Numerical and Experimental Assessment of the Effect of Residual Stresses on the Fatigue Strength of an Aircraft Blade

The work presents the results of numerical fatigue analysis of a turbine engine compressor blade, taking into account the values of initial stresses resulting from surface treatment-shot-peening. The values of the residual stresses were estimated experimentally using X-ray diffraction. The paper specifies the values of the residual stresses on both sides of the blade and their reduction due to cutting through the blade-relaxation. The obtained values of the residual stresses were used as initial stresses in the numerical fatigue analysis of the damaged compressor blade, which was subjected to resonant vibrations of known amplitude. Numerical fatigue ε-N life analysis was based on several fatigue material models: Manson’s, Mitchell’s, Baumel-Seeger’s, Muralidharan-Manson’s, Ong’s, Roessle-Fatemi’s, and Median’s, and also on the three models of cyclic hardening: Manson’s, Xianxin’s, and Fatemi’s. Because of this approach, it was possible to determine the relationship between the selection of the fatigue material ε-N model and the cyclic hardening model on the results of the numerical fatigue analysis. Additionally, the calculated results were compared with the results of experimental research, which allowed for a substantive evaluation of the obtained results. These results are of great scientific and practical importance. The problem of determining the fatigue life of blades with defects operating under resonance vibrations is one of the original tasks in the field of fracture mechanics and experimental mechanics. The results obtained are of great importance in the aviation industry and can be used during engine maintenance and inspections to assess the suitability of blades with defects in terms of the needs of further work. This aspect of engineering maintenance is of great importance from the aircraft safety point of view.     

Using the Smith-Watson-Topper Parameter and Its Modifications to Calculate the Fatigue Life of Metals: The State-of-the-Art

The Smith-Watson-Topper parameter (SWT) in its original form was designed to estimate the fatigue life of metal materials in a uniaxial load state (tension–compression) in the range up to fatigue crack initiation, with non-zero mean values. This parameter is based on the analysis of both stress and strain. Therefore, the stress–strain criterion is the focus, rather than the energy criterion. This paper presents the original SWT model and its numerous modifications. The first part presents different versions of this parameter defined by the normal parameters. Then, it presents versions defined through the tangent parameter and the most promising parameter defined through the tangent and normal parameters. It was noted that the final form of the equivalent value is defined either by stress or by an energy parameter. Therefore, the possible characteristics from which the fatigue life can be determined are also presented.     

Investigation of Surface Integrity Induced by Various Finishing Processes of AISI 52100 Bearing Rings

Surface integrity induced by finishing processes significantly affects the functional performance of machined components. In this work, three kinds of finishing processes, i.e., precision hard turning, conventional grinding, and sequential grinding and honing, were used for the finish machining of AISI 52100 bearing steel rings. The surface integrity induced by these finishing processes was studied via SEM investigations and residual stress measurements. To investigate rolling contact fatigue performance, contact fatigue tests were performed on a twin-disc testing machine. As the main results, the SEM observations show that precision hard turning and grinding introduce microstructural alterations. Indeed, in precision hard turning, a fine white layer (<1 μm) is observed on the top surface, followed by a thermally affected zone in the subsurface, and in grinding only, a white layer with 5 μm thickness is observed. However, no microstructural changes are found after sequential grinding and honing processes. White layers induced by precision hard turning and grinding possess compressive residual stresses. Grinding and sequential grinding and honing processes generate similar residual stress distributions, which are maximum and compressive at the machined surface and tensile at the subsurface depth of 15 μm. Precision hard turning generates a “hook”-shaped residual stress profile with maximum compressive value at the subsurface depth and thus contributes as a prenominal factor to the obtainment of the longest fatigue life with respect to other finishing processes. Due to the high quality of surface roughness (Ra = 0.05 μm), honing post grinding improves the fatigue life of bearing rings by 2.6 times in comparison with grinding. Subsurface compressive residual stresses, as well as low surface roughness, are key parameters for extending bearing fatigue life.     

Effects of Pre-Tensile Deformation on the Fatigue Fracture Behavior of Annealed 7005 Aluminum Alloy Plate

In the present study, the fatigue life and fatigue fracture characteristics of annealed 7005 aluminum alloy plates subjected to different pre-tensile deformations were investigated. The results obtained upon increasing the pre-tensile deformation of the alloy plate to 20% revealed that the second-phase particles did not show any obvious changes, and that the thickness of the thin strip grain slightly decreased. The dislocation distribution in the alloy matrix varied significantly among the grains or within each grain as the dislocation density gradually increased with increasing pre-tensile deformation. Moreover, the fatigue performance of the annealed 7005 aluminum alloy plate was significantly improved by the pre-tensile deformation, and the alloy plate subjected to 20% pre-tensile deformation exhibited an optimal fatigue life of ~1.06 × 10⁶ cycles, which was 5.7 times and 5.3 times that of the undeformed and 3% pre-stretched alloy plates, respectively. Two fatigue life plateaus were observed in the pre-tensile deformation ranges of 3–5% and 8–12%, which corresponded to heterogeneous dislocation distribution among various grains and within each grain, respectively. Moreover, two large leaps in the plot of the fatigue-life–pre-tensile-deformation curve were observed, corresponding to the pre-tensile deformation ranges of 5–8% and 16–20%, respectively.     

Vibration-Based Fatigue Analysis of Octet-Truss Lattice Infill Blades for Utilization in Turbine Rotors

Vibration fatigue characteristics are critical for rotating machinery components such as turbine rotor blades. Lattice structures are gaining popularity in engineering applications due to their unique ability to reduce weight and improve the mechanical properties. This study is an experimental investigation of octet-truss lattice structure utilization in turbine rotor blades for weight reduction and to improve vibration fatigue characteristics. One completely solid and three lattice infilled blades with variable strut thickness were manufactured via additive manufacturing. Both free and forced experimental vibration analyses were performed on the blades to investigate their modal and vibration fatigue characteristics. The blades were subjected to random vibration using a vibration shaker. The response was measured using a triaxial accelerometer in terms of vibration acceleration time histories in the X, Y, and Z directions. Results indicate a weight reduction of up to 24.91% and enhancement in the first natural frequency of up to 5.29% were achieved using lattice infilled blades. The fatigue life of the blades was investigated using three frequency domain approaches, namely, Lalanne, Dirlik and narrow band. The fatigue life results indicate that the 0.25 mm lattice blade exhibits the highest fatigue life, while the solid blade exhibits the lowest fatigue life of all four blades. The fatigue life of the 0.25 mm lattice blade was 1822-, 1802-, and 1819- fold higher compared to that of the solid blade, using the Lalanne, Dirlik, and narrow-band approaches, respectively. These results can serve as the first step towards the utilization of lattice structures in turbine blades, with thermal analysis as the next step. Therefore, apart from being light weight, the octet-truss lattice infilled blades exhibited superior vibration fatigue characteristics to vibration loads, thereby making them a potential replacement for solid blades in turbine rotors.     

Oxidation Damage Evolution in Low-Cycle Fatigue Life of Niobium-Stabilized Austenitic Stainless Steel

Austenitic stainless steel is a vital material in various industries, with excellent heat and corrosion resistance, and is widely used in high-temperature environments as a component for internal combustion engines of transportation vehicles or power plant piping. These components or structures are required to be durable against severe load conditions and oxidation damage in high-temperature environments during their service life. In this regard, in particular, oxidation damage and fatigue life are very important influencing factors, while existing studies have focused on materials and fracture behavior. In order to ensure the fatigue life of austenitic stainless steel, therefore, it is necessary to understand the characteristics of the fracture process with microstructural change including oxidation damage according to the temperature condition. In this work, low-cycle fatigue tests were performed at various temperatures to determine the oxidation damage together with the fatigue life of austenitic stainless steel containing niobium. The characteristics of oxidation damage were analyzed through microstructure observations including scanning electron microscope, energy-dispersive X-ray spectroscopy, and the X-ray diffraction patterns. In addition, a unified low-cycle fatigue life model coupled with the fracture mechanism-based lifetime and the Neu-Sehitoglu model for considering the influence of damage by oxidation was proposed. After the low-cycle fatigue tests at temperatures of 200–800 °C and strain amplitudes of 0.4% and 0.5%, the accuracy of the proposed model was verified by comparing the test results with the predicted fatigue life, and the validity by using the oxidation damage parameters for Mar-M247 was confirmed through sensitivity analysis of the parameters applied in the oxidation damage model. As a result, the average thickness of the oxide layer and the penetration length of the oxide intrusion were predicted with a mean error range of 14.7% and 13%, respectively, and the low-cycle fatigue life was predicted with a ±2 factor accuracy at the measurement temperatures under all experimental conditions.     

Low Cycle Fatigue Life Assessment Based on the Accumulated Plastic Strain Energy Density

The accumulated plastic strain energy density at a dangerous point is studied to estimate the low cycle fatigue life that is composed of fatigue initiation life and fatigue crack propagation life. The modified Ramberg–Osgood constitutive relation is applied to characterize the stress–strain relationship of the strain-hardening material. The plastic strain energy density under uni-axial tension and cyclic load are derived, which are used as threshold and reference values, respectively. Then, a framework to assess the lives of fatigue initiation and fatigue crack propagation by accumulated plastic strain energy density is proposed. Finally, this method is applied to two types of aluminum alloy, LC9 and LY12 for low-cycle fatigue, and agreed well with the experiments.     

Cyclic Deformation Induced Residual Stress Evolution and 3D Short Fatigue Crack Growth Investigated by Advanced Synchrotron Tomography Techniques

Diffraction and phase contrast tomography techniques were successfully applied to an austenitic–ferritic duplex stainless steel representing exemplarily a metallic material containing two phases with different crystal structures. The reconstructed volumes of both phases were discretized by finite elements. A crystal plasticity finite-element analysis was executed in order to simulate the development of the experimentally determined first and second order residual stresses, which built up due to the manufacturing process of the material. Cyclic deformation simulations showed the single-grain-resolved evolution of initial residual stresses in both phases and were found to be in good agreement with the experimental results. Solely in ferritic grains, residual stresses built up due to cyclic deformation, which promoted crack nucleation in this phase. Furthermore, phase contrast tomography was applied in order to analyze the mechanisms of fatigue crack nucleation and short fatigue crack propagation three-dimensionally and nondestructively. The results clearly showed the significance of microstructural barriers for short fatigue crack growth at the surface, as well as into the material. The investigation presented aims for a better understanding of the three-dimensional mechanisms governing short fatigue crack propagation and, in particular, the effect of residual stresses on these mechanisms. The final goal was to generate tailored microstructures for improved fatigue resistance and enhanced fatigue life.     

Simulation of the Kinematic Condition of Radial Shear Rolling and Estimation of Its Influence on a Titanium Billet Microstructure

The finite element method (FEM) computer simulation of the three-high radial shear rolling of Ti-6Al-4V alloy round billets was conducted using QForm software. The simulation was performed for the MISIS-100T rolling mill’s three passes according to the following rolling route: 76 mm (the initial billet diameter) →65 mm→55 mm→48 mm (the final billet diameter). The change in the total velocity values for the points on the radius of the 48 mm diameter billet was estimated while passing the rolls’ draft. The relative increase in the accumulated strain was estimated for the same points. Then, experimental shear rolling was performed. Grain sizes of the α- and β-phases were estimated in the cross section of the final billet at the stationary stage of rolling. The grain size distribution histograms for different phases were plotted. An area was found in the billet’s cross section in which the trend of change in the total velocity of the points changed. This area represented a neutral layer between the slowing peripheral segments of the billet and the accelerating central segments of the billet. Inside this neutral layer, the limits of the cylindrical surface radius value were estimated. Experimental radial shear rolling was performed to compare the experimental rolling results (the billet microstructure investigation) with the computer simulation results. The computer simulation obtained two estimations of the radius limits: 8–16 mm (based on the analysis of the total velocity change) and 12–16 mm (based on the accumulated strain’s relative increment change). The experimental rolling obtained two more estimations of the radius limits: 8.4–19.5 mm and 11.3–19.7 mm—based on the results of the microstructure investigation. It was confirmed that varying the kinematic and deformation parameters of radial shear rolling allows regulation of the thickness of the peripheral fine-grain layer and the diameter of the central coarse-grain layer of the rolled billets.     

Parameter Study of Interfacial Capacities for FRP–Steel Bonded Joints Based on 3D FE Modeling

This paper investigated the stress distribution of an adhesive layer for GFRP–steel bonded joints under 22.48 kN tensile loading using a three-dimensional numerical simulation. Firstly, a stress analysis of three paths was conducted, and after comparison, path II (through the middle layer of the bonding layer) was adopted as the analyzing path. Furthermore, a systemically parametric study of the effects of the FRP stiffness (i.e., elastic modulus and thickness), bonding length, adhesive thickness, and adhesive modulus was conducted. For the joints with different FRP elastic moduli, the minimum value of normal peeling stress was calculated as −3.80 MPa by the FRP for 10 GPa, showing a significantly severe stress concentration of FRP for 10 GPa. An analysis of the von Mises stresses proved that the increase in FRP stiffness could reduce the stress concentration of the adhesive layer effectively. The study of the effect of bonding lengths indicated that a more uniform peeling stress distribution could result from the longest bonding size; the largest peeling stress of 6.54 MPa was calculated for a bonding length of 30 mm. Further parameter analysis showed that the stress concentration of the adhesive layer could be influenced by the FRP thickness, bonding thickness, and elastic modulus of the adhesive layer.     

Study on the Effect of Deposited Graphene Oxide on the Fatigue Life of Austenitic Steel 1.4541 in Different Temperature Ranges

This paper presents the effect of deposited graphene oxide coating on fatigue life of austenitic steel 1.4541 at 20 °C, 100 °C, and 200 °C. The study showed a decrease in the fatigue life of samples with a deposited graphene oxide layer in comparison with reference samples at 20 °C and 100 °C. However, an increase in fatigue life of samples with a deposited graphene oxide layer in comparison with reference samples occurred at 200 °C. This relationship was observed for the nominal stress amplitude of 370 and 420 MPa. Measurements of temperature during the tensile failure of the sample and microfractographic analysis of fatigue fractures were performed. Tests have shown that graphene oxide deposited on the steel surface provides an insulating layer. A higher temperature of the samples with a deposited graphene oxide layer was observed during fracture compared to the reference samples.     

A New Creep–Fatigue Interaction Model for Predicting Deformation of Coarse-Grained Soil

Studying the creep–fatigue interaction of the coarse-grained soil (CGS) is very important for safety assessment and disaster prevention in subgrade engineering. Current research work is mainly focused on single creep or fatigue deformation. In this paper, a new creep–fatigue interaction model is established to predict the creep–fatigue interaction deformation of different gradation CGS based on the rheological mechanics and the interactive relationship between creep and fatigue complex compliance method. Triaxial creep–fatigue interaction tests of different gradations CGS under different average stresses and frequencies were conducted to verify the new creep–fatigue interaction model. Research results show that for the creep–fatigue and fatigue–creep interaction, the fatigue deformation is always larger than the creep deformation under the same stress level. For the creep–fatigue multi-interaction, the second creep and fatigue deformation are always smaller than the first creep and fatigue deformation. The results of the triaxial creep–fatigue interaction tests verify the validity of this new model.     

The Effect of Baking Heat Treatment on the Fatigue Strength and Life of Shot Peened 4340M Landing Gear Steel

In this study, the effect of baking heat treatment on fatigue strength and fatigue life was evaluated by performing baking heat treatment after shot peening treatment on 4340M steel for landing gear. An ultrasonic fatigue test was performed to obtain the S–N curve, and the fatigue strength and fatigue life were compared. The micro hardness of shot peening showed a maximum at a hardened depth of about 50 μm and was almost uniform when it arrived at the hardened depth of about 400 μm. The overall average tensile strength after the baking heat treatment was lowered by about 80–111 MPa, but the yield strength was improved by about 206–262 MPa. The five cases of specimens showed similar fatigue strength and fatigue life in high cycle fatigue (HCF) regime. However, the fatigue limit of the baking heat treated specimens showed an increasing tendency rather than that of shot peening specimens when the fatigue life was extended to the very high cycle fatigue (VHCF) regime. The effect of baking heat treatment was identified from improved fatigue limit when baking heat was used to treat the specimen treated by shot peening containing inclusions. The optimum temperature range for the better baking heat treatment effect could be constrained not to exceed maximum 246 °C.     

Stress-Shot-Peened Leaf Springs Material Analysis through Nano- and Micro-Indentations

Heat-treated and shot-peened lightweight steels with demanding requirements for durability are applied in high-performance automotive leaf springs. Due to their heat-treatment they exhibit degraded properties in the surface-near area compared to the core. This area, which may extend until 300 μm from the surface to the core, experiences the highest bending stresses at operation. The microstructure in the surface and sub-surface layers determines the mechanical performance as well as the wear resistance. The present study refers to the material properties of a stress shot-peened 51CrV4 steel at various depths from the surface. The effect of the manufacturing process has been captured both by Vickers micro-hardness measurements and nanoindentation. The latter combined with a Fine Element Method (FEM)-based algorithm enables the determination of variations in the material’s stress–strain curves over the affected layers, which translate to internal stress changes. The nanoindentation technique has been applied here successfully for the first time ever on leaf springs. The combination of microstructural analysis, microhardness and nanoindentation captures the changes of the treated material, offering insights on the material characteristics, and yielding accurate elastoplastic material properties for local, layered-based analysis of the components’ mechanical performance at operational loading scenarios, i.e., in the framework of stress shot-peening simulation models.     

Study of the Structure and Mechanical Properties after Electrical Discharge Machining with Composite Electrode Tools

Our study was devoted to increasing the efficiency of electrical discharge machining of high-quality parts with a composite electrode tool. We analyzed the chemical composition of the surface layer of the processed product, microhardness, the parameter of roughness of the treated surface, residual stresses, and mechanical properties under tension and durability with low-cycle fatigue of steel 15. Our objective was to study the effect of the process of copy-piercing electrical discharge machining on the performance of parts using composite electrode tools. The experiments were carried out on a copy-piercing electrical discharge machining machine Smart CNC using annular and rectangular electrodes; electrode tool materials included copper, graphite, and composite material of the copper–graphite system with a graphite content of 20%. The elemental composition of the surface layer of steel 15 after electrical discharge machining was determined. Measurements of microhardness (HV) and surface roughness were made. Residual stresses were determined using the method of X-ray diffractometry. Metallographic analysis was performed for the presence of microdefects. Tensile tests and low-cycle fatigue tests were carried out. The mechanical properties of steel 15 before and after electrical discharge machining under low-cycle fatigue were determined. We established that the use of a composite electrode tool for electrical discharge machining of steel 15 does not have negative consequences.     

Fatigue Behavior of Linear Friction Welded Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo-0.1Si Dissimilar Welds

The use of joints fabricated from dissimilar titanium alloys allows the design of structures with local properties tailored to different service requirements. To develop welded structures for aerospace applications, particularly under critical loading, an understanding of the fatigue behavior is crucial, but remains limited, especially for solid-state technologies such as linear friction welding (LFW). This paper presents the fatigue behavior of dissimilar titanium alloys, Ti–6Al–4V (Ti64) and Ti–6Al–2Sn–4Zr–2Mo–0.1Si (Ti6242), joined by LFW with the aim of characterizing the stress versus number of cycles to failure (S-N) curves in both the low- and high-cycle fatigue regimes. Prior to fatigue testing, metallurgical characterization of the dissimilar alloy welds indicated softening in the heat-affected zone due to the retention of metastable β, and the typical practice of stress relief annealing (SRA) for alleviating the residual stresses was effective also in transforming the metastable β to equilibrated levels of α + β phases and recovering the hardness. Thus, the dissimilar alloy joints were fatigue-tested in the SRA (750 °C for 2 h) condition and their low- and high-cycle fatigue behaviors were compared to those of the Ti64 and Ti6242 base metals (BMs). The low-cycle fatigue (LCF) behavior of the dissimilar Ti6242–Ti64 linear friction welds was characterized by relatively high maximum stress values (~ 900 to 1100 MPa) and, in the high-cycle fatigue (HCF) regime, the fatigue limit of 450 MPa at 10⁷ cycles was just slightly higher than that of the Ti6242 BM (434 MPa) and the Ti64 BM (445 MPa). Fatigue failure of the dissimilar titanium alloy welds in the low-cycle and high-cycle regimes occurred, respectively, on the Ti64 and Ti6242 sides, roughly 3 ± 1 mm away from the weld center, and the transitioning was reasoned based on the microstructural characteristics of the BMs.     

Three-Body Abrasive Wear Behavior of WC-10Cr3C2-12Ni Coating for Ball Mill Liner Application

Carbide coatings are frequently used to improve the wear resistance of industrial components in various wear environments. In this research, aiming at the service characteristics of easy wear and short service life of ball mill liners, WC–10Cr₃C₂–12Ni coatings were prepared by supersonic flame spraying technology (HVOF). The reciprocating sliding tests were conducted under four different WC particle size conditions, and the differences in the tribological behavior of the coatings and three–body abrasive wear mechanism were obtained. The findings reveal that the average nanohardness of the WC–Cr₃C₂–Ni coating is nearly five times greater than that of the steel substance. The COF of tribo-pairs decreases and then increases as the particle size increases. In the case of no particles, the surface of the coating is slightly worn, with fatigue and oxidative wear being the primary wear mechanisms. Small particles (1.5 μm and 4 μm) are crushed and coated on the coating surface, in which the extremely fine particles are plasticized to form friction layers that have a protective effect on the coatings. The protective effect of the particles disappears as the particle size increases and is replaced by a powerful chiseling effect on the coatings, resulting in serious material loss. The particle size has a direct relationship with coating wear.     

Effect of Rod-like Structure on Fatigue Life,Short Surface Crack Initiation and Growth Characteristics of Extruded Aluminum Alloy A2024 (Analysis via Modified Linear Elastic Fracture Mechanics)

In the Al alloy A2024-T3 extruded material, a rod-like structure is generated parallel to the extrusion direction. In this study, the effects of rod-like structures on fatigue crack initiation and growth behavior were comprehensively investigated. Two types of specimens were used in a fatigue experiment, in which the direction of the load stress amplitude was parallel (specimen P) and perpendicular (specimen V) to the rod-like structure. Based on the experimental and analytical results, the following findings were obtained regarding the fatigue life, location of crack initiation, and fatigue crack growth behavior. Because the fatigue life of specimen P was longer than that of specimen V, it is inferred that the rod-like structure significantly affects the fatigue life. In specimen P, fatigue cracks were generated from the grain boundaries of the Al matrix. By contrast, in specimen V, cracks were generated from the Cu–Mg-based intermetallic compound in the Al matrix. In specimen P, fatigue cracks were more likely to propagate across the rod-like structure, which decreased the fatigue crack growth rate. In specimen V, fatigue cracks did not propagate across the rod-like structure; instead, they propagated through the Al matrix. Therefore, the fatigue crack growth resistance of specimen V was lower than that of specimen P. The relationship between the fatigue crack growth rate and the modified linear elastic fracture mechanics parameter could be used to predict the S–N curve (stress amplitude vs. fatigue life) and fatigue crack growth behavior. The predicted results agreed well with the experimental results.     

Fatigue Life Prediction of Machined Specimens with the Consideration of Surface Roughness

The fatigue strength and fatigue life of high-strength steels are greatly affected by their surface roughness. This study investigates the underlying mechanisms responsible for fatigue failure of the high-strength steel 42CrMo. Bending fatigue tests of stepped shafts with different levels of surface roughness were conducted to observe the fatigue live reduction affected by surface topography. Besides, the mechanical properties of 42CrMo and its strain–life relationship were established. Moreover, the analytical formulas to describe the stress concentration factor (SCF) and fatigue notch factor (FNF) induced by surface topography were introduced. To estimate the fatigue life of machined specimens with the consideration of surface roughness, the elastic portion of the total strain–life curve of the material was revised with the proposed analytical FNF imposed by surface topography. Comparisons between the estimated fatigue lives and experimentally obtained fatigue lives show that the effect of surface roughness on fatigue lives could be estimated effectively and conveniently by the proposed procedure.     

Parameter Interval Uncertainty Analysis of Internal Resonance of Rotating Porous Shaft–Disk–Blade Assemblies Reinforced by Graphene Nanoplatelets

This paper conducts a parameter interval uncertainty analysis of the internal resonance of a rotating porous shaft–disk–blade assembly reinforced by graphene nanoplatelets (GPLs). The nanocomposite rotating assembly is considered to be composed of a porous metal matrix and graphene nanoplatelet (GPL) reinforcement material. Effective material properties are obtained by using the rule of mixture and the Halpin–Tsai micromechanical model. The modeling and internal resonance analysis of a rotating shaft–disk–blade assembly are carried out based on the finite element method. Moreover, based on the Chebyshev polynomial approximation method, the parameter interval uncertainty analysis of the rotating assembly is conducted. The effects of the uncertainties of the GPL length-to-width ratio, porosity coefficient and GPL length-to-thickness ratio are investigated in detail. The present analysis procedure can give an interval estimation of the vibration behavior of porous shaft–disk–blade rotors reinforced with graphene nanoplatelets (GPLs).