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A fatigue life assessment of 0.8%C spray-formed steel
Materials Science and Engineering A251 (1998) 157 – 165
A fatigue life assessment of 0.8%C spray-formed steel Anna A. Vakulenko * Institute for Problems of Mechanical Engineering, V.O., Bolshoj pr. 61, 199178 St. Petersburg, Russia Received 1 December 1997; received in revised form 10 March 1998
Abstract The fatigue fracture process in 0.8%C spray-formed steel has been investigated. Two groups of data result from reverse fatigue tests and from surface replicas of test samples. Fatigue life-time data demonstrate a large scatter at the same load level. The spray-formed steel has been studied as a continuous random accumulation of flaws in the material. The full morphology is related to pores and short cracks on the surface of the samples tested under different levels of stress. The microlevel background of the Markov process simulation of short crack propagation is discussed. A two-dimensional Markov process is introduced for the modelling of the short crack development in a local area and for a reliability estimation of spray-formed steel behaviour. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Pores; Short fatigue cracks; Spray-formed steel
1. Introduction Rapid manufacturing of tooling for injection moulding, stamping, composite lay-up, or similar processes where the shape of the tool is critical, is a problem with considerable commercial potential. The time needed to manufacture such a tool is about 1 week [1], and the cost of the manufactured tool is substantially less than a conventionally made steel mould or die. The main requirement for injection moulding applications is resistance to compression in the tool structure. Generally, the dominant service concern for a stamping tool is alternating load in different parts of the sprayedformed metal. Locations where the maximum principal stress is positive can be the origins of fracture. This means that the main crack may be initiated in such a region and it may progress through the entire tool structure to final fracture. Hence, the first step in the study of the fracture process in spray-formed material is the analysis of the material fracture behaviour under fatigue tensile loads. Fully reversed fatigue testing is the best approach for this purpose. Mechanical and fatigue properties of sintered steel have been investigated in several papers [2 – 13]. Each new improvement in the technology of the material
* Fax: + 7 812 217864124; e-mail: [email protected] 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00613-3
manufacturing process changes the microstructure of the material and is a new cause for study of the mechanical and fatigue behaviour of the material. The fatigue behaviour of powder metals depends very substantially on the powder type, processing, and heat treatment. An inverse relation was observed between hardness and toughness of AISI M-2 high-speed steel despite a variety of melting processes and heat treatments. It is well known that porosity significantly affects the mechanical properties of the material [2–7]. Increases in density of the material improves fatigue endurance and tensile strength. Porosity degrades a material’s resistance to the initiation of fatigue cracks that are large enough to be of engineering significance. Tensile strength, fracture toughness, resistance to fatigue crack growth all increase with density. Fatigue fracture of a high-speed, manufacturing steel has been studied in Refs. [8–13]. The range (4–10 MPa m1/2) of measured threshold stress intensities is broadly in accord with the few measurements made on powder-metallurgy steels. Fatigue crack growth rate as a function of stress intensity factor has a typical sigmoidal form and may be approximated by a power law in the middle-growth rate region. The Paris–Erdogan law is widely accepted for the description of fatigue crack propagation in different kinds of powder metals. Short crack propagation analy-
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sis analogous to the analysis for aluminium alloys and cast steel [14] is not found in the literature on arc-sprayformed steels.
2. Experimental procedure A 0.8%C tool steel is spray-deposited into the alumina mould using proprietary technology developed by Spray Forming Developments (SDL). The chemical composition (wt.%) of the material is C 0.8, P 0.04, S 0.03, Mn 0.7, Si 0.1, Fe balance. Tensile plate specimens of 63.5 mm gage length, 8.9 mm width and 2.5 mm thickness were machined from dies. Mechanical polishing was accomplished with 0.3-mm diamond paste applied to an adhesive-backed polishing cloth. The two samples were tested on an Instron testing machine at a crosshead speed of 0.13 mm min − 1. The strain was measured by means of electrical resistance strain gauges. Fatigue tests were performed on the tensile samples at room temperature in laboratory air using a servohydraulic machine under load control (810 MTS). The samples were cycled at a frequency of 3 Hz using saw-tooth-like tension – compression cycling with a minimum-to-maximum load ratio of R = −1 (reverse testing). The amplitude loads were 124, 117, 110 and 103 MPa, and four samples for each level of stress amplitude were tested. Two samples were tested under 131 and 68 MPa. The replication method was chosen for an evolution study of flaws in the spray-formed material during fatigue testing [15]. Some 10 – 20 replicas were taken from each sample surface during fatigue testing at three stress levels. Each replicated surface has been observed by image analysis with a Zeiss optical microscope and an IBS computer system (KONTRON Image Analysis System). A crack length greater than 5 mm and the area of surface pores greater than 10 mm2 were determined at 320× magnification. The individual area of pores, maximum and minimum diameter of pores, equivalent circle diameters, circularity factor, short crack length, and co-ordinates of the origin and the end of any short crack, were measured. These values were studied in an image area equal to 0.125 mm2, and 0.5 mm2 of each replicated surface was examined; this involved 100–200 pores and 70 short cracks.
all spray materials manufactured by the high-speed process, and are an artefact reflecting the violence of the ablation and atomising processes. The spray process deposits liquid droplets individually onto the solid surface. Each of these droplets undergoes thermal stressing during cooling, and these circumstances lead to decoupling (or microcracking) along the lamella/lamella interface. The mechanical properties are controlled, not by the behaviour of the material in the individual lamellae, but by the strength of the interfaces between the lamellae. The presence of an interlamellar microcrack can be detrimental under tensile loading. The scanning electron microscopy (SEM) examinations of the polished surface (234 MPa stress, life-time 65700 cycles) shows that the fracture surface contains quasi-cleavage facets, tear ridges, and dimples. The SEM study indicates that the main crack was probably formed by quasi-cleavage close to the edge of the sample. The microstructure analysis of the material before fracture and after fracture looks like the microstructure cited in Refs. [1,12]. The stress–strain diagrams measured for two samples have the following characteristic features: (i) a monotone increase of stress with increasing strain; and (ii) a very low limit of elongation (0.2–0.4%). The S–N curve for high-speed steel, as the root mean square curve from experimental data, resulting from our fatigue tests, is shown in Fig. 1 and illustrates a large scatter in the fatigue life data at the same level of stress. The curve also shows a high-gradient dependence of fatigue life with respect to the nominal stress amplitude. The sample tested at 68 MPa stress did not fracture up to 1277000 cycles. Hence, this level of stress is quite close to the fatigue limit stress. The shortest fatigue life of 1250 cycles results from testing at 262 MPa stress. The common approximation of the fatigue fracture data has the form:
3. Experimental results The microstructure of spray-formed material before testing consists of fragmented lamellae and solidified particles. The lamellae and particles probably arrived at the substrate simultaneously, the solidified particle forcing itself onto the previously deposited lamella. In the larger lamellae, pores are visible. These pores appear in
Fig. 1. Scatter of fatigue fracture data and S– N curve. The dashed curve is the predicted S – N curve.
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Fig. 2. Pore growth involving crack mechanisms (262 MPa stress): (a) 300, (b) 430, (c) 530, and (d) 1150 cycles.
Ds = C logb(tf)
(1)
where Ds (MPa) is the change in tensile stress, tf is the mean fatigue life, and C, b are constants, C = 329.3, b = − 0.036. The scatter of fatigue life data and its high gradient dependence on the stress is the cause of some difficulties in the application of the replication method to 0.8%C high-speed steel. An investigation of the surface after each replication is needed to predict the next replication moment. Two kind of defects in the material were studied by this method. These are pores and microcracks on the surface of the material (Figs. 2 – 4).
Fig. 3. A short crack near a large pore (630 cycles, 262 MPa stress).
The results of the surface replications for the 0.8%C steel under 262, 234 and 220 MPa stresses are as follows: 1. Pores and short cracks are two kinds of flaws in spray-formed material. Pores and cracks on the surface of the material at the initial moment of loading are randomly and heterogeneously distributed. 2. A stochastic heterogeneity of the porosity and cracks is maintained during fatigue testing.
Fig. 4. Short cracks trajectories (430 cycles, 262 MPa stress).
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Fig. 5. The growth of the main crack (234 MPa stress): (a) 10000, (b) 20000, (c) 32500, and (d) 64500 cycles.
3. The interaction between pores and microcracks is a very complicated process in a 0.8%C high-speed manufacturing steel. The pore–microcrack interaction has been closely observed as a major phenomena. A pore might be the initiation site for short cracks. The size of the pore is not a significant factor in determination of the initiation site of short cracks. The vicinity of the most irregular part of a pore boundary is the preferred place for crack initiation. At the same time, it provides the mechanism of initial pore growth (Fig. 2). The initiation place of short cracks is near large pores. The form of such new cracks looks like the pore boundary (Fig. 3). Short cracks may link large pores (two or three pores but not more) and the crack trajectories go across small pore systems (Fig. 4). The geometry of pore boundaries and the short crack form is very wavy and gives an appearance of zigzag cracking. A very strong tendency to form branch cracks and microcrack linkage has been observed in the last stage of the fracture process. The main crack starts at the edge of a sample at a
random point. The main crack evolution is illustrated in Fig. 5(a)–(d). Replicas demonstrate the difference in short and main cracks. The main crack has a large crack opening displacement equal to 10–20 mm during the fatigue testing (Fig. 5(d)). All these phenomena are inherent in the two (262, 234 MPa) stress levels studied. The difference is only in the rate of the evolution of the flaws. Image analysis results numerically describe the main features of the fatigue fracture process in the 0.8%C steel. Statistical data have been obtained from the treatment of six replicas from fatigue testing at 234 MPa and five replicas at 262 MPa. The linear measures of porosity are the minimum, the maximum and the equivalent circle diameter. They specify the irregular form of pores during fatigue testing. The minimum of the minimum diameters is defined as 4 mm and the maximum of the maximum diameters is found to be 296 mm at 262 MPa tensile stress. The values of 3 and 208 mm correspond to the stress level of 234 MPa. This illustrates the very large scatter in the linear measure of porosity. The probability distributions for minimum,
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Fig. 6. Evolution of the mean and standard deviation of the percentage pore area: (a) 262 MPa stress; (b) 234 MPa stress.
maximum and equivalent circle diameters change under a fatigue load, but the mean values of these parameters increase with time. The measures of porosity are the circularity factor equal to p 2/(12.56S), where p is the pore perimeter and S is the individual pore area, and the pore area percentage equal to SM i = 1 Si /A0, where A0 is the local image area containing pores of area Si, and M is the number of pores in the image area. The circularity factor has a probability density quite close to the Gaussian density, with the mean and standard values for any test conditions falling in the 0.5 – 0.65 and 0.14–0.17 ranges, respectively. The equalisation processes in the area (mean percent) start from the initial moment of the fatigue tests and last up to the final fracture of the samples. The mean of the pore area percentage increases with time during fatigue testing, as shown in Fig. 6(a) and (b), where solid lines are the mean area percentage and dashed lines are the standard deviation of the area percentage. The results of the statistical analysis of random crack length is demonstrated in Fig. 7(a) and (b). The mean (solid curves in Fig. 7) and standard deviation values (dashed curves in Fig. 7) of the crack length increase during fatigue tests. Analysis of the quasifracture state, when the time-to-lifetime ratio is equal to 2%, demonstrates that the mean critical value of the short crack length may be estimated as 140 mm for the three stress levels studied. The vertical (along the tensile axis) (Dy) and the horizontal (Dx) deflections of crack tips on the surfaces of the samples have been measured. The increment in the lateral direction has the property of increasing both the mean and the standard deviation values. The vertical co-ordinate of the crack tip has,
approximately, a zero level mean value at each observation area (0.5 mm2) during a fatigue test (Fig. 8 (a)–(d)). The curves in Fig. 8 demonstrate the mean values (——) of Dx and Dy, and corresponding values of the standard (– – –) are defined by: M
z= % zn /M
S= z −z2
(2)
n=1
where M is the number of measured cracks and z= Dx or Dy.
4. Fatigue life-time assessment The experimental observations show that microcrack growth behaviour is intermittent and irregular during a fatigue test. The continuous nucleation of fatigue microcracks has been charted on the surface of smooth samples of a 0.8%C high-speed steel. This is probably related to the high strength of the material composed of fragmented lamella while the lamellar interlayers are of lower strength. The short cracks have lengths much greater than crack lengths studied in an aluminium-matrix silicon carbide whisker composite [16]. Obviously, the crack length in sprayformed steel is enlarged by its wavy form, and the standard definition of short crack length as the smooth straight curve length is a conventional estimation. The short crack development demonstrated in Fig. 7 is followed by the kinetic dependence, having the form V= CmN − m
(3)
for mean short crack growth rate V, and the form Vd = CdN − d
(4)
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Fig. 7. Evolution of the mean (——) and standard deviation ( – – – ) of short crack lengths: (a) 262 MPa stress; (b) 234 MPa stress.
for the dispersion evolution in time, where Cm, Cd, m and d are constant values. A decrease of the mean crack growth rate and the standard of crack length during fatigue testing has been reported in [14] for cast steels and aluminium alloys. The most significant factor controlling the short crack growth rate is the nominal tensile stress. A small increase in the stress level leads to very short fatigue life and accelerates microcracking in the material. The local fracture process in the material under fatigue testing is random and heterogeneous. This microprocess is defined by a given stress range (Ds) and a random local pore area c: Dseff = Ds/(1−c)
(5)
If c is equal to the mean percentage pore area (Fig. 9), then Eq. (5) defines the known scalar damage parameter [17]. In the initial state of the material, this pore area percentage exhibits scatter from 0.5 to 6%. The alteration in the magnitude of the damage parameter c from the image analysis data is 1 – 12%. Using these data, the change in the threshold stress intensity factor (DKth) for 0.8%C spray-formed steel is evaluated. Following Ref. [18] for evaluating DKth, the standard form for the stress intensity factor of a penny-shaped crack in an infinite plate: DKth =Dseff(pac )1/2
(6)
Substituting Eq. (5) in Eq. (6), with the critical crack length value equal to 140 mm, the range of the values of DKth is evaluated as 2.85 – 3.24 MPa m1/2. These values are close to the lower limit of the threshold stress intensity factor cited in Ref. [9]; therefore, this critical value of the short crack length is chosen for the two-dimensional random analysis of the short crack development in the local area. Here we assume that the small square (140× 140 mm2) is a sufficient area for the presentation of random flaws.
The phenomenological approach to short crack evolution in spray-formed material during fatigue testing is considered in the chosen critical area. In this approach, a short crack is a plane vector with the origin lying on the boundary of the critical area with a stochastic vertex which corresponds to a crack tip. Following previous experimental observations, the Markovian random simulation is based on the following: (i) Short crack growth is a succession of small sudden jumps in the horizontal and vertical directions. These mutually perpendicular displacements are independent in an isotropic medium. (ii) The growth of a microcrack depends on the effective stress at the studied moment in the crack growth process. In the framework of (i) and (ii), the crack growth rate is equal to the sum of the mean values and the random force as followed by the Langevin approach [19]. This physical approximation results in a two-dimensional Markov process [19], which is used in modelling of the short crack tip development. The benefits of the Markov process simulation include its simple description by the Fokker–Plank equation and its ability to incorporate different experimental crack growth laws. The Fokker–Plank equation [19] describes the Markov process with the transition density f(x, y, t) equal to the relative frequency of the crack tip passage at point (x+ dx, y+ dy) from point (x, y) during time t. The drift coefficients in this equation are equal to mean rates of Dx or Dy, and diffusion coefficients are estimated as rates of evolution of the dispersions (equal to S 2) of the increments Dx and Dy during fatigue testing. Using experimental data (Fig. 9), approximations of the drift coefficients divided by the critical crack length value and the diffusion coefficients divided
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Fig. 8. Evolution of the mean and standard deviation of increments of short crack length (mm) in both the traverse and longitudinal directions: (a) 262 MPa stress; (b) 234 MPa stress.
by the square of the critical crack length value have been determined as time dependencies at each stress level. After differentiation by the time variable of these dependencies, we have, for a stress equal to 262 MPa, drift rates Au = 0.034N − 0.78 and A6 = − 0.021, and diffusion coefficients Gu =0.000516N − 0.62 and G6 = 0.027. For 234 MPa we have the corresponding system of coefficients: Au =0.041N − 0.85, A6 =0.031, Gu = 0.0121N − 0.72, G6 =0.044, where u is the lateral direction and 6 the vertical one. For these two stress levels, all these coefficients do not increase. The Gu6 coefficient is assumed to be 0 as a consequence of item (i) above. Introducing dimensionless spatial variables u and 6, where u =x/ac and 6 =y/ac , the transition density can be found by a solution of the Fokker – Plank equation with dimensionless spatial variables u and 6:
f %t = − ( fAu )%u −( fA6 )%6 + ( fGu )¦uu/2+ ( fG6 )¦66/2
(7)
where Au and A6 are the drift coefficients and Gu, G6 are the diffusion coefficients. Then, in Eq. (7) the spatial variables change in the ranges 00 u01, 00 601, and the time variable t, where t=N is the number of fatigue cycles. All trajectories of short cracks must be in the square location zone before the fatigue time to fracture t. The critical concentration of short cracks occurs before moment t and is described as the first passage problem out of the critical area. For example, in the case of a fixed origin (in any point (0, 60) from 0 0 601) for random crack development into the critical square, we have the initial condition for Eq. (7) in the form: f(w, 0) =d(u)d(6− 60)
(8)
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164
where w= (u, 6) and delta d(z) is the Dirac delta function of z. The first passage problem, when the crack tip can only cross line u = 1 at moment t and must remain inside the area 0 0u 01, 0 0 6 0 1 before this moment, dictates the following boundary conditions: Pu (0, 6, t)= 0,
P6 (u, 0, t) = 0,
f(1, 6, t)=0
P6 (u, 1, t)=0, (9)
where Pz (z=u, 6) is the normal component of the flux vector P, which satisfies the equality 9 · P(u, 6, t)= −R(u, 6, t), where R is the right side of Eq. (7). This vector P is the probability current, which is included in the local conservation equation: f %t (u, 6, t) +9 · P(u, 6, t) =0. The last boundary condition from Eq. (9) absorbs a crack tip to the boundary line u =1. The solution of Eq. (7) with the initial condition Eq. (8) and the boundary conditions of Eq. (9) defines the probability distribution of life-time before moment t in the form:
&& 1
F(w, t)=1−
0
1
5. Conclusions f(u, 6, t) du d6
(10)
0
where w=(0, 60). Its position is also a random point on the side of the critical unit square. We will approximate its probability density as a uniform density, equal to 1. In the result, we have the reliability estimation of the fatigue fracture of the material as a probability distribution Q(t), where Q(t)=
&
given by Eq. (9) and the initial conditions of Eq. (8). It was solved by the sweep method, using separation of variables. Numerical integration was used for the estimation of Eqs. (10) and (11). If the reliability measure is used as the probability of exceeding P, the fatigue life-time of a sample will be found from Eq. (11) to depend on P. Each level of these life-time estimations follows the nominal stress. The result is that after each numerical solution procedure, we have the dependence of the life-time of the sample on the probability of exceeding t for the nominal stress. This allows one to model the S– N curve. The results are shown in Fig. 9. It is demonstrated that 50, 90 and 95% of samples will be fractured before moment t, resulting from the lines which model the S–N curve with different degrees of accuracy. The two-dimensional modelling is in agreement with the experimental data of the fatigue tests.
1
F(w, t) dw
(11)
0
The numerical solution procedure was applied to the system Eqs. (7) – (11) with boundary conditions
The fracture of the spray-formed material is due to the continuous accumulation of flaws in the material. The flaws (pores and microcracks) are heterogeneously and randomly distributed in the material. Studying the stochastic behaviour of flaws under fatigue loading results in a phenomenological approach to fatigue life-time modelling. It is assumed, with some experimental background, that a two-dimensional Markov process is suitable for the description of plane short crack development. The use of two-dimensional Markov modelling for estimating fatigue life, in relation to the localization of short cracks in a critical area (a square with the sides equal to the critical crack length), is in good agreement with the experimental data determination from reverse cycling fatigue testing. The method presented for fatigue life-time assessment using a large amount of surface data of flaws gives a conservative estimation of the time to fatigue fracture for the material and the time for the initiation of the main crack in 0.8%C steel tools.
Acknowledgements
Fig. 9. Model of the S–N curve. The dashed line is the experimental S–N curve. The probabilities of exceeding the cycle to fracture are: (1) 50%, (2) 10%, and (3) 5%.
This work was completed during training on a SABIT programme in the Scientific Research Laboratory of the Ford Motor Company. The author wishes to acknowledge Mr D. Wilkosz, Mr A. Krause, Dr C. Anders, Dr J. Boileau and Mr R. Cooper for their help in connection with the experimental research, and Dr Dawn White for formulation of the problem and constant attention to the work.
A.A. Vakulenko / Materials Science and Engineering A251 (1998) 157–165 [10] [11] [12] [13] [14]
References [1] P.S. Fussell, H.O.K. Kirchner, F.B. Prinz, L.E. Weiss, J. Thermal Spray Technol. 3 (1994) 148–161. [2] N.A. Fleck, R.A. Smith, Powder Metall. 23 (1981) 126 – 130. [3] N.A. Fleck, R.A. Smith, Powder Metall. 23 (1981) 121 – 125. [4] R. Haynes, J.T. Egediege, Powder Metall. 32 (1989) 47 – 52. [5] T.J. Griffiths, A. Ghanizadeh, Powder Metall. 29 (1986) 129 – 133. [6] J.R. Moon, Powder Metall. 32 (1989) 133–139. [7] W.J. Huppman, K. Dalal, P. Wellner, G. Elssner, Powder Metall. 26 (1983) 22–30. [8] A.S. Wronski, L.B. Hussain Al-Yasiri, F.L. Jagger, Powder Metall. 22 (1979) 109–118. [9] N. Douib, I.J. Mellanby, J.R. Moon, Powder Metall. 32 (1989) 209 – 214.
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S.C. Lee, F.J. Worzala, Metall. Trans. A 12 (1981) 1477–1484. B.L. Averbach, Met. Progr. 12 (1980) 19 – 24. B. Low, B.L. Averbach, Metall. Trans. A 14 (1983) 1889–1898. J. Holmes, R.A. Queeney, Powder Metall. 28 (1985) 231–235. R.O. Ritchie, J. Lankford (Eds.), Small Fatigue Cracks, The Metallurgical Society of AIME, Warrendale, PA, 1986. M.H. Swain, Monitoring small-crack growth by the replication method, in: M. Larsen, J.E. Alison (Eds.), Small Crack Test Methods, ASME, Dearborn, MI, 1992, pp. 34 – 56. E.Y. Chen, L. Lawson, M. Meshii, Scr. Metall. Mater. 30 (1994) 737 – 742. S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, UK, 1991. D. Broek, Elementary Engineering Fracture Mechanics, Kluwer, Boston, MA, 1986. W. Feller, Introduction to Probability Theory and Its Applications, vol. 2, John Wiley and Sons, New York, 1966.
A fatigue life assessment of 0.8%C spray-formed steel Anna A. Vakulenko * Institute for Problems of Mechanical Engineering, V.O., Bolshoj pr. 61, 199178 St. Petersburg, Russia Received 1 December 1997; received in revised form 10 March 1998
Abstract The fatigue fracture process in 0.8%C spray-formed steel has been investigated. Two groups of data result from reverse fatigue tests and from surface replicas of test samples. Fatigue life-time data demonstrate a large scatter at the same load level. The spray-formed steel has been studied as a continuous random accumulation of flaws in the material. The full morphology is related to pores and short cracks on the surface of the samples tested under different levels of stress. The microlevel background of the Markov process simulation of short crack propagation is discussed. A two-dimensional Markov process is introduced for the modelling of the short crack development in a local area and for a reliability estimation of spray-formed steel behaviour. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Pores; Short fatigue cracks; Spray-formed steel
1. Introduction Rapid manufacturing of tooling for injection moulding, stamping, composite lay-up, or similar processes where the shape of the tool is critical, is a problem with considerable commercial potential. The time needed to manufacture such a tool is about 1 week [1], and the cost of the manufactured tool is substantially less than a conventionally made steel mould or die. The main requirement for injection moulding applications is resistance to compression in the tool structure. Generally, the dominant service concern for a stamping tool is alternating load in different parts of the sprayedformed metal. Locations where the maximum principal stress is positive can be the origins of fracture. This means that the main crack may be initiated in such a region and it may progress through the entire tool structure to final fracture. Hence, the first step in the study of the fracture process in spray-formed material is the analysis of the material fracture behaviour under fatigue tensile loads. Fully reversed fatigue testing is the best approach for this purpose. Mechanical and fatigue properties of sintered steel have been investigated in several papers [2 – 13]. Each new improvement in the technology of the material
* Fax: + 7 812 217864124; e-mail: [email protected] 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00613-3
manufacturing process changes the microstructure of the material and is a new cause for study of the mechanical and fatigue behaviour of the material. The fatigue behaviour of powder metals depends very substantially on the powder type, processing, and heat treatment. An inverse relation was observed between hardness and toughness of AISI M-2 high-speed steel despite a variety of melting processes and heat treatments. It is well known that porosity significantly affects the mechanical properties of the material [2–7]. Increases in density of the material improves fatigue endurance and tensile strength. Porosity degrades a material’s resistance to the initiation of fatigue cracks that are large enough to be of engineering significance. Tensile strength, fracture toughness, resistance to fatigue crack growth all increase with density. Fatigue fracture of a high-speed, manufacturing steel has been studied in Refs. [8–13]. The range (4–10 MPa m1/2) of measured threshold stress intensities is broadly in accord with the few measurements made on powder-metallurgy steels. Fatigue crack growth rate as a function of stress intensity factor has a typical sigmoidal form and may be approximated by a power law in the middle-growth rate region. The Paris–Erdogan law is widely accepted for the description of fatigue crack propagation in different kinds of powder metals. Short crack propagation analy-
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A.A. Vakulenko / Materials Science and Engineering A251 (1998) 157–165
sis analogous to the analysis for aluminium alloys and cast steel [14] is not found in the literature on arc-sprayformed steels.
2. Experimental procedure A 0.8%C tool steel is spray-deposited into the alumina mould using proprietary technology developed by Spray Forming Developments (SDL). The chemical composition (wt.%) of the material is C 0.8, P 0.04, S 0.03, Mn 0.7, Si 0.1, Fe balance. Tensile plate specimens of 63.5 mm gage length, 8.9 mm width and 2.5 mm thickness were machined from dies. Mechanical polishing was accomplished with 0.3-mm diamond paste applied to an adhesive-backed polishing cloth. The two samples were tested on an Instron testing machine at a crosshead speed of 0.13 mm min − 1. The strain was measured by means of electrical resistance strain gauges. Fatigue tests were performed on the tensile samples at room temperature in laboratory air using a servohydraulic machine under load control (810 MTS). The samples were cycled at a frequency of 3 Hz using saw-tooth-like tension – compression cycling with a minimum-to-maximum load ratio of R = −1 (reverse testing). The amplitude loads were 124, 117, 110 and 103 MPa, and four samples for each level of stress amplitude were tested. Two samples were tested under 131 and 68 MPa. The replication method was chosen for an evolution study of flaws in the spray-formed material during fatigue testing [15]. Some 10 – 20 replicas were taken from each sample surface during fatigue testing at three stress levels. Each replicated surface has been observed by image analysis with a Zeiss optical microscope and an IBS computer system (KONTRON Image Analysis System). A crack length greater than 5 mm and the area of surface pores greater than 10 mm2 were determined at 320× magnification. The individual area of pores, maximum and minimum diameter of pores, equivalent circle diameters, circularity factor, short crack length, and co-ordinates of the origin and the end of any short crack, were measured. These values were studied in an image area equal to 0.125 mm2, and 0.5 mm2 of each replicated surface was examined; this involved 100–200 pores and 70 short cracks.
all spray materials manufactured by the high-speed process, and are an artefact reflecting the violence of the ablation and atomising processes. The spray process deposits liquid droplets individually onto the solid surface. Each of these droplets undergoes thermal stressing during cooling, and these circumstances lead to decoupling (or microcracking) along the lamella/lamella interface. The mechanical properties are controlled, not by the behaviour of the material in the individual lamellae, but by the strength of the interfaces between the lamellae. The presence of an interlamellar microcrack can be detrimental under tensile loading. The scanning electron microscopy (SEM) examinations of the polished surface (234 MPa stress, life-time 65700 cycles) shows that the fracture surface contains quasi-cleavage facets, tear ridges, and dimples. The SEM study indicates that the main crack was probably formed by quasi-cleavage close to the edge of the sample. The microstructure analysis of the material before fracture and after fracture looks like the microstructure cited in Refs. [1,12]. The stress–strain diagrams measured for two samples have the following characteristic features: (i) a monotone increase of stress with increasing strain; and (ii) a very low limit of elongation (0.2–0.4%). The S–N curve for high-speed steel, as the root mean square curve from experimental data, resulting from our fatigue tests, is shown in Fig. 1 and illustrates a large scatter in the fatigue life data at the same level of stress. The curve also shows a high-gradient dependence of fatigue life with respect to the nominal stress amplitude. The sample tested at 68 MPa stress did not fracture up to 1277000 cycles. Hence, this level of stress is quite close to the fatigue limit stress. The shortest fatigue life of 1250 cycles results from testing at 262 MPa stress. The common approximation of the fatigue fracture data has the form:
3. Experimental results The microstructure of spray-formed material before testing consists of fragmented lamellae and solidified particles. The lamellae and particles probably arrived at the substrate simultaneously, the solidified particle forcing itself onto the previously deposited lamella. In the larger lamellae, pores are visible. These pores appear in
Fig. 1. Scatter of fatigue fracture data and S– N curve. The dashed curve is the predicted S – N curve.
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Fig. 2. Pore growth involving crack mechanisms (262 MPa stress): (a) 300, (b) 430, (c) 530, and (d) 1150 cycles.
Ds = C logb(tf)
(1)
where Ds (MPa) is the change in tensile stress, tf is the mean fatigue life, and C, b are constants, C = 329.3, b = − 0.036. The scatter of fatigue life data and its high gradient dependence on the stress is the cause of some difficulties in the application of the replication method to 0.8%C high-speed steel. An investigation of the surface after each replication is needed to predict the next replication moment. Two kind of defects in the material were studied by this method. These are pores and microcracks on the surface of the material (Figs. 2 – 4).
Fig. 3. A short crack near a large pore (630 cycles, 262 MPa stress).
The results of the surface replications for the 0.8%C steel under 262, 234 and 220 MPa stresses are as follows: 1. Pores and short cracks are two kinds of flaws in spray-formed material. Pores and cracks on the surface of the material at the initial moment of loading are randomly and heterogeneously distributed. 2. A stochastic heterogeneity of the porosity and cracks is maintained during fatigue testing.
Fig. 4. Short cracks trajectories (430 cycles, 262 MPa stress).
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Fig. 5. The growth of the main crack (234 MPa stress): (a) 10000, (b) 20000, (c) 32500, and (d) 64500 cycles.
3. The interaction between pores and microcracks is a very complicated process in a 0.8%C high-speed manufacturing steel. The pore–microcrack interaction has been closely observed as a major phenomena. A pore might be the initiation site for short cracks. The size of the pore is not a significant factor in determination of the initiation site of short cracks. The vicinity of the most irregular part of a pore boundary is the preferred place for crack initiation. At the same time, it provides the mechanism of initial pore growth (Fig. 2). The initiation place of short cracks is near large pores. The form of such new cracks looks like the pore boundary (Fig. 3). Short cracks may link large pores (two or three pores but not more) and the crack trajectories go across small pore systems (Fig. 4). The geometry of pore boundaries and the short crack form is very wavy and gives an appearance of zigzag cracking. A very strong tendency to form branch cracks and microcrack linkage has been observed in the last stage of the fracture process. The main crack starts at the edge of a sample at a
random point. The main crack evolution is illustrated in Fig. 5(a)–(d). Replicas demonstrate the difference in short and main cracks. The main crack has a large crack opening displacement equal to 10–20 mm during the fatigue testing (Fig. 5(d)). All these phenomena are inherent in the two (262, 234 MPa) stress levels studied. The difference is only in the rate of the evolution of the flaws. Image analysis results numerically describe the main features of the fatigue fracture process in the 0.8%C steel. Statistical data have been obtained from the treatment of six replicas from fatigue testing at 234 MPa and five replicas at 262 MPa. The linear measures of porosity are the minimum, the maximum and the equivalent circle diameter. They specify the irregular form of pores during fatigue testing. The minimum of the minimum diameters is defined as 4 mm and the maximum of the maximum diameters is found to be 296 mm at 262 MPa tensile stress. The values of 3 and 208 mm correspond to the stress level of 234 MPa. This illustrates the very large scatter in the linear measure of porosity. The probability distributions for minimum,
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Fig. 6. Evolution of the mean and standard deviation of the percentage pore area: (a) 262 MPa stress; (b) 234 MPa stress.
maximum and equivalent circle diameters change under a fatigue load, but the mean values of these parameters increase with time. The measures of porosity are the circularity factor equal to p 2/(12.56S), where p is the pore perimeter and S is the individual pore area, and the pore area percentage equal to SM i = 1 Si /A0, where A0 is the local image area containing pores of area Si, and M is the number of pores in the image area. The circularity factor has a probability density quite close to the Gaussian density, with the mean and standard values for any test conditions falling in the 0.5 – 0.65 and 0.14–0.17 ranges, respectively. The equalisation processes in the area (mean percent) start from the initial moment of the fatigue tests and last up to the final fracture of the samples. The mean of the pore area percentage increases with time during fatigue testing, as shown in Fig. 6(a) and (b), where solid lines are the mean area percentage and dashed lines are the standard deviation of the area percentage. The results of the statistical analysis of random crack length is demonstrated in Fig. 7(a) and (b). The mean (solid curves in Fig. 7) and standard deviation values (dashed curves in Fig. 7) of the crack length increase during fatigue tests. Analysis of the quasifracture state, when the time-to-lifetime ratio is equal to 2%, demonstrates that the mean critical value of the short crack length may be estimated as 140 mm for the three stress levels studied. The vertical (along the tensile axis) (Dy) and the horizontal (Dx) deflections of crack tips on the surfaces of the samples have been measured. The increment in the lateral direction has the property of increasing both the mean and the standard deviation values. The vertical co-ordinate of the crack tip has,
approximately, a zero level mean value at each observation area (0.5 mm2) during a fatigue test (Fig. 8 (a)–(d)). The curves in Fig. 8 demonstrate the mean values (——) of Dx and Dy, and corresponding values of the standard (– – –) are defined by: M
z= % zn /M
S= z −z2
(2)
n=1
where M is the number of measured cracks and z= Dx or Dy.
4. Fatigue life-time assessment The experimental observations show that microcrack growth behaviour is intermittent and irregular during a fatigue test. The continuous nucleation of fatigue microcracks has been charted on the surface of smooth samples of a 0.8%C high-speed steel. This is probably related to the high strength of the material composed of fragmented lamella while the lamellar interlayers are of lower strength. The short cracks have lengths much greater than crack lengths studied in an aluminium-matrix silicon carbide whisker composite [16]. Obviously, the crack length in sprayformed steel is enlarged by its wavy form, and the standard definition of short crack length as the smooth straight curve length is a conventional estimation. The short crack development demonstrated in Fig. 7 is followed by the kinetic dependence, having the form V= CmN − m
(3)
for mean short crack growth rate V, and the form Vd = CdN − d
(4)
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Fig. 7. Evolution of the mean (——) and standard deviation ( – – – ) of short crack lengths: (a) 262 MPa stress; (b) 234 MPa stress.
for the dispersion evolution in time, where Cm, Cd, m and d are constant values. A decrease of the mean crack growth rate and the standard of crack length during fatigue testing has been reported in [14] for cast steels and aluminium alloys. The most significant factor controlling the short crack growth rate is the nominal tensile stress. A small increase in the stress level leads to very short fatigue life and accelerates microcracking in the material. The local fracture process in the material under fatigue testing is random and heterogeneous. This microprocess is defined by a given stress range (Ds) and a random local pore area c: Dseff = Ds/(1−c)
(5)
If c is equal to the mean percentage pore area (Fig. 9), then Eq. (5) defines the known scalar damage parameter [17]. In the initial state of the material, this pore area percentage exhibits scatter from 0.5 to 6%. The alteration in the magnitude of the damage parameter c from the image analysis data is 1 – 12%. Using these data, the change in the threshold stress intensity factor (DKth) for 0.8%C spray-formed steel is evaluated. Following Ref. [18] for evaluating DKth, the standard form for the stress intensity factor of a penny-shaped crack in an infinite plate: DKth =Dseff(pac )1/2
(6)
Substituting Eq. (5) in Eq. (6), with the critical crack length value equal to 140 mm, the range of the values of DKth is evaluated as 2.85 – 3.24 MPa m1/2. These values are close to the lower limit of the threshold stress intensity factor cited in Ref. [9]; therefore, this critical value of the short crack length is chosen for the two-dimensional random analysis of the short crack development in the local area. Here we assume that the small square (140× 140 mm2) is a sufficient area for the presentation of random flaws.
The phenomenological approach to short crack evolution in spray-formed material during fatigue testing is considered in the chosen critical area. In this approach, a short crack is a plane vector with the origin lying on the boundary of the critical area with a stochastic vertex which corresponds to a crack tip. Following previous experimental observations, the Markovian random simulation is based on the following: (i) Short crack growth is a succession of small sudden jumps in the horizontal and vertical directions. These mutually perpendicular displacements are independent in an isotropic medium. (ii) The growth of a microcrack depends on the effective stress at the studied moment in the crack growth process. In the framework of (i) and (ii), the crack growth rate is equal to the sum of the mean values and the random force as followed by the Langevin approach [19]. This physical approximation results in a two-dimensional Markov process [19], which is used in modelling of the short crack tip development. The benefits of the Markov process simulation include its simple description by the Fokker–Plank equation and its ability to incorporate different experimental crack growth laws. The Fokker–Plank equation [19] describes the Markov process with the transition density f(x, y, t) equal to the relative frequency of the crack tip passage at point (x+ dx, y+ dy) from point (x, y) during time t. The drift coefficients in this equation are equal to mean rates of Dx or Dy, and diffusion coefficients are estimated as rates of evolution of the dispersions (equal to S 2) of the increments Dx and Dy during fatigue testing. Using experimental data (Fig. 9), approximations of the drift coefficients divided by the critical crack length value and the diffusion coefficients divided
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Fig. 8. Evolution of the mean and standard deviation of increments of short crack length (mm) in both the traverse and longitudinal directions: (a) 262 MPa stress; (b) 234 MPa stress.
by the square of the critical crack length value have been determined as time dependencies at each stress level. After differentiation by the time variable of these dependencies, we have, for a stress equal to 262 MPa, drift rates Au = 0.034N − 0.78 and A6 = − 0.021, and diffusion coefficients Gu =0.000516N − 0.62 and G6 = 0.027. For 234 MPa we have the corresponding system of coefficients: Au =0.041N − 0.85, A6 =0.031, Gu = 0.0121N − 0.72, G6 =0.044, where u is the lateral direction and 6 the vertical one. For these two stress levels, all these coefficients do not increase. The Gu6 coefficient is assumed to be 0 as a consequence of item (i) above. Introducing dimensionless spatial variables u and 6, where u =x/ac and 6 =y/ac , the transition density can be found by a solution of the Fokker – Plank equation with dimensionless spatial variables u and 6:
f %t = − ( fAu )%u −( fA6 )%6 + ( fGu )¦uu/2+ ( fG6 )¦66/2
(7)
where Au and A6 are the drift coefficients and Gu, G6 are the diffusion coefficients. Then, in Eq. (7) the spatial variables change in the ranges 00 u01, 00 601, and the time variable t, where t=N is the number of fatigue cycles. All trajectories of short cracks must be in the square location zone before the fatigue time to fracture t. The critical concentration of short cracks occurs before moment t and is described as the first passage problem out of the critical area. For example, in the case of a fixed origin (in any point (0, 60) from 0 0 601) for random crack development into the critical square, we have the initial condition for Eq. (7) in the form: f(w, 0) =d(u)d(6− 60)
(8)
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where w= (u, 6) and delta d(z) is the Dirac delta function of z. The first passage problem, when the crack tip can only cross line u = 1 at moment t and must remain inside the area 0 0u 01, 0 0 6 0 1 before this moment, dictates the following boundary conditions: Pu (0, 6, t)= 0,
P6 (u, 0, t) = 0,
f(1, 6, t)=0
P6 (u, 1, t)=0, (9)
where Pz (z=u, 6) is the normal component of the flux vector P, which satisfies the equality 9 · P(u, 6, t)= −R(u, 6, t), where R is the right side of Eq. (7). This vector P is the probability current, which is included in the local conservation equation: f %t (u, 6, t) +9 · P(u, 6, t) =0. The last boundary condition from Eq. (9) absorbs a crack tip to the boundary line u =1. The solution of Eq. (7) with the initial condition Eq. (8) and the boundary conditions of Eq. (9) defines the probability distribution of life-time before moment t in the form:
&& 1
F(w, t)=1−
0
1
5. Conclusions f(u, 6, t) du d6
(10)
0
where w=(0, 60). Its position is also a random point on the side of the critical unit square. We will approximate its probability density as a uniform density, equal to 1. In the result, we have the reliability estimation of the fatigue fracture of the material as a probability distribution Q(t), where Q(t)=
&
given by Eq. (9) and the initial conditions of Eq. (8). It was solved by the sweep method, using separation of variables. Numerical integration was used for the estimation of Eqs. (10) and (11). If the reliability measure is used as the probability of exceeding P, the fatigue life-time of a sample will be found from Eq. (11) to depend on P. Each level of these life-time estimations follows the nominal stress. The result is that after each numerical solution procedure, we have the dependence of the life-time of the sample on the probability of exceeding t for the nominal stress. This allows one to model the S– N curve. The results are shown in Fig. 9. It is demonstrated that 50, 90 and 95% of samples will be fractured before moment t, resulting from the lines which model the S–N curve with different degrees of accuracy. The two-dimensional modelling is in agreement with the experimental data of the fatigue tests.
1
F(w, t) dw
(11)
0
The numerical solution procedure was applied to the system Eqs. (7) – (11) with boundary conditions
The fracture of the spray-formed material is due to the continuous accumulation of flaws in the material. The flaws (pores and microcracks) are heterogeneously and randomly distributed in the material. Studying the stochastic behaviour of flaws under fatigue loading results in a phenomenological approach to fatigue life-time modelling. It is assumed, with some experimental background, that a two-dimensional Markov process is suitable for the description of plane short crack development. The use of two-dimensional Markov modelling for estimating fatigue life, in relation to the localization of short cracks in a critical area (a square with the sides equal to the critical crack length), is in good agreement with the experimental data determination from reverse cycling fatigue testing. The method presented for fatigue life-time assessment using a large amount of surface data of flaws gives a conservative estimation of the time to fatigue fracture for the material and the time for the initiation of the main crack in 0.8%C steel tools.
Acknowledgements
Fig. 9. Model of the S–N curve. The dashed line is the experimental S–N curve. The probabilities of exceeding the cycle to fracture are: (1) 50%, (2) 10%, and (3) 5%.
This work was completed during training on a SABIT programme in the Scientific Research Laboratory of the Ford Motor Company. The author wishes to acknowledge Mr D. Wilkosz, Mr A. Krause, Dr C. Anders, Dr J. Boileau and Mr R. Cooper for their help in connection with the experimental research, and Dr Dawn White for formulation of the problem and constant attention to the work.
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