Homologous Temperature Research Articles

Intergranular oxidation behaviour of Ni-Cr-Al model alloys in 480 °C hydrogenated steam

The oxidation of Ni-Cr-Al model alloys was investigated after exposure to a 480 °C hydrogenated steam environment. Low angle grain boundaries (LAGBs) and the studied coincidence site lattice grain boundaries (CSLBs) showed intergranular oxidation, while high angle grain boundaries (HAGBs) were protected by an external oxide. The intergranular oxide consisted of a central Al-rich oxide, surrounded by a Cr-rich oxide. Al contribution in intergranular and external oxidation was enhanced in Ni-Cr-Al alloys compared to Ni-Al alloys. Thus, the third element effect is applicable at low homologous temperature along short-circuit diffusion pathways, in this case grain boundaries.

Characterizing stable nanocrystalline Cu-Ta behavior and failure dynamics under extremes of strain rate, strain, temperature and pressure by modified dynamic tensile extrusion

This work presents characterization of thermo-mechanically stable, bulk nanocrystalline (NC) Cu-Ta alloys under a combination of extreme states using high-temperature, dynamic tensile extrusion experiments. The NC Cu-Ta was subjected to high dynamic strain rates up to 105/s, large plastic strains of 100–500 %, and high homologous temperatures greater than 50 %TM. This combination of thermo-mechanical conditions provides a unique look at how these extremely stable microstructures can accommodate and reconfigure under situations where dislocation activity is the most extreme, given constitutive relationships between strain rate, dislocation nucleation, and plastic work. Elevated temperature tests results reveal NC Cu-1Ta alloys are capable of forming a coherent jet, (25 mm in length by 1.8 mm in diameter) as it accelerates out of the die after being dynamically extruded. Microstructural analysis reveals that unlike previous reports on conventional un-stabilized nano or near nano-grained Cu, NC Cu-Ta alloys behave differently. They do not undergo grain coarsening via a nucleation and growth process, but rather behave identically to that of coarse-grained Cu. The NC grains elongate along the extrusion direction and at the same time shrink in diameter and develop a <111> type texture. Ultimately, the occurrence of premature dynamic failure was attributed to the inherent constraints posed by processing defects during synthesis.

Investigation on wear-resistance of nanocrystalline Pt-Au by molecular dynamics simulations

In this study, molecular dynamics simulation was used to investigate the wear-resistance behavior in nanocrystalline Pt-Au under a hard grinding ball. A comprehensive analysis was performed regarding the variation of wear volume and rate with sliding distance, frictional forces, and defect evolution at different temperatures. The effect of adding atomic concentrations of Au as solute segregation at GBs has also been investigated subsequently. It was found that the segregation of Au to the GBs of nanocrystalline Pt demonstrates improvement in the slab's anti-wear behavior compared to the pure Pt samples across all the temperatures, and it was concluded that the underlying wear-mechanism is adhesion-dominated across all the samples, and that diffusion-driven phenomenon might be an influencing factor at homologous temperature.

Development and performance study on low strength and high rockburst tendency similar simulation material of coal

The current similar simulation materials (SSM) are featured with the contradiction between low strength and high rockburst tendency (RT), thus being unable to simultaneously realize the qualitative rockburst of small size specimen and the impact damage restoration of large size model. In this case, this study developed a new SSM of low strength and high RT by using water glass (WG) as cement, talcum powder (TP) as aggregate and adding curing agent sodium silicofluoride (SSF). Under the requirements of initial set volume shrinkage rate (IVSR), long-term volume decay rate (LVDR), uniaxial compressive strength (UCS) and modulus index (MI), the optimal ratios of SSF and TP were obtained 20% and 70% by orthogonal experiment and multiple linear regression method, respectively. Micro-macroscopic experiments such as uniaxial compression, scanning electron microscope (SEM), acoustic emission (AE) and Infrared thermal imager were utilized to monitor in real time the energy-temperature-strength synergistic change characteristics and similarities between this material and the coal with a strong RT. The experimental results show that: the UCS value of this material is 6.11 MPa and its MI is 11.57, which meets the requirements for low strength and high RT. Meanwhile it satisfies a certain similar proportionality between its physical parameters and those of coal. In the whole process of uniaxial compression, there are energy and temperature homologous changes in the characteristics of consistency, and both materials conform to the stress–strain phase law. The infrared radiation temperature displays a trend of “slowly decreasing-gradually warming-continuously cooling-suddenly warming-sharply cooling”. The AE energy count rate shows a trend of “calm transition and sudden increase before and after the stress peak”. All are characterized by strong RT. There is still some difference between the two specimens during energy and temperature changes, due to their different molding characteristics and load-bearing capacity. The feasibility of the material was confirmed in terms of physical parameters, degree of rockburst and homologous energy and temperature characteristics. Meanwhile, this study revealed the multi-parameter precursor characteristics of rockburst damage in the specimen, which can lay a foundation for the prevention and study of impact ground pressure disaster.

Extraordinary superplasticity at low homologous temperature and high strain rate enabled by a multiphase nanocrystalline network

Superplasticity is a highly sought-after property of components manufactured with complex geometries in metal forming processes. However, superplasticity usually occurs at high temperatures and/or low strain rates, which entails high energy consumption, long processing time, and severe surface oxidation. Herein, we have developed a multiphase nanocrystalline network (MPNN) in a Ti6Al4V5Cu model alloy, where the grain boundary β phases promote the sliding and rotation of ultrafine α grains, while the nanosized Ti2Cu particles pin down the α/β boundaries to maintain the thermostability of the nanostructure. Results show that the onset temperature for superplasticity of the model alloy is 250 °C lower than that of the Ti6Al4V alloy at the strain rate of 10−4s − 1. Remarkably, superplasticity was also observed at an extremely high strain rate of 1 s − 1 at 750 °C, which is 2–4 orders of magnitude larger than conventional superplastic metals. The present work is of great significance in developing more economical and efficient superplastic deformation processes.

N-body potential for simulating lattice defects and diffusion in copper

We present a new classical interatomic potential for molecular dynamics simulations of copper. The potential was developed within the N-body approach, which, in addition to pair and many-body interactions, can take into account three-body ones with the required precision. Our potential accurately predicts a wide range of properties of copper: structural, elastic, point and planar, and thermal such as melting point. The latter is essential for the proper reproduction of the homologous temperature and comparison of the calculated results with the corresponding experimental data, such as diffusion coefficients. The potential predicts the intrinsic stacking fault energy within the range of experimental measurements, which indicates its ability to correctly describe the deformation behavior of copper. In this work, we performed simulations of lattice and grain-boundary diffusion of copper with our potential. It shows a moderate overestimation of the lattice diffusivity with a deviation less than one order of magnitude. To simulate the grain boundary diffusion, we built a general-type grain boundary employing the original method. The calculated grain-boundary diffusivity is in remarkable agreement with the literature data and almost coincides with them at higher temperatures in the considered temperature range. The constructed potential can be effectively used for molecular dynamics simulations of defect properties and diffusion phenomena and composing the interatomic potentials within the N-body approach for alloys containing copper.

Electrochemical Synthesis of Nanostructured Ordered Intermetallic Materials under Ambient Conditions.

ConspectusThe enhanced catalytic properties of alloy nanostructures have made them a focus of extensive research in the field of catalysis. Alloy nanostructures can be classified into two types: disordered alloys (also known as solid solutions) and ordered intermetallics. The latter are of particular interest as they possess long-range atomic scale ordering, which leads to well-defined active sites that can be used to accurately assess structure-property relationships and their impact on (electro)catalytic performance.While many ordered intermetallics (OICs) have been synthesized and evaluated as electrocatalysts, there is still a lack of understanding on how the local structure of atoms controls their catalytic performance. Ordered intermetallics are difficult to synthesize and often require high-temperature annealing for the atoms to equilibrate into ordered structures. High temperature processing results in aggregated structures (usually >30 nm) and/or contamination from the support, which can decrease their performance and preclude these materials from being used as model systems for elucidating insight into structure and electrochemical properties. Therefore, alternative methods are required to enable more efficient atomic ordering while maintaining some level of morphological control.This Account delves into the potential of electrochemical methods as a practical alternative for synthesizing ordered intermetallics at lower temperatures. Specifically, it explores the viability of electrochemical dealloying and electrochemical deposition to synthesize Pd-Bi and Cu-Zn intermetallics at room temperature and atmospheric pressure. These methods have proven useful in synthesizing phases that are typically inaccessible under ambient conditions. The high homologous temperatures at which these materials are synthesized provide the necessary atomic mobility required for equilibration and formation of ordered phases, thus making the direct synthesis of ordered intermetallic materials at room temperature by electrochemical means a reality.Beyond synthesis, the electrocatalytic performance of these intermetallics was assessed for the oxygen reduction reaction (ORR), which is an important process employed in fuel cells. The OICs displayed increased performance with respect to commercial Pd/C and Pt/C benchmarks because of lower coverages of spectator species. Furthermore, these materials exhibited improved methanol tolerance.This Account provides valuable insights into the electrochemical synthesis of ordered intermetallics and their potential use as highly effective catalysts for electrocatalytic reactions. By using electrochemical methods, it is possible to obtain ordered intermetallics with unique atomic arrangements and tailored properties, which can be optimized for specific catalytic applications. With further research, electrochemical synthesis methods may enable the development of new and improved ordered intermetallics with even higher catalytic activity and selectivity, making them ideal candidates for use in a wide range of industrial processes. Furthermore, the ability to access intermetallics under milder conditions may accelerate the ability to use these materials as model systems for revealing fundamental insight into electrocatalyst structure and function.

Sluggish dendrite growth in undercooled Fe-Co-Ni-Si multi-principal element intermetallics

The (CoFe)50Si50 and (CoFeNi)50Si50 multi-principal element intermetallics (MPEIs) with a B2 ordering structure were synthesized by arc melting, undercooled using the glass fluxing technique and observed in-situ by a high-speed high-resolution camera. A sharp increase of the dendrite growth velocity was found for (CoFe)50Si50 and (CoFeNi)50Si50 at the undercooling of ΔT = 250 K and ΔT = 225 K, respectively, related to the disorder trapping that occurs at high undercooling. As a result, low-ordered regions were found by HR-TEM in highly undercooled (CoFe)50Si50 and (CoFeNi)50Si50 alloys. Compared with the previous reported binary CoSi, the maximum dendrite growth velocity is decreased by nearly one order for the (CoFeNi)50Si50 MPEIs. Such sluggish growth phenomenon does exist in present MPEIs for both cases of the absolute and the homologous temperature scale. It is suggested that the number of components might dominate sluggish dendrite growth of the B2 crystal and the effect of specific elements cannot be omitted. This study reported the dendrite growth behavior of MPEIs, which is helpful for understanding the non-equilibrium solidification phenomena, thereby providing a reference for controlling the non-equilibrium solidification process.

Rheology of Hexagonal Close‐Packed (hcp) Iron

AbstractThe viscosity of hexagonal close‐packed (hcp) Fe is a fundamental property controlling the dynamics of the Earth's inner core. We studied the rheology of hcp‐Fe using high‐pressure and ‐temperature deformation experiments with in situ stress and strain measurements. Experiments were conducted using D111‐type and deformation‐DIA apparatuses at pressures of 16.3–22.6 GPa, temperatures of 423–923 K, and uniaxial strain rates of 1.52 × 10−6 to 8.81 × 10−5 s−1 in conjunction with synchrotron radiation. Experimental results showed that power‐law dislocation creep with a stress exponent of n = 4.0 ± 0.3, activation energy of E* = 240 ± 20 kJ/mol, and activation volume of V* = 1.4 ± 0.2 cm3/mol is dominant deformation mechanism at &gt;∼800 K, whereas a mechanism with power‐law breakdown prevails at lower temperatures. An extrapolation of the power‐law dislocation creep flow law based on homologous temperature scaling suggests the viscosity of hcp‐Fe under inner core conditions is ≥∼1019 Pa s. If this power‐law dislocation creep mechanism is assumed to be the dominant mechanism in the Earth's inner core, the equatorial growth or translation mode mechanism may be the dominant geodynamical mechanism causing the observed inner core structure.

Thermodynamic properties on the homologous temperature scale from direct upsampling: Understanding electron-vibration coupling and thermal vacancies in bcc refractory metals

We have calculated thermodynamic properties of four bcc refractory elements---V, Ta, Mo, and W---up to the melting point with full density-functional-theory accuracy, using the recently developed direct-upsampling method [J. H. Jung et al., npj Comput. Mater. 9, 3 (2023)]. The direct-upsampling methodology takes full account of explicit anharmonic vibrations and electron-vibration coupling very efficiently using machine-learning potentials. We have thus been able to compute highly converged free-energy surfaces for the PBE exchange-correlation functional, from which accurate temperature dependencies of various thermodynamic properties such as the heat capacity, thermal expansion coefficient, and bulk modulus are accessible. For all four elements, the electronic contribution is large, including a strong coupling with the thermal vibrations. The atomic forces in W are even affected by the temperature-consistent Fermi broadening, which alters the free energy by around 3 meV/atom at the melting point. Trends within group V and group VI refractory elements are observed and explained by qualitative differences in the electronic density of states. We also provide an estimate of the Gibbs energies of vacancy formation and the vacancy contribution to the thermodynamics. Lastly and most importantly, our results are analyzed in terms of the homologous temperature scale relative to theoretically predicted melting points (for the PBE functional). The homologous temperature dependencies show a remarkable agreement with experiments and reveal the predictive power of self-consistently determined ab initio thermodynamic properties.

Advanced Statistical Crystal Plasticity Model: Description of Copper Grain Structure Refinement during Equal Channel Angular Pressing

The grain structure of metals changes significantly during severe plastic deformation (SPD), and grain refinement is the main process associated with SPD at low homologous temperatures. Products made of ultrafine-grained materials exhibit improved performance characteristics and are of considerable industrial interest, which generates a need for the creation of comprehensive grain refinement models. This paper considers the integration of the ETMB (Y. Estrin, L.S. Toth, A. Molinari, Y. Brechet) model, which describes the evolution of an average cell size during deformation into the two-level statistical crystal plasticity constitutive model (CM) of FCC polycrystals. The original relations of the ETMB model and some of its modifications known from the literature were analyzed to obtain an accurate, physically admissible description of the grain refinement process. The characteristics of the grain substructure determined with the framework of the advanced ETMB model were taken into account in the CM in a hardening formula. By applying the CM with the integrated ETMB model, numerical experiments were performed to simulate the changes in the grain structure of copper during equal channel angular pressing (ECAP) at room temperature. The results obtained are in good agreement with the experimental data. The ideas about further development of the proposed model are outlined.

Reversible Cyclic Softening in HSLA Steels at Low Homologous Temperatures

Critical mechanical components employed in energy applications require certification of their mechanical response under monotonic or cyclic loading at various constant temperatures. However, components often experience combined load and temperature changes, which are more complex and time-consuming to evaluate. This work presents previously unreported reversible cyclic softening in low-alloy steel undergoing loading sequences between 293 K and 453 K (0.15 to 0.25 of the homologous temperature). Mechanical cycling at 80 pct of the yield stress results in a similar cyclic response for all temperatures, while cycling at 90 pct of the yield stress doubles the cyclic strain when preceded by cycling at 453 K. The softening can be reversed by maintaining the material at elevated temperatures and cannot be explained by dislocation recovery mechanisms or changes in the microstructure. Instead, serrated deformation at specific deformation levels suggests that Dynamic Strain Aging (DSA) contributes to softening, which is further supported by a theoretical analysis of interstitials and dislocations mean free paths at various temperatures.

Temperature dependent phase field dislocation dynamics model

In this work, we present a three-dimensional phase field-based formulation for simulating the temperature-dependent motion of discrete dislocations in crystals. For demonstration, this thermal phase field dislocation dynamics method is applied to study pyramidal-type dislocations in three exemplar hexagonal close packed materials, Mg, Ti, and Zr. Pyramidal-type dislocations are well known for temperature sensitivity and influence on the yield strength of these metals. Calculations include the predictions of activation stress, dislocation velocity, and dislocation nucleation over temperature ranges from 0 K to up to half the melting temperatures of these metals. We show that pyramidal slip is glissile, but the activation stress and stacking fault widths are asymmetric with respect to forward/backward glide. The stress to initiate glide reduces logarithmically with homologous temperature. The role of temperature in Frank-Read source operation is shown to decrease the size and time of first loop formation. The analysis identifies a master inverse relationship for the dislocation loop nucleation rate with homologous temperature followed by all three metals.

Development, Processing and Aging of Novel Zn-Ag-Cu Based Biodegradable Alloys.

The use of biodegradable materials for implants is a promising strategy to overcome known long-term clinical complications related to permanent implants. Ideally, biodegradable implants support the damaged tissue for a certain period and then degrade, while the physiological function of the surrounding tissue is restored. Although Mg-based alloys nearly ideally lend themselves to biodegradable implants, a few critical shortcomings promoted the development of alternative alloy systems. Due to their reasonably good biocompatibility, moderate corrosion rate without hydrogen evolution and adequate mechanical properties, increasing attention has been paid to Zn alloys. In this work, precipitation-hardening alloys in the system Zn-Ag-Cu were developed relying on thermodynamic calculations. After casting the alloys, their microstructures were refined by thermomechanical treatment. The processing was tracked and directed, respectively, by routine investigations of the microstructure, associated with hardness assessments. Although microstructure refinement increased the hardness, the material proved to be susceptible to aging as the homologous temperature of zinc is at 0.43 Tm. Besides mechanical performance and corrosion rate, long-term mechanical stability is another crucial factor that must be taken into consideration to ensure the safety of the implant and thus requires a profound understanding of the aging process.

High‐Temperature Fracture Growth by Constrained Sintering of Jadeite and Quartz Aggregates

AbstractFractures control heat and mass transfer and rheology in a wide range of subsurface regimes, ranging from low‐temperature diagenetic environments to high‐temperature metamorphic and magmatic systems. To investigate processes of opening‐mode fracture growth at high homologous temperature, we conducted constrained high‐temperature sintering experiments of thin layers of porous jadeite and quartz aggregate attached to a non‐sintering mullite substrate. Samples were heated stepwise at a low rate in a muffle furnace from 25°C to 1,000°C under ambient air pressure and examined for changes in mineral composition, texture, porosity, and fracture morphology using powder‐X‐ray diffraction, macro‐photography, reflected incident light microscopy, and secondary electron microscopy. Mineral reactions in the jadeite and quartz sample layer include jadeite and quartz reacting to albite at 600°C and the formation of nepheline and orthoclase at 900°C and 1,000°C. Opening‐mode fractures are first observed at 850°C coincident with the first presence of a melt phase with low aperture‐to‐length ratios similar to elastic‐brittle fractures. At 900°C, melt becomes increasingly abundant, and fractures obtain ductile morphology with high aperture‐to length ratios and blunted tips resulting from fracture growth by growth and coalescence of larger pores at the expense of smaller pores. In the absence of an externally applied mechanical load, we conclude that fracture growth is driven primarily by sintering stress resulting from differential contraction between sample layer and substrate associated with high‐temperature mineral reactions, melt formation and redistribution, and changes in pore structure. Similar fracture processes may be relevant to the segregation and migration of melt in magmatic systems.

Thermal and Radiation Stability in Nanocrystalline Cu

Nanocrystalline metals have presented intriguing possibilities for use in radiation environments due to their high grain boundary volume, serving as enhanced irradiation-induced defect sinks. Their promise has been lessened due to the propensity for nanocrystalline metals to suffer deleterious grain growth from combinations of irradiation and/or elevated homologous temperature. While approaches for stabilizing such materials against grain growth are the subject of current research, there is still a lack of central knowledge on the irradiation-grain boundary interactions in pure metals despite many studies on the same. Due to the breadth of available reports, we have critically reviewed studies on irradiation and thermal stability in pure, nanocrystalline copper (Cu) as a model FCC material, and on a few dilute Cu-based alloys. Our study has shown that, viewed collectively, there are large differences in interpretation of irradiation-grain boundary interactions, primarily due to a wide range of irradiation environments and variability in materials processing. We discuss the sources of these differences and analyses herein. Then, with the goal of gaining a more overarching mechanistic understanding of grain size stability in pure materials under irradiation, we provide several key recommendations for making meaningful evaluations across materials with different processing and under variable irradiation conditions.

Coupled chemo-mechanical modeling of point-defect diffusion in a crystal plasticity fast Fourier transform framework

Below the yield strength and at moderate-to-high homologous temperatures, the inelastic deformation of metals is mostly governed/rate-controlled by vacancy diffusion-mediated processes. As a function of grain size, stress, temperature and dislocation content, vacancies (or atoms) can adopt preferential diffusion paths across grain interiors, along grain boundaries, or towards and along dislocations, resulting in climb and self-climb. In the steady state and under constant load, grain boundary and grain bulk vacancy diffusion-mediated plasticity have been described in seminal works by Coble and by Nabarro and Herring, respectively. Yet, the interplay between all aforementioned potential diffusion pathways has not been comprehensively mapped. This work presents a thermodynamically-consistent full-field model integrated within a voxel-based elasto-viscoplastic fast Fourier transform framework, which considers the coupling between the diffusion-mediated plasticity mechanisms. In the proposed approach, the kinetics and kinematics of plastic deformation due to vacancy diffusion along grain boundaries and grain bulk, as well as the exchange between grain boundaries and bulk are described explicitly. A homogenization approach at the voxel level is further introduced to simultaneously consider bulk and grain boundary diffusion in a numerically efficient fashion. The new formulation predicts the expected strain rate dependencies and the scaling of the steady-state creep rate with respect to grain size, temperature, and stress. The model predicts the transition from grain bulk to grain boundary-dominated diffusion with reduction in grain size, a significant step towards capturing transitions in deformation behavior without any phenomenological or ad-hoc adjustments.

Correlation between plasma characteristics, morphology, and microstructure of sputtered CuAl alloy films with varied target geometry

The effect of target geometry on coating microstructure and morphology is correlated to changes in deposition conditions, plasma characteristics, and film growth during planar and hollow cathode sputtering. The sputtering plasma properties for the two target geometries were characterized via Langmuir probe analysis as a function of power density and Ar pressure to determine the evolution of ion density for each configuration. Films were then synthesized at the low (0.4 W cm−2) and high (1.2 W cm−2) power densities and characterized using x-ray diffraction, scanning electron microscopy, and electron backscatter diffraction to link changes in texturing, morphology, and microstructure with variations in ion density and sputtering deposition conditions caused by target geometry. It was observed that varying target geometry led to an over threefold increase in deposition rate, homologous temperature, and ion density, which altered the morphology and texture of the film without significant changes to the grain size.

Effect of Growth Temperature and Atmosphere Exposure Time on Impurity Incorporation in Sputtered Mg, Al, and Ca Thin Films.

Impurities can be incorporated during thin film deposition, but also can originate from atmosphere exposure. As impurities can strongly affect the composition-structure-property relations in magnetron sputter deposited thin films, it is important to distinguish between both incorporation channels. Therefore, the impurity incorporation by atmosphere exposure into sputtered Mg, Al, and Ca thin films is systematically studied by a variation of the deposition temperatures and atmosphere exposure times. Deposition temperature variation results in morphological modifications explained by considering surface and bulk diffusion as well as grain boundary motion and evaporation. The film morphologies exhibiting the lowest oxygen concentrations, as measured by energy dispersive X-ray spectroscopy, are obtained at a homologous temperature of 0.4 for both Mg and Al thin films. For Ca, preventing atmosphere exposure is essential to hinder impurity incorporation: By comparing the impurity concentration in Al-capped and uncapped thin films, it is demonstrated that Ca thin films are locally protected by Al-capping, while Mg (and Al) form native passivation layers. Furthermore, it can be learned that the capping (or self-passivation) efficiency in terms of hindering further oxidation of the films in atmosphere is strongly dependent on the underlying morphology, which in turn is defined by the growth temperature.

Evolution of the grain structure of metals and alloys under severe plastic deformation: continuum models

Products made of metals and alloys with a fine-grained structure, which have high physical, mechanical and operational characteristics, are becoming increasingly in demand in many technical and technological fields. The most common and efficient technologies for the production of parts from this class of materials are various processes of severe plastic deformation (SPD) (in general practice, at low homologous temperatures). At the same time, to achieve high degrees of deformation, a significant part of metals and alloys require increased processing temperatures, undergo significant heating during deformation processes, which may be accompanied by changes in the grain and subgrain structure due to recovery and recrystallization processes. An empirical approach to the development of SPD modes that ensure the formation of the necessary grain structure requires huge time and financial costs, and therefore considerable attention is paid by researchers in the field of solid mechanics and metal forming to approaches and methods of mathematical modeling. In connection with the foregoing, the number of publications on this subject has been intensively growing in recent years. The currently known models differ significantly in their approaches, the depth of penetration into the physics of processes, and the scope of consideration. The proposed brief review is focused on a qualitative analysis of works on this topic, a preliminary classification of existing models according to their purpose, versatility, and functionality. It seems possible to single out the two most common approaches to describing the change in the grain structure in the processes of thermomechanical processing of metals and alloys: continual (in most cases, single-level) and multilevel, based on the introduction of internal variables and physical theories. This review is devoted to the consideration of publications focused on the first of these approaches. Until now, the most common are macrophenomenological continuum models based on the analysis of experimental data conducted both in laboratories on macrosamples and in real production conditions. Models of this class are usually formulated in the form of operator relations over field quantities, they are relatively easy to implement due to their easy “embedding” into widely used commercial software packages, but they require significant costs for experiments to identify models, they are characterized by a low degree of versatility. Continuous theories are relatively less common, but still often used. These theories are based on the description of physical mechanisms and the evolution of the structure of metals and alloys in terms of continuum variables.