Elastic Viscoplastic Self-consistent Model Research Articles

Double-peak strain hardening behavior of Mg–1.2 wt.%Y alloy

In this study, the mechanical behavior and deformation mechanism of an extruded Mg–1.2 wt.%Y rod under tension and compression along the extrusion direction (ED) were systematically investigated through experiments and crystal plasticity simulations. A double-peak strain hardening behavior comprising five distinct stages was observed under compression along the ED. The five stages are as follows: a fast drop in the strain hardening rate (stage I), steady increase in the strain hardening rate (stage II), gradual decrease in the hardening rate (stage III), second increase in the strain hardening rate (stage IV), and rapid decrease in the strain hardening rate (stage V). This unique strain hardening behavior led to an ultimate compressive strength of up to 539 MPa at a high strain of 0.4. Crystal plastic simulations using an elastic viscoplastic self-consistent model revealed a high activity and a long process of {101¯2} twinning in a strain range of 0–0.35 under compression along the ED. The twinning behavior examined via electron backscattering diffraction indicated that the {101¯2} twinning was activated in both grains with relatively high and very low Schmid factors. Subsequently, the mechanism for the presence of this double-peak strain hardening was established and, finally, the significance of this double-peak strain hardening for strengthening Mg alloys was discussed.

Deformation behavior of nickel-based superalloys with bimodal γ′ size distribution studied by in-situ neutron diffraction combined with EVPSC modeling

Nickel-based superalloys with multimodal γ′ size distribution microstructure have been designed to employ in extreme environments due to their excellent mechanical performance. Whilst the effect of γ′ size on deformation mechanism has been studied before, little has been done to assess the deformation sequence and contribution to the load partition for different types of γ′ precipitates. In order to address this gap, this study focused on bimodal superalloys of GH4738 (containing the secondary and tertiary γ′ precipitates) and GH4720Li (containing the primary and tertiary γ′ precipitates) alloys. In-situ neutron diffraction assisted by a 2-site elastic-viscoplastic self-consistent (EVPSC) model was employed to elucidate the deformation mechanisms. The effects of different types of γ′ precipitates on the partitioning of interphase stress and the underlying mechanisms have been revealed. The findings indicated that bimodal superalloys exhibited distinct deformation behaviours compared to unimodal ones. Notably, premature load partitioning was observed in the microplasticity stage, attributable to Orowan looping for secondary γ′ precipitates or the emission of dislocations from the straight interface of primary γ′ precipitates. In the macroplasticity stage, considerable dislocations shear three types of γ′ precipitates, especially for the tertiary γ′ precipitates, which results in the decrease in the increment rate of interphase stress. These findings could contribute to a better understanding of deformation mechanisms in superalloys with multimodal γ′ size distributions and shine a spotlight on material design to modulate intergranular and interphase stresses.

Study on the Internal Stress and Thermal Anisotropy in Magnesium Alloys Using a Thermal Elastic Viscoplastic Self-Consistent Model.

A thermal elastic viscoplastic self-consistent model is utilized to examine the thermal stress induced by the thermal anisotropy of single crystals during heat treatments. This model considers temperature-dependent elastic constants and critical resolved shear stress associated with thermal dilation. Simulation results demonstrate that under cooling, the elastic lattice strain increases significantly when constrained compared to unconstrained cooling. The deformation mechanism observed under cooling with constraint resembles tension along the constrained direction at room temperature. Polycrystals offer more deformation mechanisms to accommodate thermal anisotropy compared to single crystals, resulting in lower applied stress at the constrained boundary. Among the various observed textures, the maximum amplitude of residual lattice strain follows the following order: rolled > extruded > random. Lower thermal anisotropy in the entire polycrystal structure leads to reduced internal stress. For a single crystal within aggregates, the {00.2} plane experiences tensile lattice strain, while the {10.0} and {11.0} planes undergo compressive lattice strain due to the greater contraction of single crystals along the <c> direction compared to the <a> direction during cooling.

On the cyclic torsion behavior of extruded AZ61A magnesium alloy tube

This work investigated the cyclic torsional behaviors of extruded AZ61A Mg alloy tubes through experiments and simulations. To capture the Bauschinger effect, the elastic viscoplastic self-consistent model with the twinning and detwinning scheme, together with a torsion-specific finite element approach was improved by adding a back-stress. A unique Swift effect with the “butterfly” shape was observed and reasonably reproduced by our model. Furthermore, the relative activities of deformation mechanisms, internal stress distributions, and twin volume fraction distributions under cyclic torsion were discussed. The findings could be beneficial to the design, manufacturing, and service of Mg alloys, particularly under cyclic torsion.

On the torsional and coupled torsion-tension/compression behavior of magnesium alloy solid rod: A crystal plasticity evaluation

Extensive attention has been paid to magnesium (Mg) alloys considering their potential as lightweight structural materials. However, it is still challenging to process and manufacture Mg materials that carry high strength and good ductility. This issue mainly arises from a lack of understanding toward the anisotropic mechanical behavior of Mg alloys in response to large deformation in a multiaxial stress state, which is unavoidably in existence under various material processing routes. In this regard, we seek to understand the torsional and coupled torsion-tension/compression behaviors of a magnesium alloy subject to large-strain deformation. The elastic viscoplastic self-consistent (EVPSC) model, which incorporates the twinning-detwinning (TDT) scheme and takes a torsion-specific finite element (TFE) approach, was employed to illuminate the inhomogeneous and multiaxial features of the torsional deformation in the AZ31 Mg alloy subjected to free/fixed-end torsion and coupled torsion-tension/compression. Experimental validation was conducted by characterizing the mechanical responses of the torsional specimens under the loading paths of free-end torsion and coupled torsion-tension. Our model successfully captures the Swift effect along with the shear texture, which is hardly predicted by conventional constitutive models. In addition, our model reveals that twinning is nearly equally active under free-/fixed-end torsion, while twinning under coupled torsion-tension and torsion-compression is promoted and suppressed, respectively. The comparison between the simple shear and pure torsion by simulation demonstrates that the significant bulk stress existing within the torsional rod is ascribed to the strong interaction between the cylindrical elements in the EVPSC model. As a final thought, we believe that the TFE-EVPSC-TDT model not only highlights the inhomogeneous and multiaxial features underlying the torsional and coupled torsion-tension/compression behaviors of the Mg alloy solid rod, but more significantly, can be used as a numerical tool for designing/tuning gradient twinning structures that may lead to optimized properties of Mg alloys.

Grain-size effect on the deformation of Mg–3Al–3Sn alloy: Experiments and elastic-viscoplastic self-consistent modeling

Grain refinement can significantly change the tensile behavior of Mg alloys. The underlying mechanisms are unclear yet. Here we report that grain refinement of a Mg–3Al–3Sn (wt.%, AT33) alloy increases the yield strength (σ0.2) following the Hall-Petch law, while decreases the strain hardening exponent (h). Tensile stress-strain curves and lattice strain evolution of four rolled AT33 samples with different grains sizes (10–97 μm) can be simulated by an elastic-viscoplastic self-consistent (EVPSC) model that takes grain size into account. The key to the model's success is that it allows both the initial critical resolved shear stress (CRSS) and the dislocation nucleation coefficient of different slip modes to be functions of the grain size. The coarse-grained samples have higher dislocation nucleation coefficient due to concurrently formed deformation twins. The model predicts that prismatic slip is the dominant deformation mode in this AT33 alloy; pyramidal <a> and <c+a> slips are more active in coarse-grained samples than in fine-grained samples.

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.

Effects of texture and twinning on the torsional behavior of magnesium alloy solid rod: A crystal plasticity approach in comparison with uniaxial tension/compression

The elastic visco-plastic self-consistent model with the twinning and detwinning scheme, in conjunction with a torsion specific finite element approach, is employed to model the deformation behavior of magnesium alloy AZ31 solid rod with different initial textures, including ideal basal, rolled, extruded, and random. Besides the loading with nearly homogenous deformation (tension and compression), loading paths with inhomogeneous deformation are investigated particularly under free-end and fixed-end torsions. Simulation reveals that the deformation mechanisms alter significantly with varying the different initial textures and different loading paths, leading to variances in stress-strain response, Swift effect, and deformation texture. In particular, our crystal plasticity model well captures the inhomogeneous distributions of internal stress and twin volume fraction within the solid rod under torsion. This is particularly significant as our torsion-specific crystal plasticity model provides a numerical tool that can be used to tune gradient twin structures via torsion of solid rod to improve the mechanical properties of magnesium alloy.

The origin of discontinuous yielding in Mg alloys under slip-dominated condition studied by in-situ synchrotron diffraction and elastic-viscoplastic self-consistent modeling

A discontinuous yielding was observed in the as-extruded Mg-7Gd-2Y-1Zn-0.6Zr (wt. %) alloy under uniaxial tension along the extrusion direction, which was eliminated by the peak-aging treatment. The deformation mechanisms of the as-extruded and peak-aged alloys were studied by in-situ synchrotron diffraction measurement and elastic-viscoplastic self-consistent (EVPSC) modeling. The in-situ synchrotron diffraction results suggest that the unpinning of basal dislocations from solute atmospheres causes the stress relaxation in the basal slip favored grains. The EVPSC modeling results indicate that the discontinuous yielding in the as-extruded alloy is associated with the interaction between basal dislocations and solute atmospheres. After peak-aging treatment, the interaction between basal dislocations and solute atmospheres is weakened due to the consumption of a number of solute atoms. Consequently, the discontinuous yielding is eliminated in the peak-aged alloy.

Study of slip activity in a Mg-Y alloy by in situ high energy X-ray diffraction microscopy and elastic viscoplastic self-consistent modeling

Slip activity from various slip modes largely determines the yield strength and ductility of Mg alloys. Solid solution elements in Mg can change the slip activity dramatically. In this paper, far-field high energy X-ray diffraction microscopy (FF-HEDM) is employed to study slip activity in a Mg-3wt%Y alloy during an in situ tensile experiment. The specimen was incrementally loaded up to 3% engineering strain along the rolling direction. At each load step, FF-HEDM data were collected to track the crystallographic orientation, center of mass, and stress tensor changes of nearly 1000 grains in the probed volume. By analyzing the change in orientation and stress tensor of individual grains at different load steps, it is possible to identify the activated slip systems and measure their critical resolved shear stress (CRSS) values. Prismatic slip and pyramidal I <a> slip are found to be very active in this alloy. The estimated CRSS values for basal slip, prismatic slip and pyramidal I <a> slip are 12 MPa, 38 MPa, and 36 MPa, respectively. These CRSS values were applied in a dislocation-based elastic viscoplastic self-consistent (EVPSC) model that successfully simulated the tensile stress-strain curve from the FF-HEDM experiment. The model also qualitatively predicted the crystal rotation in most of the selected grains, though it underestimated the internal stress and the magnitude of crystal rotation in these grains. Influence of solute Y on the strength and ductility of Mg alloys is discussed.

On the rapid hardening and exhaustion of twinning in magnesium alloy

We numerically study the rapid hardening and possible role of exhaustion of twinning in magnesium alloys under twinning dominated conditions, based on the large strain elastic visco-plastic self-consistent model. It is found that upon the exhaustion of the tensile twinning, further deformation requires the material to activate Prismatic and Pyramidal slips, which have relative high critical resolved shear stress (CRSS). However, the stress level at the exhaustion of the tensile twinning is not high enough to activate Prismatic and Pyramidal slips. Consequently, the imposed strain increments must mainly be accommodated by elastic deformations, and the strain hardening rate becomes a large fraction of the elastic modulus. Therefore, it can be concluded that the rapid hardening is a composite response more associated with elasticity of a large volume fraction of the material, than the dislocation-dislocation interactions normally held responsible for strain hardening of metals.

Deformation behavior of Mg-8.5wt.%Al alloy under reverse loading investigated by in-situ neutron diffraction and elastic viscoplastic self-consistent modeling

The EVPSC-TDT model for polycrystal plasticity and in-situ neutron diffraction have been used to investigate the behavior of a Mg-8.5wt.%Al alloy with two starting textures: 1) a typical extrusion texture in which a majority of the grains are oriented favorably for extension twinning via compression perpendicular to the basal pole, and 2) a modified texture in which extension twinning can be activated via tension parallel to the basal pole in a majority of the grains. Using a small number of adjustable parameters, and only two macroscopic tensile stress–strain curves for calibration, the model is able to capture, quantitatively, the trends in multiple data sets, including grain-level elastic lattice strains, and diffraction peak intensity changes due to lattice re-orientation associated with twinning. For twinning, the model assumes a polar critical resolved shear stress activation criterion and assigns the stress and hardening of the parent crystal to a newly formed twin. The model allows twinning to be driven either by the stress in the parent crystal (matrix reduction), in which case all of the twin transformation strain is assigned to the matrix, or by the stress in the twin (twin propagation), in which case all of the twin transformation strain is assigned to the twin. A detailed comparison between the model predictions and the neutron diffraction data reveals that assigning all of the twin transformation strain either to the matrix or to the twin is too one-sided, leading to excessive relaxation and hardening effects. A more equitable partitioning of the twin transformation strain is necessary. It is suggested that the stress and hardening assigned to a newly formed twin is of less importance to the performance of the model than the partitioning of the twin transformation strain.

Evaluation of elastic–viscoplastic self-consistent polycrystal plasticity models for zirconium alloys

Our elastic viscoplastic self-consistent (EVPSC) model is improved by including thermal strain to allow study of the behavior of a Zircaloy-2 slab under moderately large strains. The approach of introducing thermal strain effect is similar to the one developed by Turner et al. (1995) and used by Xu et al. (2008a). Various self-consistent schemes (SCSs) of the EVPSC model are then evaluated in terms of the deformation behavior of the material under different uniaxial strain paths. The material parameters for the various models are determined by fitting experimental data from uniaxial tension and compression tests along the normal direction (ND) and from uniaxial tension tests along the rolling direction (RD). The quality of the various SCS predictions is assessed based on comparisons of macroscopic deformation behavior (stress–strain curves and R-values) and microscopic mechanical response (the evolution of lattice strains and texture coefficients) between the predicted and experimental data. It is revealed that the Affine and Meff=0.1 self-consistent models give much better performance than the Secant and Tangent models.

Numerical study of the large strain behavior of extruded magnesium alloy AM30 tube by elastic viscoplastic self-consistent model

Abstract The anisotropic mechanical behaviors of the newly developed magnesium alloy AM30 are modeled by the elastic viscoplastic self-consistent (EVPSC) model. This is the first time to numerically study the mechanical behavior of AM30 based on EVPSC model. Three tests of uniaxial tension and compression along extrusion direction (ED) and ring hoop tension at temperature of 200 °C and strain rate of 0.1 s −1 are studied. It is found that extension twinning is an important deformation mechanism under both uniaxial compression along extrusion direction and ring hoop tension test. However the initial texture of the extruded AM30 tubes results in significantly different activities for these two tests. The twin volume fraction of extension twinning under ring hoop tension is less than that under uniaxial compression along extrusion direction. Numerical results are in good agreement with corresponding experimental results.

Numerical Assessment of the Role of Slip and Twinning in Magnesium Alloy AZ31B During Loading Path Reversal

Magnesium alloy AZ31B plastically deforms via twinning and slip. Corresponding to the unidirectional nature of twinning, the activity of twinning/detwinning is directly related to loading history and materials texture. Using the elastic viscoplastic self-consistent model implementing with the twinning and detwinning model (EVPSC–TDT), we revisited experimental data of AZ31B sheets under four different strain paths: (1) tension–compression–tension along rolling direction, (2) tension–compression–tension along transverse direction, (3) compression–tension–compression along rolling direction, and (4) compression–tension–compression along transverse direction, and identified the dominant deformation mechanisms with respect to the strain path. We captured plastic deformation behaviors observed in experiments and quantitatively interpreted experimental observations in terms of the activities of different deformation mechanisms and the evolution of texture. It is found that the in-plane pre-tension has slight effect on the subsequent deformation, and the pre-compression and the reverse tension after compression have significant effect on the subsequent deformation. The inelastic behavior under compressive unloading is found to be insignificant at a small strain level but pronounced at a large strain level. Such significant effect is mainly ascribed to the activity of twinning and detwinning.

Mechanical behavior and texture prediction of Ti-6Al-4V based on elastic viscoplastic self-consistent modelling

The goal of this study is to apply an elastic viscoplastic self-consistent crystal plasticity model to predict the texture evolution in a Ti-6Al-4V alloy which has a (mainly) hexagonal crystal structure. The model under consideration is an extension of the viscoplastic self-consistent model proposed by Lebensohn and Tome [1993] which has been adapted to account for elasticity and has been integrated with a new algorithm, making it more computationally efficient within an implicit FE scheme. The flow behavior of Ti-6Al-4V is strongly dependent on strain rate and temperature. To estimate the model parameters, the flow behavior of quasi-static experiments is used. A temperature sensitivity term has been introduced to correct the effects of temperature increase during the dynamic experiments. In order to have a meaningful rate sensitivity exponent, a value is calculated based on valid experimental data, rather than choosing an arbitrary large numerical value. In this way the behavior of Ti-6Al-4V is captured at different strain rates. Predictions of the model are compared to experimental data.

Investigation of deformation dynamics in a wrought magnesium alloy

In the present research, the real-time in-situ neutron diffraction measurements under a continuous-loading condition and elastic–viscoplastic self-consistent (EVPSC) polycrystal modeling were employed to study the deformation dynamics and the effect of the deformation history on plastic deformation in a wrought magnesium alloy. The experimental results reveal that pre-deformation delays the activation of the tensile twinning during the subsequent compression, mainly resulting from the residual strains. Detwinning does not occur until the applied stress exceeds the tensile yield strength during the reverse loading. It is believed that the grain rotation plays an important role in the elastic region during the reverse loading. The EVPSC model, which has been recently updated by implementing the twinning and detwinning model, was employed to characterize the deformation mechanism during the strain-path changes. The simulation result predicts well the experimental observation from the real-time in-situ neutron diffraction measurements. The present study provides a new insight of the nature of deformation mechanisms in a hexagonal close-packed (HCP) structured polycrystalline wrought magnesium alloy, which has significant implications for future work on studying the deformation mechanisms of HCP-structured materials.

Deformation behavior of solid-solution-strengthened Mg–9 wt.% Al alloy: In situ neutron diffraction and elastic–viscoplastic self-consistent modeling

In situ neutron diffraction and elastic–viscoplastic self-consistent (EVPSC) modeling have been employed to understand the deformation mechanisms of the loading–unloading process under uniaxial tension in a solid-solution-strengthened extruded Mg–9 wt.% Al alloy. The initial texture measured by neutron diffraction shows that the {00.2} basal planes in most grains are tilted around 20–30° from the extrusion axis, indicating that basal slip should be easily activated in a majority of grains under tension. Non-linear stress–strain responses are observed during unloading and reloading after the material is fully plastically deformed under tension. In situ neutron diffraction measurements have also demonstrated the non-linear behavior of lattice strains during unloading and reloading, revealing that load redistribution continuously occurs between soft and hard grain orientations. The predicted macroscopic stress–strain curve and the lattice strain evolution by the EVPSC model are in good agreement with the experimental data. The EVPSC model provides the relative activities of the available slip and twinning modes, as well as the elastic and plastic strains of the various grain families. It is suggested that the non-linear phenomena in the macroscopic stress–strain responses and microscopic lattice strains during unloading and reloading are due to plastic deformation by the operation of basal 〈a〉 slip in the soft grain orientations (e.g. {10.1}, {11.2} and {10.2} grain families).

A finite strain elastic–viscoplastic self-consistent model for polycrystalline materials

A large strain elastic–viscoplastic self-consistent (EVPSC) model for polycrystalline materials is developed. At single crystal level, both the rate sensitive slip and twinning are included as the plastic deformation mechanisms, while elastic anisotropy is accounted for in the elastic moduli. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress–strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tomé (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response due to lack of elastic deformation. It is also demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model.