Fig 9 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
Spinodal decomposition and coarsening (a) without volumetric mismatch between solute components, and with elastic accommodation of the volumetric mismatch between solute components, for mismatch strain coefficients of (b) ν 1 = 0.01, (c) ν 1 = 0.03, and (d) ν 1 = 0.05, at time (1) 500 s, (2) 800 s, (3) 6000 s, and (4) 10 000 s.
Source publication
open access:
https://www.sciencedirect.com/science/article/pii/S0045782520302139
A chemo-mechanical model for a finite-strain elasto-viscoplastic material containing multiple chemical components is formulated and an efficient numerical implementation is developed to solve the resulting transport relations. The numerical solution relies on inverti...
Contexts in source publication
Context 1
... × 10 11 6.0 × 10 10 2.8 × 10 10 evolution of the concentration field during the spinodal decomposition process, in the absence of mechanical deformation (i.e. setting ν 1 = 0.00), is shown in Fig. 9(a). At the early stage of the spinodal decomposition, the initial homogeneous mixture decomposes into two regions with different compositions, following their corresponding composition branch on the generalised multi-well Landau energy landscape Fig. 9(a 1 ). The decomposition stage is driven by the minimisation of the chemical free ...
Context 2
... on spinodal decomposition and coarsening is investigated by first considering the case of an elastically deforming material. The volumetric mismatch coefficient, ν 1 , is varied between 0.01 and 0.05, and the material parameters used are listed in Table 1. The temporal evolution of the concentration field during spinodal decomposition is shown in Fig. 9(b) to (d). It can be seen that the decomposition is minimally affected when the solute induced deformation is relatively small (i.e., ν 1 = 0.01), comparing Fig. 9(a) and (b). However, as the volumetric mismatch coefficient increases, the spinodal decomposition kinetics is significantly reduced, as shown in Fig. 9(c) and (d). This is due ...
Context 3
... 1 , is varied between 0.01 and 0.05, and the material parameters used are listed in Table 1. The temporal evolution of the concentration field during spinodal decomposition is shown in Fig. 9(b) to (d). It can be seen that the decomposition is minimally affected when the solute induced deformation is relatively small (i.e., ν 1 = 0.01), comparing Fig. 9(a) and (b). However, as the volumetric mismatch coefficient increases, the spinodal decomposition kinetics is significantly reduced, as shown in Fig. 9(c) and (d). This is due to the increasing elastic energy contribution to the driving force, which has the effect of suppressing the spinodal decomposition (Fig. 1). The binodal ...
Context 4
... decomposition is shown in Fig. 9(b) to (d). It can be seen that the decomposition is minimally affected when the solute induced deformation is relatively small (i.e., ν 1 = 0.01), comparing Fig. 9(a) and (b). However, as the volumetric mismatch coefficient increases, the spinodal decomposition kinetics is significantly reduced, as shown in Fig. 9(c) and (d). This is due to the increasing elastic energy contribution to the driving force, which has the effect of suppressing the spinodal decomposition (Fig. 1). The binodal compositions are also affected by the chemo-mechanical coupling, with the solute-rich binodal point decreasing from 0.93 to 0.82 when the volumetric mismatch ...
Context 5
... by the chemo-mechanical coupling, with the solute-rich binodal point decreasing from 0.93 to 0.82 when the volumetric mismatch coefficient increases up to 0.05. Furthermore, one can observe that the morphology of the solute-rich region is also affected by the deformation arising from volumetric mismatch between solute components, comparing Fig. 9(a) and (d). In the absence of mechanical coupling, the solute-rich regions exhibit a spherical morphology due to the isotropic nature of the interface energy ( Fig. 9(a 4 )), but with increasing volumetric mismatch between solute components, the solute-rich regions are observed to order themselves along the softer directions of the cubically ...
Similar publications
Ensuring of the durability and safety of critical structural elements leads to the necessity of increasing the accuracy of determining the stress-strain state, and, as a consequence, to the development and use of improved inelastic material models. One way for a more accurate description of the inelastic behavior of materials is to use of micromech...
This paper performs a comprehensive investigation on the high cycle fatigue (HCF) life prediction of turbine blade with film cooling holes. The modified theory of critical distance (MTCD) method is proposed to estimate the fatigue life of the specimen considering the notch sensitivity coefficient and multiaxial stress effect. Then, two types of spe...
![The orientations of the grains in the RVE [55]](publication/364349404/figure/fig1/AS:11431281100344709@1669345359260/The-orientations-of-the-grains-in-the-RVE-55_Q320.jpg)
Sn–Ag–Cu alloy, which is a solder widely used in the microelectronic industry, exhibits obvious viscoplasticity even at room temperature. Since its mechanical properties largely control the integrity of electronic packages and solder joints experience fatigue loads in most cases, the cyclic behaviors are investigated in the present work. Firstly, t...
![Fig 1 Grain and dislocation defect in microstructure of material [4]](publication/379320481/figure/fig1/AS:11431281231867586@1711551248294/Grain-and-dislocation-defect-in-microstructure-of-material-4_Q320.jpg)
![Listing of materials properties used in FEA analysis [18]](publication/379320481/figure/tbl1/AS:11431281231875966@1711551248620/Listing-of-materials-properties-used-in-FEA-analysis-18_Q320.jpg)
Multiscale modeling and simulation offer crucial insights for designing and analyzing metal forming, bending, welding, and casting processes, all of which are vital across automotive, aerospace, and construction industries. This paper overviews multiscale techniques used in these areas. Macroscopically, continuum-based methods like finite element a...
In grinding, the crystal grain size of the workpiece material is relatively same range compared to the removal depth. This raises a question if an anisotropic material model, which considers the effect of the crystal grain size and orientations, would better predict the process forces when compared to an isotropic material model. Initially, a simpl...
Citations
... A CALPHAD-informed phase-field framework developed by the current authors was applied to simulate the growth and composition evolution of the GB η-precipitates during cooling after solution treatment. Full details of this model can be found in [33,40,42]. In brief, the Gibbs free energy of the system is composed of an interfacial energy density, f surf , and chemical free energy density, f chem , ...
... Following [42,45], for a system with K components, the temporal and spatial evolution of the solute content i, ċ i , is expressed by, ...
... Nagaraja et al. Vector and tensor operators ⋅ Vector dot product × Cartesian product of sets, vector cross product ∶ Tensor double contraction ⊗ Tensor outer (dyadic) product (Anand, 2012;Shanthraj et al., 2020;Chen et al., 2014). To the best of the authors' knowledge, none of the aforementioned or other works consider variational principles and local-global solution strategies for the coupled initial boundary value problems of conserved chemo-mechanics in dissipative solids, which is a novel aspect and has been pointed out here for the first time. ...
... Furthermore, additional mathematical constraints are enforced, due to which the symmetry of the coupled problem is not preserved. A fully coupled chemo-mechanical model, including crystal plasticity, for multiple chemical components is presented in Shanthraj et al. (2020), where the numerical solution relies on the inversion of the constitutive model for the chemical driving force. Precisely, the chemical energy is split into convex and non-convex contributions, which are treated differently. ...
... This is obtained based on only two constitutive restrictions, namely the energetic storage and dissipation potential function. With respect to crystal plasticity, a point of departure from Shanthraj et al. (2020), Schmitt et al. (2014) is the use of regularised approximation of the non-smooth functions that describe the evolution of the plastic variables. Specifically, hyperbolic trigonometric functions that stabilise the algorithmic setting are used. ...
This article presents a relatively general framework for the description of the Cahn–Hilliard-type diffusion in solids undergoing infinitesimal elastic and plastic deformations. The coupled chemo-mechanical problem, characterised by phenomena such as phase segregation, microstructure coarsening and swelling, is treated using the variational framework which is governed by continuous-time, discrete-time and discrete-space–time incremental variational principles. It is shown that the governing equations of the coupled problem can be derived as Euler equations of minimisation and saddle point principles. A point of departure from the existing works is the coupling of crystal plasticity to the problem of diffusion and optimising the potential with respect to the plastic variables such that they are solved locally at the integration points. This is done using a return map algorithm which results in a reduced global problem. The variational framework results in a system of symmetric non-linear algebraic equations that are solved by Newton–Raphson-type iterative methods. This is a novel and attractive feature with respect to numerical implementation, as models resulting from the proposed variational framework are computationally less expensive in comparison with non-symmetric formulations. The numerical simulations presented at the end predict the applicability of models resulting from the proposed variational framework for multiple scenarios.
... Few works (Svoboda et al., 2006;Fischer and Svoboda, 2015;Mesarovic, 2016;Magri et al., 2020) formulated a complete thermodynamic framework for vacancy diffusion with continuous distributions of sources and sinks, but they focused mainly on the processes associated with dislocation climb. Other recent full-field models of coupled chemo-mechanical diffusion creep within a crystal plasticity framework include those from Chakraborty and Eisenlohr (2018, 2019), Shanthraj et al. (2020), which considered the diffusion strain rate to be purely dilatational. Consequently, a comprehensive full-field vacancy diffusion model is lacking, i. e., one that considers the combined effects of dislocation glide and dislocation climb, as well as, explicitly treats the interplay between grain boundary and bulk diffusion processes. ...
... This employs in particular the "weak" form [33,34] ψ wgr (c,c, ∇c; θ, ϕ) : ...
... Benefiting from the diffuse interface characteristic, the phasefield method [53][54][55][56][57][58][59][60][61][62] has emerged as an effective tool to study the twinning behaviour within the micro-elasticity [63][64][65][66] and plasticity framework [39,40,[67][68][69][70][71] . Kondo et al. [67] integrated a phase-field twinning model with a dislocation density-based crystal plasticity model, and investigated the stress relaxation and dislocation accommodation accompanying the twinning and detwinning process in a Mg single crystal. ...
Twin, dislocation, and grain boundary interaction in hexagonal materials, such as Mg, Ti, and Zr, has critical influence on the materials’ mechanical properties. The development of a microstructure-sensitive constitutive model for these deformation mechanisms is the key to the design of high-strength and ductile alloys. In this work, we have developed a mechanical formulation within the finite strain framework for modeling dislocation slip- and deformation twinning-induced plasticity. A dislocation density-based crystal plasticity model was employed to describe the dislocation activities, and the stress and strain distributions. The model was coupled with a multi-phase-field model to predict twin formation and twin-twin interactions. The coupled model was then employed to study twin, dislocation, and grain boundary interactions in Mg single- and polycrystals during monotonic and cyclic deformation. The results show that twin-twin interactions can enhance the strength by impeding twin propagation and growth. The role of dislocation accommodation on twin-twin interactions was twofold. Dislocation slip diminished twin-twin hardening by relieving the development of back-stresses, while it effectively relaxed the stress concentration near twin-twin intersections and thus may alleviate crack nucleation. The plastic anisotropy in each grain and the constraints imposed by the local boundary conditions resulted in stress variations among grains. This stress heterogeneity was responsible for the observed anomalous twinning behaviour. That is, low Schmid factor twins were activated to relax local stresses and accommodate the strain incompatibility, whereas the absence of high Schmid factor twins was associated with slip band-induced stress relaxation.
... This employs in particular the "weak" form [32,33] ψ wgr (c,c, ∇c; θ, φ) := 1 2 α (c −c) 2 + ψ gr (c, ∇c; θ) ...
Mn enrichment at dislocations in Fe-Mn alloys due to segregation and spinodal decomposition along the dislocation line is studied via modeling and experimental characterization. To model these phenomena, both finite-deformation microscopic phase-field chemomechanics (MPFCM) and Monte Carlo molecular dynamics (MCMD) are employed. MPFCM calibration is carried out with the same Fe-Mn MEAM-based potential used in MCMD, as well as CALPHAD data. Simulation results for Mn segregation to, and spinodal decomposition along, straight screw and edge dislocations as well as dislocation loops, are compared with characterization results from atom probe tomography (APT) for two Fe-Mn alloy compositions. In contrast to classical Volterra dislocation theory, both MPFCM and MCMD predict a non-zero hydrostatic stress field in screw cores. Being of much smaller magnitude than the hydrostatic stress in straight edge cores, much less solute segregates to screw than to edge cores. In addition, the segregated amount in screw cores is below the critical concentration of 0.157 for the onset of spinodal decomposition along the line. Results from MPFCM-based modeling imply that the concentration dependence of the solute misfit distortion and resulting dependence of the elastic energy density on concentration have the strongest effect. The maximum amount of Mn segregating to straight edge dislocations predicted by MPFCM agrees well with APT results. The MPFCM model for Fe-Mn predicts little or no variation in Mn concentration along a straight dislocation line, in contrast to the APT results. As shown by the example of a dislocation loop in the current work, a change in the hydrostatic stress along the line due to changing character of dislocation does lead to a corresponding variation in Mn concentration. Such a variation in Mn concentration can also then be expected along a dislocation line with kinks or jogs.
... Multi-physics approaches in these systems typically involve mechanical deformation, phase transformations, thermal and compositional fluctuations, and solid-state chemical reactions. Crystal plasticity 17,18 for modeling inelastic deformation at mesoscopic lengthscales, phase-field (PF) modeling for spatially-resolved moving boundary problems 19 , the Cahn-Hilliard model for component diffusion and spinodal decomposition phenomena 20,21 , as well as CALPHAD 22,23 for equilibrium thermodynamic phase diagram calculations are prominent examples of models and computational means for simulation of different phenomena involving microstructure and chemistry. A recent review of multi-scale methods in material science both in terms of historical development and current status can be found in ref. 24 . ...
... It is distributed with pre-and post-processing tools as a standalone software package under the GNU General Public License (GPLv3). Its modular structure allows solving of different field equations in a fully coupled way with a staggered approach, using Fast Fourier Transform (FFT) and Finite Element (FE) based solvers provided by the PETSc numerical library 21,118,171 . The pre-and post-processing of DAMASK facilitates coupling with other platforms such as DREAM.3D ...
... It is widely acknowledged that the interaction of the solute diffusion and elastoplastic deformation can strongly affect the microstructural evolution and the subsequent performance in many engineering materials 149,204,205 . Integration of the PF, Cahn-Hilliard, and crystal plasticity models enables the investigation of the complex role of plastic relaxation in chemomechanically coupled processes 21,23,[205][206][207][208][209] , such as the rafting behavior in Ni-based superalloys 149,[210][211][212][213][214] , lithiation of a phaseseparating electrode material 205,215 , or the hydrogen dissolution and permeation in the metallic components for hydrogen storage and distribution 207,216 . The influence of the misfit stress, localized creep strain, and anisotropy of plasticity on the rafting behavior of γ 0 precipitates in Ni-based and Co-based superalloys has been widely studied at varying lengthscales through coupling PF models with various plasticity theories, such as at the scale of individual dislocations within the PF theory 149,151,217 , dislocationdensity-based crystal plasticity model [211][212][213][214] , and the phenomenological mesoscale plasticity model 204,210 . ...
The complex interplay between chemistry, microstructure, and behavior of many engineering materials has been investigated predominantly by experimental methods. Parallel to the increase in computer power, advances in computational modeling methods have resulted in a level of sophistication which is comparable to that of experiments. At the continuum level, one class of such models is based on continuum thermodynamics, phase-field methods, and crystal plasticity, facilitating the account of multiple physical mechanisms (multi-physics) and their interaction during microstructure evolution. This paper reviews the status of simulation approaches and software packages in this field and gives an outlook towards promising research directions.
... A number of theoretical and numerical frameworks have be developed to model such behaviours with varying levels of physical detail and across multiple length scales: for instance, first principles-informed lattice kinetic Monte Carlo studies on solute clustering during the early-stage precipitation [3,14] , sharp interface modelling of precipitate growth kinetics [22] , and mean-field Ostwald ripening theories considering a statistically relevant number of precipitates [23] . Phase-field models are effective in describing the evolution of complex microstructures and solute distributions, where adjacent compositional enrichment and depletion zones interact and where elasto-plastic deformation fields arising from the phase transformation eigenstrains gain relevance [24][25][26][27][28][29][30][31] ; however, a challenge of applying such methods for the prediction of the precipitation kinetics in typical engineering alloys lies in dealing with the existence of multi-component elements and the complex multi-sublattice crystal structures of ordered phases. ...
... Furthermore, the Legendre transformation does not even exist for non-convex Gibbs free energy forms. More recently, a chemical-potential-based phase-field model has been constructed using a time-discrete semi-analytical inversion of the thermodynamic relations by the current authors [13,31] . Such an approach has the advantage that it is applicable to general CALPHAD Gibbs free energy forms of multi-component substitutional systems. ...
... L K i j are phenomenological chemical mobilities, calculated as the volume-average chemical mobilities inside different phases, α L K i j , (see details in [13] ). to achieve this chemical-potential inversion, applicable to a substitutional phase, has been recently proposed by the current authors [13,31] . In the following section, we will extend this algorithm to deal with ordered phases with two-sublattice structures, based on the semi-implicit inversion method, in the chemical-potentialbased solute transport equation. ...
The electrochemical properties of high strength 7xxx aluminium alloys strongly depend on the substitutional occupancy of Zn by Cu and Al in the strengthening η-phase with the two-sublattice structure, and its microstructural and compositional prediction is the key to design of new generation corrosion resistant alloys. In this work, we have developed a chemical-potential-based phase-field model capable of describing multi-component and two-sublattice ordered phases, during commercial multi-stage artificial ageing treatments, by directly incorporating the compound energy CALPHAD formalism. The model developed has been employed to explore the complex compositional pathway for the formation of the η-phase in Al-Zn-Mg-Cu alloys during heat treatments. In particular, the influence of alloy composition, solute diffusivity, and heat treatment parameters on the microstructural and compositional evolution of η-phase precipitates, was systematically investigated from a thermodynamic and kinetic perspective and compared to electron probe microanalysis validation data. The simulated η-phase growth kinetics and the matrix residual solute evolution in the AA7050 alloy indicates that Zn depletion mainly controlled the η-phase growth process during the early stage of ageing, resulting in fast η-phase growth kinetics, enrichment of Zn in the η-phase, and an excess in residual Cu in the matrix. The gradual substitution of Zn by Cu atoms in the η-phase during the later ageing stage was in principle a kinetically controlled process, owing to the slower diffusivity of Cu relative to Zn in the matrix. It was also found that the higher nominal Zn content in alloys like the AA7085 alloy, compared to the AA7050 alloy, could significantly enhance the chemical potential of Zn, but this had a minor influence on Cu, which essentially led to the higher Zn content (and consequently lower Cu) seen in the η-phase. Finally, substantial depletion of Zn and supersaturation of Cu in the matrix of the AA7050 alloy was predicted after 24 h ageing at 120∘C, whereas the second higher-temperature ageing stage at 180∘C markedly enhanced the diffusion of Cu from the supersaturated matrix into the η-phase, while the matrix residual Zn content was only slightly affected.
... However, with this approach, the grand-potentials of the bulk phases were obtained by a Legendre transformation of the Gibbs free energy, which does not exist for non-convex forms. More recently, a time-discrete semi-analytical inversion of the thermodynamic relations, that is applicable to general forms of the Gibbs free energy, was formulated by the current authors [44] . In the present work, this approach was extended to operate with a recognized CALPHAD database, enabling the investigation of GB precipitation and microchemistry evolution in a quaternary multi-component Al-Zn-Mg-Cu system, at length and time scales relevant to typical industrial heat treatment processes. ...
... Following [44,51] , a linear flux-force form is assumed for diffusion of each component, ...
... It is noteworthy that the up-hill diffusion arising from the miscibility gap is not expected for the current studies of GB ηprecipitation in Al-Zn-Mg-Cu alloys, based on an analysis of the chemical free energy of the solution matrix. Therefore, the energy contribution from the gradient of concentrations ( i.e. the forth order derivative with respect to concentrations in the Cahn-Hilliard model [44,52] ) is not considered in the current work. Based on the chemical free energy definition in Eqs. ...
The grain boundary (GB) microchemistry and precipitation behaviour in high-strength Al-Zn-Mg-Cu alloys has an important influence on their mechanical and electrochemical properties. Simulation of the GB segregation, precipitation, and solute distribution in these alloys requires an accurate description of the thermodynamics and kinetics of this multi-component system. CALPHAD databases have been successfully developed for equilibrium thermodynamic calculations in complex multi-component systems, and in recent years have been combined with diffusion simulations. In this work, we have directly incorporated a CALPHAD database into a phase-field framework, to simulate, with high fidelity, the complex kinetics of the non-equilibrium GB microstructures that develop in these important commercial alloys during heat treatment. In particular, the influence of GB solute segregation, GB diffusion, precipitate number density, and far-field matrix composition, on the growth of a population of GB
-precipitates, was systematically investigated in a model Al-Zn-Mg-Cu alloy of near AA7050 composition. It is shown that the GB solute distribution in the early stages of ageing was highly heterogeneous and strongly affected by the distribution of GB
-precipitates. Significant Mg and Cu GB segregation was predicted to remain during overageing, while Zn was rapidly depleted. This non-trivial GB segregation behaviour markedly influenced the resulting precipitate morphologies, but the overall precipitate transformation kinetics on a GB were relatively unaffected. Furthermore, solute depletion adjacent to the GB was largely determined by Zn and Mg diffusion, which will affect the development of precipitate free zones during the early stages of ageing. The simulation results were compared with scanning transmission electron microscopy and atom probe tomography characterisation of alloys of the similar composition, with good agreement.
... For instance, the dissolution of GB precipitates, whereby the η-phase, which is polarised anodically relative to the Al matrix, can generate micro-galvanic couples with the precipitate free zone (PFZ) [3,5], has mation of the Gibbs free energy, which does not exist for non-convex forms. More recently, a time-discrete semi-analytical inversion of the thermodynamic relations, that is applicable to general forms of the Gibbs free energy, was formulated by the current authors [44]. In the present work, this approach was extended to operate with a recognized CALPHAD database, enabling the investigation of GB precipitation and microchemistry evolution in a quaternary multi-component Al-Zn-Mg-Cu system, at length and time scales relevant to typical industrial heat treatment processes. ...
... Following [44,51], a linear flux-force form is assumed for diffusion of each component, ...
... It is noteworthy that the up-hill diffusion arising from the miscibility gap is not expected for the current studies of GB η-precipitation in Al-Zn-Mg-Cu alloys, based on an analysis of the chemical free energy of the solution matrix. Therefore, the energy contribution from the gradient of concentrations (i.e. the forth order derivative with respect to concentrations in the Cahn-Hilliard model [44,52]) is not considered in the current work. Based on the chemical free energy definition in Eq. (14) and Eq. ...
The grain boundary (GB) microchemistry and precipitation behaviour in high-strength Al-Zn-Mg-Cu alloys has an important influence on their mechanical and electrochemical properties. Simulation of the GB segregation, precipitation, and solute distribution in these alloys requires an accurate description of the thermodynamics and kinetics of this multi-component system. CALPHAD databases have been successfully developed for equilibrium thermodynamic calculations in complex multi-component systems, and in recent years have been combined with diffusion simulations. In this work, we have directly incorporated a CALPHAD database into a phase-field framework, to simulate, with high fidelity, the complex kinetics of the non-equilibrium GB microstructures that develop in these important commercial alloys during heat treatment. In particular, the influence of GB solute segregation, GB diffusion, precipitate number density, and far-field matrix composition, on the growth of a population of GB precipitates, was systematically investigated in a model Al-Zn-Mg-Cu alloy of near AA7050 composition. The simulation results were compared with scanning transmission electron microscopy and atom probe tomography characterisation of alloys of the similar composition, with good agreement.
