Summary. In the first part of this work, twinning is analysed by geometrical considerations in simple lattices. It is shown that all compound twins exhibit an elastic energy invariance, which holds practically for all technologically interesting twinning modes. The existence of an energy invariance for certain twinning modes has been found by Ericksen (1984b); Zanzotto (1992, 1996). However, it appears that the statement that all compound twins obey the energy invariance is new. The cases that the compound twins are crystallographically equivalent or distinct and its consequences for the elastic modelling of twinning have been discussed.
The strain energy invariance enforces a treatment of pairs of conjugate twins as one twinning mode if modelled by means of elasticity. Although not distinguishable at each material point, one can clearly recognise each of the conjugate twins by the interface alignment that is established. It is to expect that the elastic modelling works not for crystallographically distinct conjugate twins. This is because one has to treat them due to the energy invariance as one twinning mode, although they may feature different properties. The strain energy invariance may even connect a regular twinning mode to a lattice invariant shear. However, due to the high symmetry of the cubic, tetragonal and hexagonal crystals, many compound twins are crystallographically equivalent, e.g.,{112}h11¯1itwinning in the bcc, {111}h11¯2i twinning in the fcc (the TWIP-twins in manganese-alloyed steels), {¯1012}h10¯12i twinning in hcp (extension twinning),{101}h11¯1itwinning in the bct and othorhombic,{100}h00¯1i in the orthorhombic lattice, the pairs of conjugate twin systems of which are treatable as one twinning mode.
In the second half of this work, an elastic material model for twinning is developed. It consists in its core of a quadratic strain energy, which is extended by the isomorphy of the elastic law and the Ball and James-approach (Ball and James, 1987) to a piecewise quadratic nonconvex elastic energy. To obtain a continuously differentiable strain energy, a regularisation for the latter is introduced. Further, to adapt the twinning-stresses, a phenomenological model adaptation which relies on the Schmid law is introduced. In order to avoid the ill-posedness of the pseudoelastic boundary value problem, the viscous regularisation is used. The model is applied to the{10¯12}h¯1011itwinning in the hcp lattice, the twinning stress and thec/aratio are close to common magnesium alloys. As hcp crystals undergo readily crystallographic glide in the basal plane, the visco-elastic model is extended by the card glide mechanism, which allows plastic deformations by basal slip in the parent crystal.
The model is tested in various finite element simulations. It is able to predict the nucleation and prop-115
agation of the twins. The stress-drop observed shortly after the nucleation (Christian and Mahajan, 1995) and predicted by Kochmann and Le (2009) is found in the simulations as well. The predicted cusp-shape of the twin tips are in accordance with experimental findings and conclusions from the theory of transformation dislocations (Boyko et al., 1994). Moreover, it is found that the interface in-clination with respect to the shear plane is limited by the critical twinning stress, which is concluded from a stress jump analysis as well (Glüge and Kalisch, 2008). In conjunction with the basal slip mechanism, the model is able to predict the kink patterns observed by Roberts and Partridge (1966).
The model is used in a simple compression simulation of an RVE, where the orientation distribution is similar to the one that is experimentally observed in extruded magnesium. It is found that the pre-dicted twin structure is quite realistic. It is observed that due to the misfit strains the twins propagate across grain boundaries. The average twin volume fraction corresponds well to experimental findings of Jiang et al. (2007). Therefore, as the texture evolution is linked directly to the twin volume frac-tion, the texture evolution due to twinning is reproduced as well. Due to the complicated hardening behaviour owed to twin-particle interactions, the hardening rate of Magnesium alloys is underesti-mated. In the stage of extensive twinning, the model predicts a zero hardening plateau, which is in accordance to experimental observations on pure magnesium, i.e. when no twin-particle interactions are present. At the end of the stage of extensive twinning, the stresses are overpredicted in both cases.
This is due to the lack of deformation mechanisms like secondary twinning and slip inside the twins, and the lack of a damage criterion.
However, the elastic modelling induces some difficulties. The most problematic fact is that twin-ning is connected to the movement of partial dislocations. This induces a strain path-dependence and energy dissipation, which are neglected by any pseudoelastic modelling. Moreover, the strain energy invariance of conjugate twins restricts the elastic modelling to crystallographically equivalent conjugate twins. Although the conjugate twins can be distinguished clearly in the FE simulations by considering the interface orientation, the elastic modelling leaves the possibility that a twin turns over into its conjugate twin. Such behaviour is not realistic due to the kinetic process underlying the twin formation. The conclusion is that the pseudoelastic modelling cannot be applied if severe strain path changes occur.
Outlook. One disadvantage of the model, namely the necessity of the phenomenological adapta-tion for reasonable twinning stresses, comes from the Ball and James-approach. I see basically two possibilities of how the model could be advanced.
Instead of modifying the elastic law, one could think of introducing an internal variable, which evolves according to a nucleation criterion and a kinetic relation. This could be a small twinned volume at the time of nucleation, the interface of which moves according to the kinetic relation. It is to expect that such a modelling strategy is very challenging from the practical point of view.
Another method could be to derive the elastic strain energy from molecular dynamics or molecular statics simulations instead of postulating it. If one constrains the atomic arrangement to be periodic, it should be possible to derive a strain energy by summing up atomic potentials from deforming a small reference cell. If one applies Born’s rule, in a molecular statics calculation, the strain energy would emerge straightforward and display energy minima for shuffle-free twinning modes. Unfortunately, for the twinning modes involving shuffling, one has to abandon Born’s rule, which means that the motion of the atoms has to be tracked. Such a two-step homogenisation is as well challenging from the practical point of view, but it may be capable to model a variety of phenomenas observed in
117 crystals by only a few physically conclusive equations.