Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models

Abstract

The dynamics of phase transitions and hysteresis phenomena in materials with memory are described by a strongly nonlinear coupled system of partial differential equations which, in its generality, can be solved only numerically. Following principles of extended thermodynamics, in this paper we construct a new model for the description of this dynamics based on the CattaneoVernotte law for heat conduction. Models based on the Fourier law follow from this general consideration as special cases. We develop a general procedure for the solution of the resulting systems by their reduction to differential-algebraic systems. Finally, a computational code for the numerical implementation of this procedure is explained in detail, and representative numerical examples are given. IntroductionOne of the most difficult problems in computational mechanics is to quantify phase transformations in complex materials. For the so-called ''smart'' materials this problem is of enormous technological importance. Indeed, performance demands on materials and systems used in engineering applications and infrastructure necessitate the development of such ''smart'' materials that have the ability to change their properties in response to external and internal stimuli. In this way, new engineering components can be developed leading to improved efficiency and reliability of the whole structure. For many such materials phase transformations and accompanied hysteresis phenomena, demonstrated by the strain-temperature, stress-strain, and stress-temperature relations, are intrinsic parts of the material dynamics. From a mathematical point of view, one of the major difficulties come from the fact that these transformations cannot be characterised by a single-valued function. Indeed, in those solids that do not exhibit structure transitions or plastic deformations, the strain is a single-valued function of stress and temperature, which might not be the case for many ''smart'' materials exhibiting hysteresis loops. Typical to ferromagnetic, ferroelectric, plasticity effects, such loops arise due to the fact that the underlying process has more than one stable equilibrium. The free energy function for this thermodynamic process should be derived from statistical models, and any physically reasonable approximation of this function will result in a non-convex energy function. In this paper we focus on a specific type of phase transformations, so-called solid-solid phase transformations, in ''smart'' materials known as shape memory alloys (SMAs). In contrast to ferroelectricity, ferromagnetism, plasticity, where hysteresis phenomena relatively well investigated, this is not the case for pseudoelastic effects that are observed in SMA materials. At the same time, these materials have a huge potential in such fields as electronics, biomedical and environmental engineering, energy production systems, various consumer products, aerospace, and automotive industry, because they can provide functions of sensing, processing, actuation, and feedback [8]. In compa...

Simulation of Nonlinear Thermomechanical Waves with an Empirical Low Dimensional Model

“…The dynamical behavior of the SMA rod is strongly nonlinear with coupling effects between the elastic and thermal fields, first order martensitic transformations, and hysteresis [3,11,10,19]. The simulation is performed for a Au 23 Cu 30 Zn 47 rod of length L = 1cm based on Eq.(1).…”

“…The simulation is performed for a Au 23 Cu 30 Zn 47 rod of length L = 1cm based on Eq.(1). All the physical parameters for this specific material are taken here the same as in [11]. We use the following procedure.…”

“…The well known Falk model has been constructed on the basis of conservation laws for linear momentum and energy, and thermo-dynamical consistency. To model the coupled thermo-mechanical wave interactions and the first order phase transitions in the shape memory alloys, we use the following 1D mathematical model [3,11,19]:…”

“…Thermal and mechanical hysteresis effects and complicated phase transformations are the phenomena that need to be dealt with. Following our previous works [10,11,19], we re-write our original system in a way convenient for computational implementation:…”

“…At the same time, existing mathematical models for the dynamical behaviour of smart materials are typically based on a set of coupled nonlinear PDEs. For example, the mathematical models for the shape memory alloys are formulated as a system of PDEs that couples thermal and elastic fields, while the models for the piezoelectric materials are formulated as a system of PDEs that couples electric and elastic fields [11,19]. All such models are infinite dimensional and one of the ways to deal with such models is to discretize them in space and apply the method of lines to the result.…”