Material laws denoting the influence of textures due to spring back simulations

Most calculations of the spring back of sheet metals fail especially if the influence of bending processes is dominant, because at the end of a bending and a subsequent re-bending process followed by a complete un-loading, an extremely precise description of the influence of the first steps of deformation on the final behaviour is necessary. Most people try to get reasonable results using traditional material laws with yield loci within which the material behaves purely elastically. The yield locus itself may include isotropic or kinematical hardening. Also any kind of Hill’s anisotropy possibly combined with kinematical hardening is often included. But all of that is not enough since the local variations of the lattice orientations of the neighbouring crystallites induce a much softer transition from the purely elastic to the (also) plastic regime. This is extremely interesting in the never purely elastic re-bending and un-loading phase that occurs during spring back processes of sheet metals.It was often tried to denote these effects by appropriate phenomenological laws, sometimes also types that did not include any yield locus at all or had only a very small purely elastic regime, whereas a very rapidly changing kinematical (tensor) parameter denoted the smooth elastic–plastic transition. These laws have, in general, no possibility to let the basic inner process enter into the final denotation. So lots and lots of experiments are necessary.A better approach is based on the behaviour of the single crystals or crystallites within the assembly. Unfortunately, the use of reference volumes is extremely expensive and time-consuming. So – as a compromise – an enhanced Taylor’s theory for elastic plastic media including also some aspects of the Sachs’ theory is proposed. Here, it is necessary to use a manifold of representatives of the crystallites with their locally variable orientations. If their number is too high, calculations become too expensive. But with a relatively small number of representative starting orientations accompanied by a sophisticated choice of them, the results become very reasonable and can be applied in present numerical codes. The method remains to be applicable also if real yield loci do no longer exist. Those which are found according to defined bounds (offsets of deformations) at the end of a total un-loading situations will be demonstrated as well as the transitional behaviour of the material immediately after the forming process which are possibly not really comparable.

Research highlightsMaterial spring back behaviour of metals is dominated by crystal orientations. Reference volume approaches are too expensive for descriptions in Gauss points. Atomistic models are also not applicable (till now). An approach being a mixture of Taylor’s and Sachs’ theories is most considerable. Yield loci are now only defined ones and are not due to any normality rule.