Nonlinear Loading Paths 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.

Numerical fracture prediction of martensitic steel by uncoupled and coupled type fracture models

In this study, fracture of martensitic steel in hole expansion and stretch bending was predicted by finite element analysis with shell elements. Various mechanical experiments were conducted to characterize mechanical properties related to fracture as well as plasticity. Then, Swift-modified Voce and Yld2000-2d well-described plasticity properties of martensitic steel. For fracture modeling, Hosford-Coulomb (HC) and shear-modified Gurson-Tvergaard-Needleman (GTN-shear) models were selected and calibrated with load–displacement curves of three fracture tests. The hybrid numerical-experimental method was used for the HC model and inverse identification with finite element model updating was used for the GTN-shear model. In calibration, an exponent of yield function and mesh size were adjusted to match the strength of plane strain state and to maintain consistency of mesh size. After calibration, both fracture models were applied to numerical simulations of conical hole expansion and stretch bending tests. The GTN-shear model predicted well fracture in the conical hole expansion test but not in the stretch bending. The prediction results of HC model were vice versa. Through triaxiality analysis, these differences between the two models may result from a non-linear loading path and description of localized necking.

Evaluation of loading-path-dependent constitutive models for springback prediction in martensitic steel forming

In this study, the performance of conventional and advanced constitutive models to the prediction of springback for ultra-high strength steel was assessed. The elasto-plastic behavior of a martensitic steel sheet sample was characterized to assess the conventional stress-strain behavior in uniaxial tension. More advanced tests such as uniaxial compression and tension-compression experiments were conducted to assess the strength-differential (SD) effect and hardening fluctuations during non-linear loading paths (NLLP effect). Three constitutive plasticity models, namely, the conventional isotropic hardening, Yoshida-Uemori (YU) two surface kinematic hardening and pure distortional hardening (HAH20) models were calibrated using an optimization procedure. Then, these models were employed for finite element predictions of springback after forming. The isotropic hardening model does not consider both SD and NLLP effects and the YU model captures only the influence of the NLLP effect. On the other hand, the HAH20 model describes both effects simultaneously. The performance of these models with respect to springback prediction was evaluated using three forming applications: wiper bending, C-rail drawing and Roof-Side-Rail crush forming. This work indicates that the HAH20 model is the best candidate for the prediction of forming and springback for such martensitic steel because of its ability to consider both SD and NLLP effects. However, the conventional isotropic hardening model led to results in good agreement with those obtained using the HAH20 model in specific cases. The reason, discussed in detail in this article, is likely due to the compensation of the permanent softening due to load reversal by the higher flow stress due to the SD effect.

Forming Limit Analysis of Thin-Walled Extruded Aluminum Alloy Tubes under Nonlinear Loading Paths Using an Improved M-K Model

To meet the requirement of lighter weight and better performance in tube hydroforming, one of the most important tasks is to accurately predict the forming limit of thin-walled tubes under nonlinear loading paths. This work established the M-K+DF2012 model, a combination of the M-K model and the DF2012 ductile fracture criterion, for the forming limit prediction of thin-walled tubes under nonlinear loading paths. In this model, the failure of the groove is determined by the DF2012 criterion, and the corresponding strains in the uniform region are the limit strains. The limit strains of an AA6061 aluminum alloy tube under a set of linear loading paths and two typical nonlinear loading paths were tested. Parameter values of the M-K+DF2012 model for the tube were determined based on the experimental limit strains under linear loading paths, and the limit strains under the two nonlinear loading paths were predicted. Then the strain-based forming limit diagram (ε-FLD) and the polar effective plastic strain FLD (PEPS-FLD) of the tube under different pre-strains were predicted and discussed. The results show that the limit strains of the tube are obviously path-dependent, and the M-K+DF2012 model can reasonably capture the limit strains of the tube under both linear and nonlinear loading paths. The predicted ε-FLD shows a strong dependence on the pre-strain, while the predicted PEPS-FLD is weakly strain path-dependent and almost path-independent on the right-hand side for the AA6061 tube.

A Homogeneous Anisotropic Hardening Model in Plane Stress State for Sheet Metal under Nonlinear Loading Paths.

In sheet metal forming, the material is usually subjected to a complex nonlinear loading process, and the anisotropic hardening behavior of the material must be considered in order to accurately predict the deformation of the sheet. In recent years, the homogeneous anisotropic hardening (HAH) model has been applied in the simulation of sheet metal forming. However, the existing HAH model is established in the second-order stress deviator space, which makes the calculation complicated and costly, especially for a plane stress problem such as sheet metal forming. In an attempt to reduce the computational cost, an HAH model in plane stress state is proposed, and called the HAH-2d model in this paper. In the HAH-2d model, both the stress vector and microstructure vector contain only three in-plane components, so the calculation is significantly simplified. The characteristics of the model under typical nonlinear loading paths are analyzed. Additionally, the feasibility of the model is verified by the stress-strain responses of DP780 and EDDQ steel sheets under different two-step uniaxial tension tests. The results show that the HAH-2d model can reasonably reflect the Bauschinger effect and the permanent softening effect in reverse loading, and the latent hardening effect in cross loading, while the predictive accuracy for cross-loading softening remains to be improved. In the future, the HAH-2d model can be further modified to describe more anisotropic hardening behaviors and applied to numerical simulations.

Crystal Plasticity Modeling of Al Alloy under Linear and Non-Linear Loading

The crystal plasticity (CP) model is widely used in many applications to link microstructure and mechanical properties. There are varying CP constitutive laws with phenomenological or physical-based formulation to cover a large range of loading conditions. In order to predict the deformation behavior of an Al alloy during the sheet metal forming process with either linear or non-linear strain path, both phenomenological and physical-based CP constitutive laws have been chosen, and the prediction performance of both models is compared. For the linear loading condition, the uniaxial tensile tests are performed on the smooth-dog-bone (SDB) specimens along rolling and transverse directions (RD/TD). The non-linear strain path is achieved by the Marciniak testing followed by uniaxial tension. In the first stage, the Marciniak testing is performed under the stress states of RD-uniaxial, plane strain, and biaxial tension. After being loaded to a certain strain level, mini-SDB specimens are cut along RD and TD from the uniform deformation region and reloaded under RD-uniaxial tension. The digital image correlation (DIC) technique is employed to measure the strain during testing. The electron backscatter diffraction (EBSD) technique is used to characterize the initial microstructure as well as the microstructure evolution of the specimens after the first stage loading in the non-linear strain path. A phenomenological power law and a dislocation-density-based hardening law have been employed in this study. The parameters are calibrated based on the flow curve of the RD uniaxial tension. The model performance is validated by stress–strain response under all the rest loading conditions including the non-linear loading path.

Deep learning based phase transformation model for the prediction of microstructure and mechanical properties of hot-stamped parts

Hot stamping of boron steel is an effective way for weight reduction of automobiles parts, but suffers from the loss of mechanical properties due to incomplete martensitic transformation. Different from general heat treatment, phase transformation in hot stamping relates to both thermal loading path and deformation history. Unfortunately, the existing models have difficulty in capturing the history-dependent phase transformation behavior of boron steel under hot stamping conditions. In the present work, the gated recurrent unit (GRU) are combined with fully connected neural network (FCNN) to capture the coupling effect of both nonlinear strain history and thermal loading path on phase transformation. A unified deep-learning based phase transformation (DLPT) model is developed to integrate both diffusive and diffusionless transformation to cover all feasible thermo-mechanical loading path in hot stamping process. A hybrid driven thermo-mechanical-metallurgical (TMM) framework is developed by integrating the DLPT model into general TMM framework, which providing a novel approach for two-scale simulation of hot stamping process. A comprehensive database of phase transformation is constructed for model training based on well-designed thermo-mechanical loading experiments to cover hot stamping conditions, and the optimal topology of neural network in DLPT model is obtained by training on this database. The accuracy and reliability of DLPT model is systematically demonstrated by dieless hot V-bending and hot stamping of T-shaped parts besides CCT and DCCT tests. The net effect of local plastic strain on subsequent phase transformation is clarified with accuracy by dieless V-bending with the removal of disturbing factors. Non-uniform distribution in hardness of the final hot stamped parts can be predicted with the DLPT model successfully, which is better in accuracy than in case of working with the conventional model.

Effect of non-linear tension-compression loading reversal on the hardening behavior and initiation fracture strain of a cold-rolled TRIP780 steel sheet

This work examines the influences of non-linear pre-forming paths on the ductile behavior and damage evolution of a cold-rolled transformation-induced plasticity (TRIP) steel. Two non-linear pre-forming paths, i.e. tension followed by compression (TC) and compression followed by tension (CT), were designed to reveal the Bauschinger effect and fracture ductility. The experimental results demonstrated that the fracture strain was strongly affected by the loading path and magnitude of pre-forming. A single-surface plasticity model accounting for a non-associated Hill’ 48 plastic flow potential was developed by combining a Voce-Swift isotropic hardening law and a mixed non-linear kinematic (NLK) hardening rule. The experimental strain-stress curves of pre-formed specimens were utilized to calibrate the model parameters. The proposed model well predicted the hardening anisotropy, Bauschinger effect, permanent softening and transient behavior of specimens undergoing TC and CT pre-forming strategies. In addition, the influence of non-linear TC and CT pre-forming loading paths on the initiation fracture strain were modeled and investigated through a Hosford-Coulomb fracture criterion. The results showed that the proposed non-associated and mixed hardening model can describe the hardening behavior under TC and CT non-linear deformation. It is also found that the Hosford-Coulomb fracture model is capable to provide reasonable prediction on the fracture strain under different non-linear pre-forming loading paths even though a larger scatter for the TC deformation. These results also confirm that the ductility of the TRIP780 increases under a TC or CT non-linear pre-forming.

Investigation on the forming limit diagram of AA5754-O alloy by considering strain hardening model, strain path, and through-thickness normal stress

This study clarified the effects of the chosen hardening model, through-thickness normal stress, and non-linear strain path on the precision of forming limit diagram (FLD). The Marciniak-Kuczynski (M-K) instability theory was employed to calculate the FLD of AA5754-O based on the different hardening models. Swift, Voce, modified Voce, Kim-Tuan, and modified Kim-Tuan hardening models were adopted to predict the forming limits, and their constant parameters were presented. Predicted FLDs indicate that modified Kim-Tuan has a better agreement with experimental data than other models. After that the influences of through-thickness normal stress on the forming limit diagram were examined. The three-dimensional stress state was converted to plane-stress condition according to the assumption that hydrostatic stress is ineffective on the plastic deformation. The results show that formability improves when normal stress increases. However, the effect of the normal stress on FLD for various hardening models is different. Due to the complexity of sheet metal forming, the strain path of the material may change during the deformation process. The results indicated that strain path has a significant effect on the FLD, and the path dependence of the forming limit stress diagram (FLSD) will be examined in detail. Finally, the forming limits were computed to analyze the effect of the normal stress on the non-linear loading path deformation. It was observed that by increasing the amount of pre-strain, the effect of the normal stress on the formability will diminish. Moreover, results showed that normal stress will affect the path dependence of the FLSD.

Neural network model predicting forming limits for Bi-linear strain paths

The Forming Limit Curve (FLC) describes the maximum in-plane strains that can be applied to sheet metal before necking initiates. It is well known that the FLC depends on the yield locus, the hardening behavior, strain rate, temperature, superposed out-of-plane stresses and the shape of the loading path. There is an infinite number of possible paths for reaching a particular state of strain which makes the formulation of simple analytical models predicting the effect of non-linear loading paths on the FLC particularly challenging. Here, it is demonstrated that a neural network model is able to provide a computationally-efficient approach to describe the effect of non-proportional loading. A finite-element-based Marciniak-Kuczynski model is developed to characterize the forming limits of a DP780 steel through virtual experiments. Subsequently, a large data base is generated to serve as a basis for machine learning. Using a standard backpropagation algorithm with Bayesian regularization, a compact neural network model is trained and tested based on the results from virtual experiments. Even though the experimental data base includes only a finite number of random loading paths, the results show that the neural network model is able to predict the FLCs for a given proportional pre-straining history.

Modeling of ductile damage evolution in roll forming of U-channel sections

Abstract In this study, the modified Mohr-Coulomb (MMC) ductile fracture criterion was employed to investigate the ductile fracture during the roll forming process of the AA6061-T6 aluminum alloy. To this aim, the MMC fracture criterion was calibrated based on the tension tests and was integrated into the commercial finite element (FE) Abaqus/Explicit using an appropriate user subroutine. The experimental procedures were designed based on two distinct forming strategies, namely single-stage and multi-stage roll forming processes. The results indicate that the calibrated fracture model based on tension tests is capable to predict the fracture initiation in the single-stage roll forming process at an error rate of 4.47%. Meanwhile, the numerically predicted fracture initiation during the multi-stage roll forming process is inconsistent with the experiments. Hence, a nonlinear damage accumulation rule together with the additional test (multi-stage L-bending) was utilized to predict the onset of fracture throughout the multi-stage roll forming process, considering the effect of deformation mechanics and nonlinear loading path during the process. The results reveal that the fracture onset during the multi-stage roll forming process can be accurately predicted when the MMC fracture criterion with the nonlinear damage accumulation rule is used in the numerical fracture model (with an error of about 9%).

The development of a novel forming limit diagram under nonlinear loading paths in tube hydroforming

The loading paths of tubular parts containing the characteristic of variable diameter during hydroforming are nonlinear and similar to the loading paths during tube hydro-bulging tests. Therefore, the forming limit determined by hydro-bulging tests is very suitable for guiding the forming of this kind of parts. However, traditional forming limit diagram (FLD) cannot present the nonlinear loading paths. In this paper, a theoretical analysis model for tube hydro-bulging tests is developed first. And then the stress paths and strain paths during bulging with fixed-ends and free-ends are analyzed through the theoretical model and finite element simulations. The results show that as the length-diameter ratio of the bulging zone ranges from 1.0 to 3.0, stress paths and strain paths are obviously nonlinear. For hydro-bulging with fixed-ends, the axial-hoop stress ratio changes gradually from 0.5 towards 1.0, and the strain state changes from plane strain state towards equal-biaxial tension state. For hydro-bulging with free-ends, the stress ratio changes gradually from 0 towards 0.5, and the strain state changes from simple tension state towards plane strain state. In order to give a better characterization of the forming limit of tubes under nonlinear loading paths, a novel FLD combining the geometric parameter of the tubular part is proposed. Two FLDs of an AA6061 aluminum alloy seamless tube under different end conditions are determined by experiments. The results show that both in hydro-bulging with fixed-ends and with free-ends, a higher forming limit is expected when using a smaller length-diameter ratio. As the new FLD can give direct presentation of limit strains together with the end condition and the length-diameter ratio in tube hydroforming, it can be used directly for the analysis and optimization of actual tube hydroforming processes, instead of the simple evaluation of the formability of raw materials.

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.

Simulating the temporal change of the active response of arteries by finite elements with high-order time-integrators

A new numerical method to model the active response of arteries is proposed. Vasoconstrictors and vasodilators in the bloodstream diffuse from the lumen into the arterial wall through the intima and cause the smooth muscle cells, mostly in the media, to contract. We combine the diffusion process with the mechanical model in Yosibash and Priel (Comput Methods Appl Mech Eng 237–240:51–66, 2012). Finite element computations of the fully coupled field problem using time-adaptive, high-order time-integration methods based on diagonally-implicit Runge–Kutta methods are investigated with respect to their convergence behavior for linear and non-linear loading paths. Since the blood pressure is periodic, highly non-linear external loading path, the step-size estimation has to be adapted to minimize step-size rejections. An example of an artery analysis that illustrates the advantage of the proposed time-adaptive scheme is provided.

Determination of forming limits of TWIP steel sheet using linear and nonlinear strain paths

.The main goal of the present paper is investigation of the mechanical behaviour of high manganese content steel sheet which is well known in the literature as Twinning-Induced Plasticity (TWIP) under linear and nonlinear loading path. Contrary to Nakajima experiment, cruciform specimen under biaxial loading can provide linear and nonlinear strain path. This work aims to determine the numerical Forming Limit Curve (FLC) for TWIP sheet using cruciform specimen under biaxial loading. For this purpose, uniaxial tensile tests are performed on standard dog-bone samples which are extracted from rolling, transversal and diagonal directions to identify the flow curve as well as Lankford ratios. Data are used as input for numerical simulation of cruciform specimen. Results show the applicability of cruciform specimen for linear and nonlinear strain paths.

Truss-based nonlinear mechanical analysis for origami structures exhibiting bifurcation and limit point instabilities

Origami, the art of paper folding, is a technologically transformative art with applications at many length scales from material microstructure design to space deployable structures. The transformation of a two-dimensional fold pattern to a three-dimensional structure makes origami a practical basis for deployability, light-weight materials, and self-actuation. In order to predict the complex motions of origami structures including path traversal of instabilities, accurate and efficient modeling techniques are required. While many approaches consider rigid facets and/or linear approximations to rotations, a truss and hinge finite element method is presented here within a global coordinate system framework to accurately capture the geometric nonlinearities while allowing for small to moderate facet deformation. Accurately capturing these modes of deformation is critical toward understanding the elastic energetic states required for design and analysis of multistable origami structures and mechanical metamaterials. Particular formulation developments that we address to solve origami mechanics problems include formulation of a continuous and differentiable fold angle, strategy for selecting fold path off the flat state, and integration of common numerical approaches important for origami (large rotation formulation, enforcement of periodic boundary conditions for tessellations, arc-length continuation method for solving highly nonlinear loading paths, and a perturbation strategy for handling bifurcations). After formulating this truss finite element method, we verify our modified finite element method on well-studied structures. The waterbomb fold pattern is studied, and the interplay of the stretching and folding energies are considered toward design of a bistable structure that exhibits a fold based snap-through instability. This study is followed by analysis of a cylindrical network of waterbombs (axial periodicity) that exhibits a stretch driven instability. Lastly, a tessellated square twist pattern, exhibiting multiple bifurcations off the flat state is presented that employs all aspects of our numerical method.

Plastic flow of thin-walled tubes under nonlinear tension-torsion loading paths and an improved pseudo-corner model

The elastoplastic deformation behaviors of thin-walled tubes made of pure aluminum and steel were measured under various tension–torsion combined loadings. The ratio between the displacement and rotation of a grip was held constant for the linear-loading experiment. In the nonlinear-loading experiments, a specimen was subjected to uniaxial tension followed by the simultaneous application of tension and torsion. It was determined that the associated flow rule predicts the plastic flow behaviors observed in the experiments with sufficient accuracy, provided that the specimen was subjected to linear loadings. Meanwhile, under nonlinear loadings, the plastic flow behaviors were markedly dissimilar to those in the linear loadings, and the direction of plastic flow rotated toward the direction of the stress/strain rate. By analyzing the experimental results, a linear relationship between the plastic flow direction and strain rate direction was determined. Therefore, the experimental data demonstrated that the stress state and the strain rate direction are essential parameters characterizing a plastic flow rule. Eventually, a pseudo-corner model capable of reproducing experimentally observed plastic deformation behavior was proposed.

Hot stamping of AA6082 tailor welded blanks: Experiments and knowledge-based cloud – finite element (KBC-FE) simulation

A novel hot stamping technique known as ‘Solution Heat treatment, Forming and in-die Quenching (HFQ®)’ was employed to manufacture lightweight structural components from AA6082 tailor-welded blanks (TWBs) of different thickness combinations: 1.5–1.5 and 2.0–1.0mm. A finite element (FE) model was built to study the deformation characteristics during the hot stamping process. The FE model was successfully validated by comparing simulation results with experimental ones. Subsequently, the verified simulation results were analysed through a novel multi-objective FE platform known as ‘Knowledge-Based Cloud – Finite Element (KBC-FE)’. KBC-FE operates in a cloud environment and offers various advanced unique functions via functional modules. The ‘formability’ module was implemented in the current study to predict the limiting dome height and failure mode during the hot stamping process. Good agreements were achieved between the predicted and experimental results, from which studies were extended to predict the forming features of 2.0–1.5mm TWBs. The ‘formability’ module has successfully captured the complex nature of a hot stamping process, featuring a non-isothermal and non-linear loading path. The formability of TWBs was found to be dependent on forming speed and blank thickness, out of which the latter has a dominant effect.

Explicit dynamic analysis of sheet metal forming processes using linear prismatic and hexahedral solid-shell elements

PurposeThe purpose of this paper is to propose two linear solid-shell finite elements, a six-node prismatic element denoted SHB6-EXP and an eight-node hexahedral element denoted SHB8PS-EXP, for the three-dimensional modeling of thin structures in the context of explicit dynamic analysis.Design/methodology/approachThese two linear solid-shell elements are formulated based on a purely three-dimensional (3D) approach, with displacements as the only degrees of freedom. To prevent various locking phenomena, a reduced-integration scheme is used along with the assumed-strain method. The resulting formulations are computationally efficient, as only a single layer of elements with an arbitrary number of through-thickness integration points is required to model 3D thin structures.FindingsVia the VUEL user-element subroutines, the performance of these elements is assessed through a set of selective and representative dynamic elastoplastic benchmark tests, impact-type problems and deep drawing processes involving complex non-linear loading paths, anisotropic plasticity and double-sided contact. The obtained numerical results demonstrate good performance of the SHB-EXP elements in the modeling of 3D thin structures, with only a single element layer and few integration points in the thickness direction.Originality/valueThe extension of the SHB-EXP solid-shell formulations to large-strain anisotropic plasticity enlarges their application range to a wide variety of dynamic elastoplastic problems and sheet metal forming simulations. All simulation results reveal that the numerical strategy adopted in this paper can efficiently prevent the various locking phenomena that commonly occur in the 3D modeling of thin structural problems.

A New Combined Isotropic, Kinematic and Cross Hardening Model for Advanced High Strength Steel under Non-Linear Strain Loading Path

A New Combined Isotropic, Kinematic and Cross Hardening Model for Advanced High Strength Steel under Non-Linear Strain Loading Path