Modelling of aluminium sheet forming at elevated temperatures

The formability of Al–Mg sheet can be improved considerably, by increasing the temperature. By heating the\ud sheet in areas with large shear strains, but cooling it on places where the risk of necking is high, the limiting drawing ratio\ud can be increased to values above 2.5. At elevated temperatures, the mechanical response of the material becomes strain rate\ud dependent. To accurately simulate warm forming of aluminium sheet, a material model is required that incorporates the\ud temperature and strain-rate dependency. In this paper simulations are presented of the deep drawing of a cylindrical cup,\ud using shell elements. It is demonstrated that the familiar quadratic Hill yield function is not capable of describing the plastic\ud deformation of aluminium. Hardening can be described successfully with a physically based material model for temperatures\ud up to 200 �C. At higher temperatures and very low strain rates, the flow curve deviates significantly from the mode

Prediction of sheet necking with shell finite element models

In sheet forming simulations, the prediction of localised necking is an important goal. A pragmatic\ud approach is to compare calculated principal strains with a forming limit curve (FLC). However, the FLC’s\ud are known to depend on the strain path and most experimental FLC’s are determined for straight deformation\ud paths. Localisation can also be determined numerically with a Marciniak–Kuczynski analysis (M–K). It is\ud recognised that a FEM analysis with shell elements resembles the M–K analysis very much. For uniform\ud deformations a band with slightly reduced thickness is necessary to trigger localisation. In practical forming\ud conditions, however, the non-uniformity of the process automatically triggers localisation and an arbitrary initial\ud imperfection is not needed. FEM models have the additional benefit that boundary conditions, non-proportional\ud deformation and e.g. friction with the tools are completely included

Object oriented design of a thermo-mechanical FEM code

An object oriented design is presented for a computer program that can perform\ud thermo-mechanically coupled analyzes. The target of the design is a \ud exible and robust\ud computer program. It should be easy to adapt and extend, re-using existing code, without\ud interfering with already established algorithms.\ud The program uses publicly available toolkits that are currently emerging as C++ pack-\ud ages. First of all the Standard C++ Library (formerly Standard Template Library) is\ud used for packing items in container classes. Secondly the matrix and vector operations\ud are derived from the Template Numerical Toolkit (TNT) and �nally (not essentially for\ud the numerical part) a graphical user interface is made, based on the wxWindows package,\ud that can generate a GUI for Motif and MS-Windows with the same code.\ud Attention is given to the design of classes such as speci�c elements and material classes\ud based on more general classes. A hierarchy of classes is constructed where general behavior\ud is put high in the hierarchy and speci�c behavior low. The choice between inheritance and\ud aggregation is made at several levels

Evaluation of stresses in a combined plane strain-simple shear test

A biaxial testing device for sheet metal has been developed that can impose a combination of\ud plane strain and simple shear deformation. The specimen has a large width to height ratio and a small height\ud to thickness ratio. The forces in tensile and shear direction are easily measured and the tensile stress and shear\ud stress can easily be derived. For a full description of stresses, however, the stress in lateral direction should also\ud be known. This stress is a result of the constraint, imposed by the large width to height ratio and cannot be\ud measured directly. The strain in the specimen is measured on the surface. By imposing the Drucker normality\ud principle, the direction of the tangent to the yield surface is known and the unknown stress increment in lateral\ud direction can be obtained. Computer simulations are performed to test whether the intended approach can\ud recalculate all stress components from measurement of 3 in-plane strains and just 2 stresses. Without hardening,\ud good results are obtained for a complete interval between the pure shear point up to a point between uniaxial\ud stress and the plane strain point on the yield locus. With hardening, the algorithm requires a lot of data points\ud to avoid drifting from the exact solution. It is noted that, although the normality rule is used, it is not necessary\ud to have an a-priori knowledge of the yield functio

Efficient implicit FEM simulation of sheet metal forming

For the simulation of industrial sheet forming processes, the time discretisation is\ud one of the important factors that determine the accuracy and efficiency of the algorithm. For\ud relatively small models, the implicit time integration method is preferred, because of its inherent\ud equilibrium check. For large models, the computation time becomes prohibitively large and, in\ud practice, often explicit methods are used. In this contribution a strategy is presented that enables\ud the application of implicit finite element simulations for large scale sheet forming analysis.\ud Iterative linear equation solvers are commonly considered unsuitable for shell element models.\ud The condition number of the stiffness matrix is usually very poor and the extreme reduction\ud of CPU time that is obtained in 3D bulk simulations is not reached in sheet forming simulations.\ud Adding mass in an implicit time integration method has a beneficial effect on the condition number.\ud If mass scaling is used—like in explicit methods—iterative linear equation solvers can lead\ud to very efficient implicit time integration methods, without restriction to a critical time step and\ud with control of the equilibrium error in every increment. Time savings of a factor of 10 and more\ud can easily be reached, compared to the use of conventional direct solvers.\ud