Strain-dependent Constitutive Equation Research Articles

A study of hot compression behavior of an as-cast FeCrCuNi2Mn2 high-entropy alloy

In this study, hot deformation behavior of an as-cast FeCrCuNi2Mn2 high-entropy alloy (HEA) was evaluated. For this purpose, hot compression tests were carried out in the temperature range of 600–1000 oC and initial strain rate range of 0.001–0.1 s-1. Flow stresses under different deformation conditions were then friction-corrected. Work hardening rate analysis was used to evaluate the occurrence of dynamic recrystallization. Microstructures of the hot compressed specimens were characterized using optical microscopy. Strain-dependent constitutive equations were developed using the data obtained from the Sellars–Tegart–Garofalo equation. The critical and peak strain and the critical and peak stress, as dynamic recrystallization parameters, were extracted from the work hardening rate versus stress curves. The results indicated that the modeled flow stresses were generally lower than the measured ones, which could be attributed to the impact of friction on the flow stress. Microstructural observations after hot compression tests as well as evaluation of work hardening rate versus stress curves showed typical dynamic recrystallization characteristics under different deformation conditions. In addition, with increase in deformation temperature and decrease in strain rate, more DRX grains were formed along grain boundaries and shear bands, resulting in a decrease in the flow stress. The size of the DRX grains was strongly affected by deformation temperature and increased with increase in temperature. The normalized critical stress and strain for the initiation of DRX were calculated as εc/εp = 0.60 and σc/σp = 0.86, respectively. The activation energy of hot deformation was determined to be 708 kJ/mol. It was noticed that the developed constitutive equations could accurately predict the flow stress behavior of the studied alloy over a wide range of experimental conditions.

Hot deformation and dynamic recrystallization behavior of a powder metallurgy Ti‐45Al‐6Nb‐0.3W alloy

The flow behaviors of powder metallurgy (PM) Ti-45Al-6Nb-0.3W (at.%) alloy were systematically investigated in the temperature range from1050 to 1200°Cand the strain rates from 0.001 to 0.5 s−1. The effects of the temperature and the strain rate on deformation behaviors were represented by Zener–Hollomon parameter in an exponent-type equation. Microstructural observations revealed that α2/γ lamellar colonies varied into particle α2 due to dynamic recrystallization (DRX) at high temperature during the compression. The result indicated that the strain-dependent constitutive equation could lead to a good agreement between the calculated and measured flow stresses in the elevated temperature range for the PM Ti-45Al-6Nb-0.3W alloy. Subsequently, the correlation coefficient (R) and the average absolute relative error (AARE) were introduced to verify the validity of the constitutive equation, and values of R and AARE were 0.99483 and 3.956%, respectively.

Microstructure evolution and hot deformation behavior of spray-deposited TiAl alloys

Abstract

Dynamic Recrystallization Behavior of AISI 422 Stainless Steel During Hot Deformation Processes

In this work, hot compression tests were performed to investigate the dynamic recrystallization (DRX) process of a martensitic stainless steel (AISI 422) at temperatures of 950, 1000, 1050, 1100 and 1150 °C and strain rates of 0.01, 0.1 and 1 s−1. The dependency of strain-hardening rate on flow stress was used to estimate the critical stress for the onset of DRX. Accordingly, the critical stress to peak stress ratio was calculated as 0.84. Moreover, the effect of true strain was examined by fitting stress values to an Arrhenius type constitutive equation, and then considering material constants as a function of strain by using a third-order polynomial equation. Finally, two constitutive models were used to investigate the competency of the strain-dependent constitutive equations to predict the flow stress curves of the studied steel. It was concluded that one model offers better precision on the flow stress values after the peak stress, while the other model gives more accurate results before the peak stress.

Modelling the high temperature flow behaviour of sintered Cu–28Zn pre-alloyed powder compacts considering the effect of strain

ABSTRACTIn the field of the deformation process modelling, constitutive equations tend to be utilised as a calculation basis for the estimation of the materials’ flow responses. This holds also for powder metallurgy products. Hence, in this study, the flow behaviour of sintered Cu–28Zn pre-alloyed powder compacts was investigated by a collection of isothermal hot compression tests in the temperature range of 550–850°C at strain rates of 0.001, 0.01, 0.1 and 0.5 s−1 up to the true strain of 0.6. For the sintered part with a predefined porosity of 9.4%, the impacts of the temperature and strain rate on the deformation behaviour were shown by the Zener–Hollomon parameter in an exponent-type equation. As a result, a model was developed to predict the high temperature flow behaviour of the Cu–28Zn pre-alloyed powder compact. The results indicated that the strain-dependent constitutive equations are in good conformity with the experimental results.

Strain-dependent constitutive equations to predict high temperature flow behavior of AA2030 aluminum alloy

The experimental strain–stress data from isothermal hot compression tests were used to develop constitutive equations for hot deformation behavior prediction of AA2030 aluminum alloy. For this purpose, hot compression tests were carried out at the deformation temperatures from 350 to 500°C and strain rate range of 0.005–0.5s−1. The microstructure and flow stress of AA2030 alloy were evidently affected by both the deformation temperature and strain rate which the effects of these parameters on deformation behaviors were represented by Zener-Holloman parameter in an exponent type equation. The influence of strain was also incorporated in the constitutive equation by considering activation energy (Q) and all material constants as different functions of strain. The very good agreement between the measured and predicted results indicates the high accuracy of developed model and established strain-dependent constitutive equations in analyzing and predicting the hot deformation behavior of AA2030, at different temperatures and strain rates conditions.

Spectral stiffness microplane model for damage and fracture of textile composites

This contribution proposes a general constitutive model to simulate the orthotropic stiffness, pre-peak nonlinearity, failure envelopes, and the post-peak softening and fracture of textile composites.Following the microplane model framework, the constitutive laws are formulated in terms of stress and strain vectors acting on planes of several orientations within the material meso-structure. The model exploits the spectral decomposition of the orthotropic stiffness tensor to define orthogonal strain modes at the microplane level. These are associated to the various constituents at the mesoscale and to the material response to different types of deformation. Strain-dependent constitutive equations are used to relate the microplane eigenstresses and eigenstrains while a variational principle is applied to relate the microplane stresses at the mesoscale to the continuum tensor at the macroscale.The application of the model to a twill 2×2 shows that it can realistically predict its uniaxial as well as multi-axial behavior. Furthermore, the model shows excellent agreement with experiments on the axial crushing of composite tubes, this capability making it a valuable design tool for crashworthiness applications.The formulation is computationally efficient, easy to calibrate and adaptable to other kinds of composite architectures such as 2D and 3D braids or 3D woven textiles.

Analysis of flow stress and deformation mechanism under hot working of ZK60 magnesium alloy by a new strain-dependent constitutive equation

The present study investigates the variation of flow stress and microstructural evolution with strain for ZK60 magnesium alloy. A new constitutive equation was used to model the flow stress with excellent results. This constitutive analysis and the microstructural studies carried out on strained samples revealed the existence of two different regimes. At temperatures above 300°C, moderate grain growth and intragranular dislocation activity. Yet, the calculated value of the activation energy and the marked increase in the equivalent strain to fracture indicated grain boundary sliding as a dominant mechanism in this regime of strain rate and temperature, with dislocation motion playing an ancillary role. At lower temperatures, deformation was exclusively governed by dislocation motion, with the extensive occurrence of dynamic recrystallization, which started at low strains, and absence of grain growth.

Constitutive analysis to predict high-temperature flow behavior of BFe10-1-2 cupronickel alloy in consideration of strain

Isothermal compression of BFe10-1-2 cupronickel alloy was carried out on a Gleeble-3800 thermal simulation machine in a wide range of temperatures (1023–1273K) and strain rates (0.001–10s−1). Then the experimental stress–strain data from isothermal hot compression tests were used to develop the constitutive equation. The effects of temperature and strain rate on deformation behaviors a represented by Zener–Holloman parameter in an exponent-type equation. The strain-dependent constitutive equation of BFe10-1-2 cupronickel alloy was proposed. Subsequently, the correlation coefficient (R) and the average absolute relative error (AARE) were introduced to verify the validity of developed constitutive equation, and values of R and AARE were 0.988 and 5.663% respectively, which indicated that the strain-dependent constitutive equation can lead a good agreement between the calculated and measured flow stresses in the elevated temperature range for the BFe10-1-2 cupronickel alloy.

The Strain-Compensated Constitutive Equation for High Temperature Flow Behavior of an Al-Zn-Mg-Cu Alloy

In order to study flow stress behavior for hot working of a typical Al-Zn-Mg-Cu alloy, experimental stress-strain data obtained from isothermal hot compression tests at strain rates of 0.004, 0.04, and 0.4 s−1 and deformation temperatures of 400, 450, 500, and 520 °C were used to develop the constitutive equation. The peak stress decreased with increasing deformation temperature and decreasing strain rate. The effects of temperature and strain rate on deformation behavior were represented by Zener-Hollomon parameter in an exponent-type equation. Employing an Arrhenius-type constitutive equation, the influence of strain has been incorporated by considering the related material constants as functions of strain. The accuracy of the developed constitutive equations has been evaluated using standard statistical parameters such as correlation coefficient and average absolute relative error. The results indicate that the proposed strain-dependent constitutive equation gives an accurate and precise estimate of the flow stress in the relevant temperature range.

Hot deformation behavior and flow stress modeling of annealed AZ61 Mg alloys

In this study, the hyperbolic-sine type constitutive equation was used to model the flow stress of annealed AZ61 magnesium (Mg) alloys. Hot compression tests were conducted at the temperatures ranging from 250°C to 450°C and at the strain rates ranging from 1×10–3s−1 to 1s−1 on a Gleeble-3500 thermo-simulation machine. Constitutive equations as a function of strain were established through a simple extension of the hyperbolic sine constitutive relation. The effects of annealing heat treatments on the variations in constitutive parameters with strain were discussed. The hot compressive flow curves exhibited typical features of dynamic recrystallization. Multiple peak flow curves were observed in the annealed specimens upon testing at a strain rate of 1×10−1s–1 and at various temperatures. Variations in constitutive parameters with strain were related to flow behavior and dependent on the initial conditions of the test specimens. The flow stresses of annealed AZ61 Mg alloys were predicted well by the strain-dependent constitutive equations of the hyperbolic sine function under the deformation conditions employed in this study.

Comparative Analysis by the Newton Method of New Constitutive Strain Dependent Creep Equations Based on the Garofalo Equation

The improvement of optimization numerical methods for constitutive equations has been the first aim of this research. A subordinate optimization algorithm, based on Newton method, through “ad hoc” assessment, has been developed. Application to hot torsion data of AISI4145 with two grain sizes has been carried out. It is concluded that the Newton method is an excellent algorithm for the optimization of strain dependent constitutive equations. Two models are presented as alternative to the generalized Garofalo model: the normalized stress exponent model (NSE) and the Generalized Sherby model (SG). The NSE model is the most precise to restitute the experimental stress-strain curves.

Strain-Dependent Flow Stress Modeling of Hastelloy X Superalloy under Hot Compression

The compression deformation behavior and constitutive relation of Hastelloy X superalloy were investigated in the temperature range of 950 °C to 1150 °C and strain rate range of 0.001 s-1 to 1 s-1 on a Gleeble-3500 thermo-simulation machine. The obtained experimental stress-strain data were used to establish strain-dependent constitutive equations. The correlation between the strain-dependent constitutive parameters and flow behavior was analyzed. Comparisons between the experimental and predicted results showed that the developed constitutive equation could be used to simulate numerically the flow stresses of Hastelloy X superalloy at any strains under hot deformation.

El método de Newton para la optimización de una nueva ecuación constitutiva para la fluencia plástica dependiente de la deformación. Aplicación a las aleaciones de magnesio AZ80 y AZ61

A new numerical algorithm has been developed, based on Newton's method, for optimizing the parameters of a new strain dependent constitutive equation, based on the Garofalo equation. The adjustment is direct, with second order algorithms, for an equation derived from that of Garofalo with a nonlinear objective function. This new optimization algorithm has been applied to creep data of two magnesium alloys AZ80 and AZ61, having an unusual plastic behavior. A certain pseudo-stationary exists in the curves studied, in the sense that the usual deformation states are not manifested in an obvious way. The parameters of the new constitutive equation, dependent on strain, have been determined for these alloys. For analyzing the precision of the parameters and the accuracy of modeling of the stress-strain curves, a statistical treatment has been applied which allows assessing the quality of the constitutive equation proposed and the consistency of these parameters. Stress-strain curves have been compared with the modeling results, reaching a good agreement between the experimental data and the resulting modeling.

Hot deformation behavior of spray-deposited Al–Zn–Mg–Cu alloy

The flow behavior of spray-deposited Al–10.21Zn–2.76Mg–1.45Cu–0.16Zr (wt.%) alloy has been systematically investigated by thermal compression tests with temperature and strain rate ranging from 613K to 733K and 0.001–1s−1, respectively. Microstructural observations revealed that the average grain size of spray-deposited alloy was below 25μm due to the high cooling rate. Both relatively high temperature and low strain rate could promote the formation of dynamic recrystallization (DRX). The stress level of the alloy decreased with increasing deformation temperature and decreasing strain rate, which could be characterized by a Zener–Hollomon parameter in the hyperbolic-sine equation. Furthermore, the strain-dependent constitutive equation could lead to a good agreement between the calculated and measured flow stresses in the elevated temperature range for spray-deposited alloy. The deformation activation energy for spray-deposited alloy was relatively lower than that of the as-cast alloy owing to ultrafine grains and high supersaturated solid solubility.

Hot compressive flow stress modeling of homogenized AZ61 Mg alloy using strain-dependent constitutive equations

The experimental stress–strain data from hot compression tests were used to establish constitutive equations in a homogenized cast AZ61 Mg alloy. Hot compression tests were conducted using the Gleeble 3500 thermal simulation machine in the temperature range of 250–450°C and strain rate range of 1×10−3–1s−1. The constitutive analysis was performed based on the effect of strain on the constitutive parameters. Constitutive equations as a function of strain were constructed according to the hyperbolic sine constitutive law. The correlation between the strain-dependent constitutive parameters and flow behavior was analyzed. Results showed that variations in the constitutive parameters with strain were associated with the stress–strain behavior. A comparatively higher scattering was obtained at low strains based on the constitutive equation with the strain-dependent stress multiplier (α) determined by power and exponential laws. However, the constitutive analysis with a constant α determined by the hyperbolic sine constitutive equation showed better estimations between the calculated and experimental flow stresses at different temperatures and strain rate conditions used in this study.

Hot deformation behavior and processing maps of Mg–Zn–Cu–Zr magnesium alloy

The deformation behaviors of a new quaternary Mg–6Zn–1.5Cu–0.5Zr alloy at temperatures of 523–673 K and strain rates of 0.001–1 s−1 were studied by compressive tests using a Gleeble 3800 thermal-simulator. The results show that the flow stress increases as the deformation temperature decreases or as the strain rate increases. A strain-dependent constitutive equation and a feed-forward back-propagation artificial neural network were used to predict flow stress, which showed good agreement with experimental data. The processing map suggests that the domains of 643–673 K and 0.001–0.01 s−1 are corresponded to optimum conditions for hot working of the T4-treated Mg–6Zn–1.5Cu–0.5Zr alloy.

Constitutive analysis of the hot deformation behavior of Fe–23Mn–2Al–0.2C twinning induced plasticity steel in consideration of strain

In this study, constitutive analysis has been carried out on Fe–23Mn–2Al–0.2C twinning induced plasticity (TWIP) steel. For this purpose, hot compression tests were conducted on a Gleeble-3500 thermo-mechanical simulator in the temperature range of 900–1150°C and the strain rate range of 0.001–20s−1. The effects of deformation heating and friction on flow stress were analyzed and corrected. On the basis of Sellars–Tegart–Garofalo equation, the strain-dependent constitutive equations of the steel were derived. The results show that deformation heating has a significant influence on the flow stress at lower temperatures and higher strain rates, while the frictional effect is slight even at the highest strain level investigated. Comparison of the calculated flow stress with the experimental data suggests that the developed constitutive equations can adequately describe the relationships between the flow stress, strain rate, temperature and strain of the steel during hot deformation. This is supported by a high correlation coefficient (R=0.996) and a low average absolute relative error (AARE=3.31%) for the entire deformation condition range investigated.

Analyses on the Flow Stress of an Al-Mg Alloy During Dynamic Recovery

A comprehensive analysis on flow stress of a wrought Al-Mg alloy is performed to examine the effect of strain. For this study, hot compression tests were carried at different temperatures and strain rates. Corrections of friction and adiabatic heating effects lead to the true stress-true strain curves in the form of dynamic recovery, which reach to a steady-state condition. After correction, constitutive analysis at a constant strain is carried out using hyperbolic-sine equation. The effect of strain on each constitutive parameter is studied to derive a strain-dependent constitutive equation based on hyperbolic-sine equation. Some of constitutive parameters reach to the constant values at the specific strain values. Also, the relations between Zener-Holloman parameter and steady-state values of strain and stress are achieved. In order to develop a strong and general flow stress equation, stress-strain curves are normalized to their steady-state values, which results in an almost similar normalized stress-strain behavior for all of the studied deformation conditions (i.e., deformations at different temperatures and strain rates).

Flow behavior modeling of the 7050 aluminum alloy at elevated temperatures considering the compensation of strain

In order to study the workability of the 7050 aluminum alloy, the flow stress behavior has been studied by a set of isothermal hot compression tests. And the isothermal hot compression tests were carried out in the temperature range of 573–723K and strain rates of 0.001, 0.01, 0.1 and 1s−1 up to a true strain of 0.8. The exponent-type Zener–Hollomon equation was used in the effect investigation of the temperature and strain rate on hot deformation behaviors. And the influence of the strain has been incorporated in the constitutive equation by considering its effect on different material constants for the 7050 aluminum alloy. The constitutive equation of 7050 aluminum alloy considering the compensation of the strain was proposed. Subsequently, the correlation coefficient (R) and the average absolute relative error (AARE) were introduced to verify the validity of the constitutive equation, and values of R and AARE were 0.9922% and 6.28% respectively. Consequently, the result indicated that the strain-dependent constitutive equation can lead a good agreement between the calculated and measured flow stresses in the elevated temperature range for the 7050 aluminum alloy.