Keywords: finite element analysis, shell elements, stress in the thickness direction, elastic-plastic stress update, seven-parameter shell formulations, simulation of sheet metal forming operations.

A novel procedure is proposed to introduce non-vanishing stress components in the thickness direction within quadrilateral one point quadrature shell elements for applications in metal forming where this stress component leads to considerable differences between classical shell solutions and reference results obtained using continuum formulations. Thereby, the lateral stress is assumed to be linearly distributed between the top and bottom surface, respectively, while its amplitude is extracted from dynamic boundary conditions (pressures). For that matter, it is irrelevant whether the pressures are applied as external loads or are caused by normal contact conditions. Regarding normal contact pressures a new scheme is derived which extracts physically based contact stresses from nodal contact forces within an explicit time integration.

If the normal stress distribution in the thickness direction is known in all integration points a new hypo-elastic-plastic constitutive update, termed "constrained plasticity", is used to find the dependency of all other stress components on that lateral stress. It is shown that moderate to large pressures in thickness direction influence not only the damage characteristic within sheet metal forming applications but also pure mechanical properties such as force displacement histories. Special benchmark problems are introduced to prove that the predictive quality of sheet metal forming simulations is increased dramatically if the proposed procedures are applied to classical (five-parameter) shell elements.

It is a major objective of this work to adopt the lateral stress within shell elements without adding additional degrees of freedom as done by recently proposed seven-parameter formulations [1]. There, at least two extra degrees of freedom are needed to calculate a linear thickness strain distribution while referring on a three dimensional constitutive update. Consequently, those formulations suffer from two main deficiencies: Firstly, they introduce additional degrees of freedom not only on an element level but also on the global level which clearly harms the computational efficiency. Secondly, it is shown in Wall et al. [2] that seven-parameter formulations cause poor conditioning numbers if used for thin shell structures. Consequently, the computational effort of iterative solvers is heavily influenced and the stable timestep within an explicit solution procedure is decreased dramatically. Thus, higher-parameter formulations introduce higher accuracy on the cost of computational efficiency, while the present approach has only marginal influence on the computational effort.

1

M. Bischoff, E. Ramm, "On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation", International Journal of Solids and Structures, 37, 6933-6960, 2000. doi:10.1016/S0020-7683(99)00321-2

2

W. Wall, M. Gee, E. Ramm, "The challenge of a three-dimensional shell formulation - the condition number", in: Proc. IASS-IACM 2000, "Computational Methods for Shell and Spatial Structures", Athens, Greece, 2000.

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