Crystal Plasticity Finite Element Modelling of Tin Crystals Using Phase Fields
Abstract: Tin whiskers have long caused issues to electronic components and made the subject of many articles. According to NASA the oldest reports concerning whiskers date back to the 1940's. The whiskers are single-crystals growing from, for example electronic components. Despite the long history of tin whiskers the actual cause of their growth is still unknown. The hypothesis held by the author is that the whiskers' growth is stress-driven by stresses originating from the formation of an inter metallic compound (IMC) which forms due to a minimisation of the Gibbs free energy when mixing copper (Cu) and tin (Sn). Thus it is believed that the IMC will grow due to diffusion as the Cu, and Sn layers are in contact, with following transformation strains due to the volume change originating from the formation of the IMC. The need of incorporating crystal plasticity in the simulations is justified by the white-tin crystal structure which is of body centred tetragonal type showing high anisotropy for elastic properties. The chosen crystal plasticity model is a creep model, and thus plasticity is always present, even though it may be minuscule. The variables chosen to model the plasticity are the slips of the crystal planes and an increasing resistance to these slips. Dislocations due to slips are modelled through dislocation pile ups giving rise to a plastic potential within the crystal. Moreover the crystal plasticity is presented for finite deformations whereas the phase field formulations and thus also the governing equations are presented in a small strain setting. Lastly the simulations were performed using MOOSE and were able to capture the anisotropic behaviour of the tin.
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