Elastic Plastic Self-consistent Model Research Articles

Stress evolution in plastically deformed austenitic and ferritic steels determined using angle- and energy-dispersive diffraction

In the presented research, the intergranular elastic interaction and the second-order plastic incompatibility stress in textured ferritic and austenitic steels were investigated by means of diffraction. The lattice strains were measured inside the samples by the multiple reflection method using high energy X-rays diffraction during uniaxial in situ tensile tests. Comparing experiment with various models of intergranular interaction, it was found that the Eshelby-Kröner model correctly approximates the X-ray stress factors (XSFs) for different reflections hkl and scattering vector orientations.The verified XSFs were used to investigate the evolution of the first and second-order stresses in both austenitic and ferritic steels. It was shown that considering only the elastic anisotropy, the non-linearity of sin2ψ plots cannot be explained by crystallographic texture. Therefore, a more advanced method based on elastic-plastic self-consistent modeling (EPSC) is required for the analysis. Using such methodology the non-linearities of cos2φ plots were explained, and the evolutions of the first and second-order stresses were determined. It was found that plastic deformation of about 1–2% can completely exchange the state of second-order plastic incompatibility stresses.

Temperature-dependent constitutive modeling of a magnesium alloy ZEK100 sheet using crystal plasticity models combined with in situ high-energy X-ray diffraction experiment

A multiscale crystal plasticity model accounting for temperature-dependent mechanical behaviors without introducing a larger number of unknown parameters was developed. The model was implemented in elastic-plastic self-consistent (EPSC) and crystal plasticity finite element (CPFE) frameworks for grain-scale simulations. A computationally efficient EPSC model was first employed to estimate the critical resolved shear stress and hardening parameters of the slip and twin systems available in a hexagonal close-packed magnesium alloy, ZEK100. The constitutive parameters were thereafter refined using the CPFE. The crystal plasticity frameworks incorporated with the temperature-dependent constitutive model were used to predict stress–strain curves in macroscale and lattice strains in microscale at different testing temperatures up to 200 °C. In particular, the predictions by the crystal plasticity models were compared with the measured lattice strain data at the elevated temperatures by in situ high-energy X-ray diffraction, for the first time. The comparison in the multiscale improved the fidelity of the developed temperature-dependent constitutive model and validated the assumption with regard to the temperature dependency of available slip and twin systems in the magnesium alloy. Finally, this work provides a time-efficient and precise modeling scheme for magnesium alloys at elevated temperatures.

Linking constituent phase properties to ductility and edge stretchability of two DP 980 steels

Two DP 980 steels were characterized with multiple methods to determine its microstructure characteristics, tensile properties and edge stretchability. The first (DP980-B) is a commercial grade DP980 steel produced by AK Steel, while the second (DP980-T) is a modification of the first one by a tempering process. Both contain two individual phases: ferrite and martensite with body centered cubic (BCC) and tetragonal (BCT) crystal structure respectively. ASTM standard tensile tests were performed for both materials, revealing that DP980-B has higher work hardening and larger total elongation than DP980-T. Standard hole piercing and subsequent hole expansion and extrusion tests were also performed. The critical hole expansion ratio (HER) of hole pierced sheet at different nominal clearances, on the other hand, show an opposite trend: the HER of DP980-T almost doubles that of DP980-B. This indicates the tensile ductility and edge stretchability are determined by different mechanisms due to different deformation modes. The ductility of sheet metal during uniaxial tension is determined by deformation instability, i.e. necking, due to less constraint at the region of deformation and fracture will occur quickly after necking. The contribution of post-necking deformation to the total elongation is insignificant for the high strength grades such as DP980 steels. The ductility during hole-expansion, however, is mainly determined by material fracture behavior that is dependent on phase property disparity and material intrinsic fracture parameters. With the individual phase properties for both DP steels determined by in situ tensile tests under high energy X-ray diffraction, the results show that DP980-T steel has lower property disparities between the two phases than DP980-B steel, due to the tempering process. This explains why the DP980-T steel has higher HER than DP980-B steel: lower phase disparity will lead to less local deformation during loading. The higher work hardening rate for DP980-B steel contributes to its higher uniform elongation compared with DP980-T steel based on the maximum load condition of deformation instability. An integrated finite element simulation framework for studying hole expansion is also presented here, based on the calculated individual phase properties from the combined high energy x-ray diffraction (HEXRD) and elastic plastic self-consistent modeling. The simulation results correlate well with the experimental results on the HER difference between the two materials.

Neutron Diffraction Study of Phase Stresses in Al/SiCp Composite During Tensile Test

The stress partitioning between phases, phase stress relaxation as well as origins of Al/SiCp composite strengthening are studied in the present work. In this aim, the measurements of lattice strains by neutron diffraction were performed in situ during tensile test up to sample fracture. The experimental results were compared with results of elastic–plastic self-consistent model. It was found that thermal origin phase stresses relax at the beginning of plastic deformation of Al/SiCp composite. The evolution of lattice strains in both phases can be correctly simulated by the elastic–plastic self-consistent model only if the relaxation of initial stresses is taken into account. A major role in the strengthening of the studied composite plays a transfer of stresses to the SiCp reinforcement, however the hardness of Al metal matrix is also important.

A micromechanical model considering dislocation density based intra-granular backstress under cyclic loading

Cyclic loading experiments of the Sn-3.0Ag-0.5Cu solder alloy at a wide range of temperatures and strain rates are conducted. The influence of both temperature and strain rate are investigated. A micromechanical constitutive model based on the dislocation density theory is proposed for polycrystalline materials under cyclic loadings. The dislocation storage and annihilation mechanisms during strain path changes are considered by introducing the reversible dislocations. Based on Eshelby's inclusion analysis, a dislocation density related slip system level intra-granular backstress model is developed to describe the Bauschinger effect. The differences between the proposed backstress model and the traditional Armstrong-Frederick model are discussed. Combined with the small strain elastic plastic self-consistent model, the developed model is successfully applied to describe the cyclic stress-strain curves of body-centered tetragonal structured Sn-3.0Ag-0.5Cu alloy and modified 9Cr-1Mo steel with a body-centered cubic structure.

In-situ synchrotron X-ray diffraction deformation and EPSC modeling of AZ31 Mg alloy

We present a series of in-situ synchrotron X-ray diffraction deformation experiments conducted on polycrystalline AZ31 Mg alloy in a D-DIA apparatus at temperatures ranging from 154 °C to 286 °C, at ~3.1 GPa confining pressure and a strain rate of 5 × 10−6/s. For each experiment, we fit the X-ray diffraction data using elastic plastic self-consistent (EPSC) models that incorporate an isotropic deformation mechanism in addition to basal, prismatic and pyramidal slip systems. From the EPSC models, we report the sample strength and the critical resolved shear stresses (CRSS) for individual slip systems for each experiment and compared with previously reported data on this material.

Development of Intergranular Residual Stress and Its Implication to Mechanical Behaviors at Elevated Temperatures in AL6XN Austenitic Stainless Steel

Neutron diffraction was used to investigate the residual lattice strains in AL6XN austenitic stainless steel subjected to tensile loading at different temperatures, revealing the development of large intergranular stresses after plastic deformation. Elastic–plastic self-consistent modeling was employed to simulate the micromechanical behavior at room temperature. The overall variations of the modeled lattice strains as a function of the sample direction with respect to the loading axis agree in general with the experimental values, indicating that dislocation slip is the main plastic deformation mode. At 300 °C, the serrated flow in the stress–strain curve and the great amount of slip bands indicate the appearance of dynamic strain aging. Except for promoting the local strain concentration, the long-range stress field caused by the planar slip bands near the grain boundaries is also attributed to the decrease in the experimental intergranular strains. An increase in the lattice strains localized at some specific specimen orientations for reflections at 600 °C may be explained by the segregation of solute atoms (Cr and Mo) at dislocation slip bands. The evolution of full-width at half-maximum demonstrates that the dynamic recovery indeed plays an important role in alleviating the local strain concentrations during tensile loading at 600 °C.

ECCI, EBSD and EPSC characterization of rhombohedral twinning in polycrystalline α-alumina deformed in a D-DIA apparatus

Rhombohedral twinning in alumina (aluminium oxide, α-Al2O3) is an important mechanism for plastic deformation under high-temperature–pressure conditions. Rhombohedral twins in a polycrystalline alumina sample deformed in a D-DIA apparatus at 965 K and 4.48 GPa have been characterized. Three classes of grains were imaged, containing single, double and mosaic twins, using electron channeling contrast imaging (ECCI) in a field emission scanning electron microscope. These twinned grains were analyzed using electron backscatter diffraction (EBSD). The methodology for twin identification presented here is based on comparison of theoretical pole figures for a rhombohedral twin with experimental pole figures obtained with EBSD crystal orientation mapping. An 85°〈02{\overline 2}1〉 angle–axis pair of misorientation was identified for rhombohedral twin boundaries in alumina, which can be readily used in EBSD post-processing software to identify the twin boundaries in EBSD maps and distinguish the rhombohedral twins from basal twins. Elastic plastic self-consistent (EPSC) modeling was then used to model the synchrotron X-ray diffraction data from the D-DIA experiments utilizing the rhombohedral twinning law. From these EPSC models, a critical resolved shear stress of 0.25 GPa was obtained for rhombohedral twinning under the above experimental conditions, which is internally consistent with the value estimated from the applied load and Schmid factors determined by EBSD analysis.

Individual phase constitutive properties of a TRIP-assisted QP980 steel from a combined synchrotron X-ray diffraction and crystal plasticity approach

Microstructure-based constitutive models for multiphase steels require accurate constitutive properties of the individual phases for component forming and performance simulations. We address this requirement with a combined experimental/theoretical methodology which determines the critical resolved shear stresses and hardening parameters of the constituent phases in QP980, a TRIP assisted steel subject to a two-step quenching and partitioning heat treatment. High energy X-Ray diffraction (HEXRD) from a synchrotron source provided the average lattice strains of the ferrite, martensite, and austenite phases from the measured volume during in situ tensile deformation. The HEXRD data was then input to a computationally efficient, elastic-plastic self-consistent (EPSC) crystal plasticity model which estimated the constitutive parameters of different slip systems for the three phases via a trial-and-error approach. The EPSC-estimated parameters are then input to a finite element crystal plasticity (CPFE) model representing the QP980 tensile sample. The predicted lattice strains and global stress versus strain curves are found to be 8% lower that the EPSC model predicted values and from the HEXRD measurements, respectively. This discrepancy, which is attributed to the stiff secant assumption in the EPSC formulation, is resolved with a second step in which CPFE is used to iteratively refine the EPSC-estimated parameters. Remarkably close agreement is obtained between the theoretically-predicted and experimentally derived flow curve for the QP980 material.

Elastic plastic self-consistent (EPSC) modeling of plastic deformation in fayalite olivine

Elastic plastic self-consistent (EPSC) simulations are used to model synchrotron X‑ray diffraction observations from deformation experiments on fayalite olivine using the deformation DIA apparatus. Consistent with results from other in situ diffraction studies of monomineralic polycrystals, the results show substantial variations in stress levels among grain populations. Rather than averaging the lattice reflection stresses or choosing a single reflection to determine the macroscopic stress supported by the specimen, an EPSC simulation is used to forward model diffraction data and determine a macroscopic stress that is consistent with lattice strains of all measured diffraction lines. The EPSC simulation presented here includes kink band formation among the plastic deformation mechanisms in the simulation. The inclusion of kink band formation is critical to the success of the models. This study demonstrates the importance of kink band formation as an accommodation mechanism during plastic deformation of olivine as well as the utility of using EPSC models to interpret diffraction from in situ deformation experiments.

Effect of Microstructural and Morphological Parameters on the Formability of BCC Metal Sheets

The determination of forming limit strains in sheet metal forming industry is a useful way for quantifying metals in terms of formability. However, such forming limit diagrams (FLDs) remain very difficult to obtain experimentally. Therefore, the numerical prediction of forming limit strains represents a convenient alternative to replace this time consuming and expensive experimental process. Moreover, a combined theoretical‐numerical model allows investigating the impact of essential microstructural aspects (e.g., initial and induced textures, dislocation density evolution, softening mechanisms, …) and deformation mechanisms on the ductility of polycrystalline aggregates. In this paper, the impact of microstructural and morphological parameters, particularly the mean grain size, on the formability limit of BCC materials is investigated. To this end, an elastic–plastic self‐consistent (EPSC) polycrystalline model, coupled with a bifurcation‐based localization criterion, is adopted to numerically simulate FLDs. The FLDs thus determined using the Bifurcation–EPSC model for an IF–Ti single‐phase steel are compared to the FLDs given by ArcelorMittal, demonstrating the predictive capability of the proposed approach in investigations of sheet metal formability. The role of the averaging scheme is also shown to be significant by comparing the critical limit strains predicted with the self‐consistent scale‐transition scheme to those obtained with the more classical full‐constraint Taylor model. Finally, numerical simulations for different values of mean grain size are provided in order to analyze the impact of mean grain size on the formability of BCC metal sheets.

Tensile deformation behaviors of Zircaloy-4 alloy at ambient and elevated temperatures: In situ neutron diffraction and simulation study

Tensile stress–strain relationship of a rolled Zircaloy-4 (Zr-4) plate was examined in situ using a neutron diffraction method at room temperature (RT, 25°C) and an elevated temperature (250°C). Variations of lattice strains were obtained as a function of macroscopic bulk strains along prismatic (101¯0), basal (0002) and pyramidal (101¯1) planes in the hexagonal close-packed structure of the Zr-4. The mechanisms of strain responses in these three major planes were simulated using elastic–plastic self-consistent (EPSC) model based on Hill–Hutchinson method, thus the inter-granular stresses and deformation systems of each individual grain under loading were obtained. Results show that there is a good agreement between the EPSC modeling and neutron diffraction measurements in terms of macroscopic stress–strain relationship and lattice strain evolutions of the planes at RT. However, there is a slight discrepancy in the lattice strains obtained from the EPSC modeling and neutron diffraction when the specimen was deformed at 250°C. Analysis of grain structure and texture obtained using electron back-scattered diffraction suggests that dynamic recovery process is significant during the tensile deformation at the elevated temperature, which was not considered in the simulation.

Studying the effect of stress relaxation and creep on lattice strain evolution of stainless steel under tension

Due to relatively long associated count times, in situ strain measurements using neutron diffraction requires periodic interruption of the test to collect the diffraction data by holding either the stress or the strain constant. As a consequence, stress relaxation or strain creep induced by the interrupts is inevitable, especially at loads which are close to the flow stress of the material. An in situ neutron diffraction technique, which consists in performing the diffraction measurements using continuous event-mode data collection while conducting the mechanical loading monotonically with a very slow loading rate, is proposed here to avoid the effects associated with interrupts. The lattice strains in stainless steel under uniaxial tension are measured using the three techniques, and the experimental results are compared to study the effect of stress relaxation and strain creep on the lattice strain measurements. The experimental results are simulated using both the elastic viscoplastic self-consistent (EVPSC) model and the elastic plastic self-consistent (EPSC) model. Both the EVPSC and EPSC models give reasonable predictions for all the three tests, with EVPSC having the added advantage over EPSC that it allows us to address the relaxation and creep effects in the interrupted tests.

Effect of texture on load partitioning in Ti-6Al-4V

Neutron diffraction has been used to characterize the evolution of residual elastic strain in grains with different orientations due to room temperature plastic deformation in two plate product forms of Ti–6Al–4V. The evolution of lattice strains has been rationalized using a two-phase elastic–plastic self-consistent model using only the texture difference between the two product forms. It is found that the elastic properties of both the bulk and individual orientations can be reproduced quite satisfactorily, with a C′ modulus of the β phase of 15GPa. The residual microstrains produced are generally greater in the unidirectionally rolled material than the cross-rolled, but are smaller than in Ti-834. The residual strains accumulated in the (0002) orientation are near-zero, which can only be reproduced in the modelling by assuming a critical resolved shear stress for 〈c+a〉 slip only 1.5× that for 〈a〉 slip, compared to the 3× factor found for isolated single crystals. The implications of this for our understanding of deformation in these materials are discussed.

Interpreting in situ x-ray diffraction data from high pressure deformation experiments usingelastic–plastic self-consistent models: an example using quartz

We present synchrotron x-ray diffraction observations from a deformation experiment onfine-grained polycrystalline quartz using the newly developed deformation DIA apparatus.During deformation experiments we were able to observe the elastic strain of the (100),(101) and (112) lattice reflections. The elastic strains are typically converted into stressesand interpreted in terms of the differential stress supported by the specimen. Consistentlywith results from others obtained using this technique to deform monomineralicpolycrystals, our results show substantial variations in stress levels between grainpopulations. Rather than averaging the lattice reflection stresses or choosing a singlereflection to determine the macroscopic stress supported by the specimen, we explore theuse of elastic–plastic self-consistent (EPSC) models. We are able to match the measureddifferential elastic lattice strains with an EPSC model in which basal and prism slips are activated. An interesting outcome of the EPSC model is the prediction that themacroscopic stress experienced by the sample should be greater than the stress calculatedfrom any of the reflections that we observed. This observation serves as a caution againstusing reflection stresses as a proxy for the macroscopic stress in in situ deformationexperiments.

Ellipticity loss analysis for tangent moduli deduced from a large strain elastic–plastic self-consistent model

In order to investigate the impact of microstructures and deformation mechanisms on the ductility of materials, the criterion first proposed by Rice is applied to elastic–plastic tangent moduli derived from a large strain micromechanical model combined with a self-consistent scale-transition technique. This approach takes into account several microstructural aspects for polycrystalline aggregates: initial and induced textures, dislocation densities as well as softening mechanisms such that the behavior during complex loading paths can be accurately described. In order to significantly reduce the computing time, a new method drawn from viscoplastic formulations is introduced so that the slip system activity can be efficiently determined. The different aspects of the single crystal hardening (self and latent hardening, dislocation storage and annihilation, mean free path, etc.) are taken into account both by the introduction of dislocation densities per slip system as internal variables and the corresponding evolution equations. Comparisons are made with experimental results for single and dual-phase steels involving linear and complex loading paths. Rice’s criterion is then coupled and applied to this constitutive model in order to determine the ellipticity loss of the polycrystalline tangent modulus. This criterion, which does not need any additional “fitting” parameter, is used to build Ellipticity Limit Diagrams (ELDs).

Micromechanics of Ti–10V–2Fe–3Al: In situ synchrotron characterisation and modelling

The lattice strain evolution in near-β Ti–10V–2Fe–3Al during room temperature tensile loading has been characterised in the as-forged and forged and aged conditions using in situ synchrotron X-ray diffraction. As expected, the fine scale α which precipitates during aging strengthens the β-phase much more effectively than the α present in the forged sample. The behaviour has been modelled using a two-phase elastic–plastic self-consistent (EPSC) model. It is found that the constrained β-phase E 200 increases from 45 GPa in the as-forged condition to 88 GPa in the forged and aged material. C 11– C 12 increases from 12 to 47 GPa due to the increase in β-stabiliser content, in agreement with atomistic predictions. The EPSC models are reasonably successful at reproducing the observed behaviour, but do not provide a complete description of the micromechanics of these materials.

Intergranular strain accumulation in a near-alpha titanium alloy during plastic deformation

Neutron diffractometry has been used to characterise the evolution of intergranular lattice strains during the tensile loading of the near-α titanium alloy Ti-834. The measurements were made on tensile testpieces in directions both parallel and perpendicular to the applied load. To rationalise the observations, an elastic–plastic self-consistent model for the α phase is used, in which the three basal and two non-basal slip systems observed in titanium alloys are invoked; it is found that it is possible to reproduce successfully the observed trends and magnitudes of the microstrains in both the longitudinal and transverse directions. Unfortunately, no reflection was found for which intergranular strain accumulation is small, i.e. less than 1×10 -4. This situation should be recognised when making measurements of residual strains and stresses in large-scale engineering components, since intergranular strains must be distinguished from the macro-scale engineering strains. To circumvent this difficulty a novel method is proposed whereby the results of two or more peaks are combined to form a synthetic ‘actor’ which accumulates near-zero intergranular strain in both the parallel and perpendicular directions.

Lattice orientation dependence of the stored energy during cold-rolling of polycrystalline steels

During cold deformation, about 10% of the energy spent is stored in the polycrystalline material in the form of the elastic energy associated to crystal defects. The latter can be measured experimentally by calorimetry, X-ray diffraction. In this work, we propose a direct calculation based on an elastic-plastic self-consistent model. The part of the stored energy related to second order (intergranular) stresses is estimated for a cold-rolled Ti–IF steel. The stored energy as well as the accumulated plastic strain are reported as a function of the final crystallographic orientation of the grains. Along the α-fiber, the stored energy increases from {001}<110> to {111}<110> while the plastic strain decreases. This dependence with the crystallographic orientation is in good agreement with experimental results. With respect to recrystallization, experiments show that {111}-grains nucleate first. This highlights the specific role of second order (intergranular) stresses for recrystallization.

A two phase elastic–plastic self-consistent model for the accumulation of microstrains in Waspaloy

A two phase self-consistent scheme for the accumulation of elastic lattice strains in the nickel-base superalloy Waspaloy is presented. The microstructure is idealised as a set of randomly orientated anisotropic grains which are assumed to be spherical and embedded within a homogenous effective medium which is assigned the properties of the bulk. Each grain is modelled as a medium with the properties of identically aligned cubic γ-Ni with a spherical inclusion of γ′-Ni 3Al. The γ′ is treated as an elastic anisotropic solid, and the γ is modelled as an elastic–plastic single crystal according to the Taylor–Bishop–Hill plasticity theory. The diffraction elastic constants and microstrains accumulated are compared with those found experimentally using neutron diffraction, as previously reported by Stone et al. The predictions are found to be adequate, and in particular the shift of load, as reflected in the microstrain, from the matrix to the precipitate between 3 and 10% plastic strain is reproduced. The implications of these results for the development and use of Reitveld-derived schemes for the measurement of bulk residual stress at spallation sources are discussed.