#### Prof. Dr.-Ing. Andreas Menzel

Institut of Mechanics

TU Dortmund University

##### Contact

- andreas[dot]menzel[at]udo[dot]edu
- +49 231 755 5744
- personal website

##### Author IDs

- ORCID: 0000-0002-7819-9254
- Scopus: 7006329394

##### Hub

**A thermo-electro-mechanically coupled cohesive zone formulation for predicting interfacial damage**

Güzel, D. and Kaiser, T. and Menzel, A.*European Journal of Mechanics, A/Solids*99 (2023)view abstract 10.1016/j.euromechsol.2023.104935 **Machine learning-assisted parameter identification for constitutive models based on concatenated loading path sequences**

Schulte, R. and Karca, C. and Ostwald, R. and Menzel, A.*European Journal of Mechanics, A/Solids*98 (2023)view abstract 10.1016/j.euromechsol.2022.104854 **Modelling and numerical simulation of remodelling processes in cortical bone: An IGA approach to flexoelectricity-induced osteocyte apoptosis and subsequent bone cell diffusion**

Witt, C. and Kaiser, T. and Menzel, A.*Journal of the Mechanics and Physics of Solids*173 (2023)view abstract 10.1016/j.jmps.2022.105194 **Probing porosity in metals by electrical conductivity: Nanoscale experiments and multiscale simulations**

Kaiser, T. and Dehm, G. and Kirchlechner, C. and Menzel, A. and Bishara, H.*European Journal of Mechanics, A/Solids*97 (2023)view abstract 10.1016/j.euromechsol.2022.104777 **A micromechanically motivated multiscale approach for residual distortion in laser powder bed fusion processes**

Noll, I. and Koppka, L. and Bartel, T. and Menzel, A.*Additive Manufacturing*60 (2022)view abstract 10.1016/j.addma.2022.103277 **A thermo-viscoplasticity model for metals over wide temperature ranges- application to case hardening steel**

Oppermann, P. and Denzer, R. and Menzel, A.*Computational Mechanics*69 (2022)In this contribution, a model for the thermomechanically coupled behaviour of case hardening steel is introduced with application to 16MnCr5 (1.7131). The model is based on a decomposition of the free energy into a thermo-elastic and a plastic part. Associated viscoplasticity, in terms of a temperature-depenent Perzyna-type power law, in combination with an isotropic von Mises yield function takes respect for strain-rate dependency of the yield stress. The model covers additional temperature-related effects, like temperature-dependent elastic moduli, coefficient of thermal expansion, heat capacity, heat conductivity, yield stress and cold work hardening. The formulation fulfils the second law of thermodynamics in the form of the Clausius–Duhem inequality by exploiting the Coleman–Noll procedure. The introduced model parameters are fitted against experimental data. An implementation into a fully coupled finite element model is provided and representative numerical examples are presented showing aspects of the localisation and regularisation behaviour of the proposed model. © 2021, The Author(s).view abstract 10.1007/s00466-021-02103-4 **Electromechanical Coupling in Electroactive Polymers – a Visual Analysis of a Third-Order Tensor Field**

Hergl, C. and Witt, C. and Nsonga, B. and Menzel, A. and Scheuermann, G.*IEEE Transactions on Visualization and Computer Graphics*(2022)Electroactive polymers are frequently used in engineering applications due to their ability to change their shape and properties under the influence of an electric field. This process also works vice versa, such that mechanical deformation of the material induces an electric field in the EAP device. This specific behavior makes such materials highly attractive for the construction of actuators and sensors in various application areas. The electromechanical behaviour of electroactive polymers can be described by a third-order coupling tensor, which represents the sensitivity of mechanical stresses concerning the electric field, i.e., it establishes a relation between a second-order and a first-order tensor field. Due to this coupling tensor's complexity and the lack of meaningful visualization methods for third-order tensors in general, an interpretation of the tensor is rather difficult. Thus, the central engineering research question that this contribution deals with is a deeper understanding of electromechanical coupling by analyzing the third-order coupling tensor with the help of specific visualization methods. Starting with a deviatoric decomposition of the tensor, the multipoles of each deviator are visualized, which allows a first insight into this highly complex third-order tensor. In the present contribution, four examples, including electromechanical coupling, are simulated within a finite element framework and subsequently analyzed using the tensor visualization method. IEEEview abstract 10.1109/TVCG.2022.3209328 **Extremal states and coupling properties in electroelasticity**

Menzel, A. and Witt, C.*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*380 (2022)Electroelastic materials possess properties most attractive for the design of smart devices and systems such as actuators and sensors. Typical polymers show changes in shape under the action of an electric field, and vice versa, together with fast actuation times, high strain levels and low elastic moduli. This paper deals with an Ogden model inspired framework for large deformation electroelasticity which, as a special case, can also be reduced to the modelling of transversely isotropic elasticity. Extremal (local) states are elaborated based on a coaxiality analysis, i.e. extremal states of energy are considered at fixed deformation and changing direction of electric field, respectively, fixed electric field and changing principal directions of deformation. This analysis results in extremal states when stresses and strain commutate, respectively, dielectric displacements and electric field are aligned. In order to further elaborate electromechanical coupling properties, the sensitivity of stresses with respect to electric field is analysed. This sensitivity is represented by a third-order tensor which, in general, depends on deformation and electric field. To illustrate this third-order tensor, a decomposition into deviators is adopted. Related norms of these deviators, together with the electromechanical coupling contribution to the augmented energy, are investigated for different states under homogeneous deformation and changing electric field direction. The analysis is considered to contribute to a better understanding of electromechanical coupling properties and extremal states in large deformation electroelasticity and by that, as a long-term goal, may contribute to the improved design of related smart devices and systems. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'. © 2022 The Author(s).view abstract 10.1098/rsta.2021.0330 **On the determination of thermal boundary conditions for parameter identifications of thermo-mechanically coupled material models**

Rose, L. and Menzel, A.*GAMM Mitteilungen*45 (2022)Identifiability and sensitivity of thermal boundary coefficients identified alongside thermal material parameters by means of full field measurements during a simple tension test are shown empirically using a simple tension test with self heating as a proof of concept. The identification is started for 10 different initial guesses, all of which converge toward the same optimum. The solution appears to be locally unique and parameters therefore independent, but a comparison against a reference solution indicates high correlation between three model parameters and the prescribed external temperatures required to model heat exchange with either air or clamping jaws. This sensitivity is further analyzed by rerunning the identification with different prescribed external temperatures and by comparing the obtained optimal parameter values. Although the model parameters are independent, optimal values for heat conduction and the heat transfer coefficients are highly correlated as well as sensitive with respect to a change, respectively, measurement error of the external temperatures. A precise fit on the basis of a simple tension test therefore requires precise measurements and a suitable material model which is able to accurately predict dissipated energy. © 2022 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.view abstract 10.1002/gamm.202200010 **A distortional hardening model for finite plasticity**

Meyer, K.A. and Menzel, A.*International Journal of Solids and Structures*232 (2021)Plastic anisotropy may strongly affect the stress and strain response in metals subjected to multiaxial cyclic loading. This anisotropy evolves due to various microstructural features. We first use simple models to study how such features result in evolving plastic anisotropy. A subsequent analysis of existing distortional hardening models highlights the difference between stress- and strain-driven models. Following this analysis, we conclude that the stress-driven approach is most suitable and propose an improved stress-driven model. It is thermodynamically consistent and guarantees yield surface convexity. Many distortional hardening models in the literature do not fulfill the latter. In contrast, the model proposed in this work has a convex yield surface independent of its parameter values. Experimental results, considering yield surface evolution after large shear strains, are used to assess the model's performance. We carefully analyze the experiments in the finite strain setting, showing how the numerical results can be compared with the experimental results. The new model fits the experimental results significantly better than its predecessor without introducing additional material parameters. © 2021 The Author(s)view abstract 10.1016/j.ijsolstr.2021.111055 **A finite deformation electro-mechanically coupled computational multiscale formulation for electrical conductors**

Kaiser, T. and Menzel, A.*Acta Mechanica*232 (2021)Motivated by the influence of deformation-induced microcracks on the effective electrical properties at the macroscale, an electro-mechanically coupled computational multiscale formulation for electrical conductors is proposed. The formulation accounts for finite deformation processes and is a direct extension of the fundamental theoretical developments presented by Kaiser and Menzel (Arch Appl Mech 91:1509–1526, 2021) who assume a geometrically linearised setting. More specifically speaking, averaging theorems for the electric field quantities are proposed and boundary conditions that a priori fulfil the extended Hill–Mandel condition of the electro-mechanically coupled problem are discussed. A study of representative boundary value problems in two- and three-dimensional settings eventually shows the applicability of the proposed formulation and reveals the severe influence of microscale deformation processes on the effective electrical properties at the macroscale. © 2021, The Author(s).view abstract 10.1007/s00707-021-03005-5 **A finite deformation isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness**

Witt, C. and Kaiser, T. and Menzel, A.*Journal of Engineering Mathematics*128 (2021)It is a common technique in many fields of engineering to reinforce materials with certain types of fibres in order to enhance the mechanical properties of the overall material. Specific simulation methods help to predict the behaviour of these composites in advance. In this regard, a widely established approach is the incorporation of the fibre direction vector as an additional argument of the energy function in order to capture the specific material properties in the fibre direction. While this model represents the transverse isotropy of a material, it cannot capture effects that result from a bending of the fibres and does not include any length scale that might allow the simulation of size effects. In this contribution, an enhanced approach is considered which relies on the introduction of higher-gradient contributions of the deformation map in the stored energy density function and which eventually allows accounting for fibre bending stiffness in simulations. The respective gradient fields are approximated by NURBS basis functions within an isogeometric finite element framework by taking advantage of their characteristic continuity properties. The isogeometric finite element approach that is presented in this contribution for fibre-reinforced composites with fibre bending stiffness accounts for finite deformations. It is shown that the proposed method is in accordance with semi-analytical solutions for a representative boundary value problem. In an additional example it is observed that the initial fibre orientation and the particular bending stiffness of the fibres influence the deformation as well as the stress response of the material. © 2021, The Author(s).view abstract 10.1007/s10665-021-10117-3 **A finite element implementation of the stress gradient theory**

Kaiser, T. and Forest, S. and Menzel, A.*Meccanica*56 (2021)In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions. © 2021, The Author(s).view abstract 10.1007/s11012-020-01266-3 **A thermodynamically consistent modelling framework for strongly time-dependent bainitic phase transitions**

Bartel, T. and Geuken, G.-L. and Menzel, A.*International Journal of Solids and Structures*232 (2021)In this work, a thermodynamically consistent constitutive framework is introduced that is capable of reproducing the significant time-dependent behaviour of austenite-to-bainite phase transformations. In particular, the aim is to incorporate the effect of these diffusion-controlled processes by plasticity-like evolution equations instead of incorporating related global diffusion equations. To this end, a variational principle for inelastic solids is adopted and enhanced by an additional term. This term essentially contributes to the evolution equations for the phase volume fractions of several crystallography-based bainite variants. Due to the specific modifications, special attention has to be paid with respect to the fulfilment of thermodynamical consistency, which can be shown to be unconditionally satisfied for the newly proposed modelling framework. The phase transformation model itself is based on the convexification of a multi-well energy density landscape in order to provide the effective material response for possible phase mixtures. Several material parameters are determined via parameter identification based on available experimental results for 51CrV4, which also allow the quantitative evaluation of the predicted results. © 2021 The Authorsview abstract 10.1016/j.ijsolstr.2021.111172 **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 **Electrical and mechanical behaviour of metal thin films with deformation-induced cracks predicted by computational homogenisation**

Kaiser, T. and Cordill, M.J. and Kirchlechner, C. and Menzel, A.*International Journal of Fracture*231 (2021)Motivated by advances in flexible electronic technologies and by the endeavour to develop non-destructive testing methods, this article analyses the capability of computational multiscale formulations to predict the influence of microscale cracks on effective macroscopic electrical and mechanical material properties. To this end, thin metal films under mechanical load are experimentally analysed by using in-situ confocal laser scanning microscopy (CLSM) and in-situ four point probe resistance measurements. Image processing techniques are then used to generate representative volume elements from the laser intensity images. These discrete representations of the crack pattern at the microscale serve as the basis for the calculation of effective macroscopic electrical conductivity and mechanical stiffness tensors by means of computational homogenisation approaches. A comparison of simulation results with experimental electrical resistance measurements and a detailed study of fundamental numerical properties demonstrates the applicability of the proposed approach. In particular, the (numerical) errors that are induced by the representative volume element size and by the finite element discretisation are studied, and the influence of the filter that is used in the generation process of the representative volume element is analysed. © 2021, The Author(s).view abstract 10.1007/s10704-021-00582-3 **Fundamentals of electro-mechanically coupled cohesive zone formulations for electrical conductors**

Kaiser, T. and Menzel, A.*Computational Mechanics*68 (2021)Motivated by the influence of (micro-)cracks on the effective electrical properties of material systems and components, this contribution deals with fundamental developments on electro-mechanically coupled cohesive zone formulations for electrical conductors. For the quasi-stationary problems considered, Maxwell’s equations of electromagnetism reduce to the continuity equation for the electric current and to Faraday’s law of induction, for which non-standard jump conditions at the interface are derived. In addition, electrical interface contributions to the balance equation of energy are discussed and the restrictions posed by the dissipation inequality are studied. Together with well-established cohesive zone formulations for purely mechanical problems, the present developments provide the basis to study the influence of mechanically-induced interface damage processes on effective electrical properties of conductors. This is further illustrated by a study of representative boundary value problems based on a multi-field finite element implementation. © 2021, The Author(s).view abstract 10.1007/s00466-021-02019-z **Identification of thermal material parameters for thermo-mechanically coupled material models: Verification and model dependency**

Rose, L. and Menzel, A.*Meccanica*(2021)The possibility of accurately identifying thermal material parameters on the basis of a simple tension test is presented, using a parameter identification framework for thermo-mechanically coupled material models on the basis of full field displacement and temperature field measurements. Main objective is to show the impact of the material model formulation on the results of such an identification with respect to accuracy and uniqueness of the result. To do so, and as a proof of concept, the data of two different experiments is used. One experiment including cooling of the specimen, due to ambient temperature, and one without specimen cooling. The main constitutive relations of two basic material models are summarised (associated and non-associated plasticity), whereas both models are extended so as to introduce an additional material parameter for the thermodynamically consistent scaling of dissipated energy. The chosen models are subjected to two parameter identifications each, using the data of either experiment and focusing on the determination of thermal material parameters. The influence of the predicted dissipated energy of the models on the identification process is investigated showing that a specific material model formulation must be chosen carefully. The material model with associated evolution equations used within this work does neither allow a unique identification result, nor is any of the solutions for the underlying material parameters close to literature values. In contrast to that, a stable, that is locally unique, re-identification of the literature values is possible for the boundary problem at hand if the model with non-associated evolution equation is used and if cooling is included in the experimental data. © 2021, The Author(s).view abstract 10.1007/s11012-020-01267-2 **On the incorporation of a micromechanical material model into the inherent strain method—Application to the modeling of selective laser melting**

Noll, I. and Bartel, T. and Menzel, A.*GAMM Mitteilungen*44 (2021)When developing reliable and useful models for selective laser melting processes of large parts, various simplifications are necessary to achieve computationally efficient simulations. Due to the complex processes taking place during the manufacturing of such parts, especially the material and heat source models influence the simulation results. If accurate predictions of residual stresses and deformation are desired, both complete temperature history and mechanical behavior have to be included in a thermomechanical model. In this article, we combine a multiscale approach using the inherent strain method with a newly developed phase transformation model. With the help of this model, which is based on energy densities and energy minimization, the three states of the material, namely, powder, molten, and resolidified material, are explicitly incorporated into the thermomechanically fully coupled finite-element-based process model of the micromechanically motivated laser heat source model and the simplified layer hatch model. © 2021 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.view abstract 10.1002/gamm.202100015 **Simulation based prediction of compliance induced shape deviations in internal traverse grinding**

Tsagkir Dereli, T. and Schmidt, N. and Furlan, T. and Holtermann, R. and Biermann, D. and Menzel, A.*Journal of Manufacturing and Materials Processing*5 (2021)Internal traverse grinding (ITG) using electroplated cBN tools in high-speed grinding conditions is a highly efficient manufacturing process for bore machining in a single axial stroke. However, process control is difficult. Due to the axial direction of feed, changes in process normal force and thus radial deflection of the tool and workpiece spindle system, lead to deviations in the workpiece contour along the length of the bore, especially at tool exit. Simulations including this effect could provide a tool to design processes which enhance shape accuracy of components. A geometrical physically-based simulation is herein developed to model the influence of system compliance on the resulting workpiece contour. Realistic tool topographies, obtained from measurements, are combined with an FE-calibrated surrogate model for process forces and with an empirical compliance model. In quasistatic experimental investigations, the spindle deflection is determined in relation to the acting normal forces by using piezoelectric force measuring elements and eddy current sensors. In grinding tests with in-process force measurement technology and followed by measurement of the resulting workpiece contours, the simulation system is validated. The process forces and the resulting characteristic shape deviations are predicted in good qualitative accordance with the experimental results. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/jmmp5020060 **Simulation of wear and effective friction properties of microstructured surfaces**

Schewe, M. and Wilbuer, H. and Menzel, A.*Wear*464-465 (2021)Wear and friction characteristics are simulated on metal forming tools with tailored surfaces generated by micro-milling. Friction homogenisation is applied to study surface cut-outs on the meso-scale where the structures are resolved by means of finite element methods and where asperities are represented by a combined friction law appropriate for metal forming. Dissipation based and pressure based Archard wear relations are implemented in a postprocessor, and wear distributions as well as effective friction properties are investigated. Sinusoidal surface structures are able to provide anisotropic structural resistance throughout the progress of wear. A bionic surface structure shows quasi-isotropic structural resistance where sliding directions across the edge directions are benefitial with regard to the wear progress. Experimental measurements from a wear experiment give hints which support the dissipation based Archard relation while more experimental evidence is necessary. © 2020 The Authorsview abstract 10.1016/j.wear.2020.203491 **Structural optimisation of diffusion driven degradation processes**

Waschinsky, N. and Barthold, F.-J. and Menzel, A.*Structural and Multidisciplinary Optimization*(2021)In this article, we propose an optimisation framework that can contribute to the prevention of premature failure or damage to building structures and can thereby strengthen their longevity. We concentrate on structures that are contaminated by chemical substances and that have strong destructive effects on the material. The aim of this contribution is a mathematical algorithm that allows the optimisation of a structure exposed to chemical influences and thus the assurance of the static load capacity. To achieve this, we present a coupled mechanical-diffusion-degradation approach embedded in a finite element (FE) framework. Furthermore, we integrate an optimisation algorithm to reduce material degradation. In this paper, we establish shape optimisation of a structure with a gradient based optimisation algorithm. © 2021, The Author(s).view abstract 10.1007/s00158-021-02900-8 **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 **A covariant formulation of finite plasticity with plasticity-induced evolution of anisotropy: Modeling, algorithmics, simulation, and comparison to experiments**

Kaiser, T. and Lu, J. and Menzel, A. and Papadopoulos, P.*International Journal of Solids and Structures*185-186 (2020)Motivated by experimental findings on sheet-metal forming, this article concerns the modeling of evolving anisotropies in finite plasticity. A covariant formulation of plasticity is employed in conjunction with evolution equations for the structural tensors that characterize the symmetry group of the yield function. A specific model is implemented into a finite element code to simulate tension and torsion tests. The results are then compared to experiments. © 2019 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2019.08.005 **A large strain gradient-enhanced ductile damage model: finite element formulation, experiment and parameter identification**

Sprave, L. and Menzel, A.*Acta Mechanica*231 (2020)A gradient-enhanced ductile damage model at finite strains is presented, and its parameters are identified so as to match the behaviour of DP800. Within the micromorphic framework, a multi-surface model coupling isotropic Lemaitre-type damage to von Mises plasticity with nonlinear isotropic hardening is developed. In analogy to the effective stress entering the yield criterion, an effective damage driving force—increasing with increasing plastic strains—entering the damage dissipation potential is proposed. After an outline of the basic model properties, the setup of the (micro)tensile experiment is discussed and the importance of including unloading for a parameter identification with a material model including damage is emphasised. Optimal parameters, based on an objective function including measured forces and the displacement field obtained from digital image correlation, are identified. The response of the proposed model is compared to a tensile experiment of a specimen with a different geometry as a first approach to validate the identified parameters. © 2020, The Author(s).view abstract 10.1007/s00707-020-02786-5 **An electro-mechanically coupled computational multiscale formulation for electrical conductors**

Kaiser, T. and Menzel, A.*Archive of Applied Mechanics*(2020)Motivated by experimental findings on deformation induced microcracks in thin metal films and by their influence on the effective macroscopic electrical conductivity, a computational multiscale formulation for electrical conductors is proposed in this contribution. In particular, averaging theorems for kinematic quantities and for their energetic duals are discussed, an extended version of the Hill–Mandel energy equivalence condition is proposed and suitable boundary conditions for the microscale problem are elaborated. The implementation of the proposed framework in a two-scale finite element environment is shown and representative boundary value problems are studied in two- and three-dimensional settings. © 2020, The Author(s).view abstract 10.1007/s00419-020-01837-6 **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 **An isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness**

Witt, C. and Kaiser, T. and Menzel, A.*Archive of Applied Mechanics*(2020)In the modelling of fibre-reinforced composites, it is well established to consider the fibre direction in the stored energy in order to account for the transverse isotropy of the overall material, induced by a single family of fibres. However, this approach does not include any length scale and therefore lacks in the prediction of size effects that may occur from the fibre diameter or spacing. By making use of a generalised continuum model including non-symmetric stresses and couple-stresses, the gradient of the fibre direction vector can be taken into account as an additional parameter of the stored energy density function. As a consequence, the enhanced model considers the bending stiffness of the fibres and includes information on the material length scale. Along with additional material parameters, increased continuity requirements on the basis functions follow in the finite element analysis. The isogeometric finite element method provides a framework which can fulfil these requirements of the corresponding weak formulation. In the present contribution, the method is applied to two representative numerical examples. At first, the bending deformation of a cantilever beam is studied in order to analyse the influence of the fibre properties. An increasingly stiff response is observed as the fibre bending stiffness increases and as the fibre orientation aligns with the beam’s axis. Secondly, a fibre-reinforced cylindrical tube under a pure azimuthal shear deformation is considered. The corresponding simulation results are compared against a semi-analytical solution. It is shown that the isogeometric analysis yields highly accurate results for the boundary value problem under consideration. © 2020, The Author(s).view abstract 10.1007/s00419-020-01754-8 **Computational shape optimisation for a gradient-enhanced continuum damage model**

Guhr, F. and Sprave, L. and Barthold, F.-J. and Menzel, A.*Computational Mechanics*65 (2020)An isotropic gradient-enhanced damage model is applied to shape optimisation in order to establish a computational optimal design framework in view of optimal damage distributions. The model is derived from a free Helmholtz energy density enriched by the damage gradient contribution. The Karush–Kuhn–Tucker conditions are solved on a global finite element level by means of a Fischer–Burmeister function. This approach eliminates the necessity of introducing a local variable, leaving only the global set of equations to be iteratively solved. The necessary steps for the numerical implementation in the sense of the finite element method are established. The underlying theory as well as the algorithmic treatment of shape optimisation are derived in the context of a variational framework. Based on a particular finite deformation constitutive model, representative numerical examples are discussed with a focus on and application to damage optimised designs. © 2020, The Author(s).view abstract 10.1007/s00466-019-01810-3 **Continuum Damage Mechanics—Modelling and Simulation**

Menzel, A. and Sprave, L.*Solid Mechanics and its Applications*262 (2020)Continuum damage mechanics elaborates the continuum mechanics-based modelling and simulation of mechanical degradation effects. The objective of this contribution is to briefly review different aspects of continuum damage mechanics of solid continua with a focus on general modelling concepts and application to isotropic as well as anisotropic damage approaches on the one hand, and to discuss possible solution strategies in the context of finite element simulations on the other. In particular, viscous regularisation and gradient-enhanced regularisation—as a reduced form of general non-local theories—are considered. Several numerical examples including ductile damage, i.e. the coupling of damage with plasticity related phenomena, are addressed which show the applicability of the particular modelling and simulation frameworks highlighted. © 2020, Springer Nature Switzerland AG.view abstract 10.1007/978-3-030-31547-4_8 **Gradient-enhanced modelling of damage for rate-dependent material behaviour-a parameter identification framework**

Schulte, R. and Ostwald, R. and Menzel, A.*Materials*13 (2020)The simulation of complex engineering components and structures under loads requires the formulation and adequate calibration of appropriate material models. This work introduces an optimisation-based scheme for the calibration of viscoelastic material models that are coupled to gradient-enhanced damage in a finite strain setting. The parameter identification scheme is applied to a self-diagnostic poly(dimethylsiloxane) (PDMS) elastomer, where so-called mechanophore units are incorporated within the polymeric microstructure. The present contribution, however, focuses on the purely mechanical response of the material, combining experiments with homogeneous and inhomogeneous states of deformation. In effect, the results provided lay the groundwork for a future extension of the proposed parameter identification framework, where additional field-data provided by the self-diagnostic capabilities can be incorporated into the optimisation scheme. © 2020 by the authors.view abstract 10.3390/ma13143156 **On mesh dependencies in finite-element-based damage prediction: application to sheet metal bending**

Sprave, L. and Schowtjak, A. and Meya, R. and Clausmeyer, T. and Tekkaya, A.E. and Menzel, A.*Production Engineering*14 (2020)The properties of a local and a regularised gradient-enhanced continuum damage model are highlighted and both types of models are applied to the simulation of an air bending process. Constitutive relations are summarised for both Lemaitre-type models and a brief description of their implementation into Abaqus user material subroutines is given. With (several) material parameters obtained from a basic parameter identification process, an air bending experiment is simulated with different mesh densities. By means of the damage evolution as well as the distribution of representative damage and hardening variables, the mesh dependence of the local model in contrast to the mesh independence of the gradient-enhanced model is analysed for two air bending processes with different die width. © 2019, German Academic Society for Production Engineering (WGP).view abstract 10.1007/s11740-019-00937-9 **Optimisation based material parameter identification using full field displacement and temperature measurements**

Rose, L. and Menzel, A.*Mechanics of Materials*145 (2020)A material parameter identification is presented for a fully thermo-mechanically coupled material model based on full field displacement and temperature measurements. The basic theory of the inverse problem is recapitulated, focusing on the choice of the objective function, proposing a new formulation, and explaining in detail the necessary numerical treatment of experimental data during the pre-processing of an identification. This includes the handling of the intrinsically different data sets of displacement (Lagrangian type) and temperature (Eulerian type). Experimental data is obtained by means of a Digital-Image-Correlation (DIC) as well as by a thermography system and three algorithmic boxes are provided for the necessary pre-processing. The experimental setup is discussed, measured data presented and analysed. From this setup, a successive approach to the identification process is motivated. Based on the experimental observations, a thermo-mechanically coupled material model is chosen, the required constitutive relations summarised and the material parameters interpreted. For the fixed choice of model and experiments, the inverse problem is solved. A very good fit was obtained for both the displacement and the temperature field. Results are interpreted and remaining errors discussed. © 2019 Elsevier Ltdview abstract 10.1016/j.mechmat.2019.103292 **The role of microscale solid matrix compressibility on the mechanical behaviour of poroelastic materials**

Dehghani, H. and Noll, I. and Penta, R. and Menzel, A. and Merodio, J.*European Journal of Mechanics, A/Solids*83 (2020)We present the macroscale three-dimensional numerical solution of anisotropic Biot's poroelasticity, with coefficients derived from a micromechanical analysis as prescribed by the asymptotic homogenisation technique. The system of partial differential equations (PDEs) is discretised by finite elements, exploiting a formal analogy with the fully coupled thermal displacement systems of PDEs implemented in the commercial software Abaqus. The robustness of our computational framework is confirmed by comparison with the well-known analytical solution of the one-dimensional Therzaghi's consolidation problem. We then perform three-dimensional numerical simulations of the model in a sphere (representing a biological tissue) by applying a given constant pressure in the cavity. We investigate how the macroscale radial displacements (as well as pressures) profiles are affected by the microscale solid matrix compressibility (MSMC). Our results suggest that the role of the MSMC on the macroscale displacements becomes more and more prominent by increasing the length of the time interval during which the constant pressure is applied. As such, we suggest that parameter estimation based on techniques such as poroelastography (which are commonly used in the context of biological tissues, such as the brain, as well as solid tumours) should allow for a sufficiently long time in order to give a more accurate estimation of the mechanical properties of tissues. © 2020 The Authorsview abstract 10.1016/j.euromechsol.2020.103996 **A dislocation density tensor-based crystal plasticity framework**

Kaiser, T. and Menzel, A.*Journal of the Mechanics and Physics of Solids*131 (2019)The present contribution addresses a crystal plasticity formulation which incorporates hardening effects that are related to the presence of geometrically necessary dislocations. To this end, higher gradient contributions are introduced as additional arguments of the energy function based on microstructural considerations. Extending the derivations presented in Kaiser and Menzel (2019) for a purely phenomenological, associated type plasticity model to crystal plasticity, it is shown that the higher gradient contributions in terms of dislocation density tensors give rise to the balance equation of a generalised stress field together with non-ambiguous constitutive boundary conditions. This stress field can be shown to be energetically conjugated to the plastic flow and is additively composed of two parts: the classic stress field and a back-stress type stress field which is closely related to incompatibilities in the plastic deformation field and hence interpretable in terms of geometrically necessary dislocations. For a specific model which features twelve slip systems the constitutive response on material point level is studied in a first step before finite element based simulations, which are motivated by experimental findings on copper micro wires, are analysed in two- and three-dimensional settings. © 2019 Elsevier Ltdview abstract 10.1016/j.jmps.2019.05.019 **An incompatibility tensor-based gradient plasticity formulation—Theory and numerics**

Kaiser, T. and Menzel, A.*Computer Methods in Applied Mechanics and Engineering*345 (2019)In this work we discuss a gradient plasticity formulation which relies on the introduction of higher-order gradient contributions as additional arguments of the free energy function. These gradients can be interpreted in terms of the local geometrically necessary dislocation density. This gives rise to physically well-motivated kinematic-type hardening models in contrast to purely phenomenological approaches. At the same time, the framework results into a regularisation of the formulation such that localised plastic deformation processes in softening materials, e.g.the formation of shear bands, can be simulated. The employed theory is based on an extended nonlocal form of the Clausius–Duhem inequality, motivated by a possible energy exchange between particles at the microstructural level. This gives rise to the balance equation of a nonlocal stress tensor that is found to be work-conjugated to the plastic velocity gradient. The solution of the governing system of partial differential equations is approached by means of a multi-field finite element formulation with the solution of the Karush–Kuhn–Tucker conditions being addressed on a global level by means of Fischer–Burmeister complementary functions. We discuss a specific quadratic energy contribution in terms of the incompatibility, respectively dislocation density tensor that results into well-interpretable contributions to the nonlocal stress field and study the formation of shear bands induced by geometric imperfections as well as the constitutive response at a material interface where the yield limit exhibits a jump discontinuity. © 2018 Elsevier B.V.view abstract 10.1016/j.cma.2018.11.013 **Computational homogenisation of thermo-viscoplastic composites: Large strain formulation and weak micro-periodicity**

Berthelsen, R. and Menzel, A.*Computer Methods in Applied Mechanics and Engineering*348 (2019)In this article, a first order two-scale finite element framework for fully coupled thermo-mechanical problems at finite deformations is considered. The scale bridging is achieved under application of Hill–Mandel condition based boundary conditions at the lower scale. Emphasis of the present article is on the extension of the concept of weak micro-periodicity to thermo-mechanically coupled problems. With this formulation at hand, uniform traction, respectively heat flux boundary conditions as well as the classic periodic boundary conditions are captured as well as a transition between them. The governing equations are summarised with a focus on the implementation of the weak periodic boundary conditions for thermo-mechanical problems. The performance of the proposed weak periodic boundary conditions in comparison to the classic boundary conditions is shown by means of simulations of a single representative volume element as well as by means of full multi-scale simulations. © 2019 Elsevier B.V.view abstract 10.1016/j.cma.2018.12.032 **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 **Magnetostriction in magnetic gels and elastomers as a function of the internal structure and particle distribution**

Fischer, L. and Menzel, A.M.*Journal of Chemical Physics*151 (2019)Magnetic gels and elastomers are promising candidates to construct reversibly excitable soft actuators, triggered from outside by magnetic fields. These magnetic fields induce or alter the magnetic interactions between discrete rigid particles embedded in a soft elastic polymeric matrix, leading to overall deformations. It is a major challenge in theory to correctly predict from the discrete particle configuration the type of deformation resulting for a finite-sized system. Considering an elastic sphere, we here present such an approach. The method is in principle exact, at least within the framework of linear elasticity theory and for large enough interparticle distances. Different particle arrangements are considered. We find, for instance, that regular simple cubic configurations show elongation of the sphere along the magnetization if oriented along a face or space diagonal of the cubic unit cell. Contrariwise, with the magnetization along the edge of the cubic unit cell, they contract. The opposite is true in this geometry for body- and face-centered configurations. Remarkably, for the latter configurations but the magnetization along a face or space diagonal of the unit cell, contraction was observed to revert to expansion with decreasing Poisson ratio of the elastic material. Randomized configurations were considered as well. They show a tendency of elongating the sphere along the magnetization, which is more pronounced for compressible systems. Our results can be tested against actual experiments for spherical samples. Moreover, our approach shall support the search of optimal particle distributions for a maximized effect of actuation. © 2019 Author(s).view abstract 10.1063/1.5118875 **On the implementation of finite deformation gradient-enhanced damage models**

Ostwald, R. and Kuhl, E. and Menzel, A.*Computational Mechanics*64 (2019)We introduce a comprehensive framework for the efficient implementation of finite deformation gradient-regularised damage formulations in existing finite element codes. The numerical implementation is established within a thermo-mechanically fully coupled finite element formulation, where the heat equation solution capabilities are utilised for the damage regularisation. The variationally consistent, gradient-extended and geometrically non-linear damage formulation is based on an overall free energy function, where the standard local free energy contribution is additively extended by two non-local terms. The first additional term basically contains the referential gradient of the non-local damage variable. Secondly, a penalty term is added to couple the local damage variable—the evolution of which is governed by an ordinary differential equation—and the non-local damage field variable that is governed by an additional balance equation of elliptic type. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00466-019-01684-5 **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 gradient-enhanced damage model coupled to plasticity—multi-surface formulation and algorithmic concepts**

Kiefer, B. and Waffenschmidt, T. and Sprave, L. and Menzel, A.*International Journal of Damage Mechanics*27 (2018)A non-local gradient-enhanced damage-plasticity formulation is proposed, which prevents the loss of well-posedness of the governing field equations in the post-critical damage regime. The non-locality of the formulation then manifests itself in terms of a non-local free energy contribution that penalizes the occurrence of damage gradients. A second penalty term is introduced to force the global damage field to coincide with the internal damage state variable at the Gauss point level. An enforcement of Karush–Kuhn–Tucker conditions on the global level can thus be avoided and classical local damage models may directly be incorporated and equipped with a non-local gradient enhancement. An important part of the present work is to investigate the efficiency and robustness of different algorithmic schemes to locally enforce the Karush–Kuhn–Tucker conditions in the multi-surface damage-plasticity setting. Response simulations for representative inhomogeneous boundary value problems are studied to assess the effectiveness of the gradient enhancement regarding stability and mesh objectivity. © 2017, © The Author(s) 2017.view abstract 10.1177/1056789516676306 **A physics-based micromechanical model for electroactive viscoelastic polymers**

Brighenti, R. and Menzel, A. and Vernerey, F.J.*Journal of Intelligent Material Systems and Structures*29 (2018)Electroactive polymers with time-dependent behavior are considered in the present paper by way of a new physics-based micromechanical model; such viscoelastic response is described by the internal evolution of the polymer network, providing a new viewpoint on the stress relaxation occurring in elastomers. The main peculiarity of such internally rearranging materials is their capacity to locally reset their reference stress-free state, leading to a mechanical behavior that relaxes out (eases off) an induced stress state and that can thus be assimilated to a sort of internal self-healing process. Such high deformability and recoverability displayed by dynamically cross-linked polymers can be conveniently exploited when they are coupled in electromechanical problems; the deformation induced by an electric field can be easily tuned by the intensity of the electric field itself and the obtained shape can be maintained without any electric influence once the material microstructure has rearranged after a sufficient curing time. In the present paper, both features of the polymeric material, that is, internal remodeling and electromechanical coupled response, are considered and a theoretical framework is established to simulate representative boundary value problems. © The Author(s) 2018.view abstract 10.1177/1045389X18781036 **Computationally-efficient modeling of inelastic single crystal responses via anisotropic yield surfaces: Applications to shape memory alloys**

Hartl, D.J. and Kiefer, B. and Schulte, R. and Menzel, A.*International Journal of Solids and Structures*136-137 (2018)Phenomenological constitutive models of inelastic responses based on the methods of classical plasticity provide several advantages, especially in terms of computational efficiency. For this reason, they are attractive for the analysis of complex boundary value problems comprising large computational domains. However, for the analysis of problems dominated by single crystal behavior (e.g., inclusion, granular interaction problems or inter-granular fracture), such approaches are often limited by the symmetry assumptions inherent in the stress invariants used to form yield-type criteria. On the other hand, the high computational effort associated with micro-mechanical or crystal plasticity-type models usually prevents their use in large structural simulations, multi-scale analyses, or design and property optimization computations. The goal of the present work is to establish a modeling strategy that captures micro-scale single-crystalline sma responses with sufficient fidelity at the computational cost of a phenomenological macro-scale model. Its central idea is to employ an anisotropic transformation yield criterion with sufficiently rich symmetry class—which can directly be adopted from the literature on plasticity theory—at the single crystal level. This approach is conceptually fundamentally different from the common use of anisotropic yield functions to capture tension-compression asymmetry and texture-induced anisotropy in poly-crystalline SMAs. In our model, the required anisotropy parameters are calibrated either from experimental data for single crystal responses, theoretical considerations or micro-scale computations. The model thus efficiently predicts single crystal behaviors and can be applied to the analysis of complex boundary value problems. In this work we consider the application of this approach to the modeling of shape memory alloys (SMAs), though its potential utility is much broader. Example analyses of SMA single crystals that include non-transforming precipitates and poly-crystalline aggregates are considered and the effects of both elastic and transformation anisotropy in these materials are demonstrated. © 2017 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2017.12.002 **Laminate-based modelling of single and polycrystalline ferroelectric materials – application to tetragonal barium titanate**

Dusthakar, D.K. and Menzel, A. and Svendsen, B.*Mechanics of Materials*117 (2018)The present contribution deals with the development of a laminate-based model designed to study the single and polycrystalline tetragonal ferroelectric material behaviour. Laminate-based models are micromechanically motivated and consider the volume fraction of the distinct ferroelectric variants directly in their formulation. At first, a single crystal laminate-based model is established by considering the average strain and polarisation compatibility conditions. A suitable thermodynamic electric Gibbs energy and a rate-dependent dissipation equation are postulated to capture the dissipative hysteretic material response. The update of the inequality constrained volume fractions is solved by adopting a Fischer–Burmeister-type algorithm in combination with a Newton–Raphson scheme. Following the single crystal formulation, a homogenisation procedure based on random orientation of the individual grains in a polycrystalline aggregate is considered. The material properties and the polarisation switching response of the randomly oriented individual grains are averaged using a finite element framework in order to study the macroscopic polycrystalline behaviour. A parameter fitting procedure based on experimental data for single crystalline response, taken from the literature, is detailed and the material model as well as the algorithmic scheme are verified by solving representative boundary value problems. Moreover, the finite element based simulation results are compared with newly generated experimental hysteresis data for a barium titanate piezoceramic. © 2017 Elsevier Ltdview abstract 10.1016/j.mechmat.2017.10.005 **Numerical analysis of ellipticity condition for large strain plasticity**

Wcisło, B. and Pamin, J. and Kowalczyk-Gajewska, K. and Menzel, A.*AIP Conference Proceedings*1922 (2018)This paper deals with the numerical investigation of ellipticity of the boundary value problem for isothermal finite strain elasto-plasticity. Ellipticity can be lost when softening occurs. A discontinuity surface then appears in the considered material body and this is associated with the ill-posedness of the boundary value problem. In the paper the condition for ellipticity loss is derived using the deformation gradient and the first Piola-Kirchhoff stress tensor. Next, the obtained condition is implemented and numerically tested within symbolic-numerical tools AceGen and AceFEM using the benchmark of an elongated rectangular plate with imperfection in plane stress and plane strain conditions. © 2017 Author(s).view abstract 10.1063/1.5019150 **A computational framework for modelling damage-induced softening in fibre-reinforced materials – Application to balloon angioplasty**

Polindara, C. and Waffenschmidt, T. and Menzel, A.*International Journal of Solids and Structures*118-119 (2017)A computational framework for modelling damage-induced softening in fibre-reinforced materials is presented. The main aspect of this framework is the proposed non-local gradient-enhanced continuum damage formulation. At the material level, the elastic constitutive behaviour is defined by a hyperelastic functional including a volumetric and an isochoric contribution. The isochoric contribution is subdivided into three contributions associated to three different phases i=0,1,2. Phase 0 is represented by an incompressible neo-Hookean material, whereas phases 1 and 2 are represented by an exponential format that accounts for the stretching along two preferred anisotropy directions, i.e. two fibre families. Furthermore, a 1−di–type damage function, is introduced to reproduce the loss of stiffness in each phase i. Following the ideas discussed in (Dimitrijević and Hackl, 2008; Waffenschmidt et al. 2014) and references cited therein, the model is built around the enhancement of the local free energy function by means of terms that contain the referential gradients of the non-local damage variables ϕi. The inclusion of these terms ensures an implicit regularisation of the finite element implementation. A finite element implementation of the non-local gradient-enhanced continuum damage model is presented. To this end we develop an 8-noded Q1Q1P0 hexahedral element following a variational approach, in order to efficiently model the quasi-incompressible behaviour of the hyperelastic material. This element is implemented in Abaqus by means of a user subroutine UEL. Three boundary value problems are studied: an anisotropic plate with a hole, a balloon angioplasty and a full-3D artery-like tube. These computational experiments serve to illustrate the main capabilities of the proposed model. © 2017 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2017.02.010 **A finite deformation continuum modelling framework for curvature effects in fibre-reinforced nanocomposites**

Asmanoglo, T. and Menzel, A.*Journal of the Mechanics and Physics of Solids*107 (2017)Motivated by experimental findings on one-dimensional nano-materials, this contribution focusses on the elaboration of a fibre curvature based higher-order gradient contribution to the stored energy function in a finite deformation setting. The presented approach is based on the fundamental theoretical developments for fibre-reinforced composites presented by Spencer and Soldatos (2007), which take into account the fibre-bending stiffness in addition to the directional dependency induced by the fibres. A mixed-type finite element formulation is then used for the solution of the resulting system of coupled partial differential equations. A specific form of the stored energy function is introduced such that well-interpretable contributions to the stress- and the couple stress tensor are obtained. It is shown that this framework may, in principle, account for fibres of different diameters and induces a natural length scale into the model. Such continuum theory covering size-effects is of special interest since experiments for different materials suggest significant size-effects at small length scales. © 2017 Elsevier Ltdview abstract 10.1016/j.jmps.2017.06.012 **A multi-field finite element approach for the modelling of fibre-reinforced composites with fibre-bending stiffness**

Asmanoglo, T. and Menzel, A.*Computer Methods in Applied Mechanics and Engineering*317 (2017)The implementation of an enhanced modelling approach for fibre-reinforced composites is presented which may, in addition to the directional dependency induced by the fibres, allow the capturing of the fibre-bending stiffness. The theoretical framework is based on the introduction of higher-order gradients of the motion function as additional arguments of the energy function such that size effects can be taken into account. However, the application of higher-order gradients within a finite element framework requires particular care with respect to continuity requirements. In this contribution the usage of a mixed-type multi-field finite element formulation and the fulfilment of the continuity requirement only in a weak sense is proposed. Based on a particular specification of the energy function representative boundary value problems are discussed to assess the model's properties. It is then shown that a model which is based on one additional invariant compared to the classic structural tensor approach allows, in principle, to incorporate effects which are due to the fibre-bending stiffness. © 2017 Elsevier B.V.view abstract 10.1016/j.cma.2017.01.003 **A non-affine electro-viscoelastic microsphere model for dielectric elastomers: Application to VHB 4910 based actuators**

Thylander, S. and Menzel, A. and Ristinmaa, M.*Journal of Intelligent Material Systems and Structures*28 (2017)Dielectric elastomers belong to a larger group of materials, the so-called electroactive polymers, which have the capability of transforming electric energy to mechanical energy through deformation. VHB 4910 is one of the most popular materials for applications of dielectric elastomers and therefore one of the most investigated. This paper includes a new micromechanically motivated constitutive model for dielectric elastomers that incorporates the nearly incompressible and viscous time-dependent behaviour often found in this type of material. A non-affine microsphere framework is used to transform the microscopic constitutive model to a macroscopic continuum counterpart. Furthermore the model is calibrated, through both homogeneous deformation examples and more complex finite element analysis, to VHB 4910. The model is able to capture both the purely elastic, the viscoelastic and the electro-viscoelastic properties of the elastomer and demonstrates the power and applicability of the electromechanically coupled microsphere framework. © SAGE Publications.view abstract 10.1177/1045389X16651157 **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 **Computational homogenisation for thermoviscoplasticity: application to thermally sprayed coatings**

Berthelsen, R. and Denzer, R. and Oppermann, P. and Menzel, A.*Computational Mechanics*(2017)Metal forming processes require wear-resistant tool surfaces in order to ensure a long life cycle of the expensive tools together with a constant high quality of the produced components. Thermal spraying is a relatively widely applied coating technique for the deposit of wear protection coatings. During these coating processes, heterogeneous coatings are deployed at high temperatures followed by quenching where residual stresses occur which strongly influence the performance of the coated tools. The objective of this article is to discuss and apply a thermo-mechanically coupled simulation framework which captures the heterogeneity of the deposited coating material. Therefore, a two-scale finite element framework for the solution of nonlinear thermo-mechanically coupled problems is elaborated and applied to the simulation of thermoviscoplastic material behaviour including nonlinear thermal softening in a geometrically linearised setting. The finite element framework and material model is demonstrated by means of numerical examples. © 2017 Springer-Verlag GmbH Germanyview abstract 10.1007/s00466-017-1436-x **Fibre-reinforced composites with fibre-bending stiffness under azimuthal shear – Comparison of simulation results with analytical solutions**

Asmanoglo, T. and Menzel, A.*International Journal of Non-Linear Mechanics*91 (2017)In Ref. [1], Spencer and Soldatos proposed an enhanced modelling approach for fibre-reinforced composites which accounts for the fibre-bending stiffness in addition to the directional dependency induced by the fibres. Although analytical solutions for simple geometries have been derived over the past years, often subject to specific assumptions such as small deformation kinematics, the application to more general and non-academic boundary value problems is desirable. Motivated by the latter, the numerical solution of the general system of partial differential equations by means of a multi-field finite element approach is proposed in Ref. [2] and the principal model properties are studied for a specific form of the elastic energy potential. In the present contribution a comparison of the numerical solution by means of the multi-field finite element approach against the analytical solution is presented for the azimuthal shear deformation of a tube-like structure. To this end, the general deformation pattern and especially the distribution of the stress and couple stress tensor are taken into account. We find that, although the analytical solution is derived subject to the assumption of small deformations, whereas the numerical solution is based on the finite strain counterpart of the theory, the simulation results are quasi identical, which verifies the numerical framework proposed. © 2017 Elsevier Ltdview abstract 10.1016/j.ijnonlinmec.2017.01.001 **A multi-surface model for ferroelectric ceramics - Application to cyclic electric loading with changing maximum amplitude**

Maniprakash, S. and Arockiarajan, A. and Menzel, A.*Philosophical Magazine*96 (2016)Depending on the maximum amplitude of externally applied cyclic electric fields, ferroelectric ceramics show minor or major hysteresis. The materials also show asymmetric butterfly hysteresis in a prepoled material. Aiming at capturing these behaviour in a phenomenological constitutive model, a multi-surface modelling approach for ferroelectrics is introduced. In this paper, with the note on the motivation for a multi-surface model related to the results of new experimental investigations and also to experimental data reported in the literature, the constitutive relation for a rate dependent multi-surface ferroelectric model is developed. Following this, a brief graphical illustration shows how this model captures the objective phenomena. Consequently, the numerical implementation of the model to capture experimental results is demonstrated. Finally, the performance of this model to represent behaviour of decaying polarisation offset of electrically fatigued specimen is shown. © 2016 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/14786435.2016.1161861 **Analysis of viscoelastic soft dielectric elastomer generators operating in an electrical circuit**

Bortot, E. and Denzer, R. and Menzel, A. and Gei, M.*International Journal of Solids and Structures*78-79 (2016)A predicting model for soft dielectric elastomer generators (DEGs) must consider a realistic model of the electromechanical behaviour of the elastomer filling, the variable capacitor and of the electrical circuit connecting all elements of the device. In this paper such an objective is achieved by proposing a framework for reliable simulations of soft energy harvesters. In particular, a simple electrical circuit is realised by connecting the capacitor, stretched periodically by a source of mechanical work, in parallel with a battery through a diode and with an electrical load consuming the energy produced. The electrical model comprises resistances simulating the effect of the electrodes and of the conductivity current invariably present through the dielectric film. As these devices undergo a high number of electro-mechanical loading cycles at large deformation, the time-dependent response of the material must be taken into account as it strongly affects the generator outcome. To this end, the viscoelastic behaviour of the polymer and the possible change of permittivity with strains are analysed carefully by means of a proposed coupled electro-viscoelastic constitutive model, calibrated on experimental data available in the literature for an incompressible polyacrylate elastomer (3M VHB4910). Numerical results showing the importance of time-dependent behaviour on the evaluation of performance of DEGs for different loading conditions, namely equi-biaxial and uniaxial, are reported in the final section. © 2015 Elsevier Ltd.view abstract 10.1016/j.ijsolstr.2015.06.004 **Computational modelling of wear – application to structured surfaces of elastoplastic tools**

Berthelsen, R. and Wilbuer, H. and Holtermann, R. and Menzel, A.*GAMM Mitteilungen*39 (2016)Sheet bulk metal forming processes are applied in order to produce tailored workpieces at high quality. Thereby, effort is made to optimise frictional and wear behaviour by designing the structures of the contact surfaces. Numerical process analysis can evoke a deeper under-standing of the functionality of such surface structures. In this contribution, a finite element framework for the simulation of wear and the determination of effective frictional behaviour of structured surfaces at finite deformations is proposed. The modelling of wear is realised by pre- and postprocessing of an underlying representative contact problem at the meso-scale. Thereby, elasto-plastic material behaviour is assumed. A number of simulations with different parameterisations of a sinusoidal contact surface are carried out in order to demonstrate the influence of surface topology, which additionally depends on the state of wear, on the effective frictional behaviour. (© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). Copyright © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201610013 **Dynamics of a linear magnetic "microswimmer molecule"**

Babel, S. and Löwen, H. and Menzel, A.M.*EPL*113 (2016)In analogy to nanoscopic molecules that are composed of individual atoms, we consider an active "microswimmer molecule". It is made of three individual magnetic colloidal microswimmers that are connected by harmonic springs and interact hydrodynamically. In the ground state, they form a linear straight molecule. We analyze the relaxation dynamics for perturbations of this straight configuration. As a central result, with increasing self-propulsion, we observe an oscillatory instability in accord with a subcritical Hopf bifurcation scenario. It is accompanied by a corkscrew-like swimming trajectory of increasing radius. Our results can be tested experimentally, using, for instance, magnetic self-propelled Janus particles, supposably linked by DNA molecules. © Copyright EPLA, 2016.view abstract 10.1209/0295-5075/113/58003 **Experimental investigation, modelling and simulation of rate-dependent response of 1-3 ferroelectric composites**

Maniprakash, S. and Jayendiran, R. and Menzel, A. and Arockiarajan, A.*Mechanics of Materials*94 (2016)Further development and design of piezoelectric composites enhances the improved use of piezoelectric materials and devices by overcoming their brittleness. In order to engineer this class of materials and to predictably simulate its behaviour, a computationally efficient constitutive model is established in this paper. This contribution deals with the development of a model for piezoelectric composites to capture their effective behaviour. We first discuss a three-dimensional fully coupled electromechanical rate-dependent model for the response of ferroelectric ceramics. Secondly, a simple homogenisation approach is applied to capture the behaviour of composites for various volume fractions of PZT fibres under different loading frequencies. Following this, a finite element formulation is applied in order to study the behaviour of composites. Finally, these two approaches are compared with experimental results. © 2015 Elsevier Ltd.view abstract 10.1016/j.mechmat.2015.11.018 **Finite element simulation of coating-induced heat transfer: application to thermal spraying processes**

Berthelsen, R. and Tomath, D. and Denzer, R. and Menzel, A.*Meccanica*51 (2016)Thermal spraying is a widely applied coating technique. The optimisation of the thermal spraying process with respect to temperature or temperature induced residual stress states requires a numerical framework for the simulation of the coating itself as well as of the quenching procedure after the application of additional material. This work presents a finite element framework for the simulation of mass deposition due to coating by means of thermal spraying combined with the simulation of nonlinear heat transfer of a rigid heat conductor. The approach of handling the dynamic problem size is highlighted with focus on the thermodynamical consistency of the derived model. With the framework implemented, numerical examples are employed and material parameters are fitted to experimental data of steel as well as of tungsten-carbide–cobalt-coating. © 2015, Springer Science+Business Media Dordrecht.view abstract 10.1007/s11012-015-0236-7 **Finite element simulation of rate-dependent magneto-active polymer response**

Haldar, K. and Kiefer, B. and Menzel, A.*Smart Materials and Structures*25 (2016)This contribution is concerned with the embedding of constitutive relations for magneto-active polymers (MAP) into finite element simulations. To this end, a recently suggested, calibrated, and validated material model for magneto-mechanically coupled and rate-dependent MAP response is briefly summarized in its continuous and algorithmic settings. Moreover, the strongly coupled field equations of finite deformation magneto-mechanics are reviewed. For the purpose of numerical simulation, a finite element model is then established based on the usual steps of weak form representation, discretization and consistent linearization. Two verifying inhomogeneous numerical examples are presented in which a classical 'plate with a hole' geometry is equipped with MAP properties and subjected to different types of time-varying mechanical and magnetic loading. © 2016 IOP Publishing Ltd.view abstract 10.1088/0964-1726/25/10/104003 **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 **Modelling and simulation of thermal effects in internal traverse grinding of hardened bearing steel**

Biermann, D. and Holtermann, R. and Menzel, A. and Schumann, S.*CIRP Annals - Manufacturing Technology*65 (2016)Internal traverse grinding with electro-plated cBN wheels combines the advantages of both internal hard turning and internal grinding, enabling a high material removal rate along with a high surface quality. The drawback, however, is a high thermal load on the workpiece, resulting in shape and dimension errors of the finished part. This paper deals with the compensation of these manufacturing errors with a hybrid simulation system. It combines thermo-mechanically coupled finite element models on both meso- and macro-scale with a scale-bridging kinematic simulation system, enabling the prediction of the resulting temperatures and the development of corresponding simulation-based compensation strategies. © 2016 CIRPview abstract 10.1016/j.cirp.2016.04.005 **Numerical Determination of Process Values Influencing the Surface Integrity in Grinding**

Holtermann, R. and Schumann, S. and Zabel, A. and Biermann, D. and Menzel, A.*Procedia CIRP*45 (2016)Internal traverse grinding with electroplated cBN wheels using high-speed process conditions combines high material removal rates and a high surface quality of the workpiece in one single grinding stroke. In order to capture the macroscopic and mesoscopic thermo-mechanical loads onto the workpiece during internal traverse grinding, numerical simulations are conducted at the two scales. This results in a hybrid approach coupling two finite element models with a geometric kinematic simulation. The article focuses on the influence of multiple grain engagements onto a surface layer region using a two-dimensional chip formation simulation. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.view abstract 10.1016/j.procir.2016.02.072 **Simulation of balloon angioplasty in residually stressed blood vessels—Application of a gradient-enhanced fibre damage model**

Polindara, C. and Waffenschmidt, T. and Menzel, A.*Journal of Biomechanics*49 (2016)In this contribution we study the balloon angioplasty in a residually stressed artery by means of a non-local gradient-enhanced fibre damage model. The balloon angioplasty is a common surgical intervention used to extend or reopen narrowed blood vessels in order to restore the continuous blood flow in, for instance, atherosclerotic arteries. Inelastic, i.e. predominantly damage-related and elastoplastic processes are induced in the artery during its inflation resulting in an irreversible deformation. As a beneficial consequence, provided that the inelastic deformations do not exceed a specific limit, higher deformations can be obtained within the same pressure level and a continuous blood flow can be guaranteed. In order to study the mechanical response of the artery in this scenario, we make use of the non-local gradient-enhanced model proposed in Waffenschmidt et al. (2014). In this contribution, we extend this model to make use of an incompressible format in connection with a Q1Q1P0 finite element implementation. The residual stresses in the artery are also taken into account following the framework presented in Waffenschmidt (2015). From the results it becomes apparent that, when the artery is subjected to radial stresses beyond the physiological range, damage evolution is triggered in the collagen fibres. The impact of the residual stresses on the structural response and on the circumferential stress distribution along the thickness of the arterial wall is also studied. It is observed that the residual stresses have a beneficial effect on the mechanical response of the arterial wall. © 2016 Elsevier Ltdview abstract 10.1016/j.jbiomech.2016.01.037 **Towards a physics-based multiscale modelling of the electro-mechanical coupling in electro-active polymers**

Cohen, N. and Menzel, A. and DeBotton, G.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*472 (2016)Owing to the increasing number of industrial applications of electro-active polymers (EAPs), there is a growing need for electromechanical models which accurately capture their behaviour. To this end, we compare the predicted behaviour of EAPs undergoing homogeneous deformations according to three electromechanical models. The first model is a phenomenological continuumbased model composed of the mechanical Gent model and a linear relationship between the electric field and the polarization. The electrical and the mechanical responses according to the second model are based on the physical structure of the polymer chain network. The third model incorporates a neo-Hookean mechanical response and a physically motivated microstructurally based long-chains model for the electrical behaviour. In the microstructural-motivated models, the integration from the microscopic to the macroscopic levels is accomplished by the micro-sphere technique. Four types of homogeneous boundary conditions are considered and the behaviours determined according to the three models are compared. For the microstructurally motivated models, these analyses are performed and compared with the widely used phenomenological model for the first time. Some of the aspects revealed in this investigation, such as the dependence of the intensity of the polarization field on the deformation, highlight the need for an in-depth investigation of the relationships between the structure and the behaviours of the EAPs at the microscopic level and their overall macroscopic response. © 2016 The Author(s) Published by the Royal Society. All rights reserved.view abstract 10.1098/rspa.2015.0462 **Towards control of viscous effects in acrylic-based actuator applications**

Thylander, S. and Menzel, A. and Ristinmaa, M.*Smart Materials and Structures*25 (2016)Dielectric elastomers offer clear advantages over more traditional and conventional materials when soft, lightweight, noiseless actuator applications with large deformations are considered. However, the viscous time-dependent behaviour associated with most elastomers limit the number of possible applications. For this purpose, the possibility of controlling the viscous response by regulating the applied electric potential is explored. The constitutive model chosen is calibrated to fit the electro-viscoelastic response of an acrylic elastomer often used in dielectric elastomer actuators. The response of both homogeneous deformation examples and inhomogeneous finite element boundary value problems, chosen to mimic existing applications, are presented. Control of both force and displacement quantities are successfully achieved. © 2016 IOP Publishing Ltd.view abstract 10.1088/0964-1726/25/9/095034 **Towards the modelling of ageing and atherosclerosis effects in ApoE-/- mice aortic tissue**

Waffenschmidt, T. and Cilla, M. and Sáez, P. and Pérez, M.M. and Martínez, M.A. and Menzel, A. and Peña, E.*Journal of Biomechanics*49 (2016)The goal of this work consists in a quantitative analysis and constitutive modelling of ageing processes associated to plaque formation in mice arteries. Reliable information on the characteristic evolution of pressure–stretch curves due to the ageing effects is extracted from previous inflation test experiments. Furthermore, characteristic age-dependent material parameters are identified on the basis of a continuum-mechanics-based parameter optimisation technique. The results indicate that the aorta-stiffness of the healthy control mice remains basically constant irrespective of the diet-time and age. In contrast, significant differences exist within the material response and in consequence within the material parameters between the ApoE−/− and the control mice as well as for the different locations over the aorta which is underlined by our experimental observations. With regard to the temporal evolution of the material parameters, we observe that the material parameters for the ApoE−/− mice aortas exhibit a saturation-type increase with respect to age. © 2016 Elsevier Ltdview abstract 10.1016/j.jbiomech.2016.01.043 **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 **Analytical investigation of structurally stable configurations in shape memory alloy-actuated plates**

Peraza Hernandez, E.A. and Kiefer, B. and Hartl, D.J. and Menzel, A. and Lagoudas, D.C.*International Journal of Solids and Structures*69-70 (2015)Strains produced by active materials embedded in plates have been extensively used to manipulate the shape of surface-like engineering structures. Shape memory alloys (SMAs) are active materials that provide a significant amount of strain under large stresses, a characteristic of great utility in such morphing structures. In this work, an analytical approach to approximate the deformation of plates with SMA constituents is developed via the Rayleigh-Ritz method. An additive set of kinematically admissible displacement fields with unknown coefficients is used to describe the plate displacement field. The total potential energy is then calculated using the displacement field, loading conditions, and constitutive relations for the plate layer(s) composed of SMA wire meshes, dense SMA films, and/or elastic material. The unknown coefficients are then found via minimization of the total potential energy. This approach provides closed-form expressions for the approximate deformation of the plates including multistable configurations. The response of circular SMA-based plates is studied herein. The results show that temperature fields with a linear variation in the radial direction induce multistable configurations in which the plate Gaussian curvature is determined by the direction of the temperature gradient. An alternative inversion of the proposed approach is used to directly compute the temperature field required to morph a plate towards a prescribed goal shape. The obtained closed-form expressions show good agreement with detailed non-linear finite element analysis simulations. The proposed approach provides analysts with a low computational cost and relatively simple implementation to assess the potentially stable configurations of SMA-based plates under given loading conditions. Knowledge of such stable configurations is very valuable in the design of SMA-based morphing structures. © 2015 Elsevier Ltd.view abstract 10.1016/j.ijsolstr.2015.05.007 **Challenges in the Modeling of Wound Healing Mechanisms in Soft Biological Tissues**

Valero, C. and Javierre, E. and García-Aznar, J.M. and Menzel, A. and Gómez-Benito, M.J.*Annals of Biomedical Engineering*43 (2015)Numerical models have become one of the most powerful tools in biomechanics and mechanobiology allowing highly detailed simulations. One of the fields in which they have broadly evolved during the last years is in soft tissue modeling. Particularly, wound healing in the skin is one of the processes that has been approached by computational models due to the difficulty of performing experimental investigations. During the last decades wound healing simulations have evolved from numerical models which considered only a few number of variables and simple geometries to more complex approximations that take into account a higher number of factors and reproduce more realistic geometries. Moreover, thanks to improved experimental observations, a larger number of processes, such as cellular stress generation or vascular growth, that take place during wound healing have been identified and modeled. This work presents a review of the most relevant wound healing approximations, together with an identification of the most relevant criteria that can be used to classify them. In addition, and looking towards the actual state of the art in the field, some future directions, challenges and improvements are analyzed for future developments. © 2014, Biomedical Engineering Society.view abstract 10.1007/s10439-014-1200-8 **Comparison of phenomenological and laminate-based models for rate-dependent switching in ferroelectric continua**

Dusthakar, D.K. and Menzel, A. and Svendsen, B.*GAMM Mitteilungen*38 (2015)The purpose of the current work is the comparison of a phenomenological and a laminate-based model for rate-dependent switching in ferroelectric single crystals. To this end, the phenomenological model formulation of [1] is considered. In this model, the polarization vector is treated as an internal variable. The evolution of the polarization determines the remanent strain; dependence of energy storage on its direction results in generally transverse isotropic material behavior. This is compared here with a laminate-based model in which the volume fractions of ferroelectric variants are treated as internal variables. The evolution of these volume fractions determines in turn the remanent strain and polarization as volume averages of corresponding ferroelectric variant quantities. Besides a comparison of the respective model formulations, the phenomenological and laminate models are compared in the context of numerical simulation examples. It turns out that both modeling frameworks nicely recapture the underlying dissipative and rate-dependent effects, which is represented by means of simulated butterfly curves and hysteresis loops under homogeneous loading conditions as well as by finite element simulations. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201510008 **Determination of the thermal load distribution in internal traverse grinding using a geometric-kinematic simulation**

Schumann, S. and Siebrecht, T. and Kersting, P. and Biermann, D. and Holtermann, R. and Menzel, A.*Procedia CIRP*31 (2015)During grinding processes, numerous grains interact with the workpiece material producing mechanical and thermal loads on the surface. In the field of thermal simulation of grinding processes, a widely used approach is to substitute numerous cutting edges by a single moving distributed heat source of a specific geometrical shape referring to the theory of Carslaw and Jaeger. This heat source is then moved across the modelled workpiece according to the specific kinematics of the grinding process. The geometrical shape of the substituted heat source can usually be determined using different approaches, e. g., predefined distribution functions or, more precisely, based on measurements of the shear stress within the contact zone. Referring to the state of the art, it is not possible to measure the shear stress within the contact zone during internal traverse grinding with roughing and finishing zone because of its very complex engagement conditions and the non-rectangular shape of its contact zone. In this work, a novel approach to determining a heat source distribution based on a geometric-kinematic simulation for internal traverse grinding is presented. This simulation identifies the ideal geometrical interaction of workpiece and grinding wheel. For this purpose, the specific material removal rate for each grain is calculated and accumulated with respect to the contact zone resulting in a three-dimensional thermal load distribution. This heat source can be used in finite element simulations to determine the thermal load on the workpiece. © 2015 The Authors. Published by Elsevier B.V.view abstract 10.1016/j.procir.2015.03.020 **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 **Modeling of anisotropic wound healing**

Valero, C. and Javierre, E. and García-Aznar, J.M. and Gómez-Benito, M.J. and Menzel, A.*Journal of the Mechanics and Physics of Solids*79 (2015)Biological soft tissues exhibit non-linear complex properties, the quantification of which presents a challenge. Nevertheless, these properties, such as skin anisotropy, highly influence different processes that occur in soft tissues, for instance wound healing, and thus its correct identification and quantification is crucial to understand them. Experimental and computational works are required in order to find the most precise model to replicate the tissues' properties. In this work, we present a wound healing model focused on the proliferative stage that includes angiogenesis and wound contraction in three dimensions and which relies on the accurate representation of the mechanical behavior of the skin. Thus, an anisotropic hyperelastic model has been considered to analyze the effect of collagen fibers on the healing evolution of an ellipsoidal wound. The implemented model accounts for the contribution of the ground matrix and two mechanically equivalent families of fibers. Simulation results show the evolution of the cellular and chemical species in the wound and the wound volume evolution. Moreover, the local strain directions depend on the relative wound orientation with respect to the fibers. © 2015 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.jmps.2015.03.009 **Modelling and simulation of Internal Traverse Grinding: bridging meso- and macro-scale simulations**

Holtermann, R. and Menzel, A. and Schumann, S. and Biermann, D. and Siebrecht, T. and Kersting, P.*Production Engineering*9 (2015)In this work, we focus on the computational bridging between the meso- and macro-scale in the context of the hybrid modelling of Internal Traverse Grinding with electro-plated cBN wheels. This grinding process satisfies the manufacturing industry demands for a high rate of material removal along with a high surface quality while minimising the number of manufacturing processes invoked. To overcome the major problem of the present machining process, namely a highly concentrated thermal load which can result in micro-structural damage and dimension errors of the workpiece, a hybrid simulation framework is currently under development. The latter consists of three components. First, a kinematic simulation that models the grinding wheel surface based on experimentally determined measurements is used to calculate the transient penetration history of every grain intersecting with the workpiece. Secondly, an h-adaptive, plane-strain finite element model incorporating elasto-plastic work hardening, thermal softening and ductile damage is used to simulate the proximity of one cBN grain during grinding and to capture the complex thermo-mechanical material response on a meso-scale. For the third component of the framework, the results from the preceding two simulation steps are combined into a macro-scale process model that shall in the future be used to improve manufacturing accuracy and to develop error compensation strategies accordingly. To achieve this objective, a regression analysis scheme is incorporated to approximate the influence of the several cutting mechanisms on the meso-scale and to transfer the homogenisation-based thermo-mechanical results to the macro-scale. © 2015, German Academic Society for Production Engineering (WGP).view abstract 10.1007/s11740-015-0613-z **Modelling of non-linear switching effects in piezoceramics: A three-dimensional polygonal finite-element-based approach applied to oligo-crystals**

Kaliappan, J. and Menzel, A.*Journal of Intelligent Material Systems and Structures*26 (2015)A polygonal finite-element formulation is applied to solve electromechanically coupled inhomogeneous boundary value problems for polycrystalline materials. Each grain is discretised by one single polygonal finite element, which allows the efficient simulation of general boundary value problems which still resolves individual grains in space. The constitutive model proposed for switching in piezoceramics makes use of a volume-fraction-based framework, where the respective volume fractions refer to the respective crystallographic variants and take the interpretation as internal variables. Their evolution is combined with non-linear hardening-type functions which additionally account for grain-size dependencies. The computational model is shown to reproduce typical hysteresis and butterfly curves on the polycrystalline level under complex electromechanical loading conditions. The results are obtained by direct volume-averaging with respect to the discretised domain and are in good qualitative agreement with experimental data. Moreover, grain-size effects, such as an increase in the coercive electric field value with decreasing grain size, are also captured. © The Author(s) 2014.view abstract 10.1177/1045389X14554135 **Modelling of Viscoelastic Dielectric Elastomers with Deformation Dependent Electric Properties**

Ask, A. and Menzel, A. and Ristinmaa, M.*Procedia IUTAM*12 (2015)This work deals with electro-viscoelastic modelling and simulation of dielectric elastomer actuators (DEA), including the case of deformation dependent electromechanical coupling. A large deformation modelling framework is adopted, and specific thermo- dynamically consistent material models are established. The general framework is applied to VHB49 polyacrylic polymers which are commonly used in DEA applications. The effects of viscosity and deformation dependent electric permittivity are studied with regards to the stability behaviour and also in view of predicting experimentally observed electromechanical behaviour using numerical simulations. © 2014 The Authors.view abstract 10.1016/j.piutam.2014.12.015 **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 **A gradient-enhanced continuum damage model for residually stressed fibre-reinforced materials at finite strains**

Waffenschmidt, T. and Polindara, C. and Menzel, A.*Lecture Notes in Applied and Computational Mechanics*74 (2014)Themodelling of damage effects inmaterials constitutes a major challenge in various engineering-related disciplines. However, the assumption of purely local continuum damage formulations may lead to ill-posed boundary value problems and—with regard to numerical methods such as the finite element method—to meshdependent solutions, a vanishing localised damage zone upon mesh refinement, and hence physically questionable results. In order to circumvent these deficiencies, we present a non-local gradient-enhanced damage model at finite strains. We additively compose the hyperelastic constitutive response at local material point level of an isotropic matrix and of an anisotropic fibre-reinforced material. The inelastic constitutive response is characterised by a scalar [1– d]-damage model, where we assume only the anisotropic elastic part to damage. Furthermore, we enhance the local free energy by a gradient-term. This term essentially contains the gradient of the non-local damage variable which we introduce as an additional global field variable. In order to guarantee the equivalence between the local and non-local damage variable, we incorporate a penalisation term within the free energy. Based on the principle of minimum total potential energy, we obtain a coupled system of variational equations. The associated non-linear system of equations is symmetric and can conveniently be solved by standard incremental-iterative Newton-Raphson schemes or arc-length-based solution methods. As a further key aspect, we incorporate residual stresses by means of a multiplicative composition of the deformation gradient. As a three-dimensional finite element example, we study the material degradation of a fibre-reinforced tube subjected to internal pressure. This highlights the meshobjective and constitutive properties of the model and illustratively underlines the capabilities of the formulation with regard to biomechanical application such as the simulation of arteries. © Springer International Publishing Switzerland 2014.view abstract 10.1007/978-3-319-10981-7_2 **A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials**

Waffenschmidt, T. and Polindara, C. and Menzel, A. and Blanco, S.*Computer Methods in Applied Mechanics and Engineering*268 (2014)A non-local gradient-based damage formulation within a geometrically non-linear setting is presented. The hyperelastic constitutive response at local material point level is governed by a strain energy which is additively composed of an isotropic matrix and of an anisotropic fibre-reinforced material, respectively. The inelastic constitutive response is governed by a scalar [1d]-type damage formulation, where only the anisotropic elastic part is assumed to be affected by the damage. Following the concept in Dimitrijević and Hackl [28], the local free energy function is enhanced by a gradient-term. This term essentially contains the gradient of the non-local damage variable which, itself, is introduced as an additional independent variable. In order to guarantee the equivalence between the local and non-local damage variable, a penalisation term is incorporated within the free energy function. Based on the principle of minimum total potential energy, a coupled system of Euler-Lagrange equations, i.e., the balance of linear momentum and the balance of the non-local damage field, is obtained and solved in weak form. The resulting coupled, highly non-linear system of equations is symmetric and can conveniently be solved by a standard incremental-iterative Newton-Raphson-type solution scheme. Several three-dimensional displacement- and force-driven boundary value problems-partially motivated by biomechanical application-highlight the mesh-objective characteristics and constitutive properties of the model and illustratively underline the capabilities of the formulation proposed. © 2013 Elsevier B.V.view abstract 10.1016/j.cma.2013.10.013 **Bridging from particle to macroscopic scales in uniaxial magnetic gels**

Menzel, A.M.*Journal of Chemical Physics*141 (2014)Connecting the different length scales of characterization is an important, but often very tedious task for soft matter systems. Here, we carry out such a procedure for the theoretical description of anisotropic uniaxial magnetic gels. The so-far undetermined material parameters in a symmetry-based macroscopic hydrodynamic-like description are determined starting from a simplified mesoscopic particle-resolved model. This mesoscopic approach considers chain-like aggregates of magnetic particles embedded in an elastic matrix. Our procedure provides an illustrative background to the formal symmetry-based macroscopic description. There are presently other activities to connect such mesoscopic models as ours with more microscopic polymer-resolved approaches; together with these activities, our study complements a first attempt of scale-bridging from the microscopic to the macroscopic level in the characterization of magnetic gels. © 2014 AIP Publishing LLC.view abstract 10.1063/1.4901275 **Configurational forces for quasi-incompressible large strain electro-viscoelasticity - Application to fracture mechanics**

Denzer, R. and Menzel, A.*European Journal of Mechanics, A/Solids*48 (2014)This work treats theoretical and numerical aspects of configurational forces with application to fracture mechanics of electroactive polymers (EAPs) modelled in the continuum mechanics based context of large strain quasi-incompressible electro-viscoelasticity. We adopt a four-field formulation to take the quasi-incompressible behaviour in a finite element framework into account. We discuss the numerical implementation of viscosity-related evolution equations and the implications of viscous internal variables on the computation of configurational forces. As numerical examples for time depending crack driving forces we discuss a pre-cracked stacked EAP actuator as well as a pre-cracked and pre-stretched block actuator. © 2014 Elsevier Masson SAS. All rights reserved.view abstract 10.1016/j.euromechsol.2014.05.012 **Extremal states of energy of a double-layered thick-walled tube - application to residually stressed arteries**

Waffenschmidt, T. and Menzel, A.*Journal of the Mechanical Behavior of Biomedical Materials*29 (2014)Various biological tissues are designed to optimally support external loads for complex geometries and mechanobiological structures. This results in complex microstructures of such materials. The design of, for instance, (healthy) arteries, which are in the focus of this work, is characterised by a residually stressed fibre-reinforced multi-layered composite with highly non-linear elastic response. The complex interaction of material properties with the geometry and residual stress effects enables the optimal support under different blood pressures, respectively blood flow, within the vessel. The fibres reinforcing the arterial wall, as well as residual stresses present in the vessel, strongly influence its overall behaviour and performance. Turn-over and remodelling processes of the collagenous fibres occurring in the respective layers - either resulting from natural growth phenomena or from artificially induced changes in loading condition such as stent deployment - support the optimisation of the multi-layered composite structure of arteries for the particular loading conditions present in the artery. Within this contribution, the overall energetic properties of an artery are discussed by means of the inflation, bending and extension of a double-layered cylindrical tube. Different states of residual stresses and different fibre orientations are considered so that, for instance, representative fibre angles that result in extremal states of the total potential energy can be identified. In view of turn-over and remodelling processes, these orientations are considered to constitute preferred directions of fibre alignment. In summary, the main goal of this work is to calculate optimal material, structural and loading parameters by concepts of energy-minimisation. Several numerical studies show that the obtained values - such as the fibre orientations, the residual axial stretch and the opening angle - are in good agreement with respective physiological parameters reported in the literature. © 2013 Elsevier Ltd.view abstract 10.1016/j.jmbbm.2013.05.023 **Frontiers in Finite-Deformation Electromechanics**

Menzel, A. and Göktepe, S. and Kuhl, E.*European Journal of Mechanics, A/Solids*48 (2014)view abstract 10.1016/j.euromechsol.2014.05.008 **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 **On the modelling of electro-viscoelastic response of electrostrictive polyurethane elastomers**

Ask, A. and Menzel, A. and Ristinmaa, M.*IOP Conference Series: Materials Science and Engineering*10 (2014)Electroactive polymers (EAP) deform under electric fields. This effect in fact generates various new fields of engineering applications of high technological interest. As an advantage, EAP may undergo deformations much larger than those capable by electroactive ceramics - however, to the price of acting at comparatively low forces. As common for polymers, EAP exhibit time-dependent material behaviour. The model proposed in this contribution, on the one hand, captures these electro-viscoelastic effects and, on the other hand, also nicely fits into iterative finite element formulations in order to simulate general boundary value problems. While the deformation itself as well as the electric potential are introduced as global degrees of freedom, the internal variables accounting for the viscous response are incorporated at the so-called local integration point level. Apart form calibrating the model against experimental data, a simple coupled finite element example is studied to show the applicability of the finite deformation electro-viscoelastic formulation proposed. © 2010 IOP Publishing Ltd.view abstract 10.1088/1757-899X/10/1/012101 **The generalized Hill model: A kinematic approach towards active muscle contraction**

Göktepe, S. and Menzel, A. and Kuhl, E.*Journal of the Mechanics and Physics of Solids*72 (2014)Excitation-contraction coupling is the physiological process of converting an electrical stimulus into a mechanical response. In muscle, the electrical stimulus is an action potential and the mechanical response is active contraction. The classical Hill model characterizes muscle contraction though one contractile element, activated by electrical excitation, and two non-linear springs, one in series and one in parallel. This rheology translates into an additive decomposition of the total stress into a passive and an active part. Here we supplement this additive decomposition of the stress by a multiplicative decomposition of the deformation gradient into a passive and an active part. We generalize the one-dimensional Hill model to the three-dimensional setting and constitutively define the passive stress as a function of the total deformation gradient and the active stress as a function of both the total deformation gradient and its active part. We show that this novel approach combines the features of both the classical stress-based Hill model and the recent active-strain models. While the notion of active stress is rather phenomenological in nature, active strain is micro-structurally motivated, physically measurable, and straightforward to calibrate. We demonstrate that our model is capable of simulating excitation-contraction coupling in cardiac muscle with its characteristic features of wall thickening, apical lift, and ventricular torsion. © 2014 Elsevier Ltd.view abstract 10.1016/j.jmps.2014.07.015 **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 **Inverse-motion-based form finding for quasi-incompressible finite electroelasticity**

Ask, A. and Denzer, R. and Menzel, A. and Ristinmaa, M.*International Journal for Numerical Methods in Engineering*94 (2013)This work deals with inverse-motion-based form finding for electroelasticity. The inverse motion problem is formulated for the electroelastic case, and the resulting equations are implemented within a finite element framework. A four-field variational approach is adopted, taking into consideration the typically incompressible behavior of the elastomer materials commonly used in electromechanical applications. By means of numerical simulations, the inverse-motion-based form finding makes it possible to design the referential configuration so that a given set of loads and boundary conditions results in a prespecified deformed configuration. The computational finite element framework established in this work allows for such numerical simulations and testing and thereby the possibility to improve the design and accuracy in electroelastic applications such as grippers, sensors, and seals. © 2013 John Wiley & Sons, Ltd.view abstract 10.1002/nme.4462 **Modelling, simulation and experimental investigation of chip formation in internal traverse grinding**

Holtermann, R. and Schumann, S. and Menzel, A. and Biermann, D.*Production Engineering*7 (2013)We present recent developments in modelling and simulation of internal traverse grinding, a high speed machining process which enables both a large material removal rate and high surface quality. We invoke a hybrid modelling framework, including a process scale model, simulations on a mesoscale capturing the proximity of a single cBN grain and an analysis framework to investigate the grinding wheel topography. Moreover, we perform experiments to verify our simulations. Focus in this context is the influence of the cutting speed variation on the grain specific heat generation. © 2013 German Academic Society for Production Engineering (WGP).view abstract 10.1007/s11740-013-0449-3 **An electromechanically coupled micro-sphere framework: Application to the finite element analysis of electrostrictive polymers**

Thylander, S. and Menzel, A. and Ristinmaa, M.*Smart Materials and Structures*21 (2012)The number of industrial applications of electroactive polymers (EAPs) is increasing and, consequently, the need for reliable modelling frameworks for such materials as well as related robust simulation techniques continuously increases. In this context, we combine the modelling of non-linear electroelasticity with a computational micro-sphere formulation in order to simulate the behaviour of EAPs. The micro-sphere approach in general enables the use of physics-based constitutive models like, for instance, the so-called worm-like chain model. By means of the micro-sphere formulation, scalar-valued micromechanical constitutive relations can conveniently be extended to a three-dimensional continuum setting. We discuss several electromechanically coupled numerical examples and make use of the finite element method to solve inhomogeneous boundary value problems. The incorporated material parameters are referred to experimental data for an electrostrictive polymer. The numerical examples show that the coupled micro-sphere formulation combined with the finite element method results in physically sound simulations that mimic the behaviour of an electrostrictive polymer. © 2012 IOP Publishing Ltd.view abstract 10.1088/0964-1726/21/9/094008 **Anisotropic density growth of bone - A computational micro-sphere approach**

Waffenschmidt, T. and Menzel, A. and Kuhl, E.*International Journal of Solids and Structures*49 (2012)Bones are able to adapt their local density when exposed to mechanical loading. Such growth processes result in densification of the bone in regions of high loading levels and in resorption of the material in regions of low loading levels. This evolution and optimisation process generates heterogeneous distributions of bone density accompanied by pronounced anisotropic mechanical properties. While several constitutive models reported in the literature assume the growth process to be purely isotropic, only few studies focus on the modelling and simulation of anisotropic functional adaptation we can observe in vivo. Some of these few computational models for anisotropic growth characterise the evolution of anisotropy by analogy to anisotropic continuum damage mechanics while others include anisotropic growth but assume isotropic elastic properties. The objective of this work is to generalise a well-established framework of energy-driven isotropic functional adaptation to anisotropic microstructural growth and density evolution. We adopt the so-called micro-sphere concept, which proves to be extremely versatile and flexible to extend sophisticated one-dimensional constitutive relations to the three-dimensional case. In this work we apply this framework to the modelling and simulation of anisotropic functional adaptation by means of a directional density distribution, which evolves in time and in response to the mechanical loading condition. Several numerical studies highlight the characteristics and properties of the anisotropic growth model we establish. The formulation is embedded into an iterative finite element algorithm to solve complex boundary value problems. In particular, we consider the finite-element-simulation of a subject-specific proximal tibia bone and a comparison to experimental measurements. The proposed model is able to appropriately represent the heterogeneous bone density distribution. As an advantage over several other computational growth models proposed in the literature, a pronounced local anisotropy evolution is identified and illustrated by means of orientation-distribution-type density plots. © 2012 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijsolstr.2012.03.035 **Application of an anisotropic growth and remodelling formulation to computational structural design**

Waffenschmidt, T. and Menzel, A.*Mechanics Research Communications*42 (2012)A classical structural optimisation problem consists of a problem-specific objective function which has to be minimised in consideration of particular constraints with respect to design and state variables. In this contribution we adopt a conceptually different approach for the design of a structure which is not based on a classical optimisation technique. Instead, we establish a constitutive micro-sphere-framework in combination with an energy-driven anisotropic microstructural growth formulation, which was originally proposed for the simulation of adaptation and remodelling phenomena in hard biological tissues such as bones. The goal of this contribution is to investigate this anisotropic growth formulation with a special emphasis on its application to structural design problems. To this end, four illustrative three-dimensional benchmark-type boundary value problems are discussed and compared qualitatively with the results obtained by classical structural optimisation strategies. The simulation results capture the densification effects and clearly identify the main load bearing regions. It turns out, that even though making use of this conceptually different growth formulation as compared to the procedures used in a classical structural optimisation context, we identify qualitatively very similar structures or rather regions of densification. Moreover, in contrast to common structural optimisation strategies, which mostly aim to optimise merely the size, shape or topology, our formulation also contains the improvement of the material itself, which - apart from the structural improvement - results in the generation of problem-specific local material anisotropy and textured evolution. © 2012 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.mechrescom.2011.12.004 **Collective motion of binary self-propelled particle mixtures**

Menzel, A.M.*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*85 (2012)In this study, we investigate the phenomenon of collective motion in binary mixtures of self-propelled particles. More precisely, we consider two particle species, each of which consisting of pointlike objects that propel with a velocity of constant magnitude. Within each species, the particles try to achieve polar alignment of their velocity vectors, whereas we analyze the cases of preferred polar, antiparallel, as well as perpendicular alignment between particles of different species. Our focus is on the effect that the interplay between the two species has on the threshold densities for the onset of collective motion and on the nature of the solutions above onset. For this purpose, we start from suitable Langevin equations in the particle picture, from which we derive mean field equations of the Fokker-Planck type and finally macroscopic continuum field equations. We perform particle simulations of the Langevin equations and linear stability analyses of the Fokker-Planck and macroscopic continuum equations, and we numerically solve the Fokker-Planck equations. Both spatially homogeneous and inhomogeneous solutions are investigated, where the latter correspond to stripelike flocks of collectively moving particles. In general, the interaction between the two species reduces the threshold density for the onset of collective motion of each species. However, this interaction also reduces the spatial organization in the stripelike flocks. The case that shows the most interesting behavior is the one of preferred perpendicular alignment between different species. There a competition between polar and truly nematic orientational ordering of the velocity vectors takes place within each particle species. Finally, depending on the alignment rule for particles of different species and within certain ranges of particle densities, identical and inverted spatial density profiles can be found for the two particle species. The system under investigation is confined to two spatial dimensions. © 2012 American Physical Society.view abstract 10.1103/PhysRevE.85.021912 **Electrostriction in electro-viscoelastic polymers**

Ask, A. and Menzel, A. and Ristinmaa, M.*Mechanics of Materials*50 (2012)Electrostrictive polyurethane (PU) elastomers belong to the class of materials commonly referred to as electroactive polymers (EAP). These materials have their use in a variety of applications, including biomimetics and microrobotics where traditional engineering components may fall short. PU is one of the materials considered for so called dielectric actuators, where the electromechanical response is generally due to Coulomb forces on the actuator electrodes, giving rise to a compressive pressure and thereby large deformations of the polymer. On the other hand, for more moderate electric fields, which may be more attractive in certain applications, a large part of the electroactive response for PU elastomers is due to inherent electrostriction. The latter phenomena is the focus of this work. As is common in elastomers, PU elastomers are viscoelastic. A coupled electro-viscoelastic phenomenological constitutive model for electrostrictive PU is proposed and fitted to experimental data available in the literature. The possibility of performing simulations of EAP is of interest as the number of applications grow. Considering this, the computational model is embedded in a coupled finite element formulation and, based on this, representative simulations of inhomogeneous boundary value problems are presented. © 2012 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.mechmat.2012.01.009 **Frontiers in growth and remodeling**

Menzel, A. and Kuhl, E.*Mechanics Research Communications*42 (2012)Unlike common engineering materials, living matter can autonomously respond to environmental changes. Living structures can grow stronger, weaker, larger, or smaller within months, weeks, or days as a result of a continuous microstructural turnover and renewal. Hard tissues can adapt by increasing their density and grow strong. Soft tissues can adapt by increasing their volume and grow large. For more than three decades, the mechanics community has actively contributed to understand the phenomena of growth and remodeling from a mechanistic point of view. However, to date, there is no single, unified characterization of growth, which is equally accepted by all scientists in the field. Here we shed light on the continuum modeling of growth and remodeling of living matter, and give a comprehensive overview of historical developments and trends. We provide a state-of-the-art review of current research highlights, and discuss challenges and potential future directions. Using the example of volumetric growth, we illustrate how we can establish and utilize growth theories to characterize the functional adaptation of soft living matter. We anticipate this review to be the starting point for critical discussions and future research in growth and remodeling, with a potential impact on life science and medicine. © 2012 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.mechrescom.2012.02.007 **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 **Inverse-motion based modeling for electromechanics with application to electrostrictive polyurethane**

Ask, A. and Denzer, R. and Menzel, A. and Ristinmaa, M.*ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, SMASIS 2012*2 (2012)In this work the inverse motion problem for electro elasticity is considered. For given loads and boundary conditions, and a given deformed shape of the electro elastic body, the initially unknown undeformed configuration is sought. The boundary-value problem for the inverse motion is obtained by reparameterization of the forward motion equations in terms of the inverse deformation map. In order to account for incompressibility, a mixed formulation is adopted. The finite element method is used to calculate the undeformed configuration for an electro-active gripper application. Copyright © 2012 by ASME.view abstract 10.1115/SMASIS2012-8049 **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 **Phenomenological modeling of viscous electrostrictive polymers**

Ask, A. and Menzel, A. and Ristinmaa, M.*International Journal of Non-Linear Mechanics*47 (2012)A common usage for electroactive polymers (EAPs) is in different types of actuators, where advantage is taken of the deformation of the polymer due to an electric field. It turns out that time-dependent effects are present in these applications. One of these effects is the viscoelastic behavior of the polymer material. In view of the modeling and simulation of applications for EAP within a continuum mechanics setting, a phenomenological framework for an electro-viscoelastic material model is elaborated in this work. The different specific models are fitted to experimental data available in the literature. While the experimental data used for inherent electrostriction is restricted to small strains, a large strain setting is used for the model in order to account for possible applications where the polymers undergo large deformations, such as in pre-strained actuators. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijnonlinmec.2011.03.020 **Polygonal finite elements for three-dimensional Voronoi-cell-based discretisations**

Jayabal, K. and Menzel, A.*European Journal of Computational Mechanics*21 (2012)Hybrid finite element formulations in combination with Voronoi-cell-based discretisation methods can efficiently be used to model the behaviour of polycrystalline materials. Randomly generated three-dimensional Voronoi polygonal elements with varying numbers of surfaces and corners in general better approximate the geometry of polycrystalline microor rather grain-structures than the standard tetrahedral and hexahedral finite elements. In this work, the application of a polygonal finite element formulation to three-dimensional elastomechanical problems is elaborated with special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach is investigated. The applicability of the framework established is demonstrated by means of representative numerical examples. © 2012 Taylor & Francis.view abstract 10.1080/17797179.2012.702432 **Voronoi-based three-dimensional polygonal finite elements for electromechanical problems**

Jayabal, K. and Menzel, A.*Computational Materials Science*64 (2012)We discuss the combination of a hybrid finite element approach and three-dimensional Voronoi-based mesh discretisations for electromechanically coupled problems. The fluxes, i.e. the stresses and electric displacements, are defined within the volume of the polygonal finite elements, whereas the displacements and electric potential are approximated on the boundaries of the elements. A Voronoi polygon with arbitrary, but admissible, number of surfaces and nodes thereby acts as a single finite element. Representative numerical examples for electromechanical problems, in particular piezoelectric materials, are presented and discussed. © 2012 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2012.02.049 **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 **Micromechanical modelling of switching phenomena in polycrystalline piezoceramics: Application of a polygonal finite element approach**

Jayabal, K. and Menzel, A. and Arockiarajan, A. and Srinivasan, S.M.*Computational Mechanics*48 (2011)A micromechanically motivated model is proposed to capture nonlinear effects and switching phenomena present in ferroelectric polycrystalline materials. The changing remnant state of the ferroelectric crystal is accounted for by means of so-called back fields-such as back stresses-to resist or assist further switching processes in the crystal depending on the local loading history. To model intergranular effects present in ferroelectric polycrystals, the computational model elaborated is embedded into a mixed polygonal finite element approach, whereby an individual ferroelectric grain is represented by one single irregular polygonal finite element. This computationally efficient coupled simulation framework is shown to reproduce the specific characteristics of the responses of ferroelectric polycrystals under complex electromechanical loading conditions in good agreement with experimental observations. © 2011 Springer-Verlag.view abstract 10.1007/s00466-011-0595-4 **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 **Framework for deformation induced anisotropy in glassy polymers**

Harrysson, M. and Ristinmaa, M. and Wallin, M. and Menzel, A.*Acta Mechanica*211 (2010)In this paper a constitutive model for glassy polymers is developed. Glassy polymers consist of a number of polymer chains that at a microscopic level form a network. If the distribution of the polymer chains shows some preferred direction, the mechanical response at a global macroscopic level will be anisotropic. To incorporate the orientational distribution of the polymer chains, a homogenization procedure involving a chain orientation distribution function was undertaken. When polymers are exposed to external loading, the chains at the microscopic level orient in a certain manner, leading to an evolution of the macroscopic anisotropic properties. This phenomenon was modeled by use of evolution equations for the chains at a microscopic level and are then-by using the orientation distribution function-transformed to the macroscopic level. The theories involved are developed in a large strain setting in which a multiplicative split of the deformation gradient for the elastic-viscoplastic response is adopted. Various numerical experiments were conducted to evaluate the model that was developed. © 2009 Springer-Verlag.view abstract 10.1007/s00707-009-0232-x **Multi-axial behavior of ferroelectrics with two types of micro-macro mechanical models**

Jayabal, K. and Arockiarajan, A. and Menzel, A. and Sivakumar, S.M.*IUTAM Bookseries*19 (2010)Ferroelectric ceramics exhibit a significantly different nonlinear behavior with external electric and mechanical fields applied at angles to the initial poled direction. This angle dependent response of the ferroelectric polycrystals are predicted by two types of models based on irreversible thermodynamics and physics of domain switching. The first type is a uniaxial model dealing with simultaneous evolution of three variants at a given instant. The back stress and electric fields, assumed as linear functions of remnant strain and polarization developed by the domain switching process, are introduced in the model to assist or resist further switching process. The second type is a three dimensional model that considers all six variants of a tetragonal crystal in each grain and the dissipation associated with grain boundary constraints are brought into the model through switching criterion. The pressure dependent constraints imposed by the surrounding grains on the grain of interest at its boundary during domain switching process is correlated with the resistance experienced by a ferroelectric single crystal on its boundary during domain switching. Taking all the domain switching possibilities, the volume fractions of each of the variants are tracked and homogenized for macroscopic behavior. Numerical simulations were carried out for the behavior of ferroelectrics using both the models and the outcome was found to be qualitatively comparable with experimental observations given in literature. © 2010 Springer Science+Business Media B.V.view abstract 10.1007/978-90-481-3771-8-10

#### TRR 188: Damage Controlled Forming Processes (Coordination: TU Do) (concluded)

#### FOR 1509: Ferroic Functional Materials (concluded)

#### SPP 1480: Modelling, Simulation and Compensation of Thermal Effects for Complex Machining Processes (concluded)

#### SFB 708: 3D Surface Engineering of Tools for the Sheet Metal Forming (Coordination: TU Do) (concluded)

#### biological tissues

#### continuum mechanics

#### deformation

#### mechanical properties

#### modelling and simulation

#### shape memory effect

#### single crystals

#### viscoelasticity