sb 3 c ¼ c v _e ss ¼ D sdsb 3

Transcription

1 Chapter 3 Diffusional-Creep Creep at high temperatures (T T m ) and very low stresses in fine-grained materials was attributed 50 years ago by Nabarro [237] and Herring [51] to the mass transport of vacancies through the grains from one grain boundary to another. Excess vacancies are created at grain boundaries perpendicular to the tensile axis with a uniaxial tensile stress. The concentration may be calculated using [23] c ¼ c v exp sb3 1 kt ð76þ where c v is the equilibrium concentration of vacancies. Usually (sb 3 /kt ) / 1, and therefore equation (76) can be approximated by sb 3 c ¼ c v kt ð77þ These excess vacancies diffuse from the grain boundaries lying normal to the tensile direction toward those parallel to it, as illustrated in Figure 47. Grain boundaries act as perfect sources and sinks for vacancies. Thus, grains would elongate without dislocation slip or climb. The excess concentration of vacancies per unit volume is, then, (c v s/kt ). If the linear dimension of a grain is g, the concentration gradient is (c v s/ktg). The steady-state flux of excess vacancies can be expressed as (D v c v s/ktg). where g is the grain size. The resulting strain-rate is given by, _e ss ¼ D sdsb 3 ktg 2 ð78þ In 1963, Coble [52] proposed a mechanism by which creep was instead controlled by grain-boundary diffusion. He suggested that, at lower temperatures (T < 0.7 T m ), the contribution of grain-boundary diffusion is larger than that of self-diffusion through the grains. Thus, diffusion of vacancies along grain boundaries controls creep. The strain-rate suggested by Coble is _e ss ¼ a 3D gb sb 4 ktg 3 ð79þ 91

2 92 Fundamentals of Creep in Metals and Alloys Figure 47. Nabarro Herring model of diffusional flow. Arrows indicate the flow of vacancies through the grains from boundaries lying normal to the tensile direction to parallel boundaries. Thicker arrows indicate the tensile axis. where D gb is the diffusion coefficient along grain boundaries and a 3 is a constant of the order of unity. The strain-rate is proportional to g 2 in the Nabarro Herring model whereas it is proportional to g 3 in the Coble model. In recent years more profound theoretical analyses of diffusional creep have been reported [238]. Greenwood [238] formulated expressions which allow an approximation of the strain-rate in materials with non-equiaxed grains under multiaxial stresses for both lattice and grain-boundary diffusional creep. Several studies reported the existence of a threshold stress for diffusional creep below which no measurable creep is observed [ ]. This threshold stress has a strong temperature dependence that Mishra et al. [243] suggest is inversely proportional to the stacking fault energy. They proposed a model based on grainboundary dislocation climb by jog nucleation and movement to account for the existence of the threshold stress. The occurrence of Nabarro Herring creep has been reported in polycrystalline metals [ ] and in ceramics [ ]. Coble creep has also been claimed to occur in Mg [251], Zr and Zircaloy-2 [253], Cu [254], Cd [255], Ni [255], copper nickel [256], copper tin [256], iron [257], magnesium oxide [258,259], bco [242], afe [240], and other ceramics [260]. The existence of diffusional creep must be inferred from indirect experimental evidence, which includes agreement with the rate equations developed by Nabarro Herring and Coble, examination of marker lines

3 Diffusional-Creep 93 Figure 48. Denuded zones formed perpendicular to the tensile direction in a hydrated Mg-0.5%Zr alloy at 400 C and 2.1 MPa [262]. visible at the specimen surface that lie approximately parallel to the tensile axis [261], or by the observation of some microstructural effects such as precipitate-denuded zones (Figure 48). These zones are predicted to develop adjacent to the grain boundaries normal to the tensile axis in dispersion-hardened alloys. Denuded zones were first reported by Squires et al. [263] in a Mg-0.5wt.%Zr alloy. They suggested that magnesium atoms would diffuse into the grain-boundaries perpendicular to the tensile axis. The inert zirconium hydride precipitates act as grain-boundary markers. The authors proposed a possible relation between the appearance of these zones and diffusional creep. Since then, denuded zones have been observed on numerous occasions in the same alloy and suggested as proof of diffusional creep. The existence of diffusional creep has been questioned [264] over the last decade by some investigators [59,61, ] and defended by others [56 58,60,261,271,272]. One major point of disagreement is the relationship between denuded zones and

4 94 Fundamentals of Creep in Metals and Alloys diffusional creep. Wolfenstine et al. [59] suggest that previous studies on the Mg-0.5wt.%Zr alloy [273] are sometimes inconsistent and incomplete since they do not give information regarding the stress exponent or the grain-size exponent. By analyzing data from those studies, Wolfenstine et al. [59,265] suggested that the stress exponents corresponded to a higher-exponent-power-law creep regime. Wolfenstine et al. also suggested that the discrepancy in creep rates calculated from the width of denuded zones and the average creep rates (the former being sometimes as much as six times lower than the latter) as evidence of the absence of correlation between denuded zones and diffusional creep. Finally, the same investigators [59,265,266] claim that denuded zones can also be formed by other mechanisms including the redissolution of precipitates due to grain-boundary sliding accompanied by grain-boundary migration and the drag of solute atoms by grainboundary migration. Several responses to the critical paper of Wolfenstine et al. [59] were published defending the correlation between denuded zones and diffusional creep [57,58,271]. Greenwood [57] suggests that the discrepancies between theory and experiments can readily be interpreted on the basis of the inability of grain boundaries to act as perfect sinks and sources for vacancies. Bilde-Sørensen et al. [58] agree that denuded zones may be formed by other mechanisms than diffusional creep but they claim that, if the structure of the grain boundary is taken into consideration, the asymmetrical occurrence of denuded zones is fully compatible with the theory of diffusional creep. Similar arguments were presented by Kloc [271]. Recently McNee et al. [274] claim to have found additional evidence of the relationship between diffusional creep and denuded zones. They studied the formation of precipitate free zones in a fully hydrided magnesium ZR55 plate around a hole drilled in the grip section. The stress state around the hole is not uniaxial, as shown in Figure 49. They observed a clear dependence of the orientation of denuded zones on the direction of the stress in the region around the hole. Figure 49. Orientation of stresses around a hole.

5 Diffusional-Creep 95 Precipitate free zones were mainly observed in boundaries perpendicular to the loading direction at each location. They claim that this relationship between the orientation of the denuded zones and the loading direction is consistent with the mechanism of formation of these zones being diffusional creep. Ruano et al. [ ], Barrett et al. [269], and Wang [270] suggest that the dependence of the creep-rate on stress and grain size is not always in agreement with that of diffusional creep theory. A reinterpretation of several data reported in previous studies led Ruano et al. to propose that the creep mechanism is that of Harper Dorn Creep in some cases and grain-boundary sliding in others, reporting a better agreement between experiments and theory using these models. This suggestion has been contradicted by Burton et al. [64], Owen et al. [56], and Fiala et al. [272]. McNee et al. [275] have recently reported direct microstructural evidence of diffusional creep in an oxygen free high conductivity (OHFC) copper tensile tested at temperatures between 673 and 773 K, and stresses between 1.6 and 8 MPa. The temperature and stress dependencies were found to be consistent with diffusional creep. SEM surface examination revealed, first, displacement of scratches at grain boundaries and, second, widened grain boundary grooves on grain boundaries transverse to the applied stress in areas associated with scratch displacements. In principle, both diffusional creep, as well as some alternative mechanism involving grain-boundary sliding, could be responsible for the observed scratch displacements. The use of atomic force microscopy (AFM) to profile lines traversing boundaries both parallel and perpendicular to the tensile axis led to the conclusion that the scratch displacements originated from the deposition of material at grain boundaries transverse to the tensile axis and the depletion of material at grain boundaries parallel to the tensile axis. The investigators claimed that these features can only be attributed to the operation of a diffusional flow mechanism. However, a strain-rate with an order of magnitude higher than that predicted by Coble creep was found. Thus, the investigators questioned the direct applicability of the diffusional creep theory. Nabarro recently suggested that Nabarro Herring creep may be accompanied by other mechanisms (including GBS and Harper Dorn) [276,277]. Lifshitz [278] already in 1963 pointed out the necessity of grain-boundary sliding for maintaining grain coherency during diffusional creep in a polycrystalline material More recent theoretical studies have also emphasized the essential role of grain boundary sliding for continuing steady-state diffusional creep [ ]. The observations reported by McNee et al. [275] may, in fact, reflect the cooperative operation of both mechanisms. Many studies have been devoted to assess the separate contributions from diffusional creep and grain-boundary sliding to the total strain [ ]. Some claim that both diffusional creep and grain-boundary sliding contribute to the overall strain and that they can be distinctly separated [ ]; others claim that

6 96 Fundamentals of Creep in Metals and Alloys one of them is an accommodation process [ ]. Many of these studies are based on several simplifying assumptions, such as the equal size of all grains and that the total strain is achieved in a single step. Sahay et al. [282] claimed that when the dynamic nature of diffusional creep is taken into account (changes in grain size, etc., that take place during deformation), separation of the strain contributions from diffusion and sliding becomes impossible.