There are two main applications for FEA in canmaking; predicting the performance of a product and evaluating forming processes. In each case, anisotropic material models can enhance the simulations’ accuracy. For example, typical performance requirements for a beverage can include the reversal pressure of the dome and the resistance of the dome in a drop test. Typically, a can will fail under both of these test conditions by buckling of the dome. As we know, the predicted buckling load of a perfect dome under symmetric loading will usually be significantly higher than in any practical case since a perfect dome does not exist. Anisotropy in the material model is a good way of introducing physically realistic asymmetry into the model and obtaining a plausible buckling result.
The benefits of including anisotropy in the material models of a forming analysis are perhaps even greater. At Crown, all of our products are made from rolled sheet metal. The metal arrives at a rolling mill in the form of a slab about 0.2m thick, 1m wide and 10m long. Through a series of hot and cold rolling operations the slab is formed into a coil about 0.2mm thick, 1m wide and 10km long. This results in a highly distorted and directional crystal structure, with greatly differing densities of grain boundaries depending on which direction relative to the rolling direction (RD) you move. As a consequence, dislocation motion is harder in some directions than others and the plastic behaviour is highly dependent on the stress state.
Anisotropic plasticity causes some difficulties in an otherwise axisymmetric (apart from tooling misalignment) process. The resulting variation in thickness in the tangential direction is a possible trigger for wrinkling. Anisotropy also means that when a can body has been drawn, the height is not uniform. This is known as earing, and the peaks in height are called ears. A perfectly flat top is required on a body if an end is to be reliably seamed onto it. This means the body must be trimmed, and the greater the variation in the height, the more material is wasted. An anisotropy model which can predict this variation in height used in a simulation based optimisation allows us to design a non-round blank shape from which to draw a cup with as flat a top as possible.
For our anisotropy models we characterise the directional plastic behaviour using 2 quantities for a range of orientations of uniaxial tension:
- Normalised yield stress (relative to the RD value for uniaxial tension).
- The ratio of width to thickness direction strain (r-value or Lankford’s coefficient).
These data are supplemented with experimental determination of biaxial r-value (ratio of RD to transverse direction (TD) strain) and normalised biaxial yield stress along with assumptions about through thickness properties. A large number of material models are designed to fit such a data set. Hill’s 1948 model is built-in to Abaqus. More recent models published in scientific literature require a larger number of input parameters (and therefore a greater number of physical tests), and a user material for implementation in Abaqus. They do however offer a more accurate representation of the anisotropy of the alloy which can be important in accurately predicting earing.
The ears on a cup have two planes of symmetry, the first with its normal in the RD and the second with its normal in the TD. One way of evaluating an anisotropy model is by the number of ears which may be predicted in each sector. Hill’s 1948 model can only predict 1 ear per sector, translating into 2 ears per cup if the ear lies on a symmetry plane or 4 per cup if the ear lies between symmetry planes. Newer models, such as Barlat’s Yld2004 18 parameter (Yld2004-18p) model can predict a greater number of ears. The Yld2004-18p model is able to predict 3 ears per sector, one on each symmetry plane and one between the two. This translates into 8 ears per cup, as seen with many typical canmaking aluminium alloys.
There is still room for improvement though. While the Yld2004-18p model can predict enough ears, the relative heights and positions are close to reality but not always correct. This is because the complexity of the anisotropy is still too great even for this advanced model. This is one of the challenges still faced by Crown Technology in anisotropy modelling. It could be overcome using still more complex models such as a 27 parameter version of Yld2004, or multiscale techniques including the crystal structure in the models.