Distortional Hardening Model Research Articles

Thermodynamically consistent constitutive modeling of distortional hardening for wrought Mg alloys under non-proportional loading

Strong anisotropic mechanical behaviour due to texture is one of the key factors restricting the processing procedure of wrought magnesium alloys. This pronounced anisotropy, however, cannot be characterized by only classical isotropic or kinematic hardening due to the constant shape of yield surface during plastic deformation. Therefore, the shape evolution of yield surface, also known as distortional hardening is the main way to capture the anisotropic behaviour. 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 behaviour of AZ31 sheet under non-proportional loading is demonstrated after model parameters identification.

On the numerical implementation of thermomechanically coupled distortional hardening

In contrast to the by now classic isotropic and kinematic hardening, the more general framework of distortional hardening characterised by an evolution of the yield surface's shape can capture the effect of texture evolution on the macroscopic response. Although different distortional hardening models can indeed be found in the literature, efficient numerical implementations of such models are still missing. This statement proves particularly true within a thermomechanically coupled framework which is important for most technologically relevant processes – such as for deep drawing. Accordingly, this paper deals with an efficient finite element formulation for distortional hardening within a thermomechanically coupled framework. As a first step towards this target, the recently advocated isothermal distortional hardening framework Shi et al. (2014) is extended to the thermomechanically coupled setting. In order to avoid an over-estimation of the temperature increase due to plastic deformation, the initial yield stress is decomposed into a classic dissipative part and a non-classic energetic part. By doing so, the restrictions imposed by thermodynamical principles are fulfilled and simultaneously realistic temperature predictions are obtained. For the resulting model, an efficient numerical implementation is proposed. By developing a suitable time integration scheme for the evolution equations of the fourth-order tensor describing the distortional hardening, a return-mapping scheme for updating the internal variables is derived which shows the same numerical complexity as a return-mapping scheme for purely isotropic hardening. This efficient return-mapping scheme is finally incorporated into a thermomechanically coupled finite element formulation, and the resulting set of equations is fully implicitly and monolithically solved by means of a Newton-type iteration. Several numerical complex examples demonstrate the capabilities of the distortional hardening model as well as the robustness and efficiency of the numerical formulation.

Loading path dependent distortional hardening of Mg alloys: Experimental investigation and constitutive modeling

In this work, the loading path dependent distortional evolution of yield surfaces is probed experimentally on an extruded AZ31 Mg alloy. More explicitly, the initial yield surface, subsequent yield surfaces with pre-tension and with pre-tension-torsion are observed; the loading path effect on the anisotropy is discussed based on the underlying deformation mechanism in terms of microstructure evolution. In addition, experimental comparison between the yield surface and the Equivalent Plastic Work Contour (EPWC) is performed with a shared stress state. Next, a thermodynamically consistent directional distortional hardening model is employed to capture the anisotropic mechanical behavior. The directional distortional evolution of the normalized yield surfaces under non-proportional loading paths is predicted for extruded AZ31 Mg alloy.

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.

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.

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.

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.

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.

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.

Continuous strain path change simulations for sheet metal

The plastic behavior of a material undergoing continuous strain path changes, from plane strain to simple shear, was simulated for a mild steel. A distortional hardening model was employed to capture the Bauschinger effect and latent hardening. The simulations were conducted by direct application of the constitutive equations with simple boundary conditions and by using a finite element code in which the distortional hardening model was previously implemented. Depending on the rate of path change, the simulations resulted in different combinations of plastic responses including the Bauschinger effect, transient hardening, flow stress overshooting and strain hardening stagnation. The simulated stress–strain curves were found to be in good agreement with experimental results published by van Riel and van den Boogaard (2007).

On the macroscopic description of yield surface evolution by means of distortional hardening models: Application to magnesium

Texture evolution in polycrystals due to rotation of the atomic lattice in single grains results in a complex macroscopic mechanical behavior which cannot be reasonably captured only by classical isotropic or kinematic hardening in general. More precisely and focusing on standard rate-independent plasticity theory, the complex interplay at the microscale of a polycrystal leads to an evolving macroscopic anisotropy of the yield surface, also known as distortional or differential hardening. This effect is of utmost importance, if non-radial loading paths such as those associated with forming processes are to be numerically analyzed. In the present paper, different existing distortional hardening models are critically reviewed. For a better comparison, they are reformulated into the modern framework of hyperelastoplasticity, and the same objective time derivative is applied to all evolution equations. Furthermore, since the original models are based on a Hill-type yield function not accounting for the strength differential effect as observed in hcp metals such as magnesium, respective generalizations are also discussed. It is shown that only one of the resulting models can fulfill the second law of thermodynamics. That model predicts a high curvature of the yield function in loading direction, while the opposite region of the yield function is rather flat. Indeed, such a response can be observed for some materials such as aluminum alloys. In the case of magnesium, however, that does not seem to be true. Therefore, a novel constitutive model is presented. Its underlying structure is comparably simple and the model is thermodynamically consistent. Conceptually, distortional hardening is described by an Armstrong–Frederick-type evolution equation. The predictive capabilities of the final model are demonstrated by comparisons to experimentally measured data.

Thermodynamically and variationally consistent modeling of distortional hardening: application to magnesium

AbstractTo capture the complex elastoplastic response of many materials, classical isotropic and kinematic hardening alone are often not sufficient. Typical phenomena which cannot be predicted by the aforementioned hardening models include, among others, cross hardening or more generally, the distortion of the yield function. However, such phenomena do play an important role in several applications in particular, for non‐radial loading paths. Thus, they usually cannot be ignored. In the present contribution, a novel macroscopic model capturing all such effects is proposed. In contrast to most of the existing models in the literature, it is strictly derived from thermodynamical arguments. Furthermore, it is the first macroscopic model including distortional hardening which is also variationally consistent. More explicitly, all state variables follow naturally from energy minimization within advocated framework. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Comparison of different distortional hardening models suitable for the analysis of magnesium

AbstractHCP metals such as magnesium are characterized by a strong interplay between dislocation slip and deformation‐induced twinning. These micromechanical processes result in a complex macroscopic behavior. More precisely, in addition to classical isotropic and kinematic hardening, the shape of the macroscopic yield function changes during deformation as well. This effect which is frequently referred to as distortional hardening is particularly pronounced in case of non‐radial loading paths typical for most forming processes. Consequently, a physically sound distortional hardening is of utmost importance for several technically relevant applications. In the present contribution, three different of such enhanced hardening models are critically analyzed. Focus is on the modeling of magnesium. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A finite deformation model for evolving flow anisotropy with distortional hardening including experimental validation

The paper presents a new constitutive model for anisotropic metal plasticity that takes into account the expansion or contraction (isotropic hardening), translation (kinematic hardening) and change of shape (distortional hardening) of the yield surface. The experimentally observed region of high curvature (“nose”) on the yield surface in the loading direction and flattened shape in the reverse loading direction is modeled here by means of the concept of directional distortional hardening. The modeling of directional distortional hardening is accomplished by means of an evolving fourth-order tensor. The applicability of the model is illustrated by fitting experimental subsequent yield surfaces at finite plastic deformation. The simulation results were compared with test data for aluminum low and high work hardening alloys. In this introductory paper we will consider only the proportional loading case.

Modelling mechanical consequences of erosion

The process of internal erosion of fine particles from a soil progressively narrows the grading of the soil. Development of a continuum model for the mechanical consequences of erosion has taken inspiration from two-dimensional discrete element analyses of assemblies of circular discs of various gradings. The process of erosion has been described in these analyses by progressively removing the finer particles while maintaining the sample under constant stresses. The removal of particles produces an increase in specific volume because the volume of solid decreases and the volume of void increases, but the looser structure compresses under the external stresses. Parallel continuum modelling takes an existing distortional hardening model, in which the critical state line plays a central role, and adds a volumetric deformation mechanism to describe the observed compression. The discrete element modelling analyses and other results remind us that the first-order effect of narrowing grading is the raising of the critical state line in the compression plane. The model is closely similar to a model for soils with breakable particles, incorporating the same link between grading and critical states.

Convexity of Yield Surface with Directional Distortional Hardening Rules

The present paper examines the convexity of the yield surface in the directional distortional hardening models by Feigenbaum and Dafalias. In these models anisotropy develops through kinematic and directional distortional hardening, supplemented by the classical isotropic hardening, and the associative flow rule is used. However, the issue of convexity, which naturally arises due to the distortion of the yield surface, was not fully addressed. The present paper derives the necessary and sufficient conditions to ensure convexity of the yield surface for the simpler Feigenbaum and Dafalias models, but it is not as straightforward to derive corresponding conditions for convexity of the Feigenbaum and Dafalias model version which contains an evolving fourth-order tensor. In this case convexity will be addressed first in general and then at the limit state for which simple restrictions on the material constants to ensure convexity are derived. Numerical examples will show that some of the yield surfaces simulated in the original Feigenbaum and Dafalias publication will not stay convex if loaded beyond what was done in these publications. Therefore the material constants for these cases are recalibrated based on the derived relations for satisfaction of the convexity requirement, and the fitting of the yield surfaces is repeated with the new set of constants and compared with the previous case.