#### Prof. Dr.-Ing. Daniel Balzani

Institute of Mechanics

Ruhr-Universität Bochum

##### Contact

- daniel[dot]balzani[at]rub[dot]de
- +49 234 32 23080
- personal website

##### Hub

**Adaptive convexification of microsphere-based incremental damage for stress and strain softening at finite strains**

Köhler, M. and Neumeier, T. and Melchior, J. and Peter, M.A. and Peterseim, D. and Balzani, D.*Acta Mechanica*233 (2022)view abstract 10.1007/s00707-022-03332-1 **Continuum multiscale modeling of absorption processes in micro- and nanocatalysts**

Köhler, M. and Junker, P. and Balzani, D.*Archive of Applied Mechanics*92 (2022)view abstract 10.1007/s00419-022-02172-8 **Efficient and robust numerical treatment of a gradient-enhanced damage model at large deformations**

Junker, P. and Riesselmann, J. and Balzani, D.*International Journal for Numerical Methods in Engineering*123 (2022)The modeling of damage processes in materials constitutes an ill-posed mathematical problem which manifests in mesh-dependent finite element results. The loss of ellipticity of the discrete system of equations is counteracted by regularization schemes of which the gradient enhancement of the strain energy density is often used. In this contribution, we present an extension of the efficient numerical treatment, which has been proposed by Junker et al. in 2019, to materials that are subjected to large deformations. Along with the model derivation, we present a technique for element erosion in the case of severely damaged materials. Efficiency and robustness of our approach is demonstrated by two numerical examples including snapback and springback phenomena. © 2021 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.view abstract 10.1002/nme.6876 **Fluid-structure interaction simulation of tissue degradation and its effects on intra-aneurysm hemodynamics**

Wang, H. and Uhlmann, K. and Vedula, V. and Balzani, D. and Varnik, F.*Biomechanics and Modeling in Mechanobiology*(2022)Tissue degradation plays a crucial role in vascular diseases such as atherosclerosis and aneurysms. Computational modeling of vascular hemodynamics incorporating both arterial wall mechanics and tissue degradation has been a challenging task. In this study, we propose a novel finite element method-based approach to model the microscopic degradation of arterial walls and its interaction with blood flow. The model is applied to study the combined effects of pulsatile flow and tissue degradation on the deformation and intra-aneurysm hemodynamics. Our computational analysis reveals that tissue degradation leads to a weakening of the aneurysmal wall, which manifests itself in a larger deformation and a smaller von Mises stress. Moreover, simulation results for different heart rates, blood pressures and aneurysm geometries indicate consistently that, upon tissue degradation, wall shear stress increases near the flow-impingement region and decreases away from it. These findings are discussed in the context of recent reports regarding the role of both high and low wall shear stress for the progression and rupture of aneurysms. © 2022, The Author(s).view abstract 10.1007/s10237-022-01556-7 **METHOD FOR THE ANALYSIS OF EPISTEMIC AND ALEATORY UNCERTAINTIES FOR A RELIABLE EVALUATION OF FAILURE OF ENGINEERING STRUCTURES**

Miska, N. and Balzani, D.*International Journal for Uncertainty Quantification*12 (2022)view abstract 10.1615/Int.J.UncertaintyQuantification.2022042145 **Robust and Efficient Finite Element Discretizations for Higher-Order Gradient Formulations**

Riesselmann, J. and Ketteler, J.W. and Schedensack, M. and Balzani, D.*Lecture Notes in Applied and Computational Mechanics*98 (2022)view abstract 10.1007/978-3-030-92672-4_3 **Simulation of Arterial Walls: Growth, Fiber Reorientation, and Active Response**

Uhlmann, K. and Zahn, A. and Balzani, D.*Studies in Mechanobiology, Tissue Engineering and Biomaterials*24 (2022)view abstract 10.1007/978-3-030-92339-6_8 **Simulation of crack propagation based on eigenerosion in brittle and ductile materials subject to finite strains**

Wingender, D. and Balzani, D.*Archive of Applied Mechanics*(2022)In this paper, a framework for the simulation of crack propagation in brittle and ductile materials is proposed. The framework is derived by extending the eigenerosion approach of Pandolfi and Ortiz (Int J Numer Methods Eng 92(8):694–714, 2012. https://doi.org/10.1002/nme.4352) to finite strains and by connecting it with a generalized energy-based, Griffith-type failure criterion for ductile fracture. To model the elasto-plastic response, a classical finite strain formulation is extended by viscous regularization to account for the shear band localization prior to fracture. The compression–tension asymmetry, which becomes particularly important during crack propagation under cyclic loading, is incorporated by splitting the strain energy density into a tensile and compression part. In a comparative study based on benchmark problems, it is shown that the unified approach is indeed able to represent brittle and ductile fracture at finite strains and to ensure converging, mesh-independent solutions. Furthermore, the proposed approach is analyzed for cyclic loading, and it is shown that classical Wöhler curves can be represented. © 2022, The Author(s).view abstract 10.1007/s00419-021-02101-1 **Simulation of crack propagation through voxel-based, heterogeneous structures based on eigenerosion and finite cells**

Wingender, D. and Balzani, D.*Computational Mechanics*70 (2022)view abstract 10.1007/s00466-022-02172-z **A general, implicit, finite-strain FE 2 framework for the simulation of dynamic problems on two scales**

Tamsen, E. and Balzani, D.*Computational Mechanics*67 (2021)In this paper we present a fully-coupled, two-scale homogenization method for dynamic loading in the spirit of FE2 methods. The framework considers the balance of linear momentum including inertia at the microscale to capture possible dynamic effects arising from micro heterogeneities. A finite-strain formulation is adapted to account for geometrical nonlinearities enabling the study of e.g. plasticity or fiber pullout, which may be associated with large deformations. A consistent kinematic scale link is established as displacement constraint on the whole representative volume element. The consistent macroscopic material tangent moduli are derived including micro inertia in closed form. These can easily be calculated with a loop over all microscopic finite elements, only applying existing assembly and solving procedures. Thus, making it suitable for standard finite element program architectures. Numerical examples of a layered periodic material are presented and compared to direct numerical simulations to demonstrate the capability of the proposed framework. In addition, a simulation of a split Hopkinson tension test showcases the applicability of the framework to engineering problems. © 2021, The Author(s).view abstract 10.1007/s00466-021-01993-8 **A new method for the in vivo identification of degenerated material property ranges of the human eye: feasibility analysis based on synthetic data**

Muench, S. and Roellig, M. and Balzani, D.*Biomechanics and Modeling in Mechanobiology*(2021)This paper proposes a new method for in vivo and almost real-time identification of biomechanical properties of the human cornea based on non-contact tonometer data. Further goal is to demonstrate the method’s functionality based on synthetic data serving as reference. For this purpose, a finite element model of the human eye is constructed to synthetically generate full-field displacements from different data sets with keratoconus-like degradations. Then, a new approach based on the equilibrium gap method combined with a mechanical morphing approach is proposed and used to identify the material parameters from virtual test data sets. In a further step, random absolute noise is added to the virtual test data to investigate the sensitivity of the new approach to noise. As a result, the proposed method shows a relevant accuracy in identifying material parameters based on full-field displacements. At the same time, the method turns out to work almost in real time (order of a few minutes on a regular workstation) and is thus much faster than inverse problems solved by typical forward approaches. On the other hand, the method shows a noticeable sensitivity to rather small noise amplitudes rendering the method not accurate enough for the precise identification of individual parameter values. However, analysis show that the accuracy is sufficient for the identification of property ranges which might be related to diseased tissues. Thereby, the proposed approach turns out promising with view to diagnostic purposes. © 2021, The Author(s).view abstract 10.1007/s10237-021-01541-6 **An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution**

Junker, P. and Balzani, D.*Continuum Mechanics and Thermodynamics*(2021)An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials. © 2021, The Author(s).view abstract 10.1007/s00161-021-01017-z **Estimating cardiomyofiber strain in vivo by solving a computational model**

Perotti, L.E. and Verzhbinsky, I.A. and Moulin, K. and Cork, T.E. and Loecher, M. and Balzani, D. and Ennis, D.B.*Medical Image Analysis*68 (2021)Since heart contraction results from the electrically activated contraction of millions of cardiomyocytes, a measure of cardiomyocyte shortening mechanistically underlies cardiac contraction. In this work we aim to measure preferential aggregate cardiomyocyte (“myofiber”) strains based on Magnetic Resonance Imaging (MRI) data acquired to measure both voxel-wise displacements through systole and myofiber orientation. In order to reduce the effect of experimental noise on the computed myofiber strains, we recast the strains calculation as the solution of a boundary value problem (BVP). This approach does not require a calibrated material model, and consequently is independent of specific myocardial material properties. The solution to this auxiliary BVP is the displacement field corresponding to assigned values of myofiber strains. The actual myofiber strains are then determined by minimizing the difference between computed and measured displacements. The approach is validated using an analytical phantom, for which the ground-truth solution is known. The method is applied to compute myofiber strains using in vivo displacement and myofiber MRI data acquired in a mid-ventricular left ventricle section in N=8 swine subjects. The proposed method shows a more physiological distribution of myofiber strains compared to standard approaches that directly differentiate the displacement field. © 2020view abstract 10.1016/j.media.2020.101932 **Mineral-bonded composites for enhanced structural impact safety: Material level investigations [Impaktsicherheit von Baukonstruktionen durch mineralisch gebundene Komposite: Materialebene]**

Curosu, I. and Mechtcherine, V. and Vo, D.M.P. and Sennewald, C. and Cherif, C. and Wölfel, E. and Scheffler, C. and Gong, T. and Heravi, A.A. and Tamsen, E. and Balzani, D. and Shehni, A. and Häußler-Combe, U. and Fuchs, A. and Kaliske, M. and Scope, C. and Günther, E.*Beton- und Stahlbetonbau*116 (2021)Mineral-bonded composites for enhanced structural impact safety: material level investigations. The Research Training Group GRK 2250/1 „Mineral-bonded composites for enhanced structural impact safety“ aims at the development of novel strengthening materials with various types of fiber reinforcement for enhancing the impact resistance of the critical infrastructure. Adequate testing methods and evaluation protocols are developed for a fundamental material characterization including the rate effects. The experimental results form an important basis for formulating numerical models for simulations of the composites, strengthening layers and concrete elements by linking different space and time scales. Finally, methods for evaluating the sustainability and resilience of the developed strengthening solutions and materials are developed. © 2021, Ernst und Sohn. All rights reserved.view abstract 10.1002/best.202000074 **On the Potential Self-Amplification of Aneurysms Due to Tissue Degradation and Blood Flow Revealed From FSI Simulations**

Wang, H. and Balzani, D. and Vedula, V. and Uhlmann, K. and Varnik, F.*Frontiers in Physiology*12 (2021)Tissue degradation plays a crucial role in the formation and rupture of aneurysms. Using numerical computer simulations, we study the combined effects of blood flow and tissue degradation on intra-aneurysm hemodynamics. Our computational analysis reveals that the degradation-induced changes of the time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI) within the aneurysm dome are inversely correlated. Importantly, their correlation is enhanced in the process of tissue degradation. Regions with a low TAWSS and a high OSI experience still lower TAWSS and higher OSI during degradation. Furthermore, we observed that degradation leads to an increase of the endothelial cell activation potential index, in particular, at places experiencing low wall shear stress. These findings are robust and occur for different geometries, degradation intensities, heart rates and pressures. We interpret these findings in the context of recent literature and argue that the degradation-induced hemodynamic changes may lead to a self-amplification of the flow-induced progressive damage of the aneurysmal wall. Copyright © 2021 Wang, Balzani, Vedula, Uhlmann and Varnik.view abstract 10.3389/fphys.2021.785780 **Rot-free mixed finite elements for gradient elasticity at finite strains**

Riesselmann, J. and Ketteler, J.W. and Schedensack, M. and Balzani, D.*International Journal for Numerical Methods in Engineering*122 (2021)Through enrichment of the elastic potential by the second-order gradient of deformation, gradient elasticity formulations are capable of taking nonlocal effects into account. Moreover, geometry-induced singularities, which may appear when using classical elasticity formulations, disappear due to the higher regularity of the solution. In this contribution, a mixed finite element discretization for finite strain gradient elasticity is investigated, in which instead of the displacements, the first-order gradient of the displacements is the solution variable. Thus, the C1 continuity condition of displacement-based finite elements for gradient elasticity is relaxed to C0. Contrary to existing mixed approaches, the proposed approach incorporates a rot-free constraint, through which the displacements are decoupled from the problem. This has the advantage of a reduction of the number of solution variables. Furthermore, the fulfillment of mathematical stability conditions is shown for the corresponding small strain setting. Numerical examples verify convergence in two and three dimensions and reveal a reduced computing cost compared to competitive formulations. Additionally, the gradient elasticity features of avoiding singularities and modeling size effects are demonstrated. © 2020 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.view abstract 10.1002/nme.6592 **The Effect of Modeling Assumptions on the ECG in Monodomain and Bidomain Simulations**

Ogiermann, D. and Balzani, D. and Perotti, L.E.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*12738 LNCS (2021)Computing a physiologically accurate electrocardiogram (ECG) is one of the key outcomes of cardiac electrophysiology (EP) simulations. Indeed, the simulated ECG serves as a validation, may be the target for optimization in inverse EP problems, and in general allows to link simulation results to clinical ECG data. Several approaches are available to compute the ECG corresponding to an EP simulation. Lead field approaches are commonly used to compute ECGs from cardiac EP simulations using the Monodomain or Eikonal models. A coupled passive conductor model is instead common when the full Bidomain model is adopted. An approach based on solving an auxiliary Poisson problem propagating the activation field from the heart surface to the torso surface is also possible, although not commonly described in the literature. In this work, through a series of numerical experiments, we investigate the limits of validity of the different approaches to compute the ECG from simulations based on the Monodomain and Bidomain models. Significant discrepancies are observed between the common lead field and direct ECG approaches in most realistic cases – e.g., when conduction anisotropy is included – while the ECG computed via solution of an auxiliary Poisson problem is similar to the direct ECG approach. We conclude that either the direct ECG or Poisson approach should be adopted to improve the accuracy of the computed ECG. © 2021, Springer Nature Switzerland AG.view abstract 10.1007/978-3-030-78710-3_48 **Truncated hierarchical B-spline material point method for large deformation geotechnical problems**

Zhang, K. and Shen, S.-L. and Zhou, A. and Balzani, D.*Computers and Geotechnics*134 (2021)This article integrates the truncated hierarchical B-spline into the material point method (MPM) to address the large deformation problem in geotechnical engineering. The proposed approach allows the MPM to work with a locally refined hierarchical background grid, by which computational resources could be concentrated in spatial domains of concern. The truncated hierarchical B-spline forms the partition of unity property throughout the computational domain by reducing the support of the basis functions on adjacent hierarchical levels. Two auxiliary data structures beneficial to the hierarchical particle-grid mappings are introduced to facilitate the implementation of the truncated hierarchical B-splines in the framework of MPM. In addition, a particle splitting strategy is employed to eliminate numerical fracture problems that may occur in the case of extremely large deformation. Validation and application examples demonstrate the robustness and stability of the proposed method. © 2021 Elsevier Ltdview abstract 10.1016/j.compgeo.2021.104097 **A new variational approach for the thermodynamic topology optimization of hyperelastic structures**

Junker, P. and Balzani, D.*Computational Mechanics*(2020)We present a novel approach to topology optimization based on thermodynamic extremal principles. This approach comprises three advantages: (1) it is valid for arbitrary hyperelastic material formulations while avoiding artificial procedures that were necessary in our previous approaches for topology optimization based on thermodynamic principles; (2) the important constraints of bounded relative density and total structure volume are fulfilled analytically which simplifies the numerical implementation significantly; (3) it possesses a mathematical structure that allows for a variety of numerical procedures to solve the problem of topology optimization without distinct optimization routines. We present a detailed model derivation including the chosen numerical discretization and show the validity of the approach by simulating two boundary value problems with large deformations. © 2020, The Author(s).view abstract 10.1007/s00466-020-01949-4 **Architectural woven polyester fabrics: examination of possible classification of stiffness values in correlation with strength values**

Uhlemann, J. and Stranghöner, N. and Motevalli, M. and Balzani, D.*Architectural Engineering and Design Management*(2020)Down to the present day, the determination of stiffness parameters for architectural fabrics, mainly used as coated woven fabrics, is a field of intense discussion. Designers and material producers are affected by an existing uncertainty. The present contribution investigates the possibility to provide tables of tensile stiffness parameters for specific materials in which tensile stiffness values are given dependent on the material’s tensile strength. The focus is on PVC-coated polyester fabrics as the most commonly used material for textile architecture. Materials of four different producers have been tested experimentally. Furthermore, from each material producer products of different strength classes have been considered in the test series. To achieve a general statement on the possibility of material classification according to their tensile stiffness, material parameters of two different material formulations have been investigated: (1) for a simplified but widely applied linear-elastic material model based on a standardized test and evaluation method and (2) for a recently developed more advanced orthotropic hyperelastic nonlinear material model. The tensile stiffness parameters are statistically evaluated and compared. In order to achieve a required basis, standardized and project-specific biaxial test procedures are presented and analysed in combination with established evaluation procedures. A tabulation of tensile stiffness parameters would be of great help for design engineers and material producers. The results of the present work will lead to a possible classified tabulation of the investigated materials in a meaningful manner. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/17452007.2020.1793720 **Automated simulation of voxel-based microstructures based on enhanced finite cell approach**

Fangye, Y.F. and Miska, N. and Balzani, D.*Archive of Applied Mechanics*90 (2020)A new and efficient method is proposed for the decomposition of finite elements into finite subcells, which are used to obtain an integration scheme allowing to analyse complex microstructure morphologies in regular finite element discretizations. Since the geometry data of reconstructed microstructures are often given as voxel data, it is reasonable to exploit the special properties of the given data when constructing the subcells, i.e. the perpendicularly cornered shape of the constituent interfaces at the microscale. Thus, in order to obtain a more efficient integration scheme, the proposed method aims to construct a significantly reduced number of subcells by aggregating as much voxels as possible to larger cuboids. The resulting methods are analysed and compared with the conventional Octree algorithm. Eventually, the proposed optimal decomposition method is used for a virtual tension test on a reconstructed three-dimensional microstructure of a dual-phase steel, which is afterwards compared to real experimental data. © 2020, The Author(s).view abstract 10.1007/s00419-020-01719-x **Computational micro-macro analysis of impact on strain-hardening cementitious composites (Shcc) including microscopic inertia**

Tamsen, E. and Curosu, I. and Mechtcherine, V. and Balzani, D.*Materials*13 (2020)This paper presents a numerical two-scale framework for the simulation of fiber reinforced concrete under impact loading. The numerical homogenization framework considers the full balance of linear momentum at the microscale. This allows for the study of microscopic inertia effects affecting the macroscale. After describing the ideas of the dynamic framework and the material models applied at the microscale, the experimental behavior of the fiber and the fiber–matrix bond under varying loading rates are discussed. To capture the most important features, a simplified matrix cracking and a strain rate sensitive fiber pullout model are utilized at the microscale. A split Hopkinson tension bar test is used as an example to present the capabilities of the framework to analyze different sources of dynamic behavior measured at the macroscale. The induced loading wave is studied and the influence of structural inertia on the measured signals within the simulation are verified. Further parameter studies allow the analysis of the macroscopic response resulting from the rate dependent fiber pullout as well as the direct study of the microscale inertia. Even though the material models and the microscale discretization used within this study are simplified, the value of the numerical two-scale framework to study material behavior under impact loading is demonstrated. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/ma13214934 **Exasteel: Towards a virtual laboratory for the multiscale simulation of dual-phase steel using high-performance computing**

Klawonn, A. and Lanser, M. and Uran, M. and Rheinbach, O. and Köhler, S. and Schröder, J. and Scheunemann, L. and Brands, D. and Balzani, D. and Gandhi, A. and Wellein, G. and Wittmann, M. and Schenk, O. and Janalík, R.*Lecture Notes in Computational Science and Engineering*136 (2020)We present a numerical two-scale simulation approach of the Nakajima test for dual-phase steel using the software package FE2TI, a highly scalable implementation of the well known homogenization method FE2. We consider the incorporation of contact constraints using the penalty method as well as the sample sheet geometries and adequate boundary conditions. Additional software features such as a simple load step strategy and prediction of an initial value by linear extrapolation are introduced. The macroscopic material behavior of dual-phase steel strongly depends on its microstructure and has to be incorporated for an accurate solution. For a reasonable computational effort, the concept of statistically similar representative volume elements (SSRVEs) is presented. Furthermore, the highly scalable nonlinear domain decomposition methods NL-FETI-DP and nonlinear BDDC are introduced and weak scaling results are shown. These methods can be used, e.g., for the solution of the microscopic problems. Additionally, some remarks on sparse direct solvers are given, especially to PARDISO. Finally, we come up with a computationally derived Forming Limit Curve (FLC). © The Author(s) 2020.view abstract 10.1007/978-3-030-47956-5_13 **Numerical material testing for discontinuous fiber composites using statistically similar representative volume elements**

Sasagawa, T. and Tanaka, M. and Omote, R. and Balzani, D.*Scientific Reports*10 (2020)A computational method is proposed in order to predict mechanical properties of discontinuous fiber composites (DFCs) based on computational homogenization with statistically similar representative volume elements (SSRVEs). The SSRVEs are obtained by reducing the complexity of real microstructures based on statistical measures. Specifically, they are constructed by minimizing an objective function defined in terms of differences between the power spectral density of target microstructures and that of the SSRVEs. In this paper, an extended construction method is proposed based on the reformulation of the objective function by integer design variables. The proposed method is applied to the representation of a real material, namely glass fiber reinforced nylon 6. The results show that the mechanical properties computed by numerical material tests using the SSRVEs agree with experimental results. Therefore, it is found that the nonlinear mechanical properties of the DFC can be suitably predicted by the proposed method without any special calibration to experiments performed on the composites. © 2020, The Author(s).view abstract 10.1038/s41598-020-66963-1 **Saturation of the stress-strain behaviour of architectural fabrics**

Uhlemann, J. and Surholt, F. and Westerhoff, A. and Stranghöner, N. and Motevalli, M. and Balzani, D.*Materials and Design*191 (2020)The stress-strain characteristics of nonlinear visco-elastoplastic architectural fabrics show a mechanically saturating behaviour in cyclic tensile tests: stiffness changes decline, the increase of permanent strain decreases and the nonlinear material behaviour increasingly approaches a linear behaviour. Appropriate stress-strain paths for elastic material models for the structural analysis must be taken from load cycles in which the saturation processes are finished to a satisfactory degree. Generally, consistency of elastic analysis of fabric strutctures is achieved when it is based on the fabric's saturated state and the closely linked stable amount of presstress in the structure. To determine load cycles correlated to the saturated state, a new method is proposed here which considers three different inspection characteristics: irreversible strain increment, total strain increment and intensity of nonlinearity. Biaxial saturation tests with 1000 load cycles are performed on a PVC-coated polyester fabric. For each inspection characteristic, saturation development curves are generated. They are fitted by functions with horizontal asymptotes and are thereby extrapolated, revealing that for the tested polyster fabric tens of thousands of load cycles are required to achieve a satisfying state of saturation. The meaning and implications for biaxial testing and analysis of textile structures is discussed. © 2020 The Authorsview abstract 10.1016/j.matdes.2020.108584 **The elastic share of inelastic stress-strain paths of woven fabrics**

Motevalli, M. and Uhlemann, J. and Stranghöner, N. and Balzani, D.*Materials*13 (2020)Manifold variations of the mechanical behavior of structural woven fabrics appear in the first load cycles. Nevertheless, invariable states, i.e., mechanically saturated states, can be approached by multiple monotonous load cycle biaxial tests. In a state acceptably close to the ideal saturated state, the stress-strain paths reveal the elastic share of the initially inelastic stress-strain paths of woven fabrics. In this paper, the mechanical saturation behavior of two types of PTFE-coated woven glass fiber fabrics is examined and compared to the recently reported saturation behavior of a PVC-coated polyester fabric. With the help of the saturation test data, an extrapolation function is developed that facilitates an estimation of late cycle stiffness behavior based on measured early cycle behavior. Furthermore, the considerable impact of late cycle properties on structural analyses is shown exemplarily in the numerical simulation of a prestressed fabric structure by comparing results achieved from late and early load cycle stiffness parameters. © 2020 by the authors.view abstract 10.3390/MA13194243 **Three-field mixed finite element formulations for gradient elasticity at finite strains**

Riesselmann, J. and Ketteler, J.W. and Schedensack, M. and Balzani, D.*GAMM Mitteilungen*43 (2020)Gradient elasticity formulations have the advantage of avoiding geometry-induced singularities and corresponding mesh dependent finite element solution as apparent in classical elasticity formulations. Moreover, through the gradient enrichment the modeling of a scale-dependent constitutive behavior becomes possible. In order to remain C0 continuity, three-field mixed formulations can be used. Since so far in the literature these only appear in the small strain framework, in this contribution formulations within the general finite strain hyperelastic setting are investigated. In addition to that, an investigation of the inf sup condition is conducted and unveils a lack of existence of a stable solution with respect to the L2-H1-setting of the continuous formulation independent of the constitutive model. To investigate this further, various discretizations are analyzed and tested in numerical experiments. For several approximation spaces, which at first glance seem to be natural choices, further stability issues are uncovered. For some discretizations however, numerical experiments in the finite strain setting show convergence to the correct solution despite the stability issues of the continuous formulation. This gives motivation for further investigation of this circumstance in future research. Supplementary numerical results unveil the ability to avoid singularities, which would appear with classical elasticity formulations. © 2019 The Authors. GAMM - Mitteilungen published by Wiley-VCH Verlag GmbH & Co. KGaA on behalf of Gesellschaft für Angewandte Mathematik und Mechanikview abstract 10.1002/gamm.202000002 **Geometrically nonlinear simulation of textile membrane structures based on orthotropic hyperelastic energy functions**

Motevalli, M. and Uhlemann, J. and Stranghöner, N. and Balzani, D.*Composite Structures*223 (2019)New hyperelastic orthotropic models are proposed for the simulation of textile membranes used in civil engineering applications. In contrast to published models, part of the new models is polyconvex and ensures thereby a physically meaningful and mathematically sound formulation. The models are adjusted to uniaxial tension tests performed in warp and fill direction, where not only the stress-strain response in tension direction is accounted for but also the lateral contraction. Thereby, the crosswise interaction between the warp and fill direction is captured. In a series of different boundary value problems the new models as well as a competitive formulation given in literature are compared with respect to the accuracy to represent the experimental data, the mathematical properties as well as the numerical robustness. As it turns out, most formulations including the model from the literature show a loss of material stability and non-converging Newton iterations in structural simulations. Only one of the proposed polyconvex formulations works robustly in numerical simulations of realistic structural engineering problems. Thereby, this new orthotropic model enables realistic simulations of textile membranes in a fully geometrically nonlinear setting, which does not require simplifications based on linearized strains, which are currently used as standard in engineering practice. © 2019 Elsevier Ltdview abstract 10.1016/j.compstruct.2019.110908 **Mechanical damage characterization in human femoropopliteal arteries of different ages**

Anttila, E. and Balzani, D. and Desyatova, A. and Deegan, P. and MacTaggart, J. and Kamenskiy, A.*Acta Biomaterialia*90 (2019)Endovascular treatment of Peripheral Arterial Disease (PAD)is notorious for high failure rates, and interaction between the arterial wall and the repair devices plays a significant role. Computational modeling can help improve clinical outcomes of these interventions, but it requires accurate inputs of elastic and damage characteristics of the femoropopliteal artery (FPA)which are currently not available. Fresh human FPAs from n = 104 tissue donors 14–80 years old were tested using planar biaxial extension to capture elastic and damage characteristics. Damage initiation stretches and stresses were determined for both longitudinal and circumferential directions, and their correlations with age and risk factors were assessed. Two and four-fiber-family invariant-based constitutive models augmented with damage functions were used to describe stress softening with accumulating damage. In FPAs younger than 50 years, damage began accumulating after 1.51 ± 0.13 and 1.49 ± 0.11 stretch, or 196 ± 110 kPa and 239 ± 79 kPa Cauchy stress in the longitudinal and circumferential directions, respectively. In FPAs older than 50 years, damage initiation stretches and stresses decreased to 1.27 ± 0.09 (106 ± 52 kPa)and 1.26 ± 0.09 (104 ± 59 kPa), respectively. Damage manifested primarily as tears at the internal and external elastic laminae and within the tunica media layer. Higher body mass index and presence of diabetes were associated with lower damage initiation stretches and higher stresses. The selected constitutive models were able to accurately portray the FPA behavior in both elastic and inelastic domains, and properties were derived for six age groups. Presented data can help improve fidelity of computational models simulating endovascular PAD repairs that involve arterial damage. Statement of significance: This manuscript describes inelastic, i.e. damage, behavior of human femoropopliteal arteries, and provides values for three constitutive models simulating this behavior computationally. Using a set of 104 human FPAs 14–80 years old, we have investigated stress and stretch levels corresponding to damage initiation, and have studied how these damage characteristics change across different age groups. Presented inelastic arterial characteristics are important for computational simulations modeling balloon angioplasty and stenting of peripheral arterial disease lesions. © 2019 Acta Materialia Inc.view abstract 10.1016/j.actbio.2019.03.053 **Modeling of structures with polymorphic uncertainties at different length scales**

Kremer, K. and Edler, P. and Miska, N. and Leichsenring, F. and Balzani, D. and Freitag, S. and Graf, W. and Kaliske, M. and Meschke, G.*GAMM Mitteilungen*42 (2019)Multiscale analyses require to consider the scale bridging influences of uncertain parameters. In this paper, approaches for polymorphic uncertainty quantification at different length scales are presented. Especially, the effect of uncertain material parameters computed at lower structural scales is investigated with respect to the resulting macroscopic structural behavior. Also the dependencies of uncertain parameters at the macroscale on the structural response evaluated with submodels are discussed. Three examples are shown, where interval and stochastic uncertainty quantification approaches are combined. Two examples deal with material modeling of concrete and the durability of reinforced concrete structures under consideration of polymorphic uncertainties. A mesoscale model of concrete is developed based on a representative volume element containing aggregates, pores, and the cement phase, where the values of the Young's moduli for mortar and the aggregates as well as the volume fraction of the cement phase to aggregates are considered as intervals. Within the durability assessment of reinforced concrete structures, the influence of stochastic distributed loading and concrete material parameters onto the cracking behavior is analyzed by means of a submodeling strategy. In another application, the metal forming process of dual-phase steel sheets is investigated using statistical information of the microscopic material behavior in combination with epistemic uncertainties of the failure criterion and the friction coefficient between the sheet metal and the forming tools. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheimview abstract 10.1002/gamm.201900006 **Polyacrylamide Bead Sensors for in vivo Quantification of Cell-Scale Stress in Zebrafish Development**

Träber, N. and Uhlmann, K. and Girardo, S. and Kesavan, G. and Wagner, K. and Friedrichs, J. and Goswami, R. and Bai, K. and Brand, M. and Werner, C. and Balzani, D. and Guck, J.*Scientific Reports*9 (2019)Mechanical stress exerted and experienced by cells during tissue morphogenesis and organ formation plays an important role in embryonic development. While techniques to quantify mechanical stresses in vitro are available, few methods exist for studying stresses in living organisms. Here, we describe and characterize cell-like polyacrylamide (PAAm) bead sensors with well-defined elastic properties and size for in vivo quantification of cell-scale stresses. The beads were injected into developing zebrafish embryos and their deformations were computationally analyzed to delineate spatio-temporal local acting stresses. With this computational analysis-based cell-scale stress sensing (COMPAX) we are able to detect pulsatile pressure propagation in the developing neural rod potentially originating from polarized midline cell divisions and continuous tissue flow. COMPAX is expected to provide novel spatio-temporal insight into developmental processes at the local tissue level and to facilitate quantitative investigation and a better understanding of morphogenetic processes. © 2019, The Author(s).view abstract 10.1038/s41598-019-53425-6 **Quantification of uncertain macroscopic material properties resulting from variations of microstructure morphology based on statistically similar volume elements: application to dual-phase steel microstructures**

Miska, N. and Balzani, D.*Computational Mechanics*64 (2019)A method to quantify uncertain macroscopic material properties resulting from variations of a material’s microstructure morphology is proposed. Basis is the computational homogenization of virtual experiments as part of a Monte-Carlo simulation to obtain the associated uncertain macroscopic material properties. A new general approach is presented to construct a set of artificial microstructures, which exhibits a statistically similar variation of the morphology as the real material’s microstructure. The individual artificial microstructures are directly constructed in a way that a lower discretization effort is required compared to real microstructures. The costs to perform the computational homogenization for all considered SSVEs are reduced by an adapted form of the Finite Cell concept and by applying the multilevel Monte-Carlo method. As an illustrative example, the proposed method is applied to a real Dual-Phase steel microstructure. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00466-019-01738-8 **A combined growth and remodeling framework for the approximation of residual stresses in arterial walls**

Zahn, A. and Balzani, D.*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*98 (2018)A combined model of multiplicative growth and fiber remodeling in the sense of a reorientation of the collagen fibers is proposed for the simulation of adaptation processes in arterial tissues, where both mechanisms are supposed to be governed by the intensity and the direction of the principal stresses. The generalized formulation of the growth tensor includes up to three perpendicular anisotropy directions, which are defined based on the local principal stress state. Remodeling is incorporated in a straightforward manner by formulating a scalar evolution equation for the angle between the existing and the target fiber orientation vectors. In numerical examples on idealized arterial segments, the fiber remodeling algorithm is illustrated and a comparison of different approaches for the growth tensor with respect to stresses in the loaded state, fiber angles and residual stresses is conducted. © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheimview abstract 10.1002/zamm.201700273 **Construction of statistically similar representative volume elements for discontinuous fiber composites**

Sasagawa, T. and Tanaka, M. and Omote, R. and Balzani, D.*Composite Structures*203 (2018)A computational method is proposed for the construction of statistically similar representative volume elements (SSRVEs) for discontinuous fiber composites (DFCs) in order to enable an efficient calculation of material properties based on computational homogenization. The SSRVEs are obtained by solving an optimization problem which minimizes the difference between the power spectral density of a target microstructure and its simplified one. The SSRVEs are constructed for target microstructures serving as examples for DFCs, which are validated by means of comparing the mechanical properties of the target microstructures with the ones of the SSRVEs. The results show that the mechanical properties of the SSRVEs agree with the target microstructures and that the SSRVEs can extremely reduce the computational costs of finite element analyses to derive macroscopic material properties of DFCs. © 2018 Elsevier Ltdview abstract 10.1016/j.compstruct.2018.06.014 **Influence of microstructure morphology on multi-scale modeling of low-alloyed TRIP-steels**

Prüger, S. and Gandhi, A. and Balzani, D.*Engineering Computations (Swansea, Wales)*35 (2018)Purpose: The purpose of this study is to quantify the impact of the variation of microstructural features on macroscopic and microscopic fields. The application of multi-scale methods in the context of constitutive modeling of microheterogeneous materials requires the choice of a representative volume element (RVE) of the considered microstructure, which may be based on some idealized assumptions and/or on experimental observations. In any case, a realistic microstructure within the RVE is either computationally too expensive or not fully accessible by experimental measurement techniques, which introduces some uncertainty regarding the microstructural features. Design/methodology/approach: In this paper, a systematical variation of microstructural parameters controlling the morphology of an RVE with an idealized microstructure is conducted and the impact on macroscopic quantities of interest as well as microstructural fields and their statistics is investigated. The study is carried out under macroscopically homogeneous deformation states using the direct micro-macro scale transition approach. Findings: The variation of microstructural parameters, such as inclusion volume fraction, aspect ratio and orientation of the inclusion with respect to the overall loading, influences the macroscopic behavior, especially the micromechanical fields significantly. Originality/value: The systematic assessment of the impact of microstructural parameters on both macroscopic quantities and statistics of the micromechanical fields allows for a quantitative comparison of different microstructure morphologies and a reliable identification of microstructural parameters that promote failure initialization in microheterogeneous materials. © 2018, Emerald Publishing Limited.view abstract 10.1108/EC-01-2017-0009 **Local laser-strengthening: Customizing the forming behavior of car body steel sheets**

Wagner, M. and Jahn, A. and Beyer, E. and Balzani, D.*AIP Conference Proceedings*1960 (2018)Future trends in designing lightweight components especially for automotive applications increasingly require complex and delicate structures with highest possible level of capacity [1]. The manufacturing of metallic car body components is primarily realized by deep or stretch drawing. The forming process of especially cold rolled and large-sized components is typically characterized by inhomogeneous stress and strain distributions. As a result, the avoidance of undesirable deep drawing effects like earing and local necking is among the greatest challenges in forming complex car body structures [2]. Hence, a novel local laser-treatment approach with the objective of customizing the forming behavior of car body steel sheets is currently explored. © 2018 Author(s).view abstract 10.1063/1.5035023 **Numerical and Experimental Study of the Spatial Stress Distribution on the Cornea Surface During a Non-Contact Tonometry Examination**

Muench, S. and Roellig, M. and Spoerl, E. and Balzani, D.*Experimental Mechanics*(2018)The determination of biomechanical properties of the cornea by a non-contact tonometry (NCT) examination requires a precise knowledge of the air puff generated in the device, which is applied to the cornea surface. In this study, a method is proposed to identify the resulting stress profile on the surface, which may be used to numerically solve an inverse problem to obtain the material properties. This method is based on an experimental characterization of the air puff created by the Corvis ST in combination with computational fluid dynamic (CFD) simulations, which are adjusted to the experimental data. The identified nozzle inlet pressure of approximately 25 kPa (188.5mmHg) is then used for a numerical influence study of the interaction between the air puff and the cornea deformation. Therefore, eleven cornea deformation states based on measurements are implemented in the CFD model. A more realistic model is also analyzed by the geometrical reproduction of the human face, which is used for a further influence study. The outcomes showed a dependence between the cornea deformation and the pressure as well as the shear stress distribution. However, quantitatively, the shear stress component can be considered of minor importance being approximately one hundred times smaller than the pressure. The examination with consideration of the human face demonstrates that the pressure and shear stress distributions are not rotationally symmetric in measurements on real humans, which indicates the requirement to include more complex stress distributions on the eye. We present the detailed stress distribution on the cornea during a non-contact tonometry examination, which is made accessible for further investigations in the future by analytical nonlinear functions. © 2018, Society for Experimental Mechanics.view abstract 10.1007/s11340-018-00449-0 **Construction of statistically similar rves for the quantification of uncertainty associated with the material's microstructure morphology**

Balzani, D. and Miska, N. and Pruger, S.*UNCECOMP 2017 - Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering*2017-January (2017)Various engineering applications require the use of modern materials. In particular with view to automotive applications, where increased safety standards at reduced weight are important, multiphase steels are advantageous. These steels make use of a pronounced mi- crostructure in order to achieve a high ductility with high strength. The morphology of these microstructures varies over the location in the macroscopic part and over different specimen. As the macroscopic response of the steel is governed by the microstructure morphology, the randomness of the microscopic morphology implies uncertainties regarding the macroscopic material response. We propose to create statistically similar representative volume elements (SSRVE) to enable access to the incorporation of these uncertainties in numerical computations. The SSRVEs are obtained by minimizing a least-square functional consisting of higher order statistical measures, which describe the morphology of the microstructure. The resulting geometries are significantly less complex than the real microstructure and exhibit an advantage regarding meshing and computing the problem. Aside from these advantages, the method also provides a basis to construct various SSRVEs which are within predefined bounds regarding the microstructure statistics. These bounds may be obtained from measurements performed by analyzing the microstructures at different locations in one material. Rased on this variety of applicable SSRVEs multiple Finite Element (FE) simulations can be performed to obtain the homogenized response and thus, to quantify statistics regarding macroscopic material parameters. In order to automatize these numerical simulations the Finite Cell Method (FCM) can be applied such that a conforming FE mesh for each of the SSRVEs is not needed to be constructed. © 2017 The Authors. Published by Eccomas Proceedia.view abstract 10.7712/120217.5380.16923 **Method for the quantification of rupture probability in soft collagenous tissues**

Balzani, D. and Schmidt, T. and Ortiz, M.*International Journal for Numerical Methods in Biomedical Engineering*33 (2017)A computational method is presented for the assessment of rupture probabilities in soft collagenous tissues. This may in particular be important for the quantitative analysis of medical diseases such as atherosclerotic arteries or abdominal aortic aneurysms, where an unidentified rupture has in most cases fatal consequences. The method is based on the numerical minimization and maximization of probabilities of failure, which arise from random input quantities, for example, tissue properties. Instead of assuming probability distributions for these quantities, which are typically unknown especially for soft collagenous tissues, only restricted knowledge of these distributions is taken into account. Given this limited statistical input data, the minimized/maximized probabilities represent optimal bounds on the rupture probability, which enable a quantitative estimation of potential risks of performing or not performing medical treatment. Although easily extendable to all kinds of mechanical rupture criteria, the approach presented here incorporates stretch-based and damage-based criteria. These are evaluated based on numerical simulations of loaded tissues, where continuum mechanical material formulations are considered, which capture the supra-physiological behavior of soft collagenous tissues. Numerical examples are provided demonstrating the applicability of the method in an overstretched atherosclerotic artery. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.2781 **Method for the unique identification of hyperelastic material properties using full-field measures. Application to the passive myocardium material response**

Perotti, L.E. and Ponnaluri, A.V.S. and Krishnamoorthi, S. and Balzani, D. and Ennis, D.B. and Klug, W.S.*International Journal for Numerical Methods in Biomedical Engineering*33 (2017)Quantitative measurement of the material properties (eg, stiffness) of biological tissues is poised to become a powerful diagnostic tool. There are currently several methods in the literature to estimating material stiffness, and we extend this work by formulating a framework that leads to uniquely identified material properties. We design an approach to work with full-field displacement data—ie, we assume the displacement field due to the applied forces is known both on the boundaries and also within the interior of the body of interest—and seek stiffness parameters that lead to balanced internal and external forces in a model. For in vivo applications, the displacement data can be acquired clinically using magnetic resonance imaging while the forces may be computed from pressure measurements, eg, through catheterization. We outline a set of conditions under which the least-square force error objective function is convex, yielding uniquely identified material properties. An important component of our framework is a new numerical strategy to formulate polyconvex material energy laws that are linear in the material properties and provide one optimal description of the available experimental data. An outcome of our approach is the analysis of the reliability of the identified material properties, even for material laws that do not admit unique property identification. Lastly, we evaluate our approach using passive myocardium experimental data at the material point and show its application to identifying myocardial stiffness with an in silico experiment modeling the passive filling of the left ventricle. Copyright © 2017 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.2866 **A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy**

Schröder, J. and Viebahn, N. and Balzani, D. and Wriggers, P.*Computer Methods in Applied Mechanics and Engineering*310 (2016)A variety of numerical approximation schemes for boundary value problems suffer from so-called locking-phenomena. It is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables. A serious locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. The main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework. Therefore we split the strain energy into two main parts, an isotropic and an anisotropic part. For the isotropic part we can apply different well-established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right Cauchy–Green tensor. This additional constraint is attached to the strain energy function by a second-order tensorial Lagrange-multiplier, governed by a Simplified Kinematic for the Anisotropic part. As a matter of fact, for the tested boundary value problems the SKA-element based on quadratic ansatz functions for the displacements, performs excellent and behaves more robust than competitive formulations. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.06.029 **An algorithmic scheme for the automated calculation of fiber orientations in arterial walls**

Fausten, S. and Balzani, D. and Schröder, J.*Computational Mechanics*58 (2016)We propose an algorithmic scheme for the numerical calculation of fiber orientations in arterial walls. The basic assumption behind the procedure is that the fiber orientations are mainly governed by the principal tensile stress directions resulting in an improved load transfer within the artery as a consequence of the redistribution of stresses. This reflects the biological motivation that soft tissues continuously adapt to their mechanical environment in order to optimize their load-bearing capacities. The algorithmic scheme proposed here enhances efficiency of the general procedure given in Hariton et al. (Biomech Model Mechanobiol 6(3):163-175, 2007), which consists of repeatedly identifying a favored fiber orientation based on the principal tensile stresses under a certain loading scenario, and then re-calculating the stresses for that loading scenario with the modified favored fiber orientation. Since the method still depends on a highly accurate stress approximation of the finite element formulation, which is not straightforward to obtain in particular for incompressible and highly anisotropic materials, furthermore, a modified model is introduced. This model defines the favored fiber orientation not only in terms of the local principal stresses, but in terms of the volume averages of the principal stresses computed over individual finite elements. Thereby, the influence of imperfect stress approximations can be weakened leading to a stabilized convergence of the reorientation procedure and a more reasonable fiber orientation with less numerical noise. The performance of the proposed fiber reorientation scheme is investigated with respect to different finite element formulations and different favored fiber orientation models, Hariton et al. (Biomech Model Mechanobiol 6(3):163-175, 2007) and Cyron and Humphrey (Math Mech Solids 1-17, 2014). In addition, it is applied to calculate the fiber orientation in a patient-specific arterial geometry.view abstract 10.1007/s00466-016-1321-z **Analytical and numerical modelling of a sub- and supersonic moving load front along a rod’s skin**

Weber, W.E. and Fangye, Y.F. and Balzani, D. and Zastrau, B.W.*Advanced Structured Materials*60 (2016)For both civil and mechanical engineering dynamic loads of structures are a major source of inner material damage. If (fibre) reinforced composite materials are exposed to such dynamic loads a pull-out of the reinforcing elements may occur. This dynamic pull-out of reinforcing elements is characterized by, amongst others, moving boundaries between regions of (partly) damaged and perfect bonding of reinforcement and surrounding matrix. To adequately describe these moving boundaries leads to enormous challenges. Within this contribution a simplified mechanical problem is investigated, which however provides some of the main phenomena of the dynamic pull-out. In detail, the stress and displacement fields within a rod of semi-infinite extent under a distributed load are evaluated. Herein, the front of the constant longitudinal load moves along the rod in longitudinal direction. The investigations are performed both analytically and numerically thus validating the model idealization included in the analytical solution. © Springer Science+Business Media Singapore 2016.view abstract 10.1007/978-981-10-0959-4_26 **Computational modeling of dual-phase steels based on representative three-dimensional microstructures obtained from EBSD data**

Brands, D. and Balzani, D. and Scheunemann, L. and Schröder, J. and Richter, H. and Raabe, D.*Archive of Applied Mechanics*86 (2016)The microstructure of dual-phase steels consisting of a ferrite matrix with embedded martensite inclusions is the main contributor to the mechanical properties such as high ultimate tensile strength, high work hardening rate, and good ductility. Due to the composite structure and the wide field of applications of this steel type, a wide interest exists in corresponding virtual computational experiments. For a reliable modeling, the microstructure should be included. For that reason, in this paper we follow a computational strategy based on the definition of a representative volume element (RVE). These RVEs will be constructed by a set of tomographic measurements and mechanical tests. In order to arrive at more efficient numerical schemes, we also construct statistically similar RVEs, which are characterized by a lower complexity compared with the real microstructure but which represent the overall material behavior accurately. In addition to the morphology of the microstructure, the austenite–martensite transformation during the steel production has a relevant influence on the mechanical properties and is considered in this contribution. This transformation induces a volume expansion of the martensite phase. A further effect is determined in nanoindentation test, where it turns out that the hardness in the ferrite phase increases exponentially when approaching the martensitic inclusion. To capture these gradient properties in the computational model, the volumetric expansion is applied to the martensite phase, and the arising equivalent plastic strain distribution in the ferrite phase serves as basis for a locally graded modification of the ferritic yield curve. Good accordance of the model considering the gradient yield behavior in the ferrite phase is observed in the numerical simulations with experimental data. © 2015, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-015-1044-1 **Implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers**

Tanaka, M. and Balzani, D. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*301 (2016)In this paper, novel implementation schemes for the automatic calculation of internal variables, stresses and consistent tangent moduli for incremental variational formulations (IVFs) describing inelastic material behavior are proposed. IVFs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables. In the so-called Multilevel Newton-Raphson method for the inelasticity theory, this minimization problem is typically solved by using second derivatives with respect to the internal variables. In addition to that, to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required. Compared with classical formulations such as the return mapping method, the IVFs are relatively new and their implementation is much less documented. Furthermore, higher order derivatives are required in the algorithms demanding increased implementation efforts. Therefore, even though IVFs are mathematically and physically elegant, their application is not standard. Here, novel approaches for the implementation of IVFs using HDNs of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy. The proposed formulations are quite general and can be applied to a broad range of different constitutive models, which means that once the proposed schemes are implemented as a framework, any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions. These include the Helmholtz free energy function, the dissipation potential function and additional side constraints such as e.g. the yield function in the case of plasticity. Its uncomplicated implementation for associative finite strain elasto-plasticity and performance is illustrated by some representative numerical examples. © 2015 Elsevier B.V.view abstract 10.1016/j.cma.2015.12.010 **Modeling of low-alloyed trip-steels based on direct micro-macro simulations**

Prüger, S. and Gandhi, A. and Balzani, D.*ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering*2 (2016)Low-alloyed TRIP steels are often used in the automotive industry due to their favorable mechanical properties such as high ductility and strength and their moderate production costs. These steels possess a heterogeneous multiphase microstructure, initially consisting of ferrite, bainite and retained austenite which is responsible for the mechanical properties. Upon deformation, a diffusionless, stress-induced, martensitic phase transformation from austenite to martensite is observed, enhancing ductility and strength. We focus on multi-scale methods in the sense of FE2 to describe the macroscopic behavior of low-alloyed TRIP-steels, because this approach allows for a straightforward inclusion of various influencing factors such as residual stress distribution, graded material properties which can hardly included in phenomenological descriptions of these heterogeneous multiphase materials. In order to allow for efficient computations, a simplified microstructure is used in an illustrative direct micro-macro simulation. The inelastic processes in the austenitic inclusions involve the phase transformation from austenite to martensite and the inelastic deformation of these two phases. The isotropic, rate-independent, hyperelastic-plastic material model of Hallberg et al. (IJP, 23, pp. 1213-1239, 2007), originally proposed for high-alloyed TRIP steel, is adopted here for the inclusion phase. Minor modifications of the model are proposed to improve its implementation and performance. The influence of various material parameters associated with the phase transformation on the evolution of retained austenite is studied for different homogeneous deformation states. The non-monotonic stress-state dependence observed in experiments is clearly captured by the model. A numerical two-scale calculation is carried out to enlighten the ductility enhancement in low-alloyed TRIP-steels due to the martensitic phase transformation.view abstract 10.7712/100016.1959.7726 **Modeling of Microstructure Evolution with Dynamic Recrystallization in Finite Element Simulations of Martensitic Steel**

Baron, T.J. and Khlopkov, K. and Pretorius, T. and Balzani, D. and Brands, D. and Schröder, J.*Steel Research International*87 (2016)A metallurgical material description of the flow behavior for finite element (FE) simulations was developed. During hot compression tests, the dynamic microstructure evolution is modeled on the example of high-strength martensitic steel MS-W 1200. Compression tests at 900-1000 °C with a strain rate of 0.1 s-1 on fine-grain and coarse-grain samples were performed. An analysis of the flow behavior identified a strong correlation between the dynamic recrystallization kinetics and the initial microstructure. The regression analysis has been used to determine correction factors of the new model to describe the dynamic recrystallization. A good agreement between FE simulation and measurement shows the validity of the new model. A metallurgical material description of the flow behavior for finite element (FE) simulations is developed. During hot compression tests, the dynamic microstructure evolution is modeled on the example of high-strength martensitic steel MS-W 1200. An analysis of the flow behavior identifies a strong correlation between the dynamic recrystallization kinetics and the initial microstructure. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/srin.201400576 **Numerical modeling of fluid–structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains**

Balzani, D. and Deparis, S. and Fausten, S. and Forti, D. and Heinlein, A. and Klawonn, A. and Quarteroni, A. and Rheinbach, O. and Schröder, J.*International Journal for Numerical Methods in Biomedical Engineering*32 (2016)The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid–structure interactions and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid–structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid–structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple – but nonsymmetric – curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid–structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often-used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F-bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second-order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non-rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid–structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations. Copyright © 2015 John Wiley & Sons, Ltd. Copyright © 2015 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.2756 **One-way and fully-coupled FE2 methods for heterogeneous elasticity and plasticity problems: Parallel scalability and an application to thermo-elastoplasticity of dual-phase steels**

Balzani, D. and Gandhi, A. and Klawonn, A. and Lanser, M. and Rheinbach, O. and Schröder, J.*Lecture Notes in Computational Science and Engineering*113 (2016)In this paper, aspects of the two-scale simulation of dual-phase steels are considered. First, we present two-scale simulations applying a top-down oneway coupling to a full thermo-elastoplastic model in order to study the emerging temperature field. We find that, for our purposes, the consideration of thermomechanics at the microscale is not necessary. Second, we present highly parallel fully-coupled two-scale FE2 simulations, now neglecting temperature, using up to 458;752 cores of the JUQUEEN supercomputer at Forschungszentrum Jülich. The strong and weak parallel scalability results obtained for heterogeneous nonlinear hyperelasticity exemplify the massively parallel potential of the FE2 multiscale method. © Springer International Publishing Switzerland 2016.view abstract 10.1007/978-3-319-40528-5_5 **Relaxed incremental variational approach for the modeling of damage-induced stress hysteresis in arterial walls**

Schmidt, T. and Balzani, D.*Journal of the Mechanical Behavior of Biomedical Materials*58 (2016)In this paper, a three-dimensional relaxed incremental variational damage model is proposed, which enables the description of complex softening hysteresis as observed in supra-physiologically loaded arterial tissues, and which thereby avoids a loss of convexity of the underlying formulation. The proposed model extends the relaxed formulation of Balzani and Ortiz [2012. Relaxed incremental variational formulation for damage at large strains with application to fiber-reinforced materials and materials with truss-like microstructures. Int. J. Numer. Methods Eng. 92, 551-570], such that the typical stress-hysteresis observed in arterial tissues under cyclic loading can be described. This is mainly achieved by constructing a modified one-dimensional model accounting for cyclic loading in the individual fiber direction and numerically homogenizing the response taking into account a fiber orientation distribution function. A new solution strategy for the identification of the convexified stress potential is proposed based on an evolutionary algorithm which leads to an improved robustness compared to solely Newton-based optimization schemes. In order to enable an efficient adjustment of the new model to experimentally observed softening hysteresis, an adjustment scheme using a surrogate model is proposed. Therewith, the relaxed formulation is adjusted to experimental data in the supra-physiological domain of the media and adventitia of a human carotid artery. The performance of the model is then demonstrated in a finite element example of an overstretched artery. Although here three-dimensional thick-walled atherosclerotic arteries are considered, it is emphasized that the formulation can also directly be applied to thin-walled simulations of arteries using shell elements or other fiber-reinforced biomembranes. © 2015 Elsevier Ltd.view abstract 10.1016/j.jmbbm.2015.08.005 **Robust numerical schemes for an efficient implementation of tangent matrices: Application to hyperelasticity, inelastic standard dissipative materials and thermo-mechanics at finite strains**

Tanaka, M. and Balzani, D. and Schröder, J.*Lecture Notes in Applied and Computational Mechanics*81 (2016)In this contribution robust numerical schemes for an efficient implementation of tangent matrices in finite strain problems are presented and their performance is investigated through the analysis of hyperelastic materials, inelastic standard dissipative materials in the context of incremental variational formulations, and thermo-mechanics. The schemes are based on highly accurate and robust numerical differentiation approaches which use non-real numbers, i.e., complex variables and hyper-dual numbers. The main advantage of these approaches are that, contrary to the classical finite difference scheme, no round-off errors in the perturbations due to floating-point arithmetics exist within the calculation of the tangent matrices. This results in a method which is independent of perturbation values (in case of complex step derivative approximations if sufficiently small perturbations are chosen). An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of hyperelastic, finite strain elastoplastic, and thermo-elastoplastic boundary value problems, the performance of the proposed approaches is analyzed. © Springer International Publishing Switzerland 2016.view abstract 10.1007/978-3-319-39022-2_1 **A highly accurate 1st- and 2nd-order differentiation scheme for hyperelastic material models based on hyper-dual numbers**

Tanaka, M. and Sasagawa, T. and Omote, R. and Fujikawa, M. and Balzani, D. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*283 (2015)In this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models, which are derived in terms of the first and second derivatives of a strain energy function. This numerical scheme provides a compact model-independent framework, which means that once the framework is implemented, any other hyperelastic material model can be incorporated by solely modifying the energy function. The method is based on the numerical calculation of strain energy derivatives using hyper-dual numbers and thus referred to as hyper-dual step derivative (HDSD). The HDSD does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values. Furthermore, it enables the calculation of derivatives of arbitrary order. This is a great advantage compared to other numerical approaches as, e.g., the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values. Another alternative, the complex-step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values, but it only provides first derivatives and is thus not able to calculate stresses and moduli at once. In this paper, representative numerical examples using an anisotropic model are provided showing the performance of the proposed method. In detail, an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme. Furthermore, examples demonstrate the robustness and simple implementation of the HDSD scheme in finite element software. It turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time. © 2014 Elsevier B.V.view abstract 10.1016/j.cma.2014.08.020 **Comparative analysis of damage functions for soft tissues: Properties at damage initialization**

Balzani, D. and Schmidt, T.*Mathematics and Mechanics of Solids*20 (2015)In this paper several damage equations are analysed with respect to their properties at damage initialization. This is particularly important for soft biological tissues since two different loading regimes have to be clearly distinguished: the physiological domain where no damage evolution should be considered and the supra-physiological domain where damage evolves. At the transition between these two domains the behaviour of different damage models may influence the convergence of the Newton iteration when solving, for example, nonlinear finite element problems. It is shown that the model proposed by Balzani et al. (Comput Meth Appl Mech Eng 2012; 213-216: 139-151) a priori ensures smooth tangent moduli. In addition to that, a new damage function is proposed able to describe a slow damage evolution at damage initialization also providing smooth tangent moduli. Using this new damage function the approach given by Balzani et al. (Acta Biomater 2006; 2(6): 609-618) can also be modified such that smooth tangent moduli are guaranteed. Numerical analyses of a circumferentially overstretched artery are performed and show that no convergence problems are observed at the transition from the undamaged to the damaged domain, even when a model is used that has non-smooth tangent moduli. © The Author(s) 2013.view abstract 10.1177/1081286513504945 **Construction of statistically similar RVEs**

Scheunemann, L. and Balzani, D. and Brands, D. and Schröder, J.*Lecture Notes in Applied and Computational Mechanics*78 (2015)In modern engineering, micro-heterogeneous materials are designed to satisfy the needs and challenges in a wide field of technical applications. The effective mechanical behavior of these materials is influenced by the inherent microstructure and therein the interaction and individual behavior of the underlying phases. Computational homogenization approaches, such as the FE2 method have been found to be a suitable tool for the consideration of the influences of the microstructure. However, when real microstructures are considered, high computational costs arise from the complex morphology of the microstructure. Statistically similar RVEs (SSRVEs) can be used as an alternative, which are constructed to possess similar statistical properties as the realmicrostructure but are defined by a lower level of complexity. These SSRVEs are obtained from a minimization of differences of statistical measures and mechanical behavior compared with a real microstructure in a staggered optimization scheme, where the inner optimization ensures statistical similarity and the outer optimization problem controls themechanical comparativity of the SSRVE and the real microstructure. The performance of SSRVEs may vary with the utilized statistical measures and the parameterization of the microstructure of the SSRVE.With regard to an efficient construction of SSRVEs, it is necessary to consider statistical measures which can be computed in reasonable time and which provide sufficient information of the real microstructure.Minkowski functionals are analyzed as possible basis for statistical descriptors of microstructures and compared with other well-known statistical measures to investigate the performance. In order to emphasize the general importance of considering microstructural features by more sophisticated measures than basic ones, i.e. volume fraction, an analysis of upper bounds on the error of statistical measures and mechanical response is presented. © Springer International Publishing Switzerland 2015.view abstract 10.1007/978-3-319-18242-1_9 **Design of 3D statistically similar Representative Volume Elements based on Minkowski functionals**

Scheunemann, L. and Balzani, D. and Brands, D. and Schröder, J.*Mechanics of Materials*90 (2015)In this paper an extended optimization procedure is proposed for the construction of statistically similar RVEs (SSRVEs) which are defined as artificial microstructures showing a lower complexity than the associated real microstructures. This enables a computationally efficient discretization required for numerical calculations of microscopic boundary value problems and leads therefore to more efficient computational two-scale schemes. The optimization procedure is staggered and consists of an outer and an inner optimization problem. The outer problem treats different types of morphology parameterizations, different sets of statistical measures and different sets of weighting factors needed in the inner problem to minimize differences of mechanical errors that compare the response of the SSRVE with a target (real) microstructure. The inner problem minimizes differences of statistical measures describing the microstructure morphology for fixed parameterization type, statistical measures and weighting factors. The main contribution here is the analysis of new microstructure descriptors based on tensor-valued Minkowski functionals, whose numerical calculation requires less time compared to e.g. lineal-path functions. Thereby, a more efficient inner optimization problem can be realized and thus, an automated solution of the outer optimization problem becomes more practicable. Representative examples demonstrate the performance of the proposed method. It turns out that the evaluation of objective functions formulated in terms of the Minkowski functionals is almost 2000 times faster than functions taking into account lineal-path functions. © 2015 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.mechmat.2015.03.005 **Influence of isotropic and anisotropic material models on the mechanical response in arterial walls as a result of supra-physiological loadings**

Schmidt, T. and Pandya, D. and Balzani, D.*Mechanics Research Communications*64 (2015)As accepted in the literature, arterial tissues have in principle anisotropic material properties. Although some very special situations in arteries exist where isotropic constitutive models may approximate the real material behavior with sufficient accuracy, the larger part of analyses requires an anisotropic model. In particular for overstretched arteries, as e.g. a result of a balloon angioplasty, an accurate representation of the complex softening phenomena is important and then the consideration of anisotropy may be necessary. However, a variety of publications found in the literature, where such supra-physiological loading situations are analyzed to optimize e.g. stent designs, consider isotropic models. Therefore, in this contribution, the response of an isotropic and an anisotropic material model is compared in numerical calculations where arteries are subjected to supra-physiological loading. The constitutive formulations include the typical nonlinear stiffening of the fiber response as well as softening due to microscopic damage. In detail, the isotropic and the anisotropic model are adjusted to the same experimental stress-stretch curves of different arterial layers and then both models are applied to finite element simulations of overstretched arterial walls. As it turns out a significant difference is obtained for both calculations showing the importance of anisotropic models for these loading situations. © 2015 Published by Elsevier Ltd.view abstract 10.1016/j.mechrescom.2014.12.008 **Numerical calculation of thermo-mechanical problems at large strains based on complex step derivative approximation of tangent stiffness matrices**

Balzani, D. and Gandhi, A. and Tanaka, M. and Schröder, J.*Computational Mechanics*55 (2015)In this paper a robust approximation scheme for the numerical calculation of tangent stiffness matrices is presented in the context of nonlinear thermo-mechanical finite element problems and its performance is analyzed. The scheme extends the approach proposed in Kim et al. (Comput Methods Appl Mech Eng 200:403–413, 2011) and Tanaka et al. (Comput Methods Appl Mech Eng 269:454–470, 2014 and bases on applying the complex-step-derivative approximation to the linearizations of the weak forms of the balance of linear momentum and the balance of energy. By incorporating consistent perturbations along the imaginary axis to the displacement as well as thermal degrees of freedom, we demonstrate that numerical tangent stiffness matrices can be obtained with accuracy up to computer precision leading to quadratically converging schemes. The main advantage of this approach is that contrary to the classical forward difference scheme no round-off errors due to floating-point arithmetics exist within the calculation of the tangent stiffness. This enables arbitrarily small perturbation values and therefore leads to robust schemes even when choosing small values. An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of thermo-elastic and thermo-elastoplastic boundary value problems at finite strains the performance of the proposed approach is analyzed. © 2015, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-015-1139-0 **Selective enzymatic removal of elastin and collagen from human abdominal aortas: Uniaxial mechanical response and constitutive modeling**

Schriefl, A.J. and Schmidt, T. and Balzani, D. and Sommer, G. and Holzapfel, G.A.*Acta Biomaterialia*17 (2015)The ability to selectively remove the structurally most relevant components of arterial wall tissues such as collagen and elastin enables ex vivo biomechanical testing of the remaining tissues, with the aim of assessing their individual mechanical contributions. Resulting passive material parameters can be utilized in mathematical models of the cardiovascular system. Using eighteen wall specimens fromnon-atherosclerotic human abdominal aortas (55±11 years; 9 female, 9 male), we tested enzymatic approaches for the selective digestion of collagen and elastin, focusing on their application to human abdominal aortic wall tissues from different patients with varying sample morphologies. The study resulted in an improved protocol for elastin removal, showing how the enzymatic process is affected by inadequate addition of trypsin inhibitor. We applied the resulting protocol to circumferential and axial specimens from the media and the adventitia, and performed cyclic uniaxial extension tests in the physiological and supra-physiological loading domain. The collagenase-treated samples showed a (linear) response without distinct softening behavior, while the elastase-treated samples exhibited a nonlinear, anisotropic response with pronounced remanent deformations (continuous softening), presumably caused by some sliding of collagen fibers within the damaged regions of the collagen network. In addition, our data showed that the stiffness in the initial linear stress-stretch regime at low loads is lower in elastin-free tissue compared to control samples (i.e. collagen uncrimping requires less force than the stretching of elastin), experimentally confirming that elastin is responsible for the initial stiffness in elastic arteries. Utilizing a continuum mechanical description to mathematically capture the experimental results we concluded that the inclusion of a damage model for the non-collagenous matrix material is, in general, not necessary. To model the softening behavior, continuous damage was included in the fibers by adding a damage variable which led to remanent strains through the consideration of damage. © 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actbio.2015.01.003 **Construction of statistically similar representative volume elements - Comparative study regarding different statistical descriptors**

Scheunemann, L. and Schröder, J. and Balzani, D. and Brands, D.*Procedia Engineering*81 (2014)Advanced high strength steels, such as dual-phase steel (DP steel), provide advantages for engineering applications compared to conventional high strength steel. The main constituents of DP steel on the microscopic level are martensitic inclusions embedded in a ferritic matrix. A way to include these heterogeneities on the microscale into the modeling of the material is the FE2- method. Herein, in every integration point of a macroscopic finite element problem a microscopic boundary value problem is attached, which consists of a representative volume element (RVE) often defined as a segment of a real microstructure. From this representation, high computational costs arise due to the complexity of the discretization which can be circumvented by the use of a Statistically Similar RVE (SSRVE), which is governed by similar statistical features as the real target microstructure but shows a lower complexity. For the construction of such SSRVEs, an optimization problem is constructed which consists of a least-square functional taking into account the differences of statistical measures evaluated for the real microstructure and the SSRVE. This functional is minimized to identify the SSRVE for which the similarity in a statistical sense is optimal. The choice of the statistical measures considered in the least-square functional however play an important role. We focus on the construction of SSRVEs based on the volume fraction, lineal-path function and spectral density and check the performance in virtual tests. Here the response of the individual SSRVEs is compared with the real target microstructure. Further higher order measures, some specific Minkowski functionals, are investigated regarding their applicability and efficiency in the optimization process. © 2014 The Authors. Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2014.10.157 **Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations**

Balzani, D. and Scheunemann, L. and Brands, D. and Schröder, J.*Computational Mechanics*54 (2014)In this paper a method is presented for the construction of two- and three-dimensional statistically similar representative volume elements (SSRVEs) that may be used in computational two-scale calculations. These SSRVEs are obtained by minimizing a least-square functional defined in terms of deviations of statistical measures describing the microstructure morphology and mechanical macroscopic quantities computed for a random target microstructure and for the SSRVE. It is shown that such SSRVEs serve as lower bounds in a statistical sense with respect to the difference of microstructure morphology. Moreover, an upper bound is defined by the maximum of the least-square functional. A staggered optimization procedure is proposed enabling a more efficient construction of SSRVEs. In an inner optimization problem we ensure that the statistical similarity of the microstructure morphology in the SSRVE compared with a target microstructure is as high as possible. Then, in an outer optimization problem we analyze mechanical stress–strain curves. As an example for the proposed method two- and three-dimensional SSRVEs are constructed for real microstructure data of a dual-phase steel. By comparing their mechanical response with the one of the real microstructure the performance of the method is documented. It turns out that the quality of the SSRVEs improves and converges to some limit value as the microstructure complexity of the SSRVE increases. This converging behavior gives reason to expect an optimal SSRVE at the limit for a chosen type of microstructure parameterization and set of statistical measures. © 2014, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-014-1057-6 **Modeling supra-physiological loading of human arterial walls-damage, anisotropy and component-specific behavior**

Schmidt, T. and Schriefl, A.J. and Balzani, D. and Holzapfel, G.A.*Biomedizinische Technik*59 (2014)In this contribution an approach for the modeling of the mechanical behavior of arterial walls under supra-physiological loading conditions is investigated. One example of an overstretched atherosclerotic artery is provided using a finite element simulation. Therefor, the constitutive model from [1] is utilized, which reflects the anisotropic material behavior as well as damage-induced softening. This model is adjusted to the uniaxial stress response of the media and adventitia of human abdominal aortic specimens [7] for untreated control samples (with collagen and elastin intact) as well as for elastase- and collagenase-treated samples, leaving elastin-free and collagen-free tissue behind, respectively. © 2014 by Walter de Gruyter Berlin Boston.view abstract 10.1515/bmt-2014-5012 **Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis**

Tanaka, M. and Fujikawa, M. and Balzani, D. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*269 (2014)An extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard FEM software. This method is based on the complex-step derivative approximation (CSDA) approach. The CSDA is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e.g. finite difference approximation. Therefore, the CSDA can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision. Implementation details of the robust numerical approximation of tangent moduli from stress calculations using the CSDA are given and their performance is illustrated through representative examples involving finite deformations. In addition to that, we focus on the determination of material instabilities. Therefore, an accompanying localization analysis is performed, where the acoustic tensor is directly computed from the approximation of the moduli. It is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation. On the other hand, the CSDA approach provides high-accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely. © 2013 Elsevier B.V.view abstract 10.1016/j.cma.2013.11.005 **Statistical approach for a continuum description of damage evolution in soft collagenous tissues**

Schmidt, T. and Balzani, D. and Holzapfel, G.A.*Computer Methods in Applied Mechanics and Engineering*278 (2014)We propose a statistical approach to describe microscopic damage evolution in soft collagenous tissues such as arterial walls. The damage model extends a framework published by Balzani et al. (2012), Comput. Methods Appl. Mech. Engrg., 213-216:139-151, by postulating specific damage functions that result from the fibers' microstructure. Statistical distributions of three different microscopic quantities such as proteoglycan orientation, fibril length parameters and ultimate proteoglycan stretch are considered. The resulting stress-stretch response is compared with experimental data obtained from uniaxial tension tests given in the literature. In particular, the individual statistical distributions are analyzed in regard to their ability to capture the distinct softening hysteresis observed when subjecting soft tissues to cyclic loading in the supra-physiological domain. Details regarding the algorithmic implementation are provided, and the applicability of the model within a finite element framework is shown by simulating the overexpansion of simplified atherosclerotic arteries. © 2014 Elsevier B.V.view abstract 10.1016/j.cma.2014.04.011 **Two-scale modeling of DP steel incorporating distributed properties inside micro-constituents**

Schröder, J. and Gandhi, A. and Balzani, D.*Procedia Engineering*81 (2014)Advanced High Strength Steels (AHSS) are increasingly used in the industry due to their excellent strength and formability properties enabling weight savings. In this wide class of steel we restrict ourselves to the modeling of Dual Phase (DP) steels which are, at the microscale, characterized by a hard martensitic inclusion phase embedded in a soft ferritic matrix phase. During the production process the martensite transforms from austenite by rapidly cooling down the material and thereby causing a volume jump leading to initial plastic strains associated with eigenstresses of higher order. A technique to incorporate theses distributed properties in the ferrite matrix is proposed and implemented using the direct micro-macro transition approach. © 2014 The Authors. Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2014.10.162 **Computational model for the cell-mechanical response of the osteocyte cytoskeleton based on self-stabilizing tensegrity structures**

Kardas, D. and Nackenhorst, U. and Balzani, D.*Biomechanics and Modeling in Mechanobiology*12 (2013)The mechanism by which mechanical stimulation on osteocytes results in biochemical signals that initiate the remodeling process inside living bone tissue is largely unknown. Even the type of stimulation acting on these cells is not yet clearly identified. However, the cytoskeleton of osteocytes is suggested to play a major role in the mechanosensory process due to the direct connection to the nucleus. In this paper, a computational approach to model and simulate the cell structure of osteocytes based on self-stabilizing tensegrity structures is suggested. The computational model of the cell consists of the major components with respect to mechanical aspects: the integrins that connect the cell with the extracellular bone matrix, and different types of protein fibers (microtubules and intermediate filaments) that form the cytoskeleton, the membrane-cytoskeleton (microfilaments), the nucleus and the centrosome. The proposed geometrical cell models represent the cell in its physiological environment which is necessary in order to give a statement on the cell behavior in vivo. Studies on the mechanical response of osteocytes after physiological loading and in particular the mechanical response of the nucleus show that the load acting on the nucleus is rising with increasing deformation applied to the integrins. © 2012 Springer-Verlag.view abstract 10.1007/s10237-012-0390-y **Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls**

Balzani, D. and Brinkhues, S. and Holzapfel, G.A.*Computer Methods in Applied Mechanics and Engineering*213-216 (2012)In this paper a new material model is proposed for the description of stress-softening observed in cyclic tension tests of collagenous soft tissues such as arterial walls, for applied loads beyond the physiological level. The modeling framework makes use of terms known from continuum damage mechanics and the concept of internal variables introducing a scalar-valued variable for the representation of fiber damage. A principle is given for the construction of damage models able to reflect remanent strains as a result of microscopic damage in the reinforcing collagen fiber families. Particular internal variables are defined able to capture the nature of arterial tissues that no damage occurs in the physiological loading domain. By application of this principle, specific models are derived and fitted to experimental data. Finally, their applicability in numerical simulations is shown by some representative examples where the damage distribution in arterial cross-sections is analyzed. © 2011 Elsevier B.V.view abstract 10.1016/j.cma.2011.11.015 **Parallel simulation of patient-specific atherosclerotic arteries for the enhancement of intravascular ultrasound diagnostics**

Balzani, D. and Böse, D. and Brands, D. and Erbel, R. and Klawonn, A. and Rheinbach, O. and Schröder, J.*Engineering Computations (Swansea, Wales)*29 (2012)Purpose - The purpose of this paper is to present a computational framework for the simulation of patient-specific atherosclerotic arterial walls. Such simulations provide information regarding the mechanical stress distribution inside the arterial wall and may therefore enable improved medical indications for or against medical treatment. In detail, the paper aims to provide a framework which takes into account patient-specific geometric models obtained by in vivo measurements, as well as a fast solution strategy, giving realistic numerical results obtained in reasonable time. Design/methodology/approach - A method is proposed for the construction of three-dimensional geometrical models of atherosclerotic arteries based on intravascular ultrasound virtual histology data combined with angiographic X-ray images, which are obtained on a routine basis in the diagnostics and medical treatment of cardiovascular diseases. These models serve as a basis for finite element simulations where a large number of unknowns need to be calculated in reasonable time. Therefore, the finite element tearing and interconnecting-dual primal (FETI-DP) domain decomposition method is applied, to achieve an efficient parallel solution strategy. Findings - It is shown that three-dimensional models of patient-specific atherosclerotic arteries can be constructed from intravascular ultrasound virtual histology data. Furthermore, the application of the FETI-DP domain decomposition method leads to a fast numerical framework. In a numerical example, the importance of three-dimensional models and thereby fast solution algorithms is illustrated by showing that two-dimensional approximations differ significantly from the 3D solution. Originality/value - The decision for or against intravascular medical treatment of atherosclerotic arteries strongly depends on the mechanical situation of the arterial wall. The framework presented in this paper provides computer simulations of stress distributions, which therefore enable improved indications for medical methods of treatment. © Emerald Group Publishing Limited.view abstract 10.1108/02644401211271645 **Relaxed incremental variational formulation for damage at large strains with application to fiber-reinforced materials and materials with truss-like microstructures**

Balzani, D. and Ortiz, M.*International Journal for Numerical Methods in Engineering*92 (2012)In this paper, an incremental variational formulation for damage at finite strains is presented. The classical continuum damage mechanics serves as a basis where a stress-softening term depending on a scalar-valued damage function is prepended an effective hyperelastic strain energy function, which describes the virtually undamaged material. Because loss of convexity is obtained at some critical deformations, a relaxed incremental stress potential is constructed, which convexifies the original nonconvex problem. The resulting model can be interpreted as the homogenization of a microheterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one-dimensional relaxed formulation is derived, and a model for fiber-reinforced materials based thereon is given. Finally, numerical examples illustrate the performance of the model by showing mesh independency of the model in an extended truss, analyzing a numerically homogenized microtruss material and investigating a fiber-reinforced cantilever beam subject to bending and an overstretched arterial wall. © 2012 John Wiley & Sons, Ltd.view abstract 10.1002/nme.4351 **A new mixed finite element based on different approximations of the minors of deformation tensors**

Schröder, J. and Wriggers, P. and Balzani, D.*Computer Methods in Applied Mechanics and Engineering*200 (2011)Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element. © 2011 Elsevier B.V.view abstract 10.1016/j.cma.2011.08.009 **Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions**

Schröder, J. and Balzani, D. and Brands, D.*Archive of Applied Mechanics*81 (2011)For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE2-method provides a suitable numerical framework. Here, an additional microscopic boundary value problem, based on evaluations of representative volume elements (RVEs), is attached to each Gauss point of the discretized macrostructure. However, for real random heterogeneous microstructures the choice of a "large" RVE with a huge number of inclusions is much too time-consuming for the simulation of complex macroscopic boundary value problems, especially when history-dependent constitutive laws are adapted for the description of individual phases of the mircostructure. Therefore, we propose a method for the construction of statistically similar RVEs (SSRVEs), which have much less complexity but reflect the essential morphological attributes of the microscale. If this procedure is prosperous, we arrive at the conclusion that the SSRVEs can be discretized with significantly less degrees of freedom than the original microstructure. The basic idea for the design of such SSRVEs is to minimize a least-square functional taking into account suitable statistical measures, which characterize the inclusion morphology. It turns out that the combination of the volume fraction and the spectral density seems not to be sufficient. Therefore, a hybrid reconstruction method, which takes into account the lineal-path function additionally, is proposed that yields promising realizations of the SSRVEs. In order to demonstrate the performance of the proposed procedure, we analyze several representative numerical examples. © 2010 Springer-Verlag.view abstract 10.1007/s00419-010-0462-3 **FE 2-simulation of microheterogeneous steels based on statistically similar RVEs**

Balzani, D. and Schröder, J. and Brands, D.*IUTAM Bookseries*21 (2010)A main problem of direct homogenization methods is the high computational cost, when we have to deal with large random microstructures. This leads to a large number of history variables which needs a large amount of memory, and moreover a high computation time. We focus on random microstructures consisting of a continuous matrix phase with a high number of embedded inclusions. In this contribution a method is presented for the construction of statistically similar representative volume elements (SSRVEs) which are characterized by a much less complexity than usual random RVEs in order to obtain an efficient simulation tool. The basic idea of the underlying procedure is to find a simplified SSRVE, whose selected statistical measures under consideration are as close as possible to the ones of the original microstructure. © 2010 Springer Science+Business Media B.V.view abstract 10.1007/978-90-481-9195-6-2 **On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies**

Balzani, D. and Brands, D. and Klawonn, A. and Rheinbach, O. and Schröder, J.*Archive of Applied Mechanics*80 (2010)Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre-Hadamard condition automatically. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results. © 2009 Springer-Verlag.view abstract 10.1007/s00419-009-0379-x

#### SPP 1886 - Polymorphic Uncertainly Modelling for the Numerical Design of Structures (concluded)

#### SFB 837: Interaction Modelling in Mechanised Tunnelling (concluded)

#### SPP 1748: Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis (concluded)

#### SPP 1648: SPPEXA - Software for Exascale Computing - Project EXASTEEL: Bridging Scales for Multiphase Steels (concluded)

#### biomechanics

#### finite element method

#### numerical methods

#### representative volume element