Elastoplastic Phase-field Model Research Articles

Insight into element segregation mechanisms during creep in γʹ-strengthened Co-based superalloy by elastoplastic phase-field simulation

The element segregation accompanying the creep process has been shown to significantly affect the deformation resistance of the superalloys. However, the processing and mechanism of element segregation are still unclear. This paper investigated the concentration evolution of a model Co–9Al–9W (at. %) alloy during 900 ​°C/275 ​MPa using developed ternary elastoplastic phase-field model coupled with CALPHAD method and crystal plasticity model. The results of simulation show that co-depletion of Al and W element occurs in γʹ precipitate and in γ side at γ/γʹ interface, and this depletion is gradually increasing with the accumulation of plastic strain. From the perspective of changes of driving force of element diffusion, it is found that these segregation phenomena are attributed to the high elastic potential caused by the large local plastic strain. In addition, the effects of these segregations on creep property are also predicted. The current research provides a new method for exploring the mechanism of element diffusion.

Development and Calibration of a Phenomenological Material Model for Steel-Fiber-Reinforced High-Performance Concrete Based on Unit Cell Calculations.

A phenomenological material model has been developed to facilitate the efficient numerical analysis of fiber-reinforced high-performance concrete (HPC). The formulation integrates an elasto-plastic phase-field model for simulating fractures within the HPC matrix, along with a superimposed one-dimensional elasto-plasticity model that represents the behavior of the embedded fibers. The Drucker-Prager plasticity and one-dimensional von-Mises plasticity formulations are incorporated to describe the nonlinear material behavior of both the HPC matrix and the fibers, respectively. Specific steps are undertaken during the development and calibration of the phenomenological material model. In the initial step, an experimental and numerical analysis of the pullout test of steel fibers embedded in an HPC matrix is conducted. This process is used to calibrate the micro-mechanical model based on the elasto-plastic phase-field formulation for fracture. In the subsequent step, virtual experiments based on an ellipsoidal unit cell, also with the resolution of fibers (used as a representative volume element, RVE), are simulated to analyze the impact of fiber-matrix interactions and their physical properties on the effective material behavior of fiber-reinforced HPC. In the final step, macroscopic boundary value problems (BVPs) based on a cuboid are simulated on a single scale using the developed phenomenological material model. The resulting macroscopic stress-strain characteristics obtained from both types of simulations, namely simulations of virtual experiments and macroscopic BVPs, are compared. This comparison is utilized for the calibration of material parameters to obtain a regularized solution and to assess the effectiveness of the presented phenomenological material model.

An elastoplastic phase-field model for quasi-static fracture of nickel-based super-alloys

In the present study, an elastoplastic phase-field model of quasi-static fracture in ductile materials is proposed in the variational framework for J2 plasticity with isotropic hardening, which is suitable to describe the quasi-static behavior of metals as investigated in the performed experiments. These contributions include: (1) the free energy functions for coupling elastic response, plastic yielding and damage evolution are established. (2) The new elastic and plastic energy degradation functions are constructed to quantitatively describe the relationship between energy release and phase-field evolution of elastoplastic materials. (3) Damage evolution and plastic yielding criteria are derived. (4) From a numerical point of view, we derive the governing equations and the corresponding weak forms and the overall solution procedure for the phase-field model is given via the use of a return-mapping algorithm. This phase-field model was validated by a series of tensile experiments on Inconel 718 nickel-based super-alloys standard specimens. In order to compare the simulation results with the experimental results more comprehensively, the digital image correlation (DIC) technique is applied to experimentally investigate the specimen deformation information. In addition, to verify the potential of the model to capture complex cracks, we performed Nooru-Mohamed tests. The numerical simulation results are in good agreements with the results of previous experimental work.

Phase field study on phase stability at high temperatures in nanograined Fe–Ni alloy prepared by plastic deformation

The phase stability at high temperatures in nanograined Fe–Ni alloy prepared by surface mechanical attrition treatment (SMAT) and ball milling (BM) modes are drastically different. For example, the onset temperature (As) of reverse martensitic transformation (RMT) of nanocrystal prepared by SMAT is higher than that of coarse crystal, while BM exhibits the opposite result. The elastoplastic phase field model for RMT was first proposed to investigate the correlation between RMT and its forward martensitic transformation (FMT), in turn, the above experimental phenomena were revealed in this paper. The prepared nanocrystals with the same diameter (15 nm) by SMAT and BM mode were taken as representative examples. The simulation results indicated that the strain energy is relaxed by the applied strain during FMT in nanocrystalline prepared by smaller plastic deformation mode (e.g. SMAT), but it is enhanced by larger plastic deformation mode (e.g. BM). The stored strain energy in BM mode is much larger than that in SMAT mode, and hence the driving force provided for subsequent RMT is also larger in BM mode. Therefore, the phase stability of the martensitic nanocrystals prepared by BM mode at high temperatures is much lower than the counterpart by SMAT mode, which agrees well with the experimental measurements.

A Phase Field Study of the Influence of External Loading on the Dynamics of Martensitic Phase Transformation

An elastoplastic phase field model was employed for simulations to investigate the influence of external loading on the martensitic phase transformation kinetics in steel. The phase field model incorporates external loading and plastic deformation. During the simulation process, the authenticity of the phase field model is ensured by introducing the relevant physical parameters and comparing them with experimental data. During the calculations, loads of various magnitudes and loading conditions were considered. An analysis and discussion were conducted concerning the volume fraction and phase transition temperature during the phase transformation process. The simulation results prominently illustrate the preferential orientation of variants under different loading conditions. This model can be applied to the qualitative phase transition evolution of Fe-Ni alloys, and the crystallographic parameters adhere to the volume expansion effect. It is concluded that uniaxial loading promotes martensitic phase transformation, while triaxial compressive loading inhibits it. From a dynamic perspective, it is demonstrated that external uniaxial loading accelerates the kinetics of martensitic phase transformation, with uniaxial compression being more effective in accelerating the phase transformation process than uniaxial tension. When compared to experimental data, the simulation results provide evidence that under the influence of external loading, the martensitic phase transformation is significantly influenced by the applied load, with the impact of external loading being more significant than that of plastic effects.

An elastoplastic phase-field model for dynamic fracture of nickel-based super-alloys

This paper presents a elastoplastic phase-field model for predicting crack nucleation, propagation, and branching, among other things, in elastoplastic solids subjected to dynamic loading. During the entire fracture process of the nickel-based super-alloys, with the accumulation of plastic strain, to some extent the elastic response of the material is reduced and the rate of phase-field evolution is affected, this means that the elastic strength and phase-field evolution of the material will be influenced by the yield surface and the hardening function. Therefore, we construct elastic and plastic energy degradation functions to more accurately reflect the effects of mechanical fields on the damage field, which distinguish between two successive failure mechanisms: (i) elastic strain and (ii) simultaneous elastoplastic localization deformation. In a variational framework, we derive the governing equations and the corresponding weak forms. Then, damage evolution criterion combining elastoplastic and resistance energies and a plastic yielding criterion are derived. From a numerical point of view, we give model parameters’ selection criteria. Finally, the proposed model is applied to some impact scenarios with varying impact velocities or pre-existent cracks. The comparison of numerical and experimental results shows it can be easily used for identifying material gradation profiles that manipulate crack propagation.

An elastoplastic phase-field study of the precipitation behaviors of Mg17Al12 phase in Mg-Al-based alloys: Part II. Precipitation under various loadings

An elastoplastic phase-field model, developed in Part I, was employed to study stress-assisted diffusional phase transformation, taking plastic deformation into consideration. Different loading conditions were applied to a single crystal of Mg-Al-based alloys during precipitation, aiming at investigating the effects of elastoplastic deformation associated with both phase transformation and external loadings. The simulation results show that even when external loadings are below the materials’ yield limits, plastic deformation may occur due to stress accumulation triggered by the volume expansion of β-Mg17Al12 precipitates and the lattice mismatch between the precipitate and the matrix. Under both axial and shear loading conditions, plasticity plays a significant role in the morphological evolution of β-Mg17Al12 precipitate with Burgers orientation. In contrast to the lath morphology observed in the absence of external loadings, a rhombic-like precipitate is formed when applying an external tensile loading of 30 MPa along [112̅0]α direction, while a mirror-symmetrical-parallelogram-like morphology emerges under a shear loading of + 30 MPa. Additionally, plasticity plays a similar dual role during precipitation under different loading conditions, which would promote the growth of the precipitate at the initial stage but tend to retard it afterward. These findings illustrate that the plastic strain generated under external loadings has a great impact on the precipitate microstructure, which provides guidelines for designing the precipitate microstructure to obtain better mechanical properties.

An elastoplastic phase-field study of the precipitation behaviors of Mg17Al12 phase in Mg-Al-Based alloys: Part I. Modeling and the role of plasticity

An elastoplastic phase-field model for typical diffusional transformations was developed, where plastic deformation was considered through a modified time-dependent Ginzburg-Landau dynamic equation. The model was applied to investigate the precipitation in Mg-Al-based alloys, with a focus on studying the effects of elastoplastic deformation associated with phase transformation on the morphology and growth kinetics of β-Mg17Al12 precipitates (with Burgers orientation). The elastoplastic modeling results show that the precipitates adopt a lath-shaped morphology with lozenge ends, consistent with experimental observations. Although the plastic strain accumulated in the transformation front has a moderate impact on the morphology of the precipitate, it exhibits a clear dual effect on the growth of the precipitate along [112̅0]α. Specifically, plasticity promotes the growth of the precipitate during the initial stage, owing to its stress relaxation effect, and induces a solute poor zone of atomic Al with a greater concentration gradient at the transformation front. However, the plastic strain tends to retard the growth of the precipitate afterward due to its limited ability to release the increasing compressive strain between the two phases. These results shed light on understanding the role of plasticity in general diffusional transformations and provide guidelines for tailoring the microstructure of alloys.

Simulation of ductile fracture propagation using the elastoplastic phase-field damage method calibrated by ultrasonic wave velocity measurement

The phase-field damage theory has become a powerful approach to estimate crack propagation via the finite element method (FEM). The sharp crack is smeared into a diffusive fracture propagation zone within a length scale parameter (ℓ). However, the calibration of the phase-field damage model is still difficult, especially for the length scale parameter, which is difficult to directly observe and accurately measure in laboratory testing. In most numerical studies, the length scale parameter is arbitrarily determined as the minimum element size in FEM. However, the selection of the length scale parameter may largely affect the accuracy of the phase-field model. Also, most current phase-field models only consider elasticity and few studies further consider the ductile (plastic) fracture propagation, since the plastic recoverable energy does not have a convincing form. In this study, we aim to calibrate the elastoplastic phase-field damage model based on the traditional continuum damage mechanics, with the assistance of ultrasonic wave velocity measurement. The traditional damage variable can be back-calculated from the drop of P wave velocity, following the energy equivalence hypothesis. This traditional damage variable is shown to be consistent with the phase-field damage variable as calculated in our proposed elastoplastic phase-field damage model. Therefore, the length scale parameter can be determined from fitting the rock damage evolution. The plastic flow and its corresponding plastic-free energy are considered, using a phenomenological function obeying preliminary thermodynamic assumptions. The proposed model shows good consistency with laboratory observations. Numerical implementation of the proposed model was achieved by inserting it into COMSOL Multiphysics using a weak form formulation. This paper introduces a pragmatic approach to calibrate the phase-field damage model based on laboratory testing results.

Phase-field modeling of fracture in high performance concrete during low-cycle fatigue: Numerical calibration and experimental validation

The development of an elasto-plastic phase-field model is presented which is able to predict nonlinear behavior of high performance concrete (HPC) during low-cycle fatigue. An elasto-plastic damage model which follows the Drucker–Prager yield criterion is formulated. For the modeling of unsymmetric tension–compression behavior of HPC two different continuous stepwise linearly approximated degradation functions for HPC in tension and in compression are used. The experimental three-point bending tests at low-cycle using notched beams of pure HPC are performed. Load–CMOD (crack mouth opening displacement) curves are plotted using the measured experimental data and residual stiffness–CMOD curves are calculated using these experimental Load–CMOD curves. The data used for the interpolation of degradation functions are calibrated using the experimental results for three-point bending tests. For that purpose the uniaxial tension and uniaxial compression tests and three-point bending beam test for HPC are simulated numerically. The accuracy of the proposed numerical model using the presented integration algorithm is verified by comparing the degradation of stiffness in numerical and experimental results.

Effect of hydrostatic pressure on the kinetics of alpha–omega phase transformation in zirconium

A three dimensional (3D) elastoplastic phase-field model, including strain hardening, is employed to study the effect of hydrostatic pressure in the range of 4–9 GPa on the kinetics of alpha–omega phase transformation in zirconium (Zr). The input data corresponding to pure Zr are acquired from experimental studies as well as by using the CALPHAD method. A decreasing incubation time, for the formation of omega variants, with increasing pressure is observed. Avrami (JMAK) equation is used to study the transformation kinetics by analysing the phase fraction plots predicted by the phase-field simulations. The estimated activation energy is in the range of 54–59 kJ mol−1 and decreases at an average rate of 992 J mol−1 per 1 GPa increase in pressure. The analysis of Avrami exponents, based on Cahn’s approach, show that the transformation region can be divided into two distinct regions with a change in slope, which is attributed to the site saturation. It is concluded that in the first region where the exponents are above 3, the transformation proceeds by nucleation and growth. In the second region where the exponents are sub-unity, the transformation proceeds by growth of the existing variants.

Defect-induced asymmetrical mechanical behavior in shape memory zirconia: A phase-field investigation

An elastoplastic phase-field model is used to investigate the deformation mechanisms of yttria stabilized tetragonal zirconia in presence of defects. A remarkable tension-compression asymmetry is detected. A higher strength and a lower degree of transformation are observed in compression than in tension. Also, deformation mechanism is asymmetric depending on the crystal orientation. For some cases (other cases), phase transformation is absent in tension (in compression), while both transformation and plasticity are present in compression (in tension). Such tension-compression asymmetry is attributed to activation of different monoclinic variants with different Eigen strain tensors in tension versus compression. Results also reveal a higher degree of transformation and plasticity with lower onset stresses as the void size increases. Elliptic voids exhibit a directional effect with a maximum stress intensity factor of 5.6 MPa m1/2 when the long semi-axis is diagonally oriented with respect to the loading direction, and this prediction is comparable to experiments.

Martensite formation during heating from cryogenic temperatures – A phase-field study

A 3D elastoplastic phase-field model is used to study the effect of isochronal heating on martensitic transformation in stainless steel quenched to cryogenic temperature. The results show that the slower the heating, the larger is the martensite volume fraction developed on heating. The simulated microstructures show that the transformation is driven by autocatalysis during the early stages and by coarsening of existing martensite units during later stages of the transformation. The internal stresses are mainly relaxed by autocatalysis during the initial stages of the transformation, whereas they are relaxed by plastic deformation during the later stages of the transformation. The temperature for attaining a certain martensite fraction increases with increasing heating rate, which is consistent with a thermally activated transformation. Kissinger-like analysis of the simulated transformation curves provides an activation energy of 11.9 kJ/mol.

Modeling of Reversed Austenite Formation and Its Effect on Performance of Stainless Steel Components

Abstract The kinetics of reversed austenite formation in 301 stainless steel and its effect on the deformation of an automobile front bumper beam are studied by using modeling approaches at different length scales. The diffusion-controlled reversed austenite formation is studied by using the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model, based on the experimental data. The model can be used to predict the volume fraction of reversed austenite in a temperature range of 650–750 °C. A three-dimensional elastoplastic phase-field model is used to study the diffusionless shear-type reversed austenite formation in 301 steel at 760 °C. The phase-field simulations show that reversion initiates at martensite lath boundaries and proceeds inwards of laths due to the high driving force at such high temperature. The effect of reversed austenite (RA) and martensite on the deformation of a bumper beam subjected to front and side impacts is studied by using finite element (FE) analysis. The FE simulations show that the presence of reversed austenite and martensite increased the critical speed at which the beam yielded and failed. RA fraction also affects the performance of the bumper beam.

An Asymmetric Elasto-Plastic Phase-Field Model for Shape Memory Effect, Pseudoelasticity and Thermomechanical Training in Polycrystalline Shape Memory Alloys

We propose an elasto-plastic phase-field model (PFM) to conduct the first microscopic computational study of shape memory effect (SME), pseudoelasticity, stress assisted two-way memory effect (SATWME), and thermomechanical training of CuAlBe shape memory alloy (SMA). This non-isothermal PFM model considers the effects of temperature dependent properties, latent heat, grain boundaries, and asymmetric transformation and plasticity. PFM simulations demonstrate the capacity of our model to capture the thermomechanical and purely mechanical shape recovery in the SMA. When considering transforming grain boundaries, grain refinement generates a higher transformation stress, a steeper transformation hardening, and smaller hysteresis loops for both SME and pseudoelasticity. The results also point out slightly higher transformation stress when geometrical grain boundaries are used. The simulations of SATWME highlight an augmentation of the residual martensite as the hold stress increases, which is consistent with experimental observations. This is the first PFM that can mimic the thermal training within several SATWME cycles, showing an asymptotic increase of plastic strain and the related retained transformation strain until the fourth cycle, and their stabilization thereafter. The activation of plasticity occurs always after initiation of phase transformation. Although plasticity results in more stress relaxation, it deteriorates the shape recovery for both SME and pseudoelasticity by hindering the reverse transformation. Comparison between tension and compression demonstrates the capacity of this PFM to account for, for the first time, the nonsymmetrical transformation and plastic responses of SMAs.

Analysis of Martensitic Transformation Plasticity Under Various Loadings in a Low-Carbon Steel: An Elastoplastic Phase Field Study

An elastoplastic phase field model for the martensitic transformation of a Fe-0.22C-1.58Mn-0.81Si (wt pct) alloy was developed, to investigate transformation plasticity in response to uniaxial, biaxial, shear and axial-shear loadings below half the yield strength of austenite. The simulation results clearly demonstrate the preferential orientation of martensite variants as well as plastic behavior during transformation. The data also suggest that the transformation plasticity coefficient is independent of external stress. Preferential orientation can occur under both axial and shear loading conditions, and the equivalent values of transformation plastic strains are roughly the same regardless of the stress components in the combined axial and shear loadings. Similar microstructural evolution and deformation behaviors were identified in response to both uniaxial and biaxial loadings when the uniaxial stress was equal to the difference in applied stresses along both axes during biaxial loading. The Magee mechanism is considered to play a predominant role in martensitic transformation plasticity, although both the Magee and Greenwood-Johnson mechanisms can be identified through simulations. This work demonstrates that the accumulated plastic strain in martensite is primarily inherited from the parent austenite phase, with only a negligible portion due to the yielding of martensite.

A phase-field model for non-isothermal phase transformation and plasticity in polycrystalline yttria-stabilized tetragonal zirconia

We propose an elastoplastic phase-field (PF) model to investigate the mechanics of tetragonal-to-monoclinic phase transformation (TMPT) and elastoplastic deformation of polycrystalline yttria-stabilized tetragonal zirconia (YSTZ). A Landau polynomial with non-vanishing chemical energy at the equilibrium temperature is introduced to account for the actual formation energies of the phases. The effects of different grain orientations, latent heat, and temperature on TMPT and deformation mechanisms are considered. The suppressive transformation effects of the grain boundaries (GBs) is modeled using an inhomogeneous kinetic coefficient in the bulk and GBs. The simulation results for single crystals demonstrate the capability of the model to reproduce the orientation-dependent compressive deformation of YSTZ similar to atomistic simulations and micropillar experiments. The single crystal with [100] crystallographic orientation along the loading direction (SC[100]) displays both TMPT and plasticity, SC[101] experiences only phase transformation, while SC[001] undergoes only plastic yielding. The TMPT induced by compressive loading exhibits shape memory effect (SME) below the equilibrium transformation temperature and pseudoelasticity above it, while the critical transformation stress increases with increasing loading temperature. The irrecoverable plastic strain is found to trap a part of the monoclinic phase, which prevents a complete reverse transformation. The polycrystalline cases also display SME and PE at low and high temperatures, respectively. Due to the orientation differences between grains and the stress concentrations at geometric nonlinearities, plastic deformation occurs in polycrystalline YSTZ for an applied load less than the yield stress. The results suggest a possible limitation of plasticity and an improvement of the shape recovery of YSTZ if one can control the orientation of the grains and/or increase the density of stacking faults at the GBs during material processing.

Effect of carbon on elastic properties and microstructure of maraging steel: First-principles and phase-field study

The effect of the carbon concentration on the elastic properties and martensitic microstructure of Marval X12 steel is studied using first principles calculations and phase-field simulations. The density functional theory (DFT) results predict almost no changes of the elastic moduli with an increase of C content. However, there is a sizeable change of the elastic constants due to cubic to tetragonal martensitic transformation. The elastic constants are used as input data for the 3D elastoplastic phase-field model to study the martensitic microstructure evolution in Marval X12 steel. The results, showing that lath martensite is formed in this steel, are in agreement with experiments. With increasing carbon concentration, we observe formation of mixed morphology predominantly composed of martensite laths and some martensite plates. Our results also suggest that different combinations of martensite variants are formed with increasing carbon content.

Phase-field modeling of austenite grain size effect on martensitic transformation in stainless steels

A 2D elastoplastic phase-field model is developed to study the effect of prior austenite grain size on martensitic microstructure evolution in stainless steel. The effects of strain hardening and strengthening by grain size reduction (Hall-Petch effect) have been included in the model. The results show that martensite units form in different packets oriented in different crystallographic directions in simulated coarse grains, whereas uni-directional martensitic growth is observed in simulated fine grains. The number of packets and martensite block width increase with increasing grain size. With a decreasing grain size the martensitic transformation start temperature decreases, indicating strengthening of austenite. Once the transformation is initiated, at a given time, simulated fine grains give rise to higher volume fraction of martensite compared to simulated coarse grains. The von Mises equivalent stress and plastic strain are large in simulated fine grains compared to those in simulated coarse grains. The simulation results are in good agreement with experimental results.

Understanding cementite dissolution in pearlitic steels subjected to rolling-sliding contact loading: A combined experimental and theoretical study

Cementite dissolution behavior of pearlitic steels subjected to rolling-sliding contact deformation is comprehensively investigated by combining experimental characterization and phase-field modeling. An elasto-plastic phase-field model, incorporating the elastic strain-induced free energy contribution from first-principles calculations and the plastic counterpart from a rolling-sliding contact finite element model assisted with a plastic strain accumulation model, is originally proposed to simulate the real-time evolution of cementite volume fraction, cementite morphology and carbon distribution for different rolling cycles and contact depths. Upon experimental validations, the proposed model predicts more accurate and realistic results than Sauvage’s model. A three-stage behavior of cementite dissolution is also revealed, which well explains an experimentally observed significant cementite dissolution gradient along the depth direction. Besides, the effect of ferrite/cementite interface thickness and the initial lamellae thickness of cementite on cementite dissolution kinetics is studied. The proposed phase-field model can not only help understand the mechanism of cementite dissolution, but also give new sights into quantitative predictions of the mechanical properties and even the rolling contact fatigue life of pearlitic rail steels in service.