Dislocation density-based constitutive model for the mechanical behaviour of irradiated Cu

Performance degradation of structural steels in nuclear environments results from the formation of a high number density of nanometre-scale defects. The defects observed in copper-based alloys are composed of vacancy clusters in the form of stacking fault tetrahedra and/or prismatic dislocation loops that impede the motion of dislocations. The mechanical behaviour of irradiated copper alloys exhibits increased yield strength, decreased total strain to failure and decreased work hardening as compared to their unirradiated behaviour. Above certain critical defect concentrations (neutron doses), the mechanical behaviour exhibits distinct upper yield points. In this paper, we describe the formulation of an internal state variable model for the mechanical behaviour of such materials subject to these (irradiation) environments. This model has been developed within a multiscale materials-modelling framework, in which molecular dynamics simulations of dislocation–radiation defect interactions inform the final coarse-grained continuum model. The plasticity model includes mechanisms for dislocation density growth and multiplication and for irradiation defect density evolution with dislocation interaction. The general behaviour of the constitutive (homogeneous material point) model shows that as the defect density increases, the initial yield point increases and the initial strain hardening decreases. The final coarse-grained model is implemented into a finite element framework and used to simulate the behaviour of tensile specimens with varying levels of irradiation-induced material damage. The simulation results compare favourably with the experimentally observed mechanical behaviour of irradiated materials.     

A dislocation-based model for creep recovery in ice

Creep recovery strain is significant in polycrystalline ice, and its stress dependence is strongly dependent on ice type and deformation history. Although it is generally recognized that creep recovery strain is largely attributable to dislocations, a dislocation-based model that rectifies the observed differences in two important ice types (freshwater and sea ice) has not previously emerged; and the development of such a model forms the goal of the present effort. The model considers basal dislocation distributions, employs a dislocation density–stress relationship from previous work, and uses an empirical expression for the decrease in slip-line spacing with increasing stress. The dislocation processes are taken to operate over a subgrain-sized domain, and the strain associated with the relative motion of neighbouring domains is considered. The model accounts for dislocation multiplication and the decrease in slip-line spacing if the applied stress is sufficiently high. The model explains the observed differences in creep recovery strain for freshwater and sea ice and adequately reproduces both the stress dependence and the limiting value of the experimentally observed behaviour. It also reproduces the shift from nonlinear to linear behaviour that has been observed in prestrain experiments on freshwater and sea ice cores.     

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.     

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.     

A crystal plasticity finite element model for flow stress anomalies in Ni3Al single crystals

This paper presents a dislocation density-based non-Schmid constitutive model to address the anomalous thermo-mechanical behaviour of the L1₂ intermetallic single-crystal Ni₃Al. Ni₃Al is used as a strengthening precipitate (γ′ phase) in Ni-based superalloys. Addressing such anomalous behaviour by accounting for temperature-dependent flow stress and hardening evolution, as well as orientation-dependent tension–compression asymmetry, is necessary for modelling superalloys across a range of temperatures. While hardening in cube-slip systems results from statistically stored dislocations (SSDs), hardening in octahedral slip systems is due to both SSDs and cross-slip dislocations (CSDs). The constitutive model incorporates hardening evolution due to SSDs and CSDs. Experimental data for Ni₃Al-type single crystals, available in the literature, are used to calibrate material parameters. Subsequently, results of crystal plasticity FEM simulations are compared with experimental data for several orientations under constant strain rate and creep loading conditions for a wide range of temperatures. The model is able to correctly predict the response of L1₂ intermetallic single crystals including features of anomalous flow stress and non-Schmid yield behaviour.     

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.     

Deuterium accumulation in tungsten under low-energy high-flux plasma exposure

The accumulation of deuterium implanted in tungsten is simulated within the framework of kinetic diffusion theory. The influence of the tungsten microstructure (dislocation density and impurity concentration) on the process of deuterium capture and accumulation is considered. It is established that, under the chosen irradiation conditions, deuterium accumulation in the near-surface region is determined by capture at defects formed during implantation. The deuterium concentration gradient, together with the material microstructure, determines its accumulation in tungsten. Variation in the dislocation density and impurity concentration does not affect the simulation results, which is, first, related to the fact that the model used does not contain alternative mechanisms for the formation and growth of vacancy clusters under the subthreshold irradiation mode. The simulation results are compared with experimental data, and ways of improving the model are discussed in order to explain the deuterium-saturation effect for high fluences (more than 10²³ m^?2).     

Initial dislocation density effect on strain hardening in FCC aluminium alloy under laser shock peening

Abstract

The effect of initial dislocation density on subsequent dislocation evolution and strain hardening in FCC aluminium alloy under laser shock peening (LSP) was investigated by using three-dimension discrete dislocation dynamics (DD) simulation. Initial dislocations were randomly generated and distributed on slip planes for three different dislocation densities of 4.21 × 10¹², 8.12 × 10¹² and 1.26 × 10¹³ m^?2. Besides, variable densities of prismatic loops were introduced into the DD cells as nanoprecipitates to study the dislocation pinning effect. The flow stresses as a function of strain rate obtained by DD simulation are compared with relevant experimental data. The results show a significant dislocation density accumulation in the form of dislocation band-like structures under LSP. The overall yield strength in FCC aluminium alloy decreases with increasing initial dislocation density and forest dislocation strengthening becomes negligible under laser induced ultra-high strain rate deformation. In addition, yield strength is enhanced by increasing the nanoprecipitate density due to dislocation pinning effect.     

Effect of electron irradiation on creep and dislocation structure of aluminium

The effect of electron irradiation and strain on structure of technical-purity aluminium subjected to constant-load tensile creep tests was investigated. The dislocation structure was statistically analyzed. The effect of electron irradiation appears in the creep rate reduction and generation of radiation defect clusters, the density of which is approximately proportional to the dose. The effect on the dislocation distribution is dependent on the dose as well. Doses1·2×10¹⁷ cm^–2 speed up the formation of a polygonal structure characterizing the steady-state creep.     

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.     

On the homogeneous nucleation and propagation of dislocations under shock compression

The dynamic response of crystalline materials subjected to extreme shock compression is not well understood. The interaction between the propagating shock wave and the material’s defect occurs at the sub-nanosecond timescale which makes in situ experimental measurements very challenging. Therefore, computer simulation coupled with theoretical modelling and available experimental data is useful to determine the underlying physics behind shock-induced plasticity. In this work, multiscale dislocation dynamics plasticity (MDDP) calculations are carried out to simulate the mechanical response of copper reported at ultra-high strain rates shock loading. We compare the value of threshold stress for homogeneous nucleation obtained from elastodynamic solution and standard nucleation theory with MDDP predictions for copper single crystals oriented in the [0 0 1]. MDDP homogeneous nucleation simulations are then carried out to investigate several aspects of shock-induced deformation such as; stress profile characteristics, plastic relaxation, dislocation microstructure evolution and temperature rise behind the wave front. The computation results show that the stresses exhibit an elastic overshoot followed by rapid relaxation such that the 1D state of strain is transformed into a 3D state of strain due to plastic flow. We demonstrate that MDDP computations of the dislocation density, peak pressure, dynamics yielding and flow stress are in good agreement with recent experimental findings and compare well with the predictions of several dislocation-based continuum models. MDDP-based models for dislocation density evolution, saturation dislocation density, temperature rise due to plastic work and strain rate hardening are proposed. Additionally, we demonstrated using MDDP computations along with recent experimental reports the breakdown of the fourth power law of Swegle and Grady in the homogeneous nucleation regime.     

Generation and accumulation of point defects in FCC single crystals upon plastic strain

A model of point defect generation and accumulation in a FCC crystal upon strain with a constant rate at room temperature is suggested in the present work. The model results are in agreement with the available experimental data. The influence of point defects on the formation of fragmented substructures is analyzed. The model is verified using experimental data on the stress‒strain curve, dislocation density, vacancy concentration, and misorientation angle.     

Dislocation vs. production bias revisited with account of radiation-induced emission bias. I. Void swelling under electron and light ion irradiation

Early experimental data on void swelling in electron-irradiated materials disagree with the dislocation bias models based on the dislocation-point defect elastic interactions. Later, this became one of the factors that prompted the development of models based on production bias (PBM) as the main driver for swelling, which assumed that the dislocation bias was much lower than that predicted by theoretical analyses of dislocation bias. However, the PBM in its present form fails to account for important and common observations, namely, the indefinite void growth often observed under cascade irradiation and the swelling saturation observed under high-dose irradiation and in void lattices. In this paper, we show that these contradictions can be naturally resolved in the framework of the rate theory that accounts for the radiation-induced vacancy emission from extended defects, such as voids, dislocations and grain boundaries. This modification introduces a new bias type in the theory, namely, the emission bias. This modified rate theory agrees well with the experimental data and demonstrates that the original dislocation bias should be used in rate theory models along with the emission bias in different irradiation environments. The modified theory predictions include, but are not limited to, the radiation-induced annealing of voids, swelling saturation under high-dose irradiation, generally, and in void lattices, in particular.     

A dislocation-based stress-strain gradient plasticity model for strength and ductility in materials with gradient microstructures

Although metallic materials with gradient microstructure exhibit notable performance in harsh environmental conditions, they can also exhibit unusual mechanical behaviour. This is attributed to both grain size and the gradient of grain size distribution in the structure. Metallic materials with a homogenous distribution of grain size follow the traditional Hall-Petch relationship, in which strength increases with decreasing grain size at the expense of ductility. However, studies show that materials with a gradient of grain size microstructure do not follow the Hall-Petch relationship, and thus have improved strength and ductility. This suggests that with creative design and engineering of microstructure, the strength-ductility trade-off can be reduced or prevented.

In this study, we developed and implemented a dislocation density based model to investigate the mechanical behaviour of nano-microstructure. We designed a multi-scale modelling framework, coupling VPSC (Viscoplastic Self Consistent model) with CDD (Continuum Dislocation Dynamics), applying crystal plasticity equations to simulate dislocation interaction in polycrystalline metallic materials. We also developed design parameters and a model to predict the strength and ductility of materials with gradient microstructure.     

Dislocation and defect density-based micromechanical modeling of the mechanical behavior of fcc metals under neutron irradiation

A rate-independent dislocation and defect density-based evolution model is presented that captures the pre- and post-yield material behavior of fcc metals subjected to different doses of neutron radiation. Unlike previously developed phenomenological models, this model is capable of capturing the salient features of irradiation-induced hardening, including increase in yield stress followed by yield drop and non-zero stress offset from the unirradiated stress–strain curve. The key contribution is a model for the critical resolved slip resistance that depends on both dislocation and defect densities, which are governed by evolution equations based on physical observations. The result is an orientation-dependent non-homogeneous deformation model, which accounts for defect annihilation on active slip planes. Results for both single and polycrystalline simulations of OFHC copper are presented and are observed to be in reasonably good agreement with experimental data. Extension of the model to other fcc metals is straightforward and is currently being developed for bcc metals.     

Self-assembly of germanium islands under pulsed irradiation by a low-energy ion beam during heteroepitaxy of Ge/Si(100) structures

The effect of pulsed irradiation by a low-energy (50–250 eV) ion beam with a pulse duration of 0.5 s on the nucleation and growth of three-dimensional germanium islands during molecular-beam heteroepitaxy of Ge/Si(100) structures is investigated experimentally. It is revealed that, at specific values of the integrated ion flux (less than 10¹² cm^?2), pulsed ion irradiation leads to an increase in the density of islands and a decrease in their mean size and size dispersion as compared to those obtained in the case of heteroepitaxy without ion irradiation. The observed phenomena are explained in the framework of the proposed model based on the concept of a change in the diffusion mobility of adatoms due to the instantaneous generation of interstitial atoms and vacancies under pulsed ion irradiation. It is assumed that the vacancies and interstitial atoms give rise to an additional surface strain responsible for the change in the binding energy of the adatoms. Under certain conditions, these processes bring about the formation of centers of preferential nucleation of three-dimensional islands at the places where the ions impinge on the surface. The model accounts for the possibility of annihilating vacancies and interstitial atoms on the surface of the growing layer. It is demonstrated that the results obtained from the Monte Carlo calculations based on the proposed model are in good agreement with the experimental data.     

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.     

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.     

Evolution of dislocation structures having various orientations in strained single crystals of the alloy Ni3Ge

The dislocation structure of strained single crystals of Ni₃Ge with various orientations is investigated by electron microscopy. The evolution of the dislocation structure parameters is studied as a function of the degree of strain, temperature, and orientation of the single crystals. Analysis of the experimental dependences of the yield stress on the density of dislocations leads to certain conclusions about how various mechanisms for dislocation drag make temperature-dependent contributions to the deforming stress, and about the nature of the thermal hardening of Ni₃Ge. Fiz. Tverd. Tela (St. Petersburg) 40, 672–680 (April 1998)