since no other failure mechanisms act in the range of crackgrowth rates considered. As a result of detailed measurement of the associated striation profiles on the opposite surfaces of the fracture, a model was proposed in the form of a succession of stages of fatigue crack propagation (Fig. 5) explaining the reasons for such a complex profile of fatigue fracture. According to this model, the full advance of a crack in a load cycle consists of two stages and occurs in loading in the tension half-cycle or even a portion of it. The crack opened in the preceding tension half-cycle (Fig. 5a) is made sharper in the process of successive unloading and loading by compressive stresses. The main result of the action of the compression half-cycle (Fig. 5b) is the plastic zone of compression (cyclic plastic zone), inside which along the trajectories of maximum intensities of cyclic deformations there is preparation of the material for subsequent failure. At the start of the succeeding tension half-cycle failure has the character of transverse shear (crack type II) (Fig. 5c). After passage of the crack beyond the limits of the zone of strong damage of the material it starts to propagate according to the mechanism of plastic blunting (Fig. 5d). Therefore, in accordance with the described model, the starting point of crack propagation during each cycle is the point of start of a transverse shear crack (in Fig. 5 these points are shown by vertical lines separating the four fatigue striations). The model may be placed in mathematical f o r m on the basis of the fact that the growth of a c r a c k during a cycle consists of two components. The f i r s t is determined by a C o f f i n - M a n s o n type rule written for cyclic plastic deformations within the limits of the plastic zone and the second is related to plastic blunting of the c r a c k tip. F o r the model shown in Fig. 5 a number of things differentiating it f r o m other models are c h a r a c teristic. 1. In it in preparing the m a t e r i a l for the next stage of c r a c k propagation of t r a n s v e r s e s h e a r , an i m portant role no longer belongs to the p r o c e s s of unloading and the c o m p r e s s i v e half-cycle. This makes it possible to explain the fact of the f o r m a t i o n of deep d e p r e s s i o n s on the failure surface c o r r e s p o n d i n g to cycles of l a r g e loads [20, 21] in p r o g r a m loading s i m i l a r l y to the L a i r d model [19].

1258

2. In accordance with this model, it may be expected that with an increase in load the contribution of each of the two stages to the advance of a crack during a cycle may change. This must lead to a decrease in the ratio of the depth of a fatigue striation to its spacing with an increase in the amplitude of the load, which has been experimentally observed more than once [19] and was not satisfactorily explained. 3. In view of the fact that the model organically includes the mechanism of plastic blunting of the tip, all of the known explanations of failure surface relief in programmed loading remain true for it. But, in addition, the model opens the possibility of a simple and logical interpretation of the results obtained but not explained by the authors of [9] relative to programmed tests with different degrees of loading. 4. Finally, the presence in the model of the stage of failure by shear brings together stages I and II of fatigue crack propagation, the analogy between which has been mentioned more than once earlier [19] but without the necessary explanation. The fact that failure occurs in the trajectories of maximum cyclic damage of the material makes it possible to consider the model as a physical basis of the necessity of existence of a link between the fatigue limit and the threshold value of the stress intensity factor Kth. Certain additional indications of the correctness of the model shown in Fig. 5 have been obtained experimentally by Bowles [24, 30], who conducted investigations of the structure of the fracture surface and the fatigue crack tip in aluminum alloys by vacuum infiltration of a solution of plastic into the crack cavity of a sample under load. The plastic replica obtained by this method is an accurate impression of the crack cavity and may be studied on microscopes. According to observations of the author of [24, 30], in tests in air during the loading half-cycle the crack advances "brittlely, ~ but close to the peak of tensile stress the crack is blunted and acquires the shape of an ellipse. In unloading it becomes somewhat sharper, still maintaining the curvature of an ellipse. The fatigue striation formed in the n-th load cycle does not undergo any changes under the influence of the (n + l)-th load cycle (experiments were made with a cycle asymmetry of R = K I rain / K I max = 1/3). Closing of the crack in unloading occurs not continuously at all points of the free surfaces of the crack but only in individual areas of lack of contact of the opposite surfaces of the crack. A single overloading causes 'alternative ~ shear, which is shown in Fig. 4a-c. The peak of unloading is completely neutralized by the increasing pressure of the contacting areas of the opposite edges of the crack, which does not lead to marked damage of the fatigue striations but causes an increased rate of crack growth in the next loading. In tests in vacuum at different portions of the crack front there may be observed one of the following forms of failure by shear: the branching of the crack by simultaneous shear on both sides of the plane of the crack, a one-sided shear, and also mixed shear, which is a combination of the types I and III crack propagation. One of the reasons for the lower rate of crack growth in vacuum is assumed by Bowles to be separation of failure by shear at a crack tip into two branches. The other reason is the greater nonuniformity in testing in vacuum of the crack front, which is related to its propagation in different planes. According to the results of [24], the rate of crack growth in vacuum was almost an order of magnitude lower than in tests in air. Generalization of the data obtained in [24] led the author to the following model of crack propagation (Fig. 6). As may be seen, the model has two basic characteristics: the presence of separation of the material by rupture at the start of an increase in load [3] and plastic blunting of the tip [4]. Concluding the discussion of this interesting work, we should nevertheless note that concrete examples showing the correctness of joining of the profiles of opposite sides of the fatigue crack, as shown in Fig. 6, are not given. A study of this would permit an additional verification of this model. Fatigue Crack Instability. To the mechanism of fatigue crack propagation is closely related the important practical question of transition to the unstable condition or cyclic fracture toughness of the material (Kfc). In "soft" loading such a transition normally occurs in the last cycle and leads to final failure of the sample. With an increase in rigidity of the test machine, there are anincreasingnumberof jumps in the crack alternating with its slow growth according to the fatigue mechanism [31, 32]. In its nature the value of Kfc is similar to the fracture toughness of the material K c in static loading since both of these parameters characterize the transition of a crack to the unstable condition during loading of a sample or part. However, several reasons may be given which provide a basis for not completely identifying these values and lead to a numerical difference in them. The first and apparently the most important is that in determining the static and cyclic fracture toughnesses the samples have different loading histories preceding the last loading all the way to crack instability. In cyclic tests the crack is subjected to the last load cycle with a significantly different and~ as a rule, larger plastic zone at the crack tip and a higher level of structural damage of the material but with greater residual compressive stresses at the tip. Depending upon the class of the material,

1259

its capacity to be strengthened or lose strength cyclically, and certain other properties [31, 32], this may have a substantial influence both on the level of the critical local failure stress ~c [33-35] and on the macroscopic failure load, leading to different values of Kfc(Kif c) and Kc(Kic). The second reason for the possible difference between static and cyclic fracture toughness is related to the difference in actual loading rates at the crack tip [35]. Yet another reason which may cause a difference in the values of Kfc(KIfc) and Kc(Kic) , especially in high-frequency fatigue tests, is caused by local heating of the material at the crack tip. These reasons may have different influences on the measured fracture toughness, which leads to the necessity of using either Kc(KIc) or Kfc(Kife) , depending upon the character of loads to which the material is subjected during service. Paris and Erdogan [36] indicated the necessity of differentiating these two types of fracture toughness. The author of [37] established that for alloy steel Kif c
mechanical properties of unnotched samples. Obviously, we assume a similar approach in describing cyclic fracture toughness on the basis of the parameters of the cyclic work-hardening curve. LITERATURE 1. 2.

3. 4. 5. 6. 7. 8.

9. I0. ii. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21.

22. 23. 24. 25. 26.

CITED

J.C. Grosskreutz, "Fatigue mechanisms in the subcreep range," in: Metal Fatigue Damage (ASTM STP N 495), Philadelphia (1971), pp. 5-60. S.E. Gurevich and L. D. Edidovich, "The rate of crack propagation and the threshold values of the stress intensity factor in failure," in: The Fatigue and Fracture Toughness of Metals [in Russian], Moscow (1974), pp. 36-78. C.E. Riehards and T. V. Lindley, "The influence of stress intensity and microstructure on fatigue crack propagation in ferritic materials," Eng. Fract. Mech., 4, No. 4, 951-978 (1972). R. Koterazava, M. Mori, T. Mattsui, and D. Simo, "Fractographic investigation of fatigue crack propagation," Teor. Osn. Inzh. Raschetov., No. 4, 7-18 (1973). G.A. Miller, "Fatigue fracture appearance and the kinetics of striations formation in some highstrength steels," Trans. ASM, 6__22,No. 6, 651-658 (1969). D. Broek, "Some contributions of electron fractography to the theory of fracture," Int. Met. Rev., 185, 19, 135-182 (1974). S. Ya. Yarema, A. Ya. Krasovskii, O. P. Ostash, and V. A. Stepanenko, "The development of fatigue failure in low carbon sheet steel at room and low temperatures," Probl. Prochn., No. 3, 21-26 (1977). A. Ya. Krasovskii, O. P. Ostash, V.A. Stepanenko, and S. Ya. Yarema, "The influence of low temperatures on the rate and microfraetographie features of fatigue crack development in low-carbon steel," Probl. Prochn., No. 4, 74-78 (1977). J.C. McMillan and R. M. N. Pelloux, "Fatigue crack propagation under program and random loads," in: Fatigue Crack Propagation (ASTM STP N 415), Philadelphia (1967), pp. 505-532. O.M. Romaniv, Yu. V. Zima, and G. V. Karpenko, Electron Fractography of Strengthened Steels [in Ukrai,~ian], Naukova Dumka, Kiev (1974). K.D. Bichem, "Microproeesses of failure," in: Failure [in Russian] , Vol. I, Moscow (1973), pp. 265375. J.J. Gilman, "Spalling, ductility, and fracture toughness," in: The Atomic Mechanism of Failure [in Russian], Moscow (1963), pp. 220-250. B.S. Kasatkin, The Structure and Micromechanism of Brittle Failure of Steel [in Russian], Tekhnika, Kiev (1974). A. Kh. Kottrell, "Theoretical fundamentals of the failure process," in: The Atomic Mechanism of' Failure [in Russian] , Moscow (1963), pp. 30-58. D. Mac Lin, The Mechanical Properties of Metals [in Russian], MetaUurgiya, Moscow (1965). J. Friedel, Dislocations, Pergamon (1964). D.T. Khan, B. L. Averbakh, V. S. Ouen, and M. Koen, "The occurrence of spalling mieroeracks in polycrystalline iron and steel," in: The Atomic Mechanism of Failure [in Russian], Moscow (1963), pp. 109-137. T. Ekobori, The Physics and Mechanics of Strength and Failure of Solids [in Russian] , Metallurgiya, Moscow (1971). C. Laird, "The influence of metallurgical structure on the mechanisms of fatigue crack propagation," in: Fatigue Crack Propagation (ASTM STP N 415), Philadelphia (1967), pp. 131-168. C. Laird and R. De la Veaux, "Additional evidence for the plastic blunting process of fatigue crack propagation," Met. Trans., SeE A, _8, No. 4, 657-664 (1977). R.J.J. Wanhill, "Fatigue striation mechanisms in metals," in: Proceedings of the Second International Conference on the Mechanical Behavior of Materials, ICM-II, Boston, June 1976, Vol. i, Boston(1976), pp. 558-562. J. Schijve, "Discussion," in: Fatigue Crack Propagation (ASTM STP N 415), Philadelphia (1967), pp. 533-534. B. Tomkins and W. D. Biggs, "Low endurance fatigue in metals and polymers, Part 3, The mechanisms of failure," J. Mater. Sci., 4, No. 6, 544-553 (1969). C.Q. Bowles, A Study of the Crack Tip Geometry Resulting from Fatigue Crack Propagation in Air and Vacuum, Rep. LR-261, Delft Univ. of Technology, Delft (1978)o R.M.N. Pelloux, "Mechanisms of formation of ductile fatigue striations," Trans. ASM, 62, No. 2, 281-285 (1969). D. Brock and C. Q. Bowles, "Fatigue crack propagation mechanism," Int. J. Fracture Mech., 6_, 321331 (1970). 1261

27.

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29.

30. 31.

32. 33.

34.

35.

36. 37. 38. 39.

40. 41. 42.

43. 44. 45. 46. 47. 48. 49.

1262

A. Ya. K r a s o v s k i i , V. A. Stepanenko, and N. D. Bega, "The use of the scanning e l e c t r o n m i c r o s c o p e for a quantitative s t e r e o f r a c t o g r a p h i c a n a l y s i s of fatigue f r a c t u r e s , " Probl. P r o c h n . , No. 6, 35-38 (1977). A. Ya. K r a s o v s k i i and V. A. Stepanenko, "A study of the fatigue c r a c k propagation m e c h a n i s m in nickel by the method of quantitative s t e r e o s c o p i c f r a c t o g r a p h y , " Probl. P r o c h n . , No. 11, 86-94 (1978) [ P r e print, Inst. Probl. Prochn. Akad. Nauk UkrSSR, Kiev (1977)]. A. Ya. (J.) K r a s o v s k i i (Krasowsky) and V. A. Stepanenko, "A quantitative s t e r e o s c o p i c f r a c t o g r a p h i c study of the m e c h a n i s m of fatigue c r a c k propagation in nickel," Int. J. F r a c t u r e , 15, No. 3, 203-215 (1979). C . Q . Bowles, An E x p e r i m e n t a l Technique for Vacuum Infiltration of C r a c k s with P l a s t i c and Subsequent Study in the Scanning E l e c t r o n M i c r o s c o p e , Rep. LR-249, Delft Univ. of Technology, Delft (1977). V . T . T r o s h c h e n k o , V. V. P o k r o v s k i i , Yu. S. Skorenko, V. M. Koshelev, and P. V. Yasnii, "The influence of cyclic loading r a t e on c r a c k r e s i s t a n c e c h a r a c t e r i s t i c s of s t e e l s . R e p o r t I," Probl. Prochn., No. 8, 7-10 (1980). V. T. T r o s h c h e n k o and V. V. P o k r o v s k i i , "The influence of cyclic loading r a t e on c r a c k r e s i s t a n c e c h a r a c t e r i s t i c s of s t e e l s . R e p o r t II," P r o b l . P r o c h n . , No. 9 (1980). G . S . P i s a r e n k o and A. Ya. (J.) K r a s o v s k i i (Krasowsky), "Analysis of kinetics of q u a s i b r i t t l e f r a c t u r e of c r y s t a l l i n e m a t e r i a l s , " in: Mechanical Behavior of M a t e r i a l s . P r o c e e d i n g s of the 1971 International Conference on the Mechanical Behavior of M a t e r i a l s , Vol. 1, Kyoto (1972), pp. 421-432. G . S . P i s a r e n k o and A. Ya. (J.) K r a s o v s k i i (Krasowsky), "Analysis of c r i t e r i a for ultimate s t a t e of m a t e r i a l at the c r a c k tip in brittle f r a c t u r e of m e t a l s , " in: P r o c e e d i n g s of the Second International Conf e r e n c e on the Mechanical B e h a v i o r of M a t e r i a l s , Boston, 1976, ASM, Ohio (1978), s p e c i a l vol., pp. 348-376. G . S . P i s a r e n k o , A. Ya. (J.) K r a s o v s k i i (Krasowsky), and T. Yokobori, "Studies on t e m p e r a t u r e - r a t e s e n s i t i v i t y of plastic flow s t r e s s and on f r a c t u r e toughness," Rep. R e s . Inst. Strength F r a c t u r e Mater. Tohoku Univ., Sendal, Japan, 13, No. 1, 1-57 (1977). P . C . P a r i s and F. Erdogan, "A c r i t i c a l analysis of c r a c k propagation laws," T r a n s . ASME, Ser. D, 8_55, No. 3, 528-536 (1963). T . W . C r o o k e r , "Fatigue c r a c k p r o p a g a t i o n and plane s t r a i n f r a c t u r e toughness c h a r a c t e r i s t i c s of 9 N i 4 C o - 0 . 2 5 C s t e e l , " T r a n s . ASM, 61, No. 3, 568-574 (1968). T. Y o k o b o r i a n d T. A i s a w a , "A p r o p o s a l for the concept of fatigue f r a c t u r e toughness," Rep. R e s . Inst. Strength F r a c t u r e Mater. Tohoku Univ., Japan, 6, No. 1, 19-23 (1970). V . V . P o k r o v s k i i , "An investigation of the influence of low t e m p e r a t u r e s and the f o r m of loading on the r u l e s of fatigue failure of a n u m b e r of c o n s t r u c t i o n a l s t e e l s and alloys," A u t h o r ' s A b s t r a c t of Candidate's D i s s e r t a t i o n , Technical Sciences [in R u s s i a n ] , Kiev (1972). V . S . Ivanova and V. G. Kudryashov, "A method of d e t e r m i n i n g f r a c t u r e toughness (Kic) f r o m fatigue t e s t data," P r o b l . P r o c h n . , No. 3, 17-19 (1970). V . G . Kudryashov, " D e t e r m i n i n g f r a c t u r e toughness f r o m fatigue t e s t r e s u l t s , " in: P r o b l e m s of the F a i l u r e of Metals [in R u s s i a n ] , Moscow (1975), pp. 41-49. T. Kawasaki, S. Nakanishi, Y. Sawaki, K. Natanaka, and T. Yokobori, " F r a c t u r e toughness and fatigue c r a c k p r o p a g a t i o n in h i g h - s t r e n g t h s t e e l f r o m r o o m t e m p e r a t u r e t o - 180~ Eng. F r a c t . Mech., 7, 465-472 (1975). A . B . Kaplun, "The influence of load cycle p a r a m e t e r s on the growth of fatigue c r a c k s , " F i z . - K h i m . Mekh. M a t e r . , 1__44, No. 4, 58-68 (1978). V . S . Ivanova, S. E. Gurevich, I. M. Kop'ev, et al., "The fatigue and b r i t t l e n e s s of metallic m a t e r i a l s " Nauka, Moscow (1968). S. Ya. Y a r e m a , "Investigation of the growth of fatigue c r a c k s and kinematic fatigue failure c u r v e s , " F i z . - K h i m . Mekh. M a t e r . , 1_.33, No. 4, 3-22 (1977). V . T . T r o s h c h e n k o , A. V. Prokopenko, and V. V. P o k r o v s k i i , "Investigation of the c h a r a c t e r i s t i c s of f r a c t u r e toughness of m e t a l s in cyclic loading. R e p o r t I," Probl. P r o c h n . , No. 2, 8-15 (1978). S. Ya. Y a r e m a , "Some questions of the method of testing m a t e r i a l s for cyclic c r a c k r e s i s t a n c e , " F i z . Khim. Mekh. M a t e r . , 1..~4, No. 4, 68-77 (1978). L . I . D o m o z h i r o v and G. Z. Z a i t s e v , "An investigation of the development of fatigue c r a c k s in 00Kh12NZD and 15Kh2NMFA s t e e l s , " F i z . - K h i m . Mekh. M a t e r . , 1__4, No. 4, 93-98 (1978). A. Ya. K r a s o v s k i i , A. P. Fedosov, V. A. Vainshtok, et al., "An investigation of the r e s i s t a n c e to c r a c k d e v e l o p m e n t taking the s c a l e effect into c o n s i d e r a t i o n for rating the brittle s t r e n g t h of a casing," Probl. P r o c h n . , No. 4, 3-9 (1979).

50. 51. 52. 53.

O.N. Romaniv and A. N. Tkach, "Micromechanical simulation of the fracture toughness of metals and alloys," Fiz.-Khim. Mekh. Mater., I__33,No. 5, 5-22 (1977). G.S. Pisarenko and A. A. Lebedev, The Deformation and Strength of Materials in the Complex Stressed State [in Russian], Naukova Dumka, Kiev (1976). A. Ya. Krasovskii and V. A. Vainshtok, "A criterion of failure of materials taking into consideration the form of the stressed condition at a crack tip," Probl. Prochn., No. 5, 64-69 (1978). A. Ya. Krasovskii, The Brittleness of Metals at Low Temperatures [in Russian], Naukova Dumka, Kiev (1980).

ELASTOPLASTIC

DEFORMATION

AND

MATERIALS

UNDER

CONDITIONS

STRESSED

CONDITION

WITH

PROPORTIONAL N.

OF VARIOUS

FAILURE THE

OF

PLANE

METHODS

OF

LOADING S.

Mozharovskii

and

N.

I. Bobyr'

UDC

539.43

An investigation of the rules of elastoplastic deformation and failure of materials under conditions of the plane stressed condition with various methods of proportional loading was made on equipment specially built in the Department of Resistance of Materials of Kiev Polytechnic Institute [i]. The plane stressed condition was created in a thin-walled cylinder (Fig. la) as a result of the simultaneous action of a torsional moment M z and a tensile force N. The loading was done using various straightline trajectories shown in Fig. ib, e in the coordinates ofA. A. IITyushin both in the space of stresses and in the space of deformations. As investigations of materials in short-term static loading have shown, deformation curves of a material at a given temperature and with loading of it on different trajectories do not agree. The results shown in Fig. 2 indicate that the hypothesis of the existence for a given material of a "single" deformation curve independent of the form of the stressed condition is approximately confirmed. From Fig. 2a it may be seen that the deformation curves obtained for IKhI8N9T steel at T = 600~ with a load in the plane of the vector of stresses in various trajectories do not coincide. At the same

time, the angle of the direction of the vector of stresses

varied within the limits

0 ~ ~s = arctg v 31:ez ~. 2(Yzz

.

(2)

D e f o r m a t i o n c u r v e s obtained for :S10C s t e e l (T = 20~ in loading in the plane of the v e c t o r of d e f o r m a tions E'p on different t r a j e c t o r i e s [2, 3] a l s o do not a g r e e (Fig. 2b, c). In this e a s e the angle of the d i r e c t i o n of the v e c t o r of d e f o r m a t i o n s --~

70Zp

v a r i e d within the l i m i t s 0 ~o~F~ = arctg ~

2~.

(4)

ZZp

F r o m the r e s u l t s given it follows that constancy of the r e l a t i o n s h i p between intensities of s t r e s s e s and plastic d e f o r m a t i o n s is disturbed. H o w e v e r , s o m e i n v e s t i g a t o r s maintain the opinion that for pure metals the r e l a t i o n s h i p between the intensity of s t r e s s e s and the intensity of plastic d e f o r m a t i o n s does not depend upon Kiev Poiytechnic Institute. T r a n s l a t e d f r o m P r o b l e m y P r o c h n o s t i , No. 10, pp. 73-78, October, 1980. Original a r t i c l e s u b m i t t e d A p r i l 29, 1980.

0039-2316/80/1210-1263507.50

9 1981 Plenum Publishing C o r p o r a t i o n

1263