Constitutive descriptions for hot compressed low-pressure rotor steel at elevated high temperature

The constitutive equation of the material is an essential ingredient of any structural calculation. In this article, a phenomenological constitutive model is established to describe the dynamic deformation behavior of 30Cr2Ni4MoV steel in wide strain rate, strain, and temperature ranges. Also, the mathematical models to predict peak stress and corresponding strain were obtained. The stress–strain values predicted by the developed model well agree with experimental results, which confirmed that the developed constitutive equation gives an accurate and precise estimate for the flow stress of 30Cr2Ni4MoV steel.     

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.     

Analysis of Garofalo equation parameters for an ultrahigh carbon steel

Isothermal stress–strain curves data from torsion tests conducted at high temperature (950–1200 °C) and strain rates (2–26 s⁻¹) were analyzed in an ultrahigh carbon steel (UHCS) containing 1.3%C. The sine hyperbolic Garofalo equation was selected as an adequate constitutive equation for the entire range of the forming variables considered. The Garofalo parameters were assumed strain dependent allowing the prediction of stress–strain curves under transient and steady-state conditions. The average relative errors obtained were below 3% in stress. In addition, the creep deformation mechanisms in the UHCS were analyzed from the Garofalo equation parameters. For this aim, the stress exponent of the Garofalo equation was, for the first time, related to that of the power law equation. The results show that the controlled deformation mechanism at steady state is lattice diffusion-controlled slip creep.     

High density polyethylene (HDPE): Experiments and modeling

The uniaxial tension (loading and unloading), creep and relaxation experiments on high density polyethylene (HDPE) have been carried out at room temperature. The stress–strain behavior of HDPE under different strain rates, creep (relaxation) behavior at different stress (strain) levels have been investigated. These experimental results are used to compare the simulation results of a unified state variable theory, viscoplasticity theory based on overstress (VBO) and a macro-mechanical constitutive model for elasto-viscoplastic deformation of polymeric materials developed by Boyce et al. (Polymer 41:2183–2201, 2000). It is observed that elasto-viscoplasticity model by Boyce et al. (Polymer 41:2183–2201, 2000) is not good enough to simulate stress–strain, creep and relaxation behaviors of HDPE. However, the aforementioned behaviors can be modeled quantitatively by using VBO model.     

Constitutive modeling for hot deformation behavior of ZA27 alloy

The hot deformation behavior of ZA27 alloy was investigated in the temperature range of 473–523 K with the strain rates in the range of 0.01–5 s⁻¹ and the height reduction of 60 % on Gleeble-1500 thermo mechanical simulator. Based on the experimental results, constitutive equations incorporating the effects of temperature, strain rate, and strain have been developed to model the hot deformation behavior of ZA27 alloy. Material constants, α, n, ln A, and activation energy Q in the constitutive equations were calculated as a function of strain. The results showed that the stress–strain curves of ZA27 alloy predicted by the constitutive equations are in good agreement with experimental results, which validates the efficiency of the constitutive equations in describing the hot deformation behavior of the material.     

FEM simulation of viscous properties for granular materials considering the loading rate effect

The deformation and strength characteristics of sandy soils as a kind of granular materials are very complex. The experimental results show that when the strain rate suddenly changes in monotonic loading (ML) case, the stress–strain curve of sandy soils changes sharply and then gradually converges into the original inferred one that would be obtained by continuous ML at constant strain rate after having exhibited clear yielding. Similar behaviors are also observed when ML is restarted at a constant strain rate following a creep loading or stress relaxation stage. An elasto-viscoplastic constitutive model for granular materials is developed, which consists of three components. One of the most important features of the model is that it can take into account the effects of loading rate due to viscous properties on the stress–strain behavior. The stress ratio-axial strain–time relations from four drained plain strain compression (PSC) tests on the saturated Toyoura sand are successfully simulated by the finite element method (FEM) code incorporating the proposed constitutive model. It is shown that the FEM code can simulate the viscous behaviors of sand accurately under arbitrary loading history.     

Nanoindentation analysis for local properties of ultrafine grained copper processed by high pressure torsion

Severe plastic deformation (SPD) techniques have recently been developed for producing bulk ultrafine grained metallic materials. High pressure torsion (HPT) produces finer microstructures than those achieved by other SPD processes because of the higher imposed strain and hydrostatic pressure. It is known that HPT-processed metals show a highly heterogeneous microstructure not only along the radius due to the nature of torsional deformation but also through the thickness. Since the sample size for HPT is small, the local properties of HPT-processed specimens have not been investigated yet. In this paper, we propose a method to obtain stress–strain curves from nanoindenting curves by combining the finite element method and the recursion method. The nanoindentation technique was employed to elucidate the local mechanical properties, especially the stress–strain behavior. The method to extract the stress–strain curves from the load–displacement curves obtained by nanoindentation tests was applied to the edge region of the HPT-processed sample. The extracted properties correlated well with experimental results qualitatively.     

10CrNi8MoV钢的拉伸塑性应变物理本构模型

研究了10CrNi8MoV钢不同温度和应变速率下的拉伸应力-应变曲线。根据位错动力学将流变应力分解为热激活应力和非热激活应力,忽略粘拽阻力的影响。通过对塑性变形过程的分析,在Kocks热激活方程中引入位错间距演化函数,并用线性强化模型描述非热激活应力的变化,建立了10CrNi8MoV钢的物理本构模型。该模型对10CrNi8MoV钢在低应变速率和较宽的温度范围内的塑性变形行为有较好的描述结果。      

Prediction of hot deformation behavior of Al–5.9%Cu–0.5%Mg alloys with trace additions of Sn

High temperature deformation behavior of Al–5.9wt%Cu–0.5wt%Mg alloys containing trace amounts (from 0 to 0.1 wt%) of Sn was studied by hot compression tests conducted at various temperatures and strain rates. The peak flow stress of the alloys increased with increase in strain rate and decrease in deformation temperature. The peak stress could be correlated with temperature and strain rate by a suitable hyperbolic-sine constitutive equation. The activation energy for hot deformation of the alloy without Sn content was observed to be 183.4 kJ mol⁻¹ which increased to 225.5 kJ mol⁻¹ due to 0.08 wt% of Sn addition. The Zener-Hollomon parameter (Z) was determined at various deforming conditions. The tendency of dynamic recrystallization increased with low Z values, corresponding to low strain rate and high temperature. The peak flow stresses at various processing conditions have been predicted by the constitutive modeling and correlated with the experimental results with fairly good accuracy. It was possible to predict 80, 75, 100, 100, 90, and 85% of the peak stress values within an error less than ±13%, for the investigated alloys. With addition of Sn content >0.04 wt%, peak flow stress increased significantly for all strain rate and temperature combinations. Scanning electron microscope revealed two types of second phases at the grain boundary of the undeformed alloy matrix, one being an Al–Cu–Si–Fe–Mn phase while the other identified as CuAl₂. The high strength and flow stress value of the alloy with 0.06 wt% of Sn content, may be attributed to the variation in amount, composition, and morphology of the Al–Cu–Si–Fe–Mn phase, as well as to the lower value of activation energy for precipitation reaction, as revealed from differential scanning calorimetric studies.     

Hot deformation behavior of an 8% Cr cold roller steel

An 8% Cr cold roller steel was compressed in the temperature range 900–1200 °C and strain rate range 0.01–10 s⁻¹. The mechanical behavior has been characterized using stress–strain curve analysis, kinetic analysis, processing maps, etc. Metallographic investigation was performed to evaluate the microstructure evolution and the mechanism of flow instability. It was found that the work hardening rate and flow stress decreased with increasing deformation temperature and decreasing strain rate in 8% Cr steel; the efficiency of power dissipation decreased with increasing Z value; flow instability was observed at higher Z-value conditions and manifested as flow localization near the grain boundary. The hot deformation equation and the dependences of critical stress for dynamic recrystallization and dynamic recrystallization grain size on Z value were obtained. The suggested processing window is in the temperature range 1050–1200 °C and strain rate range 0.1–1 s⁻¹ in the hot processing of 8% Cr steel.     

Time-dependent creep stress redistribution analysis of thick-walled functionally graded spheres

Time-dependent creep stress redistribution analysis of thick-walled spheres made of functionally graded material (FGM) subjected to an internal pressure and a uniform temperature field is performed using the method of successive elastic solution. The material creep and mechanical properties through the radial graded direction are assumed to obey a simple power-law variation. Total strains are assumed to be the sum of elastic, thermal and creep strains. Creep strains are time, temperature and stress dependent. Using the equations of equilibrium, compatibility and stress–strain relations a differential equation, containing creep strains, for radial stress are obtained. Ignoring creep strains, a closed-form solution for initial thermoelastic stresses at zero time is presented. It has been found that the material in-homogeneity parameterβ has a substantial effect on thermoelastic stresses. From thermoelastic analysis the material identified by β=2 in which a more uniform shear stress distribution occurs throughout the thickness of the FGM sphere is selected for time-dependent stress redistribution analysis. Using the Prandtl–Reuss relations and Norton’s creep constitutive model, history of stresses and strains are obtained. It has been found that radial stress redistributions are not significant, however, major redistributions occur for circumferential and effective stresses. It has also been concluded that stresses and strains are changing with time at a decreasing rate so that there is a saturation condition beyond which not much change occurs. Indeed after 50 years the solution approaches the steady-state condition.     

A modified Johnson–Cook model for 30Cr2Ni4MoV rotor steel over a wide range of temperature and strain rate

The compressive behaviors of 30Cr2Ni4MoV rotor steel were investigated at the temperatures from 1223 to 1523 K and strain rates from 0.001 to 0.1 s⁻¹. A modified Johnson–Cook (JC) model was proposed to describe the compressive behaviors of the studied alloy steel. In the modified JC model, the coupling effects of strain, strain rate, and deformation temperature were considered. Comparisons between the predicted stress–strain values by the modified JC model and measured ones indicate a good agreement, which confirms that the modified JC model is valid for the predicting the flow stress of 30Cr2Ni4MoV rotor steel over a wide range of temperature and strain rate.     

Compressive mechanical response of a low-density epoxy foam at various strain rates

We experimentally obtained compressive stress–strain response of a low-density epoxy foam at various strain rates. In particular, compressive stress–strain behavior at intermediate strain rates that has not been previously well understood were characterized by using a modified MTS and a split Hopkinson pressure bar (SHPB). Strain rate effects on the modulus of elasticity and cell-collapse stress for this low-density epoxy foam were determined.     

Loading–unloading curves of interlocking grouted stabilised sand-flyash brick masonry

Loading–unloading and reloading stress–strain curves of interlocking grouted stabilised sand-flyash brick masonry under uniaxial cyclic compressive loading are presented. Five cases of loading at 0°, 22.5°, 45°, 67.5° and 90° to the bed joints are considered. The brick units and masonry system developed by Prof. S.N. Sinha is used in present investigation. A mathematical model to describe the Loading–unloading–reloading response of this masonry system is proposed. The model accounts for the effect of residual strain on the cyclic compressive behaviour. The model uses simple polynomial functions. The loading and unloading curves are transformed to a normalised co-ordination system on which all the curves plot within a narrow range. A parent polynomial is developed to fit the curve on this coordinate system. Then individual reloading and unloading curve are obtained by transferring the parent equation to the stress–strain coordinate system. The present model for Loading–unloading and reloading curves compare well with the experimental data for five cases of loading under consideration.     

Construction of a constitutive model in calculations of pressure-dependent material

To make constitutive modeling of materials more approaching reality, a new theory is proposed, in which a corresponding constitutive model can be constructed and characterized experimentally via two steps, one relates to the characterization of yielding behavior of material, and the second relates to the characterization of plastic flow of material deformation. The constitutive model involves two functions, yield function and plastic potential. A relationship between two functions is suggested, therefore, a corresponding plastic potential can be easily created after we have an appropriate yield function. To consider the non-isotropic hardening feature of strength differential in the constitutive model, the concept of equivalent hardening state is introduced, and then, multi-experimental flow stresses can be addressed in the model. When pressure sensitive materials are taken as an example in discussions, the Drucker–Prager yield function is employed to express the yielding behavior of material and a differently experimental characterization of the model is created as the corresponding plastic potential to describe the feature of plastic flow of material. This simple constitutive model can reproduce three sets of experimental results; including two flow-stresses and the volumetric plastic strain. The constitutive model can also well predict stress–strain relations with different pressures loaded on the material. Study shows that the feature of plastic flow is not that sensitive to the pressure loaded on the material when the yielding stress is.     

A simple orthotropic finite elasto–plasticity model based on generalized stress–strain measures

 In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress–strain measures, which reduces for one special case to the so-called Green–Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalized stress–strain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed. Received: 2 April 2002 / Accepted: 11 September 2002     

Constitutive modeling of the mechanical behavior of high strength ferritic steels for static and dynamic applications

A constitutive relation is presented in this paper to describe the plastic behavior of ferritic steel over a broad range of temperatures and strain rates. The thermo-mechanical behavior of high strength low alloy (HSLA-65) and DH-63 naval structural steels is considered in this study at strains over 40%. The temperatures and strain rates are considered in the range where dynamic strain aging is not effective. The concept of thermal activation analysis as well as the dislocation interaction mechanism is used in developing the flow model for both the isothermal and adiabatic viscoplastic deformation. The flow stresses of the two steels are very sensitive to temperature and strain rate, the yield stresses increase with decreasing temperatures and increasing strain rates. That is, the thermal flow stress is mainly captured by the yield stresses while the hardening stresses are totally pertained to the athermal component of the flow stress. The proposed constitutive model predicts results that compare very well with the measured ones at initial temperature range of 77 K to 1000 K and strain rates between 0.001 s⁻¹ and 8500 s⁻¹ for both steels.     

Fatigue and inelasticity of metals under nonuniform stressed state

Based on the review of the available publications and the findings of the original studies, the author demonstrates that with cyclic stresses in the maximum-stressed surface layer being equal the inelastic strains in the presence of stress gradients are smaller than those in the material under uniform stressed state. As a result, there is a difference in the cyclic stress–strain diagrams between these cases. An equation for the cyclic stress–strain diagram is put forward, which allows for the stress gradient effect. A model is substantiated, which provides an explanation for the difference between fatigue strength characteristics under uniform and nonuniform stressed states.     

Analysis of geopolymer concrete columns

Ordinary portland cement (OPC) has been traditionally used as the binding agent in concrete. However, it is also necessary to search for alternative low-emission binding agents for concrete to reduce the environmental impact caused by manufacturing of cement. Geopolymer, also known as inorganic polymer, is one such material that uses by-product material such as fly ash instead of cement. Recent research has shown that fly ash-based geopolymer concrete has suitable properties for its use as a construction material. Since the strength development mechanism of geopolymer is different from that of OPC binder, it is necessary to obtain a suitable constitutive model for geopolymer concrete to predict the load–deflection behaviour and strength of geopolymer concrete structural members. This article has investigated the suitability of using an existing stress–strain model originally proposed by Popovics for OPC concrete. It is found that the equation of Popovics can be used for geopolymer concrete with minor modification to the expression for the curve fitting factor, to better fit with the post-peak parts of the experimental stress–strain curves. The slightly modified set of stress–strain equations was then used in a non-linear analysis for reinforced concrete columns. A good correlation is achieved between the predicted and measured ultimate loads, load–deflection curves and deflected shapes for 12 slender test columns.     

Constitutive model for cyclic deformation of perfluoroelastomers

Observations are reported in uniaxial cyclic tensile tests with a strain-controlled program on perfluoroelastomer Hyflon MFA. A constitutive model is developed for its viscoplastic response and damage at three-dimensional deformations with finite strains. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. Numerical simulation demonstrates that the constitutive equations adequately describe the mechanical response of perfluoroelastomer in cyclic tests with complicated deformation programs.