A damage related thermo-viscoplastic constitutive model of metallic materials under high rate deformation

Abstract

A constitutive model considering the effects of strain hardening, strain rate hardening, thermal softening and material damage softening is suggested. In order to take the effect of material damage into account, a strain softening term is added in Johnson–Cook flow stress law. The model can predict the overall deformation process of metallic materials at high strain rates and a simple way is provided to determine the coefficients of softening term.     

Establishment and comparison of four constitutive relationships of PC/ABS from low to high uniaxial strain rates

The objective of this paper is to accurately predict the rate/temperature-dependent deformation of a polycarbonate (PC) and acrylonitrile-butadiene-styrene (ABS) blend at low, moderate, and high strain rates for various temperatures. Four constitutive models have been employed to predict stress–strain responses of PC/ABS under these conditions, including the DSGZ model, the original Mulliken–Boyce (M–B) model, the modified M–B model, and an adiabatic model named the Wang model. To more accurately capture the large deformation of PC/ABS under the high strain rate loading, the original M–B model is modified by allowing for the evolution of the internal shear strength. All of the four constitutive models above have been implemented in the finite element software ABAQUS/Explicit. A comparison of prediction accuracies of the four constitutive models over a wide range of strain rates and temperatures has been presented. The modified M–B model is observed to be more accurate in predicting the deformation of PC/ABS at high strain rates for various temperatures than the original M–B model, and the Wang model is demonstrated to be the most accurate in simulating the deformation of PC/ABS at low, moderate, and high strain rates for various temperatures.     

Constitutive equations to predict high temperature flow stress in a Ti-modified austenitic stainless steel

The experimental stress–strain data from isothermal hot compression tests, in a wide range of temperatures (1123–1523 K) and strain rates (10⁻³–10² s⁻¹), were employed to develop constitutive equations in a Ti-modified austenitic stainless steel. The effects of temperature and strain rate on deformation behaviors were represented by Zener-Holloman parameter in an exponent type equation. The influence of strain was incorporated in the constitutive analysis by considering the effect of strain on material constants. The constitutive equation (considering the compensation of strain) could precisely predict the flow stress only at 0.1 and 1 s⁻¹ strain rates. A modified constitutive equation (incorporating both the strain and strain rate compensation), on the other hand, could predict the flow stress throughout the entire temperatures and strain rates range except at 1123 K in 10 and 100 s⁻¹. The breakdown of the constitutive equation at these processing conditions is possibly due to adiabatic temperature rise during high strain rate deformation.     

Modeling high-temperature tensile deformation behavior of AZ31B magnesium alloy considering strain effects

The hot tensile deformation behaviors of AZ31B magnesium alloy are investigated over wide ranges of forming temperature and strain rate. Considering the effects of strain on material constants, a comprehensive constitutive model is applied to describe the relationships of flow stress, strain rate and forming temperature for AZ31B magnesium alloy. The results show that: (1) The effects of forming temperature and strain rate on the flow behaviors of AZ31B magnesium alloy are significant. The true stress–true strain curves exhibit a peak stress at small strains, after which the flow stress decreases until large strain, showing an obvious dynamic softening behavior. A considerable strain hardening stage with a uniform macroscopic deformation appears under the temperatures of 523 and 573 K. The strain hardening exponent (n) increases with the increase of strain rate or the decrease of forming temperature. There are not obvious strain-hardening stages when the forming temperature is relatively high, which indicates that the dynamic recrystallization (DRX) occurs under the high forming temperature, and the balance of strain hardening and DRX softening is easy to obtain. (2) The predicted stress–strain values by the established model well agree with experimental results, which confirm that the established constitutive equation can give an accurate and precise estimate of the flow stress for AZ31B magnesium alloy.     

Hot compressive deformation behavior of 7075 Al alloy under elevated temperature

The hot compression tests were conducted with wide strain rates and forming temperature ranges to study the high-temperature deformation behavior of 7075 Al alloy. The material flow behavior and microstructural evolution during hot-forming process are discussed. Based on the measured stress–strain data, a new constitutive model is proposed, considering the coupled effects of strain, strain rate, and forming temperature on the material flow behavior of 7075 Al alloy. In the proposed model, the material constants are presented as functions of forming temperature. The proposed constitutive model gives good correlations with the experimental results, which confirms that the proposed model can give an accurate and precise estimate of flow stress for 7075 Al alloy.     

Anisotropic viscoplasticity constitutive model for directionally solidified superalloy

Abstract

The macroscopic deformation behaviour of a Ni-based directionally solidified (DS) superalloy was experimentally investigated, and an anisotropic constitutive model of the material was developed. Monotonic and creep tests were performed on uniaxial test specimens machined from DS plates so that the angle between the loading direction and the solidified grain direction varied between 0 and 90°. Tension-torsion creep tests were also conducted to examine the anisotropic behaviour under multiaxial stress conditions. The material exhibited marked anisotropy under elastic and viscous deformation conditions, whereas it showed isotropy under plastic deformation conditions of high strain rates. Then crystal plasticity analyses were carried out to identify slip systems under creep loading conditions, assuming the anisotropic creep behaviour of the DS material. A viscoplastic constitutive model for expressing both the anisotropic elasticity-viscosity and the isotropic plasticity was proposed. The elastic constants were determined using a self-consistent approach, and viscous parameters were modelled by crystal plasticity analyses. The calculation results obtained using the constitutive model were compared with the experimental data to evaluate the validity of the model. It was demonstrated that the constitutive model could satisfactorily describe the anisotropic behaviour under uniaxial and multiaxial stress conditions with a given set of material parameters.     

Hot deformation characteristics and strain-dependent constitutive analysis of Inconel 600 superalloy

The hot deformation characteristics and constitutive analysis of Inconel (IN) 600 superalloy were investigated at elevated temperatures. Hot compressive tests were carried out in the temperature and strain rate ranging from 900 to 1150 °C and 1 × 10⁻³–10 s⁻¹, respectively. The flow behavior analyses and microstructural observations indicate that the softening mechanisms were related to dynamic recrystallization (DRX) and grain growth. DRX played a dominant role in the microstructural evolution at low temperatures (or high strain rates). DRX was the dominant softening effect at low strains on testing at high temperatures with low strain rates, whereas growth of the dynamically recrystallized grains was responsible for softening at high strains. The flow stress of IN 600 was fitted well by the constitutive equation of the hyperbolic sine function under the deformation conditions performed in this study. A constitutive equation as a function of strain was established through a simple extension of the hyperbolic sine constitutive relation.     

Adiabatic Shear in annealed and shock-hardened iron and in quenched and tempered 4340 steel

Adiabatic shear localization is a catastrophic failure mechanism which can occur in ductile metals under high strain rate loading. This mechanism is driven by a thermal instability process in which rapid temperature rise due to plastic work couples with thermal softening to cause uniform deformation to collapse into narrow bands of intense shear within which material ductility is exhausted. Adiabatic shear localization is studied in three ferrous metals: annealed Armco and as-received Remco iron, both of which are high purity alpha iron; shock-hardened Remco iron; and 4340 steel quenched and tempered to a range of hardness levels. Using a compressive split-Hopkinson bar, punching-shear experiments were performed at room and elevated initial temperatures at shear strain rates of up to 45000 s^–1. Optical and scanning electron microscopy was performed on the deformed shear specimens to determine the extent of shear localization and mode of failure. Experimental evidence showed that the tempered 4340 steels were susceptible to localization through adiabatic shear banding; however, as-received and shock-hardened Remco iron and annealed Armco iron were not. Finite element simulations of the experiments were performed utilizing a user material subroutine developed as part of this research. This constitutive routine incorporates two adiabatic shear failure criteria, namely (i) maximum shear stress with a minimum critical shear strain rate and (ii) flow localization. These criteria proved to be capable of predicting the onset of an instability; however, the deformation which follows the instability was not predicted well.     

Constitutive models for high-temperature flow behaviors of a Ni-based superalloy

The high-temperature deformation behaviors of a typical Ni-based superalloy are investigated by hot compression tests under the strain rate of 0.001–1 s⁻¹and temperature of 920–1040 °C. The experimental results show that the deformation behaviors of the studied superalloy are significantly affected by the deformation temperature, strain rate and strain. The flow stress increases with the increase of strain rate or the decrease of deformation temperature. The flow stress firstly increases with the strain to a peak value, showing the obvious work hardening behaviors. Then, the stress decreases with the further straining, indicating the dynamic flow softening behaviors. Considering the coupled effects of deformation temperature, strain rate and strain on the hot deformation behaviors of the studied Ni-based superalloy, the phenomenological constitutive models are established to describe the work hardening-dynamic recovery and dynamic softening behaviors. In the established models, the material constants are expressed as functions of the Zener–Hollomon parameter. The established constitutive models can give good correlations with the experimental results, which confirm an accurate and precise estimation of the flow stress for the studied Ni-based superalloy.     

Single and double-step stress relaxation and constitutive modeling of viscoelastic behavior of swelled and un-swelled natural rubber loaded with carbon black

A phenomenological one-dimensional constitutive model, characterizing the mechanical behavior of viscoelastic natural rubber filled with the percolation concentration of HAF carbon black is developed in this investigation. This simple differential form model is based on a combination of linear and non-linear springs with dashpots, incorporating typical polymeric behavior such as shear thinning, thermal softening and non-linear dependence on deformation. The material parameters for this model are determined for the investigated vulcanizates. The model was also developed on same samples after immersion in kerosene for different intervals of times. One step mechanism of relaxation was appeared for straining the samples to different strain levels with constant strain rate. On the other hand, two step mechanisms of relaxation were appeared on straining specimens to same strain level but with different strain rates.     

An analysis of impact-induced deformation and fracture modes in amorphous glassy polymers

The finite deformation response of a planar block of polymer material subject to impact loading is analyzed using two constitutive models for glassy polymers, a reference Drucker–Prager type model and a physics-based macromolecular model, supplemented by a phenomenological model for craze initiation and widening. Full transient finite element analyses are carried out using a Lagrangian formulation of the field equations. The analyses allow an assessment of possible failure mechanisms under dynamic loading and the ability of the different models to predict such behavior. The results highlight the effect of the stress–strain behavior of polymers, notably the post-yield softening and large strain hardening, on localization of plastic flow. This behavior is adequately captured only by the macromolecular model.     

A numerical investigation of ductile fracture initiation in a high-strength low-alloy steel

In this work, static and drop-weight impact experiments, which have been conducted using three-point bend fracture specimens of a high-strength low-alloy steel, are analysed by performing finite-element simulations. The Gurson constitutive model that accounts for the ductile failure mechanisms of microvoid nucleation, growth and coalescence is employed within the framework of a finite deformation plasticity theory. Two populations of second-phase particles are considered, including large inclusions which initiate voids at an early stage and small particles which require large strains to nucleate voids. The most important objective of the work is to assess quantitatively the effects of material inertia, strain rate sensitivity and local adiabatic temperature rise (due to conversion of plastic work into heat) on dynamic ductile crack initiation. This is accomplished by comparing the evolution histories of void volume fraction near the notch tip in the static analysis with the dynamic analyses. The results indicate that increased strain hardening caused by strain rate sensitivity, which becomes important under dynamic loading, plays a benign role in considerably slowing down the void growth rate near the notch tip. This is partially opposed by thermal softening caused by adiabatic heating near the notch tip.     

Incorporation of mean stress effects into the micromechanical analysis of the high strain rate response of polymer matrix composites

The results presented here are part of an ongoing research program, to develop strain rate dependent deformation and failure models for the analysis of polymer matrix composites subject to high strain rate impact loads. A micromechanics approach is employed in this work, in which state variable constitutive equations originally developed for metals have been modified to model the deformation of the polymer matrix, and a strength of materials based micromechanics method is used to predict the effective response of the composite. In the analysis of the inelastic deformation of the polymer matrix, the definitions of the effective stress and effective inelastic strain have been modified in order to account for the effect of hydrostatic stresses, which are significant in polymers. Two representative polymers, a toughened epoxy and a brittle epoxy, are characterized through the use of data from tensile and shear tests across a variety of strain rates. Results computed by using the developed constitutive equations correlate well with data generated via experiments. The procedure used to incorporate the constitutive equations within a micromechanics method is presented, and sample calculations of the deformation response of a composite for various fiber orientations and strain rates are discussed.     

A comparative study on the capability of Johnson–Cook and Arrhenius-type constitutive equations to describe the flow behavior of Mg–6Al–1Zn alloy

In the current study, the predictability of two phenomenological constitutive equations, Johnson–Cook (JC) and Arrhenius-type ones, for describing the flow behavior of a magnesium alloy (Mg–6Al–1Zn) under hot deformation conditions has been evaluated. Towards this end, a series of hot compression tests were performed over a temperature range of 250–450 °C, under strain rates of 0.001, 0.01 and 0.1 s⁻¹. Using the experimental results obtained through implementing the predetermined compression tests, the related parameters and material constants in the constitutive equations were calculated. In order to compare the performance of the models, the statistical parameters of correlation coefficient and absolute mean error were employed. The results imply that the predictability of the Arrhenius-type equations is much stronger in estimating the flow behavior compared to that of the JC model; although more constants are needed to be calculated when using the former equation. It is concluded that the JC model, in contrast to the Arrhenius-type equations, is not reliable for the materials possessing tangible softening stage in their stress–strain curves such as magnesium AZ series.     

Constitutive model for large strain deformation of semicrystalline polymers

A constitutive model for large strain deformation of semicrystalline polymers has been formulated to predict the complex elasto-viscoelastic-viscoplastic material response. The general form of this model can be represented by three parallel rheological components corresponding to each of the modes of deformation. It will be shown that such a configuration is well suited to the mechanical nature of polymers as observed in recent studies. The constitutive stress-strain-time relationships are drawn from continuum mechanics which are more suitable than simple linear expressions from rheology. The result is a large strain, fully three-dimensional constitutive model, derived from a thermodynamic basis. The proposed model can be fit to macroscopic experimental data and is highly suited to numerical analysis. The paper reviews the literature relevant to constitutive representation of semicrystalline polymers, provides conclusion and validation of the most suitable form of constitutive model and presents the relevant constitutive mathematics.     

High temperature flow behavior and constitutive model of alloy transition layer in composite steel tube prepared by centrifugal self-propagating high-temperature synthesis (SHS)

Centrifugal self-propagation high-temperature synthesis (SHS) is a newly developed composite preparation technique by which a ceramic-alloy-carbon steel multilayer composite tube can be prepared. The hot deformation behaviors of the alloy steel layer at 800 °C–1000 °C, strain rates of 0.01 s⁻¹, 0.1 s⁻¹, 1.0 s⁻¹ and 10 s⁻¹ were studied by Gleeble-1500 thermal simulator. Rheological curve characteristics were analyzed under different thermal compression processes and a phenomenological hyperbolic sinusoidal Arrhenius constitutive equation was established to characterize the rheological mechanics of the material. The results show that the alloy steel is sensitive to temperature and strain rate, and its value of true stress decreases with the increase of temperature and strain rate. Thermal deformation process is the interaction between work hardening and dynamic softening, which is accompanied by the increase and extinction of dislocations. Under the strain rate of 10 s⁻¹, the stress-strain curve has a significant decrease when the strain exceeds 0.5. According to the observation of microstructure, this phenomenon can be attributed to the micro-crack generated by the local instability flow in the denatured zone. With the strain rate decreases, the softening mechanism of the alloy changes from dynamic recovery to dynamic recrystallization. The calculation results of the Arrhenius constitutive equation (AARE = 6.54 %, R = 0.99452) indicate that the model can predict the flow stress of the alloy accurately.     

循环软化材料单轴时相关循环变形行为的实验研究

在室温下,对循环软化材料(调质42CrMo钢)的单轴时相关应变循环特性和时相关棘轮行为进行了实验研究。揭示了材料在不同加载速率、不同峰/谷值保持时间以及不同加载波形(三角波和正弦波)下的循环软化特性和棘轮行为特性。结果表明,在室温下,材料的循环软化和棘轮变形行为均体现出明显的时相关效应:其循环变形行为不仅依赖于加载速率,而且还明显依赖于保持时间以及加载波形的形状。研究有助于后续建立循环软化材料时相关循环本构模型。     

Experimental study on tensile property of AZ31B magnesium alloy at different high strain rates and temperatures

As the lightest metal material, magnesium alloy is widely used in the automobile and aviation industries. Due to the crashing of the automobile is a process of complicated and highly nonlinear deformation. The material deformation behavior has changed significantly compared with quasi-static, so the deformation characteristic of magnesium alloy material under the high strain rate has great significance in the automobile industry. In this paper, the tensile deformation behavior of AZ31B magnesium alloy is studied over a large range of the strain rates, from 700 s⁻¹ to 3 × 10³ s⁻¹ and at different temperatures from 20 to 250 °C through a Split-Hopkinson Tensile Bar (SHTB) with heating equipment. Compared with the quasi-static tension, the tensile strength and fracture elongation under high strain rates is larger at room temperature, but when at the high strain rates, fracture elongation reduces with the increasing of the strain rate at room temperature, the adiabatic temperature rising can enhance the material plasticity. The morphology of fracture surfaces over wide range of strain rates and temperatures are observed by Scanning Electron Microscopy (SEM). The fracture appearance analysis indicates that the fracture pattern of AZ31B in the quasi-static tensile tests at room temperature is mainly quasi-cleavage pattern. However, the fracture morphology of AZ31B under high strain rates and high temperatures is mainly composed of the dimple pattern, which indicates ductile fracture pattern. The fracture mode is a transition from quasi-cleavage fracture to ductile fracture with the increasing of temperature, the reason for this phenomenon might be the softening effect under the high strain rates.     

Behavior and modeling of flow softening and ductile damage evolution in hot forming of TA15 alloy sheets

Flow softening and ductile damage behavior of TA15 titanium alloy with initial bimodal microstructure were studied using uniaxial hot tensile tests at different temperatures (750–850 °C) and strain rates (0.001–0.1 s^− 1). SEM examination of deformed specimens shows that deformation mainly occurs in β and secondary α phase (α_s). Globularization of α_s is also observed. According to the SEM observations of the cracked specimens, the mechanism of ductile damage is attributed to the breakdown of the compatibility requirements at the α/β interface. Based on the experimental results, a set of mechanism-based unified elastic-viscoplastic constitutive equations have been formulated to model the flow behavior and the damage evolution of TA15 alloy in hot forming conditions. Dislocation density, ductile damage evolution, deformation heat, phase transformation and globularization of the α_s have been modeled. The model constants have been determined by using a Genetic Algorithm (GA)-based optimization method. Furthermore, the proposed constitutive equations were evaluated in terms of correlation coefficient (R), average absolute relative error (AARE), and root mean square error (RMSE). The results indicate that the calibrated predictions, including flow stress, volume fraction of each phase, and fracture strain, are in good agreement with experimental results.