Equivalent Stress-strain Curves Research Articles

Extraction of Equivalent Stress Versus Equivalent Plastic Strain Curve of Necking Material in Tensile Test Without Assuming Constitutive Model

Abstract An algorithm for extracting the equivalent stress versus equivalent plastic strain curve of a necking material in tensile test is proposed (the curve is called here the equivalent-plastic (EP) stress–strain (SS) curve). The presented algorithm traces the force–elongation curve via iterative finite element (FE) simulations without assuming a constitutive model and is suitable for a general-purpose FE solver available to a general audience. In the FE simulation and experiment, a slightly tapered geometry was employed around the specimen center to stably initiate necking there. The proposed algorithm and mentioned necking initiation method have been applied to extracting the EP SS curve of a high-strength steel material using an axisymmetric specimen. Because necking was initiated stably in simulation at the specimen center for a range of mesh sizes, the convergence of the extracted EP SS curve with the mesh size could be successfully verified. The EP SS curve of the tested material was extracted up to an EP strain of 0.85 with average strain intervals of approximately 2.5 mili-strain. The error values were less than 0.2 and 0.1% after four and seven FE simulations, respectively. The presented algorithm and necking initiation method in simulation can also be used for simultaneously quantifying the fracture EP strain of a necking material in the tensile test.

Tensile Test and Numerical Simulation Investigations on the Mechanical Properties of a New Type of Slightly Curved Arc HRB400 Steel Bars in Mass Concrete

The temperature stress in mass concrete structure is relatively high during construction, which usually leads to temperature cracks. To solve this problem, concrete blocks are usually placed by setting wide slots. Connecting the truncated steel bars at the position of the wide slots by welding or extruding sleeves has many disadvantages. To solve the problem of temperature-induced stress loss, a new type of slightly curved arc HRB400 (SCAHRB400) steel bars was proposed without cutting off the steel bars in this article. Tensile tests and numerical simulations were performed for five types of SCAHRB400 steel bars considering geometric and material nonlinearity. Based on the test results and numerical simulation results, the equivalent stress–strain relationships of SCAHRB400 steel bars were established, and the emergence of the plastic zone of SCAHRB400 steel bars in the tensile process were observed, the tensile properties of SCAHRB400 steel bars were analyzed and discussed. The test results indicate that SCAHRB400 steel bars are prone to local yielding near the crown of large arches and at the connection of horizontal and arc sections. The numerical simulation equivalent stress–strain curves have good regularity. The equivalent stress–strain curves of slightly curved arc HRB400 and HRB335 steel bars have the similar changing law. When the stress is small, the tensile stiffness and compressive axial stiffness of slightly curved arc HRB400 and HRB335 steel bars are similar; when the stress is large, the axial stiffness of SCAHRB400 steel bars is greater than that of slight curved arc HRB335 steel bars. Through test and numerical simulation studies, the theoretical basis can be established for the engineering application of new slightly curved arc steel bars in mass concrete.

A Forward Identification Method for High-Temperature Stress–Strain Curves of 7075 Aluminum Alloy Sheet Considering the Necking Stage

The necking phenomenon of metal sheet under high temperatures is serious and continues over a longer duration. It is difficult to describe the high-temperature mechanical properties of materials only on the basis of hardening behavior before necking. To obtain the high-temperature stress–strain curve considering diffuse necking stage, a forward identification method based on strain measurement is proposed in this study. Here, the strain field of the minimum cross-section in the necking region of the specimen is obtained using a DIC (digital image correlation) measurement technique, and the average axial true stress–strain curve is calculated. Then, the average axial true stress–strain curve is modified using the modified Bridgeman formula. Taking 7075 aluminum alloy as an example, the high temperature equivalent stress–strain curve considering the diffuse necking stage is obtained. Compared with the traditional method, the maximum effective strain range is expanded from 0.05 to 0.8 due to the consideration of the necking stage. The obtained curve is characterized by a coupled viscoplastic–damage constitutive model and embedded in ABAQUS through the user subroutine VUMAT to simulate the hot tensile process. The relative error of force–displacement between the simulation and the experiment was 2.4%, validating the ability of the presented method. This study provides theoretical guidance and a scientific basis for the application and forming control of hot stamping processes.

Determining reliable wide-strain-range equivalent stress–strain curves using 3D digital image correlation

We propose a method for obtaining reliable wide strain range equivalent stress–strain curves from tensile testing by applying 3D digital image correlation. Herein, both the front and side surfaces of the specimen were identified by the 45° tilting of the specimens. The deformation of the specimen was the most severe in the neck section, and the equivalent strain representing the entire specimen could be extracted from the neck regions on the side and front surfaces. In addition, the equivalent stress was obtained by calculating the reduction in the varying cross-sectional area from the strain information of the two surfaces. The resulting wide strain range equivalent stress–strain curves can describe nonlinear hardening behaviors up to higher strains than the conventional gauge length method. The proposed method can provide wide strain range equivalent stress–strain curves only through experiments without requiring separate finite element method simulations or correction factors.

Determination of Assumed Strain Hardening Relationship from Shear–Compression Model and Data Analysis

A new method for obtaining equivalent stress–strain curves of bilinear and Swift law material models from the numerical results of shear–compression test (SCT) is developed. This task is conducted on the basis of the approximate analytical relations for the compression and shear conditions at the gauge section and by the application of correction factors. Logarithmic correction models are developed and used to generalize the geometry of an optimum shear–compression specimen (SCS) that is calibrated by one-factor-at-time method for a reference material. Furthermore, a dataset created from 125 finite element simulations is analyzed by data analysis techniques, and a universal nonlinear strain hardening relationship is determined. This model is used to predict the stress–strain curves directly from the force–displacement curve of the SCS. The input and output stress–strain curves are in good agreement with an average error of approximately 3%. The numerical findings of this study provide a foundation to develop a general quantitative relationship to study the behavior of materials by SCT.

Influence of Crystal Plasticity Parameters on the Strain Hardening Behavior of Polycrystals

The effective mechanical properties of a polycrystal depend directly on the single-crystal properties of each grain and its crystallographic orientation with respect to the load axis. While the micromechanical approach has been used quite extensively to study the influence of grain shape and crystallographic texture on the resulting mechanical behavior of a polycrystal, the influence of the crystal plasticity parameters, which describe the constitutive behavior of the single crystal, requires to be investigated systemically because, typically, these parameters are fitted to describe a given material behavior. In the current research, this gap is filled by systemically studying the effect of changes in crystal plasticity parameters on the effective mechanical properties of polycrystals. The numerical model employed here consists of a representative volume element of 100 grains, and the material properties are described by using a non-local crystal plasticity model. A proper homogenization technique was used to homogenize the micromechanical results to an effective macroscopic material response. The equivalent stress versus equivalent plastic strain curve was obtained numerically by introducing the Voce-type hardening law, mimicking the material behavior in uniaxial tensile tests. The four parameters of the Voce-type hardening law were fitted to the macroscopic stress-strain curves, and the correlation between the crystal plasticity parameters and the Voce parameters has been studied, which is an efficient way to study the influence of microscopic material descriptions on the macroscopic behavior of polycrystals.

Ductile fracture of hydrostatic‐stress‐insensitive metals using a coupled damage‐plasticity model

AbstractIn this paper, a new coupled damage‐plasticity model is proposed to predict plastic response and ductile fracture for two hydrostatic‐stress‐insensitive metals. A convenient hybrid experiment–simulation approach is designed to calibrate model parameters. Geometry transferability of model parameters is validated to various notched tensile specimens. With the characteristics of the present constitutive model, the critical damage parameter is demonstrated to be an appropriate fracture criterion, where fracture strains of various notched tensile specimens are in good agreement with experimental data. The study also reveals that there exist complementing mechanisms for the stress‐triaxiality‐dependent damage evolution and strain hardening of effective (undamaged) stress, respectively, that result in the non‐uniform distribution of overall (damaged) equivalent stress–strain curves on the minimum cross section of tensile specimens. Then, it is the combined effect of non‐uniformly distributed overall equivalent stress–strain curves that results in the phenomenon of hydrostatic‐stress insensitivity in metals.

Layered fracture prediction of QStE700 medium-thickness steel plate under tensile loading

Layered fracture frequently occurs in the deforming process of QStE700 medium-thickness steel plates under tensile loading. In this study, the morphology of a layered fracture was observed via scanning electron microscopy, and the mechanism of the layered fracture was also analyzed. Based on the three-dimensional digital image correlation technique, a section analysis method was adopted for determining the true stress–strain curve including the necking process. A modified Bridgeman’s equation was adopted to transform the true stress–strain curve into the equivalent stress–strain curve. At the time of layered fracture occurrence, the equivalent strain and stress triaxiality of differently shaped specimens were obtained and fitted to a linear exponential relationship equation. The equation was the layered fracture criterion function and combined with the finite element method (FEM) simulations for determining the damage criterion of the layered fracture of a certain specimen. The FEM-simulated equivalent strain was consistent with the experimental equivalent strain of the layered fracture. Summarizing, the proposed method to predict the layered fracture of a QStE700 medium-thickness steel plate is effective and can be adopted in the study and control of layered fracture.

Experimental study on low-strength concrete joint core strengthened with steel meshes for connecting PFCC column and RC beam

This paper presented a new joint core strengthened with multi-layer steel meshes for connecting the PVC-FRP Confined Concrete (PFCC) column and Reinforce Concrete (RC) beam. Seven specimens were tested under concentric compression and the effects of several parameters including the height, diameter of specimen and volume ratio of steel mesh on the compressive behaviors were investigated. Test results showed that all the reinforcement yielded successively and eventually the crushing of the concrete dominated the failures of specimens. The ultimate bearing capacity and ultimate equivalent axial strain decreased as the height of specimen increased, while they increased as the diameter of specimen or the volume ratio of steel mesh increased. The ultimate strains of all the reinforcement and concrete increased as the height of specimen increased while they decreased as the diameter of specimen or the volume ratio of steel mesh increased. Considering the influence of height of specimen, a modified formula for conveniently predicting the ultimate bearing capacity of the joint core strengthened with steel meshes was proposed based on the theory of confined concrete and superposition principle of multiple confinement. The predicted results were in good agreement with the experimental data. Additionally, an equivalent stress–strain relationship model of the joint core strengthened with steel meshes was suggested based on the experimental research. The predicted curves agreed well with the measured equivalent stress–strain curves. Moreover, a validated Finite Element (FE) model for the joint core strengthened with steel meshes was developed to conduct parametric studies, which broadened the available experimental results about the mechanical performances of the joint.

Analysis and modelling of tensile and torsional behaviour at different strain rates of Ti6Al4V alloy additive manufactured by electron beam melting (EBM)

Additive manufactured Ti6Al4V alloy has become a widely used metallic material in many structural applications including aerospace components and medical implants, where its balance of mechanical performance and flexibility in assuming complex shapes is very appreciated. As a consequence, extensive efforts have been directed towards its mechanical characterization and extended data collections are reported in the literature. However, in the basic characterization, some prominent aspects related to “true” stress-strain behaviour and strain rate effects appear to have been substantially neglected. In this work the tensile behaviour of a Ti6Al4V alloy produced by Electron Beam Melting is investigated at static and dynamic rates together with the torsion response at static rates, in order to evaluate the effects of the strain rate and the Lode stress-state parameter on the elastoplastic response and the fracture initiation. The experimental procedure adopted here relied on enhanced true stress-true strain measurements based on optical measurements of the necking cross section of the specimen, continuously acquired by video recordings, as an alternative to the traditional elongation-based approach to the true stress-true strain evaluation. In the case of static torsion tests, the Nádai method has been used, delivering the equivalent stress-strain curves directly from the torque-rotation experimental data. Moreover, the real stress amplification due to high strain rates has been obtained subtracting the thermal softening effect from the flow curves of the dynamic tests. In addition to the mechanical characterization, micrographic analyses allowed to relate the microstructural properties to the additive process conditions and the fracture morphology to the various test conditions imposed to different specimens. Finally the stress-strain results from experiments are compared with literature data for Ti6Al4V alloy, manufactured by the same or different production processes, evidencing how not only manufacturing technique and process conditions, but also the approach used in the characterization affects the results of material behaviour assessment.

Strain-Based Failure Assessment Based on a Reference Strain Method for Welded Pipelines

Abstract Engineering critical assessment (ECA) is an evaluation procedure for structures with flaws and has been widely applied for assessing pipeline integrity. The standards for structural integrity assessment, including BS 7910, involve stress-based ECA, and they are known to produce overly conservative results. Therefore, strain-based ECA has been recently developed as an alternative approach. One of the effective methods for improving the accuracy of strain-based ECA is the reference strain method. However, only a limited number of studies have applied this method to welded pipelines. Therefore, a numerical analysis based on strain-based ECA was performed for girth-welded joints with a circumferentially oriented internal surface crack. Particular attention was given to the strength mismatch effects. The equivalent stress–strain curve in BS7910 was used to reflect the strength mismatch effects in the reference strain. The results of the proposed method were validated with the results of a finite element analysis (FEA) in terms of the J-integral. Previous methods and the proposed method exhibit a reasonable correlation of the J-integral in the case of over-matching (OM). In the under-matching (UM) cases, while the previous procedures tended to underestimate or excessively overestimate the elastic-plastic energy release rate in comparison with the FEA, the proposed method evaluated the J-integral of pipelines with sufficient accuracy.

A novel test method for continuous nonlinear biaxial tensile deformation of sheet metals by bulging with stepped-dies

In this paper, a novel test method named bulging with stepped-dies is proposed to overcome the difficulty of traditional test methods in realizing continuous nonlinear loading paths from initial yield up to fracture on a sheet metal. To achieve this aim, the section shape of a stepped-die cavity is varied with increasing depth. During bulging with a stepped-die, the stress state at the pole of bulging area of the sheet changes continuously with the increase in bulging height, which results in a specific nonlinear loading path. A theoretical model is established to calculate the stress components at the pole based on the assumption that the bulged surface near the pole is approximated by a rotational ellipsoid. Bulging experiments with three different stepped-dies are performed by using ST16 steel sheet. Stress and strain paths up to fracture and equivalent stress-strain curves at the pole are analyzed and compared with the results of bulging with elliptical dies. It is shown that continuous nonlinear loading paths can be effectively realized through bulging with stepped-dies and the stress ratio at the pole changes from 0.5 up to 2.0 at most in one bulging experiment. The feasibility of the novel test method is validated successfully. And the experimental data obtained are useful to determine constitutive and forming limit models suitable for complex loading conditions.

Stress–strain curves of metallic materials and post‐necking strain hardening characterization: A review

AbstractFor metallic materials, standard uniaxial tensile tests with round bar specimens or flat specimens only provide accurate equivalent stress–strain curve before diffuse necking. However, for numerical modelling of problems where very large strains occur, such as plastic forming and ductile damage and fracture, understanding the post‐necking strain hardening behaviour is necessary. Also, welding is a highly complex metallurgical process, and therefore, weldments are susceptible to material discontinuities, flaws, and residual stresses. It becomes even more important to characterize the equivalent stress–strain curve in large strains of each material zone in weldments properly for structural integrity assessment. The aim of this paper is to provide a state‐of‐the‐art review on quasi‐static standard tensile test for stress–strain curves measurement of metallic materials. Meanwhile, methods available in literature for characterization of the equivalent stress–strain curve in the post‐necking regime are introduced. Novel methods with axisymmetric notched round bar specimens for accurately capturing the equivalent stress–strain curve of each material zone in weldment are presented as well. Advantages and limitations of these methods are briefly discussed.

Determination of Constitutive Relationships of Tubular Materials at Various Strain Rates Using Hydro-Bulging Experiments

Liquid impact forming (LIF) is a rapid tube hydroforming technique. The deformation behaviour of a metal tube may be different in LIF from that in tube hydroforming. A constitutive equation or equivalently an equivalent stress-strain relationship is generally used to describe the deformation behaviour of metals. The purpose of this work is to model the deformation behaviour of tubular materials in LIF using the Johnson-Cook (JC) structural model. LIF hydro-bulging experiments combined with the analytical approach based on the membrane theory and the force equilibrium equation were used to determine the model coefficients A, B, C, and n in the equations for SS304 stainless steel tubular materials. Finite element (FE) simulations of hydro-bulging under various impact velocities were carried out to validate the resultant JC model. The relationship between the strain rates and impact velocities was determined, the bulging heights between the equivalent stress-strain curves at different impact velocities were analysed, and the bulging heights obtained by FE simulations and experimental results were compared. The results show that the proposed approach using the JC model is suitable to define the stress-strain behaviour of tubular materials in LIF.

Bauschinger effect modelled by yield surface distortions

A model for distorting the yield surface by flattening the part in the reverse of the loading direction, is suggested. As the basis for the distortion, the model applies a pair of second-order back-stress tensors of similar type as in kinematic hardening models. The yield-surface formulation provides a flattening and shrinkage of a given first-order homogeneous yield surface in the reverse directions of the back-stress tensors. The mathematical formulation is based on similar ideas as the HAH (homogeneous yield function-based anisotropic hardening) model, for which the calibration of the equivalent stress-strain curve is independent of the Bauschinger part of the model. Severe mathematical and numerical challenges of the HAH model are pointed out, but are avoided in the new model. Furthermore, the yield surface doesn't have to contain the origin, and the r-value is conserved in stress reversals.

Experimental measurement of temperature-dependent equivalent stress-strain curves of a 420 MPa structural steel with axisymmetric notched tensile specimens

Recently, the authors in this paper proposed a correction function to determine material's equivalent stress-strain curve with axisymmetric-notched tensile specimens. In this study, tensile tests were performed at room temperature, −30 °C and − 60 °C with axisymmetric notched tensile specimens to verify this method and to identify the equivalent stress-strain curves of a 420 MPa structural steel. A high-speed camera was used together with the so-called edge-tracing method to calculate average true strain. The material's equivalent stress-strain curve was also measured with extensometer and smooth round bar specimens. Experimental results show that equivalent stress-strain curve of this structural steel is sensitive to test temperature. Equivalent stress-stress curves obtained from axisymmetric notched tensile specimens by using the proposed correction function show good agreement with those from extensometer before diffuse necking and from Bridgman correction at large strain using smooth tensile specimens. Since fracture strain strongly depends on the notch geometry, it is recommended to use axisymmetric notched tensile specimens with smaller a0/R0 when applying the proposed correction function to measure material's equivalent stress-strain curve.

Dynamic recrystallization behavior of Gd-containing Mg alloy under torsion deformation

The thermal deformation behavior of Mg-11.95Gd-4.5Y-2Zn-0.37Zr alloy was investigated by torsion tests under different deformation conditions. Electron backscattered diffraction (EBSD) technique was used to study the effect of deformation temperature and strain rate on the dynamic recrystallization grain (DRX) and texture evolution during torsion deformation. The results indicate that the equivalent stress-strain curve of the alloy under hot torsion is similar to the curve under thermal compression. A predictive model for the critical strain (εc) and the Z parameter of the investigated Mg-alloy during torsion deformation has been established. The influence of deformation temperature on the DRX and grain size distribution is significantly different in different temperature regimes. We have observed that the grain size distribution is significantly wider at higher temperatures. Moreover, the distribution of pole density is relatively uniform under different torsion deformation conditions. The torsion has significantly declined the texture and the maximum texture intensity decreased with the decrease in grain size under different deformation conditions.

Cyclic simple shear test of material for cold forging

Cyclic simple shear tests were conducted to investigate Bauschinger effect in the large strain region of the material for cold forging. A specimen for the test was prepared from stainless steel drawn wire. Jigs for the test were designed. The designed simple shear test was simulated using finite element analysis to verify the test by comparing the analysis result and the work hardening law given in the simulation. Then, cyclic simple shear tests were performed in two different strain range. As a result, it was possible to observe the Bauschinger effect with cumulative equivalent strain of 1.0. Finally, the comparison of equivalent stress-strain curves by uniaxial tensile and simple shear tests as well as the discussion on the discrepancy were conducted. There was significant deviation on stress-strain relations in two tests, which was able to be explained reasonably by postulating that the material obeys the combined kinematic hardening rule.

A method for determining material's equivalent stress-strain curve with any axisymmetric notched tensile specimens without Bridgman correction

Large deformation analyses of problems such as plastic forming, ductile fracture with finite element method need a full range of material's equivalent stress-strain curve or flow stress-strain curve. The equivalent stress-strain curve determined from the smooth round bar specimen should be corrected after diffuse necking, since tri-axial stress state occurs in the neck. The well-known Bridgman correction method is a candidate, however, it is not accurate as the strain increases. Furthermore, it is impossible to measure the equivalent stress-strain curve of each individual material zone in a weldment with cross weld tensile tests. To cope with these challenges, a correction function and an associated test procedure are proposed in this study. With the proposed procedure, the true stress-strain curve from any axisymmetric notched tensile specimen can be converted to the material's equivalent stress-strain curve accurately and no Bridgman correction is needed. The proposed procedure can be applied to both perfectly plastic and strain hardening materials. The equivalent stress-strain curve of each individual material zone in a weldment can also be measured with the proposed procedure.

A study on the plasticity of soda-lime silica glass via molecular dynamics simulations.

Molecular dynamics (MD) simulations were applied to construct a plasticity model, which enables one to simulate deformations of soda-lime silica glass (SLSG) by using continuum methods. To model the plasticity, stress induced by uniaxial and a variety of biaxial deformations was measured by MD simulations. We found that the surfaces of yield and maximum stresses, which are evaluated from the equivalent stress-strain curves, are reasonably represented by the Mohr-Coulomb ellipsoid. Comparing a finite element model using the constructed plasticity model to a large scale atomistic model on a nanoindentation simulation of SLSG reveals that the empirical method is accurate enough to evaluate the SLSG mechanical responses. Furthermore, the effect of ion-exchange on the SLSG plasticity was examined by using MD simulations. As a result, it was demonstrated that the effects of the initial compressive stress on the yield and maximum stresses are anisotropic contrary to our expectations.