Predictive capabilities of the dislocation-network theory of Harper-Dorn creep

The dislocation-network theory of Harper-Dorn (H-D) creep is reformulated using a new equation for the kinetics of growth of individual dislocation links in the network. The new kinetic equation has no impact on the scaled differential equation derived previously, which predicts the distribution of link lengths. However, the new theory predicts slightly different behavior for the kinetics of static recovery and leads to a new equation for the strain rate, which is expressed in terms of parameters that can be evaluated independently. This equation is valid not only for steady-state H-D creep, but is also valid for primary creep, provided the instantaneous value of the dislocation density is known. Using data on the variation of dislocation density with time, calculated values of the creep rates for Al deformed in the H-D regime agree with experimentally measured values to within a factor of 2. Creep curves for Al are calculated with the same degree of accuracy. These calculations involve no adjustable parameters. Steady-state creep rates for many materials presumably deformed in the H-D creep regime are compared with the predictions of the new equation for the strain rate. The calculated values agree with experimentally measured data to within a factor of about 150, which compares well with the predictions of other equations proposed in the literature. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Creep at very low rates

The creep rate in a land-based power station must be less than 10⁻¹¹ s⁻¹. At these low rates of deformation the transport of matter occurs by the migration of vacancies rather than by the glide of dislocations. A quantitative understanding of these diffusional processes is, therefore, important. First type of diffusional creep (Nabarro-Herring (N-H)): the sources and sinks of vacancies are grain boundaries. The vacancies may diffuse through the bulk of the grain or along the grain boundaries (Coble (C)). Second type (Harper-Dorn (H-D)): the vacancies diffuse from edge dislocations with their Burgers vectors parallel to the major tensile axis to those with Burgers vectors perpendicular to this axis. The coherence of the polycrystalline aggregate is maintained by sliding along the grain boundaries. The three mechanisms of vacancy migration, grain boundary sliding, and dislocation glide may all interact. The theories of N-H and C creep in pure metals are established and confirmed, but H-D creep and grain boundary sliding are less well understood. Practical engineering materials are usually strengthened by precipitates that accumulate on grain boundaries and slow down creep in complicated ways. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in Dan Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Creep at low stresses: An evaluation of diffusion creep and Harper-Dorn creep as viable creep mechanisms

High-temperature creep experiments often reveal a transition at very low stresses to a region where the stress exponent is reduced to a value lying typically in the range of ∼1 to 2. This region is generally associated with the occurrence of a new creep mechanism, such as grain-boundary sliding, diffusion creep, and/or Harper-Dorn creep. Several recent reports have suggested that diffusion creep and Harper-Dorn creep may not be viable creep mechanisms. This article examines these two processes and demonstrates that there is good evidence supporting the occurrence of both creep mechanisms under at least some experimental conditions. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Understanding creep—a review

A simple model based on the Orowan equation and the dynamic evolution of the dislocation structure by generation and merging of slipped areas is used to see which experimental results on creep of pure and solute-hardened crystalline materials can or cannot be explained with regard to creep with refinement or coarsening of the dislocation structure and steady-state creep. Quantitative deficiencies of the model for pure materials are discussed; most of them are related to neglection of subgrain formation. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Some selected topics in creep cavitation

The nucleation and growth of intergranular cavitation is one of the major contributors to high-temperature failure. We focus here on three developments in this area. The first is the effect of combined creep deformation and diffusive mass transport upon void growth. Second, we discuss environmental influences upon cavity growth, principally in the form of gas-producing internal reactions. Finally, we outline some of the experimental observations concerning randomness in creep cavitation and modeling efforts intended to capture some of these random effects. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Denuded zones, diffusional creep, and grain boundary sliding

The appearance of denuded zones following low stress creep in particle-containing crystalline materials is both a microstructural prediction and observation often cited as irrefutable evidence for the Nabarro-Herring (N-H) mechanism of diffusional creep. The denuded zones are predicted to be at grain boundaries that are orthogonal to the direction of the applied stress. Furthermore, their dimensions should account for the accumulated plastic flow. In the present article, the evidence for such denuded zones is critically examined. These zones have been observed during creep of magnesium, aluminum, and nickel-base alloys. The investigation casts serious doubts on the apparently compelling evidence for the link between denuded zones and diffusional creep. Specifically, denuded zones are clearly observed under conditions that are explicitly not diffusional creep. Additionally, the denuded zones are often found in directions that are not orthogonal to the applied stress. Other mechanisms that can account for the observations of denuded zones are discussed. It is proposed that grain boundary sliding accommodated by slip is the rate-controlling process in the stress range where denuded zones have been observed. It is likely that the denuded zones are created by dissolution of precipitates at grain boundaries that are simultaneously sliding and migrating during creep. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in Dan Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

The role of impurities during creep and superplasticity at very low stresses

It is well documented that impurities play an important role in the deformation and fracture of polycrystalline materials. For example, the results of a number of studies have demonstrated that the presence of a very small of amount of impurities in polycrystalline materials can explain many phenomena such as temper embrittlement in steels, creep embrittlement, and enhancement of ductility in the intermetallic compound Ni₃Al. This article reviews the details of two high-temperature deformation phenomena whose characteristics are, according to very recent experimental evidence, influenced or controlled by impurities. The first phenomenon, micrograin superplasticity, deals with the ability of fine-grained materials (d<10 μm, where d is the grain size) to exhibit extensive neck-free elongations during deformation at elevated temperatures above 0.5 T _m, where T _m is the melting point. The second phenomenon, Harper-Dom creep, refers to the anomalous creep behavior of large-grained materials at very low stresses and temperatures near the melting point. It is shown that while these two phenomena are different in terms of the conditions of occurrence and the characteristics of deformation, they share three common features: (1) stresses applied to produce deformation are very small; (2) impurities control the deformation characteristics such as the shape of the creep curve, the value of the stress exponent, and the details of the substructure; and (3) boundaries play a key role during deformation. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Application of a modified jogged-screw model for creep of TiAl and α-Ti alloys

Stress exponents for creep, in the range of 5, are typically associated with dislocation creep processes, normally associated with a strong tendency for subgrain formation. In this article, we will demonstrate that there are several important alloy systems that have similar stress dependence and, yet, lack this tendency for subgrain formation. Specifically, dislocations in the intermetallic compound γ-TiAl and the hexagonal close-packed (hcp) α phase of the commercial Ti alloy, Ti-6242, tend to be homogeneously distributed with a tendency for alignment along screw orientation. In both alloy systems, the screw dislocations exhibit a large density of pinning points, which detailed transmission electron microscopy (TEM) investigation indicate are locations of tall jogs. These observations suggest that the jogged-screw model for creep should be appropriate after suitable modification for the presence of these tall jogs. This modified jogged-screw (MJS) model is presented, together with a discussion of the assumptions made, and the results of this model are shown to compare favorably with experiment for both alloy systems. The possible criteria for the formation of tall jogs are also described, and the potential application of this modified model to other alloy systems is discussed. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

A framework for modeling creep in pure metals

The process of creep in pure metals is modeled as the cooperative interaction of three phenomena: the thermally activated, force-dependent release of dislocation segments from obstacles; the substructural refinement of the microstructure due to plastic deformation; and the diffusion-controlled coarsening of the substructure. Key parameters are given as approximate generic values which can be varied. It is shown that for a wide range of parameters, the model reproduces the key features of the creep of pure metals: a steady-state stress exponent near 5 is recovered, and the key microstructural-length scale is related by a power law close to the reciprocal of stress (this dependence is not a strong function of temperature at a given stress). In addition, the activation energy of steady-state creep is nearly that of self-diffusion. Thus, the model reproduces the well-known phenomenology of puremetal steady-state creep. However, the present model is based on separate microstructural phenomena, which can be independently refined and studied. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Steady-state creep of <Emphasis Type="Italic">α</Emphasis>-zirconium at temperatures up to 850 °C

Cumulative zirconium creep data over a broad range of stresses (0.1 to 115 MPa) and temperatures (300 °C to 850 °C) were analyzed based on an extensive literature review. Zirconium obeys traditional power-law creep with a stress exponent of approximately 6.4 over stain rates and temperatures usually associated with the conventional “five-power-law” regime. Thus, dislocation climb, rather than the often assumed glide mechanism, may be rate controlling. Power-law breakdown occurs at values of greater than approximately 10⁹ cm⁻², consistent with most traditional five-power-law materials. The creep rate of zirconium at low values of σ/G varies proportionally to the applied stress. The rate-controlling mechanism(s) for creep within this regime is unclear. A grain-size dependency may exist, particularly at small (<90 μm) sizes, suggesting a diffusional mechanism. A grain-size independence at larger grain sizes supports a Harper-Dorn mechanism, but the low observed activation energy (∼90 kJ/mol) is not consistent with those observed at similar temperatures at higher stresses in the five-power-law regime (270 kJ/mol) where creep is also believed to be lattice self-diffusion controlled. The stress dependence in this regime is not consistent with traditional grain-boundary sliding mechanisms. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture”, which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Elevated-temperature deformation at forming rates of 10−2 to 102 s−1

In the hot working at constant strain rate ( ) of Al and α Fe alloys at 0.5 to 0.9 T _M (absolute melting temperature), steady-state deformation is achieved in similarity to creep, which is usually at constant stress. After an initial strain-hardening transient, the flow stress becomes constant in association with a substructure which remainsequiaxed and constant in the spacing of sub-boundaries and of dislocations in both walls and subgrains. All these spacings become larger at higher temperature (T) and lower values as well as with lower stress, being fully consistent with the relationships established in creep. Because hot working can proceed to a much higher true strain in torsion (∼100) and compression (∼2) as well as in extrusion (∼20) and rolling (∼5), it is possible to confirm that grains continue to elongate while the subgrains within them remain equiaxed and constant in size. When the thickness of grains reaches about 2 subgrain diameters (d _s), the grain boundaries with serrations (∼d _s) begin to impinge and the grains pinch off, becoming somewhat indistinguishable from the subgrains; this has been called geometric dynamic recrystallization (DRX). In polycrystals as at 20 °C, deformation bands form and rotate during hot working according to the Taylor theory, developing textures very similar to those in cold working. In metals of lower dynamic recoverability such as Cu, Ni, and γ Fe, new grains nucleate and grow (discontinuous DRX), leading to a steady state related to frequently renewed equiaxed grains, containing an equiaxed substructure that develops to a constant character and defines the flow stress. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

The stability of lamellar structures

The thermal stability of lamellar structures under creep or hot-working conditions is considered. It is suggested that the microstructural stability of the two-phase alloy can be altered by the introduction of boundaries formed by deformation and recovery processes. Expressions are derived for the rate of growth at triple points formed in this way. These expressions are related to the stability of unidirectionally grown eutectics under creep conditions, and spheroidization observed under hot working conditions. Under certain limiting conditions the results reduced to those of Mullins for “grooving” at grain boundary-free surface junctions by volume and surface diffusion.     

Observations of room-temperature creep recovery in titanium alloys

Conventional α(hcp) and α(hcp)/β(bcc) titanium alloys exhibit significant primary creep strains at room temperature and at stresses well below their macroscopic yield strength. It has been previously reported in various materials systems that repeated unloading during primary creep testing may either accelerate or retard the accumulation of creep strains. These effects have been demonstrated to depend on both microstructure and the applied stress. This article demonstrates that significant room-temperature recovery occurs in technologically relevant titanium alloys. These recovery mechanisms are manifested as a dramatic increase in creep rates (by several orders of magnitude) upon the introduction of individual unloading events, ranging from 1 minute to 365 days, during primary creep tests. Significant increases in both creep rate and the total accumulated creep strain were observed in polycrystalline single α-phase Ti-6Al, polycrystalline α/β Ti-6Al-2Sn-4Zr-2Mo-0.1Si, and individual α/β colonies of Ti-6242. Based on transmission electron microscopy (TEM) studies of the active deformation mechanisms, it is proposed that the presence of significant stress concentrations within the α phase of these materials, in the form of dislocation pileups, is a prerequisite for significant room-temperature recovery. M.F. SAVAGE, formerly with the Department of Materials Science and Engineering, The Ohio State University Columbus, OH. This article is based on a presentation made in the symposium entitled “Defect Properties and Mechanical Behavior of HCP Metals and Alloys” at the TMS Annual Meeting, February 11–15, 2001, in New Orleans, Louisiana, under the auspices of the following ASM committees: Materials Science Critical Technology Sector, Structural Materials Division, Electronic, Magnetic & Photonic Materials Division, Chemistry & Physics of Materials Committee, Joint Nuclear Materials Committee, and Titanium Committee.     

Subgrain formation during deformation: Physical origin and consequences

The formation of subgrains in the course of plastic deformation is explained as a result of a trend to make the deformation easier by locally reducing the number of active slip systems. Local preference of one slip system changes the crystal orientation with respect to stress (Schmid factor), thus leading to geometrical softening or hardening. The trend to subgrain formation is treated in the framework of continuum mechanics as an instability against internal bending for the simple case of a crystal originally oriented for symmetric double slip. Once formed, the boundaries of the subgrains lead to hardening as they induce long-range internal back stresses in the interior of the subgrains by forcing the mobile dislocations to take a bowed configuration. Simple dislocation-based and Cosserat models are recalled to explain the size-dependent subgrain hardening, where smaller subgrains are stronger. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.     

Observations of room-temperature creep recovery in titanium alloys

Conventional α(hcp) and α(hcp)/β(bcc) titanium alloys exhibit significant primary creep strains at room temperature and at stresses well below their macroscopic yield strength. It has been previously reported in various materials systems that repeated unloading during primary creep testing may either accelerate or retard the accumulation of creep strains. These effects have been demonstrated to depend on both microstructure and the applied stress. This article demonstrates that significant room-temperature recovery occurs in technologically relevant titanium alloys. These recovery mechanisms are manifested as a dramatic increase in creep rates (by several orders of magnitude) upon the introduction of individual unloading events, ranging from 1 minute to 365 days, during primary creep tests. Significant increases in both creep rate and the total accumulated creep strain were observed in polycrystalline single α-phase Ti-6Al, polycrystalline α/β Ti-6Al-2Sn-4Zr-2Mo-0.1Si, and individual α/β colonies of Ti-6242. Based on transmission electron microscopy (TEM) studies of the active deformation mechanisms, it is proposed that the presence of significant stress concentrations within the α phase of these materials, in the form of dislocation pileups, is a prerequisite for significant room-temperature recovery. This article is based on a presentation made in the symposium entitled “Defect Properties and Mechanical Behavior of HCP Metals and Alloys” at the TMS Annual Meeting, February 11–15, 2001, in New Orleans, Louisiana, under the auspices of the following ASM committees: Materials Science Critical Technology Sector, Structural Materials Division, Electronic, Magnetic & Photonic Materials Division, Chemistry & Physics of Materials Committee, Joint Nuclear Materials Committee, and Titanium Committee.     

Dislocation microstructure and internal-stress measurements by convergent-beam electron diffraction on creep-deformed Cu and Al

Creep experiments were conducted on aluminum single crystals and copper polycrystals deformed within the five-power-law regime. The dislocation structure of copper, which has not been extensively characterized in the past, consists of less-well-defined subgrain walls of relatively low misorientation, typically between 0.1 and 0.3 deg, with a Frank network of dislocations within the subgrains. The aluminum, as expected, consisted of well-defined subgrain boundaries with a typical misorientation between 1.0 and 2.0 deg. The subgrains were probed from one boundary to another in copper and aluminum using convergent-beam electron diffraction (CBED). This allowed a determination of any changes in the lattice parameter, which would indicate the presence of any internal stresses. Earlier investigations by others suggested that internal stresses may be high in the vicinity of the “hard” subgrain boundaries in both loaded and unloaded specimens, based on a variety of techniques including X-ray diffraction (XRD), stress-dip tests, as well as some preliminary CBED. It was determined in this work that the lattice parameter was unchanged at the equilibrium or stress-free value within the interior of the subgrains and along (within a one-beam diameter) the subgrain boundaries. This article is based on a presentation made in the workshop entitled “Mechanisms of Elevated Temperature Plasticity and Fracture,” which was held June 27–29, 2001, in San Diego, CA, concurrent with the 2001 Joint Applied Mechanics and Materials Summer Conference. The workshop was sponsored by Basic Energy Sciences of the United States Department of Energy.