#### Dr.-Ing. Thorsten Bartel

Institute of Mechanics

TU Dortmund University

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- thorsten.bartel@udo.edu
- +49 231 755 2668
- personal website

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**An energy-relaxation-based framework for the modeling of magnetic shape memory alloys—Simulation of three-dimensional effects under homogeneous loading conditions**

Bartel, T. and Kiefer, B. and Menzel, A.*International Journal of Solids and Structures*208-209 (2021)It is known from experimental findings that three-dimensional effects can have a strong influence on magnetic shape memory behavior. Such phenomena are, however, often neglected in MSMA constitutive models, as they only become meaningful under complex loading conditions. The extensions of our original modeling framework, cf. Bartel et al. (2020), to include 3D-effects is threefold: (i) vector-valued microstructural variables are now elements in R3, i.e. no longer parameterizable in polar coordinates, (ii) a third tetragonal martensite variant may form/vanish by switching from/back into both other variants, and (iii) a more general and robust algorithmic treatment is necessary. The latter includes the implementation of a staggered Augmented Lagrangian scheme to handle the now much larger and numerically more advanced sets of equality and inequality constraints. In this context, two extended model formulations are presented. The first considers a first-order, two-variant laminate approach (rank-one convexification), in which domain magnetizations, interface orientations etc. are now three-dimensional vectors. The second model is based on a convexification approach, for which the incorporation of the third martensitic variant is quite natural. Numerical examples are investigated to test the generalized modeling framework. Firstly, it is confirmed that both extended models recover the solution of the previously established two-dimensional model for a simple loading case. Secondly, response predictions for more complex loading scenarios (non-proportional bi-axial stresses, orthogonal magnetic field), motivated by experiments, are investigated. It is found that capturing the formation, elimination and mutual interaction of all martensitic variants as well as general three-dimensional magnetization vector orientations is of key importance under these conditions. The extended convexification model and modified algorithmic formulation are shown to reliably handle even such general cases. © 2020 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2020.10.024 **A computational phase transformation model for selective laser melting processes**

Noll, I. and Bartel, T. and Menzel, A.*Computational Mechanics*66 (2020)Selective laser melting (SLM) has gained large interest due to advanced manufacturing possibilities. However, the growing potential also necessitates reliable predictions of structures in particular regarding their long-term behaviour. The constitutive and structural response is thereby challenging to reproduce, due to the complex material behaviour. This motivates the aims of this contribution: To establish a material model that accounts for the behaviour of the different phases occurring during SLM but that still allows the use of (basic) process simulations. In particular, the present modelling framework explicitly takes into account the mass fractions of the different phases, their mass densities, and specific inelastic strain contributions. The thermomechanically fully coupled framework is implemented into the software Abaqus. The numerical examples emphasise the capabilities of the framework to predict, e.g., the residual stresses occurring in the final part. Furthermore, a postprocessing of averaged inelastic strains is presented yielding a micromechanics-based motivation for inherent strains. © 2020, The Author(s).view abstract 10.1007/s00466-020-01903-4 **An energy-relaxation-based framework for the modelling of magnetic shape memory alloys—Simulation of key response features under homogeneous loading conditions**

Bartel, T. and Kiefer, B. and Buckmann, K. and Menzel, A.*International Journal of Solids and Structures*182-183 (2020)In this contribution we present a constitutive modelling framework for magnetic shape memory alloys (MSMA) that builds on a global variational principle. The approach relies on concepts of energy relaxation and generalised notions of convexity to compute effective energy hulls to the non-convex energy landscape associated with the underlying multi-phase solid, from which the prediction of microstructure evolution results. In this sense it fundamentally distinguishes itself from MSMA models that essentially follow phenomenological concepts of classical plasticity (Kiefer and Lagoudas, 2005; 2009). The microstructure is not spatially resolved, but micro-scale quantities are taken into account in an effective sense by additional state variables—such as volume fractions or interface orientations—and appropriate mixture rules. The model allows all mechanisms central to MSMA behaviour—i.e. variant switching, magnetisation rotation away from easy axes, and magnetic domain evolution—to occur simultaneously. The authors have previously been able to demonstrate that such a modelling approach can quantitatively capture the key characteristics of single-crystalline MSMA response under standard loading scenarios (Kiefer et al., 2015). The modelling framework presented here is now further able to predict much more general response features, such as variant switching diagrams, magnetic field-biased pseudo-elasticity and the influence of specimen shape anisotropy. Moreover, the global variational framework is formulated in a manner that lends itself to finite element implementation. In this work, however, numerical examples are considered in which the nonlocal nature of the demagnetisation field is taken into account in an approximate sense through appropriate shape factors. © 2019 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2019.07.016 **Investigations on enhanced Fischer–Burmeister NCP functions: application to a rate-dependent model for ferroelectrics**

Bartel, T. and Schulte, R. and Menzel, A. and Kiefer, B. and Svendsen, B.*Archive of Applied Mechanics*89 (2019)This contribution deals with investigations on enhanced Fischer–Burmeister nonlinear complementarity problem (NCP) functions applied to a rate-dependent laminate-based material model for ferroelectrics. The framework is based on the modelling and parametrisation of the material’s microstructure via laminates together with the respective volume fractions. These volume fractions are treated as internal-state variables and are subject to several inequality constraints which can be treated in terms of Karush–Kuhn–Tucker conditions. The Fischer–Burmeister NCP function provides a sophisticated scheme to incorporate Karush–Kuhn–Tucker-type conditions into calculations of internal-state variables. However, these functions are prone to numerical instabilities in their original form. Therefore, some enhanced formulations of the Fischer–Burmeister ansatz are discussed and compared to each other in this contribution. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00419-018-1466-7 **Simulation of magnetised microstructure evolution based on a micromagnetics-inspired FE framework: application to magnetic shape memory behaviour**

Buckmann, K. and Kiefer, B. and Bartel, T. and Menzel, A.*Archive of Applied Mechanics*89 (2019)Microstructure evolution in magnetic materials is typically a non-local effect, in the sense that the behaviour at a material point depends on the magnetostatic energy stored within the demagnetisation field in the entire domain. To account for this, we propose a finite element framework in which the internal state variables parameterising the magnetic and crystallographic microstructure are treated as global fields, optimising a global potential. Contrary to conventional micromagnetics, however, the microscale is not spatially resolved and exchange energy terms are neglected in this approach. The influence of microstructure evolution is rather incorporated in an effective manner, which allows the computation of meso- and macroscale problems. This approach necessitates the development and implementation of novel mixed finite element formulations. It further requires the enforcement of inequality constraints at the global level. To handle the latter, we employ Fischer–Burmeister complementarity functions and introduce the associated Lagrange multipliers as additional nodal degrees-of-freedom. As a particular application of this general methodology, a recently established energy-relaxation-based model for magnetic shape memory behaviour is implemented and tested. Special cases—including ellipsoidal specimen geometries—are used to verify the magnetisation and field-induced strain responses obtained from finite element simulations by comparison to calculations based on the demagnetisation factor concept. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00419-018-1482-7 **Towards the simulation of Selective Laser Melting processes via phase transformation models**

Bartel, T. and Guschke, I. and Menzel, A.*Computers and Mathematics with Applications*78 (2019)Selective Laser Melting (SLM) – as one of a number of additive manufacturing techniques – is a promising method for the manufacturing of complex structures and may bring about significant improvements in the context of custom-made designs and lightweight constructions. However, the complex multiphysical processes occurring during SLM necessitates the establishment of appropriate constitutive and process models in order to quantitatively predict the properties of the final workpiece. In particular, the accurate determination of process-induced eigenstresses is a challenging yet important task. In this work, a constitutive modelling framework stemming from phase transformations in shape memory alloys is adopted to the modelling of the changes of state during SLM. This model is based on energy densities and energy minimisation in general and specifically serves as a basis for further enhancements such as the consideration of multiple solid phases of the underlying material. This is particularly considered important due to the fact that the cooling rates during SLM are heterogeneously distributed and that thus different solid phases may form out of the molten material pool. As a first step, the present overall model comprises three phases of the material, namely powder, molten, and re-solidified material. The thermomechanically fully coupled Finite-Element-based process model incorporates approaches for, e.g., the laser beam impact zone and the layer construction model. © 2018 Elsevier Ltdview abstract 10.1016/j.camwa.2018.08.032 **A phenomenological model for the simulation of functional fatigue in shape memory alloy wires**

Bartel, T. and Osman, M. and Menzel, A.*Meccanica*52 (2017)In this contribution, a modelling framework for functional fatigue in shape memory alloy wires is introduced. The approach is in particular designed to reproduce the effective response determined by experiments as published in, e.g., Eggeler et al. (Mat Sci Eng A 378:24–33, 2004). In this context, the decrease of transformation stresses, the increase of irreversible strains, and the occurrence of “characteristic points” with respect to the stress-strain relation is explicitly covered in the model formulation. The modelling approach for the phase transformations itself offers a large potential for further micromechanically well-motivated model extensions. © 2016, Springer Science+Business Media Dordrecht.view abstract 10.1007/s11012-016-0419-x **Active vibration isolation with a dielectric elastomer stack actuator**

Kaal, W. and Bartel, T. and Herold, S.*Smart Materials and Structures*26 (2017)This work presents the development, simulation and experimental investigation of a demonstrator for active vibration isolation with dielectric elastomers (DEs). The electromechanical behavior of the developed DE stack actuator is first characterized experimentally and a suitable simulation model is parametrized accordingly. The potential of the actuator for active vibration isolation is shown in a specially designed single axis test rig. The influence of different control strategies on the transmission behavior from the excited base to the mass is studied. A special aspect of the control strategy is the compensation of the specific nonlinearities. The analysis proves the potential of DE actuators for active vibration isolation purposes. The presented broadband active isolation could enable the use of DEs in various technical fields of application. © 2017 IOP Publishing Ltd.view abstract 10.1088/1361-665X/aa67cb **Erratum to: Modelling and simulation of cyclic thermomechanical behaviour of NiTi wires using a weak discontinuity approach**

Bartel, T. and Menzel, A.*International Journal of Fracture*202 (2016)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.). © 2016, Springer Science+Business Media Dordrecht.view abstract 10.1007/s10704-016-0169-8 **Modelling and simulation of cyclic thermomechanical behaviour of NiTi wires using a weak discontinuity approach**

Bartel, T. and Menzel, A.*International Journal of Fracture*202 (2016)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. © 2016, Springer Science+Business Media Dordrecht.view abstract 10.1007/s10704-016-0156-0 **A kinematically-enhanced relaxation scheme for the modeling of displacive phase transformations**

Bartel, T. and Kiefer, B. and Buckmann, K. and Menzel, A.*Journal of Intelligent Material Systems and Structures*26 (2015)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. © The Author(s) 2014.view abstract 10.1177/1045389X14557507 **An energy-barrier-based computational micro-sphere model for phase-transformations interacting with plasticity**

Ostwald, R. and Bartel, T. and Menzel, A.*Computer Methods in Applied Mechanics and Engineering*293 (2015)We extend a newly introduced framework for the simulation of shape memory alloys undergoing martensite-austenite phase-transformations by allowing for the evolution of individual plastic deformations in each phase considered. The goal is to obtain a generalised model which will facilitate the reflection of the characteristic macroscopic behaviour of SMA as well as TRIP steels. Particularly, we show that the incorporation of plasticity effects interacting with phase-transformations allows to capture the typical multi-cyclic stress-strain responses. As a basis, we use a scalar-valued phase-transformation model where a Helmholtz free energy function depending on volumetric and deviatoric strain measures is assigned to each phase. The incorporation of plasticity phenomena is established by enhancing the deviatoric contributions of the Helmholtz free energy functions of the material phases considered, where the plastic driving forces acting in each phase are derived from the overall free energy potential of the mixture. The resulting energy landscape of the constitutive model is obtained from the contributions of the individual constituents, where the actual energy barriers are computed by minimising parametric intersection curves of elliptic paraboloids. With the energy barriers at hand, we use a statistical physics based approach to determine the resulting evolution of volume fractions due to acting thermo-mechanical loads. Though the model allows to take into account an arbitrary number of solid phases of the underlying material, we restrict the investigations to the simulation of phase-transformations between an austenitic parent phase and a martensitic tension and compression phase. The scalar-valued model is embedded into a computational micro-sphere formulation in order to simulate three-dimensional boundary value problems. The systems of evolution equations are solved in a staggered manner, where a newly proposed, physically motivated plasticity inheritance law accounts for the inheritance of plastic deformations due to evolving phases. © 2015 Elsevier B.V.view abstract 10.1016/j.cma.2015.04.008 **Evaluation of different approaches for modeling phase transformations in machining simulation**

Schulze, V. and Uhlmann, E. and Mahnken, R. and Menzel, A. and Biermann, D. and Zabel, A. and Bollig, P. and Ivanov, I.M. and Cheng, C. and Holtermann, R. and Bartel, T.*Production Engineering*9 (2015)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. © 2015, German Academic Society for Production Engineering (WGP).view abstract 10.1007/s11740-015-0618-7 **Numerical energy relaxation to model microstructure evolution in functional magnetic materials**

Kiefer, B. and Buckmann, K. and Bartel, T.*GAMM Mitteilungen*38 (2015)This paper proposes energy relaxation-based approaches for the modeling of magnetostriction, with a particular focus on single crystalline magnetic shape memory alloy response. The theoretical development relies on concepts of energy relaxation in the context of nonconvex free energy landscapes whose wells define preferred states of spontaneous straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] represents the point of departure for the model development, and its capabilities, but also limitations, are demonstrated by means of representative numerical examples. The key features that characterize the extended approach are (i) the incorporation of elastic deformations, whose distribution among the individual phases occurs in an energy minimizing fashion, (ii) a finite magnetocrystalline anisotropy energy, that allows magnetization rotations away from easy axes, and (iii) dissipative effects, that are accounted for in an incremental variational setting for standard dissipative materials. In the context of introducing elastic strain energy, two different relaxation concepts, the convexification approach and the rank-one relaxation with respect to first-order laminates, are considered. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the pre-variant 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. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201510009 **On the thermomechanical coupling in dissipative materials: A variational approach for generalized standard materials**

Bartels, A. and Bartel, T. and Canadija, M. and Mosler, J.*Journal of the Mechanics and Physics of Solids*82 (2015)This paper deals with the thermomechanical coupling in dissipative materials. The focus lies on finite strain plasticity theory and the temperature increase resulting from plastic deformation. For this type of problem, two fundamentally different modeling approaches can be found in the literature: (a) models based on thermodynamical considerations and (b) models based on the so-called Taylor-Quinney factor. While a naive straightforward implementation of thermodynamically consistent approaches usually leads to an over-prediction of the temperature increase due to plastic deformation, models relying on the Taylor-Quinney factor often violate fundamental physical principles such as the first and the second law of thermodynamics. In this paper, a thermodynamically consistent framework is elaborated which indeed allows the realistic prediction of the temperature evolution. In contrast to previously proposed frameworks, it is based on a fully three-dimensional, finite strain setting and it naturally covers coupled isotropic and kinematic hardening - also based on non-associative evolution equations. Considering a variationally consistent description based on incremental energy minimization, it is shown that the aforementioned problem (thermodynamical consistency and a realistic temperature prediction) is essentially equivalent to correctly defining the decomposition of the total energy into stored and dissipative parts. Interestingly, this decomposition shows strong analogies to the Taylor-Quinney factor. In this respect, the Taylor-Quinney factor can be well motivated from a physical point of view. Furthermore, certain intervals for this factor can be derived in order to guarantee that fundamental physically principles are fulfilled a priori. Representative examples demonstrate the predictive capabilities of the final constitutive modeling framework. © 2015 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.jmps.2015.04.011 **A Gibbs-energy-barrier-based computational micro-sphere model for the simulation of martensitic phase-transformations**

Ostwald, R. and Bartel, T. and Menzel, A.*International Journal for Numerical Methods in Engineering*97 (2014)We introduce a material model for the simulation of polycrystalline materials undergoing solid-to-solid phase-transformations. As a basis, we present a scalar-valued phase-transformation model where a Helmholtz free energy function depending on volumetric and deviatoric strain measures is assigned to each phase. The analysis of the related overall Gibbs energy density allows for the calculation of energy barriers. With these quantities at hand, we use a statistical-physics-based approach to determine the resulting evolution of volume fractions. Though the model facilitates to take into account an arbitrary number of solid phases of the underlying material, we restrict this work to the simulation of phase-transformations between an austenitic parent phase and a martensitic tension and compression phase. The scalar model is embedded into a computational micro-sphere formulation in view of the simulation of three-dimensional boundary value problems. The final modelling approach necessary for macroscopic simulations is accomplished by a finite element formulation, where the local material behaviour at each integration point is governed by the response of the micro-sphere model. © 2014 John Wiley & Sons, Ltd.view abstract 10.1002/nme.4601 **Modeling of single crystal magnetostriction based on numerical energy relaxation techniques**

Kiefer, B. and Buckmann, K. and Bartel, T. and Menzel, A.*ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, SMASIS 2014*1 (2014)This paper presents an energy relaxation-based approach for the modeling of single crystalline magnetic shape memory 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. © 2014 by ASME.view abstract 10.1115/SMASIS20147436 **Towards the multi-scale simulation of martensitic phase-transformations: An efficient post-processing approach applied to turning processes**

Ostwald, R. and Tiffe, M. and Bartel, T. and Zabel, A. and Menzel, A. and Biermann, D.*Journal of Materials Processing Technology*214 (2014)This work presents an efficient finite element based scheme for the prediction of process properties and especially the material condition of workpiece surfaces after turning. This is achieved by using a database generated with the help of a micromechanically motivated material model - capable of simulating interactions of phase transitions and plasticity - for the efficient post-processing of a macroscopic thermo-mechanically coupled finite element simulation of the turning process. This modelling technique is applied to the martensitic part of a functionally graded workpiece which is produced by thermo-mechanically controlled forging processes. Those workpieces provide locally varying material conditions, which are tailored to the later application. The resulting pre-products have to be turned in order to achieve the desired final workpiece geometry and surfaces. Such processes strongly affect material properties such as hardness and ductility. A deterioration of the functionality of the gradation, i.e. the martensitic surface properties, may occur by generation of residual tensile principal stresses which can occur accompanied by white layer formation. These deteriorations can be avoided by adjusting the process parameters appropriately. Especially the cutting speed is supposed to be on a low level (vc < 80 m/min) to avoid thermally driven formation of a white layer and the generation of tensile residual stresses. It is shown how finite element simulations can give insight into the material interactions and thereby facilitate the support of the process parameter adjustment in order to support efficient and reliable part production in industrial applications. © 2014 Elsevier B.V.view abstract 10.1016/j.jmatprotec.2014.02.022 **An advanced energy relaxation scheme for the modeling of displacive phase transformations**

Bartel, T. and Buckmann, K. and Kiefer, B. and Menzel, A.*ASME 2013 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, SMASIS 2013*1 (2013)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 © 2013 by ASME.view abstract 10.1115/SMASIS2013-3041 **Implementation of numerical integration schemes for the simulation of magnetic SMA constitutive response**

Kiefer, B. and Bartel, T. and Menzel, A.*Smart Materials and Structures*21 (2012)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 predictorcorrector-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. © 2012 IOP Publishing Ltd.view abstract 10.1088/0964-1726/21/9/094007 **Partially relaxed energy potentials for the modelling of microstructures - Application to shape memory alloys**

Bartel, T. and Menzel, A.*GAMM Mitteilungen*35 (2012)Energy relaxation is well-established by several researchers - especially in the field of the modelling of solid-solid phase transformations. Nevertheless, critics still counter this concept by considering it as a purely mathematical tool with poor physical significance. In this contribution we aim at emphasising the significance of energy relaxation methods for the modelling of dissipative solids and especially microstructure formation and its further evolution. In particular, we shall point out aspects and advantages of this concept which are not straight forward to achieve within alternative schemes. ©c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201210005 **Phase-transformations interacting with plasticity - A micro-sphere model applied to TRIP steel**

Ostwald, R. and Bartel, T. and Menzel, A.*Computational Materials Science*64 (2012)We present an efficient model for the simulation of polycrystalline materials, particularly accounting for the interactions of solid to solid phase-transformations and plasticity. The underlying one-dimensional model is embedded into a micro-sphere formulation in order to simulate three-dimensional boundary value problems. Representative numerical examples are provided for both the micro-level and the macro-level. Moreover, a finite element implementation of the model is presented and discussed. © 2012 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2012.05.015 **Experimental and computational investigation of machining processes for functionally graded materials**

Biermann, D. and Menzel, A. and Bartel, T. and Höhne, F. and Holtermann, R. and Ostwald, R. and Sieben, B. and Tiffe, M. and Zabel, A.*Procedia Engineering*19 (2011)Experiments on dry face turning of functionally graded heat treatable steel are conducted. The workpieces have a hardened zone of approx. 60 HRC and a non-hardened zone of approx. 30 HRC. PCBN tools are used with different feeds, cutting speeds and depths of cut. Measurements of residual stresses in the surface layer reveal compressive stresses in the hardened zone and tensile stresses in the non-hardened zone. These experimental observations are compared with the results of representative simulations of the cutting process. A large-deformation thermo-elastoviscoplastic material model is used and the geometry of the cutting tool is precisely reflected by the finite element discretisation. To predict the overall response, an adaptive remeshing scheme and full thermo-mechanical coupling is accounted for. Moreover, measured residual stresses are incorporated as initial conditions within the simulation. © 2012 Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2011.11.074 **Thermodynamic and relaxation-based modeling of the interaction between martensitic phase transformations and plasticity**

Bartel, T. and Menzel, A. and Svendsen, B.*Journal of the Mechanics and Physics of Solids*59 (2011)This paper focuses on the issue plasticity within the framework of a micromechanical model for single-crystal shape-memory alloys. As a first step towards a complete micromechanical formulation of such models, we work with classical J2-von Mises-type plasticity for simplicity. The modeling of martensitic phase transitions is based on the concept of energy relaxation (quasiconvexification) in connection with evolution equations derived from inelastic potentials. Crystallographic considerations lead to the derivation of Bain strains characterizing the transformation kinematics. The model is derived for arbitrary numbers of martensite variants and thus can be applied to any shape-memory material such as CuAlNi or NiTi. The phase transition model captures effects like tension/compression asymmetry and transformation induced anisotropy. Additionally, attention is focused on the interaction between phase transformations and plasticity in terms of the inheritance of plastic strain. The effect of such interaction is demonstrated by elementary numerical studies. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.jmps.2011.02.006 **A computational micro-sphere model applied to the simulation of phase-transformations**

Ostwald, R. and Bartel, T. and Menzel, A.*Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik*90 (2010)We present an efficient model for the simulation of polycrystalline materials undergoing solid to solid phase transformations. As a basis, we use a one-dimensional, thermodynamically consistent phase-transformation model. This model is embedded into a micro-sphere formulation in order to simulate three-dimensional boundary value problems. To solve the underlying evolution equations, we use a newly developed explicit integration scheme which could be proved to be unconditionally A-stable. Besides the investigation of homogeneous deformation states, representative finite element examples are discussed. It is shown that the model nicely reflects the overall behaviour. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheimview abstract 10.1002/zamm.200900390

#### finite element method

#### martensite

#### shape memory effect

#### single crystals