Deformation Constitutive Model Research Articles

Deformation Behaviors and Constitutive Model of DP1000 under Different Strain Rates

The DP1000 cold-rolled dual phase steel, the thickness of which is 1.2 mm, was required to do the tensile test under nine different strain rates from 10-4 s-1 to 1000 s-1. The mechanical properties and morphologies of the steel were obtained and analyzed. According to the C-J model, the plastic deformation characteristics of dual phase steel under different strain rates were studied. By means of transmission electron microscope (TEM), the morphologies of ferrite and martensite in the dynamic were observed. Finally, the constitutive models of quasi-static and high strain rate were established by using the modified Johnson-Cook model. The results reveal that DP1000 dual phase steel has obvious strain rate sensitivity, and it is a relatively pure ferrite and martensite dual phase structure. There are two stage strain hardenging characteristics in DP1000. In the first stage, the strain hardening ability of ferrite is higher, and the second stage is martensite deformation stage, the strain hardening ability is lower. The modified J-C constitutive model has high fitting effect, and the experimental results are matched with the fitting values.

High‐Temperature United Constitutive Model of Non‐Oriented Electrical Steel Based on Transformation Kinetics Equation and Dislocation Density Theory

The deformation resistance of non‐oriented electrical steel not only depends on the deformation temperature, strain, and strain rate, but also closely relates to the phase transformation. This study aims to incorporate these effects into a united constitutive model and make clear the distinction of micro hardening and softening mechanisms in different phase regions. Firstly, the transformation kinetics equation is established by the thermal dilatometric tests to accurately quantify the transformation process. Then, the high‐temperature deformation constitutive models are developed for each phase region based on the hot compression experiments and the dislocation density theory. The Hardening‐Softening Ratio is also proposed to describe the combined effect of work hardening and dynamic recovery softening. Finally, the model predictions are compared to the measured data and show a good fit. The results show that yield stress has a negative tendency with temperature in the ferrite region and also in the austenite region, and it changes faster in the ferrite region. The material constant is mainly related to the phase structure but almost independent of temperature. The hardening coefficient is determined by both temperature and phase structure, and the softening coefficient mainly depends on the phase structure.

A thermomechanically coupled finite deformation constitutive model for shape memory alloys based on Hencky strain

This paper presents a new thermomechanically coupled constitutive model for polycrystalline shape memory alloys (SMAs) undergoing finite deformation. Three important characteristics of SMA behavior are considered in the development of the model, namely the effect of coexistence between austenite and two martensite variants, the variation of hysteresis size with temperature, and the smooth material response at initiation and completion of phase transformation. The formulation of the model is based on a multi-tier decomposition of the deformation kinematics comprising, a multiplicative decomposition of the deformation gradient into thermal, elastic and transformation parts, an additive decomposition of the Hencky strain into spherical and deviatoric parts, and an additive decomposition of the transformation stretching tensor into phase transformation and martensite reorientation parts. A thermodynamically consistent framework is developed, and a Helmholtz free energy function consisting of elastic, thermal, interaction and constraint components is introduced. Constitutive and heat equations are then derived from this energy in compliance with thermodynamic principles. Time-discrete formulations of the constitutive equations and a Hencky-strain return-mapping integration algorithm are presented. The algorithm is then implemented in Abaqus/Explicit by means of a user-defined material subroutine (VUMAT). Numerical results are validated against experimental data obtained under various thermomechanical loading conditions. The robustness and efficiency of the proposed framework are illustrated by simulating a SMA helical spring actuator.

A constitutive model for amorphous shape memory polymers based on thermodynamics with internal state variables

Shape memory polymers (SMPs) are a class of smart materials which can recover a large pre-deformed shape in response to external stimulus. A thermoviscoelastic finite deformation constitutive model incorporated with structural and stress relaxation to capture the material behavior in the vicinity of the glass transition is developed for thermally activated amorphous SMPs in the paper. The incorporation of the Adam–Gibbs model for structure relaxation and the modified Eying model for viscous flow into a thermoviscoelastic finite deformation modeling framework is the main feature of the model. Differing from the traditional phenomenological fictive temperature modeling approach, an internal state variable modeling approach developed recently based on thermodynamics is used in the model. Besides, the temperature dependence of the model parameters is studied in the research. Comparisons between the simulation results and the experimental data show good agreement. Furthermore, the predictability of the model is examined by a parametric study and the results also demonstrate the validity of the model.

Dynamic recrystallization behavior of Fe–20Cr–30Ni–0.6Nb–2Al–Mo alloy

Single-pass compression tests of an alumina-forming austenite (AFA) alloy (Fe–20Cr–30Ni–0.6Nb–2Al–Mo) were performed using a Gleeble-3500 thermal–mechanical simulator. By combining techniques of electron back-scattered diffraction (EBSD) and transmission electron microscopy (TEM), the dynamic recrystallization (DRX) behavior of the alloy at temperatures of 950–1100 °C and strain rates of 0.01–1.00 s−1 was investigated. The regression method was adopted to determine the thermal deformation activation energy and apparent stress index and to construct a thermal deformation constitutive model. Results reveal that the flow stress is strongly dependent on temperature and strain rate and it increases with temperature decreasing and strain rate increasing. The DRX phenomenon occurs more easily at comparably higher deformation temperatures and lower strain rates. Based on the method for solving the inflection point via cubic polynomial fitting of strain hardening rate (θ) versus strain (e) curves, the ratio of critical strain (ec) to peak strain (ep) during DRX was precisely predicted. The nucleation mechanisms of DRX during thermal deformation mainly include the strain-induced grain boundary (GB) migration, grain fragmentation, and subgrain coalescence.

Hot deformation behavior and constitutive modeling of Q345E alloy steel under hot compression

Q345E as one of typical low alloy steels is widely used in manufacturing basic components in many fields because of its eminent formability under elevated temperature. In this work, the deformation behavior of Q345E steel was investigated by hot compression experiments on Gleeble-3500 thermo-mechanical simulator with the temperature ranging from 850 °C to 1150 °C and strain rate ranging from 0.01 s-1 to 10 s-1. The experimental results indicate that dynamic softening of Q345E benefits from increasing deformation temperature and decreasing strain rate. The mathematical relationship between dynamic softening degree and deformation conditions is established to predict the dynamic softening degree quantitatively, which is further proved by some optical microstructures of Q345E. In addition, the experimental results also reveal that the stress level decreases with increasing deformation temperature and decreasing strain rate. The constitutive equation for flow stress of Q345E is formulated by Arrihenius equation and the modified Zener-Hollomon parameter considering the compensation of both strain and strain rate. The flow stress values predicted by the constitutive equation agree well with the experimental values, realizing the accurate prediction of the flow stress of Q345E steel under hot deformation.

Research on the High-Temperature Hot Compressive Deformation Behavior of Ni-Based Superalloy GH3128

Intermediate heat exchanger (IHX), which transfers the heat generated in the reactor core to the secondary loop, is one of the key structural components of the very high-temperature gas-cooled reactor (VHTR). The Ni-based superalloy GH3128 has good high-temperature strength and so is a promising main structural material for the IHX. In this paper, the flow stress behaviors and the deformation microstructure of superalloy GH3128 were investigated by high-temperature compression tests conducted at various temperatures (950–1150 °C) and strain rates (0.001–10 s−1), and the processing maps were analyzed in order to establish the hot deformation constitutive model and obtain the optimum hot forming condition. The results show that (1) both flow stresses and peak flow stresses increase along with the increase of strain rate or decrease of temperature, (2) GH3128 has excellent hot workability, (3) the dynamic recovery (DRV) plays the dominant role during the dynamic softening process due to the high stack fault energy, and (4) the optimum hot forming condition of GH3128 should be defined in the temperature of 1150 °C and strain rate range of 0.01–0.056 s−1. This work contributes to the application of GH3128 alloy on IHX structure.

Deformation Characteristic and Constitutive Modeling of 2707 Hyper Duplex Stainless Steel under Hot Compression

Hot deformation behavior and microstructure evolution of 2707 hyper duplex stainless steel (HDSS) were investigated through hot compression tests in the temperature range of 900–1250 °C and strain rate range of 0.01–10 s−1. The results showed that the flow behavior strongly depended on strain rate and temperature, and flow stress increased with increasing strain rate and decreasing temperature. At lower temperatures, many precipitates appeared in ferrite and distributed along the deformation direction, which could restrain processing of discontinuous dynamic recrystallization (DRX) because of pinning grain boundaries. When the temperature increased to 1150 °C, the leading softening behaviors were dynamic recovery (DRV) in ferrite and discontinuous DRX in austenite. When the temperature reached 1250 °C, softening behavior was mainly DRV in ferrite. The increase of strain rate was conducive to the occurrence of discontinuous DRX in austenite. A constitutive equation at peak strain was established and the results indicated that 2707 HDSS had a higher Q value (569.279 kJ·mol−1) than other traditional duplex stainless steels due to higher content of Cr, Mo, Ni, and N. Constitutive modeling considering strain was developed to model the hot deformation behavior of 2707 HDSS more accurately, and the correlation coefficient and average absolute relative error were 0.992 and 5.22%, respectively.

Experimental and numerical investigations on stress induced phase transitions in silicon

Silicon has a tremendous importance as an electronic, structural and optical material. Modeling the stress driven phase transitions during the interaction of a silicon surface with a pointed asperity at room temperature is a major step towards the understanding of various phenomena related to brittle as well as ductile regime machining of this semiconductor. In order to understand the material’s response for complex loading situations the often used J2-plasticity model is inadequate, instead dedicated constitutive models are required. We developed a novel finite deformation constitutive model set within the framework of thermodynamics with internal variables that captures the stress induced semiconductor-to-metal (cd-Si → β-Si), metal-to-amorphous (β-Si → a-Si) as well as amorphous-to-amorphous (a-Si → hda-Si, hda-Si → a-Si) transitions. The model was calibrated using load-displacement data for (111)-Si, which we show to be representative for the Berkovich indentation response of silicon. The simulation results for the residual surface topography and the size of the transformed zone agree very well with experimental data. Finally, the predictive capability of the model is demonstrated by the successful reproduction of the load displacement curve for indentation with the Knoop indenter tip. A comparison between residual stress fields computed using the phase transition model to results obtained using J2-plasticity shows significant differences, suggesting that predictions based on the latter may be unreliable.

A constitutive model for an internally balanced compressible elastic material

Many large deformation constitutive models for the mechanical behavior of solid materials make use of the multiplicative decomposition as, for example, used by Kröner in the context of finite-strain plasticity. Then describes the elastic effect by letting the potential energy of the deformation depend upon . In this paper we allow the potential energy to depend upon both portions of the multiplicative decomposition. As in hyperelasticity, energy minimization with respect to displacement gives equilibrium field equations and traction boundary conditions. The new feature, minimization with respect to the decomposition itself, generates an additional mathematical requirement that is interpreted here in terms of a principle of internal mechanical balance. We specifically consider a Blatz–Ko-type solid suitably generalized to incorporate the notion of internal balance. Conventional results of hyperelasticity are retrieved for certain limiting forms of the energy density, whereas the general form of the energy density gives rise to an overall softening response, as is demonstrated in the context of pure pressure and uniaxial loading.

Deformation behavior and characteristics of sintered porous 2024 aluminum alloy compressed in a semisolid state

Semisolid powder forming technology is a novel technology that has achieved a great progress in recent years. However, the deformation behavior and constitutive model governing semisolid powder forming remain unclear. Sintered porous 2024 aluminum alloy specimens with different initial relative densities were compressed at 580–600°C in a semisolid state at strain rates of 0.1, 1, and 5s−1. The results indicate that the solid skeleton of the sintered porous material deforms elastic-plastically in stage I (ε<εc1) without breakup of the solid skeleton and powders. In stage II (εc2>ε>εc1), the solid skeleton is broken up and the powders rupture. Then, the liquid is squeezed out, and it flows and fills the gaps/pores, resulting in higher relative density. In stage III (ε>εc2), powders are crushed into fragments and the liquid flows from the edge to the surface of the samples. Then, new pores appear, decreasing the density of the samples. The constitutive equation is modified by introducing variables of liquid fraction and relative density, and it agrees well with the measured data.

Hot Deformation Behavior and Constitutive Modeling of Alloy 800H Considering Effectsof Strain

Abstract High-temperature single-pass compression experiments were conducted on alloy 800H using a Gleeble 3500 thermal-mechanical simulation testing machine, and hot deformation behaviors at temperatures of 1,000–1,150 °C and strain rates of 0.01–1 s–1 were investigated. The results revealed that dynamic recrystallization (DRX) behavior occurred more easily under deformation conditions with relatively low strain rates and high deformation temperatures. By taking the influence of strain on the hot deformation behavior into consideration, a strain-dependent hyperbolic sine constitutive model was constructed. Based on this revised constitutive model, flow stress during deformation was predicted. The linear relation between the predicted value and the experimental result was as high as 0.99648, and the absolute average relative error was 2.019 %. Thus, it was demonstrated that the strain-dependent analysis provided a constitutive model that was able to precisely predict flow stress under experimental conditions.

Stress induced phase transitions in silicon

Silicon has a tremendous importance as an electronic, structural and optical material. Modeling the interaction of a silicon surface with a pointed asperity at room temperature is a major step towards the understanding of various phenomena related to brittle as well as ductile regime machining of this semiconductor. If subjected to pressure or contact loading, silicon undergoes a series of stress-driven phase transitions accompanied by large volume changes. In order to understand the material's response for complex non-hydrostatic loading situations, dedicated constitutive models are required. While a significant body of literature exists for the dislocation dominated high-temperature deformation regime, the constitutive laws used for the technologically relevant rapid low-temperature loading have severe limitations, as they do not account for the relevant phase transitions. We developed a novel finite deformation constitutive model set within the framework of thermodynamics with internal variables that captures the stress induced semiconductor-to-metal (cd-Si→β-Si), metal-to-amorphous (β-Si→a-Si) as well as amorphous-to-amorphous (a-Si→hda-Si, hda-Si→a-Si) transitions. The model parameters were identified in part directly from diamond anvil cell data and in part from instrumented indentation by the solution of an inverse problem. The constitutive model was verified by successfully predicting the transformation stress under uniaxial compression and load–displacement curves for different indenters for single loading–unloading cycles as well as repeated indentation. To the authors' knowledge this is the first constitutive model that is able to adequately describe cyclic indentation in silicon.

Deformation behavior and constitutive model for dual-phase Mg–Li alloy at elevated temperatures

In order to study the deformation behavior and evaluate the workability of the dual-phase Mg–9Li–3Al–2Sr alloy, isothermal hot compression tests were conducted using the Gleeble–3500 thermal-mechanical simulator, in ranges of elevated temperatures (423–573 K) and strain rates (0.001–1 s−1). Plastic instability is evident during the deformation which is in the form of serrated flow; serrated yielding is attributed to the locking of mobile dislocations by the Mg and Li atoms which diffuse during the deformation. The relationships between flow stress, strain rate and deformation temperature were analyzed and the deformation activation energy and some basic material factors at different strains were calculated using the Arrhenius equation. The effects of temperature and strain rate on deformation behavior were represented using the Zener-Hollomon parameter in an exponent-type equation. To verify the validity of the constitutive model, the predicted values and experimental flow curves under different deformation conditions were compared, the correlation coefficient (0.9970) and average absolute relative error (AARE=4.41%) were calculated. The results indicate that the constitutive model can be used to accurately predict the flow behavior of dual-phase Mg–9Li–3Al–2Sr alloy during high temperature deformation.

Constitutive Modeling of High-Temperature Flow Behavior of an Nb Micro-alloyed Hot Stamping Steel

The thermal deformation behavior and constitutive models of an Nb micro-alloyed 22MnB5 steel were investigated by conducting isothermal uniaxial tensile tests at the temperature range of 873-1223 K with strain rates of 0.1-10 s−1. The results indicated that the investigated steel showed typical work hardening and dynamic recovery behavior during hot deformation, and the flow stress decreased with a decrease in strain rate and/or an increase in temperature. On the basis of the experimental data, the modified Johnson-Cook (modified JC), modified Norton-Hoff (modified NH), and Arrhenius-type (AT) constitutive models were established for the subject steel. However, the flow stress values predicted by these three models revealed some remarkable deviations from the experimental values for certain experimental conditions. Therefore, a new combined modified Norton-Hoff and Arrhenius-type constitutive model (combined modified NH-AT model), which accurately reflected both the work hardening and dynamic recovery behavior of the subject steel, was developed by introducing the modified parameter ke. Furthermore, the accuracy of these constitutive models was assessed by the correlation coefficient, the average absolute relative error, and the root mean square error, which indicated that the flow stress values computed by the combined modified NH-AT model were highly consistent with the experimental values (R = 0.998, AARE = 1.63%, RMSE = 3.85 MPa). The result confirmed that the combined modified NH-AT model was suitable for the studied Nb micro-alloyed hot stamping steel. Additionally, the practicability of the new model was also verified using finite element simulations in ANSYS/LS-DYNA, and the results confirmed that the new model was practical and highly accurate.

Research on the High Temperature Constitutive Model of 300M Ultrahigh Strength Steel

300M ultrahigh strength steel has good mechanical properties. It has been widely used in the force bearing components of aircraft. In this paper, By using Gleeble1-500D thermal simulator, we studied the change regularity of stress-strain curve of 300M steel using hot compression deformation when temperature is from 800°C to1100°C, strain rate is from 0.001 S-1to 1 S-1 and the strain is 0.7.The experimental results showed that when the strain rate is constant, the flow stress and the peak stress decrease with the increase of deformation temperature. When the deformation temperature is constant, the flow stress and peak stress increase with the increase of strain rate. From the test, we got the true stress-strain curve, calculated the thermal deformation constants such as the deformation activation energy of 300M ultrahigh strength steel. Eventually, we built the thermal deformation constitutive model in hyperbolic sine form of 300M steel.

High temperature deformation mechanisms and constitutive modeling for Al/SiCp/45 metal matrix composites undergoing laser-assisted machining

The material removal mechanism in LAM of Al/SiCp composite. In the present study, material removal mechanisms were inferred from the analysis of the chip and chip root morphology at different cutting conditions, and the shear zone stress was determined by using three dimensional machining theory with a new approach to determining shear angles for saw-tooth chips formed in LAM of Al/SiCp composite. Based on the deformation mechanisms associated with LAM, a constitutive equation which predicted the shear zone stress was proposed, and the effects of operating conditions on the shear zone stress were determined from a systematic experimental study. A comparison of the constitutive equation predictions with the experimental results showed a smaller than the estimated uncertainty (15%) in the stress calculation.

Damage and deformation control equation for gas-bearing coal and its numerical calculation method

The dual pore structure of coal and occurrence of gas adsorption make gas-bearing coal mechanical properties different from other non-adsorptive porous materials. Considering adsorbed-gas-induced swelling stress and erosion, and pore/fracture-induced damage and failure to coal skeleton, the new effective stress equation, damage deformation control equation and constitutive model for gas-bearing coal is established. Based on principle of statistics and fractal theory, we combined the macro- and micro-structure characteristics of coal, and established the control equation of fracture field distribution. According to fracture mechanics and mesoscopic damage mechanics, we also established the expansion and damage evolution control equation of mesoscopic cracks. Furthermore, we revealed the relationship between meso-scale crack extension damage and macro mechanical characteristics, set up the macro-mesoscopic numerical model and calculation method of fractured rock mechanic constitutive, and developed the stress–strain numerical calculation program of gas-bearing coal using Comsol Multiphysics partial differential equation module and MatLab software. The model and methods were further validated in field tests. The results showed that our equations could better describe the deformation and damage process of gas-bearing coal and solve the fluid–solid coupling problems in gas and coal engineering practices.

Hot deformation behavior and constitutive modeling of homogenized 6026 aluminum alloy

Abstract The isothermal hot compression tests of homogenized 6026 aluminum alloy under wide range of deformation temperatures (673–823 K) and strain rates (0.001–10 s−1) were conducted using Gleeble-1500 thermo-simulation machine. According to the experimental obtained true stress–strain data, the constitutive equations were derived based on the original Johnson–Cook (JC) model, modified JC model, Arrhenius model and strain compensated Arrhenius model, respectively. Moreover, the prediction accuracy of these established models was evaluated by calculating the correlation coefficient (R) and average absolute relative error (AARE). The results show that the flow behavior of homogenized 6026 aluminum alloy is significantly affected by the strain rate and temperature. The original JC model is inadequate to provide good description on the flow stress at evaluated temperatures. The modified JC model and Arrhenius model greatly improve the predictability, since both of these models consider the coupled effects of deformation temperature and strain rate. However, to give more precise description, the influence of strain on the material constants should be introduced into Arrhenius model.

A multi-branch finite deformation constitutive model for a shape memory polymer based syntactic foam

A multi-branch thermoviscoelastic-themoviscoplastic finite deformation constitutive model incorporated with structural and stress relaxation is developed for a thermally activated shape memory polymer (SMP) based syntactic foam. In this paper, the total mechanical deformation of the foam is divided into the components of the SMP and the elastic glass microballoons by using the mixture rule. The nonlinear Adam-Gibbs model is used to describe the structural relaxation of the SMP as the temperature crosses the glass transition temperature (Tg). Further, a multi-branch model combined with the modified Eying model of viscous flow is used to capture the multitude of relaxation processes of the SMP. The deformation of the glass microballoons could be split into elastic and inelastic components. In addition, the phenomenological evolution rule is implemented in order to further characterize the macroscopic post-yield strain softening behaviors of the syntactic foam. A comparison between the numerical simulation and the thermomechanical experiment shows an acceptable agreement. Moreover, a parametric study is conducted to examine the predictability of the model and to provide guidance for reasonable design of the syntactic foam.