An evolving plane stress yield criterion based on crystal plasticity virtual experiments
Introduction
The plastic anisotropy, as well as many other properties of polycrystalline metals, is generally attributed to the microstructure of the material. One of the most prevailing microstructural factors that controls plastic anisotropy is crystallographic texture. There is a broad variety of attempts in literature to take this fact into account in numerical simulations of metal forming processes.
The microstructure can be explicitly dealt with if a physics-based model is employed. Crystal plasticity theories allow one not only to derive macroscopic mechanical response of the polycrystalline material, but they also provide insights on how the microscopic state evolves with an increasing deformation. Several crystal plasticity frameworks have been proposed over the last decades to answer these challenges.
A first approximation, which is attributed to Sachs (1928), is to use an iso-stress assumption for all grains in a representative volume of polycrystal. The Full Constraints (FC) Taylor–Bishop–Hill homogenization scheme, which was originally proposed by Taylor (1938) and later taken up again by Bishop and Hill (1951b), who proposed a different but equivalent solution method, assumes identical deformation throughout all the grains in the considered volume of the material. These authors proposed two different solution methods, which are both based on the Generalized Schmid Law as the constitutive model for a metallic crystal. An approximate but mathematically convenient alternative is the visco-plastic method by Asaro and Needleman (1985).
Unfortunately, the simple homogenization schemes unrealistically neglect interactions between the grains. To address this issue, several more elaborated homogenization schemes have been established. The Visco-Plastic Self-Consistent (VPSC) model, e.g. Molinari et al., 1997, Lebensohn and Tomé, 1993, Lebensohn et al., 2007 considers every crystal in the polycrystalline as an ellipsoidal inclusion embedded in an effective medium. Since the medium comprises all the grains in the considered volume, long-range interactions are captured. Another approach is used by ‘cluster’ models, which assume that the average plastic velocity gradient of the cluster is equal to the macroscopic velocity gradient. The Grain InterAction (GIA) model proposed by Crumbach et al. (2001) and extended by Engler et al. (2005), as well as the RGC model by Tjahjanto et al. (2010), take into account short-range interactions between next-neighbour grains in an aggregate consisting of eight hexahedral grains. The Advanced LAMEL (ALAMEL) was proposed by Van Houtte et al. (2005) as a generalization of the LAMEL (Van Houtte et al., 1999), which both consider interactions in clusters of two grains. Several improvements to the ALAMEL scheme have been recently proposed, for example by Mánik and Holmedal, 2013, Zhang et al., 2014, Arul Kumar et al., 2011, Mahesh, 2010.
All these frameworks have an important limitation in common: the microstructure is represented in a statistical way, thus a substantially simplified shape of the grains is generally assumed. The Crystal Plasticity Finite Element Method (CPFEM), e.g. (Peirce et al., 1982, Bate, 1999, Roters et al., 2010) takes a step far forward by explicitly treating complex microstructural morphology in a representative volume element. Recently, considerable attention has been attracted by the Crystal Plasticity Fast Fourier Transform (CP-FFT) method (Lebensohn, 2001, Prakash and Lebensohn, 2009, Roters et al., 2012, Lebensohn et al., 2012, Eisenlohr et al., 2013), which promises substantial improvement over the CPFEM in terms of calculation time, while keeping high spatial resolution in order to capture the details of complex microstructures.
The crystal plasticity models mentioned above provide homogenization framework that allows one to derive macroscopic quantities, such as macroscopic stresses, from relevant quantities at the length scale of grains or even finer. Macroscopic forming processes can in essence be simulated by means of a crystal plasticity model implemented directly into an FE code. A clear advantage of the direct coupling is that the crystal plasticity model locally follows the evolution of anisotropy due to the evolving microstructure, including texture and possibly substructure. A recent review by Roters et al. (2010) provides a detailed overview about the incorporation of crystal plasticity models into the FEM. A remarkable part of recent developments in this field has been focused on materials that pose challenges both in modelling and processing, such as hcp alloys, e.g. Segurado et al., 2012, Izadbakhsh et al., 2012, Knezevic et al., 2013, Galán et al., 2014. This kind of multi-scale computations is computationally very intensive. For detailed analyses of complex forming processes, a more robust micro-macro coupling scheme is desirable.
At the other end of the spectrum, the phenomenological models neglect the microstructural effects and focus on macroscopic mechanical response of the material. The phenomenological yield criteria consider the polycrystalline material as homogeneous at the macroscopic level, and the yield surface depends merely on the macroscopic stress, strain rate, certain strain measures as well as their rates. The microstructural features of the material, such as crystallographic texture, can be indirectly taken into account by means of extensive parametrization of these models. Numerous successful efforts have been made in the last decades to improve the macroscopic anisotropy models, for example Hill, 1948, Hosford, 1979, Barlat et al. (Barlat et al., 1991, Barlat et al., 1997a, Barlat et al., 1997b, Barlat et al., 2003b, Barlat et al., 2005, Barlat et al., 2007), Banabic et al., 2000, Aretz and Barlat, 2012, Aretz and Barlat, 2013, Banabic et al., 2003, Banabic et al., 2005, Comsa and Banabic, 2008, Banabic et al., 2010, Yoon et al., 2004, Yoon et al., 2006, Yoon et al., 2010, Yoon et al., 2014, Yoshida et al., 2013, Cazacu et al., 2006, Cazacu and Barlat, 2004, Plunkett et al., 2006, Plunkett et al., 2008, Soare et al., 2008, Soare and Barlat, 2010, Van Houtte and Van Bael, 2004, Van Houtte et al., 2009, Vegter et al., 2003, Vegter and van den Boogaard, 2006. The phenomenological yield loci are generally limited to the initial anisotropy of the material, since it is hardly possible to accurately predict the evolution of the yield surface without taking into account how the microstructure develops during the deformation. Usually it is assumed that the changes to the initial yield locus are negligible. The assumption is approximately valid if the plastic strains are not too large, which admittedly holds in some sheet metal forming processes. Nonetheless, when used in a combination with flow theories, such as the normality flow theory, the phenomenological yield loci provide an efficient technique for capturing the effects of material anisotropy during the simulation of deformation processes. This approach is nowadays commonly followed in commercial Finite Element (FE) packages dedicated for simulations of metal forming operations.
The assumption on constant anisotropy and, to put it in a broader context, a limited description of the microstructural state are particularly restricting if the deformation involves a considerable strain path change. This issue has been recently addressed by Barlat et al., 2011, Barlat et al., 2013, Barlat et al., 2014, who proposed the homogeneous yield function-based anisotropic hardening (HAH). The HAH introduces a tensor state variable, called the microstructure deviator, which evolves during plastic deformation. Yet again, this elegant mathematical framework relies on phenomenological evolution equations, although one of components of the microstructure deviator can be considered as associated to crystallographic texture, while another, which evolves rapidly during deformation, can be seen as reflecting material dislocation structure (He et al., 2013).
An extensive body of literature exists on the calibration of phenomenological yield loci by means of mechanical testing. To give only a few examples, Kuwabara et al., 2002, Kuwabara, 2007, Vegter et al., 2003, Kuwabara and Sugawara, 2013, Yin et al., 2014 use sophisticated experimental setups to measure certain sections of yield loci. Even though certain experiments, such as biaxial tension or biaxial tension-compression tests, can in principle be carried out, in practice there are several limitations that may prevent one from obtaining more thorough characterization of plastic properties. Let these few examples suffice to show common impediments: availability of the experimental facilities, calibration of the equipment to ensure consistency of the results across various types of devices used in the experiments, and, last but not least, economical reasons that may enforce constraints on the experiments (Aretz et al., 2007).
One can also find several examples that combine the strength of the two approaches mentioned above. Some of the crystal plasticity frameworks, most remarkably FC Taylor, VPSC, ALAMEL and CPFEM, have been employed to calibrate phenomenological yield loci, e.g. Savoie and MacEwen, 1996, Grytten et al., 2008, Van Houtte et al., 2009, Barlat et al., 2005, Inal et al., 2010, Plunkett et al., 2006, Kim et al., 2007, Kim et al., 2008, Kraska et al., 2009, An et al., 2011, Saai et al., 2013, Yoon et al., 2014. The hierarchical multi-scale approach was followed, in which the fine-scale model provides data needed for identification of the macroscopic one that is based on a different mathematical framework. Generally, the crystal plasticity models have to be evaluated for a huge number of possible stress or strain rate modes, sometimes exceeding one million realizations. This inspired works that aim at decreasing the computational effort related to virtual experiments. For instance, Rousselier et al. (2009) proposed a Reduced Texture Methodology, which does not aim at a complete representation of the real material texture, nor at an accurate modelling of its evolution, but it attempts to improve computational performance of virtual experimentation. It has to be emphasized that the majority of the aforementioned efforts focus on calibrating the initial yield locus, leaving the evolution of the plastic anisotropy unaddressed.
A possible way to capture the influence of microstructural changes on the anisotropic response is to use the crystal plasticity model to calculate some quantities of interest in advance and approximate these in the macroscopic simulation. This can be done by sampling followed by calculating a response surface, for instance by means of multivariate Kriging, as reported by Barton et al., 2008, Knap et al., 2008, Rouet-Leduc et al., 2014. However, the sampling is very expensive if it has to cover the evolution of the microstructural state variables in a multi-dimensional space. Alternatively, a sequence of explicit algebraic yield criteria can be pre-calculated for a finite set of strain levels and interpolated during the macroscopic simulation (Plunkett et al., 2006, Nixon et al., 2010a, Nixon et al., 2010b, Knezevic et al., 2013). Nevertheless, it appears problematic that the local state evolution may lead outside the validity range of the interpolation.
A viable method to tackle this problem is to use an adaptive hierarchical multi-scale approach. This can be expediently done by systematic updating the material state, such as texture, by applying local macroscopic deformation rates and subsequent recalibration of the phenomenological plasticity model. Each Gauss integration point of a macroscopic FE mesh can be linked with an evolving yield locus description, as it was successfully demonstrated by Gawad et al., 2013, Gawad et al., 2010, Van Bael et al., 2010, Van Houtte et al., 2011 on the Facet plastic potential (Van Houtte et al., 2009), and recently by He et al. (2014) that used the CPB06ex2 yield criterion (Cazacu et al., 2006). The evolution of the plastic anisotropy is therefore taken into account, as well as the evolution of the material state.
In order to be adopted by the sheet forming industry, this last option must provide a clear benefit in terms of improved and more complete predictions, especially for those forming simulations where the modelling error is mainly due to limitations of the material model and not to other factors (such as friction model, process details, etc.). Furthermore, it is expected that a clear advantage is gained in terms of the calculation time:
For the completeness of the overview, we mention a few attempts to develop macroscopic constitutive models that would allow deriving the current plastic anisotropy directly from instantaneous crystallographic texture or from constituents of crystal plasticity frameworks, e.g. by Arminjon and Bacroix, 1991, Kowalczyk and Gambin, 2004, Tsotsova and Böhlke, 2009.
In this paper we present a new evolving plane stress yield criterion based on crystal plasticity virtual experiments. The yield criterion is able to capture anisotropy evolution associated with changes to the crystallographic texture during the deformation. The paper is organized as follows: Section 2.1 presents the virtual experimentation framework that provides crystal plasticity data needed for identification of the yield criterion. In Section 2.2 we discuss extensions to the BBC2008 plane stress yield criterion (Comsa and Banabic, 2008), which permit it to adaptively accommodate to changes in the texture. The results are presented in Section 3. We first discuss the initial characterization of the test material. In the subsequent part we elaborate on the impact of the anisotropy evolution on the macroscopic prediction in a deep drawing of cylindrical cups. Finally, we discuss to what extent this correlates with the changes in the material texture due to the plastic deformation.
Section snippets
Hierarchical multi-scale framework for evolving plastic anisotropy
The model presented in this work adapts the notion of hierarchical modelling. Accordingly, the interacting models of different length scales exchange information: upscaling defines the flow of information from the finer-scale to the coarser-scale, while the flow in the opposite direction is called downscaling. The crucial assumption in the presented approach is that the upscaling and downscaling may involve yet another intermediate model acting as a proxy. On one hand, the proxy provides an
Characterization of the initial material
The material used in this study was an AA6016-T4 metallic sheet (1 mm nominal thickness). This material is a precipitation hardening alloy, containing aluminium, magnesium and silicon as major components. The metallic sheet has been delivered in the T4 status (solution heat treated and naturally aged).
Conclusions
- 1.
The HMS-BBC2008 model allows capturing evolution of plastic anisotropy that occurs in a sheet metal subjected to plastic deformation. To achieve this, the macroscopic FE model includes a local yield criterion in each integration point. The coefficients of the BBC2008 yield criterion co-evolve with the local crystallographic texture. The changes in the local plastic anisotropy are implemented by systematic reconstruction of the BBC2008, where the calibration data are gathered by virtual
Acknowledgements
JG, AVB, PE, PVH and DR gratefully acknowledge the financial support from the Knowledge Platform M2Form, funded by IOF KU Leuven, and from the Belgian Federal Science Policy agency, contracts IAP7/19 and IAP7/21. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Hercules Foundation and the Flemish Government – department EWI. DB, DSC and MG gratefully acknowledge the financial support from the PCCE 100 project
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