Generalised forming limit diagrams showing increased forming limits with non-planar stress states
Introduction
The forming limit diagram (FLD) is used to record estimates of the strains experienced by a particular material in one or more forming processes. The logic of the diagram is illustrated in Fig. 1 which potentially shows both thinning and thickening changes to the sheet. Once a range of surface strains before and after failure of a particular process have been identified a forming limit curve (FLC) can be drawn on the diagram. The FLC is the envelope of all recorded un-failed strain states, where failure is generally defined as the onset of necking. The hope of such curves is that they can largely be treated as a material property, so that the same curve can be used for the same sheet material being deformed in different processes.
The FLD and the many models that have been developed to predict FLCs for particular materials assume that sheet forming processes occur under loading which leads to a state of plane-stress. For some well established processes such as hydro-forming, this is not perfectly true, and a small body of work has begun to examine the effect of normal compressive stresses on the formability of sheet metals. However the principal motivation of this paper arises from recent work by the authors on the mechanics of incremental sheet forming (ISF). This process, invented in the early 1990s in Japan, is a flexible sheet forming process in which a small hemispherical tool moves under computer control over a sheet clamped around its edges. A simple schematic of the process is illustrated in Fig. 2a and a photograph of the Cambridge ISF machine (described in Allwood et al., 2005) is given in Fig. 2b. As the tool moves, it creates a small localised indent in the sheet, and by choosing an appropriate tool path, the sheet can be formed to a variety of shapes – so the process aims to be a sheet forming equivalent to a CNC machining process. In fact, the accuracy of the ISF process is limited, and it is still under development (Jeswiet et al., 2005, review developments in the area) but one feature of the process that has caused great interest is that it leads to greater forming limits than conventional sheet pressing operations. Allwood et al. (2007), reporting experiments in which lines were scribed on both upper and lower surfaces of the workpiece and measured before and after deformation, have demonstrated that the ISF process induces significant through-thickness shear strains in the sheet. The key result of these experiments is re-created in Fig. 3 showing through-thickness strain developing in the direction parallel to tool motion. In a preliminary analysis, they demonstrated that including a proportional shear stress in a model used to predict FLCs led to a significant increase in predicted formability.
The motivation of this paper is to build on this insight to propose a generalised forming limit diagram accounting for the possibility that all six components of the stress and strain tensors in sheet forming may be non-zero. Section 2 provides a review of developments in both forming limit diagrams and their analysis. In Section 3, the well known Marciniak and Kuczynski (1967) analysis of forming limits is extended to allow for sheet deformation with a six-component proportional stress tensor, and the results used to propose a generalised forming limit diagram (GFLD). Section 4 proposes a new material formability test which demonstrates the increased formability achieved by processes which lead to through-thickness shear. In Section 5, the significance of the increased forming limits demonstrated by the GFLD is discussed, both in analysing existing processes, and in developing new ones.
Section snippets
The characterisation and prediction of forming limits
The forming limit diagram (FLD) was developed by Keeler and Backhofen (1963) (although Geiger and Merklein (2003), refer to Gemsamer (1946), as the first author to draw something like an FLD) to attempt to characterise the tensile instabilities of sheet forming. The FLD is not the only possible graphical means of characterising the strains at sheet forming limits: Swift (1952) used plots of the major strain against the stress ratio in proportional loading; Ferron et al. (1994) plot effective
Predicting and reporting forming limits with a six-component proportional stress tensor
The first part of this section extends Marciniak and Kuczynski’s famous (1967) analysis to consider loading under a full six-component tensor. In the second part, the representation of the resulting forming limit predictions is discussed, leading to a proposal for a generalised forming limit diagram, and an example is given.
Experimental investigation of formability in the GFLD
The generalised forming limit diagram (GFLD) introduced in Section 3.2 is a means to represent failure strains with a full six-component tensor, and the example shown in Fig. 8 was drawn based on the modified M-K model developed in Section 3.2. What experimental evidence exists to support the increase in forming limit shown in the GFLD when normal stress or through-thickness shear exists? Section 2.4 reviewed the small number of studies that have considered the influence of normal stress on
Discussion
Starting from the observation that the incremental sheet forming process induces significant through-thickness shear strains in the workpiece, this paper has proposed a generalised forming limit diagram to represent forming limits under any possible proportional loading, has extended Marciniak and Kuczynski’s analysis to prediction of such a GFLD, and has described a first experiment to confirm the prediction. The GFLD indicates two strategies for significantly increasing the formability
Acknowledgements
The first author acknowledges the support of a Global Research Award from the UK’s Royal Academy of Engineering. The second author is supported by an EPSRC CASE studentship with Airbus UK.
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