Distortional Hardening Research Articles

Numerical simulation of sheet metal blanking based on fully coupled elastoplasticity-damage constitutive equations accounting for yield surface distortion-induced anisotropy

This paper deals with the numerical simulation of sheet metal blanking process based on fully coupled elastoplastic model accounting for the induced anisotropies due to the kinematic hardening and the yield surface distortion. The yield surface distortion is assumed to be controlled by the kinematic hardening leading additional extra hardening which enhances the predictive capabilities of the model. Series of finite element-based numerical simulations of blanking process with four kinds of assumed distortional hardening parameters have been conducted. Through the comparison between the experimentally observed responses and the numerically predicted ones with and without the yield surface distortion effect, the significance of the yield surface distortion-induced anisotropy on the estimation of the blanking edge quality has been investigated.

A comparative study between micro- and macro-mechanical constitutive models developed for complex loading scenarios

Constitutive models developed for simulating plastic response upon strain path changes are combined: 1) a macro-mechanical model based on anisotropic yield function, associated flow rule and distortional hardening using Homogeneous Anisotropic Hardening (HAH) approach; 2) a micro-mechanical model using self-consistent crystal plasticity in conjunction with crystallographic dislocation-density based hardening. The micro-mechanical model is employed to probe the yield surface in order to gain the insight required to construct empirical rules appropriate for the macro-mechanical model. Simulation results of the micro-mechanical model under various loading conditions involving strain path changes and different crystallographic textures are presented. The trends captured in the yield surface evolution predicted by the micro-mechanical model were used to validate and improve the empirical rules used in the HAH model.

Advanced constitutive modeling of AHSS sheets forapplication to springback prediction after U-draw double stamping process

The reduction of springback for a U-shaped channel using a double drawing process was investigated. In this test, the punch strokes of the 1st and 2nd stamping steps were controlled and each followed by unloading. The simulations were conducted using kinematic and distortional hardening models, which were implemented into a finite element (FE) code to describe the Bauschinger effect and its associated anisotropic hardening effects during strain path change. In addition to the usual mechanical characterization tests, in-plane compression- tension experiments were conducted on DP980 and TWIP980 to determine the constitutive parameters pertaining to load reversal. Experimental and FE simulated results of the channel shape were compared for both materials in order to understand the effect of anisotropic hardening under non-proportional loading on springback.

Evolutionary anisotropy and flow stress in advanced high strength steels under loading path changes

Chaboche kinematic hardening, Yoshida–Uemori, and homogenous anisotropic hardening (HAH) models were evaluated in terms of their prediction capability of flow stress and evolutionary r-value under loading path changes. The first two models are based on kinematic hardening, whereas the HAH model is based on distortional hardening. Continuous compression–tension (CT) tests with pre-strains for dual-phase 780 and transformation-induced plasticity (TRIP) 780 steel sheets were conducted. For the initial anisotropy described by the Yld2000-2d model, uniaxial tension tests for three material orientations and a biaxial test were performed. During the CT tests, both stress and r-value variations with plastic strain were measured. All the models were implemented into the finite element software ABAQUS using user material subroutines. Experimentally, both sheets exhibited typical anisotropic hardening features such as the Bauschinger effect, transient hardening, and long-range softening. The TRIP 780 exhibited small work-hardening stagnation. In addition, the evolutionary r-value during the second loading was significant. The three models could predict most of the features in the stress–strain curves during the second loading. However, only the HAH model could predict the experimentally observed r-value evolution. This evolutionary r-value predicted by the HAH model could be explained by distortion of the yield locus during the first loading and its rapid recovery toward the initial yield locus shape.

Evaluation of Springback for DP980 S Rail Using Anisotropic Hardening Models

The effect of anisotropic hardening models on springback of an S-rail part was investigated. Two advanced constitutive models based on distortional and kinematic hardening, which captured the Bauschinger effect, transient hardening, and permanent softening during strain path change, were implemented in a finite element (FE) code. In-plane compression–tension tests were performed to identify the model parameters. The springback of the S-rail after forming a 980 MPa dual-phase steel sheet sample was measured and analyzed using different hardening models. The comparison between experimental and FE results demonstrated that the advanced anisotropic hardening models, which are particularly suitable for non-proportional loading, significantly improved the springback prediction capability of an advanced high strength steel.

A generalized thermodynamically consistent distortional hardening model for Mg alloys

In the current work, a generalized distortional hardening model is developed by assuming each part of dissipation non-negative. More precisely, quadratic forms are assumed for dissipation and thermodynamical consistency is fulfilled naturally. In order to balance the contributions of isotropic and distortional hardening into dissipation, a weighting coefficient w is introduced in contrast to the work in Feigenbaum and Dafalias (2007). It is found that the saturation directions of distortional hardening are governed by this coefficient w, which is related with deformation modes. In particular, by setting the coefficient w = 2, the current generalized model recover to the one based on convex plastic potential, cf. Shi and Mosler (2013), and thus Armstrong–Frederick type distortional hardening law is obtained. The predictive ability of the current distortional hardening model is verified by comparison with experimental observations in three cases: stress-strain behaviours along different directions, biaxial tensile test and biaxial compression test. The evolution of normalized yield surfaces for the three cases is briefly discussed.

A particle-breakage critical state model for rockfill material

Particle breakage has a significant influence on the stress-strain and strength behavior of rockfill material. A breakage critical state theory (BCST) was proposed to describe the evolution of particle breakage. The breakage critical state line in the breakage critical state theory was correlated with the breakage factor, which was fundamentally different from that of the original critical state theory. A simple elastoplastic constitutive model was developed for rockfill in the frame of BCST. An associated flow rule was adopted in this model. Isotropic, contractive and distortional hardening rules were suggested in view of the particle breakage. It was observed that the proposed model could well represent the complex deformation behaviors of rockfill material, such as the strain hardening, post-peak strain softening, volumetric contraction, volumetric expansion, and particle breakage under different initial confining pressures.

Numerical Implementation of A Model With Directional Distortional Hardening

AbstractThe presented work outlines and tests a finite-element (FE) implementation of a simple directional distortional hardening model. The evolution equations of internal variables are based on the Armstrong–Frederick evanescent memory-type hardening rule, and the associative flow rule is adopted. The directional distortion of the yield function is governed by the contraction of the backstress tensor with the so-called unit radial tensor, and incorporates a fixed scalar distortional parameter. Therefore, the variety of possible shapes of predicted subsequent yield surfaces is limited. The tangent stiffness–radial corrector method with fine subincrementation is used. This implementation is verified by comparing numerical results with analytical solutions pertinent to proportional load cases and semianalytical solutions for nonproportional loadings. The template of the whole implementation is included in the Appendix.

Measurement and modeling of simple shear deformation under load reversal: Application to advanced high strength steels

In this paper, the stress–strain behavior under load reversal of advanced high strength steel (AHSS) sheet samples was measured using a modified simple shear (SS) apparatus. The forward–reverse loading behavior was characterized for three different grades of AHSS; namely, DP, TRIP and TWIP steels. For comparison purpose, compression–tension (CT) tests were also carried out for the same materials. For all the cases, a typical complex anisotropic hardening behavior, including the Bauschinger effect, transient strain hardening with high rate and permanent softening, was observed during load reversal. No premature localization and sheet buckling occurred in these experiments, which have been major technical hurdles in CT tests. For example, an engineering shear strain of over 40%, which corresponds to an effective strain of roughly 0.2, at reversal was achieved for DP980 although the uniform elongation of this material in uniaxial tension is only 5%. A recently developed distortional hardening model (HAH) was employed to reproduce the SS stress–strain curves. Using the coefficients determined with these SS data, the CT behavior was predicted with the HAH model and compared with experimental results. This complementary study indicated that the constitutive model determined from the SS flow curves satisfactorily reproduced the CT hardening behavior. As an application, finite element simulations of springback were carried out for 2D draw-bending of a strip sheet.

Constitutive Modeling of Magnesium Alloys with Distortional Hardening

Anisotropic mechanical behavior is one of the key factors restricting the processing procedure of magnesium alloys. This pronounced anisotropy, however, cannot be characterized by classical isotropic or kinematic hardening due to the constant shape of yield surfaces during plastic deformation. Therefore, the shape evolution of yield surfaces, also known as distortional hardening is the main way to capture the anisotropic behavior. Based on elasto-plasticity theory at finite strain, constitutive model with distortional hardening for Mg alloys is proposed. The thermodynamical consistency is proved. The anisotropic mechanical behavior of AZ31 sheet is demonstrated after material parameters calibration.

An elasto-plastic constitutive model for evolving asymmetric/anisotropic hardening behavior of AZ31B and ZEK100 magnesium alloy sheets considering monotonic and reverse loading paths

Deformation twinning and texture evolution in hexagonal close-packed (HCP) metals lead to evolving flow stress asymmetry/anisotropy and evolving plastic anisotropy. These phenomena cause a significant change in the shape of the yield surface with accumulated plastic deformation which cannot be modeled accurately with traditional hardening laws. In this paper, an anisotropic continuum-based plasticity model is proposed to capture the large strain cyclic hardening behavior of magnesium alloys. Key in the current formulation is the incorporation of distortional hardening to model the evolving asymmetric/anisotropic hardening response of magnesium alloys for both monotonic and reverse loading paths. The hardening behavior is classified into three deformation modes: Monotonic Loading [ML], Reverse Compression [RC], and Reverse Tension [RT]. The deformation modes correspond to the different loading regimes of the cyclic hardening curve. Specifically, the ML mode corresponds to the initial in-plane tension and the initial in-plane compression from the annealed state, the RC mode corresponds to the in-plane compression following previous tension and the RT mode corresponds to the in-plane tension following previous compression. Three separate hardening laws are used to define the hardening response within each deformation mode. Moreover, a multi-yield surface modeling approach is used where a CPB06 type anisotropic yield surface is assigned to each deformation mode. The evolution of the anisotropy coefficients involved in the expression of the yield function, is considered to model distortional hardening within each deformation mode. The evolving anisotropy parameters are found by minimizing the difference between the model predictions and the experiments, together with the interpolation technique proposed by Plunkett et al. (2006). The proposed model is calibrated using monotonic and reverse loading experimental data for AZ31B and ZEK100 magnesium alloys. A strain rate independent elasto-plastic formulation is used to implement the proposed constitutive model as a user material subroutine (UMAT) in the commercial finite element software LS-DYNA®. The predictions of the model are compared against the experimental monotonic and cyclic (CTC and TCT) flow stresses and r-values of AZ31B and ZEK100 sheets along different test directions. An excellent agreement is found between the simulated and experimental results.

An extended Modified Maximum Force Criterion for the prediction of localized necking under non-proportional loading

Strain localization is one of the main sources of failure in sheet forming processes. State of the art forming limit curves allow the prediction of localization for linear strain paths but fall short in case of non-proportional loading. The aim of this contribution is to revisit the Modified Maximum Force Criterion (MMFC) and extend it to accommodate distortional hardening models. This is accomplished by uncoupling its formulation from any particular yield function and thus enabling its use as a generic framework for the prediction of forming limits under arbitrary loading conditions. Furthermore a novel approach is proposed for considering strain rate sensitivity, which substantially improves the predictive capabilities of the model under plane strain tension conditions. The method is applied to steel and aluminum materials and the role of phenomena such as Bauschinger effect, latent hardening and cross-loading contraction on localization are discussed.

Isotropic to distortional hardening transition in metal plasticity

The present paper aims to discuss the transition from isotropic to distortional hardening behavior of metallic materials, based on the Homogeneous Anisotropic Hardening (HAH) model. Furthermore, the effect of yield locus distortion on the evolution of the strain increment, under the assumption of associated flow, is theoretically discussed and exemplified. Special cases, such as coaxial and orthogonal stress states, are analyzed to provide better insight into the model. Particular emphasis is put on the monotonic loading case, which is compared to isotropic hardening. Finally, the evolution equations of the state variables are examined and their properties are discussed.

Generalized distortional hardening in a thermomechanically coupled framework

AbstractIn many metals like aluminum the evolution of the microstructure leads to an evolution of the macroscopic yield surface. Clearly, isotropic and kinematic hardening models cannot capture this effect realistically. For that purpose, a new generalized distortional hardening model is proposed. This novel approach belongs to the class of so‐called generalized standard materials and is thermodynamically consistent. Although the approach is relatively simple, it shows several advantages like yield surface convexity and yield limit saturation. This novel model is embedded into a thermomechanically coupled finite strain framework. Several numerical simulations show the predictive capabilities. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Directional distortional hardening at large plastic deformations

This paper extents the directional distortional hardening model of Feigenbaum and Dafalias (2007) into the range of large plastic deformations. This model allows the yield surface to deform such that a region of high curvature develops approximately in the direction of loading and a region of flattening develops on the opposite side. To extend this model into large deformations and in order to ensure positive dissipation and objectivity, hardening rules are derived from thermodynamic conditions in terms of corotational rates. Since this model includes a fourth order tensor-valued hardening internal variable, the corotational rates for fourth order tensors are examined in this work employing the concept of plastic spin. Several choices for plastic spins are presented and used for the simulation of the response under simple shear loading up to 1000% strain.

On the thermodynamically consistent modeling of distortional hardening: A novel generalized framework

Many important physical effects of materials undergoing plasticity at the macroscale cannot be captured realistically by isotropic and kinematic hardening only. For instance, the evolution of the texture in polycrystals results macroscopically in a distorted yield surface. This paper deals with adequate hardening models for such a distortion. To be more precise, a novel general frame for finite strain plasticity models is elaborated. To the best knowledge of the authors, it is the first one combining the following features: (1) proof of thermodynamical consistency; (2) decomposition of distortional hardening into dynamic hardening (due to currently active dislocations) and latent hardening (due to currently inactive dislocations); (3) difference of the yield surface’s curvature in loading direction and in the opposite direction. The cornerstone of this model is a new plastic potential for the evolution equations governing distortional hardening. Although this type of hardening is characterized through a fourth-order tensor as internal variable, the structure of the aforementioned potential is surprisingly simple. Even though the final model is rather complex, it requires only few model parameters. For these parameters, in turn, physically sound bounds based on the convexity condition of the yield surface can be derived. Three different examples demonstrate the predictive capabilities of the novel framework.

Modeling the tension–compression asymmetric yield behavior of β-treated Zircaloy-4

Zirconium alloys such as Zircaloy-4 are used in nuclear applications due to adequate strength, ductility and resistance to radiation damage. Recent modeling efforts have focused on improvements to the predicted elastic–plastic response, complicated by the strong strength-differential (S-D) effects in HCP materials. This study develops a pressure-insensitive, continuum plasticity model, dependent on the second and third invariants of the stress deviator (J2 and J3), with an internal variable related to the plastic strain to describe the tension–compression asymmetry of a β-treated Zircaloy-4. Plastic deformation drives isotropic and distortional hardening of the non-Mises yield surface. The proposed plasticity model has been calibrated and validated using measured results from an experimental test program. Results show that the proposed model captures the complex elastic–plastic response observed in measured load–displacement and torque–rotation curves over a range of triaxiality and Lode parameter values.

Distortional hardening plasticity model for paperboard

A distortional hardening elasto-plastic model at finite strains suitable for modeling of orthotropic materials is presented. As a prototype material, paperboard is considered. An in-plane model is established. The model developed is motivated from non-proportional loading tests on paperboard where the paperboard is pre-strained in one direction and then loaded in the perpendicular direction. A softening effect is revealed in the pre-strained samples. The observed experimental findings cannot be accurately predicted by current models for paperboard. To be able to model the softening effects, a yield surface based on multiple hardening variables is introduced. It is shown that the model parameters can be obtained from simple uniaxial experiments. The model is implemented in a finite element framework which is used to illustrate the behavior of the model at some specific loading situations and is compared with strain fields obtained from Digital Image Correlation experiments.

Extension of strain-life equation for low-cycle fatigue of sheet metals using anisotropic yield criteria and distortional hardening model

The fatigue strain–life equation is in general applicable to isotropic materials. It was recently attempted to account for material anisotropy because of crystallographic texture in fatigue modelling. The proposed modification was limited to isotropic hardening. The present work is an extension of the previous work, wherein a general framework to model anisotropy using phenomenological yield criterion and anisotropic hardening is provided. Yld2004-18p yield criterion and the so-called homogenous anisotropic hardening model are used to demonstrate the anisotropic cyclic behaviour of low carbon steel. The proposed methodology can be utilized in applications including multiaxial fatigue modelling.

A New Model Describing Plastic Distortion Fully Coupled with Ductile Damage

The complex macroscopic mechanical behavior due to the polycrystalline texture evolution under complex loading paths cannot be efficiently captured by the classical isotropic and kinematic hardening constitutive equations. The complex physical interplay of different directional texture leads to an evolving shape of the yield surface due to induced anisotropy, which can be described by distortional hardening. In the present work, induced anisotropy is explicitly developed using both motion and distortion of the yield and the plastic potential surfaces. A modified François (2001) model based on “egg-shaped” subsequent yield surfaces is developed. This model is controlled by three material parameters allowing the shape change in two orthogonal directions of kinematic hardening. Two surfaces (the yield function and the plastic potential) in the framework of non-associated plasticity are used together with Continuum Damage Mechanics (CDM) framework in order to develop an elastoplastic model fully coupled with ductile damage. This model is implemented into ABAQUS/Explicit finite element code through the user subroutine/VUMAT. The capability of the developed model is briefly outlined through the comparison with Khan's experimental results with high work hardening alloy A1100.