Digital design of gas turbine engine parts based on multilevel modeling

The paper aims at rationalizing the distribution of grain structure parameters across a gas turbine engine (GTE) disk in order to minimize the disk’s mass while ensuring its safe operational conditions. For this purpose, advanced methods of digital design were improved and applied, including state-of-the-art approaches based on multilevel modeling of material structure and properties. The problem was solved using a combined approach. Macro-phenomenological models were employed to determine, during the flight cycle, the evolving fields of stress-strain state and temperature of the entire part. These results were then transferred into a multilevel model to analyze specific regions for the study of strength characteristics, namely, high-temperature strength (resistance to creep and long-term strength), fracture toughness, low-cycle fatigue strength, and thermal stability (resistance to recrystallization and grain boundary migration). Multilevel modeling was based on a comprehensive analysis of the literature data on the structure of the nickel alloy VV751P, as well as on the mechanisms of its deformation and failure. The results of digital design were obtained and analyzed, and recommendations were proposed for rationalizing the material’s grain structure to reduce the disk’s mass while maintaining strength characteristics. The developed approach proved to be effective in digital multilevel design for functionally critical components.

Gas turbine technology plays an important role in the modern energy industry due to the significant contribution of natural gas to the energy balance of developed countries (Lapina et al., 2025). Therefore, improving the efficiency and reliability of gas turbine equipment remains an important engineering task.

High-pressure turbine disks are among the most critical components of gas turbine engines, and their failure often leads to catastrophic consequences (Mishra et al., 2023). The design and material of the disk must meet requirements for strength, reliability, and durability. During operations, the disks are subjected to significant mechanical loads, i.e., peak rotational speeds of about 13,000–18,000 rpm, under high-temperature conditions at the disk rim reaching up to 550 °C–700 °C (Čerňan et al., 2019; Theska et al., 2020; Li et al., 2022; Li et al., 2023; Mishra et al., 2023; Zhang et al., 2024). A challenge is to reduce fuel consumption during aircraft operations by decreasing its total mass, as these costs constitute the main part of the life-cycle cost of an aircraft engine (Faskhutdinov et al., 2017; Zhang et al., 2024).

The operating conditions of a high-pressure turbine disk require predicting its strength characteristics: high-temperature strength (resistance to creep deformation and long-term strength), low-cycle fatigue resistance, fracture toughness, and thermal stability (resistance to recrystallization and grain boundary migration). The need to study a complex of various strength properties arises from the nonuniform fields of mechanical loads and temperatures across the disk. The disk rim and the blade-lock regions operate under cyclic loads at high temperatures and relatively low stress levels, whereas the disk hub experiences increased stresses at lower temperatures (Mishra et al., 2023). For the turbine disk rim, the material must exhibit high long-term strength, thermal stability, and a low rate of fatigue crack propagation, while the hub and web must be made of a material with high short-term strength and low-cycle fatigue resistance (Gabb et al., 2011; Jiang et al., 2023).

A promising way to reduce turbine disk mass is to employ a functionally graded distribution of grain size across the disk, larger grains near the rim and smaller grains near the hub (Zhang et al., 2024). Such a structure increases the yield strength and fatigue resistance near the hub due to the smaller grain size, while improving fracture toughness and thermal stability, and reducing creep near the rim and blade-lock regions owing to the larger grain size (Jiang et al., 2023; Sun et al., 2024).

The operational properties of components are determined by the state of the material’s internal structure (Lipiński, 2023). To improve existing methods and develop new ones for manufacturing gas turbine engine parts, it is promising to employ modern digital design tools (Zeng et al., 2025). In recent decades, multilevel material models have become widespread; these explicitly include structural variables and corresponding evolution equations (Trusov et al., 2021; Knezevic, 2024). Such models offer high accuracy and predictive capabilities but are computationally expensive. To address this issue, a combined approach has been proposed (Baldin et al., 2025), which simultaneously employs a simplified macro-phenomenological model for approximate, rapid determination of mechanical and thermal responses, and a detailed multilevel model to refine the internal structural state and stress-strain parameters in specific regions.

The problem of improving the performance of gas-turbine engine disks through the rational design of their grain structure has been known since the early 1970s (Versnyder and Shank, 1970). With the advent of modern computing technology, methods of digital design began to be employed to improve the manufacturing process. Phenomenological models have been applied actively from the 2010s to the present day (Ekhteraei Toussi and Rezaei Farimani, 2012; Mishra et al., 2023). The present study is aimed at applying a modern multilevel modelling framework to the problem of optimising the design of a turbine disk.

Despite the substantial body of research on the design of gas-turbine engine disks made of nickel-based superalloys (Ganeev et al., 2019; Sotov et al., 2019; Liu et al., 2021; Li et al., 2024), the digital design of disk components and the solution of optimisation problems aimed at improving their structural performance remain pressing issues (Čerňan et al., 2019; Mishra et al., 2023). To date, the problem of determining, through numerical modelling, the grain-structure parameters that are optimal with respect to in-service conditions of the disk has not been resolved (Faskhutdinov et al., 2017; Zhang et al., 2024).

The purpose of this study is to apply the combined approach to simulate the operation of a gas turbine engine (GTE) disk under characteristic mechanical and thermal loads during a typical flight cycle, aiming to solve the optimization problem of reducing its mass through rationalization of the grain structure.

Within this research, a literature review of the structure of the VV751P alloy, the main deformation and failure mechanisms, and the methods of their mathematical modeling was conducted (Section 2.1). A boundary-value problem describing the changing stress-strain and temperature fields under operating conditions of a turbine disk made of VV751P alloy was formulated and solved using a macro-phenomenological material model (Section 2.2). Strength criteria were applied to selected regions of the disk structure using a multilevel constitutive model (Section 3). Based on the developed model and a set of strength criteria, an optimization problem was formulated and solved to minimize the disk mass by rationalizing its grain structure (Section 3).

Analysis of the material structure of alloy VV751P and its influence on the main strength criteria of the GTE disk

Gas turbine engine disks are largely manufactured from heat-resistant nickel-based alloys (Čerňan et al., 2019; Theska et al., 2020; Li et al., 2023; Mishra et al., 2023; Zhang et al., 2024), among which the modern alloy VV751P is included. This is a Russian high-strength and high-temperature-resistant granulated nickel alloy used for many critical components of modern engines (Ber et al., 2022; Khairulin et al., 2024). The chemical composition of the alloy, in mass percentages, is given, for example, in (Ber et al., 2022; Rogozhkin et al., 2023).

The typical base structure of many heat-resistant nickel alloys is multiphase. The main phase is the γ-phase, a solid solution with a face-centered cubic (FCC) lattice based on Ni, containing intermetallic Ni 3 Al particles of the strengthening γʹ-phase. The latter possesses an ordered FCC L1 2 structure and is coherent with the γ-phase (May et al., 2025). The structure of such alloys may also include carbide, boride, and carboboride particles (Gruber et al., 2024). The alloy composition is typically complicated by alloying the γ solid solution with Cr, Co, Mo, W, Fe, V, and C, and the γʹ-phase with Ti, Nb, Si, and B (Ajay and Dabhade, 2025). Alloys similar in properties to VV751P include its later modifications VV752P and VV753P, as well as earlier developments such as EP741NP, EI698P, and VV750P (Ber L.B. and Kazberovich, A.M., 2019; Inozemtsev A.A. et al., 2021); the foreign analog of this domestic alloy is Inconel 718 (Sotov et al., 2019; Chen et al., 2020).

After a typical production cycle using powder metallurgy methods and final treatment, the VV751P alloy matrix consists of γ-phase grains with an FCC lattice and a characteristic size of about 30–50 μm (Rogozhkin S.V. et al., 2020; Zainullin et al., 2023). The volume fraction of the γʹ-phase particles ranges between 60% and 70%. Chemical composition analysis shows that the γ-phase is enriched with Co, Cr, and Mo (γ-forming elements), while the γʹ-phase is dominated by Al, Ti, Ni, and Nb (γʹ-forming elements); V, W, Fe, Si, B, and C are present in similar concentrations in both phases. The structure also contains carbides and carboborides of size 0.3–1 μm of the type M (B, C), where M represents Ti, Nb, or Hf.

Gas turbine engine disks are subjected to exceptionally high requirements for high-temperature strength, low-cycle fatigue, fracture toughness, and thermal stability. A brief analysis of each of these requirements for the VV751P alloy is provided below.

High-temperature strength, characterized by resistance to creep and long-term strength, determines the service life of the component under prolonged loading at high temperatures (about 40%–80% of the homologous temperature) and reflects the material ability to resist constant deformation caused primarily by creep (Tyagunov et al., 2019; Zaichuk et al., 2023). In the VV751P alloy and its analogues, two main creep mechanisms occur at elevated temperatures: dislocation and diffusion creep. Dislocation creep can be described as the movement of dislocations in the γ-phase matrix along octahedral {111}⟨110⟩ planes, slip systems, typical for nickel-based heat-resistant alloys. The grain size plays an important role in dislocation creep: larger grains increase the creep rate (Detrois et al., 2016) due to the greater average dislocation free path. In fine-grained alloys, grain boundaries act as barriers to dislocation motion, which slows the dislocation creep rate. The second mechanism in VV751P at temperatures above 600 °C is diffusion creep, involving vacancy and atomic mass transport both through the crystal lattice and along grain boundaries. This process arises from atomic mobility, allowing the material to deform gradually through rearrangement of atomic bonds within the lattice (Meixner et al., 2022; Dai et al., 2025). In fine-grained alloys, the total number of grain boundaries increases significantly, intensifying grain-boundary diffusion. Because grain boundaries are regions of higher free energy and defect concentration, atoms in these regions are more mobile (Detrois et al., 2016; Zainullin et al., 2023).

Another key problem in modeling disk performance is the assessment of fracture toughness, as it is necessary to determine both the incubation period for crack initiation and the crack growth rate up to the critical value (Fan et al., 2022; Xiannian et al., 2023). The fatigue crack growth period defines the residual life of the disk and may constitute a significant portion of its total cyclic life. Fracture toughness characteristics increase with grain size (Jiang et al., 2023). This is due to the effect of grain size on the planarity/reversibility of slip and on crack curvature/closure under cyclic loadings (Gruber et al., 2022; Jiang et al., 2023). The free slip length for dislocations increases with grain size, resulting in more dislocation pile-ups along slip bands, which promotes planar slip (Cruzado et al., 2018; Jiang et al., 2023). Pile-up formation continues as long as the grain boundary remains an effective barrier to dislocation motion (Kim et al., 2024). In (Kruch and Toualbi, 2024) fine grains enhance resistance to crack initiation, whereas coarse grains improve resistance to crack propagation.

The main failure mechanism under low-cycle fatigue is the growth of microcracks perpendicular to the main stress axis in the direction of tensile loading. Due to vacancy diffusion at the tips of flat microcracks, which flanks are separated by several interplanar distances, these cracks branch and propagate along slip planes (Ber L.B. and Kazberovich A.M., 2023; Guo et al., 2023). The crack path is largely transgranular and brittle-fatigue in nature, forming cleavage facets and fatigue striations (Kim et al., 2024; Zhang et al., 2024). When a microcrack reaches a critical length, it transforms into a macrocrack, leading to the formation of a fracture origin and eventual failure (Ber L.B. and Kazberovich A.M., 2023; Eckmann et al., 2024). Microcracks tend to initiate at boundaries between inclusions and the matrix (Ber L.B. and Kazberovich A.M., 2023; You et al., 2024). For disks produced by powder metallurgy, fatigue performance correlates with powder particle size, with the optimal mean size being about 40–50 μm (Higashi and Kanno, 2020; Qu et al., 2022).

Consequently, one of the key structural characteristics of the VV751P alloy is the grain size. Considering the different operational conditions between the hub and the rim of the disk, it becomes necessary to form components with a graded grain structure. Technologies have been developed to achieve such nonuniform material structures in GTE disks. Initially, powder metallurgy, hot isostatic pressing, and subsequent thermomechanical treatment are employed, producing a billet with a predominantly fine-grained structure (Gayda et al., 2004; Gabb et al., 2011; Ganeev et al., 2019; Liu et al., 2021). The fine-grained structure corresponds to the grain sizes reported in the subsolvus condition for wrought nickel-based superalloys produced by powder metallurgy: 4–5 µm and ASTM G 8–12 (∼3–10 µm) for δ-subsolvus annealed Inconel 718 (Agnoli et al., 2015; Liu et al., 2023). Proper selection of processing temperature combined with optimal powder particle size is crucial to eliminate chemical and phase heterogeneities, obtain the desired grain size, and ensure strong secondary-phase precipitation strengthening (Higashi and Kanno, 2020; Qu et al., 2022). At the next stage, two main implementation methods exist. The first uses dual microstructural heat treatment (Gayda et al., 2003), which creates a significant temperature gradient between the rim and the hub of the disk, leading to the formation of distinct microstructures (Gayda et al., 2004; Gabb et al., 2011; Li et al., 2024). The second approach involves fabricating a preform of special geometry, where plastic strain near its center is much higher than at the edge (Ganeev et al., 2019; Liu et al., 2021). Compression of the billet occurs at high temperatures, causing dynamic recrystallization in the central region (grain refinement) and static recrystallization near the edges (grain coarsening) (Liu et al., 2021).

As a result of applying the graded-grain-structure technology, the internal structure of the component remains nonequilibrium, since not all parts of the billet are fully annealed. Therefore, high operating temperatures may again activate recrystallization and grain boundary migration, causing further undesirable changes in grain size (Li et al., 2025). The driving force behind these processes is the residual heterogeneity of stored defect energy from prior plastic deformation (Humphreys and Hatherly, 2012). Thus, a thermal stability criterion is introduced into the model to verify the absence of recrystallization and grain boundary migration.

A promising method for product design in the aerospace industry is the use of digital twins, especially through advanced multilevel modeling (Zeng et al., 2025). Explicitly accounting for all structural elements and factors determining the strength behavior of the disk under operating conditions represents a complex scientific challenge. Within this study, the influence of grain size, which is the only key factor, on the identified strength characteristics is explicitly analyzed.

Building on the foregoing analysis of the alloy microstructure and on the stringent requirements identified above with respect to creep resistance, low-cycle fatigue, thermal stability and fracture toughness, a set of corresponding strength criteria has been developed for assessing the performance of the gas-turbine engine disk under service conditions. A detailed description of these criteria, together with the results of their application to the solution of the boundary-value problem governing disk operation, is presented in the following Subsection 2.2.

Modeling the operation of a turbine disk made of nickel alloy VV751P under service conditions

The study is focused on the gas-turbine engine disk made of the VV751P alloy, which typically operates at high temperatures reaching 550 °C–700 °C at the rim and rotates at high angular speeds on the order of 13,000–17,000 rpm. For the service-life analysis, a representative flight cycle of approximately 30,000 s was considered. The study used simplified disk geometry, shown in Figure 1.

The boundary-value problem for determining the transient temperature fields and the stress-strain state of the turbine disk under operating conditions was solved using the finite element method in the Abaqus software package. At this stage of modeling, a macro-phenomenological model of transient thermo-viscoelasticity was applied. Its complete mathematical formulation is presented in (Romanov et al., 2025). In this model, creep strain is described using the following relation: where E ˙ c is a creep strain rate tensor, Q a is the activation energy of the creep process, k B is the Boltzmann constant, Θ is temperature, Σ is the Cauchy stress tensor; G is the shear modulus, A, n are creep-model parameters that depend on the internal structure of the material. The use of the modulus-compensated stress Σ G instead of the stress tensor Σ in the constitutive relation is a classical approach, presented both in the original formulation of the relation employed (Mukherjee et al., 1969) and in classical works (Frost and Ashby, 1982). This approach renders the construction under the power sign dimensionless, which makes it possible to preserve the physical meaning of the parameter A for any values of the exponent n. It is assumed that parameter A depends on grain size d in the corresponding material neighborhood as follows (Mukherjee, 2002): where b is the Burgers vector length, A 0 and p are parameters of the material.

The influence of microstructure is taken into account only through the average grain size. The model is intended primarily to describe quasi-steady-state creep within the ranges of temperature, stress, and grain size used in parameter identification.

The model including equations 1, 2 was identified using data from open sources; including data for closely related alloys with similar chemical and phase composition (see Section 1). The identification of the parameters of each of the criteria presented in this section was carried out in two stages. In the first stage, a subset of the parameters was determined directly from data available in the literature. In the second stage, the multi-objective optimisation package IOSO (Novel Optimization Strategy - IOSO, 2025) was employed to minimise the norm of the residual between the computed solution and the corresponding experimentally determined value reported in the literature. The remaining parameters of the constitutive relation served as the unknowns of the optimisation problem. The parameters used in the calculations are presented in Table 1.

A detailed formulation of the boundary and initial conditions of the mathematical model describing the in-service behaviour of the gas-turbine engine disk is provided in Supplementary Appendix A.

The solution of this problem resulted in values of fields of creep strain tensor E c r, t intensity, the equivalent stress tensor Σ r, t and temperatures Θ r, t in the disk. These quantities, normalized by their respective maximum values, are shown in Figure 2.

A combined approach (Baldin et al., 2025) was used to refine the structural state and mechanical properties within selected representative volumes of the component. The strain and temperature fields obtained at the macro level for the chosen material points were used as input conditions for the multiscale models of creep (Kondratev et al., 2025) and inelastic deformation (Trusov et al., 2021). Three critical zones were analysed - near the hub, in the central section, and at the rim - as shown in Figure 3.

Under the operating conditions of the disk, both dislocation and diffusion creep mechanisms are activated. As noted above, the former is associated with dislocation glide and climb while the latter is governed by the diffusion-driven mass transport of atoms and vacancies. The multiscale creep model used here, which is based on a statistical approach and the Reis hypothesis, is presented in (Kondratev et al., 2025). The model parameters employed in the calculations are provided in Supplementary Appendix B. This model forms the basis for determining the heat-resistance criterion, which incorporates long-term strength and creep resistance.

To describe the long-term strength characteristics of the VV751P alloy (i.e., its ability to undergo creep without failure), an energy-based criterion associated with the accumulation of material damage was used (Radchenko V.P. and Tsvetkov V.V, 2018; Lee and Basaran, 2021; Ye et al., 2022). For each grain, damage-accumulation parameter W i t was calculated using the two-level creep model (Radchenko V.P. and Tsvetkov V.V, 2018): where κ i is the measure of the stress state of the individual i-th grain, d i s l is the creep strain rate due to dislocation slip, d i c l is the creep strain rate due to dislocation climb, and d i d i f is the diffusion-controlled creep rate. When the representative-volume-averaged value of the damage-accumulation parameter W i at a certain time t reaches its critical work value W *, material failure is assumed. Thus, the criterion for the material remaining intact under creep is:

The creep-resistance criterion was defined as a limit u * on the radial displacements u r r of points at the disk rim caused by irreversible deformations from dislocation and diffusion creep:

The criteria employed for describing creep failure are physically complementary: the energy-based criterion (Equation 4) controls the local energy accumulation of the damage parameter within each grain, whereas the displacement-based criterion (Equation 5) ensures the integral stability of the structure as a whole.

The universality of the energy-based criterion for damage assessment has been widely discussed in the literature (Wu, 2019; Sandström, 2024). According to the analysis presented in these works (Wu, 2019; Sandström, 2024), the applicability of such criteria may be limited. In particular, in regimes dominated by grain boundary sliding (GBS), failure occurs predominantly through an intergranular cavitation mechanism accompanied by a sharply reduced material ductility by the time of failure, and the integral work of inelastic deformation may underestimate the actual state of damage. A complementary analysis of alloys with similar chemical composition, carried out with account for the service conditions of the disk and the solutions obtained (Figure 2), indicates that in the VV751P alloy the principal operative mechanisms are intragranular dislocation glide and climb, jointly with diffusional mass transport (Frost and Ashby, 1982; Wu et al., 2012).

In Figure 4 results of multiscale modeling of the creep process based on Equation 3 -5 are presented: the evolution of the normalized accumulated creep strain ε u t and the evolution of the value W i t entering the long-term strength criterion for the investigated disk points A, B, and C over the entire flight cycle. The values shown in Figure 4 are normalized by the maximum accumulated creep strain ε max and the corresponding maximum W i obtained in the computational experiments. Diffusion creep develops most intensively during takeoff, under elevated stresses and temperatures (Figure 4), leading to a noticeable increase in inelastic deformation in regions near the disk rim (point C, see Figure 3). The disk material demonstrated resistance to creep damage accumulation, as indicated by the normalized dependence of W i t over the flight cycle (Figure 4). In absolute terms, the values of ε u t and W i t vary by 2–3 orders of magnitude between points, with the most intensive creep occurring at point C. According to the calculations, the critical values of the long-term strength criterion are reached first at the disk rim after more than 40,000 flight cycles.

To describe the strength characteristics of low-cycle fatigue (LCF) and fracture toughness of the VV751P alloy, a statistical two-level model of inelastic deformation of the polycrystal based on the Voigt hypothesis was used (Trusov et al., 2021). Statistical models are the most computationally efficient compared to direct and self-consistent models and allow for an adequately accurate description of metal processing technologies (Lebensohn et al., 2011; Trusov et al., 2021).

The model was modified to account for the influence of grain size on the yield stress according to the modified Hall-Petch relation (Hall, 1951; Petch, 1953; Mikula et al., 2024). For the application of the multilevel inelastic deformation model to high-cycle fatigue (HCF) and fracture toughness, some of its parameters were approximated as functions of temperature θ and grain size d. The identified model parameters and the aforementioned approximation are provided in Supplementary Appendix C.

As a low-cycle fatigue (LCF) parameter, the Smith-Watson-Topper (SWT) criterion was used (Smith, 1970; Lu et al., 2025), applied in its classical form (Smith, 1970) for the representative volume of the macro-level material. The physical mechanisms of failure are implicitly described in the meso-level model. In this work, according to the SWT relation, the connection between the maximum stress along the loading axis per loading cycle Σ a x i s, ⁡ max, the doubled strain amplitude along the loading axis per cycle Δ ε a x i s and number of cycles N f is taken in the following form (Łagoda et al., 2022): where A and B are material parameters determined experimentally, and E is the Young’s modulus of the material. The relationship between the SWT expression and the number of cycles N f is taken in the following form (Łagoda et al., 2022): where A and B are material parameters. To account for the influence of temperature θ (Mishra et al., 2023) and grain size d we use the linear approximation of A and B:

Parameters A i, B i are determined experimentally. To identify the low-cycle fatigue (LCF) criterion, data for the closely related VV750P alloy were used for fine (∼ 10 μ m) and coarse (∼ 50 μ m) grain sizes at temperatures of 450 and 650 ° C (Vasil’ev B.E. et al., 2018). The model parameters used in the LCF calculations are provided in Supplementary Appendix D. The results of the LCF model calculations based on Equations 6 -8 are shown in Figure 5. the LCF model demonstrates good quantitative agreement with the experimental data for the VV750P alloy (Vasil’ev B.E. et al., 2018).

Parameters A and B are approximated by linear functions of temperature and grain size and identified using data at temperatures of 450 °C and 650 °C and grain sizes of approximately 10 and 50 µm. Therefore, the model is quantitatively reliable primarily within this range, and its extrapolation to other temperatures and grain sizes requires further verification.

To determine the fracture toughness, the Paris power-law relation was used (Paris et al., 1961; Ostergaard et al., 2025): where v is the fatigue crack growth rate, l is the characteristic size (length) of the fatigue crack, N is the number of loading cycles, Δ K = K max − K min = K max 1 − R – is the stress intensity factor (SIF) range, R = ε min / ε max is the loading cycle asymmetry ratio, ε min and ε max are the minimum and maximum strain values per cycle, C, m are the Paris law coefficients. The maximum stress intensity factor K max can be expressed through the effective stresses at the crack tip (Pistorio and Clerici, 2024; Elmhaia et al., 2021; Paris et al., 1961): where I 1 σ max is the first invariant of the stress tensor corresponding to the maximum stresses within the considered cycle. From these relations, the dependence of the crack length l on the number of cycles N can be obtained as: where l 0 is the initial crack length.

To identify the parameters of the fracture toughness model (Equations 9–11), experimental data on the crack growth rate per cycle d l / d N as a function of the stress intensity factor Δ K for the VV751P alloy were used from (Garibov G.S., 2016). As a result of the identification, the fracture toughness model parameters C = 1.605 · 1 0 − 58, m = 6.738 were determined (Figure 6a). The characteristic minimum initial crack length l 0 = 750 µm, that would lead to component failure under typical technical requirements for a VV751P disk (Garibov G.S., 2016) was determined to be N = 2 · 1 0 4, σ u, ⁡ max = 1200 MPa (Figure 6b).

The typical critical defect size that leads to accelerated crack growth is on the order of 60–100 μm (Taina, 2011; Eckmann et al., 2024), therefore, the investigated GTE disk has a large safety margin in fracture toughness relative to the determined critical value l 0 = 750 μm. For the optimized gradient grain structure of the disk (see Section 3), similar estimates of the initial crack length l 0, were obtained, at which failure would occur upon reaching the number of cycles N = 2 · 1 0 4. The resulting critical sizes are significantly larger than the typical critical crack length in the disk (60–100 μm). Thus, it can be concluded that the fracture toughness requirements for the obtained gradient grain structure of the disk are satisfied with a safety margin.

Paris’s law applies primarily to the steady-state growth stage of a macrocrack under linear fracture mechanics. It does not describe the threshold region, accelerated growth before failure, the effects of short cracks, variable loading amplitude, or the influence of actual crack geometry. Therefore, the obtained results should be considered an engineering assessment of durability within the conditions used in model identification.

Within the scope of the service process of the gas-turbine engine (GTE) disk considered here, one of the governing parameters of the failure criteria is the grain size. The gradient grain structure is formed during fabrication and thermomechanical treatment through recrystallization and grain-boundary migration. The description of the manufacturing stage and of the formation of the grain structure lies beyond the scope of the present work. What is essential is the preservation of a constant grain size during service operation. In this connection, a separate problem is posed—that of determining the constraints imposed on the parameters of the formed grain structure to ensure its stability.

The selected points in the disk are analyzed for thermal stability, i.e., the ability to maintain their grain and subgrain structures, and consequently mechanical properties, at elevated temperatures (resistance to recrystallization and grain boundary migration). The recrystallization process involves the formation of new grains. In the class of alloys under consideration, the primary recrystallization mechanism is the Bailey-Hirsch mechanism, based on the bending of segments of pre-existing grain boundaries in the polycrystal (Chen et al., 2022; Galpin, 2022). A kinetic description of the high-angle grain boundary migration mechanism was proposed by J.E. Bailey (Bailey, 1962). The presence of secondary-phase particles in the alloys (Jiang et al., 2023; You et al., 2024) exerts a pinning effect. The modified Bailey-Hirsch criterion for the onset of recrystallization, accounting for the pinning effect, takes the form (Zhang et al., 2019): where d c is the critical size of a recrystallization nucleus at which the growth process begins, e s t is the stored energy in the defective structure, e g b is the grain boundary energy per unit area, and f p i n is the boundary pinning force per unit area due to the presence of second-phase particles.

Recrystallized grains formed during manufacturing and/or service can change their geometrical characteristics due to grain boundary migration. The relation describing the size change during service is given by (Agnoli et al., 2015; Wang et al., 2025): where m h a g, 0 is the pre-exponential factor, Q b is the activation energy of grain boundary diffusion, R is the universal gas constant, and θ is the absolute temperature.

To describe the pinning force f p i n in the considered alloy, the classical Zener relation is applied (Agnoli et al., 2012; Agnoli et al., 2015; Ariaseta et al., 2020) assuming spherical particle shape (Humphreys, 1997; Galpin, 2022): where f v is the volume fraction of particles, and r a v is the average particle radius. To determine the stored energy, the following well-known relation was used (Huang et al., 2024; Tang et al., 2025): where ρ ¯ is the average dislocation density, G is the shear modulus, and b is the magnitude of the Burgers vector in the material.

Thus, the thermal stability criterion is defined by Equation 12: subgrains smaller than d c will not become new grains. To this condition using Equations 13 -15, a limitation on the growth rate of recrystallized grains is added: d ˙ ≤ v c, where v c is the maximum allowable grain growth rate that does not lead to more than a 5% restructuring of the grain structure over the maximum possible number of flight cycles. From this condition, the critical dislocation density ρ с, is determined, at which it is guaranteed that the grain structure will not change beyond the specified limits.

The parameters of the thermal stability criteria for the VV751P alloy are provided in Supplementary Appendix E. In Figure 7a, graphs of the dependence of critical dislocation density ρ с on grain size d at various temperatures θ; similar dependencies for the critical subgrain size d с presented in Figure 7b. The results indicate that recrystallization and migration processes for characteristic structural values remain within allowable grain size distributions along the radial direction of the disk, i.e., the thermal stability criterion is satisfied with a safety margin.

Thus, the integrated consideration of the proposed strength criteria, based on multilevel models, can be applied for a refined investigation of the mechanical properties of the VV751P alloy, including the description of its deformation behavior and all strength characteristics, such as low-cycle fatigue and fracture toughness. The obtained model parameters and fatigue criteria allow for reliable description and prediction of the material’s durability, taking into account its microstructure and service conditions.

The developed model is not specific to the VV751P alloy and can be applied to other heat-resistant nickel alloys. This application requires an in-depth analysis of the mechanisms of deformation and failure, as well as the identification of the determining parameters.

Figure 8 presents the general concept of the proposed methodology for structural rationalization aimed at improving operational performance (reducing mass). The methodology consists of coupled analyses at the macro- and meso-scales. A specialized IOSO platform (Novel Optimization Strategy - IOSO, n.d.) was used to control the subprocesses associated with the implementation of the models. The finite element solver of initial boundary value problems from the Abaqus package (Abaqus, 2025) was employed to describe the structural response under operational loading. In turn, in-house computational tools were used to implement the multiscale material models and to evaluate the strength criteria (see Section 2.2) in selected regions of the component.

The numerical implementation of the computational scheme shown in Figure 8 makes it possible to minimize the mass of the modeled disk while ensuring that all strength criteria are satisfied. To reduce the number of degrees of freedom when varying the disk geometry, a well-known engineering approach was applied (Ekhteraei Toussi and Rezaei Farimani, 2012; Zhang et al., 2024) which consists in reducing the linear dimension g of the disk’s generatrix, as illustrated in Figure 9.

One iteration of the rationalization procedure included an Abaqus simulation of the stress, strain, and temperature fields using the macro-phenomenological model (see Section 2.2). After that, using the IOSO package and the MLS library, grain sizes in N selected in critical points of the disk were computed so as to satisfy N F specified functionality criteria F j s i g, h i where s i g, h i is the vector of histories of the physico-mechanical state, g is the vector of geometric characteristics, h i is the vector of structural characteristics j = 1, N F, i = 1, N).

At the second stage, the obtained values were extrapolated to the entire computational domain; the disk shape was adjusted (Figure 9) to reduce the mass, and the admissibility of the geometry modifications was verified, completing the iteration of the optimization procedure.

Figure 10a shows the fields of relative von Mises stresses (normalized by the maximum value) at the time corresponding to the maximum angular velocity, calculated under the assumption of a uniformly distributed grain size prior to the start of the rationalization procedure. In Figure 10b, the critical material points selected for evaluating the structure-dependent functionality criteria are indicated.

The functionality criteria (long-term strength F 1, low-cycle fatigue F 2, fracture toughness F 3, creep F 4 and thermal stability F 5) were normalized to the range [0,1], where 0 and 1 correspond to the limiting values obtained when varying the grain size within the allowable ranges. The fracture-toughness F 3, creep F 4 and thermal stability F 5 criteria were not included in the iterative optimization procedure because a sufficient safety margin (see Section 2) existed with respect to fracture toughness, creep strains, and thermal stability at the considered loading conditions and within the range of characteristic grain sizes [a, b] μm, which exceeded the range under study. The verification of F 3, F 4 and F 5 was performed after the optimization procedure to reduce computational cost and remained within the permissible limits.

Based on the values of the criteria F 1 and F 2 (Figure 11), it can be seen that they are competing in the analyzed points, except for the point near the disk hub (point A). The grain size in points B and C was determined by minimizing the arithmetic mean of the functionality criteria F 1 and F 2 (Figure 12). The distribution of the relative grain size along the radial coordinate of the disk in accordance with the functionality criteria is shown in Figure 13.

The rational distribution of the mean grain diameter along the disk radius (see Figure 13) was taken as the initial approximation for the second iteration of the optimization problem. In particular, the disk thickness was modified, which made it possible to reduce the total mass of the component by approximately ∼2.4%. The new geometric configuration reflecting this modification is shown in Figure 9 with the dashed line.

After the described adjustment of the disk geometry, a second iteration in the series of the computational experiments was performed, following the same grain-structure rationalization procedure as described above. In this iteration, five control points located along the radius from the hub to the rim were selected for the evaluation of local indicators (Figure 14). For each of these points, the problem of determining the optimal relative mean grain diameter h was solved independently. The grain size was normalized to its maximum theoretical value b μm.

The main results of this iteration for each of the five selected points (Figure 14) are summarized below. The functionality criteria are competing at all analyzed points. Similarly to the first iteration, the optimal grain size at each of these points was determined based on the obtained results (Figure 15). The rational distribution of grain size along the radial coordinate of the disk is shown in Figure 16.

The rational distribution of grain sizes determined at the second iteration is shown in Figure 17.

Thus, the second iteration of the optimization procedure, which accounted for changes in the disk geometry, demonstrated that the proposed gradient rational grain structure of the material provides a safety margin for long-term strength, low-cycle fatigue, fracture toughness, creep, and thermal stability. In other words, it is stable with respect to recrystallization and grain boundary migration processes under the determined limits on structural parameters, i.e., dislocation density and subgrain size. The performed calculations indicate the possibility of reducing the disk mass while maintaining its functional properties, highlighting the feasibility and effectiveness of applying integrated digital design to such components. In contrast to previous studies, the iterative procedure developed in the present work makes it possible to determine explicitly the rational parameters of the graded grain structure and to formulate well-founded recommendations for adjusting the geometry of the gas-turbine engine disk.

The main conclusions from the conducted study are as follows: Based on the literature data, an analysis of the microstructure of the powder alloy VV751P and alloys most closely related to it in terms of chemical composition, phase composition, and morphological structure was carried out. Using this database, constitutive models of the VV751P alloy were developed, linking the grain structure to the main mechanical properties under the operating conditions of the gas-turbine engine disk.

A combined modeling approach for the GTE turbine disk was implemented. The macro-level problem of determining the stress-strain state and temperature distribution of the disk during a characteristic flight cycle was solved. Subsequently, using a multilevel model, creep deformations and the comprehensive set of strength properties (high-temperature strength, low-cycle fatigue, fracture toughness, and thermal stability) were analyzed.

The problem of optimizing the mass of the GTE turbine disk using the developed suite of models was formulated and solved. An iterative procedure for the optimization method was created and implemented, which in computational experiments demonstrated the possibility of reducing the disk mass by decreasing its thickness through the selection of an effective grain structure distributed along the disk radius.

The scientific significance of the study lies in combining structure-sensitive multiscale models with component geometry optimization to determine a rational gradient grain structure. The practical and social significance lies in the possibility of reducing component weight while maintaining its reliability, which has the potential to reduce fuel consumption, harmful emissions, and material consumption.

Further work will be aimed at combining the optimization of the operational stage with the modeling of the thermomechanical manufacturing process, refining the spatial discretization and experimental verification of the proposed gradient structure.