PROBLEMS OF METAL FRACTURE*

J. Hollomon and C. Zener

I. INTRODUCTION

The strength or hardness of our modern metals cannot be increased by known methods without imparting brittle properties to the metals. A practical increase in strength can therefore be achieved only as a result of clarifying the problems of metal fracture. Many questions concerning the dynamics of crack propagation are still not fully understood, such as, for example, the factors determining the rate and direction of propagation. However, the solution of these questions will not lead to an increase in the useful strength of metals, since a crack, regardless of the exact path and speed of its propagation, renders a part unfit for use. The present article is therefore devoted to problems relating chiefly to the initial stage of the process of metal fracture.

II. GENERAL PICTURE OF METAL FRACTURE

It has long been known that metals, ductile under some conditions, fracture brittly under others. Thus, metals are often plastic in simple tension at moderate temperatures, but brittle at low and high temperatures or under a more complex system of stresses. It would be highly desirable to find a reliable scheme by means of which, from measurements carried out under one set of conditions, it would be possible to predict how fracture of a metal will occur under other conditions. Such a scheme was proposed in 1909 by Ludwik^1. Its basic premise is the recognition of the circumstance that the stress at the moment of fracture of a plastic material is the fracture stress of the material that has undergone a critical strain, and not of the original material. Ludwik considered two functions of strain, shown in Fig. 1. One of them is the “flow stress,” defined as the tensile stress necessary to cause further plastic deformation. The other function is the actual frac-

* Journal of Applied Physics, 17, 82 (1946), translated by N. Fuchs.

...ing stress, defined as the tensile force that would be required for fracture of the material if no further plastic deformation were to take place. Under tension the metal deforms plastically as long as the flow stress is less than the actual fracture stress. Fracture occurs at the deformation corresponding to the point of intersection of the curves of flow stress and actual fracture stress. The advantage of Ludwik’s scheme is that it reduces the influence of several factors—the distribution of stresses, temperature, and rate of deformation—on the fracture of a metal to their influence on the magnitude of the flow stress and the actual fracture stress. The success of the scheme depends on how reliably the dependence of these two functions on the various factors is determined. The drawback of Ludwik’s scheme, which prevented it from becoming generally accepted, is that the actual fracture stress can be measured directly only at one value of the deformation, namely at the deformation at the moment of fracture of the metal. For other values of deformation the magnitude of the actual fracture stress can be calculated only approximately.

Fig. 1. Ludwik’s representation of metal fracture.

The method for finding the curve of the actual fracture stress, first applied by Davidenkov and Wittman\(^{2}\), and subsequently by us\(^{3}\), comes closest to the goal. This method is applicable to all metals that are capable of undergoing considerable deformation at room temperature but are brittle at some lower temperature. According to this method, specimens are first subjected to tensile deformation of various magnitudes at room temperature; then the temperature is lowered, and the specimens are fractured without any further increase in deformation. The fracture stress measured at the low temperature is plotted as a function of the deformation at room temperature. The approximate position of the curve of the actual fracture stress at room temperature is then determined on the assumption that it is parallel to the curve found at the low temperature. The method is illustrated in Fig. 2. It can be improved by carrying out measurements of the fracture stress for a series of low temperatures and thereby partly justifying the indicated, not fully substantiated, assumption.

Pearlitic steels (containing lamellar carbides) and some age-hardening alloys are brittle at low temperatures, and for them the curves of actual fracture stress may

be found by the indicated method. Typical results for pearlitic steel are presented in Fig. 3. All specimens deformed at room temperature by more than a few percent fractured at low temperature practically without further deformation. A specimen not deformed at room temperature, however, had at low temperature an upper and lower yield point (Fig. 4) and fractured in fact at a lower stress than that which it had withstood before deformation. Hence one may conclude that, at least for this type of steel*) the curve of the true fracture stress has the form shown in Fig. 5. It is very interesting that the true fracture stress thus determined at very small strains sometimes only slightly exceeds the corresponding tensile stress. Any change in the parameters that increases the tensile stress relative to the true fracture stress is capable of making the metal more brittle.

Fig. 2. Method of finding the curve of the true fracture stress.

Fig. 3. Effect of preliminary deformation on the fracture stress at −190° C (pearlitic steel).

As already indicated, the above-described method cannot be applied to finding the curves of the true fracture stress of metals that are not brittle at low temperatures. For such metals it would be necessary to use other testing methods, in which fracture of the specimens would be achieved without deformation. It has not yet been possible to find testing conditions that would impart brittle properties to metals as conveniently as this is achieved by cooling (in the case of pearlitic steels).

*) Similar results were obtained with specimens of other steels similar to these.

The actual fracture stress may possess a high degree of anisotropy. The most common example of anisotropy is encountered in rolled metal: rolling increases the actual fracture stress in the direction of rolling and lowers it in the transverse direction. In an earlier paper one of us[^4] noted that the longitudinal rupture of certain steels can be explained only on the basis of the anisotropic effect of deformation on the fracture stress. As already indicated, tension raises the tensile stress required to rupture a specimen along a plane perpendicular to the axis of tension. In order for a specimen subjected to tension to be able to tear longitudinally, the stress required to rupture the metal along a plane parallel to the axis must decrease so much that the stress caused by necking and directed circumferentially could fracture the metal.

Fig. 4. Reduction of fracture stress due to initial deformation (pearlitic steel at —190°C).

Conversely, prior compression lowers the tensile stress necessary for the rupture of a specimen during its subsequent tension. Examples of this effect of compression are often encountered in bending. Thus, a specimen bent so far that it can still withstand further bending in the same direction may break in a brittle manner when an attempt is made to straighten it. The anisotropy of the actual fracture stress can also be demonstrated visually by stretching specimens previously subjected to plastic torsion. As Swift[^5] found, if a specimen had been twisted sufficiently strongly before tension, fracture occurred not along the plane corresponding to the maximum tensile stress, i.e., the transverse plane, but along a helical surface. Although many examples of anisotropy of the actual fracture stress are known, up to now it has not yet been studied quantitatively. In particular, no measurements have been carried out to determine the relation between the actual fracture stress and the reduction of cross section during rolling

Fig. 5. Assumed form of the curve of actual fracture stress (for pearlitic steels).

In certain frequently encountered conditions, plastic deformation in one transverse direction is impossible because of the presence of constraints. Many experiments have been carried out to determine the influence of uniaxial transverse stress on the breaking stress. However, in order to interpret the results of such experiments correctly, it is necessary, when conducting them, to allow changes only in the stress distribution, since in such tests, when the stress distribution is changed, other factors also often change—for example, the deformation at the moment of fracture. Let us consider, as an example, a thin-walled tube subjected simultaneously to longitudinal tension and internal pressure. Because of the presence of transverse stress, the tensile force required for plastic flow increases, and the deformation at the moment of fracture will also change (Fig. 6), provided only that the curve of the true breaking stress does not shift in a similar way. Moreover, if the circumferential stress in the tube is made greater than the axial stress, then instead of a transverse rupture a longitudinal one may occur. Owing to anisotropy, the stress required for rupture in the new direction may have a completely different value and, what is still more important, may be in an entirely different relation to the deformation. The tensile tests by Siebel and Maier^6 on hollow cylinders subjected simultaneously to internal pressure show that uniaxial transverse stress does not affect the magnitude of the true breaking stress, provided it is assumed that in the single-phase materials used by these authors the true breaking stress did not depend on deformation. There is also indirect confirmation of the proposition that transverse stress, when transverse deformation is constrained in one direction, does not change the true breaking stress: thus, the brittle fracture of pearlitic steels in impact tests of notched specimens can be explained only on the assumption that uniaxial transverse stress affects the true breaking stress much less than it affects the flow stress^7. It is highly desirable to have direct data on the dependence of the true breaking stress on uniaxial transverse stress.

Fig. 6. Supposed influence of transverse stress on the flow curve and on the magnitude of deformation at the moment of fracture (if the true breaking stress does not change).

Less common is the case of biaxial transverse tension. By tensile testing specimens with grooves of different depth and radius of curvature along the circumference of the specimens, Kuntze\(^8\), MacAdam\(^9\), and Sachs\(^10\) attempted to determine the influence of biaxial transverse tensile stress on the actual fracture stress, which they designated as “technical strength.” In these measurements the mean longitudinal stress at the moment of fracture was taken as the fracture stress. Attempts to interpret the dependence of this stress on the various test conditions usually encounter very great difficulties. According to the conclusions of the authors cited, by means of such measurements it is possible to determine the influence on the magnitude of the actual fracture stress of three separate factors: the amount of deformation, the character of the stresses, and their distribution throughout the entire volume of the specimen.

Fig. 7. Dependence of fracture stress on biaxial transverse tension (according to Sachs' data\(^10\)).

In one case Sachs and his collaborators succeeded in obtaining the same relative deformation \((0.026—0.028)\) at the moment of rupture of the specimen for several depths and radii of rounding of the grooves; thus the factor of the magnitude of deformation was eliminated here. If it is assumed that, for different depths and radii of rounding in the grooves, the stress distribution throughout the entire unnotched region remains essentially constant, then the observed change in fracture stress may be regarded as the effect of triaxiality of stress (the ratio of transverse stress to longitudinal stress). The results obtained by Sachs are presented in Fig. 7. The reliability of these results is evident from the good agreement of the measured fracture stress with the calculated yield stress at the moment of rupture of the specimen. The increase in the actual fracture stress with increasing biaxial transverse tension, shown in Fig. 7, confirms the earlier conclusions of MacAdam and his collaborators. The constancy of deformation at the moment of rupture means that,

that in this particular case the actual breaking stress increases with increasing biaxial transverse tension at exactly the same rate as the flow stress and, consequently, that biaxial transverse tension does not increase the brittleness of the metal. This conclusion is sharply at variance with the generally accepted views on the influence of triaxial stresses. Investigations of the type proposed by Sachs should therefore be continued.

The fracture of steel occurs most readily under the action of impact at low temperatures. It is known that the flow stress in all steels increases both when the temperature is lowered and when the rate of deformation is increased. As experiments have shown,^3 the actual breaking stress also increases with decreasing temperature, but not as rapidly as the flow stress. The relative increase of the flow stress with respect to the actual breaking stress that is thereby obtained leads, in the case of pearlitic steels, to brittleness. Owing to the comparatively small effect of the small changes in the rate of deformation attainable in ordinary types of mechanical testing machines, and owing to the comparatively large scatter of test results, no conclusions can at present be drawn about the influence of the rate of deformation on the actual breaking stress.

Under certain conditions, almost brittle fracture of metals is possible even at high temperatures. As Rosenhain and Archbutt^11 first observed, these are conditions favorable to sliding along the boundaries between grains. Under these conditions it may be assumed that the actual breaking stress decreases with time, even for a small amount of deformation. This idea of a decrease of the actual breaking stress with time suggests new and interesting experiments that have not yet been carried out. Thus, if the actual breaking stress of a specimen is first lowered by applying a load at an elevated temperature, and the temperature of the specimen is then lowered to room temperature without removing the load, the actual breaking stress should remain reduced and should thus lead to brittle fracture at room temperature.

III. MECHANISM OF FRACTURE OF METALS

It has long been known that the observed values of the breaking stress in metals and, in general, in all crystalline substances are at least two orders of magnitude smaller than the stress calculated from the forces acting between atoms. The only possible explanation of this discrepancy between theory and experiment is the assumption that fracture of a material occurs at any given moment only in individual regions in which the stress is considerably higher than the average applied stress.

Stress concentration in individual regions was first studied in detail by Inglis \(^{12}\), who pointed out that the tensile stress at the bottom of a notch may considerably exceed the average stress over the whole specimen. The stress concentration factor is, in this case,

\(1+2(a/\rho)^{1/2}\),

where \(a\) is the depth of the notch, and \(\rho\) is the radius of curvature of the bottom of the notch. Joffe \(^{13}\) and his collaborators found that the discrepancy between the usually observed and the theoretically calculated value of the breaking stress for rock-salt crystals can be fully explained by the presence of small surface cracks acting as notches. By immersing rock-salt specimens in water during testing and thereby continuously dissolving the surface layer, these authors were able to increase the actual breaking stress 400-fold. Under these conditions the breaking stress is indeed almost equal to its theoretical value.

Carefully prepared metal specimens apparently do not have the defects found in dry rock-salt crystals. The insufficient strength of metallic specimens is caused by defects within the metal itself: indeed, if only the deformation (in tension) has reached the stage of neck formation, fracture begins inside the specimen and propagates toward the periphery. The fact that fracture must begin inside specimens drawn into a neck is clear from Bridgman’s \(^{14}\) analysis of the stress distribution in the necked region. With this distribution, the tensile stress reaches a maximum along the axis and a minimum at the surface of the specimen. One should expect that metal crystals possessing the same degree of purity as the rock-salt crystals used by Joffe would exhibit the same high relative strength. However, up to the present no experiments have been carried out that could confirm the correctness of this view.

The comparatively low breaking stress of real metals is most easily explained by the presence of microcracks within the metal itself. Starting from the observation that a crack will propagate further only if the total free energy of the system thereby decreases, Griffith \(^{15}\) came to the conclusion that a circular crack of radius \(a\) can propagate only on the condition that the tensile stress in the direction perpendicular to the crack exceeds a certain critical value \(S\). In Griffith’s formula

\[ S=(6G/ca)^{1/2} \tag{1} \]

\(\sigma\) denotes the surface energy per unit surface area, \(G\) is the shear modulus, and \(c\) is a constant of order unity. Griffith derived his equation and obtained its experimental confirmation only for amorphous substances. The resistance of such substances to plastic

deformation increases rapidly as the temperature is lowered, so that at sufficiently low temperatures—for example, at room temperature for ordinary glass—the propagation of a crack may not be accompanied by plastic deformation. Thus, in amorphous substances, in which the new surface produced as a result of crack propagation has not undergone plastic deformation, the magnitude of the surface energy to be inserted in equation (1) is very close to the easily measurable surface energy of the substance in the molten state. In the case of crystalline substances the situation is more complicated. On the one hand, their resistance to plastic deformation increases comparatively slowly as the temperature is lowered, so that it is usually not possible to eliminate this deformation simply by lowering the temperature. On the other hand, the laws of plastic deformation found by macroscopic measurements are inapplicable if the stress changes appreciably over distances smaller than the linear dimensions of an individual crystal. A stress that considerably exceeds the yield point does not necessarily lead to plastic deformation if it acts over a sufficiently limited region. Until now the question has not been studied of the conditions under which crack propagation can occur without being accompanied by plastic deformation. If plastic deformation does occur, the energy associated with it must be added to the surface energy $\sigma$ in order to make equation (1) applicable in the given case.

Fig. 8. Reorientation of microcracks during deformation.

Although the unknown exact value of the surface energy $\sigma$ does not permit a quantitative application of equation (1) to metals, Griffith’s ideas can nevertheless be used qualitatively to relate various phenomena observed in the fracture of metals[^16]. These phenomena are associated with the reorientation and distortion of microcracks during deformation. Fig. 8 shows how an initially random distribution of microcracks acquires, during deformation, a highly anisotropic character. Thus, under tension the cracks lengthen in the direction of the tensile axis and shorten in the transverse direction. The critical tensile stress required in order to cause propaga-

...ing of certain cracks and thereby lead to fracture, is increased by deformation in the case of longitudinal tension and decreased in the case of transverse tension. We encounter the most common example of this phenomenon in the rolling of metal: the breaking stress increases in the direction of rolling and decreases in the transverse direction. Ordinary tensile tests of certain steels may serve as another example, in which preliminary stretching increases the breaking stress in the plane perpendicular to the axis of the specimen and decreases it in the plane parallel to the axis. Indeed, as was indicated above, the breaking stress under the action of transverse tension may decrease so much that the transverse tensile stress existing in the neck of the stretched specimen leads to a longitudinal, and not transverse, rupture of the specimen. The most striking example of the influence of reorientation of microcracks on the fracture of metal is encountered in the case when the specimen is first plastically twisted and then stretched. The spiral crack observed in this case belongs precisely to the type that could have been expected on the basis of an analysis of the reorientation effect[^16].

At the same time, while the effect of reorientation of microcracks is readily understandable, there is great uncertainty in the question of the nature and origin of microcracks. Nonmetallic inclusions such as oxides and sulfides of iron contained in steel undoubtedly lead to stress concentration and may, for certain purposes, be regarded as microcracks that are the cause of fracture. In fact, it is known that the strength of steel is the lower the “dirtier” it is, i.e., the more nonmetallic inclusions it contains. It is also known that hot working entails a reorientation of microcracks which does not disappear during heat treatment causing recrystallization of the main mass of the metal. Thus the reorientation effect can be attributed only to the action of nonmetallic inclusions. However, the fracture of metal cannot be described solely on the basis of the role played by nonmetallic inclusions; it is known that the structure of the main mass of the metal also has a substantial influence on the breaking stress. Thus, in steel, the martensitic structure after tempering (spheroidal carbides) has a higher breaking stress at small deformations than the pearlitic structure (lamellar carbides), which offers the same resistance to plastic deformation; for a definite type of structure the breaking stress is the higher, the finer the carbide particles. The sources of stress concentration must, consequently, be contained in the main mass of the metal. The stress concentration at a definite point of the metal is thus a function of two factors, determined by impurities and by the structure of the main mass of the metal. On the basis of the available experimental material it is still impossible to determine the relative role of these factors.

There is also considerable uncertainty regarding the origin of stress concentration in the main mass of the metal. If the latter contains precipitated particles, such as carbides in steel, these particles are sources of a certain stress concentration. The magnitude of the stress concentration depends on the size and shape of these particles; as already indicated, particles having the form of platelets lead to a more significant stress concentration and, consequently, to lower values of the breaking stress than spheroidal particles. A paper by one of the authors of the present article (Hollomon) will appear shortly, in which it is shown that as the size of carbide particles increases, the stress concentration associated with them increases. These data, however, are not sufficiently complete for precise quantitative conclusions.

Regardless of the presence of precipitated particles, stress concentration in the metal also appears as a result of the process of plastic deformation itself. As was first indicated by Orowan^17, stress concentration occurs at the edges of slip bands. One of the authors of the present article^18 gave examples of metal fracture that can be explained only on the assumption that such stress concentration is capable of increasing with time exactly as though the slip bands possessed viscous properties. It should be assumed that such stress concentration must increase with the widening of slip bands and, consequently, with increasing grain size. Up to the present time, the influence of grain size, as such, has not been detected. It is generally recognized that steels with a small grain size possess better mechanical properties than steels with a large grain size. However, in steels the grain size substantially affects the character of carbide precipitation, and therefore it is impossible to determine the direct influence of grain size. The stress concentration associated with plastic deformation possesses the exceptional property that it is absent until some plastic deformation has occurred. Thus, it should be expected that the breaking stress of well-annealed specimens will, before a small plastic deformation, be higher than after it. The results presented in Figs. 4 and 7 should be regarded as further confirmation of the view that the actual tearing stress of undeformed material may be very high. Still another confirmation of the close connection between the breaking stress and plastic deformation is the observation that the breaking stress changes with the rate of deformation and with temperature in approximately the same way as the resistance to plastic deformation. Up to the present time no experiments have been undertaken with the aim of determining precisely what stress an undeformed specimen can withstand without breaking.

Since stress concentration can arise as a result of plastic deformation itself, it is not excluded that the strength

of specimens can be changed by changes in the testing conditions. An example of this phenomenon was recently described by Bridgman^19. He found that, with identical tension of two identical specimens—one under atmospheric pressure and the other under high pressure—the latter then withstands, under atmospheric pressure, a considerably greater additional deformation than the former and, consequently, has a much higher fracture stress. Apparently, under high pressure the greater concentration of stresses cannot cause the opening of microcracks. A similar effect could be produced by an initial deformation at elevated temperature, provided that the temperature is not so high as to cause recovery or recrystallization. At elevated temperature a smaller stress would be required to produce a given amount of tension than at room temperature, and therefore, possibly, fewer microcracks would be formed. Experiments to prove this dependence of the strength of specimens on the temperature of preliminary deformation have not yet been carried out. It is possible that the formation of twin bands leads to a greater concentration of stresses at their edges than the formation of slip bands, since some experiments indicate a connection between twin bands and brittle fracture of metals^20.

As early as 1920, Rosenhain and his coworkers^11 convincingly showed that the boundaries between grains possess viscous properties and that, therefore, a low rate of deformation and a high temperature favor sliding along these boundaries, and that such sliding leads to premature fracture of the metal. Although Rosenhain did not state this directly, it is evident that premature fracture is caused by a concentration of stresses produced by viscous sliding along grain boundaries. Unfortunately, the concept of the viscous properties of boundaries between grains has not been sufficiently used by metallurgists in the USA. As a result, in the 25 years that have elapsed since the appearance of Rosenhain’s work, no progress has been made in investigating the factors affecting the formation of the stress concentration caused by viscous sliding along grain boundaries.

The indicated concept suggests a number of interesting experiments that have not yet been carried out. For example, the stress concentration produced by the application of a force at high temperature can be frozen in by cooling to room temperature under load, and the metal will thereby acquire brittle properties.

A specimen capable of withstanding a certain stress for an indefinitely long time may fracture if it is subjected to a stress that repeatedly changes its sign. Fracture under alternating stress, called fatigue, has been known for more than 80 years. The practical importance of fatigue has led to the accumulation of numerous experimental data obtained...

for various metals, on the relation between the number of cycles required for fracture and the maximum applied stress under various conditions. In the USA little has yet been done on the question of the mechanism of fracture caused by fatigue. At the British National Physical Laboratory a detailed descriptive study was carried out of the initiation and propagation of fatigue cracks in single and polycrystals. The earlier works, which also consider gradual changes in the hysteresis loops, are discussed in the monograph of Moore and Kommers ^21. A review of the descriptive investigation begun at the National Physical Laboratory by Orowan ^22 with collaborators and continued by Gough and collaborators, on the question of the formation of fatigue cracks and their connection with slip bands, was compiled by Gough ^23.

Fig. 9. Stress distribution near a newly formed slip band.

No attempt was made to interpret the results of these observations from the standpoint of the microstructure, with the exception of the concept of fatigue introduced by Orowan and Hempel ^22, according to which the structure of slip bands, as alternating stress is applied, becomes increasingly disordered.

A qualitative analysis of the stress distribution near a slip band leads to a qualitative explanation of the basic facts in the phenomenon of fatigue. Such a stress distribution is shown in Fig. 9 for the case when the slip band has already formed, but the applied force has not yet been removed. The shearing stress in the regions located on both sides of the slip band has had time to decrease owing to partial relaxation. Conversely, in the regions immediately adjoining the edges of the slip band, the shearing stress has reached a large value. The high tensile stress associated with this leads to the formation of small cracks, as shown in the figure. The dimensions of the cracks are too small for them to be able to propagate, under the action of the applied stress, beyond the limits of the stress-concentration zone. Since the size of this zone is comparable with the width of the slip band, the length of the cracks is also comparable with the width of the band. Upon reversal of the stress, those regions in which relaxation had previously occurred will now be subjected to a stress exceeding the applied force. If initially

the formed slip band can no longer deform; the zone adjoining it will slip in the opposite direction, again partially relieving the shearing stress in the adjacent regions. Each successive change in the sign of the stress thus increases the width of the slip band, and consequently also the length of the cracks at the edges of the band. When the crack length reaches a sufficient magnitude, fracture of the specimen will occur.

The foregoing description of the process occurring in the slip band under conditions of fatigue shows that the slip band must necessarily widen in this case. This is indeed observed\(^ {15}\). In addition, we see that the increase of fatigue is closely connected with the residual stresses in the region surrounding the slip band. If these stresses could somehow be relieved, then the accumulation of fatigue under the action of all preceding slip would thereby be eliminated. The harmful action of overload and its removal by underload can apparently also be explained on the basis of these residual stresses. The foregoing description suggests a whole series of experiments that have not yet been undertaken by anyone, such as, for example: measurement of the breaking stress before the onset of fatigue-induced fracture as a function of the number of cycles of alternating stress; elimination of the harmful influence of fatigue that has already appeared by applying a series of stresses of alternating sign with slowly decreasing amplitude.

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