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Inter- and intragranular stresses in cyclically-deformed 316 stainless steel
Materials Science and Engineering A 399 (2005) 114–119
Inter- and intragranular stresses in cyclically-deformed 316 stainless steel X.-L. Wang a,b,∗ , Y.D. Wang a , A.D. Stoica a , D.J. Horton a,c , H. Tian c , P.K. Liaw c , H. Choo c , J.W. Richardson d , E. Maxey d a
Spallation Neutron Source (SNS), Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996-2200, USA d Intense Pulsed Neutron Source, Argonne National Laboratory, Argonne, IL 60439, USA b
c
Abstract Neutron diffraction was used to study residual stresses in cyclically-deformed 316 LN stainless steel. The tension–compression high-cycle fatigue tests were conducted in air with a frequency of 0.2 Hz. Large intergranular stresses were found to develop in the stainless steel as a result of elastic and plastic anisotropy. These intergranular stresses started to decrease when micro-cracks were initiated at the surface and vanished when the sample reached failure. Cyclic loading also led to the development of intragranular stresses, as evidenced by the broadening of the diffraction peaks. Analysis of the orientation dependence of the measured peak widths indicates that the immobile dislocations generated by fatigue deformation are mostly edge rather than screw type. © 2005 Elsevier B.V. All rights reserved. Keywords: Intergranular stresses; Neutron diffraction; Stainless steel
1. Introduction When a polycrystalline material is subjected to plastic deformation, intergranular stresses are generated due to elastic and plastic anisotropy. These intergranular stresses are grain-orientation-dependent in nature, i.e., the sign and magnitude of a given stress component within a grain is dependent on the orientation of the grain relative to the specimen directions. Over the last decade, intergranular stresses have been used as fingerprints to understand the deformation in polycrystalline materials. A number of neutron diffraction measurements have been made to determine intergranular stresses in metallic alloys [1–6]. Most of the Neutron diffraction experiments were conducted with in situ uni-axial loading. Theoretical studies, based on the Taylor model [7], selfconsistent models [2–4], and the finite element method [8], have been carried out to understand the intergranular stress data obtained with Neutron diffraction. The progress in mod∗
Corresponding author. Tel.: +1 865 574 9164; fax: +1 865 574 6080. E-mail address: [email protected] (X.-L. Wang).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.02.030
eling has now produced a qualitative description of the stress states in individual grains with different orientations. However, quantitatively, the calculated strains sometimes differ from the experimental values by more than 30%, especially for samples subjected to a large degree of plastic deformation [4]. While a reasonable knowledge base is built for intergranular stresses due to uni-axial loading, little is known about the development of intergranular stresses due to cyclic loading. Whether a large intergranular stress is produced and how it develops during cyclic deformation with a small amount of monotonic strains but a large amount of accumulated strains has been an outstanding question [6,9]. Cyclic loading also produces inhomogeneous strain fields within grains, which is usually referred to as intragranular strains. The strain fluctuations about the mean strain value induce diffraction peak broadening. Thus, from the analysis of peak broadening, considerable knowledge may be gained about the intragranular stresses resulting from the immobile dislocations produced by cyclic deformation. These intragranular stresses are usually discussed in terms of the elastic energy stored in the in-
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homogeneous strain field around dislocations. To be specific, we refer this type of elastically stored energy as intragranular stored energy. We have initiated a systematic Neutron diffraction study of inter- and intragranular stresses in 316 stainless steel due to high-cycle fatigue loading. Preliminary results of an earlier experiment have been communicated previously [10]. Here we report some of the recent experimental data and provide a detailed account of the data analysis method.
2. Experimental details The specimens were commercial Type 316 LN stainless steel, with the composition 16.31Cr–10.2Ni–2.07Mo– 1.75Mn–0.23Cu–0.16Co–0.11N–0.039Si–0.029P–0.009C– 0.002S (wt.%). The 316 LN stainless steel is also the prime candidate material for the container of the liquid mercury target of the spallation neutron source being built at the Oak Ridge National Laboratory, Oak Ridge, TN [11–13]. The mechanical properties of this material have been summarized in [13]: the apparent Young’s modulus is 186 GPa whereas the 0.2% yield strength is 288 MPa. Before fatigue tests, the stainless steel was annealed at 1038 ◦ C for 1 h and cooled to room temperature in air in order to release the residual stresses produced by hot-deformation and specimen cutting. High-cycle fatigue tests with a frequency of 0.2 Hz were conducted in air (at room temperature) under a load-control mode using an R ratio of −1 (R = σ min /σ max , where σ min and σ max are, respectively, the minimum and maximum of the applied stresses). Standard cylindrical-bar specimens were used for the fatigue studies. The cylindrical specimens have a gauge length of 20.32 mm, and a diameter of 7.62 mm. The maximum applied stress was 287 MPa, just below the macroscopic 0.2% yield strength. After ∼5000 cycles, micro-cracks started to appear on the specimen surface. The specimens reached failure after 7872 cycles. The microstructures of fatigued specimens were characterized by scanning-electron microscopy (SEM). The SEM samples were cut along the longitudinal direction and observed under a HITACHI S-3500 scanning-electron microscope. The microstructure of a failed specimen after 7872 cycles is shown in Fig. 1(a). The microstructure consists of uniform austenitic grains with little segregation. Some annealing twins are also evident. Micro-cracks started to appear after 5159 cycles. The micro-cracks always start on the surface of the specimens, and there are no indications of internal cracks. The cracks are typically trans-granular in nature, as shown in Fig. 1(b), and propagate normal to the loading axis. The lengths of the cracks are typically 30–300 m. The SEM studies also indicate that on the surface of the specimens, the distances between cracks are much larger than the grain size, by at least an order of magnitude. Five fatigued specimens were studied by Neutron diffraction, after loading to 500, 3000, 5159, 5506 and 7872 cy-
Fig. 1. Scanning-electron micrographs of 316 stainless steel specimens. The arrows indicate loading direction, and the scale bar is 200 m: (a) after 7872 cycles when failure is reached. There are no indications of internal cracks: (b) after 5159 cycles when micro-cracks started to appear on the surface of the specimen.
cles, respectively. An annealed specimen was also measured and used as a stress-free reference. Neutron diffraction measurements were carried out at the Intense Pulsed Neutron Source, Argonne National Laboratory, with the general purpose powder diffractometer (GPPD). Using the time-of-flight method, a large number of reflections were measured simultaneously. The lattice strains for various reflections were determined as a function of specimen orientations by using a kappa goniometer. Because GPPD has a rather large detector coverage (54–156◦ 2θ), it is ideally suited for the determination of crystallographic texture and intergranular strains. Measurements of the orientation-dependent lattice strains followed the procedures outlined previously [10,14]. Although diffraction data were collected simultaneously in all detector banks, only those from the high-angle detector banks, with nominal scattering angles 2θ = ±126◦ , ±144◦ , were used in the present analysis in order achieve highprecision for the determination of lattice strains and peak widths.
3. Intergranular strains Representative lattice strains are illustrated in Fig. 2 as a function of tilt angle, Ψ , with regard to the load direction (LD). Evidence of large intergranular strains is readily seen. In LD (Ψ = 0), for example, the {2 0 0} strain is compressive,
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Fig. 2. Experimental intergranular strains in cyclically-deformed 316 stainless steel: intergranular strains determined with {1 1 1}, {2 0 0}, {2 2 0}, {3 1 1}, and {2 2 2} reflections as a function of tilt angle relative to the loading direction after 5159 cycles (when microcracks started to appear on the surface).
whereas the {2 2 0} strain is tensile. Both {2 0 0} and {2 2 0} strains vary strongly as a function of Ψ , reversing signs twice as Ψ increases from 0◦ to 90◦ . In the transverse direction (TD, Ψ =90◦ ), the {2 0 0} and {2 2 0} strains return to compressive and tensile values, respectively. On the contrary, the {1 1 1} and {3 1 1} reflections showed minimal intergranular strains. Lattice strains measured with {1 1 1} and {2 2 2} reflections follow each other closely, demonstrating the quality of our data. The lattice strain data at two end points, LD and TD, are consistent with previous in situ loading studies on similar stainless steels. It has been shown [2,6], for example, that tensile residual strains develop for {2 0 0} at the end of a tensile loading cycle in LD. Our fatigue test was conducted with R = −1, i.e., ending with a compressive load. Thus, a compressive residual strain is expected for {2 0 0} in LD. Several elastic–plastic self-consistent (EPSC) modeling studies [2–4,6] have been undertaken to understand how such intergranular strains developed. The underlining physics was illustrated by Holden et al. [1] with Taylor model. By considering an ensemble of grains, Holden et al. demonstrated that the 1 1 1//LD grains yielded first, despite that 1 1 1 has the highest yield strength. This unusual behavior is a result of a combination of elastic and plastic anisotropy. For stainless steel, 1 1 1 direction is the stiffest elastically. As a result, in a polycrystalline environment, 1 1 1//LD grains carry the highest load, causing 1 1 1//LD grains to reach the yield strength faster than other grains. Similarly, 1 0 0//LD grains yield last, despite having the lowest yield strength, because the 1 0 0 is the softest direction. Therefore, unloading the tensile (compressive) load leaves a tensile (compressive) strain for {2 0 0}. The tensile strain for {2 2 0} in LD can be understood in a similar way. Un-
derstanding the behavior of intergranular strains in TD is less straightforward, because the orientation of a h k l//TD grain relative to LD is not unique. This subtlety was first pointed out by Clausen et al. [2]. For 1 0 0//TD grains, for example, their orientations in LD can be any member of the h k 0 family. Thus, the {h k l} lattice strains in TD really represent an average over all grains with h k l//TD, each grain having a different {h k l} strain value along TD. Calculations of intergranular strains in TD, therefore, must rely on numerical modeling. An EPSC modeling study by Clausen et al. [2] shows that for stainless steel of similar compositions, the effect of averaging leads to a tensile strain for {2 0 0} and a compressive strain for {2 2 0} in TD after tensile loading. Thus, in our fatigue specimens with R = −1, a compressive residual strain is expected for {2 0 0}, whereas for {2 2 0}, the residual strain should be tensile in TD. Far more than the two end points, our data revealed complete profiles for the intergranular strains as a function of Ψ (from 0◦ to 90◦ ). These data should provide a more rigorous test for numerical modeling of deformation behaviors in polycrystalline materials. Fig. 3 depicts the evolution of the {2 0 0} strain (in LD) as a function of load cycles. For comparison, tensile loading data from Lorentzen et al. [6] are superimposed. Their experiment was conducted with a type of stainless steel with the composition, 18.25Cr–13.42Ni–3.66Mo–1.48Mn–0.44Si–0.02C (wt.%), which is similar to that of our Type 316 LN stainless steel (see above). The elastic and plastic properties are also similar. The specimens were incrementally loaded under load control, cycling the materials between fixed strain limits of 0.4%. Neutron diffraction data were collected at each increment in load, up to eight cycles. The maximum tensile load was 290 MPa. The combined data gave a complete picture of the evolution of intergranular strains during cyclic loading, from beginning to end. It appears that intergranular strains developed immediately and almost saturated after the first cycle of loading. The slight increase of the {2 0 0} strain during the initial cy-
Fig. 3. Evolution of the {2 0 0} lattice strain as a function of load cycles. Data in the early cycles (up to 8) were taken from Lorentzen et al. [6].
X.-L. Wang et al. / Materials Science and Engineering A 399 (2005) 114–119
cles is attributable to hardening. After micro-cracks start to appear (5159 cycles), the {2 0 0} strain vanishes. Similar behaviors have been observed for other reflections. Calculations of the elastically stored energy, which takes into account intergranular strains in all directions, confirm that the intergranular stresses began to decrease when micro-cracking was observed and vanished after failure was reached [10].
4. Peak broadening and intragranular stored energy Cyclic deformation also leads to inhomogeneous strain fields, which give rise to considerable broadening of the diffraction peaks. Fig. 4 shows experimentally determined integral breadths, βint , of the strain distribution (after correcting for instrument resolution) as a function of Ψ for the specimen after 5156 cycles. Clearly, the values of βint due to cyclic deformation are also grain-orientation dependent. Note that βint is measured in unit of strain, 10−6 . To gain insights into the sub-grain structure, we analyzed the (h k l) dependence of βint . As was generally accepted for heavily-deformed materials, the sub-grain structure, rather than the intergranular strain variations, produces the strain fluctuations responsible for peak broadening. Because the crystallographic defects in the stainless steel are mostly immobile dislocations, the deformation induced peak broadening comes primarily from the strain field around these dislocations. The sensitivity of diffraction to the dislocation strain field is modulated by the relative orientation of the diffraction vector with respect to the dislocation line orientation and the displacement vector (Burgers vector). This problem was first solved by Wilkens [15] for random-distributed screw dislocations. An analytical equation for the strain
Fourier coefficients was obtained, and an average contrast factor was defined. Later, this approach was extended to all types of dislocations including the elastic anisotropy of the material (see, for example, [16]). Following this approach, the dislocations contribution to βint can be estimated as: √ b Cρ (1) βint = √ F (Re ρ) where b is the value of the Burgers vector, C is the contrast factor, ρ is the density of dislocations, and F is an integral factor over Wilkens’ profile, which accounts for the outer cut-off radius of the dislocation strain field, Re . Although Wilkens recommended to choose Re ∼ ρ−1/2 , the accuracy of the profile analysis of the present data prevented obtaining a reliable estimation of Re . Hence, we have adopted a Re value defined by ln(Re /b) = 2π, which is often used for the calculation of elastically stored energy, W [17]: Kb2 ρ Re Kb2 ρ W= ln ≈ (2) 4π b 2 In Eq. (2), K is the energy factor defined as a molar quantity: C44 K = VA √ r(Φ); Ai
Ai =
2C44 = 3.77 C11 − C12
(3)
where VA is the molar volume, Cij are the elastic constants, and Ai is the elastic anisotropy factor of stainless steel. The ratio r(Φ) depends on the dislocation character, Φ being the angle between the dislocation line direction and the Burgers vector. This ratio is a function of Ai and N = C12 /(C11 + C12 ), as predicted by anisotropic elasticity theory of dislocations [18], and for stainless steel varies from 1 to about 1.67 when the dislocation character changes from screw to edge type. The Burgers vectors for stainless steel were considered along the 1 1 0 directions, and the dislocation lines were considered normal to the {1 1 1} planes. A total of 12 slip systems have to be accounted for the fcc structure stainless steel. The single crystal elastic constants of stainless steel with C11 = 204.6 GPa, C12 = 137.7 GPa and C44 = 126.2 GPa [19] were used for the calculation of stored energy. The contrast factors were calculated using the program ANIZC [20] and averaged over all 12 equally probable slip systems. The contrast factor, Chkl , is given by, 2 Chkl = C0 (1 − qHhkl );
Fig. 4. Experimentally determined integral breadth of the strain distribution as a function of tilt angle relative to the loading direction. The peak width has been corrected for instrument resolution.
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2 Hhkl =
h2 k2 + k2 l2 + l2 h2 2
(h2 + k2 + l2 )
(4)
The values of C0 and q depend on the dislocation type. Using ANIZC, we found C0 and q to be 0.3001 and 1.7182 for edge dislocations, and 0.3237 and 2.4695 for screw dislocations. Substituting Eq. (4) into Eq. (1), it can be seen 2 with respect to H 2 that the normalized slope, q, of βint hkl is quite different for different types of dislocations, which should allow a reliable assessment of the main dislocation type from the experimental data analysis. The average peak
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tragranular stresses did not decrease even after the initiation of cracks.
Acknowledgements
Fig. 5. Analysis of the peak broadening data: βint is the integral breadth; 2 = h2 k2 +k2 l2 +l2 h2 . The lines are fits based on edge type of dislocations, Hhkl 2 2 2 2 (h +k +l )
with q = 1.7182.
broadening data (over the tilt angle, Ψ ) presented in Fig. 5 2 are compared with a linear fit havas a function of Hhkl ing the slope q = 1.7182, which corresponds to the edge type dislocations. The excellent fit seen in Fig. 5 indicates that the plastic deformation associated with the cyclic loading induces a significant number of immobile edge dislocations. Transmission electron-microscopy studies are underway to verify the findings of Neutron diffraction experiments. The average dislocation densities calculated using Eq. (1) are 1.5 × 1014 m−2 after 5159 cycles, 2.2 × 1014 m−2 after 5506 cycles, and 1.9 × 1014 m−2 at failure, after 7872 cycles. The corresponding average intragranular stored energies, calculated using Eq. (2), are 3.6, 5.5, and 4.5 J/mol, respectively. This result shows that, in contrast to intergranular stresses, the intragranular stresses did not decrease after cracks started to appear. We want to emphasize, as shown in Fig. 4, that the broadening of diffraction peaks also shows grain-orientation dependence. This subject will be treated later in a separate publication.
5. Summary Through neutron diffraction measurements, we have established the evolution of intergranular stresses in 316 LN stainless steel due to cyclic deformation, from beginning to end. Large intergranular stresses developed immediately after a few initial cycles of deformation. The intergranular stresses started to decrease, when micro-cracks were initiated, and disappeared completely after failure was reached. From the analysis of diffraction peak widths, we suggest that the immobile dislocations in 316 LN stainless steel are mainly edge type. The elastically stored energy associated with in-
This research was sponsored by the US Department of Energy, Division of Materials Sciences and Engineering, under Contract DE-AC05-00OR22725 with Oak Ridge National Laboratory managed by UT-Battelle. The Neutron diffraction work has benefited from the use of the intense pulsed neutron source at Argonne National Laboratory, which is funded by the US Department of Energy, BES-Materials Science, under Contract No. W-31-109-ENG-38. This research was also supported in part by an appointment to the Oak Ridge National Laboratory Postdoctoral Research Associates Program administered by the Oak Ridge Institute for Science and Education, and by the US National Science Foundation, the Combined Research and Curriculum Development (CRCD) Program, under EEC-9527527 and EEC-0203415, with Ms. Mary Poats as the contract monitor, the Integrative Graduate Education and Research Training (IGERT) Program, under DGE-9987548, with Drs. Wyn Jennings and Larry Goldberg as contract monitors, and the International Materials Institute Program under DMR-0231320 with Dr. Carmen Huber as the contract monitor. H.T. and P.K.L. would like to thank Drs. Lou Mansur and Joe Strizak of the Oak Ridge National Laboratory for their kind support of H.T.’s Ph.D. thesis research related to fatigue behavior of Type 316 stainless steel used as the target container materials of the spallation neutron source.
References [1] T.M. Holden, R.A. Holt, A.P. Clarke, J. Neutron Res. 5 (1997) 241–264. [2] B. Clausen, T. Lorentzen, T. Leffers, Acta Mater. 46 (1998) 3087–3098. [3] J.W. Pang, T.M. Holden, T.E. Mason, Acta Mater. 46 (1998) 1503. [4] M.R. Daymond, C.N. Tom´e, M.A.M. Bourke, Acta Mater. 48 (2000) 553–564. [5] Y.D. Wang, R. Lin Peng, X.-L. Wang, R.L. McGreevy, Acta Mater. 50 (2002) 1717–1734. [6] T. Lorentzen, M.R. Daymond, B. Clausen, C.N. Tome, Acta Mater. 50 (2002) 1627–1638. [7] K. Van Acker, P. van Houtte, E. Aernoudt, ICOTOM-11 2 (1996) 1461. [8] P. Dawson, D. Boyce, S. MacEwen, R. Rogge, Metall. Mater. Trans. A 31 (2000) 1543. [9] A.M. Korsunsky, K.E. James, ISIS Annual report, 2001. RB No. 12180. http://www.isis.rl.ac.uk/isis2001. [10] Y.D. Wang, H. Tian, A.D. Stoica, X.-L. Wang, P.K. Liaw, J.W. Richardson, Nat. Mater. 2 (2003) 103–106. [11] H. Tian, P.K. Liaw, J.P. Strizak, L. Mansur, J. Nucl. Mater. 318 (2003) 157–166. [12] J.P. Strizak, J.R. DiStefano, P.K. Liaw, H. Tian, J. Nucl. Mater. 296 (2001) 225–230. [13] H. Tian, P.K. Liaw, H. Wang, D. Fielden, J.P. Strizak, L. Mansur, J.R. DiStefano, Mater. Sci. Eng. A 314 (2001) 140–149.
X.-L. Wang et al. / Materials Science and Engineering A 399 (2005) 114–119 [14] Y.D. Wang, X.-L. Wang, A.D. Stoica, J.W. Richardson, R. Lin Peng, J. Appl. Cryst. 36 (2003) 14–22. [15] M. Wilkens, in: J.A. Simmons, R. de Witt, R. Bullough (Eds.), Fundamental Aspects of Dislocation Theory, vol. II, Natl. Bur. Stand. US Spec., Washington, DC, 1970, pp. 1195–1221, Publication No. 317. [16] A. Borbely, J.H. Driver, T. Ungar, Acta Mater. 48 (2000) 2005–2016.
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[17] O.B. Pederson, Acta Mater. 35 (1987) 2567–2581. [18] J.W. Steeds, Introduction to Anisotropic Elasticity Theory of Dislocations, Clarendon Press, Oxford, 1973. [19] H.M. Ledbetter, Phys. Stat. Sol. A85 (1984) 89–96. [20] A. Borbely, J. Dragomir-Cernatescu, G. Ribarik, T. Ungar, J. Appl. Cryst. 36 (2003) 160–162, http://metal.elte.hu/anizc.
Inter- and intragranular stresses in cyclically-deformed 316 stainless steel X.-L. Wang a,b,∗ , Y.D. Wang a , A.D. Stoica a , D.J. Horton a,c , H. Tian c , P.K. Liaw c , H. Choo c , J.W. Richardson d , E. Maxey d a
Spallation Neutron Source (SNS), Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996-2200, USA d Intense Pulsed Neutron Source, Argonne National Laboratory, Argonne, IL 60439, USA b
c
Abstract Neutron diffraction was used to study residual stresses in cyclically-deformed 316 LN stainless steel. The tension–compression high-cycle fatigue tests were conducted in air with a frequency of 0.2 Hz. Large intergranular stresses were found to develop in the stainless steel as a result of elastic and plastic anisotropy. These intergranular stresses started to decrease when micro-cracks were initiated at the surface and vanished when the sample reached failure. Cyclic loading also led to the development of intragranular stresses, as evidenced by the broadening of the diffraction peaks. Analysis of the orientation dependence of the measured peak widths indicates that the immobile dislocations generated by fatigue deformation are mostly edge rather than screw type. © 2005 Elsevier B.V. All rights reserved. Keywords: Intergranular stresses; Neutron diffraction; Stainless steel
1. Introduction When a polycrystalline material is subjected to plastic deformation, intergranular stresses are generated due to elastic and plastic anisotropy. These intergranular stresses are grain-orientation-dependent in nature, i.e., the sign and magnitude of a given stress component within a grain is dependent on the orientation of the grain relative to the specimen directions. Over the last decade, intergranular stresses have been used as fingerprints to understand the deformation in polycrystalline materials. A number of neutron diffraction measurements have been made to determine intergranular stresses in metallic alloys [1–6]. Most of the Neutron diffraction experiments were conducted with in situ uni-axial loading. Theoretical studies, based on the Taylor model [7], selfconsistent models [2–4], and the finite element method [8], have been carried out to understand the intergranular stress data obtained with Neutron diffraction. The progress in mod∗
Corresponding author. Tel.: +1 865 574 9164; fax: +1 865 574 6080. E-mail address: [email protected] (X.-L. Wang).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.02.030
eling has now produced a qualitative description of the stress states in individual grains with different orientations. However, quantitatively, the calculated strains sometimes differ from the experimental values by more than 30%, especially for samples subjected to a large degree of plastic deformation [4]. While a reasonable knowledge base is built for intergranular stresses due to uni-axial loading, little is known about the development of intergranular stresses due to cyclic loading. Whether a large intergranular stress is produced and how it develops during cyclic deformation with a small amount of monotonic strains but a large amount of accumulated strains has been an outstanding question [6,9]. Cyclic loading also produces inhomogeneous strain fields within grains, which is usually referred to as intragranular strains. The strain fluctuations about the mean strain value induce diffraction peak broadening. Thus, from the analysis of peak broadening, considerable knowledge may be gained about the intragranular stresses resulting from the immobile dislocations produced by cyclic deformation. These intragranular stresses are usually discussed in terms of the elastic energy stored in the in-
X.-L. Wang et al. / Materials Science and Engineering A 399 (2005) 114–119
115
homogeneous strain field around dislocations. To be specific, we refer this type of elastically stored energy as intragranular stored energy. We have initiated a systematic Neutron diffraction study of inter- and intragranular stresses in 316 stainless steel due to high-cycle fatigue loading. Preliminary results of an earlier experiment have been communicated previously [10]. Here we report some of the recent experimental data and provide a detailed account of the data analysis method.
2. Experimental details The specimens were commercial Type 316 LN stainless steel, with the composition 16.31Cr–10.2Ni–2.07Mo– 1.75Mn–0.23Cu–0.16Co–0.11N–0.039Si–0.029P–0.009C– 0.002S (wt.%). The 316 LN stainless steel is also the prime candidate material for the container of the liquid mercury target of the spallation neutron source being built at the Oak Ridge National Laboratory, Oak Ridge, TN [11–13]. The mechanical properties of this material have been summarized in [13]: the apparent Young’s modulus is 186 GPa whereas the 0.2% yield strength is 288 MPa. Before fatigue tests, the stainless steel was annealed at 1038 ◦ C for 1 h and cooled to room temperature in air in order to release the residual stresses produced by hot-deformation and specimen cutting. High-cycle fatigue tests with a frequency of 0.2 Hz were conducted in air (at room temperature) under a load-control mode using an R ratio of −1 (R = σ min /σ max , where σ min and σ max are, respectively, the minimum and maximum of the applied stresses). Standard cylindrical-bar specimens were used for the fatigue studies. The cylindrical specimens have a gauge length of 20.32 mm, and a diameter of 7.62 mm. The maximum applied stress was 287 MPa, just below the macroscopic 0.2% yield strength. After ∼5000 cycles, micro-cracks started to appear on the specimen surface. The specimens reached failure after 7872 cycles. The microstructures of fatigued specimens were characterized by scanning-electron microscopy (SEM). The SEM samples were cut along the longitudinal direction and observed under a HITACHI S-3500 scanning-electron microscope. The microstructure of a failed specimen after 7872 cycles is shown in Fig. 1(a). The microstructure consists of uniform austenitic grains with little segregation. Some annealing twins are also evident. Micro-cracks started to appear after 5159 cycles. The micro-cracks always start on the surface of the specimens, and there are no indications of internal cracks. The cracks are typically trans-granular in nature, as shown in Fig. 1(b), and propagate normal to the loading axis. The lengths of the cracks are typically 30–300 m. The SEM studies also indicate that on the surface of the specimens, the distances between cracks are much larger than the grain size, by at least an order of magnitude. Five fatigued specimens were studied by Neutron diffraction, after loading to 500, 3000, 5159, 5506 and 7872 cy-
Fig. 1. Scanning-electron micrographs of 316 stainless steel specimens. The arrows indicate loading direction, and the scale bar is 200 m: (a) after 7872 cycles when failure is reached. There are no indications of internal cracks: (b) after 5159 cycles when micro-cracks started to appear on the surface of the specimen.
cles, respectively. An annealed specimen was also measured and used as a stress-free reference. Neutron diffraction measurements were carried out at the Intense Pulsed Neutron Source, Argonne National Laboratory, with the general purpose powder diffractometer (GPPD). Using the time-of-flight method, a large number of reflections were measured simultaneously. The lattice strains for various reflections were determined as a function of specimen orientations by using a kappa goniometer. Because GPPD has a rather large detector coverage (54–156◦ 2θ), it is ideally suited for the determination of crystallographic texture and intergranular strains. Measurements of the orientation-dependent lattice strains followed the procedures outlined previously [10,14]. Although diffraction data were collected simultaneously in all detector banks, only those from the high-angle detector banks, with nominal scattering angles 2θ = ±126◦ , ±144◦ , were used in the present analysis in order achieve highprecision for the determination of lattice strains and peak widths.
3. Intergranular strains Representative lattice strains are illustrated in Fig. 2 as a function of tilt angle, Ψ , with regard to the load direction (LD). Evidence of large intergranular strains is readily seen. In LD (Ψ = 0), for example, the {2 0 0} strain is compressive,
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Fig. 2. Experimental intergranular strains in cyclically-deformed 316 stainless steel: intergranular strains determined with {1 1 1}, {2 0 0}, {2 2 0}, {3 1 1}, and {2 2 2} reflections as a function of tilt angle relative to the loading direction after 5159 cycles (when microcracks started to appear on the surface).
whereas the {2 2 0} strain is tensile. Both {2 0 0} and {2 2 0} strains vary strongly as a function of Ψ , reversing signs twice as Ψ increases from 0◦ to 90◦ . In the transverse direction (TD, Ψ =90◦ ), the {2 0 0} and {2 2 0} strains return to compressive and tensile values, respectively. On the contrary, the {1 1 1} and {3 1 1} reflections showed minimal intergranular strains. Lattice strains measured with {1 1 1} and {2 2 2} reflections follow each other closely, demonstrating the quality of our data. The lattice strain data at two end points, LD and TD, are consistent with previous in situ loading studies on similar stainless steels. It has been shown [2,6], for example, that tensile residual strains develop for {2 0 0} at the end of a tensile loading cycle in LD. Our fatigue test was conducted with R = −1, i.e., ending with a compressive load. Thus, a compressive residual strain is expected for {2 0 0} in LD. Several elastic–plastic self-consistent (EPSC) modeling studies [2–4,6] have been undertaken to understand how such intergranular strains developed. The underlining physics was illustrated by Holden et al. [1] with Taylor model. By considering an ensemble of grains, Holden et al. demonstrated that the 1 1 1//LD grains yielded first, despite that 1 1 1 has the highest yield strength. This unusual behavior is a result of a combination of elastic and plastic anisotropy. For stainless steel, 1 1 1 direction is the stiffest elastically. As a result, in a polycrystalline environment, 1 1 1//LD grains carry the highest load, causing 1 1 1//LD grains to reach the yield strength faster than other grains. Similarly, 1 0 0//LD grains yield last, despite having the lowest yield strength, because the 1 0 0 is the softest direction. Therefore, unloading the tensile (compressive) load leaves a tensile (compressive) strain for {2 0 0}. The tensile strain for {2 2 0} in LD can be understood in a similar way. Un-
derstanding the behavior of intergranular strains in TD is less straightforward, because the orientation of a h k l//TD grain relative to LD is not unique. This subtlety was first pointed out by Clausen et al. [2]. For 1 0 0//TD grains, for example, their orientations in LD can be any member of the h k 0 family. Thus, the {h k l} lattice strains in TD really represent an average over all grains with h k l//TD, each grain having a different {h k l} strain value along TD. Calculations of intergranular strains in TD, therefore, must rely on numerical modeling. An EPSC modeling study by Clausen et al. [2] shows that for stainless steel of similar compositions, the effect of averaging leads to a tensile strain for {2 0 0} and a compressive strain for {2 2 0} in TD after tensile loading. Thus, in our fatigue specimens with R = −1, a compressive residual strain is expected for {2 0 0}, whereas for {2 2 0}, the residual strain should be tensile in TD. Far more than the two end points, our data revealed complete profiles for the intergranular strains as a function of Ψ (from 0◦ to 90◦ ). These data should provide a more rigorous test for numerical modeling of deformation behaviors in polycrystalline materials. Fig. 3 depicts the evolution of the {2 0 0} strain (in LD) as a function of load cycles. For comparison, tensile loading data from Lorentzen et al. [6] are superimposed. Their experiment was conducted with a type of stainless steel with the composition, 18.25Cr–13.42Ni–3.66Mo–1.48Mn–0.44Si–0.02C (wt.%), which is similar to that of our Type 316 LN stainless steel (see above). The elastic and plastic properties are also similar. The specimens were incrementally loaded under load control, cycling the materials between fixed strain limits of 0.4%. Neutron diffraction data were collected at each increment in load, up to eight cycles. The maximum tensile load was 290 MPa. The combined data gave a complete picture of the evolution of intergranular strains during cyclic loading, from beginning to end. It appears that intergranular strains developed immediately and almost saturated after the first cycle of loading. The slight increase of the {2 0 0} strain during the initial cy-
Fig. 3. Evolution of the {2 0 0} lattice strain as a function of load cycles. Data in the early cycles (up to 8) were taken from Lorentzen et al. [6].
X.-L. Wang et al. / Materials Science and Engineering A 399 (2005) 114–119
cles is attributable to hardening. After micro-cracks start to appear (5159 cycles), the {2 0 0} strain vanishes. Similar behaviors have been observed for other reflections. Calculations of the elastically stored energy, which takes into account intergranular strains in all directions, confirm that the intergranular stresses began to decrease when micro-cracking was observed and vanished after failure was reached [10].
4. Peak broadening and intragranular stored energy Cyclic deformation also leads to inhomogeneous strain fields, which give rise to considerable broadening of the diffraction peaks. Fig. 4 shows experimentally determined integral breadths, βint , of the strain distribution (after correcting for instrument resolution) as a function of Ψ for the specimen after 5156 cycles. Clearly, the values of βint due to cyclic deformation are also grain-orientation dependent. Note that βint is measured in unit of strain, 10−6 . To gain insights into the sub-grain structure, we analyzed the (h k l) dependence of βint . As was generally accepted for heavily-deformed materials, the sub-grain structure, rather than the intergranular strain variations, produces the strain fluctuations responsible for peak broadening. Because the crystallographic defects in the stainless steel are mostly immobile dislocations, the deformation induced peak broadening comes primarily from the strain field around these dislocations. The sensitivity of diffraction to the dislocation strain field is modulated by the relative orientation of the diffraction vector with respect to the dislocation line orientation and the displacement vector (Burgers vector). This problem was first solved by Wilkens [15] for random-distributed screw dislocations. An analytical equation for the strain
Fourier coefficients was obtained, and an average contrast factor was defined. Later, this approach was extended to all types of dislocations including the elastic anisotropy of the material (see, for example, [16]). Following this approach, the dislocations contribution to βint can be estimated as: √ b Cρ (1) βint = √ F (Re ρ) where b is the value of the Burgers vector, C is the contrast factor, ρ is the density of dislocations, and F is an integral factor over Wilkens’ profile, which accounts for the outer cut-off radius of the dislocation strain field, Re . Although Wilkens recommended to choose Re ∼ ρ−1/2 , the accuracy of the profile analysis of the present data prevented obtaining a reliable estimation of Re . Hence, we have adopted a Re value defined by ln(Re /b) = 2π, which is often used for the calculation of elastically stored energy, W [17]: Kb2 ρ Re Kb2 ρ W= ln ≈ (2) 4π b 2 In Eq. (2), K is the energy factor defined as a molar quantity: C44 K = VA √ r(Φ); Ai
Ai =
2C44 = 3.77 C11 − C12
(3)
where VA is the molar volume, Cij are the elastic constants, and Ai is the elastic anisotropy factor of stainless steel. The ratio r(Φ) depends on the dislocation character, Φ being the angle between the dislocation line direction and the Burgers vector. This ratio is a function of Ai and N = C12 /(C11 + C12 ), as predicted by anisotropic elasticity theory of dislocations [18], and for stainless steel varies from 1 to about 1.67 when the dislocation character changes from screw to edge type. The Burgers vectors for stainless steel were considered along the 1 1 0 directions, and the dislocation lines were considered normal to the {1 1 1} planes. A total of 12 slip systems have to be accounted for the fcc structure stainless steel. The single crystal elastic constants of stainless steel with C11 = 204.6 GPa, C12 = 137.7 GPa and C44 = 126.2 GPa [19] were used for the calculation of stored energy. The contrast factors were calculated using the program ANIZC [20] and averaged over all 12 equally probable slip systems. The contrast factor, Chkl , is given by, 2 Chkl = C0 (1 − qHhkl );
Fig. 4. Experimentally determined integral breadth of the strain distribution as a function of tilt angle relative to the loading direction. The peak width has been corrected for instrument resolution.
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2 Hhkl =
h2 k2 + k2 l2 + l2 h2 2
(h2 + k2 + l2 )
(4)
The values of C0 and q depend on the dislocation type. Using ANIZC, we found C0 and q to be 0.3001 and 1.7182 for edge dislocations, and 0.3237 and 2.4695 for screw dislocations. Substituting Eq. (4) into Eq. (1), it can be seen 2 with respect to H 2 that the normalized slope, q, of βint hkl is quite different for different types of dislocations, which should allow a reliable assessment of the main dislocation type from the experimental data analysis. The average peak
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tragranular stresses did not decrease even after the initiation of cracks.
Acknowledgements
Fig. 5. Analysis of the peak broadening data: βint is the integral breadth; 2 = h2 k2 +k2 l2 +l2 h2 . The lines are fits based on edge type of dislocations, Hhkl 2 2 2 2 (h +k +l )
with q = 1.7182.
broadening data (over the tilt angle, Ψ ) presented in Fig. 5 2 are compared with a linear fit havas a function of Hhkl ing the slope q = 1.7182, which corresponds to the edge type dislocations. The excellent fit seen in Fig. 5 indicates that the plastic deformation associated with the cyclic loading induces a significant number of immobile edge dislocations. Transmission electron-microscopy studies are underway to verify the findings of Neutron diffraction experiments. The average dislocation densities calculated using Eq. (1) are 1.5 × 1014 m−2 after 5159 cycles, 2.2 × 1014 m−2 after 5506 cycles, and 1.9 × 1014 m−2 at failure, after 7872 cycles. The corresponding average intragranular stored energies, calculated using Eq. (2), are 3.6, 5.5, and 4.5 J/mol, respectively. This result shows that, in contrast to intergranular stresses, the intragranular stresses did not decrease after cracks started to appear. We want to emphasize, as shown in Fig. 4, that the broadening of diffraction peaks also shows grain-orientation dependence. This subject will be treated later in a separate publication.
5. Summary Through neutron diffraction measurements, we have established the evolution of intergranular stresses in 316 LN stainless steel due to cyclic deformation, from beginning to end. Large intergranular stresses developed immediately after a few initial cycles of deformation. The intergranular stresses started to decrease, when micro-cracks were initiated, and disappeared completely after failure was reached. From the analysis of diffraction peak widths, we suggest that the immobile dislocations in 316 LN stainless steel are mainly edge type. The elastically stored energy associated with in-
This research was sponsored by the US Department of Energy, Division of Materials Sciences and Engineering, under Contract DE-AC05-00OR22725 with Oak Ridge National Laboratory managed by UT-Battelle. The Neutron diffraction work has benefited from the use of the intense pulsed neutron source at Argonne National Laboratory, which is funded by the US Department of Energy, BES-Materials Science, under Contract No. W-31-109-ENG-38. This research was also supported in part by an appointment to the Oak Ridge National Laboratory Postdoctoral Research Associates Program administered by the Oak Ridge Institute for Science and Education, and by the US National Science Foundation, the Combined Research and Curriculum Development (CRCD) Program, under EEC-9527527 and EEC-0203415, with Ms. Mary Poats as the contract monitor, the Integrative Graduate Education and Research Training (IGERT) Program, under DGE-9987548, with Drs. Wyn Jennings and Larry Goldberg as contract monitors, and the International Materials Institute Program under DMR-0231320 with Dr. Carmen Huber as the contract monitor. H.T. and P.K.L. would like to thank Drs. Lou Mansur and Joe Strizak of the Oak Ridge National Laboratory for their kind support of H.T.’s Ph.D. thesis research related to fatigue behavior of Type 316 stainless steel used as the target container materials of the spallation neutron source.
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