On the Sachs bound in stress-induced phase transformations in polycrystalline scalar shape-memory alloys

We consider stress–induced transformations of polycrystalline shape–memory alloys which are so–called scalar materials. In this case the Sachs bound on the phase transformation yield stress turns out to be sharp. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness for structural phase transitions in shape memory alloys

The non-linear coupled equations arising from alloy mechanism have two important features: a may take negative values and c may be degenerate. The local existence has been proved in Reference 1, but the uniqueness was open. In this paper the uniqueness is proved. For a discussion of the physical model and for the justifications of the detailed technical assumptions to be made, we refer to Reference 1.     

A variational model for the functional fatigue in polycrystalline shape memory alloys

Shape memory alloys show a very complex material behavior associated with a diffusionless solid/solid phase transformation between austenite and martensite. Due to the resulting (thermo-)mechanical properties – namely the effect of pseudoelasticity and pseudoplasticity – they are very promising materials for the current and future technical developments. However, the martensitic phase transformation comes along with a simultaneous plastic deformation and thus, the effect of functional fatigue. We present a variational material model that simulates this effect based on the principle of the minimum of the dissipation potential. We use a combined Voigt/Reuss bound and a coupled dissipation potential to predict the microstructural developments in the polycrystalline material. We present the governing evolution equations for the internal variables and yield functions. In addition, we show some numerical results to prove our model's ability to predict the shape memory alloys' complex inner processes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys

In this paper a thermomechanical model for the dynamics of structural phase transitions in the so-called ‘shape memory alloys’ is developed. These materials exhibit rather spectacular hysteresis phenomena. The resulting mathematical model consists of a coupled and highly non-linear system of partial differential equations reflecting the balance laws of linear momentum and energy. For an appropriate weak formulation the local-in-time existence of weak solutions is shown.     

Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys

Summary Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au₂₃Cu₃₀Zn₄₇ demonstrate the quality of the discrete model.Partially supported by Research Program RP.1.02Supported by DFG, SPP Anwendungsbezogene Optimierung und Steuerung     

Simulation of asymmetric effects for shape memory alloys

Experimental results of shape memory alloys show a pronounced asymmetric behaviour between tension, compression and shear. For simulation of these effects in the constitutive equations different transformation strain tensors are introduced. These are related to the different variants for the multi-variant- and detwinned-martensite as a consequence of different stress states. In the framework of plasticity the concept of stress mode dependent weighting functions is applied in order to characterize the different stress states. Verification of the proposed methodology is succeeded for simulation of the pseudoelastic behaviour of shape memory alloys with different hardening characteristics in tension, compression and shear. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Liquid-like behavior of shape memory alloys

In this Note, we prove that the identity matrix is an inner point of the quasiconvex hull K^qc of a compact set $\text{K?{X∈}\text{M}{}^{\mathrm{3,3}}\text{:}\text{det}\text{X=1}}$ whenever K^qc contains a three-well configuration. This is in particular the case for the cubic to tetragonal and the cubic to orthorhombic phase transformations, and answers a question discussed in S. Müller, Microstructures, phase transitions and geometry, in: A. Balog et al. (Eds.), Proceedings European Congress of Mathematics, Progr. Math., Birkhäuser, 1998. To cite this article: G. Dolzmann, B. Kirchheim, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A new semi-analytical method for phase transformations in binary alloys

In the present paper, a new semi-analytical method is developed to cover a wide range of phase transformation problems and their practical applications. The solution procedure consists of two parts: first, determination of the position of the moving boundary named the homogenous part and second, determination of the concentration named the non-homogenous part. The homogenous part leads to a system of homogenous linear equations, based on the mathematical fact that a homogenous system has a non-trivial solution if the determinant of the coefficient matrix equals zero. This determinant leads to an ordinary differential equation for the moving boundary, and its solution leads to a closed form formula for the position of the moving boundary. The non-homogenous part transforms the governing equations to a non-homogenous linear system of equations, having three unknowns that appear in the concentration profile assumed in the beginning of the proposed method. Solution of the non-homogenous system leads to a value of these unknowns. Once these unknowns are computed, the concentration at any time and at any point can be found easily.     

Global attractor for thermoelasticity in shape memory alloys without viscosity

We consider a nonlinear system of thermoelasticity in shape memory alloys without viscosity. The existence and uniqueness of strong and weak solutions and the existence of a compact global attractor in an appropriate space are proved. Copyright © 2015 John Wiley & Sons, Ltd.     

Control problems with state constraints for shape memory alloys

In this paper, two different control problems with state constraints for shape memory alloys are considered: in the non-isothermal case, we study boundary control problems, and in the isothermal situation, a dynamical shape optimization problem is considered. In both cases, the transverse displacement is the constrained state variable. The first-order conditions of optimality are derived.     

Weak solutions for Falk's model of shape memory alloys

In this paper we discuss the system of two partial differential equations governing the dynamics of phase transitions in shape memory alloys. We consider the one‐dimensional model proposed by Falk, in which a term containing a fourth‐derivative appears. The main purpose is to show the uniqueness for weak solutions of the problem by using the approximate dual equations for the system without growth condition for the free energy function. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal control of thermomechanical phase transitions in shape memory alloys: Necessary conditions of optimality

The necessary conditions for the optimality of an optimal control problem associated with soiid–solid phase transitions in shape memory alloys are established.     

On the interrelation between dissipation and chemical energies in modeling shape memory alloys

We use the principle of maximum dissipation for thermo-mechanically coupled modeling of poly-crystalline shape memory alloys (SMA). This modeling scheme demands approaches for both Helmholtz free energy and dissipation. For time-independent processes, dissipation is usually modeled by the norm of the internal variable's rate times a factor. We show that for SMAs this factor is not an additional modeling parameter. In contrast, it can be calculated from the Helmholtz free energy. This reduces the number of model parameters and provides furthermore an interesting effect of the model which allows to display the material behavior in an even more realistic manner. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys

This note deals with a three–dimensional model for thermal stress–induced transformations in shape–memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. The problem is formulated mathematically within the energetic framework of rate–independent processes. An existence result is proved and we study space–time discretizations and establish convergence of these approximations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermomechanical behavior of pseudoelastic NiTi shape memory alloys

The thermomechanical behavior of polycrystalline Ni-rich pseudoelastic NiTi shape memory alloys is analyzed. Special focus is on regions within the stress strain diagram which are regarded as linear elastic in common phenomenological material models, i.e. the region between zero stress and the beginning of the pseudoelastic plateau as well as the region within the hysteresis. In both cases, severe temperature changes can be observed. A possible explanation for this effect is twofold: On the one hand, it might be explained by the presence of an R-phase transformation. On the other hand, unstructured martensite of the B19' phase may form. However, the assumption of a purely thermo-elastic material behavior in those regions does not seem to hold true in general. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Materialmodelling of shape memory alloys in range of large deformations

In this contribution we present a finite deformation material model for SMA which includes the effect of pseudoelasticity. The model's structure is similiar to a Frederick-Armstrong type hardening model for elastoplasticity. A special algorithm has been developed to incorporate the concept into a FE code. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multifield coupled analysis for a harmonic movable tooth drive system integrated with shape memory alloys

In this article, a harmonic movable tooth drive system integrated with shape memory alloys is proposed. The key of the system is to integrate the shape memory alloy drive principle with the harmonic movable tooth drive principle by which a small size and a large output torque can be achieved simultaneously. Here the structure of the drive system is determined, and its operating principle is illustrated. The output torque equation for the drive is deduced and coupled dynamics equations for the system are given. With use of these equations, the forces applied to the wave generator and the output torque of the drive system under pulsed current are investigated. The results show that with reasonable pulsed current parameters, a large and steady output torque can be obtained.     

Monotone strain-stress models for shape memory alloys hysteresis loop and pseudoelastic behaviorx

To describe the behavior of Shape Memory Alloy we use a thermomechanical model, founded on a free energy which is a convex function with respect to the strain and to the martensitic volume fraction, and a concave one with respect to the temperature. The material parameters of the model are experimentally determined.Received: November 26, 2001; revised: March 20, 2002     

Weak solutions for the Falk model system of shape memory alloys in energy class

We show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23 : 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Stationary solutions to the Falk system on shape memory alloys

We study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physically closed stationary state is formulated as a nonlinear eigenvalue problem with a non‐local term. Then, some results on existence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamically stable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.