Modified Johnson Cook Constitutive Equation Research Articles

Room temperature strengthening mechanisms, high temperature deformation behavior and constitutive modeling in an Al-3.25Mg-0.37Zr-0.28Mn-0.19Y alloy

To improve the ambient strength and high temperature ductility, a novel Al-3.25Mg-0.37Zr-0.28Mn-0.19Y (wt.%) alloy has been fabricated by multidirectional forging and warm rolling. Microstructures and mechanical properties were investigated by light microscope, X-ray diffraction apparatus, transmission electron microscope, and tensile tester. The ultimate tensile strength, yield strength, and elongation of 354.67 ± 0.04, 254 ± 3 MPa, and 18.47% were achieved in the processed alloy, respectively, at room temperature. The maximum superplasticity of 412.5% was demonstrated in this fine-grained (grain size of 8.80 ± 1.06 μm) alloy at 793 K and 8.33 × 10−4 s−1. An ambient strengthening sequence was obtained: grain boundary strengthening > dislocation strengthening > Orowan strengthening > solid solution strengthening. The occurrence of Portevin–Le Chatelier effect or serrated flow in as-annealed alloy was confirmed by theoretical model estimation and experimental evidence. Microstructural evolution and flow stress curves confirmed that dynamic recovery and dynamic grain growth occurred at elevated temperatures; Sotoudeh-Bate curves were discovered at higher temperatures and lower strain rates. Solute Mg, dynamic grain growth, and flow localization were responsible for the formation of Sotoudeh-Bate curves. Modified Johnson–Cook constitutive equation considering dislocation variables and power-law constitutive equation were established. In this fine-grained alloy at 793 K and 8.33 × 10−4 s−1, the number of dislocations inside a grain was 6, the stress exponent was 2, the grain size exponent was 2, and the average activation energy for deformation was 140.10 kJ/mol. These pieces of evidence demonstrated that grain boundary sliding accommodated by intragranular dislocation controlled by lattice diffusion governs the rate-controlling process.

Simulation of micro-milling Inconel 718 considering scale effect

To study the scale effect during micro-milling process, we establish a modified Johnson-Cook (JC) constitutive model to describe the strengthening behavior of materials on microscale based on the theory of strain gradient plasticity. The VUMAT constitutive subroutine of Inconel 718 is programmed. The radius of the tool edge and its influence on the rake angle are considered. Based on the modified JC constitutive equation, the simulation of micro-milling Inconel 718 process is achieved. The micro-milling Inconel 718 experiments are conducted. The accuracy of the simulation of micro-milling Inconel 718 process considering scale effect is verified by comparing cutting force from experiments with simulation outputs based on the traditional as well as modified JC constitutive equation. It is proved that the simulation outputs of the micro-milling process based on the modified JC constitutive equation are more consistent with the experimental results.

Compressive Mechanical Properties and Shock-Induced Reaction Behavior of a Ti–29Nb–13Ta–4.6Zr Alloy

The compressive mechanical properties of a Ti–29Nb–13Ta–4.6Zr alloy were investigated at room temperature under various strain rates. The parameters of a modified Johnson–Cook constitutive equation were determined to describe the strain softening behavior under dynamic conditions. The testing results under compression show yield strength evidently increases with the increase of the strain rate. Moreover, it was found that the strain-rate effect of the yield strength upon dynamic deformation is more significant than that upon quasi-static deformation. In order to characterize the impact-initiated reaction behavior of Ti–29Nb–13Ta–4.6Zr alloy, the quasi-sealed test chamber was used to measure the pressure caused by energy released of fragments. It was obviously shown that the exothermic chemical reaction was more intense with the increase of impact velocity.

Numerical modeling and simulation of macro- to microscale chip considering size effect for optimum milling characteristics of AA2024T351

This research work deals with the numerical modeling and analysis of macro- to microscaled milling of a strain rate-sensitive alloy AA2024T351. The milling computations of a semicircular slot are performed in Abaqus/Explicit by employing the Johnson–Cook thermo-elasto viscoplastic material damage model. The Coulomb friction model is applied at the contact interfaces of work piece and cutting tool. Due to the simultaneous effect of cutting feed and angular speed (ωr), the geometry of uncut chip is modeled with variable section thickness to adapt the trochoidal path. The configuration of the uncut chip undertakes the macro- to micromilling effect. This research effort uniquely explains a comprehensive modeling and milling computations at macro- to microscales to identify the imperative milling characteristics. The presented formulation in form of the modified Johnson–Cook constitutive equation explains the size effect during micromilling. A parametric sensitivity analysis is performed with four different cutting speeds (500, 600, 700, and 800 m/min) and two contact friction coefficients (0.2 and 0.3) while keeping the feed rate constant (0.2 mm/teeth). It was observed that cutting speed and contract friction coefficient play a pivotal role in cutting reaction force and chip–tool interface temperature. A temperature in the range of 141–167 °C is predicted with different cutting speeds. The percentage deviations of the simulated cutting forces from the published experimental results are found promising. The estimated deviation may be because of compromising some imperative factors in numerical model, such as cutter vibrations, tool wear effect, tool run-out, and limitations, to develop an exact trochoidal trajectory.

Modeling and simulation of the high-speed milling of hardened steel SKD11 (62 HRC) based on SHPB technology

An easy-to-produce sawtooth chip is the main feature of the high-speed milling of hardened steel. In previous works, a theoretical geometric model was proposed for the sawtooth chip formation to predict the strain and strain rate in the shear band during chip formation; these properties are important when describing the deformation characteristics for the cutting of hardened steel materials. In the cutting process, however, the changes and distributions of stress and strain can hardly be obtained using a theoretical model. This paper modifies the conventional empirical Johnson–Cook constitutive equation by employing stress–strain curves at high temperature and a high strain rate obtained using split Hopkinson pressure bar technology and considering the negative strain rate effect and temperature effect of the material, especially for SKD11 (62 HRC) hardened steel. A thermo-mechanical coupled two-dimensional planar strain finite element model for the high-speed milling of SKD11 hardened steel with a modified Johnson–Cook constitutive equation is presented. The geometric characteristics of chip formation during the high-speed milling of SKD11 are predicted and the results are in good agreement with experimental results. Employing the modified finite element model, the stress and strain in the shear band during high-speed cutting are quantitatively analyzed and found to be in close agreement with the results of a theoretical model analysis. It is found that the cutting speed has a critical value at which the stress and strain reach a certain value and the distributions of stress and strain change in the shear band, resulting in the generation of a sawtooth chip. Moreover, the cutting force, cutting temperature, and selection of a coated tool are discussed according to results obtained with the modified finite element model. The cutting force and difference in temperature decrease while the temperature increases as the cutting speed increases. Compared with a TiAlN-coated tool, a TiSiN-coated tool performs better in cutting SKD11 hardened steel in terms of the cutting temperature.

A Modified Johnson-Cook Constitutive Equation to Predict Hot Deformation Behavior of Ti-6Al-4V Alloy

A modified Johnson-Cook constitutive equation of Ti-6Al-4V alloy is proposed based on hot compression tests performed in the temperature range of 1073-1323 K and strain rate 0.001-1 s−1. The experimental stress-strain data were employed to develop the modified Johnson-Cook constitutive equation of different phase regimes (α + β and β phase). The predicted flow stresses using the developed equation were compared with experimental data. Correlation coefficient (R) and average absolute relative error (AARE) were introduced to verify the validity of the constitutive equation. The values of R and AARE for α + β phase were 0.990 and 7.81%, respectively. And in β phase region, the values of R and AARE were 0.985 and 10.36%, respectively. Meanwhile, the accuracy, the number of material constants involved, and the computational time required of the constitutive equation were evaluated by comparing with a strain-compensated Arrhenius-type constitutive equation. The results indicate that accuracy of modified Johnson-Cook constitutive equation is higher than that of compensated Arrhenius-type model at α + β phase, while lower at single β phase region. Meanwhile, the time required for evaluating the material constants of modified Johnson-Cook constitutive equation is much shorter than that of the strain-compensated Arrhenius type ones.

A modified Johnson–Cook model of dynamic tensile behaviors for 7075-T6 aluminum alloy

The dynamic mechanical behaviors of 7075-T6 aluminum alloy at various strain rates were measured by dynamic tensile tests using the electronic universal testing machine, high velocity testing system and split Hopkinson tensile bar (SHTB). Stress–strain curves at different rates were obtained. The results show that the strain rate hardening effect of 7075-T6 aluminum alloy is significant. By modifying the strain rate hardening term in the Johnson–Cook constitutive model, a new Johnson–Cook (JC) constitutive model of 7075-T6 aluminum alloy was obtained. The improved Johnson–Cook model matched the experiment results very well. With the Johnson–Cook constitutive model, numerical simulations of tensile tests at different rates for 7075-T6 aluminum alloy were conducted. According to tensile loading and stress–strain relation of 7075-T6 aluminum alloy, calculation results were compared with experimental results. Accuracy of the modified Johnson–Cook constitutive equation was further proved.

A modified Johnson-Cook constitutive relationship for a rare-earth containing magnesium alloy

Lightweight magnesium alloy has recently attracted a considerable interest in the automotive and aerospace industries to improve fuel efficiency and reduce CO2 emissions via the weight reduction of vehicles. Rare-earth (RE) element addition can remarkably improve the mechanical properties of magnesium alloys through weakening crystallographic textures associated with the strong mechanical anisotropy and tension-compression yield asymmetry. While the addition of RE elements sheds some light on the alteration in the mechanical anisotropy, available information on the constitutive relationships used to describe the flow behavior of RE-containing magnesium alloys is limited. To establish such a constitutive relationship, uniaxial compressive deformation tests were first conducted on an extruded Mg-10Gd-3Y-0.5Zr (GW103K) magnesium alloy at the strain rates ranging from 1×10−1 to 1×10−4 s−1 at room temperature. A modified Johnson-Cook constitutive equation based on a recent strain hardening equation was proposed to predict the flow stresses of GW103K alloy. Comparisons between the predicted and experimental results showed that the modified Johnson-Cook constitutive equation was able to predict the flow stresses of the RE-containing magnesium alloy fairly accurately with a standard deviation of about 1.8%.

High-strain-rate response of ultra-fine-grained copper

The high-strain-rate response of ultra-fine-grained (UFG) copper processed by equal channel angular pressing (ECAP) was characterized by three different dynamic testing methods: reverse Taylor impact, cylindrical compression specimens, and hat-shaped specimens in Hopkinson bar experiments. Upon recovery after impact, the specimens were found to undergo dynamic recrystallization at a calculated temperature of 360 K, indicating that the UFG copper is thermally unstable. Reverse Taylor tests were conducted on as-received oxygen-free high-conductivity copper rod and ECAP specimens with 2 and 8 sequential deformation passes. The dynamic deformation of the samples was modeled using AUTODYN-2D, and a modified Johnson–Cook constitutive equation was found to closely capture the dynamic response. Both the dynamic experiments and analysis from the reverse Taylor tests indicate enhanced strain-rate sensitivity in comparison with conventional polycrystalline copper, in agreement with predictions of reduced activation volume. The shear band thickness, as obtained in forced localization tests, showed a marked decrease, in comparison to conventional polycrystalline copper; this effect is interpreted as due to an accelerated thermal softening and inherent instability exhibited by the UFG structure.

A finite element model for micro machining considering size effect of material behaviour and tool geometry

This paper presents a finite element model that is capable of describing the main characteristics in micro machining processes. By using strain gradient plasticity, a modified Johnson-Cook constitutive equation is formulated to model the size effect of material behaviour at micron level as well as the main features in macro machining processes. Based on the material model, a finite element model is developed, considering size effect, cutter edge radius and fracture behaviour. Size effect and chip formation are investigated in detail by FE simulations, applying the developed model. Two research findings can be drawn from the results: size effect of micro machining can be well explained by modelling material behaviours at micron scale using strain gradient plasticity; from the chip formation studies, the minimum chip thickness is proposed to be 0.35 times of the cutter edge radius when the rake angle is 10 degrees and the radius is 2 ?m.

고온 고변형률속도에서 SM45C의 동적 거동

A compressive split Hopkinson pressure bar (SHPB) technique is used to investigate the dynamic behavior of SM45C at high temperature. A radiant heater, which consists of one ellipsoidal reflector and one halogen lamp, is used to heat the specimen. Specimens are tested from 600℃ to 1000℃ at intervals of 100℃ at a strain-rate ranging from 1100/s to 1150/s. A critical phenomenon occurs between 700℃ and 750℃ in SM45C. This phenomenon results in the drastic drop in a flow stress. In a modified Johnson-Cook constitutive equation, a reducer function is used to take into account for the effect of the drastic drop in a flow stress. A reducer function, which is dependant on the temperature as well as the strain, is introduced and the parameters of the modified Johnson-Cook constitutive equation are determined from test results.

Modelling and analysis of micro scale milling considering size effect, micro cutter edge radius and minimum chip thickness

This paper presents mechanisms studies of micro scale milling operation focusing on its characteristics, size effect, micro cutter edge radius and minimum chip thickness. Firstly, a modified Johnson–Cook constitutive equation is formulated to model the material strengthening behaviours at micron level using strain gradient plasticity. A finite element model for micro scale orthogonal machining process is developed considering the material strengthening behaviours, micro cutter edge radius and fracture behaviour of the work material. Then, an analytical micro scale milling force model is developed based on the FE simulations using the cutting principles and the slip-line theory. Extensive experiments of OFHC copper micro scale milling using 0.1 mm diameter micro tool were performed with miniaturized machine tool, and good agreements were achieved between the predicted and the experimental results. Finally, chip formation and size effect of micro scale milling are investigated using the proposed model, and the effects of material strengthening behaviours and minimum chip thickness are discussed as well. Some research findings can be drawn: (1) from the chip formation studies, minimum chip thickness is proposed to be 0.25 times of cutter edge radius for OFHC copper when rake angle is 10° and the cutting edge radius is 2 μm; (2) material strengthening behaviours are found to be the main cause of the size effect of micro scale machining, and the proposed constitutive equation can be used to explain it accurately. (3) That the specific shear energy increases greatly when the uncut chip thickness is smaller than minimum chip thickness is due to the ploughing phenomenon and the accumulation of the actual chip thickness.

Constitutive Equation for 7050 Aluminum Alloy at High Temperatures

Accurately material constitutive model is essential to understand and predict machining process. High temperature split Hopkinson pressure bar (SHPB) test system is used to investigate flow stress behavior and characteristics of 7050-T7451 aluminum alloy. Specimens are tested from 200°C to 550°C at intervals of 50°C and room temperature, at strain-rate of 2800s-1. The experimental results show that flow stress is strongly dependent on temperature as well as strain rate, flow stress decreases with the increase of temperature, while increase with the increasing of strain rate. The material parameters are determined for both Johnson-Cook constitutive equation and modified Johnson-Cook constitutive equation. The modified JC equation is more suitable for expressing the dynamic behavior of 7050-T7451 aluminum alloy.

Constitutive equation for Ti–6Al–4V at high temperatures measured using the SHPB technique

A high temperature split Hopkinson pressure bar (SHPB) test system is used to investigate the effects of temperature as well as those of strain and strain-rate. Effects of temperature for the titanium alloy (Ti–6Al–4V) are investigated by developing a high temperature SHPB test system. In this work, high temperatures greater than 1000°C are attained in the SHPB test specimens by using two ellipsoidal radiant heating reflectors with two halogen lamps. The thermal gradients in the specimens are observed as they are heated. Methods for solving problems related to conduction between the specimens and elastic bars and techniques for measuring the temperature of the specimens are suggested. When testing with the high-temperature SHPB apparatus, care is required to prevent oxidation of the surface of the specimen, and to prevent an inhomogeneous temperature distribution from developing in the specimen. To determine the true flow stress–true strain relationship, specimens are tested from room temperature to 1000°C at intervals of 200°C and at a strain-rate of 1400 s −1 . The parameters for a Johnson–Cook constitutive equation and a modified Johnson–Cook constitutive equation are determined from the test results. The modified Johnson–Cook constitutive equation is more suitable for expressing the dynamic behavior of the Ti–6Al–4V titanium alloy in the vicinity of the recrystallization temperature.