An investigation on the size-effects on thickness-distribution of directionally rolled multi-stage deep-drawn copper cups through finite element analysis, and its validation through experimentation employed scaling law

Most sectors are moving towards downsizing products that retain high standards of quality and satisfy customer needs. Miniaturisation of components typically makes use of scaling rules. In this work, we use the idea of similarity theory to investigate the micro-and nano-scale thinning behaviours of multi-stage deep-drawn cups. Deep drawing with a limiting draw ratio (LDR) of 1.69 was used to shape ETP copper sheets into cylindrical cups via annealing and directionally rolling. Each rolling pass consists of nine incremental passes that result in a 50% reduction in thickness. The influence of scaling laws for uni and bi-directional rolling was studied, and scaled dies and punch sets were employed to create scaled cups of five different thicknesses, each 50% of the original 6 mm. For this simulation, we used Altair INSPIRE, a sort of explicit finite element analysis that uses real-world material test data. To investigate the impact of thickness variation across all scaled cups, measurements of the thickness were taken at three points on multi-stage deep-drawn cups. Bi-directional rolling produced cups with less thickness variance than uni-directional rolling. All the scaled cups had a consistency greater than the mean at the lip point and less at the cup base for both ways of rolling, it was also noticed. Physical evaluations of the drawn cups for extreme values verified that the inferred variations lie within ±15% of their original thickness for all cups, and the mean thickness likewise has 50% of its multi-stage-drawing, proving the applicability of the scaling rules.