T. Meinders

The implementation of an equivalent drawbead model in a finite-element code for sheet metal forming

Drawbeads are applied in the deep-drawing process to improve the control of the material flow during the forming operation. In numerical methods these drawbeads can be simulated with an equivalent drawbead. In this paper the implementation of an equivalent drawbead model in a finite-element code is described. The input for the equivalent drawbead consists of a drawbead restraining force, a plastic thickness strain and a drawbead lift force, which are obtained from a two-dimensional drawbead simulation or from an experiment. Two different mathematical descriptions of the implementation of the plastic thickness strain are pointed out. The correlation between the simulation results gained with the different algorithms is presented in this work.

Adaptive Through‐Thickness Integration Strategy for Shell Elements

Shell elements are commonly used in simulations of sheet metal forming using finite element analysis. Element matrices, due to their complexity, cannot conveniently be calculated in a closed form and therefore numerical integration is employed. The most commonly used rules for through-thickness integration in shell elements are Gauss quadrature and rules based on the Newton-Cotes formula. Considering the integrand in the expression for the internal force vector, one may state that it is smooth only if during a finite element analysis the material remains in the elastic regime. When the material is in the elastic-plastic regime the integrand becomes non-smooth in, for example, out-of-plane direction since the stress profile may have a point of discontinuity at the surface that separates the elastic and plastic regions. The traditional numerical schemes do not perform well when integrating a non-smooth function and integration error may increase significantly. Therefore, if a shell element is in the elastic-plastic regime, the error due to numerical integration may be large and has to be dealt with. A large number of sampling points in thickness direction may decrease the error due to numerical integration at the cost of increased computation time. Alternatively, the in- tegration error can be decreased without a drastic increase in the computation time by using an adaptive through-thickness integration. A distinguishing feature of this integra- tion scheme is that the location and/or the number of the sampling points is adapted to the through-thickness stress profile, leading to accurate numerical integration at minimal costs. An adaptive through-thickness integration strategy for shell elements is developed in this report. The strategy consists of several algorithms that locate points of discontinuity in the out-of-plane stress profile, adapt sampling points, update values of internal variables and perform numerical integration. Performance of the integration strategy is evaluated using a problem of bending of a beam under tension. It is shown that with adaptive integration it is possible to obtain accurate results with a very low number of sampling points.

Numerical Forming Simulations and Optimisation in Advanced Materials

With the introduction of new materials as high strength steels, metastable steels and fibre reinforced composites, the need for advanced physically valid constitutive models arises. In finite deformation problems constitutive relations are commonly formulated in terms the Cauchy stress as a function of the elastic Finger tensor and an objective rate of the Cauchy stress as a function of the rate of deformation tensor. For isotropic materials models this is rather straightforward, but for anisotropic material models, including elastic anisotropy as well as plastic anisotropy, this may lead to confusing formulations. It will be shown that it is more convenient to define the constitutive relations in terms of invariant tensors referred to the deformed metric. Experimental results are presented that show new combinations of strain rate and strain path sensitivity. An adaptive through- thickness integration scheme for plate elements is developed, which improves the accuracy of spring back prediction at minimal costs. A procedure is described to automatically compensate the CAD tool shape numerically to obtain the desired product shape. Forming processes need to be optimized for cost saving and product improvement. Until recently, a trial-and-error process in the factory primarily did this optimization. An optimisation strategy is proposed that assists an engineer to model an optimization problem that suits his needs, including an efficient algorithm for solving the problem.