A modified Johnson–Cook constitutive model for Mg–Gd–Y alloy extended to a wide range of temperatures

In this paper, a new phenomenological and empirically based constitutive model was proposed to change the temperature term in the original Johnson–Cook constitutive model. The new model can be used to describe or predict the stress–strain relation of the metals deformed over a wide range of temperatures even though the current temperatures were lower than the reference temperature. Based on the impact compression data obtained by split Hopkins pressure bar technique, the material constants in the new model can be experimentally determined using isothermal and adiabatic stress–strain curves at different strain rates and temperatures. Good agreement is obtained between the predicted and the experimental stress–strain curves for a hot-extruded Mg–10Gd–2Y–0.5Zr alloy at both quasi-static and dynamic loadings under a wide range of temperatures ever though the current temperatures were lower than the reference temperature.     

A comparative study on modified Johnson Cook,modified Zerilli–Armstrong and Arrhenius-type constitutive models to predict the hot deformation behavior in 28CrMnMoV steel

The true stress-strain data from isothermal hot compression tests on Gleeble-3500 thermo mechanical simulator, in a wide range of temperatures (1173–1473 K) and strain rates (0.01–10 s⁻¹), were employed to establish the constitutive equations based on modified Johnson Cook, modified Zerilli–Armstrong, and strain-compensated Arrhenius-type models respectively to predict the high-temperature flow stress of 28CrMnMoV steel. Furthermore, a comparative study has been made on the capability of the three models to represent the elevated temperature flow behavior of this steel. Suitability of the three models were evaluated by comparing the accuracy of prediction of deformation behavior, correlation coefficient, average absolute relative error (AARE) and relative errors of prediction, the number of material constants, and the time needed to evaluate these constants. The results showed that the predicted values by the modified Johnson Cook and Zerilli–Armstrong models could agree well with the experimental values except under the strain rate of 0.01 s⁻¹. However, the strain-compensated Arrhenius-type model could track the deformation behavior more accurately throughout the entire temperature and strain rate range.     

A damage related thermo-viscoplastic constitutive model of metallic materials under high rate deformation

Abstract

A constitutive model considering the effects of strain hardening, strain rate hardening, thermal softening and material damage softening is suggested. In order to take the effect of material damage into account, a strain softening term is added in Johnson–Cook flow stress law. The model can predict the overall deformation process of metallic materials at high strain rates and a simple way is provided to determine the coefficients of softening term.     

Numerical simulations of impact behaviour of thin steel plates subjected to cylindrical,conical and hemispherical non-deformable projectiles

In this paper, a numerical study of normal perforation of thin steel plates impacted by different projectile shapes is reported. The numerical simulations of this problem have been performed using a finite element code, ABAQUS-Explicit with a fixed and an adaptive mesh for the plate. To define the thermoviscoplastic behaviour of the material constituting the plate, the Johnson–Cook model has been used. This homogeneous behaviour has been coupled with the Johnson–Cook fracture criterion to predict completely the perforation process. Three kinds of projectile shape (blunt, conical and hemispherical) have been simulated with a large range of impact velocities from 190 to 600 m/s. The analysis considers the influence of adiabatic shear bands, plastic work and the gradient of temperature generated in the plate. The numerical results predict correctly the behaviour projectile-plate in agreement with experimental data published by other authors.     

A comparative study on the capability of Johnson–Cook and Arrhenius-type constitutive equations to describe the flow behavior of Mg–6Al–1Zn alloy

In the current study, the predictability of two phenomenological constitutive equations, Johnson–Cook (JC) and Arrhenius-type ones, for describing the flow behavior of a magnesium alloy (Mg–6Al–1Zn) under hot deformation conditions has been evaluated. Towards this end, a series of hot compression tests were performed over a temperature range of 250–450 °C, under strain rates of 0.001, 0.01 and 0.1 s⁻¹. Using the experimental results obtained through implementing the predetermined compression tests, the related parameters and material constants in the constitutive equations were calculated. In order to compare the performance of the models, the statistical parameters of correlation coefficient and absolute mean error were employed. The results imply that the predictability of the Arrhenius-type equations is much stronger in estimating the flow behavior compared to that of the JC model; although more constants are needed to be calculated when using the former equation. It is concluded that the JC model, in contrast to the Arrhenius-type equations, is not reliable for the materials possessing tangible softening stage in their stress–strain curves such as magnesium AZ series.     

Assessment of different ductile damage models and experimental validation

The tough fuel economy and emissions standards facing automotive industry creates the need for lightweight construction and the use of new generation of materials. However, the use of non-conventional materials leads to difficulties in the prediction of material behaviour during sheet metal forming processes, including damage and formability limits, thus challenging the numerical simulation. This paper seeks to contribute in the prediction of fracture on sheet metal alloys. Three constitutive damage models are used, GTN, Johnson Cook and Lemaitre, to simulate, as realistically as possible, the mechanical behaviour of the sheet metal material. The corresponding parameters of damage models are identified using an inverse analysis procedure, based on experimental test data. Finally, to validate and verify the applicability of the studied damage models to predict fracture, experiments are compared with FE simulations.     

Identification of Gurson's material constants by using Kalman filter

A new approach based on the inverse analysis is proposed for estimating material parameters of nonlinear constitutive equations. Using the measurable response of experimental specimens, an inverse analysis is carried out to predict most suitable values of unknown material constants. In general, the accuracy of prediction depends on geometries of specimens and types of measurements. In order to identify optimal experimental procedure, the Kalman filter technique is employed. We have chosen the Gurson model for porous elastic-plastic materials as the material model and its two parameters as the unknown constants. Gurson's constitutive model has been widely used for studying ductile fracture as well as shear localization of various metals. Detailed finite element simulations are performed to demonstrate the effectiveness of the proposed method in determination of the two parameters relating to void nucleation. In the Kalman filter procedure, it is found that the rate of convergence to the correct solutions depends on shapes of test specimens, initial estimates of the unknown parameters, and accuracy of measured data as well as computed reference data. Our analysis predicts that when two differently shaped specimens under tension are used (i.e., a plate with a center hole and another with double side notches), a significant improvement occurs in the rate of convergence.     

Stress–strain time-dependent behavior of A356.0 aluminum alloy subjected to cyclic thermal and mechanical loadings

This article presents the cyclic behavior of the A356.0 aluminum alloy under low-cycle fatigue (or isothermal) and thermo-mechanical fatigue loadings. Since the thermo-mechanical fatigue (TMF) test is time consuming and has high costs in comparison to low-cycle fatigue (LCF) tests, the purpose of this research is to use LCF test results to predict the TMF behavior of the material. A time-independent model, considering the combined nonlinear isotropic/kinematic hardening law, was used to predict the TMF behavior of the material. Material constants of this model were calibrated based on room-temperature and high-temperature low-cycle fatigue tests. The nonlinear isotropic/kinematic hardening law could accurately estimate the stress–strain hysteresis loop for the LCF condition; however, for the out-of-phase TMF, the condition could not predict properly the stress value due to the strain rate effect. Therefore, a two-layer visco-plastic model and also the Johnson–Cook law were applied to improve the estimation of the stress–strain hysteresis loop. Related finite element results based on the two-layer visco-plastic model demonstrated a good agreement with experimental TMF data of the A356.0 alloy.     

An inverse approach to determine the constitutive model parameters from axial crushing of thin-walled square tubes

Knowledge of the behaviour of structural components is essential for their design under crash consideration. Constitutive models describe their material behaviour in finite element (FE) codes. These constitutive models are in relation to the material parameters which have to be determined. The strain rates commonly observed in crash events are in the range of 0–500 s^-1. Classic experimental devices such as Hopkinson’s bars do not easily cover this range of strain rates. An inverse numerical approach based on the experimental quasi-static and dynamic axial crushing of thin-walled square tubes has therefore been developed to determine the constitutive model’s parameters. The inverse method is applied in this paper in two stages to determine the power type elastic–plastic constitutive model’s parameters and the Cowper–Symonds constitutive model’s parameters. The identified power law is compared with the results obtained by quasi-static tensile tests and shows that the identified parameters are intrinsic to the material behaviour. The Cowper– Symond’s parameters identified by this method are then used in FE simulation to predict the dynamic response of the same square tube subjected to bending loading. The results obtained show a good correlation between the experimental and numerical results.     

On the selection of constitutive equation for modeling the friction stir processes of twin roll cast wrought AZ31B

Proper numerical modeling of the Friction Stir Processes (FSPs) requires the identification of a suitable constitutive equation which accurately describes the stress-strain material behavior under an applicable range of strains, strain rates, and temperatures. While some such equations may be perfectly suitable to simulate processes characterized by low (or high) strains and temperatures, FSPs are widely recognized for their relatively moderate ranges of such state variables. In this work, a number of constitutive equations for describing flow stress in metals were screened for their suitability for modeling Friction Stir Processes of twin roll cast (TRC) wrought magnesium Mg–AL–Zn (AZ31B) alloy. Considered were 4 different reported variations of the popular Johnson–Cook equation and one Sellars–Tegart equation along with their literature–reported coefficients for fitting AZ31B stress–strain behavior. In addition, 6 variations of the (rarely used in FSPs simulations) Zerilli–Armstrong equation were also considered along with their literature–reported coefficients. The screening assessment was based on how well the considered constitutive equations fit experimental tensile stress–strain data of twin roll cast wrought AZ31B. Goodness of fit and residual sum of squares were the two statistical criteria utilized in the quantitative assessment whereas a ‘visual ’ measure was used as a qualitative measure. Initial screening resulted in the selection of one best fitting constitutive equation representing one of each of the Johnson–Cook, Sellars–Tegart, and Zerilli–Armstrong equations. An HCP–specific Zerilli–Armstrong constitutive equation (dubbed here as ZA₆ ) was found to have the best quantitative and qualitative fit results with an R² value of 0.967 compared to values of 0.934 and 0.826 for the Johnson–Cook and Sellars–Tegart constitutive equations, respectively. Additionally, a 3D thermo–mechanically coupled FEM model was built in DEFORM 3D to simulate the experimental tensile test from which the experimental load–deflection data was obtained. The three ‘finalist ’ equations were fed into the FEM simulations and were compared based on the 1) simulations’ running times and 2) goodness of fit of the simulation results to the experimental load–deflection data. It was found that the ZA₆ constitutive equation exhibited favorable run times even when contrasted against the simpler mathematical form of the Sellars–Tegart equation. On average, the ZA₆ equation showed improvements in solution time by 5.4% as compared with the Johnson–Cook equation and almost identical solution time (0.9% increase) with that of the ST equation. This result indicates that the proposed equation is not numerically expensive and can be safely adopted in such FEM simulations. Based on the favorable running times and goodness of fit, it was concluded that the HCP–specific Zerilli–Armstrong constitutive equation ZA₆ holds an advantage over all other considered equations and was, therefore, selected as most suitable for the numerical modeling of FSP of twin roll cast AZ31B.     

Constitutive modeling and prediction of hot deformation flow stress under dynamic recrystallization conditions

Simple modeling approaches based on the Hollomon equation, the Johnson–Cook equation, and the Arrhenius constitutive equation with strain-dependent material’s constants were used for modeling and prediction of flow stress for the single-peak dynamic recrystallization (DRX) flow curves of a stainless steel alloy. It was shown that the representation of a master normalized stress–normalized strain flow curve by simple constitutive analysis is successful in modeling of high temperature flow curves, in which the coupled effect of temperature and strain rate in the form of the Zener–Hollomon parameter is considered through incorporation of the peak stress and the peak strain into the formula. Moreover, the Johnson–Cook equation failed to appropriately predict the hot flow stress, which was ascribed to its inability in representation of both strain hardening and work softening stages and also to its completely uncoupled nature, i.e. dealing separately with the strain, strain rate, and temperature effects. It was also shown that the change in the microstructure of the material at a given strain for different deformation conditions during high-temperature deformation is responsible for the failure of the conventional strain compensation approach that is based on the Arrhenius equation. Subsequently, a simplified approach was proposed, in which by correct implementation of the hyperbolic sine law, significantly better consistency with the experiments were obtained. Moreover, good prediction abilities were achieved by implementation of a proposed physically-based approach for strain compensation, which accounts for the dependence of Young’s modulus and the self-diffusion coefficient on temperature and sets the theoretical values in Garofalo’s type constitutive equation based on the operating deformation mechanism. It was concluded that for flow stress modeling by the strain compensation techniques, the deformation activation energy should not be considered as a function of strain.     

45CrNiMoVA高速切削条件下本构关系建模技术研究

为建立连续介质材料高速切削的材料本构关系模型,以45Cr Ni Mo VA材料为研究对象,通过准静态扭转试验和直角自由切削试验相结合的方法,建立了满足高速切削仿真要求的45Cr Ni Mo VA材料的Johnson-Cook本构模型.采用建立的Johnson-Cook本构模型参数,利用ABAQUS有限元分析软件建立了直角自由切削的有限元模型,对切削过程中的切屑厚度、主切削力、进给抗力进行了仿真,并将仿真预测值同试验测量值进行了对比.结果表明:由于切削仿真过程中刀具不存在磨损,进给抗力的仿真误差较大;主切削力和切屑厚度的仿真预测值与试验测量值的误差在10%之内,模型的准确度较好.最后,利用VB和C语言,开发了Johnson-Cook材料本构集成建模系统,并验证了其使用效果的实用性.     

Constitutive modeling of aluminum nitride for large strain,high-strain rate,and high-pressure applications

This paper presents constitutive modeling of aluminum nitride (AlN) for severe loading conditions that produce large strains, high-strain rates, and high pressures. The Johnson–Holmquist constitutive model (JH-2) for brittle materials is used. Constants are obtained for the model using existing test data that include both laboratory and ballistic experiments. Due to the wide range of experimental data the majority of constants are determined explicitly. The process of determining constants is provided in detail. The model and constants are used to perform computations of many of the experiments including those not used to generate the constants. The computational results are used to validate the model, provide insight into the response of AlN, and to demonstrate that one set of constants can provide reasonable results over a broad range of experimental data.     

The modified temperature term on Johnson Cook model for AZ31 magnesium alloy

The quasi‐state and dynamic mechanism of AZ31 magnesium alloy at a strain rates range of 0.001 s^‐1–2500 s^‐1 under a temperature range of 20 °C–250 °C were researched by compression tests using the electronic universal testing machine and split Hopkinson pressure bar system. The true stress‐strain curves at different strain rates and evaluated temperatures were obtained. The result shows that the thermal soften effect of AZ31 magnesium alloy is significant. By modifying the temperature term of the original Johnson Cook model of AZ31 magnesium alloy, a modified Johnson Cook model of AZ31 magnesium alloy has been proposed to reveal thermal soften effect on the deformation behavior of AZ31 magnesium alloy more precisely. With the modified Johnson Cook model and fracture model, the finite element method simulation of AZ31 magnesium alloy hat shaped specimen under impacting was conducted. The numerical simulation result is consistent with the experimental result, which indicates that the modified Johnson Cook model and fracture model are greatly valid to predict the deformation and fracture behavior of the AZ31 magnesium alloy hat shaped specimen under impacting.     

A rate-dependent constitutive equation for 5052 aluminum diaphragms

In this article, both experimental and numerical approaches are conducted to present a constitutive equation for 5052 aluminum diaphragms under quasi-static strain rate loadings. For this purpose the stress–strain curves at different strain rates are obtained using tensile tests. Brittle behavior during tensile tests is observed due to samples thin thicknesses. Employing Johnson–Cook constitutive equation no yields in reasonable agreement with these tensile tests results. Therefore, developing a more suitable constitutive equation for aluminum diaphragms is taken into consideration. This equation is then implemented into the commercial finite element software, ABAQUS, via a developed user material (UMAT) subroutine utilizing von Mises plasticity theory and an own solution algorithm. A single-element pathological test method is adopted to show the well-development of the UMAT subroutine. In order to verify the proposed constitutive equation for precision predicting of mechanical behavior, a bulge test is performed in which demonstrates a good agreement between experimental and numerical results.     

A modified Zerilli–Armstrong constitutive model to predict hot deformation behavior of 20CrMo alloy steel

True stress and true strain values were obtained from isothermal hot compression tests conducted on a Gleeble thermal simulation machine, in a wide range of temperatures (1173–1373 K) and strain rates (1.5 × 10⁻³–1.5 × 10⁻² s⁻¹). The experimental data were used to develop a modified Zerilli–Armstrong constitutive model. The predicted flow stresses using the developed model were compared with experimental values. A correlation coefficient (R) of 0.989 and an average absolute relative error (AARE) of 7.71% between the measured and calculated flow stresses have been obtained. Comparing with a modified Johnson–Cook model developed in the authors’ previous study, the accuracy, the number of material constants involved and the computational time required of the model were evaluated.     

The effects of strength models in numerical study of metal plate destruction by contact explosive charge

Based on a complete mathematical model, the authors set up a problem of metal plate destruction by contacting explosive charge in highly nonlinear dynamic software AUTODYN and solved in two cases using the Johnson–Cook and von Mises strength models. The numerical simulation results were compared with the experimental results and showed a good fit of numerical calculations versus experiments by using the von Mises strength model. The study also shows that the Johnson–Cook strength model, if applied unreasonably, will lead to large errors, which would help to avoid mistakes in the future high speed impact study.     

冲击块与金属圆柱壳相互作用的缓冲吸能分析

应用Johnson Cook本构方程, 结合Singace叠缩模型,考虑冲击作用引起材料的应变强化效应、应变率强化效应和温度效应,研究了冲击物体和金属圆柱壳相互作用的能量转化过程。运用角度增量叠缩法计算了圆柱壳的塑性变形能,根据冲击块和圆柱壳所组成系统能量守恒得到了冲击块速度位移、冲击块和圆柱壳之间瞬时载荷的解析表达式,以及圆柱壳塑性变形引起的温度分布函数。通过对不同材质的金属圆柱壳在冲击作用下塑性变形过程的计算分析,并和有限元计算结果比较,证明了本文计算方法的正确性。     

Prediction of ductile failure using a local strain-to-failure criterion

In this article, we provide the details of the predictive simulations performed by the University of Texas team in response to the 2012 Sandia Fracture Challenge (Boyce et al. in The Sandia Fracture challenge: blind predictions of ductile tearing. Int J Fract. doi:10.1007/10704-013-9904-6, 2013). The material constitutive model was calibrated using the tensile test data through an optimization scheme. A modified Johnson–Cook failure criterion was also partially calibrated using the material characterization data obtained from a tension test and a compact-tension fracture test. These models are then embedded in a highly refined finite element simulation to perform a blind prediction of the failure behavior of the Sandia Fracture Challenge geometry. These results are compared with experiments performed by Sandia National Laboratories and additional experiments that were performed at the University of Texas at Austin with full-field three-dimensional digital image correlation in order to explore the different failure modes. It is demonstrated that a well-calibrated model that captures the essential elastic–plastic constitutive behavior is necessary to confidently capture the elasto-plastic response of challenging structural geometries; it is also shown that a simple ductile failure model can be used to predict ductile failure correctly, when proper calibration of the material model is established.