Constitutive description of primary and steady-state creep deformation behaviour of tempered martensitic 9Cr–1Mo steel

The model based on the coupled sine hyperbolic creep rate relation with the evolution of internal stress as a function of strain provides better understanding of primary and secondary creep behaviour of tempered martensitic 9Cr–1Mo steel. The predicted evolution of internal stress as an increase in the internal stress value (or, decrease in effective stress) with strain/time appropriately described the observed decrease in creep rate during primary creep in the steel. The applicability of the model has been demonstrated by comparing experimental and predicted creep strain–time and creep rate–strain/time data of 9Cr–1Mo steel at 793 and 873 K for quenched and tempered and simulated post-weld heat treatment conditions. Irrespective of prior heat treatment and test temperature, the optimised parameters associated with the internal stress values exhibited linear variations with applied stress. The influence of prior heat treatment on primary and secondary creep characteristics of the steel is reflected on the rate constant values associated with the model. At all temperatures and heat treatment conditions, good agreement between the experimental and predicted steady-state creep rates demonstrate the further applicability of the model.     

Description of primary creep by means of the stochastic model

Feltham's stochastic model is used to describe dislocation hardening during primary creep in metals which contain a three-dimensional irregular network of dislocations. The original stochastic equation is modified so as to fulfil a condition of volume conservation. The modified differential equation is then solved and a time-dependent distribution function of dislocation segment lengths is obtained. Since the creep strain is given by a summation of strain contributions from individual dislocation segments, the time dependence of the creep strain, i.e. the creep curve, can also be obtained.     

Crystal plasticity modeling of irradiation growth in Zircaloy-2

Abstract

A physically based reaction-diffusion model is implemented in the visco-plastic self-consistent (VPSC) crystal plasticity framework to simulate irradiation growth in hcp Zr and its alloys. The reaction-diffusion model accounts for the defects produced by the cascade of displaced atoms, their diffusion to lattice sinks and the contribution to crystallographic strain at the level of single crystals. The VPSC framework accounts for intergranular interactions and irradiation creep, and calculates the strain in the polycrystalline ensemble. A novel scheme is proposed to model the simultaneous evolution of both, number density and radius, of irradiation-induced dislocation loops directly from experimental data of dislocation density evolution during irradiation. This framework is used to predict the irradiation growth behaviour of cold-worked Zircaloy-2 and trends compared to available experimental data. The role of internal stresses in inducing irradiation creep is discussed. Effects of grain size, texture and external stress on the coupled irradiation growth and creep behaviour are also studied and compared with available experimental data.     

Effects of size on the dynamics of dislocations in ice single crystals

Single crystals of ice subjected to primary creep in torsion exhibit a softening behavior: the plastic strain rate increases with time. In a cylindrical sample, the size of the radius affects this response. The smaller the radius of the sample becomes while keeping constant the average shear stress across a section, the softer the response. The size-dependent behavior is interpreted by using a field dislocation theory, in terms of the coupled dynamics of excess screw dislocations gliding in basal planes and statistical dislocations developed through cross slip occurring in prismatic planes. The differences in the results caused by sample height effects and variations in the initial dislocation microstructure are discussed.     

Grain size effects on microstructural stability and creep behaviour of nanotwinned Ni free-standing foils at room temperature

Creep tests were performed on the high stacking fault energy (SFE) nanotwinned (NT) Ni free-standing foils with nearly the same twin thickness at room temperature (RT) to investigate the effects of grain size and loading rate on their microstructural stability and creep behaviour. The grain growth mediated by the twinning/detwinning mechanism at low applied stresses (<800 MPa) and grain refinement via the detwinning mechanism at high applied stresses (>800 MPa) were uncovered in the present NT-Ni foils during RT creep, both of which are attributed to the interactions between dislocations and boundaries. It appears that a higher initial dislocation density leads to a faster primary creep strain rate and a slower steady-state creep strain rate. Unlike the non-twinned metals in which grain growth often enhances the creep strain rate, the twinning/detwinning-mediated grain growth process unexpectedly lowers the steady-state creep strain rate, whereas the detwinning-mediated grain refinement process accelerates the creep strain rate in the studied NT-Ni foils. A modified phase-mixture model combined with Arrhenius laws is put forward to predict the scaling behaviour between the creep strain rate and the applied stress, which also predicts the transition from grain growth-reduced to grain refinement-enhanced steady-state creep strain rate at a critical applied stress. Our findings not only provide deeper insights into the grain size effect on the mechanical behaviour of nanostructured metals with high SFE, but also benefit the microstructure sensitive design of NT metallic materials.     

Deformation behaviour of body centered cubic iron nanopillars containing coherent twin boundaries

Molecular dynamics simulations were performed to understand the role of twin boundaries on deformation behaviour of body-centred cubic (BCC) iron (Fe) nanopillars. The twin boundaries varying from 1 to 5 providing twin boundary spacing in the range 8.5–2.8 nm were introduced perpendicular to the loading direction. The simulation results indicated that the twin boundaries in BCC Fe play a contrasting role during deformation under tensile and compressive loadings. During tensile deformation, a large reduction in yield stress was observed in twinned nanopillars compared to perfect nanopillar. However, the yield stress exhibited only marginal variation with respect to twin boundary spacing. On the contrary, a decrease in yield stress with increase in twin boundary spacing was obtained during compressive deformation. This contrasting behaviour originates from difference in operating mechanisms during yielding and subsequent plastic deformation. It has been observed that the deformation under tensile loading was dominated mainly by twin growth mechanism. On the other hand, the deformation was dominated by nucleation and slip of full dislocations under compressive loading. The twin boundaries offer a strong repulsive force on full dislocations resulting in the yield stress dependence on twin boundary spacing. The occurrence of twin–twin interaction during tensile deformation and dislocation–twin interaction during compressive deformation has been discussed.     

Relaxation and Creep of Dry Bulk Solids

In former investigations it has been shown that creep (constant stress, altering strain) and relaxation (constant strain, decreasing stress) can be observed with dry bulk solids. Both effects are covered when investigating the time dependent behaviour of bulk solids where time dependence can also mean an increase of the deformation resistance with increasing deformation rate. In this paper the investigated time dependent effects do not include time consolidation. The effects of creep and relaxation are often neglected for bulk solids because in many applications the influence of these time dependent behaviours is of minor importance. A deeper insight into the bulk solids flow characteristics and mechanisms can only be obtained when time dependence is taken into consideration.     

The effect of hydrogen disorder on dislocation movement and plastic deformation of ice

In the structure of ice at low pressures (ice I-h) the oxygen atoms are crystallographically arranged but the hydrogen atoms are believed to be randomly arranged consistent with the so-called Bernal-Fowler rules. The effect of this randomness is to make it impossible for a dislocation to move through the ice lattice without creating point defects (breaches of the Bernal-Fowler rules). A calculation of the energy of the defects that would have to be created gives a value so large that it would require a stress of about one tenth of the shear modulus of ice to push the dislocation through. It therefore seems likely that dislocations could not move through ice unless the hydrogen atoms are reoriented by thermally activated point defects ahead of the dislocation. This necessity will greatly slow down dislocations in ice and provides an explanation for the observed behaviour of ice single crystals in creep and constant strain-rate tests, and of the softening of ice at low temperatures produced by small concentrations of dissolved fluoride ions.     

Influence of solute mobility on dislocation motion II. Application of the basic model

In many metals containing solute atoms the dynamic strain aging phenomena have been observed at suitable temperatures and strain rates. In this paper a new model of these phenomena based on the assumption that it is the friction stress which is influenced by the dynamic strain aging is developed. The model is used to analyze the flow stress dependence on the dislocation velocity, temperature and solute concentration. The occurence of the plateau stress, the appearance of jerky flow and anomalies in the stress dependence of the strain rate sensitivity are discussed.     

Temporal dissipative structures in cyclically deformed metallic alloys

cyclically deformed metallic alloys. The model employs quasi-chemical reactions of multiplication, annihilation and positive feedback among the populations of mobile, immobile, and Cottrell-type dislocations [1]. Three major types of loading have been simulated, namely, pure sinusoidal, “creep fatigue”, and ramp loading. Computer movies of the temporal evolution of stress serrations and dislocation densities have been produced as an aide for analysis and illustration. It has been demonstrated that the model successfully reproduces strain bursts and stress serrations in fatigued metallic alloys in terms of the underlying dislocations mechanisms, thus establishing the fundamental connection between micro- and macromechanics of cyclic deformation. Received: 20 June 1996/Accepted: 6 October 1996     

Modelling and simulation of dynamic recrystallisation based on multi-phase-field and dislocation-based crystal plasticity models

ABSTRACT

The mechanical properties of metals used as structural materials are significantly affected by hot (or warm) plastic working. Therefore, it is industrially important to predict the microscopic behaviour of materials in the deformation process during heat treatment. In this process, a number of nuclei are generated in the vicinity of grain boundaries owing to thermal fluctuation or the coalescence of subgrains, and dynamic recrystallisation (DRX) occurs along with the deformation. In this paper, we develop a DRX model by coupling a dislocation-based crystal plasticity model and a multi-phase-field (MPF) model through the dislocation density. Then, the temperature dependence of the hardening tendency in the recrystallisation process is introduced into the DRX model. A multiphysics simulation for pure Ni is conducted, and then the validity of the DRX model is investigated by comparing the numerical results of microstructure formation and the nominal stress–strain curve during DRX with experimental results. The obtained results indicate that in the process of DRX, nucleation and grain growth occur mainly around grain boundaries with high dislocation density. As deformation progresses, new dislocations pile up and subsequent nucleation occurs in the recrystallised grains. The influence of such microstructural evolution appears as oscillation in the stress–strain curve. From the stress–strain curves, the temperature dependence in DRX is observed mainly in terms of the yield stress, the hardening ratio, and the change in the hardening tendency after nucleation occurs.     

Thermal dislocation depinning in a deformed crystal

The temperature dependence of the dislocation mobility threshold is investigated on the basis of a dislocation model suggested by Frenkel-Kontorova. The critical value is obtained for the stress/temperature, corresponding to the dislocation depinning from its equilibrium position. The universal behaviour of a barrier height at finite stress and temperature is revealed and investigated.     

Modelling the yield point phenomena during deformation at elevated temperatures: case study on Inconel 600

The discontinuous yield behaviour (DYB) of Inconel 600 was studied during hot compression tests at temperatures in range of 850–1150°C and strain rates of 0.001–1?s^?1. The yield point phenomena were observed in the temperature range of 850–1000°C and strain rates of 0.001–0.1 s^?1. The DYB was modelled by considering the evolution of dislocation density at the early stages of yielding. The opposite effects of dislocation multiplication, dislocation interaction (work hardening) and dynamic recovery (DRV) were considered. It was shown that the dislocation multiplication and DRV result in flow softening, while the dislocation interaction leads to work hardening. The model was established in a way to consider the effects of various microstructural evolutions on the σ(ε) function. The discontinuous flow curves were fitted by the developed model with acceptable precision. The variations of material constants with temperature and strain rate were found physically meaningful. The dislocation multiplication parameter was determined at various temperatures and strain rates. It was concluded that the rate of dislocation multiplication increases as temperature rises or strain rate declines. Accelerated dislocation multiplication leads to less drop in yield stress between the upper and lower yield points.     

Creep of multidirectionally fibre-reinforced composites

A model for the creep of metal matrix composites multidirectionally reinforced by short fibres is proposed. The reinforcement is described by the effective stiffness tensor of a multidirectional arrangement of continuous fibres and the internal damage of the composite during creep due to fibre fragmentation is introduced by assigning a heuristic nonlinear stress–strain relationship to the fibres. Based on the model, the load partitioning between matrix and fibres is computed. The macroscopic creep behaviour is simulated for composites exhibiting different fibre orientation distributions and different heuristic nonlinear stress–strain functions. The computational results rationalize the creep behaviour of multidirectional fibre-reinforced composites. For a two-dimensional random orientation distribution, a good qualitative match between simulation and experimental results is obtained for compressive loading and for in-plane tensile loading. For loading normal to the reinforcement plane, the model overestimates the creep resistance. In this case, the formation and growth of cavities seems to govern the creep deformation of the composite.     

Evaluating the effects of loading parameters on single-crystal slip in tantalum using molecular mechanics

This study is aimed at developing a physics-based crystal plasticity finite element model for body-centred cubic (BCC) metals, through the introduction of atomic-level deformation information from molecular dynamics (MD) investigations of dislocation motion at the onset of plastic flow. In this study, three critical variables governing crystal plasticity mediated by dislocation motion are considered. MD simulations are first performed across a range of finite temperatures up to 600K to quantify the temperature dependence of critical stress required for slip initiation. An important feature of slip in BCC metals is that it is not solely dependent on the Schmid law measure of resolved shear stress, commonly employed in crystal plasticity models. The configuration of a screw dislocation and its subsequent motion is studied under different load orientations to quantify these non-Schmid effects. Finally, the influence of strain rates on thermal activation is studied by inducing higher stresses during activation at higher applied strain rates. Functional dependence of the critical resolved shear stress on temperature, loading orientation and strain rate is determined from the MD simulation results. The functional forms are derived from the thermal activation mechanisms that govern the plastic behaviour and quantification of relevant deformation variables. The resulting physics-based rate-dependent crystal plasticity model is implemented in a crystal plasticity finite element code. Uniaxial simulations reveal orientation-dependent tension–compression asymmetry of yield that more accurately represents single-crystal experimental results than standard models.     

Orientation-dependent recovery in strongly deformed Al–0.1% Mn crystals

Single crystals of Al–0.1% Mn were channel-die compressed to a true strain of 2.3 and their recovery behaviour at 240–320°C investigated by microhardness measurements, electron backscatter diffraction (EBSD) microtexture mapping and X-ray line broadening analysis. The crystal orientations were the nominally stable Goss {110}?001?, brass {110}?112? and S {123}?634?. For all three orientations the microhardness decreases with a logarithmic time dependence but the instantaneous recovery rates of the brass oriented crystals are systematically lower than those of the other two orientations by a factor of about 2. The dislocation densities decrease rapidly in the first stages of recovery (<1?min) by dislocation dipole annihilation and more slowly thereafter. In the Goss and S orientations the later stage of recovery is due to sub-grain growth. The orientation dependence is ascribed to the relatively low misorientations developed by plastic straining in the brass crystals (average about 4°) compared with the Goss and S orientations (about 7–8°).     

Atomistic simulation of the stacking fault energy and grain shape on strain hardening behaviours of FCC nanocrystalline metals

ABSTRACT

Ultra-fine grained copper with nanotwins is found to be both strong and ductile. It is expected that nanocrystalline metals with lamella grains will have strain hardening behaviour. The main unsolved issues on strain hardening behaviour of nanocrystalline metals include the effect of stacking fault energy, grain shape, temperature, strain rate, second phase particles, alloy elements, etc. Strain hardening makes strong nanocrystalline metals ductile. The stacking fault energy effects on the strain hardening behaviour are studied by molecular dynamics simulation to investigate the uniaxial tensile deformation of the layer-grained and equiaxed models for metallic materials at 300?K. The results show that the strain hardening is observed during the plastic deformation of the layer-grained models, while strain softening is found in the equiaxed models. The strain hardening index values of the layer-grained models decrease with the decrease of stacking fault energy, which is attributed to the distinct stacking fault width and dislocation density. Forest dislocations are observed in the layer-grained models due to the high dislocation density. The formation of sessile dislocations, such as Lomer–Cottrell dislocation locks and stair-rod dislocations, causes the strain hardening behaviour. The dislocation density in layer-grained models is higher than that in the equiaxed models. Grain morphology affects dislocation density by influencing the dislocation motion distance in grain interior.     

A dislocation network model of creep transients after a stress dip

The transient creep behaviour which follows a sudden decrease in stress is predicted by using simple equations for the rates of creation and annihilation of dislocations. It is shown that a variety of transient types can occur, depending on the relative values of the dislocation glide and climb mobility and the magnitude of the stress dip.The work was carried out at the Berkeley Nuclear Laboratories of the Technology Planning and Research Division and the paper is published with permission of the Central Electricity Generating Board.     

Finite element modelling of creep cavity filling by solute diffusion

In recently discovered self healing creep steels, open-volume creep cavities are filled by the precipitation of supersaturated solute. These creep cavities form on the grain boundaries oriented perpendicular to the applied stress. The presence of a free surface triggers a flux of solute from the matrix, over the grain boundaries towards the creep cavities. We studied the creep cavity filling by finite element modelling and found that the filling time critically depends on (i) the ratio of diffusivities in the grain boundary and the bulk, and (ii) on the ratio of the intercavity distance and the cavity size. For a relatively large intercavity spacing 3D transport is observed when the grain boundary and volume diffusivities are of a similar order of magnitude, while a 2D behaviour is observed when the grain boundary diffusivity is dominant. Instead when the intercavity distance is small, the transport behaviour tends to a 1D behaviour in all cases, as the amount of solute available in the grain boundary is insufficient. A phase diagram with the transition lines is constructed.