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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.

The reduction of iron ore with carbon-carriers is one of the largest sources of greenhouse gas emissions in the industry, motivating global activities to replace the coke-based blast furnace reduction by hydrogen-based direct reduction (HyDR). Iron oxide reduction with hydrogen has been widely investigated both experimentally and theoretically. The process includes multiple types of chemical reactions, solid state and defect-mediated diffusion (by oxygen and hydrogen species), several phase transformations, as well as massive volume shrinkage and mechanical stress buildup. In this work, a chemo-mechanically coupled phase-field (PF) model has been developed to explore the interplay between phase transformation, chemical reaction, species diffusion, large elasto-plastic deformation and microstructure evolution. Energetic constitutive relations of the model are based on the system free energy which is calibrated with the help of a thermodynamic database. The model has been first applied to the classical core-shell (w\"ustite-iron) structure. Simulations show that the phase transformation from w\"ustite to alpha-iron can result in high stress and rapidly decelerating reaction kinetics. Mechanical stresses can contribute elastic energy to the system, making phase transformation difficult. Thus slow reaction kinetics and low metallization are observed. However, if the stress becomes comparatively high, it can shift the shape of the free energy from a double-well to a single-well case and speed up the transformation. The model has been further applied to simulate an actual iron oxide specimen with its complex microstructure, characterized by electron microscopy. The simulation results show that isolated pores in the microstructure are filled with water vapor during reduction, an effect which influences the local reaction atmosphere and dynamics.

Grain boundary (GB) characteristics play an important role in the determination and prediction of material behavior, especially when it comes to nanocrystalline metals and ceramics. The main goal of this work is to develop a general interface model to accurately incorporate grain boundary sliding as well as intergranular fracture as two main phenomena in characterizing the grain boundary. To gain a deeper insight into the behavior of different grain boundaries, molecular dynamics (MD) simulations are utilized for mode I and mode II loadings. By adding the unloading path to the MD simulations it was possible to differentiate between different active mechanisms at the GB. Current MD investigations motivate a model which accounts for anisotropic plasticity and damage within the grain boundary to capture the complex interface behavior. Therefore, a two-surface formulation is utilized in which damage and plasticity at the interface are coupled in a thermodynamically consistent way. The parameters for the introduced interface model are determined using the MD simulations based on an embedded atom potential. Finally, the calibrated interface model is implemented into a cohesive zone (CZ) element. In order to show the applicability of the proposed interface model, several numerical studies are carried out. A volume element is selected which depicts a point in an arbitrary polycrystalline material at the macroscale. The results of these studies reveal interesting behaviors of the selected volume element which can be used, e.g., to determine the parameters of a continuum damage model at the macroscale.

Materials 14(7), 1787 (2021). doi:10.3390/ma14071787 special issue: "Special Issue "Strong Coupling of Thermo-Chemical and Thermo-Mechanical States in Applied Materials" / Special Issue Editor: Prof. Ingo Steinbach, Guest Editor", Published by MDPI, Basel

The purpose of this work is the development and determination of higher-order continuum-like kinematic measures which characterize discrete kinematic data obtained from experimental measurement (e.g., digital image correlation) or kinematic results from discrete modeling and simulation (e.g., molecular statics, molecular dynamics, or quantum DFT). From a continuum point of view, such data or results are in general non-affine and incompatible, due for example to shear banding, material defects, or microstructure. To characterize such information in a (pseudo-) continuum fashion, the concept of discrete local deformation is introduced and exploited. The corresponding measures are determined in a purely discrete fashion independent of any relation to continuum fields. Demonstration and verification of the approach is carried out with the help of example applications based on non-affine and incompatible displacement information. In particular, for the latter case, molecular statics results for the displacement of atoms in and around a dislocation core in fcc Au are employed. The corresponding characterization of lattice distortion in and around the core in terms of higher-order discrete local deformation measures clearly shows that, even in the simplest case of planar cores, such distortion is only partly characterized by the Nye tensor., Comment: 30 pages, 19 figures

Combined atomic-scale characterization and simulation reveal the complexity and diversity of the chemical nature of dislocations., Dislocations are one-dimensional defects in crystals, enabling their deformation, mechanical response, and transport properties. Less well known is their influence on material chemistry. The severe lattice distortion at these defects drives solute segregation to them, resulting in strong, localized spatial variations in chemistry that determine microstructure and material behavior. Recent advances in atomic-scale characterization methods have made it possible to quantitatively resolve defect types and segregation chemistry. As shown here for a Pt-Au model alloy, we observe a wide range of defect-specific solute (Au) decoration patterns of much greater variety and complexity than expected from the Cottrell cloud picture. The solute decoration of the dislocations can be up to half an order of magnitude higher than expected from classical theory, and the differences are determined by their structure, mutual alignment, and distortion field. This opens up pathways to use dislocations for the compositional and structural nanoscale design of advanced materials.

In the continuum context, the displacements of atoms induced by a dislocation can be approximated by a continuum disregistry field. In this work, two phase-field (PF)-based approaches and their var...

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The interactions of solutes with crystal defects at near-atomic-level were investigated in five single-crystal nickel-based superalloys deformed at temperatures between 850 and 1160 °C and various deformation conditions. These interactions, and consequently the composition of a particular solute that segregates at a crystal defect, are controlled by the type of the crystal defect, the deformation conditions, i.e., temperature and stress, and the overall alloy composition. Atomistic phase-field simulations also reveal the effect of dislocation velocity on the amount of solutes that can segregate on dislocations. The observed plasticity-assisted redistribution of interacting solutes phenomena results in microstructural and chemical alterations, which are associated with recrystallization, rafting, and the formation of topologically close-packed phases. Deciphering these interactions by enabling quantitative three-dimensional imaging of solutes at crystal defects with high sensitivity and spatial resolution will allow to develop a solute–defect database that can be used as a key-design parameter for advanced superalloys.

With 1.85 billion tons produced per year, steel is the most important material class in terms of volume and environmental impact. While steel is a sustainability enabler, for instance through lightweight design, magnetic devices, and efficient turbines, its primary production is not. For 3000 years, iron has been reduced from ores using carbon. Today 2.1 tons CO2 are produced per ton of steel, causing 30% of the global CO2 emissions in the manufacturing sector, which translates to 6.5% of the global CO2 emissions. These numbers qualify iron- and steel-making as the largest single industrial greenhouse gas emission source. The envisaged future industrial route to mitigate these CO2 emissions targets green hydrogen as a reductant. Although this reaction has been studied for decades, its kinetics is not well understood, particularly during the wustite reduction step which is dramatically slower than the hematite reduction. Many rate-limiting factors of this reaction are set by the micro- and nanostructure as well as the local chemistry of the ores. Their quantification allows knowledge-driven ore preparation and process optimization to make the hydrogen-based reduction of iron ores commercially viable, enabling the required massive CO2 mitigation to ease global warming. Here, we report on a multi-scale structure and composition analysis of iron reduced from hematite with pure H2, reaching down to near-atomic scale.

The interplay of interface and bulk dislocation nucleation and glide in determining the motion of twin boundaries, slip-twin interaction, and the mechanical (i.e., stress-strain) behavior of fcc metals is investigated in the current work with the help of molecular dynamics simulations. To this end, simulation cells containing twin boundaries are subject to loading in different directions relative to the twin boundary orientation. In particular, shear loading of the twin boundary results in significantly different behavior than in the other loading cases, and in particular to jerky stress flow. For example, twin boundary shear loading along &lang, 112 &rang, results in translational normal twin boundary motion, twinning or detwinning, and net hardening. On the other hand, such loading along &lang, 110 &rang, results in oscillatory normal twin boundary motion and no hardening. As shown here, this difference results from the different effect each type of loading has on lattice stacking order perpendicular to the twin boundary, and so on interface partial dislocation nucleation. In both cases, however, the observed stress fluctuation and "jerky flow" is due to fast partial dislocation nucleation and glide on the twin boundary. This is supported by the determination of the velocity and energy barriers to glide for twin boundary partials. In particular, twin boundary partial edge dislocations are significantly faster than corresponding screws as well as their bulk counterparts. In the last part of the work, the effect of variable twin boundary orientation in relation to the loading direction is investigated. In particular, a change away from pure normal loading to the twin plane toward mixed shear-normal loading results in a transition of dominant deformation mechanism from bulk dislocation nucleation/slip, to twin boundary motion.

Acta materialia 175, 250-261 (2019). doi:10.1016/j.actamat.2019.06.008, Published by Elsevier Science, Amsterdam [u.a.]

Predicting the structural response of advanced multiphase alloys and understanding the underlying microscopic mechanisms that are responsible for it are two critically important roles that modeling plays in alloy development. The demonstration of superior properties of an alloy, such as high strength, creep resistance, high ductility, and fracture toughness, is not sufficient to secure its use in widespread applications. Still, a good model is needed to take measurable alloy properties, such as microstructure and chemical composition, and forecast how the alloy will perform in specified mechanical deformation conditions, including temperature, time, and rate. Here, we highlight recent achievements using multiscale modeling in elucidating the coupled effects of alloying, microstructure, and mechanism dynamics on the mechanical properties of polycrystalline alloys. Much of the understanding gained by these efforts relies on the integration of computational tools that vary over many length scales and time scales, from first-principles density functional theory, atomistic simulation methods, dislocation and defect theory, micromechanics, phase-field modeling, single crystal plasticity, and polycrystalline plasticity.

Steel is the most important material class in terms of volume and environmental impact. While it is a sustainability enabler, for instance through lightweight design, magnetic devices, and efficient turbines, its primary production is not. Iron is reduced from ores by carbon, causing 30% of the global CO2 emissions in manufacturing, qualifying it as the largest single industrial greenhouse gas emission source. Hydrogen is thus attractive as alternative reductant. Although this reaction has been studied for decades, its kinetics is not well understood, particularly during the wustite reduction step which is much slower than hematite reduction. Some rate-limiting factors of this reaction are determined by the microstructure and local chemistry of the ores. Here, we report on a multi-scale structure and composition analysis of iron reduced from hematite with pure H2, reaching down to near-atomic scale. During reduction a complex pore- and microstructure evolves, due to oxygen loss and non-volume conserving phase transformations. The microstructure after reduction is an aggregate of nearly pure iron crystals, containing inherited and acquired pores and cracks. We observe several types of lattice defects that accelerate mass transport as well as several chemical impurities (Na, Mg, Ti, V) within the Fe in the form of oxide islands that were not reduced. With this study, we aim to open the perspective in the field of carbon-neutral iron production from macroscopic processing towards better understanding of the underlying microscopic transport and reduction mechanisms and kinetics.

The purpose of the current work is the theoretical and computational comparison of selected models for the energetics of dislocation dissociation resulting in stacking fault and partial dislocation (core) formation in fcc crystals as based on the (generalized) Peierls–Nabarro (GPN: e.g., Xiang et al., 2008 , Shen et al., 2014 ), and phase-field (PF: e.g., Shen and Wang, 2004 , Hunter et al., 2011 , Hunter et al., 2013 , Mianroodi and Svendsen, 2015 ), methodologies (e.g., Wang and Li, 2010 ). More specifically, in the current work, the GPN-based model of Xiang et al. (2008) is compared theoretically with the PF-based models of Shen and Wang (2004) , Hunter et al., 2011 , Hunter et al., 2013 , and Mianroodi and Svendsen (2015) . This is carried out here with the help of a unified formulation for these models via a generalization of the approach of Cahn and Hilliard (1958) to mechanics. Differences among these include the model forms for the free energy density ψ ela of the lattice and the free energy density ψ sli associated with dislocation slip. In the PF-based models, for example, ψ ela is formulated with respect to the residual distortion H R due to dislocation slip (e.g., Khachaturyan, 1983 , Mura, 1987 ), and with respect to the dislocation tensor curl H R in the GPN model (e.g., Xiang et al., 2008 ). As shown here, both model forms for ψ ela are in fact mathematically equal and so physically equivalent. On the other hand, model forms for ψ sli differ in the assumed dependence on the phase or disregistry fields ϕ , whose spatial variation represents the transition from unslipped to slipped regions in the crystal. In particular, Xiang et al. (2008) and Hunter et al., 2011 , Hunter et al., 2013 work with ψ sli ( ϕ ) . On the other hand, Shen and Wang (2004) and Mianroodi and Svendsen (2015) employ ψ sli ( ϕ , ∇ ϕ ) . To investigate the consequences of these differences for the modeling of the dislocation core, dissociation, and stacking fault formation, predictions from the models of Hunter et al., 2011 , Hunter et al., 2013 and Mianroodi and Svendsen (2015) are compared with results from molecular statics (MS) for the deformation field of dissociated edge and screw dipoles in Al and Au. Particularly notable is the agreement of the MS and PF strain field results for the case of perfect screw dissociation which, in contrast to the edge case, are characterized by asymmetric displacement and strain fields. The degree of this asymmetry is apparently related to the corresponding anisotropy ratio. As well, comparison of MS and PF disregistry fields implies that the gradient dependence of ψ sli results in a broadening of the (otherwise too narrow) disregistry profile to the form predicted by MS.

The purpose of this work is the development of a two-scale phase-field-based computational model for coupled microstructure evolution and macroscopic mechanical material behavior. To this end, the mechanical behavior of the macroscopic continuum is based on a model for each of its material elements or points as a unit cell (UC) whose average properties determine those of the corresponding material point. The UC model itself is based on materially inhomogeneous elasticity, with material inhomogeneity resulting from microstructure evolution in the UC, driven in particular by stress relaxation. Since the focus of the current work is on computational aspects, microstructure evolution is restricted here for simplicity to bulk structural rearrangement (e.g., austenite–martensite transformation). Consequently, the corresponding phase fields are non-conservative. As well for simplicity, the current formulation is restricted to geometric linearity. The algorithmic formulation and numerical solution of the microscopic initial–boundary-value problem (IBVP) for mechanical equilibrium and relaxational dynamics in the UC is based on Green function and (fast) Fourier transform (FFT) methods. On the other hand, the algorithmic formulation and numerical solution of the macroscopic BVP for quasi-static macroscopic mechanical equilibrium is based on finite element (FE) methods. To demonstrate its capabilities, the current two-scale computational approach is applied in the last part of the work to the modeling and simulation of single- and polycrystalline UCs undergoing structural rearrangement from austenite into multiple martensite variants during loading of the macroscopic continuum.

We propose a new phase-field damage formulation which takes into account anisotropic damage evolution in solids. Such anisotropy projects itself in fracture energy values which depend on the direction of the crack surface. Therefore, instead of one constant scalar parameter for the fracture energy value, we use a direction-dependent fracture energy function. By incorporating a direction-dependent fracture energy function, only a single damage variable as well as a first order damage gradient need to be used within the standard phase-field damage model. This is in contrast to other available anisotropic phase-field models which typically use multiple variables or higher order gradient terms. To obtain values for the fracture energy function, atomistic calculations are performed. Here, molecular static simulations are utilized to calculate the energy of free surfaces within an Aluminum crystal. As a result, we report the fracture energy value as a function of the surface orientation. The obtained fracture energy function is passed directly to the phase-field damage formulation to investigate transgranular fracture within a single crystalline. Moreover, the grain boundary is represented via a cohesive zone model to take into account intergranular fracture in a bi-crystalline structure. The predicted crack path is in good agreement with obtained results from molecular dynamics simulations. Finally, by calibrating the length scale parameter in the phase-field damage model, it is possible to compare the reaction forces from finite element calculations with atomistic ones.

Many elementary deformation processes in metals involve the motion of dislocations. The planes of glide and specific processes dislocations prefer depend heavily on their atomic core structures. Atomistic simulations are desirable for dislocation modeling but their application to even sub-micron scale problems is in general computationally costly. Accordingly, continuum-based approaches, such as the phase-field microelasticity, phase-field dislocation dynamics (PFDD), generalized Peierls–Nabarro (GPN) models, and the concurrent atomistic–continuum (CAC) method, have attracted increasing attention in the field of dislocation modeling because they well represent both short-range cores interactions and long-range stress fields of dislocations. To better understand their similarities and differences, it is useful to compare these methods in the context of benchmark simulations and predictions. In this paper, we apply the CAC method and different PFDD variants – one of them is equivalent to a GPN model – to simulate an extended (i.e., dissociated) dislocation in Al with initially pure edge or pure screw character in terms of the disregistry. CAC and discrete forms of PFDD are also employed to calculate the Peierls stress. By conducting comprehensive convergence studies, we quantify the dependence of these measures on time/grid resolution and simulation cell size. Several important but often overlooked differences between PFDD/GPN variants are clarified. Our work sheds light on the advantages and limitations of each method, as well as the path towards enabling them to effectively model complex dislocation processes at larger length scales.

Understanding and modeling the interface behavior is an important task for predicting materials response in various applications. To formulate the behavior of an arbitrary interface, one needs to construct the relation between acting tractions and displacement jumps at the interface. In addition to capturing the correct physics of the interface, the so-called traction–separation relation must also be thermodynamically consistent and satisfy the basic balance laws. Apart from many attempts in the literature to address these issues, a new and simple method to capture the complex mechanical behavior at an arbitrary interface is proposed. The new formulation is based on introducing a new quantity called “traction density”. As a result, the traction–separation relation for any arbitrary interface is automatically computed by integrating the traction density over the interface. The traction density can be formulated based on understandings and observations from lower scales. As will be shown, the mathematical representation of the traction density is relatively simple and therefore its consistency can be verified easily. When it comes to the grain boundary (GB) behavior, the proposed methodology is able to represent not only intergranular fracture but also grain boundary sliding . For calibration and verification of the model, molecular dynamics (MD) simulations for aluminum Σ 5 GB are utilized. Interestingly, the calculations from current MD simulations show size-dependent behavior for the GB. By introducing a healing parameter in the new interface model, it is now possible to explain and predict possible GB size-dependent behavior.

Advanced engineering materials (2020). doi:10.1002/adem.201901044, Published by Deutsche Gesellschaft für Materialkunde, Frankfurt, M.

Mixed-type dislocations are prevalent in metals and play an important role in their plastic deformation. Key characteristics of mixed-type dislocations cannot simply be extrapolated from those of dislocations with pure edge or pure screw characters. However, mixed-type dislocations traditionally received disproportionately less attention in the modeling and simulation community. In this work, we explore core structures of mixed-type dislocations in Al using three continuum approaches, namely, the phase-field dislocation dynamics (PFDD) method, the atomistic phase-field microelasticity (APFM) method, and the concurrent atomistic-continuum (CAC) method. Results are benchmarked against molecular statics. The PFDD and APFM methods are advanced in several aspects such that they can better describe the dislocation core structure. In particular, in these two approaches, the gradient energy coefficients for mixed-type dislocations are determined based on those for pure-type ones using a trigonometric interpolation scheme, which is shown to provide better prediction than a linear interpolation scheme. The dependence of in-slip-plane spatial numerical resolution in FPDD and CAC is also quantified.

In a number of fcc materials such as copper or aluminum, as well as more complex materials such as twinning induced plasticity (TWIP) steels, the interaction between dislocations and other defects such as stacking faults or twins plays an important role in the hardening behavior of such materials. Interactions of dislocation and twin or stacking fault layers have been studied in this work using molecular dynamics. Depending on the material and the loading conditions, possible interaction modes include (i) penetration of the dislocation into the faulted layer, (ii) reduction of the faulted layer after interaction, (iii) growth of the faulted layer after interaction. Such studies up to this point have been performed without temperature control near zero K (0 to 2 K). In this work, we extend the previous studies to higher temperature with the help of two methods, both based on molecular dynamics (MD) modeling. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic and static fracture properties of Graphene Sheets (GSs) and Carbon nanotubes (CNTs) with different sizes are investigated based on an empirical inter-atomic potential function that can simulate nonlinear large deflections of nanostructures. Dynamic fracture of GSs and CNTs are studied based on wave propagation analysis in these nanostructures in a wide range of strain-rates. It is shown that wave propagation velocity is independent from strain-rate while dependent on the nanostructure size and approaches to 2.2 × 104 m/s for long GSs. Also, fracture strain shows extensive changes versus strain-rate, which has not been reported before. Fracture stress is determined as 115 GPa for GSs and 122 GPa for CNTs which are independent from the strain-rate; in contrast to the fracture strain. Moreover, fracture strain drops at extremely high strain-rates for GSs and CNTs. These features are considered as capability of carbon nanostructures for reinforcing nanocomposites especially under impact loadings up to high strain-rates.

The purpose of the current work is the development of a phase field model for dislocation dissociation, slip and stacking fault formation in single crystals amenable to determination via atomistic or ab initio methods in the spirit of computational material design. The current approach is based in particular on periodic microelasticity ( Wang and Jin, 2001 , Bulatov and Cai, 2006 , Wang and Li, 2010 ) to model the strongly non-local elastic interaction of dislocation lines via their (residual) strain fields. These strain fields depend in turn on phase fields which are used to parameterize the energy stored in dislocation lines and stacking faults. This energy storage is modeled here with the help of the ”interface” energy concept and model of Cahn and Hilliard (1958) (see also Allen and Cahn, 1979 , Wang and Li, 2010 ). In particular, the “homogeneous” part of this energy is related to the “rigid” (i.e., purely translational) part of the displacement of atoms across the slip plane, while the “gradient” part accounts for energy storage in those regions near the slip plane where atomic displacements deviate from being rigid, e.g., in the dislocation core. Via the attendant global energy scaling, the interface energy model facilitates an atomistic determination of the entire phase field energy as an optimal approximation of the (exact) atomistic energy; no adjustable parameters remain. For simplicity, an interatomic potential and molecular statics are employed for this purpose here; alternatively, ab initio (i.e., DFT-based) methods can be used. To illustrate the current approach, it is applied to determine the phase field free energy for fcc aluminum and copper. The identified models are then applied to modeling of dislocation dissociation, stacking fault formation, glide and dislocation reactions in these materials. As well, the tensile loading of a dislocation loop is considered. In the process, the current thermodynamic picture is compared with the classical mechanical one as based on the Peach-Kohler force.

The nonlinear vibrational properties of single layer graphene sheets (SLGSs) are investigated using a membrane model. The nonlinear equation of motion is considered for the SLGSs by including the effects of stretching due to large amplitudes. The equation of motion is numerically solved utilizing the finite difference method for SLGSs with different initial and boundary conditions, sizes and pretensions. It is concluded that the nonlinear fundamental frequency of SLGSs increases by increasing the pretension and initial velocity. In addition, it is observed that an increase in the pretension weakens the effects of the initial velocity on the fundamental frequency, such that the fundamental frequency approximately becomes independent of the initial velocity. This is an important feature of the vibrating systems consisting of SLGSs which are used in the nano-electromechanical systems (NEMS), where resonators with a specific fundamental frequency and independent of the initial velocity are of interest.