Modified Zerilli-Armstrong Model Research Articles

Optimization method of parameters inverse identification for hot deformation constitutive model of 2Cr13 martensitic stainless steel using genetic algorithm

Constitutive models are considered a key prerequisite to investigate the thermal deformation behavior of materials. Accurate identification of their parameters is crucial for enhancing the predictive precision of the models. A novel optimization method for accurate inverse identification of parameters using the genetic algorithm (GA) in the Modified Zerilli-Armstrong (MZA) model was proposed in this work. Under conditions of temperatures ranging from 1223 K to 1473 K at intervals of 50 K and strain rates of 0.01, 0.1, 1, and 5 s−1, uniaxial isothermal hot compression tests were performed on 2Cr13 martensitic stainless steel (MSS) employing a Gleeble-1500D thermal simulation test machine. Based on the experimental data, the conventional linear regression method was utilized to solve the MZA model of 2Cr13 MSS. Initial values for the material constants I1, S1, C5, and C6 to be optimized were assigned drawing from the calculated results. The GA-based iteratively optimized MZA (G-ZA) model was established by minimizing the mean squared error as an objective function between the experimental and predicted flow stress values. Compared to the MZA model, a significant enhancement in predictive performance was achieved with the G-ZA model, while good generalization was also demonstrated. Both models were successfully integrated into the Forge® finite element analysis software through the secondary development of user subroutines. Numerical simulation results indicated that the G-ZA model demonstrated a better agreement with the experimental data in predicting the load-displacement curve. This validated that a more accurate constitutive model for the hot deformation of 2Cr13 MSS can be effectively constructed using the GA-based parameter inverse identification strategy.

Modified Fields-Backofen and Zerilli-Armstrong constitutive models to predict the hot deformation behavior in titanium-based alloys

This work presents modifications for two constitutive models for the prediction of the flow behavior of titanium-based alloys during hot deformation. The modified models are the phenomenological-based Fields-Backofen and the physical-based Zerilli-Armstrong. The modifications are derived and suggested by studying the hot deformation of titanium-based alloy Ti55531. The predictability of the modified models along with the original Fields-Backofen and another modified Zerilli-Armstong models is assessed and evaluated using the well-known statistical parameters correlation coefficient (R), Average Absolute Relative Error (AARE), and Root Mean Square Error (RMSE), for the Ti55531 alloy, and validated with other two different titanium-based alloys SP700 and TC4. The results show that the modified Fields-Backofen gives the best performance with R value of 0.996, AARE value of 3.34%, and RMSE value of 5.64 MPa, and the improved version of the modified Zerilli-Armstrong model comes in the second-best place with R value of 0.992, AARE value of 3.52%, and RMSE value of 9.15 MPa for the Ti55531 alloy.

Modelling the Flow Behaviour of Al Alloy Sheets at Elevated Temperatures Using a Modified Zerilli-Armstrong Model and Phenomenological-Based Constitutive Models.

The flow behaviour of AA2060 Al alloy under warm/hot deformation conditions is complicated because of its dependency on strain rates (ε˙), strain (ε), and deformation modes. Thus, it is crucial to reveal and predict the flow behaviours of this alloy at a wide range of temperatures (T) and ε˙ using different constitutive models. Firstly, the isothermal tensile tests were carried out via a Gleeble-3800 thermomechanical simulator at a T range of 100, 200, 300, 400, and 500 °C and ε˙ range of 0.01, 0.1, 1, and 10 s-1 to reveal the warm/hot flow behaviours of AA2060 alloy sheet. Consequently, three phenomenological-based constitutive models (L-MJC, S1-MJC, S2-MJC) and a modified Zerilli-Armstrong (MZA) model representing physically based constitutive models were developed to precisely predict the flow behaviour of AA2060 alloy sheet under a wide range of T and ε˙. The predictability of the developed constitutive models was assessed and compared using various statistical parameters, including the correlation coefficient (R), average absolute relative error (AARE), and root mean square error (RMSE). By comparing the results determined from these models and those obtained from experimentations, and confirmed by R, AARE, and RMSE values, it is concluded that the predicted stresses determined from the S2-MJC model align closely with the experimental stresses, demonstrating a remarkable fit compared to the S1-MJC, L-MJC, and MZA models. This is because of the linking impact between softening, the strain rate, and strain hardening in the S2-MJC model. It is widely known that the dislocation process is affected by softening and strain rates. This is attributed to the interactions that occurred between ε and ε˙ from one side and between ε, ε˙, and T from the other side using an extensive set of constants correlating the constitutive components of dynamic recovery and softening mechanisms.

Evaluation of constitutive equations for modeling and characterization of microstructure during hot deformation of sintered Al–Zn–Mg alloy

Commercial simulation software like QFORM and DEFORM do not possess adequate material data for alloys produced through powder metallurgy. Therefore, in the present study, hot compression tests of Al-5.6%Zn–2%Mg aluminum alloy were conducted at temperature ranges (300 0C-500 °C) and strain rates (0.1 s−1-0.0001 s−1). The impact of compression temperature and strain rate on the flow curve behaviour and microstructure evolution-associated mechanisms were investigated through experimentally acquired flow curves and EBSD analysis. Four constitutive models were constructed; namely the Arrhenius-type, modified Johnson Cook (MJC), modified Zerilli-Armstrong (MZA), and an artificial neural network (ANN). The results demonstrated that among the models considered, the ANN and Arrhenius-type models exhibited the lowest AARE values of 0.486 % and 3.36 %, respectively. Conversely, the MZA and MJC models displayed higher AARE values of 8.84 % and 3.93 %, respectively. Notably, the Arrhenius-type model emerged as the most suitable prediction model due to its capability to handle nonlinear relationships between factors. However, in scenarios where material properties are unknown or experimental data is limited, the MJC model can serve as a simpler alternative. The MZA model was deemed unsuitable for accurately estimating flow stress in hot compression. Remarkably, the best-trained ANN model exhibited the highest predictive performance with an AARE of 0.486 % and an R-value of 0.99. This study offers fundamental insights to improve the accuracy of simulating hot compression procedures. EBSD analysis demonstrated that higher upsetting temperatures and lower strain rates promoted recrystallization, resulting in a more uniform microstructure.

Warm Deformation Behavior and Flow Stress Modeling of AZ31B Magnesium Alloy under Tensile Deformation.

Constitutive equations were recognized for AZ31B magnesium alloy at higher temperatures and strain rates from conventional empirical models like the original Johnson-Cook (JC), modified JC, and modified Zerilli-Armstrong (ZA) models for capturing the material warm deformation behavior. Uniaxial warm tensile tests were performed at temperatures (50 to 250 °C) and strain rates (0.005 to 0.0167 s-1) to probe AZ31 magnesium alloy flow stress values. Depending on the calculated flow stress, constitutive equations were recognized, and these established models were assessed by the coefficient of determination (R2), relative mean square error (RMSE), and average absolute relative error (AARE) metrics. The results demonstrated that the flow stress calculated by the modified JC and ZA models revealed good agreement against the test data. Thus, the outcomes confirmed that the recognized modified JC and modified ZA models could effectively forecast AZ31 magnesium alloy flow behavior by capturing the material deformation behavior accurately.

Dynamic compression behavior of TiZrNbV refractory high-entropy alloys upon ultrahigh strain rate loading

In this study, the dynamic compressive response behavior of a body-centered cubic (BCC) single-phase TiZrNbV refractory high-entropy alloy (RHEA) was investigated under impact at speeds of 313–1584 m s–1 using two-stage, gas-gun-driven, high-speed plate-impact experiments; recovery sample analysis; and theoretical calculations. The strain rate and pressure were approximately 107 s−1 and 5.07–29.37 GPa, respectively. The results showed that the TiZrNbV RHEA had a Hugoniot elastic limit of 4.12–5.86 GPa and a spall strength of 1.84–2.03 GPa. The initial yield strength of the alloy showed a strong strain-rate dependence and could be described by the modified Zerilli–Armstrong model, while the phonon-damping effect was the main reason for its high strain-rate sensitivity. Microstructural analysis showed that the dynamic deformation of the TiZrNbV RHEA was controlled by the dislocation slip, dislocation proliferation, intersection of the deformation bands, and grain refinement. The analysis also showed that the intergranular, transgranular, and mixed-type cracks dominated the spall failure of the material. The dynamic Hall–Petch effect and pinning from the lattice distortion led to high dynamic yield strength. The critical strain rate for the phonon drag effect was positively related to the relative atomic mass and local strain field of the metals. Within the experimental loading range, the RHEA showed good structural stability, and simultaneously, the theoretical calculation method for the equation of state based on a cold-energy mixture could accurately predict its shock-response behavior. The valence-electron concentration (VEC) had a direct effect on the shock-compression properties of the HEAs; higher VEC implied more difficulty in compressing the HEAs. The findings of this study provide insights into understanding the mechanical response characteristics of RHEAs under extreme conditions such as high-speed impact and ultrahigh strain-rate loading.

Interpolation and Extrapolation Performance Measurement of Analytical and ANN-Based Flow Laws for Hot Deformation Behavior of Medium Carbon Steel

In the present work, a critical analysis of the most-commonly used analytical models and recently introduced ANN-based models was performed to evaluate their predictive accuracy within and outside the experimental interval used to generate them. The high-temperature deformation behavior of a medium carbon steel was studied over a wide range of strains, strain rates, and temperatures using hot compression tests on a Gleeble-3800. The experimental flow curves were modeled using the Johnson–Cook, Modified-Zerilli–Armstrong, Hansel–Spittel, Arrhenius, and PTM models, as well as an ANN model. The mean absolute relative error and root-mean-squared error values were used to quantify the predictive accuracy of the models analyzed. The results indicated that the Johnson–Cook and Modified-Zerilli–Armstrong models had a significant error, while the Hansel–Spittel, PTM, and Arrhenius models were able to predict the behavior of this alloy. The ANN model showed excellent agreement between the predicted and experimental flow curves, with an error of less than 0.62%. To validate the performance, the ability to interpolate and extrapolate the experimental data was also tested. The Hansel–Spittel, PTM, and Arrhenius models showed good interpolation and extrapolation capabilities. However, the ANN model was the most-powerful of all the models.

Flow stress modeling of a new HSLA steel by Zerilli-Armstrong model

The selection of an appropriate constitutive model is vital for the design of thermo-mechanical treatment as it describes the flow patterns and governs the metallurgical transformations. In this research, the constitutive behavior of a newly developed high-strength steel has been studied employing a physical-based Zerilli-Armstrong model. For this purpose, the experimentally generated stress–strain data from isothermal compression tests performed over a wide range of temperatures (1073–1323 K) and strain rates (0.01–10 s−1) have been considered. The constitutive relationships of flow stress as functions of temperature, strain rate, and true strain have been developed for both original and modified Zerilli-Armstrong models. The comparison between the measured and predicted flow stresses demonstrates better predictability of the modified model with a higher correlation coefficient (R = 0.982) and a much lower average absolute relative error (AARE = 7.31%) since the values of R and AARE for the original model are 0.845 and 23.96%, respectively. The better accuracy of the modified Zerilli-Armstrong model is attributed to the fact that it additionally considers the coupled effects of temperature and strain, unlike the original Zerilli-Armstrong model.

Modified Zerilli-Armstrong and Khan-Huang-Liang constitutive models to predict hot deformation behavior in a powder metallurgy Ti-22Al-25Nb alloy

In order to establish constitutive models for a fine-grained Ti-22Al-25Nb alloy fabricated by spark plasma sintering under high vacuum, isothermal uniaxial compression tests are conducted at the different deformation conditions of 950–1070 °C and 0.001–1 s−1. Using the friction-corrected experimental data, the modified Zerilli-Armstrong model and the Khan-Huang-Liang model are developed in the α2 + β/B2 + O triple-phase and α2 + B2 two-phase fields, respectively. The determination coefficient (R2) and average absolute relative error (AARE) of the developed models are 0.9773 and 8.73%, 0.9831 and 12.36%, respectively. Based on the analyzing the reason of large deviation, the modified Zerilli-Armstrong and Khan-Huang-Liang models are improved. The prediction accuracy of the improved constitutive models is improved to 0.9896 and 6.14%, 0.9891 and 6.82%, respectively. This indicates that the improved versions of the modified ZA and KHL models are very appropriate for predicting the hot deformation behavior of a fine-grained Ti-22Al-25Nb alloy fabricated by spark plasma sintering.

Application of Four Different Models for Predicting the High-Temperature Flow Behavior of 1420 Al–Li Alloy

In this paper, the high-temperature rheological behavior of 1420 aluminum–lithium alloy under experimental conditions (temperatures of 350–475 °C and strain rates of 0.01–10 s−1) was systematically investigated using a Gleeble-3500 thermal simulation tester (temperature 350~475 °C, strain rate of 0.01~10 s−1). Based on the flow stress curves of this alloy, four different types of high-temperature constitutive models of the alloy were constructed: the Arrhenius (AR) model, the Modified Johnson–Cook (MJC) model, the Modified Zerilli–Armstrong (MZA) model, and the VOCE model. The prediction accuracy of the four constitutive models was compared, and the response of the accuracy of the four constitutive models to the deformation parameters (temperature, deformation rate, and strain) was analyzed. The results showed that the VOCE, AR, and MZA models had high overall prediction accuracy with average absolute relative error (AARE) of 1.8933%, 3.9912%, and 7.8422%, respectively. The VOCE model could achieve the prediction of large strain deformation resistance under small strain with small batch experimental conditions for the corresponding conditions. The AR model had optimal prediction accuracy for the high-rate deformation process. The MJC model had the optimal prediction accuracy for the low-temperature low-rate deformation process. The MZA model had better prediction accuracy for the low-rate high-temperature deformation process. The 1420 aluminum–lithium alloy process parameters selection area constitutive model matching diagram was constructed.

A comparative study on modified and optimized Zerilli-Armstrong and arrhenius-type constitutive models to predict the hot deformation behavior in 30Si2MnCrMoVE steel

To better understand the elevated temperature flow behaviors of 30Si2MnCrMoVE low-alloy ultra-high strength steel, hot compressive deformation tests were performed at a wide range of forming temperatures (1123–1423 K) and strain rates (0.01–1s−1). The constitutive equations of the material were established through the physical-based modified Zerilli-Armstrong (ZA) and phenomenological Arrhenius-type models, along with the optimized ZA model of strain compensation constructed based on the modified ZA model. Suitability of these models were evaluated by comparing the correlation coefficient and absolute average error between the prediction and experimental data, its ability to predict the deformation behavior, and the amount of calculation required for the material constants. Results show that the modified ZA model cannot accurately describe the flow behavior of the 30Si2MnCrMoVE steel in the abovementioned hot working zone. The optimized ZA model and Arrhenius-Type model based on strain compensation require similar computational effort, and both can effectively predict the high temperature flow behavior of 30Si2MnCrMoVE steel. Relatively, the Arrhenius type model has the highest prediction accuracy.

High strain rate and high temperature mechanical response of additively manufactured alloy 625

Alloy 625 is an important alloy in many industries, including aerospace, providing good mechanical properties in high temperature and corrosive environments. It also retains good properties when additively manufactured using Laser Powder Bed Fusion (LPBF). The LPBF process introduces complex heating and cooling cycles in the material used, thereby affecting the mechanical properties. As a result, existing constitutive models for alloy 625, are not applicable for the LPBF-fabricated material. Therefore, this study sought to establish appropriate constitutive models to simulate the mechanical response of LPBF-fabricated alloy 625 in the desired range of conditions for aerospace applications: high strain rates and temperatures. This was completed by compressing the samples at two strain rates, 700 s−1 and 1700 s−1, and temperatures ranging from 298 K to 773 K using a Split Hopkinson Pressure Bar (SHPB). The information gained from the models was reinforced with micrographs and electron backscatter diffraction (EBSD) images to examine the microstructure of alloy 625 after LPBF. The results from the SHPB testing were then used to calculate the coefficients for five constitutive models, the Johnson-Cook model, a modified Johnson-Cook model, the Hensel-Spittel model, a modified Hensel-Spittel model, and a modified Zerilli-Armstrong model. The Average Absolute Relative Error (AARE) of these models was calculated, and it was determined that the modified Zerilli-Armstrong model had the lowest AARE of the models used, 2.88 % for as-printed alloy 625 and 2.71 % for heat-treated alloy 625.

Modeling of temperature- and stress state-dependent yield and fracture behaviors for Mg-Gd-Y alloy

The research aims at characterizing and modeling the influence of temperature and stress state on the yield and fracture behaviors of an Mg-Gd-Y alloy. The mechanical experiments at 25∼300 ℃ were carried out by various designed specimens, including tension, compression and shear. The experimental result indicates that the strength of the Mg-Gd-Y alloy presents a monotonous and non-uniform downward trend as the temperature increases. The mechanical behavior at 100 ℃ and 150 ℃ are very similar, and the strength rapidly declines when the temperature is higher than 250 ℃. The tension-compression asymmetry presents the obvious nonlinear variation with the temperature and strain, which is weakened at elevated temperature. The fracture behavior belongs to the ductile fracture with tension and shear mechanisms, showing a coupling effect of temperature and stress state. A modified Zerilli-Armstrong model is proposed to characterize the plastic flow behavior with the coefficient of determination about 0.99, which is prior to the Johnson-Cook and Zerilli-Armstrong models. The Cazacu-Barlat2004 yield function is established to describe the temperature- and strain-related non-uniform evolution characteristic, and capture the fracture-related variables of all fracture tests with the acceptable prediction accuracy. The temperature-dependent parameter is introduced into the DF2016 fracture criterion to predict the onset of fracture under various temperatures and stress states with a small prediction error. The numerical simulation is conducted to validate the reliability and practicability of the established model.

A generalized non-linear flow law based on Modified Zerilli–Armstrong model and its implementation into Abaqus/Explicit FEM code

A generalized non-linear flow law based on Modified Zerilli–Armstrong model and its implementation into Abaqus/Explicit FEM code

A Generalized Non-Linear Flow Law Based on Modified Zerilli-Armstrong Model and Its Implementation into Abaqus/Explicit FEM Code

Non-linear numerical modeling, widely used in research and development to understand many complex processes such as forming or machining, does not guarantee the success of a study to be performed. Indeed, the numerical simulation uses finite element codes where the models already integrated are not based on shapes adjustable to any type of study. In this study, a new form of non-linear constitutive flow law based on the Modified Zerilli-Armstrong model, which can answer the above problem, has been developed to apply it to the numerical simulation of two different tests (a quasi-static compression test, the necking of a circular bar). This flow law is based on the modified Zerilli-Armstrong model, which, together with the new modified Johnson-Cook model, has been compared to appreciate the relevance of the proposal. For that, an implementation of this new law via the VUHARD subroutine into the Abaqus/Explicit finite element code was made to model the two tests. The comparison of the results obtained (from identification) by our proposed law with those obtained using the NMJC shows that this new law better approaches the experiments than the other one. This is also shown through the numerical results using the Abaqus software. It can be said that this way of formulating a flow law allows highlighting the great performance of the proposed approach. Although this law has been only used for quasi-static tests, we can say that it can also be used in dynamic tests.

A modified Zerilli–Armstrong constitutive model for simulating severe plastic deformation of a steel alloy

A modified Zerilli–Armstrong model has been proposed and validated in previous works for simulating distinct deformation mechanisms of continuous-shear and shear-localization during severe plastic deformation of a face centered cubic alloy. In this paper, the validity of the modified Zerilli–Armstrong model has been further tested by using it for modeling the severe plastic deformation of another face centered cubic material, a steel alloy. In particular, the modified Zerilli–Armstrong model is used as a constitutive relation for simulating behavior of AISI 1045 steel alloy while undergoing severe plastic deformation through orthogonal and plane-strain machining. Accordingly, the performance of the constitutive relation in predicting flow stress distribution along the primary shear zone is validated by comparing with forecasts made using the distributed primary zone deformation, the original Zerilli-Armstrong and Johnson-Cook models. Furthermore, finite element simulations of orthogonal cutting of this steel alloy were carried out, and good agreement was observed between the predicted chip morphology and attendant cutting forces with experimental values reported in literature for a range of cutting conditions. The force predictions also fared better compared to those predicted by using the Zerilli-Armstrong and Johnson-Cook models. These validations provide further corroboration of using the modified Zerilli–Armstrong model as a constitutive relation for simulating the behavior of face-centered cubic materials under conditions of high plastic strains and also high strain-rates.

A modified Zerilli–Armstrong model as the asymmetric visco-plastic part of a multi-mechanism model for cutting simulations

A modified Zerilli–Armstrong model as the asymmetric visco-plastic part of a multi-mechanism model for cutting simulations

Application of four different models for predicting the high-temperature flow behavior of TG6 titanium alloy

The high-temperature flow behavior of TG6 titanium alloy at strain rates from 0.001 to 10 s−1 and temperatures from 850 to 1050 ℃ was systematically studied by Gleeble-3500 thermal simulation compressor, and four different constitutive equations (classical Arrhenius (AR) model, modified Johnson-Cook (MJC) model, modified Zerilli–Armstrong (MZA) model, artificial neural network (ANN) model) were derived. EBSD results show that the deformation behavior of TG6 titanium alloy has flow softening characteristics. Furthermore, the calculated correlation coefficient (R) and average absolute relative error (AARE) are used to make an error analysis of the fit accuracy of the above models to evaluate the predictive ability of the TG6 titanium alloy flow behavior. The results of the comprehensive analysis are as follows: on the one hand, it is revealed that different predictive capabilities of the high-temperature flow behavior of TG6 titanium alloy by all the four models based on the meticulous experimental comparison and validation (AR model: R2 is 97.376 % and AARE is 14.16 %; MJC model: R2 is 97.42 % and AARE is 18.5 %; MZA model: R2 is 90.598 % and AARE is 19.32 %; ANN model: R2 is 99.94 % and AARE is 2.14 %), but MZA model is a special case. On the other hand, the MJC and AR model can be considered as the best choices for relatively accurate description and prediction of the flow stress behavior of TG6 titanium alloy when the predictability of the model and the material parameters to be considered.

Constitutive Modeling of the Flow Stress Behavior for the Hot Deformation of Cu-15Ni-8Sn Alloys

Three constitutive models, strain-compensated Arrhenius model, modified Johnson–Cook (JC) model, and modified Zerilli–Armstrong (ZA) model, were established for the hot-deformed Cu-15Ni-8Sn alloy based on hot compression tests. By introducing average absolute relative error (AARE), correlation coefficient (R), and relative error, the prediction accuracy of these three models was assessed. The results indicate that strain-compensated Arrhenius model has the highest accuracy at describing the flow stress behavior of the studied alloy, followed by modified JC model and modified ZA model. Moreover, the strain-compensated Arrhenius model established in this work has a great practicability in the hot-extrusion simulation of Cu-15Ni-8Sn alloys. This article provides a theoretical basis for optimizing hot deformation parameters in industrial production of the Cu-15Ni-8Sn alloys.

Finite element simulation of orthogonal machining of biomedical grade Co–Cr–Mo alloy

Cobalt–chromium–molybdenum (Co–Cr–Mo) alloys are used extensively within the biomedical industry for hip, knee and shoulder prostheses. These components are manufactured using a range of different processes which includes machining. However, material properties such as high hardness, high wear resistance and strain rate hardening classifies these alloys as difficult to cut materials. Finite Element (FE) Modelling of machining processes can reduce the number of machining tests required for optimisation. The aim of the present work was to develop a FE model that can predict the orthogonal forces during the machining of biomedical grade Co–Cr–Mo alloy. To achieve this, it was necessary to develop the constitutive material model for this alloy. A modified Zerilli–Armstrong model was found to have the greatest predication capability compared to a modified Johnson–Cook and a strain compensated Arrhenius-type model. An orthogonal cutting finite element model was developed using Deform 3D over a range of different feed rates and cutting speeds. The model predicted the cutting forces to a high degree of accuracy (less than 5% error) over a range of different feed rates at low cutting speeds.