Contours Of Plastic Work Research Articles

Modeling of distortional hardening including plane strain tension and pure shear for a TWIP steel

This paper focuses on calibrating and modeling of distortional hardening behaviours in twinning induced plasticity steels. True stress-strain curves for uniaxial tension, plane strain tension, and pure shear specimens are inversely identified from corresponding load-displacement curves. The study reveals that accurately predicting the hardening behaviours of TWIP980 steel under plane strain tension and pure shear stress states is challenging with an isotropic hardening model, and a negative hydrostatic effect for TWIP980 is observed through shear testing. A novel distortional hardening model is proposed to simultaneously accommodate the three stress states on the contours of plastic work. Coefficients of the distortional hardening model are calibrated at discrete levels of plastic work and then interpolated to describe the distortion of the initial yield surface. The model is then expanded to consider the true stress-strain curves under uniaxial tension along 0, 45 and 90-degree directions, as well as under the plane strain tension along the 0-degree direction simultaneously. This expansion explicitly incorporates the three true stress-strain curves under uniaxial tension, with the curve of plane strain tension captured by an evolutionary exponent related to plastic work. The developed distortional hardening models demonstrate reasonable reproduction of load-displacement curves for TWIP980 steel under uniaxial tension, plane strain tension, and pure shear stress states.

Biaxial tensile behavior of CoCrFeNi high-entropy alloy under dynamic and proportional loadings

The dynamic failure behavior of CoCrFeNi High-Entropy Alloy (HEA) under plane biaxial stress was investigated in detail. The dynamic biaxial tensile tests were conducted using an Electromagnetic Biaxial Split Hopkinson Tensile Bar (EBSHTB) system. For comparison, the quasi-static uniaxial and biaxial tensile tests, as well as dynamic uniaxial tensile tests, were performed respectively. A cruciform specimen suitable for large plastic deformation was designed and employed in the experiments. The Finite Element Method (FEM) verified that the improved cruciform specimen could satisfy the basic requirements. The feasibility of the proposed specimen was further confirmed through loading tests. Finally, the quasi-static and dynamic yield loci of the HEA in the first quadrant of the principal stress space were plotted. The results indicate that the alloy exhibits obvious strain hardening effect and strain rate strengthening effect, the yield locus and plastic work contours can be accurately described by Hill’48 criterion.

Characterization of the asymmetric evolving yield and flow of 6016-T4 aluminum alloy and DP490 steel

6016-T4 aluminum alloy and DP490 steel were systematically tested under 24 proportional loading paths, including uniaxial tensile tests with a 15° increment, uniaxial compressive and simple shear tests with a 45° increment, and biaxial tensile tests using cruciform specimens. Cruciform specimens in the rolling/transverse and 45°/135° sampling directions were tested with seven and four different stress ratios, respectively. The normal and diagonal planes plastic work contours and the yield stresses under uniaxial tension and compression were measured to investigate the anisotropic yield. Meanwhile, the normal and diagonal planes directions of plastic strain rate and the rα-values under uniaxial tension and compression were characterized to confirm the plastic flow. Several existing asymmetric yield criteria under the associated and non-associated flow rules were comprehensively evaluated to describe the asymmetric plastic anisotropy of 6016-T4 aluminum alloy and DP490 steel. The results suggest that in the investigated yield criteria, the non-associated models can predict the tension and compression asymmetry of materials more accurately than the associated models, and the function of stress triaxiality can more effectively describe the asymmetric yield behavior than the first stress invariant. In addition, the pure shear stress states are helpful in assessing the validity and applicability of advanced asymmetric yield stress functions, and the inspection of diagonal plane plastic work contours containing more pure shear stress states should prioritized over that of normal plane plastic work contours. The evaluation of plastic potential functions should not only consider the prediction accuracy of the normal plane directions of plastic strain rate, but also further check the diagonal plane directions of plastic strain rate. Expressing mechanical properties as a function of equivalent plastic strain to calibrate parameters of the yield criterion allows the continuous capture of anisotropic evolution of the asymmetric yield surface and the changes in the asymmetric plastic potential surface.

On the modeling of deformation mechanisms in a Mg-3Al-1Zn alloy under biaxial tension

Although the {10-12} twinning behavior of Mg alloys under uniaxial tension and compression has been extensively investigated, the simulations of {10-12} twinning behavior under biaxial tension have rarely been reported. In this work, the EVPSC-TDT model is first employed to systematically investigate the deformation behavior of a Mg alloy AZ31 plate under biaxial tension in the RD-TD and ND-TD planes. The RD, TD and ND refer to the rolling direction, transverse direction, and normal direction of the hot rolled plate. The measured stress-strain curves and texture evolutions are well predicted and the contours of plastic work under biaxial tension are also constructed for comparison with experiments. The plastic response has been interpreted in terms of relative activities of various deformation modes. For biaxial tension in the RD-TD plane, basal and pyramidal slips mainly contribute to the plastic deformation for stress ratios of σRD:σTD= 1:2 to 2:1. Prismatic slip becomes more active for σRD:σTD=1:4 and 4:1. Compression twinning could be activated and so cause texture reorientation at large strains, especially for σRD:σTD=1:1. The six-fold feature of {10-10} pole figure could still be observed forσRD:σTD=1:4 and 4:1 at large strain. For biaxial tension in the ND-TD plane, tensile twinning plays an important role for σND:σTD≥1:2, while prismatic slip contributes to plastic deformation for the other cases. With the increase of stress ratio from σND:σTD≥1:1 to 1:0, the predicted twin volume fractions (VFs) at a specific strain along the ND, εND, almost linearly decrease, however, it is seen that the experimental ones at given strains along the ND do not follow such a trend with the measured twin VFs within the range of stress ratios, 2:1≤σND:σTD≤6:1, clearly being overestimated, and the difference between experiments and simulations becomes most obvious at the relatively small strain of εND=0.015. The possible reasons for the observed difference are discussed.

Measurement of Biaxial Deformation Behavior and Material Modeling for HI-PVC Using Multiaxial Tube Expansion Test

This study investigates the biaxial deformation behavior of the High Impact Polyvinyl Chloride (HI-PVC) used as a water pipe material. Multiaxial tube expansion test (MTET) was performed on the HI-PVC circular tube, and the stress-strain curves were measured for nine linear stress paths. The contours of plastic work and the directions of the plastic strain rates were also measurd. The experimental results show that the test sample has a strong anisotropy. Anisotropic hardening behavior was also confirmed from the evolution of the work contours. It was confirmed that the Yld2000-2d yield function had a better agreement with the measued work contour than Hill’s quadratic and the von Mises yield functons. Moreover, it was found that the test sample follows the normality rule with the Yld2000-2d yield function.

Measurement and Analysis of Deformation Behavior of Polyethylene Tube Subjected to Linear Loading and Unloading Stress Paths with Large Strain

A material testing apparatus for measuring the biaxial deformation behavior of a polymer tube has been developed to quantitatively evaluate the deformation behavior of polymeric materials. The testing apparatus can apply axial force and internal pressure to a tubular specimen. A noncontact strain measurement system was also developed, and the biaxial strain components and the radius of curvature in the axial direction of the bulging specimen are continuously measured to control the stress path applied to the specimen. Polyethylene tube with an outer diameter of 17 mm and a thickness of 2 mm are used as a test sample. The tubular specimens were subjected to linear stress paths with stress ratios of σΦ:σθ =1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4, and 0:1, where σΦ and σθ are the axial and circumferential stress components, respectively, applied to the central area of the bulging specimen. Loading and unloading tests were performed to determine the biaxial true stress-logarithmic plastic strain curves. The strain rate was 1×10-3s-1. From these test results, contours of plastic work and the directions of the plastic strain rates were measured to identify a proper material model for the test sample using the Yld2000-2d yield function (Barlat et al., 2003).

Nonlinear biaxial tensile stress path experiment without intermediate elastic unloading for validation of material model

A linear stress path (LSP) experiment was performed using uniaxial and biaxial tensile tests with a cold-rolled mild steel sheet (SPCD; nominal thickness: 0.8 mm) as the test material. In the LSP experiment, the contours of plastic work and the directions of the plastic strain rates, β, for a plastic strain range of 0.002 ≤ ε0p≤0.234 were measured. Then, the Yld2000-2d yield function was used to identify a material model that accurately reproduces the experimental data. Stepped nonlinear stress path (NLSP) experiments were also performed; the NLSPs consisted of several linear stress paths without intermediate elastic unloading. The measured work hardening behavior and β values were compared with those calculated using the yield functions identified from the LSP experiment, namely the von Mises, Hill’s quadratic, and Yld2000-2d yield functions. For the Yld200-2d yield function, both isotropic and differential hardening models were investigated. It was found that the data measured in the NLSP experiment are consistent with calculation results obtained using the Yld2000-2d yield function identified from the LSP experiment. Thus, it can be concluded that within the range of stress paths adopted in the NLSP experiment, the deformation behavior of a test sample can be accurately predicted using the material model identified from an LSP experiment.

Material Modeling in High Strain Range and Forming Limit Analysis for 6000 Series Aluminum Alloy Sheet

This study investigates the effects of material modeling in high strain range on the forming limit analysis of a 6000-series alloy sheet 6016-T4. A servo-controlled tension-internal pressure testing machine was used to measure the multiaxial plastic deformation behavior in a strain range from initial yield to fracture. Tubular specimens were fabricated with a weld line thicker than the sheet thickness to prevent fracture from occurring in the weld zone. Proportional loading in the first quadrant of the principal stress space was applied to the specimens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC), in addition to the contours of plastic work and the directions of plastic strain rates. Moreover, forming limit strains were measured using a flat-headed punch stretching test (PST). From the multiaxial tube expansion test (MTET) data, although a differential hardening effect was observed, appropriate coefficients including the exponent of the Yld2000-2d yield function (Barlat et al, 2003) were determined and employed in the Marcińiak-Kuczyski (1967) forming limit analysis (M-K analysis). The calculated forming limit strains in the vicinity of equibiaxial tension were smaller than the measured ones. The M-K analysis predicted that the forming limit strain in the vicinity of equibiaxial tension becomes minimum when the inclination angle of the localized neck from the rolling direction (RD) is 45° (diagonal direction; DD) while the experimental fracture direction was the RD. With the additional MTET, the plastic deformation behavior in the DD was measured and compared with that calculated from the material model. The reason of the discrepancy in the fracture direction between the experiment and the M-K approach is discussed.

Measurement of Differential Hardening under Biaxial Stress of Pure Titanium Sheet

Biaxial stress tests of a commercially pure titanium sheet (JIS #1) using the cruciform specimen and the servo-controlled biaxial testing machine have been carried out in order to elucidate its anisotropic plastic deformation behavior. The geometry of the cruciform specimen is identical to that regulated by the ISO 16842. Nine linear stress paths, σx (rolling direction): σy (transverse direction) = 1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4, and 0:1 in the first quadrant of the principal stress space are applied to the cruciform specimens. Contours of plastic work in the principal stress space and the directions of plastic strain rates at selected levels of plastic work have been precisely measured. The range of the equivalent plastic strain applied to the specimens is 0.002 ≤ ≤ 0.01. The shapes of the work contours significantly change with increasing the test material exhibits differential hardening (DH). Using the data of the work contours and the directions of plastic strain rates, the applicability of selected anisotropic yield functions to the accurate prediction of the plastic deformation behavior of the test material is examined.

Influence of hardening functions on earing prediction in cup drawing of AA3104 aluminum alloy sheet

The accuracy of cup drawing simulation of aluminum alloy sheets strongly depends on the hardening and yield functions used in the simulation. In this study, stress-strain curves for an AA3104 aluminum alloy sheet were measured by uniaxial tensile and rolling tensile tests to identify parameters for various hardening functions. In addition, in-plane uniaxial tensile, biaxial tensile, and combined tension-compression tests were carried out to measure the successive contours of plastic work in the stress space and the directions of plastic strain rates under linear stress paths. The anisotropic coefficients, α1-α8, and exponent M of the Yld2000-2d yield function were determined so that the mean square error of the analytical yield locus from the measured plastic work contour was minimized. Finite element simulations of cup drawing of the aluminum alloy sheet were carried out using the determined parameters for hardening functions and the Yld2000-2d yield function implemented in LS-DYNA. By comparing the simulation results with the results of cup drawing experiments, the effects of the hardening functions on the accuracy of earing profile prediction were investigated.

Transition from convex to concave of equal plastic work contours for wrought magnesium alloy under multi-axial loading

Due to non-proportional stress state in the forming process, an extruded AZ31 magnesium alloy is subjected to combined axial loading-torsion test. Pre-elastic axial tension and compression along extruded direction is performed firstly, then torsion loading with the axial stress unchanged, denoted as TT (Torsion after Tension) and TC (Torsion after Compression) loading is implemented respectively. The results show that the shape of the Equal Plastic Work Contour (EPWC) changes remarkably from convex to concave as plastic deformation increasing. Accordingly, the EPWC cannot act instantaneously as plastic potential which take place under proportional loading path. The microstructure of failure samples were observed to clarify the reason of transition from convex to concave. It is found that the twinning behavior is different during torsion after different axial stress which results from the variation of stress state during torsion and the restricted activity of twinning with c/a ratio decreasing. For pure torsion, the stress state facilitate the growth of twins. For pre-loading samples, the activity of twinning is restricted with the c/a ratio decreasing. However, the stress to activate twinning is satisfied under TC loading with the compressive stress component increasing with pre-elastic compression. The compressive stress component decrease with pre-elastic tension under TT loading. Therefore, twinning is activated under pure torsion and TC loading. Furthermore, the barrier effect of twin boundaries and dislocation-twin interaction strengthen basal slip and prismatic slip resulting in dislocation hardening. Texture hardening takes place by reorientation of grains induced by twinning. Both dislocation hardening and texture hardening lead to high strain hardening rate under pure torsion and TC loading. Therefore, the strain hardening rate and flow stress are much higher under pure torsion and TC loading than that of TT loading. Consequently, a transition from convex to concave of EPWC is obtained in large plastic strain.

Crystal-plasticity finite-element analysis of anisotropic deformation behavior in a commercially pure titanium Grade 1 sheet

A crystal-plasticity finite-element method was used to study the deformation behavior of a commercially pure titanium Grade 1 sheet upon different strain paths. Prismatic slip, pyramidal <a> slip, basal slip, two types of pyramidal <a + c> slip, {101¯2} twinning, and {112¯2} twinning were taken into consideration. The material parameters were systematically determined considering the role of each active deformation mode. The simulation results were in good agreement with the experimental results with respect to evolution of the Lankford value, stress–strain curves, contours of plastic work, and texture evolution for the strain paths examined in this study. The mechanism of anisotropic deformation behavior was then investigated, focusing especially on the role of the activity of twinning in the plastic deformation. It was found that the twinning activity significantly affected the following characteristics: the anisotropies in the Lankford value and work hardening under compression and the tension–compression asymmetries in the stress–strain curves in the rolling direction. The detwinning activity also affected stress–strain curves upon reverse loading, in particular in the rolling direction. To systematically understand the deformation mechanism, the effect of slip activity on the deformation behavior is also discussed.

Influence of material modeling on earing prediction in cup drawing of AA3104 aluminum alloy sheet

In-plane biaxial tension and combined tension-compression tests are carried out for AA3104 aluminum alloy sheets. Linear stress paths are applied to cruciform specimens to measure the contours of plastic work in the stress space and the directions of plastic strain rates at each stress path. Coefficients α1- α8 and exponent M of the Yld2000-2d yield function are determined to minimize the mean square error of the analytical yield locus from a measured work contour. The values of the weighting coefficients in the evaluation of the error are varied to check the effect of a specific stress state on the earing behavior. The effects of the combinations of the weighting coefficients on the accuracy of earing prediction in the cup drawing process are discussed.

Constitutive modeling of evolving plasticity in high strength steel sheets

In this paper, biaxial tensile tests with cruciform specimens were conducted to obtain the experimental plastic work contours of DP590, DP780 and DP980 sheets. It was found that for high strength dual phase steel sheets, the outlines of the experimental work contours in the first quadrant evolve evidently with the increase of plastic dissipation. In order to calibrate the evolving isotropic and anisotropic plastic behaviors simultaneously, the Yld2000-2d yield criterion with variable exponent and coefficients (Yld2000-2d-Var) was introduced. In this model, the exponent m and coefficients αi of the Yld2000-2d was assumed as the function of the equivalent plastic strain. The identification procedures of the introduced parameters were then proposed based on minimizing the difference between the experimental and the model predicted yield points, using Levenberg–Marquardt optimization method. Afterwards, the proposed model was implemented into ABAQUS as user subroutine VUMAT with semi-implicit back-Euler integration algorithm. The numerical simulations of hydro-bulging and cylindrical deep drawing tests were then carried out to verify the validity of the proposed model. The results showed that the proposed model could enhance the predicting accuracy in both tests due to its flexibility.

Effect of Equivalent Stress-Strain Relation and Differential Hardening on Accuracy of Forming Limit Analyses

Multiaxial tube expansion tests (MTETs) were performed to measure the multiaxial plastic deformation behavior of a cold rolled interstitial-free (IF) steel sheet for a range of strain from initial yield to fracture. The testing machine is capable of applying arbitrary principal stress or strain paths to tubular specimens using an electrical, closed-loop servo-control system for an axial force and an internal pressure. Tubular specimens with an inner diameter of 44.6 mm were fabricated from a cold rolled IF steel sheet with a thickness of 0.7 mm by roller bending and laser welding. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the forming limit strains and forming limit stresses of the as-received sheet sample, in addition to the contours of plastic work and the directions of the plastic strain rates. It was found that the shapes of the measured work contours changed with increasing plastic work. The observed differential hardening behavior was approximated by changing the material parameters and the exponent of the Yld2000-2d yield function (Barlat et al, 2003) as functions of the reference plastic strain. The hydraulic bulge tests were also conducted to measure the forming limit strain and forming limit stress for equibiaxial tension and to determine the equivalent stress-equivalent plastic strain relation for a larger strain range. The forming limit curve and forming limit stress curve were calculated using the Marciniak-Kuczyński-type approach. The calculated results were in fair agreement with the measurement.

Measurement and Analysis of the Differential Hardening of Ultralow Carbon Steel Sheets

Multiaxial tube expansion tests were performed to precisely measure the work hardening behavior of mild steel sheets with different r-values for a range of strain from initial yield to fracture. The testing machine is capable of applying an arbitrary linear stress path to a tubular specimen using an electrical, closed-loop servo-control system for the axial force and internal pressure applied to the tubular specimen. Tubular specimens with an inner diameter of 44.6 mm were fabricated from the as-received sheet sample by roller bending and laser welding. Nine linear stress paths, σx(rolling direction) :σy (transverse direction) =1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4, and 0:1, in the first quadrant of the principal stress space were applied to the tubular specimens to measure the contours of plastic work and the directions of the plastic strain rates. It was found that the shapes of the measured work contours changed with increasing plastic work, or equivalently with increasing the reference plastic strain ; the test materials exhibited differential hardening (DH). The general trend of the DH appeared to be affected by the average r-value, or equivalently by the texture, of the materials.

Forming Limit Analyses of 590 MPa High Strength Steel Sheet Using Differential Work Hardening Model

A servo-controlled tension-internal pressure testing machine with an optical 3D deformation analysis system (ARAMIS®, GOM) was used to measure the multiaxial plastic deformation behavior of a 590MPa high strength steel sheet for a range of strain from initial yield to fracture. Tubular specimens were fabricated from the sheet sample by roller bending and laser welding. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC), in addition to the contours of plastic work and the directions of plastic strain rates. It was found that the shapes of the measured work contours changed with the increase of work hardening (plastic work). The observed differential work hardening (DWH) behavior was approximated by changing the material parameters and the exponent of the Yld2000-2d yield function (Barlat et al, 2003) as a function of the equivalent plastic strain. The FLC and FLSC calculated using the Marciniak-Kuczyński-type (M-K) approach with the DWH model were in good agreement with the measurement.

Material Modeling and Springback Analysis Considering Tension/Compression Asymmetry of Flow Stresses

In-plane tension/compression tests of a dual phase steel sheet with a tensile strength of 780 MPa were carried out using in-plane stress reversal testing machine. Remarkable tension/ compression asymmetry of flow stress (TCA) has been observed. Moreover, biaxial tensile tests using cruciform specimens were performed to measure contours of plastic work. The test material exhibited differential work hardening (DWH). In order to reproduce the TCA, an asymmetric quadratic yield function proposed by Verma et al. (2011) was used. The parameters of the yield function were changed as a function of reference plastic strain to reproduce the DWH. Furthermore, to assess the springback prediction accuracy of the developed model, a 3-point bending experiment and finite element analyses (FEA) were performed. It is concluded that the use of the material model that is capable of reproducing DWH and TCA is a must for a highly accurate FEA of springback.

Forming Simulation Considering the Differential Work Hardening Behavior of a Cold Rolled Interstitial-Free Steel Sheet

The multiaxial plastic deformation behavior of a cold rolled interstitial-free steel sheet with a thickness of 0.65 mm was measured using a servo-controlled multiaxial tube expansion testing machine for the range of strain from initial yield to fracture. Tubular specimens were fabricated from the sheet sample by roller bending and laser welding. Many linear stress paths in the first quadrant of stress space were applied to the tubular specimens to measure the contours of plastic work in stress space up to an equivalent plastic strain of 0.289 along with the directions of plastic strain rates. The test material exhibited differential hardening (DWH). A material modeling method for reproducing the DWH in a finite element simulation has been developed. Hydraulic bulge forming simulation results based on the DWH model had a closer agreement with the experimental results than those calculated using the isotropic hardening models with selected yield functions.

薄肉超小型試験片の単軸駆動式二軸引張試験方法の開発

We have established a biaxial tensile test method suitable for the measurement of the local plastic deformation characteristics of a hard aluminum alloy sheet 3104-H, which is a typical material for aluminum can bodies, as a representative of the low work hardening materials. The present method consists of the following: (1) a microminiature cruciform specimen, (2) a newly designed linkage mechanism mounted on a uniaxial tensile testing machine, and (3) the loading path diagram as an experimental method to measure the biaxial stress–strain curves accurately from the biaxial tensile tests with linear displacement paths of the chucking jigs. The following are drawn from the biaxial tensile tests for the thin test material: (i) using the proposed biaxial tensile testing apparatus, we measured a contours of plastic work, and it was in good agreement with that measured using the conventional linear stress path tests, (ii) we were successful to determine the real yield points for various biaxial tensile states using the loading path diagrams and to measure the contour of plastic work corresponding to the maximum equivalent plastic strain of 0.0012, and (iii) the contour of plastic work is well approximated using the Yld2000-2d yield function with an exponent of 7.78.