Nabarro-Herring Creep Research Articles

A Functional Coefficients Version of a Generalized Orr–Sherby–Dorn Creep Model: An Application to 2.25Cr–1Mo Steel

It is important to be able to predict the creep life of materials used in power plants. This paper illustrates the inadequacies of the Orr–Sherby–Dorn (OSD) creep model in achieving this aim for 2.25Cr–1Mo steel. This failure is explained in terms of non-constant model parameters—which in turn is the result of changing creep mechanisms. The paper introduces a semi-parametric estimation procedure for a variant of the OSD model (a structural coefficients version) that can be used to deal with such changing creep mechanisms while maintaining the structure of the model and consequently producing more reliable long-term predictions compared to the unmodified OSD model and the recently introduced LOESS technique. For 2.25Cr–1Mo steel, it was found that the model parameters varied in line with changing creep mechanisms, but in a modified way compared to that already suggested in the literature for this material. The models used suggested that with diminishing stress and increasing temperature, dislocation creep within the crystal structure morphs into grain boundary dislocation motion and finally Nabarro-Herring creep.

Distortion after solubilization treatment of X12CrNiMoV12-3 beam-like samples: A novel FE modelling technique supported by experiments

In this work, the distortion failure of mechanical components undergoing a heat treatment is dealt with. Samples made of a 9–12% Cr steel with beam-like geometry and different set-ups in the furnace were measured with laser scanner system, before and after a solubilization (air quenching) heat treatment to evaluate its permanent deflection. A numerical model, based on the Finite Element (FE) method, was then developed and a transient thermo-mechanical analysis was performed to simulate the thermal cycle. A simulation approach, which differs from the usual methodology adopted dealing with quenching, is proposed, taking into account phase changes (Sang’s semi-empirical method) but also viscoplasticity (Nabarro-Herring creep law combined with bilinear elasto-plastic kinematic model), while the material properties were related to temperature. Comparing numerical results with experiments, it has been observed that, with respect to the case of water quenching, where the distortion is mainly related to phase transformation strains and TRIP (Transformation Induced Plasticity), in the considered case the failure seems to be mainly affected by a low-stress diffusional creep due to the proper weight, when the component is held at high temperature. The influence of different set-up to support the sample in the furnace was also investigated.

Diffusion-induced stress concentrations in diffusional creep

The lattice continuum formulation for diffusional creep is implemented into the computational framework based on the finite difference method for space and the predictor–corrector algorithm for time discretization, to solve the coupled elasticity-diffusion problem with moving boundaries. The numerical scheme is implemented and tested considering 3D periodic structures without grain boundary sliding. Of primary interest in this paper are the stress non-uniformities resulting from nonuniform composition eigenstrains.We consider two regimes: the one where the rate limiting process is bulk diffusion of vacancies, and the one where the rate is controlled by vacancy nucleation/annihilation at grain boundaries (nucleation-controlled creep). We found that the stress concentration factor for diffusion-controlled creep is independent of the applied stress and grain size. No stress concentrations are present for the nucleation-controlled creep case.Stress and grain size dependence of minimum strain creep rates are determined by the present model for a variety of applied stresses, grain sizes, different driving processes (diffusion and nucleation-controlled creep) and compared with the classical theory for diffusional creep. We found that steady-state creep rates varied linearly with applied stresses for both diffusion and nucleation-controlled creep. The numerical results show good correspondence to analytical predictions for idealized diffusional Nabarro-Herring creep. Significantly lower steady-state strain rates were computed for nucleation-controlled creep. These results demonstrate the ability of the present model to reproduce the stress and grain size dependence of the steady-state strain creep rates.

Contribution of the nucleation and recovery of disconnections to shear viscosity in diffusional creep

According to the principles of diffusional creep, the normal and tangent components of the velocity jumps between adjacent grains arise from, respectively, the climbing and sliding of disconnections along grain boundaries. Stationary deformation thus implies a balance between nucleation and recovery of moving disconnections. The model considers a periodic lattice of hexagonal grains with nucleation of disconnection multipoles at triple junctions. The strain energy coupled to the population of climbing disconnections is calculated by inferring that the internal strain field associated to disconnection pile-ups brings a distribution of tractions along GBs that is consistent with the field of diffusion potential gradient that drives disconnection climb. It follows that the distribution of the density of climbing disconnections is parabolic and that the dissipation due to the nucleation and recovery of climbing disconnections is equal to 50% of the dissipation arising from diffusion fluxes. These results hold for both Nabarro-Herring creep and Coble creep. The analysis of the disconnection nucleation process highlights the sources of non-Newtonian behaviour and the existence of a threshold stress as an intrinsic feature of diffusional creep.

Contribution of the Nucleation and Recovery of Disconnections to Shear Viscosity in Diffusional Creep

According to the principles of diffusional creep, the normal and tangent components of the velocity jumps between adjacent grains arise from, respectively, the climbing and sliding of disconnections along grain boundaries. Stationary deformation thus implies a balance between nucleation and recovery of moving disconnections. The model considers a periodic lattice of hexagonal grains with nucleation of disconnection multipoles at triple junctions. The strain energy coupled to the population of climbing disconnections is calculated by inferring that the internal strain field associated to disconnection pile-ups brings a distribution of tractions along GBs that is consistent with the field of diffusion potential gradient that drives disconnection climb. It follows that the distribution of the density of climbing disconnections is parabolic and that the contribution to dissipation is equal to 50% of the contribution arising from diffusion fluxes. These results hold for both Nabarro-Herring creep and Coble creep. The analysis of the disconnection nucleation process highlights the sources of non-Newtonian behaviour and the existence of a threshold stress as an intrinsic feature of diffusional creep.

A micropolar continuum model of diffusion creep

ABSTRACT Solid polycrystalline materials undergoing diffusion creep are usually described by Cauchy continuum models with a Newtonian viscous rheology dependent on the grain size. Such a continuum lacks the rotational degrees of freedom needed to describe grain rotation. Here we provide a more general continuum description of diffusion creep that includes grain rotation, and identifies the deformation of the material with that of a micropolar (Cosserat) fluid. We derive expressions for the micropolar constitutive tensors by a homogenisation of the physics describing a discrete collection of rigid grains, demanding an equivalent dissipation between the discrete and continuum descriptions. General constitutive laws are derived for both Coble (grain-boundary diffusion) and Nabarro-Herring (volume diffusion) creep. Detailed calculations are performed for a two-dimensional tiling of irregular hexagonal grains, which illustrates a potential coupling between the rotational and translational degrees of freedom. If only the plating out or removal of material at grain boundaries is considered, the constitutive laws are degenerate: modes of deformation that involve pure tangential motion at the grain boundaries are not resisted. This degeneracy can be removed by including the resistance to grain-boundary sliding, or by imposing additional constraints on the deformation.

The role of grain boundary mobility in diffusional deformation

The model of diffusional deformation is revisited by accounting for the dependence of the diffusion potential on grain boundary curvature. The issue is developed through the analysis of two case-studies: the deformation of a lattice of columnar grains in conditions of Coble creep, and the rotation of a grain embedded in a polycrystal in conditions of either Nabarro-Herring creep or Coble creep. The analysis reveals that, unless grain boundary mobility is infinite, grain boundary curvature is dynamically induced by strain rate. A link is established between the curvature distribution and the transfer of diffusion fluxes across grain boundaries. For the two case-studies, the equation expressing the balance of grain boundary motions at steady-state is solved for calculating, within a range of grain boundary mobilities, the grain boundary profiles, the diffusion fluxes, and the contributions to power dissipation arising from curvature. The latter contributions are found to scale closely as the square of grain size. It follows that the dissipation contribution due to curvature is larger in conditions of Nabarro-Herring creep. In conditions of Coble creep, the dissipation contribution due to curvature translates into a lower bound for the apparent boundary viscosity parameter to be used in numerical simulations. This lower bound is consistent with previous identifications of the parameter in the literature. The classical model assuming flat grain boundaries with transfer of fluxes via triple junctions emerges as a particular case involving the implicit assumption of an infinite grain boundary mobility.

Fundamental principles of spark plasma sintering of metals: part III – densification by plasticity and creep deformation

ABSTRACT The mechanisms of densification during spark plasma sintering (SPS) of spherical copper particles are investigated both experimentally and analytically. Experimentally measured densification rates are compared to expected contributions to densification coming from Coble creep, Nabarro-Herring creep, power-law creep, and pressure-assisted sintering described by the two-particle model. The results indicate densification by Coble creep or a low-temperature version of power-law creep in the investigated range of temperature and pressure at a relative density of 70%. The findings are supported by (i) activation energies obtained for various conditions using the Dorn-method as well as (ii) the pressure dependence of the densification rate.

Shift of Creep Mechanism in Nanocrystalline NiAl Alloy.

We have examined the effects of temperature, stress, and grain size on the creep process including creep strain, crystal structure, dislocations and diffusions of nanocrystalline NiAl alloy through molecular dynamics simulations. A smaller grain size accelerates the creep process due to the large volume fraction of grain boundaries. Higher temperatures and stress levels also speed up this process in terms of dislocation changes and atom diffusion. In both primary creep and steady-state creep stages, atomic diffusion at the grain boundary could be seen and the dislocation density increased gradually, indicating that the creep mechanism at these stages is Coble creep controlled by grain boundary diffusion while accompanied by dislocation nucleation. When the model enters the tertiary creep stage, it can be observed that the diffusion of atoms in the grain boundary and in the crystal and the dislocation density gradually decreases, implying that the creep mechanisms at this stage are Coble creep, controlled by grain boundary diffusion, and Nabarro–Herring creep, controlled by lattice diffusion.

New and Future Developments in Microbial Biotechnology and Bioengineering: From Cellulose to Cellulase: Strategies to Improve Biofuel Production

Morphological changes in a material are a result of the evolution of its interfaces due to applied or capillary forces. Morphological changes occur by capillary forces if the motion acts to decrease the total free energy associated with the interfaces in the system. For isotropic systems with a single type of interface, the interfacial area always tends toward a minimum. On the other hand, for anisotropic surfaces, the interfacial area could increase during minimization of free energy, via a process known as faceting. The von Neumann-Mullins criterion for two-dimensional grain growth states that grains with fewer than six edges will shrink and eventually disappear, whereas grains with greater than six edges will grow. Grains with exactly six edge are dimensionally stable and will neither grow nor shrink. The von Neumann-Mullins approach has recently been generalized for systems of arbitrary dimensionality by MacPherson and Srolovitz. Diffusional creep results in permanent deformation of the material as a result of capillary and applied forces. There are two types of diffusional creep: Nabarro-Herring creep happens when the mass flux occurs via the crystalline matrix, and Coble creep occurs when the mass flux is along the grain boundaries. Sintering is the process by which a compressed powder or granular material undergoes neck growth and densification through heat treatment. This heat treatment is not hot enough to induce melting, and it is usually conducted under pressure to aid in the sintering process. The final sintered body has a lower porosity and greater mechanical integrity compared to the initial green body. Recently, cold sintering techniques have leveraged chemical reactions to promote effective sintering of ceramics at much lower temperatures compared to traditional processes.

A phase field model of pressure-assisted sintering

The incorporation of an efficient contact mechanics algorithm into a phase field sintering model is presented. Contact stresses on the surface of arbitrarily shaped interacting bodies are evaluated and built into the model as an elastic strain energy field. Energy relaxation through deformation is achieved by diffusive fluxes along stress gradients and rigid body motion of the deforming particles maintain contact between the particles. The proposed model is suitable for diffusion deformation mechanisms occurring at stresses below the yield strength of a defect-free material; this includes Nabarro-Herring creep, Coble creep and pressure-solution. The effect of applied pressure on the high pressure-high temperature (HPHT) liquid phase sintering of diamond particles was investigated. Changes in neck size, particle coordination and contact flattening were observed. Densification rates due to the externally applied loads were found to be in good agreement with a new theory which implicitly incorporates the effect of applied external pressure.

Compressive creep behavior of an oxide-dispersion-strengthened CoCrFeMnNi high-entropy alloy

Creep tests of two equiatomic CoCrFeMnNi alloys were conducted in the temperature range from 973 K to 1073 K. The alloys were prepared by milling blends of powders of pure elements in a planetary ball mill and compacting by the spark plasma technique. Two variants of the alloys were prepared: (i) without and (ii) with the dispersion of oxides. Creep resistance was substantially improved by the presence of oxides. Diffusion creep controlled by lattice diffusion was suggested as a possible mechanism at low stresses. The effective diffusion coefficient calculated for Nabarro-Herring creep was comparable to the lattice diffusion coefficient of Ni in the same high-entropy alloy. At high stresses, the creep behavior was characterized by the presence of a threshold stress invoked by oxide particles.

Lattice continuum and diffusional creep.

Diffusional creep is characterized by growth/disappearance of lattice planes at the crystal boundaries that serve as sources/sinks of vacancies, and by diffusion of vacancies. The lattice continuum theory developed here represents a natural and intuitive framework for the analysis of diffusion in crystals and lattice growth/loss at the boundaries. The formulation includes the definition of the Lagrangian reference configuration for the newly created lattice, the transport theorem and the definition of the creep rate tensor for a polycrystal as a piecewise uniform, discontinuous field. The values associated with each crystalline grain are related to the normal diffusional flux at grain boundaries. The governing equations for Nabarro-Herring creep are derived with coupled diffusion and elasticity with compositional eigenstrain. Both, bulk diffusional dissipation and boundary dissipation accompanying vacancy nucleation and absorption, are considered, but the latter is found to be negligible. For periodic arrangements of grains, diffusion formally decouples from elasticity but at the cost of a complicated boundary condition. The equilibrium of deviatorically stressed polycrystals is impossible without inclusion of interface energies. The secondary creep rate estimates correspond to the standard Nabarro-Herring model, and the volumetric creep is small. The initial (primary) creep rate is estimated to be much larger than the secondary creep rate.

Atomistic study of diffusion-mediated plasticity and creep using phase field crystal methods

The nonequilibrium dynamics of diffusion-mediated plasticity and creep in materials subjected to constant load at high homologous temperatures is studied atomistically using Phase Field Crystal (PFC) methods. Creep stress and grain size exponents obtained for nanopolycrystalline systems, $m \simeq 1.02$ and $p \simeq 1.98$, respectively, closely match those expected for idealized diffusional Nabarro-Herring creep. These exponents are observed in the presence of significant stress-assisted diffusive grain boundary migration, indicating that Nabarro-Herring creep and stress-assisted boundary migration contribute in the same manner to the macroscopic constitutive relation. When plastic response is dislocation-mediated, power law stress exponents inferred from dislocation climb rates are found to increase monotonically from $m \simeq 3$, as expected for generic climb-mediated natural creep, to $m \simeq 5.8$ as the dislocation density $\rho_d$ is increased beyond typical experimental values. Stress exponents $m \gtrsim 3$ directly measured from simulations that include dislocation nucleation, climb, glide, and annihilation are attributed primarily to these large $\rho_d$ effects. Extrapolation to lower $\rho_d$ suggests that $m \simeq 4-4.5$ should be obtained from our PFC description at typical experimental $\rho_d$ values, which is consistent with expectations for power law creep via mixed climb and glide. The anomalously large stress exponents observed in our atomistic simulations at large $\rho_d$ may nonetheless be relevant to systems in which comparable densities are obtained locally within heterogeneous defect domains such as dislocation cell walls or tangles.

High-temperature compressive creep behaviour of perovskite-type oxides SrTi1-xFexO3-δ

Compressive creep tests have been performed on mixed ionic-electronic conducting perovskite-type oxides SrTi1-xFexO3-δ (STF, x=0.3, 0.5 and 0.7). Observed activation energies and stress exponents, at 800–1000°C and in the stress range 10100MPa, indicate that the steady-state creep rate of STF under these conditions is predominantly limited by cation lattice diffusion (Nabarro-Herring creep). The effective stress exponents reflect a contribution of dislocation creep to the mechanism of creep in STF30 (x=0.3). The observed creep rates are compared with those exhibited by related perovskite-type oxides, and are discussed in view of the possible application of STF as oxygen transport membrane (OTM).

Creep of polycrystalline yttrium aluminum garnet (YAG) at elevated temperature in air and in steam

Compressive creep of high-purity polycrystalline yttrium aluminum garnet (YAG, Y3Al5O12) was investigated at 1300°C and 50–200MPa in air and in steam. Compressive creep behavior of silica-doped polycrystalline YAG (Y3Al5O12–0.14wt% SiO2) was also studied. Creep specimens were microstructurally characterized by optical microscopy and TEM before and after creep. Steam slightly increased creep rates of material with grain size less than 1μm (the undoped YAG), but otherwise had little effect. The flow stress exponent was n≈1 for both SiO2-doped YAG and undoped YAG. Creep rates and microstructural observations are consistent with the Nabarro-Herring creep mechanism, with creep rate limited by lattice diffusion of yttrium cations (Y3+). Silica-doped YAG had a larger grain size of 2.41μm and lower creep rates than undoped YAG with 0.92μm grain size. However, creep rates normalized by grain size for Nabarro-Herring creep were higher in SiO2-doped YAG. Possible effects of SiO2 doping and steam on creep of YAG are discussed.

Creep deformation mechanisms in modified 9Cr–1Mo steel

Modified 9Cr–1Mo (Grade 91) steel is currently considered as a candidate material for reactor pressure vessels (RPVs) and reactor internals for the Very High Temperature Reactor (VHTR). The tensile creep behavior of modified 9Cr–1Mo steel (Grade 91) was studied in the temperature range of 873–1023K and stresses between 35MPa and 350MPa. Analysis of creep results yielded stress exponents of ∼9–11 in the higher stress regime and ∼1 in the lower stress regime. The high stress exponent in the power-law creep regime was rationalized by invoking the concept of threshold stress, which represents the lattice diffusion controlled dislocation climb process. Without threshold stress compensation, the activation energy was 510±51kJ/mol, while after correcting for the threshold stress, the activation energy decreased to 225±24kJ/mol. This value is close to the activation energy for lattice self-diffusion in α-Fe. Threshold stress calculations were performed for the high stress regime at all test temperatures. The calculated threshold stress showed a strong dependence on temperature. The creep behavior of Grade 91 steel was described by the modified Bird–Mukherjee–Dorn relation. The rate controlling creep deformation mechanism in the high stress regime was identified as the edge dislocation climb with a stress exponent of n=5. On the other hand, the deformation mechanism in the Newtonian viscous creep regime (n=1) was identified as the Nabarro–Herring creep.

Microstructure evolution and constitutive modelling of microsize structures

Given the small dimensions to be replicated for micro components, knowledge into their microstructure evolution is crucial for dimensional and property control. Constitutive model is useful to predict the final shape of the sintered micro components. In this paper, the microstructure evolution of 316L stainless steel microsize structures of ϕ60 µm that was fabricated by micro metal injection moulding was examined via SEM and XRD. A constitutive model is established to simulate densification of the microsize structures and the contribution of lattice diffusion (Nabarro-Herring creep) is considered. The predictive capability of the model is verified by comparing the theoretical calculations with the experimental results. The influences of boundary energy on modelling results are discussed. It is found that the modelling results agree reasonably well with the experimental results.

Constitutive Modelling of Sintering of 316L Stainless Steel Microsize Structures

Constitutive model is useful to predict the final shape of sintered microsize structures. In this paper, the contribution of lattice diffusion (Nabarro-Herring creep) is considered and a constitutive model is established to simulate densification of 316L stainless steel microsize structures fabricated by micro metal injection molding. The predictive capability of the model is verified by comparing the theoretical calculations with the experimental results. The influences of boundary energy on modeling results are discussed. It is found that the modeling results agree reasonably well with the experimental results.

An Evaluation of Creep Mechanisms in Ultrafine-Grained Metals

Our earlier published creep data are analyzed for ultrafine-grained pure aluminium and copper processed by equal-channel angular pressing (ECAP). The analysis demonstrates conclusively that creep occurs in the investigated materials after ECAP by the same mechanism as in conventional coarse-grained materials with intergranular dislocation glide and climb as the dominant rate-controlling flow process. Under creep conditions examined in this work diffusion creep is not important in pure aluminium and copper because the ultrafine grains are unstable at elevated and/or high temperature creep and the grains grow sufficiently to preclude any significant contribution from Nabarro-Herring or Coble creep.