Organized by the Leipzig Team of the TMR Network <link http: www.dmsa.unipd.it tmr external>Phase Transitions in Crystalline Solids
Social programme
Friday, April 28, 20.00: Concert, Gewandhaus Orchestra, conducted by Herbert Blomstedt (Johannes Brahms, Concert for Piano and Orchestra No. 1, 2nd Symphony)
Saturday, April 29, 20.00: Dinner at "Kaiser Maximilian" in the city centre
The paper presents a macroscopic description for the simulation of the Two Way Shape Memory Effect (TWSME) of Shape Memory Alloys (SMA), and is an extension of a previous work dealing with the global thermomechanical behavior of SMA. The model is set in the framework of the thermodynamics of irreversible processes. Two internal variables are taken into account: the volume fraction of self-accommodating (pure thermal effect) and oriented (stress-induced) product phase. A term dedicated to the modeling of the trained SMA is added to the specific free energy derived for the non-trained SMA. This term fed, depends on the thermomechanical procedure of training submitted to the sample. In the present case, it accounts for the stress tensor sed applied during thermal cycling. The parameters of the model have been identified for a Cu-Zn-Al SMA, and the simulated results show good agreement with experiments.
Shape memory wires contract into the austenitic phase at high temperature and they expand into the martensitic phase at low temperature. Thus a current set through the wire will make the wire contract, because of Joule heating, and no current will make it expand, because of the cooling through the surrounding air. This behaviour may be harnessed for adapting the shape of an air foil to the existant flight conditions. This has been done and a model is shown - consisting of a "slice" of an airfoil which bends so as to minimize drag of the airfoil under the constraint of a fixed lift force.The optimal shape of the air foil for given angles of incidence and different speeds of the incoming air is stored in a data bank calculated from aerodynamics. The necessary current for the realization of the shape is calculated from the Müller-Achenbach theory of shape memory which was recently adapted by Seelecke for quantitative predictions.Basically that theory is a statistical thermodynamic theory of activated processes, where the transitions between the austenitic phase and martensitic is singled out by the direction of the elongation. This theory combines rate laws for the phase fractions with the energy balance that governs the temperature of the wires.The suitability of the theory for quantitative simulation of shape memory behaviour is shown by a feedback control problem which exhibits very good agreement between experiment and the prediction of the theory.The airfoil exhibited during the session could be used interactively by the audience, because it responded to a change of the angle of incidence that could be adjusted by a joystick.
The evolution of a single phase boundary which is hindered by dry friction can be described by a variational inequality and a rate law. The idea is to obtain existence of generalized solutions by extending the problem from pure phase distributions to phase mixtures. To relate the extended problem to the original problem a generalized relaxation concept for general rate independent evolution problems is presented. In a simple case it is possible to apply the abstract concept to a special phase transformation problem, i.e. a mathematical rigorous derivation of a coarse grained model can be established.
With the view of examining strain instability and micro-texture formation in material, lattice models are considered. More precisely, we want to understand how macroscopic behaviour of material is affected by the underlying microphysics taking place at the crystal scale. For instance, nucleation process, domain growth or phase boundary movement are often responsible for particular behaviours in solids undergoing phase transformations (hysteresis, etc.).In a previous works the emphasis has been put on a mechanism of formation of elastic microstructures or ferroelastic phase in the lattice model framework. The models thus considered include non-linear (non-convex potential) and competing interactions [1,2]. These interactions are, in fact, intimately related to specific interatomic forces that account for unstable, metastable or stable states of lattice configurations. Moreover, such interactions describe the cubic-tetragonal phase transformation in binary alloys. From 1D and 2D models some interesting, but nonetheless promising results have been obtained, among them (i) the softening of the phonon dispersion at a non-zero wave-number [2], (ii) the propagation of an array of strain solitary waves [3], (iii) the transverse instability of a strain band in a 2D lattice [4] and (iv) the instability mechanism of a modulated (periodic) strain structure [5].Along with the same model, we are continuing the study and extending it. In particular, the problem of discreteness effects seems to be crucial for phase growth and lattice instability. Accordingly, it is more appropriate to keep the discrete nature of the model (without considering the long wavelength approximation). In particular, the model enables us to examine the stability of periodic phases or the nucleation of small elastic domains. Numerical simulations should be performed directly on the discrete system and exhibiting microtexture of very rich morphology [6].The analysis of the dynamics of the localized structures in 1D an 2D systems is examined by computing the associated mass, momentum and energy. Conservative laws are deduced for the lattice and quasi-continuum models. The conserved quantities are computed as function of the soliton or localized mode parameters [7] - the velocity or the strain amplitude - in order to characterize the dynamics of the associated quasi-particle. At low velocity we recover the classical newtonian dynamics. For bigger velocities the dynamics of the localized object become more complicated and it can be described by numerical simulations.A second task for the research works should be the investigation of the influence of applied forces and dissipative effects on the formation and the dynamics of localised structures. This part should lead to a connection between applied stimuli to material and its macroscopic response. We must specify that this kind of approach has a great impact in micro-mechanics or nano-mechanics where low-dimensional materials involve few atomic planes (devices using novel or advanced materials).
[1] J. Pouget, Phase Transition 14, 251 (1989).[2] J. Pouget, Phys. Rev. B43, 3575 (1991).[3] J. Pouget, Phys. Rev. B43, 3582 (1991).[4] J. Pouget, Phys. Rev. B46, 10554 (1992).[5] J. Pouget, Phys. Rev. B48, 864 (1993)[6] J. Pouget, Meccanica 30, 449 (1995).[7] M.M. Bogdan, A.M. Kosevich and G.A. Maugin, Phys. Rev E, 1999 (in press).
We present work in progress on determining the set of recoverable strains in shape-memory single crystals from lattice parameters in a full 3D setting. More precisely, we seek the set of macroscopic deformation gradients which can be accomodated below the transition temperature by a zero energy martensitic microstructure. Mathematically, this amounts to determining the quasiconvex hull (it turns out typically to be non convex) of the set of pure martensitic phases, and is a prototype case of a homogeneization problem for a system of nonlinear partial differential equations. Via a novel, purely geometric approach it is possible to determine the desired set for cubic-to-tetragonal transformations, subject to an as yet unproved conjecture (that the quasiconvex hull in this case equals the polyconvex hull). This givex explicit predictions, e.g. for Ni-36at%Al: recoverable tensile strain of 12.7% in direction, 1.4% in , 0.74% in . It would be interesting to check these predictions experimentally, especially as geometrically linear theory (mis?)predicts the cubic-to-tetragonal recovery to be zero.
Non-weak solid-to-solid phase transitions are characterized by the absence of a finite supergroup of all the symmetry groups of the phases involved, as for example in the face-centered-cubic to body-centered-cubic of iron. Non-weak transformations necessarily involve large lattice distortions, and their properties are markedly different from those of the "weak" transitions, typical of shape-memory compounds.We show that all transitions where the parent and product phases are in a group-subgroup relation can be accomplished as weak transformations, and study a general framework for the analysis of non-weak transformations. We present a nonlinear elastic model for a square-to-hexagonal transition in planar simple lattices, which gives a two-dimensional example of non-weak phase changes. Our constitutive framework and the ensuing analysis illustrate some general propertis of non-weak transformations, showing their link to the development of dislocations and plasticity phenomena in crystalline lattices.
The crystallographic theory of martensite - developed by Wechsler, Liebermann and Read in the 1950's - is generally used to calculate the rank-one deformation gradient between aystenite and a martensitic variant which are coherently connected along a habit plane. It turns out that the orientation of the habit plane and the shear and rotation of the martensite may be related to the "Bain strain", i.e. the stretch that forms part of the overall deformation. This strain must be determined from crystallographic data.Here we attempy to extend the crystallographic theory to a loaded austenitic specimen. The Bain strain will then depend on the load, or else, the original Bain strain will be affected by the load and as a result the characteristic "wedges" formed by two martensitic variants change their angle.The new Bain strain may be calculated from the original one by exploiting the condition that the stress vectors on the habit plane are continuous. In addition the temperature for which the deformation occurs may be related to the load by the requirement that the normal component of the Eshelby tensor is continuous at the habit plane. Thus we are able to derive the analogue to the Clausius-Clapeyron equation in a liquid-vapour phase transition. In the present case this turns out to be a monotone relation between the uniaxial stress applied to the specimen and the temperature.All this - for purposes of simplicity - is demonstrated for the two-dimensional case.
In this talk we study a one dimensional model simulating the shear in a two dimensional body. We analyse the discrete system and we deduce the continuum limit of the lattice model as the lattice parameter goes to zero. Different energies are introduced and linked together.
We review the cubic-to-monoclinic transformation mechanisms proposed in the literature for CuZnAl alloys; we confirm previous arguments and add new ones in favor of a 'DO3 -> 6M1' mechanism. We show that if the strain parameters satisfy a relation that allows for non-generic twinning in the monoclinic martensite, also extra habit planes exist, at which the twinned martensite can border the austenite. For special values of the strain parameters (and thus, probably, of the compositions in the alloy), preliminary results show that also wedge-shaped microstructures involving twinned martensite may exist.
Geometrically necessary dislocations (GND) account for the incompatibility of the plastic deformation of single crystals, both in bulk and at the interface. A plethora of measures of GND has been proposed in the literature since 1952: we propose a rationale to choose among these measures, also in relation with recent experimental techiques based on electron backscattering to determine lattice orientation in single crystals.
