Developed Constitutive Equation Research Articles

Modelling of hybrid biocomposites for automotive structural applications

The demand for environmentally friendly materials is at its peak, and government legislations have become stricter and no longer tolerate violations. With biocomposites emerging as structural components instead of being hidden as non-structural applications and interiors, the automotive and motorsport sector started considering them for structural body parts. Natural fibres abundance, commercial availability, renewability, low density, low cost, and high tensile strength make biocomposite materials excellent candidates for eco-friendly vehicles. Several studies reported the utilisation of biocomposites as high-performance components and structural applications. However, current computer-aided engineering and modelling tools are insufficient to explore the vast field of possibilities during biocomposite materials selection to accurately predict the components' behaviour and analyses. Therefore, this study focuses on creating a material model to predict the behaviour of biocomposite materials. Additionally, the model is numerically implemented to illustrate the deformation and bending stiffness capabilities of a motorsport monocoque chassis structure application. A micromechanics modelling combining rate-dependant constitutive laws and multi-site interactions of inclusions is developed for studying the nonlinear response of composite materials. To avoid numerical instabilities when increments of time become very small, a regulation procedure concerning the visco-plastic function is adopted in the computation of the consistent tangent modulus. Based on the Generalised Mori–Tanaka (GMT) scheme, the effective properties are obtained for the nonlinear composite. The accuracy of the model is evaluated and validated by comparison results from the open literature. Finally, the developed constitutive equations are implemented as a user-defined material UMAT in a Finite Element code, leading to an application on a bio-based composite for the bamboo/flax fibre-reinforced epoxy hybrid composite materials.

Modeling of distribution parameters for upward steam-water boiling flows in subchannels of a vertical rod bundle

ABSTRACT In simulations of two-phase flow behavior in nuclear reactors, subchannel analysis codes are often used to evaluate the void fraction within a BWR fuel bundle in detail. When solving the momentum conservation equation averaged over a subchannel cross-section for upward two-phase flows, the distribution parameter is required to consider the void fraction and velocity distributions in the subchannel cross-section. In this paper, constitutive equations were developed for the distribution parameters for dispersed two-phase flows applicable to the inner, edge, and corner subchannels, which are typical subchannels in a fuel bundle. The distribution parameters could be calculated by giving the void fraction and the velocity distribution. Therefore, the distribution parameters were evaluated and modeled as a function of the geometrical parameters by assuming the void fraction and velocity distributions with bulk and subcooled boiling for each subchannel type. The developed constitutive equations were evaluated by comparing them with the distribution parameters estimated based on the NUPEC rod bundle void fraction test data. The developed distribution parameter model was implemented into the subchannel analysis code NASCA and compared with the measured cross-sectional average void fraction of the NUPEC rod bundle void fraction test data. In comparison with the original NASCA code, which assumed the distribution parameter to be unity, the improved NASCA with the distribution parameter model decreased the mean error of the measured cross-sectional average void fraction to less than half of the result of the original NASCA code, both in absolute and relative differences.

Dynamic recrystallization mechanism of Ti-6554 alloy during high-temperature deformation

Isothermal hot compression tests were carried out at a temperature range of 810–930 ℃ and a strain rate range of 0.001 s−1-10 s−1 to investigate the dynamic recrystallization (DRX) mechanism of Ti-6554 alloy during high-temperature deformation. The constitutive equation based on DRX was established. Optical microscopy (OM) and electron backscatter diffraction (EBSD) techniques were used to characterize the microstructure. The results showed that the developed constitutive equation based on DRX can effectively predict the flow stress of Ti-6554 alloy with a correlation coefficient of 0.995. High temperature and low strain rate are conducive to the development of DRX. The DRX mechanism is more dependent on the strain rate. At low strain rate, continuous DRX (CDRX) is dominant. With the increase of strain rate, the inhomogeneity of deformation leads to the occurrence of several DRX mechanisms simultaneously. Discontinuous DRX (DDRX) is often found near serrated grain boundaries. In addition, when the strain rate is increased to 10 s−1, the rare geometric DRX (GDRX) is detected, which is associated with the activation of deformation bands at high strain rate.

THERMAL-DEFORMATION MODEL OF A Sr-MODIFIED A356 ALUMINUM ALLOY

The hot-deformation behavior of A356 aluminum alloy with a Sr modification was investigated using a Gleeble 1500 thermal simulator. The true stress-strain curves with a deformation temperature of 300–500 °C and a strain rate of 0.01–5 s–1 were clarified. The activation energy of the A356 aluminum alloy with Sr modification was 221.474 kJ/mol. The influences of friction and temperature on the curves were investigated, and then the constitutive equation was established. The results show that the flow stress is obviously affected by temperature and strain rate. The experimental stress is lower than the theoretical stress, and the stress difference between the experimental and theoretical stress increases with the increasing strain. The maximum stress difference reaches 17.8 MPa when the sample deformed at 300 °C/5 s–1 with a reduction of 16 %. For all the deformation conditions the correlation coefficient is 0.99 and the average relative error is 4.8 %, which shows the good predictability of the current model. The developed constitutive equation can provide guidance for the study of the hot-deformation behavior of similar aluminum alloys.

Prediction of flow stress characteristics of P92 steel using a simple physically-based constitutive modelling

This study reports the flow stress behavior of three P92 steels with different compositions. Uniaxial compression tests were conducted in the deformation temperature range of 575 °C to 650 °C and strain rate range of 0.001-0.5 s-1 using a Gleeble® 3500 thermo-mechanical simulator. A simple physically-based constitutive model was used to analyse the effects of deformation conditions (temperature and strain rate) on the metal flow stress behavior during the deformation process. The method accounts for the temperature dependence of Young’s modulus and the lattice self-diffusion coefficient of Fe in the ferrite. Constitutive equations describing the flow stress behavior of the three P92 steels were developed. From the results, the stress exponent n of 26.13(steel A), 21.61(steel B) and 27.55 (steel C) were obtained using the self-diffusion activation energy in the physically-based constitutive equation. From the results, the three steels had variation in the stress exponent values, which was attributed to differences in elemental content, such as chromium and tungsten. The developed constitutive equations were verified using statistical parameters: Pearson’s correlation coefficient (R) and average absolute relative errors (AARE). Statistical analysis showed that the three steels had the same R of 0.98, while AARE was: 1.68 (steel A), 1.72 (steel B), and 1.82 (steel C). The constitutive equations developed showed a good correlation between the experimental and predicted flow stress data. Hence, the method is applicable in describing flow stress behavior in the metalworking process in the industry.

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

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

Stress–strain behavior of a selective laser melted Ti-6Al-4V at strain rates of 0.001–1/s and temperatures 20–1000 °C

Because of the lack of systematic studies of high-temperature mechanical properties of Ti-6Al-4 V fabricated using selective laser melting (SLM), the measurement of mechanical properties of SLMed Ti-6Al-4 V under thermomechanical conditions during SLM processing is necessary for accurate thermomechanical analyses of SLM processes. This study conducted compression tests of Ti-6Al-4 V fabricated by SLM and newly obtained stress–strain curves under thermomechanical conditions that are typical of those experienced during SLM processing (20–1000 °C temperatures and 0.001–1 /s strain-rates). This study also revealed that correlation between the maximum flow stress of SLMed Ti-6Al-4 V, the strain rate, and the temperature, respectively, with the Zener–Hollomon parameter (Z parameter) at temperatures of 600–1000 °C and strain rates of 0.001–1/s. A comparative study revealed that 0.2 % offset yield stress at temperatures 20–1000 °C had up to 43 % difference between the SLM-processed and conventional wrought Ti-6Al-4 V. This result indicates that not the mechanical properties of the wrought Ti-6Al-4 V but those of SLMed Ti-6Al-4 V should be used for conducting accurate thermomechanical analysis of SLM processing of Ti-6Al-4 V. Based on the obtained results, this study developed a constitutive equation using the Z parameter. It enabled calculation of the maximum flow stress of the SLM-processed Ti-6Al-4 V at strain rates of 1/s, 0.01/s, and 0.001/s and at 600–1000 °C temperatures, with average error of 15 %. The SLM-processed Ti-6Al-4 V flow stresses calculated using the developed constitutive equation are useful as mechanical properties required for the thermal stress analysis of the SLM process of Ti-6Al-4 V and are expected to contribute to improvement of the predictive accuracy of the residual stress and deformation during the SLM process of Ti-6Al-4 V.

A continuum magneto-mechanical model for magnetorheological elastomers

Magnetorheological elastomers (MREs) consist of micron-sized magnetizable particles embedded in a rubber matrix. Properties of these magneto-sensitive materials are changed reversibly upon application of external magnetic fields. They exhibit highly non-linear magneto-mechanical response which allows developing new devices and applications. However, the coupled magneto-mechanical behavior makes mathematical modeling of MREs quite complicated. So development of a reliable constitutive framework is essential for further understanding of this coupled behavior as well as simulation of the systems that utilize MREs. In this paper, a finite strain continuum model is developed for MREs where the effect of magnetization on material stiffness is directly introduced in the material shear modulus. It is shown that this approach simplifies the constitutive models and also perceives the magnetic saturation of MREs. Moreover, the coupled effects of magnetization, deformation and particle-chains orientation on the mechanical response are also taken into account in the introduced parameter. This reduces the number of material parameters, the required experimental tests for parameters identification and also simplifies the mathematical formulation of the developed constitutive equations which is beneficial for numerical formulations. A systematic two-step method is then introduced for material parameters identification which assures uniqueness of the parameters set. The predictive capabilities of the proposed model are examined via available mechanical and magneto-mechanical experimental data on both isotropic and anisotropic MRE samples at different configurations of magnetic field and loading with respect to the preferred direction of the samples. It is shown that the model can well predict the magneto-mechanical response of MREs at different deformation modes and magnetic fields.

Dynamic drags acting on moving defects in discrete dispersive media: From dislocation to low-angle grain boundary

Although continuum theory has been widely used to describe the long-range elastic behavior of dislocations, it is limited in its ability to describe mechanical behaviors that occur near dislocation cores. This limit of the continuum theory mainly stems from the discrete nature of the core region, which induces a drag force on the dislocation core during glide. Depending on external conditions, different drag mechanisms are activated that govern the dynamics of dislocations in their own way. This is revealed by the resultant speed of the dislocation. In this work, we develop a theoretical framework that generally describes the dynamic drag on dislocations and, as a result, derive a phenomenological cubic constitutive equation. Furthermore, given that a low-angle grain boundary (LAGB) can be regarded as an array of dislocations, we extend the model to describe the mobility law of LAGBs as a function of misorientation angle. As a result, we prove that both dislocations and LAGBs follow the developed constitutive equation with the same mathematical form despite their different governing drag sources. The suggested model is also supported by molecular dynamics simulations. Therefore, this work has significance for a fundamental understanding of the dynamic drag acting on defects and facilitates a general description of various drag mechanisms.

Constitutive modelling of hot deformation behaviour of a CoCrFeMnNi high-entropy alloy

ABSTRACT Models describing the constitutive flow behaviour of a metallic material are desired for appropriate process design and realization of defect-free components. In this study, constitutive equations based on the hyperbolic-sinusoidal Arrhenius-type model have been developed to define the hot deformation characteristics of a CoCrFeMnNi high-entropy alloy. The experimental true stress-true strain data were generated over a wide temperature (1023–1423 K) and strain rates (10−3–10 s−1) ranges. The impact of strain rate and temperature on deformation behaviour was further characterized through a temperature compensated strain rate parameter, i.e. Zener-Hollomon parameter. Additionally, a mathematical relation was employed to express the influence of various material constants on true-strain ranging from 0.2 to 0.75. Typical third order polynomial relations were found to be appropriate to fit the true-strain dependency of these material constants. The accuracy of the developed constitutive equations was evaluated by using the average absolute relative error (AARE) and correlation coefficient (R); the obtained values were 7.63% and 0.9858, respectively, suggesting reasonable predictions. These results demonstrate that the developed constitutive equations can predict the flow stress behaviour of the alloy with a good accuracy over a wide range of temperature and strain rate conditions and for large strains.

Constitutive modelling of high strength low alloy steel under hot compression test

The HSLA steel was studied on an MMS-200 thermal mechanical simulation in the temperature range of 800-1100 °C and a strain rate range of 0.1-1 s−1. The isothermal hot compression tests were used to investigate the flow behaviour of HSLA steel and develop a constitutive equation based on an Arrhenius equation. A modification of Zener–Hollomon parameter was considering the strain rate compensation after hot compression for improve the accuracy of the developed constitutive equation. The comparison of the results shows that the experimental flow stress data are in good agreement with the predicted flow stress data. The average absolute relative error with a correlation coefficient was found to be 6.81% and 0.96, respectively.

Hot deformation behavior of Ti-Al-Sn-Zr-Mo alloy

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.

Ductile fracture prediction using enhanced CDM model with Lode angle-dependency for titanium alloy Ti-6Al-4V at room temperature

Abstract Fully coupled ductile damage model considering strength-differential (SD) effect is proposed for simulation of metal forming processes for Ti-6Al-4V. The developed constitutive equations are based on non-associative plasticity theory under thermodynamically-consistent framework. The third stress invariant is added to Hill48 criterion to capture the SD effect. Lode angle-dependent enhanced fully coupled damage model is proposed to account for the loading path sensitivity on ductile damage initiation and growth. A detailed material parameters identification procedure is presented. The proposed model is implemented into the FE code Abaqus/Explicit through user defined subroutine VUMAT. The numerically predicted results obtained with the proposed model are compared to experimental data of Ti-6Al-4V and good agreement is observed.

Validation of a thermodynamically consistent compressible rate-type viscoelastic model with independent limits on dilation, contraction, and distortion

In the previous part of this work (see Devendiran et al. (2018)), a new non-linear rate-type compressible viscoelastic model with independent volumetric and distortional limiters was proposed. In the present work, the developed constitutive equation is validated using experimental data generated in-house for two cyclic inhomogeneous deformations which are predominantly two-dimensional: non-uniform planar extension and barreling compression of carbon-black filled elastomeric sheets and circular cylinders, respectively, under different rates of cyclic deformations. In the finite element simulations of these two boundary value problems, the two-dimensional governing equations are formulated in Lagrangian basis and the resulting system of partial differential equations are solved using COMSOL®. A good agreement is observed between the numerical results and the corresponding experimental data, including the cyclic stress softening effect (Mullins effect).

Ultra-Thin Sheet Forming Analyses Considering Size Effects

Based on robust numerical formulations and various material models, finite element (FE) analysis becomes a powerful tool in conventional sheet metal forming process. Unfortunately, the present constitutive equations irrelevant to thickness that describe well conventional sheet deformation modes have difficulties being applied directly to ultra-thin sheet deformation modes. In the present study, a constitutive equation considering size effect is established by introducing a scale factor that represents size effects through thickness and width directions. Uniaxial tensile tests were used to evaluate the scale factor of different thicknesses together with the parameter identification. The developed constitutive equation reveals that thickness is the most important factor effecting on the constitutive relation of ultra-thin sheet. 2D draw forming process of C7035 ultra-thin sheet is analyzed using JSTAMP/NV introducing the developed constitutive equation. The analysis results show that there are obvious differences in the punch forces and loading geometries according to the size effect through thickness direction. Specimen width has slight effect on the flow stress although specimen thickness has strong effect on the flow stress. It is expected that the proposed constitutive equation gives good applicability to FE analysis of micro-scale forming.

A Constitutive Equation for Predicting Elevated Temperature Flow Behavior of BFe10-1-2 Cupronickel Alloy through Double Multiple Nonlinear Regression

Abstract Constitutive analysis for elevated temperature flow behavior of BFe10-1-2 alloy was carried out by using experimental stress–strain data from isothermal hot compression tests on a Gleeble-3800 thermo-mechanical simulator, in a wide of temperature range of 1,023–1,273 K, and strain rate range of 0.001–10 s−1. A constitutive equation based on double multiple nonlinear regression (DMNR) was proposed considering the independent effects of strain, strain rate, temperature and their interrelation. The predicted flow stress data obtained from the developed equation based on DMNR was compared with the experimental data. Correlation coefficient (R), average absolute relative error (AARE) and relative errors were introduced to verify the validity of the developed constitutive equation. The results showed that the developed constitutive equation based on DMNR could predict flow stress of BFe10-1-2 alloy with good correlation and generalization.

A Strain-Compensated Constitutive Model for Describing the Hot Compressive Deformation Behaviors of an Aged Inconel 718 Superalloy

Abstract The hot deformation behaviors of an aged Inconel 718 superalloy are investigated by isothermal compression experiments at four strain rates and five deformation temperatures on a Gleeble-3500 thermo-mechanical simulator. The experimental results show that the true stresses are obviously affected by strain rate and deformation temperature. The true stress increases rapidly at the beginning of hot compressive deformation, which ascribes to the intense work hardening. The true stresses at high deformation temperatures are lower than those at lower deformation temperatures. The dynamic softening induced by DRX is weak at a relatively low deformation temperature. A strain-compensated Arrhenius-type constitutive equation linked with true stress, strain rate and deformation temperature is developed for the studied superalloy. The material constants ( α \alpha , n, Q and A) in the developed model are expressed as the functions of true strain. The flow stresses calculated by the developed constitutive equation are nicely consistent with the experimental ones, which confirms that the developed constitutive equation can accurately describe the hot deformation behaviors of the studied superalloy.

Modeling of Carbide Spheroidization Mechanism of 52100 Bearing Steel Under Warm Forming Conditions

Direct deformation spheroidization of bearing steel can produce a fine and homogeneous microstructure. Warm-compression experiments were performed to investigate the carbide spheroidization behavior of 52100 bearing steel at temperatures of 650 °C to 750 °C and strain rates of 0.1 to 10.0 s−1. A set of mechanism-based unified constitutive equations was developed using the internal state variable method to describe the carbide spheroidization and metal flow behaviors of 52100 steel under warm deformation. The carbide spheroidization fraction, dislocation density, and phase transformation were modeled and correlated with unified constitutive equations. Material parameters in the constitutive equations were determined using genetic algorithm optimization techniques. The developed constitutive equations were validated by comparing the predicted and experimental results. Good consistency between these results indicated that the carbide spheroidization and metal flow behavior could be predicted using these constitutive equations.

Stress Relaxation Behaviour and Creep Constitutive Equations of SA302Gr.C Low-Alloy Steel

AbstractIn order to study the stress relaxation behaviour of SA302Gr.C low-alloy steel, the stress relaxation experiments were performed at the temperatures of 400, 500 and 620 °C with the constant strain of 0.5 %. Based on the experimental stress relaxation curves, the relationship between creep strain rate and stress were derived and the creep strain rate–stress curves at different temperatures were obtained. Then, the creep constitutive equations at different conditions were developed and the constants in the equations were determined. In order to verify the developed creep constitutive equations, the finite-element (FE) model of stress relaxation of SA302Gr.C low-alloy steel was established with the FE software MSC.Marc. The comparison results show that the simulated stress relaxation curves agree with the experimental ones, which illustrates that the developed creep constitutive equations are appropriate for the tested steel.

Thermal Stress Analysis of Residual Stress in a Cylindrical Aluminum Casting with Cast-in GCI Liner, Taking Recovery Behavior Effect into Account

Most aluminum cylinder blocks produced using high-pressure or low-pressure die-casting processes require gray cast iron liners (GCI liners) to compensate for their insufficient wear resistance and heat resistance of the Al-Si-Cu alloys. However, the cast-in liners cause excessive residual stress at the cylinder bore region. The resultant residual stress induces distortion of the cylinder liner. These inconveniences hinder development of more efficient engines. Therefore, an accurate thermal stress analysis technique has been sought to predict the residual stress and distortion of the cylinder liner. For accurate thermal stress analysis, we have already developed an elastoplastic-creep constitutive equation for which the inelastic strain developed at high temperatures does not contribute to strain hardening that occurs at low temperatures by duplicating the recovery behavior. Our earlier investigation using this equation has already revealed that incorporation of the recovery in the alloy constitutive equation is effective for improving the prediction accuracy of the thermal stress developed during casting. However, this conclusion was obtained only for a simple shape casting with a uniaxial thermal stress state. Effects of the developed constitutive equation have not been discussed for a casting closer to an actual cylinder block. For this study, a cylindrical aluminum casting with GCI (ISO 300) insert was produced. Then, the circumferential strain of the GCI liner was measured in-situ during casting. Measurements were taken of the residual stresses of the cylindrical aluminum casting and GCI liner, and of the liner deformation at a room temperature. The experimentally obtained results supported a discussion of the predictive accuracies of the elastoplastic-creep constitutive equation and the classical elastoplastic constitutive equation. A comparison revealed that the elastoplastic-creep constitutive equation for the aluminum casting has better predictive accuracy than the classical elastoplastic equation for residual stress, liner deformation, and the circumferential strain of a GCI liner during casting. Investigation of the simulated strain components of the cylindrical aluminum casting during casting indicated incorporation of the recovery in the alloy constitutive equation as a main factor improving the predictive accuracy.