Björn Kiefer

Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys

The magnetically induced martensitic variant reorientation process under applied mechanical load in magnetic shape memory alloys (MSMAs) is considered. Of particular interest is the associated nonlinear and hysteretic macroscopic strain response under variable applied magnetic field in the presence of stress, also known as the magnetic shape memory effect (MSME). A thermodynamically consistent phenomenological constitutive model is derived which captures the magnetic shape memory effect caused by the martensitic variant reorientation process, using internal state variables, which are chosen in consideration of the crystallographic and magnetic microstructure. The magnetic contributions to the free energy function considered in this work are the Zeeman energy and the magnetocrystalline anisotropy energy. Activation functions for the onset and termination of the reorientation process are formulated and evolution equations for the internal state variables are derived. The model is applied to a two-dimensional special case in which the application of a transverse magnetic field produces axial reorientation strain in a NiMnGa single-crystal specimen under a constant compressive axial stress. It is explicitly shown how the model parameters are obtained from experimental data. Model predictions of magnetic field-reorientation strain hysteresis loops under different applied stresses are discussed. †Dedicated to Professor Gerard Maugin on the occasion of his receiving of the 2003 SES A.C. Eringen Medal.

Modeling the Coupled Strain and Magnetization Response of Magnetic Shape Memory Alloys under Magnetomechanical Loading

This article is concerned with the modeling of the magnetic shape memory alloy (MSMA) constitutive response caused by the reorientation of martensitic variants under mechanical and magnetic fields. The presented model is able to better capture the complexity of the magnetomechanical MSMA behavior by accounting not only for the mechanism of field-induced variant reorientation, but also the magnetization rotation away from magnetic easy axes and the magnetic domain wall motion at low stress and magnetic field levels. Following the general formulation of the model, reduced versions of the constitutive equations are derived for three specific loading cases: (1) magnetic-field-induced variant reorientation at constant stress; (2) stress-induced variant reorientation at constant magnetic field; (3) variant reorientation under collinear magnetic field and stress with perpendicular bias field. For each of these cases the nonlinear and hysteretic strain and magnetization response of MSMAs are predicted and compared to experimental data where available. The relation of critical stresses and magnetic fields for the activation of the reorientation process are visualized in a variant reorientation diagram. The captured loading-history-dependent macroscopic material response is explained in detail by connecting it to the evolution of the crystallographic and magnetic microstructure as represented by a set of internal state variables.

Finite element analysis of the demagnetization effect and stress inhomogeneities in magnetic shape memory alloy samples

This paper is concerned with the finite element analysis of boundary value problems involving nonlinear magnetic shape memory behavior, as might be encountered in experimental testing or engineering applications of magnetic shape memory alloys (MSMAs). These investigations mainly focus on two aspects: first, nonlinear magnetostatic analysis, in which the nonlinear magnetic properties of the MSMA are predicted by the phenomenological internal variable model previously developed by Kiefer and Lagoudas, is utilized to investigate the influence of the demagnetization effect on the interpretation of experimental measurements. An iterative procedure is proposed to deduce the true constitutive behavior of MSMAs from experimental data that typically reflect the shape-dependent system response of a sample. Secondly, the common assumption of a homogeneous Cauchy stress distribution in the MSMA sample is tested. This is motivated by the expectation that the influence of magnetic body forces and body couples caused by field matter interactions may not be negligible in MSMAs that exhibit blocking stresses of well below 10 MPa. To this end, inhomogeneous Maxwell stress distributions are first computed in a post-processing step, based on the magnetic field and magnetization distributions obtained in the magnetostatic analysis. Since the computed Maxwell stress fields, though allowing a first estimation of the influence of the magnetic force and couple, do not satisfy equilibrium conditions, a finite element analysis of the coupled field equations is performed in a second step to complete the study. It is found that highly non-uniform Cauchy stress distributions result under the influence of magnetic body forces and couples, with magnitudes of the stress components comparable to externally applied bias stress levels.

Implementation of numerical integration schemes for the simulation of magnetic SMA constitutive response

Several constitutive models for magnetic shape memory alloys (MSMAs) have been proposed in the literature. The implementation of numerical integration schemes, which allow the prediction of constitutive response for general loading cases and ultimately the incorporation of MSMA response into numerical solution algorithms for fully coupled magneto-mechanical boundary value problems, however, has received only very limited attention. In this work, we establish two algorithmic implementations of the internal variable model for MSMAs proposed in (Kiefer and Lagoudas 2005 Phil. Mag. Spec. Issue: Recent Adv. Theor. Mech. 85 4289–329, Kiefer and Lagoudas 2009 J. Intell. Mater. Syst. 20 143–70), where we restrict our attention to pure martensitic variant reorientation to limit complexity. The first updating scheme is based on the numerical integration of the reorientation strain evolution equation and represents a classical predictor–corrector-type general return mapping algorithm. In the second approach, the inequality-constrained optimization problem associated with internal variable evolution is converted into an unconstrained problem via Fischer–Burmeister complementarity functions and then iteratively solved in standard Newton–Raphson format. Simulations are verified by comparison to closed-form solutions for experimentally relevant loading cases.

Modeling of the Magnetic Field-Induced Martensitic Variant Reorientation and the Associated Magnetic Shape Memory Effect in MSMAs

This work is concerned with the magnetic field-induced rearrangement of martensitic variants in magnetic shape memory alloys (MSMAs). In addition to the variant reorientation, the rotation of the magnetization and magnetic domain wall motion are considered as the microstructural mechanisms causing the macroscopically observable constitutive response. The considered free energy terms are the elastic strain energy, the Zeeman energy and the magnetocrystalline anisotropy energy. It is shown how thermodynamic constraints on the magnetization rotation lead to only partial reorientation of the martensitic variants under higher stresses. A straightforward methodology has been devised for the calibration of model parameters based on experimental data. The presented model predictions indicate an improvement of the predictability of the nonlinear strain hysteresis and in particular the magnetization hysteresis.

Application of a magnetic SMA constitutive model in the analysis of magnetomechanical boundary value problems

A major complication in measuring material properties of ferromagnetic materials is the influence of the demagnetization effect. The resulting difference between the internal and applied magnetic field depends on the specimen geometry and the distribution of the magnetization inside the sample. This phenomenon also affects the interpretability of magnetic-field induced strain and magnetization data measured for magnetic shape memory alloys, which in turn makes the formulation of reliable constitutive models for these materials difficult. To solve this problem, the approximation of uniform magnetization is usually adopted, in which case a tabularized demagnetization factor can be used to correct the data. In this paper, the validity of this simplification is tested by explicitly solving the magnetostatic problem for relevant geometries, while taking the nonuniform magnetization of a magnetic shape memory alloy specimen into account. In addition to comparing the relation between the volume averaged internal and applied magnetic field, the local variation of the magnetic field and magnetization is analyzed.

A kinematically-enhanced relaxation scheme for the modeling of displacive phase transformations:

In this contribution, a micro-mechanically motivated, energy relaxation-based constitutive model for phase transformation, martensite reorientation and twin formation in shape memory alloys is proposed. The formulation builds on an idealized parametrization of the austenite-twinned martensite microstructure through first- and second-order laminates. To estimate the effective rank-one convex energy density of the phase mixture, the concept of laminate-based energy relaxation is applied. In this context, the evolution of the energetic and dissipative internal state variables, that describe characteristic microstructural features, is computed via constrained incremental energy minimization. This work also suggests a first step towards the continuous modeling of twin formation within the framework of energy relaxation and can be viewed as a generalization of earlier models suggested by Bartel and Hackl (2009) and Bartel et al. (2011). More specifically, in the current model the orientation of martensitic variants in space is not pre-assigned. Variants are rather left free to arrange themselves relative to the martensite-martensite interface in an energy-minimizing fashion, where, however, it is assumed that they form crystallographically-twinned pairs. The formulation also eliminates the need to introduce specific expressions for the Bain strains in each of the martensitic variants, by relating them to a master variant and utilizing the information about their absolute orientation. The predictive capabilities of the proposed modeling framework are demonstrated in several representative numerical examples. In the first part of the results section, the focus is placed on purely energetic analysis, and the particular influence of the different microstructural degrees of freedom on the relaxed energy densities and the corresponding stress-strain responses is investigated in detail. In the second part, macro-homogeneous uniaxial strain and shear loading cases are analyzed for the dissipative case. It is shown, that the proposed model, which, compared to purely phenomenological macro-scale models, has the advantage of strong micro-mechanical motivation, is capable of qualitatively predicting central features of single crystal shape memory alloy behavior, such as the phase diagram in stress-temperature space, and pseudo-elastic and pseudo-plastic responses, while simultaneously providing valuable insight into the underlying micro-scale mechanisms.

Numerical Analysis and Experimental Validation of the Thermo-Mechanical Flow Behaviour of the Hot Stamping Steel MBW® 1500

In virtual design of the hot stamping process, a reliable description of the material flow behaviour is an important input to ensure accurate estimations of the parts feasibility. Currently, to characterise the hot stamping material’s flow behaviour at elevated temperatures, tensile and upsetting tests are available. The measurement of the flow behaviour out of such tests, which is generally temperature and strain rate dependent, still remains a complex task. Therefore traditional methods to measure flow curves out of such measurements are not necessarily precise enough. In this contribution the authors focus on non-isothermal conductive tensile tests of the manganese-boron steel MBW® 1500 in order to understand its flow behaviour at elevated temperature. Numerical calculations using Finite Element Method (FEM) of the tests itself with correct boundary conditions as well as for all necessary phenomena are used to identify accurately the material’s flow curves by use of inverse optimisation. Finally, for validation purpose the identified flow curves out of the optimisation method were used to simulate the hot stamping of two different parts. Both geometries were chosen such that various strain paths are covered i.e. uniaxial tension to plane strain.

An Advanced Energy Relaxation Scheme for the Modeling of Displacive Phase Transformations

In this contribution, a micro-mechanically motivated constitutive model for phase transformation, martensite reorientation and twin formation in shape memory alloys is proposed. The formulation builds on an effective parametrization of the austenite-twinned martensite microstructure through first- and second-order laminates. To define the effective energy density of the phase mixture, the concept of energy relaxation is applied. The values of the dissipative internal state variables that describe the microstructure evolution are computed via constrained incremental energy minimization. This work also suggests a first step towards the continuous modeling of twin formation embedded into the concept of energy relaxation and can be viewed as a generalization of earlier models suggested in [1–3]. More specifically, in the current model the orientation of martensitic variants in space is not pre-assigned. Variants are rather left free to arrange in an energy-minimizing fashion and are only distinguished by their rotation in reference to a master variant. Finally, macro-homogeneous uniaxial strain and pure shear loading cases are analyzed to demonstrate the capabilities of the proposed modeling framework.Copyright