Loading Path Changes Research Articles

Application of distortional plasticity framework to EDDQ and TRIP steel sheets: Prediction of latent hardening and its influence on springback

In this study, the recently developed distortional plasticity model (HAH20) was applied to EDDQ and TRIP steel sheets to account for hardening fluctuations induced by loading path changes. Previous non-linear loading path experiments, which were tension-compression, tension-tension, and tension-shear tests, revealed intricate latent hardening in EDDQ steel and a combination of latent hardening and cross loading contraction in TRIP steel. The HAH20 model was calibrated using tension-compression and 90° tension–45° tension tests with adjustment of setting parameters for accurate latent hardening prediction. Then, a comparison between the HAH20 model and the previous distortional plasticity model (HAH14) in the other independent experiments highlights the superiority of HAH20 model due to its description of latent hardening. Furthermore, for TRIP steel, the HAH20 and HAH14 models were applied to predict springback in a U-draw bending test featuring a novel curved blank subjected to reverse and cross loading states simultaneously. The simulation results show that the HAH20 model predicts springback more accurately than the HAH14 model; however, the main reason is the accurate description of reverse loading in the HAH20 model. This is due to the intricate interplay between cross loading contraction and latent hardening in springback.

Parameter identification for the non-associated flow rules representing corner effects through the equivalent tangential shear modulus reduction after abrupt strain-path change

Some specific plastic flow rules have been proposed to describe the experimental observations of “corner effects”—the apparent violation of the normality of plastic flow after abrupt strain-path change, and rotation of the plastic strain rate direction toward the strain rate direction. In this paper, a simple parameter identification scheme for these non-associated flow rules representing corner effects is proposed from the perspective of the equivalent tangential shear modulus reduction. The nonlinear tension–torsion loading experiments with abrupt strain-path changes were performed on tube specimens made of the aluminum alloy A6063-T5, and working conditions of different prestrains and different loading path changes were included. The finite element analysis was used to verify the rationality of the determined parameters and compare the performances of different plastic flow rules in predicting the non-normality of the plastic flow. The experimental and simulation results suggest that the non-associated flow rules can well predict the plastic flow when the determined parameters are used, with the rules proposed by Yoshida and Tsuchimoto (2018) and Zhang et al. (2022) yielding the most accurate predictions. At the same time, the equivalent tangential shear modulus is apparently reduced during the transient process of strain-path change without elastic unloading; however, it does not seem to decrease to near zero as reported before.

A polycrystalline damage model applied to an anisotropic aluminum alloy 2198 under non-proportional load path changes

A ductile damage model, which fully couples classical void growth at high stress triaxiality and Coulomb ductile model at the slip system scale at low stress triaxiality, was applied for the sheet specimens in an anisotropic aluminum alloy under non-proportional load path changes. The Coulomb model combines the resolved normal and shear stresses for each slip plane and directions. A Reduced Texture Methodology (RTM) was used to provide computational efficiency and this approach involved a significant reduction in the number of representative crystallographic orientations. The 12-grain model using 12 crystallographic orientations was validated for non-proportional load path change experiments. The model was calibrated in plasticity using single element calculations under proportional tension, shear and non-proportional ‘shear to tension’ (ST) loadings. Next, the damage parameters were calibrated against the proportional loading shear and tension experimental results using 3D mesh full-size calculations. The calibrated model successfully predicted non-proportional failure. Locally, the damage indicator maxima coincided with the damage location where the damage features were observed via experimental 3D imaging (computed tomography).

Plasticity and ductility of an anisotropic recrystallized AA2198 Al-Cu-Li alloy in T3 and T8 conditions during proportional and non-proportional loading paths: simulations and experiments

The anisotropic material behaviour of a recrystallized AA2198 Al-Cu-Li alloy in T3 and T8 conditions was assessed by proportional loading of uniaxial-tension specimens in rolling (L), transverse (T) and diagonal (D) orientations. The width and longitudinal strains were measured to quantify plastic anisotropy. Notched-tension samples were tested in L and T directions. The material showed weak anisotropy in terms of stress strain curves and a moderate plastic anisotropy, consistent with its texture obtained by EBSD. An anisotropic Bron-Besson type material model was identified using this data base and a micro shear-only (SO) test. The model succeeded in predicting the behaviour of micro specimens for proportional tension-only (TO) loading and non-proportional load path changes including 'shear to tension' (ST) as well as 'tension to shear' (TS) tests. The non-proportional loading was achieved using a newly designed cross shaped sample. It was loaded in one direction, unloaded and subsequently loaded in the orthogonal direction till fracture. The average stretch to fracture of both alloys measured by a four point frame optical extensometer decreased by 29 % and 16 % for T3 and T8 respectively for the 'shear to tension' experiment compared to the proportional TO experiment. The average stretch to fracture of 'tension to shear' tests was reduced by 10 % for 2198T3 and hardly reduced for 2198T8 compared to the stretch to fracture of the SO tests, but subject to strong scatter. FE simulations showed local accumulated strain to fracture values that were similar for all loading histories for the T8 condition (0.73 − 0.84). Lower strain to fracture values were found in T3 condition (0.45 − 0.73), despite the enhanced macroscopic ductility in tension. This was attributed to larger less localized plastic zones, especially for the ST test. The ductility scatter was attributed to necking and damage development in tension that can affect strain localization, associated fracture path and ductility, as observed by DIC and fractography.

Phenomenological Modeling of Deformation-Induced Anisotropic Hardening Behaviors: A Review

Numerous studies indicate that the hardening behaviors of materials are closely related to their deformation history. In the forming processes with loading path changes, such as sheet metal forming, anisotropic hardening behaviors are universally observed. In this situation, selecting or constructing a suitable anisotropic hardening model is essential. This paper presents a review of the phenomenological modeling of the deformation-induced anisotropic hardening behaviors. At the beginning, the deformation-induced hardening behaviors are introduced together with the relevant experiments. Different from other published review works, this paper is not laid out according to the description of a series of models. Instead, the modeling is emphasized by generalizing the main mathematical modeling ideas among various hardening models and sorting out the description methods for the decomposed anisotropic hardening behaviors. Some prospective development directions for the modeling of anisotropic hardening behaviors are suggested at the end of this work. This review work tries to provide the researchers with an instruction on how modeling for the anisotropic hardening behaviors according to the materials and forming processes.

Mechanical, microstructural and in-situ neutron diffraction investigations of equi-biaxial Bauschinger effects in an interstitial-free DC06 steel

Bauschinger effects typically describe a reduction of yield strength after a load path change under uniaxial loading conditions. Here, we use in-plane loading to investigate, for the first time, Bauschinger effects in the sheet metal DC06 under well-defined equi-biaxial loading conditions: We first performed equi-biaxial tensile tests with cruciform specimens up to different maximum tensile strains. Then, smaller specimens were prepared from the equi-biaxially deformed inner part of the cruciform specimens and subjected to equi-biaxial compression. The mechanical results show that the material exhibits distinct Bauschinger effects when subjected to equi-biaxial load path changes, which differ from similar observations under uniaxial loading. Specifically, biaxial Bauschinger effect factors quickly reach a level of saturation, whereas the uniaxial Bauschinger effect factors keep decreasing. These Bauschinger effects can be rationalized by considering the results of TEM and EBSD investigations at the different stages of biaxial loading, particularly by considering the evolution of dislocation densities and the formation of substructures, which are related to intergranular and intragranular stresses. Furthermore, residual stress measurements by XRD and in-situ neutron diffraction show an increase of compressive residual stresses after equi-biaxial tensile deformation and unloading. These residual stresses facilitate yielding during subsequent equi-biaxial compression and therefore also clearly contribute to the observed Bauschinger effects.

Voltage instability curtailment and quality improvement with phasor measurement unit in power system

PurposeThe earlier methods are more resilient to improvements such as load shift and path change. This results in problems such as a voltage drop and a high reactive flux. In addition, due to the delay, congestion or interruption of the transmission, the system cannot receive all phasor measurement unit (PMU) measurements at the relevant time as well as the presence of noise in the received data.Design/methodology/approachWith the development of wide area measurement system technologies, it seems to be possible to track voltage stability online via time-stamped PMUs. As the voltage instability causes a voltage decomposition, voltage instability is one of the most important problems when monitoring the power supply.FindingsThis harmonic distortion significantly decreases the data quality in the grid. As a result, instability ascertainment based on PMU has been suggested as a method for detecting voltage instability in power systems monitored with PMU. In addition, a technique called instability amendment via load dropping has been proposed to keep the device from collapsing due to voltage failure.Originality/valueTo improve the power output, the power prominence melioration technique was developed. This proposed system has been implemented in MATLAB Simulink and compared with the recent researches.

3D in situ study of damage during a ‘shear to tension’ load path change in an aluminium alloy

A load path change (LPC) from shear to tension has been studied for a recrystallized 2198 T8 aluminium alloy sheet material by three-dimensional (3D) X-ray imaging combined with image correlation and interpreted by complementary 3D finite element (FE) simulations. A cross-shaped specimen was designed for the non-proportional loading and multiscale study. The effect of the LPC on the formability and related strain localisation, damage and failure was investigated and damage mechanisms could be clearly identified. The macroscopic tension stretch to fracture, measured by an optical extensometer during the shear to tension test, was reduced by about 20% compared to the proportional tension test. Damage, measured by in situ laminography imaging at μm-scale resolution, has interestingly already been found under shear, and was quantified as surface void fraction during the LPC. Strain was measured inside the material and at the mesoscale by 2D digital image correlation (DIC) on projected volume data, using the (natural) intermetallic particle contrast present in the 3D laminographic data. An accumulated equivalent strain definition, suited for the description of non-proportional loading, has been applied to the DIC data and FE simulations, indicating good agreement between both. On the microscopic scale, damage was seen to nucleate under shear load in the form of flat cracks, of similar width as the grain size, and in the form of cracks inside intermetallic particles. This damage subsequently grew and coalesced during tensile loading which in turn led to final fracture.

Return mapping with a line search method for integrating stress of the distortional hardening law with differential softening

A stress-update algorithm for the recently proposed distortional anisotropic hardening law is developed based on the Newton–Raphson (N–R) algorithm and line search method. The investigated yield function enables the modeling of complex path-dependent flow stress evolutions, particularly the Bauschinger effect, latent hardening/softening, and differential permanent softening. Computationally, the continuous distortion of the yield function under strain-path changes leads to numerical instability, which is overcome by a newly reformulated step-size control method in the line search algorithm. The developed algorithms were implemented in ABAQUS using the closest-point projection method and validated for extra-deep drawing quality and DP780 steel sheets in terms of numerical accuracy and stability under reverse- and cross-loading path changes. The results show that the return-mapping algorithm significantly improves the accuracy and speed of convergence when the proposed line search method is incorporated. In contrast, the conventional N–R based algorithm fails to obtain converged stresses under abrupt loading path changes owing to the sharp corners introduced by the distorted yield surface.

Fully Implicit Stress Update Algorithm for Distortion-Based Anisotropic Hardening with Cross-Loading Effect: Comparative Algorithmic Study and Application to Large-Size Forming Problem

A fully implicit stress integration algorithm is developed for the distortional hardening model, namely the e−HAH model, capable of simulating cross−hardening/softening under orthogonal loading path changes. The implicit algorithm solves a complete set of residuals as nonlinear functions of stress, a microstructure deviator, and plastic state variables of the constitutive model, and provides a consistent tangent modulus. The number of residuals is set to be 20 or 14 for the continuum or shell elements, respectively. Comprehensive comparison programs are presented regarding the predictive accuracy and stability with different numerical algorithms, strain increments, material properties, and loading conditions. The flow stress and r−value evolutions under reverse/cross−loading conditions prove that the algorithm is robust and accurate, even with large strain increments. By contrast, the cutting−plane method and partially implicit Euler backward method, which are characterized by a reduced number of residuals, result in unstable responses under abrupt loading path changes. Finally, the algorithm is implemented into the finite element modeling of large−size, S−rail forming and the springback for two automotive steel sheets, which is often solved by a hybrid dynamic explicit–implicit scheme. The fully implicit algorithm performs well for the whole simulation with the solely static implicit scheme.

Prediction of anisotropic strengths of steel plate after prior bending-reverse bending deformation: Application of distortional hardening model

In this study, the strengths of a steel pipe in different material orientations are predicted using a distortional anisotropic hardening model, namely, the HAH model, implemented in a finite element simulation. The investigated hardening can capture the transient flow behavior under bending and reverse bending strain paths. In addition, the model reproduces the material behavior under orthogonally applied tension from the prior deformation path. To improve the predictive accuracy of the existing distortional hardening model, we additionally implement the multi-component evolution rule of the Bauschinger effect and transient hardening behavior. Two-step tension tests are used to identify the constitutive parameters related to the loading path changes in the bending and reverse bending (BRB) test, which mimics the common pipe manufacturing process in a practical manner. The predicted strengths along the circumferential and transverse directions to the major bending direction agree well with the experimentally measured strengths when the distortional hardening model is employed. By contrast, the classical isotropic hardening and kinematic hardening model over- and under-estimate the yield strength of the specimens after prior BRB loading. The improved accuracy of the strength prediction with the investigated HAH is attributed to the anisotropic identification of the flow behavior under both load reversal and cross-loading conditions, whereas the isotropic-kinematic hardening only considers the back stress evolution at load reversal. In addition to the strengths, the stagnated flow behavior after BRB loading can be well predicted with the HAH model with a properly calibrated yield point elongation using an inverse identification method.

Assessment of Springback Behaviour of 800-1200 MPa Dual-Phase Steel Grades

Springback occurs in sheet metal forming due to elastic strain recovery after removal of process forces respectively after opening of the tool. For this reason, a precise description of springback requires the elastic stress-strain relationship described by the Young’s modulus as well as the internal stress distribution of the part before unloading. In this context, the Bauschinger effect influences the stress state before springback due to premature plastification during load reversal or load path change. As is well known, the stress-strain curve of a material during unloading is non-linear because of additional microplastic strain, which is reflected in a decrease of the Young’s modulus. The aim of this work is to characterize the aforementioned phenomena and their effect on springback for three dual-phase steels namely DH800, DH1000 and DP1200LY. For this purpose, cyclic tensile-compression tests as well as loading and unloading loops within uniaxial tensile tests are performed at different plastic strains. To evaluate the springback behavior of the investigated materials, two different hat-profiles geometries are investigated. By comparing the springback of dual-phase steels on part level, the significance of different material influences with regard to springback is evaluated. The results show that the investigated dual-phase steels exhibit a pronounced Bauschinger effect and a considerable amount of microplastic strain with increasing total strain. However, the comparison between the springback of the hat-profiles and the determined material parameters proves a significant influence of the elastic strain on springback, while microplastic strain and the Bauschinger effect have a minor influence.

An efficient elasto-visco-plastic self-consistent formulation: Application to steel subjected to loading path changes

A novel elasto-visco-plastic self-consistent (EVPSC) formulation based on the scheme of the homogeneous effective medium is presented. The constitutive behavior of a polycrystal is described as that of an elasto-visco-plastic effective medium interacting with grains treated as elasto-visco-plastic ellipsoidal inclusions. The formulation is based on the definition of a unique elasto-visco-plastic compliance, so avoiding the inconsistency arising from assuming superimposed elastic and visco-plastic interaction laws, as made in similar elasto-visco-plastic models. In addition, the elasto-visco-plastic constitutive equation of crystal and aggregate is formulated in terms of stress increments, which leads naturally to a semi implicit solution scheme. The superior numerical stability and computational efficiency of the new incremental EVPSC model (denoted as ΔEVPSC) are demonstrated by applying the model to a 316L austenitic stainless steel and comparing against other elasto-plastic models. The modeling capability for predicting texture, stress-strain response, and Bauschinger effect are demonstrated using a dislocation-density based hardening law applied to low carbon (LC) steel subjected to deformation histories that involve strain-path changes.

Fracture envelopes on the 3D-DIC and hybrid inverse methods considering loading history

This paper deals with the comparative investigation of fracture envelopes constructed by the 3D-DIC and hybrid inverse analyses. For the evaluation of fracture limits of the DP980 1.2t steel over a wide range of loading conditions, tensile tests were conducted using three different specimens that are expected to induce certain stress states of plane strain tension, in-plane shear, and uniaxial tension at the potential location for the fracture initiation. The modified Mohr–Coulomb ductile fracture criterion was selected in this study to investigate the influence of the fracture envelope identification based on each way of loading history evaluation not only on the level of fracture limits but also on the fracture prediction especially for the simulation of square cup deep drawing. For an in-depth understanding of non-linear loading history on the variation of ductility limit, apparent fracture with respect to strain was numerically computed according to the change of loading path. Finally, a comparison of results from the square cup deep drawing test and the FEA prediction was performed in terms of punch force and stroke to confirm the model performance at various ways of the loading history evaluation.

Characterization of forming limits at fracture for aluminum alloy 6K21-T4 sheets in non-linear strain paths using a biaxial tension/shear loading test

A good knowledge of forming limits at fracture (FLF) for aluminum sheets exhibiting poor formability at room temperature in various non-linear strain paths (N-LSPs) is essential for fully exploiting their forming ability and avoiding their fracture failure. However, the FLF over a wide range of strain states in N-LSPs with pre-strain paths between uniaxial tension and simple shear cannot be determined through the existing two-stage loading test methods. To solve the problem and fill the gap, a new two-stage loading test method employing an optimized butterfly specimen is proposed in this paper, in which strain path changes are controlled by changing biaxial loading conditions acting on the specimen. To validate the capability of the new test method to characterize the FLF in the N-LSPs and investigate the effect of the loading path changes on aluminum 6K21-T4 sheet formability at fracture, different proportional and non-proportional loading experiments using the butterfly specimen obtained from the aluminum sheets are performed. Experimental results indicate that the forming limit strains at fracture over a wide scope of strain states in the N-LSPs can be effectively determined by utilizing the proposed method, thus filling the gap mentioned above. Additionally, four newly developed ductile fracture criteria (a new model proposed in this paper, Hu−Chen 2017, Lou−Huh 2012 and MMC3) together with a non-linear damage accumulation rule (N-LDAR) are employed to forecast the FLF for the 6K21-T4 sheets in the N-LSPs. The results show that both the proposed new model with the N-LDAR and the Hu−Chen model along with the N-LDAR can provide the highest prediction accuracy for the FLF in the N-LSPs. However, the prediction accuracy of the proposed model is much higher that of the Hu−Chen 2017 model for aluminum alloy 5083-O sheets.

Neutron Diffraction and Diffraction Contrast Imaging for Mapping the TRIP Effect under Load Path Change.

The transformation induced plasticity (TRIP) effect is investigated during a load path change using a cruciform sample. The transformation properties are followed by in-situ neutron diffraction derived from the central area of the cruciform sample. Additionally, the spatial distribution of the TRIP effect triggered by stress concentrations is visualized using neutron Bragg edge imaging including, e.g., weak positions of the cruciform geometry. The results demonstrate that neutron diffraction contrast imaging offers the possibility to capture the TRIP effect in objects with complex geometries under complex stress states.

Effect of strain rate difference between inside and outside groove in M−K model on prediction of forming limit curve of Ti6Al4V at elevated temperatures

Abstract The influence of initial groove angle on strain rate inside and outside groove of Ti6Al4V alloy was investigated. Based on the evolution of strain rate inside and outside groove, the effect of strain rate difference on the evolution of normal stress and effective stress inside and outside groove was also analyzed. The results show that when linear loading path changes from uniaxial tension to equi-biaxial tension, the initial groove angle plays a weaker role in the evolution of strain rate in the M−K model. Due to the constraint of force equilibrium between inside and outside groove, the strain rate difference makes the normal stress inside groove firstly decrease and then increase during calculation, which makes the prediction algorithm of forming limit convergent at elevated temperature. The decrease of normal stress inside groove is mainly caused by high temperature softening effect and the rotation of groove, while the increase of normal stress inside groove is mainly due to strain rate hardening effect.

Calibration of Chaboche Combined Hardening Model for Large Strain Range

Variable-loading path and large strain are two typical features in multistage plastic forming process. Material’s yield surface expands and translates simultaneously as the loading path changes, leading to certain transient phenomena such as Bauschinger effect. Chaboche combined hardening model is widely used in this kind of simulation due to its capability of characterizing the movement and expansion of yield surface. This study provides a parameter calibration method for Chaboche combined hardening model with 3 back stresses in a large strain range. Uniaxial tension-compression cyclic test with constant strain control was conducted for calibrating the Chaboche model firstly. Meanwhile, the flow curve obtained from uniaxial tension test (UT) associated with Swift extrapolation model were employed to provide the stress-strain relation at large strain level. Then, both theoretical analysis and optimal fitting using NSGA-II were used to determine the parameters in Chaboche model by using the data from the small strain cyclic test and the extrapolated UT curve at large strain simultaneously. The calibration result shows a good match with the experimental and extrapolated data, proving the efficiency of the parameter determination manner put forward in this study.

Damage-induced performance variations of cold forged parts

Abstract Forming processes play a key role in the manufacturing of metal components. They allow for the economical production of geometrical shapes with reproducibly high quality. Strain hardening and residual stresses affect the performance of the produced parts. These factors are controllable and can even be utilized to increase the performance of the component. This, however, does not apply to damage. Damage in metals describes the decrease of the load-bearing capacity due to the appearance and evolution of voids. The aim is to analyse, predict, and control the evolution of damage in cold forging, to allow for a production of cold forged components with a defined, load-adapted performance. It was investigated numerically to what extent the load path, which is responsible for the damage evolution, is affected in cold forging. Subsequently, the effect of load path changes on the product performance was determined experimentally in the region of the central axis where the load path is affected most by the extrusion parameters. Hereby, a correlation between the occurring triaxiality during forming and the product performance by means of number of cycles to failure in multi-step fatigue tests, impact energy and Young’s modulus was observed.

Microstructure evolution of stainless steel subjected to biaxial load path changes: In-situ neutron diffraction and multi-scale modeling

The lattice strain and intensity evolution obtained from in-situ neutron diffraction experiments of 316L cruciform samples subjected to 45° and 90° load path changes are presented and predicted using the multi-scale modeling approach proposed in Upadhyay et al., IJP 108 (2018) 144-168. At the macroscale, the multi-scale approach uses the implementation of the viscoplastic self-consistent polycrystalline model as a user-material into ABAQUS finite element framework to predict the non-linearly coupled gauge stresses of the cruciform geometry. The predicted gauge stresses are then used to drive the elasto-viscoplastic fast Fourier transform polycrystalline model to predict the lattice strain and intensity evolutions. Both models use the same dislocation density based hardening law suitable for load path changes. The predicted lattice strain and intensity evolutions match well with the experimental measurements for all reflections studied. The simulation results are analyzed in detail to understand the role of elastic anisotropy, plastic slip, grain neighborhood interactions and cruciform geometry on the microstructural evolution during biaxial load path changes.