Development of a new quadratic shell element considering the normal stress in the thickness direction for simulating sheet metal forming

The existing degenerated shell element based on Mindlin–Reissner's theory of plates assumes that the thickness of the element remains constant and that the normal stress in the thickness direction is zero. These assumptions lead to problems in simulating sheet metal forming, especially in predicting the amount of springback. In this paper, we present a new non-linear shell element by considering the normal stress in the thickness direction for plain strain analysis, which allows for large deformation. Two pseudo nodes are introduced on the top surface and bottom surface, respectively. As a result, the incremental normal strain in the thickness direction can be calculated directly and the incremental normal stress in the thickness direction can be calculated using the constitutive relation directly. Thus, it is possible to deal with the problem that pressure acts on the top surface and the bottom surface is simultaneously constrained. Numerical examples show that the stress prediction is improved. The normal stress in the thickness direction exhibits a correspondence with boundary loads and the bending moment decreases as the normal stress in the thickness direction increases. This indicates the element can simulate the effect of bottoming, i.e. sheet metal is stamped further with a considerable punch force after the punch reaches the dead point in the process of V-type bending for the purpose of reducing springback.