Mohamed Ben Bettaieb

Localized Necking in Elastomer-Supported Metal Layers: Impact of Kinematic Hardening

The present paper deals with localized necking in stretched metal sheets using the initial imperfection approach. The first objective is to study the effect of kinematic hardening on the formability of a freestanding metal layer. To this end, the behavior of the metal layer is assumed to follow the rigid-plastic rate-independent flow theory. The isotropic (resp. kinematic) hardening of this metal is modeled by the Hollomon (resp. Prager) law. A parametric study is carried out in order to investigate the effect of kinematic hardening on the formability limits. It is shown that the effect of kinematic hardening on the ductility limit is noticeably different depending on the strain path considered. The second aim of the paper is to analyze the effect of an elastomer substrate, perfectly bonded to the metal layer, on the formability of the whole bilayer. It is found that the addition of an elastomer layer substantially enhances the formability of the bilayer, in agreement with earlier studies.

INFLUENCE OF THE YIELD SURFACE CURVATURE ON THE FORMING LIMIT DIAGRAMS PREDICTED BY CRYSTAL PLASTICITY THEORY

THE AIM OF THIS PAPER IS TO INVESTIGATE THE IMPACT OF THE MICRO-SCOPIC YIELD SURFACE (I.E., AT THE SINGLE CRYSTAL SCALE) ON THE FORMING LIMIT DIAGRAMS (FLDS) OF FACE CENTRED CUBIC (FCC) MATERIALS. TO PREDICT THESE FLDS, THE BIFURCATION APPROACH IS USED WITHIN THE FRAMEWORK OF RATE-INDEPENDENT CRYSTAL PLASTICITY THEORY. FOR THIS PURPOSE, TWO MICROMECHANICAL MODELS ARE DEVELOPED AND IMPLEMENTED. THE FIRST ONE USES THE CLASSICAL SCHMID LAW, WHICH RESULTS IN THE FORMATION OF VERTICES (OR CORNERS) AT THE YIELD SURFACE, WHILE THE SECOND IS BASED ON REGULARIZATION OF THE SCHMID LAW, WHICH INDUCES ROUNDED CORNERS AT THE YIELD SURFACE. IN BOTH CASES, THE OVERALL MACROSCOPIC BEHAVIOR IS DERIVED FROM THE BEHAVIOR OF THE MICROSCOPIC CONSTITUENTS (THE SINGLE CRYSTALS) BY USING TWO DIFFERENT SCALE-TRANSITION SCHEMES: THE SELF-CONSISTENT APPROACH AND THE TAYLOR MODEL. THE SIMULATION RESULTS SHOW THAT THE USE OF THE CLASSICAL SCHMID LAW ALLOWS PREDICTING LOCALIZED NECKING AT REALISTIC STRAIN LEVELS FOR THE WHOLE RANGE OF STRAIN PATHS THAT SPAN THE FLD. HOWEVER, THE APPLICATION OF A REGULARIZED SCHMID LAW RESULTS IN MUCH HIGHER LIMIT STRAINS IN THE RANGE OF NEGATIVE STRAIN PATHS. MOREOVER, ROUNDING THE YIELD SURFACE VERTICES THROUGH REGULARIZATION OF THE SCHMID LAW LEADS TO UNREALISTICALLY HIGH LIMIT STRAINS IN THE RANGE OF POSITIVE STRAIN PATHS.

Influence of the Non-Schmid Effects on the Ductility Limit of Polycrystalline Sheet Metals

The yield criterion in rate-independent single crystal plasticity is most often defined by the classical Schmid law. However, various experimental studies have shown that the plastic flow of several single crystals (especially with Body Centered Cubic crystallographic structure) often exhibits some non-Schmid effects. The main objective of the current contribution is to study the impact of these non-Schmid effects on the ductility limit of polycrystalline sheet metals. To this end, the Taylor multiscale scheme is used to determine the mechanical behavior of a volume element that is assumed to be representative of the sheet metal. The mechanical behavior of the single crystals is described by a finite strain rate-independent constitutive theory, where some non-Schmid effects are accounted for in the modeling of the plastic flow. The bifurcation theory is coupled with the Taylor multiscale scheme to predict the onset of localized necking in the polycrystalline aggregate. The impact of the considered non-Schmid effects on both the single crystal behavior and the polycrystal behavior is carefully analyzed. It is shown, in particular, that non-Schmid effects tend to precipitate the occurrence of localized necking in polycrystalline aggregates and they slightly influence the orientation of the localization band.

PREDICTION OF PLASTIC INSTABILITY IN SHEET METALS DURING FORMING PROCESSES USING THE LOSS OF ELLIPTICITY APPROACH

The prediction of plastic instability in sheet metals during forming processes represents nowadays an ambitious challenge. To reach this goal, a new numerical approach, based on the loss of ellipticity criterion, is proposed in the present contribution. A polycrystalline model is implemented as a user-material subroutine into the ABAQUS/Implicit finite element (FE) code. The polycrystalline constitutive model is assigned to each integration point of the FE mesh. To derive the mechanical behavior of this polycrystalline aggregate from the behavior of its microscopic constituents, the multiscale self-consistent scheme is used. The mechanical behavior of the single crystals is described by a finite strain rateindependent constitutive framework, where the Schmid law is used to model the plastic flow. The condition of loss of ellipticity at the macroscale is used as plastic instability criterion in the FE modeling. This numerical approach, which couples the FE method with the self-consistent scheme, is used to simulate a deep drawing process, and the above criterion is used to predict the formability limit of the studied sheets during this operation.

Prediction of Localized Necking Based on Crystal Plasticity: Comparison of Bifurcation and Imperfection Approaches

In the present work, a powerful modeling tool is developed to predict and analyze the onset of strain localization in polycrystalline aggregates. The predictions of localized necking are based on two plastic instability criteria, namely the bifurcation theory and the initial imperfection approach. In this tool, a micromechanical model, based on the self-consistent scale-transition scheme, is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). The mechanical behavior of the single crystals is developed within a large strain rate-independent constitutive framework. This micromechanical constitutive modeling takes into account the essential microstructure-related features that are relevant at the microscale. These microstructural aspects include key physical mechanisms, such as initial and induced crystallographic textures, morphological anisotropy and interactions between the grains and their surrounding medium. The developed tool is used to predict sheet metal formability through the concept of forming limit diagrams (FLDs). The results obtained by the self-consistent averaging scheme, in terms of predicted FLDs, are compared with those given by the more classical full-constraint Taylor model. Moreover, the predictions obtained by the imperfection approach are systematically compared with those given by the bifurcation analysis, and it is demonstrated that the former tend to the latter in the limit of a vanishing size for the initial imperfection.

Ductility prediction of substrate-supported metal layers based on rate-independent crystal plasticity theory

In several modern technological applications, the formability of functional metal components is often limited by the occurrence of localized necking. To retard the onset of such undesirable plastic instabilities and, hence, to improve formability, elastomer substrates are sometimes adhered to these metal components. The current paper aims to numerically investigate the impact of such elastomer substrates on the formability enhancement of the resulting bilayer. To this end, both the bifurcation theory and the initial imperfection approach are used to predict the inception of localized necking in substrate-supported metal layers. The full-constraint Taylor scale-transition scheme is used to derive the mechanical behavior of a representative volume element of the metal layer from the behavior of its microscopic constituents (the single crystals). The mechanical behavior of the elastomer substrate follows the neo-Hookean hyperelastic model. The adherence between the two layers is assumed to be perfect. Through numerical simulations, it is shown that bonding an elastomer layer to a metal layer allows significant enhancement in formability, especially in the negative range of strain paths. These results highlight the benefits of adding elastomer substrates to thin metal components in several technological applications. Also, it is shown that the limit strains predicted by the initial imperfection approach tend towards the bifurcation predictions as the size of the geometric imperfection in the metal layer reduces.

Modeling of void coalescence initiation and its impact on the prediction of material failure

In the present paper, Thomason’s criterion is coupled with the well-known Gurson–Tvergaard–Needleman (GTN) damage model and used for the determination of the critical void volume fraction fc, which marks the initiation of the coalescence stage. The onset of void coalescence predicted by Thomason’s criterion is compared to that obtained by using a predefined fc, which is usually fitted on the basis of experimental results, as originally proposed in the GTN model. Comparisons are made in terms of both single finite element simulations and numerical results of deep drawing of a cup.

A comparative study of Forming Limit Diagrams predicted by two different plasticity theories involving vertex effects

The main objective of this contribution is to compare the Forming Limit Diagrams (FLDs) predicted by the use of two different vertex theories. The first theory is micromechanical and is based on the use of the Schmid law, within the framework of crystal plasticity coupled with the Taylor scale-transition scheme. The second theory is phenomenological and is based on the deformation theory of plasticity. For both theories, the mechanical behavior is formulated in the finite strain framework and is assumed to be isotropic and rate-independent. The theoretical framework of these approaches will be presented in details. In the micro-macro modeling, the isotropy is ensured by considering an isotropic initial texture. In the phenomenological modeling, the material parameters are identified on the basis of micro-macro simulations of tensile tests.