Micro‐Mechanical Modelling of the Constitutive Behaviour of NiTi Shape Memory Alloys

The superelasticity and shape memory effect in NiTi alloys are examined on the basis of micromechanics within the energy minimization framework. We describe the behaviour of polycrystalline shape‐memory alloys via orientation‐distribution of the various martensite‐variants (domains) present in the material. Stress‐strain curves are presented and special attention is payed to the volume fraction of martensite for specific NiTi alloys (Nitinol) specimen under uniaxial tension. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of the Application of NiTi Alloys in Medical Technologies

Shape memory alloys are nowadays already established as a material which is able to solve exceptional tasks in practical applications. Particularly, its utilization in the field of medical technologies increases steadily. For example micro tools (staple, catheters) and implants (coronary stents) are made out of Nickel-Titanium well known as a basic shape memory alloy. Apart from the advantages like the avoidance of auxiliary components and joints in the system and to utilize the high volume specific work of shape memory alloys, NiTi alloys exhibit a good biocompatibility. This property is necessary with regard to either permanent or temporary implants. To optimize the use of NiTi alloys in the scope of medical technologies, the support of the development of applicable tools by numerical simulations is highly recommended. However the complex material behaviour containing a profoundly thermomechanical coupling poses indeed a big challenge to the material modeling and its implementation into a finite element code. Particularly, the material model proposed by Helm [1] proves to be a firm model containing the most common properties of shape memory alloys, as the pseudoelasticity, the shape memory effect and the two-way effect. In the present contribution the FE modelling of a medical staple used in foot surgery is presented by considering the model of Helm which was investigated by the authors to improve its performance in the finite element method [2]. The foot staple, produced by a group of members of the SFB 459 which is funded by the DFG, avails the shape memory effect to excite the desired clamping effect [3]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

FE Modelling of NiTi‐Composed Structures

The present contribution focuses on the finite element modelling of NiTi‐based shape memory alloys. For this purpose, the concept of Helm [1] has been further developed and implemented into a finite element formulation. The model is able to describe the multiaxial behaviour of SMA and therefore useful for the simulation of complex problems. Additional emphasis is put on the modelling of the shape memory effect in composites where the interaction between the alloy and other materials like e.g. polymers plays an important role. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudoelasticity of Shape Memory Alloys - a 1D Material Model and Finite Element Implementation

This contribution is concerned with the formulation of a 1D-constitutive model accounting for the pseudoelastic behavior of shape memory alloys. The stress-strain-relationship is idealized by a hysteresis both in the compression as in the tension loading range. It is characterized by an upper loading path, which is to be ascribed to the transformation of the lattice to a martensitic structure. Unloading the material, a lower path is described, because of the reverse transformation into austenitic lattice. The constitutive model is based on a switching criterion which serves as a potential function for the evolution of the internal state variables. The model distinguishes between local and global variables to describe the hysteresis effects for the compression and tension range. A strain driven algorithm which captures the complete nonlinear material behavior is presented. The boundary value problem is solved for a truss element applying the finite element method. A consistent linearization of the nonlinear equations is derived. Simple examples will demonstrate the applicability of the proposed model. For future developments the usage of shape memory alloys within civil engineering structures is aimed. The advantage of the material is the very good damping behavior and the potential to overcome great strains. Both properties are distinguished to be of engineering interest. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A constitutive model for shape memory alloy fibres and its applicability to prestressed composites

This contribution is concerned with a constitutive model for shape memory fibres. The 1D-constitutive model accounts for the pseudoplastic and shape memory effect (SME). The macroscopic answer of the material is determined by the evolution from a twinned martensitic lattice into a deformed and detwinned one. On the macroscopic scale these effects are responsible for the upper boundary of the hysteresis which is situated around the origin of the stress-strain-diagram. During the phase transition process inelastic strains arise. When the lattice is fully detwinned, a linear elastic branch at the end of the hysteresis is observed. The initial state of the material is recovered by unloading and heating the material subsequently. The constitutive model is derived from the Helmholtz' free energy and fulfils the 2nd law of thermodynamics. For the present model five internal state variables are employed. Two of them are used to describe the inelastic strain and a backstress. The others represent the martensitic volume fraction and are necessary to describe the SME. The latter variables are depending on the deformation state as well as on temperature. A change on temperature goes along with a reduction of the inelastic strain. The model is incorporated in a fibre matrix discretization to prestress the surrounding structure. The boundary value problem is solved for a truss element applying the finite element method. Examples will demonstrate the applicability in engineering structures. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanical model for single‐crystal shape‐memory‐alloys

This work is dealing with solid to solid phase transformations in shape‐memory‐alloys and the simulation of the corresponding characteristic phenomena, e.g. pseudoelasticity and the shape‐memory‐effect. In particular it focuses on the micromechanical behaviour of the material and the appearance of microstructures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Modelling of Shape Memory Alloys and Experimental Verification

Prestrained shape memory alloys change their length when heated above their transformation temperature. This effect can be used to generate high forces in a small workspace, which has particular advantages in actuator design. The optimization and control of the shape memory actuator requires a comprehensive simulation of the material behavior. However, many of the existing models are limited to specific load cases or offer rough approximations only. A material model for shape memory alloys from Seelecke [1] is examined in this paper. This model describes the behavior of a shape memory wire, which is heated by electric current. It is implemented in a simulation program to investigate the actuator output and to improve the performance. Finally, the parameters of the simulation are adapted to experimental results.     

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.     

Implementation of Thermomechanical Coupling for Shape Memory Alloys into the Finite Element Method

NiTi alloys open up new fields of application on the basis of their distinctive thermomechanical properties. Many options of practical application of shape memory alloys are imaginable. For example catheters or stents made of NiTi play an important role in medical technology. Thus the further development and optimisation of simulation tools for shape memory alloys (SMA) structures will play an important role in the future. Based on the powerful material model of Helm [1] the present contribution focuses on the coupling between mechanical and thermal fields. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Memory‐efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Novel memory‐efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so‐called quadratic Arnoldi method and two‐level orthogonal Arnoldi procedure methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift‐and‐invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems, which stem from boundary element discretizations of PDE eigenvalue problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Stress and deformation states in the annulus made from shape memory material

The contribution deals with the stress and deformation states in an annulus made of shape memory material (SMM). The loading‐unloading process is going on at constant temperature, which is below temperature M_f (martensite finish temperature). The research is focused on the determination of the stress and deformation states in the annulus after loading and unloading process. These results are necessary for treating process of constrained recovery in shape memory annulus. The real loading‐unloading function in the stress‐strain coordinate system of shape memory material (Ni–Ti–Cu) is included in the mathematical model.     

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)     

Spline approximation,Kronecker products and multilinear forms

Sums of Kronecker products occur naturally in high‐dimensional spline approximation problems, which arise, for example, in the numerical treatment of chemical reactions. In full matrix form, the resulting non‐sparse linear problems usually exceed the memory capacity of workstations. We present methods for the manipulation and numerical handling of Kronecker products in factorized form. Moreover, we analyze the problem of approximating a given matrix by sums of Kronecker products by making use of the equivalence to the problem of decomposing multilinear forms into sums of one‐forms. Greedy algorithms based on the maximization of multilinear forms over a torus are used to obtain such (finite and infinite) decompositions that can be used to solve the approximation problem. Moreover, we present numerical considerations for these algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Stability of the steady state for the Falk model system of shape memory alloys

In this article a stability result for the Falk model system is proven. The Falk model system describes the martensitic phase transitions in shape memory alloys. In our setting, the steady state is a nonlocal elliptic problem. We show the dynamical stability for the linearized stable critical point of the corresponding functional. Copyright © 2007 John Wiley & Sons, Ltd.     

Control the Fibre Orientation Distribution at the Outlet of Contraction

A two dimensional model of the orientation distribution of fibres in a paper machine headbox is studied. The goal is to control the fibre orientation distribution at the outlet of contraction by changing its shape. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Then, the problem is expressed as an optimal control problem governed by variational forms. By using an embedding method, the class of admissible shapes is replaced by a class of positive Radon measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this linear programming problem. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to shape variation design to a one dimensional headbox. The usefulness of this idea is that the method is not iterative and it does not need any initial guess of the solution.