Unified Viscoplastic Constitutive Equations Research Articles

A CDM-based unified viscoplastic constitutive model of FLCs and FFLCs for boron steel under hot stamping conditions

Forming limit curves (FLCs), which are constructed using the limit strains at localised necking, are the most widely used tools for the evaluation of the formability of sheet metals. Fracture forming limit curves (FFLCs) are more recently developed, complementary tools for formability evaluation which are instead constructed using the limit strains at fracture. Since the formability depends strongly on forming conditions such as strain state, temperature and strain rate, models for predicting FLCs and FFLCs are essential for the optimisation and further application of hot forming processes in which these forming conditions vary significantly with both position and time. However, no model has so far been developed to predict FFLCs either alone or in conjunction with FLCs for sheet metals such as boron steel under hot stamping conditions. In this study, a set of unified viscoplastic constitutive equations for the prediction of both FLCs and FFLCs based on continuum damage mechanics (CDM) has been formulated from a set of recently developed constitutive equations for dislocation-based hardening, in combination with two novel coupled variables characterising the accumulated damage leading to localised necking and fracture. The novel variables take into account the effects of strain state, temperature and strain rate on the formability of sheet metals. The material constants in the CDM-based constitutive equations have been calibrated using experimental data comprising true stress-true strain curves and limit strains of a 22MnB5 boron steel obtained at a range of temperatures and strain rates. Investigation of the effect of varying selected parameters in the coupled damage variables on the resulting computed FLCs and FFLCs has demonstrated the flexibility of the model in enabling curves of different shapes and numerical values to be constructed. This indicates the potential of the CDM-based constitutive model for application to other materials for warm or hot stamping processes.

A new set of physically-based unified viscoplastic constitutive equations for superplastic deformation of Al-Zn-Mg-Cu alloy

A new set of physically-based unified viscoplastic constitutive equations using internal state variable(ISV) method during the superplastic deformation of Al-Zn-Mg-Cu alloy at temperatures ranging from 480 to 530 ℃ and strain rates ranging from 3×10−4s−1 to 3×10−3s−1 has been proposed. The damage evolution equation in this set of constitutive equations takes into account the critical cavity nucleation strain. This set of constitutive equations can predict microstructure evolution and flow stress during superplastic deformation of the investigated Al-Zn-Mg-Cu alloy. The set of constitutive equations have been coupled into crystal plasticity finite element method for determining the evolution of grain size and cavity volume fraction and the lattice rotation field map. The Material parameters are determined by genetic algorithm. The influences of strain, strain rate and temperature on grain size evolution and cavitation behavior were analyzed. The predicted stress-strain behavior, grain size and damage evolution are in good agreements with the experimental results.

Modeling the anisotropic plasticity and damage of AA7075 alloy in hot forming

The theory of unified visco-plastic damage constitutive equations has been frequently used to model the thermal flow behavior and damage evolution of various materials. But the anisotropic properties of materials were generally neglected in the previous studies. In this paper, a novel damage correction formula was proposed, which was expressed as a function of loading path and loading direction and used as a multiplier of a continuum damage model. By integrating a set of unified visco-plastic damage constitutive equations, the Yld2004-18p criterion and the damage correction formula, an improved anisotropic damage constitutive model was established. Anisotropic plasticity and anisotropic damage of AA7075 alloy under hot forming conditions can be properly modeled by this model. Material constants included in the three integrated functions were determined respectively based on different experimental data. The calibrated model is capable to accurately predict the forming limit curves of AA7075 alloy for different forming temperatures, strain rates and loading directions. The model was then numerically implemented in hot Nakajima simulations. The simulation results were evaluated comprehensively from the aspects of limit strain point, punch force- displacement curve, major strain field and fracture position. For further verification of the established model, hot forming simulation of a B-pillar was conducted and compared with a real B-pillar of AA7075 alloy fabricated via an industrial hot stamping process. It is concluded the anisotropic damage constitutive model can be numerically applied to accurately predict the plastic deformation and damage-induced fracture of AA7075 alloy under hot forming conditions.

Constitutive modeling for the simulation of the superplastic forming of TA15 titanium alloy

Titanium alloy, TA15, has a high strength-to-weight ratio, high weldability, and superior creep resistance at high temperatures up to 550 °C. TA15 is difficult to deform, especially for forming complex-shaped large-scale web–rib components, due to its low plasticity, large inhomogeneous deformation and narrow processing window. The objective of this research is to model the superplastic mechanisms in TA15 alloy with equiaxed, fine grain structure, and applying the proposed constitutive model to investigate the maximum grid aspect ratio, that can be achieved in superplastic forming (SPF), for a TA15 sheet with an initial thickness of 1.2 mm. Thermo-mechanical tensile tests are conducted first to characterize the superplastic behavior of the material in the temperature range of 880– 940 °C and the strain-rate of 0.0005–0.01 s−1. A set of mechanism-based unified visco-plastic constitutive equations has been proposed and calibrated based on the results of stress-strain data. A gradient-based optimization method is applied for the calibration of constitutive equations. The constitutive model is incorporated into FEA code through creep subroutine to check the validity of the proposed material model against the experimental SPF test of a multi-box die. Predicted sheet thickness and thinning in a die entry radius region at the end of forming are examined in detail. Preliminary results show a good agreement between the computational and experimental results.

Behaviour and constitutive modelling of ductile damage of Ti-6Al-1.5Cr-2.5Mo-0.5Fe-0.3Si alloy under hot tensile deformation

In this paper, the flow softening and ductile damage of TC6 alloy were investigated using a uniaxial hot tensile test with deformation temperatures of 910 °C∼970 °C and strain rates of 0.01 s−1∼10 s−1. Scanning electron microscopy (SEM) was performed on the deformed specimens to reveal the damage mechanism. The results showed that the flow stress rapidly increases to a peak at a tiny strain, followed by a significant decrease due to flow softening and ductile damage. The ductile damage of the studied TC6 alloy can be ascribe to the nucleation, growth and coalescence of microdefects, and the microvoids preferentially nucleate at the interface of the alpha phase and beta matrix due to the inconsistent strain. Then, a set of unified viscoplastic constitutive equations including flow softening and ductile damage mechanisms was developed and determined, and this set of equations was verified by the experimental flow stress, which indicated the reliability of the prediction. Furthermore, the predicted normalized dislocation density and the adiabatic temperature rise increase with decreasing temperature and increasing strain rate. The predicted damage components show that the microdefects mainly nucleate in the initial stage, but then primarily grow and link together with continuing deformation.

Unified modelling of the flow behaviour and softening mechanism of a TC6 titanium alloy during hot deformation

In this paper, the flow behaviour and globularization kinetics of a TC6 alloy were investigated by conducting hot compression tests at different temperatures (910–970 °C), strain rates (0.01–10.0 s-1) and true strains (0.223, 0.511, 0.916). Scanning electron microscopy (SEM) measurements were performed to evaluate the phase transformation and globularization fraction. Based on the experimental results, a set of internal state variable-based unified viscoplastic constitutive equations was developed to model the flow behaviour and the globularization evolution of the TC6 alloy. The model was fully determined by an optimization method based on genetic algorithm (GA). The results show that the predicted beta phase transformation curve fits the experimental data well, and that the predicted flow stress and globularization fraction of the alpha phase are in good agreement with the experimental data, even at strain rates outside the limits of the model. Furthermore, the flow stress begins to decrease at the onset of the softening mechanism, which occurs at a true strain of approximately 0.05. While an analysis of the predicted variations in the strain hardening rate and strain rate of each phase reveals that the softer beta phase accumulates more overall strain, whereas the strain accumulation in the alpha phase decrease.

Investigation on precision and performance for hot gas forming of thin-walled components of Ti2AlNb-based alloy

Ti2AlNb-based alloys have received considerable attention as potential materials to replace the nickel alloy at 600-750 °C, depending on their advantages of high specific strength, good corrosion and oxidation resistance. To realize the precision and performance control for Ti2AlNb-based alloy thin-walled components, the microstructure evolution was analyzed for setting up the unified viscoplastic constitutive equations based on the physical variables and simulating the forming process coupled between the deformation and the microstructure evolution. Through the finite element model with coupling of microstructure and mechanical parameters, the microstructure evolution and shape fabricating can be predicted at the same time, to provide the basis for the process parameters optimization and performance control. With the reasonable process parameters for hot gas forming of Ti2AlNb thin-walled components, the forming precision and performance can be controlled effectively.

Formability evaluation for sheet metals under hot stamping conditions by a novel biaxial testing system and a new materials model

Hot stamping and cold die quenching has been developed in forming complex shaped structural components of metals. The aim of this study is the first attempt to develop unified viscoplastic damage constitutive equations to describe the thermo-mechanical response of the metal and to predict the formability of the metal for hot stamping applications. Effects of parameters in the damage evolution equation on the predicted forming limit curves were investigated. Test facilities and methods need to be established to obtain experimental formability data of metals in order to determine and verify constitutive equations. However, conventional experimental approaches used to determine forming limit diagrams (FLDs) of sheet metals under different linear strain paths are not applicable to hot stamping conditions due to the requirements of rapid heating and cooling processes prior to forming. A novel planar biaxial testing system was proposed before and was improved and used in this work for formability tests of aluminium alloy 6082 at various temperatures, strain rates and strain paths after heating, soaking and rapid cooling processes. The key dimensions and features of cruciform specimens adopted for the determination of forming limits under various strain paths were developed, optimised and verified based on the previous designs and the determined heating and cooling method [1]. The digital image correlation (DIC) system was adopted to record strain fields of a specimen throughout the deformation history. Material constants in constitutive equations were determined from the formability test results of AA6082 for the prediction of forming limits of alloys under hot stamping conditions. This research, for the first time, enabled forming limit data of an alloy to be generated at various temperatures, strain rates and strain paths and forming limits to be predicted under hot stamping conditions.

A Method of Determining Unified Viscoplastic Constitutive Equations for Hot Forging Simulations

Constitutive equations have been used extensively to accurately describe material properties over a wide range of temperatures and strain rates in numerical simulations. In this paper, an algorithmic method of determining the constants of such constitutive equations is presented. The Genetic Algorithm implementation utilising MATLAB is described, and example fits to experimental data are presented.

Microstructure evolution and constitutive equations for the high-temperature deformation of 5Cr21Mn9Ni4N heat-resistant steel

The high-temperature deformation behavior of 5Cr21Mn9Ni4N heat-resistant steel was studied using a Gleeble-1500D thermal simulation test machine for high-temperature compression tests. The deformation temperatures ranged from 1273 K to 1393 K, and the strain rates ranged from 0.1 s−1–10 s−1. The height reductions were 20%, 40%, and 60%. The flow stress of 5Cr21Mn9Ni4N increased with increasing strain rate, whereas it decreased with increasing temperature. The high-temperature deformation of 5Cr21Mn9Ni4N began from the strain-hardening stage to the steady-state deformation stage. The characteristics of the stress–strain curves were determined through the interaction of work hardening, dynamic recovery, and dynamic recrystallization. The relationship between microstructure and processing parameters was analyzed. A set of unified viscoplastic constitutive equations based on changes in dislocation density, volume fraction of dynamic recrystallization, and grain size was established to predict deformation behavior and microstructure during a high-temperature working process. The average relative error between the calculated and experimental flow stress was 4.8%. This finding indicated that the constitutive equations could be used to predict the flow behavior of 5Cr21Mn9Ni4N accurately during high-temperature deformation.

THE DEVELOPMENT OF STRESS BASED CONTINUUM DAMAGE MECHANICS MODEL FOR PREDICTING THE FORMABILITY OF MAGNESIUM ALLOYS UNDER COLD/WARM STAMPING PROCESSES

This paper presents the development and application of the novel 2D-stress based continuum damage mechanics (CDM) model for prediction of the formability of magnesium alloys under cold/warm stamping conditions. The CDM model is divided into parts; firstly, a set of uniaxial viscoplastic damage constitutive equations is determined from tensile data. Secondly, a set of multiaxial viscoplastic damage constitutive equations is formulated and calibrated from the forming limit diagram (FLD) data. The experimental uniaxial tensile data for AZ31B magnesium alloy (at different deformation conditions (temperature range of 20°C to 300°C and strain rate range of 0.001 and 0.01s-1) and FLD at the temperature of 250°C were published by Wang et al., [1] and were used to formulate and calibrate the CDM model. A good agreement has been achieved between the experimental and numerical data. Using the newly developed plane-stress unified viscoplastic damage constitutive equations, the FLD of materials can be predicted at different temperatures and strain rates with complex strain path forming conditions.

Unified constitutive modelling for two-phase lamellar titanium alloys at hot forming conditions

In this paper, a set of mechanism based unified viscoplastic constitutive equations have been established for two-phase titanium alloys with initial lamellar microstructure, which models the softening mechanisms of the alloys in hot forming conditions. The dislocation density, rotation and globularization of lamellar α-phase and their effects on flow behaviour can also be modelled. The values of material constants in the equation set have been calibrated, according to stress-strain curves and globularization fractions of lamellar α-phase obtained from compression tests at a range of temperatures and strain rates, using a genetic algorithm (GA) based optimisation method. Based on the determined constitutive equations, flow stress and globularization evolution of Ti-17 and TA15 alloys at different temperatures and strain rates were predicted. Good agreements between the experimental and computed results were obtained.

Experimental investigation of boron steel at hot stamping conditions

The effect of deformation temperature and strain rate on the fracture and strain hardening of boron steel was investigated through the analysis of large amount of experimental data tested using a Gleeble (3800) materials simulator. These features were further modelled by using a set of unified viscoplastic damage constitutive equations. To study the deformation behavior, isothermal uniaxial tension tests of 1500MPa boron steel at different strain rates of 0.01–5.0s−1 and different deformation temperatures of 550–850°C were performed on a Gleeble 3800 materials simulator. Considering the difference between the deformation of the necking cross section and the centre measuring cross section of specimen at necking stage, a correction method of measuring strain at the necking cross section was developed. In addition, by taking temperature rise during deformation into account, a correction method of measuring stress was proposed. The true stress-strain curves were obtained based on the two corrections methods. The influence of deformation temperature and strain rate on the fracture and hardening was analyzed. A set of unified constitutive equations was adopted and determined from experimental data. The correlation between the numerical-computed and experimental true stress-strain data is presented. The average relative error is within the range of allowable experimental conditions and the predicted and experimental values can almost be consistent.

Modeling of microstructural evolution and flow behavior of superalloy IN718 using physically based internal state variables

Microstructural evolution and flow behavior greatly affect the hot forming process of IN718. In this research, hot deformation behaviors of IN718 were investigated by performing hot compression tests at temperature range of 1000–1100 °C with strain rates of 0.1–20.0 s−1. By incorporating physically based internal state variables such as dislocation density, volume fraction of dynamic recrystallization, and grain size, a set of unified viscoplastic constitutive equations were developed to predict the microstructural evolution and flow behavior of IN718. The material constants were determined using a genetic algorithm (GA)-based optimization method. Comparisons of the computed and experimental results indicate that the constitutive equations established in this study can accurately describe the hot deformation behavior and microstructural evolution of IN718.

Two constitutive descriptions of boron steel 22MnB5 at high temperature

The hot deformation behavior and constitutive description of the boron steel 22MnB5 are of vital importance to the process design and numerical simulation of the steel during hot stamping. In order to establish two constitutive descriptions for 22MnB5, it was subjected to isothermal uniaxial tensile testing on a Gleeble 3500 thermomechanical simulator at temperatures ranging from 700°C to 900°C and strain rates ranging from 0.01s−1 to 10s−1. On the basis of the testing results, a strain-compensated Arrhenius-type constitutive equation and a set of unified viscoplastic equations that take into account the dislocation density were derived for describing the flow behavior of 22MnB5 during hot stamping. The material constants for the two constitutive descriptions were obtained by linear regression and genetic algorithm-based optimization, respectively. Further, the predictability of both the constitutive descriptions was assessed in terms of the correlation coefficient, the average absolute relative error, the root mean square error, and the distribution of the relative error. The results indicated that both constitutive descriptions are effective for describing the hot tensile behavior of 22MnB5, however, the prediction accuracy of the strain-compensated Arrhenius-type constitutive equation is a little higher than that of the unified viscoplastic constitutive equations.

The development of continuum damage mechanics-based theories for predicting forming limit diagrams for hot stamping applications

This paper presents a novel plane-stress continuum damage mechanics (CDM) model for the prediction of the different shapes of forming limit diagrams (FLCs) for aluminium alloys under hot stamping conditions. Firstly, a set of uniaxial viscoplastic damage constitutive equations is determined from tensile experimental data of AA5754 at a temperature range of 350–550℃ and strain rates of 0.1, 1.0 and 10 s−1. The tests were carried out on Gleeble materials simulator (3800). Based on the analysis of features of FLCs for different materials forming at different temperatures, a plane-stress damage equation is proposed to take account the failure of materials at different stress-state sheet metal forming conditions. In this way, a set of multiaxial viscoplastic damage constitutive equations is formulated. The model is calibrated from the FLC data at temperature of 350℃ and strain rate of 1.0 s−1 for AA5754. A good agreement has been achieved between the experimental and numerical data. The effect of the maximum principal stress, effective stress and hydrostatic stress on the materials failure features and on the shape of FLCs is studied individually and in combination. Using the newly developed plane-stress unified viscoplastic damage constitutive equations, the FLC of materials can be predicted at different temperatures and strain rate forming conditions.

Development and determination of unified viscoplastic constitutive equations for predicting microstructure evolution in hot forming processes

This paper reports the development of advanced unified viscoplastic constitutive equations for the modelling of microstructure evolution and their effects on viscoplastic flow of materials. The material parameters within constitutive equations are determined by minimising the difference between predicted results and the corresponding experimental data. To achieve this, firstly, new methods have been developed to overcome different scale problems in a set of constitutive equations. This enables that all the equations can be integrated accurately with a specified single tolerance. Secondly, objective functions have been formulated to enable all the data to be optimised with equal opportunity. Thus, the selection of weighting factors is eliminated. Thirdly, based on the analysis of the failure of gradient-based, genetic algorithm (GA)-based and evolutionary programming (EA)-based optimisation techniques in determining the constants within constitutive equations, a new combined method has been proposed to increase the chance to get the global minimum.

FE analysis of a notched bar under thermomechanical fatigue using a unified viscoplasticity model

Z-mat software is used to implement the multiaxial form of the Chaboche unified viscoplastic constitutive equations into the Abaqus finite element (FE) code. Complex transient stress fields within a grade P91 steel notched bar specimen are investigated in detail for different thermomechanical fatigue (TMF) loading conditions. The results of this work can be used to determine suitable test conditions for notched specimens by providing predictions of the stress-strain history local to the notch. In future, such tests will be used along with the FE analyses to develop and validate multiaxial TMF lifing models.

Modelling mechanical property recovery of a linepipe steel in annealing process

In this paper a set of mechanism-based unified viscoplastic constitutive equations has been used to model the effect of microstructural evolution on mechanical property recovery in the annealing process of a cold formed linepipe steel. Dislocation density and plasticity-induced damage accumulation during deformation, and recovery and recrystallisation of the deformed material during subsequent annealing have been modelled. The effects of annealing time on microstructural evolution also have been investigated. Tensile tests were performed on the low carbon ferritic linepipe steel before and after annealing at 700 °C with different holding times. The experimental results have been used to characterise the unified constitutive equations, using an Evolutionary Programming (EP)-based optimisation method. Using these equations, the stress–strain relationships for the interrupted constant strain rate tensile tests were predicted and close agreement between the computed and experimental results was obtained, for various annealing times and for the materials with different amounts of pre-deformation.

An investigation of descaling spray on microstructural evolution in hot rolling

High pressure water spray is used to remove oxide scales in hot rolling of steel plate. This would reduce the temperature temporarily on the workpiece surface. To investigate the effect of temperature variation due to the descaling spray on the microstructural evolution, a thermal-mechanical coupled FE model has been established. The heat loss due to the spray is experimentally measured and modelled using the established FE model. The heat transfer coefficient is determined using an inverse engineering method. A set of unified viscoplastic constitutive equations is implemented into the commercial FE solver MARC through the user-defined subroutine CRPLAW. The effect of descaling water spray on the surface temperature variation is numerically studied. Furthermore, the effect of surface temperature variation on the evolution of dislocation density, recrystallisation and grain growth is investigated for various hot rolling conditions.