Johnson-Cook Constitutive Model Research Articles

Parametric Analysis and Improvement of the Johnson-Cook Model for a TC4 Titanium Alloy

Titanium alloys are widely used in the manufacture of gas turbines’ compressor blades. Elucidating their mechanical behavior and strength under damaged conditions is the key to evaluating the equipment’s reliability. However, the conventional Johnson-Cook (J-C) constitutive model has limitations in describing the dynamic response of titanium alloy materials under the impact of a high strain rate. In order to solve this problem, the mechanical behavior of a TC4 titanium alloy under high strain rate and different temperature conditions was analyzed by combining experiments and numerical simulations. In this study, the parameters of the J-C model were analyzed in detail, and an improved J-C constitutive model is proposed, based on the new mechanism of the strain rate strengthening effect and the temperature softening effect, which improves the accuracy of the description of strain sensitivity and temperature dependence. Finally, the VUMAT subroutine of ABAQUS software was used for numerical simulation, and the predictive ability of the improved model was verified. The simulation results showed that the maximum prediction error of the traditional J-C model was 23.6%, while the maximum error of the improved model was reduced to 5.6%. This indicates that the improved J-C constitutive model can more accurately predict the mechanical response of a titanium alloy under an impact load and provides a theoretical basis for the study of the mechanical properties of titanium alloy blades under subsequent conditions of foreign object damage.

Corrected Method for Scaling the Structural Response Subjected to Blast Load

In scale-down tests of ship structures subjected to a blast load, the accuracy of the predicted response of a prototype is affected by the material substitution and geometric distortion between a scaled model and a full-size structure; this is known as incomplete similarity. To obtain a more accurate response from a prototype during small-size tests, a corrected method for scaling the response of thin plates and stiffened plates under a blast load was derived. In addition, based on numerical simulations of explosion responses by employing the elastic–plastic model and the Johnson–Cook constitutive model, it was found that using the average yield stress derived from the equivalent plastic strain energy in the ideal elastic–plastic model can obtain consistent structural responses. Moreover, a method for calculating the distortion factor caused by the yield stress of different materials was proposed. Furthermore, it was demonstrated that the average effective plastic strain between the prototype and the corrected model is equal, and based on this, a similarity prediction method was established to correct the distortions caused by yield stress and the thickness of blast loaded plates. The results indicate that the proposed correction method can compensate for the differences caused by distorted factors of yield stress and thickness, with the maximum error in the structure’s peak displacement being less than 3%.

Determination and Verification of the Johnson–Cook Constitutive Model Parameters in the Precision Machining of Ti6Al4V Alloy

Numerical simulations of the cutting process play a key role in manufacturing and cost optimization. Inherent in finite element analysis (FEA) simulations is the correct description of material behavior during machining. For this purpose, various material models are used to describe the behavior of the material in the range of high deformation, high temperature values, and high strain rates. Very often the Johnson–Cook (JC) material model is used for this purpose; however, the correct determination of the material constants of this model is a key aspect. Therefore, this paper presents a procedure for determining the material constants of the JC model using an analytical method based on normalized tensile and compression testing of the material for different strain rates over a wide temperature range. After determining the material constants, the authors conducted numerical simulations of the orthogonal turning of Ti6Al4V titanium alloy using the obtained constants. Validation of the obtained results with those obtained in experimental studies was also carried out. The outcomes demonstrated that the difference between FEM simulation and experimental tests did not exceed 0.02 mm (14%) in the case of chip thickness,. Much smaller differences were obtained for the temperature in the cutting zone, where the maximum difference was about 45 °C (4%). Comparing the components of the cutting force, we found that, in the case of the main cutting force, in most cases, the differences did not exceed 70 N (8%). After the verification of the obtained results, it was also found that the determined material constants of the Johnson–Cook model can be successfully used in FEM modeling of the cutting process of Ti6Al4V titanium alloy for the adopted range of values of technological parameters.

Analytical temperature model for spindle speed selection in additive friction stir deposition

This paper describes a physics-based, analytical model for additive friction stir deposition (AFSD) spindle speed selection to achieve a desired deposition temperature. In the model, power input to the feedstock, which enables plastic flow and deposition, is related to the material temperature rise and subsequent flow stress reduction using Fourier’s conduction rate equation. Power input is modeled as frictional heating at the deposit-surface interface and adiabatic heating due to plastic deformation. The flow stress is predicted using the strain, strain rate, and temperature-dependent Johnson-Cook constitutive model for the selected feedstock alloy. Model predictions are compared to AFSD numerical simulation results available in the literature and experiments for aluminum alloys.

Parameter identification of Johnson–Cook constitutive model based on genetic algorithm and simulation analysis for 304 stainless steel

Addressing the significant discrepancy between actual experimental cutting force and its predicted values derived from traditional constitutive model parameter identification methods, a reverse identification research of the Johnson–Cook (J-C) constitutive model for 304 stainless steel was conducted via genetic algorithm. Considering actual cutting zone and the influence of feed motion on the rake (flank) angle, an unequal division shear zone model was established to implement the theoretical calculation for shear zone stress. Through cutting experiments, the spindle speed was negatively correlated with the cutting force at first, and then became positively correlated; The empirical formula (EXP model) for turning force was corrected, revealing that the EXP model was unable to provide optimal predicted values for cutting force. The influence of the J-C constitutive parameter C on the cutting morphology was firstly investigated through simulation analysis, and determined an appropriate value for C, then obtained the precise values for the other four constitutive parameters by genetic algorithm. Moreover, the simulated values of cutting force in JC1 model (obtained from the Split Hopkinson Pressure Bar test) and JCM model (the improved model using genetic algorithm) were obtained by three-dimensional (3-D) simulation via FEM software. The results indicated that, the maximum error between actual experimental cutting force and its simulated values (by JCM model) was 14.8%, with an average error of 6.38%. These results outperformed the JC1 and EXP models, suggesting that the JCM model identified via genetic algorithm was more reliable.

Heat treatment, microstructure, texture, and mechanical properties of electron beam melted Ti6Al4V

The findings from the microstructural and textural analysis of as-printed and heat-treated electron beam melted Ti-6Al-4V (EBM-Ti64) samples are discussed in this work. The initial microstructures of both samples were characterized precisely. The as-printed microstructure exhibits aligned grain boundaries parallel to the building direction, with the presence of α platelets. In contrast, the heat-treated microstructure reveals a lamellar α/β structure with a considerably wide interlamellar spacing. This work addresses the transformations from β→α+ά and ά→α+β occurring when materials are heated up to below the β-transus temperature. While the ά→α˝+β transformation is deemed unlikely to take place, the process can still occur through spinodal decomposition of β,β→βV−poor+βV−rich→α˝+βV−rich where α˝ eventually transforms into α+β. A thorough characterization of three primary morphologies of the α phase was revealed. Due to exposure to high temperatures, as experienced in the environment of a rocket engine fan, it was necessary to conduct isothermal hot compression experiments on the homogenized EBM-Ti64 alloy. These experiments aim to investigate the hot deformation behavior within the temperature range of 250–350 °C and a strain rate of 10 s−1. The findings indicate that the modified Johnson-Cook constitutive model and its corresponding numerical simulation, based on stress-strain and work-hardening characteristics, effectively predict the flow behavior of this microstructurally complex alloy. Microstructure and mechanical properties of deformed as-printed and heat-treated samples at 250 ℃, 300 ℃, and 350 ℃ show that although dynamic recrystallization is not feasible under these conditions, we observe grain refinement and decreasing interlamellar spacing. These findings align with mechanical properties, where higher total strain and lower ultimate compression strength were noted as the testing temperature increased. Additionally, heat-treated materials exhibited inferior mechanical characteristics under each testing circumstance. Texture rotation (90 degrees) after deformation is evident in all pole figures. Considering the orientation distribution function maps, more intense <0001> fibre and {101̅0} <0001> textures were formed. Assessments of the slip systems revealed that the basal slip system has the lowest activity when considering texture orientations. Conversely, the most significant deformation contribution, particularly in the c-axis, comes from contraction twin and <c+a> 1st order pyramidal.

Inverse Identification of Constitutive Model for GH4198 Based on Genetic-Particle Swarm Algorithm.

A precise Johnson-Cook (J-C) constitutive model is the foundation for precise calculation of finite-element simulation. In order to obtain the J-C constitutive model accurately for a new cast and forged alloy GH4198, an inverse identification of J-C constitutive model was proposed based on a genetic-particle swarm algorithm. Firstly, a quasi-static tensile test at different strain rates was conducted to determine the initial yield strength A, strain hardening coefficient B, and work hardening exponent n for the material's J-C model. Secondly, a new method for orthogonal cutting model was constructed based on the unequal division shear theory and considering the influence of tool edge radius. In order to obtain the strain-rate strengthening coefficient C and thermal softening coefficient m, an orthogonal cutting experiment was conducted. Finally, in order to validate the precision of the constitutive model, an orthogonal cutting thermo-mechanical coupling simulation model was established. Meanwhile, the sensitivity of J-C constitutive model parameters on simulation results was analyzed. The results indicate that the parameter m significantly affects chip morphology, and that the parameter C has a notable impact on the cutting force. This study addressed the issue of missing constitutive parameters for GH4198 and provided a theoretical reference for the optimization and identification of constitutive models for other aerospace materials.

Research on the Mechanical Response and Constitutive Model of 18Ni300 Manufactured by SLM with Different Build Directions.

Metals manufactured by selective laser melting (SLM) with different directions exhibit different mechanical properties. This study conducted dynamic and static mechanical tests using a universal testing machine and split-Hopkinson bar (SHPB). The mechanical properties of 18Ni300 with 0° and 90° build directions manufactured by SLM were compared, and the micro-structure properties of the two build directions were analysed by metallographic tests. The Johnson-Cook (J-C) constitutive model was fitted according to the experimental results, and the obtained constitutive parameters were verified by numerical simulations. The results revealed that the constitutive model could predict the mechanical properties of 18Ni300 in a dynamic state. The build direction had little influence on the mechanical properties in a static state, but there was a significant difference in the dynamic state. The difference in the dynamic compressive yield strength of the 18Ni300 material manufactured by SLM with two build directions was 9.8%. The SLM process can be improved to produce 18Ni300 with uniform mechanical properties by studying the reasons for this difference.

Dynamic mechanical behavior of titanium matrix composites reinforced with graphene nanoplatelets

In this investigation, graphene nanoplatelets reinforced pure titanium matrix (GNPs/TA1) composites were synthesized via short-term ball milling followed by spark plasma sintering. The mechanical properties and microstructure evolution of these composites were examined under both quasi-static and dynamic compression conditions. The results indicate that as the strain rate increases from 10−3 to 3000 s−1, the yield strength rises from 436.9 MPa to 1209.6 MPa, consistent with the positive strain rate effect. The superior yield strength of GNPs/TA1 composites arises from the dynamic Hall-Petch effect, dislocation strengthening and load transfer strengthening, with the contribution from load transfer strengthening being the most significant due to the reinforcement's (TiC-GNPs-TiC) facilitation of load transfer. As the strain rate increases, interfacial debonding gradually extends along the reinforcement and grain boundaries, but no macroscopic fracture occurred in this study. At a strain rate of 3000 s−1 during compression, {112‾2}, {101‾2}, and {11 2‾ 1} twins were generated. However, at a strain rate of 10 s−1, only {112‾2} and {101‾2} twins were produced, while {11 2‾ 1} twin was inhibited. Under high strain rate loading, the plastic flow behavior of GNPs/TA1 composites was predicted using the Johnson-Cook constitutive model, modified to account for adiabatic temperature rise, and the predictions showed good agreement with experimental results.

An improved Johnson–Cook constitutive model for flow stress prediction of 92W–5Co–3Ni alloy

An improved Johnson–Cook constitutive model for flow stress prediction of 92W–5Co–3Ni alloy

A Comparative Blast Mitigation Performance Evaluation of Metallic Sandwich Panels with Honeycomb, Corrugated, Auxetic, and Foam Cores

Due to the high energy absorption capability and excellent bending strength of the sandwich panels, they are mostly preferred as protective structures against explosive blast attacks. The evaluation of their blast-mitigation characteristics through experiments is highly dangerous, costly, time-consuming, and polluting for the environment. Therefore, in this presented work, a series of numerical analyses were performed to evaluate the blast mitigation of the outstanding sandwich panels with honeycomb, corrugated, auxetic, and foam cores. The masses of sandwich panels were kept constant, and their areal densities were maintained constant throughout the study to effectively compare their performance under identical air blast loading conditions. To apply the air blast loads to the designed sandwiches, 1–3[Formula: see text]kg of spherical-shaped trinitrotoluene charges were used for a 100[Formula: see text]mm stand-off distance. The sandwiches are made of high-strength AL-6XN steel and crushable aluminum foam. The rate-dependent Johnson–Cook constitutive model of plasticity and the crushable foam model with volumetric hardening were used for the evaluation of sandwich panels’ plastic deformations. The findings of the work depicted that the honeycomb core is more efficient than the other core structures of the same masses because the sandwich panels with honeycomb core provide smaller back skin deflections, globalized core crushing, and higher core energy absorption than the other sandwich panels for extreme conditions of air blast loading.

A thermo-mechanical fully coupled model for high-speed machining of Ti6Al4V

PurposeThis study aims to further refine the model, explore the influence of cutting parameters on the machining process, and apply it to practical engineering to improve the efficiency and quality of titanium alloy machining.Design/methodology/approachThis paper establishes a comprehensive thermo-mechanical fully coupled orthogonal cutting model. This paper aims to couple the modified Johnson–Cook constitutive model, damage model and contact model to construct a two-dimensional orthogonal cutting thermo-mechanical coupling model for high-speed cutting of Ti6Al4V. The model considers the evolution of microstructures such as plastic deformation, grain dislocation rearrangement, dynamic recrystallization, as well as stress softening and hardening occurring continuously in Ti6Al4V metal during high-speed cutting. Additionally, the model incorporates friction and contact between the tool and the workpiece. It can be used to predict parameters such as cutting process, cutting force, temperature distribution, stress and strain in titanium alloy machining. The study establishes the model and implements corresponding functions by writing Abaqus VUMAT and VFRICTION subroutines.FindingsThe use of different material constitutive models can significantly impact the prediction of the cutting process. Some models may more accurately describe the mechanical behavior of the material, thus providing more reliable prediction results, while other models may exhibit larger deviations. Compared to the Tanh model, the proposed model achieves a maximum improvement of 8.9% in the prediction of cutting force and a maximum improvement of 20.9% in the prediction of chip morphology parameters. Compared to experiments, the proposed model achieves a minimum prediction error of 2.8% for average cutting force and a minimum error of 0.57% for sawtooth parameters. This study provides a comprehensive theoretical foundation and practical guidance for orthogonal cutting of titanium alloys. The model not only helps engineers and researchers better understand various phenomena in the cutting process but also serves as an important reference for optimizing cutting processes.Originality/valueThe originality of this research is guaranteed, as it has not been previously published in any journal or publication.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-05-2024-0168/

Finite element method based modeling of cutting forces and cutting temperature in micro-milling LF21

The demands for aluminum alloy LF21 micro precision parts are increasing in the fields of aerospace, high-tech electronic products, and the other fields. Micro-milling is an effective technology for machining small LF21 precision parts. Cutting forces and temperature are crucial factors in micro-milling process, directly affect tool vibration, tool wear, and surface quality of the workpiece, and even result in large deformation of the tool and workpiece. Direct measurement of cutting forces during micro-milling requires high-precision and expensive instruments. Moreover, due to the small cutting area in micro-milling, it is challenging to achieve accurate measurements of cutting area temperature. Therefore, accurate prediction of cutting forces and temperature in micro-milling is urgent and challenging. Nowadays, there are few studies on prediction of cutting forces in micro-milling LF21. The study on prediction of temperature in micro-milling LF21 is still blank. To solve the above problems, this paper proposes a finite element method based on modeling for prediction of cutting forces and temperature in micro-milling LF21. ABAQUS software is adopted. First, the geometry models of the micro-milling tool and workpiece are established. Then, the assembly and mesh division of the established models are completed. Johnson-Cook constitutive model and Johnson-Cook damage criteria are used to describe the material constitutive relationship and chip separation criteria, respectively. The suitable tool-workpiece friction models are determined. Finally, the simulation of the micro-milling LF21 process is achieved. Experiments of micro-milling LF21 are conducted and the cutting forces are measured using the dynamometer. The validity of the built process simulation model and the correctness of the cutting force prediction results are verified by the comparison of experiment and simulation cutting forces. Then, the prediction of temperature is achieved based on the verified process simulation model of micro-milling LF21.

Dynamic deformation behavior of Fe-0.22C-1.65Si-1.98Mn quenching and partitioning steel over a wide range of strain rates and constitutive model establishment

During traffic accidents, automobiles often experience rapid deformation at high strain rates spanning from 10² to 10³ s⁻¹. The dynamic deformation behavior and the constitutive model of the materials are crucial for precisely analyzing and simulating vehicle collisions and damage. This study focuses on a high-strength quenching-partitioning steel (Q&P steel) compositionally tailored for automotive applications, specifically Fe-0.22C-1.65Si-1.98Mn. Tensile tests were conducted under quasi-static (10⁻⁴–10⁻¹ s⁻¹) and dynamic (1–10³ s⁻¹) conditions to obtain stress-strain data, based on which the deformation behavior and instability mechanism of the Q&P steel were revealed. A Johnson-Cook (J-C) constitutive model was established to predict the stress-strain behavior. This model was further improved by incorporating the nonlinear influence of strain rate on stress, resulting in a significant reduction of prediction error by 60.9 % in the high strain rate range. Based on the modified J-C model, the finite element simulations of the high-speed collision for U-shaped and tubular components were conducted. The simulations exhibit excellent agreement with experimental results, thus confirming the accuracy of the model in predicting high-speed deformation behavior of the Q&P steel. This validation enables precise simulations of dynamic collision characteristics and energy absorption.

High-speed perforation of IN718 plates by spherical TC4 Ti alloy projectiles: Experiments and modeling

Data and knowledge on ballistic performance of Inconel-718 (IN718) alloy is critical for evaluating the resistance of turbine blades against foreign object impacts in aeronautical and astronautical industries. Ballistic perforation experiments (impact velocity 480 ms−1 to 1340 ms−1) with TC4 spherical projectiles (5-mm-diameter) are conducted on IN718 target plates (2-mm-thick) in this work, along with high-speed photography. The IN718 plates show multiple deformation and failure modes (bulging, dishing, petaling and plugging). The ballistic limit velocity for the investigated projectile/target combination is 823 ms−1. Postmortem material characterizations (optical photography, scanned electron microscopy and energy dispersive spectrometry) are conducted on recovered projectiles and bullet hole inner walls, and the observations indicate adiabatic shearing, surface melting and abrasion of projectile material. A finite element model based on the Johnson–Cook constitutive model and failure criterion is established, and reproduces the ballistic experiments. Dimensionless analyses are conducted on the dimensions of postmortem projectiles and bullet holes, and the projectile residual velocity/mass. On the basis of experimental observations and theoretical analysis of the projectile dynamics, the whole perforation process is divided into three stages (the entering, cratering and plugging stages). A multi-stage analytical model on perforation is then developed considering the effects of cavity expansion, projectile deformation and abrasion, and can well predict the projectile dynamics during entering and cratering stages.

Dynamic constitutive model of Fe–Cr–Ni stainless steel based on isothermal true stress-strain curves

Austenitic Fe–Cr–Ni stainless steel is widely used in aviation, chemistry, energy, due to its excellent properties of high-temperature performance. In this study, the splitting Hopkinson pressure bar with high-temperature system was employed to evaluate the dynamic mechanical properties of Fe–Cr–Ni stainless steel. The true stress-strain curves were obtained under varying conditions, including variable strains, strain rates and temperatures. The true stress increases and levels off as the true strain increases, while increases as the strain rate increases, but decreases sharply as the deformation temperature rises. The deformation temperature is consist of healing temperature and adiabatic temperature. The adiabatic temperature rise related to the specific heat capacity was calculated. The actual deformation temperatures were calculated under different strains by combining the true stress-strain curves. The true stress-strain curve under variable temperature was corrected to the stress-strain curve under isothermal state by using the thermal softening rate, which decoupled the strain and temperature. The Power-Law and Johnson-Cook constitutive models were fitted based on the real stress-strain isothermal curve. The fitting accuracy of Power-Law model was 1.61% for different strain rates at room temperature in average, 3.51% for fixed strain rate at different temperatures. While the fitting accuracy of Johnson-Cook model was 2.94% for different strain rates at room temperature in average, 6.18% for fixed strain rate at different temperatures.

Foreign object damage characteristics of a thin nickel-based superalloy plate at room and high temperatures

Turbine blades with thin-walled structures usually works in harsh environments, and foreign object damage (FOD) is one of the conditions of special concern. In this paper, the FOD characteristics of thin nickel-based superalloy plates are studied by a combination of experimental, numerical and analytical methods, considering room and high temperatures, different impact conditions and plate thicknesses. An easy-to-use test system is developed to realize high speed impact of the thin nickel-based superalloy plate under elevated temperature. Crater morphologies, internal microstructure, and residual stress are analyzed after impact with different conditions. Numerical simulation of the impact process is performed by using Johnson-Cook (J-C) constitutive model. Based on Hertz theory, an analytical method for calculating the crater length and depth is proposed considering the deformation of the impact steel sphere. Results shows that the FOD characteristics at high temperature is significantly different from that at room temperature. The crater has lager dimensions under high speed and elevated temperature. Moreover, significant grain refinement is obvious and the dislocation layer is also thicker at higher speed and higher temperature. Due to the effect of high temperature softening, hardness and residual stress after impact with elevated temperature is lower than that at room temperature. Besides, non-normal impact mainly influences Goss texture and distribution of residual stress after high temperature impact. In addition, it is found that thickness have a significant effect on the FOD characteristics especially when the plate is thinner. The validity of the numerical model and analytical method is proved by comparing with the experimental results. The present study can provide data foundation and numerical analysis support for the damage assessment and maintenance of turbine blades.

Modified Johnson-Cook Constitutive Model for Dynamic Compressive Behaviors of C250 Maraging Steel at Different Temperatures

Modified Johnson-Cook Constitutive Model for Dynamic Compressive Behaviors of C250 Maraging Steel at Different Temperatures

Modeling of laser-assisted micro-milling Inconel718

Nickel-based superalloy Inconel718 is a typical difficult-to-machine material with the characteristic of high strength and hardness. Demands for micro parts of nickel-based superalloy Inconel718 are growing in aerospace, biomedical and other fields. Laser-assisted micro-milling (LAMM) can realize precision machining of the workpiece because of its lower cutting force. However, the machining mechanism and the change law of cutting force during LAMM of nickel-based superalloy Inconel718 are still unclear. There is a crucial need for modeling of LAMM Inconel718. This paper explores LAMM of nickel-based superalloy Inconel718 and realizes the simulation of LAMM process. A three-dimensional transient heat transfer finite element (FE) model is established to predict the temperature distribution of a Gaussian heat source in LAMM. Multiple sets of simulations of laser pre-heating with various laser power are conducted to analyze the temperature fields of workpiece and determine the appropriate laser parameters. A novel 3D model of the full machining process of LAMM Inconel 718 is constructed by ABAQUS with multi-subroutine, a fully coupled thermal-mechanical process is achieved. A modified Johnson-Cook constitutive model is used to reflect the size effect and flow stress of LAMM Inconel 718. Finally, the validity of the models is verified by experiments. Based on the verified model, the appropriate laser pre-heating parameters are obtained, and the prediction of cutting forces of LAMM Inconel718 is achieved.

A hardness-based constitutive model for numerical simulation of blind rivet with gradient hardness distribution

A286 superalloy blind rivets find extensive applications in the structural connections of aerospace equipment owing to their exceptional strength and heat resistance. However, the gradient hardness distribution within the rivet sleeves complicates achieving precise numerical simulation results with traditional constitutive models. This study aims to establish a constitutive model suitable for materials exhibiting gradient hardness to support the development of A286 superalloy blind rivets. Initially, quasi-static compression experiments were conducted on A286 superalloy samples at different hardness levels (160–324 HV), strain rates (0.01–10 s⁻¹), and temperatures (25–600°C). Subsequently, the influence of material microstructural evolution on flow stress during compression was investigated using electron backscatter diffraction technique. Furthermore, a hardness-based Johnson-Cook (J-C) constitutive model was developed by introducing hardness and offset compensation onto the original J-C model and compared with its counterpart. Finally, the hardness-based model was employed to simulate the bulge compression forming of A286 blind rivets and validated through experimental trials. The findings reveal that dislocation strengthening predominantly contributes to the increased material flow stress with escalating strain rates and sample hardness. Compared to the original J-C model, the stress prediction error of the hardness-based model reduced from 26.8 % to 2.9 %, effectively depicting the variation in material flow stress under different hardness values. The simulated values of bulge morphology, dimensions, and load closely align with experimental data, affirming the efficacy of the hardness-based model in numerically simulating blind rivets with gradient hardness distribution.