slip lines which are close to, but not coincident with, the slip lines formed in

Transcription

1 FATIGUE CRACK NUCLEATION IN AIETALS* BY T. II. LIN ANI) Y. A11. ITO UNIVERSITY OF CALIFORNIA (LOS ANGELES) Communicated by T'. Y. Thomas, December 18, 1968 Abstract. One of the unanswered questions in the study of fatigue is how cracks nucleate at stresses far below the static fracture strength. Previous theories show possible qualitative mechanisms that may operate in a crystal at room and low temperatures but none provides a quantitiative theory of this phenomenon. We show here a quantitative mechanism of fatigue crack initiation. With a small initial stress field, two closely located thin slices slide in opposite directions under cyclic loading. The increase of the local plastic strain with cycles of loading is calculated. The local plastic shear strain, positive in one slice and negative in the other, reaches 100 per cent at the free surface in a few hundred cycles. These large strains clearly cause the start of an extrusion or intrusion and fatigue crack nucleation. Answers to the question of how fatigue cracks nucleate in metals under stresses far below the static fracture strength have been sought by many investigators since the beginning of the century. However, this question is far from having been clearly answered. In this paper, we attempt to answer it by showing a nucleation mechanism and a method to determine quantitatively the local plastic strain under cyclic loading. The development of the present mechanism of fatigue crack nucleation is guided by the following experimental observations.1-4 The observed fatigue in metals down to 1.70K indicates that surface corrosion, gas adsorption, or vacancy diffusion is not necessary to the nucleation mechanism. The formation of fatigue cracks is associated with slip, and slip lines appear at early stages of fatigue. As the number of loading cycles increases, these slip lines broaden into bands in which fatigue cracks ultimately form. Reverse loading produces slip lines which are close to, but not coincident with, the slip lines formed in the forward loading. This indicates that two distinct, very closely located, sliding slices intersect the free surface. One slice slides during the forward loading, producing one slip line, and the neighboring slice slides during the reverse loading, producing the other slip line. Tests have shown that single crystals, under high stress, slide along certain directions on certain crystallographic planes and that the slip depends on the shear stress along the slip direction on the slip plane (called the resolved shear stress of this slip system) and is independent of the normal pressure on the sliding plane.5 The resolved shear stress that initiates or causes further slip is known as the critical shear stress. This dependency of slip on the resolved shear stress is applied to the present analysis. Lattice imperfections exist in all metals and produce an initial heterogeneous stress field. Certain small initial stress fields can cause the above sequence 631

2 632 ENGINEERING: LIN AND ITO PROC. N. A. S. of slip. Consider an idealized distribution of two rows of edge dislocations at the free surface of a metal, as shown in Figure 1. The actual initial stress field is due to various types of imperfections. The stress field caused by the dislocation distribution can be evaluated by a method suggested by Eshelby.6 Imagine that layer S (see Fig. 1) is cut from the matrix. After being cut, S is free from the constraints of the matrix and lengthens. Both the matrix with a slot and S are now stress-free. Imagine that S is shortened to the same size as that of the slot by a set of boundary forces (mainly compressive forces at the ends) applied on S. Then, under these boundary forces, layer S is welded back to the matrix. The matrix is still stress-free. Since the surface X2 MJ 0~~~~~~~~~~ 0 S W~~ ~~K K W K Lan /~~( > Y~ ~ FIG. 1.-Dislocation distribution at free surface which produces an initial stress field. forces on the layer do not exist in the actual medium, they are relieved by applying equal and opposite forces (mainly tensile forces at the ends) at the interface of layer S and the matrix. The stress field in the matrix caused by these dislocations is the same as that caused by these equal and opposite forces. The resolved shear stress of the aj# slip system in the two regions P and Q adjacent to layer S (see Fig. 1) caused by these tensile forces can be readily shown to be positive in P' and Q' and negative in P' and Q'. Consider an initial resolved shear stress field ri of the type discussed above to exist in a most favorably oriented crystal at a free surface of a polycrystal under alternate tension and compression. Both the slip direction and the normal to the slip plane of the most favorably oriented crystal make 450 angles

3 VOL. 62, 1969 ENGINEERING: LIN AND ITO 633 with the loading axis and with the free surface, as shown in Figure 2. The elastic constants of the individual crystals are assumed to be isotropic, so that the polycrystal is elastically isotropic and homogeneous. Before slip occurs, the stress field caused by the applied load is uniform throughout the polycrystal. The resolved shear stress field caused by the applied load, denoted by ar, varies only with crystal orientation. As the resolved shear stress reaches the critical shear stress r, in some region of the aggregate, slip occurs5 to give plastic shear strain ea" that causes a residual resolved shear stress field Tr. Slip is concentrated in thin slices. Macroscopic plastic strain represents 1.0r I I I I PSI p I 2b T// 001// a A/ o=o.ipl // b=0.01fl d BOUNDARY x z ;z z -a -to CYCLES CYCLES 800 CYCLES 1.0 ± 100 PSI I I I I I FIG. 2.-Two thin slices P and Q in a most favorably oriented crystal at the free surface of a polycrystal. FIG. 3.-Plastic shear strain distribution in sliding thin slices at different cycles of loading. the average value in the crystal and hence is much less than the local strain. The rate of strain hardening in terms of local plastic strain is much less than that in terms of the macroscopic strain. To simplify the calculation, the local strain hardening is assumed to be zero. Hence rt, remains constant. During the initial tensile loading, r. is positive and the resolved shear stress Ti + Ta in P' and Q" is the highest in the polycrystal. After Tj + Ta reaches T, P' and Q' slide, giving a residual resolved shear stress field T,. In the sliding region Ts + Ta + Tr = Te. Since Tr and rc do not change with loading, T, must decrease with the increase of Ta. Initially, T, is zero, hence Tr,. is negative in

4 634 ENGINEERING: LIN AND ITO PROC. N. A. S. P' and QT. This resolved shear stress field Tr is continuous in the medium. Hence, Tr relieves the positive resolved shear stress not only in P' and Q' but also in P' and Q'. This negative Tr keeps Pa and Q' from reaching Tc during the tensile loading and also helps these regions to slide during the subsequent compressive loading. When the resolved shear stress in Q' and P' reaches - r during the compressive loading, plastic strain occurs in Q' and P' and a new residual stress field is produced. It increases the positive resolved shear stress not only in Q' and P' but also in P' and QV. This helps to keep the resolved shear stress in P' and Q' from reaching - r during the compressive loading, and also helps P' and Q' to slide in the subsequent tensile loading. This process of alternating slip is repeated for each cycle of loading and clearly explains how the local plastic strain is built up. The sequence of slip given above tends to push out region S near the free surface to start an extrusion. The reversing of the sign of the initial resolved shear stress in P and Q, such as that caused by a dislocation distribution of opposite sign to that shown in Figure 1, tends to form an intrusion which gives a crack initiation. This mechanism clearly explains why extrusion and intrusion grow monotonically as reported in many fatigue tests.7 To determine quantitatively the local plastic strain build-up for the proposed nucleation mechanism, the following method is used. The analogy8 between the plastic strain and an equivalent force, similar to Duhamel's analogy9 in thermoelasticity,'0 is applied to find the residual stress field caused by the plastic strain. The aggregate considered is of fine grain. Grain size is small as compared to that of the aggregate. Hence, the equivalent force due to slip in the most favorably oriented crystal is assumed to act in a semi-infinite solid. The two slip lines are parallel, and the thickness of each slip line is much less than its length. The plastic strain distribution and its equivalent forces are taken to be the same along the direction of the slip line. Hence the deformation in the slip regions may be considered under plane strain. The stress field caused by a point force acting in a semi-infinite plate under plane stress was given by Melan.11 Melan's solution, modified for plane strain, is applied to find the residual resolved shear stress field Tr in terms of the plastic strain distribution each In the region currently sliding, ITi + Ta + Tr Tc (1) ATa + ArJ = 0, (2) where A denotes an increment. In equation (2), A-r is a linear function of beae' in the thin sliding regions. For each increment of loading Ama, equation (2) is used to calculate Aea' in the currently sliding region that satisfies equation (1). The details of this method are given in references 12 and 13. This method satisfies the conditions of equilibrium, compatibility, and dependency of slip on the resolved shear stress in crystals. For numerical calculations, the distance a between the two thin slices P and Q is taken to be 0.1 a, which corresponds to the observed extrusion thickness

5 Voi.. 62, ENGINEERING: LIN AND ITO 65 in fatigue tests; the thickness 2b of the sliding slices, 0.02 'u; and the linear dimension d of the most favorably oriented crystals, 50 A (see Fig. 2). The critical shear stress is taken to be 53.5 psi, Poisson's ratio, 0.3, and the shear modulus, 3.85 X 106 psi. These values correspond to those of commercially pure aluminum. To simplify the numerical calculations, the initial resolved shear stress Ti is assumed to be zero everywhere except in the two thin neighboring slices P and Q (see Fig. 2) where 2xi\ Ti = psi in P. (3) \d/ vi = xiA psi in Q. (4) d Cyclic tension and compression stresses of i 100 psi are applied to the aggregate. The calculated local plastic strain eac' in the thin slices P and Q at various cycles of loading is shown in Figure 3. The local plastic strain reaches 100 per cent at the free surface in 800 cycles. The detailed calculations are given in reference 13. The large local plastic strain causes the start of a crack. For some metals, this large plastic strain indicates the start of an extrusion. The reversal of signs of the initial resolved shear stress given in equations (3) and (4) causes the start of an intrusion. A deep intrusion gives a crack initiation. The continuity of the residual resolved shear stress field caused by slip in the two closely located but distinct thin slices reveals the main mechanism of the monotonic build-up of local plastic strain under cycle loading. The proposed mechanism of the local plastic strain build-up is also applicable to a crystal in the interior as well as the free surface. However, as seen in Figure 3, the local plastic strain build-up at the interior end of P and Q is much slower (only about 20%) than that at the free surface. Hence, in general, fatigue cracks occur at the free surface before any interior crack has a chance to develop. * Research sponsored by the National Science Foundation under grant GK2198. 'Ewing, J. A., and J. W. C. Humphrey, Phil. Trans. Roy. Soc. (London), 200A, 241 (1903). 2 Forsyth, P. J. E., J. Inst. Metals, 82, 449 (1954). 3 Charsley, P., and N. Thompson, Phil. Mag., 8, 77 (1963). 4Grosskreutz, J. C., in Fatigue, An Interdisciplinary Approach, ed. J. J. Burke, N. L. Reed, and V. Weiss (Syracuse University Press, 1964). 6 Taylor, G. I., J. Inst. Metals, 62, 307 (1938). 6 Eshelby, J. D., Proc. Roy. Soc. (London), 241A, 396 (1957). 7 Kennedy, A. J., Processes of Creep and Fatigue in Metals (New York: John Wiley, 1963). 8 Lin, T. H., these PROCEEDINGS, 55, 477 (1966). " Duhamel, J. M. C., J. Acole Polytech. (Paris), 15, 1 (1837). "Timoshenko, S., and J. N. Goodier, Theory of Elasticity (New York: McGraw-Hill, 1951). 1 Melan, E., Z. Angew. Math. Mech., 12, 343 (1932). 12Lin, T. H., and Y. M. Ito, J. Appl. Phys., 38, 775 (1967). 13Lin, T. H., and Y. M. Ito, Univ. Calif. (Los Angeles) Engineering Rept