Overview of the theoretical relations between necking and strain localization criteria.

Many criteria have been developed during last decades to predict diffuse or localized necking and shear banding. The lack of confrontation of these models with each other on relevant applications makes their choice difficult for the designer. It is proposed to reformulate these plastic instability criteria in an unified framework, to compare their theoretical bases to establish links between them and then to highlighten their limitations. In the case of diffuse necking, a comparison is made between the criteria based on bifurcation analysis and on those based on maximum force principle for elastic-plastic materials. In the case of localized modes, it is shown that the predictions of the Marciniak – Kuczynski approach, based on a multizone model, tend to those of the loss of ellipticity criterion when the initial defect size tends to zero (no initial defect introduced). In the case of elasto-viscoplastic behavior, an approach based on a linear stability analysis is mentioned

Investigation of some localization criteria and their relevance to prediction of forming limit diagrams

Because prevention of forming defects has become one of the major industrial challenges, various experimental and theoretical approaches have been developed to predict sheet metal formability. The main theoretical models can be summarized as: the maximum force principle, according to which necking is associated with the maximum load in a uniaxial tensile test; Marciniak-Kuczyński (M-K) two-zone analysis, based on an initial thickness defect in the sheet; bifurcation theory, predicting diffuse necking with general bifurcation criterion or localized modes corresponding to loss of ellipticity; and more recently, linear stability analysis by means of linearization of perturbed equilibrium equations. Considering this variety of models, a careful comparison of numerical Forming Limit Diagrams (FLDs) along with in-depth understanding of their theoretical foundations is required to help select relevant localization criteria. In this paper, the theoretical bases of M-K and Rice's criteria are first reviewed, which are then applied to steels modeled by elasto-plastic constitutive equations coupled with damage. It is shown that the FLDs obtained with the M-K model tend to those yielded by Rice's criterion in the limit of vanishing initial imperfections.ANR Formef & Région Lorrain

Modélisation de critères de localisation : prédiction de l’orientation des bandes de localisation et tracé de courbes limites de formage

Différentes approches de modélisations ont été proposées pour prédire les phénomènes de localisation lors d’opérations de mise en forme de tôles métalliques par emboutissage. Des exemples de ces approches sont le principe de force maximum, les critères à deux zones et les analyses de bifurcation ou de stabilité. En l’absence de consensus relatif à leur choix et à l’étendue de leur domaine de validité, la présente étude se propose de formuler ces modélisations dans un cadre unifié, de mettre en évidence les liens théoriques pouvant exister entre elles et ensuite de valider ces observations par le tracé de courbes limites de formage et par la comparaison de l’orientation des bandes de localisation.Various modeling approaches have been proposed to predict localization phenomena during forming processes of metal sheets. Examples of these are the Maximum Force principle, two-zone criteria, and bifurcation analysis or stability studies. The main objective of this presentation is to implement these different localization formulations into a general unified framework, to point out possible theoretical links between them, and to validate these observations through numerical applications such as forming limit diagrams and the corresponding orientation of the localization bands.CNRS & Région Lorrain

Prédictions de courbes limites de formage avec les critères de localisation de Marciniak-Kuczynski et de Hill

Lors des opérations de mise en forme de structures minces, des phénomènes d’instabilité peuvent se produire. En emboutissage, ces instabilités peuvent se manifester sous différentes formes comme par exemple la striction ou encore la localisation des déformations plastiques sous forme de bandes de cisaillement. Toutes ces instabilités conduisent soit à des défauts lors des opérations de mise en forme, ce qui se traduit par la mise au rebut de pièces, soit à l’adoption de méthodes de dimensionnement trop conservatives. La maîtrise de critères de prédiction des instabilités est donc un enjeu majeur dans le domaine concurrentiel des industries de la mise en forme. Depuis la seconde moitié du 20ème siècle, de nombreux travaux expérimentaux, théoriques ou numériques ont été menés afin de comprendre et prévoir ces phénomènes d’instabilité. Les travaux expérimentaux ont permis de déterminer le domaine de formabilité pour différentes classes de matériaux avec des Courbes Limites de Formage (CLF) (Keeler (1965), Goodwin (1968)). Différents travaux théoriques et numériques ont cherché à déterminer le domaine de formabilité tout en s’affranchissant des contraintes expérimentales. Parmi ces travaux, les principaux sont basés sur le principe de force maximum (Swift (1952), Hill (1952), Hora et al. (1996)), d’analyse de bifurcation (Hill (1958), Stören et Rice (1975), Rice (1976), Valanis (1989), Bigoni et Hueckel (1991)), d’analyse par la méthode de perturbation (Dudzinski et Molinari (1991), Benallal (1997), Keryvin (1999)) ou sur l’existence d’un défaut initial (Marciniak et Kuczyński (1967), Hutchinson et Neale (1978), Kuroda et Tvergaard (2000)). Cet article porte sur l’étude des critères de Marciniak-Kuczyński ou M-K et de Hill. Un algorithme général du critère M-K, basé sur une méthode de résolution explicite, le modèle de Hill et un modèle de comportement phénoménologique sont présentés puis appliqués au tracé de courbes limites de formage.CNRS & Région Lorrain

Investigation of some localization criteria and their relevance to prediction of forming limit diagrams

Because prevention of forming defects has become one of the major industrial challenges, various experimental and theoretical approaches have been developed to predict sheet metal formability. The main theoretical models can be summarized as: the maximum force principle, according to which necking is associated with the maximum load in a uniaxial tensile test; Marciniak-Kuczyński (M-K) two-zone analysis, based on an initial thickness defect in the sheet; bifurcation theory, predicting diffuse necking with general bifurcation criterion or localized modes corresponding to loss of ellipticity; and more recently, linear stability analysis by means of linearization of perturbed equilibrium equations. Considering this variety of models, a careful comparison of numerical Forming Limit Diagrams (FLDs) along with in-depth understanding of their theoretical foundations is required to help select relevant localization criteria. In this paper, the theoretical bases of M-K and Rice's criteria are first reviewed, which are then applied to steels modeled by elasto-plastic constitutive equations coupled with damage. It is shown that the FLDs obtained with the M-K model tend to those yielded by Rice's criterion in the limit of vanishing initial imperfections.ANR Formef & Région Lorrain

Formability prediction of damageable elastic-viscoplastic media by a material stability analysis based on a linear perturbation method

The originality of the approach used here is to present stability conditions of metallic materials modeled by general phenomenological laws and to apply them to the study of the formability of thin metallic sheets. The constitutive equations are based on an elastic-viscoplastic model coupled with a classical isotropic damage model. Various classical isotropic and kinematic hardening models are considered here to reproduce the hardening behavior and the induced anisotropy, while the initial anisotropy of metallic rolled sheets is introduced by Hill’48 anisotropic yield function. Softening is taken into account by coupling the constitutive model with a scalar damage variable. This phenomenological variable is defined within the framework of continuum damage mechanics and represents the degradation of the elastic properties during the forming operations. The stability of the constitutive damageable elastic-viscoplastic system is then sought by means of linear perturbation approach. Application of this method to a mild steel and a dual phase steel leads to the determination of their formability limit diagrams. On the one hand, FLDs obtained with Rice’s criterion and with this stability criterion illustrate the theoretical relation between these criteria and on the other hand, a parametric study shows the influence of governing parameters of hardening, damage and strain rate sensitivity.CNRS & Région Lorrain

Formability prediction of damageable elastic-viscoplastic media by a material stability analysis based on a linear perturbation method

The originality of the approach used here is to present stability conditions of metallic materials modeled by general phenomenological laws and to apply them to the study of the formability of thin metallic sheets. The constitutive equations are based on an elastic-viscoplastic model coupled with a classical isotropic damage model. Various classical isotropic and kinematic hardening models are considered here to reproduce the hardening behavior and the induced anisotropy, while the initial anisotropy of metallic rolled sheets is introduced by Hill’48 anisotropic yield function. Softening is taken into account by coupling the constitutive model with a scalar damage variable. This phenomenological variable is defined within the framework of continuum damage mechanics and represents the degradation of the elastic properties during the forming operations. The stability of the constitutive damageable elastic-viscoplastic system is then sought by means of linear perturbation approach. Application of this method to a mild steel and a dual phase steel leads to the determination of their formability limit diagrams. On the one hand, FLDs obtained with Rice’s criterion and with this stability criterion illustrate the theoretical relation between these criteria and on the other hand, a parametric study shows the influence of governing parameters of hardening, damage and strain rate sensitivity.CNRS & Région Lorrain

Comparison of forming limit diagrams predicted with different localization criteria

Automotive industries are more and more subject to restrictive environmental constraints. Weight reduction of structures seems to be an interesting way to satisfy these requirements. This can be achieved either by using new materials, such as high strength steels or by adopting appropriate dimensioning methods to predict the occurrence of strain localization. Forming Limit Diagram (FLD) is a concept widely used to characterize the formability of thin metal sheets. Analytical determination of FLDs is usually based on the use of localization criteria. Some of the existing material instability criteria are for example based on empirical observations, on the maximum load principle [1-3], on the existence of an initial defect in the sheet [4], on a perturbation method [5] or on bifurcation analysis [6]. Although numerous criteria have been developed, they all have advantages but also drawbacks and limitations. Their confrontation on a wide range of materials is still insufficiently developed to compare their respective capability of accurately predicting FLDs for new materials. Adaptations of some criteria to advanced constitutive laws are also made necessary by the use of new high strength materials. The aim of this paper is to give a general formulation of some localization criteria allowing the comparison of the predicted FLDs for a wide range of materials. An implementation of these criteria coupled with different phenomenological constitutive laws is presented and compared for different materials including an aluminium alloy, a brass and a dual phase steel.CNRS & Région Lorrain

Modélisation de critères de localisation : prédiction de l'orientation des bandes de localisation et tracé de courbes limites de formage

Différentes approches de modélisations ont été proposées pour prédire les phénomènes de localisation lors d’opérations de mise en forme de tôles métalliques par emboutissage. Des exemples de ces approches sont le principe de force maximum, les critères à deux zones et les analyses de bifurcation ou de stabilité. En l’absence de consensus relatif à leur choix et à l’étendu de leur domaine de validité, la présente étude se propose de formuler ces modélisations dans un cadre unifié, de mettre en évidence les liens théoriques pouvant exister entre elles et ensuite de valider ces observations par le tracé de courbes limites de formage et par la comparaison de l’orientation des bandes de localisation

Modélisation théorique et numérique des critères d'instabilité plastique. Application à la prédiction des phénomènes de striction et de localisation lors d'opérations d'emboutissage.

The aim of this work is to contribute to the prediction of diffuse necking, localized necking and localization by the comparison of the theoretical basis of plastic instability criteria and by the numerical simulation of Forming Limit Diagrams. Four main categories of criteria may be distinguished. They are based on the maximum force principle, on the bifurcation analysis, on the stability analysis or on the multizone method. To allow a comparison, main criteria are then rewritten in a common frame. For diffuse necking, a theoretical relation is established between the force maximum criterion (MFC) and limit point bifurcation criterion. For localization, the predictions obtained with Marciniak - Kuczynski model tend to Rice criterion when M - K initial defect tends to zero. Some elements of comparison of plastic instability criteria are given by numerical simulation of FLD. The influence of the choice of hardening laws and their identification is highlighted.L'objectif est d'apporter une contribution à la prédiction des phénomènes de striction et de localisation sous forme de bandes par la modélisation des critères d'instabilité plastique, par la comparaison de leurs bases théoriques et par le tracé de Courbes Limites de Formage (CLF) numériques. Une étude bibliographique permet de distinguer quatre approches principales : le principe de force maximum, l'analyse de bifurcation, l'analyse linéaire de stabilité et les méthodes multizones. Afin de permettre une comparaison, les principaux critères sont réécrits dans un cadre commun. Pour la striction diffuse, un rapprochement entre les critères de force maximum et de bifurcation par point limite est proposé. Pour la localisation, les prédictions de CLF obtenues avec le modèle M - K tendent vers celles du critère de Rice lorsque la taille du défaut initial tend vers zéro. Des tracés de CLF illustrent ces résultats et mettent en évidence l'influence du choix de la loi d'écrouissage sur les CLF