Elastic Plastic Self-consistent Research Articles

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.

In-situ high-energy X-ray diffraction and crystal plasticity modeling to predict the evolution of texture, twinning, lattice strains and strength during loading and reloading of beryllium

Deformation behavior of beryllium during compressive loading and cross-reloading is studied using in-situ high energy synchrotron X-ray diffraction microscopy and crystal plasticity modeling. The evolution of texture, twinning, elastic lattice strains, and flow stress are measured and compared with the predictions of an advanced elastic-plastic self-consistent (EPSC) crystal plasticity model. The model is initialized with the experimentally measured texture and residual stress produced by a simulation of cooling and calibrated to establish a set of model parameters using a portion of the measured data. The rest of the measured data is used for validation of the model. It is shown that the model is sufficiently flexible to reproduce the particularities pertaining to the complex strain-path-change and strain rate sensitive deformation of the material including the evolution of texture, twinning, lattice strains, transients in the stress-strain response, and anisotropic hardening with great accuracy using a single set of model parameters. From the comparison of the experimental data and predictions, we infer that the shifts in active deformation mechanisms between the slip systems from soft to hard and vice versa as well as between twinning to de-twinning are primarily responsible for drastic changes in the flow stress from one path to another. In particular, deformation twins form during compressive in-plane loading followed either by de-twinning during compressive cross-reloading in the through-thickness direction or by forming additional twin variants with some de-twinning of the existing variants during a compressive cross-reloading in another in-plane direction. The shifts in active deformation mechanisms are a consequence of changes in texture relative to the compression direction mediated with the deformation history and strain rate dependent dislocation density evolution governing hardening. The secondary effects improving the predictions come from accounting for residual stress, slip system-level backstress, and latent hardening.

Initial Acoustoelastic Measurements in Olivine: Investigating the Effect of Stress on P‐ and S‐Wave Velocities

AbstractIt is well known that elasticity is a key physical property in the determination of the structure and composition of the Earth and provides critical information for the interpretation of seismic data. This study investigates the stress‐induced variation in elastic wave velocities, known as the acoustoelastic effect, in San Carlos olivine. A recently developed experimental ultrasonic acoustic system, the Directly Integrated Acoustic System Combined with Pressure Experiments (DIASCoPE), was used with the D‐DIA multi‐anvil apparatus to transmit ultrasonic sound waves and collect the reflections. We use the DIASCoPE to obtain longitudinal (P) and shear (S) elastic wave velocities from San Carlos olivine at pressures ranging from 3.2–10.5 GPa and temperatures from 450–950°C which we compare to the stress state in the D‐DIA derived from synchrotron X‐ray diffraction. We use elastic‐plastic self‐consistent (EPSC) numerical modeling to forward model X‐ray diffraction data collected in D‐DIA experiments to obtain the macroscopic stress on our sample. We can observe the relationship between the relative elastic wave velocity change (ΔV/V) and macroscopic stress to determine the acoustoelastic constants, and interpret our observations using the linearized first‐order equation based on the model proposed by Hughes and Kelly (1953), https://doi.org/10.1103/physrev.92.1145. This work supports the presence of the acoustoelastic effect in San Carlos olivine, which can be measured as a function of pressure and temperature. This study will aid in our understanding of the acoustoelastic effect and provide a new experimental technique to measure the stress state in elastically deformed geologic materials at high pressure conditions.

Direct diffraction measurement of critical resolved shear stresses and stress localisation in magnesium alloy

The main purpose of this work is to develop neutron diffraction methodology in order to determine stresses localised in polycrystalline grains during elastoplastic deformation, directly from experiment. As a result, for the first time, the von Mises stress for chosen grain orientations and Critical Resolved Shear Stresses (CRSS) for active slip systems were unambiguously measured without the help of crystallographic models, which introduce different theoretical assumptions. The stresses measured for groups of grains and the determined CRSS values are important characteristics of a material, which allow to study plastic deformation in textured material at different scales: slip system and grain, which play a key role in mechanical properties and formability of the material.The new method was successfully tested and applied to textured AZ31 alloy subjected to tensile deformation and the components of stress tensor were for the first time determined from measured lattice strains corresponding to chosen orientations of crystallite lattice. The obtained results positively verified hypotheses that, during plastic deformation, a large difference in the hardness as well as in the localised stresses occurs for grains having different lattice orientations. It was found directly from experiment that, the activation of basal glide, having small CRSS, does not lead to significant plastic deformation, and the activation of other non-basal systems (with higher CRSS) induces the development of plasticity at the macroscopic scale. The early plastic deformation occurring due to slip on basal system is small but it can destructively affect fatigue life, limiting applicability of the material for structural components. Finally, the comparison of experimental results with a modified version of Elastic-Plastic Self-Consistent (EPSC) model showed its capability to simulate the mechanical behaviour of such materials.

Elastic-viscoplastic self-consistent modeling for finite deformation of polycrystalline materials

Anisotropic 1-site and 2-site self-consistent models are developed to describe the elastic-viscoplastic behavior of polycrystalline materials deformed to finite strains on the basis of rate-dependent crystallographic slip and a generalized Hill-Hutchinson self-consistent approach. The choice of rate-dependent constitutive law at single crystal level implemented in the models is discussed through fitting experimental data and calibrating viscous parameters. It is found that drag-stress type Norton law works well for the 1-site elastic-viscoplastic self-consistent (EVPSC) model while threshold stress type Norton law is suitable for the 2-site EVPSC model to assure that the viscoplastic inter-granular interaction is realistic. Both models have been verified by thoroughly fitting experimental data in literatures. For the 1-site EVPSC model, selected experimental data covers both macroscopic and microscopic mechanical responses of steels during deformation with a large range of strain rate from the quasi-static (10−4s−1) to the dynamic (~104s−1). For the 2-site EVPSC model, in situ neutron diffraction data of nickel-based superalloys with various microstructures was fitted. Both models generally fit the experimental data well. A comparison between the EVPSC and elastic-plastic self-consistent (EPSC) models on the prediction of lattice strains has also been made for both the 1-site and 2-site cases, which verifies the predictability on lattice strains of the newly developed EVPSC models. A validation of the homogenization approach for the EVPSC modeling has been performed, which confirms that the proposed EVPSC models are applicable for cubic structure materials with finite deformations. Our formulation of EVPSC modeling developed in this work shines a spotlight on the way of developing a multi-functional self-consistent model to predict both macroscopic and microscopic deformation behaviors of various polycrystalline materials under different loading rates of 10−4s−1~104s−1.

In-situ investigation of the evolution of microstructure and texture during load reversal of commercially pure titanium using synchrotron X-ray diffraction

Understanding of the deformation micro-mechanism as a function of grain orientation during cyclic loading is of significant importance to have failure safe design of structural components made of commercially pure titanium (CP-Ti). The evolution of deformation microstructure and texture of commercially pure titanium samples with prismatic-pyramidal (orientation A) and near basal (orientation B) as initial texture along the loading direction has been investigated during load reversal at ±8% and ± 12% strain using in-situ synchrotron X-ray diffraction. The synchrotron X-ray diffraction results have been further complemented with Elastic-Plastic Self-Consistent (EPSC) simulation of the texture data and ex-situ Electron backscatter diffraction (EBSD) scan taken at the same region. Orientation A showed partially reversible texture whereas orientation B showed non-reversible texture with mechanical reversibility. The partial textural reversibility has been attributed to the prism to basal slip transition which toggles the c-axis between the normal and transverse direction during the tension-compression cycle. With an increase in strain from 8 to 12%, microstrain and dislocation density in the basal plane of orientation A decreases sharply. On the other hand, for the same level of strain in orientation B, microstrain increases but the dislocation density of basal plane shows insignificant change. The crystal orientation map obtained from ex-situ EBSD of deformed microstructure complements to the fact indicating, with increase in strain, deformation of basal oriented grains show non-Schmid behaviour of contraction twin propagation due to local strain incompatibility in orientation A. On the other hand, the prism oriented grains of orientation B have mostly deformed by extension twinning, which reorients them towards basal orientation. The deformation proceeds by lateral thickening of twins due to the lower elastic stiffness of the twinned region compared to the grain matrix. The extension twin boundaries get converted to contraction twin boundaries at higher strain during load reversal.

Elastic plastic self-consistent (EPSC) modeling of San Carlos olivine deformed in a D-DIA apparatus

Elastic plastic self-consistent (EPSC) modeling of San Carlos olivine deformed in a D-DIA apparatus

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.

Accounting for lattice coherency in a two-phase elastic-plastic self-consistent model for nickel-based superalloys

A 2-site elastic-plastic self-consistent (EPSC) model is developed and implemented in order to account for crystallographic texture development and grain morphology evolution under strong correlations between neighbor grains of different phases, both in space and orientation. Predictions of the model adequately fit the published in situ neutron diffraction data for nickel-based superalloys at ambient and elevated temperatures, in which γ and γ' phases exhibit exact cube-cube orientation relationship. Comparison with 2-site model (small strain algorithm, non-rotation scheme) and 1-site model (finite strain algorithm, co-rotation scheme) has been made, and the result shows that the present 2-site model (finite strain algorithm, rotation scheme) leads to better predictions in lattice strain evolution where both rotation of crystal lattice and correlation between inclusions are accounted for, especially when the applied strain is larger than 0.02 for transverse direction and 0.05∼0.18 for axial direction for the materials studied in this work. Based on a systematic study on the effects of grain-grain interaction and total grain number on the homogenized results, we found that transverse lattice strains of γ (200) and/or γ' (100) are sensitive to the interplay between γ-γ' interaction and evolution of grain orientation distribution with deformation, while that of γ (220) and γ' (110) are sensitive to the initial crystallographic texture.

Crystal Plasticity Modelling of Large Strain Behaviour of HCP Materials

In this paper, we first introduce various polycrystal plasticity models including the Visco-Plastic Self-Consistent (VPSC) model, the Elastic-Plastic Self-Consistent (EPSC) model, and the Elastic Visco-Plastic Self-Consistent (EVPSC) model, as well as twinning models including the Predominant Twin Reorientation (PTR) model and the Twinning and De-Twinning (TDT) model, with emphasizing on characteristic differences between these models. Then, we demonstrate that the EVPSC-TDT model is able to capture key features of large strain behaviour of HCP materials under various deformation processes including uniaxial tension and compression, simple shear, and torsion.

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.

Determination of the single-phase constitutive relations of α/β dual phase TC6 titanium alloy

The constitutive relations of α and β phases in a TC6 titanium alloy were determined by implementing a two-phase elastic-plastic self-consistent (EPSC) framework combined with the evolution of lattice strains; these strains were obtained via in-situ tensile loading synchrotron-based x-ray diffraction experiments. It was found that the {200}β reflection has the lowest stiffness and load partitions prior to the α phase during the elastic loading stage in this alloy. The simulated parameters including the diffraction elastic constant and initial yield stress of lattice reflections exhibited satisfactory correspondence with the experimental results. Further analysis of the characteristics of the Schmid Factor (SF) distributions of the main slip systems revealed that the elastic-plastic transition process in the α phase occurs over a prolonged period. In contrast, the β phase undergoes a transient process owing to its relatively more concentrated SF frequency distributions, than those of the α phase. In addition, the fitted stress-strain curve of each phase was compared with the measured macro stress-strain curve obtained from the in-situ experiment. It revealed Young's moduli of 110.3GPa and 104.5GPa, and yield stresses of 877.8MPa and 969MPa, for the α and β phases, respectively.

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.

An elasto-plastic self-consistent model with hardening based on dislocation density, twinning and de-twinning: Application to strain path changes in HCP metals

In this work, we develop a polycrystal mean-field constitutive model based on an elastic–plastic self-consistent (EPSC) framework. In this model, we incorporate recently developed subgrain models for dislocation density evolution with thermally activated slip, twin activation via statistical stress fluctuations, reoriented twin domains within the grain and associated stress relaxation, twin boundary hardening, and de-twinning. The model is applied to a systematic set of strain path change tests on pure beryllium (Be). Under the applied deformation conditions, Be deforms by multiple slip modes and deformation twinning and thereby provides a challenging test for model validation. With a single set of material parameters, determined using the flow-stress vs. strain responses during monotonic testing, the model predicts well the evolution of texture, lattice strains, and twinning. With further analysis, we demonstrate the significant influence of internal residual stresses on (1) the flow stress drop when reloading from one path to another, (2) deformation twin activation, (3) de-twinning during a reversal strain path change, and (4) the formation of additional twin variants during a cross-loading sequence. The model presented here can, in principle, be applied to other metals, deforming by multiple slip and twinning modes under a wide range of temperature, strain rate, and strain path conditions.

Simulation of texture evolution in magnesium alloy: Comparisons of different polycrystal plasticity modeling approaches

Three polycrystal plasticity modeling approaches: Taylor model, self-consistent (SC) model, and crystal plasticity based finite element method (CPFEM) have been mainly applied to simulate texture evolution of magnesium alloy. To examine the influence of different approaches on simulated results, comparisons between material point simulation based on Taylor model and that based on elastic–plastic self consistent (EPSC) model are conducted. Furthermore, comparisons between CPFEM based on single crystal scheme and that based on polycrystal scheme are also carried out. In addition, all the formulations are validated by comparing the predicted textures with experimental results during uniaxial compression for AZ31 magnesium alloy.

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.

Deformation behaviors of zircaloy-4 alloy under uniaxial compression

Zirconium (Zr) has a hexagonal close-packed crystal structure, which exhibits elastic and plastic anisotropy. Internal stresses can be easily generated in the rolling process and the subsequent plastic deformation process. It is critical to evaluate the internal stresses and the deformation mechanisms of Zr alloy materials. The deformation behaviors of Zr alloy influence directly its service life and safety. In this work, compression deformation behaviors of zircaloy-4 (Zr-4) alloy have been studied by the in situ neutron diffraction technique combined with the elastic-plastic self-consistent (EPSC) simulation. A compressive external load is applied along the thickness direction of the rolled plate, which is called through-thickness compression. Electron back-scattered diffraction is used to analyze the texture evolution during the plastic deformation. Transmission electron microscopy (TEM) is used to measure the distribution of the defects in the deformed sample. The EPSC simulation provides the deformation mechanism quantitatively by fitting the in situ neutron diffraction data, and the simulated results is confirmed by the TEM observations. Results show that when the true strain is small (less than 0.55%), prismatic {1010}20> (a> type) slip dominates; however, as the plastic strain is increased, the percentage of pyramidal {1011}23> (c+a> type) slip becomes larger than that of prismatic {1010}20> (a> type) slip, and the pyramidal {1011}20> (a> type) slip and pyramidal {1012}20> (a> type) slip may exist.

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.