Prediction Of Flow Stress Research Articles

Strain dependent constitutive analysis of hot deformation behaviour in 9Cr–1Mo ferritic steel

Following the sine – hyperbolic Arrhenius equation, constitutive analysis has been performed on true stress – strain data obtained from hot compression tests on plain 9Cr –1Mo steel over a wide range of temperatures (1223 – 1373 K) and strain rates (0.01 – 100 s–1). On incorporating the correction for shear modulus and diffusivity into this equation, the power-law plot exhibited a distinct deviation at higher stresses and was accounted for by considering the contribution from dislocation pipe diffusion. The correlation between stress, strain rate and temperature was found to follow the rate equation of the form /DL = constant[sinh(αLσ/G)]nh, where DL is lattice diffusivity, G is the shear modulus and, αL and nh are constants. The material constants were observed to be strain dependent and this was incorporated into the constitutive equation for predicting flow stress. The higher correlation coefficient (R = 0:996) and a lower average absolute relative error (3.264%) associated with prediction revealed that the developed strain dependent constitutive equation could predict flow stress over the investigated hot working domain.

J031021 Cr-Mo-V鋼熱間加工組織変化・降伏応力予測のための材料モデル

J031021 Cr-Mo-V鋼熱間加工組織変化・降伏応力予測のための材料モデル

Hot deformation behaviour and flow stress prediction of 7075 aluminium alloy powder compacts during compression at elevated temperatures

► The hot deformation behaviours of 7075 aluminium alloy powder compacts are investigated. ► The effect of relative green density is evaluated. ► Flow curves show a peak stress after which the flow stress remains nearly constant. ► Q and material constants are highly dependent on the relative green density. ► A constitutive equation including the variable of relative green density is developed. In the present study, the hot deformation behaviour of 7075 aluminium alloy powder compacts was studied by performing hot compression tests on a Gleeble 3800 machine. The main objectives were to evaluate the effect of the relative green density on the hot deformation behaviour and to model and predict the hot deformation flow stress of powder compacts using constitutive equations. For this purpose, powder compacts with relative green densities ranging from 83 to 95%, which were prepared by uniaxial cold pressing a commercial pre-mixed powder, were hot compressed at temperatures ranging from 350 °C to 450 °C and at true strain rates ranging from 0.01 s −1 to 10 s −1 . The true stress–true strain curves of the powder compacts exhibited a peak stress at a critical strain after which the flow stress remained nearly constant. As the deformation temperature increased or the strain rate and green density decreased, a decrease in the peak stress level was observed. The relationship between deformation temperature, strain rate, and the peak flow stress of powder compacts was described by the Zener–Hollomon parameter in an exponential equation containing relative green density compensated material constants and the deformation activation energy. The peak flow stresses calculated from the proposed formula were in good agreement with the experimental results, which confirms the applicability of the employed method for the prediction of the hot deformation flow stress of porous materials with different relative green densities.

A phenomenological constitutive model for high temperature flow stress prediction of Al–Cu–Mg alloy

Abstract The high-temperature flow behavior of Al–Cu–Mg alloy is studied by the hot compressive tests over wide range of strain rate and forming temperature. Considering the negative effects of the interfacial friction on the heterogeneous deformation of specimen, the measured flow stress was corrected. The effects of processing parameters on material flow behavior are discussed. Based on the measured stress–strain data, a phenomenological constitutive model, which considers the coupled effects of strain, strain rate and forming temperature on the flow behavior of alloy, was proposed to describe the compressive behavior of the studied Al–Cu–Mg alloy. The proposed constitutive model correlates well with the experimental results, which confirms that the proposed model can give an accurate and precise estimate of flow stress for the studied Al–Cu–Mg alloy.

A constitutive description of the thermo-viscoplastic behavior of body-centered cubic metals

The Johnson–Cook (J–C) equation, which is obtained from the phenomenological observations of experimental data at relatively low strain rates, cannot well describe the dynamic thermo-mechanical response of many materials at high strain rates, especially under the situations of high or low temperatures. This paper develops a new physics-based model for the constitutive description of BCC metals through a thermal activation analysis of the dislocation motion in the plastic deformation of crystalline materials with the use of the mechanical threshold stress (MTS) as an internal state variable. It was found that the new model can effectively reflect the plastic deformation mechanism of BCC crystals because it directly relates the macroscopic state variables in the constitutive model with the micromechanical characteristics of materials. The material parameters of the model are efficiently determined by an optimization method to guarantee that the material parameters are globally optimal in their theoretically allowed ranges. The application of the model to HSLA-65 steel and Tantalum shows that it is much easier to apply than the MTS model, that its flow stress predictions are better than the Rusinek and Klepaczko (R–K), Abed, Zerilli and Armstrong (Z–A) and J–C models, and that the present model predictions are in good agreement with the experimental data in a broad range of strain rate, temperature and strain.

Constitutive modelling of plasticity of fcc metals under extremely high strain rates

A reliable and accurate description of the constitutive behavior of metals under the coupled effect of extremely high strain rate has become more and more important. The conventional constitutive models available, however, do not apply when the strain rate is beyond 104s−1. This paper establishes a new constitutive model to describe the fcc crystalline plasticity at the extreme strain rate beyond which the material sensitivity to strain rate increases dramatically. The new model distinguishes the mobile dislocations from the total dislocations and incorporates the change of mobile dislocation density to count for the microstructural evolution of the material. A unified constitutive model is then proposed. An optimization method was used to obtain globally optimal parameters in the model. The flow stress predictions by the unified model show a very good agreement with experiments within the whole strain rate range from 1×10−4s−1 to 6.4×105s−1. The flow stress upturn phenomenon in OFHC copper was satisfactorily described.

Constitutive analysis of compressive deformation behavior of ELI-grade Ti–6Al–4V with different microstructures

In this study, a constitutive analysis of the flow responses of Ti–6Al–4V under various strain rates \( \dot{\varepsilon } \) was conducted by separately quantifying the hardening and softening effects of microstructure, interstitial solute and deformation heating on the total stress. For this purpose, a series of compression tests on an extra-low interstitial grade alloy with equiaxed, lamellar, or bimodal microstructures was performed at \( 10^{ - 3} \le \dot{\varepsilon } \le 10\;{\text{s}}^{ - 1} \) until the metal fractured, and the results were compared to those of the commercial grade alloy. The thermal stress σ* increased with an increasing interstitial solute concentration; the athermal stress increased in the order of equiaxed, lamellar, and bimodal microstructures. Load–unload–reload tests revealed that the flow softening at a relatively high \( \dot{\varepsilon } \) was likely caused by deformation heating rather than by microstructure change; thus flow softening was attributed to a decrease in σ*. Finally, a mechanical threshold stress model was extended to capture those observations; the modified model can provide a reasonable prediction of flow stress in Ti–6Al–4V with different microstructures and interstitial solute concentrations.

Prediction of flow stress in dynamic strain aging regime of austenitic stainless steel 316 using artificial neural network

Flow stress during hot deformation depends mainly on the strain, strain rate and temperature, and shows a complex nonlinear relationship with them. A number of semi empirical models were reported by others to predict the flow stress during deformation. In this work, an artificial neural network is used for the estimation of flow stress of austenitic stainless steel 316 particularly in dynamic strain aging regime that occurs at certain strain rates and certain temperatures and varies flow stress behavior of metal being deformed. Based on the input variables strain, strain rate and temperature, this work attempts to develop a back propagation neural network model to predict the flow stress as output. In the first stage, the appearance and terminal of dynamic strain aging are determined with the aid of tensile testing at various temperatures and strain rates and subsequently for the serrated flow domain an artificial neural network is constructed. The whole experimental data is randomly divided in two parts: 90% data as training data and 10% data as testing data. The artificial neural network is successfully trained based on the training data and employed to predict the flow stress values for the testing data, which were compared with the experimental values. It was found that the maximum percentage error between predicted and experimental data is less than 8.67% and the correlation coefficient between them is 0.9955, which shows that predicted flow stress by artificial neural network is in good agreement with experimental results. The comparison between the two sets of results indicates the reliability of the predictions.

Constitutive Modeling for the Prediction of Peak Stress in Hot Deformation Processing of Al Alloy Based Nanocomposite

Hot deformation tests were carried out on Al5083 – 2 %(vol) TiC nanocomposite in a temperature range of 250 – 450°C at varying strain rate of 0.01 – 1.0 sec-1. Constitutive models were developed for the prediction of peak flow stress relating strain rate, true stress, temperature and activation energy. The percentage error between measured flow stress and constitutive model values were calculated to analyse the efficacy of the model in the prediction of peak stress. Finally, a window of working of the selected nanocomposite is established for finding out the safer region of working.

An exponential material model for prediction of the flow curves of several AZ series magnesium alloys in tension and compression

This paper is concerned with flow behaviors of several magnesium alloys, such as AZ31, AZ80 and AZ81, in tension and compression. The experiments were performed at elevated temperatures and for various strain rates. In order to eliminate the effect of inhomogeneous deformation in tensile and compression tests, the Bridgeman’s and numerical correction factors were respectively employed. A two-section exponential mathematical model was also utilized for prediction of flow stresses of different magnesium alloys in tension and compression. Moreover, based on the compressive flow model proposed, the peak stress and the relevant true strain could be estimated. The true stress and strain of the necking point can also be predicted using the corresponding relations. It was found that the flow behaviors estimated by the exponential flow model were encouragingly in very good agreement with experimental findings.

A constitutive model for fiber-reinforced extrudable fresh cementitious paste

In this paper, time-continuous constitutive equations for strain rate-dependent materials are presented first, among which those for the overstress and the consistency viscoplastic models are considered. By allowing the stress states to be outside the yield surface, the overstress viscoplastic model directly defines the flow rule for viscoplastic strain rate. In comparison, a rate-dependent yield surface is defined in the consistency viscoplastic model, so that the standard Kuhn-Tucker loading/unloading condition still remains true for rate-dependent plasticity. Based on the formulation of the consistency viscoplasticity, a computational elasto-viscoplastic constitutive model is proposed for the short fiber-reinforced fresh cementitious paste for extrusion purpose. The proposed constitutive model adopts the von-Mises yield criterion, the associated flow rule and nonlinear strain rate-hardening law. It is found that the predicted flow stresses of the extrudable fresh cementitious paste agree well with experimental results. The rate-form constitutive equations are then integrated into an incremental formulation, which is implemented into a numerical framework based on ANSYS/LS-DYNA finite element code. Then, a series of upsetting and ram extrusion processes are simulated. It is found that the predicted forming load-time data are in good agreement with experimental results, suggesting that the proposed constitutive model could describe the elasto-viscoplastic behavior of the short fiber-reinforced extrudable fresh cementitious paste.

Size dependence of flow stress and plastic behaviour in microforming of polycrystalline metallic materials

Microforming which exploits the advantages of metal-forming processes has been considered to be a very promising manufacturing technology for microparts. However, the fundamental understandings on microforming have not been well established yet since the extensive findings and know-how of conventional forming cannot be simply transferred to the microscale. This article attempts to offer an effective theoretical and experimental modelling to describe the so-called ‘size effect’ on the behaviour of metals in microforming. A theoretical model was developed based on the dislocation theory to quantify the grain size and feature size effects on the flow behaviour of polycrystalline materials. Tensile tests of copper sheets of various specimen thicknesses and grain sizes were carried out, and thus the effectiveness of the theoretical model can be confirmed by comparing the predicted flow stress, dislocation density, and Hall–Petch constant with experimental measurements. The general trend of decreasing flow stress with increasing grain size and decreasing specimen thickness was observed. The ratio of feature size to grain size was identified as an important parameter to characterize the flow stress level, grain boundary strengthening, and failure characteristics in microscale deformation processes.

Hot deformation behaviour and constitutive modelling of P92 heat resistant steel

Hot compression tests are conducted in the present paper to investigate hot deformation behaviour of the newly developed heat resistant steel P92, which is used to fabricate main steam pipes for ultra supercritical power plants. Stress–strain curves at elevated temperatures and different strain rates are obtained. It is found that dynamic recrystallisation happens only when the temperature is above 1100°C and strain rate is below 0·1 s−1. Otherwise, dynamic recovery is the main softening mechanism. Constitutive modelling with the hyperbolic sine, including an Arrhenius term, is used to predict peak and saturated stresses. Material constants for this model are determined. Results show that the model can be used to predict peak and saturated stresses. However, the model would fail in predicting flow stress with respect to strain; thus, a model containing nine independent parameters and the complete form of Spittel equation are utilised to predict flow stress curves softened by dynamic recrystallisation and dynamic recovery respectively since there are no unified equations. The predicted stress–strain curves are in good agreement with experimental results, which confirmed that the models developed in the present paper are effective and accurate for P92 steel.

Prediction of high temperature flow stress in 9Cr–1Mo ferritic steel during hot compression

Constitutive analysis was performed on the experimental true stress-true strain data obtained from hot isothermal compression tests on 9Cr–1Mo steel in a wide range of temperatures (1173–1373 K, i.e. 900–1100 °C) and strain rates (0.01–100 s −1). The constitutive equation for hot deformation is represented by a hyperbolic–sine Arrhenius type equation relating flow stress, strain rate and temperature, and could be described by the Zener–Hollomon parameter in an exponential type equation. The influence of strain was incorporated in the constitutive equation by considering the variation of material constants as a function of strain. It is observed that the compensation for strain could not accurately predict the flow stress for the entire strain rate and temperature regime. The constitutive equation was revised incorporating compensation for both strain and strain rate by suitably modifying the Zener–Hollomon parameter and the modified constitutive equation is found to give good prediction of flow stresses for most strain rate and temperature combinations.

Prediction of flow stress in isothermal compression of Ti60 alloy using an adaptive network-based fuzzy inference system

Isothermal compression of Ti60 titanium alloy at the deformation temperatures ranging from 960 to 1110 °C, the strain rates ranging from 0.001 to 10 s −1 and the height reductions of 60% were carried out on a Gleeble–3800 simulator. An adaptive network-based fuzzy inference system (ANFIS) model has been established to predict the flow stress of Ti60 alloy during hot deformation process. A comparative evaluation of the predicted and the experimental results has shown that the ANFIS model used to predict the flow stress of Ti60 titanium alloy has a high accuracy. The maximum difference and the average difference between the predicted and the experimental flow stress are 13.83% and 5.15%, respectively. The comparison between the predicted results based on the ANFIS model for flow stress and those using the regression method has illustrated that the ANFIS model is more efficient in predicting the flow stress of Ti60 alloy.

Internal-state-variable based self-consistent constitutive modeling for hot working of two-phase titanium alloys coupling microstructure evolution

An internal-state-variable based self-consistent constitutive model was proposed for unified prediction of flow stress and microstructure evolution during hot working of wrought two-phase titanium alloys in both single-beta region and two-phase region. For each constituent phase of titanium alloys, a set of constitutive equations incorporating solution strengthening, Hall–Petch effect, dislocation interaction, and dynamic recrystallization were developed using internal state variable method. The effect of second phase on recystallization was modeled by considering particle stimulated nucleation and exerting drag force on boundary migration. The constitutive equations of constituent phases were implemented into a viscoplastic self-consistent scheme to predict the overall response of the aggregate. Predictions of the model are in good agreement with experimental results of the Ti–6Al–4V alloy and IMI834 alloy. The model can reproduce many features of the hot working of two-phase titanium alloys, including the dependence of flow stress on temperature, strain rate and alloying elements; the increase of strain rate sensitivity with temperature; the stress and strain partitionings between alpha and beta phases; the relatively high apparent activation energy in two-phase region, the decrease of recrystallization kinetics with temperature in two-phase region; and the decrease of recrystallized grain size with Zener–Hollomon parameter in beta working.

A new relationship between the stress multipliers of Garofalo equation for constitutive analysis of hot deformation in modified 9Cr–1Mo (P91) steel

Constitutive analysis is performed following the Garofalo sine–hyperbolic equation on the true stress–strain data obtained from isothermal hot compression tests on modified 9Cr–1Mo (P91) steel over a wide range of temperature (1123–1373 K) and strain rate (0.001–100 s −1). We propose a new relationship between the stress multipliers α g and α ER for P91 steel as ( α g / α ER) = 0.27977+0.01531( n p ) 2; where the adjustable stress multiplier α g is obtained by force fitting Garofalo equation ( ε ˙ = constant [ sinh ( α g σ ) ] n g ) and α ER is obtained as α ER = β e / n p from the data fitted to power ( ε ˙ ∝ σ n p ) and exponential ( ε ˙ ∝ exp ( β e σ ) ) laws for the entire stress/strain rate regime. α g was easily obtained knowing α ER and following this, the other constitutive parameters n g , Q and ln A g were determined and were found to be strain dependent except n g . The successful prediction of flow stress has been supported by a higher correlation coefficient ( R = 0.994) and a lower average absolute relative error (5.03%) for the entire investigated hot working domain.

A constitutive model for predicting flow stress of Al 18B 4O 33w/AZ91D composite during hot compression and its validation

The flow stress properties of 25 vol.% aluminum borate whiskers reinforced magnesium composite fabricated by gas pressure infiltration were obtained by hot compression test. According to the experimental results, a modified constitutive model was proposed and applied to description of the relationship between deformation temperature, strain, strain rate and flow stress. Based on the experimental data, the material constants of the model were identified by application of nonlinear regression method. Additionally, the derived constitutive model was embedded in a FEM software, by which the experimental hot compression process was simulated. The results shown the load–stroke values of hot compression calculated by FE simulations are in agreement with the measured results in experiments, which indicate the developed constitutive equation can provide an accurate and sound description of plastic deformation of the composite during hot forming process.

Flow stress model considering the transformation-induced plasticity effect and the inelastic strain recovery behavior

On the basis of continuum mechanics and the Mori-Tanaka mean field theory, a micro-mechanical flow stress model that considered both the transformation-induced plasticity (TRIP) effect and the inelastic strain recovery behavior of TRIP multiphase steels was presented. The relation between the volume fraction of constituent phases and plastic strain was introduced to characterize the transformation-induced plasticity effect of TRIP steels. Loading-unloading-reloading uniaxial tension tests of TRIP600 steel were carried out and the strain recovery behavior after unloading was analyzed. From the experimental data, an empirical elastic modulus expression is extracted to characterize the inelastic strain recovery. A comparison of the predicted flow stress with the experimental data shows a good agreement. The mechanism of the transformation-induced plasticity effect and the inelastic recovery effect acting on the flow stress is also discussed in detail.

Simulation of Hot Sheet Metal Forming Processes Based on a Micro-Structural Constitutive Model

A constitutive model is proposed for simulations of hot forming processes. Dominant mechanisms in hot forming including inter-granular deformation, grain boundary sliding and grain boundary diffusion are considered in the constitutive model. A Taylor type polycrystalline model is used to predict inter-granular deformation. Previous works on grain boundary sliding and grain boundary diffusion are extended to drive three dimensional macro stress-strain rate relationships for each mechanism. In these relationships, the effect of grain size is also taken into account. It is shown that for grain boundary diffusion, stress-strain rate relationship obeys the Prandtl-Reuss flow rule. The proposed model is used to simulate step strain rate tests and the results are compared with experimental data. It is concluded that the model can be used to predict flow stress for various grain sizes and strain rates. The proposed model can be directly used in simulation of hot forming processes and as an example the bulge forming process is simulated and the results are compared with experimental data.