Browsing by Author "Mielke, Alexander (Prof. Dr.)"

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A rate-independent model for phase transformations in shape-memory alloys
(2004) Mainik, Andreas; Mielke, Alexander (Prof. Dr.)
In this thesis we consider a special class of mechanical systems that is rate-independent systems. Such systems are typically driven by an external loading on a time scale much slower than any internal time scale (like viscous relaxation times) but still much faster than the time needed to find the thermodynamical equilibrium. Typical phenomena involve dry friction, elasto-plasticity, certain hysteresis models for shape-memory alloys and quasistatic delamination or fracture. The main feature is the rate-independence of the system response, which means that a loading with twice (or half) the speed will lead to a response with exactly twice (or half) the speed. The goal of this paper is to prove the existence of time evolution for such systems. In last part we use exemplary the developed existence theory to show the existence of time evolution for a simple model of phase transformations in solids. In first chapter we treat general rate-independent systems. In the literature the reader can find approaches to this kind of systems involving either differential inclusions or abstract hysteresis operators. Unfortunately the both methods require additional structures (Banach space structure, convexity, spaces of scalar-valued functions), which are often not given in mechanical models of rate-independent systems. In this paper we utilise a different approach. The main idea is to rewrite a problem in a derivative-free, energetic form. The reformulation leads to two conditions, which have a natural physical meaning. The first one requires the stability of the system at all time. It means that a jump from the actual process state to an another state is energetically disadvantageous. The second condition requires an energy balance in the system. Hence this formulation does not involve space derivatives, it is much more adequate for many mechanical systems. Moreover, the energetic formulation allows for the usage of direct methods of the calculus of variations in order to obtain existence results for the new formulation. In second chapter a short overview of the theory of functions of bounded variation is provided. The results in this chapter are more or less well known, but only for a small round of specialists. Moreover, most of them are widely scattered in the literature. For some results the author was unable to find the proofs anyway. In last chapter we use the results of the second chapter for introducing of a simple model for phase transition. The modelling of phase transition processes plays an important role in the material science. Especially in the context of shape-memory alloys such modelling has been subjected to intensive theoretical and experimental research recently. It is surely related to the importance of smart materials in the aerospace and civil engineering. There exist yet some applications to human medicine. Such smart materials are characterised by an existence of different possible atomic grids (phases) and by a strong dependence of elastic properties on the actual structure of an atomic grid. The grid with a higher symmetry mostly cubic) is referred as austenite phase while the lower-symmetrical grids (smart materials may have more than one lower-symmetrical grid) are called martensite phases. Under an external mechanical loading a smart material passes through an elastic deformation, but by attainment of a certain activation stress the phase transformation occurs. At this moment the energy, which is needed for the phase transformation, is partially dissipated to heat and partially stored in the new phase interface. Practical experiments show that the phase transformation processes can be considered, except very fast time scales, as rate-independent. This fact leads to the opportunity to treat the time evolution of phase transformation as a rate-independent process and to apply the abstract existence theory. In the model for phase transformation, which is presented in this thesis, we assume that the phase state at every material point is given by one pure crystallographic phase. It means that this model can be considered as a microscopic one. We assume also that one part of the stored energy is saved in the phase interfaces. This assumption is realised through an interface energy term of the total stored energy. This term is introduced as an integral over the phase interface of some suitable interface density function. Surely, the interface energy term forbids the experimentally observed formation of microstructure, but at the same time this additional term allows us to model nucleation effects, which were also observed in experiments. The last effect makes it reasonable to consider such interface energy term. Anyway, the introduced model seems to be interesting since it realises some experimental effects and allows to apply rigorous results in order to prove the existence of time evolution.