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Constitutive equations for the relationship between stress, strain, strain rate, and temperature are an essential input for modelling thermomechanical processing. Hot plane strain compression testing was used to deform commercial purity aluminium, Al-1Mn, and Al-1Mg alloys at temperatures of 300, 400, and 500°C and equivalent strain rates of 0.25, 2.5, and 25 s−1, to an equivalent strain of 2. Flow stress data obtained from these tests were analysed using the Sah et al. and hyperbolic sine forms of relationships. Values of constants in the constitutive equations were obtained and were shown to provide an accurate description of the experimental stress-strain curves, including the effect of temperature rise due to deformational heating and the effect of changing strain rate. The application and limitations of the relationships are discussed.

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... We identify the parameters involved in the LBL-theory by minimizing the sum of squares of the differences between the solution and the experimentally measured stresses. The comparison with the experiments conducted by Shi, McLaren, Sellars, Shahani, and Bolingbroke (1997) and Chen, Stout, Kocks, MacEwen, and Beaudoin (1998) shows again excellent agreement confirming the applicability of LBL-theory to aluminum. ...

... Stress-strain curves for aluminum at K and at three different strain rates as shown. The data points are taken from Shi et al. (1997) . For these curves = 732 MPa, = 24800 K. with respect to χ 0 , T P , σ T subjected to reasonable constraints 0 < χ 0 ≤ 0 . ...

... where ( i , σ i ), i = 1 , . . . , N correspond to the data points measured in experiment at fixed temperature and strain rate taken from Shi, McLaren, Sellars, Shahani, and Bolingbroke (1997) . We find K, K ρ , ˜ χ ini , ˜ ρ ini by minimizing function h (K, K ρ , ˜ χ ini , ˜ ρ ini ) with respect to these parameters subjected to reasonable constraints. ...

The theory of dislocation mediated plastic flow proposed by Langer, Bouchbinder, and Lookman is applied to compute the stress-strain curve of aluminum over a wide range of temperatures and strain rates. The parameter identification by the least squares method is provided leading to the excellent agreement with experiment.

... Another possible factor for the flow stress oscillations is dynamic recrystallization. Shi 36 et al. [40] among several other authors have reported that aluminium alloys have high stacking-fault energy and thus would not undergo dynamic recrystallisation [41,42]. However, empirical evidences supporting the occurrence of dynamic recrystallisation during hot working of pure aluminium, aluminium alloys and aluminium matrix composites have been widely reported [41,[43][44][45]. ...

... Shi 36 et al. [40] among several other authors have reported that aluminium alloys have high stacking-fault energy and thus would not undergo dynamic recrystallisation [41,42]. However, empirical evidences supporting the occurrence of dynamic recrystallisation during hot working of pure aluminium, aluminium alloys and aluminium matrix composites have been widely reported [41,[43][44][45]. For example, dynamic recrystallisation is usually favoured at high temperatures and low strain rates and the type of oscillations exhibited by materials during deformation is influenced by strain rate and deformation temperatures. ...

... McQueen and Ryan [63] reported that the Q hw in materials that undergo dynamic recovery is usually close to Q sd . This has been confirmed by many other authors who have worked on the hot deformation behaviour of aluminium and its alloys [39,41,57,64]. Additionally, McQueen et al. [69] and McQueen and Ryan [65] mentioned that the Q hw in materials that undergo dynamic recrystallisation could be 20% higher than activation energy for self-diffusion while it could be up to 50% higher in materials having solutes, precipitates and reinforcement additions. ...

Isothermal compression testing of BLA-SIC hybrid reinforced Aluminium composites was performed on Gleeble 3500 thermomechanical simulator under different deformation temperatures (300–400 °C) and strain rates (0.01–1 s ‑1 ). The flow behaviour and the softening mechanisms were established using the trend of the stress-strain curves, activation energy and microstructural examination. The results showed that flow stress increased with decreasing temperature; but was not entirely strain rate sensitive − a characteristic identified in some Al 6XXX based metallic systems. Also, uncharacteristic flow stress oscillations were observed at strain rates of 0.01 and 0.1 s ‑1 while steady state flow stress was observed at 1 s ‑1 . The hot working activation energy was ∼290.5 kJ/mol which was intermediate to the range of 111–509 kJ/mol reported in literature for various Al based composites. It was proposed that at strain rates of 0.01 and 0.1 s ‑1 , dynamic recrystallization and/or dislocations-reinforcements interactions were the dominant deformation mechanism(s), while at 1 s ‑1 , dynamic recovery was predominant.

... For decades, numerous researchers have worked on the development of effective constitutive models for the hot deformation behaviour of metals and many excellent works have been carried out and reported. Generally, there are three main types of constitutive models [1]: phenomenological constitutive model [3][4][5][6][7][8][9][10], physically based constitutive model [11][12][13][14][15] and neural network based constitutive model [16][17][18]. ...

... From Eqs. (9) and (39), the following equation can be derived: ...

... From Eqs. (9) and (40), the following equation can be derived: ...

The complexity of hot deformation behaviours of metallic materials is acknowledged in decades of study. The present work uses a new constitutive model by Zhu-Ou-Popov (ZOP model) and its modification to predict the hot deformation behaviours of metals. The ZOP model and its modifications are introduced firstly. The basic idea of these models is centred on a set of piecewise and transition functions: the piecewise functions are used to predict flow stress at different strain ranges, whilst the transition functions enable a smooth shifting from small strain to large strain ranges. The methods for identification of four such variant models are developed and given in detail. Hot compressive flow stress curves of 42CrMo at different strain rates and temperatures are used to show the validity of these models. Results show that, all of the developed models are able to predict the hot compressive behaviour of 42CrMo. Using the Arrhenius type equation and modified Zener-Holloman parameter in the ZOP model to predict the flow stress from the yield point, the Modified model Ⅲ gives the most favourable prediction accuracy with R2 of 0.9569 whilst other models are also effective with R2 higher than 0.91. The modifications of the ZOP model can be used to predict the yield stress and to reflect the peak stress. In addition, the applicability and advancement of the presented constitutive models are discussed, and the models are considered to be effective in reflecting the occurrence and completion of dynamic recrystallization (DRX) in hot deformation of metallic materials. It is shown that the studied models are capable of predicting the peak strain and DRX completion strain, with comparable results obtained to experimental data.

... Various research groups proposed constitutive equations of materials suggesting versions using experimental data in a single equation (161,168,172,173,157 (161). ...

... Expressions to be used to analyse the hot behaviour of metals and alloys are often derived from results obtained by combining mechanical testing and simplified models of material behaviour (157,212). Constitutive models show significant effects of temperature and strain rate on the flow stress (168,213). Flow stress is the resistance to plastic flow of the material at any instant under any strain rate. ...

... To evaluate the hot forming process and numerical simulation, a flow stress model is constituted based on the Arrhenius equation and a temperature compensated strain rate factor, the Zener-Hollomon parameter. The Arrhenius equation is widely used to describe the relationship between strain rate, flow stress and especially for elevated temperatures (156,157,171). ...

PRÉCIS
Friction Taper Stud Welding (FTSW) is a relatively new solid state welding process, developed from the concepts of friction welding, which theoretically operates below the melting temperatures of the material being welded. During friction welding, heat is generated by conversion of mechanical energy into thermal energy at the interface of the work pieces, during rotation under pressure. Quality welds are dependant on the correct selection of welding process parameters, which are currently chosen empirically, and the FTSW evaluated by mechanical testing. This method is time consuming, uneconomical and could cause that optimised conditions are overlooked. A proposed solution would be to numerically model the process, but reference to successful computational modelling of the FTSW process is currently not available and data regarding the responses during the process are limited.
The ultimate aim of the present study is to develop a finite element model to simulate the FTSW process using AISI 4140 medium carbon low alloy steel, delivering temperature profiles and hardness predictions through the Heat Affected Zone (HAZ) – using a combined experimental and numerical study.
To achieve the objectives of this study a systematic approach was adopted and conducted in several phases. A weld matrix was configured with ranging weld input parameters to determine the affect of weld input parameters on real-time responses. To provide a relationship between these factors, welding was conducted using a portable friction taper stud welding platform linked to a control and data logging system for measuring the real time axial forces, spindle speed, material displacement, torque and temperature responses as a function of time. The input process parameters applied being motor speed, axial forces, displacement and forging time.
The temperature distribution through the weld, by direct measurement, as a function of weld time and position is investigated. During the experimental welds temperature responses, as influenced by welding parameters, were recorded using embedded N-Type thermocouples at various locations in the near vicinity of the weld interface. The main hot spots during welding were identified to be close to the top surface just before weld completion and at the bottom centre surface of the plug weld at the interface line. All the welds showed similar trends and a maximum temperature of 1078°C at the bottom of the weld was reached for a rotational tool speed of 5160rpm, axial friction force of 15kN and displacement of 6.5mm, due to the heat generated by friction between the tool and weld coupon. The weld torque increase rapidly at the start of the weld and reached a peak value shortly after the start of the weld, while a peak temperature of 1366°C, for a rotational tool speed of 5160rpm, axial friction force of 10kN and displacement of 8mm was reached at the top edge of the plug weld. This position of anticipated peak temperature value is due to the heat transferred during the FTSW process together with the accumulation of expelled material forming on the surface of the weld coupon.
Statistical methods were applied to obtain knowledge of the trends and relationship between weld input parameters for various weld responses, including energy input, temperature, friction time, torque and displacement rate.
Although it was shown that no single parameter solely controls the temperature gradients in the weld, the dominant influence of the rotational speed at the bottom of the weld and that of the displacement, at the top of the weld, were evident. The peak temperatures during the weld are of interest as these temperatures, together with the subsequent cooling rates, determine the Vickers hardness, of the material, through the weld. Spindle speed was found to have the dominant effect on temperature in the bottom half of the weld with displacement having a contributive effect closer to the top of the weld. Friction force dominate the effect on friction time, displacement rate and total energy input with friction force and spindle speed having an equal effect on torque. The multiple regression analysis resulted in valid models with varied, but acceptable accuracy with the equation for friction time resulting in an R2predict value of 93.34%. These models provided a clear insight to the influence of weld input parameters on the weld responses and the model for friction time was used as an input parameter to the FTS welding simulation.
The accurate prediction of the interface temperature is fundamental for process optimisation which will allow for producing consistent, reliable plug welds. A fully coupled transient two-dimensional axi-symmetrical analysis of heat flow during the FTSW process of AISI 4140 steel and subsequent Vickers hardness profiles through the HAZ, making use of numeric simulation applied in the commercially available FEA software, COMSOL Multiphysics®, is developed and reported on. Process optimisation hinges on a better understanding of the heat distribution during welding, making a major contribution to the resultant hardness.
The thermal-plastic flow coupling of the model is such that temperature values are resolved together with that of the velocity field. The simulation utilises a Computational Fluid Dynamics (CFD) two phase laminar flow and Heat Transfer physics, applied in an Eulerian mesh-based scheme. The viscosity of the fluid is based on a constitutive law of the flow stress using the Zener-Hollomon parameter with a flow model based on the Navier-Stokes’ equations to simulate the plastic deformation. Temperature dependant thermo-physical material properties and coefficient of friction are applied and the application of viscous heating is controlled by a material state variable.
The heat source model, required for material softening, is applied as two components, frictional and shear, with the heat source moving along the z-axis delivering sufficient energy to soften the metal, causing flow. The Navier-Stokes approach is applied with solid-state material transport during the weld based on laminar, viscous flow of a non-Newtonian fluid, dependant on temperature and strain rate.
Numerically calculated values for temperature profiles and peak temperatures through to the weld as well as subsequent Vickers hardness profiles at points through the HAZ, obtained from the Finite Element model, were found to be in close agreement with values from trial welds. The largest variance was 19% for the peak temperature of weld E4W2, applying an axial friction force of 7.5kN, 6.5mm displacement and a tool rotational speed of 4080rpm – resulting in a friction time of 330 seconds. Predictions of hardness are found to be between 0% and 19% (mean 3%) of experimentally determined values with the biggest variance at the positions of peak temperatures due to the friction interfaces. The heat applied as a result of plastic deformation was found to be 5.4% of the total heat. The FTSW model predicts the temperatures at the friction interface, during the welding process, to be within the range, and frequently exceeding the solidus temperature of AISI 4140 steel. Results show that the models applied in the FTSW simulation show good agreement when compared to experimental values.
The main contribution of this thesis, towards knowledge of the FTSW process, is:
i) The relationships between weld input parameters and responses;
ii) Temperature dependant models of thermo-physical properties for AISI 4140 in the high temperature region (ranging from ambient to the solidus temperature);
iii) Successful application of the Navier-Stokes approach to simulate the plastic flow during FTSW and
A numerical finite element model for the prediction of temperature gradients and hardness profiles through a FTSW.

... The presence of metastable Al13(Fe,V)3Si; may be indicative of the very ultrafast cooling rates, (≥10 8 C/s), employed in the present investigation. The β-intermetallic phase with similar lattice parameters: a = b = 0.612 nm, c = 4.15 nm, and β =91 0 , and general composition of Al13(Fe,V)3Si; have been previously linked with a remarkable microstructural resistance that slow down the grain coarsening process [16,17]. The morphologies of intermetallic particles in the samples as extruded at 370 0 C, 400 0 C, 430 0 C, and 460 0 C are shown in Fig. 2b. ...

... Several empirical equations have been previously proposed to determine the deformation activation energy and hot deformation behaviour of alloys. However, during hot deformation, the relationship between flow stress, strain rate, and temperature described by a sine hyperbolic law in the Arrhenius type equation proposed by Sellar and Tegart (Eqn.4) gives better approximations of Z parameter and flow stress relation [13,14,15,16]. ...

Dispersoid strengthening plays an important role in improving the mechanical and dynamic properties of these high temperature A8009 aluminium alloys, consisting of Al-8.77Fe-1.27V-0.31Si elements, but the influence of extrusion degrees on the elevated temperature reliability has not been investigated. In this work, for the first time, in situ analysis of the deformation behaviour and kinetics of A8009 aluminium alloys produced by ultra-rapid cooling is systematically investigated by means of in situ tensile experiment and complemented by XRD, electrical conductivity, and microstructural analyses. This study contributes to understanding of the relationship between extrusion conditions and reliability phenomenon in complex industrial applications, enabling further optimisation of the alloy for a number of important automotive and aerospace component applications by manufacturing regulation. 1.0 Introduction For more than a century, aluminium and its alloys have attracted much interest based on their combination of impressive mechanical and dynamic properties [1], considered an attractive choice for a number of applications in automotive and aerospace component manufacture. However, the goal of addressing rising sophistication, complexity, and stringency in service requirements-while also adhering to safety, fuel efficiency, and strict emission requirements-has propelled large-scale material design initiatives and development, for weight reduction, carbon emission, and/or fuel economies. Al-Fe-V-Si alloys, which depend largely on hypereutectic AI-Fe-based compositions with lower ternary and quaternary additions of vanadium and silicon, are promising. Their performance are particularly and effectively controlled by altering the nature [1] and/or volume fraction of the microalloying element [2]. There is mounting evidence of the potential for ultra-rapidly cooled Al-Fe-Si-V alloy as a high temperature alloy with favourable characteristics. However, one major drawback of the alloy is the limited solubility of the important transition metals and some rare earth metals into the aluminium matrix [3].

... The constitutive model is usually used to describe the functional relationship among strain, strain rate, and temperature in plastic deformation of materials.The constitutive model of metal under a high strain rate can be divided into two kinds [15]. One is the 4 empirical constitutive model, including Johnson-Cook (J-C) constitutive model [16], Khan-Huang (KH) model [17], and some other empirical models [18][19][20]. The other one is the physically based constitutive model, including Zerilli-Armstrong (Z-A) constitutive model [21], Mechanical Threshold Stress (MTS) constitutive model [22], and NN-Li constitutive model [23]. ...

... The surface layer model of the micro/meso-scales cylindrical sample. 20 ...

The dynamic mechanical properties of metallic materials have been extensively investigated at the macro-scale in terms of deformation mechanisms, strain rate strengthening, and fracture mechanisms. However, the dynamic mechanical properties affected by size effects at micro/meso-scales have rarely been investigated. To explore the size effects on the dynamic mechanical properties at micro/meso-scales, the experiments of quasi-static compression and SHPB were carried out using oxygen-free, high-conductivity (OFHC) copper with different geometrical and grain sizes. The experimental results show that the quasi-static and dynamic mechanical properties of OFHC copper are affected by size effects at micro/meso-scales. In particular, OFHC copper exhibits strain rate strengthening effects at the micro/meso-scales, and the presence of micro-cracks was observed in the SHPB experimental specimens. The J-C constitutive model based on the surface layer model is proposed and the analysis of the average relative error of the modified model and the original constitutive model is performed. Finite element analysis was carried out based on the modified J-C model and the original model, and the results show that the modified J-C model was in good agreement with the experimental results.

... Besides the material composition, the hardening coefficient H Al also depends on the absolute processing temperature and the cooling rate. 34 If it is not possible to avoid temperatures during processing that would lead to interface debonding, the introduction of a textured Al-Si interface would be an option to expand the processing limits, as the critical energy release rate for debonding scales with the area of the interface. Hence, our results imply that MEMO-layers are wellsuited for low-temperature solar cell processing up to at least 210 °C, depending on the particular properties of the Al layer, and are thus compatible with the proposed module level processing concept. ...

... The maximum applicable stress values define a minimum stressorlayer thickness that we have to ensure for a successful exfoliation process. As the flow stress strongly depends on temperature and strain rate34,35 the values from the XRD measurement might not be the correct absolute stress values to define a hard limit. However, the thinnest high-purity Al layer that we could use for a successful exfoliation process had a thickness of 100 µm, in accordance with the graphin FIG. 5. ...

We report on a kerfless exfoliation approach to further reduce the costs of crystalline silicon photovoltaics making use of evaporated Al as a double functional layer. The Al serves as the stress inducing element to drive the exfoliation process and can be maintained as a rear contacting layer in the solar cell after exfoliation. The 50-70 μm thick exfoliated Si layers show effective minority carrier lifetimes around 180 μs with diffusion lengths of 10 times the layer thickness. We analyze the thermo-mechanical properties of the Al layer by x-ray diffraction analysis and investigate its influence on the exfoliation process. We evaluate the approach for the implementation into solar cell production by determining processing limits and estimating cost advantages of a possible solar cell design route. The Al-Si bilayers are mechanically stable under processing conditions and exhibit a moderate cost savings potential of 3-36% compared to other c-Si cell concepts.

... In the other words, in these models a mathematical function with some constants is fitted to the experimental flow curves of tested material. Arrhenius equation [11][12][13], exponential equation [14] and some newly developed constitutive models are some examples of this category [15][16][17][18][19]. The aim of the present work is to evaluate the modeling performance of a recently developed simple model for describing the hot flow curves of AZ91 magnesium alloy. ...

... For deriving the Arrhenius equation with strain dependent constants, the following procedure should be demonstrated to find the values of material constants. This equation for individual stresses corresponding to the strains in a predefined interval and step size (in this work, for the stresses correspond to different strains in the range of 0.05 to 0.5 with the step size of 0.05): 1) As proposed in Ref. [15], the optimum value of excess as an unknown variable of α in Eq. (8) should be determined from the following relationship: (11) where the values of and β can be obtained using the lnε-lnσ and lnε-σ plots, respectively (as the result of writing the Eqs. (9) and (10) for temperature constant and ε̇ constant conditions, respectively) [25]. ...

Modeling the flow curves of materials at elevated temperatures is the first step in mathematical simulation of the hot deformation processes of them. In this work, a comparative study was provided to examine the capability of three different constitutive equations in modeling the hot deformation flow curves of AZ91 magnesium alloy. As such, the Arrhenius and exponential equations with strain dependent constants, and a recently developed simple model (developed based on a power function of Zener-Hollomon parameter and a third order polynomial function of ε power of a constant number) were examined. Root mean square error (RMSE) criterion was used to assess the modeling performance of the examined constitutive equations. Accordingly, it was found that the Arrhenius equation with strain dependent constants has the best performance for modeling the hot deformation flow curves of AZ91 magnesium alloy. The results can be further used in mathematical simulation of hot deformation manufacturing processes of tested alloy. 1. Introduction High strength to weight ratio and excellent castability are the main characteristics of Mg alloys that make them a good candidate for transportation industries applications [1-4]. However, low workability of these alloys at room temperature is a restricting factor in their manufacturing processes. Thus, almost,all manufacturing processes of Mg alloys are conducted at high temperatures [1–4]. Modeling the flow curves of materials at elevated temperatures is the first step in mathematical simulation of the manufacturing processes of them. Consequently, various constitutive equations have been proposed to model the flow stress of different materials [5-9]. As explained by Lin and Chen [10], the constitutive equations can be divided into three categories including: phenomenological models, physical-based models (models which consider the mechanism of deformation such as dislocation dynamics and thermal activation) and artificial neural network (ANN) models [10]. Among these, phenomenological constitutive models are widely used in mathematical simulation of metal forming processes. In these models the flow stress of a material is expressed as a function of the forming temperature, strain-rate and strain. In the other words, in these models a mathematical function with some constants is fitted to the experimental flow curves of tested material. Arrhenius equation [11-13], exponential equation [14] and some newly developed constitutive models are some examples of this category [15-19]. The aim of the present work is to evaluate the modeling performance of a recently developed simple model for describing the hot flow curves of AZ91 magnesium alloy. This model has been recently developed based on a power function of Zener-Hollomon parameter and a third order polynomial function of strain power m (m is a constant)) and has been used to describe the flow stress of API X65 pipeline

... These models include some mathematical functions that can be calibrated by experimental data. In decades, many phenomenological models such as Johnson-Cook (JC), Khan-Liang-Farrokh (KLF) and Arrhenius equation have been proposed [2,[8][9][10][11][12]. The physics-based models account for physical aspects of the material behavior such as the theory of thermodynamics, thermally activated dislocation movement and kinetics [2]. ...

... Figure 15 demonstrates the comparison between the experimental flow curves and predicted ones using Eq. (12). It is obvious that by taking β 0 = f(ε), better estimation of flow stress (with R 2 = 0.9397, as shown in Fig. 16) has been obtained. ...

The ability of four constitutive equations, Johnson–Cook (JC), Zerilli–Armstrong (ZA), Arrhenius-type constitutive equations and a newly developed phenomenological model for describing the hot flow behavior of 1.4542 stainless steel was evaluated. The hot flow curves were obtained from hot compression tests in the temperature range of 900–1050 °C and at strain rates of 0.001–1 s⁻¹. The JC model was not able to predict the softening part of the flow curves owing to the separated effects of strain, strain rate and temperature on the flow stress. The original ZA model was found to be useful at large strains but the overall consistency between the experimental and the calculated flow curves was not good. The subsequent modifications of the ZA model for low strain levels resulted in acceptable predictions. It is observed that the Arrhenius-type predicted flow curves well agree with the experimental results especially at low strain rates. However, at high strain rates where the steady state condition is shifted to large strains, some deviations were observed at low strain levels. Finally, a phenomenological model was constructed based on the nonlinear estimation of work hardening. The flow curves developed by the model could predict the experimental ones with satisfactory precision.
Graphic Abstract
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... When the flow stress is high (as>1.2), Eq. (8) can be simplified as follows [31]: ...

The hot deformation behavior of a high strength aluminum alloy (Al-Zn-Mg-Cu) was studied by isothermal hot compression tests performed over a range of temperatures (350–490 °C) and strain rates (0.001–1 s⁻¹). A constitutive equation was established using experimental results to predict the flow stress of the alloy under elevated temperature. In the work hardening-dynamic recovery regime, a physically-based constitutive equation for the flow stress was obtained from the stress-dislocation relation. In the subsequent dynamic recrystallization region, the flow stress after the peak was predicted by employing the kinematics of the dynamic recrystallization in the constitutive model. The stress-strain curves of the alloy predicted by the established models were in good agreement with experimental results. The results indicate that the proposed physically-based constitutive equation can accurately predict the flow behavior of the Al-Zn-Mg-Cu alloy.

... This LBL-theory has been successfully used to simulate the stress-strain curves for copper over fifteen decades of strain rate, and for temperatures between room temperature and about one-third of the melting temperature showing the excellent agreement with the experiments conducted by Follansbee and Kocks [7]. The theory has been extended to include the interaction between two subsystems by Langer [6] and used to simulate the stress-strain curves for aluminum and steel alloy [2] which exhibit thermal softening in agreement with the experiments conducted by Shi et al. [8] and Abbot et al. [9]. It has been employed in predicting the formation of adiabatic shear band in steel HY-100 [3] that shows quantitative agreement with the experimental observations by Marchand and Duffy [10]. ...

The present paper studies non-uniform plastic deformations of crystals undergoing anti-plane constrained shear. The asymptotically exact energy density of crystals containing a moderately large density of excess dislocations is found by the averaging procedure. This energy density is extrapolated to the cases of extremely small or large dislocation densities. By incorporating the configurational temperature and the density of redundant dislocations, we develop the thermodynamic dislocation theory for non-uniform plastic deformations and use it to predict the stress-strain curves and the dislocation densities.

... There, the thermal conductivity is large enough that heat generation can be neglected and no appreciable thermal softening occurs. However, typical stress-strain curves such as the ones shown for aluminum and steel in [11,12] and discussed by us in [6] exhibit thermal softening at large strain rates even without undergoing ASB. When ASB does occur, temperatures within the bands may increase by hundreds of degrees or more, and the thermal forces become one of the principal driving mechanisms. ...

The statistical-thermodynamic dislocation theory developed in our earlier studies is used here in an analysis of the experimental observations of adiabatic shear banding in steel by Marchand and Duffy (1988). Employing only a small set of physics-based parameters, we are able to explain experimental stress-strain curves, including yielding transitions and strain hardening, over wide ranges of temperatures and strain rates. We make a simple model of weak notch-like perturbations that, when driven hard enough, trigger shear banding instabilities that are quantitatively in agreement with those seen in the experiments.

... As for hot deformation behavior of metallic materials and metal matrix composites, it is commonly accepted that the relationship between the strain rate, the peak stress (or the steady stress) and deformation temperature can be described by the empirical function using Arrhenius type equation [34][35][36]as follows： ...

The flow stress behavior of AA6061/B4C composites has been researched by compression test using Gleeble-3800 thermal simulator, in the temperature ranging from 633-783K and strain rate ranging from 0.001-1s⁻¹. Typical true stress-true strain curves showed that the peak stress levels decreased with the rising of temperature but increased with the rising of strain rates. The combined effects of temperature and strain rate on deformation were analyzed by constitutive equation which containing the Zener-Hollomon parameter (Z) in hyperbolic sine function. The effects of Z values on dynamic softening and associated microstructural evolution during hot deformation were investigated by electron back scattered diffraction technique (EBSD). It was found that with the decrease of Z values, local strain induced by deformation was released and the grain size of aluminum matrix increased gradually, which indicated that the main softening mechanism of AA6061/B4C composites was dynamic recrystallization (DRX), and the lower the Z value was, the easier the DRX occurred.

... Only one fitting parameter is required to get the full agreement with the experiment conducted by Follansbee and Kocks (1998) over a wide range of temperatures and strain rates. The theory has been extended to include the interaction between two subsystems by Langer (2017) and used to simulate the stress-strain curves for aluminum and steel which exhibit the thermal softening (Le et al., 2017) in full agreement with the experiments conducted in (Shi et al., 1997;Abbod et al., 2007). ...

The present paper extends the thermodynamic dislocation theory developed by Langer, Bouchbinder, and Lookmann to non-uniform plastic deformations. The free energy density as well as the positive definite dissipation function are proposed. The governing equations are derived from the variational equation. As illustration, the problem of plane strain constrained shear of single crystal deforming in single slip is solved within the proposed theory.

... Only a single fitting parameter is required to achieve full agreement with the experiments conducted by Follansbee and Kocks [11] over a wide range of temperatures and strain rates. The theory was extended to include the interaction between two subsystems by Langer [29] and used to simulate the stress-strain curves for aluminum [43,45] and steel [45], which capture the thermal softening in full agreement with the experiments performed in [57,1]. The extension of the LBL-theory to nonuniform plastic deformations without including the excess dislocations was proposed by Langer [30], Le et al. [39], Le and Piao [38] and was applied to the adiabatic shear band for steel and torsion of aluminum bars. ...

The present paper extends the thermodynamic dislocation theory initiated by Langer, Bouchbinder and Lookman [2010] to non-uniform finite plastic deformations. The equations of motion are derived from the variational equation involving the free energy density and the positive definite dissipation function. We also consider the simplified theory by neglecting the excess dislocations. For illustration, the problem of finite strain constrained shear of single crystals with one active slip system is solved within the proposed theory.

... The interrelation of flow stress (σ), strain (ε), strain rate ( . ε), and temperature (T) has been illustrated using constitutive equations derived by several models [25][26][27]. The Arrhenius model is mostly used because the effect of strain rate and temperature are well considered [28]. ...

The hot deformation behavior of as-cast and homogenized Al0.5CoCrFeNi high entropy alloys (HEAs) during isothermal compression was investigated as a function of temperature and strain rate. Results indicated that flow stress in a homogenized state was always higher than that in an as-cast state under the same deformation conditions. Moreover, the optimum thermo-mechanical processing (TMP) conditions for the hot working of the homogenized state were identified as 945-965 °C and 10-1.7-10-1.1 s⁻¹ and were easier to determine in practice. Constitutive equations, for both states, correlating the flow stress of Al0.5CoCrFeNi with strain rate and deformation temperature were also determined.

... It can also be shown with Zener-Hollomon parameter. The hyperbolic law in Arrhenius type equation gives better approximations between Zener-Hollomon parameter and stress [4,5,6] . ...

The hot deformation behavior of medium carbon steel has been investigated using Gleeble thermo mechanical simulator to study the effect of forging conditions like temperature and strain rate on flow behaviour. It was then compared with the simulation studies carried out using a finite element based process simulation software tool, DEFORM TM 3D V6.1. The results from both studies indicate that the flow stress increases with lowering of forging temperature and with an increase in strain rate. Flow softening was observed at all deformation conditions which was reflected by a peak followed by a drop in flow stress with further straining. Flow softening was predominantly due to dynamic recrystallization at lower strain rates of 0.2/s and 2/s. Peak stress shifted towards higher strain with an increase in strain rate at all deformation temperatures. The results from finite element based process simulation revealed an excellent agreement with the experimental results obtained using thethermomechanical simulator.

... The most widely used is Arrhenius equation which represents the relationships between the strain rate, flow stress and temperature. The effects of the temperatures and strain rate on the deformation behavior can be represented by Zener-Hollomon parameter, Z, is an exponent-type equation [17,18]. ...

The deep understanding of flow behaviors of as-extruded 7050 aluminum alloy significantly contributes to the accuracy simulation for its various plastic forming processes. In order to obtain the improved Arrhenius-type equation with variable parameters for this alloy, a series of compression tests were performed at temperatures of 573K, 623K, 673K, 723K and strain rates of 0.01s-1, 0.1s-1, 1s-1, 10s-1 with a height reduction of 60% on Gleeble-1500 thermo-mechanical simulator. It is obvious that strain rate, strain and temperature all have a significant effect on the hot flow behaviors, and the true stress-true strain curves indicate three types after the peak value: decreasing gradually to a steady state with sustaining DRX softening till a balance with work hardening, decreasing continuously with sustaining increasing DRX softening beyond work hardening and maintaining higher stress level after the peak value with a balance between work hardening and DRV softening. Based on the experimental data, the improved Arrhenius-type constitutive model was established to predict the high temperature flow stress of as-extruded 7050 aluminum alloy. The accuracy and reliability of the improved Arrhenius-type model were further evaluated in terms of the correlation coefficient (R), here 0.98428, the average absolute relative error (AARE), here 3.5%. The results indicate that the improved Arrhenius-type constitutive model presents a good predictable ability.

... The most important data for high-temperature rheological characteristics is rheological stress, while strain rate and deformation temperature have a great influence on rheological stress. The main factors affecting rheological stress are chemical composition, technological parameters and evolution of microstructure (such as work hardening, flow softening, dynamic recrystallization, etc.) [34][35][36][37]. ...

The hot deformation behavior of 21-4N heat-resistant steel was studied by hot compression test in a deformation temperature range of 1000–1180 °C, a strain rate range of 0.01–10 s−1 and a deformation degree of 60%, and the stress-strain curves were obtained. The functional relationship between flow stress and process parameters (deformation degree, deformation temperature, strain rate, etc.) of 21-4N heat-resistant steel during hot deformation was explored, the constitutive equation of peak stress was established, and its accuracy was verified. Based on the dynamic material model, the energy dissipation maps and destabilization maps of 21-4N heat-resistant steel were established at strains of 0.2, 0.4 and 0.6, and processing maps were obtained by their superposition. Within the deformation temperature range of 1060~1120°C and a strain rate range of 0.01–0.1 s−1, there is a stable domain with the peak efficiency of about 0.5. The best hot working parameters (strain rate and deformation temperature) of 21-4N heat-resistant steel are determined by the stable and instable domain in the processing maps, which are in the deformation temperature range of 1120–1180 °C and the strain rate range of 0.01–10 s−1.

... Only one fitting parameter is required to get the full agreement with the experiment conducted by Follansbee and Kocks [1998] over a wide range of temperatures and strain rates. The theory has been extended to include the interaction between two subsystems by Langer [2017] and used to simulate the stress-strain curves for aluminum and steel which exhibit the thermal softening in full agreement with the experiments conducted in [Shi et al., 1997, Abbod et al., 2007. ...

... where Z is compensation factor of temperature to strain rate. Shi et al. [34] shows that expression between Z and s is determined by the following Eq.: ...

The hydrogen was straight-forward added to the Ti–6Al–4V/(TiC + TiB) composites (TiC[dbnd]TiB=5 vol.%) by melt hydrogenation. The results of hot compression show that the peak resistance of titanium matrix composites (TMCs) decreased by 17.2% when hydrogen content was 0.035 wt% compared with the TMCs without hydrogen. Therefore, the TMCs with a hydrogen content of 0.035 wt% was performed to a thermal compression experiment. Thermal-deformation characteristics and hot processing map of TMCs with a hydrogen content of 0.035 wt% were analyzed in the light of the flow stress curve, constitutive relations, and the dynamic-material model. The computed apparent activation energy was 284.54 kJ/mol, and the corresponding strain-rate sensitivity, power dissipation, and instability parameter were calculated. The hot-processing map exhibited maximum efficiencies of power dissipation at 780–840 °C/0.005–0.06 s ⁻¹ and there was only one instable region. The microstructures corresponding to the stable and instable region were verified, confirming the optimum processing parameters of hot-working that can be used as a reference for hot deformation of hydrogenated composites.

... ermomechanical processing to fabricate these large products is very complicated and the key processes are related to two steps as follows: (1) to produce usable and intricate shapes through hot die forging; (2) to meet the high quality mechanical properties through the subsequent heat treatment [7]. Due to the hexagonal closepacked structure (HCP) with less slip systems, titanium alloys are more difficult to deform than other metallic materials, such as Fe alloys [8] and aluminium alloys [9]. Previous studies proved that β titanium alloys had a relatively narrow process window and were sensitive to the processing parameters, such as temperature, strain, and strain rate [7,[10][11][12]. ...

In the present work, the hot deformation behavior of TB18 titanium alloy was investigated by isothermal hot compression tests with temperatures from 650 to 880°C and strain rates from 0.001 to 10 s−1. The flow curves after friction and temperature correction show that the peak stress decreased with the temperature increase and the strain rate decrease. Three typical characteristics of flow behavior indicate the dynamic softening behavior during hot deformation. At a strain rate of 0.001∼0.01 s−1, the flow stress continues to decrease as the strain rate increases after the flow stress reaches the peak stress; the flow softening mechanism is dynamic recovery and dynamic recrystallization at a lower temperature and dynamic recrystallization at a higher temperature. The discontinuous yielding phenomenon could be seen at a strain rate of 1 s−1, dynamic recrystallization took place in the β single-phase zone, and flow localization bands were observed in the α + β two-phase zone. At a higher strain rate of 10 s−1, the flow instabilities were referred to as the occurrence of flow localization by adiabatic heat. Constitutive equation considering the compensation of strain was also established, and the results show high accuracy to predict the flow stress with the correlation coefficient of 99.2% and the AARE of 6.1%, respectively.

... In contrast, the higher flow stress obtained in the Ni particulate reinforced composites is indicative of higher deformation resistance which is often not desirable when forming metallic materials into different profiles. Some researchers reported that higher resistance to deformation causes tool wear during metal working and may increase the overall cost of metal working [50][51][52]. To minimise forming cost of this Ni particulate reinforced composites, the most advantageous forming parameters determined using dynamic materials modelling approach becomes desirable and is being considered as the next phase of this work. ...

The response of two different types of aluminium matrix composites (AMCs) reinforced with silicon carbide ceramic particulates or nickel metallic particulates to hot compression testing parameters was evaluated. The composites were produced via two-step stir-casting technique. Axisymmetric compression testing was performed on the samples at different deformation temperatures of 220 and 370 °Ϲ, 0.5 and 5 s ⁻¹ strain rates and total strains of 0.6 and 1.2. The initial and post-deformed microstructures were studied using optical and scanning electron microscopy. The results show that flow stress was significantly influenced by imposed deformation parameters and the type of reinforcements used in the AMCs. Nickel particulate reinforced aluminium matrix composite (AMC) showed superior resistance to deformation in comparison with silicon carbide reinforced AMC under the different testing conditions. In both AMCs, work hardening, dynamic recovery and dynamic recrystallisation influenced their response to imposed parameters. The signature of dynamic recrystallisation was very apparent in aluminium matrix composite reinforced with nickel particulates.

... The so-called LBL theory [21] deduced from these laws predicts correctly the stress-strain curves recorded by Kocks [14] and [39] during uniform plastic deformations of copper in the wide range of temperatures and strain rates. Its extension that includes thermal softening and adiabatic shear banding, proposed recently in [29,30], exhibits quantitative agreement with the experimental observations by Abbod et al. [1], Marchand and Duffy [33], and Shi et al. [40]. The extension of LBL theory to nonuniform plastic deformation that takes into account the non-redundant (geometrically necessary) dislocations [11,12,17,18,27,35,36,41], called the thermodynamic dislocation theory (TDT), was proposed in [22]. ...

This paper studies the plane constrained shear problem for single crystals having one active slip system and subjected to loading in both directions within the small strain thermodynamic dislocation theory proposed by Le (J Mech Phys Solids 111:157–169, 2018). The numerical solution of the boundary value problem shows the combined isotropic and kinematic work hardening, the sensitivity of the stress–strain curves to temperature and strain rate, the Bauschinger effect, and the size effect.

... Previous work has established that in stir zone (SZ) are achieved peak temperatures near to 80% of fusion temperature of base metal [47]. Aluminum alloy exhibits shear stress values between 10 to 15 MPa during high temperature deformation between 500 to 600 °C, respectively [50]. Numerical modelling results obtained in this work exhibit good correlation with experimental values above mentioned. ...

Shoulder geometry of tool plays an important role in friction-stir welding because it controls thermal interactions and heat generation. This work is proposed and developed a coupled rigid-viscoplastic numerical modeling based on computational fluid dynamics and finite element calculations aiming to understand these interactions. Model solves mass conservation, momentum, and energy equations in three dimensions, using appropriate boundary conditions, considering mass flow as a non-Newtonian, incompressible, viscoplastic material. Boundary conditions of heat transfer and material flow were determined using a sticking/sliding contact condition at tool / workpiece interface. Thermal history, as well as shear stress and rotational speed fields, forces and torque values for three shoulder geometry conditions were calculated. Numerical results of thermal history, torque and forces during welding showed good correlation with experimentally measured data.

... The so called LBLtheory [17] deduced from these laws predicts correctly the stress-strain curves recorded by Samanta [34], Follansbee and Kocks [11] during uniform plastic deformations of copper in the wide range of temperatures and strain rates. Its extension that includes thermal softening and adiabatic shear banding, proposed recently in [25,26], exhibits quantitative agreement with the experimental observations by Shi et al. [35], Abbod et al. [1], Marchand and Duffy [29]. The extension of LBL-theory to non-uniform plastic deformation that takes into account the non-redundant (geometrically necessary) dislocations [31,9,14,15,30,8,23,36], called the thermodynamic dislocation theory (TDT), was proposed in [19]. ...

This paper studies the plane constrained shear problem for single crystals having one active slip system and subjected to loading in both directions within the small strain thermodynamic dislocation theory proposed by Le (2018). The numerical solution of the boundary value problem shows the combined isotropic and kinematic work hardening, the sensitivity of the stress-strain curves to temperature and strain rate, the Bauschinger effect, and the size effect.

... This LBL-theory has been successfully used to simulate the stress-strain curves for copper over fifteen decades of strain rate, and for temperatures between room temperature and about two third of the melting temperature showing the excellent agreement with the experiments conducted by Follansbee and Kocks [6]. The theory has been extended to include the interaction between two subsystems [7] and used to simulate the stress-strain curves for aluminum and steel alloy [8] which exhibit thermal softening in agreement with the experiments conducted in [9,10]. It was again extended and employed to predict the formation of an adiabatic shear band in rapidly loaded HY-100 steel [11] that shows quantitative agreement with the experimental observations by Marchand and Duffy [12]. ...

This paper presents the thermodynamic dislocation theory containing several modifications over its first version which was originally proposed by Langer, Bouchbinder, and Lookman (2010). Employing a small set of physics-based material parameters identified by the large scale least squares analysis, we show that the theory can fit the stress-strain curves of bcc crystals niobium, tantalum, tungsten, and vanadium over a wide range of temperatures and strain rates.

... The application of the approach of hyperbolic sine (equation (3)) in the Arrhenius-type equation results in a better coincidence between the flow stress and the Zener-Hollomon parameter [8][9][10]17], being also more universal and effective when applied in a wide range of stresses. Moreover, a large number of works describing the constitutive equations or the plastic deformation activation energy (Q) of titanium alloys [18][19][20][21][22] is exclusively based on hyperbolic sine equations. ...

In the first part of this research Ti-Al-Sn-Zr-Mo alloy compression tests were performed on Gleeble 3800 thermomechanical simulator, in order to characterize the material behavior under various temperature-strain-strain rate conditions. The tests were carried out at strain rates from 0.01 to 100 s-1 , and at the temperatures of 800, 900, 950, 1000, and 1100 °C, to a true strain of 1. The data obtained from the compression tests were converted into true stress-true strain curves. Based on the stress-strain curves, the constitutive equation describing the flow behavior of the investigated alloy for all stress levels in the hot deformation process was developed as Arrhenius-type equation. The deformation activation energy and material constants for constitutive equation developed for the investigated Ti-Al-Sn-Zr-Mo alloy were calculated. The distribution of the Zener-Hollomon parameter in dependence on the flow stress values was close to linear, with high linear regression correlation coefficient, what confirmed the accuracy of developed constitutive equation describing the behavior of the Ti-Al-Sn-Zr-Mo alloy during hot forming. The distribution of deformation activation energy at different deformation temperatures and strain rates, supporting better understanding of the microstructural changes during deformation, was also elaborated. The obtained results allowed better understanding of the deformation mechanisms in Ti-Al-Sn-Zr-Mo alloy during hot working.

... The prediction of strain/stress distribution and microstructural evolution of hot deformed AISI 1035 steel is of great interest to the material designers, to study the workability and optimize the hot forming processing parameters. With the increasing use of Finite Element Method to characterize the work-piece behaviour under the different stages of forging, knowledge of the constitutive relationships relating process variables such as strain rate and temperature to the flow stress of the deforming material is required, and hence it is important to evaluate the flow stress [3]. Flow stress can be defined as the resistance of a material against plastic deformation and expressed as function of temperature, strain, strain rate and microstructure [4]. ...

... Where A, α and n are the material constants, and the value of σ is regarded as σ p . The Eq. (2) has two simple representations as [29,30]: ...

The hot deformation behavior of a multi-direction forged (MDFed) T2 copper was investigated by the isothermal compression test at deformation temperatures between 673 and 1173 K and strain rates between 0.001 and 10 s− 1. The results reveal that the deformation characteristics of the flow stress are sensitive to the hot deformation parameters. The deformation activation energy of the MDFed copper under the test conditions was calculated as 195.601 kJ/mol. The constitutive behavior was described by a two-stage constitutive model, which is based on the stress–dislocation relation and kinetics of dynamic recrystallization (DRX). Based on the dynamic material model, the three-dimensional (3D) processing maps were established to identify the instability regions and optimal hot processing parameters. It is found that DRX occurred in all the stability and instability regions, and the optimal processing conditions are in the temperature range of 923–1023 K and strain rate range of 0.1–1 s− 1, which indicates a feature of incomplete DRX. Moreover, the grains overgrew at high deformation temperatures and low strain rates, resulting in a poor workability for the MDFed copper.

... Constitutive equations may include different parameters depending on the scale and aims of a study; if the goal is to analyse, for J o u r n a l P r e -p r o o f with the Johnson-Cook model for Finite Element simulations; Ref. [33] used an artificial neural network model in which no mathematical model exists; Refs. [34,32,35,23,36,37,38,39,40] studied the Garofalo-Arrhenius model for an AA6061 or an Al-1Mg alloy without including strain dependence. In this work, strain dependence was included in the applied models, along with strain rate and temperature. ...

Constitutive models were built based on the results of isothermal hot compression tests for a 6061 aluminium alloy at temperatures of 400, 450, 500, and 550 °C and strain rates of 0.1, 1, and 10/s, which reproduced conditions of the hot rolling forming process for this alloy. The Garofalo-Arrhenius, Johnson-Cook, and Hensel-Spittel material models, modified versions of the latter two, as well as a newly proposed Johnson-Cook model were applied based on the experimental data. The predictive power of the constitutive models was assessed for a wide range of plastic strains, from the start of the plastic region up to a strain value of 1, including strain hardening at the beginning of the flow curve. Comparisons between experiments and models by means of the Pearson correlation coefficient and relative errors, considering different strain ranges, showed that the goodness of the models depends strongly on the considered strain range. Results revealed that the Garofalo-Arrhenius model provided the highest accuracy at any strain range, followed by the Hensel-Spittel models and the newly proposed Johnson-Cook model, which performed more accurately than its commonly employed modified version.

... In this work, a kinetic rate equation given by Zener, Hollomon and Shi et al..were utilized. [32,33] ...

The intrinsic properties and the damage behavior of powder metallurgy (P/M) connecting rod preform have significant effects on its metal flow behavior during flashless forging into its final complex shape with substantial densification. The P/M material constitutive equation were established using isothermal compression tests, and then used to study the metal flow behavior of a P/M connecting rod preform during flashless forging based on finite element modelling (FEM). Moreover, the preform geometry was designed based on these metal flow mechanisms. Experiments and flashless forging of P/M connecting rod preforms were performed in order to verify the accuracy of the simulations. The simulated results are well consistent with the experimental results. Results showed that the shank is much more prone to cracking due to the higher deformation rate and the faster cooling rate. The optimal dimensions of the P/M preform were obtained. When the preform geometry are optimized, the average density of the connecting rod increases homogeneously, and becomes superior to that of the original shape. This work suggests that the geometry of the preform can be designed efficiently based on our models. This work can help to derive a P/M connecting rod preform optimization methodology, which can offer the possibility of improving the quality of the connecting rods efficiently.

Hot compression behaviours of Ag-12 wt% SnO2 composite were studied on Gleeble−3500 machine at temperatures of 700–850 °C and strain rates of 0.001–1 s⁻¹, the flow stress constitutive equation of the composite during hot compression was established and the microstructure evolution was characterized by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The results showed that the flow stress decreased with an increase of the deformation temperature, but increased with an increase of the strain rate, and the serrated flow stress fluctuation happened on the true stress-true strain curves at a strain rate of 1 s⁻¹. SnO2 particles were mainly distributed along the extrusion direction before hot compression, and were dispersively distributed in the Ag matrix after hot compression. Formation of the twins in the Ag matrix and some SnO2 particles was mainly responsible for the serrated flow stress fluctuation. The combined action of dynamic recrystallization and twinning was the main deformation mechanism during hot compression of the Ag-12 wt% SnO2 composite.

The tensile flow behavior of 6061 aluminum alloy was investigated on Gleeble3500 thermal-mechanical simulator over a range of temperature 365°C~565°C and strain rate 0.01 s-1~1 s-1. The results demonstrate that 6061 aluminum alloy is a positive-strain-rate sensitive material. The flow stress increases with the increase of strain rate and decreases with the increase of deformation temperature. The stress exponent n and the deformation activation energy Q were evaluated by the linear regression analysis. And the tensile flow stress constitutive equation of 6061 aluminum alloy during high-temperature was obtained.

Hot compression experiments of 316LN stainless steel were carried out on Gleeble-3500 thermo-simulator in deformation temperature range of 1 223-1 423 K and strain rate range of 0.001-1 s-1. The flow behavior was investigated to evaluate the workability and optimize the hot forging process of 316LN stainless steel pipes. Constitutive relationship of 316LN stainless steel was comparatively studied by a modified Arrhenius-type analytical constitutive model considering the effect of strain and by an artificial neural network model. The accuracy and effectiveness of two models were respectively quantified by the correlation coefficient and absolute average relative error. The results show that both models have high reliabilities and could meet the requirements of engineering calculation. Compared with the analytical constitutive model, the artificial neural network model has a relatively higher predictability and is easier to work in cooperation with finite element analysis software.

The hot deformation behavior of Ti-25Al-14Nb-2Mo-1Fe alloy was investigated at deformation temperatures 950-1100℃ with strain rate 0.001-1 s-1, and the height reduction of 50% by using isothermal simulative compression testing. The results show that the flow stress of the alloy is sensitive to the hot working parameters, such as deformation temperature and strain rate. The true stress-true strain curves exhibit a peak flow stress, flow softening and steady state flow behavior. The stress exponent (n) and the apparent activation energy (Q) for hot deformation is calculated based on a hyperbolic-sine Arrhenius-type equation and multiple regression method. Constitutive equation is established for high temperature deformation of the alloy. ©, 2015, Chinese Mechanical Engineering Society of Heat Treatment. All right reserved.

20CrMnTiH steel was studied on a Gleeble-3500 thermo-simulation in the temperature range of 850-1150°C and astrain rate range of 0.1-1 s21. Plastic deformation behaviour was investigated, and to formulate a thermoplastic constitutive model, the Arrhenius equation was utilised. In addition, a modified Zener-Hollomon parameter considering the compensation of strain rate during hot compression was employed to improve the accuracy of the developed constitutive equation. Analysis results indicate that the constitutive model agrees well with experimental values and reveals the relationship of the flow stress, strain, strain rate and temperature, which provides the basis of numerical simulation of 20CrMnTiH forging process.

In order to research the flow behavior of 20CrMnTi and obtain its constitutive equation, the isothermal compression tests of 20CrMnTi were carried out using the Gleeble—3500 thermo-simulation machine, up to a 60% height reduction of the sample at strain rate range from 0.01 s⁻¹ to 10 s–1 and deformation temperature range from 1123 K to 1273 K. According to the experimental results, the constitutive equation of 20CrMnTi was established based on Arrhenius model. In addition, the compensation of strain was taken into account and a new method of modifying the constitutive equation was proposed by introducing a coefficient K related to the deformation temperature and stain rate, which effectively improved the prediction accuracy of the developed constitutive equation. The results show that the flow stress decreases with increasing deformation temperature and decreasing strain rate, and the proposed constitutive equation well predicts the flow stress of 20CrMnTi during the high temperature deformation.

Necking defects have long troubled the application of cross-wedge rolling technology in aluminium alloy shaft parts. To accurately predict necking defects, new judgement conditions are established based on the thermal performance of 6082 aluminium alloy. The limit-sectional shrinkage without necking defects is achieved by combining theoretical calculation and finite-element model analysis, which couples heat transfer and deformation. In this paper, a 6082 aluminium alloy extruded rod with a 40 mm diameter rolled at a preheated temperature of 500 °C and a rolling angular velocity of 1 rad/s is taken as an example. The simulation and experimental results show that necking defects do not occur on the rolled pieces if the sectional shrinkage is below the limit-sectional shrinkage but will occur when the sectional shrinkage is above it. The results prove that the prediction model of necking defects in cross-wedge rolling of 6082 aluminum alloy is feasible, and this research provides a theoretical basis for the qualified aluminum alloy shafts produced by the cross-wedge rolling.

Al-W alloy billets were produced by powder pressing at room temperature and subsequent hot pressing. Quantities of billets were compressed at constant strain rate and temperature with 60% height reduction on Gleeble-3800 thermal simulation testing machine to study the plastic flow behaviors of the test alloy. The temperature of the compression processes ranged from 450 to 570ºC. The strain rate was varied between 0.001 and 1-1. The regularity of flow stress for the test alloy varied at elevated temperatures was studied. The activation energy during hot deformation is 757.943 kJ/mol by calculated, and the Arrhenius constitutive relation model was established.

The flow stress behavior of the Al-Mg-Sc alloy during hot compression deformation conditions was studied by isothermal hot compression with thermal simulation test at deformation temperature range of 300-450°C and strain rate range of 0.001-1 s-1. The experimental results indicate that the flow stress of Al-Mg-Sc alloy increases with increasing strain and tends to be constant after a peak value at 300°C and strain rates range of 0.01-1 s-1, showing dynamic recovery. The flow stress falls down after a peak value with the increase of strain in other conditions, showing dynamic recrystallization. A hyperbolic sine relationship is found to correlate well the flow stress with the strain rate, and an Arrhenius relationship with the temperature. The strain hardening coefficient n and deformation activation energy Q are evaluated by linear regression analysis. And the flow stress constitutive equation of the alloy during hot compression is obtained.

The hot compression deformation test of extruded AZ80 magnesium alloy was conducted using a Gleeble-1500 stimulator, with the temperature ranging from 250°C to 450°C and the strain rate varying from 0.001 s-1 to 10 s-1. After the correction of the flow stress at different deformation conditions caused by deformation heat, the strain rate sensitivity of as-extruded AZ80 magnesium alloy was investigated, the constitutive equation was established and the critical parameters for dynamic recrystallization were determined. The results show that the strain rate sensitivity is relatively high at low strain rate and low temperature and relatively low at high strain rate and low temperature, and that the critical strain for dynamic crystallization in as-extruded AZ80 magnesium alloy increases with increasing strain rate or decreasing temperature.

The high temperature compression test of Cu-Cr0.5-Sn0.31-Zn0.15-Y0.054 alloy was carried out by Gleeble-1500D thermal simulation test machine. The strain rates were 0.01, 0.1, 1, 5 s-1. The deformation temperatures were 600, 700, 800°C. The maximum deformation degree was 0.6. The results show that the flow stress declines with deformation temperature rising. The flow stress increases with strain rate increasing. The hot compressive flow stress of the alloy has a noticeable peak stress at a low strain rate, 700 and 800°C, and the continuous dynamic recrystallization characteristic is obvious. The thermal deformation activation energy (Q) and the flow stress equation are obtained by the correlation of flow stress, strain rate and temperature. The dynamic recrystallization microstructure of the alloy is influenced by the deformation conditions. The effect of deformation on the hardness and conductivity of the alloy after cooling are significant. Copyright © 2014 Northwest Institute for Nonferrous Metal Research. Published by Elsevier BV. All rights reserved.

The effects of squeeze casting process on microstructure and flow stress behavior of Al-17.5Si-4Cu-0.5 Mg alloy were investigated and the hot-compression tests of gravity casting and squeeze casting alloy were carried out at 350–500 °C and 0.001–5 s–1. The results show that microstructures of Al-17.5Si-4Cu-0.5 Mg alloys were obviously improved by squeeze casting. Due to the decrease of coarse primary Si particles, soft α-Al dendrite as well as the fine microstructures appeared, and the mechanical properties of squeeze casting alloys were improved. However, when the strain rate rises or the deformation temperature decreases, the flow stress increases and it was proved that the alloy is a positive strain rate sensitive material. It was deduced that compared with the gravity casting alloy, squeeze casting alloy (solidified at 632 MPa) is more difficult to deform since the flow stress of squeeze casting alloy is higher than that of gravity casting alloy when the deformation temperature exceeds 400 °C. Flow stress behavior of Al-17.5Si-4Cu-0.5Mg alloy can be described by a hyperbolic sine form with Zener-Hollomon parameter, and the average hot deformation activation energy Q of gravity casting alloy and squeeze casting alloy is 278. 97 and 308.77 kJ/mol, respectively.

The dynamic recrystallization (DRX) process of hot compressed aluminium alloy 7050 was predicted using cellular automaton (CA) combined with topology deformation. The hot deformatation characteristics of aluminium alloy 7050 were investigated by hot uniaxial compression tests in order to obtain the material parameters used in the CA model. The influences of process parameters (strain, strain rate and temperature) on the fraction of DRX and the average recrystallization grain (R-grain) size were investigated and discussed. It is found that larger stain, higher temperature and lower strain rate (less than 0.1 s–1) are beneficial to the increasing fraction of DRX. And the deformation temperature affects the mean R-grain size much more greatly than other parameters. It is also noted that there is a critical strain for the occurrence of DRX which is related to strain rate and temperature. In addition, it is shown that the CA model with topology deformation is able to simulate the microstructural evolution and the flow behavior of aluminium alloy 7050 material under various deformation conditions.

To better understand the hot deformation behaviors of Hastelloy C-276 alloy under elevated temperatures, hot tensile tests were carried out in the temperature range of 1223−1423 K and the strain rate range of 0.01−10 s⁻¹, respectively. Based on the modified Zerilli−Armstrong, modified Johnson-Cook, and strain-compensated Arrhenius-type models, three constitutive equations were established to describe the high-temperature flow stress of this alloy. Meanwhile, the predictability of the obtained models was evaluated by the calculation of correlation coefficients (r) and absolute errors (Δ), where the values of r for the modified Zerilli−Armstrong, Johnson−Cook, and Arrhenius-type constitutive models were computed to be 0.935, 0.968 and 0.984, and the values of Δ were calculated to be 13.4%, 10.5% and 6.7%, respectively. Moreover, the experimental and predicted flow stresses were compared in the strain range of 0.1−0.5, the results further indicated that the obtained modified Arrhenius-type model possessed better predictability on hot flow behavior of Hastelloy C-276.

Reducing the weight of vehicles through lightweight construction is an effective method to extend the range of electric vehicles as well as to reduce emissions of conventional motor vehicles. The joining technology plays a crucial role in both the constructive and the material lightweight designs. High-strength welding of low-alloy ferritic steels, as used in body construction, is nowadays mastered by various fusion and pressure welding processes, e. g. laser or resistance spot welding.
When welding high-strength aluminum materials using present-day prevalent welding processes, however, significant reductions in strength can occur at the joint. Aluminum’s strengthening mechanisms are reduced or lost through the high heat input when fusing. These mechanisms cannot or only to a lesser extend be activated during formation of new microstructures accompanied by the solidification. Moreover, depending on the chemical composition of the aluminum alloy, solidifica¬tion cracks and in the particular case of resistance spot welding the high wear of electrodes cause general problems.
In order to solve or rather avoid the problems associated with the fusing and solidi-fication of high-strength aluminum alloys, The Welding Institute (GB) developed friction stir welding in 1991. Friction stir welding is a special pressure welding method with which the material remains completely in solid state. Contrary to the conventional friction welding processes, such as linear or rotary friction welding, no relative movement between the components or materials to be joined is needed. In fact, the friction work is applied by a rotating welding tool, which is pressed into the joint gap with the appropriate process force and running longitudinally along it, and thus the welding seam is produced.
Because of this specialty, that the material remains in a solid state, not only high-strength aluminum joints but also dissimilar joints can be produced which are not, or only to a limited extend possible, in melting metallurgy.
These include, in particular, material-bonded aluminum-steel hybrid compounds which are of special interest for the economical hybrid lightweight construction of the car body. However, the strength of such compounds can be severely limited by brittle, intermetallic compounds. This is one of the basic technological challenges of this work.
Therefore, this work shall contribute to establish the friction stir welding process as an industrial manufacturing method for high-strength aluminum and aluminum-steel-hybrid-compounds, especially for body construction with its specific require-ments. For this, mainly experimental as well as numerical approaches are developed in this work in order to better understand the process and to be able to quantify the effects on the resulting strength properties. Moreover, the aim is to utilize the gained findings for industrial processes by means of expanding, improving or modifying processes. As the solutions developed in this context are in part signifi¬cantly beyond the state for the art, a high number of inventions is made in this work with following patent applications. See Tabelle 8.1.
In the first part of the work, geometrically novel welding seam configurations, including the appropriate manufacturing process, are developed to be able to join aluminum and steel sheets of various thicknesses with high strength. Here, the requirements for later use of the hybrid joints in hybrid tailor welded blanks (TWB) are explicitly addressed. This is, in particular, the requirement to execute the welding seam as a butt joint with one smooth side. Another important requirement is that the TWB can be formed in deep-drawing processes without tearing in the area of the welding seam.
Two different solutions are developed for this purpose: With the first one, the higher-strength, yet thinner steel sheet is folded along the welding seam to realize an extended joining cross section. As this requires an additional process step and considering the fact that high-strength steels in particular cannot be folded without cracking, a second solution is developed in the course of this work. Thereby, a friction stir welding tool with a stepped welding pin is used to create a combined lap-and-butt joint. In this process, the lower cylindrical section of the welding pin executes a butt welding between steel and aluminum. The circular ring section of the pin simultaneously creates an overlap joint between the two materials.
The comparison of both solutions developed with the state of the art joints is performed using the material combination EN AW-6016-T4 2.0 mm (aluminum-magnesium-silicon alloy) / HC340LAD 1.0 mm (microalloyed fine-grained steel) typical for automobiles. A significant superiority of the combined lap-and-butt joint compared to state-of-the-art technology is revealed, especially in fatigue strength tests.
Combinations of aluminum and steel sheets where the product of thickness and strength of the aluminum sheet is larger than that of the steel sheets show results in cupping tests without tearing of the weld. Material combinations where the product of the sheet thickness and the strength of the steel sheet is larger the joints showed even after optimization of the welding parameters strain localization and failure in the heat affected zone of the aluminum side of the weld.
For this case of strain localization in the welding seam, a novel heat treatment method is developed for precipitation hardening alloys based on the aluminum-magnesium-silicon ternary system (6000 series). Starting point of the development are systematic analyses of the base material’s ageing behavior at different ageing temperatures, ageing times and pre-ageing times.
Furthermore, the limits for the occurrence of recrystallization for the base material, prestrained material as well as welded aluminum joints have been examined experimentally. Moreover, the growth of the intermetallic phases is analyzed in annealing tests of aluminum-steel friction stir welded joints as well as the effect on the joint strength. Here it can be demonstrated that the thickness-dependent, strength-limiting effect of this boundary layer can be explained very well with the theory developed by Weibull. The quantitative description of this correlation shows that conventional solution annealing processes are not expedient due to the time required to heat the components.
The newly developed heat treatment method therefore uses the welding process itself as a local solution annealing process. The basic requirement for this is that the welding process is performed fast enough so that there is no overageing of the strength-enhancing precipitates. Due to the significantly longer, logistically conditioned pre-ageing at room temperature of the base material compared to the welding seam, the base material responds much more slowly to artificial ageing. This means that, with this method, the strength of the welding seam can be enhanced by artificial ageing without the base material experiencing a significant increase in strength.
Process diagrams are developed for alloy EN AW-6016 to determine the minimum artificial ageing time needed. The diagrams take the ageing temperature, the natural ageing time of the welding seam as well as the seam undercut of the friction stir welding seams into account. The diagrams are systematically validated with aluminum welding seams and also by some aluminum-steel hybrid seams.
The third and final part of this work deals with the numerical modeling of the friction stir welding process in order to be able to perform numerical process optimizations in the future to further enhance the strength. At first, by means of literature research, the largest potential for an increase in the validity of continuum mechanical simulation models is identified for the friction stir welding process with the material models used. Secondly, material models known from literature are analyzed on how well they approximate the yield behavior of the material in the broad ranges of strain rate, temperature and strain which can occur during friction stir welding. Since well-known thermomechanical material models were developed for other applications, such as ballistic impacts or hot forming, the need for a new development became apparent. The new development deliberately only focuses on effects which are already known in literature and can be rated as relevant for the process scope of friction stir welding.
The newly developed model is implemented as a user subroutine for Abaqus/Explicit taking into account various assumptions regarding material behavior at changes in temperature. In order to determine the model parameters needed, compression tests are performed using a Gleeble 2000 at a broad spectrum of temperature as well as strain rates for the materials Al 99.5, EN AW-5182, AlSi10Mg and EN AW-6016. The material model significantly reduces the model error when approximating the test results compared to already established material models. This considerably increases the validity of process simulations using this material model as against the established Johnson-Cook model.

This paper addresses the development of a neural network (NN) model assisted with nonlinear data transformation for flow stress prediction of an aluminium alloy. The stress-strain data were collected from a plane strain compression (PSC) test machine, which is capable of testing under a wide range of strain rates. A straightforward NN modelling failed to give satisfactory stress predictions across the whole range of the input conditions. Careful analysis of the error behaviour led to the finding that the poor performance was caused by the large variation of the strain rates. To improve the model performance, a nonlinear data transformation was adopted for the strain rate, and the flow stress prediction was improved significantly.

The hot deformation behavior of TC27 titanium alloy at the temperatures of 900-1150 °C and the strain rate of 0.01-10 s ⁻¹ , the height reduction of 70%, was investigated in the isothermal compression test to identify the optimal extrusion parameters. The processing-map of TC27 titanium alloy was constructed based on dynamic materials model (DMM) and principle of Prasad*s instability. The conclusion shows that temperature and strain rate of deformation had a great influence on flow stress. At the beginning of deformation, the flow stress increased quickly with the augment of true strain and decreased slowly after flow stress reaching to the maximum value. Finally, flow stress tended to relatively stable condition. The flow stress decreased with the increase of temperature and increased with the increase of strain rate. The TC27 titanium alloy was sensitive to temperature and strain rate. Processing-map exhibited two peak efficiencies of power dissipation; one peak was 49% at 900°C/0.01 s ⁻¹ , which dynamic recovery occured. The other peak was also 49% at 1050 °C /0.01s ⁻¹ , which dynamic recrystallization occured in the domain. Besides, there were two instability areas in the processing-map which should be avoided during the extrusion. Therefore, in order to obtain the satisfactory properties, the parameters that 1050 °C and 0.01 s ⁻¹ were selected in the extrusion.

Hot deformation behavior of cast-and-homogenized GH706-ingot material was studied in this work. Isothermal uniaxial compression tests were performed at temperatures(°C): 990, 1020, 1050, 1080 and 1100 with strain rates (s-1): 0.01, 0.1 and 1. The stress-strain curves as well as changes in microstructures of various hot deformed specimens were analyzed. Inhomogeneous microstructures were found in the specimens and the flow curves were resulted from the comprehensive functions of microstructures change in all part of the specimens. The constitutive relationship of alloy GH706 has been established by linear regression analysis of the experimental data taken from the Arrhenius equations as a model. Then hot compression tests were carried out to estimate the allowable reductions, and 52.5% and 50.6% are suggested as the allowable reductions during cogging processing of GH706 alloy in each blow at 1110°C and 1130°C.

A phenomenological treatment of plastic deformation is proposed which makes it possible to describe in a unified way the plastic behavior of a material both under dynamic loading and in creep. The treatment is based on the notion of a unique structure parameter that determines the mechanical state of the material. By combining an evolution equation for this structure parameter with a kinetic equation, which relates the strain rate to the stress at a fixed value of the structure parameter, a complete description of plastic deformation is achieved. The evolution is viewed as that towards a steady state defined by a dynamic equilibrium of athennal work hardening (associated with the storage of dislocations) and strain-rate and temperature dependent work softening (associated with the annihilation of dislocations). Analytical solutions of the resulting set of equations are given for two different models based on physically most plausible assumptions concerning the hardening and competing softening processes. The corresponding expressions for the two deformation modes provide a basis for converting stress-strain data into creep data without any adjustable parameters. This is exemplified by a good agreement between a measured creep curve and the one predicted from stress-strain data in the case of room temperature deformation of polycrystalline copper. Commonly used techniques for evaluating creep and work hardening data are critically discussed.

Plane strain compression tests have been carried out on a ferritic stainless steel at a nominally constant temperature of 917°C and at strain rates of 0.15-5 s-1, and on commercially pure aluminium and an Al-1% Mg alloy at 450°C and strain rates in the range 0.4-12 s-1. Tests have been conducted both at constant strain rate and with strain rate increasing or decreasing from one constant rate to another in a controlled manner after steady state conditions had been established. A wide range of rates of change of strain rate have been studied, with initial and final strain rates differing by up to one and a half orders of magnitude.The ferritic stainless steel and the Al-1% Mg alloy follow a mechanical equation of state under all test conditions of changing strain rate in the sense that the flow stress is dependent only on the instantaneous strain rate and not on the way this strain rate is reached. On the other hand the commercial purity aluminium shows deviations from a mechanical equation of state, which increase systematically with the rate of change of strain rate but are the same magnitude for increasing and decreasing strain rate.

The axisymmetric deformation behavior of 0.9999 Cu is investigated at strain rates from 10-4 to 104 s-1. The variations of the flow stress and of the mechanical threshold stress (the flow stress at 0 K), which is used as an internal state variable, with strain rate and strain are measured. The experimental results are analyzed using a model proposed by Kocks and Mecking: results at constant structure are described with thermal activation theory; structure evolution (strain and strain rate evolution of the mechanical threshold stress) is treated by the sum of dislocation generation and dynamic recovery processes. A significant result is that the athermal dislocation accumulation rate, or Stage II hardening rate, becomes a strong function of strain rate at strain rates exceeding 103 s-1. This leads to the apparent increased strain rate sensitivity seen in a plot of flow stress at a given strain vs the logarithm of strain rate. An explanation is proposed for the strain rate dependence of this initial strain hardening rate based on the limiting dislocation velocity and average distance between obstacles.