A COMPARISON OF THE MECHANICAL PROPERTIES OF Fe-Cr-Si-B METALLIC GLASS WIRES AND HT STEEL WIRES J. A. Verduzco, R. J. Hand and H. A. Davies Department of Engineering Materials, University of Sheffield Mappin Street, Sheffield, S1 3JD, UK ABSTRACT The mechanical properties of melt-spun of Fe78-xCrxSi10B12 amorphous alloy wires are compared with those of HT steel wires. Fracture toughness was measured for melt spun ribbon. A new double-pulley tensioning machine has been developed to facilitate tension/compression fatigue testing of the Fe78-xCrxSi10B12 amorphous alloys and the HT steel wires. Some tension/tension fatigue testing was also performed using a single pulley testing machine. The tensile strength and microhardness increased and the fatigue performance improved with the substitution of Fe by Cr. The fatigue performance of the glassy alloys was generally similar when expressed on the basis of bend strain but inferior when expressed on the basis of bend stress. Fracture toughness was largely independent of Cr content. Fractography revealed differences in the fracture surface features for the amorphous wires tested in the two different modes. INTRODUCTION It is well known that the mechanical strengths of some Fe-based amorphous alloys produced in the rotating water bath melt spinning process [1] are higher than those of high tensile (HT) steel wires [2,3]. It is difficult to measure the mechanical properties of very fine amorphous wires by conventional methods and thus novel methods have been developed to study and to analyse them (see, for example [3,4]). It has been found that the fatigue limit for the Fe-Cr-Si-B quaternary alloy system increases on replacing Fe by Cr, reaching a maximum at 10 at% Cr; above this Cr content the as-cast wires are brittle rather than ductile. The enhancement in the fatigue limit has been attributed to the formation of a protective film composed of Cr/Si hydrated oxide. These enhancements in fatigue properties were observed using tension/tension testing only. In this paper we consider the behaviour of the Fe78-xCrxSi10B12 compositional series under tension/compression and tension/tension loading. Microhardness, tensile strength and fracture toughness measurements have also been undertaken; for practical reasons the latter tests were conducted on ribbons rather than wires. EXPERIMENTAL Glassy alloy wires (diameter 90 to 130 μm), from the compositional series Fe78-xCrxSi10B12, (0 ≤ x ≤8), were directly cast using a rotating water bath melt spinning process, similar to that described by Onhaka [1]. The wires were fully bend ductile and x-ray diffractometry (Co Kα radiation) confirmed that they were amorphous. HT steel wires tested were standard brass plated tyre reinforcement filament (245 μm diameter) supplied by Goodyear, Luxembourg. Vickers microhardness tests were performed on the wires before and after fatigue tests using an indentation load of 100 g for 20 s; around 50 measurements were used in each case in order to determine the average microhardness. Tensile tests were performed on a Hounsfield tensometer at a strain rate of 1x10-4 s-1; a Camscan S2 Scanning Electron
Figure 1.- Schematic illustration of the new twin pulley fatigue testing machine. Microscope (SEM) was used to determine the cross sectional area at fracture in each case in order to calculate the ultimate tensile strength. Fatigue tests on wires of the Fe78-xCrxSi10B12 alloy series were carried out using both an existing single pulley bend type fatigue testing machine [3] (SP) which produces only tension/tension loading and a modified fatigue machine which uses two pulleys (DP) which can perform tension/compression loading, figure 1. To facilitate comparison between the two testing machines, the single pulley tests were carried out using a constant mean stress of 1550 MPa rather than the constant minimum stress utilised previously [3,5]. The double pulley tests were carried out at constant mean stresses of 200, 900 and 1550 MPa. The HT steel wires were tested in the DP machine at a constant mean stress of 200 MPa. The load frequency for all the fatigue tests was 3.4 Hz and the tests were carried out in ambient humidity (RH~60%). The stress range, Δσa, experienced by the outer surface of the bent wire sample in the SP machine is given by: p w w a d d d E + = ⋅ Δσ (1) where E is the Young’s Modulus (extrapolated from Donald et al [6]) and dw and dp are the wire and pulley diameters, respectively. Thus, for equivalently sized pulleys the stress range in the DP machine is twice that in the SP machine. Figure 2 shows schematically the surface stress profile generated by each machine. In order to determine the fracture toughness for glassy metallic alloys, ribbons of thickness ~45 μm were cast using the chill block melt spinning quenching technique at a roll speed ~30 ms-1 under an atmosphere of Figure 2.- Schematic illustration for the surface bending stress cycles for the single (SP) (a) and (b), and double (DP) (c) and (d) pulley machines; where Δσa, σm, σt, σmin and σmax are the stress range, mean stress, tensioning load, minimum and maximum stresses respectively. a a b b b b c c c d c ΔσD σm BENDING STRESS (+) (-) σmax σ t σmin = Cycle (a) t Cycle (c) ΔσD 2 σmax f σ σ m= σmin (+) (-) a a b b b b c c c c d d d d e e e e B ENDING STRESS E E a b c d e f (d) E a c d b (b)
helium. A notch was cut in the ribbons which were tested at a strain rate of 1.66x10-4 s-1 in a Hounsfield tensometer. The fracture toughness was obtained from the resultant failure load using: K C a I σ π = (2) where σ is the fracture strength, a is the crack length and C is given by: 4 3 2 W a 30.39 W a 21.72 W a 10.55 W a C 1.12 0.231 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ + ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ ⎟+ ⎠ ⎞ ⎜ ⎝ ⎛ = − (3) where W is the ribbon width. The fracture surfaces of the wire and ribbon samples were examined by SEM operated at 20 kV. RESULTS AND DISCUSSION Tensile strength and Vickers hardness Figure 3 shows the microhardness, Hv, and tensile strength, σf, for the Fe78-xCrxSi10B12 alloy wires and HT steel wires. It can be seen that both Hv and σf increase with increasing Cr content. These increases in Hv and σf are broadly consistent with those observed previously [3] for the Fe75-xCrxSi10B15 alloy series, although the absolute values are significantly smaller, consistent with the lower B content in the present case. The Cr containing alloy wires proved to be easier to cast with a uniform cross section than the Fe78Si10B12 base alloys because, it is considered, of the formation of a surface film of Cr/Si hydrated oxide (confirmed by XPS studies [3,5]), which aided in stabilising the liquid jet. The Fe78Si10B12 wire was subsequently characterised by a shallow necking which could adversely influence the measured strength and fatigue life. The fracture toughness, KI, for Fe78-xCrxSi10B12 system is almost constant over the range of Cr content (Figure 4). The values are similar to those reported for NiCrB glassy ribbon and X-200 steel [7]. Fracture in both wires and ribbons occurred by shear-induced deformation following inhomogeneous flow, similar to previously reported fractures in these glassy alloys [7-9]. The fracture occurred at ~45° to the tensile axis (i.e. plane of maximum critical resolved shear stress), implying that fracture was shear induced. The fracture morphologies for wires and ribbons consisted of two distinctive regions; one uniform, and the other veined, as has been reported previously [9-11], (figure 5). The smooth regions represent the initial shear displacement. The fracture veins resulted from the existence of a “liquid-like” or softened layer formed during shear displacement, involving extensive plastic flow, corresponding to instability phenomena characteristic of ductile glassy alloys [8]. It has been suggested [10] that the thin layer of softened material must result from local heating occurring under close to adiabatic conditions. The fracture morphology in the Figure 3.- Tensile strength and Vickers microhardness results for the Fe78-xCrxSi10B12 wires tested. 0 1 2 3 4 0 2 4 6 8 X at% Cr 500 1000 1500 2000 σf (GPa) Hv (VHN) σf HT steel wire Hv HT steel wire σf Hv
Figure 4.- Fracture Toughness, KI results on Fe78-xCrxSi10B12 ribbons. HT steel wire was characteristic of this type of high strength steel. Fatigue The fatigue performance of these wires is shown in figure 6. It can be seen that, in general, increasing the Cr content leads to improved fatigue performance, especially under high cycle conditions. In particular, there is a noticeable difference in performance between the 0 and 3.5 at% Cr samples and the 6.25 and 8 at% Cr samples; corrosion pits, that could act as crack nucleating sites were observed for both the 0 and 3.5 at% Cr alloys. This is in line with previously reported enhancements in the fatigue performance of Fe77.5-xCrxSi7.5B15 and Fe75-xCrxSi10B15 amorphous alloys with increasing Cr content [3-5]. Previous studies [3-5] have shown that the initiation of the fatigue crack is environmentally dependent. The presence of corrosion pits at fracture initiation sites observed in the present study lends support to this conclusion. In addition, wire smoothness is expected to play an important role; alloys with 0 and 3.5 at% Cr showed periodic shallow necking along their lengths, leading to stress concentrations and premature fatigue failure. Olofinjana and Davies [3] suggested that the enhancement of fatigue strength due to Cr addition could not be attributed to the effect of the Cr on the intrinsic mechanical properties alone, since the fatigue life does not scale with the static strength, whose increase with Cr content is relatively modest. It was assumed [3,4] that fatigue life enhancement is probably largely due to a surface phenomenon and that fatigue crack initiation is suppressed by a Cr rich film. XPS studies [3,5] demonstrated that the spontaneous formation of a passive thin film formed of a Cr/Si hydrated oxide on the outside surface of the wires appeared to accelerate the improvement of fatigue strength through the enhancement of corrosion resistance. No change occurred in microhardness for any of the wires tested, which suggests that there was no softening or hardening during the process of fatigue. Fatigue experiments with σm = 200 and 900 and 1550 MPa were carried out on the DP machine only since tensioning load stresses of 900 MPa or lower could not be used in the SP machine. In contrast, the DP (a) (b) Figure 5.- Tensile fracture morphologies for (a) wire and (b) ribbon amorphous alloys. 0 2 0 4 0 6 0 8 0 0 2 4 6 8 X at % Cr K I (MPa m 1/2 )
Figure 6.- Fatigue results for Fe78-xCrxSi10B12 glassy and HT steel wires tested in function of stress. machine can be operated with σm as low as 150 MPa. Figure 6 also shows the maximum number of cycles reached for each alloy wire. Although the DP machine can produce tension/compression loading, in practice, only the glassy wires tested at σm = 200 MPa in the DP machine were subjected to tension/compression loading for all the stress ranges used. Under high mean stress conditions ( σm = 900 and 1550 MPa), the wires were subjected to tension/compression stress for large stress ranges (low cycle fatigue), but only tension/tension loading for low values of the stress range (figure 2.a). Figure 6.(a) suggests that HT steel wires have superior fatigue performance to the amorphous alloy wires. In terms of stress this is correct; however, higher stresses are generated in HT steel wires for a given strain because of the higher modulus of HT steel compared to the amorphous alloys. Thus, if the data are reported in terms of strain range, the amorphous alloy wires have generally similar performance to HT steel wires (see figure 7). Figure 7.- Fatigue results for Fe78-xCrxSi10B12 glassy and HT steel wires tested in function of strain.
Fractographic studies of the amorphous wires indicate that similar fracture surfaces are obtained on samples tested in the SP machine and in the DP machine if the samples tested in the DP machine were only subjected to tension/tension loading (high mean stress, low stress range conditions). In this case a relatively rough region surrounds the crack initiation site which develops at a shear displacement. This rough region is believed to be due to repeated shear events during the cyclic loading process. Surrounding this relatively rough region of fatigue crack growth is a smoother region which becomes progressively rougher on moving further away from the crack initiation site. This is believed to be the region of fast fracture. For larger stress ranges in the DP machine where the samples are significantly rougher, (see figure 8). In this case, there is no clear evidence of a single fracture emanating from a region of cyclic crack growth. This is because two opposing regions of the wire see tensile stress whereas in tension/tension loading one region of the wire sees tensile stresses (which lead to cyclic crack growth and eventual failure) whereas the opposing region only sees compressive stress. Thus, in the tension/compression case, fatigue cracking can emanate from two different regions of the wire and, judging by the shear events visible on the exterior surface of the wire, may potentially start from several points within those regions. However, as a fatigue crack becomes dominant on one side of the wire the stress across the remaining section of the wire in the vicinity of crack is increased and thus a growth of nearby fatigue crack on the other side of the wire will be enhanced. As a result, the final fracture may be due to the interception of at least two fatigue cracks growing separately from each other, but whose growth enhances the stresses on the other crack. In this case, as most of the fracture is due to fatigue crack growth, the entire fracture surface is very rough (Figure 8). Of course, a fatigue crack may not initiate at similar points on the two sides of the wire. In this case, one crack will tend to dominate and a situation such as that shown in figure 9 is observed. The precise features are related to the number of shear events generated which depends on the stress range used. Large stress range (low cycle).- Large numbers of shear bands are generated on the surfaces of wires subjected to the tensile and compressive bending stresses. The potential for the nucleation of cracks on these shear bands is high; the fatigue crack would be expected to initiate at the root of one or more shear steps after an incubation period. Figure 8.b shows evidence that there were at least two cracks initiated for fracture, resulting in 100% ductile failure. Medium stress range (medium cycle).- The bending stresses generate a few shear bands. The process of crack initiation could occur in a manner similar to that suggested by Inoue et al [12]: (a) the glassy wire usually contains a rather high residual stress. (b) repetitive bending load generates shear steps in the high tensile residual stress region on the wire surface. (c) a small corrosion pit generates at the shear event in a high surface energy state. (d) the adsorption and dissociation of moisture in air become easier near the corrosion pit, resulting in the generation and dissolution of hydrogen. The increases of internal stress and embrittlement sensivity due to the dissolution of H give rise to easy initiation and propagation of the fatigue crack, leading to final fracture. The diffusion of H down the shear band would be accelerated by the additional free volume generated by the shear event [3]. The ductile fracture, i.e. from crack initiation to the initiation of the brittle fracture extends to 50 to 80 % of the sample diameter. (a) (b) (c) Figure 8.- Fracture morphology of Fe74.5Cr3.5Si10B12 glassy wire tested in the DP machine, failed at Δσα = 5.42 GPa, using σm = 900 MPa. Crack growth
Figure 9.- Fe61.75Cr6.25Si10B12 glassy wire fractured in the DP fatigue machine at 2132 cycles applying Δσa = 3.34 GPa, using σa = 200 MPa. Low stress range (high cycle).- Here, the mechanism of fracture is similar to that for the medium stress range. The difference is that the fraction of rough length is between 15 and 50 % of the sample diameter, see figure 10. The occurrence of pitting corrosion would only be expected for alloys with 0 and 3.5 at% Cr. For the alloys with 6.25 and 8 at% no pitting corrosion was found at any of the points of crack initiation for medium and high cyclic stresses. Thus, in this case, a fatigue crack is initiated at the root of a shear step after a long incubation period. The fraction of rough length from the crack initiation to fracture is approximately 15 %, (figure 11). Fracture in HT steel wire initiated at shear bands generated on both surfaces of the wire. The initial morphology of the fracture was smooth with small peaks of about 3 μm. The final part of the fracture was ductile, characteristic of crystalline wires (figure 12). Figure 10.- Fatigue fracture for Fe73.5Cr3.5Si10B12. Figure 11 .- Fe61.75Cr6.25Si10B12 glassy wire. Figure 12.- HT steel wire fractured in the DP fractured in the DP machine. machine. CONCLUSIONS Only compressive to tensile bending stresses in the DP machine were performed at σm = 200 MPa for all stress ranges used. Under high cycle conditions at σm = 900 and 1550 MPa, the machine imparts bending stresses similar to the SP machine. Region of rapid fracture Region of cyclic fatigue Crack initiation Shear events
Fe70Cr8Si10B12 glassy alloy wire yielded the highest fatigue strength for both fatigue bending machines at all σm values used. Fracture morphologies in the SP machine were similar to those obtained previously, with a region of fatigue crack growth followed by fast fracture. Fracture morphologies in the DP machine at large stress ranges were different from those obtained with the SP machine but similar for low stress ranges. Fractures were similar for the four glassy alloy compositions studied and for all σm values, with cracks initiating at shear events. Fractures obtained in the medium and low stress ranges resulted from the nucleation of cracks at corrosion pits formed on surface shear steps for the alloys containing 0 and 3.5 at% Cr. ACKNOWLEDGEMENTS One of us, Jorge Verduzco, would like to acknowledge the financial support received from CONACyT, México. REFERENCES: 1.- Onhaka, I. (1984-85) Int. J. of Rap. Sol., 4, 219. 2.- Hagiwara, M., Inoue, A. and Masumoto, T. (1982) Met. Trans. 13A, 373. 3.- Olofinjana, A. and Davies, H. A. (1994) Mater. Sci. and Engng. A186, 143. 4.- Hagiwara, M., Inoue, A. and Masumoto, T. (1985). In Rap. Quench. Met., pp. 1779-1782, Steeb, S. and Warlmont, H. (Eds.), Elsevier Science Publishers, New York. 5.- Olofinjana, A., Nurminen, J. and Davies, H. A. (1994). In Proceedings of the Fourth International Workshop on Non-Crystalline Solids, pp. 72-78, Vázquez, M. and Hernando, A. (Eds.), Madrid, Esp. 6.- Donald, I. W., Whang, S. H., Davies, H. A. and Giessen, B. C. (1981). In Proceedings of the 4th International Conference on Rapidly Quenched Metals, pp. 1377-1380, Sendai. 7.- Masumoto, T. and Maddin, R. (1975) Mater. Sci. and Engng. 19, 1. 8.- Spaepen, F. and. Taub, A. I (1983). In Amorphous Metallic Alloys, pp 231-255, Luborsky, F. E Editor. Butterworths, London. 9.- Olofinjana, A. and Davies, H. A. (1994) Int. J. of Rap. Sol. 8, 225. 10.- Spaepen, F. (1975) Acta Met. 23, 615. 11.- Hagiwara, M., Inoue, A. and Masumoto, T. (1982) Mater. Sci. and Engng. 54, 197. 12.- Inoue, A., Hagiwara, M. and Masumoto, T. (1988) Sci. Rep. RITU, Series A 34,48.
A CRACK AND A DEBONDING INITIATION FROM A CIRCULAR RIGID INCLUSION UNDER UNIFORM TENSION OR COMPRESSION N. Hasebe1 and Y. Yamamoto2 1 Department of Civil Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan 2 Metropolitan Express Way Public Cooperation, Kasumigaseki 1-4-1, Chiyoda-ku, Tokyo 100, Japan ABSTRACT A model of debonding or debonding and crack at the interface of a circular rigid inclusion existing in an infinite elastic body is analyzed under uniaxial uniform loading in the y directions. It is investigated how the debonding develops along the interface of the inclusion under applied loading and when a crack occurs from the tip of debonding. The angle at which the debonding develops and the position of crack occurrence are determined. As the criterion for fracture, the strain energy release rates of debonding and crack are used. Moreover, the normal stress to the circular inclusion at the tip of debonding and the stress intensity factor of mode I of the crack are used as the restricting condition. The analysis is carried out as a mixed boundary value problem of plane elasticity. The rational mapping function of the sum of fractional expressions and complex stress functions are used for the analysis. INTRODUCTION It is well known that the fracture of the material containing inclusion occurs as debonding and crack due to the local stress concentration at the interface of the inclusion. According to the results of the experimental observation of materials, in which the rigidity of inclusion is larger than that of the base material and the adhesion of the base material with the inclusion is weak such as concrete and high strength steel, it was reported that first, a debonding occurs at the interface of inclusion and develops. It is also found that the debonding develops from initial defects at the interface. And it has been investigated that at the tip of debonding which is the singular point of stress, large stress concentration occurs, a crack arises, and thus fracture advances further. Thereupon, when the fracture originating at the inclusion is investigated, it is necessary to consider the problem, in which the debonding and crack are coupled. Already a number of problems on the inclusion accompanied by debonding on the interface of inclusion and crack existing near the inclusion have been analyzed [1], but the analytical solutions on the inclusion problem accompanied by both a debonding and a crack on the inclusion seem not many [3]. In this present paper, at the interface near Point I at a circular rigid inclusion existing in an infinite elastic body as shown in Fig.1a, models of the development of debonding (Fig. 1b), and of a debonding and a crack (Fig.1c) are analyzed under uniaxial tension or compression. Under the applied load, the conditions under which a debonding develops and under which a crack arises at a certain size of the debonding are investigated. Particularly when there are both possibilities of the debonding development and of the crack occurrence from the tip of the debonding, it can be decided which phenomenon actually occurs. Moreover, when the acting load is increased gradually from zero, the phenomena of fracture are also investigated. The
strain energy release rate obtained by this analysis and the fracture toughness value are used as the fracture criterion. Also the load at which a debonding develops or a crack arises can be determined. Cracking is analyzed as the case when it occurs at one tip of the debonding (Point B) shown in Fig.1b, but it is also the condition when cracking occurs simultaneously at another debonding tip (Point A). The analysis under tension in the x direction has been reported by [2]. The rational mapping function of the sum of fractional expressions and the complex stress functions are used for the analysis, which is carried out as a mixed boundary value problem of plane elasticity. The inclusion is a rigid body, and then the analysis of stress and stress singular value in the state that both a debonding and a crack exist is feasible [3, 4, 5]. For the shape that the rational mapping function represents, the exact solution is obtained. The stress intensity of debonding expressing the magnitude of the stress singularity at the tip of the debonding and the stress intensity factor immediately after a crack initiation at the tip of the debonding are calculated. Using these stress singular values, the strain energy release rate of the debonding development and that of the crack occurrence are obtained. And by using these strain energy release rates as the fracture criteria, the phenomena of fracture due to the debonding and crack at a circular rigid inclusion are elucidated. ANALYTICAL METHOD, STRESS INTENSITY FACTOR AND STRESS INTENSITY OF DEBONDING As the stress analytic method, complex stress functions and a conformal mapping function are used. The mapping function which maps the infinite region of the outside of the circular hole with a crack as shown in Fig.1c into the outside region of the unit circle (Fig.1d) is formed as the rational function. And the mixed boundary value problem of the plane elasticity is solved, where the displacements on the rigid inclusion are zero and the stress is free on the debonding and crack surface. The solution was reported [3, 4]. The stress intensity factors KICand KII are obtained from the stress function and the following nondimensional stress intensity factor are defined, q a K iK F iF II I II I S ( q > 0 ) (1) “q” is a uniform tension and ”a” is a radius of the circular rigid inclusion. Also the strain energy release rate Gc of the crack is expressed by using stress intensity factors as follows: 2 2 2 2 2 8 1 8 1 F F q G a K K G G II I II I c S N N (2) where G is the shear modulus. N is 3-4Q for plane strain state and (3-Q)/(1+Q) for plane stress state, and Q is Poisson’s ratio. In this paper, the very short crack length of b/a=0.0005 and 0.001 normal to the boundary is used, and its (a) z-plane (b) z-plane (c) z-plane (d) ] –plane Figure 1 : Analytical region (z-plane) and a unit circle(] -plane )
length is identified as the crack length just after the crack occurrence. Gc obtained by using these stress intensity factors is adopted as the strain energy release rate in crack occurrence. As shown in Fig.2, when the origin is set at the tip of debonding, and the x-axis is taken as the direction of the debonding, and its normal direction is taken as the y-axis, the stress components on the bonded surface of the distance r away from the tip of the debonding are expressed as follows [6]: > @ r r y ln cos 1 ~ 0 0.5 0 E T J N V > @ r r x ln cos 3 ~ 0 0.5 0 E T J N V (3) > @ r r xy ln sin 1 ~ 0 0.5 0 E T J N W where T0=arg 0 ~ E and J NS log /2 . Similarly to the stress field near the crack tip, these stress components possess the singularity of –0.5 power in relation to the distance r from the tip of the debonding. 0 ~ E in Eqn. 3 represents the magnitude of singularity at the tip of debonding. In the present case, in order to distinguish it from the stress intensity factor of the crack in homogeneous case, 0 ~ E is named “stress intensity of debonding”. 0 ~ E is calculated by the stress function. In this paper, the dimensionless stress intensity of debonding defined by the following expression is used:. q a Fd 0 ~ E ( q > 0 ) (4) The strain energy release rate in debonding development Gd is expressed by using the stress intensity of debonding as follows [7]: 2 2 2 0 2 1 ~ 2 1 F q G a G G d d S N N E SN N (5) CRITERIA FOR FRACTURE As shown in Fig.1, the cases that (a) the debonding exists at Point I at the circular rigid inclusion under loading q and (b) the debonding develops to a certain position expressed by angle T, or the initial debonding expressed by angle T is considered. (c) At this time, which behavior arises, the further debonding development or a crack occurrence at the tip of the debonding, is investigated. By this means, how the fracture phenomena due to the debonding and the crack occur can be determined. In order to examine under what condition debonding develops or cracking occurs, the strain energy release rate in the debonding development Gd and that in the crack occurrence Gc are used as fracture criteria. The fracture toughness Figure 2 : Coordinates at the debonding tip
Figure 3 : Strain energy release rate of crack occurrence under uniaxial tension in the y direction Figure 4 : Strain energy release rate of debonding development under uniaxial tension in the y direction Figure 5 : Ratio of strain energy release rates of debonding development and crack occurrence under uniaxial tension in the y direction
value related to the strength of adhesion of base material and inclusion generally is different from the fracture toughness value for the crack occurrence in the base material. The fracture toughness in the debonding development expressed by the strain energy release rate is denoted by Gd0 and that in the crack occurrence is denoted by Gc0. When the restricting condition mentioned later is satisfied, according to the relative magnitude of the values of Gd and Gd0 and the values of Gc, and Gc0, the following matters for the fracture behavior can be stated: (A) Case of Gd < Gd0 and Gc. < Gc0: neither debonding develops nor cracking occurs. (B) Case of Gd < Gd0 and Gc > Gc0: debonding does not develop but cracking occurs. (C) Case of Gd > Gd0 and Gc < Gc0: debonding develops but cracking dose not occur. (D) Case of Gd > Gd0 and Gc > Gc0: there are both possibilities of debonding development and crack occurrence. In (A), (B) and (C) mentioned above, respective fracture behaviors can be specified, but in the case of (D), it becomes necessary to judge further which actually occurs, debonding or cracking. For this purpose, the ratio of the strain energy release rates of debonding and cracking Gd / Gc and the ratio of the fracture toughness values of debonding and cracking Gd0 / Gc0 are considered. According to the relation of the magnitude of the values of Gd / Gc and Gd0 / Gc0, the following matters can be stated: (a) Case of (Gd / Gc) < (Gd0 / Gc0), i.e. Gd / Gd0 < Gc / Gc0:cracking occurs. (b) Case of (Gd / Gc) > ( Gd0 / Gc0), i.e. Gd / Gd0 > Gc / Gc0:debonding develops. The analysis is carried out with the model that a crack occurs from one tip of debonding (see Fig.1c). Figure 6 : Nondimensional Mode I stress intensity factor under uniaxial tension in the y direction Figure 7 : Normal stress r V and tangential stress T W r near the debonding tip under uniaxial tension in the y direction
However, this is also the model that cracks occur simultaneously from both tips of debonding, because the cracks are minute just after they occur, and their mutual effect can be neglected and the inclusion is rigid. However, the state of crack development after the crack has occurred is not referred to in this paper. In this paper, as the restricting condition of crack occurrence, the condition of the stress intensity factor of mode I just after crack occurrence being positive, namely the condition of F1 > 0 is considered. However, strictly speaking, the condition to which also the stress intensity factor of mode II is related may be necessary. Moreover, the restricting condition of the debonding development is also used, and in this case, the condition that the normal stress r V at the interface near the tip of debonding is positive, namely r V > 0, is used. Strictly speaking, the restricting condition may be the function of r V and T W r , which is the tangential stress near the tip of the debonding. RESULTS OF ANALYSIS AND INVESTIGATION The strain energy release rate in the crack occurrence Gc is shown in Fig.3 for uniform tensile load in the y direction, and =1, 2, 3. On the ordinate, Gc and on the abscissa, T, the angle of circumference representing the size of debonding are taken. The value of Gc is shown for the micro crack lengths b/a=0.0005 and 0.001. There are some differences between them. The strain energy release rate for debonding development Gd is shown in Fig. 4, and the ratio Gd / Gc is shown in Fig.5. The mode I stress intensity factor F1 of dimensionless form is shown in Fig.6. In this paper, F1 > 0 is used as the restricting condition of crack occurrence. The normal stress r V / q and the shearing stress T W r / q at the interface of the location GT= (0.1/180)a apart from the tip of debonding are shown in Fig.7 with solid lines and broken lines, respectively. The sign of r V is positive in tension and negative in compression to the normal. Shearing stress is positive when T W r tends to turn in counterclockwise direction. In this paper, for simplicity, r V >0 is used as the restricting condition of the debonding development. The restricting condition for the case of =2 is used for the account. According to Fig.6, F1 is positive in 100Û<T< 360Û, therefore, a cracking can occur in this range of T. However in the range of 0Û<T< 100Û in which F1 is negative, a cracking cannot occur. According to Fig.7, r V is positive in 0Û<T< 315Û, Therefore, debonding can develop in this range of T. However, in the range of 315Û<T< 360Û in which r V is negative, the debonding cannot develop. In relation to an arbitrary initial debonding angle T, and under the condition (A) of the criteria for fracture mentioned in previous section, neither debonding development nor crack occurrence arises, Namely if the respective values of Gd0 and Gc0 are larger than the respective values of Gd and Gc in Figs.3 and 4, neither debonding nor cracking occurs. If Gd0 and Gc0 are larger than the maximum values of Gd and Gc, respectively, debonding and cracking never occur for arbitrary values of T. Under the condition (B) of the criteria for fracture, the debonding does not develop. Accordingly, the value of Gd0 is larger than Gd in Fig.4 for the initial debonding angle T being considered. A crack occurs at the debonding tip of the angle T with the restricting condition F1 > 0 and GctGc0. It is known that a crack is most apt to occur when the initial debonding is around T= 260Û for =2, at which the curve of Gc for =2 takes the maximum value. Namely, a crack is occured by the smallest q. Under the condition (C) of the criteria for fracture, a cracking does not occur. Accordingly, the value of Gc0 is larger than that of Gc in Fig.3 for the debonding angle T being considered. The debonding of the angle T that satisfies the restricting condition r V > 0 and Gd t Gd0 begins to develop, and develops in the range in which r V > 0 and Gd t Gd0 are satisfied. On the other hand, when the strain energy release rate in the cracking Gc also becomes larger together with debonding development, the case of Gc exceeding Gc0 during debonding development is conceivable. Then the fracture condition changes from the condition (C) to (D). In this way, the phonomenon of the fracture is not constant, but sometimes changes together with debonding development. Under the condition (D) of the criteria for fracture, there are possibilities of debonding development and crack occurrence. Therefore, the judgment is made by using the ratio of the strain energy release rates in the debonding and cracking Gd / Gc in Fig.5. According to the ratio of fracture toughness value Gd0 / Gc0, the
following three cases can be classified. As example by taking some values of Gd0 / Gc0 and using the case of =2, the concrete explanation is made. (i) Case of the value of Gd0 / Gc0 being larger than the maximum value of Gd / Gc, for =2 in Fig.5: the condition (a) in the previous section is always satisfied. Accordingly, a crack occurs at the debonding tip of the initial angle T satisfying the restricting condition Fl > 0 which is known from Fig.6. (ii) Case of the value of Gd0 / Gc0 being between the maximum value and the minimum value for =2: Explanation is made, for example, by assuming Gd0 / Gc0 =2.0 (Line { in Fig.5). According to Fig.5, in the case that there is the initial debonding for 0Û<T<60Û and 140Û<T< 360Û ( 60Û and 140Û are at the intersection of Line { and the curve Gd /Gc), the condition (a) in the previous section is satisfied, and Fl > 0 for 100Û<T<360Û. Accordingly, a crack occurs at the debonding tip with the initial angle T for 100Û<T<140Û. In the case of 60Û<T< 140Û in Fig.5, it comes under the condition (b). Since r V > 0 between 0Û and 315Û, the debonding develops till 140Û. Therefore, a crack arises at T=140Û if Gc > Gc0 is still satisfied. (iii) Case of the value of Gd0 / Gc0 being smaller than the minimum value for =2: Explanation is made, for example, by assuming Gd0 / Gc0 =0.25 (Line | in Fig.5). In this case, the condition (b) is always satisfied. Accordingly, in the case that the initial debonding angle is 0Û<T<315Û where r V >0, the debonding develops up to around T=315Û at which r V <0, and stops there. And if Gc tGc0 is still satisfied there ( Gc changes in the range of T), a crack occurs. In the above description, the fracture phenomena in relation to the criteria (A) ~ (D) were investigated when an arbitrary debonding angle T is given, and a constant load q is applied. Next, the case of an applied load q gradually increasing from zero is considered. As known from Eqn. 2 and Eqn. 5, Gc and Gd increase in proportion to q 2. The fracture phenomenon occurs by either Gc or Gd has reached first to the fracture toughness value Gc0 or Gd0 when the load q increases. Which reaches first to the fracture toughness value can be known by the relation of the magnitude in the ratio Gd / Gc and the ratio Gd0 / Gc0. The value of Gd / Gc is not dependent on the magnitude of load q(see Eqns. 2 and 5). On the other hand, also the value of Gd0 / Gc0 does not change due to the variance in the magnitude of load q. From these facts, when the debonding tip is at the position of angle T, the following matters can be said under the increase of load: Case of (Gd / Gc) < (Gd0 / Gc0) i.e. c0 c d0 d G G G G : When load q increases, and Gc = Gc0 is attained, Gd is still satisfying Gd < Gd0. For example, in the case of =2, Line { in 0Û<T<60Û and 140Û<T<360Û in Fig.5 comes under the present case. For example, under this condition and when an initial debonding angle T is in 140Û<T<360Û, in which F1>0 is satisfied, a crack occurs at the tip of the debonding. The magnitude of q at this time is determined by Gc = Gc0. When the initial debonding is T=260Û at which the Gc curve for =2 takes the maximum value, a crack occurs at the smallest value of q. Case of (Gd / Gc) > (Gd0 / Gc0) : When load q increases, and Gd = Gd0 is attained, Gc is still satisfying Gc < Gc0. Accordingly, Gd first reaches Gd0. For example, in the case of =2, Line | in 0 Û<T<360Û or Line { in 60Û<T< 140Û in Fig.5 comes under the present case. Under this condition and when an initial debonding angle T is in 0Û<T< 315Û, in which r V > 0 is satisfied, the debonding starts to develop. The magnitude of q at this time is determined by Gd = Gd0. The debonding develops till T=315Û for Line | and till T=140Û for Line { while Gd > Gd0 is satisfied. When Gc = Gc0 is attained with increasing load, a crack occurs at those angles, respectively. CONCLUSIONS The phenomena regarding the development of debonding at the interface of a circular rigid inclusion and the occurrence of a crack from the tip of the debonding were investigated. The considered load is the uniaxial uniform tension or compression of arbitrary magnitude in the y direction. The results of compressive load must be necessary for investigation of the compressive fracture. When q < 0, the sign of FI, r V and T W r are changed. Then the range of FI > 0, and r V > 0 has meaning. The strain energy release rate in the debonding development Gd determined by Eqn. 5 corresponds to the square of stress intensity of debonding.
As the criteria for fracture in debonding development and crack occurrence, the strain energy release rates Gd and Gc were used. By the relation of the magnitude of the fracture toughness value with that of the strain energy release rate, how the fracture phenomena occur due to debonding and cracking is determined. In particular, in the case of the condition (D) in the criteria for fracture, it is determined by the relation of the magnitude in the ratio Gd / Gc and Gd0 / Gc0 in the debonding and the cracking. Moreover, the load when a debonding develops or a crack occurs can be determined. This way can be applied to not only the case of constant load but also a load becoming gradually larger from zero. Besides, when a cyclic load acts, this way may be also applicable. In this paper, as the restricting condition of debonding development, the condition of the normal stress r V , being positive at the interface near the debonding tip was used. As the restricting condition of the crack occurrence, the condition of the stress intensity factor Fl being positive immediately after crack occurrence was used. When other criteria for fracture and restricting conditions must be used, the investigation of the fracture may be carried out by using them in a way as well as that has been followed in this paper. Since the strain energy release rates in the debonding development and the crack occurrence possess the maximum values, the debonding development and the crack occurrence can be prevented by using fracture toughness values larger than these values. The strain energy release rates, stress intensity factors and stresses depend on the value of , which depends on Poisson's ratio, and affects the characteristics of debonding development and the position of crack occurrence. The direction of the crack initiation has been assumed as the normal direction to the boundary for calculating Gc and FI. Accutually the direction of the crack initiation must be determined by using the proper criterion for calculating Gc and FI. References 1. Mura, T., (1988) “Inclusion Problems”, Applied Mechanics Reviews, 41, pp.15-20. 2. Hasebe, N. and Yamamoto, Y. (1999) Eighth Int. con. on the Mechanical Bechaviour of Materials (ICM8), pp. 1021-1025. 3. Hasebe, N., Okumura, M., and Nakamura, T. (1987) Int. J. of Fracture, 32, pp.169-183. 4. Hasebe, N., Okumura, M., and Nakamura. T. (1987) Int. J. of Fracture 33, pp.195-208. 5. Hasebe, N., Ueda, M., Yamamoto, Y., and Nakamura, T. (1992) J. of Structural Engineering, 38A, pp.369-382 (In Japanese). 6. Tsutsui, S., and Nakamura, T. (1988) J. of Appl. Mech., ASME, 110, pp.574-579. 7. Hasebe, N., Okumura, M., and Nakamura, T. (1990) J. of the Society of Materials Science, 39, 445, pp. 1405-1410 (In Japanese).
A LIFING PROCEDURE TO PREDICT LIFE FOR SHOT-PEENED CAST COMPONENTS SUBJECTED TO HIGH CYCLE FATIGUE I. Chantier,1 V. Bobet-Denier2 and F. Hild1 1LMT-Cachan, ENS de Cachan / CNRS / Université Paris 6 61 avenue du Président Wilson, 94235 Cachan Cedex, France 2RENAULT Technocentre / Direction de la Recherche / Sce 64230 TCR RUC 4 29 1 rue du Golf, 78288 Guyancourt Cedex, France ABSTRACT In many studies on high cycle fatigue, failure occurs at the free surface or close to it. Consequently, the effect of the environment and the surface treatment are of utmost importance. In the case of structures made of spheroidal graphite cast iron, the presence of different kinds of initial flaws within the material explains the scattered fatigue lives. An expression for the cumulative failure probability depending of these initial flaws is derived. For tested suspension arms, this probabilistic approach is used to assess the fatigue lives by taking into account a flaw size distribution, a crack propagation law, the environment and the surface treatment. INTRODUCTION Nowadays, the automotive industry is one of the main fields that want to improve the life assessment of safety components made of Spheroidal Graphite (SG) cast iron. Designing components better, faster and cheaper is one of their challenges. In cars, cast components are subjected to High Cycle Fatigue (HCF). On the one hand, the fatigue strength is generally reduced by the presence of initial casting flaws randomly distributed within the material; on the other hand, scattered results are observed in S-N (Woehler) diagrams. The life of cast components depends on two key elements: the state of their surface (as-cast, machined or shot-peened), and the nature of the material microstructure. Lastly, the size and location of flaws can be detrimental to the fatigue resistance and therefore the flaws need to be taken into account by a probability density function. For any state of the free surface, scattered results of fatigue tests on structures show the need for a probabilistic approach. The cumulative probability of a structure, written for different cycling conditions, depends on an initial flaw size distribution and a microcrack propagation law. The main point of the present
study is the integration of the effects of shot-peening and the environment on the reliability of structures subjected to HCF. The reliability of shot-peened suspension arms is evaluated and compared with experimental data. MATERIAL The material studied herein is an SG cast iron essentially ferritic (i.e., containing less than 5% of pearlite). Its composition (wt%) is: C = 3.8, Si = 3.21, Mg = 0.01, S = 0.04, P = 0.007. The main microstructure characteristics are: mean graphite nodule diameter = 17 μm, ferritic grain size = 15/22 μm, mean distance between nodules = 80 μm [1]. For machined samples, the tensile properties are: yield stress = 365 MPa, tensile strength = 500 MPa, elongation = 16%, Young’s modulus = 150 GPa. In the present study, typical casting flaws leading to failure were observed on machined samples designed for HCF. One class of flaws was characterised: microshrinkage cavities within the material (Fig. 1). Their size vary up to 400 μm (i.e., the maximum flaw size aM). Consequently, these flaws are physically short cracks [2], modelled by penny-shaped cracks whose size is 2a, and normal is oriented along the direction of the local maximum principal stress σ. The corresponding Stress Intensity Factor (SIF) can be described by K Y a σ = where Y is a geometric dimensionless parameter. Microscopic observations lead us to introduce a flaw size distribution f0 that describes the flaw population [3] ( ) t (t 1) dt B B a a (a a) f a 1 0 1 M M 0 ∫ − = − = + + β α αβ α β αβ β α with (1) It is assumed that the representative volume V0 of this material is equal to the gauge volume of tested samples, i.e., 340 mm3. Systematic SEM observations allows one the measurement of the flaw sizes observed on fractured surfaces, and therefore the identification of the distribution parameters: α = 2.3, β = 18 and aM = 400 μm. The associated propagation law is a modified Paris’ law [3] g(R) K K K g(R) K K C K dN da th c th max eff n eff − − Δ = = Δ with (2) and max min R mR R g(R) σ σ = − − = with 1 1 (3) Under cyclic loading, the initiation period is neglected compared to the stable microcrack propagation period. In HCF, the number of cycles to failure NF mainly depends on the load ratio R, the threshold SIF Kth, the maximum flaw size Ma and the fatigue limit of the largest flaw ( ) / S K Y a g R th M th = . The next section introduces the cumulative failure probability on a complex structure. A PROBABILISTIC APPROACH IN HCF FAILURE By using the previous analysis, a flaw size distribution f0 was identified. Under cyclic loading conditions, f0 evolves with the number of cycles N to become equal to fN. By considering a representative element Ω0 of
volume V0 that contains the initial flaws, the cumulative failure probability PF0 is the probability of finding flaws greater than ac after N cycles 2 max c c a N F0 Y K a f (a)da P c ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = = ∫ +∞ σ with (4) No nucleation of flaws is assumed during the whole load history. By introducing the initial crack size ac0 that becomes critical after N = NF cycles, PF0 can be rewritten as ∫ +∞ = c0 a 0 F0 f (a)da P (5) In other words, more and more critical flaws appear when NF increases. The framework of the weakest link theory [4] allows one to express the cumulative failure probability PF of a complex structure Ω of volume V as a function of PF0, by neglecting the flaw interactions ) ln(1 1 1 exp 0 0 ∫∫∫ Ω − = − P dV V P F F (6) Figure 2 shows the identification of tension tests (R = 0.1) on machined samples with the present model [3]. In the next section, the different surface effects are explained in detail. SURFACE EFFECTS The surface effects can be summarised as follows: the SIF is a function of the distance of flaws (i.e., cracks) from the free surface, the compressive residual stress field induced by shot-peening slows down the crack propagation rate in the vicinity of the free surface and the environment influences the crack propagation law. Mechanical layer As often observed, HCF is typically a surface problem which can be partly explained by mechanical concepts. The size a and the location (depth d from the free surface) of flaws within the material are important to describe the critical flaw for the fatigue life. This surface effect is analysed with the dependence of the maximum SIF max K on the crack size a and the depth d by means of a geometric dimensionless parameter Y [5] a K Y(a,d ) max max σ = (7) The closer the defect to the free surfaces, the greater the SIF. The limits are 2 (at the surface) and 1.46 (in the bulk) [5]. By integrating the propagation law with this last expression, the evolution of the flaw size leading to failure with the depth can be described [6]. It follows that the fatigue life is different if the mechanical layer is accounted for or not (Fig. 3). Propagation law To introduce the effect of the environment, the distinction between crack propagation in air or in vacuum is made. The crack propagation rate in air is ten times higher than in vacuum for the same SIF range [7]. For
the same flaw size, a crack at the free surface is more critical for the fatigue life than an internal crack subjected to the same stress level. The first parameter identification concerns the crack propagation law in air. Two methods can be followed: • first, by considering a constant cumulative failure probability in tension/compression (say 50%), the parameters of the propagation law can be identified [3]: C/aM = 5.9 10-5, Kth/Kc = 0.33, m = 0.59, n = 2.1 (Fig. 4); • second, by considering the propagation data for the relevant flaw sizes, the modified Paris’ law given by Eqn. 2 is fitted with a constant threshold SIF Kth and a constant critical SIF Kc [7]. Similar parameters are found (Fig. 4). The second parameter identification concerns the crack propagation law in vacuum [7]. In that case, only crack propagation data are available. The following parameters are obtained: C/aM = 2.76 10 -3, Kth/Kc = 0.163, m = 0.59, n = 4.23. Surface treatment The shot-peening process allows one to create an initial compressive stress field in the vicinity of the free surface of as-cast components. This stress field induces crack closure, which is beneficial to the fatigue life [8]. By means of X-ray diffraction, the in-plane stresses are measured on samples and suspension arms, which underwent the same double shot-peening treatment. The following results are obtained: • an initial equibiaxial stress state is obtained whose level is −450±25 MPa for samples and −350±22 MPa for suspension arms; • after cycling of samples in tension (R = 0.1) for a stress level (295 MPa) corresponding to the endurance limit of shot-peened samples when PF0 = 50%, the longitudinal residual stress increases by an amount of 200 MPa. Therefore, the stabilised residual stress is −250±20 MPa at the free surface; • lastly, measurements and Ref. [8] confirm the fact that the compressive stress is present in a surface layer 0.5 mm in depth. All these effects need to be considered to assess the reliability of structures in HCF. To carry out numerical computations, structures are divided into two entities, viz. a surface layer (the shot-peened ‘skin’) and the remaining volume, to distinguish their respective properties (i.e., SIF, propagation law and stabilised initial stress state). The next section aims at analysing the experiments on suspension arms. In particular, observations of fractured surfaces of arms are carried out to compare them to fractured surfaces of as-cast samples. The reliability of a suspension arm is then evaluated. APPLICATION TO STRUCTURES Samples To analyse the three previous surface effects, computations are carried out on samples 10 mm in diameter. For these calculations, a stabilised compressive stress level of −250 MPa on a 0.5 mm thick surface layer is considered in the case of rotary bending (R = –1) with a maximum applied stress Fσ of 350 MPa. The
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