A Cohesive-Zone-Model based fracture criterion for silicon microstructures with sharp notches Z. L. Zhang SINTEF Materials Technology, Trondheim, Norway E-mail: zhiliang.zhang@sintef.no ABSTRACT – Anisotropically etched silicon microstructures may contain atomically sharp notches, which quite often become the origin of mechanical failure. For sharp notches a relation between the notch stress intensity factor and the microscopic surface separation energy similar to the one for classical crack problems, does not exist. A general fracture criterion independent of notch angle has been sought in the literature. In this study, by postulating the existence of a cohesive zone in front of the notch tip and using a simple Barenblatt cohesive zone model, a relation between the critical macroscopic notch stress intensity factor and material’s microscopic separation energy is derived. Being different to the relation for classical crack problems, this general relation is also a function of material's cohesive strength. The criterion has been applied to silicon microstructures with sharp notches, where experimental results are available. INTRODUCTION Fracture of brittle solids with sharp notches becomes a particular concern in recent technology development. For example, micromachined microstructures made of elastic and brittle single crystal silicon, may contain notches which are down to atomically sharp. The sharp notches are often the origin of mechanical failure [1-5]. The sharp notches invalidate the stress based design criterion and a fracture mechanics approach must be utilized. For a crack, a stress singularity exists close to the crack tip and the degree of singularity ( λ-1) (see Eq. (1)) is -0.5. For structures with sharp notches, a stress singularity also exists close to the notch tip. However, the degree of singularity ( λ-1) becomes dependent on the notch opening angle and is generally weaker (less) than –0.5. Because of this, the well known relation for cracks, 0 2 / G K E c c does not hold for notches, where cG is the critical energy release rate, cK is the critical stress intensity factor, 0 is the microscopic surface separation work, 2 /(1 ) E E ν in plane strain and E E in plane stress, E is the Young’s modulus and ν the Poisson ratio.
For notches, the energy release rate is a function of crack size increment and it vanishes when the crack size increment approaches to zero. Further, the value of the critical notch stress intensity factor becomes dependent on the notch geometry. So does its unit. General criteria that link the macroscopic toughness for different notch angles with intrinsic microscopic parameters have been discussed in the literature [2,3], and no solution was provided. In this paper, we postulate that a small size cohesive zone exists in front of the notch tip. Based on a simple Barenblatt’s cohesive zone model for brittle fracture, we obtained a relation between the macroscopic notch stress intensity factor (notch toughness) and materials microscopic surface separation work. With this relation, a unified failure criterion for both cracks and notches is obtained. Figure 1: a) Notch fracture problem, b) cohesive zone driving force, c) cohesive zone resistance and d) a cohesive law. COHESIVE ZONE MODEL SOLUTION For isotropic elastic materials with traction free notch edges and mode I loading, the stress and displacement fields close to the notch tip can be written [6]: ( ) ( ) 1 θ θ σ λ λ Ni N i Nij N ij u K r h K r g (1) l ( ) N ij T K α β b) r ( )N ij T K x y β a) α θ c) CZ σ 0 σ u2 Separation CZ σ d)
where N K is the notch stress intensity factor, (r, θ) are the polar coordinates (see Fig. 1), Nij g and Nih are the load-independent but notch geometry dependent functions. Please note that the definition for N K used here is different to the widely used K in liner elastic fracture mechanics by a factor π2 : π β( 0 ) / 2 K KN o , where β is the notch opening angle. Barenblatt cohesive zone model [7] has been used to derive the fracture criterion. There are two parts that contribute to the elastic singularity at the end of the cohesive zone: the cohesive zone driving force, Fig. 1b, and the material cohesive resistance, Fig. 1c. The cancellation of stress singularity at the end of the cohesive zone can be written ( ) 0 N CZ K K K K (2) In order to derive the cohesive zone model solution, we need to calculate the stress intensity factors at the end of the cohesive zone caused by the remote N K field and by the cohesive resistance force. In addition, we also need the corresponding crack opening displacements by the driving force and resistance. In the following, we treat the driving force and resistance separately. Cohesive Zone Driving Force For an elastic isotropic solid within a notch N K annulus with a small size crack (cohesive zone) at the notch tip, Fig. 1b, Dunn et al. [8] have recently presented an approximate, but analytical relation between K and N K for anti-plane shear case. In this study we will derive a similar relation for mode I case using a more direct formulation. By using the solution by Tada et al. [9] for point-loading at the crack face in an infinite mode I notch, and integrating the point-load solution along the whole cohesive zone length for the virtual notch opening stress acting over the cohesive zone plane ( 1 λ σ x K xN yy ), the K at the end of the cohesive zone caused by the remote N K can be written: ( ) 2 1 β λ K K l m N (3) where m is a dimensionless constant that depends only on the notch opening angle, l is the cohesive zone length. Fig. 2a plots m as a function of the notch opening angle β, which has been fitted by the following polynomial, 3 2 0.02419 0.006533 ( ) 2.503 0.05302 β β β β m (4)
where β π 0 The crack tip opening displacement solution given by Tada et al [9] is accurate for a crack, but is very poor for large notch angles. Finite element analyses have been carried out to determine the crack opening displacement at the notch tip under a given remote N K field. In general, the crack opening displacement can be expressed by the asymptotic equation multiplied by a correction factor, f, which is a function of the β: π β β π β δ λ 2 8 2 8 ( 0) f m l E l K E K f N K x . (5) Fig. 2b shows the f versus the notch opening angle. Fig. 2b has been fitted by the following polynomial, 3 2 0.002234 0.000163 1 0.003926 β β β β f . (6) where β π 0 . 1,9 2 2,1 2,2 2,3 2,4 2,5 2,6 0 30 60 90 120 150 180 Notch opening angle m 0,9 0,92 0,94 0,96 0,98 1 1,02 0 30 60 90 120 150 180 Notch opening angle f a) b) Figure 2: a) m versus notch opening angle β, b) f versus β. Cohesive Zone Resistance Fig. 1d shows a simple cohesive law chosen in this paper. In Fig. 1d, 0 is the surface separation work, 0 0 2 u du CZ σ . For the simple cohesive zone model, a semi-analytical solution for the stress intensity factor and the crack opening displacement from Tada et al [9] can be directly used: ˆ ( ) 0.5 σ β l n K CZ (7) ( ) 4 ˆ 1 β σ δ F E l CZ (8)
where n and 1F are dimensionless parameters from Tada et al. [9]. Failure Criterion By using Eqs. (3) and (7), Eq. (2) can be re-written ( ) ˆ ( ) 0 0.5 0.5 β σ β λ l n K l m N (9) From the above equation, we can obtain the cohesive zone length as λ σ 1 1 ˆn K m l N . Obviously, the cohesive zone length is a function of the notch opening angle. The final crack opening displacement at the original notch tip (x=0) can be obtained by superposing the opening displacements by the driving force (positive, Eq. (5)), and the resistance (negative, Eq. (8): F l K mfl E N σ π δ λ 4 ˆ 2 1 8 1 (10) By substituting the cohesive zone length into the above equation, we get λ λ λ σ σ δ 1 2 1 1 1 ˆ ( ) ˆ 1 NK E A (11) where 1 1 1 4 2 8 F n mf m n m A λ λ π (12) The above A has been fitted by the following polynomials: 2 3 4 ( ) 6.27 0.3223 0.3122 0.07354 0.06644 A β β β β β (13) It must be noted that A does not depend on the loading and material. Fig. 3a shows the A versus the notch opening angle. For cracks A is π2 , and for straight notch ( β=180) the stress singularity disappears, and A=0. Eq. (11) shows that the crack opening displacement at x=0, not only depends on the notch stress intensity factor, but also is a function of the cohesive strength. For the simple cohesive zone model shown in Fig. 1d, the work done at the notch tip (x=0) to separate the material is σˆ δ. Failure by fracture initiation will naturally occur when the σˆ δ approaches to the critical value 0 . Therefore, a failure criterion for notches based on the cohesive zone model can be written 0 ˆ σδ . (14)
By substituting Eq. (11) into the above equation and by rearrangement we finally obtain the critical notch stress intensity factor 2 1 1 0 ˆ λ λ σ A E KN c . (15) With the above equation, the macroscopic notch toughness has been linked to the materials intrinsic separation work. Eq. (15) shows that notch fracture depends on the microscopic surface separation work, cohesive strength, Young’s modulus and notch geometry. According to Eq. (15), fracture will occur when N c N K K .Eq. (15) returns to the well-known Griffith equation for a crack, and we get σˆ N cK for a tensile specimen. APPLICATIONS Fracture toughness of single crystal silicon with cracks and notches at room temperature already exists in the literature. For a crack, average toughness for {111} is 0.33 MPa m and 0.36 MPa m for {110} [10]. The value for {111} has been used here for calculation of the 0 in Eq. (15). The data for notch opening angle 70.53o and 125.26o have been presented by Suwito et al. [2]. The averaged critical notch stress intensity factors for the 70.53o and 125.26o are 0.76 0.481 MPam and 6.5 0.374 MPam , respectively. A T-type structure with notch opening angle 90o has also been tested by Suwito [2]. The reported critical mode I notch stress intensity factor is 2.1 0.452 MPam . Because the T specimens have very thin thickness and are subject to mixed mode loading. The results are not compared in the paper. 2 3 4 5 6 7 0 30 60 90 120 150 Notch opening angle A 0 1 2 3 4 5 6 7 8 9 0 30 60 90 120 150 Notch opening angle KN 0,003 0,025 0,05 0,078 0,16 EXP Figure 3: a) A versus β, b) Critical notch stress intensity factors ( λ1 MPam ) versus β. The numbers in the legend are for E ˆ /σ .
Fig. 3b plots the predicted critical notch toughness versus notch opening angle, together with the experimental data. Various values of E ˆ /σ have been tested in Fig. 3b. It can be seen that E ˆ /σ =0.08 gives a reasonably good prediction of the experimental results. Fig. 3b shows that with the 0 used, the new failure criterion, Eq. (15) underestimates the notch toughness for small notch angles, irrespective of the E ˆ /σ values used. For single crystal silicon, the data for 70.53o is less reliable than the data for 126.26o , becausethe etch stop may not necessarily results in a sharp notch [11]. DISCUSSIONS A criterion based on the so-called “equivalent average stress” concept has been introduced by Seweryn for notch fracture [12]. This criterion assumes that fracture is controlled by the average stress of the critical region over a critical crack size increment, irrespective of the stress gradient. For the notch problem, the criterion can be written: λ λ σ σ π λ 2 2 1 2 y Ic y N c K K (16) where yσ is the yield stress and Ic K the fracture toughness for a cracked specimen. By using the Griffith criterion and substituting Ic K by the separation work 0 , a similar form as Eq. (15) can be obtained. Even though the final form of the two criteria are similar, the present criterion, Eq. (15) is truly based on microscopic energy consideration, while Eq. (16) is from macroscopic average stress concept. During the preparation of this paper, a recent paper by Leguillon [13] on a similar topic was made aware to the author. In contrast to the criterion presented in the paper the criterion by Leguillon is also a macroscopic one. The materials considered in this study are isotropic. For anisotropic materials Stroh’s formulism can be used for constructing the notch tip fields where the eigenvalue solution involves a 6x6 eigen matrix [2,14]. The final structure still preserves the form of Eq. (1), with an additional mode II term. The mode II term for anisotropic elastic materials will vanish when the notch opening angle is larger than about 100. For single crystal silicon which is slightly elastically anisotropic, the mode I eigenvalue for isotropic andanisotropic calculations differs by about 3% [2].
CONCLUDING REMARKS Based on a simple Barenblatt cohesive zone model, a unified failure criterion for brittle fracture of both cracks and notches is presented. Unlike the macroscopic criterion, this fracture criterion does not involve a sudden jump of crack size from zero to a critical value. With this criterion, the macroscopic critical notch stress intensity factor has been directly linked to materials microscopic separation work. Surface separation work and cohesive strength are identified as the two main material parameters for notch fracture of brittle solids. The critical notch stress intensity factor is of course also a function of the notch opening angle and Young’s modulus. Acknowledgements: The work in this study was carried out during the author’s sabbatical stay at Harvard University. Prof. J. Hutchinson’s interest in this work and his support and advice are greatly appreciated. REFERENCES 1. Pourahmadi F., Gee D. and Petersen K. (1991), Proceedings of the IEEE, 197. 2. Suwito W., Dunn M. L., Cunningham S. J., and Read T. (1999) J. Applied Physics, 85, 3519. 3. Mazza E. and Dual J., (1999) J. Mech. Phys. Solids 47, 1795. 4. Busch M., Kuna M. and Heizelmann M. (1994) in Proceedings of European Conference on Fracture, V 1.1. 5. Zhang Z. L., N. Vitorovich, E. Westby, and D. T. Wang, Proceedings of the 10th ICF at Hawaii, Dec. 2-8, 2001. 6. Williams, M. L. (1952) J. Applied Mech. 74, 526. 7. Lawn B., Fracture of Brittle Solids, Cambridge University Press, 1993. 8. Dunn M. L., Hui C. Y., Labossiere P. E. W. and Lin Y. Y. (2001), Int. J. Fracture. 9. Tada H., Paris P. C. and Irwin G. R. (2000), The stress analysis of cracks handbook, 3rd edition, ASME Press. 10. Chen C. P. and Leipold M. H. (1980) Am. Ceram. Soc. Bull, 59, 469. 11. Dunn M. L., Private communications. 12. Seweryn A. (1994) Engineering Fracture Mechanics, 47, 673-681. 13. Leguillon D. Strength or toughness – a criterion for crack onset at a notch, to be published. 14. Stroh A. N. (1962) J. Math. Phys. 41, 77.
A Constraint based Statistical Approach to Failure from Complex Defects Bostjan Bezensek* and John W. Hancock University of Glasgow, Mechanical Engineering Department, Glasgow, UK ABSTRACT: Structural failure from complex defects is examined by including the constraint effects within a statistical weakest link analysis. The method allows the failure probabilities of arbitrary geometries to be compared and identifies the most likely failure initiation site for a complex defect. The analysis can be generalised to problems in which toughness varies with the spatial position of the crack front, as in the case of a crack located in a temperature gradient or subject to embrittlement. 1. INTRODUCTION Defects in engineering structures frequently have complex shapes. Assessment procedures re-characterise these defects as simpler shapes, which are more amenable to analysis. This is conservative as long as the re-characterised defect is more detrimental than the original defect. This process was developed for defects extending by fatigue, but is frequently applied to monotonic loading. In the current work the re-characterisation procedure is applied to defects which extend by cleavage [1]. Cleavage failures are inherently statistical, and are frequently discussed using weakest link arguments in which the local fracture toughness is considered to be a statistically distributed material property (Wallin [2], Slatcher et al [3]). The work extends methods developed by Slatcher et al [3] for defect assessment problems in which both the resistance and the crack driving force are spatial functions of the crack tip position. Although more general aspects can be incorporated into the argument, the paper focuses on in-plane constraint effects. The loss of crack tip constraint can be quantified [4] by the non-singular term in the Williams expansion [5], which Rice [6] has denoted the T-stress. The constraint enhanced fracture toughness is quantified by an expression suggested by Wallin [7]: * On leave from University of Maribor, Faculty of Mechanical Engineering, Slovenia
^ ` T (T) IC 10 (T) IC K 20 (K 20) exp 0.0019( ) , for T<0 K K , forT 0 t (1) 2. WEAKEST LINK STATISTICS FOR SPATIALLY VARYING CRACK PARAMETERS Geometry and the crack shape determine the distribution of the applied stress intensity factor, K(t), along the crack front. Let the crack front be divided into m incremental segments of length dt. The probability that a small crack front segment (i) of length dt, will fail at a locally applied stress intensity less than or equal to K is F(i). The survival of the crack front requires survival of all segments, giving: (i) m (K) F 1 1 F (2) where m is a large number. Reasonable agreement with experimental cleavage data (Landes and Shafer [8]) can be achieved using a two parameter Weibull function, which can be expressed for a straight crack of length, s, under applied stress intensity, K, as: n (K) o o s K F 1 exp s K ° § · ½° ® ¨ ¸ ¾ °¯ © ¹ °¿ (3) Ko and so are curve fitting parameters, and n is the shape factor which Wallin [9] argues to be 4 in J-dominant fields. Without loss of generality Ko can be replaced with a mean value of the local toughness, 1 o n K K (1 ) * , where 1 n (1 ) * is the Gamma function. This allows Eq. (3) to be rewritten with a modified constant, osc : n (K) o 1 K F 1 exp s K ½ ° § · ° ® ¨ ¸ ¾ °¯ c © ¹ °¿ (4) In present problem the mean fracture toughness K is taken to be a function of constraint, as given by Eq. (1). In a complex defect the
constraint, and thus mean local toughness vary with spatial position, t. This allows Eq. (4) to be written more generally as: n (t) (K') o (T,t) s 1 K F 1 exp dt s K § · § · ¨ ¸ ¨ ¸ ¨ ¸ c ¨ ¸ © ¹ © ¹ ³ (5) Under any given loading, the probability of failure, and the most likely site of failure initiation can be determined if the crack front is divided into two segments, A and B, as shown schematically in Figure 1. The probability that failure initiates from segment A requires segment A to fail, while the other segment survives. The total probability of failure of the entire crack is given by moving the boundary, from one extremity of the crack to the other. The most likely (modal) failure initiation site can be determined by differentiating the cumulative probability function (CPF) of (5) along the crack front. The most likely site of failure initiation is the location with the greatest value of the probability density function (pdf). This procedure is used to analyse cleavage failures from complex defects, and allows a comparison of the failure probabilities of real and re-characterised defects. 3. EXPERIMENTAL The experimental work focused on a complex defect, formed by the coalescence of two surface breaking cracks which were developed by fatigue. Experiments were performed on a plain carbon-manganese 50D steel (BS 4360), using three point bend specimens with the geometry shown in Figure 2. Surface breaking defects were produced by fatigue tests on plate
containing two co-planar starter notches, cut with a circular slitting wheel. This procedure resulted in a family of complex defects with pronounced reentrant sectors, as shown in Figure 3(a) and eventually a bounding semielliptical defect, as shown in Figure 3(b). These were tested to failure at –196oC, (representative of the lower shelf) and at –100oC, (representative of the ductile-brittle transition). At –196oC the configurations with pronounced re-entrant sector failed at 75 kN and 85 kN, whereas bounding defect failed at 90 kN and 98 kN [1]. In the tests at –100 oC failure loads were 210 kN and 93 kN for re-entrant and bounding geometries. Figure 3: Fracture surface of test specimens with a complex defect and a bounding semi-elliptical defect (a) (b)
4. RESULTS AND DISCUSSION 4.1 Deterministic analysis The crack profiles were digitised and analysed using the line spring technique of Rice and Levy [10] as implemented in ABAQUS [11] to determine the local stress intensity factor and T-stress. Defects were re-characterised following BS 7910 [12], with a bounding semi-elliptical defect. Results for the re-entrant and re-characterised geometries are presented in Figure 4. Figure 4(a) shows the crack profiles. The normalised stress intensity factors are given in Figure 4(b), and the T-stress normalised with the outer fibre stress is shown in Figure 4(c). Compressive T-stresses and amplified stress intensity factors develop in re-entrant sectors. Limit loads were determined numerically as 208 kN at –196 oC for the defect with the pronounced re-entrant sector and 225 kN for re-characterised defects and 197 kN and 108 kN for the two defects tested at –100 oC.
4.2 Probabilistic analysis The cumulative probability of failure along the crack front of the complex and re-characterised geometries is given in Figure 5 for the applied failure load of the real defect. For failures at small fractions of the limit load, the constraint effects in pronounced re-entrant sectors are not significant, giving higher failure probability for the real defect with re-entrant sector, as shown in Figure 5(b) over the re-characterised defect. Conversely for failures close to the limit load (–100oC) there is significant constraint loss in the shallow re-entrant sector, decreasing the failure probability of the real defect compared to the re-characterised defect, as shown in Figure 5(c). This is consistent with the failure loads observed in the experiments.
4.3 The origin of failure The most likely site of failure is indicated by the probability density function, shown in Figure 6. The most likely site of failure initiation at small fractions of the limit load is the re-entrant sector, as shown by failure at –196oC, and illustrated in Figure 6(b). For failures close to the limit load the pdf (Figure 6(c)) indicates the most likely failure site are the deeper crack segments close to the re-entrant sector. Although these sites do not have the greatest stress intensity, there are only small constraint enhanced toughness effects.
5. CONCLUSIONS The paper presents a statistical method for assessing the probability of failure for complex defects incorporating in-plane constraint effects. The method has been extended to indicate the most likely failure site from complex crack profiles. The defect re-characterisation process is shown to be conservative in ductile-brittle transition, when in-plane constraint effects are invoked. A potential non-conservative assessment occurs when re-characterisation procedures developed for fatigue are applied to cleavage failures from complex defects [1] on the lower shelf. 6. REFERENCES 1 Bezensek, B. and Hancock, J.W. (2002), To be published 2 Wallin, K., (1985), Engineering Fracture Mechanics, 22, p.149 3 Slatcher, S. and Oystein, E., (1986), Engineering Fracture Mechanics, 24, No.4, p. 495 4 Betegón, C. and Hancock, J.W., (1991), Journal of Applied Mechanics, 58, p:104 5 Williams, M.L., (1957), ASME Journal of Applied Mechanics, 24, p.111 6 Rice, J.,R., (1974), Journal of The Mechanics and Physics of Solids, 22, p: 17 7 Wallin, K., (2000), Proceedings ASME PVP2000 conference, 412, USA, July 2000 8 Landes, J.D. and Shaffer, D.H., (1980), In “Fracture Mechanics: Twelfth Conference”, ASTM STP 700 9 Wallin, K., (1984), Engineering Fracture Mechanics, 19, p.1085 10 Rice, J.R. and Levy, N., (1972), Journal of Applied Mechanics, 39, No. 1, March 1972, p: 185 11 HKS, (1998), ABAQUS/Standard, v5.8, Hibbitt, Karlsson and Sorensen, inc, Providence, Rhoad Island 12 BS 7910, (1999), Guidance on methods for assessing the acceptability of flaws in metallic structures, British Standard Institution, London, UK
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A Creep Fatigue Crack Propagation Study of the PM Nickel Based Superalloy UDIMET 720 Li Branco, C.M. 1, Brito, A.S. 1 and Byrne, J. 2 1 ICEMS/IST, Lisbon University of Technology, Lisbon, Portugal. 2 University of Portsmouth, Dep. of Mech. Engineering, Portsmouth, UK. ABSTRACT: In this paper, the fatigue and creep crack growth behaviour was studied at temperatures of 650 and 725ºC in compact tension specimens, and the results were compared. Fatigue tests were carried out with a constant stress ratio (R=0.1) at two different frequencies (5 and 0.25Hz) in order to study the effect of frequency in crack propagation. The results obtained in these tests were converted, based on time, and compared with the creep results. For fatigue, da/dN vs. ΔK curves were used, and for creep, da/dt vs. Kmax curves, since, for this material, the stress intensity factor has shown to be the most appropriate parameter for the study of crack propagation. The FCGR results have shown that the effect of frequency is predominant over the temperature influence, that is, da/dN has a more significant increase when the frequency is lowered from 5 to 0.25Hz than when the temperature is increased from 650 to 725ºC. An analysis of the cracking mechanisms will be presented in the paper, specially for creep, where mixed load type crack propagation was found (I+II) for loading under mode I only. The main failure mechanism detected in the material was mixed mode transgranular – intergranular. However, in creep intergranular and secondary cracking was predominant when the crack propagated at inclined angles (I+II) with the crack propagation direction of mode I. The crack growth rates, da/dN and da/dt, where mainly related with the stress intensity factor component in mode I. 1. INTRODUCTION The nickel base superalloy Udimet 720Li was developed bearing in mind its use in turbine discs and applications in energy production. The combination of the alloying elements, as well as its fabrication process and sintering, were devised with the aim of improving its behaviour in service, where the fatigue-creep interaction is predominant and the temperatures are between 650 and 700ºC. It is generally accepted [1-2] that this improvement is essentially due to the refinement of the grain size, associated to a higher number of alloying elements and a greater dispersion of the γ' phase. In the fabrication process, the powder material was isostatically heated, pressed and extruded; a solution treatment was subsequently applied, from
which a uniform structure of fine grain was obtained, with an average size of 7μm. The composition of the microstructure (Table 1) is, approximately, 18% of phase (γ'), associated with a small amount of carbo-nitrites. The thermal treatment consisted in four hours of preparation with a temperature between 1080 and 1100ºC, followed by cooling in oil up to 650ºC and maintenance of this temperature for 24 hours and an air cooling. Finally, the material was exposed to a temperature of 760ºC for 16 hours, followed by cooling at room temperature. TABLE 1: Chemical composition of the sinterized nickel base superalloy Udimet 720Li. Ni Cr Mo Ti Al Co Zr W Fe B C 59.3 16.2 3.2 5.1 2.6 14.5 0.035 1.7 0.072 0.022 <0.1 There is a limited amount of work published in the literature presenting fatigue and crack growth data on Udimet 720Li. The effect of stress ratio on FCGR was analysed in [3]. However, all this data were essentially obtained at temperatures below or up to 600ºC. Byrne and co-workers [4] carried out more extensive work on the FCGR mechanisms, at temperatures between 600 and 650ºC. Additional results obtained at 725ºC were published recently by the authors [5]. Creep crack growth rate data is also available by the authors [5], but no other creep data was found in the literature for temperatures above 650ºC. This paper presents a summary of the fatigue and creep crack growth rate results mainly obtained at 650ºC. The obtained fatigue and creep crack growth rate data is compared with published data obtained at 650ºC for other nickel based superalloys of the cast and wrought type. 2. EXPERIMENTAL DETAILS AND RESULTS All the tests were made in CT specimens (26x13mm), in a servohydraulic test machine. The specimens were taken from a turbine disc and tested in a high temperature furnace. The monitorization of the crack growth was made with a pulsed potential drop system. By using the valid results for each test zone, calibration curves were obtained, as well as da/dN; ΔK cracking diagrams. Results were correlated with a Paris law type relation, da/dN=CΔKm; linear regression type correlations were applied to the results. The values of the C constant and the m exponent were determined for all the test situations. The values of
da/dN were obtained from the numerical differentiation of the pairs of values a and N obtained in each specimen, according with the finite difference method. Fig. 1 shows the results and correlations da/dN; ΔK for 650ºC at 5 and 0.25Hz. Results show a significant influence of the frequency in the crack speed, which increases from a factor of 2 in the lower zone of the values of ΔK(≅20 MPa.m1/2) until a factor of about one order of magnitude for ΔK÷60 MPa.m1/2, when the frequency lowers form 5 to 0.25Hz, sinusoidal. (da/dN)/Δ K 0,25Hzsni 650 ºC 5H zsin650 ºC 1,00E-07 1,00E-06 1,00E-05 1,00E+01 1,00E+02 Δ K [MPa*m1 /2] da/dN[m/ciclo] Refª 9 ,10 Figure 1: Results and correlations da/dN; ΔK. R=0.1. CT specimens. 650ºC (5 and 0.25Hz). PM nickel base superalloy Udimet 720Li. The fractographic analysis indicates that the fatigue crack mode of this material, at 650ºC, changes from mixed with transgranular dominance, at 5Hz (Fig. 2a), to mixed with intergranular dominance, at 0.245Hz (Fig. 2b). This change in the behaviour of the material according to frequency is due to an increase in the time of exposure to the maximum load and, consequently, to a higher tensile field, which provokes a greater damage due to the oxidation and creep effect [6, 7].
a) b) Figure 2: SEM fractographic image of the fracture surface of the material. a) 650ºC, 5Hz.ΔK≅30 MPa.m1/2.b) 650ºC, 0.25Hz. ΔK≅30 MPa.m1/2. Figure 3: Curved crack in CT specimen. Creep at 650ºC. In the case of curved cracks in mode I loading, the following reference relation is applicable for the stress intensity factor, K: KIl=cos 3( θ/2.KIn) (1) where KII is the component of the local stress intensity factor in mode I at the tip of the curved crack with θ angle, and KIn the nominal remote stress intensity factor in mode I, given only by the stress intensity factor equation for the CT specimen [8]. TABLE 2 : Values of C, m of the Paris law. Creep tests. Udimet 720Li. Da/dt=CKIn m Da/dt=CKII m Ref. T C m r2 C m r2 ºC mm/hour mm/hour 1892 650 1.45E-4 2.87 0.94 6.5E-9 5.86 0.92 1898 725 1.10E-11 7.15 0.94 6.91E-16 10.9 0.92 1897 725 1.09E-05 3.73 0.97 1.99E-11 7.97 0.97 TOTAL 725 1.80E-06 4.28 0.85 3.73E-08 5.61 0.68
The creep tests performed are indicated in Table 2. At the creep crack tests it was shown that the crack propagated with a curved trajectory, inflecting towards the direction of the applied load (Fig. 3). The direction of the propagation occurred, thus, in mixed mode I+II, where the Mode II component increased as the crack length increased. Apparently, this type of behaviour is peculiar to the nickel base superalloys, and has been very briefly referred to in the literature [4,5], although not in detail. In Fig. 4, the da/dt; Kmax data, quoted in [3,4] and obtained for Udimet 720Li, are also plotted. For creep lower da/dt values were obtained in [3,4], which could be due to the testing method, which was constant K testing in [4,8] under a decreasing stress field. For the computation of KIn and KIi, the projected equivalent depth of the crack, aef, was used in equation (1). aef is, therefore, the crack depth in the mode I direction of crack growth, normal to the loading axis. The equation for aef is: aef=ApI/B (2) where ApI is the projected area of the curved crack (Fig. 3) in the mode I direction. To use the calibration line methodology, the real area, A, of the curved crack should be used. Assuming that a uniform crack front is obtained (no variation of A and a in the thickness direction), the relation between A an ApI is given by cos pI A A θ = (3) where θ is the tangent angle to the curved crack in each point. It was found that θ varied + 20% along the crack front (through the thickness), and the mean value of θ through the thickness was used in equations (1-3). Since this approach is approximate, further work is in progress to obtain K or J for the curved cracks, using a 2D or 3D FE approach, which will provide a more adequate solution. Non-uniform crack fronts and tilted cracks could be taken into account in the FE simulations, in the follow-up of previous work for tunnelling cracks [9]. It was verified that, even for Kmax, the highest crack speed, da/dt, is obtained for a frequency of 5Hz, which means that, in this type of material, the plastic cyclic deformation seems to induce a larger damage within the same time, t. As previously observed, the fractographic analysis has shown that the failure mode in this material, for fatigue at 650ºC, is essentially mixed, intergranular/transgranular.
(da/dt)/Κmax 5 Hz 0,25 Hz 650 ºC creep Ki l 650 ºC creep Kin 1,00E-01 1,00E+00 1,00E+01 1,00E+02 1,00E+01 1,00E+02 Kmax[MPam1/2] da/dt[mm/h] Refª 9,10 creep Ref 9,10 Figure 4: Results and correlations da/dt; Kmax. Udimet 720Li. Creep at 650ºC. Fig. 5 is a fractography of the creep crack at 650ºC. The failure is now intergranular, with a great number of empty voids (cavities), some of them interconnected by coalescence. The slippering between grain boundaries is reduced, as a consequence of the smaller plasticity of the material and the reduced time of loading. The roughness of the fracture surface is quite high, due to the intergranular character of the failure. The main mechanism of the creep damage, in this case, is the coalescence of the microcavities. The present data for Udimet 720Li at 650ºC was compared with da/dt and da/dN data for other Ni base superalloys. The plots da/dt and da/dN against frequency are shown in Figs. 6a) and 6b). For da/dt under fatigue loading, the present results are only marginally higher than the results obtained for the other alloys (Fig. 6a)). Under creep (Fig. 6a)), there is a sudden increase in daq/dt due to the change of failure mode to a basic intergranular fracture (Fig. 5). The obtained results da/dN against are in good agreement with those obtained in the other Ni base superalloys of similar composition (Rene 95 and Waspalloy) but, for Udimet 720Li, da/dN
is slightly higher than for Inconel 718, which is a Ni-Fe cast and wrought alloy. Within the range of frequencies analysesd, a unique straight line may be fitted to the entire set of da/dN against frequency data (Fig. 6b)), indicating a similar cracking mechanism of mixed mode at this temperature. Figure 5 : SEM fractographic image of the fracture surface of Udimet 720Li. 650ºC, creep/Kmax≅30 MPa.m1/2. da/dt;f 1,00E-02 1,00E-01 1,00E+00 1,00E+01 1,00E+02 0,0001 0,001 0,01 0,1 1 10 100 f da/dt [mm/h] UDIMET 720LI UDIMET 720LI (UK) INCONEL RENE’95 Waspaloy da/dN;f 1,00E-04 1,00E-03 1,00E-02 1,00E-01 0,001 0,01 0,1 1 10 f da/dN[mm/cycle] INCONEL UDIMET 720LI UDIMET 720LI (UK) RENE’95 Waspaloy a) b) Figure 6 : a) da/dt (log) against the frequency logf. ΔK=30MPa.m1/2. b) da/dN against the frequency logf. ΔK=30MPa.m1/2 3. CONCLUSIONS • The fatigue failure mode detected and observed at the SEM was essentially mixed, transgranular/intergranular, even at room temperature. • The creep crack growth was in mixed mode (I+II), with a curved crack propagating in loading conditions typical of the mode I. • The conversion of the fatigue results to da/dt has shown that the crack speed is higher for higher frequencies, that is to say, the plastic cyclic deformation contributes for a higher crack speed of the material. • CGR on Udimet 720Li at 650ºC was foud to be slightly higher than Inconel 718, but very close to other similar grades Ni base superalloys
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