Eduard Oberaigner

Continuum mechanical aspects of phase transformations in solids

SummarySome aspects of “classical” thermodynamics of phase transformations are discussed. Then typical solid state transformations as the displacive and diffusional transformation in metals are explained. A general formulation of the Gibbs free energy is presented including all energy terms required. Based on the “classical” nucleation, the “triggering-off” and the “dissipation” condition, various transformation conditions are formulated taking into account the elasto-plastic deformation of both phases. Transformation conditions presented in the literature over the last 40 years are reviewed and compared to the transformation conditions derived here. The transformation conditions for a spherical region growing under a certain volume change in an elasto-plastic matrix are studied as an example. The relevant analytical expressions are presented and discussed.ÜbersichtEinige Aspekte der „klassischen” Thermodynamik von Phasenumwandlungen werden behandelt. In der Folge werden typische Festkörperumwandlungen wie die displazive und die diffusive Umwandlung in Metallen erörtert. In allgemeiner Formulierung wird die Gibbs Energie unter Berücksichtigung aller erforderlichen Energieterme hergeleitet Basierend auf der klassischen Keimentwicklungsbedingung, einer Formulierung über das Wachstum von Keimen und einem auf der Dissipationsleistung beruhenden Konzept werden verschiedene Umwandlungsbedingungen hergeleitet. Dabei wird in beiden Phasen elasto-plastisches Materialverhalten vorausgesetzt. Die hier formulierten Umwandlungsbedingungen werden mit einigen der in den letzten 40 Jahren publizierten Beziehungen verglichen und bewertet. In einem Beispiel werden diese Bedingungen für eine in einer elasto-plastischen Matrix wachsende Kugel bei einer umwandlungsbedingten Volumsdehnung angewandt. Analytische Ausdrücke werden präsentiert und näher erläutert.

TRANSFORMATION INDUCED PLASTICITY REVISED: AN UPDATED FORMULATION

Abstract An externally stressed specimen in the process of a phase transformation may show a significant nonlinear behavior which is known as transformation-induced plasticity (TRIP). The TRIP-strain can be irreversible as in the case of steels, or reversible with a certain hysteresis as in the case of shape memory alloys. The basic mechanisms contributing to this nonlinear phenomena are the accommodation process of the transformation strain and the orientation process of the transforming microregions. TRIP strain formulations carried out so far do not meet both effects. A thermodynamical concept is presented in this paper to find a TRIP strain rate which takes into account the coupling of phase transformation and microplasticity. The start and progress of the transformation condition, during which the plastic behavior governs the transformation/plastic processes, are derived by solving a conditional extremum problem, composed of the dissipation inequality and the constraint conditions: the transformation condition and the yield condition. The thermomechanical and calorimetric constitutive equations are derived in rate form when the transformation and the plastic processes are fully coupled. Some illustrative examples are discussed by assuming a concrete form of the Gibbs free energy and the transformation/yield conditions. The cross-coupling effect of the transformation and plasticity is well understood in the TRIP strain rate and the transformation kinetics.

Investigation of the damping behavior of a vibrating shape memory alloy rod using a micromechanical model

A micromechanical model developed previously by the authors is adapted to investigate the damping behavior of a shape memory alloy rod. The model delivers a kinetic equation and a stress - strain temperature-transformed volume - fraction relation. These equations are coupled with the equations for the heat conduction and the (free) vibration of a rod. Phase change leads to energy dissipation and thus to damping. From the kinetic equation and the constitutive law, the dissipation rate and the dissipated energy can be derived at a given temperature. They attain a maximum at a certain fixed temperature between the martensite start and the martensite finish temperatures. Damping would be maximized when this temperature was reached everywhere in the specimen simultaneously. Since this ideal temperature cannot be realized instantaneously everywhere, a realistic configuration of sudden cooling/heating of the fixed end of a vibrating rod is simulated. Applying this model, a working temperature for global optimal damping can be found which lies between the martensite start and the martensite finish temperatures. The damping itself is represented by the ratio of the sum of the actual kinetic energy and strain energy to the strain energy of the initial prestressed rod.

A micromechanical approach to constitutive equations for phase changing materials

Abstract When in a microregion the change of the Gibbs free energy (due to stress and temperature change) reaches a critical value, a so called threshold value, then this microregion jumps from one into the other phase. Phase change in polycrystals can be described adequately using statistical methods, e.g. by averaging over representative elementary volumes. Through this averaging one gets physical quantities in the mesodomain, such as volume fractions, transformation strains, etc. Kinetic laws for these values follow from extremization of the energy dissipation rate under constraints, such as an averaged transformation condition. The present paper delivers kinetic laws for the martensitic volume fraction and the transformation strain as the constitutive equations for the phase changing materials. The results are in good qualitative agreement with experiments. Quantitative agreement follows from matching of system parameters, which do not follow from the theory.

A new micromechanical formulation of martensite kinetics driven by temperature and/or stress

SummaryMartensitic transformation behavior of alloys is studied under the arbitrary action of a thermal and/or a triaxial mechanical load-stress state by solving a transformation kinetic equation presented recently by the same authors. Numerical and analytical solutions reveal that the transformation behavior is almost path-independent. Lines of constant volume fraction of martensite are nearly parallel in the stress-temperature plane. Some new analytical formulae for martensitic transformation kinetics are presented.ÜbersichtEs wird das Verhalten der martensitischen Umwandlung von Legierungen unter beliebiger thermischer und/oder dreiachsiger mechanischer Spannungsbelastung untersucht, indem eine neue Gleichung der Umwandlungskinetik gelöst wird. Diese Gleichung wurde vor kurzem von denselben Autoren vorgestellt. Die numerischen und die analytischen Lösungen zeigen, daß das Transformationsverhalten nahezu pfadunabhängig ist. Die Linien gleicher Volumsfraktion von Martensit sind nahezu parallel im Spannungs-Temperatur-Diagramm. Es werden auch einige neue analytische Formeln für die Umwandlungskinetik von Martensit präsentiert.

On the Optimal Damping of a Vibrating Shape Memory Alloy Rod

Vibration damping through phase transformation is one major area of application of shape memory alloys in smart systems and structures. The authors of this study have shown in earlier publications, how damping of vibrating rods can be accomplished. This paper is an extension and generalization. On the one side it uses the proper description of the stress-wave phenomenon instead of a quasi-static approximation, on the other side it describes, how the damping could be optimized. The basic equations of the underlying mathematical model are the stress-wave equation, the heat conduction equation, a kinetic and a constitutive law as well as a condition to ensure maximal damping. The major results are the heating history, which governs the phase transformation, and the domain splitting along the rod into elastic and inelastic regions.

Damping of a Vibrating Sma Rod Through Phase Transformation

Vibration damping is an important task in structural engineering. Several ways exist to accomplish it. One possibility is to ‘weaken’ the material by introducing a phase transformation through cooling or heating and to force the material to dissipate its mechanical energy during the transformation. The vibration damping due to phase transformation can be formulated mathematically by taking into account a micromechanical model of the authors [1, 2, 3] on the behavior of shape memory alloys (SMA’s). This model incorporates a kinetic law to describe the stress-temperature-transformed volume fraction-relation and a constitutive law (stress-strain-temperature-transformed volume fraction-relation). These are nonlinear ordinary differential equations, which are coupled with two partial differential equations, the heat conduction equation and the wave equation.

A multi-block-spin approach for martensitic phase transformation based on statistical physics

Current strategies in modeling shape memory alloy (SMA) behavior follow either the concept of classical irreversible thermodynamics or the methodology of phenomenological approaches at the micro as well as at the macro space scale. The objective of the present study is to show a new approach in modeling SMAs by using a statistical physics concept without the requirement of evolution equations for internal variables. Thermodynamic principles in connection with the mathematical apparatus of statistical physics allow deriving relevant system properties in analogy to the formalism used for paramagnetic-ferromagnetic systems. As a result the macroscopic strains and the volume fractions of the martensitic variants and their rates are obtained. The multi-block-spin approach further maps the tension compression asymmetry of multivariant SMAs.

A space‐time concept for martensitic phase transformation based on statistical physics

Understanding martensitic phase transformation (MPT) is of crucial importance for many engineering applications. Especially in polycrystalline shape memory alloys and steels one can observe phase transformations on several length and time scales. Those are firstly the atomistic length scale (nano scale, nm) and the scale of the crystallites (micro scale, μm), which, in turn, have a certain size and orientation distribution. The transformation kinetics is described on the mesoscale (mm), where an averaging of physical properties is useful and possible within the representative volume element (RVE). A proper handling of the relevant physical properties within the RVE allows to incorporate effective material laws for computations on the macroscale (m). The present study focuses mainly on the aspect of deriving the relevant physical properties on the mesoscale from atomistic and single crystal properties, i.e., on closing the gap in modelling MPT between the nano‐ and microscale resp., and the macroscale. It ...

Diffusional Transformation and Creep in Steels

The influence of creep on a material under diffusional transformation represented by a sphere is investigated. Some new results for an elasto-plastic and creeping hollow sphere are presented. It can be concluded that the transformation kinetics and, therefore, the time dependend deformation behavior is not strongly influenced by creep.