Thorsten Bartel

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.

Evaluation of different approaches for modeling phase transformations in machining simulation

Presently, the main mechanism for phase transformations in machining of steels is not absolutely clear and is still subject to research. This paper presents, three different approaches for modeling phase transformations during heating in machining operations. However, the main focus lies on two methods which can be classified into a stress related method and a thermal activation related method for the description of austenitization temperature. Both approaches separately showed very good agreements in the simulations compared to the experimental validation but were never compared in a simulation. The third method is a pre-calculated phase landscape assigning the transformation results based on a micro-mechanically motivated constitutive model to the workpiece in dependence on the temperature and strain history. The paper describes all three models in detail, and the results are also presented and discussed.

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.

Modelling and simulation of cyclic thermomechanical behaviour of NiTi wires using a weak discontinuity approach

In this contribution, a thermodynamically consistent and mathematically canonical modelling framework for the investigation of the cyclic thermomechanical behaviour of Nickel-Titanium shape memory alloy wires is developed. Particular focus is placed on the self-heating of the material subjected to multiple load cycles. The relatively high load rates necessitates the consideration of inertia terms, the applied load amplitudes of six percent strain motivates the use of a non-linear, Hencky-type strain measure. Comparisons of the results with experimental data on the one hand reveal reasonable results and on the other hand underline the necessity of further model enhancements.

Erratum to : Modelling and simulation of cyclic thermomechanical behaviour of NiTi wires using a weak discontinuity approach (Int J Fract, 10.1007/s10704-016-0156-0)

Due to an unfortunate turn of events, the following equations were published with erroneous parameters. Please find in this erratum the correct version of the equations that should be regarded as the final version by the reader: (Formula Presented.).

A thermodynamically consistent finite strain micro-sphere framework for phase-transformations

We extend a thermodynamically consistent finite strain micro-sphere framework elaborated by Carol et al. towards the modelling of phase-transformations to allow for the simulation of polycrystalline solids such as, e.g., shape memory alloys and shape memory polymers undergoing large deformations. The considered phase-transformation mechanism is based on statistical physics and allows the consideration of an arbitrary number of solid material phases. The specifically constructed, non-quadratic Helmholtz free energy functions considered in every micro-plane of the micro-sphere framework are extended to include individual Bain-type transformation strains for each of the phases. The total strains acting in each material phase are multiplicatively decomposed into elastic strains and transformation strains. (Less)

Modeling of Single Crystal Magnetostriction Based on Numerical Energy Relaxation Techniques

This paper presents an energy relaxation-based approach for the modeling of single crystalline magnetic shape memor)) alloy response under general two-dimensional magnetomechanical loading. It relies on concepts of energy relaxation in the context of non-convex free energy landscapes whose wells define preferred states of straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] forms the basis for the model development. The key features that characterize the extended approach are (i) dissipative effects, accounted for in an incremental variational setting, and (ii) finite magnetocrystalline anisotropy energy. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the prevariant reorientation regime, and the stress dependence of the maximum field induced strain, can be captured, which are prohibited by the inherent assumptions of the constrained theory. The enhanced modeling capabilities of the extended approach are demonstrated by several representative response simulations and comparison to experimental results taken from literature. These examples particularly focus on the response of single crystals under cyclic magnetic field loading at constant stress, and cyclic mechanical loading at constant magnetic field. (Less)

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