Potential of the Cross Biaxial Test for Anisotropy Characterization Based on Heterogeneous Strain Field
- Shunying Zhang, Lionel Leotoing, +4 authors THUILLIER Sandrine
The optimization of sheet metal forming processes requires an accurate prediction of material behavior and forming abilities, especially for aluminum alloys which exhibit generally a low formability compared with typical mild steels. This study presents an original technique based on the use of a cruciform shape for experimental characterization and numerical prediction of forming limit curves (FLCs). By using a cruciform shape, the whole forming limit diagram is covered with a unique geometry thanks to the control of the displacements in the two main directions of the specimen. The test is frictionless and the influence of linear and non-linear strain paths can be easily studied since the strain path is controlled by the imposed displacements, independently on the specimen geometry. The influence of strain paths is first studied by introducing a linear prestrain (uniaxial, plane strain or equi-biaxial), in rolling direction. Afterwards non-linear prestrain paths are also tested. Introduction In sheet metal forming operations, the sheet can be deformed only up to a certain limit, defined as its formability. Many factors, like temperature, strain rate or strain paths can affect considerably the material formability. The optimization of forming operations with numerical tools needs accurate predictions of material formability in order to fully exploit its forming abilities. Thus, understanding and characterizing the formability of metal sheets are essential for controlling final product quality and then evaluating the success of the sheet forming operations. The most popular tool to quantify the formability of sheet metals is the forming limit diagram (FLD). A FLD is a strain diagram built with the in-plane principal strains in which the forming limit curve (FLC) can distinguish between safe points and necked points. The determination of FLDs has always been the subject of extensive experimental, analytical or numerical studies. Experimentally, two conventional tests exist, the so-called out-of-plane stretching (e.g. Nakajima test) and the inplane stretching (e.g. Marciniak test). The main drawbacks of these tests are the use of a high number of specimens with various geometrical specifications, the influence of friction and the description of formability by following simplistic linear strain paths. Many predictive tools have been established for FLCs. The Marciniak and Kuczynski model (known as the M-K model) is a widely used analytical tool but its initial geometrical imperfection factor is uncertain. The value can be adjusted by making the best fit with experimental results or by making a microstructural analysis of the metallic sheet. Moreover, the choice of an appropriate constitutive law is a key to obtaining the practical prediction of FLCs. The use of a cruciform shape to characterize and predict forming limit curves can be an interesting alternative to overcome the major drawbacks of the conventional methods. The test is frictionless and the main advantage of this shape is that the strain path at the onset of necking is directly imposed by the control of the testing machine, independently on the specimen geometry. A unique geometry is then sufficient to cover the whole forming limit diagram, the influence of strain path can be easily studied by applying linear or non-linear loadings. Numerically, the use of the finite element method to model the cruciform shape permits the implementation of complex mechanical behaviors in order to evaluate the influence of operating conditions like temperature or strain rate. Moreover, the calibrating step of the initial geometrical imperfection factor which is essential for M-K models is unnecessary. The present study focuses on the potential of the cruciform shape to study the effect of strain path on the formability. Many authors , have demonstrated that non-linear loadings, frequently encountered in industrial processes, have a great influence on level and shape of FLCs. In literature, most of the studies employs the M-K model [2,3] with the limits previously mentioned. Experimentally, it is rather difficult to apply a prestrain with the conventional tests and very few experimental data exist . In this work, the finite element model of the cruciform shape is first presented and its ability to evaluate the influence of strain path on the formability of the aluminum alloy 5086 is clearly demonstrated. Linear or non-linear prestrains effects can be observed for different modeling of the hardening law. The cruciform shape In order to control the strain path of the necking zone thanks to the displacements of the four arms of the cruciform shape, the onset of necking must be observed in the central zone. A promising specimen shape (Fig. 1) following this requirement has been previously optimized by present authors [5,6]. The onset of necking is detected when the equivalent plastic strain increment ratio between two zones (necking and adjacent zone) reaches a critical value . Fig. 1: Optimized cruciform shape Due to the symmetrical properties of the specimen, only one-quarter is considered in the finite element model (Fig. 2). For meshing, tetrahedral elements are applied and a refined mesh is adopted where strain localization may appear (central zone, fillet, grooves). Fig. 2: One-quarter meshed of the cruciform shape Firstly, the classical power law of Ludwick (Eq. 1) is used to model the hardening behavior of the material. n 0 Kε + σ = σ (1) In equation (1), σ and ε are respectively the equivalent stress and the equivalent plastic strain. The three constants are given by: = σ0 125.9 MPa, = K 447.1 MPa and = n 0.41. The hardening curve associated with the Ludwick’s power law is given by Fig. 3. Fig. 3: Hardening curves for Ludwick’s and Voce’s law The anisotropy of the material is described with the Hill48 yield criterion in which the equivalent stress is expressed by a quadratic function (Eq. 2): ( ) ( ) ( ) 2 xy 2 zx 2 yz 2 y x 2 x z 2 z y 2 N 2 M 2 L 2 H G F 2 σ + σ + σ + σ − σ + σ − σ + σ − σ = σ (2) with F = 0.7, G = 0.637, H = 0.363, L = 1.5, M =1.5, and N = 1.494, constants specific to the state of anisotropy of the material. The direction x is the rolling direction, y the transverse direction and z the normal direction. Effect of linear prestrains The material is prestrained to two different strains levels in the rolling direction (RD) under uniaxial (TU), plane (PS) and equibiaxial (BS) strain state. All the prestrain paths are linear.