Harper-dorn creep and specimen size

Under conditions typical of Harper-Dorn (H-D) creep the statistical slip-length may become comparable to, or even exceed, the specimen diameter (a size effect). It is demonstrated that a consequence of such a size effect is that the rates of dislocation storage and dynamic recovery are reduced and the static recovery rate will exceed the dynamic one. Under such conditions, the analysis shows that the creep rate will scale linearly with the applied stress, a characteristic of H-D 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 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.     

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.     

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.     

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.     

Micromechanics effects in creep of metal-matrix composites

The creep of metal-matrix composites is analyzed by finite element techniques. An axisymmetric unit-cell model with spherical reinforcing particles is used. Parameters appropriate to TiC particles in a precipitation-hardened (2219) Al matrix are chosen. The effects of matrix plasticity and residual stresses on the creep of the composite are calculated. We confirm (1) that the steady-state rate is independent of the particle elastic moduli and the matrix elastic and plastic properties, (2) that the ratio of composite to matrix steady-state rates depends only on the volume fraction and geometry of the reinforcing phase, and (3) that this ratio can be determined from a calculation of the stress-strain relation for the geometrically identical composite (same phase volume and geometry) with rigid particles in the appropriate power-law hardening matrix. The values of steady-state creep are compared to experimental ones (Krajewskiet al.). Continuum mechanics predictions give a larger reduction of the composite creep relative to the unreinforced material than measured, suggesting that the effective creep rate of the matrix is larger than in unreinforced precipitation-hardened Al due to changes in microstructure, dislocation density, or creep mechanism. Changes in matrix creep properties are also suggested by the comparison of calculated and measured creep strain rates in the primary creep regime, where significantly different time dependencies are found. It is found that creep calculations performed for a timeindependent matrix creep law can be transformed to obtain the creep for a time-dependent creep law. This article is based on a presentation made in the symposium entitled “Creep and Fatigue in Metal Matrix Composites” at the 1994 TMS/ASM Spring meeting, held February 28–March 3, 1994, in San Francisco, California, under the auspices of the joint TMS-SMD/ASM-MSD Composite Materials Committee.     

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.     

Deformation mechanisms of a rapidly solidified Al-8.8Fe-3.7Ce alloy

The mechanisms of deformation of a rapidly solidified and compacted Al-8.8Fe-3.7Ce (wt pct) alloy were investigated in the stress range 20 to 115 MPa and temperature range 523 to 623 K. The stress dependence of the steady state strain rates indicated a transition from diffusional creep to power law creep, the transition stress decreasing with increasing temperature from 70 MPa (σ/G = 3.1 × 10^-3) at 523 K to 40 MPa (σ/G = 1.9 × 10^-3) at 623 K. The activation energy in the power law creep regime was close to that of bulk self-diffusion in aluminum, while the activation energy in the diffusional creep regime was close to that of grain boundary self-diffusion in Al. The creep strain rates in the power law creep regime were found to be predicted much better by the substructure-invariant creep law (Sherby, 1981) than by the semi-empirical Dorn equation for Al, with the inclusion of a “threshold” stress. In the Coble creep regime, it was found that the cell/subgrain boundaries are inefficient vacancy sources/sinks and that their contribution to Coble creep is totally suppressed in this alloy. The Coble creep rates could be explained by using the average diameter of the powder particles as the effective grain size in the Coble creep equation.     

Newtonian viscous creep in metals

All materials exhibit Newtonian viscous creep behavior at low stresses and high temperatures. We review here such creep behaviors in metals comprising of pure metals and alloys. The underlying creep mechanism(s) depends mainly on the grain size and test temperature while other factors such as the initial dislocation density might also be a factor. Coble creep due to diffusion of point defects through grain boundaries is known to be the dominant creep mechanism in metals with very small grain sizes and relatively low temperatures while Nabarro-Herring creep becomes important for intermediate grain sizes and/or high temperatures. Large grain size and bulk single crystalline metals exhibit Harper-Dorn creep due to dislocation motion rather than point defect diffusion-dominated mechanisms albeit the underlying mechanism is still unclear. Microstructural studies of the specimens deformed in the Harper-Dorn regime have provided some insights. Recent studies suggest that microstructural characterization of deformed specimens is necessary for accurate determination of the rate controlling mechanism. The aim of this paper is two fold namely, to first review the viscous creep mechanisms and to present recent results on Ti3Al2.5V alloy emphasizing the importance of post creep microstructural characterization in establishing the rate controlling mechanism(s).     

Grain boundary sliding in the presence of grain boundary precipitates during transient creep

A constitutive rate equation for grain boundary sliding (GBS), in the presence of grain boundary precipitates, is developed. Langdon’s GBS model is modified by incorporating physically de-fined back stresses opposing dislocation glide and climb and by modifying the grain size de-pendence of creep rate. The rate equation accurately predicts the stress dependence of minimum creep rate and change in activation energy occurring as a result of changing the grain boundary precipitate distribution in complex Ni-base superalloys. The rate equation, along with the math-ematical formulations for internal stresses, is used to derive a transient creep model, where the transient is regarded as the combination of primary and secondary stages of creep in constant load creep tests. The transient creep model predicts that the transient creep strain is dependent on stress and independent of test temperature. It is predicted that a true steady-state creep will only be observed after an infinitely long time. However, tertiary creep mechanisms are expected to intervene and lead to an acceleration in creep rate long before the onset of a true steady state. The model accurately predicts the strain vs time relationships for transient creep in IN738LC Ni-base superalloy, containing different grain boundary carbide distributions, over a range of temperatures.     

Harper-Dorn creep in alpha-zirconium

Harper-Dorn (H-D) creep in alpha-zirconium has been investigated at homologous temperatures 0.35 to 0.48 (773 to 1023 K) and stresses ranging from 4.10⁻⁶ to 9.10⁻⁵G (G is the shear modulus) by the helicoid specimen technique. It has been shown that H-D creep takes place at intercept grain sizes larger than about 125μ, while at smaller grain sizes Coble creep operates under the same external conditions. The H-D creep is most probably dislocation core diffusion controlled. A threshold stress for steady state creep has been detected increasing with decreasing temperature. The existence of threshold stress has been qualitatively accounted for by high dislocation density (~10¹²m⁻²) in the specimens tested. The conservative motion of jogs on screw dislocations dependent on dislocation core diffusion has been suggested as the creep rate controlling mechanism. The energy of jog formation has been estimated to which a mean distance between jogs ~ 5b corresponds at 1000 K, where b is the Burgers vector. Also the observed transient creep has been briefly discussed. The transient components of the creep strain cannot be accounted for exclusively by anelastic bowing out of links of dislocation network.     

Creep characteristics of single crystalline Ni3Al(Ta,B)

The creep characteristics, including the nature of the creep transient after a stress reduction and activation energy for creep of single crystalline Ni₃Al(Ta,B) in the temperature range 1083 to 1388 K, were investigated. An inverse type of creep transient is exhibited during stress reduction tests in the creep regime where the stress exponent is equal to 3.2. The activation energy for creep in this regime is equal to 340 kJ mol⁻¹. A normal type of creep transient is observed during stress reduction tests in the regime where the stress exponent is equal to 4.3. The activation energy for creep in this regime is equal to 530 kJ mol⁻¹. The different transient creep behavior and activation energies for creep observed in this investigation are consistent with the previous suggestion that then = 4.3 regime is associated with creep controlled by dislocation climb, whereas then = 3.2 regime is associated with a viscous dislocation glide process for Ni₃Al at high temperatures.     

Characteristics of a new creep regime in polycrystalline NiAl

Constant-load creep tests were conducted on fine-grained (≈23 μm) Ni-50.6 (at. pct) Al in the temperature range of 1000 to 1400 K. Power-law creep with a stress exponent,n ≈ 6.5, and an activation energy,Qc ≈ 290 kJ mol, was observed above 25 MPa, while a new mechanism withn~1 andQ _c ≈ 100 kJ mol dominates when σ < 25 MPa, wherea is the applied stress. A comparison of the creep behavior of fine- and coarse-grained NiAl established that the mech-anism in then ≈ 2 region was dependent on grain size, and the magnitude of the grain-size exponent was estimated to be about 2. Transmission electron microscopy (TEM) observations of the deformed specimens revealed a mixture of dislocation tangles, dipoles, loops, and sub-boundary networks in the power-law creep regime. The deformation microstructures were in-homogeneous in then~ 2 creep regime, and many grains did not reveal any dislocation activity. However, bands of dislocation loops were observed in a few grains, where these loops appeared to have been emitted from the grain boundaries. The observed creep characteristics of the low-stress region suggest the dominance of an accommodated grain-boundary sliding (GBS) mech-anism, although the experimental creep rates were lower than those predicted by theoretical models by over seven orders of magnitude. The low value ofQ _c in this region, which is ap-proximately one-third that for lattice self-diffusion, is attributed to the possible existence of interconnected vacancy flow channels, or “nanotubes,” at the grain boundaries.     

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.     

Elevated temperature deformation behavior of a dispersion-strengthened Al-Fe,V,Si alloy

The deformation behavior of a rapidly solidified, dispersion-strengthened Al alloy containing 11.7 pct Fe, 1.2 pct V, and 2.4 pct Si was studied at test temperatures up to 450 °C using constantstress creep and constnt strain-rate tensile tests. Apparent stress exponents (n) up to ∼24 and an activation energy of 360 kJ/mol were obtained with the standard Arrhenius type power-law creep equation, which also suggested a change in behavior at ∼300 °C. Substructure-invariant and dislocation/dispersoid interaction models were found to be inadequate for explaining the behavior. When the data were replotted as vs σ, two regimes were found between 350 °C and 450 °C. A model with a pseudothreshold stress (σ _Th′ ) for the higher stress regime resulted inn ∼3, indicating solute drag in this regime. Transmission electron microscopy (TEM) showed departureside pinning of dislocations at higher stresses. In the lower stress regime, TEM showed dislocation subgrain structures. Here, the model resulted in a stress exponent of ∼4.5 indicating the dislocation climb mechanism. At temperatures below ∼300 °C, a single regime was found along with lower activation energies and a stress dependence of ∼3. Dislocation pipe diffusion is proposed to explain the lower activation energy. The origin ofσ _Th′ has been tied to dislocation generation at the grain boundaries.     

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.     

Features and effect factors of creep of single-crystal nickel-base superalloys

The creep behavior of two single-crystal nickel-base superalloys with [001] orientation has been studied by measuring the creep curves, internal friction stress of dislocation motion, transmission electron microscopy (TEM) observation and energy-dispersive X-ray analysis (EDAX) composition analysis. The results show that over the stress and temperature range, there are different creep activation energies, time exponents, and effective stress exponents in two alloys at different creep stages. The size and volume fraction of the γ′ phase in the tantalum-free alloy is obviously decreased with the elevated temperature. This results in the decrease of the internal friction stress during steady-state creep. Higher levels of tungsten in the alloy result in a smaller strain value and lower directional-coarsening rate during primary creep. During steady-state creep, the primary reason for the better creep resistance of the other alloy is that it contains more Al and also Ta, which maintains a high volume fraction of γ′ phase. The dislocation climb over the γ′ rafts is the major deformation mechanisms during steady-state tensile creep. The fact that the strain rate is decreased with the increase of the size and volume fraction of the γ′ rafts may be described by a modified constitutive equation that is based on a model of the rate of dislocation motion.     

<Emphasis Type="Italic">In Situ</Emphasis> Neutron-Diffraction Studies on the Creep Behavior of a Ferritic Superalloy

Precipitate strengthening effects toward the improved creep behavior have been investigated in a ferritic superalloy with B2-type (Ni,Fe)Al precipitates. In situ neutron diffraction has been employed to study the evolution of the average phase strains, (hkl) plane-specific lattice strains, interphase lattice misfit, and grain-orientation texture during creep deformation of the ferritic superalloy at 973 K (700 °C). The creep mechanisms and particle-dislocation interactions have been studied from the macroscopic creep behavior. At a low stress level of 107 MPa, the dislocation-climb-controlled power-law creep is dominant in the matrix phase, and the load partition between the matrix and the precipitate phases remains constant. However, intergranular stresses develop progressively during the primary creep regime with the load transferred to 200 and 310 oriented grains along the axial loading direction. At a high stress level of 150 MPa, deformation is governed by the thermally activated dislocation glide (power-law breakdown) accompanied by the accelerated texture evolution. Furthermore, an increase in stress level also leads to load transfer from the plastically deformed matrix to the elastically deformed precipitates in the axial direction, along with an increase in the lattice misfit between the matrix and the precipitate phases.