Elasto-plastic Self-consistent Model Research Articles

Study of stress localisation in polycrystalline grains using self-consistent modelling and neutron diffraction

The time-of-flight neutron diffraction technique and the elastoplastic self-consistent model were used to study the behaviour of single and multi-phase materials. Critical resolved shear stresses and hardening parameters in austenitic and austenitic–ferritic steels were found by analysing the evolution of the lattice strains measured during tensile tests. Special attention was paid to the changes of the grain stresses occurring due to transition from elastic to plastic deformation. Using a new method of data analysis, the variation of the stress localisation tensor as a function of macrostress was measured. The experimental results were successfully compared with model predictions for both phases of the duplex steel and also for the austenitic sample.

Deformation of high β-phase fraction Zr–Nb alloys at room temperature

A series of in situ compression tests has been carried out at room temperature for three dual-phase Zr–Nb alloys with different Nb contents. Changes in the Nb content change the relative fraction of β-phase present in the alloy. The evolution of interphase and intergranular strain was monitored during deformation by neutron diffraction. Surprisingly the strength difference between the three alloys is caused principally by changes in the α-phase strength rather than being due to changes in the β-phase volume fraction; the yield strength of the Nb-rich β-phase is ∼400MPa and is almost unchanged in the three alloys. The strength of the α-phase significantly changed between alloys due to the combined effects of increasing oxygen content and decreasing grain size. The change in strengthening mechanisms in the α-phase is crystallographically anisotropic, e.g. more change was observed for prismatic slip than for pyramidal slip. As a result, the mechanical anisotropy of the α-phase decreases with increasing strength. Depending on the β/α strength ratio, the β-phase can be either weaker or stronger than the average material strength. A soft β-phase helps to produce compressive residual stresses in the α-phase, while a strong β-phase produces tensile residual stresses in α-phase. A combined finite-element method and elastoplastic self-consistent multiscale model is used to interpret the experimental results. Despite the inability to capture the rapid stress relaxation observed, the combined method is shown to be effective in interpreting the deformation behavior of these dual-phase Zr–Nb alloys. The results are relevant to the study of other two-phase alloys, particularly titanium-based alloys.

Texture and elastic strains in hcp-iron plastically deformed up to 17.5 GPa and 600 K: experiment and model

Internal elastic strains and textures are measured using monochromatic x-ray diffraction in ϵ-Fe plastically deformed up to 17.5 GPa and 600 K in the deformation-DIA. We observe the development of a strong 0 0 0 1 compression texture along with a strongly non-linear behavior of 0 0 0 2 lattice strains. We then use an elastoplastic self-consistent polycrystal model to simulate the macroscopic flow curves, internal strain and texture development within the sample. Input parameters are single-crystal elastic moduli, critical resolved shear stresses, and hardening behavior of the slip and twinning mechanisms. The model is found to reproduce the experiment data with basal slip as the easiest and most active deformation mechanism. Other active mechanisms are tensile twinning, prismatic and pyramidal 〈c + a〉 slip. Tensile twinning is most active at lower temperatures (e.g. 400 K) and a higher strain rate. In most cases, the twinning activity occurs early in the deformation and later saturates. It is also responsible for the non-linear behavior of 0 0 0 2 lattice strains. Later in the deformation, the plastic activity of ϵ-Fe is controlled by basal, prismatic and pyramidal 〈c + a〉 slip.

Modelling the elastic–plastic transition of polycrystalline metals with a distribution of grain sizes

A number of recent studies have indicated that the extent of the elastoplastic stage in nanocrystalline polycrystals, with grain sizes smaller than several hundred nanometres, is appreciably larger than that for coarse-grained polycrystals (i.e. grain sizes larger than ≈1 µm). The so-called tangent modulus approach has, accordingly, been used in order to identify the extent of this stage and to define the stress at which all the grains of a polycrystal become plastic. In this work, the applicability of this methodology to single-phase polycrystals, where deformation is accommodated by dislocation slip, is examined in the context of a grain size dependent elastoplastic self-consistent model. Assuming the size distribution is the primary cause of inhomogeneous yielding among the grains of a polycrystal, the stress–strain behaviour of a number of lognormal, fcc polycrystals with varying mean grain sizes and standard deviations is simulated. The true yield strength of the polycrystals is determined by monitoring the evolution of the volume fraction of yielded grains as a function of imposed deformation. It is found that yielding is essentially complete when the tangent modulus (i.e. the work hardening rate) of the polycrystal drops below ∼E/20, with E being Young's modulus. A simple statistical model for the simulation of the elastoplastic response of polycrystals is introduced. Using this model, the offset strain corresponding to the onset of macro-plasticity in copper polycrystals having different grain size distributions is determined.

Neutron time-of-flight diffraction used to study aged duplex stainless steel at small and large deformation until sample fracture

Owing to its selectivity, diffraction is a powerful tool for analysing the mechanical behaviour of polycrystalline materials at the mesoscale (phase and/or grain scale).In situneutron diffraction during tensile tests and elastoplastic self-consistent modelling were used to study slip phenomena occurring on crystallographic planes at small and large deformation. The critical resolved shear stresses in both phases of duplex stainless steel were found for samples subjected to different thermal treatments. The evolution of grain loading was also determined by showing the large differences between stress concentration for grains in ferritic and austenitic phases. It was found that, for small loads applied to the sample, linear elastic deformation occurs in both phases. When the load increases, austenite starts to deform plastically, while ferrite remains in the elastic range. Finally, both phases undergo plastic deformation until sample fracture. By using an original calibration of diffraction data, the range of the study was extended to large sample deformation. As a result, mechanical effects that can be attributed to damage processes initiated in ferrite were observed.

A slip system-based kinematic hardening model application to in situ neutron diffraction of cyclic deformation of austenitic stainless steel

Accurate prediction of the Bauschinger effect is considered a litmus test for the validity of strengthening theories. The effect is known to arise from ‘backstresses’ having intergranular and intragranular sources. Polycrystal plasticity models inherently capture intergranular effects but typically neglect intragranular (dislocation-based) sources of backstress. The negative impact of this omission is made apparent by comparisons of model predictions with in situ neutron diffraction measurements of the hystereses and internal stresses within a sample subjected to fully-reversed tension–compression cyclic deformation. An elasto-plastic self-consistent (EPSC) model is modified to include a Voce-type non-linear kinematic hardening rule, similar to the phenomenological Armstrong–Frederick–Chaboche model, but implemented at the slip system level. This additional physically-based hardening evolution enables the polycrystal model to account for hardening due to reversible, geometrically necessary dislocation structures, such as pile-ups, as well as the more isotropic hardening effect due to forest dislocations. The model accurately predicts the macroscopic hysteresis loops and internal strains observed during the aforementioned in situ low cycle fatigue tests.

Evolution of internal strains in a two phase zirconium alloy during cyclic loading

In situ cyclic loading experiments were carried out on a hot rolled Zr–2.5 Nb plate material at the ISIS pulsed neutron facility of the Rutherford Appleton Laboratory. The cyclic strain was controlled to a fixed total of ±2.4% and the lattice strain evolutions in both the loading and the Poisson’s directions were monitored by neutron diffraction. Experimental results show that the large Bauschinger effect during reverse loading is caused by a combination of the interphase, intergranular and intragranular stresses. Experimental data were interpreted by a combined finite element method (FEM) and elasto-plastic self-consistent (EPSC) model. An overall good agreement was achieved. However, several discrepancies were seen due to deficiencies with the mechanism of modelling the hardening behaviour during stress reversal. Further improvements require functions that can relate micromechanical properties to the evolution of the intragranular microstructure in a more realistic way.

Influence of prior dislocation structure on anisotropy of thermal creep of cold-worked Zr–2.5Nb tubes

Normally, creep anisotropy of hcp metals is thought to be controlled by the crystallographic texture. Here, we show that the creep anisotropy of cold-worked Zr–2.5Nb tubes is also very dependent on the anisotropic dislocation structures introduced by cold-work. The contribution of each slip system to the creep deformation of an individual grain orientation depends upon the activity of that slip system during prior cold-work. This conclusion is reached by comparing the self-consistant visco-plastic polycrystalline models with thermal creep tests performed on internally pressurized thin-wall capsules with different textures under a transverse stress of 300 MPa at 350 °C, where dislocation creep is the dominant operating mechanism. The non-uniform dislocation distributions prior to creep were derived by simulating the cold-work process of Zr–2.5Nb tubes from an Elasto-Plastic Self-Consistent (EPSC) model.

In-Situ Neutron Diffraction Studies of Micromechanical Behavior in a Friction Stir Welded AA7475-T761

An in-situ neutron diffraction technique was used to investigate the lattice strain distributions and micromechanical behavior in a friction stir welded (FSW) sheet of AA7475-T761. The neutron diffraction experiments were performed on the spectrometer for material research, STRESS-SPEC, at FRM II (Garching, Germany). The lattice strain profiles around the weld center were measured as a function of the applied strain during the tensile loading and unloading. The anisotropic elastic and plastic properties of the FSW aluminum alloy were simulated by elasto-plastic self-consistent (EPSC) model to predict the anisotropic deformation behaviors involving the grain-to-grain interactions. Material parameters used for describing the constitutive laws of each test position were determined from the measured lattice strain distributions for different diffraction hkl planes as well as the macroscopic stress-strain curve of the FSW aluminum alloy. A good agreement between experimental results and numerical simulations was obtained. The present investigations provided a reliable prediction of the anisotropic micromechanical behavior of the FSW aluminum alloy during tensile deformation.

In-Situ Neutron Diffraction Study of the Bauschinger Effect in B2 Structured CoZr

A combination of in-situ neutron diffraction and elastoplastic self-consistent (EPSC) modeling have been used to elucidate the role played by intergranular stresses in the Bauschinger effect in B2 structured CoZr at room temperature and 423 K (150 °C). It is shown that, when insufficient slip modes are present to accommodate arbitrary strains, the large intergranular stresses built up due to inhomogeneous plastic deformation are responsible for the observed Bauschinger effect. Upon the onset of secondary deformation mechanism(s), the stresses are more uniformly distributed among the grains and the influence of intergranular stresses on the Bauschinger effect diminishes. On the other hand, it is speculated that the contribution of intragranular (dislocation-based) stresses is responsible for the persistent Bauschinger effect past the transition point. Similar results are obtained at both room temperature and 423 K (150 °C), and while the yield strength decreases with temperature, the high-temperature stress-strain curve progressively becomes harder than the room temperature one. In light of this, the previously characterized yield strength anomaly in CoZr has been re-examined.

X-ray measurement of residual stresses and texture development during a rolling sequence of zirconium alloy cladding tubes – influence of plastic anisotropy on mechanical behaviour

Texture and residual stress analysis using X-ray diffraction have been carried out on zirconium alloy cladding tubes after cold pilgering, an industrial mechanical process. The final rolling pass has been completely characterized by X-ray diffraction. An interpretation of the effect of intergranular stresses on the development of analysed stress has been made using a modified elastoplastic self-consistent model in order to account for the effect of the high intrinsic plastic anisotropy of hexagonal close-packed crystals. The contribution and magnitude of the first- and second-order residual stresses were correctly evaluated using information from the model. The main features of the rolling sequence were qualitatively reproduced by the simulations, considering prismatic slip as the main active deformation mode in this alloy under large strain.

Modeling lattice strain evolution at finite strains and experimental verification for copper and stainless steel using in situ neutron diffraction

An elasto-plastic self-consistent (EPSC) polycrystal model is extended to account, in an approximate fashion, for the kinematics of large strains, rigid body rotations, texture evolution and grain shape evolution. In situ neutron diffraction measurements of the flow stress, internal strain, texture and diffraction peak intensity evolutions were performed on polycrystalline copper and stainless steel, up to true tensile strains of ε = 0.3. Suitably adjusted slip system hardening model parameters enable the model to quantitatively describe the flow stress of the polycrystalline aggregate. Quantitative predictions of the texture evolution and the internal strain evolution along the stress axis are good, while predictions of transverse internal strains (perpendicular to the tensile loading direction) are less satisfactory. The latter exhibit a large dispersion from grain to grain around a macroscopic average, and the implications of this finding for the interpretation of in situ neutron diffraction method are explored. Finally, as a demonstration of the applicability of the model to problems involving finite rotation, as well as deformation, simulations of simple shear were conducted which predict a texture evolution in agreement with published experimental data, and other modeling approaches as well.

Mapping of crack tip strains and twinned zone in a hexagonal close packed zirconium alloy

This paper describes the use of synchrotron X-ray diffraction to map crack tip strain field evolution and initiation of twinning under applied load in a fatigue pre-cracked zirconium alloy. Compact tension specimens were machined in two orientations from a textured rolled plate and loaded along the normal direction and the transverse direction to rolling. Residual strains due to fatigue and the strains introduced at applied K I of approximately 10 and 30 MPa√m were measured for the two specimen orientations. Maps of the region of twinned material were determined. The experimental data is compared to finite element (FE) calculations for the crack tip strain fields and a prediction of the size of the twinned region using a multiscale model based on combining an FE approach with an elastoplastic self-consistent polycrystalline plasticity model.

On the correlation between deformation twinning and Lüders-like deformation in an extruded Mg alloy: In situ neutron diffraction and EPSC.4 modelling

The current work focuses on the yielding and immediate post-yielding deformation of fine-grained and coarse-grained ZM20 Mg alloys obtained by extrusion. Compressive deformations along the extrusion direction, known to be governed by profuse twinning are examined in detail. It is shown that the fine-grained alloy exhibits Lüders-like plateaux suggesting heterogeneous transition from elastic to plastic deformation. This is due to the cooperative twinning of neighbouring grains which is promoted in the fine-grained alloy by the high internal stresses borne by the parent grain families in the vicinity of yielding, and the auto-catalytic nature of twin nucleation. The elasto-plastic response of tested alloys was also simulated using version 4 of the Elasto-Plastic Self-Consistent (EPSC) model. The finite initial fraction (FIF) assumption is employed to account for the stress relaxation related to the twin nucleation process. It is shown that the new EPSC.4 model is superior to its previous version as it enables realistic predictions of the development of elastic lattice strains in variously oriented grain families and the macroscopic stress–strain response of a polycrystalline aggregate undergoing profuse twinning.

Mechanisms of Ductility in CoTi and CoZr B2 Intermetallics

The present research is conducted in order to elucidate the operative deformation mechanisms allowing ductility in the B2 (CsCl) intermetallics CoTi and CoZr. A twofold approach combines in-situ neutron diffraction during uniaxial compression with elastoplastic self-consistent (EPSC) polycrystal modeling. Tensile and compression tests of CoZr and CoTi confirm that both intermetallics are ductile at room temperature. Low compressive yield points of −62 and −50 MPa are identified for CoTi and CoZr, respectively. Analysis of stress-strain curves, in-situ neutron diffraction internal strain measurements, and EPSC modeling confirm that the initial plasticity is accommodated via the established $$ \left\langle { 100} \right\rangle \left\{ {0 1 1} \right\} $$ slip mode. However, a sudden decrease in the hardening rate observed in the macroscopic stress-strain curve and apparent in the internal stress developments signals the activation of a secondary mechanism, which helps to explain the anomalous ductility. The EPSC simulations involving either $$ \left\langle { 1 10} \right\rangle \left\{ {\overline{1} 10} \right\} $$ or $$ \left\langle { 1 1 1} \right\rangle \left\{ {1\overline{1} 0} \right\} $$ slip mechanisms coupled with $$ \left\langle { 1 0 0} \right\rangle $$ slip can reproduce the observed transitions at −350 and −250 MPa for CoTi and CoZr, respectively. Implications related to previous observations of a yield strength anomaly and the possible influence of kink banding are discussed.

Elastic and plastic properties of β Zr at room temperature

Room temperature elastic and plastic properties of a single phase β Zr have been studied by in-situ neutron diffraction compression testing. The measured macroscopic Young’s modulus is ∼60 GPa and the yield strength is ∼500 MPa. Dislocation slip is the major mode of plastic deformation. An Elasto-Plastic Self-Consistent (EPSC) model was used to interpret the experimental results and was shown to be effective in extracting the single crystal properties from the polycrystalline data. The single crystal elastic constants of the β-phase are determined as: C 11 = 145.9 ± 2.6 GPa, C 12 = 117.4 ± 2.5 GPa and C 44 = 29.8 ± 0.2 GPa. The calculated elastic modulus of 〈1 0 0〉, 〈1 1 0〉, 〈1 1 1〉, 〈2 1 1〉 and 〈3 1 0〉 directions was ∼41.2, 66.2, 82.9, 66.2 and 47.7 GPa, respectively. Pencil glide on the {110}, {112} and {123} planes was used in the EPSC model and gave a good simulation to the early part of the plastic deformation. The average β-phase strain is best represented by the peak average method, while in cases where only a limited number of diffraction peaks are available, the {211} grain family is a good candidate for estimation of the average β-phase strain.

Modeling analysis of the influence of plasticity on high pressure deformation of hcp-Co

Previously measured in situ x-ray diffraction is used to assess the development of internal elastic strains within grains of a sample of polycrystalline cobalt plastically deformed up to a pressure of 42.6 GPa. An elastoplastic self-consistent polycrystal model is used to simulate the macroscopic flow curves and internal strain development within the sample. Input parameters are single-crystal elastic moduli and their pressure dependence, critical resolved shear stresses, and hardening behavior of the slip and twinning mechanisms which are active in Co crystals. At 42 GPa, the differential stress in hcp-Co is $1.9\ifmmode\pm\else\textpm\fi{}0.1\text{ }\text{GPa}$. The comparison between experimental and predicted data leads us to conclude that: (a) plastic relaxation plays a primary role in controlling the evolution and ordering of the lattice strains; (b) the plastic behavior of hcp-Co deforming under high pressure is controlled by basal and prismatic slip of $⟨a⟩$ dislocations, and either pyramidal slip of $⟨c+a⟩$ dislocations, or compressive twinning, or both. Basal slip is by far the easiest and most active deformation mechanism. Elastoplastic self-consistent models are shown to overcome the limitations of models based on continuum elasticity theory for the interpretation of x-ray diffraction data measured on stressed samples. They should be used for the interpretation of these experiments.

Modeling the room temperature deformation of a two-phase zirconium alloy

A new modeling approach, which combines the finite element method (FEM) and elasto-plastic self-consistent model (EPSC), was used to model the deformation behavior of hot-rolled Zr–2.5Nb plate material and was shown to be an effective method for describing the deformation of multiphase materials. The macroscopic mechanical properties and the phase interaction stresses were well captured by a unit cell FEM model. The average phase stress–strain histories simulated by FEM were used as boundary conditions in an EPSC model to calculate the evolution of intergranular strains in both phases. Taking the asymmetric character of the < c + a> pyramidal slip during tension and compression into account in EPSC, the agreement between the experimental and the modeling results was considerably improved.

Investigation of deformation mechanisms involved in the plasticity of AZ31 Mg alloy: In situ neutron diffraction and EPSC modelling

In situ neutron diffraction and Elasto-Plastic Self-Consistent (EPSC) polycrystal modelling have been employed to investigate which deformation mechanisms are involved in the plasticity of extruded AZ31 Mg alloy during the tensile loading along the extrusion direction. On the basis of this study we were able to determine the relative activity of the slip and twinning deformation modes. By tuning the parameters of the EPSC model (i.e. the critical resolved shear strengths and hardening parameters), excellent agreement with the experimental data has been achieved. It is shown that the strain in the crystallographic 〈 c〉 direction is accommodated mainly by 〈 c + a〉 dislocation slip on second-order pyramidal planes. The results further indicate that either slip of 〈 a〉 dislocations occurs on {10.1} pyramidal planes or cross-slip from basal and prismatic planes takes place.

Modeling lattice strain evolution during uniaxial deformation of textured Zircaloy-2

An elastoplastic self-consistent model was used to interpret the experimental lattice strain evolution previously reported for testing in three directions of a thick polycrystalline Zircaloy-2 slab. The model was used to infer the underlying deformation mechanisms. The influences of prism 〈 a〉 slip, basal 〈 a〉 slip, pyramidal 〈 c + a〉 slip and tensile twinning were considered. The critical resolved shear stresses and hardening parameters for each mode were obtained by simultaneously fitting the macroscopic flow curves, Lankford coefficients and internal elastic strain development for all diffraction peaks, for the combination of three measurement directions and three loading directions, for compression and tension. The effects of dislocation interactions during deformation and hardening between deformation modes were considered. Tensile twinning inferred from the intensity changes of the diffraction peaks and its activity was qualitatively reproduced by the simulations for compression in the plate rolling and transverse directions and tension in the plate normal direction.